In the next section we address the "personal" part of the problem-the specific individuals choice of the best risk-return combination from the set of feasible combinations. Suppose the investor has already decided on the composition of the risky portfolio. Now the concern is with the proportion of the investment budget, y, to be allocated to the risky portfolio, P. The remaining proportion, 1 y, is to be invested in the risk-free asset, F. Denote the risky rate of return by rP and denote the expected rate of return on P by E(rP) and its standard deviation by P. The rate of return on the risk-free asset is denoted as rf. In the numerical example we assume that E(rP) 15%, P 22%, and that the risk-free rate is rf 7%. Thus the risk premium on the risky asset is E(rP) rf 8%. With a proportion, y, in the risky portfolio, and 1 - y in the risk-free asset, the rate of re- turn on the complete portfolio, denoted C, is rC where   rC yrP (1 y)rf   Taking the expectation of this portfolios rate of return,   E(rC) yE(rP) (1 y)rf (7.1) rf y[E(rP) rf] 7 y(15 7)   This result is easily interpreted. The base rate of return for any portfolio is the risk-free rate. In addition, the portfolio is expected to earn a risk premium that depends on the risk premium of the risky portfolio, E(rP) rf, and the investors position in the risky asset, y. Investors are assumed to be risk averse and thus unwilling to take on a risky position with- out a positive risk premium. As we noted in Chapter 6, when we combine a risky asset and a risk-free asset in a port- folio, the standard deviation of the resulting complete portfolio is the standard deviation of the risky asset multiplied by the weight of the risky asset in that portfolio. Because the stan- dard deviation of the risky portfolio is P 22%, C y P 22y (7.2) II. Portfolio Theory 7. Capital Allocation between the Risky Asset and the Risk−Free Asset The McGraw−Hill Companies, 2001         188 PART II Portfolio Theory     Figure 7.2 The investment opportunity set with a risky asset and a risk-free asset in the expected return-standard deviation plane.