The conversion factor for a bond is equal to the quoted price the bond would eave 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 bond maturity and the times to the coupon payment dates are rounded down to the nearest 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. 345 by the most recent futures price and adding accrued interest, Problem A Arteriolar futures price changes from 36. 76 to 9682. What is the gain or loss to an investor who is long two contracts? The Arteriolar futures price has increased by 6 basis points. The investor makes again per contract of or $300 in total. Problem 6. 5. What is the purpose Of the convexity adjustment made to Arteriolar futures rates? Why is the convexity adjustment necessary? Suppose that a Arteriolar 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 he 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. 80th of these lead to the futures rate being greater than the forward rate, problem 6,6, The 350-day LABOR rate is 3% with continuous compounding and the forward rate calculated from a Arteriolar futures contract that matures in 350 days is 3. 2% with continuous compounding. Estimate the 440-day zero rate. From equation (6. ) the rate is or 3. 0409%. Problem 6. 7. It is January 30. YOU are managing a bond portfolio worth 56 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 cheapest-to-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 The number of contracts that should be shorted is Rounding to the nearest whole number, 60 contracts should be shorted. The position should be closed out at the end of July. Problem 6. , The price of a 90-day Treasury bill is quoted as 10. 00. What continuously compounded return (on an actual/ASS basis) does an investor earn on the Treasury bill for the 90-day period? The cash price of the Treasury bill is The annulled continuously compounded return is Problem 6. 9. It is May 5, 2011. The quoted price off government bond with a 12% coupon that matures on July 27, 2014, is 110-17. What is the cash price? The number Of days between January 27, 2011 and May 5, 2011 is 98. The number of days between January 27, 201 1 and July 27, 2011 is 181. The accrued interest is therefore The quoted price is 110. 312. The cash price is therefore or $113. 78. Problem 6. 10. Suppose that the Treasury bond futures price is 101-12. Which of the following tour bonds is cheapest to deliver? Bond Price Conversion Factor 125-05 1 _2131 142-15 1. 3792 115-31 1. 1149 144-02 1. 4026 The cheapest-to-deliver bond is the one for which is least. Calculating this factor for each of the 4 bonds we get Bond 4 is therefore the cheapest to deliver. Problem 6,11_ It is July 30, 2013. The cheapest-Weedier bond in a September 2013 Treasury bond futures contract is a 13% coupon bond, and delivery is expected to be made on September 30, 2013.
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 IS. 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: The rate of interest with continuous compounding is or 11. 65% per annum. A coupon of 6. 5 will be received in 5 days years) time.
The present value of the coupon is The futures contract lasts for 62 days years). The cash tortures price it the contract were written on the bond would be 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 Taking the conversion factor into account the quoted futures price should be: 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 old 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 shorts bonds and goes long an equivalent number of bond futures contracts. Uncertainty as to which bond will be delivered introduces complications. The bond that appears cheapest-to- deliver now may not in fact be cheapest-to-deliver at maturity. In the case where the futures price is too high, this is not a major problem since the party with the short position (i. . , the arbitrageur) determines which bond is to be delivered, In he case where the futures price is too low, the arbitrageurs position is tar more difficult since he or she does not know which bond to short; it is unlikely that a profit can be locked in for all possible outcomes. Problem 6. 13. Suppose that the nine-month LABOR interest rate is 8% per annum and the six- month LABOR interest rate is 7 per annum (both with actual/ASS and continuous compounding). Estimate the three-month Arteriolar futures price quote for a contract maturing in six months.
The forward interest rate for the time period between months 6 and g is 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 or 9. 102%. This assumes that the day count is actual/actual. With a day count of actual/360 the rate is The three-month Arteriolar quote for a contract maturing in six months is therefore Problem 6. 14.
Suppose that the 300-day LABOR zero rate is and Arteriolar quotes for contracts maturing in 300, 398 and 489 days are 95. 83, 95. 62, and 95. 48 Calculate 98-day and 489-day LABOR zero rates, Assume no difference between forward and futures rates for the purposes of your calculations. The forward rates calculated form the first two Arteriolar futures are 4. 17% and 4. 38%_ These are expressed with an actual/ICC day count and quarterly compounding. With continuous compounding and an actual/365 day count they are and 4. 4167%. It follows from equation (6. 4) that the 398 day rate is (4000+4. 060x38V 398=4. 0507 or 4. 0507%. The 489 day rate is (4. 0507098+4. 4167* 188 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 12-year rate is less volatile than the tour-year rate? Duration-based hedging procedures assume parallel shifts in the yield curve.
Since the 12-year rate tends to move by less than the 4-year rate, the portfolio manager may find that he or she is over-hedged. Problem 6. 16. Suppose that it is February 20 and a treasurer realizes that on July 1 7 the many 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 (In other words, the company would receive $4,820,000 for its paper and have to redeem it at in 180 days’ time. ) The September Arteriolar 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 Arteriolar futures contracts. The Arteriolar futures position leads to a profit if rates rise and a loss if they fall. The duration Of the commercial paper is nice that of the Arteriolar deposit underlying the Arteriolar futures contract. The contract price of a Arteriolar futures contract is 980,000. The number of contracts that should be shorted is, therefore, 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. Years, The December Treasury bond futures price is currently 91-12 and the cheapest-to-deliver bond will have a duration of 8. 8 years at maturity. How should the portfolio manager minimize he 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 shorted is Rounding to the nearest whole number 88 contracts should be shorted. Problem 6. 18. HOW can the portfolio manager change the duration Of the portfolio to 3. Years in Problem 6. 17? The answer in Problem 6. 17 is designed to reduce the duration to zero. To reduce the duration from 7. 1 to 3. 0 instead of from 7. 1 to O, the treasurer should short or 51 contracts. Problem 6. 9. Between October 30, 2012, and November l, 2012, you have a choice between owning a LIES. Government bond paying a 12% coupon and a LIES. Corporate bond paying a coupon. Consider carefully the day count conventions discussed in this chapter and decide which of the two bonds you would prefer to own. Ignore the risk of default, You would prefer to own the Treasury bond.
