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How bond pricing works

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.

On this page

In short

  • Price is the present value of every coupon plus the face. A $1,000.00 bond, 5 percent coupon, two payments a year, 10 years, 5 percent market rate, prices at $1,000.00. Each coupon is $25.00.
  • Current yield is the annual coupon over the price. At par that is 5.00 percent, equal to the coupon rate. Par is the only price where those two agree, on a bond that pays a coupon at all.
  • When the market wants 7 percent, the same $25.00 coupons and $1,000.00 face price at $857.88. Current yield rises to 5.83 percent, because the same coupons cost less to buy.
  • When the market will accept 3 percent, the price is $1,171.69, a premium of $171.69. Current yield falls to 4.27 percent. A buyer at this price takes a known loss at maturity, when the bond repays $1,000.00.
  • Rates and prices move in opposite directions because the coupons are fixed. How bonds work is the claim. This page is the price.

Par is the identity check

A coupon bond is a stack of present values:

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 coupon each period, FF the face, ii the market rate for one period, nn the number of periods left.

A $1,000.00 bond paying 5 percent in two payments a year for 10 years has a $25.00 coupon each half-year. At a 5 percent market rate, i=0.025i = 0.025 and n=20n = 20. Discount the twenty coupons and the face at that rate and they add to $1,000.00.

When the coupon rate equals the market rate, the bond prices at par. Current yield is 5.00 percent, the annual coupon over $1,000.00. The bond price calculator on this page is that sum.

How present value works is the discounting. How bonds work is what you hold. How bond duration works is how far the price moves when the yield moves.

Price is the denominator of duration. Once you have PP, Macaulay against modified duration is the next split: a weighted-average wait, then that wait divided by 1+y/k1 + y/k so it can move the price.

A higher market rate is a lower price

Nothing about the bond changes. Buyers of comparable bonds now want 7 percent. The coupon is still $25.00 twice a year. Only ii moves, to 0.035. The price falls to $857.88. Current yield rises to 5.83 percent, because the same $25.00 coupons now cost less to buy.

The payments were fixed. Newly issued bonds of the same risk now pay more, so the old one can only clear by trading below the $1,000.00 it will repay. That gap is what trading at a discount means.

The three sheets on this page are three points on that curve. The bond price and yield explorer lets you drag the market rate and watch the price move, which is the same identity as a picture.

Current yield against yield to maturity is the pair that gap creates. Current yield vs yield to maturity is why 5.83 percent of coupon cash on a $857.88 price is not the 7 percent the market is paying.

A lower market rate is a premium

The same bond, buyers now accepting 3 percent. Price $1,171.69, a premium of $171.69 over face. Current yield 4.27 percent. A buyer at $1,171.69 is repaid $1,000.00 at maturity, a known loss that is offset, in the yield, by the $25.00 coupons along the way.

Current yield is not yield to maturity. Current yield counts the coupons and nothing else. Yield to maturity is the ii that makes the present-value sum equal the price, face included. At a premium, yield to maturity sits below current yield because the known loss at maturity has to be averaged in.

What this page is not doing

It is not a duration, not a credit spread, and not accrued interest between coupon dates. The three sheets are a $1,000.00 par bond at a 5 percent market rate, the same coupons at 7 percent ($857.88), and at 3 percent ($1,171.69). This is educational material, not financial advice.

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.

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 $25.00 twice a year on $1,000.00 of face, whatever rates do next. If newly issued bonds of the same risk pay 7 percent, the old one can only compete by trading at $857.88.

Is current yield the same as yield to maturity?

No. Current yield is the annual coupon divided by the price. At $857.88 that is 5.83 percent. Yield to maturity is the discount rate that makes the present value of the coupons and the face equal the price. At a discount, yield to maturity sits above current yield.

What do par, discount and premium mean?

Par is a price equal to face value, $1,000.00 on this sheet, and it happens when the market rate matches the coupon rate. A discount is $857.88 here, when the market rate is higher than the coupon rate. A premium is $1,171.69, when the market rate is lower.

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.