How U.S. Treasury FRN Coupon Algorithms Reset Weekly: 13-Week Bill Index Rates, Fixed Spreads, Daily Actual/360 Accrual and Quarterly Interest

Reader question: A U.S. Treasury floating-rate note pays interest quarterly, but its rate can change every week. How does the Treasury convert a weekly 13-week bill auction into a precise daily accrual and then one quarterly cash payment?

The algorithm has two rate components. The index rate is tied to the most recent 13-week Treasury bill auction and can reset weekly. The spread is established when a new FRN is first auctioned and remains fixed for the life of that security. For each accrual day, Treasury combines the applicable index rate and spread, applies a zero-percent minimum, divides by 360 and accrues interest on par. The quarterly payment is the sum of those daily accrual amounts.

This makes the FRN a useful example of a piecewise-constant rate process: one quarterly coupon is not generated by one quarterly rate. It is built from a sequence of daily accruals whose underlying index can change during the period.

What this page owns — and what it does not

This page owns:

13-week bill auction result + fixed FRN spread + daily calendar → daily accrual rates → quarterly interest payment.

It does not replace Treasury-bill yield conversions, SOFR floating-rate loan calculations, day-count fractions, or date-engine algorithms. The Treasury FRN has its own legally specified index and reset process.

This is public fixed-income mathematics, not a recommendation to buy an FRN or Treasury security.

The security structure

TreasuryDirect currently describes U.S. Treasury FRNs as:

• maturing in two years;
• paying interest every three months;
• having an interest rate that can float over time;
• issued electronically;
• sold in $100 minimums and $100 increments.

Original two-year FRN issues currently occur in January, April, July and October, with reopenings in the intervening months.

The rate has two pieces

For an accrual day j:

FRN Rate_j = Index Rate_j + Spread.

The two terms behave differently:

• Index rate: changes as 13-week Treasury bill auction results change;
• Spread: fixed for the life of the FRN once the new issue is auctioned.

This separation is essential. A reopening can have a different purchase price from the original issue while retaining the original security’s spread.

The index rate comes from the 13-week bill auction

31 CFR Appendix B to Part 356 states that the FRN index rate is based on the High Rate of the 13-week Treasury bill auction announced in the auction results, converted into a simple-interest money-market yield on an Actual/360 basis and rounded to nine decimal places.

Treasury auctions 13-week bills weekly, so the FRN’s index can update weekly.

This is more precise than saying “the FRN follows three-month Treasury rates.” It follows a specific auction-derived index under Treasury’s formula.

The spread comes from the original FRN auction

When Treasury auctions a new FRN, competitive bidders submit discount margins.

TreasuryDirect states that the spread for the security is the highest accepted discount margin in that original auction.

Once set:

Spread_t = constant for the life of the FRN.

Later reopenings sell more of the same security and retain that spread.

A reopening is not a new floating-rate formula

TreasuryDirect defines a reopening as an additional issuance of an existing security. A reopened FRN has the same maturity date and spread as the original security, but a different issue date and usually a different purchase price.

A reference-data system that assigns a new spread to a reopening has created a different instrument mathematically even if it reuses the same security identity.

Daily accrual, not weekly lumping

Although the index changes weekly, interest accrues daily.

For $100 par, a stylised daily accrual is:

Daily Interest_j = 100 × max(r_j + s, 0) / 360.

where:

• r_j is the applicable index rate for day j;
• s is the fixed spread.

The zero-percent floor means the daily accrual rate cannot fall below zero even if index plus spread would otherwise be negative.

Quarterly interest is a sum of daily pieces

If an interest period contains days j = 1…N:

Quarterly Interest per $100 = Σ [100 × max(r_j + s, 0) / 360].

For a constant rate across all days this simplifies to:

100 × N/360 × max(r+s,0).

But in a real quarter, the index can reset several times, so the daily-sum form is the safer conceptual model.

A two-week teaching example

Suppose:

• par = $100;
• fixed spread = 0.120%;
• week 1 index = 4.800%;
• week 2 index = 4.700%;
• seven accrual days at each rate.

Week 1 daily rate:

4.920% / 360.

Seven-day interest:

100 × 0.04920 × 7/360 ≈ $0.09567.

Week 2:

100 × 0.04820 × 7/360 ≈ $0.09372.

Total for the 14-day teaching interval:

≈ $0.18939 per $100 par.

A real Treasury payment follows the exact daily index schedule and regulatory precision rules.

