*Article* **Application of Di**ff**usion Models in the Analysis of Financial Markets: Evidence on Exchange Traded Funds in Europe**

#### **Adam Marszk \* and Ewa Lechman**

Faculty of Management and Economics, Gdansk University of Technology, Narutowicza 11/12, 80-233 Gdansk, Poland; eda@zie.pg.gda.pl

**\*** Correspondence: amarszk@zie.pg.gda.pl

Received: 14 January 2020; Accepted: 10 February 2020; Published: 14 February 2020

**Abstract:** Exchange traded funds (ETFs) are financial innovations that may be considered as a part of the index financial instruments category, together with stock index derivatives. The aim of this paper is to explore the trajectories and formulates predictions regarding the spread of ETFs on the financial markets in six European countries. It demonstrates ETFs' development trajectories with regard to stock index futures and options that may be considered as their substitutes, e.g., in risk management. In this paper, we use mathematical models of the diffusion of innovation that allow unveiling the evolutionary patterns of turnover of ETFs; the time span of the analysis is 2004–2015, i.e., the period of dynamic changes on the European ETF markets. Such an approach has so far rarely been applied in this field of research. Our findings indicate that the development of ETF markets has been strongest in Italy and France and weaker in the other countries, especially Poland and Hungary. The results highlight significant differences among European countries and prove that diffusion has not taken place in all the cases; there are also considerable differences in the predicted development paths.

**Keywords:** financial innovations; diffusion; exchange traded funds; stock index futures; stock index options; stock market indexes

#### **1. Introduction**

Over the last decades, dynamic changes across financial markets have included the introduction of innovative financial instruments that contribute to global financial diversity. The category of innovative financial instruments is highly heterogeneous, e.g., in terms of the rate of their expansion; exchange-traded funds (ETFs) are among the most rapidly expanding financial instruments. ETFs are funds structured to mimic the performance of selected financial assets, usually stock indexes. The difference between ETFs and conventional investment products (such as mutual funds) is that units of ETFs resemble, financial instruments such as listed equities or bonds because they are purchased and sold through stock exchanges. The growing popularity of ETFs, the increase in the sums involved and the rate of turnover are predominantly enhanced by low trading costs, low tracking errors, high liquidity and (in some countries) high tax efficiency (Agapova 2011; Madhavan 2016; Ben-David et al. 2017; Lettau and Madhavan 2018).

Until recently, ETFs were mainly considered as substitutes for index funds in passive investing strategies because of their similar features and users. However, a rising recognition and complexity of the products offered has resulted in increasing demand from different types of players in the financial markets. As a result, the shares of innovative funds have become substitutes not only for index funds but also for derivatives. To the best of our knowledge, there have been almost no empirical works covering the subject of switching between ETFs and stock index derivatives, although a theoretical background was provided by the framework suggested by Gastineau (2010). The only exception is the analysis of the Asia-Pacific markets presented in Marszk et al. (2019)—due to different geographical coverage, it cannot be compared directly to the current study. In the recent years, there have been a number of studies devoted to the relationships between ETFs and futures but they have focused on particular attributes of selected instruments rather than market-broad analysis (e.g., Liu and Tse 2017; Oztekin et al. 2017; Chang et al. 2018; Wang et al. 2018; Chang et al. 2019; Jiang et al. 2019; Wallace et al. 2019; Liu et al. 2020).

Even though the diffusion of financial innovations has been discussed in a number of publications in recent decades, most studies have focused on the banking sector (e.g., Persons and Warther 1997; Hayashi and Klee 2003; Frame and White 2004, 2012; Akhavein et al. 2005). Earlier studies of innovations in the capital markets mostly concentrated on asset-backed securities or junk bonds (see, e.g., McConnell and Schwartz 1992; Molyneux and Shamroukh 1996; DeMarzo and Duffie 1999). Since the global financial crisis, this category of research has been marginalized due to a decreasing popularity of these instruments (Philippas and Siriopoulos 2012). The studies that have considered the diffusion of innovative financial products traded on exchanges (such as ETFs) have been relatively rare (e.g., Lechman and Marszk 2015; Hull 2016; Marszk et al. 2019). However, the current analysis differs from the previous studies because we empirically examine the diffusion of ETFs in relation to the other stock index instruments in a highly heterogeneous group of countries in two regions. Some attempts to outline the theory of ETF adoption have also been made by Diaz-Rainey and Ibikunle (2012), Awrey (2013) and Blocher and Whaley (2016), but the process of diffusion has not been discussed in these publications.

According to data published by the ETF research company, ETFGI, the value of assets that are globally managed by ETFs reached a value of almost 4.7 trillion USD at the end of 2018, and there were circa 6500 such funds such funds available worldwide. In comparison, the global value of assets was 2.9 trillion USD at the end of 2015, and there were circa 6100 such funds; at the end of 2009 the corresponding figures were just over 1 trillion USD and fewer than 2000 ETFs. Nonetheless, in case of Europe, the substantial changes had taken place over 2004–2015, as it had been the period of the launch and rapid expansion of ETFs in terms of number, assets and turnover. However, the growth dynamics of the ETF markets (understood as increasing values of assets accompanied by an increased turnover rate) in different countries differed significantly. In Europe (here understood as the EU member countries; in other European countries, except for Switzerland, the ETF markets remain underdeveloped) their use is still low compared to other advanced economies such as the United States, Japan and South Korea. It is important to note that comparing most European ETF markets to their American or Far Eastern counterparts is problematic because ETFs have a longer history and wider recognition in these countries.

The main aim of this paper is to explore the trajectories and formulate forecasts regarding the spread of ETFs on the financial markets in selected European countries between 2004 and 2015: Germany, France, Spain, Italy, Poland and Hungary. Consequently, we exclusively analyze the EU member states: four countries with the most developed ETF markets in the examined time period (except for the United Kingdom, due to a lack of necessary data) and, for comparison, two countries in the Central and Eastern Europe with substantially underdeveloped ETF markets—we can thus examine radically different patterns of diffusion of ETFs. In order to reach the stated aim, we use a novel methodological approach to the analysis of financial markets—mathematical models of the diffusion of innovation based on the turnover data. We thus contribute to the present state of knowledge not only by addressing the role of ETFs among the index financial instruments in Europe, but also by showing the financial application of the diffusion models. Moreover, this is one of the first studies to present and examine ETFs and stock index derivatives as substitutes for risk management.

More specifically, we aim to:


In order to do the above, we use monthly time series with ETF data from the economies selected between 2004 and 2015. This time period was selected due to the substantial pace of changes on the ETF markets in Europe, following their launch in the analyzed countries (the next years, since 2016 onwards, can generally be characterized as the period of stabilization and slower growth in terms of turnover of ETFs—moreover, in the case of some countries such as France or Spain, substantial decline could be noted; there were almost no ETFs traded in Europe prior to 2004).

This paper comprises five sections. Section 2 outlines the theoretical setting and presents some issues with regard to ETFs: their fundamentals, and how they compare with stock index derivatives. Section 3 outlines the methodological framework and presents the sources of data. Section 4 is devoted to the discussion of the results of the conducted empirical study; it is further divided into two parts: a presentation of introductory descriptive evidence related to the ETF market development, and a discussion of our key results obtained using diffusion models. Section 5 concludes the paper.

#### **2. Theoretical Background**

#### *2.1. Basic Features of ETFs*

In their basic form, exchange traded funds (ETFs) may be defined as baskets of securities (or other assets) traded on a stock exchange (comparable to equities of public corporations), typically with the intermediation of brokerage companies (Ferri 2009). ETFs are innovative financial instruments and they were introduced in the 1990s and 2000s. The units of ETFs closely replicate (i.e., track) the financial performance of certain financial assets, usually blue-chip or broad market indexes of the stock markets (Hehn 2005; Hill 2016). ETF shares (units) can be purchased and sold on the stock exchanges during their trading hours at prices determined by the interaction of demand and supply (Abner 2016; Investment Company Institute 2017). The prices of ETF shares remain typically at levels close to their net asset value (which is related to the prices of the assets tracked). There are two parts of the ETF market: the primary and secondary segments (Hill et al. 2015; Ben-David et al. 2018; Box et al. 2019). The shares of ETFs can be created and redeemed exclusively on the primary segment, as a result of operations including both the company that manages the fund (its sponsor) and financial institutions that act as authorized participants. In the case of physical ETFs, they involve the delivery of the assets underlying the fund in exchange for the shares of ETFs, and in the case of synthetic ETFs (the funds that employ derivatives, popular above all on the European markets) transactions involve cash. As a result of transactions on the primary market, which are a part of the arbitrage mechanism, ETF tracking errors (deviations in the returns on the units of ETFs from the returns on the tracked assets) are low in most cases. The secondary segment of the ETF market involves transactions on stock exchanges conducted by market participants (institutional or individual investors)—they include in particular the sale or purchase of ETF shares without any interaction with the managing company.

The rising recognition of ETFs in the last decade has generally been the result of the attributes they provide to investors compared to conventional financial instruments, particularly the sub-category of mutual funds with similar aims—index funds. These advantages stem from the mechanisms for the creation and distribution of ETFs. Their key benefits relative to index funds include: lower tracking errors and lower tracking costs (in some circumstances, index funds are more cost-efficient—this depends on the trading frequency and the investment period), higher liquidity (units of index funds are usually priced once a day and have daily buying/purchasing cycles) and greater tax efficiency in some countries (e.g., in the USA) (Agapova 2011; Investment Company Institute 2017; Piccotti 2018). It should not be forgotten that the expansion of ETFs in Europe can affect not only the financial sector but also non-financial companies—potential effects are not limited to the countries with the highest assets of locally listed ETFs, as other economies may also be affected through, e.g., cross-listing of the shares of ETFs (Alderighi 2020) or foreign assets held in the portfolios of ETFs (Baltussen et al. 2019); one of the possible consequences is the impact on the probability of bankruptcy of the companies (Kovacova et al. 2019) as the level of the development of the local ETF market can become one of the

possible determinants of insolvency. However, this issue remains substantially understudied, both theoretically and empirically.

#### *2.2. ETFs Compared to Stock Index Derivatives*

Exchange traded funds, stock index futures and stock index options may be regarded as competing products within the category of (portfolio basket) index financial instruments. Together with a few other instruments, they constitute the equity index arbitrage complex—a group of related financial instruments based on common underlying assets (usually a basket of assets). This is a group of instruments with related values because of the similarity of their underlying financial assets (Gastineau 2010). The underlying assets are usually stock market indexes or stock baskets determined by the index rules. The equity index arbitrage complex consists of three instrument categories (less commonly used instruments have been omitted):


In this classification, ETFs are included in the first category because they are combinations and extensions of the underlying traditional assets, not because they lack innovative features. The values of symmetric instruments are straightforward functions of the prices of the underlying assets, whereas the prices of convex instruments do not move proportionately.

The following discussion regards three groups of instruments traded on exchanges: ETFs, stock index futures and stock index options. Stocks, the most basic instruments, are not discussed. Instruments which belong to an arbitrage complex are perceived by investors as substitutes, not only because of the similarity of the underlying assets but also because of the potential for (usually limited) arbitrage profits. This means that their prices are related. Treating the arbitrage complex as an object of analysis is a suitable way to perform research concerning modern financial markets, as feedback between increasing trading volumes and decreasing trading costs on the one hand and arbitrage complexes on the other has been observed on most of the world's stock exchanges (Gastineau 2010).

Before the current dynamic development of the ETF market, these innovative instruments were considered as alternatives to futures or options, mostly for short- and long-term risk management by large investors. Gastineau (2010) presents the results of a preliminary comparison based on data from the US market (the assets tracked were S&P 500 stocks). The key characteristic compared is the cost of these two alternatives. The costs of ETFs for risk managers result from the cost of gathering the stocks in a creation basket (it is assumed that transactions are conducted on the primary ETF market due to their size) or opposite transactions—commission fees, management fees and market impact. In the case of futures, the main costs are roll risk (the cost of extending the contracts after they end) and market impact. As a result, futures seem to be a better choice for short-term risk management, whereas ETFs are beneficial in the long term due to their lack of rolling expenses.

In recent years, ETFs have become increasingly popular alternatives to futures and options, not only as risk management tools for specific categories of investors but also for a wider group of market participants. The reasons for this change in the financial landscape can be traced back to the financial crisis of 2008 and regulatory decisions made in its aftermath, which were aimed at reducing systematic shock risks (Goltz and Schröder 2011; Arnold and Lesné 2015). As a result of the increased cost of capital for investment banks, growing operational (e.g., improved transparency) and capital requirements, and liquidity constraints—mainly linked to the Basel III regulations (Madhavan et al. 2014; Madhavan 2016), the cost of traditional instruments such as futures or options grew and ETFs became relatively more cost-effective, for example in obtaining long-term exposure. Moreover, because of the high level of competition among ETF providers and economies of scale, the costs of investments in ETFs, especially in equity index ETFs (the closest substitutes for index futures and options), have been significantly declining (Arnold and Lesné 2015). To sum up, ETFs and stock index derivatives can be perceived as alternatives in certain fields of risk management (Hill and Mueller 2001; Madhavan 2016; Arunanondchai et al. 2019).

The differences between ETFs and stock index futures and in particular their relative advantages and disadvantages will now be described. Despite their different features, which make direct comparisons difficult, most of the relative advantages and disadvantages of futures with respect to ETFs which are discussed below also apply to options (as derivatives traded on regulated exchanges, which in many cases may be alternatives to futures, and even more importantly to ETFs (Thomsett 2016)).

The similarities between ETFs and stock index futures include (Goltz and Schröder 2011; Arnold and Lesné 2015; Marszk et al. 2019):


Table 1 presents some selected main features which distinguish ETFs from stock index futures. The key difference, which influences the relative costs of these two categories of instruments, lies in the rolling costs of futures contracts, i.e., the costs of entering a new contract after the expiry of the previous one, which involve both explicit costs (trading commissions and bid-ask spreads) and potential mispricing (Madhavan et al. 2014; Arnold and Lesné 2015). The main relative advantages of futures can be observed in the following features: the capital required, leverage, and short sale possibilities. The strengths of ETFs are higher accessibility, wider product ranges, minimal management requirements prior to exiting, no predefined maturity and easier foreign investment. To sum up, similarly to the use in risk management discussed in the preceding paragraphs, even for the broad investing audience, ETFs may be considered as more efficient long-term investment instruments, whereas futures are regarded as more suitable short-term choices (Eurex 2016). It should be noted, however, that the final choice depends not only on the holding period but also on the investment strategy. According to the results of a study conducted by the CME Group (2016), in the case of leveraged or short sale positions index futures are relatively more beneficial, regardless of the holding period.



It should be underlined that the framework presented above only applies to equity ETFs, and many more types of these instruments are currently available, such as fixed income and commodity ETFs. However, despite the increasing heterogeneity of ETFs, equity ETFs (based on the equity market, usually stock market indexes) are still by far the largest category.

#### **3. Materials and Methods**

#### *3.1. Innovation Di*ff*usion Models*

To achieve the main aims of this study, we adopted a methodological framework allowing for examination of the evolution over time of the processes observed across the financial systems including, e.g., ETF diffusion. For that reason, apart from the usual descriptive statistics, we employed innovation diffusion models (presented in, inter alia, Geroski 2000; Rogers 2010; Lechman 2015); they may be used to approximate ETF diffusion trajectories and modeled projected future ETF development patterns. A similar approach is presented in a study by Lechman and Marszk (2015), who study ETF diffusion paths in chosen countries.

In the main part of our empirical analysis, in order to reveal ETF market development patterns, we follow the approach of, among others, Mansfield (1961) and Dosi and Nelson (1994), who adopted the concept of evolutionary dynamics. It can be expressed mathematically in the form of a logistic growth function, which may further be presented as an ordinary differential equation (Meyer et al. 1999):

$$\frac{d\mathbf{Y}\_{\mathbf{x}}(\mathbf{t})}{d\mathbf{t}} = \alpha \mathbf{Y}\_{\mathbf{x}}(\mathbf{t}).\tag{1}$$

If Y(t) denotes the level of variable x, α is a constant growth rate and t denotes time, then Equation (1) explains the time pattern of Y(t). Moreover, the introduction of e to Equation (1) leads to its reformulation as:

$$\mathbf{Y}\_{\mathbf{x}}(\mathbf{t}) = \beta \mathbf{e}^{\alpha \mathbf{t}},\tag{2}$$

or alternatively:

$$\mathbf{Y}\_{\mathbf{x}}(\mathbf{t}) = \; \mathbf{a} \exp \boldsymbol{\beta} \mathbf{t},\tag{3}$$

with notation parallel to the one in Equation (1) and β representing the starting level of x at t = 0. Presented simple growth model is due to its assumptions being pre-defined as exponential. Therefore, it assumes that x will continue to grow infinitely in a geometric progression. Arbitrary extrapolation of Yx(t) within an exponential growth model can result in improbable predictions, since systems do not grow infinitely because of their constraints (Meyer 1994). For that reason, to address the issue of 'infinite growth', Equation (1) can be extended by adding a 'resistance' parameter (Kwasnicki 2013). This change imposes an upper 'limit' to the model, thus giving the original exponential growth curve a shape that is sigmoid. Consequently, the revised version of Equation (1) becomes a logistic differential function:

$$\frac{d\mathcal{Y}(\mathbf{t})}{d\mathbf{t}} = \alpha \mathcal{Y}(\mathbf{t}) \Big(\mathbf{1} - \frac{\mathcal{Y}(\mathbf{t})}{\kappa}\Big) \tag{4}$$

with following notation—the parameter labeled as κ denotes the upper asymptote imposed, which arbitrarily restricts the increase of the variable Y.

As shown above, introducing a resistance parameter to exponential growth leads to trajectory that can be described as S-shaped. Equation (4), the 3-parameter logistic differential equation, can also be presented in another way, using a logistic growth function which takes non-negative values:

$$\mathbf{N}\_{\mathbf{x}}(\mathbf{t}) = \frac{\kappa}{1 + \mathbf{e}^{-\alpha \mathbf{t} - \beta}},\tag{5}$$

or, alternatively:

$$N\_{\mathbf{x}}(\mathbf{t}) = \frac{\kappa}{1 + \exp(-\alpha(\mathbf{t} - \boldsymbol{\beta}))},\tag{6}$$

where Nx(t) represents the level of variable x in certain time period t. Other parameters in Equations (5) and (6) stand for the following: κ is the upper asymptote, which determines the limit of growth and is also labeled 'carrying capacity' or 'saturation'; α is the growth rate, which determines the speed of diffusion; and β stands for the midpoint, which shows the exact moment in time (Tm) when the logistic pattern reaches 0.5κ. Nonetheless, interpretation may be facilitated by replacing <sup>α</sup> with a parameter known as 'specific duration', and defined as <sup>Δ</sup><sup>t</sup> = ln(81) <sup>α</sup> . With Δt, it becomes easy to approximate the time necessary for x to increase from 10%κ to 90%κ. Moreover, the midpoint (β) denotes the point in time at which the logistic growth begins to level off. Finally, mathematically, it is the inflection point of the logistic curve. Incorporating Δt and Tm into Equation (6) produces:

$$\mathbf{N\_{x}(t)} = \frac{\mathbf{x}}{1 + \exp\left[-\frac{\ln(81)}{\Delta t}(\mathbf{t} - \mathbf{T\_{m}})\right]}.\tag{7}$$

In our research, we aim to use the methodological framework for innovation diffusion models briefly presented above. In the first part of the analysis, we assume that the growing value of ETF unit turnover can be regarded as diffusion of ETFs on financial markets in the examined countries. Nevertheless, in the core part of our study, we assume that the process of the growing ETF share of the total turnover of comparable investment options (in the equity index arbitrage complex) is comparable to the process of diffusion of innovative products and services across heterogeneous socio-economic systems. We assume that ETFs are innovations which due to a 'word of mouth' effect (Geroski 2000) and emerging network effects are progressively adopted by growing numbers of investors (who may also be described as 'users' of ETFs). We also rely on a basic assumption that investors (users) in certain type of financial innovations (here, ETFs) may freely contact 'non-investors' ('non-users'), i.e., people either not using ETFs before or previously choosing other similar options, which leads to adoption by this group.

In short, we assume that ETFs diffuse on financial markets in the analyzed countries and gain a growing share of the total turnover of similar investment options (apart from ETFs, stock index futures and stock index options (Gastineau 2010; Madhavan 2016; Arunanondchai et al. 2019; Marszk et al. 2019)). In the fundamental specification of the 3-parameter logistic growth model as defined in Equation (6), we presume that Nx(t) = ETFi(t) represents changes in the ETF share of the total turnover of comparable investment options over time t in country i. Put differently, it describes changes in country i's level of ETF financial market penetration. The parameter κ is represented as κETF <sup>i</sup> , which is the ceiling (upper asymptote/system limit) on the process of ETF diffusion on financial markets. The estimated parameter κETF <sup>i</sup> denotes the potential ETF share of the total turnover of comparable investment options on the financial market in country i. However, there is the strict assumption that the trajectory of ETF diffusion (development) follows the sigmoid pattern generated by the logistic growth equation.

Next, the parameter α (as in Equation (6)) is represented as αETF <sup>i</sup> , which is the speed of ETF diffusion on the financial market in country i. Hence, the estimated parameter αETF <sup>i</sup> shows how fast the ETF share of the total turnover of comparable investment options is increasing on the financial market selected. Moreover, using parameter αETF <sup>i</sup> we calculate the 'specific duration,' defined as <sup>Δ</sup><sup>t</sup> <sup>=</sup> ln(81) αETF i , which represents the time needed to pass from κETF <sup>i</sup> <sup>=</sup> 10% to <sup>κ</sup>ETF <sup>i</sup> = 90%.

The parameter β is expressed as βETF <sup>i</sup> , and its estimated value denotes the midpoint TmETF <sup>i</sup> , which indicates the exact time when 50% of κETF <sup>i</sup> is reached. Hence, TmETF <sup>i</sup> represents the time (year/month) when the process of ETF diffusion reaches the half-way point if we assume that it is heading toward κETF <sup>i</sup> .

Thus, the modified specification of Equation (6) is:

$$\text{ETF}\_{\text{i}}(\text{t}) = \frac{\kappa\_{\text{i}}^{\text{ETF}}}{1 + \exp\left(-\alpha\_{\text{i}}^{\text{ETF}} \left(\text{t} - \beta\_{\text{i}}^{\text{ETF}}\right)\right)},\tag{8}$$

with notation as explained above.

The parameters in Equation (8) can be estimated using not only ordinary least squares (OLS) but also maximum likelihood (MLE), algebraic estimation (AE) or nonlinear least squares (NLS). Nonetheless, as Satoh (2001) suggests, NLS returns the best predictions, as its estimates of standard errors (of κETF <sup>i</sup> , <sup>α</sup>ETF <sup>i</sup> , <sup>β</sup>ETF <sup>i</sup> ) are more valid than those returned using the other methods. Adopting NLS allows time-interval biases, which occur in the case of OLS estimates (Srinivasan and Mason 1986), to be avoided. However, NLS has the disadvantage that estimates of the parameters may be sensitive to the initial values of the time-series adopted. Finally, it should be emphasized that the construction of the utilized model hinders inclusion of the explanatory variables. However, the issue of the factors that affect the diffusion of ETFs was analyzed using different methodologies—the results were presented in, inter alia, Lechman and Marszk (2015) and Marszk et al. (2019).

#### *3.2. Data*

Our research covers stock exchanges in six European countries: two countries in the Central and Eastern Europe (CEE) region—Poland and Hungary; another four EU countries with the longest history of ETF trading—France, Italy, Germany and Spain. Our analysis covers the Euronext exchange considered as a whole (due to data availability), and thus (in addition to France) also includes The Netherlands, Belgium and Portugal. However, most of the turnover is reported to be in the French segment and so we decided to consider this exchange as if it was located in France. Consequently, we also used other indicators for France.

The time span of the analysis is 2004–2015. It was selected due to the high rate of changes on the ETF markets in Europe, following their launch in the analyzed countries. The beginning of this period was chosen due to the fact that there were almost no ETFs traded in Europe in 2003 or earlier. The selected end of the time period of analysis is 2015 as since 2016, the changes have been less significant. The time coverage is also a result of data availability. For the period 2004–2015 a balanced data set is available for most of the countries included in the analysis, while for the CEE countries, the time span of the analysis is shorter as ETFs were launched there later than in the advanced European economies.

The financial instrument databases used in the study are the dataset provided by the World Federation of Exchanges (World Federation of Exchanges 2017), datasets provided by the selected stock exchanges and reports published by these institutions. The most important financial indicators used are the turnover values (in USD millions) on the stock exchanges of the instruments selected: ETFs, stock index options and stock index futures. Monthly data are used.

Due to a lack of reliable data on the turnover of stock index futures and options on the main stock exchange in the United Kingdom (caused by changes in the organizational structure of the London Stock Exchange Group), it was excluded from the analysis.

#### **4. Results**

#### *4.1. Exchange Traded Fund Market Development: Preliminary Evidence*

Our investigation of the development of the ETF markets starts with an analysis of summary statistics on the key changes in two measures: the turnover value and the percentage share of the total turnover of index financial instruments (see Table 2).

**Table 2.** Summary statistics for exchange traded funds, stock index options, stock index futures and total index financial instruments. Monthly data for 2004–2015. For ETFs, the periods of analysis are: Poland, 2010m9–2015m12; Hungary, 2007m1–2015m12; Italy, 2004m1–2015m12; Spain, 2006m7–2015m12; Germany, 2004m1–2015m12; and France, 2004m1–2015m12. The number of ETFs varies across time periods and countries.





**Table 2.** *Cont.*

In the time period analyzed, there were only two countries in the CEE region where ETFs were listed on the local stock exchanges: Poland and Hungary. In Poland, the highest values of ETF turnover were reached several months after their launch in September 2010, in August 2011 (see Table 2 and Figure A1 in the Appendix A). However, in 2012 turnover severely declined and reached a minimum level of only \$0.9 million USD in November 2012. From 2013 to 2015, trading in ETFs was still at a rather low level. However, in April 2015, ETFs reached their maximum share of the total market: 0.39%, which was mostly caused by a one-month spike in ETF trading (yet it was still one of the lowest shares among the countries considered). In Hungary, ETFs were launched much earlier than in Poland (in 2007) but their turnover was significantly lower (a mean monthly value of \$0.6 million USD compared to \$5.5 million USD in Poland). As in Poland, the highest turnover values were observed soon after their introduction. However, in terms of ETF market share, the highest value in Hungary was reached

(as in the Polish market) near the end of the time period analyzed, in May 2015 (see Figure 1). In contrast to the Polish exchange, the turnover of other related financial instruments (stock index futures and options) on the Hungarian market was extremely low: in most months there were almost no transactions in options and the value of futures trading was steadily declining. As a result, the mean turnover values in Hungary were minimal in comparison to the other stock exchanges considered. The very low turnover of ETFs in both Poland and Hungary was mostly caused by the low number of such financial products. In Poland, the number of ETFs grew from 1 to 3 (yet only one of them was listed exclusively in Poland and it accounted for the majority of turnover; the other two ETFs were cross-listed). In Hungary, there was only one ETF listed between 2007 and 2015 and it had a minimal turnover. The lack of further development was caused by a number of factors, including a lack of awareness of ETF features among market participants and the relatively small size of the financial markets, which limited the possibility of gaining benefits from the larger scale of offerings provided by ETF managers.

