Bias decomposition in the value-at-risk calculation by a GARCH(1,1)

Abstract

The recent researches show that value-at-risk (VaR) estimations are biased and is calculated conservatively. Bao and Ullah (2004) proved the bias of an ARCH(1) model for VaR can be decomposed in to two parts: bias due to the returns' misspecification distributional assumption for GARCH(1,1), i.e., (Bias1) and bias due to estimation error, i.e., (Bias2). Using quasi maximum likelihood estimation method this paper intends to find an analytical framework for the two sources of bias. We generate returns from Normal and t-student distributions, then estimate the GARCH(1,1) under Normal and t-student assumptions. Our findings reveal that Bias1 equals to zero for the Normal likelihood function, but Bias2 ≠ 0. Also, Bias1 and Bias2 are not zero for the t-student likelihood function as analytically were expected, however, all the biases become modest, when the number of observations and degree of freedom gets large.