4

The Characteristics of the Opportunity

Set Under Risk

In Chapter 1 we introduced the elements of a decision problem under certainty. The

same elements are present when we recognize the existence of risk; however, their

formulation becomes more complex. In the next two chapters we explore the nature

of the opportunity set under risk. Before we begin the analysis we present a brief

summary or roadmap of where we are going. The existence of risk means that the

investor can no longer associate a single number or payoff with investment in any

asset. The payoff must be described by a set of outcomes and each of their

associated probability of occurrence, called a frequency function or return

distribution. In this chapter we start by examining the two most frequently employed

attributes of such a distribution: a measure of central tendency, called the expected

return, and a measure of risk or dispersion around the mean, called the standard

deviation. Investors shouldn't and in fact don't hold single assets; they hold groups or

portfolios of assets. Thus a large part of this chapter is concerned with how one can

compute the expected return and risk of a portfolio of assets given the attributes of

the individual assets. One important aspect of this analysis is that the risk on a

portfolio is more complex than a simple average of the risk on individual assets. It

depends on whether the returns on individual assets tend to move together or

whether some assets give good returns when others give bad returns. As we show in

great detail there is a risk reduction from holding a portfolio of assets if assets do not

move in perfect unison.

We continue this discussion in Chapter 5. Initially we examine portfolios of only two

assets. We present a detailed geometric and algebraic analysis of the

characteristics of portfolios of two assets under different estimates of how they

covary together (how related their returns are to each other). We then extend this

analysis to the case of multiple assets. Finally, we arrive at the opportunity set facing

the investor in a world with risk. Let us begin by characterizing the nature of the

opportunity set open to the investor.

In the certainty case the investor's decision problem can be characterized by a

certain outcome. In the problem analyzed in Chapter 1, the 5% return on lending (or

the 5% cost of borrowing) was known with certainty. Under risk, the outcome of any

action is not known with certainty and outcomes are usually represented by a

frequency function. A frequency function is a listing of all possible outcomes along

with the probability of the occurrence of each. Table 4.1 shows such a function. This

investment has three possible returns. If event 1 occurs, the investor receives a

return of 12%; if event 2 occurs, 9% is received; and if event 3 occurs, 6% is

received. In our examples each of these events is assumed to be equally likely.

Table 4.1 shows us everything there is to know about the return possibilities.

Table 4.1

Return Probability Event

12 1/3 1

9 1/3 2

6 1/3 3

Usually we do not delineate all of the possibilities as we have in Table 4.1. The

possibilities for real assets are sufficiently numerous that developing a table like

Table 4.1 for each asset is too complex a task. Furthermore, even if the investor

decided to develop such tables, the inaccuracies introduced would be so large that

he or she would probably be better off just trying to represent the possible outcomes

in terms of some summary measures. In general, it takes at least two measures to

capture the relevant information about a frequency function: one to measure the

average value and one to measure dispersion around the average value.

DETERMINING THE AVERAGE OUTCOME

The concept of an average is standard in our culture. Pick up the newspaper and you

will often see figures on average income, batting averages, or average crime rates.

The concept of an average is intuitive. If someone earns $11,000 one year and

$9,000 in a second, we say his average income in the two years is $10,000. If three

children in a family are age 15, 10, and 5, then we say the average age is 10. In

Table 4.1 the average return was 9%. Statisticians usually use the term "expected

value" to refer to what is commonly called an average. In this book we use both

terms.

An expected value or average is easy to compute. If all outcomes are equally likely,

then to determine the average, one adds up the outcomes and divides by the

number of outcomes. Thus, for Table 4.1 the average is . A

second way to determine an average is to multiply each outcome by the probability

that it will occur. When the outcomes are not equally likely, this facilitates the

calculation. Applying this procedure to Table 4.1 yields .

It is useful to express this intuitive calculation in terms of formula. The symbol

should be read sum. Underneath the symbol we put the first value in the sum and

what is varying. On the top of the symbol we put the final value in the sum. We use

the symbol to denote the th possible outcome for the return on security .

Thus,

Using the summation notation just introduced and a bar over a variable to indicate

expected return, we have for the expected value of the M equally likely returns for

asset

If the outcomes are not equally likely and if is the probability of the th return on

the th asset, then expected return is1

We have up to this point used a bar over a symbol to indicate expected value. This is

the procedure we adopt throughout most of this book. However, occasionally, this

notation proves awkward. An alternative method of indicating expected value is to

put the symbol in front of the expression for which we wish to determine the

expected value. Thus should be read as the expected value of just as

is the expected value of .

Certain properties of expected value are extremely useful. These properties are:

The expected value of the sum of two returns is equal to the sum of the expected

value of each return, that is,

The expected value of a constant "C" times a return is the constant times the

expected return, that is,

1 This latter formula includes the formula for equally likely observations as a special case. If we have

M observations each equally likely, then the odds of anyone occurring are 1/M. Replacing the in

the second formula by 1/M yields the first formula.

These principles are illustrated in Table 4.2. For any event the return on asset 3 is

the sum of the return on assets 1 and 2. Thus, the expected value of the return on

asset 3 is the sum of the expected value of the return on assets 1 and 2. Likewise,

for any event the return on asset 3 is three times the return on asset 1.

Consequently, its expected value is three times as large as the expected value of

asset 1.

These two properties of expected values will be used repeatedly and are worth

remembering.

