Reader question: A Treasury bill may be quoted at a 4.50% discount rate, yet the investor’s annualised return can be higher than 4.50%. How can both numbers be correct?
Because a Treasury bill’s bank discount rate and its investment rate (coupon-equivalent yield) are different mathematical conventions. The discount rate measures the discount from face value, annualised on a 360-day year. The investment rate measures return relative to the actual purchase price and uses a 365- or 366-day year under Treasury’s prescribed formulas.
So the conversion chain is:
discount rate + days to maturity → price → discount amount → investment-rate convention.
The formulas are deterministic. Most errors come from using the wrong denominator, wrong day basis, wrong maturity branch or rounding too early.
What this page owns — and what it does not
This page owns short-term Treasury-bill quote conversion. It does not replace general day-count algorithms, coupon-bond yield-to-maturity inversion, or Treasury auction allocation.
This is fixed-income arithmetic education, not a recommendation to purchase Treasury bills or any other security.
Treasury bills are zero-coupon securities
U.S. Treasury bills mature in one year or less. TreasuryDirect currently lists regular terms of 4, 6, 8, 13, 17, 26 and 52 weeks, with cash-management bills issued for variable terms.
A bill generally pays no periodic coupon. Instead:
Interest earned = face value − purchase price.
If a bill costs 98.50 per 100 of face value and pays 100 at maturity, the investor earns 1.50 before considering reinvestment or transaction effects.
The bank discount rate uses face value as the denominator
For price P per 100 face value, discount rate d and r days to maturity, Treasury’s price formula is:
P = 100 × [1 − d × r / 360].
Rearranging:
d = [(100 − P) / 100] × (360 / r).
Notice the denominator:
100 face value.
The discount rate therefore annualises the discount relative to what Treasury will repay, not relative to the cash the investor actually pays.
A 90-day example
Assume:
- face value = 100;
- days to maturity = 90;
- discount rate = 4.50%.
Then:
P = 100 × [1 − 0.045 × 90 / 360]
P = 100 × (1 − 0.01125) = 98.875.
The discount amount is:
100 − 98.875 = 1.125.
Why the investor’s simple return is higher than the discount rate
The investor paid 98.875, not 100.
The holding-period return is therefore:
HPR = (100 − P) / P.
For the example:
HPR = 1.125 / 98.875 ≈ 1.1378%.
The bank discount calculation instead measured:
1.125 / 100 = 1.125%.
The difference looks small over 90 days, but it is systematic.
The 360-day quote creates another difference
The discount rate annualises on a 360-day basis:
annualisation factor = 360 / r.
An investment-rate convention uses the actual year length prescribed by Treasury—normally 365, but 366 when the relevant one-year period contains February 29.
So the investment-rate conversion changes both:
- the denominator from face value to purchase price;
- the annualisation basis from 360 to 365/366.
Short-bill investment-rate intuition
For bills of no more than one half-year to maturity, the coupon-equivalent investment-rate formula can be understood as annualising the return on purchase price:
i ≈ [(100 − P) / P] × (y / r),
where y is 365 or 366 under Treasury’s leap-year rule.
Using the 90-day example and y = 365:
i ≈ (1.125 / 98.875) × (365 / 90) ≈ 4.614%.
So a 4.50% bank discount quote corresponds to a higher simple annualised investment return because the investor earns the discount on less than face value and the year basis is longer.
Why 52-week bills need a different formula branch
A common implementation error is to apply the short-bill simple annualisation formula to every Treasury bill.
Treasury’s Appendix B separates bills of no more than one half-year from longer bills. For longer bills, the official investment rate (coupon-equivalent yield) formula is designed to make the bill comparable with coupon-bearing securities that use semiannual yield conventions.
The longer-bill conversion therefore uses a different equation rather than simply multiplying holding-period return by 365/r.
A production engine should select the correct maturity branch from the current Treasury regulation instead of inventing one universal formula.
Coupon-equivalent does not mean the bill pays a coupon
The phrase coupon-equivalent yield is a comparison convention.
The Treasury bill remains a zero-coupon security: it pays face value at maturity rather than semiannual coupons.
The conversion merely expresses its return in a form intended to be comparable with coupon-bearing Treasury notes and bonds.
Leap years are an algorithmic input
Treasury’s Appendix B specifies that the year length y is normally 365, but can be 366 if the one-year period following the issue date contains February 29.
This is subtler than checking whether the calendar year number is divisible by four.
For example, an issue late in one year can have a one-year comparison period that crosses February 29 even though the issue year itself is not a leap year.
The date engine should test the actual one-year interval.
Price per 100 is the computational anchor
Treasury’s auction regulations calculate price on a per 100 face-value basis.
Once price per 100 is known, the dollar purchase price for par amount F is:
Dollar Purchase Price = (F / 100) × P.
For 10,000 par at P = 98.875:
Purchase Price = 100 × 98.875 = 9,887.50.
Rounding belongs at specified boundaries
Treasury’s regulations prescribe six-decimal price calculations per 100 and normal rounding procedures at defined stages.
This matters because bill prices are close to par and large par amounts magnify tiny price differences.
A system that rounds discount rate, price and yield after every arithmetic step can drift away from the official settlement amount even if each individual approximation appears harmless.
Discount amount and interest are the same cash difference from two viewpoints
For a bill held to maturity:
Discount Amount = Face Value − Purchase Price.
TreasuryDirect describes this same difference as the bill’s “interest”.
The cash amount is identical. The yield convention determines how that cash difference is annualised and reported.
Discount yield can become misleading for extreme prices
Because the bank discount formula divides by face value, it is a quote convention rather than a compounding return model.
If prices move far from par, the gap between discount yield and return on invested cash widens.
