Risks, Volume 6, Issue 3 (September 2018) – 41 articles

Variable annuities, as a class of retirement income products, allow equity market exposure for a policyholder’s retirement fund with optional guarantees to limit the downside risk of the market. Management fees andguarantee insurance fees are charged respectively for the market exposure and for the protection from the downside risk. We investigate the pricing of variable annuity guarantees under optimal withdrawal strategies when management fees are present. We consider from both policyholder’s and insurer’s perspectives optimal withdrawal strategies and calculate the respective fair insurance fees. We reveal a discrepancy where the fees from the insurer’s perspective can be significantly higher due to the management fees serving as a form of market friction. Our results provide a possible explanation of lower guarantee insurance fees observed in the market than those predicted from the insurer’s perspective. Numerical experiments are conducted to illustrate the results. Full article

A fluctuation theory and, in particular, a theory of scale functions is developed for upwards skip-free Lévy chains, i.e., for right-continuous random walks embedded into continuous time as compound Poisson processes. This is done by analogy to the spectrally negative class of Lévy processes—several results, however, can be made more explicit/exhaustive in the compound Poisson setting. Importantly, the scale functions admit a linear recursion, of constant order when the support of the jump measure is bounded, by means of which they can be calculated—some examples are presented. An application to the modeling of an insurance company’s aggregate capital process is briefly considered. Full article

A quantum mechanics approach is proposed to model non-life insurance risks and to compute the future reserve amounts and the ruin probabilities. The claim data, historical or simulated, are treated as coming from quantum observables and analyzed with traditional machine learning tools. They can then be used to forecast the evolution of the reserves of an insurance company. The following methodology relies on the Dirac matrix formalism and the Feynman path-integral method. Full article

The study considers the product life cycle in the stages of technological innovation, and focuses on how to evaluate the optimal investment strategy and the project value. It applies different product stages (three stages including production innovation, manufacture innovation, and business innovation) factors to different risks to build a technology innovation strategy model. This study of option premiums aims for the best strategy timing for each innovation stage. It shows that the variation of business cycle will affect the purchasing power under the uncertainty of Gross Domestic Product (GDP). In application, the compound binomial options for the manufacture innovation will only be considered after the execution of the production innovation, whereas the operation innovation will only be considered after the execution of the manufacture innovation. Thus, this paper constructs the dynamic investment sequential decision model, assesses the feasibility of an investment strategy, and makes a decision on the appropriate project value and option premiums for each stage under the possible change of GDP. Numerically, the result shows the equity value of the investment is greater than 0. Therefore, this paper recommends the case firm to invest in its innovation project known as one-time passwords. Sensitivity analysis shows when the risk-adjusted discounted rate r increases, the risk of the investment market increases accordingly, hence the equity value must also be higher in order to attract the case firm’s investment interest. Also, the average GDP growth rate u sensitivity analysis results in different phenomena. The equity value gradually decreases when the average GDP growth rate rises. When the average GDP growth rate u rises to a certain extent, however, its equity value is gradually growing. The study investigates the product life cycle innovation investment topic by using the compound binomial options method and therefore provide a more flexible strategy decision compared with other trend forecast criteria. Full article

We utilize the data of a very large UK automobile loan firm to study the interaction of the characteristics of borrowers and loans in predicting the subsequent loan performance. Our broader findings confirm the earlier research on the issue of subprime auto loans. More importantly, unmarried borrowers living with furnished tenancy agreements who have relatively new jobs have a probability of defaulting of more than 60% compared to an average 7% default rate in overall subprime borrowers in the dataset. Also, in the above category are those who live in a less prosperous part of the UK such as the north-west, are full-time self-employed, have other large loan arrears, fall into the bottom 25% percentile of monthly income, secure loans with high loan to total value (LTV), purchase expensive automobiles with shorter loan duration payment plans, and have a high dependency on government support. This in fact is also true of those who go into arrears, except that the highest probability in this context is around 40% compared to 6% for an overall sample. These findings shall help in the understanding of subprime auto loans performance in relation to borrowers and loan features alongside helping auto finance firms improve predictive models and decision-making. Full article

