A Note on Optimal Estimation from a Risk-Management Perspective under Possibly Misspecified Tail Behavior
Lucas, Andre, Journal of Business & Economic Statistics
Many financial time series show leptokurtic behavior--that is, fat tails. Such tail behavior is important for risk management. In this article I focus on the calculation of Value-at-Risk (VAR) as a downside-risk measure for optimal asset portfolios. Using a framework centered on the Student-t distribution, I explicitly allow for a discrepancy between the fat-tailedness of the true distribution of asset returns and that of the distribution used by the investment manager. As a result, numbers for the overestimation or underestimation of the true VaR of a given portfolio can be computed. These numbers are used to rank several well-known estimation methods for determining the unknown parameters of the distribution of asset returns. Minimizing the absolute (percentage) mismatch between the nominal and actual or true VaR leads to the choice of a Gaussian maximum quasilike-lihood estimator--that is, a least squares type of estimator. The maximum likelihood estimator has less satisfactory behavior. Outlier-robust estimators perform even worse if the required confidence level for the VaR is high. An explanation for these results is provided.
KEY WORDS: Downside-risk; Leptokurtosis; Minimax optimality; Model misspecification; Optimal-asset allocation; Quasi-likelihood; Risk management; Robustness; Value-at-Risk.
Uncertainty is the key ingredient of most financial and economic decision problems. For example, an investment manager trying to design a solid investment policy has to come up with a consistent set of forecasts for future returns on alternative investment opportunities. In certain cases, predictions of macroeconomic developments are needed, as well--for example, inflation rates in case of pension funds with liabilities that are defined in real terms. Other sources of risk affecting the performance of financial policies include interest-rate risk, exchange-rate risk, credit risk, and so forth.
To characterize the risk associated with decision making under uncertainty, different risk measures have been proposed. In the framework of asset management, the most familiar risk measure is the standard deviation of portfolio returns (e.g., see Markowitz 1959). The main drawback of the standard deviation as a measure of risk is that it is symmetric: Extreme positive returns are treated the same way as extreme negative returns. As an alternative to the standard deviation, several downside-risk measures have been proposed. By far the most popular downside-risk measure used today is the Value-at-Risk (VaR) as introduced by J. P. Morgan. The VaR measures the maximum (dollar) loss on a portfolio over a given period of time given a certain confidence level. For example, if the VaR of a portfolio is 10 million dollars with a confidence level of 99% and a holding period of 10 days, this means that there is only a 1% probability that the portfolio will produce a loss of more than 10 million dollars if it cannot be liquidated within a 10-days period. The popularity of VaR is enhanced by the fact that regulatory institutions have adopted the use of VaR as a measure of risk in their supervisory policies. The Basle Committee on Banking Supervision (1996a,b), for example, proposed to require banks to report VaR figures to the supervisory institution--for example, the Central Bank--on a regular basis. Within certain limits, banks are allowed to construct their own models for computing such VaR measures.
Given the popularity of VaR and its importance for supervision, it is interesting to study the properties of alternative methods for computing VaR. All methods for calculating VaR have to deal with the fact that it is difficult to obtain reliable estimates of the (lower) quantiles of a distribution from a limited number of data points. Different methods have been proposed, ranging from the use of parametric models (often the normal distribution), via the use of seminonparametric models to the employment of fully nonparametric methods like historical simulation. …