Question

please do the entire thing A B and C, im stuck, thanks!

1. You are given the following information: Stock Expected return (in %) o (in %) А 10 10 B The covariance between these returns is 16%. The risk-free rate is 6%. (a) Find the expected return and standard deviation of the following portfolios: i. 50% in A, 50% in B ii. 50% in A, 50% in the risk-free asset iii. 150% in A, financed by borrowing at the risk-free rate iv. -100% in A, rest invested at the risk-free rate v. 20% in A, 20% in B, rest invested at the risk-free rate (Hint: you can use what you calculated in part (a)). (b) Which portfolio of A and Rf has: i. An expected return of 10% ii. An expected return of 7.5% iii. A standard deviation of 7.5% (C) Which portfolio of A and B has: i. An expected return of 10% ii. An expected return of 7.5% iii. A standard deviation of 7.5% (Don't be too rigorous, just try to find one portfolio with this SD. You might need Excel for this one).

Answer 1

The return of a portfolio is the weighted return of the two stocks and

The standard deviation of a portfolio is given by

Op =

Where Wi is the weight of the security i,

is the standard deviation of returns of security i.

and is the correlation coefficient between returns of security i and security j

a) i) Expected Return = 0.5*10%+0.5*5% =7.5%

Standard Deviation = (0.5^2*0.1^2+0.5^2*0.05^2+2*0.5*0.5*0.0016)^0.5 =0.06265 = 6.265%

ii) As the standard deviation of Risk free asset is 0

Expected Return = 0.5*10%+0.5*6% =8%   

Standard Deviation = (0.5^2*0.1^2+0.5^2*0.00^2+2*0.5*0.5*0.0)^0.5 =0.05 =5%

iii) Weight of A will be 1.5 and weight of Risk free Asset will be -0.5

Expected Return = 1.5*10%+ (-0.5)*6% =12%   

Standard Deviation = (1.5^2*0.1^2+(-0.5)^2*0.00^2+2*0.5*(-0.5)*0.0)^0.5 =0.15 =15%

iv) Weight of A will be -1 and weight of Risk free Asset will be 2

   Expected Return = (-1)*10%+2*6% =2%   

Standard Deviation = ((-1)^2*0.1^2+0.5^2*0.00^2+2*0.5*0.5*0.0)^0.5 =0.10 =10%

v)  Expected Return = 0.2*10%+0.2*5% +0.6*6% =6.6%   

Standard Deviation = (0.2^2*0.1^2+0.2^2*0.05^2+0 + 2*0.2*0.2*0.0016 + 0 + 0)^0.5 =0.02506 =2.506%

b)

i) As A has a return of 10% and Risk free Asset 6%

Investing 100% in A and 0% in Risk free Asset gives a return of 10%

ii) Let w be the weight of A and (1-w) be the weight of Risk free Asset

w*10%+(1-w)*6% = 7.5%

=> w= 1.5%/4% = 0.375

So, 37.5% invested in A and 62.5% invested in Risk free Asset gives a return of 7.5%

iii) As A has a standard deviation of 10% and Risk free Asset has a standard deviation of 0%

Weight of A * Standard deviation of A = standard deviation of portfolio of A and Risk free Asset

=> w*10% = 7.5%

=> w =0.75

So, 75% invested in A and 25% invested in Risk free Asset gives a standard deviation of 7.5%

c)

i) Let w be the weight of A and (1-w) be the weight of B

w*10%+(1-w)*5% = 10%

=> w= 5%/5% = 1

So, 100% invested in A and 0% invested in B gives a return of 10%

ii)  Let w be the weight of A and (1-w) be the weight of B

w*10%+(1-w)*5% = 7.5%

=> w= 2.5%/5% = 0.5

So, 50% invested in A and 50% invested in B gives a return of 7.5%

iii)  Let w be the weight of A and (1-w) be the weight of B

w^2* 0.1^2+ (1-w)^2*0.05^2+ 2*w*(1-w) * 0.0016 = 0.075^2 = 0.005625

Multiplying by 1000

=> 10*w^2+ 2.5*w^2 - 3.2*w^2 -5*w+3.2*w=5.625-2.5

=> 9.3*w^2-1.8*w-3.125 =0

Solving this quadratic equation

w= (1.8 + (1.8^2-4*9.3*(-3.125))^0.5)/ (2*9.3)

= (1.8 + 10.93115)/18.6

=0.68447 or -0.490922

Taking only positive value

68.447% invested in A and 31.553% invested in B gives a standard deviation of 7.5%