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1、CHAPTER 6: CAPITAL ALLOCATION TO RISKY ASSETSPROBLEM SETS1.averse investor would avoid In addition, higher or for risk. The Sharpe ratio unit of risk.(e) The first two answer choices are incorrect because a highly risk portfolios with higher risk premiums and higher standard deviations, lower Sharpe
2、 ratios are not an indication of an investor's tolerance is simply a tool to absolutely measure the return premium earned per2. (b) A higher borrowing rate is a consequence of the risk of the boorrwers ' defa. ult In perfectmarkets with no additional cost of default, this increment would equ
3、al the value of the borrower 'opstion to default, and the Sharpe measure, with appropriate treatment of the default option, would be the same. However, in reality there are costs to default so that this part of the increment lowers the Sharpe ratio. Also, notice that answer (c) is not correct be
4、cause doubling the expected return with a fixed risk-free rate will more than double the risk premium and the Sharpe ratio.3. Assuming no change in risk tolerance, that is, an unchanged risk-aversion coefficient (A), higher perceived volatility increases the denominator of the equation for the optim
5、al investment in the risky portfolio (Equation 6.7). The proportion invested in the risky portfolio will therefore decrease.4. a. The expected cash flow is: (0. 5x$70, 000) + (0. 5 x 200, 000) = $135, 000.With a risk premium of 8% over the risk-free rate of 6%, the required rate of return is 14%. Th
6、erefore, the present value of the portfolio is:$135, 000/1. 14 = $118, 421b. If the portfolio is purchased for $118, 421 and provides an expected cash inflow of $135, 000, then the expected rate of return E(r) is as follows:$118,421 xl + E(r) = $135, 000Therefore, E(r) = 14%. The portfolio price is
7、set to equate the expected rate of return with the required rate of return.c. If the risk premium over T-bills is now 12%, then the required return is:6% + 12% = 18%The present value of the portfolio is now:$135, 000/1. 18 = $114, 407d. For a given expected cash flow, portfolios that command greater
8、 riskpremiums must sell at lower prices. The extra disco unt from expected value is a pen alty for risk.5.6.When we specify utility by U = E(r) -0. 5A c the utility level for T-bills is: 0. 07The utility level for the risky portfolio is:U = 0. 12 -o. 5 从 X0. 18)2 = 0. 12 -0. 0162 A<In order for t
9、he risky portfolio to be preferred to bills, the following must hold:0. 12 -o. 0162A > 0. 07 A < 0. 05/0. 0162 = 3. 09A must be less than 3. 09 for the risky portfolio to be preferred to bills.Points on the curve are derived by solving foE(r) in the following equation: U = 0. 05 = E(r) -0. 5AC
10、 = E(r) T. 5*The values of E(r), given the values of C, are therefore:E(r)0. 000. 00000. 050000. 050. 00250. 053750. 100.01000. 065000. 150. 02250. 083750. 200. 04000. 110000. 250. 06250. 14375The bold line in the graph on the n ext page (labeled Q6, for Questi on 6) depicts the in differe nee curve
11、.7. Repeati ng the an alysis in Problem 6, utility is now:U = E(r) -o. 5A 2 = E(r) -2. 0 2 = 0. 05The equal-utility comb in ati ons of expected return and sta ndard deviati on are presented in the table below. The indifference curve is the upward sloping line in the graph on the n ext page, labeled
