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Unformatted text preview: CHAPTER 6
Interest Rate Futures Problem 6.1. A US. ﬁeasury bond pays a 7% coupon on January 7' and July 7. How much interest accrue per $100 of principal to the bond holder between July 7, 2009 and August 9, 2009?
How would your answer be diﬁ‘erent if it were a corporate bond? Problem 6.2. It is January 9, 2009. The price of a Treasury bond with a 12% coupon that matures
on October 12, 2020, is quoted as 102—07. What is the cash price? There are 89 days between October 12, 2009, and January 9, 2010. There are 182 days
between October 12, 2009, and April 12, 2010. The cash price of the bond is obtained by adding the accrued interest to the quoted price. The quoted price is 102% or 10221875.
The cash price is therefore 89 '
102.21875 + Tea—2 x 6 ___$105.15 Problem 6.3. How is the conversion factor of a bond calculated by the Chicago Board of Trade?
How is it used? The conversion factor for a bond is equal to the quoted price the bond would have
per dollar of principal on the ﬁrst day of the delivery month on the assumption that the
interest rate for all maturities equals 6% per annum (with semiannual compounding). The three months for the purposes of the calculation. The conversion factor deﬁnes how much
an investor with a short bond futures contract receives when bonds are delivered. If the
conversion factor is 1.2345 the amount investor receives is calculated by multiplying 1.2345
by the most recent futures price and adding accrued interest. Problem 6.4. A Eurodollar futures price changes from 96.76 to 96.82. What is the gain or loss to
an investor who is long two contracts? 42 The Eurodollar futures price has increased by 6 basis points. The investor makes a
gain per contract of 25 x 6 = $150 or $300 in total. Problem 6.5. What is the purpose of the convexity adjustment made to Eurodollar futures rates?
Why is the convexity adjustment necessary? Suppose that a Eurodollar futures quote is 95.00. This gives a futures rate of 5% for
the three—month period covered by the contract. The convexity adjustment is the amount
by which futures rate has to be reduced to give an estimate of the forward rate for the
period The convexity adjustment is necessary because a) the futures contract is settled
daily and b) the futures contract expires at the beginning of the three months. Both of
these lead to the futures rate being greater than the forward rate. Problem 6.6. The 350day LIBOR rate is 3% with continuous compounding and the forward rate
calculated from a Eurodollar futures contract that matures in 350 days is 3.2% with con
tinuous compounding. Estimate the 440day zero rate. From equation (6.4) the rate is 3.2x90+3x350 440 = 3.0409 or 3.0409%. Problem 6.7. It is January 30. You are managing a bond portfolio worth $6 million. The duration
of the portfolio in six months will be 8.2 years. The September Treasury bond futures
price is currently 108—15, and the cheapestto—deliver bond will have a duration of 7.6
years in September. How should you hedge against changes in interest rates over the next
six months? The value of a contract is 108% x 1,000 = $108,468.75. The number of contracts
that should be shorted IS 6, 000, 000 8. 2 108,468.75 X it = 59'? Rounding to the nearest whole number, 60 contracts should be shorted. The position
should be closed out at the end of July. Problem 6.8.
The price of a 90day Treasury bill is quoted as 10.00. ‘What continuously compounded
return (on an actual/365 basis) does an investor earn on the Treasury bill for the 90day period?
The cash price of the Treasury bill is 90
__ .. = _ 0
100 360 X 10 $975 43 The annualized continuously compounded return is 365 2.5
__]. ——— =
90 n (1 + 97.5) 10.27% Problem 6.9. It is May 5, 2008. The quoted price of a government; bond with a 12% coupon that
matures on July 27, 2011, is 11017. What is the cash price? The number of days between January 27, 2008 and May 5, 2008 is 99. The number of
days between January 27, 2008 and July 27, 2008 is 182. The accrued interest is therefore 99
6 x m — 3.2637 .1 The quoted price is 110.5312. The cash price is therefore
110.5312 + 3.2637 = 113.7949 or $113.79. Problem 6.10. Suppose that the ﬂeasury bond futures price is 10112. Which of the following four
bonds is cheapest to deliver? _—"_'——————_‘__._________________ Bond Price Conversion Factor
———————________._______
1 12505 1.2131
2 1421 5 1.3 7'92
3 11531 1.1 149
4 14402 1.4026 ——————~___________________
The cheapestto—deliver bond is the one for which Quoted Price — Futures Price x Conversion Factor is least. Calculating this factor for each of the 4 bonds we get Bond 1: 125.15625 — 101.375 x 1.2131 2 2.178
Bond 2: 142.46875  101.375 X 1.3792 = 2.652
Bond 3: 11596875 — 101.375 x 1.1149 = 2.946
Bond 4: 144.06250  101.375 x 1.4026 = 1.874 Bond 4 is therefore the cheapest to deliver. 44 Problem 6.11. It is July 30, 2009. The cheapest—to—deliver bond in a September 2009 Treasury bond
futures contract is a 13% coupon bond, and delivery is expected to be made on September
30, 2009. Coupon payments on the bond are made on February 4 and August 4 each year.
