# Chapter 7: Optimal Risky Portfolios Essay

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 most likely zero.) (a) Answer (a) is valid

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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