Introduction to Corporate Finance

7: Risk, Return and the Capital Asset Pricing Model

7-2 RISK AND RETURN FOR PORTFOLIOS

In Chapter 6, we saw that investors can reduce risk dramatically by holding diversified portfolios rather

than individual shares. An investor who chooses to diversify will be more concerned with how her

portfolio performs than with the performance of each individual security in the portfolio. Therefore, we

need a way to measure risk and return for portfolios.

7-2a PORTFOLIO EXPECTED RETURN

Suppose an individual has $10,000 to invest, and she decides to divide that money between two different

assets. Asset 1 has an expected return of 8%, and Asset 2 has an expected return of 12%. Our investor

puts $4,000 in Asset 1 and $6,000 in Asset 2. What is the expected return on the portfolio?

To begin, we must calculate the fraction of the individual’s wealth invested in each asset, known

as the portfolio weights. The fraction invested in Asset 1 equals 0.40 ($4,000/$10,000), and the fraction

invested in Asset 2 equals 0.60 ($6,000/$10,000). Notice that the portfolio weights add up to 1.0.

The portfolio’s expected return equals the weighted average of the expected returns of the securities

in the portfolio. In this case, the expected return equals:

Expected return = (0.40)(8%) + (0.60)(12%) = 10.4%

We can write a more general expression describing a portfolio’s expected return. Suppose a portfolio

contains different securities. The expected returns on these securities are E(r1), E(r2),...., E(rn). The

portfolio weights are w 1 , w 2 , ....wn. The portfolio expected return E(rp) is given by this equation:

Eq. 7.1 E(rp) = w 1 E(r 1 )+ w 2 E(r 2 )+...+ wn E(rn)

w 1 +w 2 +...+wn = 1

example

Calculate the expected return on the portfolio

described in the following table.

Share E(r) $ invested

Telstra 10% $ 2,500

Billabong 12% 5,000

Woolworths 8% 2,500

Cochlear 14% 10,000

First, calculate the portfolio weights. The total

dollar value of the portfolio is $20,000. The weights

for the investments in Telstra and Woolworths are

0.125 ($2,500/$20,000). The fraction invested in

Billabong is 0.25, and the weight associated with

Cochlear is 0.50. Now multiply those weights times

the expected return for each share and add up:

E(rp) = (0.125)(10%) + (0.25)(12%)

+ (0.125)(8%) + (0.5)(14%) = 12.25%

Short Selling

We noted that the portfolio weights must add up to 1. It is natural to assume that these weights also

fall in a range between zero and 1, meaning that an investor can invest nothing or everything in any

particular asset. However, a more exotic arrangement is possible, one that results in a negative portfolio

weight for a particular asset. A negative portfolio weight means that, rather than investing in the given

asset, an individual is borrowing that asset, selling it and using the proceeds to invest more in something

LO7.2

portfolio weights

The percentage invested in

each of several securities in

a portfolio. Portfolio weights

must sum to 1.0 (or 100%)