Question

CHAPTER 2-4

Exercise 2.24 In this exercise, we examine the effect of combining investments with positively correlated risks, negatively correlated risks, and uncorrelated risks. A firm is considering a portfolio of assets. The portfolio is comprised of two assets, which we will call ''A" and "B." Let X denote the annual rate of return from asset A in the following year, and let Y denote the annual rate of return from asset B in the following year. Suppose that E(X) = 0.15 and E(Y) = 0.20, SD(X) = 0.05 and SD(Y) = 0.06, and CORR(X, Y) = 0.30.

		(a) What is the expected return of investing 50% of the portfolio in asset A and 50% of the portfolio in asset B? What is the standard deviation of this return?

		(b) Replace CORR(X, Y) = 0.30 by CORR(X, Y) = 0.60 and answer the questions in part (a). Do the same for CORR(X, Y) = 0.60, 0.30, and 0.0.

		(c) (Spreadsheet Exercise). Use a spreadsheet to perform the following analysis. Suppose that the fraction of the portfolio that is invested in asset B is f, and so the fraction of the portfolio that is invested in asset A is (1 f). Letting f vary from f = 0.0 to f = 1.0 in increments of 5% (that is, f = 0.0, 0.05, 0.10, 0.15, . . . ), compute the mean and the standard deviation of the annual rate of return of the portfolio (using the original data for the problem). Notice that the expected return of the portfolio varies (linearly) from 0.15 to 0.20, and the standard deviation of the return varies (non-linearly) from 0.05 to 0.06. Construct a chart plotting the standard deviation as a function of the expected return.

		(d) (Spreadsheet Exercise). Perform the same analysis as in part (c) with CORR (X, Y) = 0.30 replaced by CORR(X, Y) = 0.60, 0.0, 0.30, and 0.60.

Exercise 2.38

Ninety percent of residential gas customers in Illinois use gas for residential heating. Sixteen residential gas customers are randomly selected to participate in a panel discussion for a state energy fair. A gas industry executive is hopeful that at least twelve of the panel members, i.e., 75%, will come from homes in which gas is used for residential heating. If you were the executive's assistant, what degree of assurance could you give the executive that her 75% goal might be reached or exceeded?

Answer 1

ANS

a) E(X) 0.15 sd(X) 0.05 0.2 sd(Y) 0.06 E(Y) 0.3 r a) expected return 0.175 0.044441 sd formula E(X) 0.15 sd(X) 0.05 E(Y) 0.2 sd(Y) 0.06 0.3 r expected return 0.5*B1+0.5*B2 -SQRT(0.5^2 D1^2 0.5^2* D2^2 2* 0.5 *0.5 * B4*D1*D2) sd + +

b) 0.6 expected return 0.175 sd 0.049244289 0 expected return 0.175 sd 0.039051248

w1 w2 sd mean 0 1 0.2 0.06 0.057799 0.05 0.95 0.1975 0.055705 0.1 0.9 0.195 0.053728 0.15 0.85 0.1925 0.2 0.8 0.19 0.051884 0.050187 0.25 0.75 0.1875 0.048652 0.3 0.7 0.185 0.1825 0.35 0.65 0.047294 0.04613 0.4 0.6 0.18 0.045175 0.45 0.55 0.1775 0.5 0.5 0.175 0,044441 0.1725 0.04394 0.55 0.45 0.6 0,4 0.17 0.043681 0.1675 0,043666 0.65 0.35 0.043898 0.7 0.3 0.165 0.75 0.25 0.1625 0.044371 0,8 0.2 0.16 0.045078 0.04600 0.85 0.15 0.1575 0.9 0.1 0.155 0.047149 0.95 0.05 0.1525 0.048485 1 0.15 0.05

formula E(X) 0.15 sd(X) 0.05 0.2 0.06 E(Y) sd(Y) r 0.3 w1 w2 sd mean -SQRT(A6^2*SD$1^2 + B6^2 *SD$2^2+ 2*A6*B6*o.3*SD$1 SD$2) =A6*SB$1+B6* SB$2 1- A6 C -SQRT(A7^2*SD$1^2 B7^2 *SDS2^2+ 2*A7*B7*0.3*SD$1 SD$2) = 1-A7 =A7*SBS1+B7*SBS2 0.05+A6 =A8*SB$1+B8*SB$2 =0.05+A7 1- A8 -SQRT(A8^2*SD$1^2+ B8^2 *SD$2^2+ 2*A8*B8*0.3*SD$1*SD$) 1 -A9 =A9*SB$1+B9*SB$2 =SQRT(A9^2*SD$1^2 + B9^2 *SD$2^2+ 2*A9*B9*0.3*SD$1*SD$2) =0,05+A8 =A10*SBS1 B10*SBS2 SQRT(A10^2*SD$1^2 B10 2 *SD$2^2+ 2*A10*B10*0.3*SD$1*SD$2) =0,05+A9 =1-A10 -SQRT(A11 2*SD$1^2 B11^2 *SD$2^2+ 2*A11*B11*0.3*SD$1 SD$2) =A11 SB$1+B11*$B$2 =0,05+A10 =1-A11 =SQRT(A12 2*SD$1^2 + B12^2 *SD$2^2+ 2*A12*B12*0.3*SD$1*SD$2) =A12*SB$1+B12*SB$2 =0,05+A11 =1-A12 =A13*SB$1+B13*SBS2 SQRT(A13^2*SD$1^2 B13^2 *SD$2^2+ 2*A13*B13*0.3*SD$1*SD$2) =0,05+A12 =1-A13 SQRT(A14^2*SD$1^2 B14^2 *SD$2^2+ 2*A14*B14*0.3*SD$1 SD$2) =A14*SB$1+B14*SB$2 =0,05+A13 =1-A14 =A15*SBS1 B15*SB$2 =SQRT(A15^2*SD$1^2 + B15^2 *SD$2^2+ 2*A15*B15*0.3*SD$1*SD$2) =0.05+A14 =1-A15 =A16*SB$1 B16*SB$2=SQRT(A16^2*SD$1^2 B16^2 *SD$2^2+ 2*A16*B16*0.3*SD$1*SD$2) =0,05+A15 -1-A16

