Every Treasury bill result comes with two rates, and they are not two different deals. The bank discount rate is the convention Treasury uses to bid and price bills; the investment rate restates the same dollars as a yield on the money you actually pay. Below are both formulas side by side, Treasury's separate formula for bills longer than six months, worked examples from real 4-, 13- and 52-week auctions, and how to turn either figure into something you can set against a savings APY. The latest rates for every term are on T-bill rates today.
What is the difference between a T-bill's discount rate and investment rate?
The discount rate measures a bill's discount against face value over a 360-day year, while the investment rate measures it against the price paid over a 365-day year.
| Feature | Bank discount rate | Investment rate |
|---|---|---|
| Other names | Discount yield, discount basis | Coupon equivalent, bond-equivalent yield |
| Measured against | Face value, the amount repaid | Purchase price, the amount paid |
| Year length | 360 days | 365 days, or 366 when the year after issue includes February 29 |
| Compounding | None | None up to six months; one half-year step for longer bills |
| Used for | Bidding at auction and setting the price | Comparing a bill with notes, bonds, CDs and savings |
| Where Treasury shows it | High discount rate in auction results; bank discount column in the daily bill table | Investment rate in auction results; coupon equivalent column in the daily bill table |
Treasury has sold bills only on a discount-rate basis since April 18, 1983. Bidders name a discount rate, the auction's high rate sets the price, and Treasury then reports the investment rate so the result can be compared with securities that pay interest. Neither rate changes what you receive: the price and the face value are fixed once the auction closes. The basics of buying at a discount are covered in what is a Treasury bill.
How do you calculate a T-bill's price from the discount rate?
Multiply the discount rate by the days to maturity divided by 360, subtract the result from 1, and multiply by the face value.
Here d is the discount rate as a decimal and r is the number of days from issue to maturity. Treasury's regulations round the price per $100 to six decimal places, which is why auction results show prices such as 99.688889. Run backward, the formula gives the discount rate from a price: d = (100 − P) ÷ 100 × 360 ÷ r. TreasuryDirect's own pricing page uses the same 360-day formula.
Worked example: A 4-week bill auctioned October 15, 2026
Treasury sold 28-day bills, issued October 20 and maturing November 17, 2026, at a high discount rate of 4.000%.
- Price per $100: 100 × (1 − 0.04 × 28 ÷ 360) = 99.688889
- Price of $10,000 of face value: $9,968.89
- Interest at maturity: $10,000 − $9,968.89 = $31.11
- Investment rate: 0.311111 ÷ 99.688889 × 365 ÷ 28 = 4.068%
How is the investment rate calculated for bills of six months or less?
Divide the dollar discount by the price paid, then annualize it over 365 days, or 366 when the year after issue contains February 29.
P is the price per $100, r the days to maturity and y the days in the year that starts on the issue date. Treasury's rule is specific: y is 366 if the period from the issue date to the same date a year later includes February 29, and 365 otherwise. Its own illustration uses 26-week bills: one issued February 28, 2019 used 365, while one issued March 1, 2019 used 366, because the following year included February 29, 2020. The formula covers every bill of not more than a half-year, which means the 4-, 6-, 8-, 13-, 17- and 26-week terms.
Worked example: A 13-week bill auctioned October 13, 2026
The bill was issued October 15, 2026 and matures January 14, 2027, 91 days later, at a high discount rate of 4.085%.
- Price per $100: 100 × (1 − 0.04085 × 91 ÷ 360) = 98.967403
- Discount per $100: 100 − 98.967403 = 1.032597
- Investment rate: 1.032597 ÷ 98.967403 × 365 ÷ 91 = 4.185%, the rate Treasury published
- On $10,000 of face value you pay $9,896.74 and receive $10,000, earning $103.26
What formula does Treasury use for bills longer than six months?
For bills longer than a half-year, Treasury solves a quadratic equation that assumes the interest compounds once at the half-year, like a note paying semiannual coupons.
