How to calculate the fair price of a bond
A bond is a loan split into securities. The issuer, a government or a company, pays a fixed interest, the coupon, every year and repays the face value at maturity. Buying a bond means buying a series of known future payments. Valuing it answers one question: what is that series of payments worth today, given the return the market requires for this kind of investment?
A dollar received in five years is worth less than a dollar today, because today's dollar could be invested in the meantime. Each payment is therefore discounted to its present value by dividing it by (1 + r) raised to the number of years to wait. The bond price is the sum of these present values:
P = Σt=1..n C ÷ (1 + r)t + F ÷ (1 + r)n
F is the face value, C the annual coupon (F × coupon rate), r the market yield, also called yield to maturity, and n the number of years remaining.
A worked example
Take a bond with a $1,000 face value, a 5% coupon ($50 a year) and 5 years to maturity. If the market requires 4%, the five coupons are worth $222.59 today and the final repayment $821.93: the bond is worth $1,044.52, or 104.45% of face value. It trades at a premium because it pays 5% while the market only asks for 4%. If the market requires 6% instead, the same bond is worth only $957.88: it trades at a discount. At 5%, exactly its coupon rate, it is worth $1,000.00: it trades at par.
The inverse relationship between rates and bond prices
This is the fundamental rule of the bond market: when interest rates rise, the prices of existing bonds fall, and vice versa. Imagine you hold the 5% bond and new issues now pay 6%. Nobody will pay you $1,000 for your bond when they can earn more elsewhere. To find a buyer, its price must fall until the buyer also earns 6% a year, counting the gain at maturity: that is the $957.88 price calculated above.
The chart shows this relationship for yields from 0% to 15%. For our example, the price moves from $1,250.00 at 0% (simply the sum of the cash flows) to $664.78 at 15%. The curve is not a straight line but convex: a 1-point fall in rates gains more (+$47.08) than a 1-point rise loses (-$44.52). This asymmetry, convexity, works in the bondholder's favor.
Duration: measuring interest rate risk
Modified duration measures how strongly the price reacts. It is derived from Macaulay duration, the average time until the cash flows are received, weighted by their present value, divided by (1 + r). In our example, Macaulay duration is 4.56 years and modified duration 4.38: a 1-point change in rates moves the price by about 4.38%. The calculator also shows the exact change, recalculated without approximation.
Two features increase duration. Maturity first: with a 3% coupon and a 4% market yield, a 2-year bond is worth $981.14, a 10-year bond $918.89 and a 30-year bond $827.08. The coupon second: the lower it is, the more the value depends on the final repayment. A 10-year zero-coupon bond has a modified duration of 9.62, its maturity divided by (1 + r).
What the calculator does not include
- Accrued interest. Between coupon dates, the buyer also pays the seller the interest accrued since the last coupon. Bond prices are usually quoted "clean", without it.
- Semi-annual coupons. US Treasuries and most US corporate bonds pay twice a year; UK gilts too. Payment frequency changes the price slightly.
- Credit risk. A weaker issuer must offer a higher yield. Include that risk premium in the yield you enter; in a default, payments are not made.
- The yield curve, call features, inflation-linked or floating coupons, fees and taxes.
To project reinvested coupons, use the compound interest calculator.
Table: example bond price by market yield
| Market yield | Bond price | % of face value |
|---|---|---|
| 0% | $1,250.00 | 125% |
| 1% | $1,194.14 | 119.41% |
| 2% | $1,141.40 | 114.14% |
| 3% | $1,091.59 | 109.16% |
| 4% | $1,044.52 | 104.45% |
| 5% | $1,000.00 | 100% |
| 6% | $957.88 | 95.79% |
| 7% | $918.00 | 91.8% |
| 8% | $880.22 | 88.02% |
| 9% | $844.41 | 84.44% |
| 10% | $810.46 | 81.05% |
| 11% | $778.25 | 77.82% |
| 12% | $747.67 | 74.77% |
| 13% | $718.62 | 71.86% |
| 14% | $691.02 | 69.1% |
| 15% | $664.78 | 66.48% |
Frequently asked questions
How do you calculate the price of a bond?
Add up the present value of every remaining payment: each annual coupon and the repayment of face value at maturity, each discounted at the yield the market requires today. The formula is P = Σ C / (1 + r)^t + F / (1 + r)^n, where C is the coupon, F the face value, r the market yield and n the years to maturity.
Why do bond prices fall when interest rates rise?
A fixed-rate bond pays the same coupon whatever happens to rates. If new bonds pay more, an existing bond only finds a buyer if its price drops until its yield matches the market. When rates fall, its fixed coupon becomes more attractive and its price rises.
What do premium, discount and par mean?
A bond trades at par when its price equals its face value (100%), which happens when its coupon rate equals the market yield. It trades at a premium above 100% when its coupon is higher than the market yield, and at a discount below 100% when its coupon is lower.
What is duration?
Macaulay duration is the weighted average time until you receive the bond’s cash flows, in years. Modified duration (Macaulay duration divided by 1 + yield) estimates the percentage price change for a 1-point move in rates: a modified duration of 4.4 means a bond loses about 4.4% if rates rise by 1 percentage point.
Does this match the price quoted by my broker?
Not exactly. The calculator values the bond on a coupon date, without accrued interest, with annual coupons and a single yield for all maturities. US Treasuries and most corporate bonds pay semi-annual coupons, and quoted prices also reflect credit risk, liquidity and the full yield curve. Use it as a close estimate and to understand how prices react to rates.
Educational tool: not financial, tax or investment advice. Results are estimates based on the assumptions you enter and on official rates at the date shown. Investments can fall as well as rise. For decisions about your own situation, speak to a qualified adviser or tax professional. Read the full disclaimer.