Background, or systematic, risks are integral parts of many systems and models in insurance and finance. These risks can, for example, be economic in nature, or they can carry more technical connotations, such as errors or intrusions, which could be intentional or unintentional. A most natural question arises from the practical point of view: is the given system really affected by these risks? In this paper we offer an algorithm for answering this question, given input-output data and appropriately constructed statistics, which rely on the order statistics of inputs and the concomitants of outputs. Even though the idea is rooted in complex statistical and probabilistic considerations, the algorithm is easy to implement and use in practice, as illustrated using simulated data. Full article

The paper addresses three objectives: the first is a presentation and overview of some important developments in quantile times series approaches relevant to demographic applications—secondly, development of a general framework to represent quantile regression models in a unifying manner, which can further enhance practical extensions and assist in formation of connections between existing models for practitioners. In this regard, the core theme of the paper is to provide perspectives to a general audience of core components that go into construction of a quantile time series model. The third objective is to compare and discuss the application of the different quantile time series models on several sets of interesting demographic and mortality related time series data sets. This has relevance to life insurance analysis and the resulting exploration undertaken includes applications in mortality, fertility, births and morbidity data for several countries, with a more detailed analysis of regional data in England, Wales and Scotland. Full article

Almost sure bootstrap consistency of the blockwise bootstrap for the Average Value at Risk of single risks is established for strictly stationary $\beta$ -mixing observations. Moreover, almost sure bootstrap consistency of a multiplier bootstrap for the Average Value at Risk of collective risks is established for independent observations. The main results rely on a new functional delta-method for the almost sure bootstrap of uniformly quasi-Hadamard differentiable statistical functionals, to be presented here. The latter seems to be interesting in its own right. Full article

We propose a novel credit risk measurement model for Corporate Default Swap (CDS) spreads that combines vector autoregressive regression with correlation networks. We focus on the sovereign CDS spreads of a collection of countries that can be regarded as idiosyncratic measures of credit risk. We model CDS spreads by means of a structural vector autoregressive model, composed by a time dependent country specific component, and by a contemporaneous component that describes contagion effects among countries. To disentangle the two components, we employ correlation networks, derived from the correlation matrix between the reduced form residuals. The proposed model is applied to ten countries that are representative of the recent financial crisis: top borrowing/lending countries, and peripheral European countries. The empirical findings show that the contagion variable derived in this study can be considered as a network centrality measure. From an applied viewpoint, the results indicate that, in the last 10 years, contagion has induced a “clustering effect” between core and peripheral countries, with the two groups further diverging through, and because of, contagion propagation, thus creating a sort of diabolic loop extremely difficult to be reversed. Finally, the outcomes of the analysis confirm that core countries are importers of risk, as contagion increases their CDS spread, whereas peripheral countries are exporters of risk. Greece is an unfortunate exception, as its spreads seem to increase for both idiosyncratic factors and contagion effects. Full article

This study examines the linkages between Brazil, Russia, India, and China (BRICS) stock market returns, country risk ratings, and international factors via Non-linear Auto Regressive Distributed Lags models (NARDL) that allow for testing the asymmetric effects of changes in country risk ratings on stock market returns. We show that BRICS countries exhibit quite a degree of heterogeneity in the interaction of their stock market returns with country-specific political, financial, and economic risk ratings. Positive and negative rating changes in some BRICS countries are found to have significant implications for both local stock market returns, as well as commodity price dynamics. While the commodity market acts as a catalyst for these emerging stock markets in the long-run, we also observe that negative changes in the country risk ratings generally command a higher impact on stock returns, implying the greater impact of bad news on market dynamics. Our findings suggest that not all BRICS nations are the same in terms of how they react to ratings changes and how they interact with global market variables. Full article

There exist several estimators of the regression line in the simple linear regression: Least Squares, Least Absolute Deviation, Right Median, Theil–Sen, Weighted Balance, and Least Trimmed Squares. Their performance for heavy tails is compared below on the basis of a quadratic loss function. The case where the explanatory variable is the inverse of a standard uniform variable and where the error has a Cauchy distribution plays a central role, but heavier and lighter tails are also considered. Tables list the empirical sd and bias for ten batches of one hundred thousand simulations when the explanatory variable has a Pareto distribution and the error has a symmetric Student distribution or a one-sided Pareto distribution for various tail indices. The results in the tables may be used as benchmarks. The sample size is $n=100$ but results for $n=\infty$ are also presented. The error in the estimate of the slope tneed not be asymptotically normal. For symmetric errors, the symmetric generalized beta prime densities often give a good fit. Full article