Under the 30/360 day count convention there is one day between October 30 and November 1. Under the actual/actual (in period) day count convention, there are two days, Therefore you would earn approximately twice as much interest by holding the Treasury bond. Problem 6. 20. Suppose that a Arteriolar futures quote is 88 for a contract maturing in 60 days. What is the LABOR poniard rate for the 60- to 150-day period? Ignore the difference between futures and forwards for the purposes Of this question. The Arteriolar futures contract price Of 88 means that the Arteriolar futures rate is 12% per annum with quarterly compounding.
This is the forward rate for the 60. To 150. Day period with quarterly compounding and an actual/360 day count convention. Problem 6. 21. The three-month Arteriolar futures price for a contract maturing in six years is quoted as 95. 0, The standard deviation of the change in the short-term interest rate in one year is 1. 1%. Estimate the forward LABOR interest rate tort the period between 6. 00 and 6. 25 years in the future, Using the notation of Section 6,3, , , and The convexity adjustment is or about 23 basis points. The futures rate is with quarterly compounding and an actual,BIB day count. This becomes with an actual/actual day count.
It is faith continuous compounding. The forward rate is therefore with continuous compounding. Problem 6. 22. Explain Why the forward interest rate is less than the corresponding futures interest rate calculated from a Arteriolar futures contract. Suppose that the contracts apply to the interest rate between times and . 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 . The second is that without daily settlement a futures contract would be settled at time not .
Both reasons tend to make the futures rate greater than the forward rate. Further Questions problem 6,23 The December Arteriolar futures contract is quoted as 98. 40 and a company Lana to borrow $8 million for three months starting in December at LABOR plus 0. 5%. (a) What rate can then company lock in by using the Arteriolar futures contract? (b) What position should the company take in the contracts? (c) Fifth actual three-month rate turns out to be 1. 3%, what is the final settlement price on the futures contracts, Ignore timing mismatches between the cash flows from the Arteriolar futures contract and interest rate cash flows. A) The company can lock in a 3-month rate of 100 – 98. 4=1. 60%. The rate it pays is therefore locked in at 1. 6 * O. S = 2. 1%. (b) The company should sell (i. . , short) 8 contracts. If rates increase, the futures quote goes down and the company gains on the futures. Similarly, if rates decrease, the futures quote goes up and the company loses on the futures. (c) The final settlement price is 100 – 1. 30 = 98. 70. Problem 624 A Arteriolar futures quote for the period been 5. 1 and 5. 35 year in the future is 97. 1. The standard deviation of the change in the short. Term interest rate in one year is 1. %. Estimate the foamed interest rate in an FAR. The futures rate is 2. 9%. The foamed rate can be estimated using equation (6. 3) as 0. 029-0. 5* 0. 0142 0. 263 or 2. 63%. Problem 6,25 It is March 10, 2011. The cheapest-to-deliver bond in a December 2011 Treasury bond futures contract is an coupon bond, and delivery is expected to be made on December 31 31, 2011. Coupon payments on the bond are made on March I and September I each year _ The term structure is flat, and the rate of interest with continuous compounding is 5% per annum. The conversion factor for the bond is 1. 2191.
The current quoted bond price is $137. Calculate the quoted futures price for the contract. The cash bond price is currently A coupon of a will be received after 1 75 days or 0. 795 years. The present value Of the coupon on the bond is e-C. Ox. 4795=3. SASS. The futures contract lasts 296 days or 0. 8110 years. The cash futures price it were written on the 8% bond would therefore be (137. 1957 – 3. 9053)e. Sox. 8110 -138. 8061 At delivery there are 121 days of accrued interest. The quoted futures if the contract were written on the 8% bond would therefore be The quoted price should therefore be problem 6. 6. Assume that a bank can borrow or lend money at the same interest rate in the LABOR market, The go-day rate is per annum, and the 180-day rate is 10. 2% re annum, both expressed with continuous compounding The Arteriolar futures price for a contract maturing in 91 days is quoted as 89,S, What arbitrage opportunities are open to the bank? The Arteriolar futures contract price of 83. 5 means that the Arteriolar futures rate is 108% per annum with quarterly compounding and an actual/360 day count This becomes with an actual/actual day count. This is or 1051 % with continuous compounding.
The forward rate given by the 90-day rate and the 180-day rate is 10. 4% with continuous compounding. This suggests the following arbitrage opportunity: 1. Buy Arteriolar futures. 2. Borrow 180-day money. 3. Invest the borrowed money for 90 days. Problem 6. 27. A Canadian company wishes to create a Canadian LABOR futures contract from a U. S. Arteriolar futures contract and forward contracts on foreign exchange. Lasing an example, explain how the company should proceed. For the purposes of this problem, assume that a futures contract is the same as a forward contract. The U. S.