Table 4.2 Return on Various Assets

Event Probability Asset 1 Asset 2 Asset 3 A 1/3 14 28 42 B 1/3 10 20 30 C 1/3 6 12 18 Expected Return 10 20 30

A MEASURE OF DISPERSION

Not only is it necessary to have a measure of the average return, it is also useful to

have some measure of how much the outcomes differ from the average. The need

for this second characteristic can be illustrated by the old story of the mathematician

who believed an average by itself was an adequate description of a process and

drowned in a stream with an average depth of 2 inches.

Intuitively a sensible way to measure how much the outcomes differ from the

average is simply to examine this difference directly; that is, examine .

Having determined this for each outcome, one could obtain an overall measure by

taking the average of this difference. Although this is intuitively sensible, there is a

problem. Some of the differences will be positive and some negative and these will

tend to cancel out. The result of the canceling could be such that the average

difference for a highly variable return need be no larger than the average difference

for an asset with a highly stable return. In fact, it can be shown that the average value

of this difference must always be precisely zero. The reader is encouraged to verify

this with the example in Table 4.2. Thus, the sum of the differences from the mean

tells us nothing about dispersion.

Two solutions to this problem suggest themselves. First, we could take absolute

values of the difference between an outcome and its mean by ignoring minus signs

when determining the average difference. Second, since the square of any number

is positive, we could square all differences before determining the average. For ease

of computation, when portfolios are considered, the latter procedure is generally

followed. In addition, as we will see when we discuss utility functions, the average

squared deviations have some convenient properties. 2 The average squared

deviation is called the variance, the square root of the variance is called the standard

deviation. In Table 4.3 we present the possible returns from several hypothetical

assets as well as the variance of the return on each asset. The alternative returns on

any asset are assumed equally likely. Examining asset 1, we find the deviations of its

returns from its average return are (15-9), (9-9), and (3-9). The squared deviations

are 36, 0, and 36, and the average squared deviation or variance is (36+0+36)/3=24.

To be precise, the formula for the variance of the return on the th asset (which we

symbolize as ) when each return is equally likely is

Table 4.3 Returns on Various Investmentsa

Market Condition Returna Returna

Asset 4 Asset 1 Asset 2 Asset 3 Asset 5

Good 15 16 1 16 Rainfall Plentiful

Average Poor

16 Average 9 10 10 10 10 Poor 3 4 19 4 4 …………………………………………………………………………………………………………… Mean return 9 10 10 10 10 Variance 24 24 54 24 24 Standard deviation 4.9 4.9 7.35 4.90 4.9 a

The alternative returns on each asset are assumed equally likely and, thus, each has a probability of 1/3.

2 Many utility functions can be expressed either exactly or approximately in terms of the mean and

variance. Furthermore, regardless of the investor's utility function, if returns are normally distributed, the mean and variance contain all relevant information about the distribution. An elaboration of these points is contained in later chapters.

If the observations are not equally likely, then, as before, we multiply by the

probability with which they occur. The formula for the variance of the return on the

th asset becomes

Occasionally, we will find it convenient to employ an alternative measure of

dispersion called standard deviation. The standard deviation is just the square root

of the variance and is designated by . In the examples discussed in this chapter

we are assuming that the investor is estimating the possible outcomes and the

associated probabilities. Often initial estimates of the variance are obtained from

historical observations of the assets return. In this case, many authors and programs

used in calculators multiply the variance formula given above by . This

produces an estimate of the variance that is unbiased but has the disadvantage of

being inefficient (i.e., it produces a poorer estimate of the true variance). We leave it

to readers to choose which they prefer. In our examples in this book, we will not

make this correction.3

The variance tells us that asset 3 varies considerably more from its average than

asset 2. This is what we intuitively see by examining the returns shown in Table 4.3.

The expected value and variance or standard deviation are the usual summary

statistics utilized in describing a frequency distribution.

3 As stated, sometimes the formula is divided by and sometimes it is divided by

. The choice is a matter of taste. However, the reader may be curious why

some choose one or the other. The technical reason authors choose one or the other

is as follows.

Employing as the denominator gave the best estimate of the true value or the so-called maximum

likelihood estimate. Although it is the best estimate as gets large, it does not converge to the true

value (it is too small). Dividing by produces a that converges to the true value as gets

large (technically unbiased) but is not the best estimate for a finite . Some people consider one of these properties more important than the other, whereas some use one without consciously realizing why this might be preferred.

There are other measures of dispersion that could be used. We have already

mentioned one, the average absolute deviation. Other measures have been

suggested. One such measure considers only deviations below the mean. The

argument is that returns above the average return are desirable. The only returns

that disturb an investor are those below average. A measure of this is the average

(overall observations) of the squared deviations below the mean. For example, in

Table 4.3 for asset 1 the only return below the mean is 3. Since 3 is 6 below the

mean, the square of the difference is 36. The other two returns are not below the

mean so they have 0 deviations below the mean. The average of (0)+(0)+(36) is 12.

This measure is called the semivariance.

Semivariance measures downside risk relative to a benchmark given by expected

return. It is just one of a number of possible measures of downside risk. More

generally, we can consider returns relative to other benchmarks, including a risk-free

return or zero return. These generalized measures are, in aggregate, referred to as

lower partial moments. Yet another measure of downside risk is the so-called Value

at Risk measure, which is widely used by banks to measure their exposure to

adverse events and to measure the least expected loss (relative to zero, or relative

to wealth) that will be expected with a certain probability. For example, if 5% of the

outcomes are below - 30% and if the decision maker is concerned about how poor

the outcomes are 5% of the time, then - 30% is the value at risk.

