Test Bank for
Derivatives: Principles & Practice
Chapter 6: Interest-Rate Forwards & Futures
Rangarajan K. Sundaram
Sanjiv R. Das
August 1, 2010
1. Eurodollar deposits are
(a) Deposits that may be made in euros or dollars.
(b) Euro denominated deposits that are redeemable in dollars.
(c) Dollar denominated deposits made in banks in Europe.
(d) Euro denominated deposits made in the US.
Answer c.
2. Eurodollar deposits follow the money-market day-count convention. Suppose a deposit
is made for 92 days at a Libor rate of 4% on a notional amount of $100. The interest
amount is
(a) 1.0082
(b) 1.0099
(c) 1.0101
(d) 1.0222
Answer d. The eurodollar money-market convention is Actual/360, so the interest is
100×4×92/360 = 1.0222.
3. Consider a 6×12 FRA where the underlying six-month period is 183 days and the
notional is $100. The FRA fixed rate is 5%. At maturity of the contract the underlying
Libor for six months is 7%. What is the settlement amount on the FRA? Assume the
Actual/360 convention.
(a) 0.9683
(b) 0.9687
(c) 0.9817
(d) 1.0167
Answer c. Straightforward computation using the FRA payoff formula. The
calculations are as follows:
> k=0.05
> L=0.07
> d=183
> payoff = (L-k)*d/360*100/(1+L*d/360)
> payoff
[1] 0.9817333
4. The payoff of the FRA has the following property
(a) It is convex in the Libor rate.
(b) It is linear in the Libor rate.
(c) It is concave in the Libor rate.
(d) None of the above.
Answer c. See the segment on the “convexity bias” in the final part of Section 6.4.
5. You are given the following data concerning a 6×12 FRA. The first six-month period
is 182 days and the second is 183 days. The Libor rate for six months is 5% and for
one year is 6%. The arbitrage-free price of the 6×12 FRA, assuming the Actual/360
day-count convention, is
(a) 5.50%
(b) 6.22%
(c) 6.55%
(d) 6.82%
Answer d. Straightforward computation using the formulae in the text. The
calculations are as follows:
> d1=182
> d2=183
> r1=0.05
> r2=0.06
> B1 = 1/(1+d1*r1/360)
> B2 = 1/(1+(d1+d2)*r2/360)
> B1
[1] 0.9753454
> B2
[1] 0.9426551
> f = function(x) { B2*(1+d2*x/360)-B1 }
> uniroot(f,c(0,1))
$root
[1] 0.06822088
6. A $100 notional 6×12 FRA has the following features at inception. The first six month
period is 182 days and the second is 183 days. The locked-in rate on the FRA is 6%.
After one month the 5×11 FRA is trading at a fair strike of 6.2% with 151 days in the
first five month period. What is the value of the FRA to the buyer if the five-month
Libor rate at this point is is 5%?
(a) +0.0965
(b) -0.0965
(c) 0.0986
(d) –0.0986
Answer a. The buyer of the FRA is long the FRA so gains on a marked-to-market
basis when FRA rates go up. The calculations are as follows:
> d1=151
> d2=183
> L1=0.05
> L2=0.062
> k=0.06
> payoff = (L2-k)*100*d2/360/(1+d2*L2/360)
> discounted_payoff = payoff/(1+L1*d1/360)
> discounted_payoff
[1] 0.0965358
7. ABC Inc. has to borrow money to undertake a seasonal business expansion in six
months time. They will need additional working capital funding for six months and
wish to hedge themselves against a rise in interest rates in six month’s time. They
should
(a) Take a short position in a 6×12 FRA.
(b) Take a long position in a 6×12 FRA.
(c) Lend the notional amount for one year and borrow the same amount for six
months, both at the spot rates prevailing today.
(d) Lend the notional for one year, wait six months, and borrow the same amount for
six months at the spot rate prevailing then.
Answer b.
8. You are long 5 eurodollar futures contracts. If the Libor rate underlying the contract
increases by 5 basis points, your position gains the following value:
(a) -$25
(b) +$25
(c) -$125
(d) +$125
Answer c. Each basis point is worth $25. An increase in the interest rate reduces the
price of the futures contract, so the “gain” is negative.