**Figure 1.** ETFs, stock index options and stock index futures—share of total turnover of index financial instruments. 2004–2015 (monthly time series). Left-hand Y axis—ETF share of total index financial instruments; right-hand Y axis—stock index options and stock index futures share of total index financial instruments.

In the four advanced EU countries selected, the only country with no ETFs listed at the beginning of the time period analyzed was Spain (ETFs were launched in Spain in July 2006). In terms of ETF turnover, in Italy growth of the ETF market was somewhat stable and the highest values were reached near the end of the time period. In France and Germany, ETF turnover grew until 2011, when it sharply declined, which may be explained by the eurozone crisis and falling stock prices (in the other three advanced EU economies a decline in ETF turnover in 2011 was also observed but it was relatively weaker (see Figure A1 in the Appendix A)). After 2011, turnover in France began to grow, whereas in Germany it was stable. The Spanish ETF market developed in a different way. After much variability until 2011, it entered a stage of stability between 2012 and 2013, and from the end of 2013 it started growing. This shows that the development of the ETF markets in these countries was to some extent shaped by similar determinants (e.g., the euro-zone crisis), although there were also some country-specific factors despite the high level of financial market integration.

Regarding ETF market shares, some substantial differences between the four countries can be noticed (see Table 2 and Figure 1). In Spain and Germany, the market share of ETFs was very low over the whole period. The case of Germany is particularly interesting. The mean value of ETF turnover in this country was the highest among all the countries analyzed and one of the highest in the world. Nevertheless, their average market share was the second-lowest (it was only lower in Poland), which shows that the role of ETFs in Germany was negligible compared to that of other index financial instruments. In both France and Italy, the market share of ETFs increased considerably: in France particularly from 2014, and in Italy from 2009. The mean market share value of ETFs in Italy was the highest of all the countries under study (5.6%), yet was still much lower than the shares of the other index instruments. The rapid development of the Italian ETF market may to a large extent be explained by the acquisition of the Italian stock exchange by its British counterpart, which is one of the largest in Europe in terms of the number and turnover of ETFs. The two markets have been integrated in some areas, which considerably boosted the Italian stock exchange's growth opportunities.

In the remainder of this study, we will use the market share as the indicator of ETF market development, as changes occurring in ETF markets should not be viewed in isolation but instead put in a broader context, thus showing the position of these innovative financial products in the financial system. Our preliminary analysis of changes occurring in the ETF markets will be expanded in the next sections—we will attempt to analyze the main features of the ETF diffusion process and predict its trajectories.

#### *4.2. Exchange Traded Funds: Di*ff*usion Models*

As an aim of this study is to provide in-depth insight into the development process of ETFs across countries, we adopt a logistic growth model (for details, see Section 3) because use of this type of model allows the development trajectories of different variables in economic systems to be approximated and evaluated. Moreover, it allows the characteristic phases of the process of diffusion to be distinguished, such as the early diffusion phase, take-off, the exponential growth phase and saturation (maturity phase). Through the early diffusion stage, the number of contacts between adopters and non-adopters of a given innovation is still small, which may hinder its dissemination and so in this stage of diffusion the process is still reversible. However, under favorable conditions, easy contacts allow a domino effect to come into play and hence diffusion may speed up. Driven by various market forces, reductions in the cost of adopting innovations and multiple applications and uses of them, the number of new-users can rapidly increase and the curve takes off. It then enters a fast diffusion phase, when the diffusion process usually proceeds exponentially. Finally, a maturity (stabilization) stage is reached, during which the pace of diffusion again becomes slow and no substantial growth in the number of new users of the innovation is reported. In addition to revealing these phases, a simple logistic growth model returns good forecasts of future development (Kucharavy and Guio 2011).

Following the above-mentioned approach and using monthly time series for the period 2004–2015, we develop logistic growth patterns and estimate parameters (see Section 3) representing the ETF share of total index financial instrument turnover for each country individually. The results of our analysis are presented in Figure A2 in the Appendix A, which shows that the current and predicted ETF share diffusion paths, and in Table 3, which summarizes the country-wise logistic growth model estimates.

The graphical evidence presented in Figure 1 suggests that ETF diffusion patterns in some countries (i.e., growing ETF shares) may be well described by the logistic (sigmoid) growth trajectory. In some cases, the characteristic phases of the S-shaped path can be distinguished (also see the analysis for other countries in Lechman and Marszk (2015)). Initially slow changes in the ETF share of total turnover of index financial instruments are followed by a sudden take-off and then the ETF share pattern enters the rapid growth phase. However, it is important to note that the shapes of the ETF share diffusion paths across the countries examined are different and so they need special attention.


**Table 3.** Diffusion of exchange traded funds (as share of total turnover of index financial instruments). Logistic growth model estimates. 2004–2015 (monthly time series). Poland—data from 2010m9; Hungary—data from 2005m1; Spain—data from 2006m7.

The picture which emerges from analysis of the ETF share in the two selected CEE countries—Poland and Hungary—differs radically from that for other countries. As already mentioned in the previous section, in neither Poland nor Hungary did ETFs gain much popularity and their share of total turnover remained extremely low over the time period analyzed. In Hungary, the growth of the ETF share of total turnover was minimal and its role in shaping the financial market was negligible. In Poland, a diffusion of ETFs across the domestic financial market was reported but still their role and share of the total turnover was marginal. It should be noted that between 2004 and 2015, the ETF share of the total turnover was close to zero. This leads to the conclusion that in both Hungary and Poland a diffusion of ETFs did not take place and so logistic growth models should not be applied. Table 3 presents the estimates of logistic growth models for Poland and Hungary, but as in both cases the R<sup>2</sup> of the models is zero, the parameters returned are misleading and inconclusive.

Finally, we discuss the results of the analysis of ETF diffusion for the four developed financial markets selected: Italy, Germany, France and Spain. In Germany, a diffusion of ETFs on the domestic financial market was not observed and ETF market penetration remained below 1%. As in the cases of Hungary and Poland, the logistic growth model estimates are not reliable. Despite the fact that the R<sup>2</sup> of the model is 0.27 (see Table 3), the value returned for the midpoint (Tm) is negative and so cannot be treated as valid. The situation in Spain is analogous, with a very low ETF share of total turnover during the time period examined. At the end of 2015, Spain was still located in the early diffusion stage, and as a result reliable estimates of a logistic growth model are not possible (the logistic growth parameters returned cannot be treated as valid).

In the other two advanced European economies—France and Italy—the ETF share was relatively high between 2004 and 2015. In both cases, the ETF diffusion patterns take off into self-sustaining growth after the early diffusion stage, during which increases in the ETF share were slow. In the case of Italy, the specific take-off occurred relatively early compared to the other economies examined. It should be noted that between June and July 2008 the ETF share almost doubled (from 1.8% to 3.4%) and the take-off took place shortly afterwards—between the middle of December 2008 and January 2009, when the ETF share increased from 3.7% to 8.0%. All the parameters returned from the logistic growth model estimates for Italy are statistically significant. The R<sup>2</sup> of the model is about 0.76, which implies a good fit between the empirical data and the theoretical model. Even though the R2 of the model is low, there are no obvious misspecifications as the diffusion of ETFs is relatively well described by the logistic growth trajectory. The upper asymptote is estimated as κETF <sup>i</sup> = 8.56%. The estimated midpoint

is TmETF <sup>i</sup> <sup>=</sup> 52.9. The rate of diffusion is <sup>α</sup>ETF <sup>i</sup> = 0.113 and Δt ETF <sup>i</sup> = 38.7, which can be interpreted as the number of months required to pass from 10% to 90% of κETF <sup>i</sup> .

For France too, the diffusion of ETFs is well described by the logistic growth trajectory, despite the fact that in this case the early diffusion stage was relatively long. The take-off into the exponential growth phase did not happen until between the middle of December 2013 and January 2014, when the ETF share of total turnover grew abruptly. Even though the diffusion of ETFs (in terms of market share) on the French financial market is well approximated by the logistic growth pattern, the parameters estimated for the logistic growth model are not valid. The upper asymptote (ceiling) is reported as κETF <sup>i</sup> = 7,755,333, which is a definite overestimation.

Regarding the process of ETF diffusion in our country sample, the eight economies can be divided into two groups. The first group encompasses two countries—France and Italy—where an early diffusion stage was followed by a take-off into an exponential growth phase along a sigmoid trajectory. These two countries managed to leave the early diffusion stage, during which ETF share growth was slow and spasmodic, and take off into rapid growth. In the other four countries, the ETF share did not leave the early diffusion stage and remained virtually locked at a low level.

This empirical analysis of ETF diffusion trajectories can be enriched by providing additional specifications of the predicted development of ETFs across the economies selected. Table 4 summarizes the predicted country-specific ETFs diffusion paths, and Figure A2 in the Appendix A portrays them graphically. Fixing the critical level of the upper asymptote (κETF <sup>i</sup> ) at 5%, 7.5%, 10%, 15%, 20%, 25% and 30%, we predict logistic growth model parameters under the strict assumption that ETF market development will follow an S-shaped trajectory.

For Hungary, with κETF <sup>i</sup> fixed at 5% the predicted TmETF <sup>i</sup> is June 2027 and the 'specific duration' forecast is about 320 months, i.e., more than 26 years. The predicted rate of diffusion is 0.014, which implies that the speed of ETF diffusion will be rather low in Hungary. The forecasts for higher κETF i show even more distant midpoints and they cannot be treated as being very reliable (and also because of the low R<sup>2</sup> of the models).

Italy has already reached the levels of κETF <sup>i</sup> <sup>=</sup> 5%, 7.5% and 10%. With <sup>κ</sup>ETF <sup>i</sup> fixed at 15%, the predicted TmETF <sup>i</sup> is April 2012 if the Italian ETF market follows an S-shaped trajectory. The predicted rate of diffusion is similar to that in Hungary, i.e., much lower than in, e.g., France.

Regarding Spain, with κETF <sup>i</sup> fixed at 5% the predicted TmETF <sup>i</sup> is July 2021 (considerably sooner than in the case of Hungary) and the 'specific duration' forecast is about 300 months. The rate of diffusion predicted is 0.015, which is consistent with the results obtained for Hungary and Italy. Finally, for France with κETF <sup>i</sup> fixed at 7.5%, the predicted TmETF <sup>i</sup> is July 2015 if the French ETF market follows the S-shaped trajectory. The rate of diffusion predicted for this level of κETF <sup>i</sup> is 0.028, but for higher levels it is slightly lower, which suggests that the diffusion of ETFs on the French market will be much faster than in other European countries.

The ETF diffusion paths predicted for Germany and Poland are not valid and so they will not be discussed. It should be emphasized that all these forecasts are tentative and should be treated with caution. The projected future diffusion paths are not entirely random but rather assume an S-shaped trajectory and all the predictions show a high level of sensitivity to historical data. Special caution is urged regarding the predictions referring to relatively high fixed ceilings like 20%, 25% and 30%, where the accuracy of the forecasts is questionable and they are to some extent misleading and inconclusive.


**Table 4.** Predicted ETFs diffusion patterns (as share of total turnover of index financial instruments). Hungary—outliers excluded. Italics = misspecifications.

#### **5. Conclusions**

This extensive research was designed to analyze the development paths and dynamics of financial innovations introduced on stock exchanges in France, Germany, Spain, Italy—which have been treated as economies with relatively well developed stock exchanges—Hungary and Poland—two European economies where financial innovations have relatively short histories.

We examined the development of ETF markets using descriptive statistics and diffusion models. Graphical evidence on the ETF markets shows that ETF diffusion patterns in some countries may be described as a logistic growth trajectory—characteristic phases of the S-shaped path can be distinguished, which justifies the application of diffusion models. In Hungary and Poland, the level of ETF market development was very low and no significant changes are expected in the future unless the market environment is deeply transformed (which cannot be predicted). The trajectory of ETF market development in the more advanced European economies differed considerably. In Spain and Germany, the ETF market share remained very low and no meaningful predictions could be obtained using diffusion models. In France and Italy, significant development of ETF markets was identified and the predictions indicate potential further growth.

In our research we claimed that ETFs are financial innovations and thus it would be justifiable to analyze their development paths analogously to the process of diffusion of other tangible or intangible innovations. Following the latter, we have proposed to use the mathematical diffusion models, traditionally used to approximation of diffusion patterns of innovations, to draw the trajectories of ETFs diffusion across the financial markets. Our empirical evidence has demonstrated applicability of these diffusion models to the numerical analysis of ETFs diffusion process, allowing for detecting their in-time behavior, case-specific dynamics and development patterns, as well as providing long-term predictions. We believe that this approach to the analysis of financial markets development paves avenues for further and more profound research in this field (similar in-kind conclusions were reached in Marszk et al. (2019)).

Notably, ETFs as innovative financial instruments are still a poorly explored area, including our knowledge on what determines their development, or simply—what enhances or hinders their fast diffusion across financial markets. Apparently, in some countries, ETFs have rapidly gained popularity, while in other their development is negligible. The main limitation of the research method used in this study is that it is derived from the logistic growth function that is based on S-shaped trajectory of the diffusion of innovation which may be inconsistent with the attributes of the financial innovations. Moreover, our analysis has not addressed (with the exception of some preliminary suppositions) the factors that have influenced the diffusion processes. Detecting major determinants of ETFs diffusion in relation to other stock index instruments, including legal and institutional regulations that enable or stop this process, constitutes the direction of possible future research.

**Author Contributions:** Both authors contributed equally. Conceptualization, A.M. and E.L.; methodology, A.M. and E.L.; software, A.M. and E.L.; formal analysis, A.M. and E.L.; investigation, A.M. and E.L.; writing—original draft preparation, A.M. and E.L.; writing—review and editing, A.M. and E.L.; project administration, A.M. and E.L.; funding acquisition, A.M. and E.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by project no. 2015/19/D/HS4/00399 financed by the National Science Centre, Poland. It was also supported by a grant from the CERGE-EI Foundation under a program of the Global Development Network.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **Appendix A**

**Figure A1.** *Cont.*

**Figure A1.** ETFs, stock index options, stock index futures and total index financial instruments—diffusion trajectories. 2004–2015 (monthly time series). Values in USD millions.

**Figure A2.** Current and predicted ETF diffusion patterns. Graphs and forecasts prepared using IIASA software.

#### **References**


Hamid, Nizam, and Jessica Edrosolan. 2009. A comparison: Futures, swaps, and ETFs. *ETFs and Indexing* 1: 39–49.

Hayashi, Fumiko, and Elizabeth Klee. 2003. Technology adoption and consumer payments: Evidence from survey data. *Review of Network Economics* 2: 175–90. [CrossRef]

Hehn, Elizabeth. 2005. Introduction. In *Exchange Traded Funds: Structure, Regulation and Application of a New Fund Class*. Edited by Elizabeth Hehn. Berlin/Heidelberg: Springer, pp. 1–5. [CrossRef]


Goltz, Felix, and David Schröder. 2011. Passive Investing before and after the Crisis: Investors' Views on Exchange-Traded Funds and Competing Index Products. *Bankers, Markets and Investors* 110: 5–20.


World Federation of Exchanges. 2017. *WFE Database*. London: World Federation of Exchanges.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Heads and Tails of Earnings Management: Quantitative Analysis in Emerging Countries**

**Pavol Durana 1,\*, Katarina Valaskova 1, Darina Chlebikova 1, Vladislav Krastev <sup>2</sup> and Irina Atanasova <sup>2</sup>**


Received: 29 April 2020; Accepted: 27 May 2020; Published: 1 June 2020

**Abstract:** Earnings management is a globally used tool for long-term profitable enterprises and for the apparatus of reduction of bankruptcy risk in developed countries. This phenomenon belongs to the integral and fundamental part of their business finance. However, this has still been lax in emerging countries. The models of detections of the existence of earnings management are based on discretionary accrual. The goal of this article is to detect the existence of earnings management in emerging countries by times series analysis. This econometric investigation uses the observations of earnings before interest and taxes of 1089 Slovak enterprises and 1421 Bulgarian enterprises in financial modelling. Our findings confirm the significant existence of earnings management in both analyzed countries, based on a quantitative analysis of unit root and stationarity. The managerial activities are purposeful, which is proven by the existence of no stationarity in the time series and a clear occurrence of the unit root. In addition, the results highlight the year 2014 as a significant milestone of change in the development of earnings management in both countries, based on homogeneity analyses. These facts identify significant parallels between Slovak and Bulgarian economics and business finance.

**Keywords:** business finance; earnings management; EBIT; financial modelling; homogeneity; stationarity; time series methods; unit root

#### **1. Introduction**

The issues of riskmanagement have been analyzed and discussed for along time (Hudakova et al. 2018). The managements of the enterprises must select the best solutions for future development in any conditions (Kral et al. 2019). Spuchlakova and Cug (2015) argue that a structural approach is necessary to reduce and model their business risk. Meyers et al. (2019) highlight big data-driven algorithmic decision-making related to risk management. Vagner (2017) adds that the practical benefits connected with cost controlling and costs optimization and earnings management may be very beneficial for applying to enterprises to risk. Earnings management is an accounting technique to manage financial reports that shows a mostly positive view of business finance and the financial situation. Earnings management means the transformation into a new accounting regime, in a lot of cases (Hoang and Joseph 2019). The reasons for managers to do a manipulation of earnings is good looking for investors and potential investors (Susanto et al. 2019), moreover, Khanh and Thu (2019) declare a positive correlation between earnings management and leverage management. This phenomenon of earnings modification is an increasingly important topic, obviously in the area of the assessment of efficiency—which is a fundamental part of the corporate rational behavior that aims to survive in a challenging competitive

environment in the long term (Balcerzak et al. 2017)—as well as in the areas of financial accounting, financial risk and financial modelling. The most relevant researches have been conducted in the developed markets, but this topic is very rarely investigated in emerging countries, as they still adhere to conventional approaches. The European market significantly varies from other global markets (Rahman et al. 2017).

The number of publications concerned with earnings management changes according to the country. From all the countries, earnings management is the most discussed topic in the USA. Almost five thousand research papers have their origins in this country. We may highlight that developed European countries, such as the United Kingdom (Iatridis and Kadorinis 2009; Pina et al. 2012), Spain (Ferrer Garcia and Lainez Gadea 2013; Rodriguez-Perez and van Hemmen 2010), Germany (Christensen et al. 2015; Velte 2019), Italy (Cimini 2015), the Netherlands (Kempen 2010), Belgium (Andries et al. 2017), France (Bzeouich et al. 2019; Ben Amar and Chakroun 2018) and many others, focus on earnings management phenomenon from different perspectives. However, the issue of earnings management is not developed and investigated properly in emerging countries, with various explanations for this. The identified research gap was analyzed in the case of Slovakia (SK) and Bulgaria (BG). Both countries, former Soviet-controlled Eastern bloc countries, have experienced a massive transformation of their economies over the past decades, and their significant development has been emphasized by their participation in the European Union. Following their economic development (Figure 1) using the gross domestic product (GDP) index, it is evident that the same development trend in the 10-year horizon can be indicated.

**Figure 1.** Gross domestic product (GDP) index. Source: Trading Economics.

Moreover, both countries established mutual cooperation in several areas, one of them is focused on the support of local communities and society by the establishment of the Environmental Partnership Association (EPA) Consortium; they expand the bilateral cooperation in the high-tech sector. The development of mutual cooperation in different sectors of economy forced authors to aim the research at these two countries. Slovakia is the largest producer of cars per capita, with highly developed automobile and electronics exports accounting for more than 80 per cent of national gross domestic product. However, Bulgaria changes its sectoral orientation from agricultural to an industrial economy, which makes the situation in the countries easier to compare. Despite the fact that, according to the rating of the World Bank Country and Lending Groups, Bulgaria is an open upper-middle-income market economy, contrary to Slovakia, which is an open high-income market economy, the importance of the research on the earnings management phenomenon is of a vital importance in both countries. The detection and revelation of manipulation with earnings needs to be portrayed, as it is a relevant measure of investors' and business partners´ protection against risks which may occur if distorted and

incomplete information is presented by the enterprises; thus, it is a helpful tool to solve the basic issues of risk management.

It is evident that the earnings management phenomenon plays an important role in financial reports managing and should be properly investigated in conditions of national economies. However, the fact is that this issue has not been explored properly in both countries analyzed. In Slovakia, the first researches on earnings management were published in 2019, highlighting the importance of this issue in unique country samples. Nonetheless, no relevant research has been published yet on the conditions of Bulgaria. Thus, this study investigates earnings management in both countries and determines its presence by quantitative methods of time series.

The main aim and the essence of the study is to investigate the question of earnings management in Bulgarian and Slovak environments, where the motivation is to detect the existence of earnings management by time series analysis, as this topic is only rarely searched for in emerging countries. The investigation of the presence of manipulation with earnings may help to reveal the reasons for earnings management occurrence. As the issue of manipulation with earnings in both countries is unexplored, the significance of the analysis of unique country samples has to be underlined.

The manuscript is structured as follows. In the introduction, the purpose of the study and the significance of the issue of earnings management are provided. Then, the literature review is presented, concentrating on the analyses of different approaches and investigations of the solved topic. The next chapter depicts the materials used and appropriate methods of mathematical statistics to fulfil the aim. These quantitative methods are: the Dixon test, Jarque–Bera test, Box–Pierce test, Dickey–Fuller tests, Kwiatkowski–Phillips–Schmidt–Shin test, Von Neumann's test, and Standard Normal Homogeneity test. The outcomes of the investigation, as well as the results of the hypotheses, are portrayed in the Results section. This part confirms the existence of the earnings management of Slovak and Bulgarian enterprises and marks the year 2014 as a significant milestone in the development of earnings management in both countries. In the Discussion section, the connection of ease of doing business, annual growth rate of gross domestic product (GDP), long-term unemployment rank and Standard & Poor's outlook to the results is implicated and previous studies from emerging markets are compared. The limitations and weaknesses of our study are noted, and possible avenues of future research are determined in the conclusions of the research.

#### *1.1. Literature Review*

#### 1.1.1. Graphic Modelling of Specific Accruals

The first mention of earnings management is captured in a study of Hepworth (1953), which was focused on balancing periodic income. The author captured several tactics, e.g., methods of balancing income through specific accruals that can be used to move net profit to subsequent accounting periods. Hepworth (1953) did not capture a way to identify the transfer of profits itself.

The initial disclosure of corporate earnings management is based on graphical methods based on data set in the time series. Gordon (1964) examines whether managers choose accounting principles and reporting rules that allow them to balance reported earnings. For each of the enterprises examined, he establishes a curve showing the profit calculated in two ways—excluding and including the dependent variables. If the discrepancies in the observations are smaller in the latter case, the earnings adjustment is due to movements in the account. Dopuch and Drake (1966) create a group of enterprises. For each enterprise, they record the total income and income from the given investment shares. The authors argue that adjusting the earnings with this approach does not pose a serious problem for the enterprise in the group, a certain part of the observed enterprise apparently acts purposefully. Archibald (1967) investigates how and why the set of enterprises has shifted from accelerated depreciation of fixed assets to straight-line depreciation for financial and tax reasons.

#### 1.1.2. Mathematical Modelling of Specific Accruals

Gordon et al. (1966) use mathematical modelling to test the profit equalization. The authors choose the investment credit as a variable to test whether enterprises are trying to balance profits. Copeland (1968) empirically tests the use of more than one variable in revealing the existence of earnings management through additional scrutiny of government financial statements. White (1970) applies other tests, using profits from a decade. He includes several dependent variables in the tests and, for the first time, uses regression as a method to detect enterprises that balance earnings. Dascher and Malcom (1970) perform a test applying data from a six- and eleven-year time interval and draw conclusions about the reduction in semi-logarithmic trend variability attributable to discretionary balance variables. Barefield and Comiskey (1972) use data from a ten-year time series to identify variability and average absolute profit increase in enterprises that may use earnings from non-consolidated subsidiaries to balance.

#### 1.1.3. Modelling of Total Discretionary Accruals with Application of Cross-Sectional Data

Burgstahler and Dichev (1997) detect earnings management on a cross-sectional analysis. In their research, they verify whether the managers of the tested enterprises are trying to avoid a decline in profits or losses. They choose binomial tests to verify the hypotheses in their research and present the results graphically using histograms. Degeorge et al. (1999) focus on exceeding threshold values. The authors conclude that thresholds artificially evoke specific forms of earnings management, with positive thresholds being the most dominant.

#### 1.1.4. Modelling Using Manipulation Score

Beneish (1997) proposes a model detecting earnings manipulation similar to the Altman's bankruptcy model. Variables called M-score capture both the distortion of financial statements and the factors that can stimulate enterprises to manipulate. Beneish (1997) and Young (1999) independently express doubts about the involvement of depreciation in the measurement of total accruals.

#### 1.1.5. Cross-Sectional Earnings Analysis and Accrual Modelling

Peasnell et al. (2000) provide a new approach for approximation of abnormal accruals, labelled as the Margin model, which applied cross-sectional data to mitigate the weaknesses of the Jones model (Jones 1991). The authors take a two-step approach from previous models but use the working capital accrual and different explanatory variables—sales and cash from customers—as an estimate of the total accrual. The authors are criticized for assuming a linear relationship between cash flow and accruals.

#### 1.1.6. Detection of Real Earnings Management

Burgstahler and Dichev (1997) find that enterprises often use cash flow gained by operating activities and working capital to earnings management. Headquarters pursuing specific goals thus change their economic performance and their decisions in order to make a profit. Dechow and Skinner (2000) point out that head officers can modify earnings by shifting revenue differentiation, changing the timing of deliveries, or postponing research and development to keep costs at the desired level. Graham et al. (2005) note that the most commonly used earnings management method is the modification of discretionary accruals thanks to its simplicity, inexpensiveness and difficulty to identify by recipients of financial statements. Roychowdhury (2006) finds that many enterprises stop earnings management through discretionary accruals. The author proves that the modification of discretionary accruals is no more the core way of earnings management. Penman and Zhang (2002) argue that enterprises increase earnings by reducing capital investment. Gunny (2010) states that real earnings management involves changes in the underlying operations and activities of the enterprises to increase earnings in the current period. Eldenburg et al. (2011) run their study in the environment of non-profit organizations, proving the existence of real operational decisions in order to manage earnings.

#### 1.1.7. Modelling Using Neural Networks

Hoglund (2012), because of the insufficient results of previous approaches, applies an alternative way to deal with the nonlinearity of accrual processes through neural networks. He designs models based on self-organizing maps, multilayer networks and general regression.

