Bond Yield Calculator

Market price = Σ [ coupon × (1 + i)^−k ] + face value × (1 + i)^−n, solved for i (k = 1 … n; coupon = face value × coupon rate ÷ periods per year; n = years to maturity × periods per year)

Market price

What the bond costs today, and the one quantity this page is given instead of solving for

Coupon

Face value × coupon rate ÷ periods per year — 20 on the default bond, paid twice a year

Face value

The amount repaid at maturity, and the base the coupon is calculated from

Years to maturity

How long is left, entered in years and multiplied by the periods per year to give the number of payments still to come

Yield to maturity (i)

The unknown this page solves for: the annual rate that discounts every remaining payment down to the market price

Use it whenever a bond's price is quoted to you and its coupon is not enough to compare it with anything. The coupon rate is fixed for the life of the bond and says nothing about what you are paying for it: an old 4 percent bond and a new 4 percent bond have the same coupon rate and completely different yields if one of them costs 922.05 and the other costs 1,000. Read the yield to maturity as the return you would actually earn by holding to maturity and reinvesting every coupon at that same rate, and read the current yield next to it as the honest cash-on-cash figure for this year alone — if the two are far apart, the bond's price is far from par, and most of your return is coming from the price moving to par rather than from the coupons. The case that makes the page worth using is a price near or above the sum of everything the bond will ever pay, where the current yield still looks like a coupon and the yield to maturity is at or below zero.

1. 1,000 face value, 4% coupon, 10 years, semiannual, priced at 922.05

   1. The coupon is 1,000 × 4% ÷ 2 = 20, paid 20 times over the ten years
   2. At a yield of 5 percent the rate per period is 5% ÷ 2 = 2.5 percent
   3. Present value of the 20 coupons: 20 × (1 − 1.025⁻²⁰) ÷ 0.025 = 311.78
   4. Present value of the face value: 1,000 × 1.025⁻²⁰ = 610.27
   5. Those two add to 311.78 + 610.27 = 922.05, which is the price you entered, so 5 percent is the yield
   6. Current yield: the 40 paid each year ÷ 922.05 = 4.34 percent — the 77.95 discount this bond repays on top of its coupons is exactly what it leaves out

   This is the bond price page's default case read backwards, and running it that way is the cheapest check that both pages are doing the same arithmetic: he priced the bond at 5 percent and got 922.05, this page is handed 922.05 and gives back 5. Note also that the two yields differ by 0.66 of a point on a bond whose price is only 7.8 percent below par — the distance grows fast as the price falls, which the next example shows.

2. The same bond at 728.19 — an 8% yield

   1. The coupon is still 20 every six months, so the price is the only thing that moved
   2. At a yield of 8 percent the rate per period is 4 percent
   3. Present value of the 20 coupons: 20 × (1 − 1.04⁻²⁰) ÷ 0.04 = 271.8066
   4. Present value of the face value: 1,000 × 1.04⁻²⁰ = 456.3869
   5. Together 271.8066 + 456.3869 = 728.1935, which rounds to the 728.19 you entered
   6. Current yield: 40 ÷ 728.19 = 5.49 percent, against a yield to maturity of 8 percent

   The 2.51 point gap between 5.49 and 8 is the 271.81 this bond repays above what it costs, and it is the whole reason a deep-discount bond cannot be judged by its coupon or by its current yield. Reading only the current yield here understates the return by about a third.

3. A zero-coupon bond: 1,000 face value, no coupon, priced at 610.27

   1. Set the coupon rate to zero: the bond pays nothing at all until maturity
   2. 1,000 × 1.025⁻²⁰ = 610.27, so at a 5 percent yield the entire price is the discounted face value
   3. There is no coupon to divide, so the current yield is zero — a zero-coupon bond has no current yield in any useful sense
   4. The 389.73 discount is the whole return, and it arrives all at once at maturity

   The cleanest proof that the yield is not the coupon and not the current yield either: this bond pays 0 percent and returns 5, and no column other than the yield to maturity says so. It is also the reason the page can report a yield for a bond whose current yield is meaningless, which is most of what a bond desk does all day.

