Final Exam Fall 2014

1. The one-year spot rate of interest is 4%. A 2-year bond that pays a $50 (annual payments)
currently sells for $1,000.46. What is the 2-year spot rate?

2. Todays date is November 15, 2014. Spot rates in the market right now look like this:

1-year

3.5%

4-year

4.5%

2-year

4.0%

5-year

5.0%

3-year

4.25%

6-year

5.75%

A bond pays a 5.5% coupon rate (annual payments) and has a par value of $1,000. It matures in
six years and makes payments on November 14 each year (so the first payment is due one year
from now). Assuming the expectations hypothesis is true, what is the expected price of this bond
on November 15, 2017?

3. Refer back to the bond in problem #2. Suppose that the expectations hypothesis is not true,
and instead there is a liquidity premium built into interest rates such that the expected return on
long-term bonds is generally higher than the expected return on a series of short-term bonds.
Given this, the price you estimated in question 2 is

a. correct
b. too high
c. too low

4. Which of the two bonds below would you rather buy if you think interest rates at all
maturities are about to fall (i.e, the entire yield curve is shifting down)? Explain your answer.

A zero-coupon Treasury bond maturing in 6 years, priced to yield 5%, with a $1,000 par value.

A Treasury bond with a $1,000 par value, paying an annual coupon rate of 6.5% (assume annual
payments), priced to yield 5%, and maturing in 7 years.

5. You can only invest in the two stocks below, and there is no risk-free asset.

Stock #1

Stock #2

Expected return

8%

12%

Std. deviation

20%

40%

Correlation between #1 and #2 = 0.3

a. Calculate the expected return and standard deviation of a portfolio consisting of 90% stock #1
and 10% stock #2.

b. The portfolio you examined in part (a) is
i. efficient
ii. inefficient
iii. impossible to say

6. The risk-free rate is 4%. You are allowed to borrow or lend at the risk-free rate, and you can
invest in one of the following mutual funds, but only one. Which one would be part of your
portfolio (assume that you want to take at least some risk, so investing 100% of your money in
the risk-free asset is not an option).

Expected

Standard

Fund

Return

Deviation

1

8%

14%

2

11%

22%

3

13%

28%

4

17%

43%

7. A stock with a beta of 1.0 will almost surely have a standard deviation that is

a. equal to the markets standard deviation
b. greater than the markets standard deviation
c. less than the markets standard deviation

8. The risk-free rate is 0.1% per month (thats 0.001 in decimal form). The monthly expected
return and standard deviation for five different assets appear below. Calculate the Sharpe ratio
(use monthly datadont convert to annual) for each asset.
Expected

Standard

Return

Deviation

#1

0.5%

6.0%

#2

0.7%

6.5%

#3

0.9%

7.0%

#4

1.1%

7.5%

#5

1.3%

8.0%

9. Open the portfolio optimizer file that I provided. This file has the data above already input,
along with the matrix of correlation coefficients between every pair of stocks.

a. Asset #1 has the 2nd highest weight in the optimal portfolio. Is this what you would have
expected given this assets Sharpe ratio from the previous problem? If yes, why? If no, explain
why it gets a different weighting than you would have expected based solely on the Sharpe ratio.

b. Why does Asset #4 get the heaviest weighting in the optimal portfolio?

c. Suppose you have a friend who says that he is very tolerant of risk, and therefore prefers to
invest in riskier assets to earn higher returns. Your share the portfolio optimizer with your friend,
who says that he would prefer to invest in asset #5 rather than the optimal portfolio. He says,
Its true that asset #5 is riskier than the optimal portfolio, but Im willing to take the extra risk to
earn the extra return. What would your response be? How would you advise your friend? You
dont need to do any math to answer this question.

10. In the portfolio optimizer file, there is a tab labeled telecom. On that tab, column B lists
monthly returns on the S&P500 stock index for the past 36 months. Column C lists monthly
returns on a Fidelity mutual fund that invests exclusively in stocks in the telecommunications

industry. I looked up the beta of this fund online (ticker is FTUIX). According to Yahoo
Finance, the beta is 0.54.

a. Calculate the monthly standard deviation of both the fund and the S&P500. Which investment
has greater volatility? Is this surprising given the funds beta as reported by Yahoo? Why or
why not?

b. Estimate a single-index model regression for the fund. This means running a regression where
the funds return is the Y variable and the markets return is the X variable. What is the funds
beta? What is its R2?
11. Construct a gross payoff diagram for the following portfolio.

Buy a call with X = $35, sell 2 calls with X = $40, buy a call with X = $45.

In this portfolio, you are spending money to buy two options, but you are receiving money from
two options that you are selling. Based solely on a no arbitrage argument, which is greater, the
cost of the options you are buying or the revenue of the options you are selling? Another way to
frame this question is thiswhen you construct this portfolio, do you have to pay money up
front or do you receive money up front? You do not need to attempt to calculate option prices to
answer this question, and in fact, if you base your answer on a numerical example that you
create, that answer will be insufficient because one numerical example cannot prove the general
point here.

12. The price of a stock today is $71. In three months, the stock may go up to $80, or it may fall
to $64. Lets assume that you believe these outcomes are equally likely. The risk-free rate is 2%
per year.

a. What is the arbitrage-free price of a put option that expires in three months and has a strike
price of $75?

b. Suppose all of the facts are the same as in part (a), except that you now believe that the
probability that the stock will go up is 60%, and the probability that it will go down is 40%.
Does this change the arbitrage-free price of the put option? Why or why not? There is no need to
do any math to answer this question.