#### 1.1.8. Modelling of Total Discretionary Accruals with Application of Time Series

Healy (1985) applies average total accruals as an estimate of discretionary accruals, and thus an estimate of earnings management. Healy's model clearly assumes the non-existence of non-discretionary accruals during estimation periods. The author concludes that the accrual policy of managers is related to incentive bonuses, which are enshrined in their contracts, and thus the shift in accounting practices is related to changes of the extra payment schedule. Kaplan (1985) criticized Healy (1985). DeAngelo (1986) supplemented Healy's model with an accrual from the previous period. The model does not assume the existence of non-discretionary accruals in the present interval and uses the non-discretionary accruals from the previous period to estimate them. McNichols and Wilson (1988) add to the DeAngelo model capturing discretionary accruals as measures of earnings management, replacing the total accruals applied by Healy (1985) and DeAngelo (1986).

Jones (1991) investigates earnings management using two-step models during a government investigation of import relief in the United States. It is used an enterprise-specific model, based on data from at least fourteen-year time series. Discretionary accrual, which represents the remainder, prediction error, calculated as the difference among the current total accruals found in the financial statements and the expected non-discretionary accruals. Dechow et al. (1995) modified the original Jones model by supplementing the year-on-year change in receivables, thus eliminating the error of the discretionary accrual estimate. Guay et al. (1996) criticize both the original and the modified Jones model but does not suggest any other alternatives.

Our study also continues in approaches of detecting of earnings management with application of time series. We consider the new gap to disclose the earnings manipulation of the enterprises through unit root and stationarity analysis, supported by homogeneity analyses. The time series analysis allows us to formulate the following hypotheses:


#### **2. Materials and Methods**

The secondary sources are observations of earnings before interest and taxes (EBIT) of the enterprises from the chosen emerging countries (Slovakia and Bulgaria). In the context of historical development, we may add Slovakia to the Soviet-controlled Eastern bloc countries and Bulgaria to the Soviet-controlled Balkans countries. In total, 1347 Slovak enterprises and 1839 Bulgarian enterprises were extracted from the Amadeus database over the period 2010 to 2018 and involved in the analysis. The variable earnings before interests and taxes (EBIT) is selected to eliminate different tax and interest policies of these countries. We require three conditions to be met by the analyzed business units:


These criteria were used to analyze only the companies with stable financial situation and the same financial and economic background to mitigate the problems of the classification of enterprises by their size or the years of their operation.

Following methodological steps were used:

#### 1. The elimination of missing cases.

The database Amadeus provides a large sample of data, but there are some missing cases involved. If we have a sufficiently large data file, we may afford a simple solution in the form of removing those units from the file that have missing values (Svabova and Michalkova 2018). Thus, these observations are necessary to be found and eliminated.

#### 2. The removal of inconsistent cases.

An outlier in a sample is an observation far away from most or all other observations (Ghosh and Vogt 2012). Different methods and tests are used to determine the existence of outliers in raw samples. Svabova and Michalkova (2018) recommend in pre-processing of data in earnings management to use Dixon or Grubbs test. Both tests provide satisfying results in identification of the outlying values (Garcia 2012). "Masking phenomenon" (several observations are close together, but the group of observations is still outlying from the rest of data (Berti-Équille et al. 2015) could occur in our case that is why the Dixon statistics *r*<sup>22</sup> is chosen. This test is designed to be used in situations where additional outliers may occur to minimize the effect of these outliers arise because of masking (Garcia 2012). An avoiding of additional outliers allows using conventional Dickey–Fuller tests in further analysis and prevents the spurious rejection of *Ho* of these tests (Leybourne et al. 1998). The test statistic of Dixon is defined as:

$$r\_{22} = \frac{y\_3 - y\_1}{y\_{n-2} - y\_1} \text{ or } r\_{22} = \frac{y\_n - y\_{n-2}}{y\_n - y\_3} \tag{1}$$

where *y* is an analyzed variable and numbers mean the places in the order.

Nagy (2016) highlights the possibilities after the detection of outliers: do not consider/ignore outliers, exclude outliers or exclude only extreme values (far outliers). We decide to apply the possibility of removal of all inconsistent cases to robust statistics and results insensitive to the outliers which is also supported by the study of Svabova and Durica (2019). They argue that it may be useful to eliminate outlined enterprises from the analyzed group because of the fact that outliers may generate discrepancies of conclusions of statistical tests and procedures. We run test and its *p*-values are estimated with a Monte Carlo simulation using 1,000,000 replicates.

#### 3. The verification of normal distribution.

Normally distributed sample is a required assumption in the estimation of attributes of the times series (Bai and Ng 2005). There are nearly 40 tests of normality in the statistical literature (Dufour Jean-Marie et al. 1998). Bai and Ng (2005) recommend testing the normality of time series of financial data by the Jarque–Bera test. Jarque and Bera (1980) and Bera and Jarque (1981) show their test statistics as follows:

$$JB = \frac{n}{6} \left( \mathbf{S}^2 + \frac{1}{4} (\mathbf{K} - \mathbf{3})^2 \right) \tag{2}$$

$$S = Skewness = \frac{\widehat{\mu}\_3}{\widehat{\sigma}\_3} = \frac{\frac{1}{n} \sum\_{i=1}^{n} \left( y\_i - \overline{y} \right)^3}{\left( \frac{1}{n} \sum\_{i=1}^{n} \left( y\_i - \overline{y} \right)^2 \right)^{\frac{3}{2}}} \tag{3}$$

$$K = Kurtosis = \frac{\hat{\mu}\_4}{\hat{\sigma}\_4} = \frac{\frac{1}{n} \sum\_{i=1}^n \left( y\_i - \overline{y} \right)^4}{\left( \frac{1}{n} \sum\_{i=1}^n \left( y\_i - \overline{y} \right)^2 \right)^2} \tag{4}$$

where *y* is an analyzed variable, *n* means all amount of observations, μˆ <sup>3</sup> and μˆ <sup>4</sup> mean the approximations of third and fourth central moments, *y* means the average of the sample, σˆ <sup>2</sup> is the approximation of the second central moment, the variance. Jarque–Bera is asymptotically χ2 distributed with two degrees of freedom because test statistics of Jarque–Bera test is just the sum of squares of two asymptotically independent standardized normals (Bowman and Shenton 1975).

#### 4. The proof of no serial correlation.

The occurrence of no serial correlation means that the data are independently distributed, and it is a recommended assumption for financial time series after testing normality. The Box–Pierce and Ljung–Box tests are generally run to test the required independence in time series. Box and Pierce (1970) perform the test of the randomness at each distinct lag in their study. Ljung and Box (1978) modify this test to overall randomness. We prefer the robustness of the Box–Pierce Q statistic to test if the analyzed sample of financial data is uncorrelated without assuming statistical independence.

$$Q = n \sum\_{k=1}^{h} r\_k^2 \tag{5}$$

*Q* is the Box–Pierce test statistic, which is compared with the χ2 distribution; *n* means all amount of observations; *h* is the maximum lag we are considering (Box and Pierce 1970).

5. The determination of unit root and disproof of stationarity.

A time series is stationary if its statistical properties do not change in the process of time. A stationary time series means that the mean and variance are constant over time. The white noise is an example of a stationary time series. The determination that a series is not stationary enables to study where the non-stationarity comes from. Stationarity tests may determine whether a series is stationary or not. There are different approaches on how to test stationarity (unit root or stationarity tests). Unit root tests, as the Dickey–Fuller test and its augmented version, for which *H*<sup>0</sup> is that the series possesses a unit root and thus is not stationary. On the other hand, there are stationarity tests as the parametric Kwiatkowski–Phillips–Schmidt–Shin test or nonparametric Phillips–Perron test, for which *H*<sup>0</sup> is that the series is stationary. Standard Dickey–Fuller tests can have very low power and can lead to a very serious problem of spurious rejection of the unit root *H*<sup>0</sup> (Leybourne et al. 1998) and thus we support the tests by Kwiatkowski–Phillips–Schmidt–Shin test. Dickey and Fuller (1979) show three different equations to test the occurrence of unit root:

$$
\Delta y\_t = \gamma y\_{t-1} + \varepsilon\_t \tag{6}
$$

$$
\Delta y\_t = a\_0 + \gamma y\_{t-1} + \varepsilon\_t \tag{7}
$$

$$
\Delta y\_t = a\_0 + \gamma y\_{t-1} + a\_2 t + \varepsilon\_t \tag{8}
$$

where Δ*yt* is first order linear differential of equation, γ is unit root, ε*<sup>t</sup>* is white noise. The difference between these deterministic elements is *a*<sup>0</sup> and *a*2*t*. Under the null hypothesis, Equation (6) represents a pure model of random walk, Equation (7) adds the intercept *a*0, and Equation (8) contains both the *a*<sup>0</sup> as well as linear time trend *a*2*t*. Hacker and Hatemi-J (2010) argue that it is difficult to choose from the three Dickey–Fuller equations for unit root testing. According to Elder and Kennedy (2001), if the trend is mistakenly included, the strength of the test drops. On the contrary, if the trend is not included, there is only one way to capture the trend—to use the intercept to detect the trend. All three tests are computed to compare their results and strength in our analysis.

The Kwiatkowski–Phillips–Schmidt–Shin test verifies if a time series is stationary around a mean or linear trend or is non-stationary due to a unit root (Kwiatkowski et al. 1992). Time series is divided into the sum of the random walk *rt*, deterministic trend ξ*t*, and stationary errors ε*t*:

$$y\_t = r\_t + \xi t + \varepsilon\_t \tag{9}$$

where *rt* is random walk:

$$r\_t = r\_{t-1} + u\_t \tag{10}$$

where *ut* are independent and identically distributed random variables 0, σ<sup>2</sup> *u* . *Risks* **2020**, *8*, 57

It is used ω statistics for testing:

$$
\omega = \frac{\sum\_{t=1}^{T} \mathcal{S}\_t^2}{T^2 \mathfrak{d}\_\varepsilon^2} \tag{11}
$$

where

$$S\_t = \sum\_{i=1}^t c\_i \tag{12}$$

and σˆ <sup>2</sup> <sup>ε</sup> is the estimate of long-term variance *et*:

$$\left\|\boldsymbol{\hat{\sigma}}\_{t}^{2} = \lim\_{T \to \infty} \frac{E}{T} \left[ \left( \sum\_{t=1}^{T} \boldsymbol{\varepsilon}\_{t} \right)^{2} \right] \tag{13}$$

#### 6. The determination of heterogeneity.

Homogeneity tests allow detecting if time series may be considered as homogeneous during the analyzed time period, or if there is any date at which significant change in a mean of data occurred. Kanovsky (2018) and Agha et al. (2017) recommend selecting from von Neumann test, standard normal homogeneity test, Buishand tests, and Pettitt's test. We apply the von Neumann test to detect the existence of significant changepoint in the earnings management and parametric standard normal homogeneity test to determine a year when a significant change occurs. Von Neumann's test is a test using the ratio of mean square successive (year to year) difference to the variance (Von Neumann 1941). The test statistic is shown as follows:

$$N = \frac{\sum\_{i=1}^{n-1} (y\_i - y\_{i+1})^2}{\sum\_{i=1}^{n-1} (y\_i - \overline{y})^2} \tag{14}$$

The null hypothesis is that the data are dependent. If the value of *N* is equal to 2, it means that the sample is homogeneous while the values of *N* less than 2 indicate that the sample has a breakpoint (Buishand 1982). This test gives no information about the break point.

The standard normal homogeneity test is a method created by Alexandersson (1986) and assumes if a times series is normally distributed (Kang and Yusof 2012). Then the following model with a single change can be proposed according to Pohlert (2016) as:

$$y\_i = \begin{cases} \mu + \epsilon\_i & i = 1, \dots, m \\ \mu + \Delta + \epsilon\_i & i = m + 1, \dots, n \end{cases} \tag{15}$$

ε ≈ *N* (0, σ). The null hypothesis Δ = 0 is tested against the alternative hypothesis Δ - 0. The test statistic is:

$$T\_k = kz\_1^2 + (n-k)z\_2^2 \quad (1 \le k \le n) \tag{16}$$

where

$$z\_1 = \frac{1}{k} \sum\_{i=1}^{k} \frac{y\_i - \overline{y}}{\sigma} \quad z\_2 = \frac{1}{n-k} \sum\_{i=k+1}^{n} \frac{y\_i - \overline{y}}{\sigma} \tag{17}$$

The critical value is:

$$T\_0 = \max\_{1 \le y \le k} T\_k \tag{18}$$

The *p*-value is estimated by a Monte Carlo simulation using *m* replicates. We run test and its *p*-values are estimated with a Monte Carlo simulation using 1,000,000 replicates.

#### **3. Results**

This part consists of pre-processing data, testing of assumptions and processing results.

#### *3.1. Pre-Processing of Data*

The samples were very wide but consisted of significant amount of missing values. These values of EBIT were found and eliminated from the Slovak sample of enterprises as well as the Bulgarian one. Table 1 involves the number of missing values.

**Table 1.** Investigate samples.


The detection of inconsistent data (outliers) follows the identification of missing values. Dixon test is used in the analysis. Testing is run for every observation for each year from the analyzed nine-year period. The outlying cases are detected for every year. The enterprise is removed from the analysis for all periods if only one value is detected as an outlier. The Dixon test is created for small sample despite this fact we use it for its robustness. The *p*-value was computed using 1,000,000 Monte Carlo simulations. The existence of minimal one outlying value of EBIT for Slovak and Bulgarian samples in every analyzed year is confirmed based on *p*-value computed in Table 2, which portrays the amount of outlying cases of enterprises for both sides and the final sample as well.


**Table 2.** Dixon test.

Source: own research.

Based on annual values of EBIT of 1089 Slovak enterprises and 1421 Bulgarian enterprises, annual average EBIT is calculated for the analyzed period from 2010 to 2019 (Table 3). The development of both countries in time is very similar, which is shown in Figure 2. The similarities of the development of EBIT is also supported by 5% error bars (calculated based on standard deviation) which show almost identical coverage of EBIT development in seven years from the nine-year analyzed period.


**Table 3.** Annual average EBIT of enterprises.

Source: own research.

**Figure 2.** The values of average EBIT with error bars. Source: own research.

#### *3.2. Testing of Assumptions*

It is necessary to prove the assumption concerned with the normality (Jarque–Bera test) on the one hand and on the other hand to prove the assumption of serial correlation (Box–Pierce test). These tests are run for the series of EBIT of enterprises before testing any significant occurrence of earnings management or occurrence of the significant year of the change in the earnings management in Slovak or Bulgarian enterprises.

As the computed *p*-value is greater than the significance level alpha, one cannot reject the null hypothesis *H*<sup>0</sup> in Slovak case as well as in Bulgarian case, based on Table 4. It is not rejected based on Jarque–Bera test that the sample of EBIT of extracted Slovak and Bulgarian enterprises follows a normal distribution. The test of the randomness of the sampling process is running after proving the normality.


**Table 4.** Jarque–Bera test.

Source: own research.

As the computed *p*-value is greater than the significance level alpha, one cannot reject the null hypothesis *H*0, based on Table 5. It is not rejected based on Box–Pierce test that the data of EBIT of Slovak enterprises, as well as Bulgarian enterprises, exhibit no serial correlation.


Source: own research.

#### *3.3. Processing of Results*

After proving normality and confirmation of no serial correlation significant, the occurrence of earnings management and significant year of the change in the earnings management in Slovak or Bulgarian enterprises are testing. These investigations are realized by stationarity tests and homogeneity tests.

Firstly, Dickey–Fuller test of unit root for No intercept, Intercept, and lastly Intercept + Trend is used. Null hypothesis indicates that the series possesses a unit root and hence it is not stationary. It means, statistical properties of EBIT of Slovak and Bulgarian enterprises vary with time. The earnings management exists, the managerial activities are not random, but the managers of the enterprises purposefully manipulate earnings within the legal barriers.

• **H1A.** *There is no unit root for the series of EBIT. There is no significant existence of the earnings management.*

As the computed *p*-value is greater than the significance level alpha, one cannot reject the null hypothesis *H*0*A*, based on Table 6. It is not rejected following the Dickey–Fuller test of unit root for No intercept, Intercept, and Intercept + Trend, that there is a unit root for the series of EBIT. There is significant existence of the earnings management of Slovak and Bulgarian enterprises.



Source: own research.

Secondly, the result of the Dickey–Fuller test of unit root is recommended to be supported by the Kwiatkowski–Phillips–Schmidt–Shin test of stationarity for Level and Trend. The null hypothesis is, on the contrary to Dickey–Fuller test, that the verification of series of EBIT is stationary.

• **H1B.** *The series of EBIT is not stationary. There is significant existence of the earnings management.*

As the computed *p*-value is lower than the significance level alpha, one should reject the null hypothesis *H*0*B*, and accept the alternative hypothesis *H*1*B*, based on Table 7. It is rejected following the KPSS test of stationarity for Level and Trend that the series of EBIT is stationary. These results confirm the conclusions of previous tests of no stationarity in managerial activities but significant managing earnings in Slovak and Bulgarian enterprises.



Thirdly the existence of a significant change in the earnings management is detected. After the identification of significant occurrence of earnings management in both countries, it is required to determine if the mean of the development is homogenous all the time or if the heterogeneity exists (significant change of mean). Von Neumann's test is run to detect the homogeneity of the series of EBIT.

• **H1C.** *The series of EBIT is heterogeneous. There is a significant change in the earnings management.*

The *p*-value of von Neumann's test was computed using 1,000,000 Monte Carlo simulations. As the computed *p*-value is lower than the significance level alpha, one should reject the null hypothesis *H*0*C*, and accept the alternative hypothesis *H*1*C*, based on Table 8. It is rejected following the von Neumann test of homogeneities of EBIT. Thus, there is a significant change in the earnings management of Slovak and Bulgarian enterprises.


**Table 8.** von Neumann's test.

Source: own research.

#### **4. Discussion**

Von Neumann's test indicates heterogeneity in the series of EBIT, but not a year of the significant change. This situation may mark the occurrence of the year that divides the development of EBIT of Slovak and Bulgarian enterprises into two homogenous groups. These groups are differentiated by the year of the change, they do not have only one mean of the development, but each has own central mean line of the development. The standard normal homogeneity test is run to detect a year of a significant change in the earnings management and the values of both central mean lines of development labelled mu.

• **H1D.** *The series of EBIT is heterogeneous. There is a year of a significant change in the earnings management.*

The *p*-value of SNHT was computed using 1,000,000 Monte Carlo simulations. As the computed *p*-value is lower than the significance level alpha, one should reject the null hypothesis *H*0*D*, and accept the alternative hypothesis *H*1*D*, based on Table 9. It is rejected following the SNHT that no year of a significant change of EBIT does exist. Table 9 and Figures 3 and 4 involve indicated year as well as indicated central mean lines of development. The year 2014 is the year of significant change in the earnings management of Slovak and Bulgarian enterprises. This year divides the development of EBIT and determines the individual central line. The difference between calculated central lines of EBIT of Bulgarian and Slovak enterprises is not very noticeable. It is EUR 92,000 until 2014 and EUR 110,000 since 2014. The finding also confirms some parallel of these emerging economies.


**Table 9.** Standard normal homogeneity test (SNHT).

Source: own research.

**Figure 3.** Significant change in the earnings management of Slovak enterprises. Source: own research.

**Figure 4.** Significant change in the earnings management of Bulgarian enterprises. Source: own research.

Our findings concerned with the significance of the year 2014 for Slovak and Bulgarian enterprises are supported by Figure 5. It shows the rank of ease of doing business in Slovakia and Bulgaria. The lowest value of this index, the better for business. This year also divides the development of ease

of doing business as in the case of earnings management into two periods with very homogenous development within own group but very heterogeneous comparing the groups. The year 2014 saw a breaking point in the improvement of business conditions in both countries. These results of assessment of the business environment were expertly evaluated by global monitoring of the entrepreneurship (Madgerova et al. 2019).

**Figure 5.** Ease of doing business rank. Source: own research.

The positive macroeconomic environment influences earnings manipulation of Slovak and Bulgarian enterprises and implicates significant change (heterogeneity) in the earnings management. Despite the ease of doing business rank, the positive macroeconomic environment in 2014 was represented by the annual growth rate of GDP, since this year it has been above zero and the annual growth rate has had rapid upward tendency (Figure 6). This year was the milestone also in decreasing of the long-term unemployment rank in both countries (Figure 7). Last but not least, Standard & Poor's set positive outlook related to the credit rating for Slovakia and stable outlook for Bulgaria in the year 2014 (Trading Economics 2020) as well as the cohesion policy has changed new development environment within the European Union after the year 2014, based on the new programming period (Marin and Dimitrov 2018).

**Figure 6.** Annual growth rate of GDP. Source: Trading Economics.

**Figure 7.** Long-term unemployment rank. Source: Trading Economics.

Our results are discussed with the last studies and investigations from emerging markets. Cugova et al. (2019) analyze various forms of earnings and the subsequent analysis of the profitability indicators of the engineering companies operating in the Slovak Republic. They analyze the period of 2012–2017. Both profits and profitability indicators show an increasing tendency, and in the last few years, they achieved impressive results. The performed analyses are only an elementary basis for earnings management. Orazalin (2019) focuses on earnings management activities of enterprises from Kazakhstan. This study indicates that enterprises with larger boards adopt a more restrained concept to earnings management manipulation. However, the conclusions deliver not significant proof of the connection among board independence and earnings quality. The study of Orazalin and Akhmetzhanov (2019) an identify the force of earnings management and audit quality on the cost of debt in Kazakh enterprises. The result portrays that earnings management is negatively depended on the cost of debt. Their conclusions highlight higher audit quality means to a lower cost of debt and confirm no significant influence of audit quality on earnings management. Valaskova et al. (2019) evaluate the robustness of selected models in automotive of Slovakia. They analyze Jones' model and the modified Jones model, and find that the original Jones model is the most appropriate in identifying the earnings management in that environment. Pavlovic et al. (2019) investigate if the board of directors' age impact earnings management practices. The sample consists of all Serbian agriculture enterprises from Belgrade Stock Exchange for the interval of years 2013 to 2016. To detect the earnings management the modified Jones Model is used, which is shown as the most appropriate. The results indicate that there is no impact of board of directors' age on earnings management practices. They also find no evidence of the impact of the chairman's age on earnings management practices. Relationship between gender diversity and earning management practices has not been found. Pavlovic et al. (2018) suggest that there is an insignificant negative linear relationship between the number of women in the board and earnings management. These findings are supported by the studies which indicate that the reasons for earnings management should be found in different factors, like cultural and political factors or religious attitude or age of the members of the boards but not on the gender differences. Piosik and Genge (2019), analyze enterprises from the Warsaw Stock Exchange in Poland and detect the negative dependency among total upward real earnings management and managerial ownership. Their study argues that specific tools of real earnings management are connected to the ownership assembly and managerial ownership in individual cases. Sosnowski and Wawryszuk-Misztal (2019) also used a sample from the Warsaw Stock Exchange and they reveal that some attributes of the supervisory board raise the effectivity of forward-focusing financial data connecting the initial public offering (IPO) prospectus, as some of boards attributes have the impact on the assessment of the earnings approximation credibility at the realizing of the IPO. Sosnowski (2018) confirms no proof of the existence of private equity fund between the shareholders of the enterprise in the time of preceding first listing of stocks on a market constrains the applying of earnings management prior to the IPO. He does not reject that any significant discrepancy exists between the discretionary accruals in private equity backed and matched enterprises, when controlling for the market value and book-to-market ratio. Lizinska and Czapiewski (2018) disclose positive and significant discretionary accruals in the IPO year that can be considered as an indication of weak earning quality. They depict that analyzed accruals are indirectly depended on the subsequent long-term market value for IPOs realized before the global recession. Istrate (2019) confirms the increased rounding of earnings in a limited amount of units, even if the amplitude of identified gaps is really significant. The development of the accounting regulation tends to the state when it has begun preferring decreased modifications. The International Financial Reporting Standards (IFRS) transition does not tend to a limitation of the gap among the real occurrence and the normal one. This study finds out that smaller enterprises modify the net income not so significantly upward than the larger enterprises. Turlea et al. (2019) provide results from Romania concerned to the impact of granted by the auditor when the value of discretional accruals is encountered and approximate the impact on the mandatory implementation of IFRS. They estimate the value of discretional accruals by the value of residuals from two equations as regression models that calculate and detect the value of total accruals. The paper of Tanchev and Todorov (2019) examines the long-run and short-run tax buoyancies. They empirically test the impact of the buoyancy on income, profit, and consumption increases in Bulgaria.

#### **5. Conclusions**

The effective business finance is a key core of the success of all enterprises to be profitable in short as well as long-term period. The globally used phenomenon of earnings management allows a legal opportunity for the enterprises to make a purpose-built decision in the profit policy. Earnings management is widely realized in developed countries and the occurrence is comprehensively mapped. However, the aim of this paper was to detect the existence of earnings management in emerging countries by the times series analysis. Our results confirm that also managers of Slovak and Bulgarian enterprises are not static but significantly manage their earnings during the analyzed nine-year period. Earnings management creates an important part of coherent business finance. It supports the annual prosperity of the enterprises and presents a substantial tool of reducing risk in analyzed emerging countries.

The weakness of the provided research is the use of annual average values of EBIT. Panel data for the whole analyzed period may be used in further research. The analysis could be extended for all Soviet-controlled countries to disclose a comprehensive view on the issue of earnings management in these countries with similar historical and political development. We run only Dickey–Fuller test to detect the existence of significant change in the earnings management of Slovak and Bulgarian enterprises, not its modified versions. Further research may support these results by additional use of these tests. The standard normal homogeneity test was used to determine the year of a significant change in the earnings management of Slovak and Bulgarian enterprises, but this test is very sensitive when detecting the breaks near the beginning and the end of the series. Our results focus a priori on the parametric test. In the future investigations, the results of Dickey–Fuller tests may be compared with a nonparametric Phillips–Perron test, Kwiatkowski–Phillips–Schmidt–Shin test with the nonparametric test for stationarity in continuous-time Markov processes, von Neumann's test and standard normal homogeneity test with nonparametric Pettitt's test.

**Author Contributions:** Conceptualization, P.D. and V.K.; methodology, P.D.; software, P.D.; validation, D.C., V.K., K.V. and I.A.; formal analysis, V.K., K.V. and I.A.; investigation, P.D. and K.V.; resources, D.C., K.V. and I.A.; data curation, K.V.; writing—original draft preparation, P.D.; writing—review and editing, K.V.; visualization, P.D.; supervision, D.C., V.K., K.V. and I.A.; project administration, K.V.; funding acquisition, K.V. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** This research was financially supported by the Slovak Research and Development Agency—Grant NO. APVV-17-0546: Variant Comprehensive Model of Earnings Management in Conditions of the Slovak Republic as an Essential Instrument of Market Uncertainty Reduction and VEGA 1/0210/19: Research of innovative attributes of quantitative and qualitative fundaments of the opportunistic earnings modeling.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


Trading Economics. 2020. Available online: https://tradingeconomics.com/indicators (accessed on 20 April 2020).


