# Variance-Covariance Method for Calculating Value at Risk

Instructor: James Walsh

M.B.A. Veteran Business and Economics teacher at a number of community colleges and in the for profit sector.

Investors get excited about the profit opportunities for investments, but they need to consider the risk of big losses too. We will consider the variance-covariance method of calculating value at risk, which quantifies the chances for big losses.

## Value at Risk

Carl is getting excited about a new company he wants to invest in. He believes robotics is the future, and QRS Corp has the right products to be a big winner. To keep things balanced, though, he also needs to consider the risk of big losses.

Value at Risk (VaR) tools give him a way to quantify that risk and get a truer picture of the investment. There are three primary ways to calculate value at risk. In this lesson, we will consider the variance-covariance method and watch Carl apply it to calculating value at risk for his investment ideas.

## Variance-Covariance Method

The variance-covariance method is an analytical way to calculate VaR. To use it you need different information than the other methods because of the assumptions it makes.

• The variance-covariance method assumes that a stock investment's returns will be normally distributed around the mean of a normal or bell-shaped probability distribution.
• Since returns are distributed in a normal or bell curve format, we need the standard deviation of the returns. These can be looked up or computed for most traded stocks.
• A complicating factor of this method is that stocks can have a tendency to move up and down together, usually caused by some external factor. That means we need the covariance of returns for all of the stocks in a portfolio against all of the other stocks.

## Value at Risk for One Stock

Computing the variance-covariance for a one-stock portfolio is straightforward. Carl will need the stock's price and its standard deviation, along with a confidence level. Most value at risk calculations use either a 95% or 99% confidence level. From a statistics table, he can look up the z value that corresponds to his desired confidence level. In the example below, the z value for a 95% confidence level is 1.645. Then the numbers go into the formula:

Value at Risk = Stock price or investment amount * standard deviation * z value

Carl wants to calculate VaR for an investment in QRS Co. The price for QRS Co. stock is \$100, its standard deviation for monthly returns is 10%, and we would like a 95% confidence level for the greatest monthly losses for this stock. The calculation is:

\$100 * 0.10 * 1.645 = \$16.45

This means that 95% of the time, Carl will not have a monthly loss greater than \$16.45 per share.

## Value at Risk for Two Stocks

The variance-covariance method works for a portfolio of two stocks, too, but it gets complicated. We need two additional things: the portfolio volatility and the covariances for the stocks against each other. Because there is only one covariance for a two-stock portfolio, we can calculate it.

To get the portfolio volatility we use a long formula:

Where:

• W1 = The weight for stock 1
• W2 = The weight for stock 2
• Sd1 = The standard deviation for stock 1
• Sd2 = The standard deviation for stock 2
• C1,2 = The covariance for the two stocks

Once we have that information, we can just plug the numbers into the same formula we used for one stock to get the portfolio VaR, only now we use our portfolio volatility measure instead of standard deviation.

Carl is going to add a second, safer stock to his holdings, XYZ Corp. He will invest a total of \$50,000 and wants to calculate VaR for monthly returns at 95% confidence for the two-stock portfolio.

Company Weight Standard deviation
QRS 60% 10%
XYZ 40% 5%

The covariance for the two stocks is 30%. The first thing Carl needs to do is calculate the portfolio volatility:

The VaR is \$50,000 * 1.645 * 0.0687 = \$5,650.

That means that 95% of the time, Carl's monthly losses will not exceed \$5,650. Now he can make a decision about his new asset mix.

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