Reader question: A Treasury bill has one purchase price and one maturity payment. Why can the same bill appear to have several different “yields” at the same time?
Because a yield quote is not merely a return. It is a return transformed by a quotation convention. Short-term discount instruments are especially good examples: the same price discount can be annualized using face value or purchase price, a 360-day or 365-day basis, simple or compounded logic, and different market naming conventions.
The mathematics is not difficult once the conventions are separated. The danger comes from comparing numbers that look like percentages but do not use the same denominator or time basis.
What this page owns — and what it does not
This page owns the conversion problem:
(face value, purchase price, settlement date, maturity date, quotation basis) → annualized short-term yield measure.
It does not replace bond yield-to-maturity inversion, which solves a multi-cash-flow present-value equation; the day-count engine, which owns the exact accrual fraction rules; or yield-curve construction, which fits rates across many maturities.
This is public fixed-income mathematics, not a recommendation to buy Treasury bills or any other security.
The economic object is simpler than the quotation
A conventional Treasury bill is a discount security. It does not pay periodic coupons. The investor pays a price P today and receives face value F at maturity.
The dollar gain at maturity is:
Gain = F − P.
The holding-period return on money actually invested is:
HPR = (F − P) / P.
Everything that follows is a way of annualizing or re-expressing that basic economic relationship.
Current U.S. Treasury bill mechanics
TreasuryDirect currently describes Treasury bills as securities with maturities of one year or less that are sold at face value or at a discount and redeem at face value. As of 2026, regular bill tenors include 4-, 6-, 8-, 13-, 17-, 26- and 52-week bills, with cash-management bills issued when needed.
TreasuryDirect also publishes the auction pricing formula for bills:
P = F × [1 − d × t / 360]
where d is the annualized discount rate and t is the number of days to maturity under the applicable Treasury bill convention.
This equation immediately reveals an unusual feature: the discount rate uses face value as its price denominator, not the amount actually invested.
Bank discount yield
Let D = F − P. A standard bank-discount-style quotation is:
BDY = (D / F) × (360 / t).
Two choices define the measure:
- the discount is divided by face value;
- the fraction is annualized on a 360-day basis.
That is why the quotation is useful as a market convention but should not be mistaken for the investor’s simple annualized return on cash invested.
A numerical example
Suppose:
- face value F = 100;
- purchase price P = 97.50;
- days to maturity t = 90.
The discount is 2.50.
Bank discount yield:
BDY = (2.50 / 100) × (360 / 90) = 10.00%.
But the investor actually invested 97.50, not 100. The 90-day holding-period return is:
HPR = 2.50 / 97.50 ≈ 2.5641%.
Annualizing that same return on different conventions produces different percentages.
Money-market or add-on style yield
A common simple money-market transformation uses purchase price in the denominator while retaining a 360-day basis:
MMY = [(F − P) / P] × (360 / t).
For the example:
MMY ≈ (2.50 / 97.50) × 4 = 10.2564%.
The instrument has not changed. Only the quotation denominator changed from face value to invested price.
Bond-equivalent or investment-style simple annualization
A simple 365-day investment-rate style transformation is:
Investment Rate ≈ [(F − P) / P] × (365 / t).
Using the same 90-day example:
≈ 2.5641% × 365/90 ≈ 10.3996%.
Again, there is no new cash flow. The apparent yield rose because the denominator is purchase price and the annualization year is longer.
Actual Treasury auction reporting should follow the Treasury’s published official formulas rather than a generic textbook label. A production algorithm therefore stores the exact quotation-definition identifier, not just a field called yield.
Why “bond equivalent yield” can be a naming trap
Different textbooks, data vendors and markets can use names such as “bond equivalent yield,” “investment rate,” “coupon-equivalent yield” or “money-market yield” with subtly different formulas.
The safe software design is not:
if label contains “bond equivalent,” use remembered formula.
It is:
quote_definition = explicit formula + day basis + compounding rule + source.
The name is metadata. The formula is the computation.