This is the clean version of a bond yield, and real settlement is messier in four specific ways. It prices a bond whose next coupon is a full period away, so it does not handle accrued interest: between coupon dates the seller is owed the part of the current coupon that has already elapsed, and a real invoice adds that on top of the quoted price. It is a clean price for that reason, not a dirty one. It assumes every remaining coupon is a full period, so a bond with an odd first or last period is out of scope. It treats the coupon rate as an annual rate split evenly across the year's payment periods rather than accrued by an actual day count, and it makes no distinction between the 30/360 and actual/actual conventions that govern real settlement. The yield to maturity it reports assumes every coupon is reinvested at that same rate until maturity, which is the standard definition and also the standard caveat: reinvest at a lower rate and the realised return is lower. The precision of the answer is bounded by the price you typed — a two-decimal price pins the yield to roughly a thousandth of a point, and changing the price by one cent moves it. On the other side, a price above the sum of everything the bond will ever pay has no positive yield at all, and the page says so instead of printing a negative number. Nothing here models default risk, liquidity, taxes, a call feature or a sinking fund, and no currency is attached to any figure.

What is yield to maturity?

It is the single annual rate at which every payment the bond still owes you — each remaining coupon and the face value at maturity — discounts back to the price the bond costs today. It is the return you would earn by buying at that price and holding to maturity, and it is the only yield that lets you compare bonds with different coupons and different prices. On the default case, a 1,000 face value bond paying 4 percent twice a year with ten years left and a market price of 922.05 has a yield to maturity of 5 percent.

How do I calculate the yield to maturity of a bond?

Write down the pricing equation — the market price equals the discounted coupons plus the discounted face value — and then find the rate that makes it true. There is no way to rearrange that equation for the rate once more than one period is left, so the answer is found by trying rates and narrowing in. For the default bond, 5 percent gives 311.78 of coupons plus 610.27 of face value, which is 922.05, the price exactly.

Why is the yield to maturity different from the current yield?

Because they measure different things. The current yield is the annual coupon divided by the price and stops there: 40 ÷ 922.05 = 4.34 percent. The yield to maturity also counts the 77.95 this bond repays above its price, spread across the ten years to maturity, which is why it reads 5 percent. The gap is small for a bond near par and large for one far below it: at a price of 728.19 the current yield is 5.49 percent against a yield to maturity of 8.

What is a discount bond?

One that costs less than the face value it repays, which happens when the yield the market demands is above the coupon rate the bond pays. The default bond is a discount bond: it costs 922.05 and repays 1,000, so it carries a discount of 77.95, and part of your return arrives as that price difference rather than as coupon income. The deeper the discount, the more of the total return comes from it — at 728.19 the bond repays 271.81 more than it costs.

What does it mean when a bond's yield is higher than its coupon rate?

That you are buying it below par, so the coupon rate understates what you will earn. A 4 percent bond priced to yield 8 percent pays exactly the same 40 a year as a 4 percent bond priced to yield 4 percent; the difference is that the first one costs 728.19 and repays 1,000. Whenever the yield to maturity sits above the coupon rate the price must be below face value, and whenever it sits below the coupon rate the price must be above it.

Can the yield to maturity be negative?

Not on this page, and not for a bond you would buy. A negative yield means paying more today than the bond will ever return — more than the face value plus every coupon still to come — and that is a real thing in some government markets but it is not something this page reports. If you enter a market price above the sum of everything the bond will pay, it stops and says so rather than printing a negative rate, because the number would be arithmetically defined and financially misleading.

Which yield should I use to compare two bonds?

The yield to maturity, because it is the only one that accounts for what you pay. Two bonds with the same 4 percent coupon rate and the same maturity are not the same investment if one costs 922.05 and the other costs 1,000, and the coupon rate cannot tell you that. The current yield is useful as a check on how much of the return is cash in hand this year rather than a price gain — and when the two figures are far apart, that gap is the answer to a question you should ask before buying.

• 31 CFR Part 356, Appendix B — Formulas and Tables: the present value factors and the price formula this page solves in reverse — U.S. Department of the Treasury rule, via the Electronic Code of Federal Regulations (United States)
• Understanding Pricing and Interest Rates — why the price of a note or bond moves in the opposite direction to its yield — U.S. Department of the Treasury, Bureau of the Fiscal Service (TreasuryDirect), United States
• Current Yield — Investor.gov glossary (the ratio of the interest paid on a bond to its market price, which is the figure printed beside the yield to maturity) — U.S. Securities and Exchange Commission, Investor.gov (United States)