13.3 Interest Rate Futures

Interest Rate Futures

FMP–19 applies futures mechanics to short-term and long-term interest-rate contracts. Exam success hinges on day-count discipline, T-bill quote conversion, the Treasury futures delivery system (conversion factors and cheapest-to-deliver), and hedge sizing with duration—plus knowing when that hedge breaks.

Day-Count Conventions

Interest accrual depends on the market’s day-count:

Mis-applying ACT/360 versus ACT/ACT is a classic FRM trap when computing accrued interest or comparing instruments.

Worked accrued interest (Treasury-style sketch)

A Treasury note with semiannual coupon $3 per 100 par: if 45 actual days have elapsed in a 182-day coupon period (ACT/ACT), accrued ≈ 3 × (45/182) ≈ $0.7418 per 100. Under a wrong 30/360 assumption with 45 days in a 180-day half-year, accrued = 3 × (45/180) = $0.75—close but not identical; larger gaps appear near irregular periods.

T-Bill Discount and Price

U.S. T-bills are quoted on a bank discount basis:

Yd = [(Face − Price) / Face] × (360 / t)

Price = Face × [1 − Yd × (t / 360)]

where t = days to maturity. The discount yield understates the true investment rate because it uses Face in the denominator and a 360-day year. Bond-equivalent yield (semiannual) and effective annual yield require converting from price.

Worked T-bill price

Face $1,000,000; 90 days; discount quote Yd = 4.00%.

Price = 1,000,000 × [1 − 0.04 × (90/360)] = 1,000,000 × [1 − 0.01] = $990,000.

Holding-period return = 10,000 / 990,000 ≈ 1.0101%. Annualize with 365 days for a simple BEY-style measure: (10,000 / 990,000) × (365/90) ≈ 4.10%. Note 4.10% > 4.00% discount quote.

Reverse: price → discount

Price 992,500; t = 60; Face 1,000,000.

Yd = [(1,000,000 − 992,500)/1,000,000] × (360/60) = 0.0075 × 6 = 4.50%.

Clean vs Dirty Price and Accrued Interest

Treasury notes and bonds (and T-bond futures invoice amounts) separate:

• Clean price: quoted price without accrued
• Dirty price: clean + accrued = amount paid in the cash market

Futures invoice price for delivery ≈ (futures settlement price × conversion factor) + accrued interest on the delivered bond. Understanding clean versus dirty prevents double-counting accrued in hedge P&L.

Conversion Factor

The conversion factor (CF) scales bonds of different coupons and maturities into the futures’ 6% standard (for CBOT U.S. T-bond/note futures; confirm contract specs on the exam if a vignette gives a different standard).

Roughly, CF is the clean price per $1 face that the bond would have if its YTM equaled the futures’ standard rate (6%), with maturity rounded per exchange rules. High-coupon bonds have CF > 1; low-coupon bonds have CF < 1.

Invoice amount ≈ futures quote × CF × contract size scaling + accrued interest

Delivery Cost and Cheapest to Deliver (CTD)

Shorts in T-bond futures may deliver any eligible bond. They choose the cheapest-to-deliver (CTD)—the bond that maximizes the short’s profit (minimizes the cost of buying the bond and delivering).

A common ranking metric:

Delivery cost basis ≈ Dirty cash price − (futures settlement × CF + accrued)

Equivalently, traders watch basis = cash clean price − futures × CF. The CTD is typically the bond with the lowest basis (most negative / cheapest) after financing (forward basis / implied repo).

Worked CTD comparison (illustrative)

Futures settlement = 120-00 (120% of par). Bond A: clean 118, CF = 0.9750 → futures×CF = 117.00; basis = 118 − 117 = 1.00. Bond B: clean 131, CF = 1.1000 → futures×CF = 132.00; basis = 131 − 132 = −1.00. Bond B looks cheaper on this clean basis snapshot (more negative basis); financing and accrued can change the ranking—exam questions usually give enough numbers to compute which bond maximizes the short’s proceeds net of purchase cost.

Theoretical T-Bond Futures Price

For the CTD, in a frictionless market without delivery options:

Forward clean price of CTD ≈ (Cash dirty price − PV of coupons during forward period) × e^(rT) − accrued at forward date

(or the equivalent with discrete repo financing). Then:

Theoretical futures ≈ Forward clean price of CTD / CF_CTD

Actual futures trade below this forward/CF value by the value of the short’s delivery options (quality option among bonds, timing options). When the question says “ignore delivery options,” use Forward/CF. When it emphasizes options, futures < Forward/CF.

Worked theoretical futures (simplified)

CTD forward clean price = 99.00; CF = 0.9000. Theoretical futures (no options) = 99.00 / 0.9000 = 110.00. If delivery options are worth 0.40 futures points, futures ≈ 109.60.

Duration-Based Hedge Ratio

To hedge a bond portfolio with interest-rate futures:

N_f ≈ − (P × D_P) / (F × D_CTD / CF) or commonly N_f ≈ − (D_P P) / (D_F F)

where D_P is the portfolio’s duration, P portfolio value, F futures contract value (per contract), and D_F is the duration of the futures underlying (often approximated by CTD duration / CF, or given directly).

More precisely, many texts use:

Hedge ratio = (P_target × ModDur_target) / (P_CTD × ModDur_CTD) × CF

with sign negative for a short futures hedge of a long bond inventory.

Worked hedge ratio

Portfolio value $10,000,000; modified duration 7.0. CTD dirty price value corresponding to one futures ≈ $120,000; CTD modified duration 9.0; CF = 1.05. Approximate number of contracts to short:

N ≈ (10,000,000 × 7) / (120,000 × 9) × 1.05 ≈ 70,000,000 / 1,080,000 × 1.05 ≈ 64.81 × 1.05 ≈ 68 contracts (short).

(Exact formulas vary by whether F is futures invoice value or CTD value; follow the vignette’s definition.)

Limitations of Duration Hedges

Duration-based futures hedges assume:

1. Parallel yield shifts — curve twists (2s10s steepeners) leave residual P&L
2. Stable CTD — if the CTD switches when yields move, futures duration jumps
3. Linear (duration-only) response — convexity differences between portfolio and CTD matter in large moves
4. Spread risk — corporate or MBS portfolios hedged with Treasury futures retain OAS/spread exposure
5. Delivery option effects — futures DV01 ≠ CTD DV01 / CF exactly when options are valuable

Exam Synthesis

Convert T-bill discount quotes to prices before comparing returns. For bond futures, always ask: which bond is CTD, what is CF, and is the question ignoring delivery options? Size hedges with duration, then list why the hedge can fail—especially CTD switches and curve reshaping.