Round-trip conversion is a powerful invariant
Given face value, days and a discount rate, TreasuryDirect’s pricing formula gives:
P = F[1 − d t/360].
If the algorithm then recomputes:
d* = [(F − P)/F] × 360/t,
it should recover the original d within the expected rounding tolerance.
This round trip is one of the easiest ways to catch a denominator or basis error.
Quote-space versus cash-space
It is useful to separate two layers:
- cash-space: price paid, maturity cash received, actual gain;
- quote-space: the convention used to annualize and report that gain.
A quote conversion should never alter the cash-space object.
If converting from discount yield to investment rate causes the stored purchase price to change, the implementation has confused economic state with display convention.
Why bank discount yield understates a purchase-price-based annualized return when the bill trades below par
For a discount bill, P < F. Therefore:
(F − P)/F < (F − P)/P.
So, holding the same day basis constant, the face-value-denominator yield is lower than the purchase-price-denominator yield.
This is a mathematical consequence, not a market opinion.
Negative-yield and premium-price edge cases
A discount security is usually introduced with P < F. But market conditions can create prices at or above par.
If P = F, the discount and simple holding-period return are zero.
If P > F, then F − P is negative and simple yield measures become negative.
An implementation that clamps every bill yield to zero because “bills are discount securities” is imposing a historical intuition on the algebra.
52-week bills expose annualization assumptions
Short-term conventions are easiest to ignore when maturity is only a few weeks. Near one year, basis choices become especially visible.
A 365-day annualization applied to a period near one year behaves differently from a 360-day money-market basis. Leap years and exact settlement-to-maturity day counts can also matter depending on the official formula.
This is why the day-count engine and quotation engine should be separate modules with explicit interfaces.
Simple annualization is not compounding
Multiplying a holding-period return by 365/t creates a simple annualized rate. It does not assume that the investment can be repeatedly reinvested at the same rate throughout the year.
An effective annual yield would instead use a compounding transformation such as:
EAY = (1 + HPR)365/t − 1
under a chosen 365-day convention.
That number can be higher than the simple annualized return because it assumes repeated compounding at the same per-period return. For a one-off Treasury bill, that is a hypothetical comparison metric, not a guaranteed realized annual return.
The comparison problem: normalize before ranking
Suppose Instrument A quotes 4.90% on a bank discount basis and Instrument B quotes 4.88% on a purchase-price 365-day basis.
It is invalid to rank them simply by comparing 4.90 and 4.88.
The algorithm should convert both instruments into one common economic comparison basis using:
- their actual prices;
- settlement and maturity dates;
- cash flows;
- a common annualization or discount-factor convention.
This is a miniature version of the broader rule in fixed income: normalize representation before comparing value.
Discount factors are a cleaner common language
A Treasury bill price naturally implies a discount factor:
DF(0,T) = P / F
for a zero-coupon instrument with no intermediate cash flows, before considering transaction-specific settlement details.
Different yield conventions can all be derived from this same underlying price ratio.
This is why yield-curve algorithms often use discount factors internally even when market instruments arrive in different quotation formats.
Inputs and outputs
A robust short-term yield conversion engine can require:
- instrument type;
- face or redemption value;
- purchase/settlement price;
- settlement date;
- maturity date;
- official day-count basis;
- source quotation convention;
- target quotation convention;
- rounding precision;
- currency and market metadata.
Outputs should include not merely “yield” but a labelled set such as:
- holding-period return;
- bank discount yield;
- money-market/add-on yield;
- investment-rate or 365-day simple annualization;
- effective annualized comparison rate where requested;
- discount factor;
- calculation basis and exact source formula.
Evidence polarity: what supports confidence?
Evidence for a correct conversion includes exact reproduction of TreasuryDirect price examples, round-trip recovery of the source quote, explicit distinction between face-value and purchase-price denominators, correct settlement-to-maturity day counts, stable decimal precision and agreement with an independent implementation using the same official formula.
