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Bond price calculator and current yield

A bond is worth the present value of its coupons plus its face value, discounted at the market rate. A $1,000.00 bond paying a 5 percent coupon twice a year for 10 years is worth $1,000.00 when the market rate is 5 percent, $857.88 when it is 7 percent, and $1,171.69 when it is 3 percent.

Price today

$1,000.00

$25.00 per coupon, twice a year, then $1,000.00 back at maturity. It trades at par.

Coupon per payment
$25.00
Coupon rate, on face value
5.00%
Current yield, on price
5.00%
Premium or discount
$0.00
$

What the issuer repays at maturity. Coupons are quoted on this, never on the price.

%

The annual rate, fixed when the bond is issued, so this one does not move. It is split across the coupons in a year.

%

What buyers want today on the same risk and maturity. This is the yield to maturity, quoted as an annual rate and split the same way.

yr

Priced on a coupon date, so this leaves a whole number of coupons still to come.

The formula

P=∑t=1nC(1+i)t+F(1+i)nP = \sum_{t=1}^{n} \frac{C}{(1+i)^t} + \frac{F}{(1+i)^n}

PP is the price, CC the coupon paid each period, FF the face value repaid at maturity, ii the market rate for one period, and nn the number of periods left.

What this calculator works out

Enter the face value, the coupon rate, the market rate, the years left to maturity and how often a coupon arrives. It returns what the bond is worth today, the cash each coupon pays, and the current yield that price implies.

Two rates go in and they do different jobs. The coupon rate is written into the bond and never changes: it fixes the cash. The market rate is what buyers can get today on bonds of the same risk and the same maturity, and it moves constantly. The price is the number that reconciles a fixed payment stream with a moving rate.

In the United States most Treasury and corporate bonds pay a coupon twice a year, so the calculator opens on two payments a year. Prices here are quoted on a coupon date, with a whole number of payments still to come.

The bond price formula

A bond is a set of dated payments, and its price is the present value of all of them at the market rate:

P=∑t=1nC(1+i)t+F(1+i)nP = \sum_{t=1}^{n} \frac{C}{(1+i)^t} + \frac{F}{(1+i)^n}

CC is the cash coupon, FF the face value repaid at maturity, ii the market rate for one period, and nn the number of periods left. With two coupons a year, ii is the annual market rate divided by 2 and nn is the years to maturity times 2.

The coupon is settled when the bond is issued: it is the face value times the coupon rate, split across the payments in a year. Nothing in the numerator moves after that. Between one quote and the next the only thing that moves is ii, which is why a bond's price and the rate the market wants are two ways of saying the same thing. Over the bond's life nn falls as well, by one at every coupon date, and that is what walks a premium or a discount back to face value by the time the bond matures.

Why price moves the opposite way to rates

The payments a bond makes are fixed on the day it is issued and never move again. So when the going rate changes, the only thing left that can adjust is the price.

Say the market rate rises above the coupon rate. A buyer can now get better payments from a newly issued bond at face value, so nobody pays face value for the older, smaller coupons. The older bond's price falls until those fixed payments deliver the going rate to whoever buys at the lower price. That is a discount, and it is the second example below.

When the market rate falls below the coupon rate the same logic runs backwards. The old bond's coupons beat anything newly issued, buyers bid for it, and the price rises above face value until the return on the larger sum they put in comes back down to the market rate. That is a premium, and it is the third example.

  • Market rate above the coupon rate: price below face value, a discount.
  • Market rate equal to the coupon rate: price at face value, at par.
  • Market rate below the coupon rate: price above face value, a premium.

How far the price moves usually depends on how long the bond has left, because a longer bond has more fixed payments being repriced. That is a strong tendency rather than a law: on a deeply discounted bond so much of the value already sits in the final repayment that adding years can move the price less, not more. Rates also differ by maturity in the first place, which is what the yield curve describes.

Current yield, coupon rate and yield to maturity

Three rates get attached to the same bond and they answer different questions.

  • The coupon rate is the annual coupon divided by face value. It is fixed when the bond is issued and never changes.
  • The current yield is the annual coupon divided by the price a buyer actually pays. It moves whenever the price moves.
  • The yield to maturity is the one discount rate that makes the present value of every remaining payment add up to the price. It is the market rate box in this calculator.

Current yield counts the coupons and stops there. A buyer of the discounted bond below pays $857.88 and, on holding it to maturity, also collects $1,000.00, a gain the current yield says nothing about. That is why the current yield on that bond, 5.83 percent, sits under its 7 percent yield to maturity, and why a premium bond's current yield sits above its yield to maturity.

In the United States a bond's yield is quoted as a nominal annual rate, in step with how the coupons arrive, so a 7 percent yield on a semiannual bond pays 3.5 percent per half year and compounds to 7.12 percent across the full year. Other markets quote some bonds on an annually compounded basis instead, where 7 percent means 7 percent for the year and nothing has to be halved. The APR against APY calculator covers what that convention does to the rate a number really stands for. The net present value calculator runs the same discounting on any set of dated cash flows, not only on a bond's.

Worked examples

A bond priced at par

A $1,000.00 bond pays a 5 percent coupon in two payments a year and matures in 10 years. Bonds of the same risk and maturity are yielding 5 percent. What is it worth?

