Chapter 6: Interest Rate Futures

Day Count Conventions #

Different instruments use different day count conventions:

Converting between conventions: the same interest rate expressed under different conventions will give different dollar amounts of interest.

Treasury Bond Futures #

Contract: Deliverable on any US Treasury bond with more than 15 years to maturity (if callable, more than 15 years to first call).

Conversion Factors #

Since bonds have different coupons and maturities, conversion factors standardize them. The cash received by the short position at delivery:

$\text{Cash}=(\text{Futures settlement price}\times \text{Conversion factor})+\text{Accrued interest}$

The conversion factor equals the price per $1 of principal the bond would have at a yield of 6% per annum.

Cheapest-to-Deliver Bond #

The short position can choose which bond to deliver. They will choose the cheapest-to-deliver bond — the one that minimizes:

$\text{Quoted bond price}-(\text{Futures quotation}\times \text{Conversion factor})$

Determining the Futures Price #

Assuming the cheapest-to-deliver bond and delivery date are known:

$F_0=\frac{(S_0+A{I}_0)e^{rT}-A{I}_T-C}{CF}$

where $A{I}_0$ = accrued interest now, $A{I}_T$ = accrued interest at delivery, $C$ = coupon during the period.

Delivery Options (Wild Card) #

The short position has timing options (can deliver any day in the delivery month), quality options (choose which bond to deliver), and the wild card play (can announce intention to deliver after futures market closes but before bond market closes).

Eurodollar Futures (Historical) and SOFR Futures #

Eurodollar Futures #

Settled in cash based on 3-month LIBOR. Contract price = $100-3$-month LIBOR. Tick = 0.01% = $25 per contract. Now being phased out.

SOFR Futures #

Based on the Secured Overnight Financing Rate (SOFR). One-month SOFR futures settle based on the arithmetic average of daily SOFR rates during the delivery month. Three-month SOFR futures settle similarly.

Convexity Adjustment #

Forward rates implied by futures need a convexity adjustment because futures are settled daily (marked to market) while forward rates are settled at maturity:

$\text{Forward rate}=\text{Futures rate}-\frac{1}{2}{\sigma }^2{t}_1{t}_2$

where $t_1$ is time to futures maturity, $t_2$ is time to the end of the rate period, and $\sigma$ is the volatility of the short rate.

Duration-Based Hedging #

To hedge a bond position against parallel shifts in the yield curve using Treasury bond futures, the number of contracts needed is:

$N^{\ast }=\frac{D\cdot V}{D_F\cdot V_F}$

(assuming continuous compounding)

When hedging using interest rate futures, we need to consider the duration of the bond underlying the cheapest-to-deliver bond rather than the futures contract itself.

Hedging Portfolios of Assets and Liabilities #

Banks often use interest rate futures to match the duration of assets and liabilities, mitigating interest rate risk in their balance sheet. This is a key component of asset-liability management (ALM).