Time Value of Money (TVM) and Annuities
Algebra
Work out what a lump sum needs to be to buy a fixed income for life, using the mathematics of annuities.
Where this idea comes from
Start here — this is the source that inspired this exploration.
Introduction
An annuity converts a lump sum of money into a fixed income paid out over time, and it is the mechanism behind most private pensions. The price of an annuity comes down to one question: what is a promise of future money worth today? Because money now can be invested and grow, money later is worth less, and that difference can be calculated exactly. This exploration builds the present-value formula for an annuity from a geometric series, then tests it against a real product.
Guiding Questions
- An annuity pays a fixed amount each year. Why is money later worth less than money now?
- Write the present value of an annuity as a geometric series and derive the formula.
- Price a real annuity from an advert. What interest rate is the seller assuming?
- How does the deal change with inflation or living longer than expected? Stress-test your model.