#### *Risks* **2020**, *8*, 57


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **A Raroc Valuation Scheme for Loans and Its Application in Loan Origination**

#### **Bernd Engelmann and Ha Pham \***

Faculty of Finance-Banking, Ho Chi Minh City Open University, 35-37 Ho Hao Hon, Dist 1, Ho Chi Minh City 700000, Vietnam; bernd.engelmann@ou.edu.vn

**\*** Correspondence: ha.p@ou.edu.vn

Received: 27 April 2020; Accepted: 4 June 2020; Published: 10 June 2020

**Abstract:** In this article, a risk-adjusted return on capital (RAROC) valuation scheme for loans is derived. The critical assumption throughout the article is that no market information on a borrower's credit quality like bond or CDS (Credit Default Swap) spreads is available. Therefore, market-based approaches are not applicable, and an alternative combining market and statistical information is needed. The valuation scheme aims to derive the individual cost components of a loan which facilitates the allocation to a bank's operational units. After its introduction, a theoretical analysis of the scheme linking the level of interest rates and borrower default probabilities shows that a bank should only originate a loan, when the interest rate a borrower is willing to accept is inside the profitability range for this client. This range depends on a bank's internal profitability target and is always a finite interval only or could even be empty if a borrower's credit quality is too low. Aside from analyzing the theoretical properties of the scheme, we show how it can be directly applied in the daily loan origination process of a bank.

**Keywords:** loan pricing; RAROC; loan origination

**JEL Classification:** C69; C19

#### **1. Introduction**

A loan is probably the most traditional banking product. However, when different people in different countries or even different people in the same country working in different customer segments speak about a loan, and they probably do not speak about the same product. The only common feature is that a lender gives money to a borrower and hopes to get back more than he has lent. Besides that, differences can be substantial. Critical drivers of product structure are the availability of funding and collateral. In some countries, only short-term funding is available to a bank. For this reason, interest rates of loans are rarely fixed over a long-term horizon but can be adjusted by a bank on short notice. In other countries, long-term funding is available, and loans are often fixed-rate or floating-rate loans, where the floating rate follows an objective rule like 6M Ibor plus a spread.1 The most prominent type of loans that is linked to a particular collateral type is the mortgage. Here, often long maturities up to 30 years are observed. However, still, there are some differences between countries. For instance, in the US, a borrower can pass the key of a house to a bank when the house loses in value while in Germany defaulting on a mortgage is not that easy, and the borrower still is responsible for the residual amount between the loan balance and house value. There are a lot more differences. In some countries, borrowers have prepayment rights on their loans; in other countries, prepayment rights

<sup>1</sup> In this article, Ibor stands for all kinds of official floating rates like Libor, Euribor, etc.

are less popular, but floating-rate loans are embedded with caps and floors. A lot more covenants can be included, like interest rates that increase with rating downgrades or minimum requirements on collateral value.

In this article, we develop a loan pricing scheme based on risk-adjusted return on capital (RAROC) as a performance measure. The purpose of this article is twofold. First, we propose a scheme that is directly implementable in banking practice drawing on input data that is readily available in most banks. This data consists of quotes from the interbank market, like deposit and swap rates, internal costs for funding and operation, and credit risk parameters related to borrower and collateral, i.e., statistical default probabilities and loss rates. We explicitly assume that no market information related to borrower credit quality, like bond or CDS spreads, is observable.

The proposed scheme is most valuable during loan origination where it provides a bank not only with the loan performance related to a particular offer of the loan's interest rate but, in addition, with a decomposition of the interest rate into cost components associated with different bank operations related to lending. These are the funding of a bank loan by deposits, the management of loss risks, the hedging of interest rate risks, and the coverage of operational costs. Therefore, the scheme could be helpful in determining fund transfer prices between a bank's operational units.

In addition, the scheme delivers a loan valuation. This could be valuable in price negotiations when loan portfolios are sold to investors. In economies with negative interest rate like the European Union, life insurers and pension funds are increasingly attracted by residential mortgages (European Central Bank 2019) where they are still able to get a positive return in contrast to most government bonds. To determine a price during sales negotiations and for the investor's ongoing reporting, our proposed model could be applied.

The second contribution of this article is an analysis of the RAROC scheme's theoretical properties. Roughly speaking, RAROC is defined as (interest income − costs) / capital. On first glance this means if a bank would raise interest rates, leaving all other quantities unchanged, profitability would increase. However, it is intuitively clear that raising interest payments beyond the income of a borrower or the profit of a firm will lead to a default. This means, an economically meaningful performance measure should reflect this by having a low value in this scenario. We will show that, when properly relating borrower default probability and loan interest rate, this is indeed reflected by RAROC, i.e., that there is only a finite interval of interest rates where it is economically sensible for a bank to originate a loan. The market power of banks in a particular loan segment will then determine whether a bank can originate a loan at the lower or the upper end of this interval. This interval might be empty in cases where a borrower's credit quality is too low, meaning that a bank should not originate a loan to this borrower regardless of the interest rate.

The general pricing framework presented in this article is not linked to a particular loan segment and is applicable both for corporate and retail lending. In the next Section, we will provide a literature review. In Section 3, the loan pricing formula and its parameterization will be explained. In Section 4, the RAROC pricing scheme will be developed, and the calculation of all its cost components will be derived. After that, in Section 5 the theoretical properties of this scheme will be analyzed by linking the level of interest rates to the default rates of a borrower, and it will be shown how meaningful loan acceptance rules can be derived. In Section 6, numerical examples are presented for illustration. The final Section concludes.

#### **2. Literature Review**

Our aim is developing a RAROC scheme that is applicable in banks world-wide drawing on inputs that are readily available in most banks. This is data related to internal costs and risk parameters measured statistically possible amended by expert judgment. As already outlined in the introduction, we explicitly assume that there is no market information of a borrower available, i.e., the borrower did not issue any bonds but all his debt consists exclusively of bank loans. This implies that the stream of literature using risk-neutral probabilities for asset pricing, e.g., as in Jarrow et al. (1997) or

Choi et al. (2020), which are based on trading strategies in arbitrage-free markets, is not applicable in this context. A recent article on fair value approaches for loans is Skoglund (2017) where the utilization of market data for loan valuation is discussed. In most loan markets world-wide this is not feasible since the required market data is available only in countries with developed capital markets. Even in these countries, market valuations can only be applied to a small segment of the loan market, typically large corporates.

The RAROC concept dates back to the 70s, where it has been developed by practitioners. Often, it is applied in a one-period model analyzing one year only even if the loan maturity is longer (Crouhy et al. 1999). The earliest extensions of RAROC to a multi-period framework are Aguais et al. (1998); Aguais and Forest (2000); Aguais and Santomero (1998). However, the description is very sketchy and loan pricing aspects that became important after the financial crisis are, of course, not included since these articles were written well before. Besides that, there is no split of the interest rate into cost components related to different operational units of a bank provided. Closing this gap is one purpose of this article.

When applying a RAROC scheme in practice, a number of aspects have to be considered that are not covered by our work but are required as inputs. These are the definition of capital for performance measurement, the allocation of total bank capital to sub-units, and the determination of target equity returns. As outlined in the introduction, RAROC is broadly defined as (interest income − costs)/capital. There are two definitions of capital, regulatory capital that is defined by supervisory rules Basel Committee on Banking Supervision (2006, 2011), and economic capital. Economic capital is measured by a credit portfolio model that should reflect economic reality more closely than the regulatory rules, e.g., by quantifying concentration risks. Most economic capital models are based on Gupton et al. (1997) and CSFB (1997). Once the loss distribution of a credit portfolio is computed a risk measure is derived. Usually expected shortfall is used to compute portfolio risk and the risk is distributed among the single credits in a process called capital allocation. For details see Kalkbrener et al. (2004), Kalkbrener (2005) and Balog et al. (2017). Both academia and practice have used predominantly economic capital models for performance measurement. A recent study is Chun and Lejeune (2020) where a multi-period model for loan pricing and profit maximization in an economic capital framework is developed. Compared to our work they do not model the interaction of default probabilities and interest rates but link the interest rate to a probability that a borrower accepts a loan offer. Besides that, their framework cannot be used for fund transfer pricing as they do not provide a loan's cost components.

As pointed out by Klaassen and Van Eeghen (2018), economic capital usually exceeded regulatory capital in the past. For this reason, measuring loan performance by economic capital did not interfere with the regulatory rules since it was more conservative. However, this has changed with the recent Basel reforms resulting in higher minimum requirements of regulatory capital. When regulatory capital is higher than economic capital, performance measurement based on economic capital no longer works because it is overstating true performance. This motivated banks to move from economic to regulatory capital for performance measurement, as confirmed by a survey in (Ita 2016, p. 38), and recent empirical research (Akhtar et al. 2019; Oino 2018). The RAROC scheme in this article does not depend on a particular capital definition and will work with either economic or regulatory capital. We will use regulatory capital for illustration and concrete examples because it is more tractable and increasingly more common in practice.

An important process where RAROC (or the related alternative RORAC) is applied, is the allocation of bank capital to business units. Here, a bank faces the problem that the managers of business units, who have only limited information about the total bank, should optimize the performance of their unit in a way leading to the maximum performance of the organization. The first article, demonstrating the usefulness of RAROC for this purpose is Stoughton and Zechner (2007). Extensions of this work are Buch et al. (2011), Baule (2014), Turnbull (2018) and Kang and Poshakwale (2019) where the latter even empirically demonstrate the usefulness of this approach. In this article, we do not analyze capital allocation on the business unit level but assume

that this is done already. Instead, we focus on the loan-level and the costs and risks associated with each individual loan.

To evaluate loan performance, banks have to define a profitability target for their equity capital. This could be done by an application of the CAPM (Capital Asset Pricing Model). As pointed out by Crouhy et al. (1999) a uniform hurdle rate for all banking activities could lead to erroneously rejecting low-risk and accepting high-risk loans. Miu et al. (2016) propose a framework where target profitability can be defined on a granular basis for loan segments more accurately reflecting their riskiness. However, the application of their framework requires traded equity of borrowers, which limits its usefulness to a small part of the global loan markets. The determination of profitability targets is not part of this article. We assume this has been defined either quantitatively or by expert judgment, bank-wide or individually for loan segments. Since we focus on the individual loan, the scheme will work with either definition of profitability target.

Finally, we point out that we do not model the interaction of borrowers and lenders or the loan market as a whole. Modeling loan prices using equilibrium models is done, for instance, in Greenbaum et al. (1989); Repullo and Suarez (2004) and Dewasurendra et al. (2019). While these models shed some insights on the effect of regulatory rules on loan prices, they are of limited practical value as they miss out important aspects like loan structure or costs for interest rate risk management. An interesting example of modeling the borrower–lender interaction is Zhang et al. (2020) analyzing the special case of retailers loan decision in inventory financing under different bank loan offerings depending on the regulatory regime. In this article, we entirely focus on the bank and the costs associated with a loan. We will derive a set of interest rates under which it is profitable for a bank to offer a loan. Under what conditions or if at all a prospective borrower will accept an offer, is not part of this work.

#### **3. Loan Pricing Formula**

In this Section, the loan pricing formula and its parameterization are outlined. This formula builds the basis of the RAROC pricing scheme, which is explained in the next Section. Some aspects of the parameterization might not be apparent immediately but will be justified in the next Section. The value of a loan will be defined as the expected present value of all future cash flows. These are the interest rate payments, the amortization payments of the loan's notional, and the liquidation proceeds of collateral in the case of a borrower default. The general expression for a loan's value *V* at time *t* is given as

$$V(t) = \sum\_{t < T\_i} \left( N\_i \mathbf{z}\_i \mathbf{r}\_i + A\_i \right) \delta(T\_i) \upsilon(T\_i) + \sum\_{t < T\_i} N\_i R\_i \delta(T\_i) \left( \upsilon(T\_{i-1}^\*) - \upsilon(T\_i) \right), \tag{1}$$

where *Ti* is the interest rate payment time in period *i*, *T*<sup>∗</sup> *<sup>i</sup>*−<sup>1</sup> <sup>=</sup> max(*Ti*−1, *<sup>t</sup>*), *<sup>τ</sup><sup>i</sup>* is the year fraction of interest period *i*, *Ni* the outstanding notional in each period, *Ai* the amortization payments, *Ri* the recovery rate in case of a default in period *i*, *δ*(*Ti*) is the discount factor corresponding to time *Ti*, and *v*(*Ti*) the survival probability of the borrower up to time *Ti*. It is assumed that default is recognized in payment times only and that the recovery rate summarizes the liquidation proceeds in case of default discounted back to default time.

The quantities *Ni*, *Ai*, and *zi* are defined by the loan terms. The amortization payments depend on the loan structure, i.e., whether the loan is a bullet loan, an installment loan or an annuity loan. The interest rate *zi* could be fixed or floating. In the case of a fixed-rate loan, the interest rate in each period is *y*, where we assume that *y* is fixed and period-independent. In the case of a floating-rate loan, the interest rate is (*Li* + *s*), where *Li* is an Ibor rate, which is fixed at the beginning of each interest rate period *i* and *s* is a spread that is assumed constant throughout the loan's lifetime. We will use the notation *zi* for the interest rate in period *i* with

$$z\_i = \begin{cases} \quad y\_\prime & \text{if the loan's interest rate is fixed,} \\ \quad f\_i + s\_\prime & \text{if the loan's interest rate is floating.} \end{cases} \tag{2}$$

where *fi* is the forward rate corresponding to the floating rate *Li*. To compute forward rates a second discount curve is needed, which will be denoted with *δM*(*t*). Forward rates are computed as

$$f\_i = \frac{1}{\pi\_i} \left( \frac{\delta\_M(T\_{i-1})}{\delta\_M(T\_i)} - 1 \right). \tag{3}$$

It remains to explain how the parameters discount factors, survival probabilities, and recovery rates are determined. This is done in a separate subsection for each parameter.

#### *3.1. Discount Factors*

The pricing Formula (1) requires two discount curves, the discount curve for discounting cash flows and the forward curve for computing forward rates for floating-rate loans. The forward curve is computed from the money market and the swap market. Usually, the forward curve up to one year is computed from deposit rates and forward rate agreements. Swap rates exist for maturities from one year up to 30 years in some currencies. A swap rate *sfix* is the fixed-rate of a swap, which periodically exchanges the fixed-rate with a Ibor rate *Ls* with tenor Λ*s*. If the loan is linked to the same Ibor rate the discount factors *δ<sup>M</sup>* can be computed from the relation

$$s\_{fix} = \frac{\delta\_{M,\Lambda\_s}(\mathcal{U}\_0) - \delta\_{M,\Lambda\_s}(\mathcal{U}\_{\tilde{\text{m}}})}{\sum\_{j=1}^{\text{fl}} \eta\_j \delta\_{M,\Lambda\_s}(\mathcal{U}\_j)},\tag{4}$$

where *U*<sup>0</sup> is the start date of the swap, *Uj* are the payment times of the fixed leg and *η<sup>j</sup>* are the day count fractions of the fixed leg. Usually, a bootstrap algorithm is applied in computing *δ<sup>M</sup>* starting from the swap rate with the lowest maturity and moving forward in swap maturities iteratively using the results of the previous calculation to compute the discount factors corresponding to higher maturities.

If the loan is linked to a different Ibor rate *Ll* with tenor Λ*l*, the above curve cannot be used for computing forward rates. The spread of a basis swap exchanging periodically Ibor payments with tenor Λ*<sup>s</sup>* for Ibor payments with tenor Λ*<sup>l</sup>* has to be added to *sfix* before the bootstrapping starts. We assume the basis swap exchanges *Ls* + *sB* for *Ll*, where *sB* is the basis swap spread which depends on the maturity of the basis swap and can be negative.2 This changes Equation (4) to

$$s\_{fix} + s\_B = \frac{\delta\_{M,\Lambda\_l}(\mathcal{U}\_0) - \delta\_{M,\Lambda\_l}(\mathcal{U}\_{\mathcal{H}})}{\sum\_{j=1}^{\text{in}} \eta\_j \delta\_{M,\Lambda\_l}(\mathcal{U}\_j)}. \tag{5}$$

The discount curve *δ* has to reflect the funding conditions of a bank. It is computed from the fund transfer prices that are provided by a bank's treasury. Typically fund transfer prices are given for a grid of standardized tenors like, 1Y, 2Y, ..., 10Y and are provided as Ibor + spread. This means, that internally the credit department buys a bond from the treasury department with a notional equal to the amount they intend to lend to a borrower. The coupon of this funding instrument is linked to a Ibor rate *Lf* with tenor Λ*<sup>f</sup>* plus a spread *sf* depending on a loan's maturity. The discount curve *δ<sup>f</sup>* can be computed from the relation

$$1 = \sum\_{j=1}^{\text{th}} (f\_{j, \Lambda\_f} + s\_f) \zeta\_i \delta\_f(\mathcal{W}\_i) + \delta\_f(\mathcal{W}\_{\text{th}}),\tag{6}$$

where *Wi* are interest rate payment times of the funding bond and *ξ<sup>i</sup>* are the year fractions of the interest rate periods. The forward rates *fj*,Λ*<sup>f</sup>* are computed by (3) using the swap curve linked to *Lf* . For the calculation of discount factors, the notional is normalized to 1. Similar to the bootstrapping of the swap curve, a bootstrapping of the funding curve can be performed starting from the lowest

<sup>2</sup> This means that in reality, *Ll* + *sB* is exchanged for *Ls*.

maturity and working iteratively up to the highest. If a loan has a fixed rate of interest or a floating rate linked to the Ibor rate *Lf* we get the discount factors in (1) as *δ* = *δ<sup>f</sup>* .

If the loan's interest rate is floating and its Ibor's tenor Λ*<sup>l</sup>* = Λ*<sup>f</sup>* again an adjustment by basis swap spreads is needed. Assume that the basis swap for *Ll* and *Lf* exchanges *Ll* + *sB*<sup>ˆ</sup> for *Lf* where again *sB*<sup>ˆ</sup> might be negative. Equation (6) has to be adjusted to

$$1 = \sum\_{j=1}^{m} (f\_{j, \Lambda\_l} + s\_f + s\_{\mathcal{B}}) \tau\_i \delta(T\_i) + \delta(T\_m), \tag{7}$$

where the payment times and year fractions in (7) are the same as for the loan. Forward rates *fj*,Λ*<sup>l</sup>* are computed from the swap curve corresponding to the Ibor rate *Ll*. Bootstrapping this relation results in the discount curve needed for discounting a loan's cash flows. Why this is a sensible discount curve for cash flow discounting will become clear in Section 4.

#### *3.2. Survival Probabilities*

Equivalent to the calculation of a survival probability *v*(*T*) up to time *T* is the calculation of a default probability *p*(*T*) = 1 − *v*(*T*). Default probabilities with a time horizon of one year are typically one outcome of a bank's rating system. We assume that a bank's rating system has *n* grades where the *n*-th grade is the default grade. Again, remember that one key assumption of this article was the absence of market information like bond or CDS spread. Therefore, a bank has to rely on statistical information which is derived using balance sheet information for corporate clients, personal information of retail clients, and expert judgment as inputs. There are typically two ways of how banks could extract statistical information about defaults from their rating systems to estimate multi-year default probabilities.

In the first approach, a one-year transition matrix is estimated from the rating transitions that are observed in a bank's rating system. The resulting matrix is denoted with **P**(1). The entries of the matrix are denoted with *pkl*, *k*, *l* = 1, ... , *n* where *pkl* is the probability that a borrower in rating grade *k* moves to grade *l* within one year. The matrix **P**(1) has the following properties:


If we assume that rating transitions are Markovian, i.e., they depend on a borrower's current rating grade only, and that transition probabilities are time-homogeneous, i.e., the probability of a rating transition between two-time points depends on the length of the time interval only, then it is possible to apply the theory of Markov chains to construct transition matrices **P**(*h*) for an arbitrary full-year *h* just by multiplying **P**(1) with itself:

$$\mathbf{P}(h) = \underbrace{\mathbf{P}(1)\cdot\ldots\mathbf{P}(1)}\_{\text{h times}}.\tag{8}$$

Once **P**(*h*) is computed, the default probability *pk*(*h*) can be read from the last column in the *k*-th row. Interpolating the values *pk*(*h*) gives the term-structure of default probabilities for rating grade *k*.

In the second approach, banks directly estimate a term-structure of default probabilities, i.e., for each rating grade *k* a function *pk*(*T*) is estimated where *pk*(*T*) is the probability that a borrower in rating grade *k* will default within the next *T* years. From the term structure of default probabilities given today, conditional default probabilities for future times *U* can be computed easily. The probability

*pk*(*T*|*U*) that a borrower in rating grade *k* will default up to time *T* conditional that he is alive at time *U* is given by

$$p\_k(T|\mathcal{U}) = 1 - \frac{1 - p\_k(T)}{1 - p\_k(\mathcal{U})}, \mathcal{U} < T. \tag{9}$$

One way of estimating *pk*(*T*) is by using techniques from survival analysis, where the Cox proportional hazard model has been successfully applied in a credit risk context by numerous authors. Examples are Banasik et al. (1999) and Malik and Thomas (2010). In this model, *p*(*T*) is parameterized as

$$p(T) = 1 - \exp\left(-\exp\left(\beta\_0 + \sum\_{i=1}^{l} \beta\_i K\_i\right) \int\_0^T h(u) du\right),\tag{10}$$

where *β<sup>i</sup>* are model coefficients, *Ki* borrower-dependent risk factors like balance sheet ratios for companies or personal data for retails clients and *h*(*u*) a borrower-independent baseline hazard function. Borrowers with similar *p*(1) can be summarized into a rating category *k* and are then represented by the curve *pk*(*T*).

Throughout this article, we will assume that *pk*(*T*) is estimated by a Cox proportional hazard model as in (10). However, this is by no means the only way to estimate a default probability (PD) term-structure. A good overview of available methodologies is provided in Crook and Bellotti (2010).

#### *3.3. Recovery Rates*

Recovery rates reflect the degree of collateralization of a loan. They can be period-dependent. For instance, if a loan is amortizing and the collateral value stays the same over a loan's lifetime, a loan becomes less risky over the years. This should be reflected in an increasing recovery rate. One pragmatic way to include collateral in a loan pricing framework is to provide the collateral value as input. This collateral value should not be the current market value of collateral but include the loss given default (LGD) of the collateral, i.e., the expected loss in value in the case of a borrower default. This loss can stem from price reductions in a distressed sale or reflect the costs of a liquidation process, e.g., for lawyers. Overall, the collateral valuation and LGD estimation process is complex and beyond the scope of this article. Some ideas can be found in articles on LGD estimation in Engelmann and Rauhmeier (2006).

For the purpose of loan pricing, we assume that such a process exists and that the outcome is a collateral cash value *C*. For the unsecured part of a loan, a bank estimates a recovery rate *Ru*. From these data, the recovery rate *Ri* in each period is computed as

$$R\_i = \min\left(100\%, \frac{\mathbb{C} + R^u \max(N\_i - \mathbb{C}, 0)}{N\_i}\right) \tag{11}$$

In (11) a cap of 100% was introduced. It depends on the specific legal environment of a country's loan market, whether recovery rates of more than 100% are possible or not. In case recovery rates can be larger than 100%, this assumption can be relaxed.

Note, that when applying this approach, consistency is an important issue. In (11) the recovery rate is related to the outstanding notional. In internal risk parameter estimation processes, recovery rates (or, equivalently, LGD values) are often estimated with respect to outstanding notional plus one interest payment. Since loan pricing and risk parameter estimation is usually done in different departments of a bank, one has to take some care to ensure that consistent definitions and assumptions are used throughout a bank. Where this is not the case, appropriate transformations have to be defined.

#### **4. RAROC Scheme**

The main purpose of this Section is the derivation of a RAROC scheme for calculating the interest rate of a loan which covers all costs and adequately compensates for the risks associated with a loan. For bank internal purposes, it is important to split the interest rate into its components, i.e., which part of the interest rate reflects funding costs, which part expected losses, or which part basis swap hedging costs. For this reason, a RAROC scheme is derived step-by-step using the general valuation Formula (1). Before we start, we have to make an assumption on the disbursement of a loan's notional. This is not reflected in the valuation equation (1). The assumption in this article is that a loan's notional is disbursed on disbursement dates *Dj* and that on the date *Dj* the amount *N<sup>D</sup> <sup>j</sup>* is paid out to the borrower. The total notional *<sup>N</sup><sup>D</sup>* is the sum over all disbursements *<sup>N</sup><sup>D</sup>* = <sup>∑</sup>*Dj <sup>N</sup><sup>D</sup> j* .

The first component of the proposed RAROC scheme is the base swap rate. It is only relevant for a fixed-rate loan. For a floating-rate loan, this component is zero. The base swap rate is the fixed-rate that has to be charged by a bank that leads to an identical present value as the stream of Ibor payments. This rate is needed as a reference point to make floating-rate and fixed-rate loans comparable. The base swap rate *ys* at valuation time *t* is computed from

$$y\_s = \frac{\sum\_{t < T\_i} N\_i f\_{i, \Lambda\_l} \tau\_i \delta(T\_i)}{\sum\_{t < T\_i} N\_i \tau\_i \delta(T\_i)}. \tag{12}$$

Alternatively, *ys* can be interpreted as the interest rate that has to be charged for fixed-rate loans to make assets equal to liabilities in a bank's balance sheet under the assumption that all other costs and risks can be ignored. Using this as a starting point, we will add all other relevant cost components of a loan to *ys* in the following steps. To simplify the notation, we will use the abbreviation

$$PV(t; N^D, \delta) = \sum\_{D\_j < t} N\_j^D + \sum\_{t < D\_j} N\_j^D \delta(D\_j). \tag{13}$$

This is the sum of all parts of the total loan balance that are already paid out at time *t* and the present value of the loan parts that still have to be disbursed.