The simple formula would overstate a 52-week bill's yield next to a note's, because a note investor collects a coupon after six months and can reinvest it. Treasury's appendix therefore sets the price equal to the present value of the face amount under half-year compounding:
Rearranged, that is a quadratic in the investment rate i, solved with the standard formula:
Worked example: A 52-week bill auctioned September 29, 2026
The bill was issued October 1, 2026 and matures September 30, 2027, 364 days later, at a high discount rate of 4.400%. The year after issue has no February 29, so y = 365.
- Price per $100: 100 × (1 − 0.044 × 364 ÷ 360) = 95.551111
- a = 364 ÷ 730 − 0.25 = 0.248630; b = 364 ÷ 365 = 0.997260; c = (95.551111 − 100) ÷ 95.551111 = −0.046560
- b² − 4ac = 0.994528 + 0.046305 = 1.040833, whose square root is 1.020212
- i = (−0.997260 + 1.020212) ÷ (2 × 0.248630) = 0.022952 ÷ 0.497260 = 4.616%, matching Treasury's published investment rate
The simple formula would have given 4.669%, 0.053 point higher. Treasury's appendix runs the same steps on a 52-week bill at a 7.65% discount rate, priced at 92.265, and arrives at an investment rate of 8.237%.
What do the formulas give for 4-, 13- and 52-week bills?
In three fall 2026 auctions, the investment rate ran 0.068 point above the discount rate on a 4-week bill, 0.100 on a 13-week and 0.216 on a 52-week.
| Bill | Auction | Days | Discount rate | Price per $100 | Cost of $10,000 | Interest | Investment rate | Gap |
|---|---|---|---|---|---|---|---|---|
| 4-week | Oct 15, 2026 | 28 | 4.000% | 99.688889 | $9,968.89 | $31.11 | 4.068% | 0.068 |
| 13-week | Oct 13, 2026 | 91 | 4.085% | 98.967403 | $9,896.74 | $103.26 | 4.185% | 0.100 |
| 52-week | Sep 29, 2026 | 364 | 4.400% | 95.551111 | $9,555.11 | $444.89 | 4.616% | 0.216 |
Those auctions cleared at different rates, so the gaps mix two effects. Holding the discount rate at 4.00% for every term isolates the effect of the term alone:
| Bill | Days | Price per $100 | Interest on $10,000 | Investment rate | Gap over 4.00% |
|---|---|---|---|---|---|
| 4-week | 28 | 99.688889 | $31.11 | 4.068% | 0.068 |
| 6-week | 42 | 99.533333 | $46.67 | 4.075% | 0.075 |
| 8-week | 56 | 99.377778 | $62.22 | 4.081% | 0.081 |
| 13-week | 91 | 98.988889 | $101.11 | 4.097% | 0.097 |
| 17-week | 119 | 98.677778 | $132.22 | 4.110% | 0.110 |
| 26-week | 182 | 97.977778 | $202.22 | 4.139% | 0.139 |
| 52-week | 364 | 95.955556 | $404.44 | 4.183% | 0.183 |
The gap widens with term because a longer bill's discount is a bigger share of its face value, so the price it is measured against is lower. The 52-week row would read 4.226% under the simple formula; the half-year compounding step brings it to 4.183%. Auction days, rollover limits and liquidity for each term are compared in T-bill terms compared.
Why is the investment rate always higher than the discount rate?
Two adjustments both push it up: it divides by the price, which is below face value, and it uses a 365-day year instead of 360 days.
For bills of six months or less, the two rates are tied by a simple identity:
The 365 ÷ 360 factor alone lifts the rate by about 1.4%, so a 4.000% discount rate becomes 4.056% before any price effect. The 100 ÷ P factor then grows with the discount: 1.003 for a 4-week bill at 4.00%, where P is 99.69, and 1.021 for a 26-week bill, where P is 97.98. The higher the rate and the longer the term, the wider the gap; it closes only at a zero rate, where the price equals face value. For 52-week bills the compounding step trims the gap but leaves the investment rate above the discount rate.
