Treasury-Bill Discount Rates and Investment Yields Use Different Bases

Why is a Treasury-bill discount rate not the investor's return?

Under the bank-discount convention, the discount is measured against face value and annualized on a 360-day basis. The investor's holding-period gain is measured against the price actually paid. These denominators differ whenever price differs from redemption. TreasuryDirect publishes the bill price formula using the discount rate. Its formula describes a quotation convention, not a promise that a short holding-period gain can be earned repeatedly over an entire year.

A bill normally pays no periodic coupon; the contractual maturity receipt is its face amount. Selling earlier introduces a different sale price and holding period. Start with those cash flows rather than inserting a bill's quoted rate into a coupon-bond model. A label such as yield is incomplete until the price base, year basis and any compounding treatment are identified. Do not rank quotations that answer different questions.

Write the three calculations separately

Let F be redemption face, P the purchase price and d the actual days remaining. The holding-period return is (F minus P) divided by P, assuming payment as promised and ignoring costs. The bank-discount rate is (F minus P) divided by F, multiplied by 360 divided by d. A simple 365-day investment yield instead uses (F minus P) divided by P, multiplied by 365 divided by d.

The Department of Finance Canada's technical guide explicitly describes a purchase-price-based 365-day simple bill yield. It provides a primary example of a different convention, not a statement that every U.S. auction investment-rate field uses that formula for every tenor. Longer bill maturities and particular market definitions can require additional conventions. Keep the calculation named by its actual method instead of relabeling an approximation as an official auction result.

Reconcile one synthetic ninety-day bill

Assume an invented USD 100,000 face amount, USD 99,000 purchase price and 90 days to redemption. The discount is USD 1,000. Holding-period return is 1,000 divided by 99,000, or 1.010101% rounded. The bank-discount rate is 1,000 divided by 100,000 times 360 divided by 90, or 4%. The simple 365-day investment yield is 4.096521% rounded. All three describe the same stipulated cash flows.

An independent reverse check inserts 4% into the discount-price formula: 100,000 times (1 minus 0.04 times 90 divided by 360) returns USD 99,000. The difference between annualized rates does not create extra dollars at maturity; the stated gain remains USD 1,000 before costs. These inputs are synthetic, not auction observations, current market yields or a projection of reinvestment opportunities after the ninety-day period.

Distinguish simple annualization from compounding

Multiplying a period return by a year fraction produces a simple annualized rate. It does not calculate compound wealth. An effective annual rate would require an explicit repeated-investment assumption and its own exponentiation. Actual reinvestment after maturity may occur at different rates, with different available terms and costs. Do not use the word earned for an annualized number when the only stipulated investment lasts for a shorter period.

A comparison with a coupon-bond yield also needs compatible compounding and payment timing. Converting quotation conventions does not make the instruments identical. They may differ in maturity, liquidity, tax treatment and payment risk. This guide isolates arithmetic, not a choice between investments. If a data source reports several yield fields, preserve their definitions rather than assuming the largest percentage represents the most favorable economic outcome.

Check days, rounding and transaction cash

Use days from the relevant settlement date to maturity, not the issue's original advertised term when the purchase occurs later. Confirm which endpoints enter the count and whether the official quotation uses a particular year convention. Rounding the displayed price before computing a yield can create a small reconciliation difference. Retain the unrounded price where available and document the required cash and rate rounding independently.

Transaction charges, taxes and an earlier sale can change the investor's net result without changing the published quotation method. Record those items in their own cash ledger instead of adjusting face value to hide them. Useful questions are whether the price is per unit of face, whether it includes all purchase cash and whether the yield assumes redemption rather than sale. Without those answers, a precise percentage remains ambiguous.

Keep bills outside an unsupported maturity shortcut

The site's zero-coupon calculation uses whole-year regular-bond conventions, not arbitrary-day bill quotations. Entering one year and copying a discount rate changes timing and discounting. Use the separate formulas here to inspect the quotation; they do not add bill-pricing or auction-replication capability to that tool.

Check the denominator before comparing rates, and never label simple annualization a realized annual return. A reverse-price check can catch arithmetic errors but not a mislabeled convention. Preserve inputs and formulas together. The synthetic example supplies no current rate, execution assumption or reinvestment recommendation.

Primary references

1. TreasuryDirect: Understanding Pricing and Interest Rates, public bill discount-price formula
2. TreasuryDirect: Treasury Bills, public payment description
3. Department of Finance Canada: Determining Bond and Treasury Bill Prices and Yields, public simple-yield convention