Reader question: A U.S. Treasury floating-rate note pays interest every three months, but its reference rate resets every week. How does a computer turn weekly 13-week Treasury-bill auction results into a precise daily accrual and then one quarterly cash payment?
The answer is a small state machine. A Treasury FRN has a fixed spread determined when the security is first auctioned and a variable index rate tied to the highest accepted discount rate from the most recent 13-week Treasury-bill auction. After each qualifying bill auction, the index rate changes. Each calendar day’s interest accrual uses the applicable index rate plus the fixed spread, divided by 360, subject to Treasury’s minimum daily accrual rule. The daily accruals are accumulated until the quarterly interest payment date.
The mathematics is simple; the calendar logic is not. Lockout periods before settlement and coupon payment dates deliberately delay some index-rate changes, reopenings preserve the original spread, and the current cash payment is an accumulation of potentially several weekly rates.
What this page owns — and what it does not
This article owns the transformation:
13-week bill auction results + fixed FRN spread + calendar rules → daily FRN interest rate → accrued interest → quarterly payment.
It does not replace Treasury-bill yield conventions, day-count fractions, financial date engines, or Treasury auction allocation. Those are adjacent owners.
This is public fixed-income mathematics, not a recommendation to buy or sell Treasury securities and not personalized financial advice.
The instrument first: what is a Treasury FRN?
TreasuryDirect currently describes U.S. Treasury floating-rate notes as two-year marketable securities that pay interest four times per year. Original issues are auctioned in January, April, July and October, with reopenings in the intervening months.
The key difference from a fixed-rate Treasury note is that the FRN’s interest rate changes over its life.
The governing relationship is:
FRN interest rate = index rate + spread.
The spread is fixed at the original auction
For a new FRN, the competitive auction is expressed in terms of discount margin. Treasury defines the security’s spread as the highest accepted discount margin at the original auction.
The spread is quoted to tenths of a basis point under the auction regulations and remains fixed for the life of the security.
That means a reopening does not create a new spread. It creates additional principal amount of the same security under current market pricing.
The index rate comes from the 13-week bill
TreasuryDirect states that the FRN index rate is tied to the highest accepted discount rate from the most recent 13-week Treasury-bill auction.
Because the 13-week bill is normally auctioned weekly, the FRN index rate normally resets weekly.
Conceptually:
IndexRatet = HighRate of most recent qualifying 13-week bill auction before day t.
The word qualifying matters because lockout rules can postpone an otherwise scheduled reset.
Do not confuse the bill high rate with a bond-equivalent yield
The 13-week bill auction produces several commonly reported return measures. Treasury FRNs specifically reference the bill auction’s high discount rate under the governing rules.
A programmer who substitutes a bond-equivalent yield or investment rate because it “looks like the more intuitive annual yield” will create the wrong FRN rate.
This is exactly why the Treasury-bill yield article is a separate owner.
Daily accrual uses an Actual/360-style denominator
Under Appendix B to 31 CFR Part 356, daily accrued interest generally uses:
Daily rate = max(IndexRate + Spread, 0) / 360.
The zero in that expression captures Treasury’s minimum daily interest-accrual rate: the daily accrual rate cannot be negative.
If par amount is P, then a stylised daily interest amount is:
Daily Interest = P × Daily rate.
Quarterly interest is accumulated across all calendar days in the accrual period using the applicable daily rates.
A small weekly-reset example
Suppose:
- par = 1,000,000;
- fixed spread = 0.080%;
- 13-week bill high rate for week 1 = 4.100%;
- 13-week bill high rate for week 2 = 4.160%.
Week 1 FRN rate:
4.100% + 0.080% = 4.180%.
Week 2 FRN rate:
4.160% + 0.080% = 4.240%.
If each rate applies for seven calendar days in this simplified example, the interest contribution is approximately:
Week 1 = 1,000,000 × 0.04180 × 7/360.
Week 2 = 1,000,000 × 0.04240 × 7/360.
The quarterly payment continues accumulating day by day as later weekly index rates become applicable.
