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

Abstract

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.

Keywords: Conditional variance, Conditional standard deviation, Non-linear transformation, Small sample bias, EWMA, GARCH, Value at risk

JEL Classification: C10, G10