Risk measures in the neutral approach will always be symmetrical measures of risk, such as standard deviation and other statistical measures of dispersion. We shall refer to them in what follows as symmetrical risk measures. To express quantitatively the negative approach to risk, it is necessary to apply asymmetrical measures, accounting only for the left-hand side of the distribution of rates of return; an example of such measures is lower partial moments. Both the measures applying to results of risk and risk causes can be divided into symmetrical (classical) and downside ones.
2.1. Market Risk Measures
The classical risk measures are variance (
) and standard deviation (
). These statistical measures of dispersion were first applied to measure risk by
Markowitz (
1952). In that case, the random variable is the rate of return
. The higher the value of these measures, the higher the risk that the achieved rate of return will differ from the expected one.
A value equal to zero means the absence of risk. Calculations of variance and standard deviation lead to attributing relatively more importance to large than to small deviations from the mean. This approach is suitable for investors with an aversion to risk.
A certain drawback of variance and standard deviation as risk measures is that negative and positive deviations from the expected rate of return are treated identically. In reality, negative deviations are undesirable while positive ones create an opportunity to gain a higher profit (
Pla-Santamaria and Bravo 2013;
Klebaner et al. 2017;
Rutkowska-Ziarko and Pyke 2017). Measures devoid of these defects are semi-variance (
and standard semi-deviation (
). Their application is in accord with the so-called negative concept of risk. These measures were first defined by
Markowitz (
1959) with the help of the formulas:
where:
is the number of time units;
is the rate of return achieved in the time t by the i-th company; and
is the target rate of return. It can equal the average rate of return or the rate of return arbitrarily chosen by the investor; often a risk-free rate is taken as a reference point.
Risk measures whose values take into account consequences of investment decisions are total risk measures. In essence, this second group of measures accounts for sources of risk, which are a derivative of systematic cyclical and economic factors. These measures, which are referred to in the literature as sensitivity measures, reflect the impact of the analyzed causes of risk on prices or rates of return generated by financial instruments. The higher the sensitivity of prices or rates of return to changes in systematic factors, the higher the so-called systematic risk of a given instrument. The single-index model by Sharpe assumes that rates of return of securities are in line with the tendency that dominates a given capital market.
In the classical CAPM model (a model for evaluation of capital assets), systemic risk is measured with a beta coefficient (
which is usually calculated as follows (
Barucci and Fontana 2017, p. 220):
where:
is the covariance of the rate of return of the i-th company and market portfolio; and
is the variance of the market portfolio.
The beta coefficient for this index equals 1 ). Securities that have the value of this coefficient higher than 1 are referred to as aggressive. During a bull market, the achieved profitability of such stocks is higher than that of the market index; during a bear market, a decline in the profitability of these stocks is greater than a decrease in the profitability of investment in the market index. Defensive stocks, i.e., the ones for which the beta coefficient is within the range (0;1), respond to growths and fall on the market to a lesser degree than the market index. If security is characterized by a negative value of the beta coefficient, it means that the profitability rates it achieves are a reverse reaction to changes in the rates of return on the market index.
Starting from the risk measures relating to risk effects in the downside approach, it is possible to characterize sensitivity measures that account for sources of risk, especially the market risk (
Rutkowska-Ziarko et al. 2019). Quantification of the risk-to-profit ratio in the context of semi-metrics is based on lower partial moments, which were defined in several articles independently (
Hogan and Warren 1974;
Bawa and Lindenberg 1977;
Fishburn 1977).
Downside risk measures account for only a certain deviation on the left side of the distribution of the rate of return. The so-called assumed benchmark rate of return γ (target point) is a significant concept for the downside beta coefficient. Reaching the rate of return below the threshold rate is understood as the realization of downside risk. Considering the risk of an investment as a possibility of having a loss, the proper measure of the sensitivity of a given security is the downside beta coefficient. In theory, there are many variants of downside beta coefficients, distinguished in terms of their formulas and target points. Relationships between different downside beta coefficients have been presented, inter alia, by
Markowski (
2018). The formulas for downside beta coefficients used in this study are written as given by
Galagedera (
2007).
Bawa and Lindenberg (
1977) took assumed rate of return as a risk-free rate of return, and the downside beta, in terms of expected values, is expressed as follows:
where:
is the rate of return achieved in the time t by the market portfolio; and
is the risk-free rate.
One of the options is to take the risk-free rate (the minimum required rate of return) as the point where there are no losses and no profits (
Żebrowska-Suchodolska and Karpio 2020). A special case of the above coefficient (10) where the target rate is zero (
Nurjannah et al. 2012) can be written as follows:
In another approach, market players treat downside risk as deviations from the average rate of return of the market portfolio as opposed to a risk-free rate. This approach was proposed by
Harlow and Rao (
1989), who defined the downside beta coefficient in the following way:
where:
are, respectively, the average rates of return of the ith company and the market portfolio.
2.2. Accounting Risk Measures
To measure accounting profitability, return on equity (ROE) and return on assets (ROA) were used. The advantage of the ROA and ROE ratios is the possibility to compare the effectiveness of the company’s operation between entities of different sizes. The disadvantage of ROE is that it is not possible to calculate this ratio when an enterprise has a negative value of equity. On the other hand, the new emission can make the ROE lower just by increasing the equity. ROA is sensitive to changes in the level of assets. For example, selling the ground or building makes the ROA higher when the ability to generate profit is on the same level.
In this study, accounting risk measures were considered in the context of total risk and systematic risk. Symmetric and downside measures of dispersion were adopted as risk measures. The systematic risk measures are accounting beta coefficients.
