Question

Paul is considering investing in a portfolio with two assets a and b. The following is his prediction about future return of a and b in the next year:

probability Asset a. Asset b

20% 12% 15%

30% 10% 12.5%

40% 7.5% 8%

10%. 5%. 4.5%

a. Calculate the expected return and standard deviation of asset a and b

b. If Paul is going to invest 40% of his wealth on asset a and the remaining to asset b, what are the expected return and standard deviation of the portfolio?

c. If the risk-free rate in the market is 1.25% and market return is 8.75%. Calculate required return of the portfolio based on camp. Should Paul invest in this portfolio or not? Why?

Answer 1

A)

Expected return= P* R

P is the probability

R is the return

for asset A E(r)= 0.20(0.12) + 0.3(0.1) + 0.4(0.075) + 0.1(0.05)= 8.9%

for asset B E(r)= 0.20(0.15) + 0.3(0.125) + 0.4(0.08) + 0.1(0.045)= 10.4%

Variance= P*(R-E(r))^2

Here E(r)= expected return

Variance=0.20(15% - 10.4%)^2 + 0.3(10% - 10.4%)^2 + 0.4(8% -10.4%)^2 + 0.1(4.5% - 10.4%)^2 =0.0010065

Standard deviation= SQRT(variance)= SQRT(0.0010065)= 3.17%

Variance=0.20(15% - 8.9%)^2 + 0.3(12.5% - 8.9%)^2 + 0.4(7.5% - 8.9%)^2 + 0.1(5% - 8.9%)^2 = 0.000459

Standard deviation= SQRT(variance)= SQRT(0.000459)= 2.14%

B)

Exp. return for portfolio= W1*Ra + (1-W1)*Rb

W1 is the weight of asset A

Ra is the return on asset A

Rb is the return of asset B

Exp.return for A= 0.4*8.9% + (1-0.4)*10.4%= 9.8%

W2=1-W1=1-0.4=0.6

Cov(A,B)= summation(Probability*( Ra-E(Ra))*(Rb-E(Rb)))

Covariance = 0.2*(12%-8.9%)*(15%-10.4%)+ 0.3*(10%-8.9%)*(12.5%-10.4%)+ 0.4*(7.5%-8.9%)*(8%-10.4%)+ 0.1*(5%-8.9%)*(4.5%-10.4%)

COV(A,B)= 0.00072

Note: S.D. is standard deviation

Variance of portfolio= (W1*S.D.a)^2 + (W2*S.D b)^2 +(2*Wa*Wb*cov(A,B))

var= (0.4*3.17%)^2+(0.6*2.14%)^2+(2*0.4*0.6*3.17%*2.14%*0.00072)= 0.000326

Standard deviation of portfolio = sqrt(0.000326) = 1.81%

C)

First we have to determine portfolio Beta

Beta_p = beta _A W_a +Beta _B* W_b

Where W_a, W_b are the weights

Beta a= COV(A,B)/ VAR (A)= 0.00072/0.0010065= 0.715

Beta b= COV(A,B)/VAR(B)= 0.00072/0.000459= 1.568

Beta_p = 0.4*0.715 +0.6*1.568= 1.22

Given Rf=1.25%, Rm=8.75%

Rf is risk free rate

Rm is market return

Exp return= Rf +Beta*(Rm-Rf)

=0.0125+1.22*(0.0875-0.0125)

=10.4%

He should invest because he can get more return if he is in the market considering the systematic risk