In the next step, funding costs are computed. This is a bit awkward because funding might be linked to a Ibor tenor that is different from the payment frequency of the loan. To separate funding costs from basis swap hedging costs, we have to use the discount curve *δ<sup>f</sup>* and, if the loan is a floating rate loan, compute forward rates from the swap curve corresponding to the Ibor rate *Lf* . For a floating-rate loan, this leads to the condition

$$PV(\mathbf{t}; \mathbf{N}^D, \delta\_f) = \sum\_{T\_i < t} A\_i + \sum\_{t < \mathcal{W}\_i} \hat{\mathcal{N}}\_l \mathbf{f}\_{i, \Lambda\_f} \mathbb{E}\_i \delta\_f(\mathcal{W}\_i) + \sum\_{t < T\_i} \left(\mathcal{N}\_i \mathbf{s}\_f \tau\_i + A\_i\right) \delta\_f(T\_i), \tag{14}$$

where *Wi* are the payment times of the funding bonds in (6), *N*ˆ is the average notional in an interest rate period and *sf* is the spread over Ibor that has to be paid by a borrower to cover funding costs. To give an example for clarification, suppose Λ*<sup>f</sup>* = 6M and Λ*<sup>l</sup>* = 1M. Since a loan might be amortizing, in each 6M period, the notional might change from month to month. Assuming that a repayment of the notional immediately leads to a reduction in the outstanding bonds for funding, the interest paid on the funding bonds has to be reduced with the amortizations. The mismatch in interest tenors is reflected in the averaging of the loan's outstanding notional. If the mismatch is the other way round, i.e., Λ*<sup>f</sup>* < Λ*l*, this problem does not exist. Solving (14) for *sf* gives

$$s\_f = \frac{PV(t; N^D, \delta\_f) - A\_{\text{past}} - \sum\_{t < W\_i} \hat{N}\_i f\_{i, \Lambda\_f} \mathbb{1}\_{\delta} \delta\_f(W\_i) - A\_{PV}}{\sum\_{t < T\_i} N\_i \pi\_i \delta\_f(T\_i)},\tag{15}$$

where we used the abbreviations *Apast* = ∑*Ti*<*<sup>t</sup> Ai* and *APV* = ∑*t*<*Ti Aiδf*(*Ti*). For a fixed-rate loan, the solution can be derived from (15) by setting all forward rates *fi*,Λ*<sup>f</sup>* to zero and replacing *sf* by the fixed-rate *yf* . After solving for *yf* the funding cost margin *sf* can be computed as *sf* = *yf* − *ys*.

When the payment frequencies of funding bonds Λ*<sup>f</sup>* and the loan Λ*<sup>l</sup>* are different, a basis swap is needed for hedging the mismatch in Ibor payments. These hedging costs can be computed from (14) by replacing the discount curve *δ<sup>f</sup>* with the loan's discount curve *δ* and going back to the loan's payment frequency. This leads to

$$PV(t; N^D, \delta) = A\_{\text{past}} + \sum\_{t < T\_i} N\_i \left. f\_{i, \Lambda\_i} \tau\_i \delta(T\_i) + \sum\_{t < T\_i} \left( N\_i s\_{b\_t f} \tau\_i + A\_i \right) \delta\_f(T\_i) \right. \tag{16}$$

where *sb*, *<sup>f</sup>* is the interest margin covering both funding and swap costs. Solving (16) leads to

$$s\_{b,f} = \frac{PV(t; N^D, \delta) - A\_{past} - \sum\_{t < T\_i} N\_i f\_{i, \Lambda\_l} \tau\_i \delta(T\_i) - A\_{PV}}{\sum\_{t < T\_i} N\_i \tau\_i \delta(T\_i)} \tag{17}$$

from which the margin for hedging costs *sb* can be computed as *sb* = *sb*, *<sup>f</sup>* − *sf* . Again, the case of a fixed-rate loan is covered by setting *fi*,Λ*<sup>l</sup>* = 0 and replacing *sb*, *<sup>f</sup>* by the fixed rate *yb*, *<sup>f</sup>* . The margin *sb* associated with basis swap costs is computed as *sb* = *yb*, *<sup>f</sup>* − *yf* .

So far, we have considered cost components that are independent of a loan's default risk. The next step is taking default risk into account. To derive a margin *sEL* reflecting expected loss risk, the condition expected assets equals liabilities is applied. We use the abbreviations

$$V\_D(t) = \sum\_{t < T\_i} N\_i R\_i \delta\left(T\_i\right) \left(\upsilon(T\_{i-1}^\*) - \upsilon(T\_i)\right) \tag{18}$$

and

$$PV(t; \mathbf{N}^D, \delta, \upsilon) = \sum\_{D\_{\dot{\jmath}} < t} N\_{\dot{\jmath}}^D + \sum\_{t < D\_{\dot{\jmath}}} N\_{\dot{\jmath}}^D \upsilon(D\_{\dot{\jmath}}) \delta(D\_{\dot{\jmath}}) \tag{19}$$

In (19), the survival probabilities reflect the fact that a bank will only pay out future tranches of a loan when the borrower is still alive. Using these abbreviations and (1), the interest rate spread *sEL*,*b*, *<sup>f</sup>* containing expected loss risk and the already computed funding and hedging costs is computed from the condition

$$PV(t; N^D, \delta, v) = A\_{\text{past}} + \sum\_{t < T\_i} \left( N\_i \left( f\_{i, \Lambda\_l} + s\_{\text{EL}, b, f} \right) \tau\_i + A\_i \right) \delta(T\_i) v(T\_i) + V\_D(t), \tag{20}$$

where again the special case of a fixed-rate loan is included by setting *fi*,Λ*<sup>l</sup>* = 0 and replacing *sEL*,*b*, *<sup>f</sup>* by a fixed-rate *yEL*,*b*, *<sup>f</sup>* . Solving (20) for *sEL*,*b*, *<sup>f</sup>* gives

$$s\_{\rm EL,b,f} = \frac{PV(t;N^D,\delta,\upsilon) - A\_{\rm past} - \sum\_{t < T\_i} \left(N\_i f\_{i,\Lambda\_i} \tau\_i + A\_i\right) \delta(T\_i)\upsilon(T\_i) - V\_D(t)}{\sum\_{t < T\_i} N\_i \tau\_i \delta(T\_i)\upsilon(T\_i)}\tag{21}$$

The expected loss margin *sEL* is computed as *sEL*,*b*, *<sup>f</sup>* − *sb*, *<sup>f</sup>* for the floating-rate loan and as *yEL*,*b*, *<sup>f</sup>* − *yb*, *<sup>f</sup>* for the fixed-rate loan.

When calculating *sf* , *sb*, and *sEL* the interest margins were motivated from balance sheet considerations. In all the calculation steps, the assets of a bank and its liabilities were matched exactly or in expectation depending on the assumptions in each step. If default risks were independent, the calculations would be finished at this step because, by the law of large numbers, the variance in a loan portfolio's losses will become arbitrarily small if the number of loans is sufficiently large and without any volume concentration. This would result in deterministically matched assets and liabilities. Credit risks, however, are not independent since all borrowers are affected by macroeconomic risk resulting in dependent defaults. In a bad macroeconomic environment, credit losses are higher than expected, while in a benign scenario, they are lower. To avoid bankruptcy in recession years, banks have to hold an equity capital buffer than could absorb losses beyond expectation.

There are minimum requirements on the size of the capital buffer from regulators in Basel Committee on Banking Supervision (2006, 2011). For less sophisticated banks, a simple approach using fixed weights like 8% of outstanding loan balance are applied in the standardized Approach. Here, the *Risks* **2020**, *8*, 63

calculation of the capital buffer *E* is simple and *E* is independent of credit risk parameters like PD and LGD. Most of the internationally active banks, however, apply the Internal Ratings Based Approach which allows banks to compute minimum capital buffers from internal estimates of PD and LGD using the formula

$$E = N \, LGD \left( \Phi \left( \frac{\Phi^{-1} \left( PD \right) + \sqrt{\rho} \Phi^{-1} \left( 0.999 \right)}{\sqrt{1 - \rho}} \right) - PD \right), \tag{22}$$

where *PD* = 1 − *v*(1) is the one-year default probability of the borrower, *LGD* can be computed from the collateralization at the loan's start, and *ρ* is the asset correlation which is defined in Basel Committee on Banking Supervision (2006) depending on the borrower segment.3 As already outlined in Section 2, using regulatory capital for performance measurement becomes increasingly more popular in banking practice due to tightened capital requirements. However, the scheme would also work for *E* derived from a more complex credit portfolio model.

The capital *E* is allocated to the loan and cannot be used for other investments. A bank defines a target return *wt* on its equity capital. As already outlined in Section 2 this could be done by some modeling approach or by expert decision. The equity capital *E* is not lying in a safe but invested in assets like government bonds where it generates a return *wr*. The difference between *wt* and *wr* has to be generated by the interest income of the loan. This leads to an additional interest rate margin *sUL*, the unexpected loss margin, which is computed as

$$s\_{UL} = \left(w\_t - w\_r\right) \frac{E}{N^D}.\tag{23}$$

Finally, the operating costs of a bank, like staff salaries or office costs, have to be covered by a loan's interest income. These costs are summarized in an additional cost margin *c*. The calculation of *c* depends on the institutional details of a bank, and there is no general rule that is applicable to any bank. To include these costs into RAROC an adjustment is required reflecting the fact that only surviving borrowers can cover the costs. This leads to a cost margin *sc* which is computed as

$$s\_{\mathbb{C}} = \mathcal{c} \frac{\sum\_{t \prec T\_i} N\_i \tau\_i \delta\_i(T\_i)}{\sum\_{t \prec T\_i} N\_i \tau\_i \delta\_i(T\_i) \upsilon(T\_i)} \tag{24}$$

Note, that this assumption is not required for economic capital. The reason is that by construction the expected loss margin *sEL* should be sufficient to cover expected loss and economic capital is only a buffer against unexpected events. Once a borrower defaults and a loss provision is built, the capital is freed and can be used for other investments of the bank.

Putting all cost components together gives the hurdle rate *zh* of a loan, i.e., the interest rate that covers all costs and profitability targets of a bank. It is computed as

$$z\_h = y\_s + s\_f + s\_b + s\_{EL} + s\_{UL} + s\_c.\tag{25}$$

Note, that this calculation is true only if the PD of a borrower does not depend on the interest rate *z*. If this is not the case (25) has to be replaced by a numerical algorithm as discussed in the following two sections.

If the interest rate *z* is given, the return on equity capital, or, equivalently, a loan's RAROC can be computed as

$$\text{RAROC} = \frac{z - y\_s - s\_f - s\_b - s\_{EL} - s\_c}{E/N^D} + w\_r. \tag{26}$$

<sup>3</sup> Note that for calculating *E* under Basel II, a different set of *PD* and *LGD* is applied than for expected loss calculations. For regulatory purposes, *PD* is a long-term average reflecting average default risk over an economic cycle while *LGD* is computed under worst-case assumptions. We do not go into these details here since the precise calculation of *E* does not affect the structure of the RAROC scheme.

This equation allows a bank to measure the impact of interest rates different from the hurdle rate *zh* on the return on economic capital. Furthermore, (26) can be used to measure the performance of already existing loans.

#### **5. Properties of RAROC**

When looking at (26) it seems that if *z* becomes arbitrarily large, so does RAROC. This means that this performance measure suggests that banks should charge as high as possible interest rates to maximize profitability. Obviously, this reasoning is flawed since, at a certain interest rate level, a borrower is unable to service his debt and will default. In order to make RAROC realistic, we have to link default probabilities and interest rates.

When building internal models for the default risk of a borrower, banks often include a variable known as the debt service ratio (DSR) into the list of explanatory risk factors. DSR computes the ratio of annual interest and amortization payments on all loans of a borrower and the available funds to pay interest. In the case of a company, these funds are net profit before interest and taxes. In the case of a retail client, it is net annual income. The interest rate *z* of a loan enters DSR linearly as DSR = *β*<sup>0</sup> + *z* · *β*1. The coefficient *β*<sup>0</sup> contains payments on other existing credit products a borrower might have, while *β*<sup>1</sup> is the ratio of the loan's balance divided by available funds. If *β*<sup>1</sup> is small, then PD can be approximately considered as independent of *z*. The larger *β*1, however, the more this approximation leads to wrong conclusions.

To analyze the properties of RAROC when default risk and interest rates are coupled, we use a simplified setup to maintain analytical tractability. We assume a bullet loan with a balance of one that pays interest annually at times *Ti* = 1, ... , *m* and *t* = 0. Furthermore, we assume funding costs, hedging costs and operational costs of a bank are zero. In addition, we assume the interbank curve is flat, and all zero rates are zero, i.e., all discount factors are equal to one. Finally, we assume *wr* equals zero and *Ri* is a constant *R*. Concerning economic capital, we assume that a bank follows the Basel Standardized Approach, i.e., that economic capital *E* is independent of PD. This leads to a simplified RAROC formula

$$\text{RAROC} = \frac{z - s\_{EL}}{E},\tag{27}$$

with a simplified *sEL* as

$$s\_{EL} = \frac{(1 - R) \cdot (1 - v(m))}{\sum\_{i=1}^{m} v(i)}.\tag{28}$$

For the latter equation, note that in (21) *Ai* is zero for all *i* < *m* for a bullet loan and one for *i* = *m*. The default part *VD* simplifies because *Ri* is constant and discount factors are one which results in a telescoping sum that can be simplified to 1 − *v*(*m*).

To model survival probabilities, we assume DSR is part of the risk factors in (10) and we condense all other risk factors into the constant *β*0. Furthermore, we assume a constant *h*. This leads to the survival probability *v*(*i*) at time *i* of

$$\begin{aligned} v(i) &=&\exp\left(-\exp(\beta\_0 + z \cdot \beta\_1) \cdot \int\_0^i h \, ds\right) \\ &=&\exp\left(-\exp(\beta\_0 + z \cdot \beta\_1) \cdot h i\right) \\ &=&\exp\left(-\exp(\beta\_0 + z \cdot \beta\_1)\right)^{hi} = q^{hi} \end{aligned} \tag{29}$$

$$\eta \quad := \ -\exp(-\exp(\beta\_0 + z \cdot \beta\_1)).\tag{30}$$

The properties of RAROC under these assumptions are summarized in Theorem 1.

**Theorem 1.** *Under the assumptions of (27)–(29) where the constants β*1*, h and R are required to fulfill β*<sup>1</sup> > 0*, h* > 0*, and* 0 < *R* < 1*, RAROC(z) as a function of the loan interest rate z has the properties:*

$$1. \quad \lim\_{z \to -\infty} \text{RAROC}(z) = -\infty$$


The proof of Theorem 1 is provided in the Appendix A. The economic interpretation of Theorem 1 is quite intuitive. The first relation means that if a bank pays huge interest until the verge of bankruptcy, RAROC becomes arbitrarily small. If, on the other hand, the borrower pays huge interest which brings him close to bankruptcy, RAROC becomes arbitrarily small, too. Therefore, if a loan brings either the borrower or the bank into trouble, this is adequately reflected by RAROC. Somewhere in between these extreme cases, there is an optimum from the perspective of the bank which allows the bank to generate high income while keeping default risk manageable if the client is creditworthy.

Theorem 1 has direct consequences on the loan origination process of a bank. There exists only a finite range of interest rates that should be considered as acceptable from a bank's perspective. Given the profitability target of a bank *wt* only loans should be accepted with a RAROC greater or equal *wt*. This translates directly into a set of acceptable interest rates which we call the profitability range.

**Theorem 2.** *Define the profitability range P for a loan as the set of interest rates leading to a RAROC greater or equal wt*

$$P = \{ z : \text{RAROC}(z) \ge w\_t \}. \tag{31}$$

*Then exactly one of the three cases is true:*


The proof of Theorem 2 follows directly from Theorem 1. In the case of RAROC(*z*max) < *wt*, *P* is empty and if RAROC(*z*max) = *wt* then *P* = {*z*max}. Finally, if RAROC(*z*max) > *wt*, there exists an interval [*zl*, *zu*] where RAROC(*z*) ≥ *wt*. The lowest interest rate of this interval is the hurdle rate *zh* which is the minimum interest rate that covers all costs and risks associated with the loan. Note, when *PD* is a function of *z*, the hurdle rate can no longer be determined by (25) but has to be computed by a numerical algorithm finding min*<sup>z</sup>* RAROC(*z*) <sup>≥</sup> *wt*. Although there exists interest rates *<sup>z</sup>* with *z* > *z*max and RAROC(*z*) > *wt* it does not make sense for a bank to charge them. It can achieve the same profitability at a lower interest rate making it more likely that the client will accept the offer from the bank and not from a competitor.

To conclude this Section, we remark that we suppose that Theorem 1 holds in more general setup. In numerical examples, when using (22) for calculating economic capital *E* we still get a unique maximum RAROC in numerical examples. While it is quite easy to show that Parts 1 and 2 of Theorem 1 still hold, the third part becomes rather complex since the most general Basel formula includes PD as a function of *z* and the asset correlation *ρ* as a function of PD and, therefore, as a function of *z* which makes the analytical treatment of RAROC in this case rather difficult. Yet from our numerical experiments, we suppose that Theorem 1 holds in the more general setup.

#### **6. Numerical Example**

To illustrate the RAROC pricing scheme, we consider fixed-rate loans with or without amortization, and with or without collateralization. We consider a ten-year fixed-rate loan paying an interest rate of 4% with quarterly interest payments. The loan's notional is *N* = 1, 000, 000 which is paid in one tranche at the loan's start date. We consider a bullet loan, i.e., a loan without amortization payments and an installment loan with an amortization rate of 5% annually. This means that in addition to the interest payment, the installment loan pays back 1.25% of the initial notional, i.e., *Ai* = 12, 500 every quarter. Furthermore, the impact of collateral is illustrated. We assume that in this

case, collateral with a cash equivalent value of *C* = 600, 000 is available. For the unsecured parts of the loan, we assume a recovery rate *R<sup>u</sup>* = 20%. This leads to a total of four different loans. For these loans RAROC is computed in the first part and hurdle rates and maximum RAROC in the second part.

To carry out these calculations, information about interest rate markets and institutional details of the bank is required. In the first step, the information on funding and interest rate markets is collected. We assume that the funding of a bank is expressed as a spread over 12M Ibor rates, i.e., the bank funds itself by issuing bonds paying annual interest linked to a 12M Ibor rate. Furthermore, swap rates of fixed-to-floating swaps and basis swaps have to be included to account for the tenor mismatch in funding and lending. Assuming the European conventions, we have quotes for swaps exchanging a fixed-rate against a 6M Ibor rate. Furthermore, we need the spreads of basis swaps exchanging a 6 M Ibor rate against a 12M Ibor rate because of the funding tenor Λ*<sup>f</sup>* = 12M, and we need the spreads of basis swaps exchanging a 3M Ibor rate against a 6M Ibor rate because of the loan's tenor Λ*<sup>l</sup>* = 3M. The data is summarized in Table 1.

**Table 1.** Quotes of fixed-to-floating swaps, 3M Ibor against 6M Ibor basis swaps, 6M Ibor against 12M Ibor basis swaps, and funding spreads. All quotes are in percent.


The front part of the discount curves that is bootstrapped from fixed-to-floating swaps is built from deposit rates. In the example of Table 1 the 3M and 6M deposit rate are used for computing the front part of *δM*,6*M*. The data in Table 1 are not real market quotes but serves for illustration only.

For the evaluation of default risk, we assume that a bank has established a rating system with six grades and uses a Cox proportional hazard model (10) to estimate term-structures of default probabilities. We assume that the loan's interest rate is part of one risk factor, all other risk factors are summarized in the coefficient *β*<sup>0</sup> and *h* is a constant as in (29). The parameters for each rating grade are summarized in Table 2 while the default probabilities for each rating grade are illustrated in Figure 1 using the interest rate of the example, 4%.

**Table 2.** Parameters for the Cox proportional hazard model for each rating grade.


**Figure 1.** Term-structure of default probabilities for each rating grade.

It remains to define economic capital and the operating costs of the bank. We assume an annual operating cost margin *c* = 0.50%. Economic capital is computed following the regulatory rules for corporate clients with an annual turnover above 50 million EUR where we use both the Standardized and the Internal Ratings Based Approach in our examples. In the case of the Standardized Approach, we assume that the company does not have an external rating. Finally, we assume a target RAROC *wt* of 10%.

Cost components and RAROC are computed for the collateralized bullet loan (Loan I), the unsecured bullet loan (Loan II), the collateralized installment loan (Loan III), and the unsecured installment loan (Loan IV). The borrower rating is "3", i.e., we assume a borrower with a one-year default probability of roughly 1%. The results are summarized in Table 3 when *E* is computed as 0.08 · *<sup>N</sup><sup>D</sup>* and in Table <sup>4</sup> when *<sup>E</sup>* is computed by (22).


**Table 3.** Cost components RAROC for the four example loans assuming rating grade 3 and using the Standardized Approach for computing *E*. All results are percentage values.


**Table 4.** Cost components RAROC for the four example loans assuming rating grade 3 and using the Internal Ratings Based Approach for computing *E*. All results are percentage values.

The quantities *ys* and *sf* show the effect of the amortization rate. Both the swap curve and the funding spreads curve are steep. Since an amortization rate reduces the effective maturity of a loan both quantities are lower for amortizing loans. This effect is not seen in *sb* because both basis swap spread curves are flat. The expected loss margin *sEL* is considerably higher for the unsecured loan. For the amortizing collateralized loan, the expected loss margin is lowest because this loan becomes less risky when the outstanding balance is reduced due to the amortizations. This effect is not seen in *E* in Table 4 because economic capital is based on a one-year horizon in the Basel II setup. We see that in both tables, only the collateralized loans pass the RAROC target of 10%. The unsecured loans show a RAROC below 10% and should be rejected if a bank strictly sticks to its profitability target.

In the second example, we compute *zh*, *z*max and maximum RAROC for Loan IV. Again we present the results for both regulatory regimes. The outcome for the Standardized Approach is displayed in Table 5 while the numbers for the Internal Ratings Based Approach are shown in Table 6.


**Table 5.** Hurdle rate, maximum RAROC and *z*max for the unsecured installment loan Loan IV under the Standardized Approach. All results are percentage values.

**Table 6.** Hurdle rate, maximum RAROC and *z*max for the unsecured installment loan Loan IV under the Internal Ratings Based Approach. All results are percentage values.


We see that for the high-risk clients, no hurdle rate *zh* exists. This means that it is not possible for a bank to set an interest rate that makes the loan profitable. Therefore, a loan application of these clients should be rejected. We see that for Rating "3" in both cases, the hurdle rate is above 4%. This is consistent with the results in Tables 3 and 4 where RAROC was below the profitability target of 10% for Loan IV when an interest rate of 4% was used. Consistent with intuition, in both cases *zh* is increasing with borrower default risk while *z*max and RAROCmax are decreasing.

#### **7. Discussion**

In this article, a loan pricing scheme is developed using the performance measure RAROC. Motivated by balance sheet considerations, i.e., the desire to match assets and liabilities, a calculation scheme is proposed which explicitly decomposes a loan's interest rate into relevant cost components: Funding costs, costs for hedging interest rate risks, expected loss costs, target return on economic capital, and internal bank costs. For fixed-rate loans, a formula for the base swap rate was given in addition. These cost components are essential for internal fund transfer pricing processes between separate functions in a bank.

The proposed pricing scheme is applicable for loans with the deterministic interest rate, i.e., fixed-rate loans and floating-rate loans linked to Ibor rates. We have analyzed the scheme mainly for the case where term-structures of default probabilities are estimated using a Cox proportional hazard model. This was motivated by the analytical tractability of this model. However, the scheme does not depend on this modeling assumption and could work with any term-structure of default probabilities regardless of its determination.

In a theoretical analysis, it was shown in a slightly simplified setup that if a borrower's default probability increases with a loan's interest rate then RAROC becomes −∞ in the limiting cases of arbitrarily large negative and positive interest rates which means that both the cases of bank and borrower bankruptcy are treated within economic intuition by RAROC. It was further shown that RAROC has a unique maximum and that at most a finite interval of interest rates exists at which a bank should accept a loan application. In cases where interest rates a borrower is willing to accept are outside this interval or when the acceptance range is empty, a bank should reject a loan application.

Numerical examples illustrated the application of this loan pricing framework. The examples suggest that the main results of the article hold in a more general setup than we were able to prove formally. The main challenge of applying this framework in practice is finding a link between a loan's interest rate and borrower default rates empirically. In real data sets important information for determining this relationship like the total interest a borrower is paying on all his existing loan products or timely income information is often missing in retail data sets which makes the parameters *β*<sup>0</sup> and *β*<sup>1</sup> of our examples very hard to estimate.

The benefits of implementing this approach in practice are threefold. First, the scheme delivers a split of a loan's interest rate into cost components for internal fund transfer pricing. Second, when interest rate costs are properly included in a credit scorecard, the scheme allows the calculation of the profitability range for a loan. Only rates within this range a bank should offer when originating a loan. Finally, since the scheme delivers a loan valuation, it could, in addition, be valuable in the price determination of loan portfolio transactions when loans are sold to investors.

One shortcoming of the profitability range is that this interest rate interval models only the perspective of the bank. These are the interest rates that ensure that the profitability requirements of the bank are met, but there is no view from the borrower's perspective included. It would be very helpful if a bank would have, in addition to the profitability range, some information about the likelihood that a borrower will accept a loan offer and how this likelihood changes within the profitability range. Chun and Lejeune (2020) use the probability that a borrower accepts a loan offer in their model but merely on a theoretical basis using several distribution functions without giving any suggestion on empirical verification. Complementing the profitability range by including the borrower perspective would further increase the value of the RAROC scheme. We leave this challenging task for future research.

**Author Contributions:** Conceptualization, B.E. and H.P.; methodology, B.E.; validation, B.E. and H.P.; writing—original draft preparation, B.E.; writing—review and editing, H.P.; supervision, H.P.; project administration, H.P.; funding acquisition, H.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Ministry of Education and Training in Vietnam, Ho Chi Minh City Open University and Open Source Investor Services B.V. grant number B2019-MBS-03.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **Abbreviations**

The following abbreviations are used in this manuscript:

CAPM capital asset pricing model


#### **Appendix A. Proof of Theorem 1**

Theorem 1 consists of three parts which we each proof in a separate step. We start with the proof of 1. For the survival probabilities *vi*, it is easy to see that lim *<sup>z</sup>*→−<sup>∞</sup> *<sup>v</sup>*(*i*) = 1. This means that lim *<sup>z</sup>*→−<sup>∞</sup> *zEL* <sup>=</sup> <sup>0</sup> and lim *<sup>z</sup>*→−∞(*<sup>z</sup>* <sup>−</sup> *zEL*)/*<sup>E</sup>* <sup>=</sup> <sup>−</sup>∞.

