FMP 19: Interest Rate Futures
Interest on a bond arrives in lumps on coupon dates, so whenever a bond changes hands between two of those dates somebody must decide how much of the approaching coupon the seller has earned. That amount is the accrued interest, and the rule measuring it is the day count convention.
Every convention is written in the form X/Y. The first part sets how days inside the period of interest are counted, the second how days inside a reference period are counted, and for a bond that reference period runs from one coupon date to the next, since bonds commonly pay interest semi-annually in arrears. Conventions travel badly across borders: money market instruments in Australia, Canada, and New Zealand use actual/365, while corporate issues denominated in euros and in British pounds usually take actual/actual.
| Convention | Days in the period | Days in the reference period | Typical market | Accrual in Example 1 |
|---|---|---|---|---|
| actual/actual | Real calendar days | Real days between coupon dates | U.S. Treasury bonds and notes | 2.9348 |
| 30/360 | 30 per month | 360 per year | U.S. corporate bonds and mortgages | 2.9444 |
| actual/360 | Real calendar days | 360 per year | U.S. money market, including Treasury bills | Not applicable |
| actual/365 | Real calendar days | 365 per year | Money market in Australia, Canada and New Zealand | Not applicable |
Source: chapter conventions; the final column is computed below.
A bond carries a coupon rate of 10% per year and pays on March 15 and September 15. It is bought on July 1. Each semi-annual payment on a face value of USD 100 is USD 5, half of the annual 10% rate.
The odd corners of 30/360
Pretending that months are uniform produces strange arithmetic. Counting from February 28 to March 1 gives three days under 30/360, so a bond can pick up three days of interest in one calendar day, while the same convention pays nothing at all on January 31, March 31, May 31, July 31, August 31, October 31 or December 31. It also governs mortgage payments, so monthly mortgage interest ignores the calendar length of the month.
Money market instruments state interest per year, so the Y describes how many days the market treats as a year and the X describes how the accrual period is counted. Treasury bills in the United States run on actual/360: the quoted rate applies to 360 days, and the holding period is the actual number of calendar days.
That mismatch inflates the return relative to what the quoted rate suggests. Take 6% per year on an actual/360 basis. A full calendar year of 365 days earns 365/360 x 6% = 6.0833%, because the investor collects 365 daily accruals from a rate built to cover 360. Over 90 days the same rate earns 1.5%.
The discount basis
Treasury bills carry a second complication on top of the day count. The quoted rate is expressed on a discount basis, meaning it is a percentage of the final value the investor receives rather than of the amount initially paid, and for a bill that value is the face value. Writing Q for the quote, C for the cash price per USD 100 of face value, and n for the days to maturity, the conversion runs both ways.
Take a bill of USD 100,000 face value with 120 days left to run, quoted 4.08 bid against 4.05 ask. A quote of 4.08 says that interest accrues at 4.08% over 360 days on the face amount. Across the 120 days that remain, the dollar interest comes to 100,000 x 0.0408 x 120/360, which is 1,360, and stripping that off the face value leaves a bid price of 98,640. So the bill could be sold for USD 98,640, while the ask quote of 4.05 puts the purchase price at USD 98,650. Per USD 100 the formula reaches the same place: with n = 120 and Q = 4.08, C = 100 – (120/360) x 4.08 = 98.64.
Why the bid quote sits above the ask quote
A bid of 4.08 above an ask of 4.05 looks like a market quoting backwards. Both sides of a Treasury bill quote are interest rates rather than prices, and prices move inversely to rates, so the higher rate is the cheaper price at which the dealer buys.
Government issues in the United States carrying an original maturity of ten years or less are Treasury notes, while longer ones are Treasury bonds, though the two are usually lumped together as bonds. Corporate issuers slice the spectrum differently: under five years is a short-term note, five to 12 years a medium-term note, beyond 12 years a long-term bond.
Reading a Treasury bond quote
Treasury bond prices are quoted per USD 100 of principal, in dollars and thirty seconds of a dollar. A quote of 105-07 means 105 and 7/32, so USD 100,000 of face value costs USD 105,218.75. The most heavily traded issues run to the nearest 1/64 instead, and a plus sign appended to a quote adds 1/64, so 105-07+ stands for 105 and 15/64.
