Floaters, index-linked and structures

Coupon reset mechanics

A floating coupon is defined by five separate terms: the reference rate, the observation method, the timing of observation relative to the interest period, the quoted margin, and any cap, floor or multiplier. Changing any one of them changes the coupon on the same instrument over the same period.

Field	Value
Formula	Coupon rate for period j = multiplier * R_j + QM, bounded by any floor and cap, where R_j is the reference rate determined by the documented observation procedure over the period's observation window
Reference rate	The index, such as compounded SOFR, 30-day Average SOFR, Term SOFR, or an inflation index
Observation method	Compounded, simple average, or a single fixing
Timing	In arrears over the interest period, in arrears over a shifted window, or in advance from the prior period
Quoted margin	An additive spread; the illustrative floater elsewhere in this corpus carries 80 basis points
Bounds	A floor, typically at zero, and sometimes a cap; a floor is an option the issuer has written and is not free
Worked spread of outcomes	On the illustrative interest period 2026-09-04 to 2026-09-11, six documented procedures give rates from 4.229642 to 4.365423 percent

• The observation procedure is where the money is and it is the least prominent term in the documentation. Two notes from the same issuer referencing the same index can pay different coupons for the same period, and the difference is in a definitions section rather than on the term sheet.
• A zero floor on the reference rate is a strip of options written by the issuer to the holder, and it has value whenever the forward curve implies any probability of a negative index. Pricing a floored floater as an unfloored one understates its value.
• The multiplier is rarely one for structured notes and rarely anything else for plain floaters. A multiplier above one creates leverage in the coupon and correspondingly larger rate duration, which the usual near-zero-duration reasoning about floaters does not apply to.
• Reset frequency and payment frequency need not match. A note can observe daily, compound over a quarter and pay quarterly, and a note can observe once and pay monthly. The three frequencies are three separate terms.

Also described at: Wikipedia · Wikidata · ARRC: A User's Guide to SOFR

SOFR compounded in arrears

The standard convention for a SOFR-referencing floating leg: each overnight fixing is compounded daily across the interest period, weighted by the number of calendar days it applies for, and the result is annualised on an ACT/360 basis.

Field	Value
Formula	Compounded rate = [product over i of (1 + r_i * n_i/360) - 1] * 360/d_c, where r_i is the fixing on business day i, n_i the calendar days until the next business day, d_b the number of business days and d_c the calendar days in the period
Setup	Interest period 2026-09-04 to 2026-09-11, d_c = 7 calendar days, d_b = 4 business days, with 7 September 2026 a holiday
Daily factors	(1 + 4.3500/100 * 4/360) x (1 + 4.3500/100 * 1/360) x (1 + 4.4000/100 * 1/360) x (1 + 4.4000/100 * 1/360)
Product	1.000848832161
Worked, compounded rate	(1.000848832161 - 1) * 360/7 = 0.000848832161 * 51.428571429 = 4.365423 percent
Interest on 10,000,000	notional * (product - 1) = 8,488.32, which equals notional * rate * 7/360
Weighting check	Calendar days weighted: 4 + 1 + 1 + 1 = 7 = d_c

• The interest amount is the product minus one, multiplied by the notional. Annualising to a rate and then re-applying the day-count fraction returns the same number and adds two rounding opportunities, which is why the compounded-balance formulation is preferred in operations.
• The n_i weights are what make a Friday fixing count three times and a pre-holiday fixing count four. A calculation that weights every business day equally is a simple average of fixings, not a compounded rate, and the difference is largest around long weekends.
• The calendar is the US Government Securities Business Day calendar, which is not the New York banking calendar. Using the wrong one changes the number of observations and therefore the rate.
• The final rate is not known until one business day before the period ends, which is the whole operational problem in arrears creates and the reason lookbacks, lockouts and payment delays exist.
• The rates used here are illustrative, chosen so the compounding is visible in the digits. They are not observed fixings and no inference about the level of SOFR should be drawn from them.

Source: ARRC recommended conventions for syndicated and bilateral business loans and for floating-rate notes; ISDA 2021 Interest Rate Derivatives Definitions

Also described at: Wikipedia · Wikidata · Federal Reserve Bank of New York: SOFR · ARRC: A User's Guide to SOFR

Simple average against compounded average

Two ways of turning a series of overnight fixings into one period rate. The simple average weights each fixing by its calendar days and divides; the compounded average multiplies daily growth factors. Compounding is larger whenever rates are positive.

