# Annuity calculator (growth and payout)

> Calculate an annuity's future value from regular payments, the payout a lump sum buys over N years, or how long it lasts at a set withdrawal.

Versión interactiva: https://www.calcopenly.com/es/finance/annuity-calculator
Tema: Calculadoras financieras

An annuity is a series of equal payments at a fixed interest rate. Saving into one, the future value is the payment times ((1 + i)^n − 1) ÷ i, where i is the rate per period and n the number of payments; paying out of one, the payment a lump sum supports is the lump sum times i ÷ (1 − (1 + i)^−n). Payments at the start of each period (an annuity due) earn one extra period of interest. A third mode solves for n, how long a lump sum lasts.

With the defaults, 500 at the end of each month for 20 years at 5% grows to 205,516.83 from 120,000 of payments. Paid out instead, 500,000 at 5% supports 3,299.78 a month for 20 years, and 3,000 a month lasts 23.76 years: 285 full withdrawals and a final 427.00.

The rate is a nominal yearly rate divided by the number of payments a year. Insurance annuity fees, surrender charges, taxes and inflation are not included.

## Datos

- **Quiero calcular** (opciones: Future value of regular payments, Payout from a lump sum over a set time, How long a lump sum lasts)
- **Payment each period**
- **Starting balance**
- **Lump sum at the start**
- **Withdrawal each period**
- **Term**
- **Term in** (opciones: Años, Meses)
- **Interest rate (per year)**: Nominal yearly rate, compounded once per payment period
- **Payments every** (opciones: Week, Two weeks, Mes, Quarter, Half-year, Año)
- **Payments made at** (opciones: End (ordinary), Start (annuity due))

## Resultados

- Future value — resultado principal
- Payment each period
- Money lasts (years)
- Number of payments
- Final partial payment
- Present value of the payments
- Total of all payments
- Interest earned
- Rate per period

## Fórmula

$$
FV = S(1+i)^n + P\,\frac{(1+i)^n-1}{i}(1+i)^{\delta},\qquad P = \frac{PV\,i}{\big(1-(1+i)^{-n}\big)(1+i)^{\delta}},\qquad n = -\frac{\ln\!\big(1 - \frac{PV\,i}{W(1+i)^{\delta}}\big)}{\ln(1+i)}
$$

## Ejemplos resueltos

### Future value of 1,000 a year for 10 years at 5%

- Quiero calcular: Future value of regular payments
- Payment each period: 1000
- Starting balance: 0
- Term: 10
- Term in: Años
- Interest rate (per year): 5%
- Payments every: Año
- Payments made at: End (ordinary)
- **Future value: 12,577.89**
- **Total of all payments: 10,000.00**
- **Interest earned: 2,577.89**
- Fuente de comprobación: Future value annuity factor table, 5% and 10 periods: 12.5779; Python decimal 12,577.8925355…

### Same payments as an annuity due

- Quiero calcular: Future value of regular payments
- Payment each period: 1000
- Starting balance: 0
- Term: 10
- Term in: Años
- Interest rate (per year): 5%
- Payments every: Año
- Payments made at: Start (annuity due)
- **Future value: 13,206.79**
- Fuente de comprobación: 12,577.89 × 1.05 (annuity-due factor); Python decimal 13,206.7871623…

### Excel FV example: 500 now plus 200 at the start of each month, 6%, 10 months

- Quiero calcular: Future value of regular payments
- Payment each period: 200
- Starting balance: 500
- Term: 10
- Term in: Meses
- Interest rate (per year): 6%
- Payments every: Mes
- Payments made at: Start (annuity due)
- **Future value: 2,581.40**
- Fuente de comprobación: Microsoft FV documentation: =FV(0.06/12, 10, -200, -500, 1) returns $2,581.40

### Excel PV example reversed: 59,777.15 paid out monthly for 20 years at 8%

- Quiero calcular: Payout from a lump sum over a set time
- Lump sum at the start: 59,777.15
- Term: 20
- Term in: Años
- Interest rate (per year): 8%
- Payments every: Mes
- Payments made at: End (ordinary)
- **Payment each period: 500.00**
- Fuente de comprobación: Microsoft PV documentation: =PV(0.08/12, 12*20, 500, , 0) returns ($59,777.15)

