CHAPTER 6: PORTFOLIO SELECTION

Transcription

1 CHAPTER 6: PORTFOLIO SELECTION 6-1

2 21. The parameters of the opportunity set are: E(r S ) = 20%, E(r B ) = 12%, σ S = 30%, σ B = 15%, ρ =.10 From the standard deviations and the correlation coefficient we generate the covariance matrix [note that Cov(r S,r B ) = ρσ S σ B ]: Bonds Stocks Bonds Stocks The minimum-variance portfolio is found by applying the formula: w Min (S) = = σ 2 B Cov(B,S) σ 2 S + σ2 B 2Cov(B,S) =.1739 w Min (B) =.8261 The minimum variance portfolio mean and standard deviation are: E(r Min ) = = 13.39% σ Min = [W 2 S σ2 S + W2 B σ2 B + 2W S W B Cov(S,B)]1/2 = [ ] 1/2 = 13.92% 6-2

3 22. % in stocks % in bonds Exp. return Std. Dev 0.00% % % 82.61% minimum variance 20.00% 80.00% % 60.00% % 54.84% tangency portfolio 60.00% 40.00% % 20.00% % 0.00% INVESTMENT OPPORTUNITY SET CML Tangency Pf Efficient frontier of risky assets Min.Var

4 E(r) σ The graph approximates the points: Min. Variance Portf. 13.4% 13.9% Tangency Portfolio The proportion of stocks in the optimal risky portfolio is given by: W S = [E(r S ) r f ]σ 2 B [E(r B ) r f ]Cov(B,S) [E(r S ) r f ]σ 2 B + [E(r B ) r f ]σ2 S [E(r S ) r f + E(r B ) r f ]Cov(B,S) = (20 8)225 (12 8)45 (20 8)225 + (12 8)900 [ ]45 =.4516 W B =.5484 The mean and standard deviation of the optimal risky portfolio are: E(r p ) = = 15.61% σ p = [ ] 1/2 = 16.54% 25. The reward-to-variability ratio of the optimal CAL is: E(r p ) r f = =.4601 σ p a. If you require your portfolio to yield a mean return of 14% you can find the corresponding standard deviation from the optimal CAL. The formula for this CAL is: E(r C ) = r f + E(r p) r f σ p σ C = σ C Setting E(r C ) equal to 14% we find that the standard deviation of the optimal portfolio is 13.04%. b. To find the proportion invested in T-bills we remember that the mean of the complete portfolio, 14%, is an average of the T-bill rate and the optimal combination of stocks and bonds, P. Let y be the proportion in this portfolio. The mean of any portfolio along the optimal CAL is: E(r C ) = (l y) r f + y E(rp) = r f + y [E(rp) r f ] = 8 + y ( ) 6-4

5 Setting E(r C ) = 14% we find: y =.7884, and 1 y =.2116, the proportion in T-bills. To find the proportions invested in each of the funds we multiply.7884 by the proportions of the stocks and bonds in the optimal risky portfolio: Proportion of stocks in complete portfolio = =.3560 Proportion of bonds in complete portfolio = = Using only the stock and bond funds to achieve a portfolio mean of 14% we must find the appropriate proportion in the stock fund, w S, and w B = 1 w S in the bond fund. The portfolio mean will be: 14 = 20w S + 12(1 w S ) = w S w S =.25 So the proportions will be 25% in stocks and 75% in bonds. The standard deviation of this portfolio will be: σ p = ( ) 1/2 = 14.13%. This is considerably larger than the standard deviation of 13.04% achieved using T-bills and the optimal portfolio. 28. With no opportunity to borrow you wish to construct a portfolio with a mean of 24%. Since this exceeds the mean on stocks of 20%, you will have to go short on bonds, which have a mean of 12%, and use the proceeds to buy additional stock. The graphical representation of your risky portfolio is point Q on the following graph: OPPORTUNITY SET WITH NO RISK-FREE ASSET Q P Standard Deviation (%) Point Q is the stock/bond combination with mean of 24%. Let w S be the proportion of stocks and 1 w S be the proportion of bonds required to achieve the 24% mean. Then: 6-5