Intuitively, these alternative measures of downside risk are reasonable and some

portfolio theory has been developed using them. However, they are difficult to use

when we move from single assets to portfolios. In cases where the distribution of

returns is symmetrical, the ordering of portfolios in mean variance space will be the

same as the ordering of portfolios in mean semivariance space or mean and any of

the other measures of downside risk discussed above. For well-diversified equity

portfolios, symmetrical distribution is a reasonable assumption so variance is an

appropriate measure of downside risk. Furthermore, since empirical evidence shows

most assets existing in the market have returns that are reasonably symmetrical,

semivariance is not needed. If returns on an asset are symmetrical, the

semivariance is proportional to the variance. Thus, in most of the portfolio literature

the variance, or equivalently the standard deviation, is used as a measure of

dispersion.

In most cases, instead of using the full frequency function such as that presented in

Table 4.1, we use the summary statistics mean and variance or equivalent mean and

standard deviation to characterize the distribution. Consider two assets. How might

we decide which we prefer? First, intuitively one would think that most investors

would prefer the one with the higher expected return if standard deviation was held

constant. Thus, in Table 4.3 most investors would prefer asset 2 to asset 1. Similarly,

if expected return were held constant, investors would prefer the one with the lower

variance. This is reasonable because the smaller the variance the more certain an

investor is that she will obtain the expected return and the fewer poor outcomes she

has to contend with.4 Thus in Table 4.3 the investor would prefer asset 2 to asset 3.

VARIANCE OF COMBINATIONS OF ASSETS

This simple analysis has taken us partway toward an understanding of the choice

between risky assets. However, the options open to an investor are not to simply

pick between assets 1,2,3,4, or 5 in Table 4.3 but also to consider combinations of

these five assets. For example, an investor could invest part of his money in each

asset. While this opportunity vastly increases the number of options open to the

investor and hence the complexity of the problem, it also provides the raison d'être of

portfolio theory. The risk of a combination of assets is very different from a simple

average of the risk of individual assets. Most dramatically, the variance of a

combination of two assets may be less than the variance of either of the assets

themselves. In Table 4.4 there is a combination of asset 2 and asset 3 that is less

risky than asset 2.

4 We will not formally develop the criteria for making a choice from among risky opportunities until the

next chapter. However, we feel we are not violating common sense by assuming at this time that investors prefer more to less and act as risk avoiders. More formal statements of the properties of investor choice will be taken up in the next chapter.

Let us examine this property. Assume an investor has $1 to invest. If he selects

asset 2 and the market is good, he will have at the end of the period $1+0.16 = $1.16.

If the market's performance is average, he will have $1.10, and if it is poor $1.04.

These outcomes are summarized in Table 4.4 along with the corresponding values

for the third asset. Consider an alternative. Suppose the investor invests $0.60 in

asset 2 and $0.40 in asset 3. If the condition of the market is good, the investor will

have $0.696 at the end of the period from asset 2 and $0.404 from asset 3, or $1.10.

If the market conditions are average, he will receive $0.66 from asset 2, $0.44 from

asset 3, or a total of $1.10. By now the reader might suspect that if the market

condition is poor the investor still receives $1.10, and this is, of course, the case. If

the market condition is poor the investor receives $0.624 from his investment in 2

and $0.476 from his investment is asset 3, or $1.10. These possibilities are

summarized in Table 4.4.

Table 4.4 Dollars at Period 2 Given Alternative Investments

Condition of Market

Combination of Asset 2 (60%) and Asset 3 (40%) Asset 2 Asset 3

Good $1.16 $1.01 $1.10 Average 1.10 1.10 1.10 Poor 1.04 1.19 1.10

This example dramatically illustrates how the risk on a portfolio of assets can differ

from the risk of the individual assets. The deviations on the combination of the assets

was zero because the assets had their highest and lowest returns under opposite

market conditions. This result is perfectly general and not confined to this example.

When two assets have their good and poor returns at opposite times, an investor can

always find some combination of these assets that yields the same return under all

market conditions. This example illustrates the importance of considering

combinations of assets rather than just the assets themselves and shows how the

distribution of outcomes on combinations of assets can be different than the

distributions on the individual assets.

The returns on asset 2 and asset 4 have been developed to illustrate another

possible situation. Asset 4 has three possible returns. Which return occurs depends

on rainfall. Assuming that the amount of rainfall that occurs is independent of the

condition of the market, then the returns on the assets 2 and 4 are independent of

one another. Therefore, if the rainfall is plentiful we can have good, average, or poor

security markets. Plentiful rainfall does not change the likelihood of any particular

market condition occurring. Consider an investor with $1.00 who invests $0.50 in

each asset. If rain is plentiful he receives $0.58 from his investment in asset 4, and

anyone of three equally likely outcomes from his investment in asset 2: $0.58 if the

market is good, $0.55 if it is average, and $0.52 if the market is poor. This gives him

a total of $1.16, $1.13, or $1.10. Similarly, if the rainfall is average, the value of his

investment in asset 2 and 4 is $1.13, $1.10, or $1.07, and if rainfall is poor, $1.10,

$1.07, or $1.04. Since we have assumed that each possible level of rainfall is equally

likely as is each possible condition of the market, there are nine equally likely

outcomes. Ordering then from highest to lowest we have $1.16, $1.13, $1.13, $1.10,

$1.10, $1.10, $1.07, $1.07, and $1.04. Compare this to an investment in asset 2 by

itself, the results of which are shown in Table 4.3. The mean is the same. However,

the dispersion around the mean is less. This can be seen by direct examination and

by noting that the probability of one of the extreme outcomes occurring ($1.16 or

$1.04) has dropped from

to

.

