Value at risk (VAR) is one measure of the market risk associated with a portfolio. The main reason for its popularity is its conceptual simplicity – it is an amount of money that might be lost over a given period, a single number that has direct importance.

VAR can be a useful tool for investment and pensions practitioners, providing a means of assessing how much risk exposure managers are taking on to achieve their portfolio returns.

Traditional investment asset allocation strategies have been based on investing in a mixture of assets – for example, 55% equities, 35% bonds and 10% cash. Maintaining these proportions in their portfolios, investment managers ignore the impact of fluctuating market risk. Risk allocation strategies are now more popular. Investment managers first stipulate the risk that they wish to take on. Allocations to asset classes are then made with the target risk in mind.

Portfolio risk is commonly measured using either a relative or absolute measure. For the former, the portfolio returns are compared to a benchmark (such as the DJ Euro Stoxx). In this case, risk is generally measured by the tracking error. In addition, portfolio tilts can be assessed using techniques such as reverse optimisation. One measure of absolute risk is the amount of money that the portfolio is expected to lose more than a given percentage of the time over a specified time horizon. This measure is VAR.

A variety of investment portfolios can be assessed in the VAR framework, for example:

q Total return funds Total return funds are not judged relative to a benchmark. Hence, the inherent risk

cannot be judged according to a relative risk measure such as tracking error. VAR is useful in gauging the overall risk level. It can be decomposed into the respective country, industry or sector components.

q Balanced funds and total plan risk When a portfolio contains bonds, commodities or other assets in addition to equities, there is often not a suitable benchmark or model to assess the total risk. VAR can be used. In situations where the risk is too large (or too small), portfolio adjustments can be made. VAR is also useful at the higher level of a collection of portfolios aggregated together.

q Long/short portfolios Many long/short portfolios are designed to be ‘market neutral’, that is, uncorrelated with the underlying market. Long/short portfolios are generally compared to an absolute benchmark (such as cash returns). Hence VAR is appropriate. VAR analysis can be used to decide if a long-short portfolio is exhibiting more risk than a fund manager desires.

q Portfolios with options When a portfolio contains options, tracking error as a means of risk assessment becomes suspect even if the portfolio has a benchmark. Tracking error is dependent on the distribution of possible gains and losses being symmetric. Options cause the distribution to be asymmetric. If the portfolio has a benchmark, then VAR can be relative to the benchmark. In this case you could also consider downside risk, which is an asymmetric version of tracking error.

VAR methodologies

To calculate VAR, a fund manager needs to specify the duration of the holding period and the probability level. For example, a fund manager may be interested in the expected maximum loss over a two-week period with a 5% probability.

The VAR estimation procedure can generally be considered to be a three-step process:

q Evaluate the portfolio returns.

q Estimate the distribution of gains and losses over the given timeframe.

q Calculate the VAR from this distribution.

Understanding VAR requires knowledge of the distribution of the portfolio returns. Once this distribution has been determined, either analytically or by simulation, a variety of statistical measures can be used to measure the risk.

The different methods of estimating VAR are chosen as a trade-off between simple assumptions, and hence ease of calculation, and more complex models, which emulate the markets more accurately and can handle more sophisticated financial instruments.

Though the concept of VAR is quite easy, there are technical difficulties in the estimation process. There is no perfect model. The model is simply being used as a guide to assess the risk. It is important to understand the structure and hence the limitations of whichever model is being used.

An important aspect of calculating VAR is to understand which assets contribute most to risk. Armed with this information, a fund manager can alter stock positions in the portfolio to modify VAR most effectively. For this purpose an individual security’s volatility is not sufficient. Volatility measures the uncertainty of an asset in isolation, ignoring the correlations between securities. When the asset belongs to a portfolio, what matters is the contribution to the overall portfolio risk. This is often referred to as the incremental or marginal contribution to risk.