9. The September eurodollar contract is trading at 95. You have a 90-day borrowing
commencing in September for $500,000,000 that you wish to hedge using futures.
How many eurodollar futures contracts should you buy (rounded off to the nearest
integer)?
(a) 490
(b) 492
(c) 494
(d) 500
Answer c. The hedge ratio is 1/(1+0.5×90/360) = 0.9876543. Since each contract has
face value 1 million, the total contracts needed are 0.9876543×500 = 493.83  494.
10. The convexity bias between FRAs and eurodollar futures implies that
(a) The futures results in greater cash outflows or smaller cash inflows than the FRA.
(b) The futures settlement amount is convex in Libor rates.
(c) The futures results in greater cash inflows or lower cash outflows than the FRA.
(d) The FRA payoff minus the futures payoff is convex in Libor rates.
Answer c. All other options are wrong.
11. In satisfaction of a US Treasury bond futures contract, the short position delivers a
15-year 9% bond instead of the standard bond. What is the conversion factor on this
delivery? Assume the last coupon on the bond was just paid.
(a) 1.255
(b) 1.294
(c) 1.354
(d) 1.446
Answer b.
1  4.5
4.5
1 

 
= 1.294
2

100 1.03 1.03
1.0330 
12. The quoted price on a 91-day Treasury bill is 5. What is the cash price of the bill?
(a) 95.00
(b) 98.50
(c) 98.74
(d) 98.75
Answer c. P = 100  5  (91/360) = 98.73611 .
13. All else being equal, a bond with a higher coupon has a duration that is ________
than that of a bond with a lower coupon.
(a) greater than.
(b) less than.
(c) equal to.
(d) undetermined in relation to.
Answer a.
14. Bonds A and B both have a duration of exactly one year. An equally-weighted
portfolio of these bonds will have a duration of
(a) Greater than 1 year because duration is additive.
(b) Equal to one year because the average duration is still one year.
(c) Less than one year, because duration is a measure of risk, and combining two
bonds into a portfolio diversifies away risk.
(d) Cannot say because the outcome depends on the interaction of specific cash flows
of both bonds.
Answer b. Here is a simple proof of this. Let cit denote cash flow at time t (in years)
from bond i , and Pi , Di its price and duration, respectively. For the two bonds, we
1
may write Di =  tt cit dt = 1, i = A, B . Note that the duration of bonds is not
Pi
affected by how much we invest in the bond, because both cit and Pi scale together.
And, since duration is additive, the combined duration of both bonds assuming we
1
1
invest equal amounts in each one is just DA  DB = 1 .
2
2
15. Your bond portfolio has a value of $10,600,000 with a duration of 2.2 years. How
many 90-day US Treasury bill futures contracts do you need to hedge this exposure if
the futures contract is priced at $995,000? Assume you are carrying out durationbased hedging.
(a) 1.21
(b) 5.86
(c) 10.65
(d) 93.75
Answer d. The hedge ratio is h = [10, 600, 000  2.2]/[995, 000  0.25] = 93.7487 .
16. You plan to borrow $1,000,000 for six months (183 days) in six months’ time (182
days). The current Libor rate for six months is 6%. You want to hedge your interestrate exposure by using 90-day eurodollar futures contracts that mature in six months.
Using PVBP analysis, how 90-day eurodollar futures contracts are needed for this
hedge?
(a) 1.79
(b) 1.85
(c) 1.92
(d) 2.00
Answer c. Use PVBP analysis to Answer this. A change of 1 bp increases borrowing
costs by
1, 000, 000  0.0001
1  0.06 
365
360
183
360 = $47.9183
Since the PVBP of a 90-day future is $25, the hedge ratio is
h = 47.9183/25 =1.916732 .
17. Ceteris paribus, as interest rates rise, which of these statements is most likely to be
true?
(a) The duration of bonds rises.
(b) The duration of bonds falls.
(c) Newly issued bonds have a higher duration than bonds issued some time ago.
(d) The volatility of bonds increases.
Answer b. For all bonds, note that a plot of the pricing function (with interest rates on
the x-axis and price on the y-axis) shows that bond prices are convex in the interest
rate, and decrease monotonically when rates rise. These two features imply that at
low rates the slope of this function (duration) will be steeper than at higher rates. This
provides the intuition for why duration falls when interest rates rise.
