SOFR Compounding in Arrears Conventions

Under LIBOR, a borrower on a three-month floating loan knew the exact interest rate for the whole quarter on day one — it was fixed in advance and paid in arrears. SOFR is an overnight rate with no term structure of its own, so a three-month SOFR loan has to build its quarterly rate out of roughly ninety separate overnight fixings, most of which have not even happened yet when the period begins. The market's answer is compounding in arrears: let the rate accrue day by day using the actual overnight SOFR prints, and only know the total once the period is nearly over.

The formula, in words first

Each business day's overnight SOFR applies for that day, or for three days over a weekend, because Friday's rate covers Friday through Sunday. Compounding in arrears means treating every overnight period as a mini-deposit that rolls into the next: $1 grows by each day's rate, one day at a time, and the total growth over the period, annualized back down, is the rate actually paid.

$$R = \left[\prod_{i=1}^{n} \left(1 + r_i \times \frac{d_i}{360}\right) - 1\right] \times \frac{360}{D}$$

In words: for each of the $n$ SOFR fixings $r_i$, grow a dollar by that day's rate for $d_i$ days (the number of calendar days that fixing applies to, usually 1 but 3 over a weekend); multiply all those growth factors together; subtract the 1 you started with to get total growth over the whole period; then annualize that total growth over the $D$ actual days in the period, using the standard money-market actual/360 convention.

Worked example: a five-day compounding period

A short accrual period covers Monday through Friday of one week, with these overnight SOFR fixings (Friday's rate applies for 3 days, covering the weekend):

Day	Rate	Days it applies
Mon	4.30%	1
Tue	4.31%	1
Wed	4.29%	1
Thu	4.30%	1
Fri	4.32%	3

Total calendar days $D = 7$. Compute each day's growth factor and multiply:

$$\left(1+\tfrac{0.0430}{360}\right)\left(1+\tfrac{0.0431}{360}\right)\left(1+\tfrac{0.0429}{360}\right)\left(1+\tfrac{0.0430}{360}\right)\left(1+\tfrac{0.0432 \times 3}{360}\right)$$

The first four factors are each about $1.0001194$, $1.0001197$, $1.0001192$, $1.0001194$; the Friday factor, covering 3 days, is $1 + (0.0432\times3)/360 = 1.00036$. Multiplying all five together gives a compounded growth of roughly $1.000836$. Annualizing:

$$R = 0.000836 \times \frac{360}{7} \approx 4.30\%$$

The five-day compounded rate lands almost exactly at the simple, days-weighted average of the daily fixings (4.31%) because rates this small barely compound — the effect of compounding versus simple-averaging only becomes visible over long periods or high-rate regimes, but the mechanics are identical either way.

Mon 4.30 Tue 4.31 Wed 4.29 Thu 4.30 Fri 4.32 ×3d

compounded balance grows day by day R ≈ 4.30%

Lookback, lockout and the payment-date problem

Because the true compounded rate isn't final until the accrual period ends, but payments need to be calculated and confirmed a few days ahead of the payment date, most SOFR instruments use a lookback (shift the rate observation window a few business days earlier than the accrual dates, so the last few days' rates are known in time) or a lockout (freeze the rate at its most recent value for the final few days of the period, rather than shifting dates). Both are approximations that trade a small amount of accuracy for the operational certainty of knowing the payment amount before it is due.

Where it shows up

Nearly every SOFR-linked floating-rate note, business loan, and OTC swap since the LIBOR transition compounds in arrears by convention (SOFR OIS swaps compound daily as standard). It is also the calculation underlying SOFR itself as a published index — the same daily-compounding logic, one level up, aggregates individual overnight repo transactions into the published overnight rate that then gets compounded again by borrowers.

Key terms

• Compounding in arrears — accruing interest using actual daily overnight fixings, known only near the end of the period.
• Actual/360 — the day-count convention where each day's rate accrues on a 360-day year, standard in US money markets.
• Lookback / lockout — conventions that shift or freeze the rate observation window so a payment amount is known before it is due.
• Term SOFR — a forward-looking SOFR rate, known at the start of a period, used in limited applications like business loans.