Econ175 Answer Key to Homework3
6. Answer:
Given Conditions:
Funds
Standard Deviation (%)
Expected Return (%)
Stock fund (S)
32
22
Bond fund (B)
23
13
T-bill fund
0
9
Proportion
Standard Expected
Deviation Return
(%) p (%) E(rp)
WeightB
WB
WeightS
Ws
WB*SDB
Ws*SDs
CorrelBS

100%B+0*S
23.00
13
100%
0
23
0
0.15
80%B+20%S
20.37
14.8
80%
20%
18.4
6.4
0.15
60%B+40%S
20.18
16.6
60%
40%
13.8
12.8
0.15
40%B+60%S
22.50
18.4
40%
60%
9.2
19.2
0.15
20%B+80%S
26.68
20.2
20%
80%
4.6
25.6
0.15
0*B+100%S
32.00
22
0%
100%
0
32
0.15
P192: (7.2)
E(rp) =WBE(rB) + WSE(rS)
(7.3)
2p=(WBB)2+(WSS)2+2(WBB)(WSS)
For the minimum variance portfolio: E(rp) =16.6; p= 20.18
Investment opportunity set
25
Standard deviation
20
15
10
5
0
-
5.00
10.00
15.00
20.00
Expected Return
Investment opportunity set
25.00
30.00
35.00
Funds
Standard
Deviation
(%) p
Expected E(rorp)-rf R/V Ratio
Return
(%) E(rp)
T-bill Fund
0
9
20%B+80%S
26.68
20.20
extension point
35
23.69
T-bill Fund
0
9
40%B+60%S
22.50
18.40
Comments
20%B+80%S is the optimal risky portfolio
for increments of 20%.
11.20
0.419783
9.40
40%B+60%S is just a candidate for the
optimal risky portfolio. It can be easily
0.417769
seen that its R/V ratio is the lowest one
among the three.
7. Answer:
For the optimal risky portfolio: E(rp) =20.20; p= 26.68
The Tangent to the opportunity set
25
Expected Return
20
15
10
5
0
-
5.00
10.00
15.00
20.00
25.00
30.00
35.00
40.00
Standard Deviation
Investment opportunity set
Tangent
8. Answer: reward-to-variability ratio = (20.20-9)/26.68 = 0.419783
(by 6.11 on page 172)
9. Answer: a) p1= (15-9)/ 0.419783 = 14.29 (by (6.11) on page 172)
b) By Wf + Worp =1 and (7.2) E(rp) =Wf*rf + Worp*E(rorp)
15= Wf*9 + (1- Wf)*20.20  Wf = 46%.
WB= (1- Wf)*20% = 11% WS = (1- Wf)*80% = 43%
10. Answer: By WS + WB =1 and (7.2) E(rp) =WSE(rS) + WBE(rB)
15= WB*13 + (1- WB)*22  WB = 78%
WS= 1- WB = 22%
By (7.3) 2p=(WBB)2+(WSSS)2+2(WBB)(WSS), p2= 20.22
Conclusion: p2 > p1, i.e. the standard deviation is higher than the one
of the previous optimal portfolio, i.e. it is riskier than the previous
optimal portfolio which yields the same level of expected return.
12. Answer: The Equilibrium rf could not be greater than 10%. First, we can
replicate a risk-free asset by a portfolio composed of 60% stock A and 40% stock B.
Since stock A and B have correlation = -1, the variance of the portfolio become zero. You
can use (7.3) on page 192 of the textbook to check the result. This portfolio will yield a
risk-free rate of 10%. If the market is in equilibrium, rf can’t be any number other than
10%. To see this, when rf >10%, you will only want to hold stock B and the risk-free
asset because the portfolios along this CAL give higher return than any other portfolio
involving stock A. Then, no one want to hold stock A in the market, the price of A will
fall since people are selling it. That means the expected return will become higher for A.
And the risk-free rate will be lower since people are buying the risk-free asset then its
price will go up. This will go on until the market moves back into equilibrium again. It is
also true that when rf <10%, disequilibrium will occur.
14. Answer: It is impossible to get such a diagram if your assistant did his work
correctly, so you would not trust him. Look at Figure 7.9 on page 204 in the textbook
for a concrete example. Intuitively, since the correlation ρ is always less than or equal to
1, the efficient frontier formed by A and B can only bulge to the left. This is because
when ρ = 1, i.e. ρ achieves its highest possible value, the efficient frontier of A and B is a
straight line passing them. For any given level of expected return between the expected
return of A and B, the lower value of ρ, i.e. ρ < 1, the lower the standard deviation of the
portfolio composed with A and B, so the efficient frontier bulges to the left.
17. Answer: E(rp) =  + E(rM)  E(rp)= + E(rM)
12%=  + 1.2*10%   = 0  E(rp) = 1.2*8% = 9.6%
18. Answer: a) Stock B: In a diversified portfolio, stock B’s beta is greater than that of
stock A, so its systematic risk is not diversifiable in the portfolio, thus stock B is the
riskiest one.
Part b): Since the picture is too ambiguous to draw meaningful conclusion, we just
neglect this part.