In this chapter, we cover the most popular approach for calculating value at risk (VaR) and expected shortfall (ES) for market risk. It is known as historical simulation. It involves using the day-to-day changes in the values of market variables that have been observed in the past in a direct way to estimate the probability distribution of the change in the value of the current portfolio between today and tomorrow.
After describing the mechanics of the historical simulation approach, the chapter explains a number of extensions that can improve accuracy. It covers stressed VaR and stressed ES, which are used (or will soon be used) by regulators to determine capital for market risk. Finally, it covers extreme value theory. This is a tool that can be used to improve VaR and ES estimates and to increase the confidence level for these estimates.
All the models covered in this chapter are illustrated with a portfolio consisting of an investment in four different stock indices. Historical data on the indices and VaR calculations can be found at www-2.rotman.utoronto.ca/∼hull/RMFI/VaRExample.
Historical simulation involves using past data as a guide to what will happen in the future. Suppose that we want to calculate VaR for a portfolio using a one-day time horizon, a 99% confidence level, and 501 days of data. (The time horizon and confidence level are those typically used for a market risk VaR calculation; ...