180 PART • Producers, Consumers, and Competitive Markets of the portfolio invested in the risky asset times the standard deviation of that asset:15 sp = bsm (5.2) The Investor’s Choice Problem In §3.2 we explain how a budget line is determined from an individual’s income and the prices of the available goods We have still not determined how the investor should choose this fraction b To so, we must first show that she faces a risk-return trade-off analogous to a consumer’s budget line To identify this trade-off, note that equation (5.1) for the expected return on the portfolio can be rewritten as R p = R f + b(R m - R f) Now, from equation (5.2) we see that b ϭ ␴p/␴m, so that Rp = Rf + • Price of risk Extra risk that an investor must incur to enjoy a higher expected return (R m - R f) sm sp (5.3) RISK AND THE BUDGET LINE This equation is a budget line because it describes the trade-off between risk (␴p) and expected return (Rp) Note that it is the equation for a straight line: Because Rm, Rf, and ␴m are constants, the slope (Rm − Rf)/ ␴m is a constant, as is the intercept, Rf The equation says that the expected return on the portfolio Rp increases as the standard deviation of that return ␴p increases We call the slope of this budget line, (Rm − Rf)/␴m, the price of risk, because it tells us how much extra risk an investor must incur to enjoy a higher expected return The budget line is drawn in Figure 5.6 If our investor wants no risk, she can invest all her funds in Treasury bills (b ϭ 0) and earn an expected return Rf To receive a higher expected return, she must incur some risk For example, she could invest all her funds in stocks (b ϭ 1), earning an expected return Rm but incurring a standard deviation ␴m Or she might invest some fraction of her funds in each type of asset, earning an expected return somewhere between Rf and Rm and facing a standard deviation less than ␴m but greater than zero RISK AND INDIFFERENCE CURVES Figure 5.6 also shows the solution to the investor’s problem Three indifference curves are drawn in the figure Each curve describes combinations of risk and return that leave the investor equally satisfied The curves are upward-sloping because risk is undesirable Thus, with a greater amount of risk, it takes a greater expected return to make the investor equally well-off Curve U3 yields the greatest amount of satisfaction and U1 the least amount: For a given amount of risk, the investor earns a higher expected return on U3 than on U2 and a higher expected return on U2 than on U1 15 To see why, we observe from footnote that we can write the variance of the portfolio return as s 2p = E[brm + (1 - b)R f - R p]2 Substituting equation (5.1) for the expected return on the portfolio, Rp, we have s 2p = E[brm + (1 - b)R f - bR m - (1 - b)R f]2 = E[b(rm - R m)]2 = b 2s 2m Because the standard deviation of a random variable is the square root of its variance, sp = bsm