This is one reason analysts convert different money-market instruments onto a common yield basis before comparing them.
Negative rates and prices above par
A negative bill discount rate implies a price above face value under the price equation:
P = 100 × [1 − d r/360].
If d is negative, the bracket can exceed 1.
That means an investor can pay more than face value and receive less at maturity in nominal terms. The formula itself handles negative rates; systems should not hard-code “bill price must be below 100” as an invariant.
Money-market basis comparisons require normalization
A commercial paper quote, Treasury-bill discount rate, certificate-of-deposit money-market yield and bond-equivalent yield can all use different combinations of:
- price versus face-value denominator;
- 360 versus 365/366 day basis;
- simple versus compounded annualisation;
- actual versus standardized days.
Comparing the raw percentages directly can therefore rank investments incorrectly.
Inputs and outputs
A Treasury-bill yield engine can require:
- issue date;
- settlement date;
- maturity date;
- days remaining to maturity;
- par amount;
- quoted discount rate or price;
- Treasury regulation/formula version;
- leap-year determination;
- rounding convention.
Outputs can include price per 100, dollar purchase price, discount amount, bank discount rate, holding-period return, investment rate/coupon-equivalent yield and diagnostic flags.
Evidence polarity: what supports confidence?
Evidence for a correct implementation includes exact reproduction of TreasuryDirect examples, agreement with Appendix B price formulas, correct day counts, six-decimal price handling, correct 365/366 selection and the proper short-versus-long bill investment-rate branch.
Evidence against confidence includes using purchase price in the discount-rate denominator, using face value in the investment-return denominator, treating every year as 365 days, applying the <=half-year formula to a 52-week bill, or producing different prices depending on whether the calculation starts from rate or price.
Counterexample: 4.50% discount rate is not a 4.50% return on cash invested
The 90-day example produced a 4.50% discount quote but about 4.61% simple annualised investment return.
The difference is structural, not a data error.
Counterexample: same price, different annualised yield
A 1% holding-period return earned over 30 days and the same 1% earned over 180 days have the same cash return but very different annualised rates.
Days to maturity are therefore part of the yield definition, not merely descriptive metadata.
Counterexample: same discount rate, different price
A 4% discount rate on a 30-day bill produces a smaller discount amount than the same 4% quote on a 180-day bill.
The quote is annualised; price still depends on remaining time.
Counterexample: same calendar year, different y
Two bills issued in the same calendar year can have different 365/366 treatment if one one-year comparison interval contains February 29 and the other does not.
A simple “if issue year is leap year” test can therefore fail.
Weak links in implementation
Basis confusion. Discount and investment yields are stored in one field without a quote type.
Denominator error. Face value and purchase price are interchanged.
Day-count error. Settlement-to-maturity days are off by one.
Leap-year shortcut. Calendar-year logic replaces Treasury’s one-year-period test.
Maturity-branch error. 52-week bills use the short-bill formula.
Premature rounding. Intermediate values are rounded before the official boundary.
Percent/decimal error. 4.50 is treated as 4.50 rather than 0.045.
Negative-rate rejection. Valid above-par prices are blocked by an outdated assumption.
Diagnostics: how to test the engine
- Treasury example replay: reproduce TreasuryDirect’s published bill price example.
- round trip: rate → price → rate should recover the input within required rounding.
- zero-rate test: d = 0 must produce P = 100.
- negative-rate test: d < 0 should be able to produce P > 100.
- maturity sensitivity: for a positive discount rate, longer time to maturity should lower price, all else equal.
- denominator test: investment-rate result should exceed discount rate for an ordinary positive-rate bill because P < 100 and year basis differs.
- leap test: run issue dates immediately before and after the one-year interval begins to include February 29.
- half-year boundary test: verify the correct investment-rate formula branch around Treasury’s half-year cutoff.
- large-par test: verify six-decimal price rounding scales correctly to dollar purchase price.
- independent source test: compare calculated auction price with the official Treasury result for a real bill.
What would falsify confidence?
Confidence should be withdrawn if the engine cannot reproduce Appendix B examples; if rate-to-price and price-to-rate round trips diverge materially; if a 0% rate does not produce par; if the same bill receives different yields solely because of server timezone; or if the short and long bill formulas are not distinguished.
Alternatives and limits
For portfolio return comparison, analysts can convert bills to effective annual yields or a common money-market basis. For discounting, a zero-coupon discount factor may be more useful than a quoted bill yield. For market valuation, continuously compounded zero rates can be derived from bill prices and integrated into a yield curve.
No one yield convention is “the true yield” outside its definition. The correct convention depends on the question being asked.
How this connects to the surrounding knowledge estate
The date count belongs to day-count algorithms. The auction’s accepted discount rate comes from the Treasury auction algorithm. Bill prices can provide short-end inputs to yield-curve bootstrapping. Exact cents and scaling depend on banking money arithmetic.
Verification and update triggers
Preserve the Treasury formula version, issue/settlement/maturity dates, day count, quote type, price precision and leap-year branch. Revalidate after Treasury auction-rule changes, bill-term changes, rounding changes or any disagreement with an official auction settlement price.
Primary and high-quality references
- U.S. Treasury, TreasuryDirect, Understanding Pricing and Interest Rates.
- U.S. Treasury, TreasuryDirect, Treasury Bills.
- 31 CFR Part 356 Appendix B, Formulas and Tables, Section VI: Treasury-bill purchase price, discount rate and investment rate.
- 31 CFR §356.2, Definitions, including bank discount rate and FRN index-rate conventions.
- 31 CFR §356.20, Treasury auction awards and price calculations.
Educational boundary: This article explains Treasury-bill quote conversion mathematics. It does not recommend any security or yield strategy and does not provide personalized financial advice.