Fin 3710 Investment Analysis Professor Rui Yao CHAPTER 6: EFFICIENT DIVERSIFICATION

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1 HW 4 Fin 3710 Investment Analysis Professor Rui Yao CHAPTER 6: EFFICIENT DIVERIFICATION 1. E(r P ) = (0.5 15) + (0.4 10) + (0.10 6) = 1.1% 3. a. The mean return should be equal to the value comuted in the sreadsheet. The fund's return is 3% lower in a recession, but 3% higher in a boom. However, the variance of returns should be higher, reflecting the greater disersion of outcomes in the three scenarios. b. Calculation of mean return and variance for the stock fund: (A) () (C) (D) (E) (F) (G) Rate of from Exected quared Col. G Recession Normal oom Exected = 10.0 Variance = 96.4 tandard = 17. c. Calculation of covariance: (A) () (C) (D) (E) (F) from Mean Col. D Col. E tock ond Recession Normal oom Covariance = -13 Covariance has increased because the stock returns are more extreme in the recession and boom eriods. This makes the tendency for stock returns to be oor when bond returns are good (and vice versa) even more dramatic.

2 4. a. One would exect variance to increase because the robabilities of the extreme outcomes are now higher. b. Calculation of mean return and variance for the stock fund: (A) () (C) (D) (E) (F) (G) Rate of from Exected quared Col. G Recession Normal oom Exected = 9.0 Variance = 9.8 tandard = c. Calculation of covariance: (A) () (C) (D) (E) (F) from Mean Col. D Col. E tock ond Recession Normal oom Covariance = -15 Covariance has increased because the robabilities of the more extreme returns in the recession and boom eriods are now higher. This makes the tendency for stock returns to be oor when bond returns are good (and vice versa) more dramatic. 6. The arameters of the oortunity set are: E(r ) = 15%, E(r ) = 9%, = 3%, = 3%, ρ = 0.15,r f = 5.5% From the standard deviations and the correlation coefficient we generate the covariance matrix [note that Cov(r, r ) = ρ ]: onds tocks onds tocks The minimum-variance ortfolio roortions are:

3 w Min () = + w Min () = Cov(r, r ) Cov(r, r ) = = ( 110.4) The mean and standard deviation of the minimum variance ortfolio are: E(r Min ) = ( %) + ( %) = 10.89% 1 [ w + w + w w Cov(r,r ] Min = ) = [( ) + ( ) + ( )] 1/ = 19.94% % in stocks % in bonds Ex. return td dev minimum variance tangency ortfolio Investment oortunity set for stocks and bonds CAL min var tandard (%) 40

4 The grah aroximates the oints: E(r) Minimum Variance Portfolio 10.89% 19.94% Tangency Portfolio 13.5% 4.57% 8. The reward-to-variability ratio of the otimal CAL is: E(r ) r f = = a. The equation for the CAL is: E(r E(r ) r f C ) = rf + C = etting E(r C ) equal to 1% yields a standard deviation of 0.61%. b. The mean of the comlete ortfolio as a function of the roortion invested in the risky ortfolio (y) is: E(r C ) = (l y)r f + ye(r P ) = r f + y[e(r P ) r f ] = y( ) etting E(r C ) = 1% y = (83.87% in the risky ortfolio) 1 y = (16.13% in T-bills) From the comosition of the otimal risky ortfolio: Proortion of stocks in comlete ortfolio = = Proortion of bonds in comlete ortfolio = = C 10. Using only the stock and bond funds to achieve a mean of 1% we solve: 1 = 15w + 9(1 w ) = 9 + 6w w = 0.5 Investing 50% in stocks and 50% in bonds yields a mean of 1% and standard deviation of: P = [( ) + ( ) + ( )] 1/ = 1.06% The efficient ortfolio with a mean of 1% has a standard deviation of only 0.61%. Using the CAL reduces the D by 45 basis oints. 11. a. Although it aears that gold is dominated by stocks, gold can still be an attractive diversification asset. If the correlation between gold and stocks is sufficiently low, gold will be held as a comonent in the otimal ortfolio.

5 b. If gold had a erfectly ositive correlation with stocks, gold would not be a art of efficient ortfolios. The set of risk/return combinations of stocks and gold would lot as a straight line with a negative sloe. (ee the following grah.) The grah shows that the stock-only ortfolio dominates any ortfolio containing gold. This cannot be an equilibrium; the rice of gold must fall and its exected return must rise TOCK 8 6 GOLD tandard (%) ince tock A and tock are erfectly negatively correlated, a risk-free ortfolio can be created and the rate of return for this ortfolio in equilibrium will always be the risk-free rate. To find the roortions of this ortfolio [with w A invested in tock A and w = (1 w A ) invested in tock ], set the standard deviation equal to zero. With erfect negative correlation, the ortfolio standard deviation reduces to: P = Abs[w A A w ] 0 = 40 w A 60(1 w A ) w A = 0.60 The exected rate of return on this risk-free ortfolio is: E(r) = (0.60 8%) + ( %) = 10.0% Therefore, the risk-free rate must also be 10.0%.