Why do a Treasury bill’s discount rate and investment yield differ?

Identify the convention before comparing percentages

TreasuryDirect gives the bill pricing identity as price = face value × [1 − discount rate × days / 360]. The Treasury’s rate notes distinguish a bank discount rate based on par from a coupon-equivalent or investment yield based on purchase price and a 365- or 366-day year.

The current offering-circular formulas separate bills of not more than one half-year from longer bills. This worksheet deliberately uses the short branch and stipulates a 365-day year. It does not apply the simple short-bill conversion to a 52-week bill, and it does not present a current auction quote.

Give one fictional bill three precisely labeled measures

Assume a bill costs $9,875, redeems for $10,000 after 91 days, and has no fees or taxes in the exercise. The cash difference is $125. Its unannualized holding-period return is 125 / 9,875 = 1.265823%. That is the gain relative to the amount paid over this particular 91-day interval.

The bank discount rate is (125 / 10,000) × (360 / 91) = 4.945055% per quoted annual basis. The short-bill investment rate is (125 / 9,875) × (365 / 91) = 5.077201% on the stipulated annual basis. The $125 payoff difference has not changed; the measurement convention has.

Reverse the calculation rather than guessing which rate is right

Putting the unrounded discount rate back into TreasuryDirect’s price formula gives 10,000 × [1 − (125 / 10,000 × 360 / 91) × 91 / 360] = $9,875. Reversing the investment-rate calculation gives a 91-day gain of 9,875 × [(125 / 9,875 × 365 / 91) × 91 / 365] = $125.

Using the rounded six-decimal percentages can produce tiny reconciliation differences. Preserve precision internally, then round the presentation. A quotation shown to fewer decimals may also not reproduce an invoice exactly, particularly when the purchase amount is small.

Do not turn annualization into a reinvestment promise

Neither annualized number says that the same 91-day bill can be bought repeatedly at an unchanged rate. The worksheet contains one purchase and one redemption. Compounding a sequence of future bills would require separate future purchase prices and timing assumptions.

When comparing reported yields, retain maturity, purchase date, denominator, day basis and instrument type. A short-bill investment yield is not interchangeable with an effective annual deposit yield or a long bond’s realized return. Selling before maturity introduces a different sale-price cash flow; taxes and transaction charges would change the investor’s net result.

Method, sources, and limits

All prices, amounts and rates here are invented teaching inputs, recomputed on 27 October 2026. Current primary sources were opened for conventions, not to select a security or infer future rates. AI assisted the draft; a separate AI verification pass is required before publication and is not human financial-expert approval.