Value at Risk (VaR)

Value at risk (VaR) is the maximum potential loss expected on a portfolio over a given time period, using statistical methods to calculate a confidence level. (VaR is capitalized differently to distinguish it from VAR, which is used to denote variance.) VaR is widely used by financial institutions, portfolio managers, and regulators to forecast potential losses. If an unleveraged investor's portfolio declines in value, he can simply wait it out, in many cases, until it rises again. Banks, on the other hand, being highly leveraged, cannot suffer substantial losses over extended periods without affecting its operations. Therefore, banks need to both measure and manage risk continually. One of the main methods is the VaR, sometimes called earnings at risk.

The VaR is applied to portfolios, using models and statistical methods to calculate what the probability is that the value of a certain portfolio will decline below a given value. For instance, the VaR may yield a maximum loss of $1 million over a 30-day period, calculated with a confidence level of 95%, meaning that there is a 5% probability that losses will exceed the VaR or that, in 1 month out of 20 months, losses will exceed the VaR.

The time horizon for the VaR depends on the liquidity of the portfolio: greater liquidity allows for shorter time periods, since the underlying securities can be sold quickly to offset risk.

Normal Distributions, Means and Standard Deviations

Many outcomes of life depend on chance, and it is commonly observed that they fall into a common distribution according to the probability of each possible outcome, called the normal distribution. VaR calculates the variance of return distributions for a portfolio as a measure of its risk.

Normal distributions are symmetric and completely described by 2 parameters: the mean and the standard deviation. The other benefit of the normal distribution is that the weighted average of variables that are normally distributed will also be normally distributed. So different stocks with normal distributions can be combined with the other stocks or securities with normal distributions, and the entire portfolio will be normally distributed.

The normal distribution has a mean equal to the average value. Values close to the mean have a higher probability of occurring than those that are further from the mean. Regarding investments, the mean is the expected return.

The variance, or the dispersion, of the portfolio is calculated by subtracting the mean from actual outcomes and squaring them to eliminate negative numbers, then dividing by n – 1, where n = number of samples. The standard deviation is equal to the square root of the variance. Hence, the standard deviation is the average amount of deviation and is commensurate with the dispersion of the outcomes about the mean. The wider the dispersion, the greater the standard deviation. Converting the normal distributions of sample returns to the standard normal distribution changes the standard deviation to integer values, so that if an outcome has an average deviation, then its standard deviation will be equal to 1. If an outcome deviates by twice the average from the mean, then it is 2 standard deviations from the mean. For a normal distribution, the probability of an event occurring within 1 standard deviation of the mean is 68%, 2 standard deviations represent 95% of the values in the distribution and 99.7% of the values fall within 3 standard deviations of the mean.

The mean and standard deviations of investment returns are determined using historical outcomes. Different financial instruments will have different means and different standard deviations, meaning different amounts of volatility.

The VaR is a quantile of a distribution, meaning that part of the distribution where a certain percentage of the values lie. For instance, the median equals the 50% quantile. VaR usually uses the 5% or the 1% quantile. One property of the normal distribution is that it symmetrical. However, since risk is defined as the probability of a loss, the VaR is calculated by subtracting the portion of the left tail, which represents extreme losses. The remaining area under the normal distribution curve represents the probability that a portfolio will not exceed those losses.