Estimating Value at Risk and Expected Shortfall: A Kalman Filter Approach

Abstract

Value at Risk (VaR) estimates the maximum loss a portfolio may incur at a given confidence level over a specified time, while expected shortfall (ES) determines the probability weighted losses greater than VaR. VaR has recently been replaced by (but remains a crucial step in the computation of) ES by the Basel Committee on Banking Supervision (BCBS) as the primary metric for banks to forecast market risk and allocate the relevant amount of regulatory market risk capital. The aim of the study is to introduce a more accurate approach of measuring VaR and hence ES determined using loss forecast accuracy. VaR (hence ES) is unobservable and depends on subjective measures like volatility, more accurate (loss forecast) estimates of both are constantly sought. Modelling the volatility of asset returns as a stochastic process, so a Kalman filter (which distinguishes and isolates noise from data using Bayesian statistics and variance reduction) is used to estimate both market risk metrics. A variety of volatility estimates, including the Kalman filter's recursive approach, are used to measure VaR and ES. Loss forecast accuracy is then computed and compared. The Kalman filter produces the most accurate loss forecast estimates in periods of both calm and volatile markets. The Kalman filter provides the most accurate forecasts of future market risk losses compared with standard methods which results in more accurate provision of regulatory market risk capital.

Keywords

• Kalman filter
• Value-at-Risk
• Expected Shortfall