6 24 = 20 w S + 12 (1 w S ) = w S w S = 1.50, and 1 w S =.50 Therefore, you would have to sell short an amount of bonds equal to.50 of your total funds, and invest 1.50 times your total funds in stocks. The standard deviation of this portfolio would be: σ Q = [ (.50) (1.50) (.50) 45] 1/2 = 44.87% If you were allowed to borrow at the risk-free rate of 8%, the way to achieve the target 24% would be to invest more than 100% of your funds in the optimal risky portfolio, moving out along the CAL to the right of P, up to R, on the following graph INVESTMENT OPPORTUNITY SET R Q Tangency Pf CML Efficient frontier of risky assets Min.Var R is the point on the optimal CAL which has the mean of 24%. Using the formula for the optimal CAL we can find the corresponding standard deviation: E(r C ) = σ C = 24 Setting E(r C ) = 24, we get: σ C = 34.78%, which is considerably less than the 44.87% standard deviation you would get without the possibility of borrowing at the risk-free rate 6-6

7 of 8%. What is the portfolio composition of point R on the optimal CAL? The mean of any portfolio along this CAL is: E(r C ) = r f + y[e(r P ) r f ] where y is the proportion invested in the optimal risky portfolio P and r P is the mean of that portfolio, which is 15.61%. 24 = 8 + y( ) y = This means that for every $1 of your own funds invested in portfolio P, you would borrow an additional $ and invest it also in portfolio P. 29. a Optimal CAL P Stocks Gold Standard Deviation(%) Even though gold seems dominated by stocks, it still might be an attractive asset to hold as a part of a portfolio. If the correlation between gold and stocks is sufficiently low, it will be held as an element in a portfolio -- the optimal tangency portfolio. 6-7

8 b. If gold had a correlation coefficient with stocks of +1, it would not be held. The optimal CAL would be comprised of bills and stocks only. Since the set of risk/return combinations of stocks and gold would plot as a straight line with a negative slope (see the following graph), it would be dominated by the stocks portfolio. Of course, this situation could not persist. If no one desired gold, its price would fall and its expected rate of return would increase until it became an attractive enough asset to hold Stocks 10 Gold 5 r f Standard Deviation(%) 30. Since A and B are perfectly negatively correlated, a risk-free portfolio can be created and its rate of return in equilibrium will be the risk-free rate. To find the proportions of this portfolio (with w A invested in A and w B = 1 w A in B), set the standard deviation equal to zero. With perfect negative correlation, the portfolio standard deviation reduces to σ P = Abs[ w A σ A w B σ B ] 0 = 5w A 10(1 w A ) w A =.6667 The expected rate of return on this risk-free portfolio is: E(r) = = 11.67% Therefore, the risk-free rate must also be 11.67%. 6-8

9 31. False. If the borrowing and lending rates are not identical, then depending on the tastes of the individuals (that is, the shape of their indifference curves), borrowers and lenders could have different optimal risky portfolios. 32. False. The portfolio standard deviation equals the weighted average of the componentasset standard deviations only in the special case that all assets are perfectly positively correlated. Otherwise, as the formula for portfolio standard deviation shows, the portfolio standard deviation is less than the weighted average of the component-asset standard deviations. The portfolio variance will be a weighted sum of the elements in the covariance matrix, with the products of the portfolio proportions as weights. 33. The probability distribution is: Probability Rate of Return.7 100%.3 50% Mean = ( 50) = 55% Variance =.7 (100 55) ( 50 55) 2 = 4725 Standard deviation = /2 = 68.74% 34. σ P = 30 = yσ = 40y y =.75 E(r p ) = (30 12) = 25.5% Note that each correlation is based on only seven observations, so we cannot really arrive at any statistically significant conclusion. Looking at the numbers, however, it appears that there is persistent serial correlation with the exception of large-company stocks. This conclusion changes when we turn to real rates in the next problem. 6-9