Force Of Interest Calculator

Q: When is δ used in practice?

In actuarial and derivative pricing work, where cash flows and mortality are modelled continuously rather than at discrete year ends.

Work out force of interest instantly with clear inputs, formula shown and shareable results.

About Force Of Interest Calculator

The force of interest is the instantaneous rate of growth, δ = ln(1 + i). It is the continuous-time counterpart of an effective annual rate and is the natural rate to use whenever interest is credited without interruption.

Formula

Force of interest

δ = ln(1 + i); A(t) = e^(δt)

i: Effective annual rate
t: Time in years

Worked example

With the default values below, Force Of Interest Calculator returns Force Of Interest (δ) of 5.826891%. Change any input above and every figure here is recalculated in your browser.

Inputs used

Effective Annual Rateaccepts 0–100 %: 6 %
Yearsaccepts 0.25–100: 5

Results

Force Of Interest (δ): 5.826891%
Accumulation Factor e^(δt): 1.338226
Total Growth Over Period: 33.8226%
Equivalent Nominal Rate (Monthly): 5.841061%

Step by step

1Values used
2Force of interest
3Force Of Interest (δ)= 5.826891
4Accumulation Factor e^(δt)= 1.338226
5Total Growth Over Period= 33.8226
6Equivalent Nominal Rate (Monthly)= 5.841061

Assumptions and limits

The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax or fees or charges into account.
Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Is the force of interest smaller than the effective rate?

Yes, for positive rates. Because ln(1+i) < i, a 6% effective rate corresponds to a force of about 5.83%.

When is δ used in practice?