Why quarterly payment is not “quarter-end rate × quarter fraction”
Suppose the reference bill rate moves substantially during a quarter.
If software waits until the payment date and applies only the latest index rate to the entire three-month period, it erases the weekly reset history.
The correct concept is:
Interest Payment = Σ daily accrual amounts over the accrual period.
Each day uses the rate applicable to that day under the Treasury rules.
The lockout period is the high-information edge case
Treasury’s regulations provide a special rule for the two-business-day period before an auction settlement date or interest payment date.
A 13-week bill auction occurring in that lockout period is excluded from the accrued-interest calculation used for that settlement or payment. The index-rate change that would otherwise have occurred is delayed until the first calendar day after the lockout ends.
This prevents a very late reference-rate reset from entering a settlement amount that is already operationally being finalised.
A lockout example
Suppose an FRN interest payment is on Wednesday and a 13-week bill auction falls inside the two-business-day lockout period before that payment.
The new bill high rate does not suddenly rewrite the interest amount being settled for Wednesday.
Instead, the prior applicable index rate continues through the lockout for that payment calculation, and the new index rate becomes effective after the lockout according to Treasury’s rule.
A date engine that knows only “reset after every bill auction” but not the lockout will fail precisely near payment dates.
The zero-percent daily floor
The regulation states that the daily interest-accrual rate is subject to a minimum of zero percent.
So if:
IndexRate + Spread < 0,
the applicable daily interest-accrual rate is floored at zero.
This is a contractual floor on daily coupon accrual, not a guarantee that the market price of the FRN cannot decline.
Reopenings preserve the original spread
TreasuryDirect explains that a reopened FRN has the same maturity date and spread as the original FRN but a different issue date and usually a different purchase price.
This is important for reference data:
same security identity → same maturity + same spread + same interest-payment schedule.
If a system assigns the reopening’s high discount margin as a new contractual spread, it has created a different security that does not exist.
Reopening settlement can include accrued interest
A purchaser of a reopened FRN may pay accrued interest from the relevant dated/accrual period to the new issue settlement date. Treasury then returns that accrual through the next interest payment.
Therefore the settlement amount can contain:
- par amount;
- premium or discount from the auction price;
- accrued interest.
This is a settlement accounting effect, not an extra yield gift.
Price and coupon are different state variables
An FRN can auction or trade above or below par even though the coupon resets weekly.
Why? The contractual spread is fixed. If market-required spread changes after issuance, the security’s price can move away from par even though the index component keeps updating.
A floating coupon reduces some interest-rate duration compared with a fixed-rate note, but it does not make price exactly constant.
Discount margin is a pricing variable, not the same as contractual spread
For a new FRN, the auction high discount margin becomes the fixed contractual spread.
Later, a market or reopening discount margin can be used as a pricing/required-return measure. That current discount margin can differ from the security’s fixed spread.
So:
spread = contractual constant;
market discount margin = valuation variable.
Inputs and outputs
A robust Treasury FRN accrual engine can require:
- FRN identifier and original issue terms;
- par amount;
- fixed spread;
- 13-week bill auction dates and high discount rates;
- interest-payment schedule;
- settlement dates;
- lockout-calendar rules;
- day-count denominator and precision rules;
- reopening/original-issue linkage.
Outputs can include:
- daily applicable index rate;
- daily all-in FRN rate;
- daily accrued interest;
- cumulative accrued interest;
- quarterly payment amount;
- lockout status;
- calculation provenance to the source 13-week bill auction.
Evidence polarity: what supports confidence?
Evidence for a correct implementation includes agreement with Treasury’s daily FRN index data, preservation of the original spread across reopenings, correct weekly resets, correct lockout behaviour, daily accruals that sum to the published coupon amount and a zero floor that activates only when necessary.
Evidence against confidence includes use of a bill investment yield instead of high discount rate, a reopening that changes spread, a rate reset during the lockout when Treasury would defer it, quarterly interest calculated from one end-of-quarter rate, or negative daily coupon accrual.
