title

Chapter 07 - Optimal Risky Portfolios

CHAPTER 7: OPTIMAL RISKY PORTFOLIOS
PROBLEM SETS
1.

(a) and (e).

2.

(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.)

3.

(a) Answer (a) is valid because it provides the definition of the minimum variance portfolio. 4.

The parameters of the opportunity set are:
E(rS) = 20%, E(rB) = 12%, σS = 30%, σB = 15%, ρ = 0.10
From the standard deviations and the correlation coefficient we generate the covariance matrix [note that Cov(rS, rB) = ρσSσB]:
Bonds
Stocks

Bonds
225
45

Stocks
45
900

The minimum-variance portfolio is computed as follows: wMin(S) =

σ 2 − Cov(rS , rB )
225 − 45
B
=
= 0.1739
2
2 σ S + σ B − 2Cov(rS , rB ) 900 + 225 − (2 × 45)

wMin(B) = 1 − 0.1739 = 0.8261
The minimum variance portfolio mean and standard deviation are:
E(rMin) = (0.1739 × 20) + (0.8261 × 12) = 13.39%
2 2 σMin = [ w S σ S + w 2 σ 2 + 2w S w B Cov(rS , rB )]1 / 2
B B

= [(0.17392 × 900) + (0.82612 × 225) + (2 × 0.1739 × 0.8261 × 45)]1/2
= 13.92%

7-1

Chapter 07 - Optimal Risky Portfolios

5.
Proportion
in stock fund
0.00%
17.39%
20.00%
40.00%
45.16%
60.00%
80.00%
100.00%

Proportion in bond fund
100.00%
82.61%
80.00%
60.00%
54.84%
40.00%
20.00%
0.00%

Expected return 12.00%
13.39%
13.60%
15.20%
15.61%
16.80%
18.40%
20.00%

Standard
Deviation
15.00%
13.92%
13.94%
15.70%
16.54%
19.53%