Answer Key, 3/4/04
Derivatives and Risk Management, Midterm, 2/17/04
The attached answer key shows in detail the answer(s) to each question. The answers are
lengthy—much more than what you are expected to write—and are intended to help you
understand what you missed and how. Where applicable, the answers illustrate different but
equivalent ways to approach the problem although, again, you are not expected to solve a
problem using all ways, just one.
Besides each question you will see in bold font, and within parentheses, the number of points that
a student could miss by answering incorrectly. The total that can be missed is 105 points out of a
total credit of 135. That is, answering all questions incorrectly will still leave the student with 20
points of credit out of 135—a standard scaling approach used in exams where it is possible for
some raw scores to be much lower than the intended average for the class. Naturally, the effect of
such a scale will be more pronounced for the lowest of the raw scores, but will never affect the
ordering of the grades. Students in the A and B ranges will continue to be in these very ranges
and those in the C and D ranges will also stay where they are. Only the dispersion of the grades
will be reduced.
On the first page of your exam, you will find two number grades, a raw negative number
representing total missed points (for mistakes made) and a percent grade. The percent grade is
calculated as follows:
% grade = 100*(135 – lost points)/135 + an adjustment factor,
where the adjustment factor is a way to further reduce the dispersion of grades, especially at the
lower end. The adjustment factor is not the same for all; it is a decreasing function of the grade
that, nevertheless, does not affect the ordering of the grades. The average percent grade for the
midterm is 81 and the median is 82 (half the class above and half below), just where they
should be for a second year elctive. The highest grade in the midterm was 100%. By the way, the
target average and median for the final course grade are 82, which means that they will
practically be anywhere between 80 and 84.
If you‘re curious about your ―raw‖ score, you can compute it as 100*(105-lost points)/105. Let
me warn you though that this score does not mean much since the final grade assigned is a
relative not an absolute grade. As a benchmark, for example, the average raw score in last year‘s
midterm for Finance 1 (MBA 503) was 94/150, or 63% (for the four sections) while the grades
granted had an average of 78%. In comparison, the average raw score in our midterm is around
72%, which is significantly higher, and the curving is much more modest. Good for us!
As I had told few of you, the grading scheme in a derivatives class has to be highly non-linear,
with option features, and not easily understood. It is fairly ‗priced‘, however, with no arbitrage
Review the exam at your leisure and think a bit before rushing to see me. Grading mistakes are
certainly possible but are unlikely to be significant. Please do not contest a few points here and
there, since a difference of few points in the raw score is unlikely to generate a meaningful
difference for the final percent grade. Furthermore, if you feel you‘ve been wronged in some
parts, chances are that you‘ve been treated generously in other parts. A re-grading of the whole
exam, which I do when you contest a part, is not likely to lead to a meaningful change in grade. If
it does, it could be in either direction. Remember that!
Since grading was done by determining the points lost to mistakes, it would be in your favor had
we forgotten to grade a question (as if you‘ve answered that question correctly).
If you decide to ask me to review your exam, please type on a separate page which parts you
want reviewed and why (i.e., describe the grading mistake you suspect).
Answer Key, 3/4/04
Derivatives and Risk Management Midterm, 2/17/04
Read all questions carefully. Answer all questions. Show your work but be precise and concise.
Remember that short selling an asset means borrowing the asset and selling it.
1. Consider the following market conditions. Interest rates are quoted with continuous compounding.
Spot rate CA$1.30/US$
3-month forward rate CA$1.31/US$
One-year interest on CA$ 2.4% per annum
One-year interest on US$ 0.9% per annum
It costs you nothing now to buy or sell a currency forward. There are no transaction costs.
a. (5 pts.) Do these rates provide any arbitrage opportunity? Why?
Fair forward rate = 1.30 x e(2.4%-0.9%)*1/4 = 1.3050 < 1.31. Therefore there is an arbitrage
b.(10 pts.) If yes, illustrate in detail how you can exploit this opportunity. Outline the trades you will have to
make and show the riskless profit in CA$ that you will end up with. You are authorized to borrow in US$ or
in CA$ the equivalent of CA$10 million.