12、Q7 (for Questi on 7).E(rl 0. 000. 00000. 05000. 050. 00250. 05500. 100.01000. 07000. 150. 02250. 09500. 200. 04000. 13000. 250. 06250. 1750The in differe nee curve in Problem 7 differs from that in Problem 6 in slope. When A in creases from 3 to 4, the in creased risk aversi on results in a greater
13、slope for the indifference curve since more expected return is needed in order to compe nsate for additi onal . ccorresponding utility is equal to the portfolio, Thexcorfesp一 (Cstdmng8.The coefficie nt of risk avers ion for a risk n eutral inv estor is zero. Therefore, thein differe nee curve in the
14、 expected return-sta ndard deviatio n pla ne is a horiz ontal lin e, labeled Q8 in the graph above (see Problem 6).9. A risk lover, rather than penalizing portfolio utility to account for risk, derives greater utility as varia nee in creases. This amounts to a n egative coefficie nt of risk aversi o
15、n. The corresp onding in differe nee curve is dow nward slop ing in the graph above (see Problem 6), and is labeled Q9.10. The portfolio expected return and varianee are computed as follows:(1)WBills(2)rBills(3)Wndex(4)rIndexPPortfolio (1) (2) + (3) (4)Portfolio(3)20%7Portfolio0.05%1.013. 0%13. 0% =
16、 0. 13020%=0. 200. 04000.250.813.011.4% =0. 11416%=0. 160. 02560.450.613.09. 8% =0. 09812%=0. 120. 01440.650.413.08. 2% =0. 0828% : =0. 080. 00640.850. 213.06. 6% =0. 0664% : =0. 040. 00161.050.013.05. 0% =0. 0500% : =0. 000. 000011. Computing utilityfrom U =:E(r) 0. 5xA d- = E(r)一占 we arrive at the
17、 valuesin the columnlabeledU(A = 2) in the following table:WBillsWndexPPortfolioPortfolio2U(A = 2)U(A 二 3)0. 01.00. 1300. 200. 04000. 0900.07000. 20.80. 1140. 160. 02560. 0884.07560.40.60. 0980. 120. 01440. 0836.07640.60.40. 0820. 080. 00640. 0756.07240. 80.20. 0660. 040. 00160. 0644.06361.00.00.050
18、0. 000. 00000. 0500.0500The columnlabeledU(A =2) impliesthat investorswith A 二:2 prefer aportfolio that is invested 100% in the market in dex to any of the other portfolios in the table.12. The column labeledU(A 二 3) in the table above is computed from:U = E(r) -0. 5A62 = E(r) -1.5 d2The more risk a
19、verse inv estors prefer the portfolio that is in vested 40% in the market, rather than the 100% market weight preferred by investors wit A = 2.13. Expected return 二(0. 7x 18%) + (0. 3 x 8%)= 15%Standard deviation = 0.7x28% = 19. 6%14. Inv estme nt proporti ons:30. 0% in T-bills0. 725%=17. 5% in Stoc
20、k A0. 732% =22. 4% in Stock B15.0. 7 43%=30. 1% in Stock CYour reward-to-volati1ity ratio:.18 .08U. 35/1oo1508Client s rewardtovolatility ratio： S0. 357116.30E(r)%25201510510203040CAL (Slope = 0.3571)8)17. a. E(rc) = rf + y xE(rP) rf = 8 + y X(18If the expected return for the portfolio is 16%, then:
21、.16 .0816% = 8% + 10% 浮 y0. 8.10Therefore, in order to have a portfolio with expected rate of retur n equal to 16%, the die nt must in vest 80% of total funds in the risky portfolio and 20% in T-bills.Client 's investment proportions:20. 0%inT-bills0.8X5% =20. 0%inStock A0.832% =25. 6%inStock B0
22、.843% =34. 4%inStock Cc.cC = 0. 8 x cP = 0. 8 x28% = 22. 4%8 a. oc = y x28%b.19. a.If your die nt prefers a sta ndard deviati on of at most 18%, the n: y 二 18/28 = 0. 6429 =64. 29% inv ested in the risky portfolio.E(rc) . 08 . 1 y . 08 (0. 6429 . 1) 14. 429%E.(g)二 o. 18 0. 080. 100.3644a4>3. 5 0.
