## Value at risk

VaR is a very intuitive measure to evaluate market risk because it indicates the maximum potential loss at a given level of confidence (a) for a portfolio of financial assets over a specified time horizon (h).

In practice, the value of a portfolio is expressed as a function of K risk factors, xT = X^w wi,tPi,r(.X1,r,..., XK,r). The factors influencing the portfolio value are usually identified with some market variables such as interest rates, exchange rates or stock indexes. If their distribution is known in a closed form, we need to estimate the distribution of the future value of the portfolio conditional on the available information and the VaR is then the solution to:

Different parametric models can be used to forecast the portfolio return distribution. The simple way to calculate VaR involves assuming that the risk factor returns follow a multi-variate normal distribution conditional on the available information. If the portfolio return is linearly dependent on them, its probability distribution is also normal and the VaR is simply the quantile of this analytic distribution. If the linear assumption is inappropriate, the portfolio return can be approximated as a quadratic function of the risk factor returns.

An alternative way to handle the non-linearity is to use Monte Carlo simulation. The idea is to simulate repeatedly the random processes governing the risk factors. Each simulation gives us a possible value for the portfolio at the end of our time horizon. If enough of these simulations are considered, it is possible to infer the VaR, as the relevant quantile of the simulated distribution.

Since market risk factors usually have fatter tails than the normal distribution, it is also possible to use historical simulation rather than a parametric approach. The idea behind this technique is to use the historical distribution of returns to the assets in the portfolio to simulate the portfolio's VaR, on the hypothetical assumption that we held this portfolio over the period of time covered by our historical data set. Thus, the historical simulation involves collecting historic asset returns over some observation period and using the weights of the current portfolio to simulate the hypothetical returns we would have had if we had held our current portfolio over the observation period. It is then assumed that this historical distribution of returns is also a good proxy for the portfolio return distribution it will face over the next holding period and VaR is calculated as the relevant quantile of this distribution.

The advantage of the parametric approach is that the factors variance-covariance matrix can be updated using a general model of changing or stochastic volatility. The main disadvantage is that the factor returns are usually assumed to be conditionally normal, losing the possibility to take into account non-linear correlations among them. Historical simulation has the advantage of reflecting the historical multivariate probability distribution of the risk factor returns, avoiding ad hoc assumptions. However the method suffers a serious drawback. Its main disadvantage is that it does not incorporate volatility updating. Moreover, extreme quantiles are difficult to estimate, as extrapolation beyond past observations is impossible. Finally, quantile estimates tend to be very volatile whenever a large observation enters the sample and the database is not sufficiently large.

The advantage of the parametric approach to update the volatility suggests the simplest utilisation of the SV models for the VaR computation. Having chosen the asset or portfolio distribution (usually the normal one), it is possible to use the forecasted volatility to characterise the future return distribution. Thus, oT+1/T can be used to calculate the VaR over the next period.

A different approach using the SV model is to devolatilise the observed returns series and to revolatilise it with an appropriate forecasted value, obtained with a particular model of changing volatility. This approach is considered in several recent works (Barone-Adesi et al., 1998; Hull and White, 1998) and is a way of combining different methods and partially overcoming the drawbacks of each.

To make the historical simulation consistent with empirical findings, the log-normal SV model and the regime switching model may be considered to describe the volatility behaviour. Past returns are standardised by the estimated volatility to obtain standardised residuals. Statistical tests can confirm that these standardised residuals behave approximately as an iid series which exhibits heavy tails. Historical simulation can then be used. Finally, to adjust them to the current market conditions, the randomly selected standardised residuals are multiplied by the forecasted volatility obtained with the SV model.

Table 8.2 VaR at different confidence levels for the FTSE100 index return

Confidence level Log-normal SV model Regime switching model

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