In: Finance
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:
a. 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?
b. What is the standard deviation of your portfolio in (a)?
c. 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?
d. What do these weights mean: What are you doing with the risk free asset and what are you doing with the market portfolio?
e. What is the standard deviation of the portfolio in c?
f. What is your conclusion about the effect of leverage on the risk of the portfolio?
PLEASE SHOW ALL WORK
a] | Let the weight of the market portfolio be w. | |
Then, the weight of the risk free asset = 1-w | ||
Now, the expected return of 0.15 = 0.2*w+0.05*(1-w) | ||
Solving for w, we have | ||
0.15 = 0.2*w+0.05-0.05*w | ||
0.15-0.05 = 0.2*w-0.05*w | ||
0.1 = 0.15*w | ||
w = 0.1/0.15 = 66.67% | ||
So, weight of the market portfolio = 66.67%, and | ||
weight of the risk free asset = 1-66.67% = 33.33% | ||
b] | As the SD of the risk free asset is 0 and its correlation | |
with the market portfolio is also 0, the SD of the new | ||
portfolio = (weight of the market portfolio^2*SD of the market portfolio^2)^0.5 = weight of the market portfolio*SD of the market portfolioi = 20%*66.67% = | 13.33% | |
c] | Using similar equation as in [a] | |
0.25 = 0.2*w+0.05*(1-w) | ||
0.25 = 0.2*w+0.05-0.05*w | ||
0.20 = 0.15*w | ||
w = 0.25/0.15 = 1.67 | ||
Weight of the risk free asset = 1-1.6667 = -0.6667 | ||
The weight of the risk free asset = -66.67%. | ||
d] | It implies that borrowings are made at the risk free | |
rate of 5% and then the borrowings along with equity | ||
are invested in the market portfolio. The market | ||
portfolio will have a weight of 166.67%. | ||
e] | The SD [as in b] will be 20%*166.67% = | 33.33% |
f] | Leverage will increase the risk of the portfolio. |