投资学第10版课后习题答案

CHAPTER 7: OPTIMAL RISKY PORTFOLIOS

PROBLEM SETS

1. (a) and (e). Short-term rates and labor issues are factors that

are common to all firms and therefore must be considered as market risk factors. The remaining three factors are unique to this corporation and are not a part of market risk.

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 (r S ) = 20%, E (r B ) = 12%, σS = 30%, σB = 15%, ρ =

From the standard deviations and the correlation coefficient we generate the covariance matrix [note that (,)S B S B Cov r r ρσσ=⨯⨯]: Bonds Stocks Bonds 225 45 Stocks 45 900

The minimum-variance portfolio is computed as follows:

w Min (S ) =1739.0)

452(22590045

225)(Cov 2)(Cov 2

22=⨯-+-=-+-B S B S B S B ,r r ,r r σσσ w Min (B ) = 1 =

The minimum variance portfolio mean and standard deviation are:

E (r Min ) = × .20) + × .12) = .1339 = %

σMin = 2/12222)],(Cov 2[B S B S B B S S

r r w w w w ++σσ = [ 900) + 225) + (2 45)]1/2

= %

5.

Proportion in Stock Fund Proportion

in Bond Fund Expected

Return

Standard Deviation

% % % %

minimum

tangency

Graph shown below.

0.00

5.00

10.00

15.00

20.00

25.00

0.00 5.00 10.00 15.00 20.00 25.00 30.00

Tangency Portfolio

Minimum Variance Portfolio

Efficient frontier of risky assets

CML

INVESTMENT OPPORTUNITY SET

r f = 8.00

6. The above graph indicates that the optimal portfolio is the

tangency portfolio with expected return approximately % and

standard deviation approximately %.

7. The proportion of the optimal risky portfolio invested in the stock

fund is given by:

2

22[()][()](,)

[()][()][()()](,)S f B B f S B S S f B B f S

S f B f S B E r r E r r Cov r r w E r r E r r E r r E r r Cov r r σσσ-⨯--⨯=

-⨯+-⨯--+-⨯

[(.20.08)225][(.12.08)45]

0.4516[(.20.08)225][(.12.08)900][(.20.08.12.08)45]

-⨯--⨯=

=-⨯+-⨯--+-⨯

10.45160.5484B w =-=

The mean and standard deviation of the optimal risky portfolio are:

E (r P ) = × .20) + × .12) = .1561 = % σp = [ 900) +

225) + (2

× 45)]1/2

= %

8. The reward-to-volatility ratio of the optimal CAL is:

().1561.08

0.4601.1654

p f

p

E r r σ--=

=

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:

()().080.4601p f

C f C C P

E r r E r r σσσ-=+

=+

If E (r C ) is equal to 14%, then the standard deviation of the portfolio is %.

b. To find the proportion invested in the T-bill fund, 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 invested in the portfolio P . The mean of any portfolio along the optimal CAL is:

()(1)()[()].08(.1561.08)C f P f P f E r y r y E r r y E r r y =-⨯+⨯=+⨯-=+⨯-

Setting E (r C ) = 14% we find: y = and (1 − y ) = (the proportion invested in the T-bill fund).