Chapter 6 Interest Rate Futures Geng Niu

Motivation Increased volatility in interest rates after late 1970s Pose significant risks for bond issuers, financial intermediaries, and investors. Interest rate futures were introduced around 1980.

Growth Rate Calculations | US recession dates

Day Count Convention Defines: the period of time to which the interest rate applies The period of time used to calculate accrued interest (relevant when the instrument is bought or sold) Usually expressed as X/Y X: defines the way in which the number of days between the two dates is calculated, Y defines the way in which total number of days in the reference period is measured. The interest then is X/Y*interest earned in reference period. We know the interest earned over some reference period, such as semiannual coupons; but we are interested in calculating the interest earned over some other period. Usually expressed as X/Y X: defines the way in which the number of days between between the two dates is calculated, Y defines the way in which total number of days in the reference period is measured. The interest then is X/Y*interest earned in reference period. 4 4

Day Count Conventions in the U.S. Treasury Bonds: Actual/Actual (in period) Corporate Bonds: 30/360 Money Market Instruments: Actual/360 5 5

Examples 1. Bond: 8% Actual/ Actual in period. 4% is earned between coupon payment dates. Accruals on an Actual basis. When coupons are paid on March 1 and Sept 1, how much interest is earned between March 1 and April 1? Answer: Assume the principal is $100. There are 184 actual days in the March 1st –Sep 1st reference period There are 31 actual days between March 1st and Apr 1st The interest accrued/earned is: $4*(31/184)=$0.67 Bond: 31/184*4=0.67 30/180*4=0.67 6 6

Examples continued 2. T-Bill: 8% Actual/360: 8% is earned in 360 days. Accrual calculated by dividing the actual number of days in the period by 360. How much interest is earned between March 1 and April 1? Answer There are 31 actual days in the March 1st – Apr 1st period The interest accrued/earned is $100 *0.08 *(31/360)=$0.69 31/360*8=0.69 For a whole year: 365/360 For money market funds 7 7

Examples continued 3. Corporate Bond: 8% 30/360 Assumes 30 days per month and 360 days per year. When coupons are paid on March 1 and Sept 1, how much interest is earned between March 1 and April 1? Answer There are 180 days in the March 1st –Sep 1st reference period There are 30 days between March 1st and Apr 1st The interest accrued/earned is $4 * (30/180)=$0.67

The February Effect (Business Snapshot 6.1) How many days of interest are earned between February 28, 2013 and March 1, 2013 when day count is Actual/Actual in period? day count is 30/360? Under 30/360 count, there are 3 days, Under actual/actual, only one day count. You should earn three times as much as the second one. 9 9

Quote and price: Treasury Bill The price of money market instruments are sometimes quoted using a discount rate. (as a percentage of the final face value rather than on the initial price). Example: 91-day Treasure Bills quote at 8. means 8% interest is earned on the face value per 360 days. Suppose face value is 100, then 100*0.08X91/360=2.022 is earned over 91 day period Cash price = 100-2.022=97.978 Then true rate is 2.022/(100-2.022)=2.064% Bond equivalent yield (BEY)=2.022/(100-2.022)*365/91

Quote and price: Treasury Bill General Cash price and quote price following the relationship: 100-P*n/360=Y Then in general we have: The price of money market instruments are sometimes quoted using a discount rate. (as a percentage of the final face value). Example: 91-day Treasure Bills quote at 8. means 8% interest is earned on the face value per 360 days. Suppose face value is 100, then 100*0.08X91/360=2.022 is earned over 91 day period. Then true rate is 2.022/(100-2.022)=2.064% General Cash price and quote price following the relationship as indicated above Bond equivalent yield (BEY)=(100-P)/P*365/n*100 11 11

Quotes and yield: T-bill http://online. wsj

Quotes and yield: T-bill n= Days between 4/1/2016 and 4/72016=6 Quoted (bid) price: 0.215 Cash (bid) price: 100- 0.215*6/360= 99.99642 This is the price a buyer is willing to pay to get the t-bill. Quoted (asked) price: 0.205 Cash (asked) price: 100- 0.205*6/360= 99.99658 This is the price at which a seller is willing to sell the t-bill. Asked yield: (100-99.99658)/99.99658*365/6*100=0. 208

Quote and price:Treasury note and treasury bond Quoted in Dollars and thirty-seconds of a dollar. The quoted price is for a bond with face value of $100. Example: a quote of 90-05 indicates that if the bond has a face value of $100,000 its price will be (90+5/32)*1000=$90156.25 The quoted price is also known as clean price. Cash price (invoice price) = Quoted price + Accrued Interest

