# 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 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.

The VaR is based on models that assume normal market conditions and that a correlation can be calculated for the covariance between financial instruments. To reduce risk, banks diversify their financial assets so that there is less correlation in prices when market conditions change, since a diversified portfolio is less risky than an undiversified one. There are 3 widely used methods to calculate VaR: variance/covariance approach; Monte Carlo simulation; and the historical approach.

## Historical Approach

The historical approach is the simplest way to calculate VaR because no assumptions need to be made about the probability distribution or about the variances and covariances, since they will be accurately represented by the historical data. This is the primary difference between the historical approach and the Monte Carlo simulation. Instead of changes in financial data being randomly generated, it is based on historical changes for the financial instruments that compose the portfolio. The time period for the selected historical information is known as the **observation period**. As with Monte Carlo simulation, the variation in profit or loss is recorded for each set of data based on different time periods, then the worst results are excluded, yielding a VaR equal to the worst outcome remaining in the set.

## 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, etc.
- 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, etc, 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 3^{rd} 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 3^{rd} moment is scaled by dividing it by the cubed standard deviation, called the **skew** of the distribution, a measure of the asymmetry.