A B C D E F G H I J

1 Stock Fund Bond Fund

2 Deviation Deviation

3 Rate from Column B Rate from Column B

4 of Expected Squared x of Expected Squared x

5 Scenario Prob. Return Return Deviation Column E Return Return Deviation Column I

6 Recession 0.3 -11 -21 441 132.3 16 10 100 30

7 Normal 0.4 13 3 9 3.6 6 0 0 0

8 Boom 0.3 27 17 289 86.7 -4 -10 100 30

9 Variance = SUM 222.6 Sum: 60

10 Standard deviation = SQRT(Variance) 14.92 Sum: 7.75

S P R E A D S H E E T 6.3

Performance of the portfolio of stock and bond funds

A B C D E F G

1 Portfolio of 60% in stocks and 40% in bonds

2 Rate Column B Deviation from Column B

3 of x Expected Squared x

4 Scenario Probability Return Column C Return Deviation Column F

5 Recession 0.3 -0.2 -0.06 -8.60 73.96 22.188

6 Normal 0.4 10.2 4.08 1.80 3.24 1.296

7 Boom 0.3 14.6 4.38 6.20 38.44 11.532

8 Expected return: 8.40 Variance: 35.016

9 Standard deviation: 5.92

The low risk of the portfolio is due to the inverse relationship between the performance of

the two funds. In a recession, stocks fare poorly, but this is offset by the good performance of

the bond fund. Conversely, in a boom scenario, bonds fall, but stocks do well. Therefore, the

portfolio of the two risky assets is less risky than either asset individually. Portfolio risk is re-

duced most when the returns of the two assets most reliably offset each other.

The natural question investors should ask, therefore, is how one can measure the tendency

of the returns on two assets to vary either in tandem or in opposition to each other. The statis-

tics that provide this measure are the covariance and the correlation coefficient.

The covariance is calculated in a manner similar to the variance. Instead of measuring

the typical difference of an asset return from its expected value, however, we wish to measure

the extent to which the variation in the returns on the two assets tend to reinforce or offset

each other.

We start in Spreadsheet 6.4 with the deviation of the return on each fund from its expected

or mean value. For each scenario, we multiply the deviation of the stock fund return from its

mean by the deviation of the bond fund return from its mean. The product will be positive if

both asset returns exceed their respective means in that scenario or if both fall short of their

respective means. The product will be negative if one asset exceeds its mean return, while the

other falls short of its mean return. For example, Spreadsheet 6.4 shows that the stock fund

return in the recession falls short of its expected value by 21%, while the bond fund return

exceeds its mean by 10%. Therefore, the product of the two deviations in the recession is

21 10 210, as reported in column E. The product of deviations is negative if one as-

set performs well when the other is performing poorly. It is positive if both assets perform well

or poorly in the same scenarios.

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6 Efficient Diversification

S P R E A D S H E E T 6.4

Covariance between the returns of the stock and bond funds

A B C D E F

Deviation from Mean Return Covariance

1

Scenario Probability Stock Fund Bond Fund Product of Dev Col. B Col. E

2

Recession 0.3 21 10 210 63

3

Normal 0.4 3 0 0 0

4

Boom 0.3 17 10 170 51

5

Covariance: SUM: 114

6

Correlation coefficient = Covariance/(StdDev(stocks)*StdDev(bonds)): 0.99

7

If we compute the probability-weighted average of the products across all scenarios, we ob-

tain a measure of the average tendency of the asset returns to vary in tandem. Since this is a

measure of the extent to which the returns tend to vary with each other, that is, to co-vary, it is

called the covariance. The covariance of the stock and bond funds is computed in the next-to-

last line of Spreadsheet 6.4. The negative value for the covariance indicates that the two assets

vary inversely, that is, when one asset performs well, the other tends to perform poorly.

Unfortunately, it is difficult to interpret the magnitude of the covariance. For instance, does

the covariance of 114 indicate that the inverse relationship between the returns on stock and

bond funds is strong or weak? It™s hard to say. An easier statistic to interpret is the correlation

coefficient, which is simply the covariance divided by the product of the standard deviations

of the returns on each fund. We denote the correlation coefficient by the Greek letter rho, .

Covariance 114

Correlation coefficient .99

14.92 7.75

stock bond

Correlations can range from values of 1 to 1. Values of 1 indicate perfect negative cor-

relation, that is, the strongest possible tendency for two returns to vary inversely. Values of 1

indicate perfect positive correlation. Correlations of zero indicate that the returns on the two

assets are unrelated to each other. The correlation coefficient of 0.99 confirms the over-

whelming tendency of the returns on the stock and bond funds to vary inversely in this sce-

nario analysis.

We are now in a position to derive the risk and return features of portfolios of risky assets.

<

1. Suppose the rates of return of the bond portfolio in the three scenarios of Spread- Concept

sheet 6.4 are 10% in a recession, 7% in a normal period, and 2% in a boom. The

CHECK

stock returns in the three scenarios are 12% (recession), 10% (normal), and 28%

(boom). What are the covariance and correlation coefficient between the rates of

return on the two portfolios?

Using Historical Data

We™ve seen that portfolio risk and return depend on the means and variances of the component

securities, as well as on the covariance between their returns. One way to obtain these inputs

is a scenario analysis as in Spreadsheets 6.1“6.4. As we noted in Chapter 5, however, a com-

mon alternative approach to produce these inputs is to make use of historical data.