Trades Cash Flow now Cash Flow in 3 months
Borrow CA$ 10 mm for 3 CA$ 10,000,000 -CA$10,060,180
Convert to US $ at spot rate US$ 7,692,307.629
- CA$ 10,000,000
Invest US$ for 3 months @ US$ 7,709,620
Sell US$ 7,709,620 - US$ 7,709,620
forward for 3 month 10,099,602.2
delivery @1.31 CA$/US$
Net -0- CA$ 39,422
2. (10 pts.) Consider the following futures prices for the S&P 500 contract at the end of trading on Thursday,
February 12, 2004, as they appeared on WSJ.com. The index closed the day at a spot price of 1,152.1. The last
two columns indicate the trading day and the closing time, respectively.
S&P 500 futures
Data retrieved at 02/12/04 21:19:55 • All quotes are in exchange local time • Data provided by
Contract Closing Price Date Time
S&P 500 Mar '04 1151.00 02/12/04 16:39:31
S&P 500 Jun '04 1150.00 02/12/04 16:39:31
S&P 500 Sep '04 1149.10 02/12/04 16:39:31
S&P 500 Dec '04 1148.70 02/12/04 16:39:31
S&P 500 Mar '05 1150.20 02/12/04 16:39:31
S&P 500 Jun '05 1153.70 02/12/04 16:39:32
S&P 500 Sep '05 1157.70 02/12/04 16:39:32
S&P 500 Dec '05 1161.70 02/12/04 16:39:32
You are currently managing a stock mutual fund with market capitalization of $7.7 million and Beta of 0.75
(relative to the S&P index). You are concerned about stock market volatility during this presidential election
year in the US and would like to shield your fund from market risk until after the election in November (2004).
Using the information above, describe how you could use S&P index futures to achieve this hedging objective.
In your answer, indicate whether you will buy or sell futures, for which maturity, and how many contracts do
you buy or sell. Note that each futures contract is written on $250 times the index.
We have to short the S&P 500 futures to hedge. We will choose the futures of maturity
nearest to Nov ‘04 but later, i.e Dec‘04.
Value of index covered in one contract = 250*1152.21 = $288,052.5 (also ok to compute
the cash value of one contract, $250*1148.70 = $287,175)
Since =0.75 relative to S&P, we need to short
($7.7 million x 0.75)/$288,052.5 = 20 Dec‘04 contracts
3. (5 pts.) You can see from the Table above that the futures price was below the spot for contracts with maturities
up to March 2005, but above the spot and increasing with maturity for contracts maturing after March 2005.
How could you explain this pattern?
We know that F=S e(r-q)T, where r is the zero rate for maturity T and q is the dividend yield
per year on the index. The current (upward sloping yield curve) is such that zero rates for
maturities up to March 2005 are below the dividend yield (currently around 1.45% per
annum), explaining why the S&P futures prices for contracts up to March ‗05 are below
the spot level of the index. Zero rates for 18 month terms and longer are currently above
1.45%, explaining why the futures prices for the June ‘05 contract and beyond are higher
than the spot level of the index and increasing with T.
4. (5 pts.) Your company had entered a long time ago a forward contract to sell 1000 ounces of gold at $405 per
ounce. The contract now still has 6 months to maturity. Gold is trading now at $400 per ounce and the six-month
risk-free rate is 5% with continuous compounding. What is the total value of the contract to your company at
the present time? (Show work to get any credit.)
The fair 6-month forward price at this point in time is: F = S erT = 400 e(5% x 0.5) = 410.126.
Existing contract obliges us to sell at $405 in 6 months, $5.126 per ounce below what we
could have locked in now. Hence,
Value of the existing contract to us = -1000(410.126 – 405)e –(5% x 0.5) = -$5,000 .
5. An interest rate swap has a remaining life of 4 years. Under the terms of the swap, company ABC pays 6-month
LIBOR and receives 10% per annum fixed interest. Settlements are made every 6 months. A settlement date
has just passed and the LIBOR for the next settlement is to be reset now. What is the effect of the following
changes in market conditions on the value of the swap to company ABC? Say increase, decrease, no effect, or
a. (3 pts.) The fixed rate that is being exchanged for 6-month LIBOR in swaps of all maturities has just
increased to 11% per annum. Decrease
b. (3 pts.) The current level of LIBOR has increased by almost 1% per annum relative to what it was six
months ago. No effect
6. (4 pts., 1 each for a, b, c, and f; d and e are not independent) A company has entered a 10 year swap
according to which the company pays interest at 6% per annum in Canadian Dollars (CA$) and receives interest
at 5% per annum in American Dollar (US$). Interest payments are exchanged once a year, and the principal
amounts (to be exchanged at maturity) are comparable at current exchange rates. During the life of the swap,
which changes in market conditions will lead to an increase in the value of the swap to the company (circle one
or more if appropriate)?
