Introduction to Corporate Finance

Ross et al.: Fundamentals

of Corporate Finance, Sixth

Edition, Alternate Edition

V. Risk and Return 12. Some Lessons from

Capital Market History

© The McGraw−Hill^427

Companies, 2002

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 devi-

ations, by the number of returns less 1. Let Var(R), or 2 (read this as “sigma squared”),

stand for the variance of the return:

Va r(R) ^2 .027/(4 
1) .009

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

for the standard deviation of return:

SR(R)  .09487

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 percent-

age, 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 Thistorical returns, where Tis some number, we can write the historical variance

as:

Va r(R)  [(R 1 
 )^2    (RT
 )^2 ] [12.3]

This formula tells us to do just what we did above: take each of the Tindividual returns

(R 1 , R 2 ,... ) and subtract the average return, ; square the results, and add them all up;

and finally, divide this total by the number of returns less 1 (T
1). The standard devi-

ation is always the square root of Var(R). Standard deviations are a widely used measure

of volatility. Our nearby Work the Webbox gives a real-world example.

R

R R

1

T
1

.009

398 PART FIVE Risk and Return

Calculating the Variance and Standard Deviation

Suppose the Supertech Company and the Hyperdrive Company have experienced the follow-

ing returns in the last four years:

What are the average returns? The variances? The standard deviations? Which investment

was more volatile?

To calculate the average returns, we add up the returns and divide by 4. The results are:

Supertech average return .70/4 .175

Hyperdrive average return .22/4 .055

To calculate the variance for Supertech, we can summarize the relevant calculations as

follows:

R

R

Year Supertech Return Hyperdrive Return

1999 
.20 .05

2000 .50 .09

2001 .30 
.12

2002 .10 .20

EXAMPLE 12.2