Variance/Covariance Approach

Also, called the mean/variance method, the variance/covariance approach requires that the probability distribution of investment returns is well represented by the normal distribution, using standard deviations and correlations between different financial instruments based on historical data.

To simplify the mathematics, the variance and covariance is calculated for each financial instrument and then weighted according to the weight of the financial instrument in the portfolio. Often the relationship between underlying variables is simplified. For instance, one assumption is that there is a direct relationship between option prices and prices of the underlying assets.

With a normal distribution, the 95% confidence level will be 1.645 standard deviations below the mean, and the 99% confidence level is at 2.33 standard deviations below the mean. If volatility is doubled, then VaR doubled; if the time horizon is doubled, then the VaR is multiplied by the square root of 2.

When calculating the regulatory capital requirements for banks using the VaR model, international regulations requires that the observation period be at least 10 working days and that the confidence level must be at least 99%.

Monte Carlo Simulation

The Monte Carlo simulation uses computers to simulate random changes in prices, and then applies them to the portfolio to calculate the effect:

·         A probability distribution is selected for each risk factor that would affect the portfolio: exchange rates, interest rates, volatility, and so on.

·         Pricing models are selected for each financial instrument.

·         Variances and covariances are calculated from historical data.

·         The computer model generates random changes in prices, volatilities, interest rates, and so on, but in such a way that the randomness conforms to the normal probability distribution, then applies those changes to the pricing models to calculate the portfolio return.

The pricing models are run many times using different random data to see how profits and losses will vary. For a 95% confidence level, 5% of the worst results are excluded. The VaR will then equal the worst result within the 95%.

The advantages of Monte Carlo simulation include using different probability distributions for different securities, using relevant pricing models for the security, such as option pricing models for options, and using different volatilities for different securities that more realistically reflects their actual volatility. The Monte Carlo simulation is often used to calculate the VaR for options.

Disadvantages of VaR

There are several disadvantages to using VaR models:

·         historical information does not predict future outcomes

·         models may not account for extreme circumstances, since VaR models are based on normal market conditions

·         the choice of observation period could be crucial, since too short of a period may be unrepresentative of the data, while choosing a long time period may include periods when variances and correlations differed significantly from the present

Another problem is that the probability distributions of actual outcomes have fatter tales, i.e., there is a greater chance of large losses or profits than a normal probability distribution model would suggest. The actual cumulative probability of monthly returns of large and small stocks has historically yielded more extreme values, especially at the lower end, which are the losses.

For portfolios with large negative deviations, the lower partial standard deviation (LPSD) can be calculated. A large LPSD indicates greater risk than what would be forecast by a normal distribution.

Another way to quantify the asymmetry of a distribution is using the 3rd moment of the distribution, which is the cubed deviation, instead of just being squared. This preserves the sign, and shows a greater contrast for extreme values. The 3rd moment is scaled by dividing it by the cubed standard deviation, called the skew of the distribution, a measure of the asymmetry.