The term structure is ﬂat, and the rate of interest with semiannual compounding is 12%
per annum. The conversion factor for the bond is 1.5. The current quoted bond price is
$110. Calculate the quoted futures price for the contract. There are 176 days between February 4 and July 30 and 181 days between February
4 and August 4. The cash price of the bond is, therefore: 176
110 + 331' X 6.5 — 116.32 The rate of interest with continuous compounding is 21m 1.06 = 0.1165 or 11.65% per
annum. A coupon of 6.5 will be received in 5 days (= 0.01370 years) time. The present value of the coupOn is
6.56—0.01370X0.1165 = 6.490 The futures contract lasts for 62 days (= 0.1699 years). The cash futures price if the
contract were written on the 13% bond would be (116.32 — 6.490)e°‘1699x°'1165 = 112.03 At delivery there are 57 days of accrued interest. The quoted futures price if the contract
were written on the 13% bond would therefore be 57
112.03 — 6.5 X m — 110.01 . Taking the conversion factor into account the quoted futures price should be: 110.01 “F = 73.34 Problem 6.12. An investor is looking for arbitrage opportunities in the Treasury bond futures market.
What complications are created by the fact that the party with a short position can choose
to deliver any bond with a maturity of over 15 years? If the bond to be delivered and the time of delivery were known, arbitrage would be
straightforward. When the futures price is too high, the arbitrageur buys bonds and shorts
an equivalent number of bond futures contracts. When the futures price is too low, the
arbitrageur sells bonds and goes long an equivalent number of boud futures contracts. Uncertainty as to which bond will be delivered introduces complications. The bond that appears cheapestto—deliver now may not in fact be cheapesttodeliver at maturity.
In the case where the futures price is too high, this is not a. major problem since the party 45 with the short position (i.e., the arbitrageur) determines which bond is to be delivered. In
the case where the futures price is too low, the arbitrageur’s position is far more difﬁcult
since he or she does not know which bond to buy; it is unlikely that a proﬁt can be locked
in for all possible outcomes. Problem 6.13.
Suppose that the ninemonth LIBOR interest rate is 8% per annum and the six
mOnth LIBOR interest rate is 7.5% per annum (both with actual/365 and continuous compounding). Estimate the threemonth Eurodoilar futures price quote for a contract
maturing in six months. The forward interest rate for the time period between months 6 and 9 is 9% per
annum with continuous compounding. This is because 9% per annum for three months
when combined with 7%% per annum for six months gives an average interest rate of 8%
per annum for the nine—month period. With quarterly compounding the forward interest rate is
4(3009/4 — 1) = 0.09102 or 9.102%. This assumes that the day count is actual/actual. With a day count of
actual/360 the rate is 9.102 x 360/365 = 8.977. The threemonth Eurodollar quote for a contract maturing in six months is therefore
100 — 8.977 = 91.02 This assumes no difference between futures and forward prices. Problem 6.14. Suppose that the 300day LIBOR zero rate is 4% and Eurodoiiar quotes for contracts
maturing in 300, 398 and 489 days are 95.83, 95.62, and 95.48. Calculate 398day and
489day LIBOR zero rates. Assume no diﬂ'erence between forward and futures rates for
the purposes of your caicuiations. The forward rates calculated form the ﬁrst two Eurodollar futures are 4.17% and
4.38%. These are expressed with an actual/360 day count and quarterly compound
ing. With continuous compounding and an actual/365 day count they are (365/90) ln(1 +
0.0417 /4) = 4.2060% and (365/90) ln(1 + 0.0438/4) = 4.4167%. It follows from equation
(6.4) that the 398 day rate is 4 x 300 + 4.2060 x 98 = 4.
