How to use the forward rate from spot rates calculator
- Enter the spot rates, one per line, as the number of years and the rate in percent, with years going up from line to line.
- Choose how the rates are compounded: annually, twice a year, quarterly, monthly or continuously.
- Enter the start and end of the forward period you want. Each must be a year from your list, and the start can be 0 for today.
- Read the forward rate, the compounding check and the table of forward rates between neighboring maturities.
Formula
growth to t at spot rate s with m periods a year = (1 + s / m)^(m x t); continuous: e^(s x t)
forward growth from t1 to t2 = growth(t2) / growth(t1)
forward rate f = m x ((growth(t2) / growth(t1))^(1 / (m x (t2 - t1))) - 1); continuous: (s2 x t2 - s1 x t1) / (t2 - t1)
The idea is that investing to t1 and then at the forward rate to t2 must give the same growth as investing straight to t2. The page checks this and shows both growth figures.
Between year 2 and year 5
With a 2-year spot rate of 3.4% and a 5-year spot rate of 4.0%, compounded annually, 1.04^5 = 1.216653 and 1.034^2 = 1.069156. Their ratio is 1.137935 and its cube root (the gap is 3 years) is 1.044019, so the forward rate is 4.4019%.
Between years 1 and 2 the forward rate is 1.034^2 / 1.03 - 1 = 3.8016%, and from today to year 1 it equals the 1-year spot rate of 3.0000%. With continuous compounding the 1-to-2 year forward rate is 3.4 x 2 - 3.0 x 1 = 3.8000%.
What a forward rate means
A forward rate is the rate for a future stretch of time that is implied by today's rates for longer and shorter periods. If the curve slopes upward, forward rates lie above spot rates. They are the rates at which an investor would be indifferent between a long investment and a shorter one followed by a roll-over, so they carry no promise about what rates will really be.
Spot rates here are zero coupon rates, with one payment at the end. Rates from coupon bonds must be turned into zero rates first. For yields from bond prices see the Bond Yield Calculator, to discount cash flows the Present Value Calculator, and for the price sensitivity of bonds the Bond Duration and Convexity Calculator.
Checking your input
Years must be positive and increase down the list. Rates are typed as percentages, so 3.4 means 3.4%, and the page asks again if a rate looks like a fraction typed by mistake. The compounding you choose must match how your rates were quoted, or the forward rates will be slightly off.
Assumptions and limits
- The spot rates are zero coupon rates for the maturities given, with no bid and ask difference.
- The same compounding convention applies to all the spot rates and the forward rate.
- Up to 40 maturities are accepted and the start of a forward period can be 0.
- Rates are shown to four decimal places of a percent and calculated with ordinary floating point numbers, checked in the tests against 60 digit decimals.
- Taxes, fees and day count conventions are not modeled.
Frequently asked questions
Is a forward rate a forecast?
No. It is the break-even rate that follows from today's rates. Actual rates later can be higher or lower.
Why does the compounding matter?
The same quote means a slightly different growth under annual and monthly compounding, so the implied forward rate differs a little. Choose the one your quotes use.
Why can a forward rate be negative?
If the longer spot rate is far below the shorter one, growth to the later date is less than growth to the earlier date plus any gain, which means a negative implied rate.
What does the growth check show?
It compounds the shorter spot rate and then the forward rate. The result must equal compounding the longer spot rate, and the two numbers should match.