The annuity payment formula is used to calculate the regular payment on an annuity – a series of payments received at a future date. This is the exact same formula used in loan payments.
P = \frac{r(PV)}{1 - (1 + r)^{-n}}Where P = the payment, PV = the present value, r = the rate per loan period, and n = the number of periods.
The present value part of the formula is the initial payment, for example the original payout on an amortized loan.
As with the loan payment formula, the annuity payment formula assumes the rate and payments stay the same, and that the first payment is one period away – which means it can only be used for ordinary annuities. If the annuity grows at a proportional rate, then the growing annuity payment formula can be used.
An annuity where the payment or rate changes can be calculated with this formula, but it should be calculated for each period separately, using the most recent values.
The annuity payment formula can be used for:
- amortized loans
- income annuities
- lottery payouts (you can also see annuity due formula if the first payment starts immediately)
- structured settlements
- any constant period payment (i.e loan payments)
Annuity payment formula example
Jack is entitled to receive $100,000 after his retirement from his company. However, the company offers him to receive fixed yearly payments for the next 20 years if he agrees to purchase an annuity of this amount at an interest rate of 3%. Find the fixed periodic payment that he is expected to receive for the next 20 years.
- Present Value (PV) = $ 100, 000
- Interest Rate (r) = 3%
- Total number of periods (n) = 20
The total amount received by Jack in 20 years would be $134,431.42.So why is the total interest $34,431.42 in 20 years? The answer lies in the time value of money which states that a dollar today is worth more than a dollar from future. The fixed payment of annuity received by Issuer is invested in different projects with a guaranteed return. As a result, some portion of that return is paid to the purchaser of the annuity by annuity issuer.
Alternate annuity payment formulas
Another type of annuity requires periodic payments, and against those periodic payments, the purchaser is entitled to a certain sum of series of fixed payments. The formula for Future Value of an Annuity predicts the future value of an annuity to be paid to the purchaser as a result of a series of periodic payments. This formula assumes that the rate, periodic payment does not change, and the first payment is just one period away.
FV_{Annuity} = \frac{P((1+r)^{n}-1)}{r}Where P = the periodic payment, r = the rate per loan period, and n = the number of periods.