# Bond Price

> The bond price formula adds up what each future payment from a bond is worth today, and a higher market yield makes every one of those payments worth less.

Page: https://formulabro.yunusemrevurgun.com/finance/bond-price/  
Category: Finance  
Published: 2026-10-23  
Updated: 2026-10-23

## Formula

$$
P = \sum_{t=1}^{n} \frac{C}{(1 + y)^{t}} + \frac{F}{(1 + y)^{n}}
$$

Read it out loud: "P equals the sum, for t from 1 to n, of C divided by 1 plus y to the power t, plus F divided by 1 plus y to the power n. P is the price of the bond, C is the yearly coupon, y is the yield, n is the number of years, and F is the face value."

## Before you start

- **Percent and decimal**: A percent is a part out of 100, so 5% means 5 parts in 100, which is written as the decimal 0.05. In the formula, the yield is always a decimal. So a 5% yield gives $1 + y = 1.05$, and a 6% yield gives $1 + y = 1.06$.
- **Exponents like $(1 + y)^{3}$**: The small raised number is the **exponent**. It says how many times to multiply a number by itself. So $(1.05)^{3}$ means $1.05 \times 1.05 \times 1.05 = 1.157625$. A raised number of 1 leaves the number as it is, and a raised number of 2 means the number multiplied by itself once.
- **The sum sign $\sum$**: The Greek capital letter sigma, written $\sum$, means **add up**. The letter and number under it, $t = 1$, say where to start, and the letter or number above it, $n$, says where to stop. So $\sum_{t=1}^{3}$ means add up the terms for t = 1, t = 2 and t = 3.
- **Bond, coupon, face value and yield**: A **bond** is a loan made to a government or a company. The borrower pays a fixed **coupon**, an interest payment, each year, and repays the **face value**, often 1000 dollars, at the end. The **yield** is the rate of return the market asks for today, per year.
- **Present value**: The **present value** of a future payment is what it is worth today. You find it by dividing the payment by $(1 + y)$ once for each year until it arrives. A payment of 50 dollars due in one year, at a 5% yield, has a present value of $50 \div 1.05 = 47.62$ dollars.

More notation help: https://formulabro.yunusemrevurgun.com/math-basics/

## What each symbol means

- $P$: The **price** of the bond today, in dollars. The formula gives this number, and it is the sum of the present values of all the bond's payments.
- $\sum_{t=1}^{n}$: The **sum sign**. It means add up the terms for every year from $t = 1$ to $t = n$. The sum is over the coupons only, and the face value is added separately.
- $C$: The **coupon**, the fixed interest payment the bond makes each year, in dollars. A bond with 1000 dollars face value and a 5% coupon pays 50 dollars a year.
- $y$: The **yield** per year, as a decimal. It is the rate of return the market requires today on this kind of bond. A value of 0.05 means 5% a year. When this number goes up, the price goes down.
- $t$: The **year number** of a payment, counting 1, 2, 3 and so on up to n. It also sets the exponent, so a payment in year 3 is divided by $(1 + y)$ three times.
- $F$: The **face value** of the bond, the amount repaid at the end, in dollars. A common face value is 1000 dollars.
- $n$: The **number of years** until the bond matures, when the last payment and the face value are made. A 3-year bond has $n = 3$.

## The formula piece by piece

- $\frac{C}{(1 + y)^{t}}$: **One coupon in today's money.** The coupon is divided by $(1 + y)$ once for each year until it arrives, so a coupon that arrives later is worth less today.
- $\sum_{t=1}^{n} \frac{C}{(1 + y)^{t}}$: **All the coupons in today's money.** The same division is done for year 1, year 2, up to year n, and the results are added. Together they give the value today of every interest payment.
- $\frac{F}{(1 + y)^{n}}$: **The face value in today's money.** The repayment of the face value arrives at the end, in year n, so it is divided by $(1 + y)$ n times.
- $P = \sum_{t=1}^{n} \frac{C}{(1 + y)^{t}} + \frac{F}{(1 + y)^{n}}$: **The bond's price equals the present value of all its coupons plus the present value of its face value.**

## What the formula is doing, and why

A bond is a loan that you can buy and sell. The borrower pays a coupon each year, then returns the face value at the end. The formula adds up the value today of every one of those payments. The answer is the price that a buyer should pay today to earn the yield $y$ on the bond.

**Why divide by $(1 + y)$?** Money in hand today can be invested to earn the yield, so a payment that arrives later is worth less than the same amount today. If the yield is 5%, a payment of 50 dollars in one year is worth $50 \div 1.05$, about 47.62 dollars today. Every extra year of waiting means one more division by $(1 + y)$, which is why the formula uses a power.

**Why does the price fall when the yield rises?** A higher yield makes the divisor $(1 + y)$ larger in every term. Each payment then shrinks more when it is turned into today's money, so the sum gets smaller. The price falls. In the other direction, when the yield drops, the same fixed payments are worth more, and the price rises. Bond prices and yields always move in opposite directions.

**Why does the face value have its own term?** A bond makes two kinds of payment: coupons every year, and the face value once at the end. The formula gives the coupons a sum and the face value its own fraction. The face value is usually the biggest single payment, so it has a large effect on the price, especially for long bonds.

