Post on 08-Nov-2014
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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