Fall 2014
Final Exam
Closed Book.
Econ 180-367
Formula Sheet Provided. Calculators OK.
Time Allowed: 3 hours
Please write your answers on the page below each question
1. (5 points) Assume that the risk-free interest rate is 1% and that gold is costless to store. The spot price of gold is $1,000 per ounce. According to spot-futures parity, what must the futures price for delivery of gold one year hence be?
2. (5 points) A factory costs $10 million to build. To finance the factory, the firm borrows $5 million at a
3% interest rate. The profits from the factory are $500,000 per year. What is the return on equity?
3. (10 points) Mergenthaler Enterprises stock sells for $80. After one year, it will either go up to $94
(with probability 0.75) or down to $70 (with probability 0.25). The risk-free rate is 10%. Find the price of a European call option to buy one share in Mergenthaler Enterprises for $82 one year from today.
4. (5 points) Suppose that the risk-free interest rate is 10 percent. Consider the following possible configuration of prices of European options and current share prices on a non-dividend paying stock with a maturity of one year hence. Say if it is possible or impossible. If it is impossible, please say why.
Call Option
Put Option
Stock
Strike Price
60
60
-
Value
33
10
32
5. (10 points) What is the duration of a bond that sells at par, has a coupon rate of 8% per annum (paid annually) and a remaining time to maturity of 3 years?
6. (10 points) Suppose that there are two potential risky assets, A and B. The expected return on A is 6% and it’s standard deviation is 15%. The expected return on B is 8% and it’s standard deviation is 20%.
The covariance (not correlation) between the two assets is -0.005. The risk-free rate is 4%. You want to form a portfolio of assets A and B alone so as to maximize the Sharpe ratio. What weight will you put in asset A?
7. (10 points) Please state whether each of the following statements is true or false. No explanation needed.
(i) The price of a call option on the S&P 500 index with an exercise price of 1,330 is greater than the price of a call option on the S&P 500 index with an exercise price of 1,340.
(ii) Callable bonds have a higher yield to maturity than otherwise identical bonds that are not callable.
(iii) In fall 2008, the oil futures curve was in normal backwardation.
(iv) Fixed rate mortgages had higher delinquency rates than adjustable rate mortgages during the financial crisis.

(v) American options cannot be worth less than European options.
8. (5 points) Suppose that the risk-free rates in the United States and in the United Kingdom are 2% and
0%, respectively. The spot exchange rate between the dollar and the pound is 1 pound = $1.50. What should the futures price of the pound for a one-year contract be to prevent arbitrage opportunities?
9. (10 points) Suppose that a stock is trading for $30. Its volatility is
σ =
0.3
. The stock pays no dividends and the risk-free interest rate is zero. You buy a European call option on this security at a strike price of $30 with an expiration of 1 year hence.
(a) According to the Black-Scholes formula, what should this call option be worth?
(b) What is the delta of this option (i.e. how many stocks would you need to buy to hedge it)?
10. (10 points) The Japanese one-year interest rate is 0, and the Australian one-year interest rate is 4%.
The current exchange rate is that 1 Australian Dollar = 90 Yen. Suppose that in one years time, you think that there are three possible values for the Australian Dollar-Japanese Yen interest rate as in the table below
Exchange Rate
1 AUD = 75 Yen
1 AUD = 85 Yen
1 AUD = 95 Yen
Probability
10%
20%
70%
Consider the “carry trade” strategy of starting with a fixed amount of yen, converting to Australian dollars, investing in Australia at the one-year interest rate, and converting back at the end of the year. What is the
Sharpe ratio on this strategy from the perspective of a Japanese yen investor?
11. (10 points) Consider the following data for a one-factor economy. All portfolios are well-diversified.
Portfolio
A
Expected Return
12%
Beta
2
B
F 6%
1
0
(a) According to APT, what is the expected return on portfolio B?
(b) Suppose that another portfolio, portfolio E, is well-diversified with a beta of 4 and an expected return of 20%. Would an arbitrage opportunity exist? If so, what would be the arbitrage strategy?
12. (10 points) (a) To test the CAPM, researchers have run cross-sectional regressions of the return for stock i ( r i
) onto the book-to-market ratio of that stock ( BM i
), the size of that stock ( S i
) and the beta of that stock (
β i
): r i
= γ
0
+ γ
1
BM i
+ γ
2
S
(a) According to the CAPM, what should the coefficients
+ i
γ
,
0
γ β
γ
,
1 i
γ
+
2
ε i
and
γ
be?
3
(b) When the regression in (a) was run by Fama and French and other researchers, what results did they find? Please be very specific about the signs of the coefficients that they found for
γ
1
and
γ
2
.

