How is a T-bill priced if it never pays a coupon?
A Treasury bill has no coupon, no reinvestment schedule, and no complexity — which is why it is the cleanest possible instrument for learning how short-term yield conventions work, and why almost everyone gets those conventions wrong the first time.
The instrument
A T-bill is short-dated government debt sold at a discount to face value and redeemed at face. Your entire return is the pull to par. US tenors are 4, 6, 8, 13, 17, 26 and 52 weeks, plus irregular cash management bills. The 17-week became a regular benchmark in October 2022 and the 6-week in 2025, and the Treasury publishes rates for all seven every business day.
Two yields, two conventions, and they never agree
Discount rate — the quoted convention:
d = (F − P) / F × 360 / t
Note what it does: it divides the gain by face value, and it uses a 360-day year.
Coupon-equivalent yield (also called the investment rate):
y = (F − P) / P × 365 / t
This one divides by the price, which is what you actually invested, and uses 365 days. It uses 366 when the year following the bill's issue date contains 29 February: the Treasury's own test, not a test on the bill's own term.
And it has a limit. That simple form is the Treasury's published investment rate only for bills with 182 days or less to maturity — everything up to and including the 26-week. Beyond half a year the return has to account for semi-annual compounding, so Treasury solves a quadratic instead (31 CFR Part 356, Appendix B). On the 52-week bill the simple formula runs about 3 basis points rich against the published figure. Do not learn the quadratic; learn where the simple one stops.
The worked example
A 91-day bill, face 100, priced at 98.75. The gain is 1.25.
Discount rate:
(1.25 / 100) × (360 / 91) = 0.0125 × 3.9560 = 0.04945 → 4.945%
Coupon-equivalent yield:
(1.25 / 98.75) × (365 / 91) = 0.012658 × 4.0110 = 0.05077 → 5.077%
About 13 basis points apart, on the same bill, on the same day. Neither is wrong. The discount rate is a quoting convention inherited from an era of paper and mental arithmetic; the coupon-equivalent yield is the one you compare with a coupon-paying bond, because it measures the return on the money you actually parted with.
Going the other way
From a quoted discount rate to a price:
P = F × (1 − d × t / 360)
At d = 4.945% and t = 91:
100 × (1 − 0.04945 × 91 / 360) = 100 × (1 − 0.0125) = 98.75
The arithmetic closes exactly, which is a useful check whenever you are unsure which convention a screen is showing you.
Why bills matter beyond their own market
Bills are the closest thing to risk-free cash with a maturity date. They are the benchmark short rate, the raw material of money market funds in the next lessons, and — as repo collateral — the plumbing of Unit 2.
Their supply and demand also carries information. When bills are scarce relative to demand for safe short assets, they trade rich and their yields fall below comparable overnight-indexed swap rates. When the government issues heavily to rebuild its cash balance, the reverse happens and bill yields cheapen. Both are descriptions of a supply-demand balance, not forecasts of anything.
In the data
The Treasury publishes every bill in both conventions side by side:
On 24 September 2026 the 4-week bill was quoted at a 3.86% discount rate and a 3.93% coupon-equivalent yield, and the 52-week at 4.27% and 4.47%. The discount rate is always the smaller of the two, because it is measured against face value while the coupon-equivalent is measured against the price actually paid, and the gap widens with maturity. Set a bill's discount rate beside a Treasury note's yield and the bill looks as if it pays less than it does: the coupon-equivalent is the comparable number.
Try it now
- Below are the latest published rates for the same three bills, with each bill's maturity date. Check that the pattern above still holds on the newest day: the discount rate the smaller of the two on every row, and the gap widest on the 52-week bill.
- Take the 13-week bill. Its days to maturity, t, are the maturity date minus the quote date: around 90, comfortably inside the 182-day limit. Measure the gap between its two published yields, then reproduce that gap from the formulas above (price it from the discount rate first). They should agree to a basis point or so. Try the same on the 52-week bill and you will miss by a few basis points: that is the limit showing up, not a mistake in your arithmetic.
- Compare the 4-week and 52-week discount rates on the same date and describe the shape in one neutral sentence — no inference about where either is heading.