Accrued Interest Suppose now is March 5th, 2010. There is an 11% coupon bond matures on July 10th, 2018 with quoted price 95-16. Semi-annual coupon payment, the most recent coupon date is Jan 10th, 2010. and the next date is July 10th, 2010. 1/10/2010 3/5/2010 7/10/2010 7/10/2018 Last coupon day Today Next coupon day Maturity

Accrued Interest Jan 10th and March 5th , 2010 : 54 days in between; Jan 10th and July 10th, 2010: 181 days in between; Hence, the accrued interests on March 5, is the share of the July 10th coupon accruing to bondholders on March 5th. Coupon: 100*11%/2=$5.5 Under actual/actual: 54/181*5.5=$1.64. Hence the cash price per $100 face value=$95.5+$1.64=$97.14 The cash price of a $100,000 face value bond is $97,140 This is the price you have to pay to get the bond.

Note and Bond Quotes Coupon rate: 6 1/2 percent=6.5% Maturity date: 8/15/05 N: note, an initial maturity of two to 10 years Referred to as "the 6 1/2s of August 2005.“ Bid quote : 105.08 Asked quote : 12 105.12 (shows only the 32nds of a dollar) Bid clean price: 105+ 8/32=$105.25 Asked clean price: 105+ 12/32=105.375

Treasury Bond Futures Underlying asset: a 6% coupon treasury bond Treasure bond: long-term government debt instrument Quoted in Dollars and 30-seconds of a dollar per $100 face value. 124-150, means 124+15.0/32 Bond quote and Futures quote are similar but not entirely identical.

Bond and Futures quote The trailing “2”: 2/8ths of 1/32nd = 1/128th The trailing “5”: 0.5*1/32nd = 1/64th The trailing “7”: 0.75*1/32nd = 3/128th

Delivery Practices The Treasury bond futures allows the party with the short position to choose to deliver any government bond that has more than 15 years to maturity on the 1st day of the delivery month and is not callable within 15 years from that day A significant number of securities, ranging widely in terms of coupon and maturity, may be eligible for delivery. If only one bond were permitted for delivery, it would be easy for one or two institutions to corner the supply of that security and create a scarcity: market manipulation

Invoicing System High-coupon bonds will naturally command a greater price than comparable low-coupon bonds. These differences must be reflected in the futures contract. the futures contract permits the delivery of a wide range of bonds at the discretion of the short. The invoice price must be adjusted to reflect the specific pricing characteristics (e.g. coupon rate) of the bond that is delivered. Invoice price: the cash price received by the short for delivering the underlying asset

Treasury Bond Futures Pages 132-136 Cash price received by party with short position = Most recent settlement price × Conversion factor + Accrued interest A typical treasure bond future contract traded by the CME group: Any government bond that has more than 15 years to maturity on the 1st day of the delivery month and is not callable within 15 years from that day can be delivered. If 10 year treasure notes-any government bond maturity between 6.5 and 10 years can be delivered. 5 year notes future contract: bond must have a remaining life of between 4.167 and 5.25 years. Quoted in Dollars and 30-seconds of a dollar per $100 face value. 124-150, means 124+15/32 23 23

Example Most recent settlement price = 90.00 Conversion factor of bond delivered = 1.3800 Accrued interest on bond =3.00 Price received for bond is 1.3800×90.00+3.00 = $127.20 per $100 of principal 24 24

Conversion Factor (CF) The underlying asset for treasury bond futures: a 6% coupon The bond delivered by the short: not necessarily 6% coupon Conversion factor: the price ( per dollar par) at which the bond will provide a yield to maturity of 6%. If the short indeed deliver a 6% coupon bond: CF=1

Conversion Factor The bond maturity and the times to the coupon payment dates are rounded down to the nearest 3 months for the purposes of calculation. After rounding, if the bond lasts for an exact number of sic-month periods, the first coupon is assumed to be made in 6 months If does not, the first coupon is assumed to be paid after 3 months, and accrued interest is subtracted (for the 3 months preceding the present) Bond maturity and times to the coupon dates are rounded down to the nearest 3 months for the purpose of the calculation. If after rounding, the bond lasts exactly 6-months to mature, the 1st coupon is assumed to be paid in 6 months. If not, then the 1st coupon is assumed to be paid after 3 month and the accrued interest is subtracted. Consider: a 10% coupon with 20 years and 2 months to maturity. For the purpose of calculation, the bond is assumed to be mature in 20 years. The first coupon is assumed to be paid in 6 months. Coupon payments are then assumed to be paid each 6-months until the end of year 20 when the principal is paid. Assuming the face value is 100. With 6% coupon paid semi-annually, the value of the bond is: sum(5/1.03^i)i=1to40 +100/1.03^40=146.23 Dividing by the value of $100 gives a conversion factor of 1.4623. Another example: 8% coupon bond with 18 years and 4 months to maturity. For the purpose of calculating, the bond is assumed to have exactly 18years and 3 months to maturity. Discounting all the cash flows to a point in time 3 months from today at 6% per annum: 4+sum(4/1.03^i)+100/1.03^36=125.83 The interest rate for a 3 month period sqrt(1.03)-1 or 1.4889%. Hence discounting back to today will give 125.83/1.014889=$123.99. subtracting the accrued interest of 2.0=$121.99. 26 26

