# Analytical Method, Monte Carlo Simulation and Historical Simulation: the Benefits and Pitfalls

## Alternative Approaches

Three alternative but complementary techniques have been deployed in market risk analysis. The oldest and still most widely used approach is an analytical method, known as the variance-covariance model. In this approach, approximate factor sensitivities are computed for the portfolios, and the portfolio Value at Risk numbers are then computed simply by multiplying the factor sensitivities by the relevant shifts in the risk factors. Approximate factor sensitivities can be based on first, second or higher order approximations.

While higher order approximations allow the variance-covariance approach to be applied to a wider range of portfolios, the Basle Committee still recommends that the full valuation principle is used for analyzing the market risk of certain positions. In particular, full valuation should be applied to positions containing non-linear instruments such as options, and therefore these positions should be analyzed using a simulation-based method.

The two primary simulation approaches implemented for market risk analysis are Monte Carlo and historical. Monte Carlo simulation involves three stages. First, the multidimensional probability distribution that represents the (random) behavior of the market risk factors is estimated. Secondly, a large number (5 000 – 10 000) of random samples are drawn from the distribution of exchange rates, interest rates, stock prices, commodities prices etc to represent the randomly chosen scenarios that the markets may follow. This task is not simple, as most banks are exposed to several tens or even hundreds of market rates and prices (or risk factors). Thirdly, all portfolios are evaluated against these random scenarios, and the evaluation procedure produces the probability distribution of the P/L changes. On the basis of the P/L distributions, the VaR numbers and other statistical risk measures may be computed.

The implementation steps for historical simulation are very similar to those for Monte Carlo simulation, except that the process is far simpler. The probability distribution of the risk factors is obtained directly from historical changes in market prices and rates. The history used can vary from a few months to several years, and different durations of history may be used for different purposes. The market price changes represent the changes in risk factors and therefore can be used to derive the exact joint probability distribution of risk factors that occurred during the historical period. Historical market movements are then sampled randomly from the joint distribution. Portfolios are then evaluated and the P/L changes constructed in the same way as in Monte Carlo simulation approach.

## Benefits and Pitfalls

The main advantages of the variance-covariance approach are its relatively simple structure and the speed of calculations. Positions are evaluated once only in this approach, so that intra day calculations for even large portfolios are possible within a realistic time scale. If the institution using the analytical approach is trading only regular, linear instruments then the level of accuracy obtained is reasonably good. However, results become unreliable when the portfolios include significant numbers of non-linear instruments. Using higher order analytical approximations can also be problematic. For example, a delta-gamma approximation for a strangle option can be less accurate than using a standard delta approach. Careful analysis of portfolio composition is required, to ensure that the appropriate analytical method is chosen.

Both simulation approaches (historical and Monte Carlo) share the strength of the full valuation principle and can therefore provide a reliable result for all types of portfolio. Historical simulation is relatively fast since only a few hundred observations are typically used. If a longer period of historical data is used then market price fluctuations might contain changes that are due to structural shifts in the markets (such as changes in taxation, new financial innovations etc.). It is unlikely that similar structural changes will be repeated in the market in the future. A further benefit of historical simulation is that the distribution used in simulations is the actual historical distribution recently observed including any anomalies (such as fat tails, skewness, kurtosis etc.). As long as historical fluctuations are repeated, then the predicted risk numbers and statistics represent accurately the true distribution with its non-normalities. The main problem with historical simulation is that a relatively small number of simulation rounds are typically run. As a result, the various statistics like the Value at Risk numbers are rather inaccurate estimates of the true numbers, despite of the true nature of the distribution. For example, if a hundred business days of historical data are used then the 1% VaR number is estimated using only one observation for each portfolio. This is very inaccurate.

In Monte Carlo simulation the statistical estimates are highly accurate in terms of the distribution of risk factors, and the reliability of the VaR figure. Monte Carlo simulation usually assumes that risk factors are normally distributed but nothing in principle prevents other distributions from being substituted e.g. fat tailed distributions. The distribution of market rates is estimated from historical data, rather than derived directly from past rate changes, as in the historical method. This means that a large a number of simulations can be run (which is necessary to obtain an accurate VaR figure), without using a long historical time period that would risk irrelevant structural shifts in the data. If, for example, 10 000 rounds simulation studies are run, then the 1% VaR estimates are each supported by 100 evaluations, which gives a very robust estimation. The main problem with Monte Carlo simulation is that implementing large scale realistic simulation models is very computationally intensive. A 10 000 round Monte Carlo simulation involves 10 000 times the number of evaluations of an analytical model and 100 times the number of evaluations of a 100 round historical simulation. However, there are various ways of making the calculations faster, and using the highly developed models which are now available, intra-day applications for sizeable positions are becoming a realistic option.

The three approaches each have their benefits and pitfalls and they can be used both as alternatives to each other and also as complementary approaches.

CD Financial Technology Risk Engine product suite supports all these methods, providing the user with a flexible and sophisticated approach to risk management.