13.3 Interest Rate Futures
Key Takeaways
Money-market day-count conventions (ACT/360, ACT/365, 30/360) change accrued interest and discount-instrument pricing
T-bill quotes are discount rates; convert carefully between discount, price, and bond-equivalent yield
T-bond/note futures use clean prices, accrued interest, and conversion factors; the CTD minimizes net delivery cost
Theoretical bond futures price links the CTD forward price to the conversion factor, adjusted for delivery options
Duration-based hedge ratios size futures overlays but fail when yields curve-twist, CTD switches, or convexity differs
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:
| Convention | Typical use | Year basis |
|---|---|---|
| ACT/360 | U.S. money markets, T-bills, LIBOR-era deposits, many FRAs | Actual days / 360 |
| ACT/365 | Some sterling markets, certain gilts conventions | Actual days / 365 |
| 30/360 | Many U.S. corporates and agencies | 30-day months / 360 |
| ACT/ACT | U.S. Treasuries (notes/bonds) | Actual / actual |
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).
| Factor favoring CTD status | Intuition |
|---|---|
| Duration / coupon vs 6% standard | Low-coupon long bonds often CTD when yields > 6%; high-coupon when yields < 6% (classic rule of thumb) |
| Liquidity and repo specialness | Hard-to-finance bonds may not be CTD even if on paper |
| Delivery-option value | Timing, wild-card, and end-of-month options affect futures vs forward |
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:
- Parallel yield shifts — curve twists (2s10s steepeners) leave residual P&L
- Stable CTD — if the CTD switches when yields move, futures duration jumps
- Linear (duration-only) response — convexity differences between portfolio and CTD matter in large moves
- Spread risk — corporate or MBS portfolios hedged with Treasury futures retain OAS/spread exposure
- Delivery option effects — futures DV01 ≠ CTD DV01 / CF exactly when options are valuable
| Limitation | Practical implication |
|---|---|
| Nonparallel shifts | Use key-rate durations or multiple contracts |
| CTD switch | Rebalance; watch yield vs 6% region |
| Spread risk | Add CDS or credit futures overlays |
| Convexity | Options or barbell/bullet structure management |
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.
A 180-day T-bill with face 100 is quoted at a 5% discount. Its price is closest to:
95.00
97.53
97.50
95.00 on an ACT/365 basis only
The cheapest-to-deliver bond into T-bond futures is the eligible bond that:
Has the highest coupon regardless of price
Always has the longest maturity by exchange rule
Must be the on-the-run 10-year note
Maximizes the short’s net advantage in delivery (typically lowest basis after financing)
Ignoring delivery options, if the CTD forward clean price is 108 and CF = 1.20, the theoretical futures price is:
90.00
129.60
108.00
1.20
A duration hedge of corporates using T-bond futures is most likely to underperform when:
Yields make a small parallel shift and spreads are unchanged
Credit spreads widen sharply while Treasury yields are stable
The hedge ratio is recomputed daily and the CTD is unchanged
Portfolio and CTD modified durations are matched by construction
Sections you finish are checked off in the contents.