PORTFOLIO ANALYSIS

Section 1

Mean Variance

Portfolio Theory

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 road map 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 probabilities of occurrence, called a fre- quency 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 stan- dard deviation. Investors should not and, in fact, do not 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 indi- vidual 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 port- folios 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

42

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.

Usually we do not delineate all of the possibilities, as we have in Table 4.1. The possi- bilities 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 bet- ter 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 fre- quency 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 out- comes. Thus, for Table 4.1, the average is (12 ⫹9 ⫹6)/3 ⫽9. A second way to determine an average is to multiply each outcome by the probability that it will occur. When the out- comes are not equally likely, this facilitates the calculation. Applying this procedure to Table 4.1 yields ᎏ1

3ᎏ(12) ⫹^ᎏ13ᎏ(9) ⫹^ᎏ13ᎏ(6) ⫽9.

It is useful to express this intuitive calculation in terms of a formula. The symbol 兺 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 R_ijto denote the jth possible outcome for the return on security i. Thus

R

R R R

ij

j= i i i

∑

¹ = + + = + +

3

1 2 3

3 3

12 9 6 3 Table 4.1 Data on Three Hypothetical Events

Return Probability Event

12 ᎏ1

3ᎏ 1

9 ᎏ1

3ᎏ 2

6 ᎏ1

3ᎏ 3

CHAPTER 4 THE CHARACTERISTICS OF THE OPPORTUNITY SET UNDER RISK 43

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 i:

If the outcomes are not equally likely and if Pijis the probability of the jth return on the ith 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 E in front of the expression for which we wish to determine the expected value. Thus E(Ri) should be read as the expected value of Rij, just as R–

iis the expected value of Rij.

Certain properties of expected value are extremely useful:

1. The expected value of the sum of two returns is equal to the sum of the expected value of each return, that is,

2. The expected value of a constant C times a return is the constant times the expected return, that is,

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 3 times the return on asset 1. Consequently, its expected value is 3 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 ᎏ1

3ᎏ 14 28 42

B ᎏ1

3ᎏ 10 20 30

C ᎏ1

3ᎏ 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 but it is also useful to have some measure of how much the outcomes differ from the average. The need for this second

E C R

[ ( )

1j

]

⁼CR1

E R

(

1j+R2j

)

⁼R1⁺R2

Ri P Rij ij j

M

=

∑

= 1

R R

i M

ij j

M

=

∑

= 1

1This latter formula includes the formula for equally likely observations as a special case. If we have M obser- vations each equally likely, then the odds of any one occurring are 1/M. Replacing the P_ijin the second formula with 1/M yields the first formula.

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 R_ij⫺R–

i. 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 can- celing 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, because the square of any number is positive, we could square all differences before determining the average. For ease of computation, when port- folios 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.²The average squared deviation is called the variance; the square root of the vari- ance is called the standard deviation. In Table 4.3 we present the possible returns from sev- eral 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 ith asset (which we sym- bolize as ␴²i) when each return is equally likely is

σi

ij i

j

M R R

M

2

2

1

=

(

−

)

∑

=

CHAPTER 4 THE CHARACTERISTICS OF THE OPPORTUNITY SET UNDER RISK 45

2Many 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 vari- ance contain all relevant informaton about the distribution. An elaboration of these points is contained in later chapters.

Table 4.3 Returns on Various Investments^a

Market Return^a Return^a

Condition Asset 1 Asset 2 Asset 3 Asset 5 Rainfall Asset 4

Good 15 16 1 16 Plentiful 16

Average 9 10 10 10 Average 10

Poor 3 4 19 4 Poor 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

aThe alternative returns on each asset are assumed equally likely, and thus each has a probability of ᎏ1 3ᎏ.

If the observations are not equally likely, then, as before, we multiply by the probabil- ity with which they occur. The formula for the variance of the return on the ith 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 ␴i.

In the examples discussed in this chapter we are assuming that the investor is estimat- ing the possible outcomes and the associated probabilities. Often initial estimates of the variance are obtained from historical observations of the asset’s return. In this case, many authors and programs used in calculators multiply the variance formula given earlier by M/(M⫺1). This produces an estimate of the variance that is unbiased but has the disad- vantage 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.

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 meas- ure considers only deviations below the mean. The argument is that returns above the aver- age 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. Because 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 deviation 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 con- cerned about how poor the outcomes are 5% of the time, then ⫺30% is the value at risk.

σ_i ij ij i j

M

P R R

2 2

1

= ^⎡_⎣⎢

(

−

)

^⎤_⎦⎥

∑

=

3As stated, sometimes the formula is divided by M, and sometimes it is divided by M⫺1. The choice is a mat- ter 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 M as the denominator gave the best estimate of the true value or the so-called maximum likeli- hood estimate. Although it is the best estimate as M gets large, it does not converge to the true value (it is too small). Dividing by M⫺1 produces a S²_ithat converges to the true value as M gets large (technically unbiased) but is not the best estimate for a finite M. Some people consider one of these properties more important than the other, whereas some use one without consciously realizing why this might be preferred.

Intuitively, these alternative measures of downside risk are reasonable, and some portfo- lio 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 sym- metrical, the ordering of portfolios in mean variance space will be the same as the order- ing of portfolios in mean semivariance space or mean and any of the other measures of downside risk discussed earlier. For well-diversified equity portfolios, symmetrical distri- bution is a reasonable assumption, so variance is an appropriate measure of downside risk.

Furthermore, because 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 meas- ure 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 devi- ation 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.⁴Thus, 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 simply to pick between assets 1, 2, 3, 4, or 5 in Table 4.3 but also to consider combinations of these five assets. For exam- ple, an investor could invest part of her money in each asset. While this opportunity vastly increases the number of options open to the investor and hence the complexity of the prob- lem, 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 dra- matically, 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.

Table 4.4 Dollars at Period 2 Given Alternative Investments

Combination of

Condition of Asset 2 (60%)

Market Asset 2 Asset 3 and Asset 3 (40%)

Good $1.16 $1.01 $1.10

Average 1.10 1.10 1.10

Poor 1.04 1.19 1.10

CHAPTER 4 THE CHARACTERISTICS OF THE OPPORTUNITY SET UNDER RISK 47

4We 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 out- comes 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 aver- age, 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 asset 2 and $0.476 from his investment in asset 3, or $1.10. These possibilities are summarized in Table 4.4.

This example dramatically illustrates how the risk of a portfolio of assets can differ from the risk of the individual assets. The deviations on the combination of the assets were 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 combina- tion 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 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 any one 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 assets 2 and 4 is $1.13, $1.10, or

$1.07, and if rainfall is poor, $1.10, $1.07, or $1.04. Because we have assumed that each possible level of rainfall is equally likely as each possible condition of the market, there are nine equally likely outcomes. Ordering them 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 ᎏ1 3ᎏto ᎏ1

9ᎏ.

This example once again shows how the characteristics of the portfolio can be very different than the characteristics of the assets that compose 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 had he invested the entire