Approximating the distributions of estimators of financial risk under an asymmetric Laplace law

Explicit expressions are derived for parametric and nonparametric estimators (NPEs) of two measures of financial risk, value-at-risk (VaR) and conditional value-at-risk (CVaR), under random sampling from the asymmetric Laplace (AL) distribution. Asymptotic distributions are established under very general conditions. Finite sample distributions are investigated by means of saddlepoint approximations. The latter are highly computationally intensive, requiring novel approaches to approximate moments and special functions that arise in the evaluation of the moment generating functions. Plots of the resulting density functions shed new light on the quality of the estimators. Calculations for CVaR reveal that the NPE enjoys greater asymptotic efficiency relative to the parametric estimator than is the case for VaR. An application of the methodology in modeling currency exchange rates suggests that the AL distribution is successful in capturing the peakedness, leptokurticity, and skewness, inherent in such data. A demonstrated superiority in the resulting parametric-based inferences delivers an important message to the practitioner.

[1]  Philippe Artzner,et al.  Coherent Measures of Risk , 1999 .

[2]  S. Uryasev.  Probabilistic constrained optimization : methodology and applications , 2000 .

[3]  D. Tasche,et al.  Expected Shortfall: a natural coherent alternative to Value at Risk , 2001, cond-mat/0105191.

[4]  Tomasz J. Kozubowski,et al.  A CLASS OF ASYMMETRIC DISTRIBUTIONS , 1999 .

[5]  S. Stigler.  The Asymptotic Distribution of the Trimmed Mean , 1973 .

[6]  Andrew T. A. Wood,et al.  Laplace approximations for hypergeometric functions with matrix argument , 2002 .

[7]  G. Pflug.  Some Remarks on the Value-at-Risk and the Conditional Value-at-Risk , 2000 .

[8]  Samuel Kotz,et al.  The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance , 2001 .

[9]  R. Rockafellar,et al.  Optimization of conditional value-at risk , 2000 .

[10]  R. Rockafellar,et al.  Generalized Deviations in Risk Analysis , 2004 .

[11]  E. Ronchetti,et al.  General Saddlepoint Approximations with Applications to L Statistics , 1986 .

[12]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[13]  J. McFarland,et al.  The Distribution of Foreign Exchange Price Changes: Trading Day Effects and Risk Measurement--A Reply , 1982 .

[14]  K. Podgórski,et al.  Asymmetric laplace laws and modeling financial data , 2001 .

[15]  Chris Field,et al.  Small sample confidence intervals , 1990 .

[16]  Khursheed Alam,et al.  Distribution of a Sum of Order Statistics , 1979 .

[17]  Stanislav Uryasev,et al.  Conditional Value-at-Risk for General Loss Distributions , 2002 .

[18]  H. Daniels.  Saddlepoint Approximations in Statistics , 1954 .

[19]  S. Rice,et al.  Saddle point approximation for the distribution of the sum of independent random variables , 1980, Advances in Applied Probability.

[20]  Samuel Kotz,et al.  Asymmetric Laplace Distributions , 2001 .