  1. Work out the cash coupon, which is face value times the coupon rate: 1000×0.05=501000 \times 0.05 = 50 a year, paid as two coupons of $25.00.
  2. Set the period rate and the number of periods: i=0.05/2=0.025i = 0.05/2 = 0.025 and n=10×2=20n = 10 \times 2 = 20.
  3. Discount the 20 coupons: 25×1−1.025−200.025=25×15.589162=389.7325 \times \frac{1 - 1.025^{-20}}{0.025} = 25 \times 15.589162 = 389.73.
  4. Discount the face value: 1000×1.025−20=1000×0.610271=610.271000 \times 1.025^{-20} = 1000 \times 0.610271 = 610.27.
  5. Add the two present values: 389.73+610.27=1000.00389.73 + 610.27 = 1000.00.
  6. Current yield is the annual coupon over the price: 50/1000=0.0550 / 1000 = 0.05, which is 5.00 percent.

The bond is worth $1,000.00, exactly its face value, so it trades at par. The premium or discount is $0.00. The current yield is 5.00 percent, the same as the coupon rate, and on a bond that pays a coupon at all, par is the only price where those two agree.

The same bond when the market wants 7 percent

Nothing about the bond changes. Rates move, and buyers of comparable bonds now want 7 percent. What happens to the price?

  1. The coupon is untouched at $25.00 twice a year, because the coupon rate is fixed against face value.
  2. Only the discount rate moves: i=0.07/2=0.035i = 0.07/2 = 0.035, with n=20n = 20 as before.
  3. Discount the coupons: 25×1−1.035−200.035=25×14.212403=355.3125 \times \frac{1 - 1.035^{-20}}{0.035} = 25 \times 14.212403 = 355.31.
  4. Discount the face value: 1000×1.035−20=1000×0.502566=502.571000 \times 1.035^{-20} = 1000 \times 0.502566 = 502.57.
  5. Add them: 355.31+502.57=857.88355.31 + 502.57 = 857.88.
  6. Measure the gap against face value: 1000−857.88=142.121000 - 857.88 = 142.12.
  7. Current yield: 50/857.88=0.058350 / 857.88 = 0.0583, or 5.83 percent.

The price falls to $857.88. The bond changes hands for 142.12 less than the $1,000.00 it repays at maturity, which is what trading at a discount means. The current yield rises to 5.83 percent, because the same $25.00 coupons now cost less to buy.

The same bond when the market wants 3 percent

Same bond once more, but this time buyers of comparable bonds will accept 3 percent. What is it worth now?

  1. The coupon is unchanged again at $25.00 twice a year.
  2. i=0.03/2=0.015i = 0.03/2 = 0.015 and n=20n = 20.
  3. Discount the coupons: 25×1−1.015−200.015=25×17.168639=429.2225 \times \frac{1 - 1.015^{-20}}{0.015} = 25 \times 17.168639 = 429.22.
  4. Discount the face value: 1000×1.015−20=1000×0.742470=742.471000 \times 1.015^{-20} = 1000 \times 0.742470 = 742.47.
  5. Add them: 429.22+742.47=1171.69429.22 + 742.47 = 1171.69.
  6. The gap against face value: 1171.69−1000=171.691171.69 - 1000 = 171.69.
  7. Current yield: 50/1171.69=0.042750 / 1171.69 = 0.0427, or 4.27 percent.

The price rises to $1,171.69, a premium of $171.69 over face value. The current yield falls to 4.27 percent, because the same $25.00 coupons cost more to buy. A buyer at this price also takes a known loss at maturity, when the bond repays $1,000.00 rather than the $1,171.69 they paid for it.

The mistake that costs the most

Reading the coupon rate as the return.

The coupon rate is fixed against face value, so it equals the return only when the price equals face value. The premium bond in the third example pays a 5 percent coupon, hands its buyer 4.27 percent in cash on the price they actually paid, and still earns them only 3 percent, because $171.69 of the $1,171.69 they paid is never coming back: the bond repays $1,000.00 at maturity and not a cent more.

The error runs the other way on a discount bond, where the coupon rate understates what a buyer earns. Yield to maturity is the measure that puts two bonds on the same footing, and it is the market rate box in this calculator. The coupon rate is the rule that decides how much cash arrives and when, not a measure of return.

Common questions

Why does a bond price fall when interest rates rise?

Because the payments are fixed. A bond issued with a 5 percent coupon keeps paying 5 percent of face value whatever rates do next. If newly issued bonds of the same risk pay 7 percent, the old one can only compete on price, so its price falls until a buyer at that price earns the going rate. Longer bonds usually move further on the same change in rates, because more of their payments are being repriced. That is a tendency rather than a law: on a deeply discounted bond so much of the value sits in the final repayment that adding years can move the price less.

Is current yield the same as yield to maturity?

No. Current yield is the annual coupon divided by the price, so it counts the coupons and nothing else. Yield to maturity is the discount rate that makes the present value of the coupons and the face value repayment together add up to the price. On the discount bond above the current yield is 5.83 percent while the yield to maturity is 7 percent, because a buyer at $857.88 also collects $1,000.00 at maturity.

What do par, discount and premium mean?

Par is a price equal to face value, and it happens when the market rate matches the coupon rate. A discount is a price below face value, which happens when the market rate is higher than the coupon rate. A premium is a price above face value, which happens when the market rate is lower. The issuer repays face value at maturity in all three cases, so the price a buyer pays is what decides the return.

Keep reading

This page is educational material, not financial advice. The figures come from the formula shown and assume the inputs you enter hold for the whole term. Your own rate, fees, taxes and timing will differ, so treat the output as arithmetic to check a decision against, not as a recommendation.