A justification of the Basel liquidity formula for risk capital in the trading book is given under the assumption that market risk-factor changes form a Gaussian white noise process over 10-day time steps and changes to P&L (profit-and-loss) are linear in the risk-factor changes. A generalization of the formula is derived under the more general assumption that risk-factor changes are multivariate elliptical. It is shown that the Basel formula tends to be conservative when the elliptical distributions are from the heavier-tailed generalized hyperbolic family. As a by-product of the analysis, a Fourier approach to calculating expected shortfall for general symmetric loss distributions is developed. Full article

In this article we introduce the stability analysis of a compound sum: it consists of computing the standardized variation of the survival function of the sum resulting from an infinitesimal perturbation of the common distribution of the summands. Stability analysis is complementary to the classical sensitivity analysis, which consists of computing the derivative of an important indicator of the model, with respect to a model parameter. We obtain a computational formula for this stability from the saddlepoint approximation. We apply the formula to the compound Poisson insurer loss with gamma individual claim amounts and to the compound geometric loss with Weibull individual claim amounts. Full article

We study a toy model of linear-quadratic mean field game with delay. We “lift” the delayed dynamic into an infinite dimensional space, and recast the mean field game system which is made of a forward Kolmogorov equation and a backward Hamilton-Jacobi-Bellman equation. We identify the corresponding master equation. A solution to this master equation is computed, and we show that it provides an approximation to a Nash equilibrium of the finite player game. Full article

The paper analyzes the relationship between the credit default swaps (CDS) spreads for 5-year CDS in Europe and US, and fundamental macroeconomic variables such as regional stock indices, oil prices, gold prices, and interest rates. The dataset includes consideration of multiple industry sectors in both economies, and it is split in two sections, before and after the global financial crisis. The analysis is carried out using multivariate regression of each index vs. the macroeconomic variables, and a Granger causality test. Both approaches are performed on the change of value of the variables involved. Results show that equity markets lead in price discovery, bidirectional causality between interest rate, and CDS spreads for most sectors involved. There is also bidirectional causality between stock and oil returns to CDS spreads. Full article

Co-risk measures and risk contribution measures have been introduced to evaluate the degree of interaction between paired risks in actuarial risk management. This paper attempts to study the ordering behavior of measures on interaction between paired risks. For various co-risk measures and risk contribution measures, we investigate how the marginal distributions and the dependence structure impact on the level of interaction between paired risks. Also, several numerical examples based on Monte Carlo simulation are presented to illustrate the main findings. Full article

With the purpose of introducing dependence between different types of claims, multivariate collective models have recently gained a lot of attention. However, when it comes to the evaluation of the corresponding compound distribution, the problems increase with the dimensionality of the model. In this paper, we consider a multivariate collective model that generalizes a model already studied from the point of view of recursive and FFT evaluation of its distribution, and we extend the same study to the general model. With the intention to see which method works better for this general model, we compare the recursive method with the FFT technique, and emphasize the advantages and drawbacks of each one, based on numerical examples. Full article

In this paper, we study the discounted renewal aggregate claims with a full dependence structure. Based on a mixing exponential model, the dependence among the inter-claim times, the claim sizes, as well as the dependence between the inter-claim times and the claim sizes are included. The main contribution of this paper is the derivation of the closed-form expressions for the higher moments of the discounted aggregate renewal claims. Then, explicit expressions of these moments are provided for specific copulas families and some numerical illustrations are given to analyze the impact of dependency on the moments of the discounted aggregate amount of claims. Full article

In this short paper, we study a VaR-type risk measure introduced by Guérin and Renaud and which is based on cumulative Parisian ruin. We derive some properties of this risk measure and we compare it to the risk measures of Trufin et al. and Loisel and Trufin. Full article