When does a new bill-auction rate take effect?

The Treasury regulation states that the FRN interest rate can be adjusted on the calendar day following each 13-week bill auction.

So the auction event and the accrual-rate effective date are distinct states:

bill auction result → index determination → next-calendar-day FRN rate effect.

A date engine that applies the new rate on the auction day creates an off-by-one accrual error.

The two-business-day lockout around settlement and payment

Appendix B contains an important operational rule. For accrued interest used in a settlement amount or interest payment, a new 13-week bill rate determined during the two-business-day period immediately before the relevant settlement or interest-payment date is excluded from that calculation.

The rate change that would otherwise have occurred is deferred to the first calendar day after that period.

This prevents the final amount from changing too close to settlement or coupon processing.

Why the lockout matters

Suppose a 13-week bill auction occurs one business day before an FRN coupon payment.

A naive engine immediately switches to the new index for the last day of the interest period.

The Treasury rule can require the prior rate to remain in the payment calculation through the protected two-business-day period.

One day at the wrong rate looks small, but across large outstanding principal it can create material reconciliation differences.

Actual/360 is a contractual convention

The daily accrual uses a 360-day denominator.

This connects to the day-count article.

A calendar year can have 365 or 366 days, but the FRN accrual denominator remains 360 under the specified convention.

Why dividing a quarterly rate by four is wrong

An FRN does not determine one quarter-end annual rate and divide it by four.

The rate path inside the quarter matters.

If the 13-week bill index falls gradually over 13 weeks, the quarter contains higher accrual rates early and lower rates later. The correct payment aggregates each daily rate period.

The index rate is not the bill’s bank-discount quote copied directly

The regulation specifies conversion of the 13-week bill auction High Rate into a simple-interest money-market yield on an Actual/360 basis.

This distinction connects to the existing Treasury-bill yield-conversion article.

A system that copies the wrong bill yield convention into the FRN index can be consistently wrong every week.

FRN price can still move away from par

A floating coupon reduces some duration relative to a fixed-rate note, but it does not force the market price to equal 100.

TreasuryDirect explicitly notes that an FRN’s auction price can be above, below or equal to par.

Market price can respond to:

• the fixed spread relative to current market required margins;
• liquidity;
• time to next reset;
• interest-rate expectations;
• supply and demand;
• accrued interest and settlement timing.

A new FRN’s spread can make its issue price differ from par

The original auction’s accepted discount margin helps determine the spread, while the security’s purchase price follows Treasury’s auction-pricing rules.

A floating-rate design does not imply every auction clears exactly at par.

This is another reason to separate:

coupon reset algorithm ≠ market price algorithm.

FRNs and SOFR loans use different benchmarks

A floating-rate corporate or syndicated loan can reference SOFR and apply daily compounding, lookbacks, floors and credit spreads.

A Treasury FRN instead uses the Treasury-specified 13-week bill auction index and its own fixed spread.

Both are floating-rate instruments, but the benchmark, reset rule and accrual mechanics differ.

FRNs cannot be stripped

TreasuryDirect states that FRNs are not eligible for the Treasury STRIPS program.

This provides a clean boundary with the STRIPS article.

A security-master flag that marks every Treasury coupon-bearing instrument as strip-eligible is therefore wrong.

Inputs and outputs

An FRN coupon engine can require:

• FRN CUSIP and original/reopening status;
• par amount;
• dated date and maturity date;
• quarterly interest-payment dates;
• fixed spread from original auction;
• 13-week bill auction High Rates;
• conversion to the prescribed index yield;
• index effective dates;
• Actual/360 convention;
• two-business-day lockout logic;
• zero-percent minimum accrual rate;
• calendar and settlement data.

Outputs can include:

• daily applicable index rate;
• daily FRN accrual rate;
• daily accrued interest;
• quarter-to-date accrued interest;
• quarterly coupon payment;
• settlement accrued interest;
• source auction and rate provenance.

Evidence polarity: what supports confidence?

Evidence for confidence includes index rates tied to the correct 13-week bill auctions, correct next-calendar-day reset timing, a spread matching the original FRN auction, daily Actual/360 accruals that sum to the quarterly payment and correct handling of the two-business-day protected period.

Evidence against confidence includes a reopened FRN receiving a new spread, rate changes applied on the auction day, use of a raw bill discount-rate field when the regulation requires the converted money-market index, negative daily coupon accrual despite the zero floor, or quarter interest computed from only the final weekly rate.

