Portfolio Solution
Problem 1
- A) Portfolio A:
Annual Expected Return = Rf + Risk PremiumA1*βA1 + Risk PremiumA2*βA2
Or 0.19 = 0.06 + Risk PremiumA1*1 + Risk PremiumA2*2
Or Risk PremiumA1*1 + Risk PremiumA2*2 = 0.13 ----------------------------(i)
Portfolio B:
Annual Expected Return = Rf + Risk PremiumA1*βA1 + Risk PremiumA2*βA2
Or 0.22 = 0.06 + Risk PremiumA1*2 + Risk PremiumA2*2
Or Risk PremiumA1*2 + Risk PremiumA2*2 = 0.16 ----------------------------(ii)
By equating both the equation, we get:
Risk Premium of F1 = 0.03 or 3%
Risk Premium of F2 = 0.05 or 5%
- B) Construction of unit portfolio by using Factor 1:
Portfolio using Factor 1:
|
Weight of resulting unit portfolio: | |
|
Weight of Portfolio A |
60% |
|
Weight of Portfolio B |
0% |
|
Weight of Risk-free asset |
40% |
- Expected Return: (Return on Portfolio A*Weight of Portfolio A)+ (Return on Portfolio B*Weight of Portfolio B)+ (Return on Portfolio Risk-free asset*Weight of Risk-free asset)
= (19%*60%)+(22%*0%)+(6%*40%)
= 13.80%
- Portfolio beta = (Beta on Portfolio A*Weight of Portfolio A)+ (Beta on Portfolio A*Weight of Portfolio B)+ (Beta on Portfolio A*Weight of Risk-free asset)
=(1*60%)+(2*0%)+(0*40%)
= 0.60
- C) Construction of unit portfolio by using Factor 2:
Portfolio using Factor 2:
|
Weight of resulting unit portfolio: | |
|
Weight of Portfolio A |
0% |
|
Weight of Portfolio B |
70% |
|
Weight of Risk-free asset |
30% |
- Expected Return: (Return on Portfolio A*Weight of Portfolio A)+ (Return on Portfolio B*Weight of Portfolio B)+ (Return on Portfolio Risk-free asset*Weight of Risk-free asset)
= (19%*0%)+(22%*70%)+(6%*30%)
= 17.20%
- Portfolio beta = (Beta on Portfolio A*Weight of Portfolio A)+ (Beta on Portfolio A*Weight of Portfolio B)+ (Beta on Portfolio A*Weight of Risk-free asset)
=(1*0%)+(2*70%)+(0*30%)
= 1.40
- D) Portfolio C:
Required Return = Rf + Risk PremiumC1*βC1 + Risk PremiumC2*βC2
= 0.06 + (0.03*2) + (0.05*0)
= 0.06 + 0.06 + 0
= 0.12 or 12%
Annual expected return = 16%
Since, actual return is less than annual expected return, hence portfolio is overvalued.
- E) Since, the beta of portfolio C is 2 and return is 12% which is lower than the return of portfolio of A, B and risk-free asset i.e, it provides an expected return of 22% with same beta> hence, we should short sell the portfolio C to earn an income of 10% with no investment.