398 0507 or 4.0507%. The 489 day rate is 4.0507 x 398 + 4.4167 x 91
489 46 = 4.1188 or 4.1188%. We are assuming that the ﬁrst futures rate applies to 98 days rather than the
usual 91 days. The third futures quote is not needed. Problem 6.15. Suppose that a bond portfolio with a duration of 12 years is hedged using a futures
contract in which the underlying asset has a duration of four years. What is likely to be
the impact on the hedge of the fact that the 12year rate is less volatile than the fouryear
rate? Durationbased hedging schemes assume parallel shifts in the yield curve. Since the
12year rate tends to move by less than the 4year rate, the portfolio manager may ﬁnd
that he or she is over—hedged. Problem 6.16. Sappose that it is February 20 and a treasurer realizes that on July 17' the company
will have to issue $5 million of commercial paper with a maturity of 180 days. If the
paper were issued today, the company would realize $4,820,000. (In other words, the
company would receive $4,820,000 for its paper and have to redeem it at $5,000,000 in 180
days’ time.) The September Eurodollar futures price is quoted as 92.00. How should the
treasurer hedge the company’s exposure? The company treasurer can hedge the company’s exposure by shorting Eurodollar
futures contracts. The Eurodollar futures position leads to a proﬁt if rates rise and a loss if they fall.
The duration of the commercial paper is twice that of the Eurodollar deposit under lying the Eurodollar futures contract. The contract price of a Eurodollar futures contract
is 980,000. The number of contracts that should be shorted is, ”therefore, i 4, 820, 000 980,000 x 2 = 9‘84 Rounding to the nearest whole number 10 contracts should be shorted. Problem 6.17.
On August 1 a portfolio manager has a bond portfolio worth $10 million. The duration of the portfolio in October Will be 7.1 years. The December Treasury bond futures price
is currently 9112 and the cheapesttodeliver bond will have a duration of 8.8 years at maturity. How should the portfolio manager immunize the portfolio against changes in
interest rates over the next two months? The treasurer should short Treasury bond futures contract. If bond prices go down,
this futures position will provide offsetting gains. The number of contracts that should be h .
S “ted ls 10,000,000 x 7.1
91,375 x as Rounding to the nearest whole number 88 contracts should be shorted. = 88.30 47 Problem 6.18. or 51 contracts. Problem 6.19. y bond. Under the 30/360 day count convention there is one day between October 30, 2009 and November 1, 2009. Under the actual/ actual (in period) day count convention,
. ' assumes that the quoted
prices of the two bonds are the same. Problem 6.20. Suppose that a Eurodoﬂar futures quote 1'3 88 for a contract maturing in 60 days.
What is the LIBOR forward rate for the 60— to 150 day period? Ignore the diﬁerence
between futures and forwards for the purposes of this question. The Eurodollar futures contract price of 88 means that the Eurodollar futures rate is 12% per annum. This is the forward rate for the 60 to 150day period with quarterly
compounding and an actual/360 day count convention, Problem 6.21 . h fu period between 6.00 and 6.25
years in t e ture. Using the notation of Section 6.4, a = 0.011, T1 = 6, and T2 = 6.25. The convexity
adjustment is or about 23 basis points. The futures rate is
actual/360 clay count. (365/90)ln(1.012)
ing and actual/365 day count. The forw
continuous compounding. 4.8% with quarterly compounding and an
= 0.0484 or 4.84% with continuous compound
ard rate is therefore 4.84 — 0.23 = 4.61% with 48 Problem 6.22.
Explain why the forward interest rate is less than the corresponding futures interest rate calculated from a Eurodoliar futures contract. Suppose that the contracts apply to the interest rate between times T1 and T2. There
are two reasons for a difference between the forward rate and the futures rate. The ﬁrst is that the futures contract is settled daily whereas the forward contract is settled once at
time T2. The second is that without daily settlement a futures contract would be settled at time T1 not T2. Both reasons tend to make the futures rate greater than the forward
rate. 49 ...
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 Spring '08
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