This example once again shows how the characteristics of the portfolio can be very

different than the characteristics of the assets that comprise the portfolio. The

example illustrates a general principle. When the returns on assets are independent

such as the returns on assets 2 and 4, a portfolio of such assets can have less

dispersion than either asset.

Consider still a third situation, one with a different outcome than the previous two.

Consider an investment in assets 2 and 5. Assume the investor invests $0.50 in

asset 2 and $0.50 in asset 5. The value of his investment at the end of the period is

$1.16, $1.10, or $1.04. These are the same values he would have obtained if he

invested the entire $1.00 in either asset 2 or 5 (see Table 4.3). Thus, in this situation

the characteristics of the portfolios were exactly the same as the characteristics of

the individual assets, and holding a portfolio rather than the individual assets did not

change the investor's risk.

We have analyzed three extreme situations. As extremes they dramatically

illustrated some general principles that carry over to less extreme situations. Our first

example showed that when assets have their good and bad outcomes at different

times (assets 2 and 3), then investment in these assets can radically reduce the

dispersion obtained by investing in one of the assets by itself. If the good outcomes

of an asset are not always associated with the bad outcomes of a second asset, but

the general tendency is in this direction, then the reduction in dispersion still occurs

but the dispersion will not drop all the way to zero as it did in our example. However,

it is still often true that appropriately selected combinations of the two assets will

have less risk than the least risky of the two assets.

Our second example illustrated the situation where the conditions leading to various

returns were different for the two assets. More formally, this is the area where returns

are independent. Once again, dispersion was reduced but not in as drastic a fashion.

Note that investment in asset 2 alone can result in a return of $1.04 and that this

result occurs

of the time. The same result can occur when we invested an equal

amount in asset 2 and asset 4. However, a combination of asset 2 and 4 has nine

possible outcomes, each equally likely, and $1.04 occurs only

of the time. With

independent returns, extreme observations can still occur. They just occur less

frequently. Just as the extreme values occur less frequently, outcomes closer to the

mean become more likely so that the frequency function has less dispersion.

Finally, our third example illustrated the situation where the assets being combined

had their outcomes affected in the same way by the same events. In this case, the

characteristics of the portfolio were identical to the characteristics of the individual

assets. In less extreme cases this is no longer true. Insofar as the good and bad

returns on assets tend to occur at the same time, but not always exactly at the same

time, the dispersion on the portfolio of assets is somewhat reduced relative to the

dispersion on the individual assets.

We have shown with some simple examples how the characteristics of the return on

portfolios of assets can differ from the characteristics of the returns on individual

assets. These were artificial examples designed to dramatically illustrate the point.

To reemphasize this point it is worthwhile examining portfolios of some real

securities over a historical period.

Table 4.5 Monthly Returns on IBM, Alcoa, and GM (in percent)

Month IBM Alcoa GM

1 12.05 14.09 25.20 13.07 19.65 18.63 2 15.27 2.96 2.86 9.12 2.91 9.07 3 -4.12 7.19 5.45 1.54 6.32 0.67 4 1.57 24.39 4.56 12.98 14.48 3.07 5 3.16 0.06 3.72 1.61 1.89 3.44 6 -2.79 6.52 0.29 1.87 3.41 -1.25 7 -8.97 -8.75 5.38 -8.86 -1.69 -1.80 8 -1.18 2.82 -2.97 0.82 -0.08 -2.08 9 1.07 -13.97 1.52 -6.45 -6.23 1.30

10 12.75 -8.06 10.75 2.35 1.35 11.75 11 7.48 -0.70 3.79 3.39 1.55 5.64 12 -.94 8.80 1.32 3.93 5.06 0.19

2.95 2.95 5.16 2.95 4.05 4.05

7.15 10.06 6.83 6.32 6.69 6.02 Correlation Coefficient: IBM and Alcoa = 0.05;

GM and Alcoa = 0.22; IBM and GM = 0.48

Three securities were selected: IBM, General Motors, and Alcoa Aluminum. The

monthly returns, average return, and standard deviation from investing in each

security is shown in Table 4.5. In addition, the return and risk of placing one half of

the available funds in each pair of securities is shown in the table. Finally, we have

plotted the returns from each possible pair of securities in Figure 4.1. In this figure we

have the return from each of two securities as well as the return from placing one half

of the available funds in each security. Both Figure 4.1 and Table 4.5 make it clear

how diversification across real securities can have a tremendous payoff for the

investor. For example, a portfolio composed of 50% IBM and 50% Alcoa had the

same return as each stock but less risk than either stock over the period studied.

Earlier we argued that an investor is better off working with summary characteristics

rather than full frequency functions. We used two summary measures: average

return and variance or standard deviation of return. We will now examine analytically

how the summary characteristics of a portfolio are related to those of individual

assets.

Figure 4.1 Securities and predetermined portfolios.

