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CHAPTER 7: OPTIMAL RISKY PORTFOLIOS

CHAPTER 7: OPTIMAL RISKY PORTFOLIOS

PROBLEM SETS 1. 2. (a) and (e). (a) and (c). After real estate is added to the portfolio, there are four asset classes in the portfolio: stocks, bonds, cash and real estate. Portfolio variance now includes a variance term for real estate returns and a covariance term for real estate returns with returns for each of the other three asset classes. Therefore, portfolio risk is affected by the variance (or standard deviation) of real estate returns and the correlation between real estate returns and returns for each of the other asset classes. (Note that the correlation between real estate returns and returns for cash is ...view middle of the document...

00% 13.92% 13.94% 15.70% 16.54% 19.53% 24.48% 30.00%

minimum variance

tangency portfolio

Graph shown below.

25.00

INVESTMENT OPPORTUNITY SET

20.00

CML

Tangency Portfolio

15.00

Efficient frontier of risky assets

10.00

rf = 8.00

5.00

Minimum Variance Portfolio

0.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00

6.

The above graph indicates that the optimal portfolio is the tangency portfolio with expected return approximately 15.6% and standard deviation approximately 16.5%.

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CHAPTER 7: OPTIMAL RISKY PORTFOLIOS

7.

The proportion of the optimal risky portfolio invested in the stock fund is given by:

wS [ E ( rS ) r f ] [ E ( rS ) r f ]

2 B 2 B

[ E ( r B ) r f ] C o v ( rS , r B )

2 S

[ E ( rB ) r f ]

[ E ( rS ) r f E ( r B ) r f ] C o v ( r S , r B )

0 .4 5 1 6

[(.2 0 .0 8 ) 2 2 5 ] [(.1 2 .0 8 ) 4 5 ] [(.2 0 .0 8 ) 2 2 5 ] [(.1 2 .0 8 ) 9 0 0 ] [(.2 0 .0 8 .1 2 .0 8 ) 4 5 ]

w B 1 0 .4 5 1 6 0 .5 4 8 4

The mean and standard deviation of the optimal risky portfolio are: E(rP) = (0.4516 × .20) + (0.5484 × .12) = .1561 = 15.61% σp = [(0.45162 900) + (0.54842 225) + (2 0.4516 0.5484 × 45)]1/2 = 16.54% 8. The reward-to-volatility ratio of the optimal CAL is:

E ( rp ) r f

.1 5 6 1 .0 8 .1 6 5 4

0 .4 6 0 1 .4601

should be .4603 (rounding)

p

9.

a.

If you require that your portfolio yield an expected return of 14%, then you can find the corresponding standard deviation from the optimal CAL. The equation for this CAL is:

E ( rC ) r f E ( rp ) r f

C .0 8 0 .4 6 0 1 C

.4601 should be .4603 (rounding)

P

If E(rC) is equal to 14%, then the standard deviation of the portfolio is 13.03%. b. To find the proportion invested in the T-bill fund, remember that the mean of the complete portfolio (i.e., 14%) is an average of the T-bill rate and the optimal combination of stocks and bonds (P). Let y be the proportion invested in the portfolio P. The mean of any portfolio along the optimal CAL is:

E ( rC ) (1 y ) r f y E ( rP ) r f y [ E ( rP ) r f ] .0 8 y ( .1 5 6 1 .0 8 )

Setting E(rC) = 14% we find: y = 0.7881 and (1 − y) = 0.2119 (the proportion invested in the T-bill fund). To find the proportions invested in each of the funds, multiply 0.7884 times the respective proportions of stocks and bonds in the optimal risky portfolio: Proportion of stocks in complete portfolio = 0.7881 0.4516 = 0.3559 Proportion of bonds in complete portfolio = 0.7881 0.5484 = 0.4322

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CHAPTER 7: OPTIMAL RISKY PORTFOLIOS

10.

Using only the stock and bond funds to achieve a portfolio expected return of 14%, we must find the appropriate proportion in the stock fund (wS) and the appropriate proportion in the bond fund (wB = 1 − wS) as follows: .14 = .20 × wS + .12 × (1 − wS) = .12 + .08 × wS wS = 0.25 So the...

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