What that quote gives is the clean price. The cash amount the purchaser hands over is the dirty price, and the two differ by the interest built up since the last coupon.
A trade in a U.S. Treasury bond is settled on May 8, 2021. The bond pays 8% per year in two coupons, falling on April 1 and October 1. Its quoted price is 105-08 and the position runs to USD 100,000 of face value. Settlement sits two trading days behind the trade date, and accrued interest is measured to that settlement date.
Interest rate futures on the Chicago Mercantile Exchange come in a family, graded by the maturity of the instrument deliverable against them. Delivery may be made on any day within the delivery month, and the short position picks both the bond that goes and the day it goes. Similar contracts on non-U.S. interest rates trade in many other countries.
| Contract | What may be delivered | Face value per contract | Rounding step for the conversion factor |
|---|---|---|---|
| Ultra Treasury Bond | Remaining maturity greater than 25 years | USD 100,000 | Nearest three months |
| Treasury Bond | Any bond with remaining maturity between 15 and 25 years | USD 100,000 | Nearest three months |
| Ten-Year Treasury Note | Remaining maturity between 6.5 years and ten-years | USD 100,000 | Nearest three months |
| Five-Year Treasury Note | Original maturity capped at five years and three-months, current maturity at least four-years and two-months as of day one of the delivery month | USD 100,000 | Nearest month |
| Two-Year Treasury Note | Original maturity capped at five-years and three-months, remaining maturity at least one-year and nine-months as of day one of the delivery month and no more than two-years on the final day | USD 200,000 | Nearest month |
Source: chapter contract specifications, reordered by underlying maturity; the last two columns are added here.
What the short position gets paid
Because many bonds satisfy the delivery grade, the exchange needs a way of putting them on comparable terms, and that job belongs to the conversion factor. Roughly speaking, it is the hypothetical clean price of the bond per one dollar of face value, assuming all interest rates are 6% with semi-annual compounding.
Suppose the most recent settlement price is USD 105.50, the conversion factor of the chosen bond is 1.1542, and the accrued interest at delivery is USD 1.12 per USD 100 of face value. The cash received by the party with the short position is 105.50 x 1.1542 + 1.12 = USD 122.888. For nearly all Treasury bonds and notes one futures contract covers a face value of USD 100,000, so the short position delivers USD 100,000 of bonds and receives USD 122,888. The two-year note contract is the exception, covering USD 200,000.
For the Ten-Year Treasury Note, the Treasury Bond and the Ultra Treasury Bond the conversion factor comes from three steps. Start on day one of the delivery month and measure forward to the date the bond matures, round that span down to the nearest three months, then price the bond at the rounded maturity using a yield of 6% per annum with semi-annual compounding, per one dollar of face value. The two-year and five-year note contracts round down to the nearest month instead.
Rounding is what makes the calculation branch. A rounded maturity on a whole number of half years puts the bond immediately after a coupon payment, so there is no accrued interest and the clean price equals the dirty price. An extra three months means valuing at a coupon date, discounting back over that quarter, then stripping out accrued interest.
A bond carrying a 5% coupon and maturing on May 15, 2040, is deliverable into both the September 2021 and the December 2021 Treasury bond futures contracts. The conversion factor differs between them because time to maturity is measured from a different starting point.
Bond futures prices are quoted in the same dollars-and-32nds style as bond prices, but the shorter-maturity contracts carry finer resolution. Quotes run to 1/256 on the two-year contract, to 1/128 on the five-year contract, and to 1/64 on the ten-year contract.