Field	Value
Formula	Simple average = sum of (r_i * n_i) / d_c. Compounded = [product of (1 + r_i * n_i/360) - 1] * 360/d_c. The difference is the interest-on-interest term and grows with the level of rates and the length of the period
Worked, simple average	(4.3500x4 + 4.3500x1 + 4.4000x1 + 4.4000x1) / 7 = 4.364286 percent
Worked, compounded	4.365423 percent
Difference	0.1137 basis points over 7 days
Interest difference on 10,000,000	8,488.32 against 8,486.11, a difference of 2.21
Scaling	The gap is second order in the rate and roughly quadratic in the period length, so it is negligible over a week and material over a year

• Over a 7-day period at these levels the difference is a small fraction of a basis point. Over a three-month period at the same levels it is a few basis points, and over a year it is close to the difference between a nominal and an effective rate.
• SOFR Averages published by the Federal Reserve Bank of New York are compounded averages, not simple averages, despite the name. Substituting a published SOFR Average into a simple-average formula double-counts the compounding.
• US loan markets have historically favoured simple averaging for operational reasons and derivatives markets compounding, so a loan hedged with a swap can carry a small systematic basis that is pure convention.
• Because the gap is a function of the rate level, a convention mismatch that was immaterial in a near-zero-rate period becomes visible when rates rise. It is worth re-checking rather than relying on a historical judgement.

Also described at: ARRC: A User's Guide to SOFR

In advance against in arrears

In arrears determines the coupon from fixings inside the interest period, so it reflects the period's actual rates and is not known until the period is nearly over. In advance uses the preceding period's rate, so it is known at the start and reflects a window that has already passed.

Field	Value
Formula	In arrears: R_j = compounded rate over period j itself. In advance: R_j = compounded rate over period j-1, or a term rate fixed at the start of period j
Worked, in arrears	Period 2026-09-04 to 2026-09-11: 4.365423 percent, known 2026-09-10
Worked, in advance	Same period paying the prior period's compounded rate over 2026-08-28 to 2026-09-04: 4.244147 percent, known at the start
Worked, difference	12.13 basis points on an illustrative rising rate path
Interest on 10,000,000	8,488.32 in arrears against 8,252.51 in advance
Bias	On a rising path in advance underpays and on a falling path it overpays, systematically, because it is always looking backward

• The choice is a trade between economic accuracy and operational certainty. In arrears tracks the period's actual funding cost; in advance lets the borrower know its payment in advance, which some borrowers require.
• Term SOFR is an in-advance rate derived from derivatives markets rather than from realised overnight fixings, so it is forward-looking rather than backward-looking. That makes it operationally similar to LIBOR and economically different from compounded in arrears.
• An in-advance leg hedged with an in-arrears swap carries a timing basis that is not a spread and cannot be hedged away with a spread. It is a one-period lag and it shows up as tracking error whenever the rate path is not flat.
• The illustrative rate path used here is monotonically rising, which maximises the gap and makes the direction of the bias legible. A flat path would make the two conventions coincide, which is exactly why the bias is easy to overlook in a calm market.

Also described at: ARRC: A User's Guide to SOFR

Lookback, and the difference between a rate shift and an observation shift

A lookback moves the observation window earlier so the rate is known before the period ends. There are two ways to do it and they give different answers: shifting only which rates are read, or shifting the whole observation period including the calendar-day weights.

Field	Value
Formula	Rate shift: use the fixing from k business days before each date, weighted by the interest period's own n_i and d_c. Observation shift: shift the entire period back k business days and use that period's n_i and d_c
Interest period	2026-09-04 to 2026-09-11, d_c = 7, business day weights 4, 1, 1, 1
Worked, rate shift of five business days	rates from 2026-08-28, 2026-08-31, 2026-09-01, 2026-09-02, weighted 4, 1, 1, 1, d_c = 7, giving 4.229642 percent
Worked, observation shift of five business days	observation period 2026-08-28 to 2026-09-03, d_c = 6, weights 3, 1, 1, 1, giving 4.234333 percent
Difference in rate	0.4691 basis points
Difference in interest on 10,000,000	7,057.22 against 8,224.30, a difference of -1,167.08
Why they differ here	The interest period spans a holiday and the shifted observation period does not, so the interest period has 7 calendar days and the observation period 6

