Estimation of Distortion Risk Measures
Hideatsu Tsukahara
Faculty of Economics, Seijo University, Tokyo, Japan

The concept of risk measure is introduced to quantify the financial risk in Artzner et al. [2]. They listed some properties, called axioms of 'coherence', that any good risk measure should possess, and studied the (non-)coherence of widely-used risk measure such as Value-at-Risk (VaR) and expected shortfall (also known as tail conditional expectation or tail VaR).

The class of distortion risk measures with convex distortions coincides with the set of coherent risk measures that are law invariant and comonotonically additive. We note that the essentially same class of risk measures has been introduced in Acerbi [1] under the name spectral measures of risk, and in Cherny [3] under the name weighted VaR. The most well-known example of distortion risk measure is the expected shortfall; there are also many other families of distortion risk measures, for example, given in Tsukahara [6].

To implement the risk management/regulatory procedure using these risk measures, it is necessary to statistically estimate the values of such risk measures. A natural estimator takes a simple form of L-statistics, and we investigate the large sample properties of the estimator based on a strictly stationary sequence. Under certain regularity conditions which are slightly weaker than those given in the literature (see Mehra and Sudhakara Rao [4] and Puri and Tran [5]), we prove that it has the strong consistency and asymptotic normality. We also give a consistent estimate of the asymptotic variance. The conditions involve the strong mixing property, and are satisfied, for example, by GARCH sequences and stochastic volatility model which are often used for modelling financial time series data.

We perform comparison of several estimators under inverse-gamma autoregressive stochastic volatility model and GARCH(1,1) by Monte Carlo simulation. Related issues such as semiparametric estimation with the extreme value theory and backtesting are briefly addressed.


[1] Acerbi, C. (2002). Spectral measures of risk: A coherent representation of subjective risk aversion, Journal of Banking & Finance, 26, 1505-1518.

[2] Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D. (1999). Coherent measures of risk, Mathematical Finance, 9, 203-228.

[3] Cherny, A. S. (2006). Weighted V@R and its properties, Finance and Stochastics, 10, 367-393.

[4] Mehra, K. L. and M. Sudhakara Rao (1975). On functions of order statistics for mixing processes, Annals of Statistics, 3, 874-883.

[5] Puri, M. L. and L. T. Tran (1980). Empirical distribution functions and functions of order statisitics for mixing random variables, Journal of Multivariate Analysis, 10, 405-425.

[6] Tsukahara, H. (2009), One-parameter Families of Distortion Risk Measures, Mathematical Finance, 19, 691-705.

Keywords: risk measure; distortion; asymptotic normality; L-statistics

Biography: He received a Bachelor of Economics from the Univeristy of Tokyo in 1988, and a Ph.D. in Statistics from the University of Illinois at Urbana-Champaign in 1996. He joined the faculty of Economics at Seijo University in 1998, and is now a Professor there.