Due date: Thursday October 31st

1. Calculate EAR and APR for the following questions.

a. You have an APR of 7.5% with continuous compounding. What is the EAR? b. You have an EAR of 9%. What is the equivalent APR with continuous compounding? c. The buyer of a new home is quoted a mortgage rate of 0.5% per month. What is the APR on the loan? d. A loan for a new car costs the borrower 0.8% per month. What is the EAR?

Answer:

a.

1 + EAR = eAPR => EAR = eAPR - 1 = e.075 - 1 = 7.79%

b.

1 + 9% = eAPR => Ln(1.09) = Ln(eAPR) = APR

APR = Ln(1.09) = 8.62%

c.

0.5% x 12 = 6%

d.

(1.008)12 - 1 = 10.03%

2. Treasury bills are paying a 4% rate of return. A risk-averse investor with a risk aversion of A = 3 should invest entirely in a risky portfolio with a standard deviation of 24% only if the risky portfolio’s expected return is at at least ______.

Answer:

3 = (E[rQ] - .04)/(.242) => E[rQ] = (.242) x 3 + .04 = 21.28%

3. You invest $10,000 in a complete portfolio. The complete portfolio is composed of a risky asset with an expected rate of return of 15% and a standard deviation of 21% and a Treasury bill with a rate of return of 5%. a. How much money should be invested in the risky asset to form a portfolio with an expected return of 11%? b. How much money should be invested in the risky asset to form a portfolio with a standard deviation of 9%?

Answer:

a.

E[rC] = yE[rp] + (1 - y)rf

11% = y x 15% + (1 - y) x 5%

y = 60%.

Invest $10,000 x 60% = $6,000 in the risky asset.

b.

𝛔C = y𝛔p

9% = y x 21%

y = 45%.

Invest $10,000 x 45% = $4,500 in the risky asset.

4. You have $500,000 available to invest. The risk-free rate, as well as your borrowing rate, is 8%. The return on the risky portfolio is 16%. The standard deviation on the risky portfolio is 50%. a. If you wish to earn a 22% return, how much money should you borrow? b. If the standard deviation on the complete portfolio is 25%, what is the expected return on the complete portfolio?

Answer:

a.

E[rC] = yE[rp] + (1 - y)rf

22% = y x 16% + (1 - y) x 8%

y = 1.75 and 1 - y = -0.75

Invest 1.75 x $500,000 = $875,000 in the risky portfolio by borrowing 0.75 x $500,000 = $375,000.

b.

𝛔C = y𝛔p

25% = y x 50% => y = .5

E[rC] = yE[rp] + (1 - y)rf = .5 x 16% + .5 x 8% = 12%

5. You are considering investing $1,000 in a complete portfolio. The complete portfolio is composed of Treasury bills that pay 5% and a risky portfolio, P, constructed with two risky securities, X and Y. The optimal weights of X and Y in P are 60% and 40%, respectively. X has an expected rate of return of 14%, and Y has an expected rate of return of 10%. To form a complete portfolio with an expected rate of return of 11%, what percent of your complete portfolio should you invest in Treasury bills?

Answer:

E[rp] = wxE[rx] + wyE[ry] = .6 x 14% + .4 x 10% = 12.4%

E[rC] = yE[rp] + (1 - y)rf = y x 12.4% + (1 - y) x 5% = 11%

=> y = 81% and (1 - y) = 19%

Invest 19% in Treasury bills.

6. XYZ stock price and dividend history are as follows:

Year

Beginning-of-Year Price

Dividend Paid at Year-End

2010

$100

$4

2011

$110

$4

2012

$90

$4

2013

$95

$4

An investor buys three shares of XYZ at the beginning of 2010, buys another two shares at the beginning of 2011, sells one share at the beginning of 2012, and sells all four remaining shares at the beginning of 2013. a. What are the arithmetic and geometric average rates of return for the investor? b. What is the dollar-weighted rate of return?

Answer:

HPR = Capital gain yield + Dividend yield

HPR in 2010 = ($110 - $100)/$100 + $4/$100 = 14%

HPR in 2011 = ($90 - $110)/$110 + $4/$110 = -14.55%

HPR in 2012 = ($95 - $90)/$90 + $4/$90 = 10%

a.

Arithmetic average rate of return = (14% - 14.55% + 10%)/3 = 3.15% Geometric average rate of return = [(1 + .14) x (1 - .1455) x (1 + .1)]1/3 - 1 = 2.33%

b.

Net Cash Flow = Value of shares sold −...