Answer is (b) and (c)
a. An increase in US$ interest rate.
b. An increase in CA$ interest rate.
c. An appreciation of the value of the US$ relative to the CA$.
d. (a) and (b) above.
e. (a) and (c) above.
f. None of the above.
7. Company X wishes to borrow £10 million at a fixed rate for 5 years. Company Y wishes to borrow US$18
million at a fixed rate, also for 5 years. The amounts required by the two companies are roughly the same at the
current $/£ exchange rate. The companies have been quoted the following annual borrowing rates (with annual
Company X 6.4% 5%
Company Y 5% 4.6%
a. (5 pts.) Calculate the potential gain from a swap arrangement.
Potential gain = 1.4% - 0.4% = 1%
b. (10 pts.) Design a swap that will net the intermediary bank 20 basis points (0.2%) per annum, that is equally
attractive to both firms, and that ensures that all exchange risk is borne by the bank. Your answer should
make clear what each firm receives and pays in the swap. (Show the net borrowing cost for the firms, the
savings to each, and the profit to the bank—all in % per annum.)
Borrow $18 million
6% £ 5% £ Pay 5% £
X Bank Y
Pay 5% $ 5% $ 4.2% $
Borrow £10 million m
Company X Bank Company Y
Borrow $18 million @ 5% Gain 1% on £10 m Borrow £10 million @ 5%
Loose 0.8% on $ 18m
Swap: Net : Gain 0.2% (but in 2 currencies) Swap:
Rec: 5% in $ Bank bears exchange risk Rec: 5% in £
Pay 6% in £ Pay: 4.2 % in $
Net: Pay 6 % in £ (net CF only in £) Net: Pay 4.2% in $ (CF only in $)
Save: 0.4% Save 0.4%
c. (5 pts.) What are the net annual cash flows to the bank?
1% of £ 10 million – 0.8% of $ 18 million = £100,000 - $144,000
d. (5 pts.) How can the bank hedge the currency risk inherent in the cash flows you identified in (c) above? Be
very specific and brief.
The bank can enter a series of 1 to 5 year currency forward contracts to sell £100,000 for $ or, alternatively,
enter into contracts to buy $144,000 with £. The first will lock in profits in $; the second in £.
The bank can enter into a 5-year swap, where it pays £100,000 and receives $, or where it receives $144,000
and pays £. The first locks in profits in $; the second in £.
8. It is January 2nd now. A bank with seasonal liquidity needs expects to borrow in the interbank market $100
million for a three-month period starting July 1st. The following are the spot interbank rates at the present time:
3-month LIBOR is 1.5%, 6-month LIBOR is 2%, and 9-months LIBOR is 2.5% (all with continuous
compounding). The bank is expecting continued economic recovery in the US in the next two quarters, and is
concerned that the Federal Reserve Bank might raise interest rates in the mean time.
a. (5 pts.) If neither interest rate futures nor forward rate agreements are available, how can the bank hedge
against rising interest rates. Demonstrate (be specific as to the bank‘s actions or trades) and show the
borrowing rate that the bank can effectively lock. (Express this rate with quarterly compounding.)
The bank should essentially ―lock in‖ a borrowing rate for $100 million for the three month period starting
July 1 (six months from now). If neither futures nor FRAs are available, the bank has to achieve this
synthetically, in the following manner:
1. Borrow for 9 months an amount that would grow to $100 million in 6 months if invested at the
current 6 month LIBOR. That is, borrow $100e– 0.02 x.5 = $99,004,983.38 for 9 months (until October
1) at 2.5%. In October 1, the bank has to pay back 99,004,983.38 e 0.025 x .75 = $100.878,839.3.
2. Invest 99,004,983.38 for six month at 2%. This will bring in a cash inflow of $100 million on July 1 st.
Actions now CF now CF July 1 CF October 1
Borrow $99.005 million for 9 months +$99.005 -$100.879
Invest $99.005million for 6 months -$99.005 +$100
Net cash flows 0 +$100 -$100.879
The net cash flows look exactly like borrowing $100 million in July first for three months. The effective
borrowing rate that is locked is:
4*ln(100,878,839.3/100,000,000) = 3.50% with continuous compounding
3.5154% with quarterly compounding.
b. (5 pts.) If forward rate agreements (FRA) are available, how much should the bank be willing to pay for an
agreement that locks in a borrowing rate of 3% (with quarterly compounding) for the $100 million
principal? (Note that interest computation for the FRA is done on a quarterly basis.)
