Sofr And Rfr Compounding

SOFR and RFR compounding

A compounded overnight rate is not an average of the fixings. It is a product of daily accrual factors, each weighted by the number of calendar days that fixing applies for, on a 360-day year for SOFR and a 365-day year for SONIA — and then a lookback, lockout or observation shift changes which fixings are in the window at all.

Every figure below is printed by scripts/sofr.py (runs in 0.3 s; QuantLib optional). ✅ Measured means this file produced it on 2026-09-09 with QuantLib 1.43, numpy 2.2.6, Python 3.11.3. The worked coupon is 2026-04-01 → 2026-07-01 (62 business-day fixings, 91 calendar days), 100,000,000 notional, on a seeded synthetic SOFR path around 4.30% with month-end spikes and a 25 bp policy step on 2026-06-18.

The rule: a compounded RFR is prod(1 + r_i × d_i / basis), weighted by CALENDAR days — ACT/360 for SOFR and ESTR, ACT/365 for SONIA and TONA — and "lookback" names two different windows until you say whether it shifts.

1. ✅ The NY Fed's own SOFR Index example, reproduced exactly

✅ Source-verified at the NY Fed, "Additional Information about Reference Rates Administered by the New York Fed" (read 2026-09-09): the SOFR Index starts at 1.00000000 on 2018-04-02 and compounds by (1 + SOFR × d/360) where d is the number of calendar days applicable.

🔑 The last row is the whole point: a Friday fixing carries three calendar days. ✅ The NY Fed also states the weighting around a holiday: if the start date is a Wednesday and Thursday is a holiday, that Wednesday's rate applies for 2/360, and the following Friday's for 3/360.

2. 🚨 Four ways to be a few basis points wrong

✅ Measured on the worked coupon:

🔑 The basis appears TWICE — once inside every daily factor (1 + r·d/basis) and again in the annualization × basis/D. ✅ Changed in both places it nearly cancels (−0.03 bp); changed in one it is the full 365/360 (+6.05 bp). That is why "we use ACT/365, it is only 1.4%" is either harmless or a 6 bp error depending on a line you did not look at.

🚨 Compounding vs simple averaging is not a rounding choice. ARRC and ISDA fallback conventions compound; the gap is roughly r²·T/2 and grows with the square of the rate and the length of the period. At 4.35% over a quarter it is 2.32 bp; it was under 0.01 bp in 2021.

3. 🚨 Lookback, lockout and observation shift are different windows

✅ Measured, 5-day conventions on the same coupon:

• 🚨 The two "5-day lookbacks" differ by 0.30 bp (758 per 100mm) and both are called a 5-day lookback. They use the same rates; what differs is the day weights — a fixing that is a Friday in the observation window can map to a Wednesday in the accrual window, and 3/360 becomes 1/360.
• 🚨 The lookback itself is the bigger number: −2.24 bp. It pushes the 25 bp policy step five business days later inside the accrual period, so fewer days of the quarter see the higher rate.
• Lockout freezes the last k fixings at the last observable one. It is the smallest of the three here because it only affects the tail, but it turns the coupon into a step function of one fixing.
• Payment delay does not change the rate at all — only when the cash moves — which is why it is the one convention people get right.

⚠️ The magnitudes depend on where the rate moves sit relative to the window. The invariant is the sign structure, not the size: a lookback pulls the window backwards in time, a lockout freezes the tail, and an observation shift additionally re-weights.

4. ✅ The index is the only thing you need to store

✅ Measured on the same coupon:

rate = (I_end / I_start − 1) × basis / days. Two index values and a day count reproduce any tenor, which is why the NY Fed publishes an index at all. ⚠️ The NY Fed notes that averages built from the rounded index may differ from the published averages in the fifth decimal place — true, and 4.0e-09 here, i.e. below a hundredth of a basis point.

5. Five currencies, two bases

✅ Day counts read out of QuantLib 1.43's own index definitions (ql.Sofr(...).dayCounter()):

🚨 The two index families do not even start at the same number. ✅ The SOFR Index is 1.00000000 from 2018-04-02 (NY Fed); ✅ the SONIA Compounded Index is 100.00000000 from 2018-04-23, is rounded to eight decimal places, carries the previous day's value internally at 18 decimals, and — unlike SOFR's — is published at 9am on the same London business day (Bank of England, "SONIA key features and policies", read 2026-09-09). ✅ SONIA itself is a trimmed mean rounded to four decimal places over the central 50% of the volume-weighted distribution; ✅ SOFR is a volume-weighted median rounded to the nearest basis point (NY Fed).

⚠️ QuantLib's UnitedStates::SOFR fixing calendar is not UnitedStates::GovernmentBond. ✅ Measured: they differ in 2026 by Good Friday, 2026-04-03 — SIFMA recommends a full close and no SOFR is published. Adding a fixing on that date to ql.Sofr raises At least one invalid fixing provided, which is the only place in this file where a convention mismatch actually errors.

6. ✅ Checked against QuantLib on all four window conventions

The reference implementation reproduces QuantLib's lookbackDays and applyObservationShift exactly, and QuantLib's RateAveraging.Compound vs RateAveraging.Simple reproduces §2's +2.32 bp. Two independent implementations, one arithmetic.

7. What the script gives you

scripts/sofr.py — numpy only at import; QuantLib inside two functions.

Where this sits

• ../libor-transition-and-fallbacks/SKILL.md — 🚨 which SOFR a fallback contract actually gets: ISDA's compounded Fallback Rate for derivatives, CME Term SOFR for non-consumer cash, and the five statutory spreads that are the same for both.
• ../ois-discounting-and-multi-curve/SKILL.md — turning these fixings and forwards into a curve, and 🚨 why discounting at the projection curve leaves the par rate unchanged while moving the annuity 1.461%.
• ../duration-convexity-and-dv01/SKILL.md — a floater's rate duration is the time to its next reset; this skill defines what "reset" means.
• ../bond-conventions-and-accrued/SKILL.md — the ACT/360 vs ACT/365 argument on the bond side, and 🚨 ACT/ACT ICMA without its schedule.
• ../../../fin-models/skills/term-structure-models/SKILL.md — compounding conventions on a zero curve, where the same discount factor is six different rates.
• ../../../fin-core/skills/fundamental-and-macro-data/SKILL.md — FRED SOFR, SOFR30DAYAVG and SOFRINDEX, and their revision behaviour.
• ../../../fin-libraries/skills/lib-quantlib/SKILL.md — addFixing, clearFixings and the global fixing history that survives between objects.