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Portfolio Solution

Problem 1

  1. A) Portfolio A:

Annual Expected Return = Rf + Risk PremiumA1A1 + Risk PremiumA2A2

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 PremiumA1A1 + Risk PremiumA2A2

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%

  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 = Rf + Risk PremiumC1C1 + Risk PremiumC2C2

                           = 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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