代做RSM332、代写MSFT留学生、代写CS/Python, C/C++、代做Java/C++设计

日期：2018-11-20 09:10

UNIVERSITY OF TORONTO

Joseph L. Rotman School of Management

RSM332 PROBLEM SET #3

1. Consider three securities with the following payoffs for different states of the economy:

Economic State Probability R1 R2 R3

Recession 0.4 30% ?30% 10%

Normal 0.4 10% 30% 15%

Expansion 0.1 10% 50% 25%

Boom 0.1 0% 120% 45%

(a) What is the expected return and standard deviation on each security?

(b) What is Cov(R1, R2), Cov(R1, R3), and Cov(R2, R3)? What is Corr(R1, R2),

Corr(R1, R3), and Corr(R2, R3)?

(c) What is the expected return, μp, and standard deviation, σp, of a portfolio which

has its funds invested equally in (i) securities #1 and #2, or (ii) securities #1 and #3,

or (iii) securities #2 and #3?

(d) What is μp and σp, of a portfolio which has its funds invested equally in securities

#1 to #3?

(e) What is the correlation of the return of the portfolio in part (d) with the return of

an equally weighted portfolio of securities #1 and #2?

2. An investor has an opportunity to invest in two risky assets and a risk-free asset. The

expected return of the two risky assets are μ1 = 0.12, μ2 = 0.15. Their standard

deviations are σ1 = 0.05 and σ2 = 0.1, and the correlation coefficient between their

return is 0.2. The risk-free rate is 0.05. Suppose the investor has $1000 and he wants

to hold a portfolio with expected return of 0.1. If the investor is risk averse, how much

should he invest in the two risky assets and the risk-free asset?

3. The Sharpe ratio of a portfolio p is defined as

SR(p) = E[Rp] ? RF

σp

,

where σp is the standard deviation of the portfolio. Suppose a portfolio T is the

tangency portfolio (the tangency portfolio is the portfolio which has the maximum

Sharpe ratio among all the portfolios) and it has a Sharpe ratio of 0.6. Also consider a

portfolio p which may or may not be on the mean-variance efficient frontier. You find

out that the correlation between the returns of portfolios p and T is 0.5. What is the

Sharpe ratio of portfolio p? You need to show the work to support your answer.

1

4. hw3.xlsx contains the monthly return data for Microsoft (MSFT), IBM (IBM), Apple

(AAPL) and Hewlett-Packard (HPQ) for the period 2008/1–2017/12.

(a) Report the average return and sample covariance matrix of the four stocks based

on the monthly data.

(b) Using the estimates from part (a) as the expected returns and covariance matrix

for the four stocks, find out the global minimum variance portfolio of the four stocks

assuming (i) short-selling is allowed, and (ii) short-selling is not allowed.

(c) Using the estimates from part (a) as the expected return and covariance matrix for

the four stocks, find out the tangency portfolio of the four stocks assuming (i) shortselling

is allowed, and (ii) short-selling is not allowed. Assume a monthly risk-free rate

of 0.1%/month.

Note: To do optimization in Excel, you may want to use “Tools, Solver.”