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Variance of returns


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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Essentials of Investments, Companies, 2003
Fifth Edition




175
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
Bodie’Kane’Marcus: II. Portfolio Theory 6. Efficient Diversification © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition




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.
Bodie’Kane’Marcus: II. Portfolio Theory 6. Efficient Diversification © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition




177
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
Bodie’Kane’Marcus: II. Portfolio Theory 6. Efficient Diversification © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition




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

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