That precision is carried in a third digit encoding a fraction of one thirty second: a 5 indicates 0.5/32, a 7 indicates 0.75/32, a 2 indicates 0.25/32, a 1 indicates 0.125/32, a 3 indicates 0.375/32, a 6 indicates 0.625/32, and an 8 indicates 0.875/32. So a quote of 108-125 reads as 108 plus 12.5/32, or 108.390625.
| Contract | Quoted price | Thirty seconds | Decimal price |
|---|---|---|---|
| Ultra Treasury Bond | 200-26 | 26/32 | 200.8125 |
| Treasury Bond | 167-12 | 12/32 | 167.375 |
| Ten-Year Treasury Note | 132-305 | 30.5/32 | 132.9531 |
| Five-Year Treasury Note | 121-007 | 0.75/32 | 121.0234 |
| Two-Year Treasury Note | 108-125 | 12.5/32 | 108.390625 |
Source: www.cmegroup.com quoted prices, reordered by underlying maturity; the last two columns are computed here.
Why the quoted prices climb with the maturity of the deliverable
Reading up the table from the two-year contract, the quoted prices rise steadily with the maturity of the deliverable instrument. The conversion factor system is the reason. A bond paying 6% per year has a conversion factor of 1.0000, so the underlying asset behaves like a 6% coupon bond. On February 26, 2020, rates across every maturity sat far below 6%, and in that situation a 6% coupon bond is worth more the longer it runs, since the above-market coupon keeps arriving.
Since the short position may deliver any bond meeting the grade, it will deliver whichever costs least, cost being the difference between what the bond costs in the cash market and what the futures contract pays for it.
The bond with the smallest value of Q minus Sf is the cheapest-to-deliver bond. Nothing depends on which candidate looks expensive in absolute terms, because the conversion factor rescales them all.
Three bonds are eligible for delivery into a Treasury bond futures contract. All prices are quoted clean, per USD 100 of face value, and accrued interest on an actual/actual basis drops out of the comparison.
Which bond wins follows from the fixed 6% yield built into the conversion factor system, so the level of market yields and the slope of the curve both push the answer in a predictable direction. When bond yields are greater than 6%, low-coupon long-maturity bonds tend to be the cheapest to deliver. When yields are less than 6%, high-coupon short-maturity bonds win instead. Slope acts separately: an upward-sloping yield curve tends to favour long-maturity bonds, while a downward-sloping yield curve tends to favour short-maturity bonds.
The wild card play and the other delivery options
Choosing the bond is only one of the choices the short position holds. A second is timing, since any day inside the delivery month will serve. A third is the wild card play, which comes out of a gap in the trading day. The settlement price is struck on trading at 2 p.m. Chicago time, yet a notice of intention to deliver may be lodged some hours afterwards. So the short position can watch for bond prices falling after 2 p.m., buy at the 3:30 p.m. price, then hand the bond over at a delivery value fixed off the earlier 2 p.m. futures price.
Every one of these choices belongs to the short position and none to the long position. A contract loaded with options in one direction is worth more to the side holding them, so that party becomes willing to accept a lower price. The cheapest-to-deliver option, the timing option and the wild card play together push the futures price down.
All those delivery choices make an exact futures price hard to pin down. Assume the cheapest-to-deliver bond and the delivery date are known, though, and the theoretical futures price follows from a familiar case: a contract on an asset providing a known income, the coupons paid before delivery.
Delivery of a bond is set for 250 days from today under a futures contract. Its last coupon was paid 50 days ago, the next falls due in 133 days, and the one after that arrives 315 days from now, which is 65 days past delivery. Every maturity carries a risk-free rate of 5% with continuous compounding. The bond pays a coupon of 7% semi-annually, its quoted clean price is USD 108.00, and its conversion factor is 1.0400.
Short-term rate exposure is handled by a different family of contracts. The three-month Eurodollar futures contract settled in cash on the three-month USD Libor fixing set two days ahead of the third Wednesday in the contract month, and it traded for maturities up to ten years out. Libor has been discontinued in the United States and replaced by the Secured Overnight Funding Rate, so Eurodollar futures have given way to SOFR futures.
The two rates differ in one respect that matters for settlement. Libor was fixed at the start of the period it covered, while SOFR is an overnight repo rate that stays unknown until the period ends. A three-month SOFR futures contract settles on whatever an investment rolled forward day by day at the overnight SOFR rate would have returned across the preceding three months. Contracts lasting up to five years trade actively.
