Portfolio Solution

Problem 1

1. A) Portfolio A:

Annual Expected Return = R_f + Risk Premium_A1*β_{A1 +} Risk Premium_A2*β_A2

Or 0.19 = 0.06 + Risk Premium_A1*1 ₊ Risk Premium_A2*2

Or Risk Premium_A1*1 ₊ Risk Premium_A2*2 = 0.13 ----------------------------(i)

Portfolio B:

Annual Expected Return = R_f + Risk Premium_A1*β_{A1 +} Risk Premium_A2*β_A2

Or 0.22 = 0.06 + Risk Premium_A1*2 ₊ Risk Premium_A2*2

Or Risk Premium_A1*2 ₊ Risk Premium_A2*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%

1. 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%

1. 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%

1. 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

1. 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%

1. 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

1. D) Portfolio C:

Required Return = R_f + Risk Premium_C1*β_{C1 +} Risk Premium_C2*β_C2

                           = 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.

1. 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 10% = (Beta of Portfolio P)*(SD of market) 10% = 0.60*SD of market SD of market = 10%/0.60 SD of market = 16.67%
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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