23、 282 o. 2744Therefore, the clients optiorabpsoare: 36. 44% invested in the riskyportfolio and 63. 56% in vested in T-bills.b. E(rc) = 0. 08 + 0. 10 y* =>0. 08 + (0. 3644 x 0. 1) = 0. 1164 or 11. 644% c = 0. 3644 x28 二10. 203%20. a. If the period 1926012 is assumed to be represe ntative of future
24、expected performa nee, the n we use the followi ng data to compute the fracti on allocated to equity: A = 4, E(rM) - rf = 8. 10%, o = 20. 48% (we use the standard deviati on of the risk premium from Table 6. 7). The n y* is give n by:0.0810如 That is, 4828% of the portfolio should be allocated to equ
25、ity and 51.72% should be allocated to T-bills.b. If the period 1968-988 is assumed to be represe ntative of future expected performa nee, the n we use the follow ing data to compute the fract ion allocated to equity: A = 4, E(rw) - rf = 3. 44%, cm = 16. 71% andy* is given by:E( M rfo. 0344y*0.30804
26、0. 16712Therefore, 30.80% of the complete portfolio should be allocated to equity and 69.20% should be allocated to T-bills.c. In part (b), the market risk premium is expected to be lower tha n in part (a) and market risk is higher. Therefore, the reward-to-volati1ityratio is expected to be lower in
27、 part (b), which expla ins the greater proporti on inv ested in T-bills.21. a.E(rc)= 8% = 5% + y >(11% 5%) 08 .050.5.11.05b. de - y Xop = 0. 50 )1596 = 7. 5%c. The first die nt is more risk averse, preferri ng in vestme nts that have less risk as evide need by the lower sta ndard deviati on.22. J
28、oh nson requests the portfolio sta ndard deviati on to equal one half the market portfolio standard deviation. The market portfolio x 20% , which impliesp 10% . The intercept of the CML equals rf0. 05 and the slope of the CMLequals the Sharpe ratio for the market portfolio (35%). Therefore using the
29、 CML:E(M)rfE (r?) rf- p 0. 05 0. 35 0. 10 0. 085 8. 5%M23. Data: rf = 5%, E(rw) = 13%, d = 25%, and r： = 9%The CML and in differe nee curves are as follows:24. For y to be less tha n 1. 0 (that the inv estor is a len der), risk aversioiA( must be large eno ugh such that ：0. 13 0. 050. 25°1.28E(
30、r-) JA dMFor y to be greater tha n 1 (the inv estor is a borrower), A must be small eno ugh:E(r-) J0. 13 0. 090. 2520. 64For values of risk aversion within this range, the client will neither borrow nor lend but will hold a portfolio composed only of the optimal risky portfolio:y = 1 for 0. 64 献 <
31、;1. 2825. a. The graph for Problem 23 has to be redraw n here, with: E(rP) = 11% and(P = 15%15有0. 110. 05b. For a lending positi on： A,2.670. 15-0.110.09For a borrow ing positi on: A0, 890. 152Therefore, y = 1 for 0. 89 <A <2. 6726. The maximum feasible fee, denotef , depe nds on the reward-1o
32、-varlability ratio.For y < 1, the lending rate, 5%, is viewed as the releva nt risk-free rate, and we solve for f as follows：n 05 f,1305.15.251508.06.012, or 1.2%.25The n we no tice that,For y > 1, the borrow! ng rate, 9%, is the releva nt risk-free rate.eve n without a fee,the active fund is
33、in feriorto the passive fund because:13 - 09.15 .25More risk tolera nt in vestors (who are more in cli ned to borrow) will not be die nts of the fund. We find that f is negative： that is, you would need tcpay investors to choose your active fund. These inv estors desire higher ris-iigher return comp
34、leteportfolios and thus are in the borrowi ng range of the releva nt CAL. In this ran ge, the reward-1o-varlability ratio of the index (the passive fund) is better than that of the man aged fund.27.a. Slope of the CML13080. 20The diagram follows.CML and CALJSta ndard Deviati onb. My fund allows an i
