What this calculator does
If a five year rate is 6 per cent and a three year rate is 3 per cent, the market is implying something quite specific about years four and five. Investing for five years must equal investing for three and then rolling into a two year rate, and that two year rate is the implied forward.
The implied rate can be far above either spot rate. A 6 per cent five year rate against a 3 per cent three year rate implies 10.7 per cent for the intervening two years, because the later period has to carry all the extra return.
The formula
Compound the longer rate over its full term and the shorter rate over its term, divide one by the other, then take the root corresponding to the gap between them and subtract one.
| Term | Meaning |
|---|---|
| Spot rate | The rate for an investment running from today to a given maturity. |
| Forward rate | The rate implied for a future period between two maturities. |
| No-arbitrage | The principle that the two routes to the same date must give the same return. |
The inputs explained
| Field | What to enter |
|---|---|
| Spot rate for the longer period (%) | The spot rate for the longer period. |
| Longer period (years) | The longer period in years. |
| Spot rate for the shorter period (%) | The spot rate for the shorter period. |
| Shorter period (years) | The shorter period in years. |
When to use it
Reading the yield curve
Forward rates show what the market implies about future short-term rates.
Pricing a forward rate agreement
The implied forward is the fair fixed rate for the contract.
Deciding between maturities
Comparing the implied forward against your own expectation says which maturity is better value.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What does each long rate imply for years 3 to 5?
Four five year spot rates against the same three year rate.
| Five year spot rate | Implied forward rate | Applies to the period from year |
|---|---|---|
| 4% | 5.52% | 3.0 to year 5.0 |
| 5% | 8.07% | 3.0 to year 5.0 |
| 6% | 10.7% | 3.0 to year 5.0 |
| 7% | 13.3% | 3.0 to year 5.0 |
Questions
Why is the implied forward so much higher than the long rate?
Because the early years are locked in at the lower short rate. To average out to 6 per cent over five years when the first three earn 3 per cent, the remaining two must earn far more than 6 to compensate.
Is the forward rate a forecast of future rates?
Only loosely. It is what the current curve implies under no-arbitrage, but it also embeds a term premium, so it tends to sit above the rate the market genuinely expects.
What is an inverted yield curve?
When shorter rates exceed longer ones, which produces implied forwards below current short rates. It has historically preceded recessions, which is why it attracts so much attention.
Does compounding convention matter?
Yes. This uses annual compounding. Continuous compounding, common in derivatives work, gives slightly different figures, and the two should not be mixed within one calculation.
For the currency version of the same logic, see the interest rate parity calculator. For a bond's yield to maturity, see the bond yield to maturity calculator.