18. You borrow money at Libor with a floating-rate note for one year with two semiannual payments. What position do you need to add to this note to fix the cost of
borrowing for the entire year?
(a) Sell a 6 12 FRA.
(b) Buy a 6 12 FRA.
(c) Buy a one-year zero-coupon bond and short a 1.5-year zero-coupon bond.
(d) Buy a six-month zero-coupon bond and short a one-year zero-coupon bond.
Answer b.
19. A $100,000,000 3  6 FRA has a fixed rate of 4%. The first three-month period is for
91 days and the second one for 92 days. The 91-day Libor rate is 3% and the 183-day
Libor rate is 3.5%. The value of this contract to the buyer of the FRA is
(a) $8913
(b) $4822
(c) $3289
(d) $5433
Answer a. The calculations are as follows:
> d1=91
> d2=92
> r1=0.03
> r2=0.035
> B1=1/(1+r1*d1/360)
> B2=1/(1+r2*(d1+d2)/360)
> f = (B1/B2-1)*360/92
>f
[1] 0.03964501
> profit=(f-0.04)*d2/360/(1+f*92/360)*B1
> profit
[1] -8.913362e-05
> profit*100000000
[1] -8913.362
20. When you are short a position in a 3  6 FRA, you are effectively
(a) Long the three-month zero-coupon bond, and long the six-month zero-coupon
bond.
(b) Long the three-month zero-coupon bond, and short the six-month zero-coupon
bond.
(c) Short the three-month zero-coupon bond, and long the six-month zero-coupon
bond.
(d) Short the three-month zero-coupon bond, and short the six-month zero-coupon
bond.
Answer c.
21. A long position in a 6  9 FRA can be replicated using
(a) A six-month borrowing combined with a 9-month investment.
(b) A six-month investment combined with a 9-month borrowing.
(c) A six-month investment combined with a 3-month borrowing.
(d) A six-month borrowing combined with a 3-month investment.
Answer b. See Appendix 6A.
22. You anticipate a three-month borrowing in 6 months’ time. To hedge the interest-rate
exposure you can go either
(a) Long a 6  9 FRA or long a eurodollar futures contract maturing in 6 months.
(b) Short a 6  9 FRA or long a eurodollar futures contract maturing in 6 months.
(c) Long a 6  9 FRA or short a eurodollar futures contract maturing in 6 months.
(d) Short a 6  9 FRA or short a eurodollar futures contract maturing in 6 months.
Answer c. A long position in an FRA pays off when interest rates increase as does a
short position in a eurodollar futures position.
23. A long position in a eurodollar futures contracts expiring in June may be used to
hedge interest-rate exposure resulting from a planned
(a) 90-day borrowing ending in June.
(b) 90-day borrowing beginning in June.
(c) 90-day investment ending in June.
(d) 90-day investment beginning in June.
Answer d.
24. You are long an 3  6 FRA and long a eurodollar futures contract expiring in 3
months. Assume the fixed rate in the FRA is the same as the rate locked-in via the
eurodollar futures contract. If interest rates jump down by 100 basis points,
(a) There is no net cash flow consequence because you are perfectly hedged.
(b) You will lose more on the FRA than you will make on the eurodollar futures.
(c) You will make more on the FRA than you will lose on the eurodollar futures.
(d) You will lose less on the FRA than you will make on the eurodollar futures.
Answer b. This is the “convexity bias.”
25. You are short an 3  6 FRA and short a eurodollar futures contract expiring in 3
months. Assume the fixed rate in the FRA is the same as the rate locked in via the
eurodollar futures contract. If interest rates jump up by 100 basis points,
(a) You will lose money on both the FRA and the eurodollar futures.
(b) You make money on the FRA but lose on the eurodollar futures.
(c) You make money on both the FRA and the eurodollar futures.
(d) You lose money on the FRA but make money on the eurodollar futures.
Answer d.
26. Suppose the duration of a bond portfolio is 2. This means
(a) The final cash flow from the portfolio will occur in two years.
(b) The weighted-average maturity of the portfolio’s cash flows is 2 years.
(c) The portfolio is fully equivalent to a 2-year zero-coupon bond.
(d) The portfolio is fully equivalent to a 2-year par-coupon bond.
Answer b.