Income = Return on combined portfolio -Return on Portfolio C
= 22% - 12% = 10%
Weight are given below (in both conditions):
Weight of Portfolio A : 0%
Weight of Portfolio B : 100%
Weight of Portfolio Risk free asset : 0%
Problem 2
|
A) |
Assets |
Expected Return |
SD |
Correlation with P |
Market beta | ||||
|
Stock A |
21% |
20% |
95% |
1.14 | |||||
|
Stock B |
34% |
40% |
80% |
3.20 | |||||
|
Portfolio P |
8% |
10% |
100% |
0.60 | |||||
|
T-Bill |
2% |
0% |
0% |
- | |||||
|
Here, | |||||||||
|
Rf = 2% | |||||||||
|
Portfolio P | |||||||||
|
By using CAPM, calculate Rm: | |||||||||
|
RR = Rf + (Rm - Rf)*β | |||||||||
|
8% = 2% + (Rm - 2%)*0.60 | |||||||||
|
Rm - 2% = 6%/0.60 | |||||||||
|
Rm - 2% = 10% | |||||||||
|
Rm = 12% | |||||||||
|
Standard Deviation of market portfolio: Systematic Risk of market = SD of portfolio P
| |||||||||
|
B) |
Expected Return of Stock A: | ||||||||
|
By using CAPM, calculate RR: | |||||||||
|
RR = Rf + (Rm - Rf)*β | |||||||||
|
RR = 2% + (12% - 2%)*1.90 | |||||||||
|
RR = 2% + 19% | |||||||||
|
RR = 21% | |||||||||
|
C) |
Market beta of Stock B: | ||||||||
|
Beta = (SD of Stock A/ SD Portfolio P)*Correlation between Stock B and Portfolio P | |||||||||
|
Beta = (40%/10%)*0.80 | |||||||||
|
Beta = 3.20 | |||||||||
|
D) |
Systematic Risk of Stock B = SD of portfolio*β | ||||||||
|
Systematic Risk of Stock B = 10%*3.20 | |||||||||
|
Systematic Risk of Stock B = 32% | |||||||||
Problem 3
|
A) |
Fund |
Expected Return |
SD |
Beta |
(RRsec - Rf)/βsec |
Ranking |
|
Fund A |
8% |
20% |
0.50 |
0.12 |
1 | |
|
Fund B |
18% |
60% |
2.00 |
0.08 |
3 | |
|
Fund C |
16% |
40% |
1.50 |
0.09 |
2 | |
|
T-bill |
2% | |||||
|
She should invest in Fund A. | ||||||
|
B) |
Risk Aversion coefficient = 1.50 | |||||
|
Utility score of investment = Rf - 0.5*A*SD^2 | ||||||
|
= 0.08 - 0.5*1.50*(0.20)2 | ||||||
|
= 0.08 - 0.03 | ||||||
|
= 0.05 or 5% | ||||||
|
Now, to get the expected return of 5%, the proportion of Fund A in portfolio can be calculated as follows: | ||||||
|
Expected return = (RR of Fund A)*(Weight of Fund A) + (RR of T-bill)*(Weight of T-bill) | ||||||
|
Expected return = (0.08*Wa) + (0.02*Wt) | ||||||
|
Expected return = (0.08*Wa) + (0.02*(1-Wa)) | ||||||
|
0.05 = 0.08Wa + 0.02 - 0.02Wa | ||||||
|
0.05 = 0.06Wa +0.02 | ||||||
|
0.03 = 0.06Wa | ||||||
|
Wa = 0.03/0.06 | ||||||
|
Wa = 0.50 or 50% | ||||||
|
Weight of this fund in his portfolio = 50% | ||||||
C)
|
Calculation of Portfolio Beta | |||
|
Fund A |
0.33 |
0.5 |
0.165 |
|
Fund B |
0.33 |
2 |
0.66 |
|
Fund C |
0.33 |
1.5 |
0.495 |
|
1.32 | |||
|
Calculation of Expected Return | |||
|
Fund A |
0.33 |
0.08 |
0.0264 |
|
Fund B |
0.33 |
0.18 |
0.0594 |
|
Fund C |
0.33 |
0.16 |
0.0528 |
|
0.1386 | |||
|
Basis |
Portfolio | ||
|
Alpha |
AR - RR | ||
|
Calculation |
0.1386 - [0.02+(0.10-0.02)*1.32] | ||
|
|
0.013 | ||
|
Remarks |
Under-priced | ||
|
Basis |
Portfolio | ||
|
Sharpe Ratio |
(RRport - Rf)/βport | ||
|
(13.86%-2%)/1.32 | |||
|
0.08985 | |||
Problem 4
Variance of portfolio = + Covariance [Covariance = Beta of stock * Variance of Market]
= + 1*20*20
= 25+400
= 425
Standard deviation of portfolio =
= 20.62%
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