• The two methods coincide whenever the interest period and the shifted observation period have the same weekend and holiday structure, which is most of the time. They diverge around holidays and month ends, which is where the exposure sits.
• Under observation shift the calendar-day count comes from the observation period, so an interest period of one length can pay interest calculated over a different length. That is not an error and it does surprise people the first time.
• Whether the resulting rate is then applied over the interest period's day count or the observation period's is a further documented term, and the two produce different cash amounts. Confirm it against the note's calculation provisions.
• ISDA's observation period shift and the ARRC's recommended lookback with observation shift are the same construction described in different documents. A lookback without observation shift is the ARRC's alternative and is also in use.
• The rate path and calendar used here are illustrative. The structural point is that the two conventions are not interchangeable, and the size of the difference depends entirely on where the holidays fall.

Source: ARRC recommended conventions for floating-rate notes; ISDA 2021 Interest Rate Derivatives Definitions (Observation Period Shift)

Also described at: ARRC: A User's Guide to SOFR

Lockout, or the suspension period

The rate for the final few business days of an interest period is frozen at the last observed fixing before the lockout begins. It buys the same operational certainty as a lookback by a different route, and introduces a different distortion.

Field	Value
Formula	For the final k business days of the period, r_i = the fixing on the (k+1)th-to-last business day. All other terms are unchanged, and d_c is still the interest period's calendar days
Worked, two-business-day lockout	The 2026-09-09 and 2026-09-10 fixings are replaced by the 2026-09-08 rate of 4.3500 percent
Worked, result	4.351126 percent against 4.365423 percent with no lockout
Difference	1.4296 basis points
Interest on 10,000,000	8,460.52 against 8,488.32
Bias	On a rising path the lockout underpays and on a falling path it overpays, because it extends a stale rate over the end of the period

• A lockout is directionally biased in a way a lookback is not: it freezes the most recent rate rather than reading an older one, so it always propagates the level at the lockout date forward. Over a period containing a policy move that lands inside the lockout, the move is simply not reflected.
• The distortion is concentrated at the end of the period, which is where a hedging swap is most likely to be using the actual fixings. That mismatch is not hedgeable with a spread.
• Lockouts are more common in the securitisation and structured markets than in vanilla notes, where lookbacks have become the norm. The ARRC's recommended conventions favour a lookback.
• A long lockout on a short period is close to an in-advance rate. A five-business-day lockout on a weekly period leaves almost no live observations at all.

Also described at: ARRC: A User's Guide to SOFR

Payment delay

The payment date is set a stated number of business days after the end of the interest period, leaving the calculation entirely alone. It is the only one of the operational fixes that does not distort the rate.

Field	Value
Formula	Payment date = the interest period end date plus k business days. The rate, the observation window and the day-count fraction are all unchanged
Worked	Interest period ends 2026-09-11; a two-business-day payment delay pays on 2026-09-15
Worked, rate	4.365423 percent, identical to the no-delay case
Interest	8,488.32, identical
What it costs the holder	The time value of the delay, which is the interest on the payment for k business days and is not compensated
Illustrative cost	Two days at 4.365423 percent on 8,488.32 is 2.06

• Because payment delay leaves the rate untouched, it is the cleanest solution to the in-arrears timing problem from a valuation standpoint. Its cost is a small, uncompensated financing item rather than a distortion of the coupon.
• It creates a mismatch on the final period, where the payment falls after maturity. Documentation has to say explicitly that the final payment is made after the redemption, and some structures shorten the final period instead.
• For a note held against a swap with no payment delay, the delay produces a genuine funding mismatch on each payment date. It is small and it is real.
• The delay interacts with the settlement of a sale: a bond sold between the period end and the payment date has a payment in flight, and who receives it is determined by the record date rather than by the trade.

Also described at: ARRC: A User's Guide to SOFR

Statutory LIBOR spread adjustments under Regulation ZZ

For contracts that lacked a workable fallback when USD LIBOR ceased, the Adjustable Interest Rate (LIBOR) Act substituted a Board-selected benchmark replacement by operation of law. That replacement is a SOFR-based base rate plus a fixed tenor spread adjustment, and the five adjustment values are set in the regulation itself.

