Interest Rate Futures

Q: What are the primary day-count conventions used for U.S. Treasury bonds, corporate bonds, and money market instruments?

U.S. Treasury bonds use Actual/Actual , corporate and municipal bonds use 30/360 , and money market instruments use Actual/360 .

Q: How is the dollar price per \$100 face value calculated from a quoted discount rate for a Treasury bill?

$ P = 100 - \left( \frac{n}{360} \times Y \right) $, where $ n $ is the actual days to maturity and $ Y $ is the annualized discount rate.

Q: What is the relationship between a bond's clean price and dirty price?

$ \text{Dirty Price} = \text{Clean Price} + \text{Accrued Interest} $

Q: How does the short position identify the Cheapest-to-Deliver (CTD) bond in a Treasury futures contract?

The CTD is the bond that minimizes the net delivery cost: $ \text{Quoted Bond Price} - (\text{Futures Price} \times \text{CF}) $.

Q: How does market yield relative to 6% influence which bond tends to be Cheapest-to-Deliver (CTD)?

When yields exceed 6%, the CTD tends to be a long-maturity, low-coupon bond; when yields are below 6%, it tends to be a short-maturity, high-coupon bond.

Q: Why does the futures interest rate exceed the equivalent forward rate in short-term interest rate markets?

Due to the convexity adjustment arising from daily marking-to-market and the positive correlation between interest rates and futures margin cash flows.

Q: What is the duration-based hedge ratio formula for hedging a bond portfolio using Treasury bond futures?

$ N^* = -\frac{P \times D_P}{F_c \times D_{\text{CTD}}} \times \text{CF}_{\text{CTD}} $

Q: What are the key limitations and sources of basis risk inherent in duration-based hedging?

The key limitations are the assumptions of parallel yield-curve shifts , small yield changes (omitting convexity) , and a constant CTD bond .

FRM Part I · FRM · 8 min read · Updated: 8 Nov 2026

Introduction: Mastering the World of Interest Rate Futures

Welcome! Today we are diving into one of the most practical and frequently tested areas of the FRM Part I curriculum: Interest Rate Futures. If you’ve ever wondered how banks and corporations protect themselves from the "rollercoaster" of changing interest rates, you’re in the right place. These instruments allow market participants to lock in a rate today for a transaction that will happen in the future.

We will break it down step-by-step, from day-count conventions and price conversions to cheapest-to-deliver dynamics and duration-based portfolio hedging. Let’s get started!

1. The Starting Line: Day Count Conventions

Before calculating interest or accrued interest, we must know how to count the days between two dates. Different debt instruments use different conventions:

Actual/Actual (ACT/ACT): Used for U.S. Treasury Bonds and Notes. You count the actual number of days elapsed between dates and divide by the actual number of days in the coupon period.
30/360: Used for Corporate and Municipal Bonds. Each full month is assumed to contain 30 days, and the year is assumed to contain 360 days.
Actual/360: Used for Money Market Instruments (including U.S. Treasury bills and short-term interbank instruments). You count actual days elapsed and divide by a 360-day year.

Quick Review: Why does this matter?

Using the incorrect day-count convention will distort accrued interest and cash settlement amounts. In multi-million-dollar transactions, minor day-count differences lead to substantial dollar misallocations. Always check the bond or instrument type before performing calculations.

2. Treasury Bill Discount-to-Price Conversion

Treasury bills (T-bills) are zero-coupon instruments that pay face value at maturity. Rather than quoting a price, the market quotes a discount rate based on an Actual/360 convention.

Let:

$ P $ = Quoted price per \$100 face value
$ Y $ = Annualized discount rate (in percent)
$ n $ = Actual number of calendar days remaining to maturity

Discount-to-Price Formula:

$ P = 100 - \left( \frac{n}{360} \times Y \right) $

Example: A 90-day T-bill is quoted at a discount rate of 4.00%. The dollar price per \$100 face value is:

$ P = 100 - \left( \frac{90}{360} \times 4.00 \right) = 100 - 1.00 = 99.00 $

To convert this into a money-market yield (cash yield) or bond equivalent yield (BEY), recognize that the return is earned on the cash invested (\$99.00), not the face value (\$100.00).

3. Bond Pricing: Clean vs. Dirty

Between coupon dates, a bond accumulates interest. This necessitates a distinction between two prices:

Clean Price (Quoted Price): The price quoted on exchange screens and financial media. It excludes accrued interest.

Dirty Price (Cash Price): The actual cash handed from buyer to seller upon settlement. It represents the clean price plus accrued interest.

