Modeling Hedge Fund Risk with a Multi-State Normal Distribution
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Date: 11-21-2005
Start Time:
6:00pm
End Time: 7:30pm
Speaker: Robert Litzenberger, Azimuth
Location: Interschool Lab, 750 Schapiro CEPSR
ABSTRACT
The limitations of variance of asset returns as a sufficient measure of risk are well known. A half a century ago, at the dawn of modern portfolio theory, Markowitz argued that when returns are asymmetrically distributed, semi-variance is a better measure of risk. Nevertheless, the analytical tractability of mean-variance portfolio theory for measuring portfolio risk based on variance and covariance of historical returns on individual assets has resulted in its wide use in risk management. The two most popular approaches to risk management are Value-at-Risk (VaR) based on portfolio variance, or VaR based on historical simulation. The former approach does not account for fat tails, and the latter approach is unable to accurately reflect fat tails because of small sample problems. Therefore, when there are limited observations, prior parametric restrictions on the distribution may be reasonable. Expected Tail Loss (ETL) is a better measure of downside risk than VaR since it accounts for the distribution of losses in the lower tail. However, the use of ETL under a normal distribution is merely using a different multiplier of the standard deviation to generate the risk metric and does not reduce the small sample problems associated with historical simulation. While extreme value theory correctly focuses on the lower tail, and some results are available for linear combinations of random variables, it does not have the same analytical tractability for aggregating asset returns. It is also well known that a normal distribution with a stochastic variance will result in fatter tails than a normal distribution. However, the resulting distributions are not stable under addition, and when the mixing probabilities are independent across assets, under the central limit theorem the return distribution on a portfolio will approach normality as the number of assets in the portfolio increases.
In periods of financial crisis, standard deviations of affected trades increase by a multiple of their non-crisis standard deviations, correlations between such trades increase, and liquidation pressure causes trades that are relatively uncorrelated in non-crisis periods to lose money together in crisis periods. Many firms use scenario analysis to analyze impacts of crises on their portfolio, but have difficulty aggregating such risk across asset classes.
The current study uses a multi-state normal distribution that maintains stability under addition within each state, and the ETL may be aggregated across assets. The states represent the set of different combinations of capital market crises. Individual assets impacted by a given type of crisis have their state-dependent standard deviations increase by a crisis multiplier, and suffer a crisis-dependent downward mean shift equal to a Z-value times their state-dependent standard deviation. This downward mean shift is equivalent to a perfectly correlated component of return. Under the assumption of normality within each state, the solution for the state-contingent ETL is similar to the well known formulation for the value of a put option when asset returns are normally distributed. The overall ETL is expressed as the probability weighted average of state contingent put options. Since a portfolio’s risk premium and ETL per dollar of equity are proportional to leverage, the “unlevered” portfolio with the highest ratio of risk premium to ETL when combined with borrowing or lending is optimal. The contribution of individual assets to the portfolio’s ETL are used to derive the conditions for maximizing this reward to risk ratio. Equilibrium risk premiums are defined as the risk premiums that would result in market capitalization weights being optimal. Finally, the Black-Litterman asset allocation approach is extended to the mean-ETL framework.
BIO
Robert Litzenberger is currently an Executive Director at Azimuth Trust where he
manages the Select Trust a fund of hedge funds. He previously served as Firmwide
Risk Manager at Goldman, Sachs & Co. where he was responsible for the
development, implementation and monitoring of Goldman's global risk management
system as well as for setting and monitoring risk limits firm wide. Robert was
also charged with meeting with Goldman's trading leaders (encompassing a large
and diverse grouping of financial strategies, instruments and markets including
equities, fixed income and commodities), assessing and monitoring their market
risk for each, and subsequently making comprehensive recommendations regarding
his findings to Goldman's Risk Committee. Prior to this, Robert Litzenberger
served as Director of Derivative Research and Quantitative Modeling at Goldman.
Previously, he served as Director of Research and Chief Economist at
AIG-Financial Products where he worked on the development, pricing, and hedging
of customized derivative products, developed valuation and hedging models, and
determined all risk management procedures for new activities.
Robert
Litzenberger is co-author of Foundations of Financial Economics (1988), and has
published more than 50 articles in leading academic finance journals. Since
1986, he has taught finance at the Wharton School of the University of
Pennsylvania where he held the Edward Hopkinson Chair in Investment Banking. He
is currently a Professor Emeritus. Before joining the Wharton faculty, Robert
was the C.O.G. Miller Distinguished Professor of Finance at the Stanford
Graduate School of Business. He is a former President of the American Finance
Association.
Robert Litzenberger holds a Ph.D. from University of North
Carolina, an M.B.A from University of Pennsylvania, and a B.A. from Wagner
College.