Question

Given that the risk-free rate is 5%, the expected return on the market portfolio is 20%, and the standard deviation of returns to the market portfolio is 20%, answer the following questions:

1. You have $100,000 to invest. How should you allocate your wealth between the risk free asset and the market portfolio in order to have a 15% expected return?

2. What is the standard deviation of your portfolio in (a)?

3. Now suppose that you want to have a portfolio, which pays 25% expected return. What is the weight in the risk free asset and in the market portfolio?

4. What do these weights mean: What are you doing with the risk free asset and what are you doing with the market portfolio?

5. What is the standard deviation of the portfolio in c?

6. What is your conclusion about the effect of leverage on the risk of the portfolio?

Answer 1

a..Rp=w1*R1+w2*R2

Rp=Return of the portfolio after combining risk free asset=15%

R1=Return of asset 1(market portfolio)=20%

w1=Weight of asset 1 (Market folio) after allocation

R2=Return of asset 2(Risk Free asset)=5%

w2=Weight of asset2(Risk free asset) in the combined portfolio

w1+w2=1

w2=(1-w1)

w1*20+w2*5=15

w1*20+(1-w1)*5=15

w1*(20-5)=15-5=10

w1=10/15=2/3$

w2=1-w1=1/3

Amount to be allocated to Market Portfolio =100000*(2/3)=$66,666.67

Amount to be allocated to Risk Free Asset =100000*(1/3)=$33,333,33

b)Standard Deviation of the portfolio:

Variance of the portfolio =Vp=(w1^2)*(S1^2)+(w2^2)*(S2^2)+2*w1*w2*Cov(1,2)

w1=(2/3)

w2=(1/3)

S1=20%

S2=Standard Deviation of risk less asset =0

Cov(1,2)=Covariance of Asset 1 and 2 =0

Variance of the Portfolio =((2/3)^2)*(20^2)=177.7778%%

Standard Deviation of the portfolio =Square Root of Variance =SQRT(177.7778)=13.33%

c: Rp=w1*R1+w2*R2

R1=Market Portfolio return=20%

R2=Return of Risk free asset=5%

w1+w2=1

Rp=w1*20+(1-w1)*5

Required Portfolio return =Rp=25%

25=20w1+5-5w1

15w1=20

w1=(20/15)=1.3333

w2=1-w1=1-1.3333=-0.3333

Weight of Risk Free Asset =-0.3333

Weight of Market Portfolio =1.3333

d.The above weights means you sell risk free assets (Negative Weight) and Buy excess Market portfolio

e. Standard Deviation of Portfolio :

Variance =(1.3333^2)*(20^2)=711.11%%

Standard Deviation of the portfolio=Square Root of Variance =SQRT(711.11)=26.66%

f. Leverage has increased expected return , but it also increases the risk(measured in terms of standard deviation)