## Info

In the first column, we write down the four actual returns. In the third column, we calculate the difference between the actual returns and the average by subtracting out

Ross et al.: Fundamentals I V. Risk and Return I 12. Some Lessons from I I © The McGraw-Hill of Corporate Finance, Sixth Capital Market History Companies, 2002

Edition, Alternate Edition

398 PART FIVE Risk and Return

4 percent. Finally, in the fourth column, we square the numbers in the third column to get the squared deviations from the average.

The variance can now be calculated by dividing .0270, the sum of the squared deviations, by the number of returns less 1. Let Var(R), or ct2 (read this as "sigma squared"), stand for the variance of the return:

The standard deviation is the square root of the variance. So, if SD(R), or ct, stands for the standard deviation of return:

The square root of the variance is used because the variance is measured in "squared" percentages and thus is hard to interpret. The standard deviation is an ordinary percentage, so the answer here could be written as 9.487 percent.

In the preceding table, notice that the sum of the deviations is equal to zero. This will always be the case, and it provides a good way to check your work. In general, if we have T historical returns, where T is some number, we can write the historical variance as:

Var(R) = T—y [(R - R)2 + • • • + R - R)2] [12.3]

This formula tells us to do just what we did above: take each of the T individual returns (R1, R2, . . . ) and subtract the average return, R; square the results, and add them all up; and finally, divide this total by the number of returns less 1 (T - 1). The standard deviation is always the square root of Var(R). Standard deviations are a widely used measure of volatility. Our nearby Work the Web box gives a real-world example.

Calculating the Variance and Standard Deviation

Suppose the Supertech Company and the Hyperdrive Company have experienced the following returns in the last four years:

Year

Supertech Return

Hyperdrive Return

0 0