Evidence against confidence includes a field simply called “yield” with no convention, mixing 360 and 365 annualization, treating a discount quote as a purchase-price return, inconsistent prices after a quote conversion, unexplained one-basis-point differences near maturity, or an algorithm that cannot reproduce the official auction price from the official discount rate.
Counterexample: the highest quoted percentage need not represent the highest economic return
Two instruments can use different quotation rules. The larger displayed percentage may simply be generated by a more aggressive annualization basis or a different denominator.
Economic comparison requires normalization, not label comparison.
Counterexample: a perfectly calculated discount yield can be a poor cross-asset comparison
Bank discount yield is designed for discount-market quotation. Comparing it directly with a coupon bond’s YTM, a deposit APY or an OIS zero rate mixes conventions with different reinvestment assumptions and cash-flow structures.
The calculation can be correct and the comparison can still be wrong.
Counterexample: rounding the price too early changes the reconstructed yield
TreasuryDirect notes that auction calculations can be carried to greater precision before the investor’s dollar payment is rounded to cents.
If a pricing engine rounds price to two decimals per 100 before converting back into yield, the reconstructed rate can differ from the official quote.
Therefore the algorithm should preserve internal precision and round only at the specified reporting or settlement boundary.
Diagnostics: how to test the engine
- TreasuryDirect example replay: reproduce an official published bill pricing example.
- round-trip test: discount rate → price → discount rate.
- denominator test: verify bank-discount and purchase-price yields diverge exactly as expected below par.
- zero-return test: set price equal to face and require every simple yield to return zero.
- negative-yield test: set price above face and ensure the engine does not clamp the result.
- basis test: compare 360- and 365-day annualizations on the same bill.
- precision test: calculate with high internal precision, then vary display rounding.
- date test: shift settlement by one day and verify the annualized quote moves consistently.
- normalization test: convert two differently quoted discount instruments to one common basis before ranking.
- label test: reject ambiguous input such as “yield = 5%” when the quotation convention is missing.
What would falsify confidence?
Confidence should be withdrawn if the engine cannot reconstruct official Treasury bill prices; if yield changes without any change to price, dates or target convention; if a quote conversion alters the underlying cash flows; if ambiguous yield labels are accepted silently; or if different modules disagree about which denominator or day basis a named quotation uses.
Alternatives and limits
For broad cross-market work, discount factors or continuously compounded zero rates can be cleaner internal representations. For investor-facing statements, official market quotation conventions remain important because they match trading and auction practice. For realized return analysis, actual purchase price, maturity proceeds, fees and reinvestment outcomes matter more than an annualized quote.
No conversion formula can remove liquidity, tax, settlement or reinvestment differences between instruments. Quote normalization solves a representation problem, not every economic comparison problem.
How this connects to the surrounding knowledge estate
Day-count algorithms supply the time basis. Money arithmetic preserves exact prices and rounding. Yield-curve construction consumes discount instruments alongside coupon instruments. YTM inversion handles the different problem of a coupon bond with multiple future cash flows.
Verification and update triggers
Preserve the quotation-definition source, Treasury formula version, date-count implementation, precision policy, price rounding policy and instrument metadata. Revalidate after Treasury auction-rule changes, market-data vendor convention changes, financial-library upgrades, new bill tenors, negative-yield handling changes or any persistent discrepancy between internal calculations and official auction results.
Primary and high-quality references
- U.S. Treasury, TreasuryDirect, Understanding Pricing and Interest Rates, including the current Treasury-bill discount-price formula.
- U.S. Treasury, TreasuryDirect, Treasury Bills In Depth, describing bill maturities, discount issuance and auction mechanics.
- U.S. Treasury, TreasuryDirect, General Auction Timing, for the current bill-tenor auction pattern.
- Federal Reserve Board, H.15 Selected Interest Rates, for current Treasury-bill secondary-market rate publication context.
Educational boundary: This article explains quotation mathematics for short-term discount securities. It does not recommend any Treasury security or provide personalized financial advice.