 # 3.2 Methods to measure VAR

Various methods are possible to compute Value-at-Risk. These methods basically differ by:
- distributional assumptions for the risk factors (e.g. normal versus other distributions) and
- linear vs full valuation, where linear valuation approximates the exposure to risk factors by a linear model.

## (1) Delta-Normal Method

The delta-normal method assumes that all asset returns are normally distributed. As the portfolio return is a linear combination of normal variables, it is also normally distributed. This method consists of going back in time, e.g. over the last 5 years, and computing variances and correlations for all risk factors. Portfolio risk is then generated by a combination of linear exposures to many factors that are assumed to be normally distributed, and by the forecast of the covariance matrix.
Required:
(1) for each risk factor, forecasts of volatility and correlations
(These data can be downloaded from the RiskMetrics site, originally developed by JP Morgan)
(2) positions on risk factors.

## (2) Historical-Simulation Method

This method consists of going back in time, e.g. over the last 5 years, and applying current weights to a time-series of historical asset returns. This return does not represent an actual portfolio but rather reconstructs the history of a hypothetical portfolio using the current position. Of course, if asset returns are all normally distributed, the VAR obtained under the historical-simulation method should be the same as that under the delta-normal method.
Required:
(1) for each risk factor, a time-series of actual movements, and
(2) positions on risk factors.

## (3) Monte Carlo Method

Monte Carlo simulations proceed in two steps.
First, the risk manager specifies a stochastic process for financial variables as well as process parameters; the choice of distributions and parameters such as risk and correlations can be derived from historical data.
Second, fictitious price paths are simulated for all variables of interest. At each horizon considered, which can go from one day to many months ahead, the portfolio is marked-to-market using full valuation. Each of these ``pseudo'' realizations is then used to compile a distribution of returns, from which a VAR figure can be measured.

Required:
(1) for each risk factor, specification of a stochastic process (i.e., distribution and parameters),
(2) valuation models for all assets in the portfolio, and
(3) positions on various securities.

## Comparison of Methods

(1) Delta-Normal Method: This is the simplest method to implement. Drawbacks, however, are the assumptions of normal distributions for all risk factors, and that all securities are linear in the risk factors (e.g. no options).
(2) Historical-Simulation Method: This is also relatively simple to implement. We just keep a historical record of previous price changes; distributions can be non-normal, and securities can be non-linear. One drawback is that only one sample path is used, which may not adequately represent future distributions.
(3) Monte Carlo Method: This is the most sophisticated method. It allows for any distribution and non-linear securities. The method, unfortunately, requires computer time and a good understanding of the stochastic process used.

Each method requires software to combine risk measures with internal positions. A list of software providers is at the gloriamundi site. Back to Case
Contents (c) 1996 - Philippe Jorion
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