Field	Value
Formula	Board-selected benchmark replacement = base rate + tenor spread adjustment, where the base rate depends on the contract type and the adjustment depends only on the LIBOR tenor the contract referenced
Overnight LIBOR	0.00644 percent, or 0.644 basis points
One-month LIBOR	0.11448 percent, or 11.448 basis points
Three-month LIBOR	0.26161 percent, or 26.161 basis points
Six-month LIBOR	0.42826 percent, or 42.826 basis points
12-month LIBOR	0.71513 percent, or 71.513 basis points
LIBOR replacement date	The first London banking day after 30 June 2023
Worked	A contract referencing three-month LIBOR that is not a consumer loan, an FHFA-regulated-entity contract or FFELP ABS becomes three-month CME Term SOFR plus 0.26161 percent

• The values above were checked against the current text of 12 CFR Part 253 and match the statutory definition at 12 U.S.C. 5802(20) word for word. They are exact decimal figures, not rounded basis-point approximations, and reproducing them as 11.4, 26.2, 42.8 and 71.5 basis points loses precision that the regulation specifies.
• The base rate is not uniform. Derivative transactions and Federal Home Loan Bank advances take the ISDA Fallback Rate (SOFR); FHFA-regulated-entity contracts other than FHLB advances take 30-day Average SOFR even for the term tenors; FFELP asset-backed securities take 30-day Average SOFR for one-month LIBOR; and everything else takes the corresponding tenor of CME Term SOFR. The same tenor spread adjustment applies on top of all of them.
• Consumer loans are handled differently again: the spread transitions linearly each business day across the one year following the LIBOR replacement date, starting from the actual observed difference on the day before, and only reaches the full statutory adjustment after that year.
• The ARRC's recommended spread adjustments carry the same five numbers but rest on a different legal basis: they apply where the contract's own fallback language incorporates them, whereas Regulation ZZ operates by statute on contracts that had no workable fallback. Citing one when the other applies is a legal error even though the arithmetic is identical.
• The statute provides a safe harbour for using the Board-selected replacement, which is why the values matter operationally rather than merely historically: they are still the governing rate on a large population of long-dated contracts.

Source: 12 CFR 253.4(c) (Regulation ZZ); Adjustable Interest Rate (LIBOR) Act, 12 U.S.C. 5802(20); final rule at 88 FR 5220, 26 January 2023

Also described at: 12 CFR Part 253 (Regulation ZZ) · Federal Reserve final rule implementing the LIBOR Act

Rate duration and spread duration of a floating-rate note

A floater has almost no interest-rate duration, because its coupon resets, and close to full-maturity spread duration, because its margin does not. Carrying one duration number for a floater misstates its risk in both directions at once.

Field	Value
Formula	Rate duration is approximately t_1/(1 + (I + DM)*t_1), where t_1 is the time to the next reset, because only the already-fixed coupon is exposed. Spread duration is the sensitivity to the discount margin and runs to maturity
Setup	Illustrative floater: 12 quarterly periods, index assumed flat at 4.000 percent, quoted margin 80 basis points, quoted price 99.500, discount margin 98.0461 basis points, 45 days to the next reset
Worked, rate duration	0.125000/(1 + 0.049805 * 0.125000) = 0.124227
Worked, spread duration	2.776718, from bumping the discount margin by one basis point
Ratio	Spread duration is 22.4 times the rate duration on the same instrument
Against the reference bond	The fixed-rate reference bond has a modified duration of 6.706168 and a spread duration of 6.698219, which are close to each other

• The ratio in the fourth field is the whole point: on this floater a report that used one duration figure would either overstate rate risk by a factor of 22 or understate spread risk by the same factor, depending on which number it chose.
• The near-zero rate duration is why floaters are held for capital preservation in a rising-rate environment, and the full spread duration is why that protection does not extend to a credit selloff. The two risks are separable in a floater and largely are not in a fixed-rate bond.
• A floater with a coupon floor is not duration-free. Once the index is at or below the floor the coupon has stopped resetting and the instrument behaves like a fixed-rate bond, so its rate duration jumps discontinuously as the index approaches the floor.
• A leveraged floater with a multiplier above one has rate duration in the opposite direction from intuition, because the coupon overshoots. The near-zero-duration reasoning is specific to a multiplier of one.

