Please show your work and write solutions in details, including formulas and how
ID: 2651484 • Letter: P
Question
Please show your work and write solutions in details, including formulas and how you plug numbers, given in each question, into these formulas. Please, write short VERBAL conclusion at the end of each problem summarizing your answer in words. Verbal answers are not substitute for the quantitative solution.
Question 4
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) Suppose that the market pays either 40% or 0% each with probability one half. You alter your portfolio to a more risky level by borrowing $50,000 at the risk free rate and investing it and your own $100,000 in the market portfolio. Give the probability distribution of your wealth (in dollars) next period.
Explanation / Answer
a) Let the weightage of risk free asset be 'X' such that the weighatge of market portfolio is '(1-X)'.
Therefore, Expected return = Weightage of risk free asset * Return of risk free asset + Weightage of market portfolio * Return of market portfolio
=> 15% = X * 5% + (1 - X) * 20%
=> X = 0.3333
Therefore, the asset allocation should be such tha it consists of 33.33% of risk free asset and 66.67% of market portfolio.
b) The standard deviation of risk free asset is zero, as such the standard deviatiopn of the portfolio is entirely determined by that of the market portfolio.
Standard deviation of the portfolio = Weightage of market portfolio * Standard deviation of market portfolio
= 0.6667 * 20%
= 13.34%
c) In case the market pays return of 40%, then
The wealth during next period = Total sum invested in market * (1 + Return of market) - Total sum borrowed * (1 + Cost of risk free asset)
= $150,000 * (1 + 40%) - $50,000 * (1 + 5%)
= $157,500
In case the market pays return of 0%, then
The wealth during next period = Total sum invested in market * (1 + Return of market) - Total sum borrowed * (1 + Cost of risk free asset)
= $150,000 * (1 + 0%) - $50,000 * (1 + 5%)
= $97,500
Scenario Wealth Value Probability Market pays 40% in return $157,500 0.5 Market pays 0% in return $97,500 0.5 Overall expected $127,500