Comparative Analyses of Expected Shortfall and Value-at-Risk (3): Their Validity under Market Stress

In this paper, we compare value-at-risk (VaR) and expected shortfall under market stress. Assuming that the multivariate extreme value distribution represents asset returns under market stress, we simulate asset returns with this distribution. With these simulated asset returns, we examine whether market stress affects the properties of VaR and expected shortfall. Our findings are as follows. First, VaR and expected shortfall may underestimate the risk of securities with fat- tailed properties and a high potential for large losses. Second, VaR and expected shortfall may both disregard the tail dependence of asset returns. Third, expected shortfall has less of a problem in disregarding the fat tails and the tail dependence than VaR does.

Comparative Analyses of Expected Shortfall and Value-at-Risk (2): Expected Utility Maximization and Tail Risk

We compare expected shortfall and value-at-risk (VaR) in terms of consistency with expected utility maximization and elimination of tail risk. We use the concept of stochastic dominance in studying these two aspects of risk measures. We conclude that expected shortfall is more applicable than VaR in those two aspects. Expected shortfall is consistent with expected utility maximization and is free of tail risk, under more lenient conditions than VaR.

On the Validity of Value-at-Risk: Comparative Analyses with Expected Shortfall

Value-at-risk (VaR) has become a standard measure used in financial risk management due to its conceptual simplicity, computational facility, and ready applicability. However, many authors claim that VaR has several conceptual problems. Artzner et al. (1997, 1999), for example, have cited the following shortcomings of VaR. (1) VaR measures only percentiles of profit-loss distributions, and thus disregards any loss beyond the VaR level ("tail risk"), and (2) VaR is not coherent since it is not sub-additive. To alleviate the problems inherent in VaR, the use of expected shortfall is proposed. In this paper, we provide an overview of studies comparing VaR and expected shortfall to draw practical implications for financial risk management. In particular, we illustrate how tail risk can bring serious practical problems in some cases.

Analytical Solution for Expected Loss of a Collateralized Loan: A Square-root Intensity Process Negatively Correlated with Collateral Value

In this study, we derive an explicit solution for the expected loss of a collateralized loan, focusing on the negative correlation between default intensity and collateral value. Three requirements for the default intensity and the collateral value are imposed. First, the default event can happen at any time until loan maturity according to an exogenous stochastic process of default intensity. Second, default intensity and collateral value are negatively correlated. Third, the default intensity and collateral value are non-negative. To develop an explicit solution, we propose a square-root process for default intensity and an affine diffusion process for collateral value. Given these settings, we derive an explicit solution for the integrand of the expected recovery value within an extended affine model. From the derived solution, we find the expected recovery value is given by a Stieltjes integral with a measure-changed survival probability.stochastic recovery, default intensity model, affine diffusion, extended affine, survival probability, measure change

Analytical solutions for expected and unexpected losses with an additional loan

We evaluate expected and unexpected losses of a bank loan, taking into account the bankfs strategic control of the expected return on the loan. Assuming that the bank supplies an additional loan to minimize the expected loss of the total loan, we provide analytical formulations for expected and unexpected losses with bivariate normal distribution functions.There are two cases in which an additional loan decreases the expected loss: i) the asset/liability ratio of the firm is low but its expected growth rate is high; ii) the asset/liability ratio of the firm is high and the lending interest rate is high. With a given expected growth rate and given interest rates, the two cases are identified by two thresholds for the current asset/liability ratio. The bank maintains the current loan amount when the asset/liability ratio is between the two thresholds. Given the bankfs strategy, the bank decreases the initial expected loss of the loan. On the other hand, the bank has a greater risk of the unexpected loss.Probability of default (PD), Loss given default (LGD), Exposure at default (EaD), Expected loss (EL), Unexpected loss (UL), Stressed EL (SEL)

Model Risk and Its Control

In this paper, we analyze model risks separately in pricing models and risk measurement models as follows. (1) In pricing models, model risk is defined as "the risk arising from the use of a model which cannot accurately evaluate market prices, or which is not a mainstream model in the market." (2) In risk measurement models, model risk is defined as " the risk of not accurately estimating the probability of future losses." Based on these definitions, we examine various specific cases and numerical examples to determine the sources of model risks and to discuss possible steps to control these risks. Sources of model risk in pricing models include (1) use of wrong assumptions, (2) errors in estimations of parameters, (3) errors resulting from discretization, and (4) errors in market data. On the other hand, sources of model risk in risk measurement models include (1) the difference between assumed and actual distribution, and (2) errors in the logical framework of the model. Practical steps to control model risks from a qualitative perspective include improvement of risk management systems (organization, authorization, human resources, etc.). From a quantitative perspective, in the case of pricing models, we can set up a reserve to allow for the difference in estimations using alternative models. In the case of risk measurement models, scenario analysis can be undertaken for various fluctuation patterns of risk factors, or position limits can be established based on information obtained from scenario analysis.

Comparative Analyses of Expected Shortfall and Value-at-Risk: Their Estimation Error, Decomposition, and Optimization

We compare expected shortfall with value-at-risk (VaR) in three aspects: estimation errors, decomposition into risk factors, and optimization. We describe the advantages and the disadvantages of expected shortfall over VaR. We show that expected shortfall is easily decomposed and optimized while VaR is not. We also show that expected shortfall needs a larger size of sample than VaR for the same level of accuracy.

Analytical Solution for the Loss Distribution of a Collateralized Loan under a Quadratic Gaussian Default Intensity Process

In this study, we derive an analytical solution for expected loss and the higher moment of the discounted loss distribution for a collateralized loan. To ensure nonnegative values for intensity and interest rate, we assume a quadratic Gaussian process for default intensity and discount interest rate. Correlations among default intensity, discount interest rate, and collateral value are represented by correlations among Brownian motions driving the movement of the Gaussian state variables. Given these assumptions, the expected loss or the m-th moment of the loss distribution is obtained by a time integral of an exponential quadratic form of the state variables. The coefficients of the form are derived by solving ordinary differential equations. In particular, with no correlation between default intensity and discount interest rate, the coefficients have explicit closed form solutions. We show numerical examples to analyze the effects of the correlation between default intensity and collateral value on expected loss and the standard deviation of the loss distribution.default intensity, stochastic recovery, quadratic Gaussian, expected loss, measure change