| Cornish-Fisher VaR 101 |
|
|
| 25/11/2007 | |
|
Brian Peterson, knowledge leader at US-based Diamond Management & Technology Consultants, answers this week's questions on Cornish-Fisher VaR. 1. What is Cornish-Fisher VaR and how does it differ from VaR? Value-at-Risk (VaR) is a measure of the likely loss at a given confidence level (quantile). Historical VaR is calculated by simply ranking the observed returns and looking at the loss at the desired confidence. J.P. Morgan's RiskMetrics parametric mean-VaR was published in 1994 and this methodology for estimating parametric mean-VaR has become what most literature generally refers to as "VaR" (J.P.Morgan/Reuters, 1996). Parametric VaR works by assuming a distribution to more precisely estimate the unobserved risk at the tails of the distribution; the RiskMetrics approach assumes a Gaussian normal distribution. In this case, estimation of VaR requires the mean return R, and the standard deviation of the returns ó. In the most common case, parametric VaR is thus calculated by  where qp is the p% confidence quantile of the distribution. Zangari (1996), Campbell et al. (2001) and Favre and Galeano (2002) provide a modified VaR calculation that takes the higher moments (skewness, the directionality or tilt of the returns and kurtosis, the measurement of the "fat-tailed" nature of the returns) of non-normal distributions into account through the use of a Cornish and Fisher (1937) expansion, better approximating the shape of the true distribution. They arrive at their modified VaR calculation in the following manner:  where S is the skewness and K is the kurtosis of the return series. In a portfolio context, the moments may be calculated utilizing either the historical returns of the whole portfolio (i.e. univariate), or by using a multivariate estimate of the moments for a more accurate representation of the portfolio VaR. Cornish-Fisher VaR will give a larger loss estimate than traditional VaR when returns are negatively skewed or highly kurtotic fat-tailed),and, conversely, will give a smaller loss magnitude when returns are positively skewed or leptokurtotic. Cornish-Fisher VaR collapses to traditional mean-VaR when returns are normally distributed. This measure is now widely cited and used in the literature, and is usually referred to as "Modified VaR" or "Modified Cornish-Fisher VaR". 2. What are the advantages and constraints of using VaR as part of a trading or investment strategy? VaR is one of the most widely used measures of downside risk. In general, a good VaR estimate should provide a manager with a view on the likely loss at a given confidence level. This is why calculating VaR is required by most regulatory regimes such as Basel II to set capital reserves. When you have a good estimate of what your likely loss may be, you can reserve enough capital to cover those losses and remain solvent. VaR estimates can inform a portfolio manager whether observed losses are within normal or expected bounds. On the other hand, VaR is typically not a "coherent risk measure". Artzner et al. (1997) defines the properties of a coherent risk measure, significantly including the property of sub-additivity: the risk of an individual component should add up to the risk of the portfolio. To overcome this problem, Artzner and others developed Expected Shortfall (sometimes also called Conditional VaR). Expected Shortfall measures the mean loss when the loss is greater than the VaR. Another subadditive measure based on VaR is called Component VaR, which examines the contribution to the VaR of the portfolio by each individual component of the portfolio. Another limitation of both traditional mean-VaR and Cornish-Fisher VaR in its main form (above) is that it is not suitable for modelling the risk of complex structured products with non-continuous return-generating functions. 3. Is Cornish-Fisher VaR superior to the nonparametric method in estimating Expected Shortfall and Tail Risk? Why? They measure different things in very different ways. Historical measures are always limited to the observed returns, and will not deviate from observations already seen. Parametric methods such as Cornish-Fisher VaR attempt to mathematically predict the shape of the tail, even when such extreme returns have not been previously observed. Many managers use Expected Shortfall as a replacement for VaR because of the perceived inaccuracy of traditional historical or parametric VaR. Cornish-Fisher VaR will produce superior estimates to traditional parametric VaR. Asset managers should still use nonparametric (historical) analyses as part of the examination of scenario risk around specific historical events (Russian bond default, Black Tuesday, the recent credit crunch and volatility spike, etc.). Expected Shortfall (either parametric or nonparametric)or some other coherent VaR measure will be more useful in portfolio optimization than univariate Cornish-Fisher VaR. Cornish- Fisher VaR, in general, will be more useful for risk measurement and prediction for individual assets. 4. Finally, what kind of investors should use Cornish Fisher VaR? VaR estimates are estimates of downside risk. Because Cornish-Fisher VaR will give the same answer as traditional VaR when returns are normally distributed, Cornish-Fisher VaR may be substituted in all cases where VaR would normally be used, and may be expected to give a more accurate estimate of downside risk in almost all cases. Use of Cornish- Fisher VaR can inform a manager when an asset is riskier than might be apparent by examining only the standard deviation of the returns. Cornish-Fisher VaR will give a larger loss estimate than traditional VaR when returns are negatively skewed or highly kurtotic (fat-tailed) and, conversely, will give a smaller loss magnitude when returns are positively skewed or leptokurtotic. In this way Cornish-Fisher VaR more accurately accounts for expected investor preferences for positive skew and predictable ranges of returns. References: Artzner, Philippe, Freddy Delbaen, Jean-Marc Eber, and David Heath. 1997. Thinking Coherently. RISK 10(10):68–71. Campbell, Rachel, Ronald Huisman, and Kees Koedijk. 2001. Optimal Portfolio Selection in a Value at Risk Framework. Jornal of Banking and Finance 25:1789–1804. Cornish, Edmund A., and Ronald A. Fisher. 1937. Moments and Cumulants in the Specification of Distributions. Revue de l'Institut International de Statistique 5(4):307–320. Favre, Laurent, and Jose-Antonio Galeano. 2002. Mean-Modified Value-at- Risk Optimization with Hedge Funds. Journal of Alternative Investment Fall, 5(2):2–21. J.P.Morgan/Reuters. 1996. RiskMetrics Technical Document. Tech. Rep. Fourth Edition, J.P. Morgan/Reuters, New York. Zangari, Peter. 1996. A VaR Methodology for Portfolios that include Options. RiskMetrics Monitor First Quarter:4–12. |
|
Articles of the same Serie : 101 |
|
© Copyright 2008 bfinance. This document is for your personal non-commercial use. Any further copying, reproduction, distribution is strictly prohibited. To obtain permission please contact This e-mail address is being protected from spam bots, you need JavaScript enabled to view it