4.4Variances and Standard Deviations

Example 4.6 illustrates that it is possible to generate arbitrarily large expected returns

by buying the investment with the largest expected return and selling short the invest-

⁴If investors arerisk neutral, which means they do not care about risk, they would select portfolios

solely on the basis of expected return. However, most investors are concerned with risk as well as

expected returns.

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105

ment with the lowest expected return.⁴The force that prevents investors from doing

this is risk. When leveraging a portfolio to achieve higher expected returns, the investor

also increases the risk of the portfolio.

Return Variances

The concern investors have for losses is known as risk aversion. Afundamental

research area of finance theory is how to define and quantify risk. Mean-variance analy-

sis defines the risk of a portfolio as the variance of its return.

To compute return variances, the examples below assume that there is a finite num-

ber of possible future return outcomes, that we know what these outcomes are, and that

we know what the probability of each outcome is. Given these possible return outcomes:

1.Compute demeaned returns. (Demeaned returnssimply subtract the mean

return from each of the possible return outcomes.)

2.Square the demeaned returns.

3.Take the probability-weighted average of these squared numbers.

The varianceof a return is the expected value of the squared demeaned return out-

comes, that is

var(˜) E [(r˜)²

rr]

where ˜,the return of the investment, is a random variable, and r,the expected return of

r

the investment, is a statistic that helps summarize the distribution of the random variable.

Example 4.7:Computing Variances

Compute the variance of the return of a $400,000 investment in the hypothetical company

SINTEL.Assume that over the next period SINTEL earns 20 percent 8/10 of the time, loses

10 percent 1/10 of the time, and loses 40 percent 1/10 of the time.

Answer:The mean return is

11% .8(20%).1(10%).1(40%)

Subtract the mean return from each of the return outcomes to obtain the demeaned returns.

11%	9%	.09
20%

10%11% 21%

.21

40%11% 51% .51

The variance is the probability-weighted average of the square of these three numbers

var .8(.09)²²²

.1(.21).1(.51) .0369

Auseful property of variances is that the variance of a constant times a return is

the square of that constant times the variance of the return, that is

var (x˜) x2˜)	(4.5)
rvar(r

Estimating Variances: Statistical Issues

The variances in Example 4.7 are based on return distributions computed with a

“forward-looking approach.” With this approach, the analyst estimates the variances

and covariances by specifying the returns in different scenarios or states of the econ-

omy that are likely to occur in the future. We have simplified this process by giving

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Strategy, Second Edition

106Part IIValuing Financial Assets

you the outcomes and probabilities. In practice, however, this involves a lot of guess-

ing, and is quite difficult to implement. More commonly, the variance is computed by

averaging squared historical demeaned returns, as outlined in Example 4.8.

Example 4.8:Estimating Variances with Historical Data

Estimate the variance of the return of the S&P 500.Recall from Example 4.5 that the annual

returns of the S&P 500 from 1995 to 1999 were 37.43 percent, 23.07 percent, 33.36 per-

cent, 28.58 percent, and 21.04 percent, respectively, and that the average of these five num-

bers was 28.70 percent.

Answer:Subtracting the average return of 28.70 percent from each of these five returns

results in demeaned returns of 8.73 percent for 1995, 5.63 percent for 1996, 4.66 percent

for 1997, 0.12 percent for 1998, and 7.66 percent for 1999.Thus, the average squared

demeaned return is:

.0087²(.0056)²²²²

(.0466)(.0012)(.0766)

.0038

5

It is important to stress that the number computed in Example 4.8 is only an esti-

mate of the true variance. For instance, a different variance estimate for the return of

the S&P500, specifically .0416, would result from the use of data between 1926 and

1995. Recent years have been good for holding stocks, given their exceptionally high

average return and low variance of their annual return.

Several fine points are worth mentioning when estimating variances with histori-

cal data on returns. First, in contrast with means, one often obtains a more precise esti-

mate of the variance with more frequent data. Hence, when computing variance esti-

mates, weekly returns would be preferred to monthly returns, monthly returns to annual

returns, and so forth.⁵Daily data often present problems, however, because of how trad-

ing affects observed prices.⁶

If the data are sufficiently frequent, a year of weekly data

can provide a fairly accurate estimate of the true variance. By contrast, the mean return

of a stock is generally estimated imprecisely, even with years of data.

Some statisticians recommend computing estimated variances by dividing the

summed squared demeaned returns by one less than the number of observations instead

of by the number of observations.⁷In Example 4.8, this would result in a variance esti-

mate equal to 5/4 times the existing estimate, or .0047.

In some instances—for example, when valuing real estate in Eastern Europe—his-

torical data may not be available for a variance estimation. In this case, the forward-

looking approach is the only alternative for variance computation. Chapter 11 discusses

how to do this in a bit more detail.

Standard Deviation

Squaring returns (or demeaned returns) to compute the variance often leads to confu-

sion when returns are expressed as percentages. It would be convenient to have a mea-

sure of average dispersion from the mean that is expressed in the same units as the

⁵To obtain an annualized variance estimate from weekly data, multiply the weekly variance estimate

by 52; to obtain it from monthly data, multiply the monthly estimate by 12, and so on.

⁶This stems from dealers buying at the bid and selling at the ask, as described in Chapter 3, which

tends to exacerbate variance estimates.

⁷This altered variance estimate is unbiased, that is, tending to be neither higher nor lower than the

true variance. The variance computations here do not make this adjustment.

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Markets and Corporate			Companies, 2002
Strategy, Second Edition

Chapter 4

Portfolio Tools

107

variable itself. One such measure is the standard deviation. The standard deviation

(denoted or (r)), is the square root of the variance. One typically reports the stan-

˜

dard deviation of a return in percent per year whenever returns are reported in units of

percent per year. This makes it easier to think about how dispersed a distribution of

returns really is. For example, a stock return with an expected annual return of 12 per-

cent and an annualized standard deviation of 10 percent has a typical deviation from

the mean of about 10 percent. Thus, observing annual returns as low as 2 percent or

as high as 22 percent would not be unusual.

The standard deviation possesses the following useful property: The standard devi-

ation of a constant times a return is the constant times the standard deviation of the

return; that is

(x˜) x(˜)	(4.6)
rr

This result follows from equation (4.5) and the fact that the square root of the product

of two numbers is the product of the square root of each of the numbers.