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Commutation Function Calculator

Work out commutation function instantly with clear inputs, formula shown and shareable results.

About Commutation Function Calculator

Commutation functions pre-combine interest and mortality so that annuity and assurance values become simple ratios. Dₓ = v^x·lₓ is the discounted number of survivors and Nₓ is the running total of Dₓ from age x upwards, giving äₓ = Nₓ/Dₓ.

Formula

Commutation columns

Dₓ = v^x·lₓ; Nₓ = ΣDₓ₊ₖ; äₓ = Nₓ/Dₓ

v
1/(1+i)
lx
Lives at age x

Worked example

With the default values below, Commutation Function Calculator returns Discounted Lives Dₓ of 10,239.28. Change any input above and every figure here is recalculated in your browser.

Inputs used

Age xaccepts 0–110
45
Lives At Age xaccepts 1–10,000,000
92,000
Interest Rateaccepts 0.01–25 %
5 %
Annual Decrement Rateaccepts 0.01–40 %
1.5 %

Results

Discounted Lives Dₓ
10,239.28
Commutation Column Nₓ
165,403.73
Life Annuity-Due Factor äₓ
16.1538 ×
Discount Factor v^x
0.11129651

Step by step

  1. 1Values used
  2. 2Commutation columns
  3. 3Discounted Lives Dₓ= 10,239.28
  4. 4Commutation Column Nₓ= 165,403.73
  5. 5Life Annuity-Due Factor äₓ= 16.1538 ×
  6. 6Discount Factor v^x= 0.11129651

Assumptions and limits

  • The result is derived only from the 4 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 8 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

Why were commutation functions invented?

Before computers they turned every annuity valuation into one division, using columns tabulated once per mortality basis.

What does the decrement rate do here?

It models survivors falling by a constant proportion each year, which makes Nₓ a geometric sum and keeps the illustration self-contained.

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