### Excel PMT example: 10,000 over 10 months at 8%

- Quiero calcular: Payout from a lump sum over a set time
- Lump sum at the start: 10,000
- Term: 10
- Term in: Meses
- Interest rate (per year): 8%
- Payments every: Mes
- Payments made at: End (ordinary)
- **Payment each period: 1,037.03**
- Fuente de comprobación: Microsoft PMT documentation: =PMT(8%/12, 10, 10000) returns ($1,037.03)

### How long 100,000 lasts at 6% with 1,000 a month

- Quiero calcular: How long a lump sum lasts
- Lump sum at the start: 100,000
- Withdrawal each period: 1000
- Interest rate (per year): 6%
- Payments every: Mes
- Payments made at: End (ordinary)
- **Number of payments: 138.98**
- **Money lasts: 11.58 years**
- **Final partial payment: 975.78**
- Fuente de comprobación: Python decimal: −ln(1 − 100,000 × 0.005/1,000)/ln 1.005 = 138.9757 months; period-by-period simulation for the final payment

## Preguntas

### How do you calculate the future value of an annuity?

Multiply the payment by ((1 + i)^n − 1) ÷ i, where i is the rate per period and n the number of payments. 1,000 at the end of each year for 10 years at 5% grows to 1,000 × 12.5779 = 12,577.89; paid at the start of each year (an annuity due) it grows to 13,206.79, 5% more, because every payment earns one more year.

### How much does an annuity pay per month?

Divide the lump sum by the present-value factor (1 − (1 + i)^−n) ÷ i. 500,000 at 5% a year (0.4167% a month) paid over 20 years gives 3,299.78 a month, 791,946.89 in total. Microsoft's PV documentation gives the reverse: 500 a month for 20 years at 8% costs 59,777.15 today.

### How long will a lump sum last with monthly withdrawals?

Solve n = −ln(1 − P·i ÷ W) ÷ ln(1 + i), with P the lump sum, i the monthly rate and W the withdrawal. 100,000 at 6% with 1,000 taken at the end of each month lasts 138.98 months: 138 full payments and a final 975.78, about 11.6 years. If W is no more than P·i (500 here), the balance never runs out.

### What is the difference between an ordinary annuity and an annuity due?

An ordinary annuity pays at the end of each period, like loan repayments; an annuity due pays at the start, like rent. An annuity due is worth (1 + i) times the ordinary one, both in future value and in present value. At 5% a year, a 10-year annuity due of 1,000 accumulates 13,206.79 against 12,577.89.

### ¿Qué precisión tiene «Annuity calculator (growth and payout)»?

La precisión depende de tus datos y de los supuestos del método. El cálculo decimal usa 50 cifras significativas, pero las estimaciones, los métodos numéricos y los datos de origen pueden ser menos precisos; el redondeo mostrado no elimina esos límites. Ejemplos resueltos comprobados con fuentes independientes: 8. Por ejemplo, «Future value of 1,000 a year for 10 years at 5%» se comprueba con Future value annuity factor table, 5% and 10 periods: 12.5779; Python decimal 12,577.8925355….

### ¿De dónde procede el método?

Microsoft Excel FV, PV, PMT and NPER functions (type 0 = end, 1 = start of period); Kellison, The Theory of Interest, 3rd ed., ch. 3 (annuities-immediate and annuities-due).

## Fuentes

- [Microsoft Excel FV, PV, PMT and NPER functions (type 0 = end, 1 = start of period)](https://support.microsoft.com/office/pv-function-23879d31-0e02-4321-be01-da16e8168cbd)
- Kellison, The Theory of Interest, 3rd ed., ch. 3 (annuities-immediate and annuities-due)

_Solo para planificar. Prestamistas, autoridades fiscales y mercados aplican sus propios redondeos, comisiones y reglas; confirma las cifras con ellos antes de comprometerte._