Bond Equivalent Yield Calculator

Securities that pay no interest at all

Not every debt security pays interest along the way. A whole family of short-term instruments, Treasury bills chief among them, works differently: there are no coupons, no periodic payments, nothing arriving in your account every six months. Instead you buy the security for less than its face value, and when it matures you are paid the full face value. The gap between the two is your entire return.

These are called discount securities, and their simplicity is appealing. Pay 9,750, collect 10,000 six months later, and you have earned 250. But that simplicity creates a problem the moment you try to compare such a security against an ordinary coupon-paying bond, and solving that problem is precisely what the bond equivalent yield does. This calculator computes it from three things: the face value, what you paid, and how many days until it matures.

The problem this measure exists to fix

Here is the awkward bit. Treasury bills are quoted in the market using a long-standing convention called the bank discount rate, and that convention does not describe your actual return. It is a quoting habit inherited from an era of hand calculation, and it is misleading in two specific ways.

First, it measures your gain against the security's face value rather than against what you actually paid. That is backwards, because your return should be measured against the money you genuinely put in, which is the lower purchase price. Dividing by the bigger number makes the return look smaller than it is. Second, it annualises using a 360-day year, a convention that survives from the days when dividing by 360 was easier arithmetic than dividing by 365. Using a short year understates how many times over your return would repeat in a real calendar year. Both quirks push in the same direction, so the quoted rate always sits below the return you truly earn. Compare that quoted number against a bond's yield and you would be comparing two things measured on different rulers.

The two corrections

The bond equivalent yield repairs both flaws, which is the whole of its purpose. It measures the gain against the purchase price, the money you actually committed, and it annualises over a 365-day year, matching the real calendar and the convention that coupon bonds use.

So the calculation takes the gain, the face value minus what you paid, divides it by what you paid to get the true return over the holding period, then scales that up to a full year by the ratio of 365 to the days you held it. In short: bond equivalent yield = (face value − purchase price) ÷ purchase price × 365 ÷ days to maturity. The result is a rate stated the same way a bond's yield is stated, which is what the name is telling you: it puts a discount security's return on a bond's terms, so the two can finally be compared fairly. This is also why it is sometimes called the coupon equivalent yield, and it is the measure the US Treasury itself publishes for its bills.

A worked example

Take a Treasury bill with a face value of 10,000, bought for 9,750, with 182 days to run.

Your gain is 250, the difference between the 10,000 you will receive and the 9,750 you paid. Measured against what you actually invested, that is a return of about 2.56% over the 182 days you hold it. Now annualise: 182 days is almost exactly half a year, so the 365-over-182 scaling roughly doubles it, giving a bond equivalent yield of about 5.14%. That figure is directly comparable to the yield quoted on a coupon-paying bond, so if a bond of similar quality and maturity offers 4.9%, you now know the bill is the better deal, a judgement you simply could not make from the raw quoted rate.

Seeing the gap

It is worth seeing how much difference the two corrections make, because it is larger than people expect for a security this simple.

The same bill, the same 250 of profit, and yet the two figures differ by nearly 0.2 percentage points, with the quoted rate the lower of the two. Neither number is wrong; they are answers to different questions. The bank discount rate is a market quoting convention, useful for comparing one bill against another bill because they all share the same distortion. The bond equivalent yield is the one that tells you what you are really earning and lets you hold a bill up against a bond. When you are deciding where to put your money, the second is the number you want.

Where you meet this measure

Treasury bills are the most common place to encounter a bond equivalent yield, but the measure applies to any security sold at a discount and redeemed at face value with nothing paid in between.

That includes commercial paper, the short-term debt companies issue to cover their day-to-day funding, along with banker's acceptances, some certificates of deposit, municipal notes, and zero-coupon bonds. All of them share the same shape: one price going out, one larger amount coming back, no coupons along the way. Whenever you want to line one of these up against a conventional bond, or against each other on consistent terms, the bond equivalent yield is the translation that makes the comparison honest. For a coupon-paying bond, the comparable measure is the yield to maturity, which our bond yield to maturity calculator works out. And since Treasury bill interest is exempt from state and local income tax in the US, our tax equivalent yield calculator is worth a look when you are weighing a bill against a fully taxable alternative.

A note on longer maturities

The calculation here uses simple interest: it takes the return over the holding period and scales it straight up to a year, with no compounding along the way. That is exactly right for the short-term instruments this measure was designed for, which is most of them, since Treasury bills run to a year at most and commercial paper is usually far shorter.

Worth knowing, though, is that for bills with more than about six months to run, the US Treasury switches to a slightly more involved formula that accounts for the semi-annual compounding a comparable bond would enjoy. The difference is modest, but it exists, so for a longer-dated bill treat the simple figure here as a close and very usable approximation of the official coupon equivalent yield rather than an exact match to Treasury's published number. For the short maturities where these securities mostly live, the straightforward calculation this tool performs is the standard one.

Questions people ask

What is the bond equivalent yield?

It is the annualised return on a discount security, such as a Treasury bill, stated on a basis that can be compared directly with a coupon-paying bond's yield. It divides the gain by the purchase price and annualises using a 365-day year.

Why is it needed?

Because discount securities are quoted using the bank discount rate, which measures the gain against face value and uses a 360-day year. Both choices understate the true return, so the quoted figure cannot be fairly compared with a bond yield until it is converted.

How does it differ from the discount rate?

In two ways. The bond equivalent yield divides by the price you actually paid rather than the face value, and it uses 365 days rather than 360. Both corrections raise the figure, so the bond equivalent yield is always higher than the quoted discount rate for the same security.

Which securities is it used for?

Any security sold at a discount and redeemed at face value with no interest paid in between: Treasury bills, commercial paper, banker's acceptances, some certificates of deposit, municipal notes, and zero-coupon bonds.

References

The coupon equivalent yield formula, dividing the gain by the purchase price and annualising over a 365-day year, along with the bank discount rate convention it corrects, follows the US Treasury's published calculations and the fixed-income text by Fabozzi below.

1. United States Department of the Treasury (TreasuryDirect). Price, Yield and Rate Calculations for a Treasury Bill. treasurydirect.gov
2. Fabozzi, F. J. Bond Markets, Analysis, and Strategies (money market instruments and yield conventions). Pearson.