ConverterHub

Force of Interest Calculator

Enter the effective annual rate to find the force of interest.

How it is calculated

Enter the effective annual interest rate i as a percentage. The force of interest is δ = ln(1 + i), with i written as a decimal (6% is 0.06). It is shown as a percentage and as a decimal to six places. The tool also gives the equivalent nominal annual rate compounded monthly, i(12) = 12 × ((1 + i)1/12 − 1). A negative rate is allowed as long as it stays above −100%; at −100% or below the logarithm does not exist and there is no result. Because ln(1 + i) is always smaller than i for a positive rate, δ comes out below the effective rate, and the monthly nominal rate lies between the two.

Worked example

An account pays an effective annual rate of 6%. Write it as i = 0.06, so 1 + i = 1.06. The natural logarithm of 1.06 is 0.058269, so δ = 0.058269, which is 5.83% as a percentage. For monthly compounding, 1.06 raised to the power 1/12 is about 1.004868; subtracting 1 and multiplying by 12 gives 0.0584, shown as 5.84%. So continuous growth at 5.83% a year gives the same result after one year as 6% effective: e0.058269 = 1.06.

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Common mistakes

  • Typing the rate as a decimal (0.06) instead of a percentage (6). The tool then treats it as 0.06% and the force of interest is about 0.06%.
  • Giving a nominal rate where an effective rate is expected. If the quoted 6% is already compounded monthly, convert it to an effective annual rate first, otherwise δ is slightly off.
  • Expecting δ to be larger than the effective rate. Continuous compounding at δ equals the effective rate after one year, so δ must be smaller for a positive rate.
  • Confusing the force of interest with the simple or the nominal rate. Only the monthly figure is a nominal rate and it is not equal to δ.
  • Entering −100% or less for a total-loss scenario. 1 + i would be zero or negative, so the logarithm is undefined and the tool returns nothing.

Sources

About this calculator

The force of interest, denoted δ (delta), is the continuously compounded equivalent of a stated effective interest rate, defined as δ = ln(1 + i) where i is the effective annual rate. It represents the instantaneous rate at which money grows at every single moment rather than at discrete annual, monthly, or daily compounding intervals, and it's the rate that appears inside continuous-compounding formulas like A = Pe^(δt).

This calculator converts an effective annual interest rate into its force of interest. Actuaries use δ constantly in life contingency and annuity valuation formulas where continuous compounding simplifies the math, and finance students and quants use it to move between nominal, effective, and continuously compounded rate conventions when pricing bonds or derivatives.

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