The Formula:

$ \text{Cash (Dirty) Price} = \text{Quoted (Clean) Price} + \text{Accrued Interest} $

Where accrued interest since the last coupon payment date is:

$ \text{Accrued Interest} = \text{Coupon} \times \frac{\text{Days accrued since last coupon}}{\text{Total days in current coupon period}} $

Analogy:

Imagine purchasing a vehicle that comes with a full tank of fuel. The Clean Price is the sticker price of the car itself. The Accrued Interest is the fuel already inside the tank. The Dirty Price is the total amount of money paid at closing.

4. Treasury Bond Futures

A Treasury Bond (T-Bond) futures contract permits the short position to deliver any government bond from a basket of eligible bonds meeting specified maturity criteria (e.g., bonds with remaining maturity greater than 15 or 25 years).

Conversion Factors (CF)

Because eligible deliverable bonds have varying coupons and maturities, the exchange establishes a Conversion Factor (CF) for each bond. The CF equals the dirty price per dollar of face value of the bond on the first day of the delivery month if its yield to maturity were 6% per annum (with semiannual compounding).

The cash paid by the buyer (long) and received by the seller (short) upon delivery is:

$ \text{Cash Received} = (\text{Futures Settlement Price} \times \text{CF}) + \text{Accrued Interest} $

Cheapest-to-Deliver (CTD) and Delivery Cost

The party with the short position decides which bond to purchase in the spot market and deliver against the futures contract. The short seeks to minimize the net cost of delivery:

$ \text{Cost of Delivery} = \text{Cash (Dirty) Bond Price} - [(\text{Futures Price} \times \text{CF}) + \text{Accrued Interest}] $

Because $ \text{Cash Bond Price} - \text{Accrued Interest} = \text{Clean Bond Price} $, this simplifies to minimizing:

$ \text{Net Cost} = \text{Quoted (Clean) Bond Price} - (\text{Futures Price} \times \text{CF}) $

The bond that yields the lowest net cost (or most negative value) is the Cheapest-to-Deliver (CTD) bond.

Yield-Curve Shape and Cheapest-to-Deliver

The 6% benchmark yield embedded in the conversion factor system produces predictable delivery tendencies depending on current market yields and curve shape:

Market Yields > 6%: The conversion factor system penalizes low-coupon, long-maturity bonds least. Thus, the CTD tends to be a long-maturity, low-coupon bond (highest duration).
Market Yields < 6%: The CTD tends to be a short-maturity, high-coupon bond (lowest duration).
Upward-Sloping Yield Curve: Longer-maturity bonds generally yield more than shorter-maturity bonds, giving long-maturity bonds an advantage to be CTD.
Downward-Sloping (Inverted) Yield Curve: Shorter-maturity bonds tend to be favored as the CTD.

Theoretical Futures Price

Assuming the CTD bond is known and does not change, the theoretical cash futures price $ F_0 $ is derived using the standard cost-of-carry model for an asset providing discrete income:

$ F_0 = (S_0 - I) e^{r T} $

Where $ S_0 $ is the cash (dirty) price of the CTD bond today, $ I $ is the present value of coupon payments received during the life of the contract, and $ r $ is the risk-free rate for time $ T $.

To convert this into the theoretical quoted futures price:

$ \text{Quoted Futures Price} = \frac{1}{\text{CF}_{\text{CTD}}} \times \left( F_0 - \text{Accrued Interest at Delivery} \right) $

5. Eurodollar and Money Market Futures

Short-term interest rate futures (such as Eurodollar futures and 3-month SOFR futures) are quoted on an index basis:

$ P = 100 - R $

Where $ R $ is the annualized settlement interest rate. When interest rates rise, the futures price falls. A 1 basis point (0.01) shift in price on a standard \$1,000,000 3-month contract represents a change of \$25:

$ \Delta \text{Value} = \$1,000,000 \times 0.0001 \times \frac{90}{360} = \$25 $

The Convexity Adjustment

Forward rate agreements (FRAs) settle in arrears or at inception with a single settlement, whereas futures contracts undergo daily marking-to-market. Because of this daily cash flow reinvestment feature and the positive correlation between rates and futures margin flows, the futures rate exceeds the forward rate by a convexity adjustment:

$ \text{Futures Rate} = \text{Forward Rate} + \text{Convexity Adjustment} $

6. Hedging with Interest Rate Futures

To hedge an existing bond portfolio against rising interest rates, an investor takes a short position in interest rate futures, locking in an exit yield and offsetting portfolio losses with futures gains.