Counterexample: weekly reset does not mean weekly cash payment

The index can change weekly, but Treasury pays FRN interest quarterly.

The weekly changes alter the daily accrual path that is accumulated into the quarter-end payment.

Counterexample: a reopening’s auction margin is not a new permanent spread

A reopened FRN is the same security. Its existing spread remains fixed.

A reopening auction can clear at a different price and discount margin, but the contract does not replace the original fixed spread.

Counterexample: lower bill rates do not instantly rewrite previously accrued interest

Interest already accrued under earlier daily rates remains part of the quarter’s accumulated amount.

A new weekly index affects future accrual days under the effective-date rules; it does not retrospectively reprice the earlier part of the quarter.

Counterexample: a floating coupon does not mean zero duration

The reset mechanism reduces sensitivity to broad rate changes, but price can still respond to fixed-spread value, time to reset, market liquidity and other factors.

“Floating” is not the same as “market value cannot move.”

Weak links in implementation

wrong bill source. A 26-week or secondary-market bill yield is substituted for the 13-week auction index.

yield-convention error. The bill High Rate is not converted as the FRN regulation requires.

effective-date error. The new index is applied on the auction day instead of the next calendar day.

spread reset bug. The fixed spread changes at a reopening.

lockout omission. A late bill auction changes settlement/payment accrued interest improperly.

day-count error. 365 is used instead of 360.

rate-floor omission. index plus spread below zero creates negative accrual.

quarter shortcut. One weekly rate is used for the entire interest period.

Diagnostics: how to test the engine

• spread persistence test: original issue and both reopenings use the same spread.
• weekly reset test: change the 13-week bill auction result and verify the FRN index changes on the next calendar day.
• daily sum test: sum independent daily accruals and compare with quarterly interest.
• Actual/360 test: reproduce a constant-rate period with N/360.
• zero-floor test: force index + spread below zero and require zero daily accrual rather than negative interest.
• lockout test: schedule a bill auction inside the two-business-day protected period and verify deferred effect.
• holiday test: move bill auctions and payment dates around holidays using the authoritative calendar.
• source-provenance test: every daily index maps to a specific Treasury bill auction result.
• reopening test: issue date and purchase price change while maturity and spread remain the same.
• payment-reconciliation test: quarterly payment equals the accumulated daily ledger.

What would falsify confidence?

Confidence should be withdrawn if the quarterly payment cannot be reproduced from daily accruals; if the spread differs between original and reopened CUSIPs of the same security; if index provenance is missing; if lockout-period handling disagrees with 31 CFR Appendix B; or if server date/time settings change which auction rate applies.

Alternatives and limits

Fixed-rate Treasury notes lock the coupon rather than resetting it. Treasury bills are discount instruments. TIPS adjust principal for CPI. Private-sector floating-rate debt can use SOFR or other benchmarks and different spread/reset conventions.

The Treasury FRN algorithm is contract-specific. Its simplicity comes from a public weekly benchmark and fixed spread, not from one universal floating-rate formula used by all debt instruments.

How this connects to the surrounding knowledge estate

The Treasury-bill yield engine explains why different bill yield conventions must not be confused. Day-count algorithms supply the Actual/360 fraction. Date engines control auction, effective and payment dates. STRIPS provides the contrast that FRNs are explicitly not strip-eligible.

Verification and update triggers

Preserve Treasury regulation version, FRN identity, fixed spread, source 13-week bill auctions, index conversion method, effective dates, daily accrual records, day-count rule and lockout logic. Revalidate after Treasury rule amendments, auction-schedule changes, bill-rate methodology changes, date-engine upgrades or any coupon-payment reconciliation break.

Primary and high-quality references

• U.S. Treasury, TreasuryDirect, Floating Rate Notes (FRNs).
• U.S. Treasury, TreasuryDirect, Understanding Pricing and Interest Rates, describing the FRN index rate and fixed spread.
• 31 CFR Appendix B to Part 356, Formulas and Tables, including the FRN indexing, daily accrual, Actual/360 and two-business-day rules.
• U.S. Treasury, TreasuryDirect, Treasury Reopenings, including the two-year FRN reopening schedule and same-spread rule.
• U.S. Treasury, TreasuryDirect, Treasury Auction Regulations — Uniform Offering Circular.

Educational boundary: This article explains Treasury FRN coupon mechanics. It does not recommend a Treasury security or provide personalized financial advice.