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

CHAPTER • 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 not know what particular outcome will occur The risky asset will have a higher expected return than the risk-free asset (Rm 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 percent (Rf ϭ 04), the stock market’s expected return is 12 percent (Rm ϭ 12), and b ϭ 1/2 Then Rp ϭ 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