Financial Risk Management

CHAPTER 11Shortcomings of Risk Metrics

THE PROBLEM OF STATIONARITY

To compute a Value‐at‐Risk (VaR) and thus an Expected Shortfall (ES)—since both are interrelated—over a given holding period or time horizon (e.g., the next 24 hours or 10 days), a very demanding assumption is required. This assumption is called stationarity.

We recall that VaR is nothing more than a quantile (theoretical or empirical) of a portfolio Profit and Loss (P&L) calculated for a specified confidence level α. With this aim, we use historical data to estimate the potential changes in value of this portfolio in the near future—the next 24 hours according to the Basel III regulatory framework.

Unfortunately, we do not know the future and thus what could occur in financial markets over the next 24 hours. As a result, to deliver a VaR figure, we must assume the stationarity of the portfolio change in value over the holding period. In other words, we consider that the near future will behave like the past. If so, it makes sense to derive a 24‐hour VaR estimate from what happened over the past trading year in terms of P&L fluctuations.

Of course, such an assumption is unrealistic knowing what may occur in financial markets in just 5 minutes. However, without this stationarity assumption, the VaR calculation is not possible, simply because the best ever forecast model is unable to predict the future. Therefore, there is no other choice than to compute statistical parameters (e.g., the mean and standard deviation ...