CHARACTERISTICS OF PORTFOLIOS IN GENERAL

The return on a portfolio of assets is simply a weighted average of the return on the

individual assets. The weight applied to each return is the fraction of the portfolio

invested in that asset. If is the th return on the portfolio and is the fraction of

the investor's funds invested in the th asset, and is the number of assets, then

The expected return is also a weighted average of the expected returns on the

individual assets. Taking the expected value of the expression just given for the

return on a portfolio yields

But we already know that the expected value of the sum of various returns is the sum

of the expected values. Therefore, we have

Finally, the expected value of a constant times a return is a constant times the

expected return, or

This is a perfectly general formula, and we use it throughout the book. To illustrate its

use, consider the investment in assets 2 and 3 discussed earlier in Table 4.3. We

determined that no matter what occurred, the investor would receive $1.10 on an

investment of $1.00. This is a return of .

Let us apply the formula for expected return. In the example discussed earlier, $0.60

was invested in asset 2 and $0.40 in asset 4; therefore, the fraction invested -in

asset 4 is 0.40/1.00. Furthermore, the expected return on asset 2 and asset 4 is

10%. Applying the formula for expected return on a portfolio yields

The second summary characteristic was the variance. The variance on a portfolio is

a little more difficult to determine than the expected return. We start out with a

two-asset example. The variance of a portfolio P, designated by ( , is simply the

expected value of the squared deviations of the return on the portfolio from the mean

return on the portfolio, or

. Substituting in this expression the

formulas for return on the portfolio and mean return yields in the two-security case

where stands for the expected value of security with respect to all possible

outcomes. Recall that

Applying this to the previous expression we have

Applying our two rules that the expected value of the sum of a series of returns is

equal to the sum of the expected value of each return, and that the expected value of

a constant times a return is equal to the constant times the expected return, we have

[( )( )] has a special name. It is called the covariance and will be

designated as .5 Substituting the symbol for [( )( )] yields

5 Note that when all joint outcomes are equally likely, the covariance can be expressed as

where M is the number of equally likely joint outcomes" Once again when estimates are based on a sample of data such as actual historical returns it is traditional to divide by T-1 rather than T where T

is the number of periods in the sample.

Notice what the covariance does. It is the expected value of the product of two

deviations: the deviations of the returns on security, 1 from its mean ( ) and

the deviations of security 2 from its mean ( ). In this sense it is very much

like the variance. However, it is the product of two different deviations. As such it can

be positive or negative. It will be large when the good outcomes for each stock occur

together and when the bad outcomes for each stock occur together. In this case, for

good outcomes the covariance will be the product of two large positive numbers,

which is positive. When the bad outcomes occur, the covariance will be the product

of two large negative numbers, which is positive. This will result in a large value for

the covariance and a large variance for the portfolio. In contrast, if good outcomes for

one asset are associated with bad outcomes of the other, the covariance is negative.

It is negative because a plus deviation for one asset is associated with a minus

deviation for the second and the product of a plus and a minus is negative. This was

what occurred when we examined a combination of assets 2 and 3.

The covariance is a measure of how returns on assets move together. Insofar as

they have positive and negative deviations at similar times, the covariance is a large

positive number. If they have the positive and negative deviations at dissimilar times,

then the covariance is negative. If the positive and negative deviations are unrelated,

it tends to be zero.

Table 4.6 Calculating Covariances

Condition Deviations Deviations Product of Deviations Deviations Product of of Market Security 1 Security 2 Deviations Security 1 Security 3 Deviations

Good (15-9) (16-10) 36 (15-9) (1-10) -54 Average (9-9) (10-10) 0 (9-9) (10-10) 0

Poor (3-9) (4-10) 36 (3-9) (19-10) -54 72 -108

For many purposes it is useful to standardize the covariance. Dividing the

covariance between two assets by the product of the standard deviation of each

asset produces a variable with the same properties as the covariance but with a

range of -1 to +1. The measure is called the correlation coefficient. Letting stand

for the correlation between securities and the correlation coefficient is defined

as

Dividing by the product of the standard deviations does not change the properties of

the covariance. It simply scales it to have values between -1 and +1. Let us apply

these formulas. First, however, it is necessary to calculate covariances. Table 4.6

shows the intermediate calculations necessary to determine the covariance between

securities 1 and 2 and securities 1 and 3. The sum of the deviations between

securities 1 and 2 is 72. Therefore, the covariance is 72/3=24 and the correlation

coefficient is √ √ . For assets 1 and 3 the sum of the deviations is -108. The

covariance is -108/3=-36 and the correlation coefficient is √ √ . Similar

calculations can be made for all other pairs of assets, and the results are contained

in Table 4.7.

Table 4.7 Covariance and Correlation Coefficients (in Brackets) Between Assets

1 2 3 4 5

1 24 -36 0 24 (+1) (-1) (0) (+1)

2 -36 0 24 (-1) (0) (+1)

3 0 -36 (0) (-1)

4 0 (0)

S

Earlier we examined the results obtained by an investor with $1.00 to spend who put

$0.60 in asset 2 and $0.40 in asset 3. Applying the expression for variance of the

portfolio we have

This was exactly the result we obtained when we looked at the combination of the full

distribution. The correlation coefficient between securities 2 and 3 is -1. This meant

that good and bad returns of assets 2 and 3 tended to occur at opposite times. When

this situation occurs, a portfolio can always be constructed with zero risk.

Our second example was an investment in securities 1 and 4. The variance of this

portfolio is

In this case where the correlation coefficient was zero, the risk of the portfolio was

less than the risk of either of the individual securities. Once again, this is a general

result. When the return patterns of two assets are independent so that the

correlation coefficient and covariance are zero, a portfolio can be found that has a

lower variance than either of the assets by themselves.

