Modern portfolio theory makes frequent use of the idea that investment opportunities can be

evaluated using expected return as a measure of reward and variance of return as a measure

of risk. The calculation and interpretation of portfolio expected return and variance of return

are fundamental skills. In this section, we will develop an understanding of portfolio expected

return and variance of return.10 Portfolio return is determined by the returns on the individual

holdings. As a result, the calculation of portfolio variance, as a function of the individual asset

returns, is more complex than the variance calculations illustrated in the previous section.

We work with an example of a portfolio that is 50 percent invested in an S&P 500 Index

fund, 25 percent invested in a U.S. long-term corporate bond fund, and 25 percent invested in

a fund indexed to the MSCI EAFE Index (representing equity markets in Europe, Australasia,

and the Far East). Table 4-5 shows these weights.

TABLE 4-5 Portfolio Weights

Asset Class Weights

S&P 500 0.50

U.S. long-term corporate bonds 0.25

MSCI EAFE 0.25

10Although we outline a number of basic concepts in this section, we do not present mean–variance

analysis per se. For a presentation of mean–variance analysis, see the chapter on portfolio concepts, as well

as the extended treatments in standard investment textbooks such as Bodie, Kane, and Marcus (2001),

Elton, Gruber, Brown, and Goetzmann (2003), Reilly and Brown (2003), and Sharpe, Alexander, and

Bailey (1999).

Chapter 4 Probability Concepts 153

We first address the calculation of the expected return on the portfolio. In the previous

section, we defined the expected value of a random variable as the probability-weighted average

of the possible outcomes. Portfolio return, we know, is a weighted average of the returns on the

securities in the portfolio. Similarly, the expected return on a portfolio is a weighted average of

the expected returns on the securities in the portfolio, using exactly the same weights. When

we have estimated the expected returns on the individual securities, we immediately have

portfolio expected return. This convenient fact follows from the properties of expected value.

• Properties of Expected Value. Let wi be any constant and Ri be a random variable.

- The expected value of a constant times a random variable equals the constant times the

expected value of the random variable.

E(wiRi) = wiE(Ri) - The expected value of a weighted sum of random variables equals the weighted sum of

the expected values, using the same weights.

E(w1R1 + w2R2 +···+ wnRn) = w1E(R1) + w2E(R2) +···+ wnE(Rn) (4-13)

Suppose we have a random variable with a given expected value. If we multiply each outcome

by 2, for example, the random variable’s expected value is multiplied by 2 as well. That is

the meaning of Part 1. The second statement is the rule that directly leads to the expression

for portfolio expected return. A portfolio with n securities is defined by its portfolio weights,

w1, w2, … , wn, which sum to 1. So portfolio return, Rp, is Rp = w1R1 + w2R2 +···+ wnRn.

We can state the following principle:

• Calculation of Portfolio Expected Return. Given a portfolio with n securities, the

expected return on the portfolio is a weighted average of the expected returns on the

component securities:

E(Rp) = E(w1R1 + w2R2 +···+ wnRn)

= w1E(R1) + w2E(R2) +···+ wnE(Rn)

Suppose we have estimated expected returns on the assets in the portfolio, as given in Table 4-6.

We calculate the expected return on the portfolio as 11.75 percent:

E(Rp) = w1E(R1) + w2E(R2) + w3E(R3)

= 0.50(13%) + 0.25(6%) + 0.25(15%) = 11.75%

TABLE 4-6 Weights and Expected Returns

Asset Class Weight Expected Return (%)

S&P 500 0.50 13

U.S. long-term corporate bonds 0.25 6

MSCI EAFE 0.25 15

154 Quantitative Investment Analysis

In the previous section, we studied variance as a measure of dispersion of outcomes

around the expected value. Here we are interested in portfolio variance of return as a

measure of investment risk. Letting Rp stand for the return on the portfolio, portfolio

variance is σ2(Rp) = E{[Rp − E(Rp)]2} according to Equation 4-8. How do we implement

this definition? In the chapter on statistical concepts and market returns, we learned how to

calculate a historical or sample variance based on a sample of returns. Now we are considering

variance in a forward-looking sense. We will use information about the individual assets in the

portfolio to obtain portfolio variance of return. To avoid clutter in notation, we write ERp for

E(Rp). We need the concept of covariance.

