What Does Value at Risk (VaR) Say About the "Tail" of the Loss Distribution?

The value at risk (VaR) is a statistical measure that assesses, with a degree of confidence, the financial risk associated with a portfolio or a firm over a specified period. The VaR measures the probability that a portfolio will not exceed or break a threshold loss value. The VaR is based solely on potential losses in an investment and does so by determining the loss distribution. However, the tail loss of the distribution is not thoroughly assessed in the typical VaR model.

The VaR assesses the worst-case scenario of a firm or an investment portfolio. The model uses a confidence level, such as 95% or 99%, a time period and a loss amount. For example, an investor determines that the one-day, 1% VaR of his investment portfolio is $10,000. The VaR determines that there is a 1% probability that his portfolio will have a loss greater than $10,000 over a one-day period. He has 99% confidence that his worst daily loss will not exceed $10,000.

The VaR can be calculated by using historical returns of a portfolio or firm and plotting the distribution of the profit and losses. The loss distribution negates the profit and loss distribution. Therefore, under this convention, the profits will be negative values, and the losses will be positive.

For example, a firm calculates its daily returns for all of its investment portfolios over a one-year period. The VaR describes the right tail of the loss distribution. Suppose the alpha level selected is 0.05. Then the corresponding confidence level is 95%. The 95% confidence interval of the daily returns range from 5% to 10%. Therefore, with 95% confidence, the firm concludes that the expected worst daily loss will not exceed 5%. However, this is a probabilistic measure and is not certain because losses can be much greater depending on the heaviness, or fatness, of the tail of the loss distribution.

Value at risk does not assess the kurtosis of the loss distribution. In the VaR context, a high kurtosis indicates fat tails of the loss distribution, where losses greater than the maximum expected loss may occur. Extensions of VaR can be used to assess the limitations of this measure, such as the conditional VaR, also known as tail VaR. The conditional VaR is the expected loss conditioned on the loss exceeding the VaR of the loss distribution. The conditional VaR thoroughly examines the tail end of a loss distribution and determines the mean of the tail of the loss distribution that exceeds the VaR.

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