As an additional check on the accuracy of the formula just derived, we calculate the

variance directly. Earlier we saw there were nine possible returns when we

combined assets 2 and 4. They were $1.16, $1.13, $1.13, $1.10, $1.10, $1.10,

$1.07, $1.07, and $1.04. Since we started with an investment of $1.00, the returns

are easy to determine. The return is 16%, 13%, 13%, 10%, 10%, 10%,7%,7%, and

4%. By examination it is easy to see that the mean return is 10%. The deviations are

6, 3, 3, 0, 0, 0, -3, -3, -6. The squared deviations are 36, 9, 9, 0, 0, 0, 9, 9, 36, and the

average squared deviation or variance is 108/9=12. This agrees with the formula

developed earlier.

The final example analyzed previously was a portfolio of assets 1 and 5. In this case

the variance of the portfolio is

As we demonstrated earlier, when two securities have their good and bad outcomes

at the same time, the risk is not reduced by purchasing a portfolio of the two assets.

The formula for variance of a portfolio can be generalized to more than two assets.

Consider first a three-asset case. Substituting the expression for return on a portfolio

and expected return of a portfolio in the general formula for variance yields

Rearranging,

Squaring the right-hand side yields

Applying the properties of expected return discussed earlier yields

Utilizing for variance of asset and for the covariance between assets

and j, we have

This formula can be extended to any number of assets. Examining de expression for

the variance of a portfolio of three assets should indicate how. First note that the

variance of each asset is multiplied by the square of the proportion invested in it.

Thus, the first part of the expression for the variance of a portfolio is the sum of the

variances on the individual assets times the square of the proportion invested in

each, or

The second set of terms in the expression for the variance of a portfolio is covariance

terms. Note that the covariance between each pair of assets in the portfolio enters

the expression for the variance of a portfolio. With three assets the covariance

between 1 and 2, 1 and 3, and 2 and 3 entered. With four assets, covariance terms

between 1 and 2, 1 and 3, 1 and 4, 2 and 3, 2 and 4, and 3 and 4 would enter. Further

note that each covariance term is multiplied by two times the product of the

proportions invested in each asset. The following double summation captures the

covariance terms:

The reader concerned that a 2 does not appear in this expression can relax. The

covariance between securities 2 and 3 comes about both from j=2 and k=3 and from

j=3 and k=2. This is how the term "2 times the covariance between 2 and 3" comes

about. Furthermore, examining the expression for covariance shows that order does

not matter; thus . The symbol means should not have the same value

as . To reemphasize the meaning of the double summation, we examine the

three-security case. We have

Since the order does not matter in calculating covariance and thus we

have

Putting together the variance and covariance parts of the general expression for the

variance of a portfolio yields

This formula is worth examining further. First, consider the case where all assets are

independent and, therefore, the covariance between them is zero. This was the

situation we observed for assets 2 and 4 in our little example. In this case

and the formula for variance becomes

Furthermore, assume equal amounts are invested in each asset. With assets the

proportion invested in each asset is . Applying our formula yields

The term in the brackets is our expression for an average. Thus our formula reduces

to ⁄

, where represents the average variance of the stocks in the

portfolio. As N gets larger and larger, the variance of the portfolio gets smaller and

smaller. As N becomes extremely large, the variance of the portfolio approaches

zero. This is a general result. If we have enough independent assets, the variance of

a portfolio of these assets approaches zero.

In general, we are not so fortunate. In most markets the correlation coefficient and

the covariance between assets is positive. In these markets the risk on the portfolio

cannot be made to go to zero but can be much less than the variance of an individual

asset. The variance of a portfolio of assets is

Once again, consider equal investment in N assets. With equal investment, the

proportion invested in anyone asset is and the formula for the variance of a

portfolio becomes

Factoring out from the first summation and from the second yields

Both of the terms in the brackets are averages. That the first is an average should be

clear from the previous discussion. Likewise the second term in brackets is also an

average. There are values of and values of . There are

values of since cannot equal so that there is one less value of than . In

total there are covariance terms. Thus the second term is the summation

of covariances divided by the number of covariances and it is, therefore, an average.

Replacing the summations by averages, we have

This expression is a much more realistic representation of what occurs when we

invest in a portfolio of assets. The contribution to the portfolio variance of the

variance of the individual securities goes to zero as N gets very large. However, the

contribution of the covariance terms approaches the average covariance as N gets

large. The individual risk of securities can be diversified away, but the contribution to

the total risk caused by the covariance terms cannot be diversified away.

Table 4.8 Effect of Diversification

Number of Securities

Expected Portfolio Variance

1 46.619 2 26.839 4 16.948 6 13.651 8 12.003 10 11.014 12 10.354 14 9.883 16 9.530 18 9.256 20 9.036 25 8.640 30 8.376 35 8.188 40 8.047 45 7.937 50 7.849 75 7.585

100 7.453 125 7.374 150 7.321 175 7.284 200 7.255 250 7.216 300 7.190 350 7.171 400 7.157 450 7.146 500 7.137 600 7.124 700 7.114 800 7.107 900 7.102

1000 7.097 Infinity 7.058

Table 4.8 illustrates how this relationship looks when dealing with U.S. equities. The

average variance and average covariance of returns were calculated using monthly

data for all stocks listed on the New York Stock Exchange. The average variance

was 46.619. The average covariance was 7.058. As more and more securities are

added, the average variance on the portfolio declines until it approaches the average

covariance. Rearranging the previous equation clarifies this relationship even

further. Thus,

The first term is times the difference between the variance of return on

individual securities and the average covariance. The second term is the average

covariance. This relationship clarifies the effect of diversification on portfolio risk.

