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# Confidence Level and Confidence Interval in Value at Risk (VaR)

The value at risk (VaR) uses both the confidence level and confidence interval. A risk manager uses the VaR to monitor and control the risk levels in a company's investment portfolio. VaR is a statistical metric measuring the amount of the maximum potential loss within a specified period with a degree of confidence. The VaR indicates that a company's losses will not exceed a certain amount of dollars over a specified period with a certain percentage of confidence. While confidence level and confidence interval are interconnected and can be part of a risk assessment, they are not exactly alike.

### Key Takeaways

• Value at risk (VaR) is a statistic that quantifies the amount of potential loss that could occur within an investment, a portfolio of investments, or a firm over a specified time period.
• A VaR assessment helps financial institutions identify high-risk investments and determine the cash reserves they will need to cover potential portfolio losses.
• The VaR uses both the confidence interval and confidence level to build a risk assessment model.
• A confidence interval is two set values that probability indicates a parameter will fall between.
• The confidence level reflects the level of probability (expressed as a percentage) that the confidence interval would contain the population parameter.

## Value at Risk (VaR)

VaR is a useful statistic because it helps financial institutions determine the level of cash reserves they need to cover potential portfolio losses. Risk managers traditionally use volatility as a statistical measurement for risk. However, investment and commercial banks frequently use VaR to determine cumulative risks from highly correlated positions held by different departments within the institution.

The VaR analysis helps the institution estimate with a high confidence level the maximum amount or percentage that could potentially be lost on an investment over a given time. With VaR modeling, managers can identify investments that have higher-than-acceptable risks, allowing them to reduce or exit positions if needed.

One criticism of VaR and other risk assessment metrics is their potential for understating risks and their inability to account for black swan events.

## Confidence Level

The confidence level determines how sure a risk manager can be when they are calculating the VaR. The confidence level is expressed as a percentage, and it indicates how often the VaR falls within the confidence interval. If a risk manager has a 95% confidence level, it indicates he can be 95% certain that the VaR will fall within the confidence interval.

For example, assume that a risk manager determines the 5% one-day VaR to be \$1 million. This means that he has a 95% confidence level that the worst daily loss will not exceed \$1 million.

Although a risk manager can choose any number of probabilities, it is most common to use a 95% or 99% confidence level.

## Confidence Interval

Conversely, the confidence interval is a statistical measure that produces an estimated range of values that is likely to include an unknown population parameter. For example, suppose a risk manager is measuring the confidence interval of an investment portfolio. He finds it to be \$2 million and \$10 million at the endpoints.

Bridging the confidence interval and confidence level, the risk manager can calculate the value at risk. The confidence level of the VaR estimate is the quantile the risk manager uses to calculate the VaR. This, however, should not be confused with the confidence interval. The confidence interval is an interval that has a probability of including the VaR estimate.

## VaR Portfolio Comparison

Suppose a risk manager is evaluating the VaR of two different investment portfolios. The first portfolio has a one-day value at risk of \$11 million and a confidence interval of \$6 million to \$17 million, whereas the second portfolio has a one-day VaR of \$5 million with a confidence interval of \$3 million to \$7 million. The first portfolio has a 95% confidence level, and the second portfolio has a 99% confidence level. The first portfolio is riskier and has a higher level of uncertainty because the confidence interval and the VaR are much larger.

The confidence interval of the first portfolio includes the VaR of \$11 million at 95% of the time. On the other hand, the confidence interval for the second portfolio includes the VaR of \$5 million at 99% of the time.

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