Cherif Guermat

Forecasting value at risk allowing for time variation in the variance and kurtosis of portfolio returns

Abstract A common approach to forecasting the value at risk (VaR) of a portfolio is to assume a parametric density function for portfolio returns, and to estimate the parameters of the density function by maximum likelihood using historical data. In order to allow for volatility clustering in short horizon returns, this approach is typically combined with a conditional variance model such as EWMA or GARCH. However, these models implicitly assume that while the volatility of returns may be time-varying, the kurtosis of the return distribution is constant, at least over the estimation sample. In this paper, we show that the EWMA variance estimator can be obtained as a special case of a more general, exponentially weighted maximum likelihood (EWML) procedure that potentially allows for time-variation not only in the variance of the return distribution, but also in its higher moments. We use EWML to forecast VaR allowing for time-variation in both the variance and the kurtosis of daily equity returns. Our results show that the EWML based VaR forecasts are generally more accurate than those generated by both the EWMA and GARCH models, particularly at high VaR confidence levels.

Robust Conditional Variance Estimation and Value-at-Risk

A common approach to estimating the conditional volatility of short horizon asset returns is to use an exponentially weighted moving average (EWMA) of squared past returns. The EWMA estimator is based on the maximum likelihood estimator of the variance of the normal distribution, and is thus optimal when returns are conditionally normal. However, there is ample evidence that the conditional distribution of short horizon financial asset returns is leptokurtic, and so the EWMA estimator will generally be inefficient in the sense that it will attach too much weight to extreme returns. In this paper, we propose an alternative EWMA estimator that is robust to leptokurtosis in the conditional distribution of portfolio returns. The estimator is based on the maximum likelihood estimator of the standard deviation of the Laplace distribution, and is a function of an exponentially weighted moving average of the absolute value of past returns, rather than their squares. We employ the robust EWMA estimator to forecast the VaR of aggregate equity portfolios for the US, the UK and Japan using historical simulation. We find that the robust EWMA estimator offers an improvement over the standard EWMA estimator. In particular, the VaR forecasts that it generates are as accurate as those generated by the standard EWMA estimator, but are more efficient in the sense that the average level of capital required to cover against unexpected losses is lower and the root mean square deviation between the VaR forecast and actual returns is smaller. Moreover, the volatility of the VaR forecast itself is substantially lower with the robust EWMA estimator than with the standard EWMA estimator, reflecting its lower sensitivity to extreme returns.

Rules versus Discretion in Committee Decision Making: An Application to the 2001 RAE for UK Economics Departments

The question of rules versus discretion has generated a great deal of debate in many areas of the social sciences. Recently, much of the discussion among academics and stakeholders about the assessment of research in UK higher education institutions has focused on the means that should be used to determine research quality. We present a model of committee decision-making when both rules and discretion are available. Some of the predictions of the model are tested empirically using the UK RAE 2001 results.

Bias in the Estimation of Non-Linear Transformations of the Conditional Variance of Returns

Many applications in finance use a non-linear transformation of the variance of returns. While the sample variance is an unbiased and consistent estimator of the population variance of returns, non-linear transformations of the sample variance will be consistent but biased. For estimates of non-linear transformations of the unconditional variance, this will rarely be a problem in practice, since sample sizes employed in finance are typically large. However, estimators of the conditional variance typically use sample sizes that are effectively much smaller, particularly those that apply an exponential weighting to returns such as GARCH or EMWA. Consequently, the bias is likely to be more important in estimating non-linear transformations of the conditional variance. In this paper, we derive a simple analytical approximation for the unconditional bias in estimators of non-linear transformations of the conditional variance, under the assumption that returns are conditionally normally distributed, and that the true conditional variance is generated by an arbitrary stochastic volatility model. As an illustration, we estimate the bias inherent in the RiskMetrics approach to the calculation of value at risk.