A functional ARMA-GARCH model for predicting the value-at-risk of the EURUSD exchange rate is introduced. The model implements the yield curve differentials between EUR and the US as exogenous factors. Functional principal component analysis allows us to use the information of basically the whole yield curve in a parsimonious way for exchange rate risk prediction. The data analyzed in our empirical study consist of the EURUSD exchange rate and the EUR- and US-yield curves from 15 August 2005–30 September 2016. As a benchmark, we take an ARMA-GARCH and an ARMAX-GARCHX with the 2y-yield difference as the exogenous variable and compare the forecasting performance via likelihood ratio tests. However, while our model performs better in one situation, it does not seem to improve the performance in other setups compared to its competitors. Full article

In this paper, we study the problem of misrepresentation under heavy-tailed regression models with the presence of both misrepresented and correctly-measured risk factors. Misrepresentation is a type of fraud when a policy applicant gives a false statement on a risk factor that determines the insurance premium. Under the regression context, we introduce heavy-tailed misrepresentation models based on the lognormal, Weibull and Pareto distributions. The proposed models allow insurance modelers to identify risk characteristics associated with the misrepresentation risk, by imposing a latent logit model on the prevalence of misrepresentation. We prove the theoretical identifiability and implement the models using Bayesian Markov chain Monte Carlo techniques. The model performance is evaluated through both simulated data and real data from the Medical Panel Expenditure Survey. The simulation study confirms the consistency of the Bayesian estimators in large samples, whereas the case study demonstrates the necessity of the proposed models for real applications when the losses exhibit heavy-tailed features. Full article

We present a stochastic simulation forecasting model for stress testing that is aimed at assessing banks’ capital adequacy, financial fragility, and probability of default. The paper provides a theoretical presentation of the methodology and the essential features of the forecasting model on which it is based. Also, for illustrative purposes and to show in practical terms how to apply the methodology and the types of outcomes and analysis that can be obtained, we report the results of an empirical application of the methodology proposed to the Global Systemically Important Banks (G-SIB) banks. The results of the stress test exercise are compared with the results of the supervisory stress tests performed in 2014 by the Federal Reserve and EBA/ECB. Full article

(1) Background: Health insurance and social protection in Myanmar are negligible, which leaves many citizens at risk of financial hardship in case of a serious illness. The aim of this study is to explore the views of healthcare consumers and compare them to the views of key informants on the design and implementation of a nationwide health insurance system in Myanmar. (2) Method: Data were collected through nine focus group discussions with healthcare consumers and six semi-structured interviews with key health system informants. (3) Results: The consumers supported a mandatory basic health insurance and voluntary supplementary health insurance. Tax-based funding was suggested as an option that can help to enhance healthcare utilization among the poor and vulnerable groups. However, a fully tax-based funding was perceived to have limited chances of success given the low level of government resources available. Community-based insurance, where community members pool money in a healthcare fund, was seen as more appropriate for the rural areas. (4) Conclusion: This study suggests a healthcare financing mechanism based on a mixed insurance model for the creation of nationwide health insurance. Further inquiry into the feasibility of the vital aspects of the nationwide health insurance is needed. Full article

This paper examines the impact of volatility-based fund classification on portfolio performance. Using historical data on equity indices, we find that a strategy based on long-term portfolio volatility, as is imposed by the Synthetic Risk Reward Indicator (SRRI), yields better Sharpe Ratios (SR) and Buy and Hold Returns (BHR) than passive investments. However, accounting for the Fama–French factors in the historical data reveals no significant alphas for the vast majority of the strategies. Further analyses conducted by running a simulation study based on a GJR(1,1)-model show no significant difference in mean returns, but significantly lower SRs for the volatility-based strategies. This evidence suggests that neither the higher leverage induced by the SRRI, nor the potential protection in downside markets pay off on a risk adjusted basis. Full article