The minimum variance is obtained for very large portfolios and is equal to the

average covariance between all stocks in the population. As securities are added to

the portfolio, the effect of the difference between the average risk on a security and

the average covariance is reduced.

Table 4.9 Percentage of the Risk on an Individual Security that Can Be Eliminated by Holding a Random Portfolio of Stocks within Selected National Markets and among National Markets [13].

United States 73 U.K. 65.5 France 67.3 Germany 56.2 Italy 60.0 Belgium 80.0 Switzerland 56.0 Netherlands 76.1 International stocks 89.3

Figures 4.2 and 4.3 and Table 4.9 illustrate this same relationship for common

equities in a number of countries. In Figure 4.3 the vertical axis is the risk of the

portfolio as a percentage of the risk of an individual security for the U.K. The

horizontal axis is the number of securities in the portfolio. Figure 4.2 presents the

same relationship for the United States. Table 4.9 shows the percentage of risk that

can be eliminated by holding a widely diversified portfolio in each of several

countries as well as an internationally diversified portfolio. As can be seen, the

effectiveness of diversification in reducing the risk of a portfolio varies from country

to country. From the previous equation we know why. The average covariance

relative to the variance varies from country to country. Thus, in Switzerland and Italy

securities have relatively high covariance, indicating that stocks tend to move

together. On the other hand, the security markets in Belgium and the Netherlands

tend to have stocks with relatively low covariances. For these latter security markets,

much more of the risk of holding individual securities can be diversified away.

Diversification is especially useful in reducing the risk on a portfolio in these markets.

Figure 4. 2 The effect of number of securities on

risk of the portfolio in the United States [13].

Figure 4. 3 The effect of securities on risk in the

U.K. [13].

TWO CONCLUDING EXAMPLES

We will close this chapter and several chapters that follow with realistic applications

of the principles discussed in the chapter. These applications serve both to review

the concepts presented and to demonstrate their usefulness. The two examples that

follow are applications to the asset allocation decision. The first application analyzes

the allocation between stocks and bonds; the second analyzes the allocation

between domestic and foreign stocks.

Bond Stock Allocation

One of the major decisions facing an investor is the allocation of funds between

stocks and bonds. In order to make this allocation one needs to have estimates of

mean returns, standard deviations of return, and either correlation coefficients or

covariances. In order to estimate these variables it is useful to begin by looking at

historical data. Even in allocating among managed portfolios it is useful to start by

assuming that the stock and bond portfolio managers you are allocating between

have performance similar to that of broad representative indexes.

The principal index used to represent common stock portfolios is the Standard and

Poor's index. As described in Chapter 2, the Standard and Poor's index is a value

weighted index of 500 large stocks. Value weighting means that the weight each

stock represents of the portfolio is the market value of that stock (price times number

of shares) divided by the aggregate market value of all shares in the index. Thus

large stocks are weighted more heavily.

The version of the Standard and Poor's index reported in the newspapers is a capital

appreciation index and as such doesn't include the return from dividends. In order to

get total return one has to add dividend income. We will use the S&P index plus

dividends for examining the characteristics of stock returns.

The standard index used to represent bond performance is the Lehman Brothers

aggregate bond index. It is a value weighted index of almost all bonds in the market,

and includes both capital appreciation and interest income. Thus it is a total return

index.

Table 4.10 Historical Data on Bonds and Stocks

Standard Deviations

Date Bonds Stocks Correlation Coefficients

77-81 9.70% 14.54% 0.34 82-86 6.63% 14.66% 0.41 87-91 4.72% 15.40% 0.49 77-91 7.46% 14.87% 0.41

Table 4.11 Mean Return and Standard Deviation for Combinations of Stocks and Bonds

Proportion Proportion Standard Stocks Bonds Mean Return Deviation

1 0 12.5 14.90 0.9 0.1 11.85 13.63 0.8 0.2 11.2 12.38 0.7 0.3 10.55 11.15 0.6 0.4 9.9 9.95 0.5 0.5 9.25 8.80 0.4 0.6 8.6 7.70 0.3 0.7 7.95 6.69 0.2 0.8 7.3 5.82 0.1 0.9 6.65 5.16 0 1 6 4.80

In Table 4.10 we report the standard deviation and correlation coefficients calculated

using monthly data but expressed in annual terms. The data is for a 15-year period

and three 5-year periods. The month of the major market crash, October 1987, was

omitted in the belief that it was atypical. Examining Table 4.10 shows that the

standard deviation over each of the five-year periods is fairly constant for the S&P

index; thus, using the overall average is a reasonable estimate and we will use

14.9%. Because the standard deviation for bonds has declined as markets have

become less volatile, an estimate closer to the latest five-year results is probably

appropriate and we will use 4.8%. The correlation coefficient has risen over time.

Placing more emphasis on recent data, 0.45 is a reasonable estimate. At the time of

the revision of this book the average forecast by security analysts surveyed was a

return of 12.5% for the S&P index and 6% for the Lehman Brothers aggregate index.