• Definition of Covariance. Given two random variables Ri and Rj, the covariance between

Ri and Rj is

Cov(Ri, Rj) = E(Ri − ERi)(Rj − ERj)

Alternative notations are σ(Ri, Rj) and σij.

Equation 4-14 states that the covariance between two random variables is the probabilityweighted average of the cross-products of each random variable’s deviation from its own

expected value. We will return to discuss covariance after we establish the need for the concept.

Working from the definition of variance, we find

σ2

(Rp) = E[(Rp − ERp)

2]

= E{[w1R1 + w2R2 + w3R3 − E(w1R1 + w2R2 + w3R3)]2}

= E{[w1R1 + w2R2 + w3R3 − w1ER1 − w2ER2 − w3ER3]

2

}

(using Equation 4−13)

= E{[w1(R1 − ER1) + w2(R2 − ER2) + w3(R3 − ER3)]2

} (rearranging)

= E{[w1(R1 − ER1) + w2(R2 − ER2) + w3(R3 − ER3)]

×[w1(R1 − ER1) + w2(R2 − ER2) + w3(R3 − ER3)]}

(what squaring means)

= E[w1w1(R1 − ER1)(R1 − ER1) + w1w2(R1 − ER1)(R2 − ER2)

+w1w3(R1 − ER1)(R3 − ER3) + w2w1(R2 − ER2)(R1 − ER1)

+w2w2(R2 − ER2)(R2 − ER2) + w2w3(R2 − ER2)(R3 − ER3)

+w3w1(R3 − ER3)(R1 − ER1) + w3w2(R3 − ER3)(R2 − ER2)

+w3w3(R3 − ER3)(R3 − ER3)] (doing the multiplication)

= w2

1E[(R1 − ER1)

2

] + w1w2E[(R1 − ER1)(R2 − ER2)]

+w1w3E[(R1 − ER1)(R3 − ER3)] + w2w1E[(R2 − ER2)(R1 − ER1)]

+w2

2E[(R2 − ER2)

2

] + w2w3E[(R2 − ER2)(R3 − ER3)]

+w3w1E[(R3 − ER3)(R1 − ER1)] + w3w2E[(R3 − ER3)(R2 − ER2)]

Chapter 4 Probability Concepts 155

+w2

3E[(R3 − ER3)

2

] (recalling that the wi terms are constants)

= w2

1σ2(R1) + w1w2Cov(R1, R2) + w1w3Cov(R1, R3)

+w1w2Cov(R1, R2) + w2

2σ2

(R2) + w2w3Cov(R2, R3)

+w1w3Cov(R1, R3) + w2w3Cov(R2, R3) + w2

3σ2(R3) (4-15)

The last step follows from the definitions of variance and covariance.11 For the italicized

covariance terms in Equation 4-15, we used the fact that the order of variables in covariance

does not matter: Cov(R2, R1) = Cov(R1, R2), for example. As we will show, the diagonal

variance terms σ2(R1), σ2(R2), and σ2(R3) can be expressed as Cov(R1, R1), Cov(R2, R2), and

Cov(R3, R3), respectively. Using this fact, the most compact way to state Equation 4-15 is

σ2(RP) =

3

i=1

3

j=1

wiwjCov(Ri, Rj). The double summation signs say: ‘‘Set i = 1 and let j run

from 1 to 3; then set i = 2 and let j run from 1 to 3; next set i = 3 and let j run from 1 to 3;

finally, add the nine terms.’’ This expression generalizes for a portfolio of any size n to

σ2(Rp) = n

i=1

n

j=1

wiwjCov(Ri, Rj) (4-16)

We see from Equation 4-15 that individual variances of return (the bolded diagonal

terms) constitute part, but not all, of portfolio variance. The three variances are actually

outnumbered by the six covariance terms off the diagonal. For three assets, the ratio is 1 to

2, or 50 percent. If there are 20 assets, there are 20 variance terms and 20(20) − 20 = 380

off-diagonal covariance terms. The ratio of variance terms to off-diagonal covariance terms is

less than 6 to 100, or 6 percent. A first observation, then, is that as the number of holdings

increases, covariance12 becomes increasingly important, all else equal.

