How the Bond Calculator Works
A bond is a loan that you make to a government or a company. In return, the issuer promises to pay you regular interest, called coupons, and to repay the face value on a set date, the maturity. Bonds can be bought and sold before they mature, and their prices move up and down with interest rates. The Bond Calculator works out the price of a bond from its yield, or the yield from its price, and shows how sensitive the bond is to changes in rates.
What you enter
- Face value: the amount the bond repays at maturity, also called par. It is often $1,000.
- Coupon rate: the yearly interest the bond pays, as a percentage of its face value. A zero-coupon bond pays none.
- Years to maturity: how long until the bond repays its face value.
- How often interest is paid: once a year, twice a year, quarterly or monthly. Twice a year is the most common.
- Yield to maturity, or price: enter the yield you want to see the price, or enter the price to see the yield.
How the price is worked out
A bond is worth the value today of everything it will pay. Each payment is discounted back to today at the yield. With a coupon payment C, a yield per payment period of i, n payments in all, and a face value F:
price = C / (1 + i) + C / (1 + i)^2 + ... + C / (1 + i)^n + F / (1 + i)^n
The yield to maturity is the yearly return you earn if you buy at that price and hold the bond until it matures, with all the payments made on time. There is no simple formula for it, so the calculator finds it by trial and refinement.
A worked example
A bond has a face value of $1,000, a 5% coupon paid twice a year, and 10 years to maturity. Each payment is $25. The market yield for similar bonds is 6%.
| Result | Amount |
|---|---|
| Price | $925.61 |
| Coupon payment | $25.00 every 6 months ($50 a year) |
| Current yield ($50 / $925.61) | 5.402% |
| Total coupons over 10 years | $500.00 |
| Gain when it matures ($1,000 - $925.61) | $74.39 |
| Yield to maturity | 6% |
The bond pays only 5%, but buyers want 6%, so its price falls below its face value. It is sold at a discount. The discount, plus the coupons, gives the buyer the 6% return. If the market yield is lower than the coupon rate, the bond sells at a premium, above its face value. If the two are equal, it sells at par. At a yield of 4%, this bond would cost $1,081.76.
Why bond prices move opposite to rates
When interest rates rise, new bonds pay more, so older bonds with lower coupons become less attractive, and their prices fall. When rates fall, older bonds with higher coupons become more attractive, and their prices rise. The same bond at different yields:
| Market yield | Price | Change from $925.61 |
|---|---|---|
| 4% | $1,081.76 | +16.9% |
| 5% | $1,000.00 | +8.0% |
| 6% | $925.61 | 0% |
| 7% | $857.88 | -7.3% |
| 8% | $796.15 | -14.0% |
Chart: the price at different yields
4% ████████████████████████████████████████ $1,081.76 5% █████████████████████████████████████ $1,000.00 6% ██████████████████████████████████ $925.61 7% ██████████████████████████████████ $857.88 8% ███████████████████████████████ $796.15
The calculator draws this curve for your own bond. Notice that it bends: a fall in yields raises the price by a little more than the same rise in yields lowers it.
Duration: how sensitive is the bond?
Duration measures how long it takes you to get your money back, on average, with each payment weighted by its size and timing. The Macaulay duration is in years. The modified duration is the useful one: it tells you roughly how much the price falls for each 1% rise in yields. In the example, the Macaulay duration is 7.90 years, and the modified duration is 7.67. So a 1% rise in yields lowers the price by about 7.7% (the exact drop is 7.3%, because of the bend in the curve). Longer bonds and bonds with low coupons have a longer duration, so their prices move more. A zero-coupon bond's duration equals its maturity: 10 years for a 10-year bond.
Tips
- If you hold to maturity, price changes along the way do not matter, as long as the issuer pays. You get the face value back.
- Longer bonds are riskier when rates change, and they usually pay a higher yield to make up for it.
- Credit risk matters too. A bond from a weaker issuer pays a higher yield, because there is a bigger chance that it will not pay in full.
- Compare yields, not coupons. The yield to maturity counts the price you pay, so it is the better way to compare bonds.
What this calculator does not include
- Accrued interest when you buy between coupon dates. The calculator assumes you buy on a coupon date.
- Bonds that the issuer can pay off early (callable bonds), floating-rate bonds, and inflation-linked bonds.
- Taxes on the interest and on any gain, and fees or commissions.
- The chance that the issuer does not pay.
This tool gives an estimate for general information only. It is not financial or investment advice. Bond prices can fall, and you can lose money if you sell before maturity.