The Duration-Based Hedge Ratio

To align the interest-rate sensitivity of the futures position with that of the bond portfolio, we match their dollar durations (or DV01s):

$ N^* = -\frac{P \times D_P}{F \times D_F} $

Where:

$ N^* $ = Optimal number of futures contracts (negative sign denotes a short hedge for a long bond portfolio)
$ P $ = Market value of the bond portfolio
$ D_P $ = Modified duration of the bond portfolio
$ F $ = Quoted price of the futures contract multiplied by contract size
$ D_F $ = Modified duration of the bond underlying the futures contract (typically the CTD bond divided by its CF, or $ D_{\text{CTD}} $)

Expressed directly in terms of the cheapest-to-deliver bond:

$ N^* = -\frac{P \times D_P}{F_c \times D_{\text{CTD}}} \times \text{CF}_{\text{CTD}} $

Where $ F_c $ is the contract face value times the futures price.

Limitations of Duration-Based Hedging

While widely used, duration-based hedging rests on strong assumptions that introduce real-world basis risk:

Parallel Shifts Only: Modified duration assumes that interest rate changes are uniform across all maturities. If the yield curve twists, flattens, or steeples, the hedge will under- or over-perform.
Small Yield Changes (Linearity): Duration is a first-order linear approximation. For large yield changes, bond convexity causes non-linear gains/losses that duration alone cannot capture.
Cheapest-to-Deliver Switching Risk: As market yields shift, the identity of the CTD bond can change. A new CTD bond alters both $ D_{\text{CTD}} $ and $ \text{CF}_{\text{CTD}} $, requiring the hedge to be rebalanced.
Credit and Basis Risk: The spread between corporate or non-Treasury bond yields and Treasury futures yields may widen or narrow unpredictably.

Summary and Common Pitfalls

Key Takeaways:

Day Counts: Treasuries use ACT/ACT; corporate/muni bonds use 30/360; money market instruments use ACT/360.
T-Bills: Quoted on a discount basis: $ P = 100 - (n/360 \times Y) $.
Clean vs. Dirty: Quoted price is Clean; actual cash paid upon settlement is Dirty (Clean + Accrued Interest).
CTD Rule: If yields > 6%, long-maturity/low-coupon bonds are favored; if yields < 6%, short-maturity/high-coupon bonds are favored.
Hedging Limitations: Duration-based hedges assume small, parallel curve movements and constant CTD assignments.

Common Mistakes to Avoid:

1. Day-Count Confusion: Never use 30/360 for a Treasury note or bond (they use ACT/ACT) or for a T-bill (which uses ACT/360).

2. Direction of the Hedge: Long asset exposure requires a Short futures position to hedge against rate increases.

3. Forgetting Conversion Factors: The futures invoice price always multiplies the futures settlement price by the bond's specific Conversion Factor before adding accrued interest.

References

2026 FRM Part I Learning Objectives — Global Association of Risk Professionals (GARP)
2026 FRM Candidate Guide — Global Association of Risk Professionals (GARP)

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What are the primary day-count conventions used for U.S. Treasury bonds, corporate bonds, and money market instruments?

U.S. Treasury bonds use Actual/Actual, corporate and municipal bonds use 30/360, and money market instruments use Actual/360.

How is the dollar price per \$100 face value calculated from a quoted discount rate for a Treasury bill?

$ P = 100 - \left( \frac{n}{360} \times Y \right) $, where $ n $ is the actual days to maturity and $ Y $ is the annualized discount rate.

What is the relationship between a bond's clean price and dirty price?

$ \text{Dirty Price} = \text{Clean Price} + \text{Accrued Interest} $

How is the total cash invoice price calculated upon delivery of an eligible bond against a Treasury bond futures contract?

$ \text{Cash Received} = (\text{Futures Settlement Price} \times \text{CF}) + \text{Accrued Interest} $

How does the short position identify the Cheapest-to-Deliver (CTD) bond in a Treasury futures contract?

The CTD is the bond that minimizes the net delivery cost: $ \text{Quoted Bond Price} - (\text{Futures Price} \times \text{CF}) $.

How does market yield relative to 6% influence which bond tends to be Cheapest-to-Deliver (CTD)?

When yields exceed 6%, the CTD tends to be a long-maturity, low-coupon bond; when yields are below 6%, it tends to be a short-maturity, high-coupon bond.

Why does the futures interest rate exceed the equivalent forward rate in short-term interest rate markets?

Due to the convexity adjustment arising from daily marking-to-market and the positive correlation between interest rates and futures margin cash flows.

What is the duration-based hedge ratio formula for hedging a bond portfolio using Treasury bond futures?

$ N^* = -\frac{P \times D_P}{F_c \times D_{\text{CTD}}} \times \text{CF}_{\text{CTD}} $

What are the key limitations and sources of basis risk inherent in duration-based hedging?

The key limitations are the assumptions of parallel yield-curve shifts, small yield changes (omitting convexity), and a constant CTD bond.

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