What exactly is the effect of covariance on portfolio variance? The covariance terms

capture how the co-movements of returns affect portfolio variance. For example, consider

two stocks: One tends to have high returns (relative to its expected return) when the other

has low returns (relative to its expected return). The returns on one stock tend to offset the

returns on the other stock, lowering the variability or variance of returns on the portfolio. Like

variance, the units of covariance are hard to interpret, and we will introduce a more intuitive

concept shortly. Meanwhile, from the definition of covariance, we can establish two essential

observations about covariance. - We can interpret the sign of covariance as follows:

Covariance of returns is negative if, when the return on one asset is above its expected

value, the return on the other asset tends to be below its expected value (an average

inverse relationship between returns).

Covariance of returns is 0 if returns on the assets are unrelated.

11Useful facts about variance and covariance include: 1) The variance of a constant times a random

variable equals the constant squared times the variance of the random variable, or σ2(wR) = w2σ2(R);

2) The variance of a constant plus a random variable equals the variance of the random variable, or

σ2(w + R) = σ2(R) because a constant has zero variance; 3) The covariance between a constant and a

random variable is zero.

12When the meaning of covariance as ‘‘off-diagonal covariance’’ is obvious, as it is here, we omit the

qualifying words. Covariance is usually used in this sense.

156 Quantitative Investment Analysis

TABLE 4-7 Inputs to Portfolio Expected Return and Variance

A. Inputs to Portfolio Expected Return

Asset A B C

E(RA) E(RB) E(RC )

B. Covariance Matrix: The Inputs to Portfolio Variance of Return

Asset A B C

A Cov(RA, RA) Cov(RA, RB) Cov(RA, RC)

B Cov(RB, RA) Cov(RB, RB) Cov(RB, RC)

C Cov(RC, RA) Cov(RC, RB) Cov(RC, RC)

Covariance of returns is positive when the returns on both assets tend to be on the

same side (above or below) their expected values at the same time (an average positive

relationship between returns). - The covariance of a random variable with itself (own covariance) is its own variance:

Cov(R, R) = E{[R − E(R)][R − E(R)]} = E{[R − E(R)]2} = σ2(R).

A complete list of the covariances constitutes all the statistical data needed to compute portfolio

variance of return. Covariances are often presented in a square format called a covariance

matrix. Table 4-7 summarizes the inputs for portfolio expected return and variance of return.

With three assets, the covariance matrix has 32 = 3 × 3 = 9 entries, but it is customary

to treat the diagonal terms, the variances, separately from the off-diagonal terms. These

diagonal terms are bolded in Table 4-7. This distinction is natural, as security variance

is a single-variable concept. So there are 9 − 3 = 6 covariances, excluding variances. But

Cov(RB, RA) = Cov(RA, RB), Cov(RC, RA) = Cov(RA, RC), and Cov(RC, RB) = Cov(RB, RC).

The covariance matrix below the diagonal is the mirror image of the covariance matrix above

the diagonal. As a result, there are only 6/2 = 3 distinct covariance terms to estimate. In

general, for n securities, there are n(n − 1)/2 distinct covariances to estimate and n variances

to estimate.

