(8th edition) (the pearson series in economics) robert pindyck, daniel rubinfeld microecon 204

CHAPTER 5 • Uncertainty and Consumer Behavior 179

TABLE 5.8

INVESTMENTS—RISK AND RETURN (1926–2010)
AVERAGE RATE OF
RETURN (%)

AVERAGE REAL
RATE OF RETURN
(%)

RISK (STANDARD
DEVIATION)

11.9

8.7

20.4

Long-term corporate
bonds

6.2

3.3

8.3

U.S. Treasury bills

3.7

0.7

3.1

Common stocks
(S&P 500)

Source: Ibbotson® SBBI® 2001 Classic Yearbook: Market results for Stocks, Bonds, Bills, and Inflation 1926–2010.
© 2011 Morningstar.

The Trade-Off Between Risk and Return
Suppose a woman wants to invest her savings in two assets—Treasury bills,
which are almost risk free, and a representative group of stocks. She must
decide how much to invest in each asset. She might, for instance, invest only
in Treasury bills, only in stocks, or in some combination of the two. As we will
see, this problem is analogous to the consumer’s problem of allocating a budget
between purchases of food and clothing.
Let’s denote the risk-free return on the Treasury bill by Rf . Because the
return is risk free, the expected and actual returns are the same. In addition, let the expected return from investing in the stock market be Rm and the
actual return be rm. The actual return is risky. At the time of the investment
decision, we know the set of possible outcomes and the likelihood of each,
but we do not know what particular outcome will occur. The risky asset will
have a higher expected return than the risk-free asset (Rm 7 Rf). Otherwise,
risk-averse investors would buy only Treasury bills and no stocks would
be sold.
THE INVESTMENT PORTFOLIO To determine how much money the investor
should put in each asset, let’s set b equal to the fraction of her savings placed

in the stock market and (1 - b) the fraction used to purchase Treasury bills. The
expected return on her total portfolio, Rp, is a weighted average of the expected
return on the two assets:14
R p = bR m + (1 - b)R f

(5.1)

Suppose, for example, that Treasury bills pay 4 percent (Rf ϭ .04), the stock
market’s expected return is 12 percent (Rm ϭ .12), and b ϭ 1/2. Then Rp ϭ 8
percent. How risky is this portfolio? One measure of riskiness is the standard
deviation of its return. We will denote the standard deviation of the risky stock
market investment by ␴m. With some algebra, we can show that the standard
deviation of the portfolio, ␴p (with one risky and one risk-free asset) is the fraction
14

The expected value of the sum of two variables is the sum of the expected values. Therefore
R p = E[brm] + E[(1 - b)R f] = bE[rm] + (1 - b)R f = bR m + (1 - b)R f