The most frequently used VAR estimation techniques are as follows:

q Closed form VAR Closed form VAR assumes that the equity returns are normally distributed and that any options in the portfolio can be well approximated by their deltas. The delta reflects the rate of change of the option’s price with respect to the price of the underlying security.

Given these assumptions, profit or loss is also normally distributed. VAR can then be calculated directly from the volatilities and correlations of the applicable risk factors.

The closed form model is very simple. Two key assumptions that it makes are that returns follow the normal distribution, and that the distribution does not change over time. Neither assumption is correct, so closed form VAR estimates are often not very accurate.

q Historical VAR In historical VAR, scenarios are drawn directly from historical market data. We consider how the portfolio’s return is affected by the relevant risk factors (eg, stock prices or interest rates). We assume that the changes in these risk factors are independent and are equally likely to occur.

Historical VAR is easy to implement. Because it uses real data, it often reflects the non-normality of market data, including large price swings when a market falls or rises sharply. However, this technique is based on just one sample and it does not account for the volatility clustering prevalent in market data.

q Garch Volatility measures are important as they directly impact on the VAR. An increase in the volatility causes an increase in the VAR. Garch (Generalised autoregressive conditional heteroscedasticity) models attempt to capture the ‘fat tails’ that exist in the return distribution as well as the volatility clustering that is apparent in equity returns. Garch techniques therefore provide a more accurate measure of the risk distribution.

Another feature of the Garch model is that it captures the mean reversion that drives changes in volatility and correlation. The model also has the ability to incorporate recent price behaviour when forecasting covariance since recent observations are given more weight than older ones.

However, Garch models are complex, requiring a long history of data for model evaluation, which can be computationally demanding.

q Monte Carlo VAR Monte Carlo VAR is a technique that allows risk measures to be obtained for complex portfolios that have significant gamma or convexity. The gamma reflects the second order rate of change of the portfolio’s value to the price of the underlying securities. The impact of large gamma is that it skews the risk distribution. Hence, Monte Carlo VAR is particularly useful for calculating the risk profile of a portfolios containing derivatives. Because of their flexibility, Monte Carlo techniques are very powerful.

The portfolio’s risk variables are simulated many times over a specified time horizon and the risk distribution is constructed. Monte Carlo simulation can take a long time for complex situations.

Experience of using Monte Carlo procedures and prior knowledge of the portfolio being analysed allows successful application of this type of VAR technique.

Typically, a model will be determined which specifies a stochastic process and process parameters. Risk and correlations can be determined by historical or option data. Price paths are then simulated for all variables of interest over the time horizon considered. The realisation is then used to compile a distribution of returns, from which a VAR figure can be determined.

Challenges of VAR estimation

Although a simple measure, VAR modelling is not without its complications. Here we highlight just a few of the most common problems encountered in VAR estimation.

The most important input to a VAR analysis process is often the estimation of the correlations between all the assets involved in the analysis in the form of a correlation matrix. The estimation of the correlation matrix of asset returns would seem initially to be a simple task. However, there are a number of practical considerations that make it challenging to obtain a good estimate.

The most vexing problem is that the covariance matrix does not remain constant – both the volatility of individual stocks and the correlation between stocks are known to change over time.

Portfolios containing options or other derivatives create difficulties when evaluating VAR. The change in a stock’s value is a linear function of the change in price. However, for an option, it has a curved price function. That curved price function skews the option price distribution, causing it to be asymmetric.

Also, changes in the values of the option position depend not only on the underlying spot values, but also on the level of the spot values. At-the-money options, for instance, display a high level of convexity, which leads to a highly skewed risk distribution.

If a portfolio is global, then the problem of asynchronous data will almost certainly be encountered. The problem is that not all markets are open at the same time, so markets may react to the same news on different days. The result is that correlations are actually higher than they appear to be. If the asynchrony is not taken care of, then VAR estimates are likely to be too small.

Michael Benjamin is assistant vice president and Stavros Siokos is managing director of global portfolio trading strategies at Schroder Salomon Smith Barney in London

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