Solutions
1. $1,000*1.01=$1,010 per ounce.
2. Equity is $5,000,000. Profits net of interest costs are $500,000-$150,000=$350,000. So return on equity
350, 000 is
5, 000, 000
=
7% .
3. If I buy the call option, the payoff will be $12 on the upper branch and $0 on the lower branch.
Suppose I buy the stock and borrow $63.64. This costs me $16.36. On the lower branch, the payoff will be $70-$70=$0. On the upper branch, the payoff will be $94-$70=$24. So this portfolio is effectively 2 call options. The price of the call option should be $8.18.
4. Impossible. It violates the arbitrage condition C
≤
S
0
.
5. The price of the bond is $100 and the yield is 8%. The duration is
1 8
(
100 1.08
+
1 8
*1) (
100 1.08
2
+
1 108
* 2) (
100 1.08
3
*3)
=
2.78
years.
6. The expected excess returns on A and B are 2% and 4%, respectively. So the weight on asset A is
0.02 * 0.2
+
2 −
−
−
+ −
=
0.001
=
1
0.02 * 0.2
2
0.04 * 0.15
2
(0.02 0.04) * ( 0.005) 0.002
2
.
7. (i) True.
(ii) True.
(iii) False.
(iv) False.
(v) True.
8. By covered interest parity, the futures price F satisfies 1.5*1.02
=
F *1 . Solving this gives the futures price of 1 pound=$1.53.
9. (a) In the Black-Scholes formula,
From the normal distribution tables, d
N
1
=
30
(ln( )
+
0.3
30 2
(0.15)
=
0.5596
2
) / 0.3
and
=
0.15
and d
2
=
N ( 0.15)
=
0.4404
0.15 0.3
= −
0.15
.
. Hence the call price should be C
=
(30 * 0.5596) (30 * 0.4404)
=
$3.58
(b) The hedge ratio is N (0.15)
=
0.5596.
.
10. The returns in the three scenarios are:
Exchange Rate Return
1 AUD = 75 Yen
1 AUD = 85 Yen
1 AUD = 95 Yen
-0.13333
-0.01778
0.097778
Probability
10%
20%
70%
The mean return is 0.05156 and the variance is 0.005875, so the Sharpe ratio is 0.67.

11. (a) The key equation is ( )
= r f
+ β λ
. Because the expected return on F is 6%, we know that the risk-free rate must be 6%. Because the expected return on A is 12%, we know that
λ =
0.03
. Hence the expected return on B must be 9%.
(b) With a beta of 4, the expected return on E ought to be 18%. So here is the strategy.
•
Invest $100 in E. Payoff is $100*(1.20+4F).
•
Invest -$400 in B. Payoff is $-400*(1.09+F)
•
Invest $300 at the risk-free rate. Payoff is $300*1.06
Adding the pieces up, there is no cost today, but my payoff is $2. This is an arbitrage.
12. (a)
γ
0
should be the risk-free rate, excess return.
(b)
γ
1
is positive,
γ
2
is negative, and
γ
1
and
γ
2
should be zero, and
γ
3
should be the expected market
γ
3
is smaller than the average market excess return.

Grading Rubric for Final
1. 5 points for correct answer. 3 points for correct answer apart from visible algebra error.
2. 5 points for correct answer. 3 points for correct answer apart from visible algebra error.
3. 6 points for constructing the replicating portfolio including working out the cost of the portfolio and payoffs on the two branches. 10 points for complete answer. 3 points off for any algebra error.
4. No partial credit.
5. 7 points for writing down the formula. 10 points for the correct answer (with or without formula).
6. 3 points off for using returns instead of excess returns; 3 points off for mixing up standard deviation with variance; 3 points off for algebra mistake. No credit if does not have answer and does not write down the correct formula (apart from errors above).
7. No partial credit.
8 5 points for correct answer. 3 points with algebra error. No points for an answer of 1.5/1.02.
9. (a) 6 points. 2 points off for algebra mistake, failure to look up normal distribution tables correctly, mixing up variance and standard deviation or other specific mis-applications of the Black-Scholes formula.
Must have the elements of the Black-Scholes formula for any credit.
(b) 4 points. 2 points for identifying that the hedge ratio is N(d1) and 2 more points for working this out. Full credit for this part if got the wrong d1 in (a) but worked out N(d1) correctly.
10. 2 points for computing returns, 2 points for computing the mean, 3 points for computing the variance and 3 points for computing the Sharpe ratio. No points on any part that contains an algebra error. But an algebra error on one part should not affect mark on other parts. For example, if student computes returns incorrectly, but gets everything else right with those incorrect returns, then (s)he would get 8 points.
11 (a). 4 points for correct answer. 2 points for writing down formula correctly but making an algebra error.
(b). 6 points for correct answer. If student has the right arbitrage strategy, there is no need to identify what the expected return on E should be. If student does not have the right arbitrage strategy, then 2 points for identifying that the expected return on E ought to be 18%. No need to compute that the arbitrage strategy has a payoff of $2. But no credit for a student who gets the direction of the arbitrage backwards.
12 (a). 4 points. 1 point for each of the four coefficients.
(b) 6 points. 3 points for saying that
γ
1
is positive, and 3 more points for saying that
Students who correctly identify the signs of
γ
1
and
γ
2
γ
2
is negative.
get full credit, because that is what the question specifically asked for. Students who do not correctly identify the signs of
γ
1 correct statements about
γ
0
or
γ
3
get one point for each correct statement.
and
γ
2
, but who do make