How to Calculate Bond Price and Interest Rate
For a bond paying a fixed coupon forever, price equals the coupon divided by the interest rate, so bond prices and interest rates move in opposite directions.
The Bond Price formula
Calculator
Enter a bond's fixed annual coupon and two interest rates to see how far its price falls when rates rise.
Coupon divided by the interest rate as a decimal, for a coupon paid forever.
- Price at the new rate
- $500
- Change in price
- −50%
Rates up, price down: the inverse relationship behind open market operations.
How to calculate Bond Price, step by step
- 1Identify the annual coupon payment. The fixed dollar amount the bond pays its holder each year. This amount is set when the bond is issued and does not change.
- 2Identify the current interest rate. The going market interest rate on comparable bonds, written as a decimal (5% is 0.05).
- 3Divide the coupon by the interest rate. Price = Coupon ÷ Interest rate. This treats the bond as a perpetuity, a bond that pays its coupon forever, which is the version tested in AP Macro.
- 4Reprice at a new rate to see the relationship. Divide the same coupon by a new interest rate. A higher rate in the denominator gives a lower price; a lower rate gives a higher price.
Worked example: Bond Price
A bond pays a fixed coupon of $50 a year forever and the interest rate is 5%. Price = 50 ÷ 0.05 = $1,000. If the interest rate rises to 10%, price = 50 ÷ 0.10 = $500, so a five percentage point rate rise cuts the price in half.
Why the price has to move, not just the return
Think of two bonds sitting side by side. Bond A was issued earlier and pays a fixed $50 a year. Bond B is issued today, when the interest rate is 10%, and also pays $50 a year, priced at 50 divided by 0.10, which is $500.
If Bond A still sold for its old price of $1,000, nobody would buy it. Paying $1,000 for a $50 annual payment is a 5% return, while Bond B pays the same $50 for only $500, a 10% return. Money moves to Bond B until Bond A's price falls to $500 too, at which point both bonds offer the same return to a new buyer.
That is the whole mechanism behind the formula. The interest rate is the return a dollar can earn elsewhere. A bond's price adjusts until its fixed coupon, divided by that price, matches the rate available everywhere else. Rearranging price = coupon ÷ rate into rate = coupon ÷ price makes this explicit: the price and the rate are two ways of describing the same trade, so one cannot move without the other.
Why this matters for open market operations
The Federal Reserve does not set interest rates by decree. It buys and sells government bonds, and the price formula is the reason that changes rates at all.
When the Fed buys bonds in the open market, it adds a large buyer to the market, and bond prices rise the same way any price rises when demand goes up. Look at the formula rearranged as rate = coupon ÷ price. The coupon on existing bonds is fixed, so if the price in the denominator rises, the rate it implies falls. That fall in interest rates is the point of the purchase: it is meant to make borrowing cheaper and push up spending on investment and durable goods.
Selling bonds runs the same logic in reverse. The Fed adds supply to the market, bond prices fall, and coupon divided by a smaller price gives a higher rate. That is how open market sales tighten monetary policy.
This is also why a free-response question that asks you to trace an open market purchase through to interest rates expects you to state the bond-price step explicitly: purchase raises bond prices, and higher bond prices mean lower interest rates, not the other way around.
Bond Price questions
Why do bond prices fall when interest rates rise?
A bond already in circulation pays a fixed coupon set at issue. When the interest rate on new bonds rises, that fixed coupon looks less attractive next to what new bonds now pay. Buyers will only hold the old bond if its price falls enough that the coupon represents the same return the new, higher rate offers.
Does a higher price change the coupon a bond pays?
No. The coupon payment is fixed in dollar terms when the bond is issued and never changes. It is the price that moves to keep the bond's return in line with current interest rates, not the payment.
Why does the AP exam use price = coupon divided by rate instead of a present value sum?
That formula is for a perpetuity, a bond with no maturity date that pays its coupon every year without end. Summing the present value of a coupon paid forever reduces to exactly coupon divided by rate, so AP Macro uses this shortcut rather than a multi-year present value calculation.
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