Counterexample: a weekly reset does not mean seven-day interest periods
The bill auction schedule can interact with holidays and lockouts. A reference rate can therefore apply for more or fewer than seven calendar days.
The correct unit is the calendar day, not an assumption of exactly 52 equal weekly blocks.
Counterexample: floating rate does not mean market-value stability
If the market demands a much wider discount margin than the FRN’s fixed spread, the security can trade below par even though its index rate updates correctly.
The coupon algorithm can be perfect while the market price moves.
Counterexample: a negative bill rate does not force negative cash interest
Under the Treasury minimum daily accrual rule, a sufficiently negative index-plus-spread combination is floored at zero for daily interest accrual.
That contractual floor is part of the security’s cash-flow algorithm.
Counterexample: the most recent bill auction is not always the applicable one
Near an FRN settlement or interest-payment date, a bill auction inside the lockout period can be chronologically most recent but still excluded from that settlement calculation.
“Most recent” must therefore be implemented with the qualifying-auction rule, not a simple max(date) query.
Weak links in implementation
wrong bill field. High discount rate is confused with investment rate.
spread reset error. A reopening creates a new spread.
lockout omission. A late weekly reset changes a payment incorrectly.
business-day/calendar mismatch. The two-business-day window is counted as two calendar days.
weekly-block simplification. Every reference rate is assumed to apply exactly seven days.
negative-rate bug. Daily accrual falls below zero.
quarter-end shortcut. One current rate is applied to the whole quarter.
premature rounding. Daily interest is rounded too aggressively and cumulative drift appears.
Diagnostics: how to test the engine
- Treasury daily-index replay: reproduce published FRN daily index records for a live or historical CUSIP.
- weekly reset test: change the 13-week bill high rate and verify the index transition on the correct calendar day.
- holiday test: move a bill auction schedule around a holiday and verify day coverage.
- lockout test: place a bill auction inside the two-business-day period before payment and verify deferred application.
- zero-floor test: force index + spread below zero and verify daily rate equals zero.
- reopening test: confirm spread and maturity are inherited from the original issue.
- quarterly-sum test: sum daily accruals and compare with the payment calculation.
- precision test: increase internal precision and verify the reported amount is stable.
- provenance test: every daily rate should trace to one source bill auction plus one fixed spread.
- settlement test: calculate premium/discount and accrued-interest components separately.
What would falsify confidence?
Confidence should be withdrawn if the system disagrees with Treasury’s daily FRN index data; if a reopening changes the spread; if the same daily rate cannot be traced to its 13-week bill auction; if lockout dates are wrong; if daily accrual can be negative; or if quarterly cash interest cannot be reproduced by summing daily accruals.
Alternatives and limits
A simple floating-rate model can approximate the FRN as “current bill rate plus spread”, which is fine for intuition. Production settlement requires daily accrual, lockout and reopening rules.
The accrual algorithm does not forecast future bill rates. Projecting future FRN cash flows requires a forward-rate or scenario model. Pricing also requires a market discount-margin assumption and settlement conventions.
How this connects to the surrounding knowledge estate
The reference input is rooted in Treasury-bill rate conventions. The calendar and lockout logic relies on the financial date engine. Daily accrual uses the ACT/360-style day-count layer. The fixed spread originates in the Treasury auction mechanism.
Verification and update triggers
Preserve the auction-regulation version, original FRN terms, spread, 13-week bill auction data, payment calendar, lockout logic and precision policy. Revalidate after Treasury rule changes, auction-calendar changes, reference-data migrations or any discrepancy with Treasury’s daily FRN index database.
Primary and high-quality references
- U.S. Treasury, TreasuryDirect, Floating Rate Notes.
- 31 CFR Part 356 Appendix B, FRN formulas and tables, including index rate, spread, daily accrual and lockout treatment.
- 31 CFR §356.20, Treasury auction award and FRN interest-rate rules.
- TreasuryDirect, FAQs About Floating Rate Notes.
- TreasuryDirect, Understanding Pricing and Interest Rates.
Educational boundary: This article explains Treasury FRN cash-flow mathematics. It does not recommend any Treasury security or provide personalized financial advice.