The FRA is offering the bank a borrowing rate that is below what the bank can lock on its own (3.51545%),
which means the bank will save interest by locking in the 3% rate. The present value of this interest saving,
which is the current value of the FRA, is what the bank should be willing to pay for the agreement.
FRA = 100,000,000 *0.25*(3.5154% - 3%) e-0.25 x0.75= +$126,446
Or you could answer the question regardless of your answer to (a). Two ways to do that:
You could compute the 3-month forward rate for the July-Oct period (which would be 3.5%), convert it to
quart. compounded rate (3.5156%), and then compute the interest saving and value of FRA as we did above.
You could value the FRA from the bank‘s point of view simply by discounting the cash flows from the
agreement at the current zero LIBOR rates. Keep in mind that the FRA locks in 3% rate for three months,
which locks in an interest expense to the bank of $100 million*(3%/4) = $750,000.
FRA = $100e– 0.02 x.5 – 100.750 e-0.25 x0.75 = +$126,446
c. Assume the following interest rate futures are also available and for all maturities: Treasury bond futures, 3-
month Treasury bill futures, and 3-month Eurodollar futures. How can the bank hedge its interest rate
exposure with futures?
(3 pts.) Which futures contract is the most appropriate? Why?
Eurodollar futures are most appropriate. They are a short duration instrument similar to the bank‘s
borrowing horizon and the 3-month LIBOR (3-month Eurodollar rate) will be very close to what the bank
will end up paying when it borrows in the interbank market.
(2 pts.) Should the bank short or long these futures?
The bank should short the futures.
d. (5 pts.) Using the information given in this problem, can you guess what the futures cash price would be
now for the contract you chose in (c)? (compute the price per $100 par)
Whether the bank does it itself, enter a FRA, or sell Eurodollar futures, the outcome should be the same: lock
in the forward rate of 3.5% (continuous compounding). Hence, the futures price now for the Eurodollar
futures contract that matures in July (which will then be a 3-month instrument that matures in October)
should be such that the implied rate on the instrument is 3.5%. The Eurodollar instrument is a discounted
instrument with face (or par) value of $1 million. The futures price, F, per $100 should be such that:
4*ln(100/F) = 3.50% ln(100/F) = 0.875% 100/F = e0.875% F = 99.1288 per 100 par, or $991,288.
9. (5 pts.) We always claim that the forward rate, F, has to be equal to S*e rT (assuming no dividend yield) for there
to be no arbitrage. However, for consumption commodities we often see F < S*erT. How can this ‗violation‘
If this inequality existed for a financial asset, investors who hold the asset would sell it spot and buy it
forward, and those who don‘t hold the asset will short sell it and buy it forward. In the case of a consumption
commodity, this is not possible, however. Parties who hold the commodity do so because they use it for
production or processing purposes. Hence, the commodity brings them a ‗convenience yield‘, which will be
lost if they sell the commodity spot or lend it to someone else. Parties who don‘t hold the commodity cannot
short sell it because no one will lend it to them. Hence, the inequality persists. Essentially, the inequality
disappears once we adjust the forward price to take into account the convenience yield, y: F = Se(r-y)T.
10. A small regional bank operates in a location where its deposits are mostly 5-year CDs while its loans are mostly
lines of credit to small businesses, that carry variable, i.e., adjustable, interest rates.
a. (5 pts.) What would hurt the bank‘s net income, an interest rate rise or drop? Explain.
A drop in interest rates will hurt the bank‘s net income since the rates it pays on its deposits will not drop as
fast as the rates it charges on its loans.
b. (5 pts.) How could the bank hedge its interest rate exposure without disrupting its core business and without
effecting its capital requirements?
It could swap its fixed interest payments for variable payments (or its variable receipts for fixed receipts) by
entering into a swap where it receives a fixed rate and pays a floating rate. Swaps would not disrupt the
bank‘s core business since the bank can continue making the same loans it always made and offering the same
CDs it always offered. Neither would they affect much the bank‘s capital requirements since the risk from
swaps is much less than the risk from loans with the same principal.