# 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.

직접 계산할 수 있는 페이지: https://www.calcopenly.com/ko/finance/annuity-calculator
분야: 금융 계산기

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.

## 입력

- **계산할 항목** (선택 항목: 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** (선택 항목: 년, 개월)
- **Interest rate (per year)**: Nominal yearly rate, compounded once per payment period
- **Payments every** (선택 항목: Week, Two weeks, 월, Quarter, Half-year, 년)
- **Payments made at** (선택 항목: End (ordinary), Start (annuity due))

## 결과

- Future value — 주요 결과
- 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

## 공식

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

## 계산 예제

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

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

### Same payments as an annuity due

- 계산할 항목: Future value of regular payments
- Payment each period: 1000
- Starting balance: 0
- Term: 10
- Term in: 년
- Interest rate (per year): 5%
- Payments every: 년
- Payments made at: Start (annuity due)
- **Future value: 13,206.79**
- 검증 출처: 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

- 계산할 항목: Future value of regular payments
- Payment each period: 200
- Starting balance: 500
- Term: 10
- Term in: 개월
- Interest rate (per year): 6%
- Payments every: 월
- Payments made at: Start (annuity due)
- **Future value: 2,581.40**
- 검증 출처: 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%

- 계산할 항목: Payout from a lump sum over a set time
- Lump sum at the start: 59,777.15
- Term: 20
- Term in: 년
- Interest rate (per year): 8%
- Payments every: 월
- Payments made at: End (ordinary)
- **Payment each period: 500.00**
- 검증 출처: 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%

- 계산할 항목: Payout from a lump sum over a set time
- Lump sum at the start: 10,000
- Term: 10
- Term in: 개월
- Interest rate (per year): 8%
- Payments every: 월
- Payments made at: End (ordinary)
- **Payment each period: 1,037.03**
- 검증 출처: Microsoft PMT documentation: =PMT(8%/12, 10, 10000) returns ($1,037.03)

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

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

## 자주 묻는 질문

### 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.

### “Annuity calculator (growth and payout)”의 정확도는 어느 정도인가요?

정확도는 입력값과 계산 방법의 가정에 따라 달라집니다. 십진 연산은 유효숫자 50자리를 사용하지만, 추정값·수치해석 방법·원본 데이터의 정밀도는 더 낮을 수 있습니다. 표시값을 반올림해도 이러한 한계는 사라지지 않습니다. 독립적인 출처의 풀이와 대조한 계산 예시: 8. 예를 들어 “Future value of 1,000 a year for 10 years at 5%”은 Future value annuity factor table, 5% and 10 periods: 12.5779; Python decimal 12,577.8925355…와 대조해 확인합니다.

### 이 계산 방법의 출처는 무엇인가요?

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).

## 출처

- [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)

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