=A17*SB$1+B17*SB$2 -SQRT(A17^2*SD$1^2+ B17^2 *SD$2^2+ 2*A17*B17*0.3*SD$1*SDS2) =0,05+A16 =1-A17 -SQRT(A18^2*SD$1^2 0.05+A17 =A18*SB$1+B18*SB$2 B18^2 *SD$2^2+ 2*A18*B18*0.3*SD$1*SD$2) =1-A18 -SQRT(A19 2*SD$1^2 =0,05+A18 1- A19 =A19*SB$1+ B19*SB$2 B19^2 *SD$2^2+ 2*A19*B19*0.3*SD$1*SD$2) =1-A20A20SBS1 B20 SBS2 SQRT(A20 ^ 2 *SD$1^2 + B20^2 *SDS2^2+ 2*A20*B20*0.3*SD$1*SDS2 =0.05+A19 -SQRT(A21^2*SD$1^2 B21^2 *SDS2^2+ 2*A21*B21*0.3*SDS1 SD$2) 1- A21 A21 SBS1 B21 SB$2 =0.05+A20 =A22*SB$1+B22*SB$2 SQRT(A22^2*SD$1^2 B22^2 *SDS2^2+ 2*A22*B22*0.3*SD$1*SD$2) =1- A22 =0.05+A21 =A23*SB$1 B23*SB$2 SQRT(A23^2*SD$1^2 B23^2 *SDS2^2+ 2*A23*B23*0.3*SD$1*SD$2) =0.05+A22 = 1 - A23 -1-A24 |A24*SB$1+B24*SBS2 SQRT(A24^2*SD$1^2 + B24^2 *SD$2^2+ 2*A24*B24*0.3*SD$1*SD$2) =0,05+A23 =A25*SB$1 B25*SB$2=SQRT(A25^2*SD$1^2 B25^2 *SD$2^2+ 2*A25*B25*0.3*SD$1*SD$2) 0.05+A24 =1-A25 -1-A26 |A26*SB$1+B26*SB$2 SQRT(A26^2*SD$1^2+ B26^2 *SD$2^2+ 2*A26*B26*0.3*SD$1*SD$2) 0.05+A25

C w1 w2 sd mean 0.06 0.2 0.057054798 0.05 0.95 0.1975 0.9 0.1 0.195 0.054230987 0.15 0.85 0.1925 0.051548521 0.8 0.19 0.2 0.049030603 0.25 0.75 0.1875 0.046703854 0.044598206 0.3 0.7 0.185 0.042746345 0.35 0.65 0.1825 0.041182521 0.4 0.6 0.18 0,039940581 0.45 0.55 0.1775 0.039051248 0.5 0.5 0.175 0.55 0.45 0.1725 0,038538941 0,038418745 0.6 0.4 0.17 0.038694315 0.65 0.35 0.1675 C

0.7 0.3 0.165 0.039357337 0.75 0.25 0.1625 0.040388736 0.8 0.2 0.16 0.041761226 0.85 0.15 0.1575 0.043442491 0.9 0.1 0.155 0.045398238 0.95 0.047594643 0.05 0.1525 1 0 0.15 0.05

0.6 w1 w2 sd mean C 1 0.2 0.06 0.1975 0.05 0.95 0.058534178 0.1 0.9 0.195 0.057140179 0.15 0.85 0.1925 0.055823382 0.2 0.8 0.19 0.054589376 0.053443896 0.25 0.75 0.1875 0.7 0.3 0.185 0.052392748 0.35 0.65 0.1825 0.051441715 0.4 0.6 0.18 0.050596443 0.45 0.55 0.1775 0.04986231 0.5 0.5 0.175 0.049244289

0.55 0.45 0.1725 0.048746795 0.6 0.4 0.17 0.048373546 0.65 0.35 0.1675 0.048127435 0.048010416 0.7 0.3 0.165 0.75 0.25 0.1625 0.048023432 0.8 0.2 0.16 0.048166378 0.157 0.85 0.15 0.048438105 0.9 0.1 0.155 0.048836462 0.95 0.05 0.1525 0.049358383 1 C 0.15 0.05