Thus our inputs are

The means and standard deviation of return for combinations of stocks and bonds

varying from 100% in the S&P, which is and to 0% in the S&P are

presented in Table 4.11. Note that the expected return varies linearly from 12.5% to

6% as we decrease the amount in the S&P and increase it in bonds. Also the risk

decreases as we put more in the bonds, but not linearly. Figure 4.4 shows the

various choices diagrammatically.

Figure 4.4 Combinations of U.S. stocks and international stocks.

Domestic Foreign Allocation

As a second example consider the allocation decision between domestic and foreign

stocks. In Chapter 12 we will review the characteristics of foreign portfolios in some

detail. In that chapter we will show that on average foreign stock portfolios are

somewhat less risky than domestic. Thus, if we are assuming domestic portfolios

have a standard deviation of 14.9%, foreign portfolios can reasonably be assumed

to have a standard deviation of 14%. Furthermore, a reasonable correlation

coefficient is 0.33. This was the average correlation between a U.S. mutual fund and

a foreign mutual fund for the most recent five years (as shown in Table 12.11). At the

time of this revision analysts were more pessimistic about foreign markets than U.S.

markets and were estimating returns 2% lower. Thus our inputs are

Table 4.12 Mean Return and Standard Deviation for Combinations of Domestic and International Stocks

Proportion Proportion Standard S&P International Mean Return Deviation

1 0 12.5 14.90 0.9 0.1 12.3 13.93 0.8 0.2 12.1 13.11 0.7 0.3 11.9 12.46 0.6 0.4 11.7 12.01 0.5 0.5 11.5 11.79

0.45 0.55 11.4 11.76 0.4 0.6 11.3 11.80 0.3 0.7 11.1 12.04 0.2 0.8 10.9 12.50 0.1 0.9 10.7 13.17 0 1 10.5 14.00

The expected return and standard deviation of return for all combinations of the two

portfolios is shown in Table 4.12 and is plotted in Figure 4.5. Note that investment in

the two portfolios combined substantially reduced risk. This is a powerful

demonstration of the effect of diversification.

Figure 4.5 Combinations of U.S. stocks and international stocks.

CONCLUSION

In this chapter we have shown how the risk of a portfolio of assets can be very

different from the risk of the individual assets comprising the portfolio. This was true

when we selected assets with particular characteristics such as those shown in

Table 4.3. It was also true when we simply selected assets at random such as those

shown in Tables 4.8 and 4.9.

In the following chapter we examine the relationship between the risk and return on

individual assets in more detail. We then show how the characteristics on

combinations of securities can be used to define the opportunity set of investments

from which the investor must make a choice. Finally, we show how the properties of

these opportunities taken together with the knowledge that the investor prefers

return and seeks to avoid risk can be used to define a subset of the opportunity set

that will be of interest to investors.

QUESTIONS AND PROBLEMS

1. Assume that you are considering selecting assets from among the following four

candidates:

Asset 1 Asset 2

Market Market Condition Return Probability Condition Return Probability

Good 16 ¼ Good 4 ¼ Average 12 ½ Average 6 ½

Poor 8 ¼ Poor 8 ¼

Asset 3 Asset 4

Market Condition Return Probability Rainfall Return Probability

Good 20 ¼ Plentiful 16 1/3 Average 14 ½ Average 12 1/3

poor 8 ¼ Light 8 1/3

2. Assume that there is no relationship between the amount of rainfall and the

condition of the stock market.

A. Solve for the expected return and the standard deviation of return for each

separate investment.

B. Solve for the correlation coefficient and the covariance between each pair of

investments.

C. Solve for the expected return and variance of each of the portfolios shown

below.

Portions Invested in Each Asset Portfolio Asset 1 Asset 2 Asset 3 Asset 4

a 1/2 1/2 b 1/2 1/2 c 1/2 1/2 d 1/2 ½ e 1/2 ½ f 1/3 1/3 1/3 g 1/3 1/3 1/3 h 1/3 1/3 1/3 i 1/4 1/4 1/4 1/4

D. Plot the original assets and each portfolio from part C in expected return

standard deviation space.

Security A Security B Security C

time Price Dividend Price Dividend Price Dividend

1 57 6/8 333 106 6/8 2 59 7/8 368 108 2/8 3 59 3/8 0.725a 368 4/8 1.35 124 0.40 4 55 4/8 382 2/8 122 2/8 5 56 2/8 386 135 4/8 6 59 0.725 397 6/8% 1.35 141 6/8 0.42 7 60 2/8 392 165 6/8

a A dividend entry on the same line as a price indicates that the return between that time period and

the previous period consisted of a capital gain (or loss) and the receipt of the dividend.

a. Compute the rate of return for each company for each month.

b. Compute the average rate of return for each company.

c. Compute the standard deviation of the rate of return for each company.

d. Compute the correlation coefficient between all possible pairs of securities.

e. Compute the average return and standard deviation for the following

portfolios:

3. Assume that the average variance of return for an individual security is 50 and

that the average covariance is 10. What is the expected variance of an equally

weighted portfolio of 5, 10, 20, 50, and 100 securities?

4. In Problem 3 how many securities need to be held before the risk of a portfolio is

only 10% more than minimum?

5. For the Italy data and Belgium data of Table 4.9, what is the ratio of the difference

between the average variance minus average covariance and the average

covariance? If the average variance of a single security is 50, what is the

expected variance of a portfolio of 5, 20, and 100 securities?

6. For the data in Table 4.8, suppose an investor desires an expected variance less

than 8. What is the minimum number of securities for such a portfolio?

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