In this approach, we use realized returns to estimate mean returns and volatility as well as

the tendency for security returns to co-vary. The estimate of the mean return for each security

is its average value in the sample period; the estimate of variance is the average value of the

squared deviations around the sample average; the estimate of the covariance is the average

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176 Part TWO Portfolio Theory

value of the cross-product of deviations. As we noted in Chapter 5, Example 5.5, the averages

used to compute variance and covariance are adjusted by the ratio n/(n 1) to account for the

“lost degree of freedom” when using the sample average in place of the true mean return, E(r).

Notice that, as in scenario analysis, the focus for risk and return analysis is on average re-

turns and the deviations of returns from their average value. Here, however, instead of using

mean returns based on the scenario analysis, we use average returns during the sample period.

We can illustrate this approach with a simple example.

More often than not, variances, covariances, and correlation coefficients are estimated from

past data. The idea is that variability and covariability change slowly over time. Thus, if we es-

6.1 EXAMPLE timate these statistics from a recent data sample, our estimates will provide useful predictions

for the near future”perhaps next month or next quarter.

Using

The computation of sample variances, covariances, and correlation coefficients is quite easy

Historical Data

using a spreadsheet. Suppose you input 10 weekly, annualized returns for two NYSE stocks,

to Estimate

ABC and XYZ, into columns B and C of the Excel spreadsheet below. The column averages in

Means,

cells B15 and C15 provide estimates of the means, which are used in columns D and E to com-

Variances, and pute deviations of each return from the average return. These deviations are used in columns

Covariances F and G to compute the squared deviations from means that are necessary to calculate vari-

ance and the cross-product of deviations to calculate covariance (column H). Row 15 of

columns F, G, and H shows the averages of squared deviations and cross-product of deviations

from the means.

As we noted above, to eliminate the bias in the estimate of the variance and covariance we

need to multiply the average squared deviation by n/(n 1), in this case, by 10/9, as we see in

row 16.

Observe that the Excel commands from the Data Analysis menu provide a simple shortcut

to this procedure. This feature of Excel can calculate a matrix of variances and covariances di-

rectly. The results from this procedure appear at the bottom of the spreadsheet.

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6 Efficient Diversification

The Three Rules of Two-Risky-Assets Portfolios

Suppose a proportion denoted by wB is invested in the bond fund, and the remainder 1 wB,

denoted by wS, is invested in the stock fund. The properties of the portfolio are determined by

the following three rules, which apply the rules of statistics governing combinations of ran-

dom variables:

Rule 1: The rate of return on the portfolio is a weighted average of the returns on the

component securities, with the investment proportions as weights.

rP wBrB wSrS (6.1)

Rule 2: The expected rate of return on the portfolio is a weighted average of the expected

returns on the component securities, with the same portfolio proportions as weights.

In symbols, the expectation of Equation 6.1 is

E(rP) wBE(rB) wSE(rS) (6.2)

The first two rules are simple linear expressions. This is not so in the case of the portfolio

variance, as the third rule shows.

Rule 3: The variance of the rate of return on the two-risky-asset portfolio is

2 2 2

(wB B) (wS S) 2(wB B)(wS S) BS (6.3)

P

where is the correlation coefficient between the returns on the stock and bond

BS

funds.

The variance of the portfolio is a sum of the contributions of the component security vari-

ances plus a term that involves the correlation coefficient between the returns on the compo-

nent securities. We know from the last section why this last term arises. If the correlation

between the component securities is small or negative, then there will be a greater tendency

for the variability in the returns on the two assets to offset each other. This will reduce port-

folio risk. Notice in Equation 6.3 that portfolio variance is lower when the correlation coeffi-

cient is lower.

The formula describing portfolio variance is more complicated than that describing port-

folio return. This complication has a virtue, however: namely, the tremendous potential for

gains from diversification.

The Risk-Return Trade-Off with Two-Risky-Assets Portfolios

Suppose now that the standard deviation of bonds is 12% and that of stocks is 25%, and as-

sume that there is zero correlation between the return on the bond fund and the return on the

stock fund. A correlation coefficient of zero means that stock and bond returns vary inde-

pendently of each other.

Say we start out with a position of 100% in bonds, and we now consider a shift: Invest 50%

in bonds and 50% in stocks. We can compute the portfolio variance from Equation 6.3.

Input data:

E(rB ) 6%; E(rS ) 10%; 12%; 25%; 0; wB 0.5; wS 0.5

B S BS

Portfolio variance:

2

12)2 25)2

(0.5 (0.5 2(0.5 12) (0.5 25) 0 192.25

P

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178 Part TWO Portfolio Theory

The standard deviation of the portfolio (the square root of the variance) is 13.87%. Had we

mistakenly calculated portfolio risk by averaging the two standard deviations [(25 12)/2],

we would have incorrectly predicted an increase in the portfolio standard deviation by a full

6.50 percentage points, to 18.5%. Instead, the portfolio variance equation shows that the

addition of stocks to the formerly all-bond portfolio actually increases the portfolio standard

deviation by only 1.87 percentage points. So the gain from diversification can be seen as a full

4.63%.

This gain is cost-free in the sense that diversification allows us to experience the full con-

tribution of the stock™s higher expected return, while keeping the portfolio standard deviation

below the average of the component standard deviations. As Equation 6.2 shows, the port-

folio™s expected return is the weighted average of expected returns of the component securities.

If the expected return on bonds is 6% and the expected return on stocks is 10%, then shifting

from 0% to 50% investment in stocks will increase our expected return from 6% to 8%.

Suppose we invest 75% in bonds and only 25% in stocks. We can construct a portfolio with an

expected return higher than bonds (0.75 6) (0.25 10) 7% and, at the same time, a