An annuity factor is the present value of receiving one unit of payment each period for a fixed number of periods, and it is used to convert between a lump sum and equal periodic payments.
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Core formula
For level annual payments over (n) periods at interest rate (r):
$$ a_{\overline{n}|r} = \sum_{t=1}^{n} \frac{1}{(1+r)^t} = \frac{1 - (1+r)^{-n}}{r} $$
If the payment each period is (PMT), then:
$$ PV = PMT \times a_{\overline{n}|r} $$
That is why the annuity factor appears in mortgage payments, loan amortization, and pension valuation.
Economic intuition
The factor is larger when:
- the interest rate is lower, because future payments are discounted less,
- the payment horizon is longer, because there are more payments to value.
So a higher annuity factor means a given lump sum can support only a smaller periodic payment.
Related Terms
Knowledge Check
### What does the annuity factor convert?
- [x] a lump sum and a stream of equal payments
- [ ] an exchange rate and inflation
- [ ] profits and tax rates
- [ ] labor and capital shares
> **Explanation:** The annuity factor is the bridge between present value and equal periodic payments.
### Holding the payment horizon fixed, the annuity factor is larger when:
- [x] the interest rate is lower
- [ ] the interest rate is higher
- [ ] there is no time value of money
- [ ] the number of payments falls to zero
> **Explanation:** Lower discounting makes future payments more valuable in present-value terms.
### Why does the annuity factor matter for retirement economics?
- [x] It helps translate accumulated wealth into sustainable periodic income
- [ ] It fixes stock returns by law
- [ ] It removes longevity risk automatically
- [ ] It determines GDP growth
> **Explanation:** Retirement planning depends on converting assets into income over time.