The proof of Part 2 is a bit more evolved. We have to show that lim *<sup>z</sup>*→+<sup>∞</sup> *<sup>z</sup>* <sup>−</sup> *zEL* <sup>=</sup> <sup>−</sup>∞. From (29) we see that lim *<sup>z</sup>*→+<sup>∞</sup> *<sup>v</sup>*(*i*) = 0. Therefore, lim *<sup>z</sup>*→+<sup>∞</sup> *zEL* = +<sup>∞</sup> and lim *<sup>z</sup>*→+<sup>∞</sup> *<sup>z</sup>* = +<sup>∞</sup> which makes it not obvious to see how RAROC will behave in the limit. A reformulation of *z* − *zEL* leads to

$$z - z\_{EL} = z - \frac{(1 - R)(1 - v(m))}{\sum\_{i=1}^{m} v(i)} = \frac{z \sum\_{i=1}^{m} v(i) - (1 - R)(1 - v(m))}{\sum\_{i=1}^{m} v(i)}$$

For each product *zv*(*i*), we can show that lim *<sup>z</sup>*→+<sup>∞</sup> *zv*(*i*) = 0:

$$\begin{aligned} \lim\_{z \to +\infty} zv(i) &= \lim\_{z \to +\infty} zq^{hi} = \lim\_{z \to +\infty} \frac{z}{q^{-hi}} = \lim\_{z \to +\infty} \frac{1}{-hi \cdot q^{-hi-1} \cdot q \log(q)\beta\_1} \\ &= \lim\_{z \to +\infty} -\frac{q^{hi}}{hi \cdot \log(q)\beta\_1} = 0. \end{aligned}$$

At the third equality sign we have used the rule of L'Hospital and the derivative of q with respect to *z*

$$\frac{dq}{dz} = \frac{d}{dz} \exp(-\exp(\beta \mathfrak{o} + z \mathfrak{k}\_1)) = -\exp(-\exp(\beta \mathfrak{o} + z \mathfrak{k}\_1)) \exp(\beta \mathfrak{o} + z \mathfrak{k}\_1) \mathfrak{k}\_1 = q \log(q) \mathfrak{k}\_1.$$

This results allows us to compute (we use a bit sloppy notation in the end)

$$\lim\_{z \to +\infty} z - z\_{EL} = \lim\_{z \to +\infty} \frac{z \sum\_{i=1}^{m} v(i) - (1 - R)(1 - v(m))}{\sum\_{i=1}^{m} v(i)} = \frac{0 - (1 - R)}{0} = -\infty.$$

For the proof of Part 3, we will show that the first derivative of RAROC with respect to *z* is between 1 and −∞ and it is monotonically decreasing which implies that there exists exactly one root of *d*RAROC(*z*)/*dz* which proofs the Theorem. We start with *d*RAROC(*z*)/*dz* where we use the abbreviations *L* := 1 − *R* and *D* := *z* − *zEL*:

$$\begin{split} \frac{dD}{dz} &= 1 - \frac{d}{dz} \frac{L(1 - q^{hm})}{\sum\_{i=1}^{m} q^{li}} \\ &= 1 - L \frac{-\sum\_{i=1}^{m} q^{li} \cdot lm \cdot q^{hm-1} \frac{dq(z)}{dz} - (1 - q^{hm}) \sum\_{i=1}^{m} l i \cdot q^{hi-1} \frac{dq(z)}{dz}}{\left(\sum\_{i=1}^{m} q^{li}\right)^2} \\ &= 1 + L \log(q) \beta\_1 \frac{\sum\_{i=1}^{m} q^{hi} l m \cdot q^{hm} + (1 - q^{hm}) \sum\_{i=1}^{m} l i \cdot q^{hi}}{\left(\sum\_{i=1}^{m} q^{li}\right)^2} \end{split}$$

*Risks* **2020**, *8*, 63

$$=1+L\log(q)\beta\_1\frac{\sum\_{i=1}^{m}hi\cdot q^{hi}+h(m-i)q^{h(i+m)}}{\left(\sum\_{i=1}^{m}q^{hi}\right)^2}$$

We have lim *<sup>z</sup>*→−<sup>∞</sup> *<sup>q</sup>*(*z*) = 1 and, therefore, lim *<sup>z</sup>*→−<sup>∞</sup> *d*(*z*−*zEL*) *dz* <sup>=</sup> 1. On the other end, we have lim*z*→<sup>∞</sup> *<sup>q</sup>*(*z*) = 0 which leads to lim*z*→<sup>∞</sup> *d*(*z*−*zEL*) *dz* <sup>=</sup> <sup>−</sup>∞. The reason for the latter is that lim*z*→<sup>∞</sup> log(*q*) = <sup>−</sup><sup>∞</sup> and

$$\lim\_{q \to 0} \frac{\sum\_{i=1}^{m} hi \cdot q^{hi} + h(m-i)q^{h(i+m)}}{\left(\sum\_{i=1}^{m} q^{hi}\right)^2} = +\infty,$$

which can be seen after applying the rule of L'Hospital 2*m* − 1 times. Note that the highest exponent of *<sup>q</sup>* in the numerator is 2*<sup>m</sup>* − 1 because the coefficient of *<sup>q</sup>*2*<sup>m</sup>* is zero and the highest exponent of the denominator is 2*m*. This leaves one *q* in the denominator after 2*m* − 1 times applying L'Hospital's rule while there is none in the numerator.

Since *<sup>d</sup>*(*z*−*zEL*) *dz* is continuous, it must have at least one root. To show that this root is unique, we show that *<sup>d</sup>*(*z*−*zEL*) *dz* is decreasing monotonically by proving that *<sup>d</sup>*2*<sup>D</sup> dz*<sup>2</sup> <sup>=</sup> *<sup>d</sup>*2(*z*−*zEL*) *dz*<sup>2</sup> is negative for all *z*.

*d*2*D dz*<sup>2</sup> <sup>=</sup> *<sup>L</sup>β*<sup>1</sup> 1 *q dq dz* ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *hi* · *<sup>q</sup>hi* <sup>+</sup> *<sup>h</sup>*(*<sup>m</sup>* <sup>−</sup> *<sup>i</sup>*)*qh*(*i*+*m*) - ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *<sup>q</sup>hi*<sup>2</sup> + *L* log(*q*)*β*<sup>1</sup> ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *<sup>q</sup>hi*<sup>2</sup> ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *h*2*i* <sup>2</sup> · *<sup>q</sup>hi*−<sup>1</sup> + *<sup>h</sup>*2(*m*<sup>2</sup> − *<sup>i</sup>* <sup>2</sup>)*qh*(*i*+*m*)−<sup>1</sup> - ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *<sup>q</sup>hi*<sup>4</sup> *dq dz* − *L* log(*q*)*β*<sup>1</sup> 2 ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *<sup>q</sup>hi* · ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *hi* · *<sup>q</sup>hi*−<sup>1</sup> · ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *hi* · *<sup>q</sup>hi* <sup>+</sup> *<sup>h</sup>*(*<sup>m</sup>* <sup>−</sup> *<sup>i</sup>*)*qh*(*i*+*m*) - ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *<sup>q</sup>hi*<sup>4</sup> *dq dz* = *Lβ*<sup>2</sup> <sup>1</sup> log(*q*) ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *hi* · *<sup>q</sup>hi* <sup>+</sup> *<sup>h</sup>*(*<sup>m</sup>* <sup>−</sup> *<sup>i</sup>*)*qh*(*i*+*m*) - ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *<sup>q</sup>hi*<sup>2</sup> + *L* log(*q*)2*β*<sup>2</sup> 1 ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *<sup>q</sup>hi* ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *h*2*i* <sup>2</sup> · *<sup>q</sup>hi* + *<sup>h</sup>*2(*m*<sup>2</sup> − *<sup>i</sup>* <sup>2</sup>)*qh*(*i*+*m*) - ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *<sup>q</sup>hi*<sup>3</sup> <sup>−</sup> *<sup>L</sup>* log(*q*)2*β*<sup>2</sup> 1 2 · ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *hi* · *<sup>q</sup>hi* · ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *hi* · *<sup>q</sup>hi* <sup>+</sup> *<sup>h</sup>*(*<sup>m</sup>* <sup>−</sup> *<sup>i</sup>*)*qh*(*i*+*m*) - ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *<sup>q</sup>hi*<sup>3</sup> = *Lβ*<sup>2</sup> <sup>1</sup> log(*q*) ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *hi* · *<sup>q</sup>hi* <sup>+</sup> *<sup>h</sup>*(*<sup>m</sup>* <sup>−</sup> *<sup>i</sup>*) · *<sup>q</sup>h*(*i*+*m*) - ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *<sup>q</sup>hi*<sup>2</sup> + *L* log(*q*)2*β*<sup>2</sup> 1 ∑*<sup>m</sup> <sup>i</sup>*,*j*=<sup>1</sup> *<sup>q</sup>hj h*2*i* <sup>2</sup> · *<sup>q</sup>hi* + *<sup>h</sup>*2(*m*<sup>2</sup> − *<sup>i</sup>* <sup>2</sup>) · *<sup>q</sup>h*(*i*+*m*) - ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *<sup>q</sup>hi*<sup>3</sup> <sup>−</sup> *<sup>L</sup>* log(*q*)2*β*<sup>2</sup> 1 <sup>2</sup> · <sup>∑</sup>*<sup>m</sup> <sup>i</sup>*,*j*=<sup>1</sup> *hj* · *<sup>q</sup>hj* · *hi* · *<sup>q</sup>hi* <sup>+</sup> *<sup>h</sup>*(*<sup>m</sup>* <sup>−</sup> *<sup>i</sup>*) · *<sup>q</sup>h*(*i*+*m*) - ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *<sup>q</sup>hi*<sup>3</sup> = *Lβ*<sup>2</sup> <sup>1</sup> log(*q*) ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *hi* · *<sup>q</sup>hi* <sup>+</sup> *<sup>h</sup>*(*<sup>m</sup>* <sup>−</sup> *<sup>i</sup>*) · *<sup>q</sup>h*(*i*+*m*) - ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *<sup>q</sup>hi*<sup>2</sup> <sup>−</sup> *<sup>L</sup>* log(*q*)2*β*<sup>2</sup> 1 ∑*<sup>m</sup> <sup>i</sup>*,*j*=<sup>1</sup> *<sup>h</sup>*2(2*ij* − *<sup>i</sup>* <sup>2</sup>) · *<sup>q</sup>h*(*i*+*j*) <sup>+</sup> *<sup>h</sup>*2(2*j*(*<sup>m</sup>* <sup>−</sup> *<sup>i</sup>*) <sup>−</sup> *<sup>m</sup>*<sup>2</sup> <sup>+</sup> *<sup>i</sup>* <sup>2</sup>) · *<sup>q</sup>h*(*i*+*j*+*m*) - ∑*<sup>m</sup> <sup>i</sup>*=<sup>1</sup> *<sup>q</sup>hi*<sup>3</sup>

Since log(*q*) < 0) the first term of this expression is negative. To prove that the full expression is negative, it is sufficient to prove that each coefficient of *qhi* in the numerator of the second term is non-negative and at least one is strictly positive. In total there are 3*<sup>m</sup>* − 1 terms *<sup>q</sup>hk* with *<sup>k</sup>* = 2, ... , 3*m*. For *k* = 2, ... , *m* + 1 only the first part of the double sum is relevant, from *k* = *m* + 2, ... , 2*m* both parts contribute, while from *k* = 2*m* + 1, . . . , 3*m* only the last part counts.

We start with *k* = 2, ... , *m* + 1. For fixed *k*, the index *i* can run from 1 to *k* − 1 while *j* is set to *k* − *i*. Then *i* + *j* = *k* and all possible combinations leading to *i* + *j* = *k* are covered. Summing over the coefficients that contribute to *qhk* yields

$$\sum\_{i=1}^{k-1} 2\left(i(k-i)\right) - i^2 = \sum\_{i=1}^{k-1} 2ki - 3i^2$$

Using the relations ∑*<sup>k</sup> <sup>i</sup>*=<sup>1</sup> *<sup>i</sup>* <sup>=</sup> *<sup>k</sup>*(*k*+1) <sup>2</sup> and <sup>∑</sup>*<sup>k</sup> <sup>i</sup>*=<sup>1</sup> *i* <sup>2</sup> = *<sup>k</sup>*(*k*+1)(2*k*+1) <sup>6</sup> leads to

$$\sum\_{i=1}^{k-1} 2ki - 3i^2 = 2k \frac{(k-1)k}{2} - 3 \frac{(k-1)k(2k-1)}{6} = \frac{k(k-1)}{2} > 0.1$$

Next, we look at *k* = *m* + 2, ... , 2*m*. We parametrize this as *m* + *k*, *k* = 2, ... , *m*. In this case we have contributions from both terms of the double sum and the coefficient of *qm*+*<sup>k</sup>* is computed as

$$\sum\_{i=k}^{m} \left( 2i(m+k-i) - i^2 \right) + \sum\_{i=1}^{k-1} \left( 2(k-i)(m-i) - m^2 + i^2 \right) = \frac{1}{2}(m^2 - m) + k(m-k+1) > 0.$$

The derivation of the above result is a bit lengthier as in the first case but uses the same reasoning.

Finally, we report the result for *k* = 2*m* + 1, ... , 3*m*. Similar as before, we run *k* from 1 to *m* and ensure that *i* + *j* = 2*m* + *k*. Summing over all combinations of *i* and *j* fulfilling this condition leads to

$$\sum\_{i=k}^{m} \left( 2(m-i+k)(m-i) - m^2 - i^2 \right) = \frac{1}{2} \left( m^2 + k^2 \right) - km + \frac{1}{2} \left( m - k \right) \ge 0.$$

Here, the sum can become zero in the case of *k* = *m*. In all other cases, it is strictly positive. This concludes the proof that the second derivative of RAROC is always strictly negative.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Modelling Australian Dollar Volatility at Multiple Horizons with High-Frequency Data**

**Long Hai Vo 1,2 and Duc Hong Vo 3,\***


Received: 1 July 2020; Accepted: 17 August 2020; Published: 26 August 2020

**Abstract:** Long-range dependency of the volatility of exchange-rate time series plays a crucial role in the evaluation of exchange-rate risks, in particular for the commodity currencies. The Australian dollar is currently holding the fifth rank in the global top 10 most frequently traded currencies. The popularity of the Aussie dollar among currency traders belongs to the so-called three G's—Geology, Geography and Government policy. The Australian economy is largely driven by commodities. The strength of the Australian dollar is counter-cyclical relative to other currencies and ties proximately to the geographical, commercial linkage with Asia and the commodity cycle. As such, we consider that the Australian dollar presents strong characteristics of the commodity currency. In this study, we provide an examination of the Australian dollar–US dollar rates. For the period from 18:05, 7th August 2019 to 9:25, 16th September 2019 with a total of 8481 observations, a wavelet-based approach that allows for modelling long-memory characteristics of this currency pair at different trading horizons is used in our analysis. Findings from our analysis indicate that long-range dependence in volatility is observed and it is persistent across horizons. However, this long-range dependence in volatility is most prominent at the horizon longer than daily. Policy implications have emerged based on the findings of this paper in relation to the important determinant of volatility dynamics, which can be incorporated in optimal trading strategies and policy implications.

**Keywords:** exchange-rate risk; long-range dependency; wavelets; multi-frequency analysis; AUD–USD exchange rate

**JEL Classification:** F31; G32; C58

#### **1. Introduction**

In his seminal work advocating for a system of flexible exchange rates, (Friedman 1953) envisaged that speculative forces would have stabilising effects that cause exchange rates to adjust smoothly over time, moving from one equilibrium to another. However, since the breakdown of the Bretton Woods system in 1973, high volatility has been one of the few persistent characteristics of exchange rates. Nevertheless, (Friedman 1953)'s prediction is not without merit, as exchange rates are shown to be less volatile over the longer term, and tend to revert to an equilibrium value that is in close association with relative prices (Lothian 2016); (Marsh et al. 2012). Consider, for example, the values of the US dollar in terms of the Australian dollar. Over the last 45 years, as presented in Panel C of Figure 1, more than 70% of the annual absolute changes of the AUD were greater than 3%, whereas the corresponding number for large relative price changes (Australian price level relative to that of the US) was only approximately 30%. The excess volatility is also apparent for other major currencies, such as the British

Pound, the Deutsch Mark or the Japanese Yen, over the same period, as presented in Panels A, B and D of this figure.

**Figure 1.** Price and Exchange-Rate Volatilities in the Post Bretton-Woods Era. Notes: Each panel presents the densities of *xc*,*<sup>t</sup>* = 100 × *logXc*,*<sup>t</sup>* − *logXc*,*t*−<sup>1</sup> (*t* = 1974, ... , 2018) *Xc*,*<sup>t</sup>* = *Sc*,*<sup>t</sup> or Xc*,*<sup>t</sup>* = *Rc*,*<sup>t</sup>* = *Pc*,*t*/*PUS*,*<sup>t</sup>* , where *Sc*,*t* denotes the cost of 1 USD in country c's currency and *Rc*,*t* denotes relative prices, defined as the price level, proxied by the CPI of country c (*Pc*,*t*), deflated by that of the US (*PUS*,*t*). The sample period for the Deutsch Mark ends in 1998. To aid presentation, extreme values are not shown. The figures presented in these plots indicate the proportion of relative-price and exchange-rate changes that are greater than 3%, respectively. Source: (Vo 2019); (International Monetary Fund 2019) and authors' calculations.

An extensive body of research has been devoted to the understanding of this excess volatility since the global adoption of floating-rate regimes. See (James et al. 2012) for a recent collection of papers on this topic and other themes of exchange-rate economics. Concurrent with the development of this literature, another strand of research illustrates the stylised fact of long-range dependence, or long memory, in the volatility of multiple financial asset classes including currencies (e.g., Akgül and Sayyan 2008; Aloy et al. 2011). In particular, the response to new information is dragged over a long period due to inefficiency in currency markets (Caporale et al. 2019; Vo and Vo 2019b) leading to long-range dependence of exchange rates. However, the standard statistical techniques proposed to examine long-range dependence, such as the rescaled range method, are inadequate to capture both shortand long-range dependence (Lo 1991). The former is a result of high-frequency autocorrelation or heteroscedasticity. Furthermore, it is distinct from the martingale property. This implies that we need to account for dependence at higher frequencies when drawing empirical inferences of long-range behaviour. This feature may not be sufficiently captured by short-range dependence models (such as an AR(1)).

Another weakness of the existing approaches to modelling long memory is the limited scope of the trading horizon considered. To gauge the importance of this idea, (Vo and Vo 2019a) considered the diverse group of participants operating over different horizons. First, the long-run trends of foreign exchange rates are the primary concern of a group of market makers whose objective is to ensure currency values are kept consistent with long-term economic fundamentals. Second, at the high end of the frequency spectrum are intraday traders who seek to exploit the volatile currency-value movements and other market inefficiencies to obtain short-term abnormal returns. A wide range of traders exists in between these two extremes. However, when it comes to the examination of long-run dependence in exchange-rate volatility, the conventional two-scale approach reflecting these extremes, that is, short- and long-run, offers a limited solution. Answering the critical questions of "How long is the long-run?" and "What is a possible transition path from short- to long-run?" is important. This is due to the true volatility dynamics being shrouded in the observed data that aggregate the heterogeneous decision-making processes of different traders.

In light of these developments, in this study, we aim to reinvigorate the investigation of the long-run dynamics of exchange rate volatility, with the focus on the Australian dollar. The main justification for the choice of this currency is its interesting relationship with the movements of the main exports of the issuing country. Specifically, when the demand for Australian goods increases, Australia's terms-of-trade improve. This means that the prices of the primary exports increase relative to those of the imports. Then, the value of the AUD appreciates relative to the currencies of Australia's trading partners, which is generally termed as a "commodity currency" (Cashin et al. 2004; Chen and Rogoff 2003). As observed in the recent period of the mineral price boom (2003–2013), the appreciation of the AUD has exerted a significant adverse effect on the Australian non-mining exports, such as agricultural and manufacturing products. This is considered as a classic example of the "Dutch disease" or "resource-induced de-industrialisation" phenomenon (Downes et al. 2014). The risk involved in the fluctuation of the AUD value is clear. Understanding the dynamics of exchange-rate volatility is therefore of much interest to policymakers in a small and very open economy like Australia. This insight is crucial for: (i) navigating exchange-rate risks to different sectors of the economy; (ii) planning the government budgets and forecast mining revenues and (iii) evaluating the performance of related forecasts.

As discussed above, though the existing literature has reached a near-consensus agreement on the existence of long memory in exchange-rate volatility, research on the extent to which this dynamic behaviour relates to trading horizons remains inconclusive. In this study, we aim to fill in this gap by examining extensively what trading frequency the risk of a volatile exchange rate persists at. In particular, we are interested in capturing the long-memory behaviour of the Australian dollar at multiple trading horizons with the help of a wavelet decomposition analysis.1 Specifically, the role of lower-frequency (or longer-horizon) trading activities is documented to be crucial in determining exchange-rate volatility. We also contribute to the literature by providing a clear picture of the long-memory transition path from the short- to long-run.

The remaining paper has the following structure: First, we surveyed and outlined vital studies in the literature regarding the implications and tests of long-memory in foreign-exchange time series in Section 2. Then in Section 3, a brief overview of wavelet methodology is provided, followed by a discussion of the testing procedure for possible structural breaks in the volatility process, which is a potential source for long-range dependency. Section 4 describes our high-frequency data, which serves as the basis for the empirical analyses presented subsequently. Conclusions and implications of our study are provided in Section 5.

<sup>1</sup> The scope for the development of a wavelet-based application on volatility modelling is expected to present significant potential for financial, economic research. As discussed in Section 3, wavelet-based methodology in the field of volatility modelling has been on the rise as a means of filling the gap between short- and long-run analyses.

#### **2. Related Literature**

An understanding of the long-memory characteristic of financial processes is crucial in determining optimal investment strategies and asset-portfolio management because of its relevance to market efficiency (Mensia et al. 2014). Specifically, as the presence of long memory in asset returns implies the existence of significant correlations between price observations that are separated in time, this directly contradicts the validity of the Efficient Market Hypothesis (EMH), which suggests the unpredictability of prices and the impossibility of abnormal returns generation. From a different angle, evidence of long memory in the volatility process implies volatility persistence, which suggests that uncertainty is an inherent aspect of the behaviour of exchange rates.

Since long memory affects the riskiness of exchange-rate changes, it also has important implications for the effectiveness of exchange-rate risk hedging. According to (Coakley et al. 2008), the optimal hedge ratio (OHR), also known as the minimum-variance hedge ratio, can be estimated using several well-established methodologies2. The prolonged debate on which approach generates the best hedging performance yields mixed evidence (e.g., Moosa 2003; Wen et al. 2017; Jitmaneeroj 2018; Maples et al. 2019; Xu and Lien 2020). This lack of a consensus is partly attributable to the workings of long memory. Specifically, the above approaches assume that the futures premium process generated as the difference between contemporaneous futures and spot prices is stationary. However, in the context of an integrated process of order *d* (0 < *d* < 1), (Lien and Tse 1999) show theoretically that this will affect the OHR and thus renders the assessment of its relevance for hedging effectiveness difficult.

Adding to the difficulties described above is the fact that conventional statistical tests for stationarity, such as the Dickey–Fuller and Phillips–Perron tests, often falsely lead to non-rejection of the unit-root null hypothesis for exchange rates. This is because the long-memory characteristic of these time series could lower the power of these tests. Relatedly, it can be shown that fractionally integrated processes exhibiting long-range dependence, as opposed to a random walk, can still be stationary (Jiang et al. 2018; Peng et al. 2018). In addition, mean-stationary (with long memory) integrated processes of an order close to unity can be misspecified as fully integrated processes (*d* = 1), because they typically yield indistinguishable unit-root test results. Additionally, the lack of power in unit-root tests can be a source of mixed results for hypotheses relying on such tests such as the purchasing power parity theory (Drine and Rault 2005). This highlights the importance of the careful examination of the long-memory characteristic in time series.

Concerning long-memory research, a method to detect and estimate long-run dependence in the form of the "rescaled range" statistic *R*/*S*(*n*), where *n* denotes the sample size, was developed by (Hurst 1951). The long-range dependence relationship is implied by *E*[*R*/*S*(*n*)] ∼ *CnH*, when *n* → ∞. This method aims to estimate the so-called "Hurst exponent" *H*. As shown by (Vo and Vo 2019b), the parameter *H* is related to the "fractional" degree of integration *d* of stochastic processes via the simple expression: *d* = *H* − 0.5. When modelling volatility, we are mostly interested in the case where 0.5 < *H* < 1, which corresponds to a long-memory process. The conventional rescaled-range approach was first adopted by (Booth et al. 1982) to account for long memory in exchange-rate data. However, a weakness of this early developed technique is that it is not robust to the short-range persistence and heteroscedasticity (Caporale et al. 2019; Gil-Alana and Carcel 2020; Ouyang et al. 2016; Youssef and Mokni 2020).

The above discussions show that though the long-memory regularity is important to both investors and forecasters, the related literature has not reached a consensus on how to examine it at different horizons. We add to this debate by exploring the application of an advanced wavelet-based technique recently developed to simultaneously analyse both the time and frequency domains of a

<sup>2</sup> These methodologies include a least-squares approach whereby the OHR is given by the slope coefficient of the regression line of spot exchange-rate returns against futures returns. Other alternatives such as the error-correction model and the generalised autoregressive conditional heteroscedasticity model can be used to estimate time-varying OHRs.

data generating process. In this study, we combined the strengths of well-established long-memory estimators and wavelet methodology to capture the short-term and long-term dependence structure of financial volatility. A wavelet maximum likelihood estimator is shown to provide superior accuracy in estimating the long-memory parameter compared with the *R*/*S*(*n*) estimator and its variations (Vo and Vo 2019b). In the next section, we describe the particular wavelet-based approach adopted in our analyses.

#### **3. Methodology**

#### *3.1. Wavelet Multi-Scale Decomposition*

Here, we briefly review the wavelet decomposition methodology. Detailed treatments of the approach can be found in (Mallat 2009). Several of its applications in economics and finance are discussed by (Gencay et al. 2002; In and Kim 2013). First, according to (Baqaee 2010), the "mother" wavelet function ψ(*t*) satisfies:

$$\int\_{-\infty}^{\infty} \left| \psi(t) \right|^2 dt = 1; \int\_{-\infty}^{\infty} \psi(t) dt = 0.$$

These two fundamental conditions constitute the main features of a "small wave", with unity energy and oscillations dissipating quickly. We used the Discrete Wavelet Transform (DWT) technique developed by (Mallat 2009) in this study, following most applications with finite time series. Specifically, the translated and dilated wavelet function can be expressed as:

$$
\psi\_{j,k}(t) = 2^{j/2} \psi(2^j t - k).
$$

where *j* and *k* denote scale and location parameters, respectively. Then, with the combination of the mother wavelet and a complementary component (called the "father" wavelet) ϕ, we can represent any function *f* as follows:

$$f(t) := \sum\_{l \in \mathbb{Z}}  \phi\_l(t) + \sum\_{j=0}^{\infty} \sum\_{k \in \mathbb{Z}}  \psi\_{j,k}(t),$$

where < ., . > denotes the convolution or inner product between the signal and the filters. The DWT wavelet coefficients can then be computed as *W*(*j*, *k*) = 2*j*/2 *<sup>t</sup> xt*ψ 2*j t* − *k* and scaling coefficients as *V*(*j*, *k*) = 2*j*/2 *<sup>t</sup> xt*φ 2*j t* − *k j* = 1, ... , *J*; *k* = 1, ... , *n*/2*<sup>j</sup>* (Gencay et al. 2002).

