An alternative to the historical simulation approach for calculating risk measures such as VaR and expected shortfall (ES) is the *model-building approach*, sometimes also referred to as the *variance–covariance approach*. This involves assuming a model for the joint distribution of changes in market variables and using historical data to estimate the model parameters.

The model-building approach is ideally suited to a portfolio consisting of long and short positions in stocks, bonds, commodities, and other products. It is based on Harry Markowitz's pioneering work in portfolio theory (see Section 1.1). The mean and standard deviation of the value of a portfolio can be calculated from the mean and standard deviation of the returns on the underlying products and the correlations between those returns. If, and it is a big if, daily returns on the investments are assumed to be multivariate normal, the probability distribution for the change in the value of the portfolio over one day is also normal. This makes it very easy to calculate value at risk.

As we shall see, the model-building approach is much more difficult to use when a portfolio involves nonlinear products such as options. It is also difficult to relax the assumption that returns are normal without a big increase in computation time.

We start by considering how VaR is calculated using the model-building approach in a very simple situation where the portfolio ...

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