35、nv estor to achieve a higher mea n for any give n sta ndard deviati on tha n would a passive strategy, i. e. , a higher expected retur n for any give n level of risk.28.expeca. With 70% of his money invested in my fund' s portfolio, the clientretur n is 15% per year with a sta ndard deviati on o
36、f 19. 6% per year. If he shifts that money to the passive portfolio (which has an expected retur n of 13% and sta ndard deviati on of 25%), his overall expected retur n becomes:E(rc) = rf + 0. 7 fE (rw) - rf = . 08 + 0. 7 x (. 1308) = . 115, or 11. 5%The sta ndard deviati on of the complete portfoli
37、o using the passive portfolio would be：cC = 0. 7 x cm = 0. 7 x25% = 17. 5%Therefore, the shift en tails a decrease in mean from 15% to 11. 5% and a decrease in standard deviation from 19. 6% to 17. 5%. Since both mean return and sta ndard deviati on decrease, it is not yet clear whether the move is
38、beneficial. The disadvantage of the shift is that, if the client is willing to accept a mean return on his total portfolio of 11. 5%, he can achieve it with a lower sta ndard deviati on using my fund rather tha n the passive portfolio.To achieve a target mea n of 11. 5%, we first write the mea n of
39、the completeportfolio as a function of the proportion invested in my fund (y):E (rc) = . 08 + y (. 108) = . 08 + . 10 y xOur target is: E(rc) = 11. 5%, Therefore, the proporti on that must be inv ested in my fund is determined as follows：.115 .08.115 = . 08 + . 10 y Xy0. 35.10The sta ndard deviati o
40、n of this portfolio would be：0 = y x28% = 0. 35 X 28% = 9. 8%Thus, by using my portfolio, the same 11. 5% expected return can be achieved with a sta ndard deviati on of on ly 9. 8% as opposed to the sta ndard deviati on of 17. 5% using the passive portfolio.b. The fee would reduce the reward-to-vola
41、ti1ity ratio, i. e., the slope of the CAL. The client will be indifferent between my fund and the passive portfolio if the slope of the after-fee CAL and the CML are equal. Lef denote the fee:1808 f.10 fSlope of CAL with fee .28.281308Slope of CML (which requires no fee)0. 20.25Setti ng these slopes
42、 equal we have： .10 f0.200.0444.4% per year.2829. a. The formula for the optimal proporti on to inv est in the passive portfolio is:E(rM)20My*Substitute the followi ng: E(rM)= 13%; rf = 8%; cm = 25%; A = 3. 5:0. 13 0. 08y*20.2286, or 22.86% in the passive portfolio3.5 0. 25b. The an swer here is the
43、 same as the an swer to Problem 28 (b). The fee that you can charge a die nt is the same regardless of the asset allocati on mix of the client ' s portfolio. You charge a fee that will equate the reward-to-volatility ratio of your portfolio to that of your competiti on.CFA PROBLEMS1. Utility for
44、 each investment = E(r) -0 5 4< X?"We choose the inv estme nt with the highest utility value, Inv estme nt 3.Expected Sta ndardreturndeviatio nUtility UInv estme ntE(r)10. 120. 30-0. 060020. 150. 50-0. 350030.210. 160. 158840. 240.210. 15182.When investors are risk neutral,thenA = 0; theinve
45、stmentwith the highest utility is Inv estme nt4 because it has the highest expected return.3. (b)4. In differe nee curve 2 because it is tangent to the CAL.5. Poi nt E6. (0.6$50, 000) + 0. 4 ( $30, 000)$5, 000 = $13,0007. (b) Higher borrowing rates will reduce the total return to the portfolio and t
46、his results in a partof the line that has a lower slope.8. Expected return for equity fund = T-bill rate + Risk premium = 6% + 10% = 16%Expected rate of return of theclient ' s portfolio =(0. 66%) + (0. 4 X6%)= 12% Expected return of the client' s portfolio = 0. 12X $100,000 = $12,000(which implies expected total wealth at the end of the period = $112,000)Standard deviation of clients overall portX14% 0. (8. 4%0.719. Reward-to-volati1ity ratio =CHAPTER 6: APPENDIX1. By year-e nd, the $50,000 in vestme nt will