Next, following the procedure outlined by (Gencay et al. 2010), the "detail" and "smooth" coefficient vectors, (*Dj*) and (*Sj*), were derived, which refer to information about a particular observation and its neighbours. An important characteristic of the DWT is that we can reconstruct *Xt* from (*Dj*) and (*Sj*). Additionally, the signal energy is preserved by the summation of the variances of the components:

$$X\_t = \sum\_{j=-1}^{J} D\_{j,t} + S\_{J,t}; \ \|\mathbf{X}\|^2 = \sum\_{j=-1}^{J} \|D\_j\|^2 + \|S\_I\|^2.$$

In its simplest form, this multi-scale transform is nothing more than taking the "difference of difference" and "average of average" to move from the finest to the coarsest representation levels of the original signal (*Xt*), while preserving information that is localised in time (Nason 2008). Specifically, at level *j*, all fluctuations associated with the frequency band <sup>1</sup> <sup>2</sup>*j*+<sup>1</sup> , <sup>1</sup> 2*j* are captured by *Dj* while all other activities (which are associated with frequencies lower than ( <sup>1</sup> <sup>2</sup>*j*+<sup>1</sup> ) are reflected in *Sj*.

As an illustration, Table 1 links the interpretation of detail levels, various frequency bands and period bands defined in terms of minutes and days (which are computed by dividing the minute column (3) by 1440—the total number of minutes in a day). In particular, in this study, we focused on frequencies corresponding to periods of 5 up to 40,960 minutes, that is, up to 28.5 days, or approximately one month.


**Table 1.** Frequency bands of the first 13 decomposition level.

Source: Authors' computations.

We can examine our data by means of the DWT described above. The multi-horizon nature of this approach gives it the name "multi-resolution analyses" (MRA). In the empirical application presented in Section 5, we employed several well-established long-memory parameter estimators on the exchange-rate data decomposed using MRA. These include the R/S estimator (Mandelbrot and Van Ness 1968), aggregated variance estimator (Dieker and Mandjes 2003), differenced variance and absolute moment estimators (Teverovsky and Taqqu 1997) and Higuchi estimator (Higuchi 1981). See (Vo and Vo 2019b) for discussions regarding these estimators.

Over the last decade, the wavelet-based methodology has gained considerably more attention in the financial volatility modelling literature thanks to its ability to offer powerful insights with respect to horizon-specific dynamics of data generating processes. Recently, (Boubaker 2020) carried out Carlo simulations to compare several wavelet-based estimators and concluded that the Wavelet Exact Local Whittle estimator outperforms the Wavelet OLS and Wavelet Geweke–Porter-Hudak estimators and generates more accurate results to identify the fractional integration parameter for symmetric heavy-tailed distributions. One of the most important applications of this novel line of research is the analyses of co-movement patterns among asset classes at different investment frequencies that could potentially offer hedging strategies to mitigate market-wise and sector downside risks. Recent related studies include (Ghosh et al. 2020), who adopted a wavelet-based time-varying dynamic approach for estimating the medium- and long-range conditional correlation among various financial and energy assets to determine their hedge ratios. Along a similar vein, (Kang et al. 2019) found strong evidence of volatility persistence, causality and phase differences between Bitcoin and gold futures prices. In addition, wavelet-filtered data are used to capture movements of Bitcoin returns at various investment horizons, and form the basis for the examination of Bitcoin's ability to hedge global uncertainty (Bouri et al. 2017).

To the best of our knowledge, applications of wavelet-based methodology to the currency markets are much more limited compared to other financial markets, a fact we seek to change with the contributions of this paper.

#### *3.2. Testing for Structural Breaks in the Presence of Long Memory*

In this section; we describe a procedure with which we can test for the existence of possible multiple structural breaks; which is a source of long memory in volatility. Previous research has documented that structural breaks in the mean can partly explain the persistence of realised volatility (Choi et al. 2010), but the effect of structural breaks could also mask that of true long memory and thus lead to misspecifications (Sibbertsen 2004). Therefore; we need to account for the possibility of structural breaks in our data. Specifically; we were firstly interested in fitting a univariate GARCH model to our data using several alternative specifications that are prominent in the literature. In GARCH models, the normalised density function is often written in terms of the location and scale parameters as:

$$\alpha\_{\mathfrak{t}} = (\mu\_{\mathfrak{t}\_{\mathfrak{t}}} \, \sigma\_{\mathfrak{t}\_{\mathfrak{t}}} \, \, \alpha)\_{\mathfrak{t}}$$

where the conditional mean and variance are given by:

$$\mu\_t = \mu(\theta, r\_t) = E(v\_t | r\_t); \sigma\_t^2 = \sigma^2(\theta, r\_t) = E[(v\_t - \mu\_t)^2 | r\_t]\_{\text{vol}}$$

with *rt* and *vt* denoting returns and volatility, and ω = ω(θ,*rt*) is the remaining parameters of the distribution.

Here, for simplicity, we assumed an ARIMA (1,1) model for the mean equation, normal distribution of the error terms and different GARCH (1,1) specifications for the variance equation. These include the standard GARCH (Bollerslev 1986), the exponential GARCH/eGARCH (Nelson 1991) and the GJRGARCH (Glosten et al. 1993), which account for asymmetric volatilities and the fractionally-integrated GARCH/fiGARCH (Baillie et al. 1996), which accounts for fractionally integrated (long-memory) processes.<sup>3</sup> For example, the specific equations for the standard GARCH (1,1) models are:

$$
\Phi(L)(1 - L)(\upsilon\_t - \mu\_t) = \Theta(L)\varepsilon\_t; \sigma\_t^2 = \omega + \alpha\_1 \varepsilon\_{t-1}^2 + \beta\_1 \sigma\_{t-1}^2,
$$

with *L* denoting the lag operator and ε*<sup>t</sup>* the residual from the mean filtration process. The asymmetric GARCH models (eGARCH and GJRGARCH) have an additional parameter γ capturing the degree of asymmetry. In contrast, the fiGARCH model has an additional parameter *d* that captures the degree of long memory.

After fitting these GARCH models, we performed tests for structural breaks by applying a Change Point Model (CPM) on the corresponding model residuals. We aimed to detect multiple change points in a sequence of observations of the volatility process, with different CPMs such as the *t*-tests proposed by (Hawkins et al. 2003); the Bartlett test (Hawkins and Zamba 2005a); the Generalised Likelihood Ratio test (Hawkins and Zamba, Statistical process control for shifts in mean or variance using a changepoint formulation (Hawkins and Zamba 2005b); the Mann–Whitney test (Ross, Tasoulis, & Adams, Nonparametric monitoring of data streams for changes in location and scale, (Ross et al. 2011); the Mood test (Ross et al. 2011) and the Kolmogorov–Smirnov test (Ross and Adams 2012). While the first three methods are designed to capture change points in Gaussian processes, the others are for non-Gaussian processes.

#### **4. Results**

Our five-minute USD/AUD nominal exchange rates (measured as the AUD cost of 1 USD, instead of the default AUD/USD rate) are provided by the commercial data vendor Bloomberg. The data coverage period is from 18:05, 7th August 2019 to 9:25, 16th September 2019—a total of *T* = 8481 intervals/observations. This period was selected when the analysis was conducted. In addition, we

<sup>3</sup> Note that when *d* = 0, the FIGARCH (1,d,1) model collapses to the standard GARCH(1,1), while when *d* = 1, it collapses to iGARCH(1,1).

considered that a total of 8481 observations is sufficient for the analysis using our selected technique. We selected the closing ask USD/AUD rate as our subject of study. Figure 2 presents MRA plots for this time series, starting with the original level in the top-left plot and ends with the coarsest smoothed series in the bottom-right plot. In between these cases are detailed series corresponding to the 13 decomposition levels presented in Table 1. Note that the original data can be reconstructed by the direct summation of all the components. It can be seen clearly that noisy fluctuations are captured by higher-frequency detail series (*D*<sup>1</sup> to *D*8) while these noises can be filtered out in lower-frequency details (*D*<sup>9</sup> to *D*13) and in the smooth component (*S*13).

**Figure 2.** Multi-resolution plots of USD/AUD time series. Notes: This figure presents the MRA for the full data range from 7th August 2019, to 9:25, 16th September 2019. In each panel, the horizontal axis indicates the corresponding days of the two months.

The continuously compounded exchange rate returns are computed as the differences of logarithmic five-minute exchange rates: *rit* = log *pit* − log *pi*,*t*−<sup>1</sup> (*t* = 1, ... , *T*). Exchange rate volatility is proxied by the 288-interval (or one-day) rolling standard deviations of returns, that is, σ*it* = (1/288) 289 *j* = 2 log *pi*,*t*+*<sup>j</sup>* <sup>−</sup> log *pi*,*t*+*j*−<sup>1</sup> <sup>−</sup> *rit*<sup>2</sup> , where *rit* = (1/288) 289 *j* = 2 log *pi*,*t*+*<sup>j</sup>* − log *pi*,*t*+*j*−<sup>1</sup> is the rolling average return. This means the first 288 return observations are set aside for the computation of the first realized volatility value, leaving us with 8193 observations.

Summary statistics of the volatility and return series are presented in Table 2. Overall, the return series distribution resembles normality, albeit having high kurtosis. On the other hand, the volatility is left-skewed, as, by construction, it only contains positive values. To examine the long-range dependence patter of our data, Figure 3 illustrates the corresponding five-minute auto-correlograms, or visualised autocorrelation function (ACF), for the two series. As can be seen, the return series exhibit no significant autocorrelation pattern after the first lag, while the volatility series clearly demonstrates long-range dependence.


**Table 2.** Summary statistic of five-minute USD/AUD returns and volatilities.

Notes: Returns of the USD/AUD exchange rate are computed as the log-change of the corresponding five-minute spot USD/AUD: *rit* = log *pit* − log *pi*,*t*−<sup>1</sup> (*t* = 1, ... , *T*), where *pit* denotes the nominal exchange rate. *T* denotes the number of observations (8193 five-minute intervals). Volatilities are defined as the one-day rolling standard deviations of *rit*. JB and LB denote the Jarque–Bera and the Ljung–Box statistics, respectively. *p*-values are in parentheses. Source: Authors' computations.

**Figure 3.** Auto-correlograms of USD/AUD high-frequency returns and volatilities. Notes: Exchange-rate returns are computed as the log-change of the corresponding five-minute closing spot rates: *rit* = log *pit* − log *pi*,*t*−<sup>1</sup> (*t* = 1, ... , 8193). Volatilities are defined as the one-day rolling standard deviations of *rit*. Source: Authors' computations.

#### *4.1. A Multi-Resolution Analysis*

Our main question of interest is "At which particular horizons does the long-memory behaviour of the USD/AUD exchange rate persist?" To answer this, Table 3 presents estimates of the Hurst index as applied to the original series as well as the 12 levels of smooth components which capture all activities at frequencies lower than <sup>1</sup> <sup>2</sup>*j*+<sup>1</sup> and thus preserve the underlying trends of the volatility.4 As can be seen from this table, the volatility measures of five-minute AUD/USD returns exhibit a very persistent pattern of long memory, at all levels of decomposition, where the Hurst index estimated using all methods is significantly larger than 0.5. This means that there are no frequencies that are solely responsible for the long-range dependence characteristic of the Australian dollar. This could be explained by the fact that the trader base of this open-economy currency is quite diverse and active, who tend to switch trading

<sup>4</sup> On the other hand, "detail" components reflect the fluctuations (or differences) of volatility series and thus are not representative of the long-range dependent behaviour.

horizons frequently via diversification/rebalancing operations. This makes disentangling the impacts of activities at a particular frequency from those at other frequencies difficult.5


**Table 3.** Long-memory parameter estimates of exchange-rate volatility at different horizons.

Notes: Nomenclatures: (1) R/S: Rescaled range; (2) aggVar: Aggregated variance; (3) diffVar: Differenced variance; (4) AbsVar: Absolute moments; (5) Higuchi: Higuchi's method. Heteroskedasticity-robust standard errors are in parentheses. Refer to Table 2 for interpretation of the decomposition levels/time-scales. Source: Authors' computations.

#### *4.2. Horizon-Based Power Decomposition*

Given the highly persistent pattern of long memory observed in AUD/USD volatility, it is now fruitful to analyse the multi-scale composition of power (or variations) of the original nominal five-minute exchange rate. To do this, in Figure 4 we present a "heat map" representation of the MRA as proposed by (Torrence and Compo 1998), which illustrates the power scale of the original series through both time and frequencies. Stronger colours (i.e., red or orange) at any frequency and time represent higher power scales and stronger cyclical behaviour.6 We performed wavelet decomposition only up to the horizon corresponding to 512 five-minute intervals. This design allowed us to investigate

<sup>5</sup> Additionally, the behaviour of exchange rates can be related to the dynamic long-memory properties of other economic variables, such as the aggregated price levels (via the purchasing power parity relationship) or the interest rates (via the uncovered interest parity relationship).

<sup>6</sup> The computation is done this time with a continuous Morlet wavelet transform, rather than a DWT. Due to some issues with this operator, certain information outside the region outlined by the parabolic curve (the "cone of influence") should be ignored. (e.g., (Daubechies 1992) for details.)

the interaction dynamics of the intraday volatility process. The map reveals features that are in close conjunction with the cyclical behaviour of the series, which is not easy to discern without the map. Specifically, frequencies corresponding to the periods of 256 to 512 five-minute intervals are observed to exhibit the highest power while no strong cyclical pattern can be observed at higher frequencies. These results corroborate those of (Caporale et al. 2019).

In agreement with Figure 2, though at shorter horizons there are only small intraday noises, there exist large (but infrequent) movements at the longer horizons in the AUD/USD exchange-rate dynamics. Interestingly, it can also be seen that there are two episodes of volatility spillover between the low frequencies and the high frequencies in this sample: The August 13th and the September 10th. These days are also associated with episodes of relatively high volatility. The former effect is tied in with the release of the statement on monetary policy by the Reserve Bank of Australia on August 9th, while the more prominent effect on the second date could be a result of market anticipation during the week leading to the meetings of the Federal Open Market Committee (US) and Reserve Bank Board (Australia) meetings, both of which are on September 17th. In the next subsection, we investigated possible breaks in the volatility process in more details.

**Figure 4.** Wavelet heat map of five-minute nominal USD/AUD exchange rate. Notes: Horizontal axis ranges from 0 to 8193, the number of five-minute intervals in our sample. The vertical axis indicates the horizons (in five-minute) corresponding to the frequencies at which the underlying time series fluctuates. The power meter is located beneath the graph. The area within the parabolic region indicates the "zone of influence". The plots were drawn using functions provided in the R package *dplR* (Bunn 2008). Source: Authors' computations.

#### *4.3. Sources of Long Memory: Structural Breaks*

Table 4 presents the estimated results for the alternative GARCH (1,1) models discussed in Section 3. The likelihood value and information criteria are in agreement that the most appropriate model is the eGARCH, followed by the standard GARCH, then the fiGARCH and GJRGARCH. Interestingly, the long-memory parameter estimates implied by fiGARCH (*d* = 0.85) again confirm the long-memory characteristic of the volatility process.


**Table 4.** GARCH (1,1) models estimates.

Notes: This table presents estimation results for different GARCH (1,1) specifications described in Section 4. The last row of parameter estimates refers to the asymmetric parameter γ for the eGARCH and GJRGARCH models, while for the fiGARCH model this refers to the fractional differential parameter *d*. *p*-Values based on robust standard errors in parentheses.

Based on these estimates, we were able to extract the residuals of these models and apply the CPMs described in Section 4 to test for breakpoints of the (conditional) volatility process. Test results are presented in Table 5. We can see that the number of structural breaks detected is substantial, given the high frequency of our data.<sup>7</sup> Importantly, the breaks exist for models exclusively designed to capture long memory, such as fiGARCH, regardless of the assumption of the underlying distribution.


**Table 5.** Number of breakpoints detected using different test statistics.

Notes: The tests listed are applied to residuals from the GARCH models described in Table 5. Nomenclatures: GLR (Generalised Likelihood Ratio), MW (Mann–Whitney), M (Mood) and KS (Kolmogorov-Smirnov). Sources: Authors' examinations and computations.

#### **5. Discussions, Conclusions and Implications**

#### *5.1. Discussions*

We contribute to the existing literature by providing a careful examination of the time series characteristics of the AUD/USD exchange rate at a very high frequency, which has important economic implications. As a final note, we conjecture that long memory is observed for this series and is persistent throughout the trading horizons, which implies that investors should be wary of such changes when managing their portfolios. Secondly, rather than focusing on short-term fluctuations and gains, an optimal trading horizon would preferably be longer than half-day, as this could capture more fundamental trend information of the exchange-rate return processes.

<sup>7</sup> The exact time periods when the breaks are detected are not presented here to conserve space but are available upon request.

Our study complements the findings of (Caporale et al. 2019), who documented the persistence of both returns and volatility processes of the EUR/USD and USD/JPY exchange rates at lower trading frequencies. In agreement with this paper, we concur that such evidence against random-walk behaviour implies predictability and is inconsistent with the Efficient Market Hypothesis since abnormal profits can be made using trading strategies based on trend analysis. We also extended this research by introducing the wavelet-based long-memory estimator, as opposed to relying on conventional tools such as the R/S statistic or the fractional integration analysis. Another recent study related to ours is (Boubaker 2020) whose Monte Carlo simulation results suggest that when it comes to estimating the long-memory parameter in stationary time series, the Wavelet Exact Local Whittle estimator outperforms the Wavelet OLS and Wavelet Geweke–Porter-Hudak estimators in terms of smaller bias. It would be interesting to extend this comparison exercise to include our Wavelet MLE approach and apply these estimators on actual data (rather than on simulations).

This study is subject to two qualifications. First of all, due to our limited access to high-frequency exchange-rate data, we were unable to examine further the implication of our results for the Australian dollar for other (commodity) currencies. A possibly more general conclusion can be drawn when more of these valuable data are available to us.8 Secondly, our research is limited to the currency market. Applying the same approach to other financial markets such as stocks, bonds or commodity futures to examine their volatility persistence and the workings of the EMH offers an interesting future research venue.

#### *5.2. Conclusions and Implications*

Existing literature indicates that the choice of an appropriate statistical tool for analysing exchange-rate dynamics should ultimately be made based on the long-memory properties of the underlying data generating process, which varies across different trading horizons. The Australian dollar is generally considered as a representative commodity currency given the performance of the Australian economy is mainly driven by commodities and the Australian dollar is one of the top 10 most frequently traded currencies in the world. As such, this study was conducted to examine the Australian dollar-US dollar exchange rates—one of the most popular and frequently traded pairs of currencies. This study covers the period from 18:05, 7th August 2019 to 9:25, 16th September 2019 with a total of 8481 observations—a sufficient number of observations required for our analysis. In this paper, we used a wavelet-based approach that allows for modelling long-memory characteristics of this important currency pair at different trading horizons.

The high-frequency behaviour of exchange rates observed from our study would be valuable for designing and evaluating exchange-rate models and/or forecasts. More generally, these insights can potentially be used to evaluate the currency risks related to the Australian trade balance, trade flows, terms-of-trade, prices of foreign-exchange futures (or options) and/or international asset portfolio formation.

**Author Contributions:** Theoretical frameworks surveyed conducted are done by L.H.V. Both authors conduct reviews of empirical analyses. The original draft is prepared by D.H.V. Reviewing and editing are done by both authors. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Ho Chi Minh City Open University, Vietnam [E2020.14.1].

<sup>8</sup> Nevertheless, in a recent study (Vo and Vo 2019b) have applied wavelet-based estimators on daily data of six heavily traded currencies, including AUD, and have shown cross-currency results that are similar to ours.

**Acknowledgments:** Part of this research was completed when Long Vo was a Master student at the School of Economics and Finance, Victoria University of Wellington, where he received financial support from a New Zealand-ASEAN Scholar Award provided by the New Zealand Ministry of Foreign and Trade. Duc Vo acknowledges the financial assistance from Ho Chi Minh City Open University. We thank the editor and three anonymous referees whose comments helped improve the paper. All remaining errors are our own.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


Moosa, Imad. 2003. The sensitivity of the optimal hedging ratio to model specification. *Finance Letter* 1: 15–20. Nason, Guy. 2008. *Wavelet Methods in Statistics with R*. Berlin: Springer.


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **A Note on Simulation Pricing of** *π***-Options**

#### **Zbigniew Palmowski \*,† and Tomasz Serafin †**

Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, ul. Wyb. Wyspia ´nskiego 27, 50-370 Wrocław, Poland; tomaszserafin.96@gmail.com

**\*** Correspondence: zbigniew.palmowski@pwr.edu.pl; Tel.: +48-71-320-2012

† These authors contributed equally to this work.

Received: 8 July 2020; Accepted: 24 August 2020; Published: 28 August 2020

**Abstract:** In this work, we adapt a Monte Carlo algorithm introduced by Broadie and Glasserman in 1997 to price a *π*-option. This method is based on the simulated price tree that comes from discretization and replication of possible trajectories of the underlying asset's price. As a result, this algorithm produces the lower and the upper bounds that converge to the true price with the increasing depth of the tree. Under specific parametrization, this *π*-option is related to relative maximum drawdown and can be used in the real market environment to protect a portfolio against volatile and unexpected price drops. We also provide some numerical analysis.

**Keywords:** *π*-option; American-type option; optimal stopping; Monte Carlo simulation

**MSC:** 49L20; 60G40

**JEL Classification:** G13; C61

#### **1. Introduction**

In this paper, we analyze *π*-options introduced by Guo and Zervos (2010) that depends on so-called relative drawdown and can be used in hedging against volatile and unexpected price drops or by speculators betting on falling prices. These options are the contracts with a payoff function:

$$\mathcal{S}(\mathcal{S}\_T) = (M\_T^a \mathcal{S}\_T^b - K)^+ \tag{1}$$

in case of the call option and

$$\lg(S\_T) = (K - M\_T^a S\_T^b)^+ \tag{2}$$

in the case of put option, where

$$S\_t = S\_0 \exp\left( \left( r - \frac{\sigma^2}{2} \right) t + \sigma B\_t \right) \tag{3}$$

is an asset price in the Black-Scholes model under martingale measure, i.e., *r* is a risk-free interest rate, *σ* is an asset's volatility and *Bt* is a Brownian motion. Moreover,

$$M\_t = \sup\_{w \le t} S\_w$$

is a running maximum of the asset price and *T* is its maturity. Finally, *a* and *b* are some chosen parameters.

A few very well-known options are particular cases of a *π*-option. In particular, taking *a* = 0 and *b* = 1 produces an American option and by choosing *a* = 1 and *b* = 0 we derive a lookback option. Another interesting case, related to the concept of drawdown (see Figure 1), is when −*a* = *b* = 1

and *<sup>K</sup>* = 1. Then the pay-out function (*<sup>K</sup>* − *<sup>M</sup><sup>a</sup> TS<sup>b</sup> <sup>T</sup>*)<sup>+</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> *<sup>S</sup><sup>b</sup> T M*−*<sup>a</sup> T* = *MT*−*St MT* <sup>=</sup> *<sup>D</sup><sup>R</sup> <sup>T</sup>* equals the relative drawdown *D<sup>R</sup> <sup>t</sup>* , defined as a quotient of the difference between maximum price and the present value of the asset and the past maximum price. In other words, *D<sup>R</sup> <sup>t</sup>* corresponds to the percentage drop in price from its maximum. We take a closer look at this specific parametrization of the *π*-option in the later sections, starting from Section 3.2.

**Figure 1.** A sample drawdown for the Microsoft Corporation stock is marked with black arrows and dashed lines. Data is taken from www.finance.yahoo.com.

Monte Carlo simulations are widely used in pricing in financial markets they have proved to be valuable and flexible computational tools to calculate the value of various options as witnessed by the contributions of Barraquand and Martineau (1995); Boyle (1977); Boyle et al. (1997); Broadie et al. (1997); Caflisch (1998); Clément et al. (2002); Dyer and Jacob (1991); Geske and Shastri (1985); Glasserman (2004); Jäckel (2002); Joy et al. (1996); Longstaff and Schwartz (2001); Niederreiter (1992); Raymar and Zwecher (1997); Rogers (2002); Tilley (1993); Tsitsiklis and van Roy (1999, 2001); Resenburg and Torrie (1993); Villani (2010).One of the first attempts of Monte Carlo simulation for American options is by Tsitsiklis and van Roy (1999) where the backward induction algorithm was introduced. However, as appears later, Tilley method suffers from exponentially increasing computational cost as the number of dimensions (assets) increases. Broadie et al. (1997) to overcome this problem offered a non-recombining binomial simulation approach instead combined with some pruning technique to reduce computation burden and other variance reduction techniques to increase precision. In the same year Broadie and Glasserman (1997) construct computationally cheap lower and upper bounds to the American option price. This method is used in this paper. An alternative way to formulate the American option pricing problem is in terms of optimal stopping times. This is done in Carriere (1996), where it was proved that finding the price of American option can be based on a backwards induction and calculating several conditional expectations. This observation gives another breakthrough in pricing early exercise derivatives by Monte Carlo done by Longstaff and Schwartz (2001). They propose least square Monte Carlo (LSM) method which has proved to be versatile and easy to implement. The idea is to estimate the conditional expectation of the payoff from continuing to keep the option alive at each possible exercise point from a cross-sectional least squares regression using the information in the simulated paths. To do so we have to then solve some minimization problem. Therefore, this method is still computationally expensive. Some improvements of this method have been also proposed; see also Stentoft (2004a, 2004b) who gave theoretical foundation of LSM and properties of its estimator.

There are other, various pricing methods in the case of American-type options; we refer Zhao (2018) for review. We must note though that not all of them are good for simulation of prices of general *π*-options as it is a path-dependent product. In particular, in pricing *π*-options one cannot use finite difference method introduced by Brennan and Schwartz (1978); Schwartz (1977) which uses a linear combination of the values of a function at three points to approximate a linear combination of the values of derivatives of the same function at another point. Similarly, the analytic method of lines of Carr and Faguet (1994) is not available for pricing general *π*-options. One can use though a binomial tree algorithm (or trinomial model) though which goes backwards in time by first discounting the price along each path and computing the continuation value. Then this algorithm compares the former with the latter values and decide for each path whether or not to exercise; see Broadie and Detemple (1996) for details and references therein. It is a common belief that Monte Carlo method is more efficient than binomial tree algorithm in case of path-dependent financial instruments. It has another known advantages as handling time-varying variants, asymmetry, abnormal distribution and extreme conditions.

In this paper, we adapt a Monte Carlo algorithm proposed in 1997 by Broadie and Glasserman (1997) to price *π*-options. This numerical method replicates possible trajectories of the underlying asset's price by a simulated price tree. Then, the values of two estimators, based on the price tree, are obtained. They create an upper and a lower bound for the true price of the option and, under some additional conditions, converge to that price. The first estimator compares the early exercise payoff of the contract to its expected continuation value (based on the successor nodes) and decides if it is optimal to hold or to exercise the option. This estimation technique is one of the most popular ones used for pricing American-type derivatives. However, as shown by Broadie and Glasserman (1997), it overestimates the true price of the option. The second estimator also compares the expected continuation value and early exercise payoff, but in a slightly different way, which results in underestimation of the true price. Both Broadie–Glasserman Algorithms (BGAs) are explained and described precisely in Section 2. The price tree that we need to generate is parameterized by the number of nodes and by the number of branches in each node. Naturally, the bigger the numbers of nodes and branches, the more accurate price estimates we get. The obvious drawback of taking a bigger price tree is that the computation time increases significantly with the size of the tree. However, in this paper we show that one can take a relatively small price tree and still the results are satisfactory.

