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Solutions to Problems at End of Chapter
Forward Contracts and Forward-Spot Parity.
1. Suppose that you are planning a trip to England. The trip is a year from now, and you have reserved a
hotel room in London at a price of ₤ 50 per day. You do not have to pay for the room in advance. The
exchange rate is currently $1.50 to the pound sterling.
a. Explain several possible ways that you could completely hedge the exchange rate risk in this situation.
b. Suppose that r₤=.12 and r$=.08. Because S=$1.50, what must the forward price of the pound be?
c. Show that if F is $0.10 higher than in your answer to part b, there would be an arbitrage opportunity.
SOLUTION:
a. Ways to hedge the exchange rate risk:
Pay for the room in advance
Buy the pounds you will need in the forward market.
Invest the present value of the rental payments in a pound-denominated riskless asset.
b. F = S (1+r$)/(1+r£) = $1.50 x 1.08/1.12 = $1.4464 per pound
c. If F is $1.55 then arbitrage profits can be made by borrowing dollars, investing in pounds and selling them
forward at the inflated forward price. After paying off principle and interest on the dollars borrowed, you would
have pure arbitrage profits left over. For example,
Borrow $1.50,
Convert it into 1 pound,
Invest it in pound-denominated bonds to have 1.12 pounds a year from now,
Sell 1.12 pounds forward at $1.55 per pound to have $1.736 a year from now,
After 1 year, pay off the principle and interest on the loan ($1.50x 1.08 = $1.62).
This series of transactions leaves you with $.116 a year from now with no initial outlay of your money.
Arbitrage Position
Borrow $1.50
Buy pound-denominated bond
Sell 1.12 pounds forward at $1.55
per pound
Net Cash Flows
Immediate Cash Flow
$1.50
-$1.50
0
Cash Flow 1 Year From Now
-$1.62
S1
$1.736-S1
0
$1.736-$1.62 = $.116
Forward-Spot Parity Relation with Known Cash Payouts
2. Suppose that the Treasury yield curve is flat at an interest rate of 7% per year (compounded
semiannually).
a. What is the spot price of a 30-year Treasury bond with an 8% coupon rate assuming coupons are paid
semiannually?
b. What is the forward price of the bond for delivery six months from now?
c. Show that if the forward price is $1 lower than in your answer to part b, there should be an arbitrage
opportunity.
SOLUTION:
a. The spot price of the 30-year Treasury is $1,124.724:
n
i
PV
60
3.5
?
b.
FV
1000
The forward price for delivery six months from now is $1,124.089:
F = S(1+r) - C = $1,124.724 x 1.035 - 40 =$1,124.089
PMT
40
Result
PV =1124.724
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c.
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If the forward price is only $1,123.089, then arbitrage profits can be made by selling the bond short and buying
it forward at the low forward price. It can be described as follows:
Sell short a bond at $1,124.724; buy it forward at $1,123.089; invest the proceeds of the short sale to earn 3.5%
for 6 months
After 6 months, take delivery of the bond and cover your short sale
Arbitrage Position
Sell short a 30-year T-bond
Buy 6-month T-bills paying 3.5%
Buy a forward contract for a 30year T-bond
Net Cash Flows
Immediate Cash Flow
$1,124.089
-$1,124.089
0
Cash Flow 1 Year From Now
-(S1 + $40)
$1,163.432
S1-$1,123.089
0
$1,163.432- ($1,123.089 + $40) =
$.343
Forward-Spot Parity Relation with Uncertain Dividends
3. A stock has a spot price of $100; the riskless interest rate is 7% per year (compounded annually), and the
expected dividend on the stock is $3, to be received a year from now.
a. What should be the one-year futures price?
b. If the futures price is $1 higher than your answer to part a, what might that imply about the expected
dividend?
SOLUTION:
a. S = $100, r = .07, D = $3. F = S ( 1+r) - D = $104
b. If F is $105, that might imply that D is really only $2.
Storage Costs versus Dividend Yield
4. Compare the forward-spot price-parity relation for gold to the one for stocks. Is it fair to say that stocks
have a negative storage cost equal to the dividend yield?
SOLUTION
One could definitely say that stocks have a negative storage cost equal to the dividend.
5. Suppose you are a distributor of canola seed and you observe the spot price of canola to be $7.45 per
bushel while the futures price for delivery one month from today is $7.60. Assuming a $.10 per bushel
carrying cost, what would you do to hedge your price uncertainty?
SOLUTION
We see that F> S+C. If you short the futures contract, you can sell your seed at $7.60 per bushel.
6. Infer the spot price of an ounce of gold if you observe the price of one ounce of gold for forward delivery
in three months is $435.00, the interest rate on a 91-day Treasury bill is 1% and the quarterly carrying cost
as a percentage of the spot price is .2%.
SOLUTION
Deduce from the futures price parity condition for gold that F = S 0 (1 + r + s) so that S0 = $429.84.