Reading a quote and valuing a tick
A quote is 100 minus the rate, so a SOFR futures quote of 98.36 indicates a three-month futures rate of 1.64%. A one basis point move in the quote is worth USD 25 on one contract.
| Contract month | Final settlement price | Contracts traded | Implied three-month rate |
|---|---|---|---|
| June 2023 | 94.8750 | 288,402 | 5.1250% |
| December 2023 | 95.4650 | 200,941 | 4.5350% |
| December 2024 | 96.9000 | 98,460 | 3.1000% |
| December 2025 | 97.0550 | 56,563 | 2.9450% |
Source: settlement prices and volumes as reported in the chapter; the implied rate is derived here as 100 minus the price.
A hedge follows from that arithmetic. Late in April 2021 a borrower commits to a three-month loan of $100 million starting in December 2021, priced at the three-month SOFR rate with a spread of 200 basis points on top. The December 2021 quote stands at 99.94, implying a SOFR rate of 6 basis points, or 0.06%, and shorting 100 December contracts locks the borrowing rate at 2.06%. Suppose final settlement arrives at 99.20. The three-month SOFR rate is then 0.8%, the loan costs 2.8%, and the interest bill is $100,000,000 x 0.028 x 0.25, or $700,000. The contract has fallen 74 basis points, so the gain is 100 x $25 x 74 = $185,000, leaving net interest of $515,000, a rate of 2.06%.
Why a futures rate is not a forward rate
A SOFR futures contract is settled daily, and that daily settlement makes the rate implied by the contract differ from the corresponding forward rate. Corrections that convert a futures rate into a forward rate are called convexity adjustments, and once forward rates are in hand a zero curve can be built from them.
A zero rate for one maturity plus the forward rate beyond it delivers the next zero rate. Suppose the zero rate stands at 4% out to two years, while the forward rate covering the stretch from two years to 2.5 years is 4.4%. The 2.5-year zero rate then comes out at (0.04 x 2 + 0.044 x 0.5)/2.5 = 0.0408, or 4.08%. Repeating the step builds a complete term structure.
A hedger rarely finds a futures contract whose underlying maturity matches the exposure being hedged. Duration bridges the gap, provided only small parallel shifts in the yield curve are contemplated. Write B for the value of a bond portfolio and D for its modified duration: a parallel shift of size delta y then moves the portfolio by approximately minus B D delta y. A hedge ratio needs two sensitivities on the same scale, both measured against a downward parallel shift of one basis point in the zero curve. EV is the gain on the position of the trader under that shift, and EF is the gain on one futures contract.
Take a trader who holds USD 2 million of a money market instrument with nine months to run. Nothing is paid before maturity, so the duration is 0.75, and the gain under the one basis point shift is 0.0001 x 0.75 x 2,000,000 = 150. That same shift is worth USD 25 on one three-month SOFR futures contract, making EF equal to 25. Dividing 150 by 25 gives six, so a short position in six contracts provides an approximate hedge.
Now take a fund manager with a USD 5 million position in bonds of duration 12, hedging with a bond futures contract whose cheapest-to-deliver bond is expected to carry a duration of 16 by the time the contract matures, at a futures price of 103.25. For the portfolio, EV works out at 0.0001 x 12 x 5,000,000 = 6,000. One contract covers a face value of USD 100,000, so it is worth USD 103,250, and the futures price moves in the same proportion as the delivered bond, giving EF = 0.0001 x 16 x 103,250 = 165.2. The ratio 6,000 divided by minus 165.2 is minus 36.3, so 36 contracts should be shorted.
What a duration-based hedge does not do
The protection is narrow. It covers small parallel movements in the term structure of interest rates and nothing else, so a twist or a steepening of the curve is left uncovered. The calculation assumes that the rate to which the hedger is exposed moves in step with the rate underlying the futures contract, and, where bond futures are used, that the identity of the cheapest-to-deliver bond is settled in advance. Smaller approximations sit underneath: the 0.75 used above is not strictly a modified duration, the relationship is exact only for continuously compounded rates, and day count conventions have been set aside.