The Monte Carlo simulation presented in this paper can be used in corporate finance and especially in portfolio management and personal finance planning. Having American-type options in the portfolio, the analyst might use the Monte Carlo simulation to determine its expected value even though the allocated assets and options have varying degrees of risk, various correlations and many parameters. In fact determining a return profile is a key ingredient of building efficient portfolio. As we show in this paper portfolio with *π*-options out-performs typical portfolio with American put options in hedging investment portfolio losses since it allows investors to lock in profits whenever stock prices reaches its new maximum.

In this paper, we use BGA to price the *π*-option on relative drawdown for the Microsoft Corporation's (MSFT) stock and for the West Texas Intermediate (WTI) crude oil futures. Input parameters for the algorithm are based on real market data. Moreover, we provide an exemplary situation in which we explain the possible application of the *π*-option on relative drawdown to the protection against volatile price movements. We also compare this type of option to an American put and outline the difference between these two contracts.

This paper is organized as follows. In the next section we present the Broadie–Glasserman Algorithm. In Section 3 we use this algorithm to numerically study *π*-options for the Microsoft Corporation's stock and WTI futures. Finally, in the last section, we state our conclusions and recommendations for further research in this new and interesting topic.

#### **2. Monte Carlo Algorithm**

Formulas identifying the general price of *π*-option are known in some special cases and they are given in terms of so-called scale functions and hence in terms of the solution of some second order ordinary differential equations; see for example, (Christensen 2013, chp. 5) and Egami and Oryu (2017) for details and further references. Still, the formulas are complex, and a Monte Carlo method of pricing presented in this paper is very efficient and accurate alternative method. In this section we present a detailed description of the used algorithm. In particular, we give formulas for two estimators, one biased low and one biased high, that under certain conditions converge to the theoretical price of the option.

#### *2.1. Preliminary Notations*

We adapt the Monte Carlo method introduced by Broadie and Glasserman (1997) for pricing American options. In this algorithm, values of two estimators are calculated on the so-called price tree that represents the underlying's behavior over time. This tree is parametrized by the number of nodes *n* and the number of branches in each node—denoted by *l*. For example, the tree with parameters *n* = 2, *l* = 3 is depicted in Figure 2.

**Figure 2.** An example of the *price tree*. Underlying's price is marked with circles and corresponding maximums are marked with rectangles.

*Risks* **2020**, *8*, 90

To apply the numerical algorithm, we must discretize the price process given in (3), by considering the time sequence *t*<sup>0</sup> = 0 < *t*<sup>1</sup> < ... < *tn* = *T* with *ti* = *i <sup>T</sup> <sup>n</sup>* for *i* = 0, . . . , *n*. By

$$S\_{t\_i^{l\_1 \dots l\_i}}$$

we denote the asset's price at the time *ti* = *iT <sup>n</sup>* . The upper index *l*1,..., *li*, associated with *ti*, describes the branch selection (see Figure 3) in each of the tree nodes and allows us to uniquely determine the path of the underlying's price process up to time *ti*. Similarly, we define

$$\mathcal{M}\_{\mathfrak{t}\_i^{l\_1,\ldots,l\_i}} = \max\_{k \le i} \mathcal{S}\_{\mathfrak{t}\_k^{l\_1,\ldots,l\_k}}.$$

We introduce the state variable *S t l* 1,...,*l <sup>i</sup> <sup>i</sup>* = (*S t l* 1,...,*l <sup>i</sup> <sup>i</sup>* , *Mt l* 1,...,*l <sup>i</sup> <sup>i</sup>* ) as well.

**Figure 3.** Branch selecting.

We relate with it the payoff of an immediate exercise (for *π*-put) at time *ti* in the state *S t l* 1,...,*l <sup>i</sup> <sup>i</sup>* given by

$$h\_{\mathfrak{l}\_i}(\widetilde{\boldsymbol{S}}\_{\mathfrak{l}\_i^{l\_1,\dots,l\_i}}) = (\boldsymbol{K} - \boldsymbol{S}^{a}\_{\mathfrak{l}\_i^{l\_1,\dots,l\_i}} \boldsymbol{\mathcal{M}}^{b}\_{\mathfrak{l}\_i^{l\_1,\dots,l\_i}})^{+} $$

and the expected value of holding the option from *ti* to *ti*+1, given asset's value *S t l* 1,...,*l <sup>i</sup> <sup>i</sup>* at time *ti* defined via

$$\log\_{t\_i}(\check{\mathcal{S}}\_{t\_i^{l\_1,\ldots,l\_i}}) = \mathbb{E}\left[e^{-\frac{\mathsf{r}}{\mathsf{m}}} f\_{t\_{i+1}}(\check{\mathcal{S}}\_{t\_{i+1}^{l\_1,\ldots,l\_{i+1}}}) \Big| \check{\mathcal{S}}\_{t\_i^{l\_1,\ldots,l\_i}}\right] \prec$$

where

$$f\_{t\_i}(\check{\mathcal{S}}\_{t\_i^{l\_1,\dots,l\_i}}) = \max \{ h\_{t\_i}(\check{\mathcal{S}}\_{t\_i^{l\_1,\dots,l\_i}}), g\_{t\_i}(\check{\mathcal{S}}\_{t\_i^{l\_1,\dots,l\_i}}) \} $$

is the option value at time *ti* in state *S t l* 1,...,*l <sup>i</sup> <sup>i</sup>* . Please note that

$$f\_{t\_n}(\check{\mathcal{S}}\_{t\_n^{l\_1,\dots,l\_n}}) = f\_T(\check{\mathcal{S}}\_{T^{l\_1,\dots,l\_n}}) = h\_T(\check{\mathcal{S}}\_{T^{l\_1,\dots,l\_n}}) = (\mathcal{K} - \mathcal{S}^{a}\_{T^{l\_1,\dots,l\_n}} \mathcal{M}^{b}\_{T^{l\_1,\dots,l\_n}})^+.$$

#### *2.2. Estimators*

We will now give the formulas for the estimators Θ and Φ which overestimate and underestimate the true price of the option, respectively. Then, we will state the main theorem showing that both estimators are asymptotically unbiased and that they converge to the theoretical price of the *π*-option. We also provide a detailed explanation of the estimation procedure based on the exemplary price tree. In all calculations we consider a *π*-put option with parameters *a* = −1, *b* = 1 and *K* = 1. Additionally, we assume that the risk-free rate used for discounting the payoffs equals 5%.

#### 2.2.1. The Θ Estimator

The formula for the estimator is recursive and given by:

$$\Theta\_{l\_i} = \max \left\{ h\_{l\_i}(\widetilde{S}\_{l\_i^{l\_1,\ldots,l\_i}}), e^{-\frac{L}{n}} \frac{1}{l} \sum\_{j=1}^l \Theta\_{l\_{i+1}^{l\_1,\ldots,l\_i;j}} \right\}, \qquad i = 0, \ldots, n-1.$$

At the option's maturity, *T*, the value of the estimator is given by

$$
\Theta\_T = f\_T(\mathcal{S}\_T).
$$

The Θ estimator, at each node of the price tree, chooses the maximum of the payoff of the option's early exercise at time *ti*, *hti* (*S t l* 1,...,*l <sup>i</sup> <sup>i</sup>* ), and the expected continuation value, i.e., the discounted average payoff of successor nodes. Figure 4 shows how the value of Θ estimator is obtained given the certain realization of a price tree. All calculations are also shown below:

$$\begin{array}{ll} \bullet & \mathfrak{g} \begin{cases} \text{Holding value:} \frac{0 + \frac{10}{15} + 0}{3} e^{-0.05} \approx \underline{0.028} \\ \text{Early exercise:} \
0 \\ \text{ $\mathfrak{g}$ } \begin{cases} \text{Holding value:} \frac{5 + 0 + \frac{10}{100}}{3} e^{-0.05} \approx \, 0.052 \\ \text{Early exercise:} \frac{10}{170} \approx \underline{0.091} \\ \text{Holding value:} \frac{\frac{10}{150} + \frac{15}{100} + \frac{5}{100}}{3} e^{-0.05} \approx \, 0.086 \\ \text{Early exercise:} \frac{20}{170} \approx \underline{0.182} \\ \text{Holding value:} \frac{0.028 + 0.091 + 0.182}{3} e^{-0.05} \approx \underline{0.095} \\ \text{Early exercise:} \frac{10}{110} \approx 0.091 \end{cases} \end{array}$$

**Figure 4.** Explanation of Θ estimator.

#### 2.2.2. The Φ Estimator

The Φ estimator is also defined recursively. Before we give the formula we need to introduce an auxiliary function *ξ* by

$$\mathbb{Z}\_{t\_i^{l\_1,\ldots,l\_i}}^j = \begin{cases} h\_{l\_i}(\widecheck{S}\_{t\_i^{l\_1,\ldots,l\_i}}), \text{ if } h\_{l\_i}(\widecheck{S}\_{t\_i^{l\_1,\ldots,l\_i}}) \ge e^{-\frac{r}{\pi}} \frac{1}{l-1} \sum\_{\substack{k=1\\k\neq j}}^l \Phi\_{l\_{i+1}^{l\_1,\ldots,l\_i}} \\\\ e^{-\frac{r}{\pi}} \Phi\_{l\_{i+1}^{l\_1,\ldots,l\_i;j}}, \text{ if } h\_{l\_i}(\widecheck{S}\_{t\_i^{l\_1,\ldots,l\_i}}) < e^{-\frac{r}{\pi}} \frac{1}{l-1} \sum\_{\substack{k=1\\k\neq j}}^l \Phi\_{l\_{i+1}^{l\_1,\ldots,l\_i;k}} \end{cases} \tag{4}$$

for *j* = 1, . . . , *l*. Now we can define the Φ estimator in the following way:

$$\begin{cases} \Phi\_{\boldsymbol{t}\_{i}^{l\_{1},...,l\_{i}}} = \frac{1}{l} \sum\_{j=1}^{l} \xi\_{\boldsymbol{t}\_{i}^{l\_{1},...,l\_{i}}}^{j} \\ \Phi\_{T} = f\_{T}(\widetilde{S}\_{T}). \end{cases} \tag{5}$$

The formula for this estimator is more complicated. Therefore, we provide a detailed explanation of the mechanism behind the algorithm in the following part of this section. In our explanation we refer to Figure 5. Please note that in the following example, underlined numbers correspond to the final values associated with the specific branches of the tree.

**Figure 5.** Explanation of Φ estimator.

Early exercise: 0

⎧

• **a** ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ Holding value for branch *<sup>j</sup>* <sup>=</sup> 1: 0.087+<sup>0</sup> <sup>2</sup> *<sup>e</sup>*−0.05 <sup>≈</sup> 0.041 <sup>&</sup>gt; <sup>0</sup> <sup>→</sup> <sup>0</sup> Holding value for branch *j* = 2: <sup>0</sup>+<sup>0</sup> <sup>2</sup> *<sup>e</sup>*−0.05 = <sup>0</sup> ≤ <sup>0</sup> = *hi*(*Sti* ) → 0 Holding value for branch *j* = 3: <sup>0</sup>+0.087 <sup>2</sup> *<sup>e</sup>*−0.05 ≈ 0.041 > <sup>0</sup> → <sup>0</sup>

For the branch *j* = 1 we look at the two remaining ones to determine whether early exercising (payoff = 0) or holding the option (payoff = 0.087+<sup>0</sup> <sup>2</sup> *<sup>e</sup>*<sup>−</sup>0.05) is more profitable. Obviously, early exercise is not optimal, so we hold the option and thus, as the value of *ξ*<sup>1</sup> *t* 1 1 we take the payoff of the branch *j* = 1 which is 0.

For the branch *j* = 2 both early exercise value and holding value from two other branches equals 0. Thus, from (4) the value of *ξ*<sup>2</sup> *t* 1 1 equals the payoff of early exercise, which is 0.

For the third branch, again holding the option is a more profitable decision (based on the payoffs of the two remaining branches). Thus, *ξ*<sup>3</sup> *t* 1 1 takes the value corresponding to the branch *j* = 3 and it is 0.

Now the value of the estimator for node **a** is the sum of *ξ j t* 1 across all branches:

$$
\Phi\_{\mathfrak{t}\_1^1} = \frac{1}{3} \sum\_{j=1}^3 \xi\_{\mathfrak{t}\_1^1}^j = 0.
$$

1

Similarly, we have the following values of our estimator.

⎧ Early exercise: <sup>10</sup> <sup>110</sup> ≈ 0.091

• **b** ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ Holding value for branch *j* = 1: <sup>0</sup>+0.118 <sup>2</sup> *<sup>e</sup>*−0.05 ≈ 0.056 < 0.091 → 0.091 Holding value for branch *j* = 2: 0.045+0.118 <sup>2</sup> *<sup>e</sup>*−0.05 ≈ 0.078 < 0.091 → 0.091 Holding value for branch *<sup>j</sup>* <sup>=</sup> 3: 0.045+<sup>0</sup> <sup>2</sup> *<sup>e</sup>*−0.05 <sup>≈</sup> 0.021 <sup>&</sup>lt; 0.091 <sup>→</sup> 0.091

In this case, the value of the estimator for the **b** node equals 0.091. ⎧

$$\text{Early exercise: } \frac{20}{170} \approx \underline{0.182}$$

• **c** ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ Holding value for branch *j* = 1: 0.045+0.136 <sup>2</sup> *<sup>e</sup>*−0.05 ≈ 0.086 < 0.182 → 0.182 Holding value for branch *j* = 2: 0.091+0.045 <sup>2</sup> *<sup>e</sup>*−0.05 ≈ 0.065 < 0.182 → 0.182 Holding value for branch *j* = 3: 0.091+0.136 <sup>2</sup> *<sup>e</sup>*−0.05 ≈ 0.108 < 0.182 → 0.182

For node **c** the value of the estimator is 0.182. ⎧

Early exercise: <sup>10</sup> <sup>110</sup> ≈ 0.091

• **d** ⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩ Holding value for branch *j* = 1: 0.091+0.182 <sup>2</sup> *<sup>e</sup>*−0.05 ≈ 0.13 > 0.091 → <sup>0</sup> Holding value for branch *j* = 2: 0.0+0.182 <sup>2</sup> *<sup>e</sup>*−0.05 ≈ 0.087 < 0.091 → 0.091 Holding value for branch *j* = 3: <sup>0</sup>+0.091 <sup>2</sup> *<sup>e</sup>*−0.05 ≈ 0.043 < 0.091 → 0.091

The value of the estimator for this node equals <sup>0</sup>+0.091+0.091 <sup>3</sup> <sup>=</sup> 0.061. This is also the (under)estimated value of the option.

Following arguments of Broadie and Glasserman (1997), one can easily prove the following crucial fact.

**Theorem 1.** *Both* Θ *and* Φ *are consistent and asymptotically unbiased estimators of the option value. They both converge to the true price of the option as the number of price tree branches, l, increases to infinity. For a finite l:*

• *The bias of the* Θ *estimator is always positive, i.e.,*

$$\mathbb{E}[\Theta\_0(I)] \ge f\_0(\tilde{S}\_0).$$

• *The bias of the* Φ *estimator is always negative, i.e.,*

$$\mathbb{E}[\Phi\_0(l)] \le f\_0(\check{S}\_0).$$

*On every realization of the price tree, the low estimator* Φ *is always less than or equal to the high estimator* Θ*, i.e.,*

$$\mathbb{P}(\Phi\_{t\_i^{l\_1,\ldots,l\_i}} \le \Theta\_{t\_i^{l\_1,\ldots,l\_i}}) = 1.$$

#### **3. Numerical Analysis**

In this section, we will present results of the numerical analysis. First, we use the algorithm described above to price the American option with arbitrary parameters. This will allow us to confirm that our Monte Carlo algorithm produces precise estimates of options' prices. We focus on options related to Microsoft Corporation stock. Next, we price *π*-options for several combinations of parameters. We also consider *π*-option on drawdown using the real market data and we compare it with an American put, which is one of the most popular tool for protecting our portfolio against price drops.

#### *3.1. American Options*

First of all, we decided to check the robustness of the Monte Carlo pricing algorithm. We estimate prices of the American call options with different strike prices. In the example, the uderlying asset price *S*<sup>0</sup> equals 100, *σ* = 20%, risk-free rate *r* = 5% and the maturity is 30 days. In Table 1 we present the results of the estimation. Please note that when using the Broadie–Glasserman algorithm, we obtain the upper and the lower boundaries of the option price. To obtain the American option price estimate we average both values.

**Table 1.** Comparison of the estimated and 'real' American option prices with different strikes. Absolute percentage errors are also included.


#### *3.2. π-Options*

We will analyze put *π*-option for various combinations of parameters *a* and *b*. We assume that parameter *a* is varying from −1.1 to −0.9 and *b* parameter between 0.9 and 1.1. The ranges of these parameters have been chosen arbitrarily for illustrative purposes. All input parameters for options pricing, *S*0, *M*0, volatility and interest rate are taken from the real market data for the Microsoft Corporation stock (MSFT) and are given in Table 2. The numerical results are presented in Figure 6.

**Table 2.** Input parameters for pricing *π*-put option on the Microsoft Corporation stock.


**Figure 6.** *π*-option price estimations for varying *a* and *b*—Microsoft Corporation stock.

#### *3.3. π-Options on Relative Drawdown*

Recall that for *a* = −1 and *b* = 1 the payoff of the *π*-option equals

$$\left(K - \frac{S\_t}{M\_t}\right)^+,\tag{6}$$

where *St*/*Mt* is the current value of the relative drawdown of the underlying asset. We believe that such contracts could be very efficiently used for hedging and managing portfolio risk against the volatile drops in underlying's price (see Section 3.4). One can adjust the payoff function (6) by the appropriate choice of the strike *K*. The choice is arbitrary and solely dependent on the risk management goals of the option's buyer. It allows the setting of the minimal size of drawdown we would like to protect against and let the buyer adjust and full control of the level of our exposure at risk associated with unexpected price drops. For example by setting *K* = <sup>9</sup> <sup>10</sup> , the payoff of our option becomes greater than zero only if the drop in the price of the underlying from its maximum exceeds 10%. Of course, the bigger the value of *K*, the more expensive the option is.

We take a closer look at the impact of *Mt* and *K* on the price of this special case of *π*-option. Here, we assume that the maximum price *Mt* is between 100 and 120 and *K* ranges between 0.8 and 1. This time, the remaining parameters, namely *S*0, *r* and *σ*, have been arbitrarily chosen for illustrative purposes and are given in Table 3. The results are shown in Figure 7.

**Table 3.** Input parameters for pricing *π*-option on relative drawdown.


**Figure 7.** *π*-option on relative drawdown price estimations for varying *K* and *M*<sup>0</sup> parameters.

#### *3.4. π-Options on Relative Drawdown - Application*

We now focus on the potential application of *π*-options and compare the prices of American put and *π*-option on relative drawdown. We compare these particular instruments due to the fact that their values increase with the decrease of the underlying asset's price. As an exemplary environment for the options comparison we choose two time series containing daily closing prices of the Microsoft Corporation's stock (see Figure 8) as well as daily closing prices of the West Texas Intermediate (WTI) crude oil futures (see Figure 9). Both datasets are taken from *www.finance.yahoo.com* and span approximately one year, from 6 November 2017 to 9 November 2018. We use the first 9 months (from 6 November 2017 to 3 August 2018) to calibrate the historical volatility for both assets, which is one of the input parameters in our pricing algorithm.

Then, using the historical volatility, we compute prices of *π* and American options (using assets' prices from 3 August 2018), both expiring 3 months after the end of calibration period. Please note that the parameters for the *π*-option on a relative drawdown are *a* = −1, *b* = 1 and *K* = 1. Input parameters for calculation and estimated options prices for both assets are given in Tables 4 and 5.

**Figure 8.** Daily closing prices of the Microsoft Corporation's stock. Data spans from 6.11.2017 to 9.11.2018. Vertical dashed line indicates the end of volatility calibration period. Option prices are calculated based on the volatility and the stock's price on 3.08.2018.

**Figure 9.** Daily closing prices of the West Texas Intermediate crude oil futures contracts. Data spans from 6.11.2017 to 9.11.2018. Vertical dashed line indicates the end of volatility calibration period. Option prices are calculated based on the volatility and the asset's price on 3.08.2018.

Since the payoff of *π*-option on relative drawdown with *K* = 1 is always less than 1, to compensate against the drop in underlying's price, we need a certain number of these contracts per each unit of stock in our portfolio. This number must be equal to *M*0. Please note that in Tables 4 and 5, the real price of the single *π*-option on relative drawdown contract should be 0.0735 for MSFT and 0.0949 for WTI. However, in order to be able to compare the results to the American put values, we initially need to make the instruments pay the same amount in case of a price drop, therefore we multiply the price of single *π*-option on drawdown by *M*<sup>0</sup> (110 and 74 for MSFT and WTI respectively). That is why in Tables 4 and 5 the price of *π*-option equals 0.0735 · 110 = 8.09 for the stock and 0.094 · 74 = 6.95 for the oil futures contract.


**Table 4.** Input parameters for computation and estimated options' prices for the MSFT dataset.

**Table 5.** Input parameters for computation and estimated options' prices for the WTI dataset.


It turns out that *π*-option is more expensive than vanilla put in case of both assets, which is not a surprise as it initially pays the amount equivalent to the present maximum drawdown. However, since the difference in price between these instruments is rather significant, a question emerges whether there exists a situation in which purchasing *π*-option on relative drawdown is more profitable than buying a simple vanilla put. To answer this question, let us focus on the dashed part of the Microsoft Corporation and WTI futures data from the beginning of this section. In Figures 10 and 11 we show the amount each instrument would pay (on each day) throughout the whole 3-month period until options' maturity.

**Figure 10.** Microsoft Corporation stock closing prices (**top**) and the corresponding payoffs of *π*-option on relative drawdown and American put (**bottom**) with the parameters from Table 4.

**Figure 11.** WTI crude oil futures contract closing prices (**top**) and the corresponding payoffs of *π*-option on relative drawdown and American put (**bottom**) with the parameters from Table 5.

To display the difference more clearly, we construct two portfolios *V*American and *Vπ*, both consisting of an underlying asset (a single Microsoft Corporation stock or a barrel of the WTI crude oil) and an option (American put and *π*-option on relative drawdown, respectively). We observe them at the end of the volatility calibration period. Assets' prices and options prices are taken from Tables 4 and 5. In Tables 6 and 7 we show the initial net values of both *V*American and *V<sup>π</sup>* portfolios.


**Table 6.** Portfolios and their initial net values for the MSFT dataset.


Then we analyze the behavior of the constructed portfolios, by calculating the net value of each portfolio for each day until options' maturity; see Figures 12 and 13.

Based in Figures 12 and 13 we can observe that the maximum value of portfolio *V*American is greater than the one for *Vπ*. Thus, when focusing purely at the possible maximum profit over some period of time, then the portfolio containing American option performs better. However, we can notice that *V*American's value over time is much more volatile compared to *V<sup>π</sup>* and it directly follows the behavior of underlying asset (it increases when asset's price rises and decreases in the opposite situation). The value *Vπ* of *π*-option portfolio is most of the time non-decreasing. Moreover, *Vπ* increases its value every time the asset's price reaches a new maximum and essentially does not decrease in case of any price drop. In other words, combining the underlying asset and *π*-option on drawdown allow us to lock in our profit whenever the price reaches its new maximum.

This brings us to the conclusion that the purpose of using *π*-option on relative drawdown and an American put is completely different. Vanilla American option protects us from asset price drops and ensures us that the current worth of our portfolio will not be less than its initial value. Unfortunately, in this case our portfolio's value is more volatile and reflects the volatility of the underlying asset. This may result in bigger gains when compared to the use of *π*-option on relative drawdown if the price of the underlying rises significantly and stays on that level until option's maturity. However, in case of a drop in asset price after the upswing, we do not benefit from the fact that the new maximum has been reached and thus the value of our portfolio decreases together with the price of the underlying asset. When looking at the value of *Vπ* over time one can notice that combining stock or a commodity and *π*-option on relative drawdown protects us against price drops as well but the volatility of our portfolio is reduced significantly. Additionally, the contract allows us to benefit from the underlying's price upswings and locks in the profit every time new maximum is reached.

**Figure 12.** Microsoft Corporation stock closing prices (**top**) and payoffs of portfolios with parameters from Table 4 (**bottom**).

**Figure 13.** WTI crude oil futures contract closing prices (**top**) and payoffs of portfolios with parameters from Table 5 (**bottom**).

We have analyzed two datasets, MSFT and WTI, and the above analysis shows that the behavior of a portfolio based on *π*-option is similar for various choices of underlying assets.

#### **4. Conclusions**

In this paper we focus on the numerical pricing of the new derivative instrument—a *π*-option. We adapted the Monte Carlo algorithm proposed by Broadie and Glasserman (1997) to price this new option. We focused on a specific parametrization of this option which we call the *π*-option on drawdown. We observed that this specific financial instrument is related to so-called relative maximum drawdown. We obtained prices of the *π*-option on relative drawdown for the Microsoft Corporation stock with different parameters to examine the influence of those parameters on option's premium. Our next step involved the analysis of two portfolios: first one based on a *π*-option on relative drawdown and the second one based on an American put. We used the Microsoft Corporation data as well as the West Texas Intermediate crude oil futures dataset. It turned out that the portfolios behave in a completely different manner. The value of the portfolio containing the American put was highly correlated with the underlying's price movements and thus had an unpredictable and volatile behavior. On the other hand, combining *π*-option on relative drawdown with the underlying asset not only ensures that the worth of the portfolio will not drop below the initial level, but it also allows us to take advantage of price upswings and to reduce the portfolio's volatility at the same time. Similar analysis could be carried out for a geometric Lévy process of asset price. One can also consider the regime-switching market.

**Author Contributions:** Methodology, Z.P. and T.S.; Formal Analysis, Z.P., T.S.; Investigation, Z.P. and T.S.; Writing—Original Draft Preparation, T.S.; Writing—Review and Editing, Z.P.; Visualization, T.S.; Supervision, Z.P.; Project Administration, Z.P.; Funding Acquisition, Z.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This paper is supported by the National Science Centre under the grant 2016/23/B/HS4/00566 (2017–2020).

**Conflicts of Interest:** The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**


Glasserman, Paul. 2004. *Monte Carlo Methods in Financial Engineering*. New York: Springer.

Guo, Xin, and Mihail Zervos. 2010. *π* options. *Stochastic Processes and their Applications* 120: 1033–59. [CrossRef] Jäckel, Peter. 2002. *Monte Carlo Methods in Finance*. Chichester: John Wiley.


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