There are five basic steps to calculating value at risk. They are as follows:

The dollar value at risk equation is as follows (see note 7):

^{[1]}

Additionally, there is another presentation of the dollar VaR equation that utilizes slightly different inputs (see note 8):

^{[2]}

As mentioned previously, percentage VaR can also be calculated. The steps are similar to those for the dollar VaR, with a few differences:

• Obtain the periodic log returns and then calculate the average log return.

• Calculate the volatility of the log returns. The variance can be found using VARP in Microsoft Excel and then taking the square root to find the standard deviation. Alternatively, STDEVP in Microsoft Excel can be used to find the standard deviation directly. Note: Both the VARP and STDEVP functions utilize the population version of the formula.

♦ Calculate VaR using the following equation once the desired confidence level has been selected.

The percentage VaR equation is:

The equations just presented are for individual assets. A similar approach can be used for portfolios of multiple assets. Simply return to theories surrounding portfolios by finding the weighted average portfolio return, and obtain the volatility of the portfolio. Matrix applications are quite helpful in calculating portfolio volatility utilizing the concept of covariance. The portfolio VaR equation is:

The standard deviation of the portfolio can be computed using percentage weights or dollar values of each asset within the portfolio. The end results are identical with either approach.

Confidence levels are self-selected in the VaR model. Obviously, choosing higher confidence levels will yield more precise VaR estimates; however, lower confidence levels provide estimated VaRs that are broader and more informative. Exhibit 24.1 contains several different confidence levels as well as the associated normal distribution factor or z value.

Dollar VaR: One-Asset Example

The mark-to-market value of the investment is $85 million. The standard deviation (variability) of the asset is 20 percent. Using a holding period of seven business days and a confidence level of 99 percent, what is the value at risk for this investment?

Exhibit 24.1 Various Confidence Levels and Associated Alphas as Well as the z Value or Normal Distribution Factor Utilized in VaR Calculations

Confidence Level

Alpha (a)

z Value or Normal Distribution Factor

99.9%

0.10%

±3.09

99.5%

0.50%

±2.58

99.0%

1.00%

±2.33

97.5%

2.50%

±1.96

95.0%

5.00%

±1.65

90.0%

10.00%

±1.23

The interpretation: You are 99 percent confident that the loss will not exceed about $6.6 million.

Percentage VaR: One-Asset Example

Using the following information for stock XYZ, calculate VaR at the 95 percent level. Note: A short horizon is used for illustrative purposes only. A longer horizon should be used for actual calculations.

Date

Adjusted Closing Price

Periodic ROR

Dec. 9

$714.84

Dec. 10

$718.42

0.50%

Dec. 11

$699.20

-2.71%

Dec. 12

$699.35

0.02%

Dec. 13

$694.05

-0.76%

Dec. 14

$689.96

-0.59%

First, the periodic rate of return (ROR) is found by taking the natural logarithm of the daily return. Next, the average daily periodic ROR is obtained by finding the simple average. It is -0.7085 percent. Third, the volatility must be calculated. This can be obtained by taking the square root of the variance (VARP in MS Excel) or by calculating the standard deviation directly (STDEVP in MS Excel). The variance is 0.0001, and the standard deviation is 1.0974 percent. Using the confidence level provided, VaR can be calculated. The normal distribution factor for the 95 percent level is 1.65.

The interpretation: You are 95 percent confident that the worst loss will not exceed about 1.81 percent.

Exhibit 24.2 illustrates the distribution of VaR with 99 percent and 90 percent confidence levels as well. VaR at the 99 percent level is 2.56 percent, and it is 1.35 percent at the 90 percent confidence level.

[2] Here and throughout the remainder of this chapter, σ represents volatility (variability), which is proxied by standard deviation, and μ is the mean or expected mean return. Additionally, z is the zα value or normal distribution factor. Exp represents the exponential function.

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