Conversion Factor: Example 1 Consider: a 10% coupon with 20 years and 2 months to maturity. For the purpose of calculation, the bond is assumed to be mature in 20 years. The first coupon is assumed to be paid in 6 months. Coupon payments are then assumed to be paid each 6-months until the end of year 20

Conversion Factor Assuming the face value is 100. With 6% coupon paid semi-annually, the value of the bond is: Dividing by the value of $100 gives a conversion factor of 1.4623.

Conversion Factor: Example 2 coupon 8% coupon bond with 18 years and 4 months to maturity. The bond is assumed to have exactly 18 years and 3 months to maturity. Discounting all the cash flows to a point in time 3 months from today : now 1st coupon Maturity -3m 3m T=0 18 yr+ 3m

Conversion Factor The interest rate for a 3 month period: Hence discounting back to today will give 125.83/1.014889=$123.99. Accrued interest for the 3 months preceding the present: $2 Conversion factor: (123.99-2)/100= 1.2199.

Cheapest-to-Deliver Bond The short position trader receives: Most recent settlement price X Conversion factor + Accrued Interest The cost of purchasing a bond: Quoted bond price + Accrued Interest The cheapest to deliver is to minimize: Quoted bond price – most settlement price X conversion factor

Cheapest-to-Deliver Bond Quoted bond price($) Conversion factor 1 99.5 1.0382 2 143.5 1.5188 3 119.75 1.2615 Assume settlement price is $93.25. Cost of delivering : Bond1: 99.50-(93.25*1.0382)=$2.69 Bond2: 143.50-(93.25*1.5188)=$1.87 Bond3: 119.75-(93.25*1.2615)=$2.12

CBOT T-Bonds & T-Notes Exact theoretical futures prices are difficult to determine because there are many factors that affect the futures price Delivery can be made any time during the delivery month Any of a range of eligible bonds can be delivered If we assume that both the cheapest-to-deliver bond and the delivery date are known, the futures price is F0 = (S0 – I)ert Cheapest-to-Deliver Bond: The party with the short position receives: most recent settlement price X Conversion factor + Accrued Interest: The cost of purchasing a bond: Quoted bond price + Accrued Interest The cheapest to deliver is to minimize Quoted bond price – most settlement price X conversion factor Example: 3 bonds to choose for a short position, assuming the most settlement price is 93-08 or 93.25 Bond 1 2 3 Quoted bond price 99.05 143.05 119.75 Conversion factor 1.0382 1.5188 1.2615 Then cost of delivering each of the bond is: 99.05-93.25X1.0382=2.69 143.05-93.25X1.5188=1.87 119.75-93.25X1.2615=2.12 Bond 2 will be chosen. Wild Card Play: CME group treasury bond futures cease at 2pm Chicago time; Treasury bond themselves continue trading in the spot until 4pm A trader with the short position has until 8pm to issue a notice of intention to deliver. If the notice is issued, invoice price is the settlement price at 2pm 33 33

Examples of T-bond Futures It is March 10, 2011. The cheapest-to-deliver bond is an 8% coupon bond Delivery is expected to be made on Dec 31, 2011. Coupon payments on the bond are made on Mar 1 and Sep 1 each year. The rate of interest (cont. comp.) is 5% per annum. The conversion factor is 1.2191. The current quoted bond price is $137. Determine the quoted futures price for the contract.

3/1/2011 3/10/2011 9/1/2011 12/31/2011 3/1/2012 9d 175d 121d 61d Coupon payment Now Coupon payment Maturity Coupon payment The cash bond price is currently S0=137+9/184*4=137.1957 A coupon of 4 will be received after 175 days or 0.4795 years. The present value of the coupon on the bond (I) = 4e-0.05×0.4795=3.9053. The futures contract lasts 296 days or 0.8110 years: t=0.8110 The cash futures price if it were written on the 8% bond: (137.1957 − 3.9053)e0.05×0.8110 =138.8061

138.8061 is the theoretical cash price at which the short can deliver the 8% bond (the price that the long can obtain the 8% bond) at maturity. However, the quoted price in a T-bond futures contract is the clean price at which the short can deliver a 6% bond. At delivery there are 121 days of accrued interest. The accrued interest = 4*121/182=2.6593 The clean price at which the short can deliver the 8% bond :138.8061-2.6593=136.1468 The quoted price should therefore be 136.1468/1.2191=111.68