There is a great variety of concepts presented in the literature of how information from financial reports can be employed to analyze systematic risk. A broad review of different notions can be found in articles by
Amorim et al. (
2012) and
Latif and Shah (
2021). The accounting-based risk (ABR) model is also worth mentioning. This model, which was proposed by
Toms (
2012), replaces the market rate of return that features in the conventional CAPM with the accounting rate of return. An interesting compilation of downside risk with accounting information is the concept of accounting-based downside risk (
Huang et al. 2021).
An important accounting measure of risk is Z-score. This measure is a different concept of risk compared with accounting beta; accounting beta and market beta are measures of sensitivity, while Z-score indicates the possibility of insolvency of the bank. Z-score is dedicated especially to financial institutions. The Z-score compares ROA ratios extended by the ratio of equity to total assets with a standard deviation of ROA. The higher the Z-score for a bank, the lower the risk of bankruptcy (
Martínez-Malvar and Baselga-Pascual 2020). It would be interesting to compare the Z-score with the accounting risk measures used in this article. Although some limitations make it inapplicable, all accounting measures used in this study have their market equivalents; for example, market beta and accounting beta. This study aimed to compare the market and accounting measures of risk. There is no market equivalent for a Z-score. Additionally, the analysis was performed using two approaches based on ROA and ROE. The Z-score is defined only for ROA. In another dimension of the research, measures were considered in the downside and symmetrical approaches. However, the Z-score exists only in the symmetrical approach.
In this paper, the term ‘accounting beta’ is understood as a measure of sensitivity describing a change in the accounting profitability of a given company induced by a change in the accounting profitability of the appropriate sector or the market. According to
Hill and Stone (
1980), each beta coefficient calculated on the basis of a market price (
,
,
) can be named a market beta. By analogy, beta coefficients determined on the basis of accounting information are referred to as accounting betas. In our study, accounting betas were determined based on the rate of return on equity (
,
,
,
,
). Measures of total risk in the standard and downside approaches were determined for these profitability ratios. The semi-deviation of the return on assets (
) was calculated as follows (
Rutkowska-Ziarko 2015):
where:
is the return on assets ratio of the ith company in time t; and
is the long-term average rate of return on assets on the market (in the sector).
where:
is the return on assets ratio on the market (in the sector) in time t; and
is the long-term average rate of return on assets on the market (in the sector).
The profitability ratio of assets on the market (in the sector) in time
t was determined as:
where:
is the market value of the i-th company; and is the number of companies,
.
The semi-deviation for a given profitability ratio can be calculated in the same way.
The accounting beta coefficient for the profitability ratio of assets (
) was derived from the formula:
where:
is the covariance of the profitability ratio of the i-th company and the market portfolio (in practice, the index of the market or sector based on the given profitability ratio); and
is the variance of the profitability ratio for the market (sector).
The notion of an accounting beta coefficient can be linked to the concept of a downside beta coefficient. The outcome is a downside risk measure, calculated based on accounting information, called a downside accounting beta (DAB) (
Rutkowska-Ziarko and Pyke 2017).
By adapting Bawa and Lindberg’s Formula (4) to the calculation of a downside accounting beta, risk-free rates should be replaced by another point of reference suitable for a given accounting measure. It could be a long-term average for a given profitability ratio in the sector or market. In addition, the market index for a given profitability ratio is needed. The basic solution is to use the average of this ratio (
Hill and Stone 1980) and the easiest solution is to use the simple average of a given accounting variable (Kim and Ismail). This average can be weighted in a few ways.
Beaver et al. (
1970) suggested using the arithmetic mean or weighted average according to the market value. Another possibility is to build weights based on the volume of assets, equity, or volume of sales.
When using the formula developed by Bawa and Lindenberg for determination of an accounting beta for the profitability ratio of assets (
, the risk-free rate (
) was replaced by the average long-term return on assets for the market or sector (
. This can be written as follows (
Rutkowska-Ziarko 2020a;
Rutkowska-Ziarko and Markowski 2020):
Hallow and Rao’s formula for the downside market beta was transformed in a similar manner to obtain the downside accounting beta
, as given below (
Rutkowska-Ziarko 2020a):
where:
The downside accounting beta for any profitability ratio can be determined analogously.
Other concepts combining downside risk with accounting data can be found in earlier papers by other scholars (see
Kim and Ismail 1998;
Konchitchki et al. 2016). The most recent concept of accounting-based downside risk alongside an empirical analysis can be found in a paper authored by
Huang et al. (
2021).
Konchitchki et al. (
2016) proposed a downside risk measure using accounting data and named it the accounting-based downside risk (ABDR). The measure of downside risk earnings is calculated using the ratio of the root of lower partial moments for ROA to the root of upper partial moments for ROA. In the cited work, the authors also used accounting beta coefficients for risk measured with variance. However, they did not use any sensitivity measures based on downside risk.
Huang et al. (
2021) employed the measure proposed by
Konchitchki et al. (
2016) to predict stock price falls on the Chinese market. They discovered a negative relation between ABDR and future stock price crash risk.
The model put forth by
Kim and Ismail (
1998) is closest to the concept of DAB. In the model suggested by
Kim and Zumwalt (
1979), the market rates of return were replaced by information from accounting reports. Earnings and cash flows were used as accounting variables. These researchers determined accounting betas separately for the up-market and down-market. Thus, they took advantage of the classical beta calculation, but divided accounting data time series into two subsets: up-market and down-market (see
Kim and Ismail 1998;
Kim and Zumwalt 1979), where ‘up-market’ refers to a situation when an increase in the market index determined for a given accounting variable (in this case, earnings or cash flow) reached a positive value, and ‘down-market’ refers to periods when this increase was negative. Betas are determined separately from the up-and-down markets.