Also described at: Wikipedia · Wikidata

Index ratio for an inflation-linked bond

The factor by which principal and coupons are scaled to reflect cumulative inflation since the bond's dated date. It is a ratio of two reference index values, each interpolated linearly within the settlement month from published index levels lagged three months.

Field	Value
Formula	Reference Index(D) = Index(M-3) + (day(D) - 1)/(days in month M) * [Index(M-2) - Index(M-3)]. Index Ratio = Reference Index(settlement) / Reference Index(dated date), conventionally rounded to five decimal places
Worked, interpolation	Settlement 2026-08-27: 300.00000 + (27 - 1)/31 * (301.20000 - 300.00000) = 300.00000 + 0.838709677 * 1.20000 = 301.00645
Worked, ratio	301.00645 / 250.00000 = 1.20403
Adjusted principal on 1,000,000 par	1,204,030.00
Coupon on the adjusted principal	1,204,030.00 * 0.04500/2 = 27,090.67
Accrued on the adjusted principal	1,204,030.00 * 0.04500/2 * 104/184 = 15,312.12
Interpolation on the first of the month	The weight (day - 1)/days is zero on the first, so the reference index on the first of any month is exactly the published level for three months earlier

• The three-month lag exists because the index is published with a lag and the bond needs a known reference value at settlement. It means an inflation-linked bond's index ratio today reflects price levels from two to three months ago, so the instrument has a built-in indexation lag that is a genuine economic feature, not an artefact.
• The lag also means the near-term index ratio path is already fully determined by published data. Two to three months of a linker's accretion is known with certainty at any moment, which is the basis of the carry calculation on the asset.
• The interpolation weight is (day - 1) over days in the month, not day over days in the month. The off-by-one produces a small error that reverses each month and is therefore hard to spot in aggregate.
• Rounding to five decimal places is the US Treasury convention for the index ratio. Other markets round differently or not at all, and the rounding is applied before scaling the principal, so it is not a display convention.
• The index values used here are illustrative round numbers. They are not published index levels and imply nothing about actual inflation.

Source: US Treasury Inflation-Protected Securities offering circular, 31 CFR Part 356 Appendix B

Also described at: 31 CFR Part 356, Appendix B (formulas and tables) · TreasuryDirect: TIPS

Inflation accretion and the deflation floor

The inflation adjustment applies to principal, so it flows into both the coupon and the redemption. A deflation floor guarantees that redemption is not below original par even if the cumulative index ratio falls below one, which makes the floor an option written to the holder.

Field	Value
Formula	Coupon payment = par * IR * (C/f). Redemption = par * max(IR_maturity, 1) where a deflation floor applies, and par * IR_maturity where it does not
Worked, coupon accretion	On 1,000,000 par at an index ratio of 1.20403, the semiannual coupon is 27,090.67 rather than 22,500.00
Uplift	4,590.67 per coupon, a 20.403 percent increase
Redemption with a floor	par * max(IR, 1), so a fall in the index below the dated-date level does not reduce the redemption below 1,000,000
Redemption without a floor	par * IR, so cumulative deflation reduces the principal repaid
Where the floor has value	Only where cumulative deflation from the dated date is possible, which for a seasoned bond with a high accumulated index ratio is remote and for a new issue is not

• US Treasury inflation-protected securities carry a deflation floor on the redemption but not on the coupons: coupons are paid on the deflated principal even when it is below par, and only the final redemption is floored. That asymmetry is easy to miss and changes the cash-flow profile in deflation.
• The floor is worth more on a newly issued linker than on a seasoned one, because a seasoned bond's accumulated index ratio is far above one and would need years of deflation to breach it. The option is therefore a new-issue feature economically, whatever the documentation says.
• Tax treatment of accretion is jurisdiction-specific and can require the holder to recognise income on the inflation uplift before receiving it in cash. That is a real cash-flow consideration for a taxable holder and is one reason linkers are disproportionately held in tax-deferred accounts in some markets.
• The real yield of a linker and the nominal yield of a comparable conventional bond differ by a breakeven inflation rate, which is a market-implied quantity and not a forecast. Comparing a real yield with a nominal yield directly, without that adjustment, compares two incompatible numbers.

Also described at: Wikipedia · Wikidata · TreasuryDirect: TIPS