One way to formulate a multivariate probability distribution with dependent univariate margins distributed gamma is by using the closure under convolutions property. This direction yields an additive background risk model, and it has been very well-studied. An alternative way to accomplish the same task is via an application of the Bernstein–Widder theorem with respect to a shifted inverse Beta probability density function. This way, which leads to an arguably equally popular multiplicative background risk model (MBRM), has been by far less investigated. In this paper, we reintroduce the multiplicative multivariate gamma (MMG) distribution in the most general form, and we explore its various properties thoroughly. Specifically, we study the links to the MBRM, employ the machinery of divided differences to derive the distribution of the aggregate risk random variable explicitly, look into the corresponding copula function and the measures of nonlinear correlation associated with it, and, last but not least, determine the measures of maximal tail dependence. Our main message is that the MMG distribution is (1) very intuitive and easy to communicate, (2) remarkably tractable, and (3) possesses rich dependence and tail dependence characteristics. Hence, the MMG distribution should be given serious considerations when modelling dependent risks. Full article

Insurers issuing segregated fund policies apply dynamic hedging to mitigate risks related to guarantees embedded in such policies. A typical industry practice consists of using fund mapping regressions to represent basis risk stemming from the imperfect correlation between the underlying fund and its corresponding hedging instruments. The current work discusses the implications of using fund mapping regressions when the joint dynamics of the underlying and hedging assets is a regime-switching process. The potential underestimation of capital requirements stemming from the use of a fund mapping regression under such dynamics is discussed. The magnitude of the latter phenomenon is quantified through simulations calibrated on market data. Full article

Most of the models leading to an analytical expression for option prices are based on the assumption that underlying asset returns evolve according to a Brownian motion with drift. For some asset classes like commodities, a Brownian model does not fit empirical covariance and autocorrelation structures. This failure to replicate the covariance introduces a bias in the valuation of calendar spread exchange options. As the payoff of these options depends on two asset values at different times, particular care must be taken for the modeling of covariance and autocorrelation. This article proposes a simple alternative model for asset prices with sub-exponential, exponential and hyper-exponential autocovariance structures. In the proposed approach, price processes are seen as conditional Gaussian fields indexed by the time. In general, this process is not a semi-martingale, and therefore, we cannot rely on stochastic differential calculus to evaluate options. However, option prices are still calculable by the technique of the change of numeraire. A numerical illustration confirms the important influence of the covariance structure in the valuation of calendar spread exchange options for Brent against WTI crude oil and for gold against silver. Full article

The aim of this paper is to provide several examples of convex risk measures necessary for the application of the general framework for portfolio theory of Maier-Paape and Zhu (2018), presented in Part I of this series. As an alternative to classical portfolio risk measures such as the standard deviation, we, in particular, construct risk measures related to the “current” drawdown of the portfolio equity. In contrast to references Chekhlov, Uryasev, and Zabarankin (2003, 2005), Goldberg and Mahmoud (2017), and Zabarankin, Pavlikov, and Uryasev (2014), who used the absolute drawdown, our risk measure is based on the relative drawdown process. Combined with the results of Part I, Maier-Paape and Zhu (2018), this allows us to calculate efficient portfolios based on a drawdown risk measure constraint. Full article

We study the dynamics of the one-year change in P&C insurance reserves estimation by analyzing the process that leads to the ultimate risk in the case of “fixed-sum” insurance contracts. The random variable ultimately is supposed to follow a binomial distribution. We compute explicitly various quantities of interest, in particular the Solvency Capital Requirement for one year change and the Risk Margin, using the characteristics of the underlying model. We then compare them with the same figures calculated with existing risk estimation methods. In particular, our study shows that standard methods (Merz–Wüthrich) can lead to materially incorrect results if the assumptions are not fulfilled. This is due to a multiplicative error assumption behind the standard methods, whereas our example has an additive error propagation as often happens in practice. Full article

This paper provides a rationale for the macro-prudential regulation of insurance companies, where capital requirements increase in their contribution to systemic risk. In the absence of systemic risk, the formal model in this paper predicts that optimal regulation may be implemented by capital regulation (similar to that observed in practice, e.g., Solvency II ) and by actuarially fair technical reserve. However, these instruments are not sufficient when insurance companies are exposed to systemic risk: prudential regulation should also add a systemic component to capital requirements that is non-decreasing in the firm’s exposure to systemic risk. Implementing the optimal policy implies separating insurance firms into two categories according to their exposure to systemic risk: those with relatively low exposure should be eligible for bailouts, while those with high exposure should not benefit from public support if a systemic event occurs. Full article