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7. You are a dealer in kryptonite and are contemplating a trade in a forward contract. You observe that the
current spot price per ounce of kryptonite is $180.00, the forward price for delivery of one ounce of
kryptonite in one year is $205.20, and annual carrying costs of the metal are 4% of the current spot price.
a. Can you infer the annual return on a riskless zero-coupon security implied by the Law of One Price?
b. Can you describe a trading strategy that would generate arbitrage profits for you if the annual return on
the riskless security is only 5%? What would your arbitrage profit be, per ounce of kryptonite?
SOLUTION
a. By no-arbitrage, we require that the riskless rate r satisfy:
F = S0 (1 + r + s)
205.2 = 180 (1 +r +.04) = 187.2 + 180r
r = 18/180 = .10 or 10%
b. The implicit risk-free rate that you can earn by buying kryptonite, storing it, and selling it forward at $205.2 per
ounce is 10%. If the riskless borrowing rate is five percent, you should borrow at that rate and invest in hedged
kryptonite. If you buy an ounce of kryptonite for $180, you will get $205.2 for it for sure a year from now. If
you borrow the $180, you will have to pay principal and interest of $180 x 1.05 plus another .04 x $180 in
storage costs. This totals $196.2, thus leaving you with $9 in arbitrage profits.
8. Calculate the implicit cost of carrying an ounce of gold and the implied storage cost per ounce of gold if
the current spot price of gold per ounce is $425.00, the forward price of an ounce of gold for delivery in 273
days is $460.00, the yield over 91 days on a zero-coupon Treasury bill is 2% and the term structure of interest
rates is flat.
SOLUTION
First, we solve it assuming a simple compounding method for the risk free interest rate. Over 273 days, the Risk free
rate is 2%*3=6%. Therefore we have,
F = S (1 + r + s )
460 = 425 (1.06 + s)
s = (460 - 450.5)/425 = 9.5/425 = .02235 for 273 days
Thus the carrying costs are roughly 8.24% for 273 days or 10.98% per year.
Second, we solve it assuming we need to compound the interest rates. The risk free rate over 273 days will be
(1+2%)3-1=6.12%.
plug in the above formulae we get s=.021145 for 273 days.
Thus the carrying costs are roughly 8.23% for 273 days or 11.13% per year.
9. The forward price for a share of stock to be delivered in 182 days is $410.00, whereas the current yield on
a 91-day T-bill is 2%. If the term structure of interest rates is fiat, what spot price for the stock is implied by
the Law of One Price?
SOLUTION
F = $410; r = .02 per quarter.
S = F/(1+r)2 = $394.08
10. You observe that the one-year forward price of a share of stock in Kramer, Inc., a New York tour-bus
company and purveyor of fine clothing, is $45.00 while the spot price of a share is $41.00. If the riskless yield
on a one-year zero-coupon government bond is 5%:
a. What is the forward price implied by the Law of One Price?
b. Can you devise a trading strategy to generate arbitrage profits? How much would you earn per share?
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SOLUTION
a. The no-arbitrage value of the forward price is F = $43.05.
b. The observed forward price is excessive. Consider short-selling a forward contract and taking a long position in
a portfolio consisting of one stock and the sale of a bond with face value of F. Future liabilities for this position
are zero, while the current cash inflow is $1.86.
11. Infer the yield on a 273-day, zero-coupon Japanese government security if the spot price of a share of
stock in Mifune and Associates is 4,750 yen whereas the forward price for delivery of a share in 273 days is
5,000 yen.
SOLUTION
The implied yield over the 273 day term is r = 5.26%.
12. On your first day of trading in Vietnamese forward contracts, you observe that the share price of Giap
Industries is currently 54, 000 dong while the one-year forward price is 60, 000 dong. If the yield on a oneyear riskless security is fifteen percent, are arbitrage profits possible in this market? If not, explain why not.
If so, devise an appropriate trading strategy.
SOLUTION
Arbitrage profits would seem to be possible, since the no-arbitrage forward price implied by these parameters is
F = $62,100.
The futures contract is underpriced, relative to this no-arbitrage value. Consider taking a long position in the forward
contract and simultaneously selling a share of Giap stock and buying a riskless bond with a face value equal to the
observed forward price. The liabilities from these joint positions are zero, while the current cash inflow is $1826.09.
13. The share price of Schleifer and Associates, a financial consultancy in Moscow, is currently 10, 000
roubles whereas the forward price for delivery of a share in 182 days is 11,000 roubles. If the yield on a
riskless zero-coupon security with term to maturity of 182 days is 15%, infer the expected dividend to be paid
by Schleifer and Associates over the next six months.
SOLUTION
The implied dividend is 500 roubles.
14. The spot rate of exchange of yen for Canadian dollars is currently 113 yen per dollar but the one-year
forward rate is 110 yen per dollar. Determine the yield on a one-year zero-coupon Canadian government
security if the corresponding yield on a Japanese government security is 2.21%.
SOLUTION
The implied Canadian rate over this term is approximately 5.00%.
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