A Note on Optimal Estimation from a Risk-Management Perspective under Possibly Misspecified Tail Behavior

Abstract

Many financial time-series show leptokurtic behavior, i.e., fat tails. Such tail behavior is important for risk management. In this paper I focus on the calculation of Value-at-Risk (VaR) as a downside-risk measure for optimal asset portfolios. Using a framework centered around 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 over-estimation or under-estimation 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 quasi-likelihood estimator, i.e., a least-squares type estimator. The maximum likelihood estimator has a 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.

Evaluating Interest Rate Covariance Models Within a Value-at-Risk Framework

“…It is interesting to find that the standard normal distribution generates VaR estimates that perform well at such a low quantile, especially since portfolio returns are commonly found to be distributed non-normally, in particular with fatter tails. This result appears to be in line with those of Lucas (2000), who found that the upward bias in the estimated dispersion measure under such models appears to partially offset the neglected leptokurtosis.…”

“…The reasons for these findings range from the issue of parameter uncertainty, as per Skintzi, Skiadopoulos, and Refenes (2003), to overfitting the data in-sample to the sensitivity (i.e., the first derivative) of relevant economic loss functions to the volatility forecast inputs. An interesting paper that addresses this question more directly is Lucas (2000) This table presents RMSPE for the daily interest rate changes and foreign exchange rate geometric returns covariances using alternative model specifications. The minimum value in each column is in bold and underlined, and the second smallest value is just in bold.…”

Evaluating Interest Rate Covariance Models within a Value-at-Risk Framework

“…The reasons for these findings range from the issue of parameter uncertainty, as per Skintzi, Skiadopoulos, and Refenes (2003), to overfitting the data in-sample to the sensitivity (i.e., the first derivative) of relevant economic loss functions to the volatility forecast inputs. An interesting paper that addresses this question more directly is Lucas (2000), who finds that VaR models based on simple measures of portfolio variance and the normal distribution generate smaller discrepancies between actual and postulated VaR estimates than more sophisticated VaR models. He argues that this outcome is based on offsetting biases in the variance and VaR estimates of simple models that cannot be captured by more sophisticated models that attempt to capture the actual (but unknown) degree of leptokurtosis in the portfolio returns.…”

Thinly traded securities and risk management

“…We used a t-student distribution for the security returns to calculate the VaR following the work of Wilson (1993) and Lucas (2000). The t-student distribution has the advantage of better adjustments to tails than normal distributions.…”