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

Answer (a);

Let us assume amount invested in risk free asset = A

Hence:

=> A * 5% + (100000 - A) * 20% = 100000 * 15%

=> 5%A + 20000 - 20% A = 15000

=> -15%A = 15000 - 20000 = -5000

=> A = 5000 / 15% = $33,333.33

As such:

Allocation to risk free asset = $33,333.33 ( 1/3rd of $100,000)

Allocation to market = 100000 - $33333.33 = $66,666.67 (2/3rd of $100,000)

Answer (b);

Standard deviation of risk free asset = 0

Hence:

Standard deviation of your portfolio in (a) = Standard deviation of market portfolio = 20%

Standard deviation of your portfolio in (a) = 20%

Answer (c):

Let us assume weight of risk free asset = W

Hence:

=> W * 5% + (1 - W) * 20% = 25%

=> 5%W + 20% - 20%W = 25%

-> -15%W = 25% - 20% = 5%

=> W = 5% / -15% = -33.33%

Weight of risk free asset = -33.33%

Weight of market portfolio = 1 - (-33.33%) = 133.33%

Hence:

Weight of risk free asset = -33.33%

Weight of market portfolio = 133.33%

Answer (d):

These weight means:

Borrow 33.33% of the investment amount from @ 5% and invest in market portfolio.

For example:

If you have $30000 to invest, borrow another (30000 * 33.33%=) $10,000 @5% and invest (30000 * 133.33%=) $40,000 in market portfolio.

Answer (e):

Standard deviation of risk free asset = 0

Hence:

Standard deviation of your portfolio in (c) = Standard deviation of market portfolio = 20%

Standard deviation of your portfolio in (c) = 20%

Answer (f):

To an extent increase in leverage increases return on equity. As we observe from example in part c above. However increase in leverage beyond optimal level, exposes the portfolio to high risks and return will decrease.