A third measure sits between the two. The money-market yield divides by the price, like the investment rate, but keeps the 360-day year; Treasury uses this actual/360 conversion of the 13-week high rate as the index for its floating rate notes. For the October 13, 2026 13-week bill it works out to 4.128%, between the 4.085% discount rate and the 4.185% investment rate.
How do you compare a T-bill rate with an APY?
Start from the bill's price, compound one term's return over a year to get an APY, then allow for state income tax, which bill interest does not pay.
Bank deposit APYs follow Regulation DD, the Truth in Savings rule, whose formula is APY = 100 × [(1 + interest ÷ principal)^(365 ÷ days in term) − 1]. Applied to a bill, interest is face value minus price and principal is the price:
Using the discount rate in place of an APY understates the bill by the whole gap shown above. Using the investment rate as if it were an APY still understates it slightly, because the investment rate leaves out compounding for bills of six months or less.
Worked example: The 13-week bill against a 4.20% APY savings account
Take the October 13, 2026 13-week bill, priced at 98.967403 for 91 days, and a savings account paying 4.20% APY.
- Bill APY: (100 ÷ 98.967403)^(365 ÷ 91) − 1 = 4.251%
- On its 4.085% discount rate the bill looks worse than the account; as an APY it is 0.05 point ahead.
- In a state with a 6% income tax, the account keeps 4.20% × 0.94 = 3.95% after state tax, while the bill keeps all 4.251%, since its interest is exempt from state and local income tax.
- The account would need 4.251% ÷ 0.94 = 4.52% APY to match.
The bill's APY assumes it could be rolled at the same rate every 91 days, which is never certain. The compounding idea is explained in APY vs APR, and the tax side in are T-bills taxable.
What mistakes do people make with T-bill rates?
The most common mistake is comparing a bill's discount rate with a bank APY, which understates the bill by roughly 0.15 to 0.27 point at late-2026 rates.
- Mixing conventions. A discount rate, an investment rate, a money-market yield and an APY can all describe one bill. Broker screens may show yet another, such as a yield to maturity on the ask price, so check the label before comparing.
- Using 365 days in the price formula. Bills are priced on 360 days. For the 13-week bill above, a 365-day price would be $9,898.15 for $10,000 of face value instead of $9,896.74.
- Applying the simple formula to a 52-week bill. It overstates the investment rate, by 0.053 point on the September 29, 2026 auction.
- Assuming exact weeks. The 6-week bill auctioned October 13, 2026 runs 43 days, not 42, because it matures on Friday, November 27, after Thanksgiving. Use the issue and maturity dates.
- Treating a daily quote as your rate. Treasury's daily table shows secondary-market bid quotes, and its 3-month par yield on the Treasury yield curve is a third, constant-maturity measure. An auction buyer gets that auction's high rate, as explained in how Treasury auctions work.
To skip the arithmetic, the Treasury bill calculator takes a face value, discount rate and term and returns the price, the interest and the investment yield using the formulas above.
What are the risks of relying on a T-bill's stated rate?
The stated rate is locked in only if you hold to maturity; selling early, reinvesting at a new rate, inflation and taxes all change what you actually keep.
- Price risk before maturity. If you sell early, the price depends on the discount rate the market then demands. The 52-week bill bought at 95.551111 would be worth 95.916667 per $100 with 300 days left at a 4.90% discount rate, against 96.333333 had its rate stayed at 4.40%: $41.67 less per $10,000, though still $36.56 above the purchase price. Selling Treasuries before maturity covers how a sale works.
- Reinvestment risk. A bill's APY assumes rolling at the same rate, which holds only if rates stand still.
- Inflation risk. Neither rate allows for inflation; a 4.185% investment rate is a nominal return.
- Tax. Bill interest is subject to federal income tax, so the after-tax return is below either rate, though it is exempt from state and local income taxes.
Bills are backed by the full faith and credit of the US government, so these risks concern timing and purchasing power rather than repayment; are Treasuries risk-free? puts them in context, and the US Treasuries hub shows where bills fit among Treasury's other securities.