How to use the zero rate bootstrapping calculator
- Enter the par yield for year 1 on the first line, the par yield for year 2 on the second line, and so on.
- Use percent a year, with annual coupons. You can enter up to 30 years.
- Read the discount factor and the zero rate for each year in the table.
- The headline shows the zero rate for the last year you entered.
Formula
DF(n) = (1 - y(n) x (DF(1) + ... + DF(n-1))) / (1 + y(n))
Zero rate z(n) = DF(n) ^ (-1 / n) - 1
The first year has no earlier discount factors, so DF(1) = 1 / (1 + y(1)) and the zero rate equals the first par yield.
Par yields of 3.00%, 3.40%, 3.75%, 4.00% and 4.20%
Year 1 has a discount factor of 1 / 1.03 = 0.970874 and a zero rate of 3.0000%. For year 2, DF = (1 - 0.034 x 0.970874) / 1.034 = 0.935194, which gives a zero rate of 3.4068%.
Continuing the same way gives discount factors of 0.894961, 0.853807 and 0.812377 for years 3 to 5, and zero rates of 3.7684%, 4.0304% and 4.2434%. The zero rates sit above the par yields because the later coupons are discounted at rising rates. A flat 4% curve gives a flat 4% zero curve.
Why zero rates are useful
A par yield describes a bond that pays coupons, so it mixes rates for several years into one number. A zero rate is the rate for a single payment at a single date, which is what you need to discount any cash flow on its own. Bootstrapping builds the zero rates one year at a time from the par yields: each new par bond has one new unknown, the final discount factor, and the earlier ones are already known.
Once you have discount factors you can price any set of annual cash flows by multiplying each by its factor and adding. The forward rate between two years follows from the ratio of their discount factors, as the Forward Rate from Spot Rates Calculator shows.
Limits of the method
The method needs a par yield for every year with no gaps. Real markets quote only some maturities, so curves are filled in with interpolation, and real bonds pay on specific dates with day counts. This tool uses annual coupons and bonds priced at par. If yields are too different from year to year, a discount factor can become negative, and the tool says so. For the price sensitivity of bonds, use the Bond Duration and Convexity Calculator, for compounding see the Compound Interest Calculator, and for settlements on rate contracts the Forward Rate Agreement Calculator.
Assumptions and limits
- Coupons are paid once a year and every bond is priced at par.
- Yields are consecutive years starting at year 1 with no gaps.
- Zero rates compound once a year.
- The yields shown are examples and not market data.
Frequently asked questions
What is a discount factor?
It is the value today of one dollar paid at a future date. A factor of 0.9 means a dollar in the future is worth 90 cents now.
Why is the first zero rate the same as the first par yield?
With one payment at the end of the first year, a par bond and a zero coupon bond are the same thing.
Can I enter negative yields?
Yes, down to -100%, as long as the discount factors stay positive.
Does it handle semiannual coupons?
No. Convert semiannual yields to an annual equivalent first, or use a tool built for those conventions.