**The 1000 dollar landmark.** When the yield equals the coupon rate, the bond pays exactly what the market asks for, so its price equals its face value. In the example below, a 5% coupon bond sells for exactly 1000 dollars at a 5% yield. A price above face value means the coupon rate is higher than the yield, and a price below face value means it is lower.

## An everyday picture

Think of a bond as a promise to pay you a steady allowance and then a final lump sum. To decide what the promise is worth today, ask how much money you would need to set aside now, at the going rate, so that it produces each payment when it is due. If the going rate jumps up, you need to set aside less money, so the promise is worth less to a buyer. Bond prices move opposite to interest rates for exactly this reason.

## Worked example: Price of a 3-year bond at a 5% yield and at a 6% yield

A made-up bond has a face value of 1000 dollars, pays a 5% coupon, which is 50 dollars a year, and matures in 3 years. We find its price first when the market yield is 5%, then when the yield rises to 6%. In decimal form, $C = 50$, $F = 1000$ and $n = 3$, and the yield is $y = 0.05$ or $y = 0.06$.

1. **Write the bond's payments as numbers.** The coupon of 50 dollars is paid at the end of years 1, 2 and 3, and the face value of 1000 dollars is repaid at the end of year 3. The formula then becomes $P = \frac{50}{(1 + y)} + \frac{50}{(1 + y)^{2}} + \frac{50}{(1 + y)^{3}} + \frac{1000}{(1 + y)^{3}}$, with one term for each payment.
2. **Find the price at a 5% yield.** Here $1 + y = 1.05$, so the powers are $1.05$, $1.05^{2} = 1.1025$ and $1.05^{3} = 1.157625$. The pieces are $\frac{50}{1.05} = 47.62$, $\frac{50}{1.1025} = 45.35$, $\frac{50}{1.157625} = 43.19$ and $\frac{1000}{1.157625} = 863.84$ dollars.
3. **Add the pieces at 5%.** Adding them gives $47.62 + 45.35 + 43.19 + 863.84 = 1000.00$ dollars. At a yield equal to the 5% coupon rate, the bond is worth exactly its face value, 1000 dollars.
4. **Raise the yield to 6%.** Now $1 + y = 1.06$, so the powers are $1.06$, $1.06^{2} = 1.1236$ and $1.06^{3} = 1.191016$. Each payment is worth a little less than before, because each division is by a slightly larger number.
5. **Discount each payment at 6%.** The coupon pieces are $\frac{50}{1.06} = 47.17$, $\frac{50}{1.1236} = 44.50$ and $\frac{50}{1.191016} = 41.98$ dollars. The face value piece is $\frac{1000}{1.191016} = 839.62$ dollars, the largest piece by far.
6. **Add the pieces at 6%.** Adding them gives $47.17 + 44.50 + 41.98 + 839.62 = 973.27$ dollars. The price has fallen from 1000 dollars to 973.27 dollars, a drop of $1000 - 973.27 = 26.73$ dollars, which is about 2.7% of the price.

**Answer:** At a 5% yield the bond is worth **1000 dollars**, exactly its face value. At a 6% yield it is worth **about 973.27 dollars**. The yield went up, so the price went down, which is the inverse relationship the formula shows.

## Common mistakes

- Typing the yield as 5 instead of 0.05. With $1 + y = 6$ instead of $1.05$, every payment is divided by a number far too large, and the price comes out wrong by a huge amount.
- Forgetting to divide the later payments by the extra years. Each payment must be divided by $(1 + y)$ once for each year it waits, so a year 3 payment uses the power 3.
- Leaving out the face value. The repayment of the 1000 dollars at the end is usually the largest single payment, and it must be added in its own term.
- Expecting the price to rise when the yield rises. A higher yield means a larger divisor in every term, so the sum gets smaller and the price falls.

## Where you will meet it

- Pricing government and corporate bonds when they are issued or traded.
- Working out the yield to maturity, the yield that makes the formula equal the market price.
- Measuring how much a bond portfolio would gain or lose if interest rates change.
- Comparing bonds with different coupons and lengths on a like-for-like basis.

## Questions people ask

### Why is the coupon written as a sum but the face value is not?

The coupon arrives every year, so it needs one term for each year, added together by the sum sign. The face value arrives only once, at the end of year n, so it needs a single fraction with the exponent n.

### What is a bond yield exactly?

The yield is the rate of return that the market requires on a bond, per year, as a decimal. It is set by buyers and sellers, not by the bond's coupon, so it can differ from the coupon rate. When the bond is priced at its face value, the yield and the coupon rate are equal.

### What does the price look like at a 4% yield?

The same 3-year bond sells for about 1027.75 dollars at a 4% yield. Because the yield is lower, each payment is worth more today, so the price is above its face value. That is the mirror image of the 6% case.

## Related formulas

- [Present Value](https://formulabro.yunusemrevurgun.com/finance/present-value/)
- [Present Value of an Annuity](https://formulabro.yunusemrevurgun.com/finance/present-value-of-annuity/)
- [Macaulay Duration](https://formulabro.yunusemrevurgun.com/finance/macaulay-duration/)

## Sources

- [Bond valuation, Wikipedia](https://en.wikipedia.org/wiki/Bond_valuation)
- [Yield to maturity, Wikipedia](https://en.wikipedia.org/wiki/Yield_to_maturity)

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