What this calculator does
Treasury bills and other discount instruments pay no coupon. They are bought below face value and redeemed at face, and the gain is the return. Bond equivalent yield converts that into an annualised figure comparable with a coupon-paying bond.
The conversion matters because the conventional quote uses a different basis. A bill bought at $9,750 against $10,000 face over 180 days has a bond equivalent yield of 5.20 per cent but a quoted bank discount rate of only 5.00 per cent, and the two are not interchangeable.
The formula
The gain is divided by the purchase price, not the face value, then annualised over a 365 day year. The bank discount rate, by contrast, divides by face value over a 360 day year, which always makes it look lower.
| Term | Meaning |
|---|---|
| Bond equivalent yield | The return annualised on a 365 day basis against the price paid. |
| Bank discount rate | The conventional quote, using face value and a 360 day year. |
| Holding period return | The unannualised gain over the actual period held. |
The inputs explained
| Field | What to enter |
|---|---|
| Face value ($) | The face value repaid at maturity. |
| Purchase price ($) | The purchase price, which is below face value. |
| Days to maturity | Days remaining to maturity. |
When to use it
Comparing a bill against a bond
The two are quoted on different conventions, and only the bond equivalent yield is comparable.
Evaluating a short-dated investment
Annualising makes a 90 day return comparable with anything else on offer.
Understanding a quoted discount rate
The quoted rate understates the true return, and the gap widens as maturity shortens.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the term affect the annualised yield?
The same $250 gain earned over three different periods.
| Days to maturity | Bond equivalent yield | Bank discount rate (360-day) |
|---|---|---|
| 90 days | 10.4% | 10.0% |
| 180 days | 5.20% | 5.00% |
| 365 days | 2.56% | 2.47% |
Questions
Why does the bank discount rate understate the return?
Two reasons compound. It divides the gain by face value rather than by the smaller price actually paid, and it annualises over 360 days rather than 365. Both push the quoted figure below the true return.
Why is a 90 day return annualised so high?
Because the same gain earned four times faster represents a much higher annual rate. Annualising is a comparison device, not a promise that the rate could be repeated four times over.
Does this account for compounding?
No. Bond equivalent yield is a simple annualisation. The effective annual yield, which assumes the proceeds are reinvested each period, would be slightly higher for short maturities.
Where is this convention used?
Mainly for Treasury bills, commercial paper and other money market instruments sold at a discount. Anything that pays no coupon but redeems at face value fits the same arithmetic.
For a coupon-paying bond's return, see the bond yield to maturity calculator. For comparing a tax-free yield, see the taxable equivalent yield calculator.