Risk Measurement Performance of Alternative Distribution Functions

This paper evaluates the performance of three extreme value distributions, i.e., generalized Pareto distribution (GPD), generalized extreme value distribution (GEV), and Box-Cox-GEV, and four skewed fat-tailed distributions, i.e., skewed generalized error distribution (SGED), skewed generalized t (SGT), exponential generalized beta of the second kind (EGB2), and inverse hyperbolic sign (IHS) in estimating conditional and unconditional value at risk (VaR) thresholds. The results provide strong evidence that the SGT, EGB2, and IHS distributions perform as well as the more specialized extreme value distributions in modeling the tail behavior of portfolio returns. All three distributions produce similar VaR thresholds and perform better than the SGED and the normal distribution in approximating the extreme tails of the return distribution. The conditional coverage and the out-of-sample performance tests show that the actual VaR thresholds are time varying to a degree not captured by unconditional VaR measures. In light of the fact that VaR type measures are employed in many different types of financial and insurance applications including the determination of capital requirements, capital reserves, the setting of insurance deductibles, the setting of reinsurance cedance levels, as well as the estimation of expected claims and expected losses, these results are important to financial managers, actuaries, and insurance practitioners.

[1]  Paul H. Kupiec,et al.  Techniques for Verifying the Accuracy of Risk Measurement Models , 1995 .

[2]  G. C. Tiao,et al.  A Further Look at Robustness via Bayes's Theorem , 1962 .

[3]  James B. McDonald,et al.  A generalization of the beta distribution with applications , 1995 .

[4]  James B. McDonald,et al.  Risky Loss Distributions and Modeling the Loss Reserve Payout Tail , 2004 .

[5]  H. Lilliefors.  On the Kolmogorov-Smirnov Test for Normality with Mean and Variance Unknown , 1967 .

[6]  Yildiray Yildirim,et al.  The Dynamics of Operational Loss Clustering , 2008 .

[7]  Turan G. Bali.  An Extreme Value Approach to Estimating Volatility and Value at Risk , 2003 .

[8]  Thomas A. Aiuppa.  Evaluation of Pearson Curves as an Approximation of the Maximum Probable Annual Aggregate Loss , 1988 .

[9]  W. N. Street,et al.  Financial Data and the Skewed Generalized T Distribution , 1998 .

[10]  C. Klüppelberg,et al.  Modelling Extremal Events , 1997 .

[11]  Peter F. Christoffersen.  Evaluating Interval Forecasts , 1998 .

[12]  A. S. Paulson,et al.  A Practical Approach to Measuring the Distribution of Total Annual Claims , 1985 .

[13]  Daniel B. Nelson.  CONDITIONAL HETEROSKEDASTICITY IN ASSET RETURNS: A NEW APPROACH , 1991 .

[14]  J. Pickands.  Statistical Inference Using Extreme Order Statistics , 1975 .

[15]  Henrik Ramlau-Hansen.  A solvency study in non-life insurance , 1988 .

[16]  Henrik Ramlau-Hansen.  A solvency study in non-life insurance: Part 1. Analyses of fire, windstorm, and glass claims , 1988 .

[17]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[18]  C. R. Sprecher,et al.  Large Losses, Risk Management and Stock Prices , 1983 .

[19]  Peng Shi,et al.  Long-tail longitudinal modeling of insurance company expenses , 2010 .

[20]  F. Diebold,et al.  How Relevant is Volatility Forecasting for Financial Risk Management? , 1997, Review of Economics and Statistics.

[21]  Francis Galton F.R.S..  IV. Statistics by intercomparison, with remarks on the law of frequency of error , 1875 .

[22]  D. Cox,et al.  An Analysis of Transformations , 1964 .

[23]  B. Gnedenko.  Sur La Distribution Limite Du Terme Maximum D'Une Serie Aleatoire , 1943 .

[24]  N. L. Johnson,et al.  Systems of frequency curves generated by methods of translation. , 1949, Biometrika.

[25]  F. Longin,et al.  From value at risk to stress testing : The extreme value approach Franc ß ois , 2000 .

[26]  Anthony M. Santomero,et al.  Financial Risk Management by Insurers: An Analysis of the Process , 1997 .

[27]  Kevin Dowd,et al.  After VAR: The Theory, Estimation, and Insurance Applications of Quantile-Based Risk Measures , 2006 .

[28]  B. Hansen.  Autoregressive Conditional Density Estimation , 1994 .

[29]  A. McNeil,et al.  Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach , 2000 .

[30]  Dongsae Cho.  The Impact of Risk Management Decisions on Firm Value: Gordon's Growth Model Approach , 1988 .

[31]  Turan G. Bali.  The generalized extreme value distribution , 2003 .

[32]  Stuart A. Klugman,et al.  Loss Models: From Data to Decisions , 1998 .

[33]  J. McDonald,et al.  Applications of the GB2 family of distributions in modeling insurance loss processes , 1988 .

[34]  A. Jenkinson.  The frequency distribution of the annual maximum (or minimum) values of meteorological elements , 1955 .

[35]  J. Cummins.  Risk Management and the Theory of the Firm , 1976 .

[36]  James B. McDonald,et al.  Partially Adaptive Estimation of Regression Models via the Generalized T Distribution , 1988, Econometric Theory.