Suppose we have the covariance matrix shown in Table 4-8. Taking Equation 4-15 and

grouping variance terms together produces the following:

σ2

(Rp) = w2

1σ2

(R1) + w2

2σ2

(R2) + w2

3σ2

(R3) + 2w1w2Cov(R1, R2)

+2w1w3Cov(R1, R3) + 2w2w3Cov(R2, R3) (4-17)

= (0.50)2(400) + (0.25)2(81) + (0.25)2(441) + 2(0.50)(0.25)(45)

+2(0.50)(0.25)(189) + 2(0.25)(0.25)(38)

= 100 + 5.0625 + 27.5625 + 11.25 + 47.25 + 4.75 = 195.875

The variance is 195.875. Standard deviation of return is 195.8751/2 = 14 percent. To

summarize, the portfolio has an expected annual return of 11.75 percent and a standard

deviation of return of 14 percent.

Let us look at the first three terms in the calculation above. Their sum, 100 + 5.0625 +

27.5625 = 132.625, is the contribution of the individual variances to portfolio variance. If

the returns on the three assets were independent, covariances would be 0 and the standard

Chapter 4 Probability Concepts 157

TABLE 4-8 Covariance Matrix

U.S. Long-Term MSCI

S&P 500 Corporate Bonds EAFE

S&P 500 400 45 189

U.S. long-term corporate bonds 45 81 38

MSCI EAFE 189 38 441

deviation of portfolio return would be 132.6251/2 = 11.52 percent as compared to 14 percent

before. The portfolio would have less risk. Suppose the covariance terms were negative. Then

a negative number would be added to 132.625, so portfolio variance and risk would be even

smaller. At the same time, we have not changed expected return. For the same expected

portfolio return, the portfolio has less risk. This risk reduction is a diversification benefit,

meaning a risk-reduction benefit from holding a portfolio of assets. The diversification benefit

increases with decreasing covariance. This observation is a key insight of modern portfolio

theory. It is even more intuitively stated when we can use the concept of correlation. Then we

can say that as long as security returns are not perfectly positively correlated, diversification

benefits are possible. Furthermore, the smaller the correlation between security returns, the

greater the cost of not diversifying (in terms of risk-reduction benefits forgone), all else

equal.

• Definition of Correlation. The correlation between two random variables, Ri and Rj,

is defined as ρ(Ri, Rj) = Cov(Ri, Rj)/σ(Ri)σ(Rj). Alternative notations are Corr(Ri, Rj)

and ρij.

Frequently, covariance is substituted out using the relationship Cov(Ri, Rj) = ρ(Ri, Rj)σ(Ri)

σ(Rj). The division indicated in the definition makes correlation a pure number (one without

a unit of measurement) and places bounds on its largest and smallest possible values. Using the

above definition, we can state a correlation matrix from data in the covariance matrix alone.

Table 4-9 shows the correlation matrix.

For example, the covariance between long-term bonds and MSCI EAFE is 38, from

Table 4-8. The standard deviation of long-term bond returns is 811/2 = 9 percent, that

of MSCI EAFE returns is 4411/2 = 21 percent, from diagonal terms in Table 4-8. The

correlation ρ(Return on long-term bonds, Return on EAFE) is 38/(9%)(21%) = 0.201,

rounded to 0.20. The correlation of the S&P 500 with itself equals 1: The calculation is own

covariance divided by its standard deviation squared.

TABLE 4-9 Correlation Matrix of Returns

U.S. Long-Term MSCI

S&P 500 Corporate Bonds EAFE

S&P 500 1.00 0.25 0.45

U.S. long-term corporate bonds 0.25 1.00 0.20

MSCI EAFE 0.45 0.20 1.00

158 Quantitative Investment Analysis

• Properties of Correlation. - Correlation is a number between −1 and +1 for two random variables, X and Y :

−1 ≤ ρ(X , Y ) ≤ +1 - A correlation of 0 (uncorrelated variables) indicates an absence of any linear (straightline) relationship between the variables.13 Increasingly positive correlation indicates

an increasingly strong positive linear relationship (up to 1, which indicates a perfect

linear relationship). Increasingly negative correlation indicates an increasingly strong

negative (inverse) linear relationship (down to −1, which indicates a perfect inverse

linear relationship).14