When evaluating risk exposure, many organizations have adopted the value-at-risk, or VaR, metric, which is a statistical risk management technique measuring the maximum loss that an investment portfolio is likely to face within a specified time frame with a certain degree of confidence.

VaR modeling determines the potential for loss in the entity being assessed and the probability of occurrence for the defined loss. One measures VaR by assessing the amount of potential loss, the probability of occurrence for the amount of loss, and the timeframe.

For example, a financial firm may determine an asset has a 3% one-month VaR of 2%, representing a 3% chance of the asset declining in value by 2% during the one-month time frame. The conversion of the 3% chance of occurrence to a daily ratio places the odds of a 2% loss at one day per month.

### Key Takeaways

- Value-at-risk (VaR) is a statistical method for judging the potential losses an asset, portfolio, or firm could incur over some period of time.
- The parametric approach to VaR uses mean-variance analysis to predict future outcomes based on past experience.
- The parametric VaR calculation is straightforward, but makes the assumption that possible outcomes are normally distributed about the mean.

## Parametric vs. Non-Parametric VaR

The nonparametric method does not require that the population being analyzed meet certain assumptions, or parameters. This gives analysts a great deal of flexibility and allows for qualitative or ordinal variables to be included. Although nonparametric statistics have the advantage of having to meet few assumptions, they are less powerful than parametric statistics. This means that they may not show a relationship between two variables when in fact one exists. As a result, most risk managers prefer a more quantitative approach.

The parametric method, also known as the variance-covariance method, is a risk management technique for calculating the VaR of a portfolio of assets that first identifies the mean, or expected value, and standard deviation of an investment portfolio. The parametric method looks at the price movements of investments over a look-back period and uses probability theory to compute a portfolio's maximum loss. The variance-covariance method for the value at risk calculates the standard deviation of price movements of an investment or security. Assuming stock price returns and volatility follow a normal distribution, the maximum loss within the specified confidence level is calculated.

## Example with One Security

Consider a portfolio that includes only one security, stock ABC. Suppose $500,000 is invested in stock ABC. The standard deviation over 252 days, or one trading year, of stock ABC, is 7%. Following the normal distribution, the one-sided 95% confidence level has a z-score of 1.645.

The value at risk in this portfolio is

$57,575 = ($500000*1.645*.07).

Therefore, with 95% confidence, the maximum loss will not exceed $57,575 in a given trading year.

## Example with Two Securities

The value at risk of a portfolio with two securities can be determined by first calculating the portfolio's volatility. Multiply the square of the first asset's weight by the square of the first asset's standard deviation and add it to the square of the second asset's weight multiplied by the square of the second asset's standard deviation. Add that value to two multiplied by the weights of the first and second assets, the correlation coefficient between the two assets, asset one's standard deviation, and asset two's standard deviation. Then multiply the square root of that value by the z-score and the portfolio value.

For example, suppose a risk manager wants to calculate the value at risk using the parametric method for a one-day time horizon. The weight of the first asset is 40%, and the weight of the second asset is 60%. The standard deviation is 4% for the first and 7% for the second asset. The correlation coefficient between the two is 25%. The z-score is -1.645. The portfolio value is $50 million.

The parametric value at risk over a one-day period, with a 95% confidence level, is:

$3.99 million = ($50,000,000*-1.645)*√(0.4^2*0.04^2)+(0.6^2*0.07^2)+[2(0.4*0.6*0.25*0.04*0.07*)]

## The Bottom Line

If a portfolio has multiple assets, its volatility is calculated using a matrix. A variance-covariance matrix is computed for all the assets. The vector of the weights of the assets in the portfolio is multiplied by the transpose of the vector of the weights of the assets multiplied by the covariance matrix of all of the assets.

In practice, the calculations for VaR are typically done through financial models. Modeling functions will vary depending on whether the VaR is being calculated for one security, two securities, or a portfolio with three or more securities.