T-Bill Discount Rate vs Investment Yield: Compare the Cash, Not the Headline

A Treasury bill quoted at 5.00% can offer a higher return than another quoted at 5.10%. That is possible when the first number is a discount rate and the second is an investment yield. They use different denominators and annualization conventions. Comparing the two percentages directly can reverse the purchase decision.

The useful comparison starts with cash paid and cash received on matching dates. In the hypothetical 91-day example below, the 5% discount quote becomes a 5.1343% simple annual return on the purchase price, slightly above 5.10%. Yet an extra dollar of purchase cost on USD 10,000 face value reverses that result. The difference is small enough that execution details matter more than the headline.

First identify what the displayed rate measures

A US Treasury bill has no periodic coupon. In the held-to-maturity calculation used here, the investor pays a price below face value and receives face value at maturity. A bank-discount quote annualizes the discount relative to face value using a 360-day basis. The short-bill investment rate instead relates that gain to the purchase price, using the applicable 365- or 366-day year.

Neither label is a promise that the displayed quote is available for your order. Record the security, settlement date, maturity date, price side, quotation time and available quantity. For a purchase, use the executable offer price, not a dealer's purchase bid. If the screen instead displays a fund yield, a deposit APY or a different bond-yield convention, do not silently insert it into a bill formula.

Turn both quotations into the same future cash payment

Define face value F, purchase price P, days to maturity t, discount rate d and investment rate i. Rates enter the equations as decimals. The discount-price equation is P = F × (1 − d × t/360). For the short-bill investment convention, i = (F/P − 1) × y/t, so P = F/(1 + i × t/y). The annualization basis y is 365 or 366 as applicable.

Our two fictional bills have the same settlement and maturity, t = 91 days, F = USD 10,000, and y = 365. This is an issue-to-maturity example whose following one-year period does not include February 29. It excludes taxes, FX, funding costs, interim cash flows and trading costs initially. Both prices are assumed executable for that face amount. No current auction or dealer quotation is being reported.

Bill A has d = 0.05: P = 10,000 × (1 − 0.05 × 91/360) = 9,873.6111. Bill B has i = 0.051: P = 10,000/(1 + 0.051 × 91/365) = 9,874.4457. Both pay 10,000 on the same date if held to maturity with payment made as assumed. A buys that cash flow for slightly less.

The higher headline loses after the denominators are aligned

For A, the gain is 126.3889, but the denominator for the holding return is 9,873.6111, not 10,000. Its 91-day return is 1.2801%, and multiplying the unrounded result by 365/91 gives 5.1343%. B's corresponding numbers are 1.2715% and 5.1000%. These are simple annualized returns, not a forecast of a full year's compound earnings.

The table rounds money to four decimal places to make the model traceable; calculations retain full precision. A real order must use its actual rounded settlement amount. The comparison fixes face value, not invested capital. Someone investing a fixed cash budget must also account for permitted denominations and residual cash rather than treating these two outlays as identical.

A cost of about 83 cents is enough to remove the advantage

With equal maturity proceeds and dates, A and B break even when their all-in outlays match. If B has no extra cost, A's maximum additional cost is 9,874.4457 − 9,873.6111, approximately USD 0.8346 before settlement rounding. That is the economic size of the initial advantage, not the 0.10 percentage-point gap between the two incompatible headline rates.

Charge A an additional USD 1 at purchase and leave B unchanged. A's outlay becomes 9,874.6111 and its maturity gain 125.3889. The simple annual return falls to 5.0932%, below B's 5.1000%. This is an invented fee, not a claim about a broker's tariff. Include actual fees in the initial outlay and any maturity deductions in final cash. Do not count a spread twice if the executable offer already includes it.

Selling after 30 days is a different calculation

The maturity return assumes the bill is held until the stated payment. Suppose instead that A, without the extra dollar cost, is sold 30 days after settlement, leaving 61 days. Use three separate hypothetical executable bid discount rates to reconstruct sale proceeds with the same discount-price equation. There are no sale costs in this comparison.

At an unchanged 5% discount rate, less time to maturity raises the price and produces a 41.6667 gain. At 6%, the gain is only 24.7222. At 8%, the investor loses 9.1667 despite having bought a bill below par. The no-cost break-even exit discount rate is 5% × 91/61, approximately 7.4590%. An exit rate above that threshold produces a loss in this particular model.

These are 30-day holding returns, not maturity yields or price forecasts. Actual bid size, spreads and fees change the boundary. A money need before maturity therefore cannot be assessed solely from the initial held-to-maturity yield.

Keep the short-bill convention inside its valid range

The US Treasury investment-rate formula for bills longer than one half-year is different; the short formula is not a universal shortcut for every maturity. Its year basis is also not simply “366 whenever the issue year is a leap year.” The relevant official definition examines the year following the issue date and whether that interval contains February 29. For secondary-market quotations, check the provider's convention and actual settlement dates rather than assuming the auction label fixes your purchase return.

Annualizing a 91-day return does not lock the reinvestment rate for the remaining year. Nor does a USD return establish the outcome in another currency. Taxes, currency conversion and liquidity needs require separate cash flows. For coupon-paying instruments, the additional payment dates raise the distinct issues covered in yield to maturity and reinvestment assumptions.

Use a five-check cash comparison

1. Match the instrument, currency, settlement date, maturity and executable trading side.
2. Identify face-value discount versus purchase-price return, the day count and any compounding convention.
3. Reconstruct the price and maturity cash; calculate holding return before annualizing.
4. Add actual costs once and solve the all-in price at which the ranking changes.
5. If cash may be needed early, separately test sale bids and funding needs rather than substituting maturity yield.

The result is dependable only within those matching conditions. A different payment date, a non-executable quote or an omitted cost invalidates the simple ranking. A rate becomes useful when it can be reconciled to the money leaving and returning to the account.