# 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.

Phiên bản tương tác: https://www.calcopenly.com/vi/finance/annuity-calculator
Chủ đề: Máy tính tài chính

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.

## Dữ liệu đầu vào

- **Tôi muốn tính** (lựa chọn: 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** (lựa chọn: Năm, Tháng)
- **Interest rate (per year)**: Nominal yearly rate, compounded once per payment period
- **Payments every** (lựa chọn: Week, Two weeks, Tháng, Quarter, Half-year, Năm)
- **Payments made at** (lựa chọn: End (ordinary), Start (annuity due))

## Kết quả

- Future value — kết quả chính
- 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

## Công thức

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

## Ví dụ có lời giải

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

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

### Same payments as an annuity due

- Tôi muốn tính: Future value of regular payments
- Payment each period: 1000
- Starting balance: 0
- Term: 10
- Term in: Năm
- Interest rate (per year): 5%
- Payments every: Năm
- Payments made at: Start (annuity due)
- **Future value: 13,206.79**
- Nguồn đối chiếu: 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

- Tôi muốn tính: Future value of regular payments
- Payment each period: 200
- Starting balance: 500
- Term: 10
- Term in: Tháng
- Interest rate (per year): 6%
- Payments every: Tháng
- Payments made at: Start (annuity due)
- **Future value: 2,581.40**
- Nguồn đối chiếu: 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%

- Tôi muốn tính: Payout from a lump sum over a set time
- Lump sum at the start: 59,777.15
- Term: 20
- Term in: Năm
- Interest rate (per year): 8%
- Payments every: Tháng
- Payments made at: End (ordinary)
- **Payment each period: 500.00**
- Nguồn đối chiếu: 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%

- Tôi muốn tính: Payout from a lump sum over a set time
- Lump sum at the start: 10,000
- Term: 10
- Term in: Tháng
- Interest rate (per year): 8%
- Payments every: Tháng
- Payments made at: End (ordinary)
- **Payment each period: 1,037.03**
- Nguồn đối chiếu: Microsoft PMT documentation: =PMT(8%/12, 10, 10000) returns ($1,037.03)

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

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

## Câu hỏi

### 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)” chính xác đến mức nào?

Độ chính xác phụ thuộc vào dữ liệu nhập và giả định của phương pháp. Phép tính thập phân dùng 50 chữ số có nghĩa, nhưng ước lượng, phương pháp số và dữ liệu nguồn có thể kém chính xác hơn; làm tròn khi hiển thị không loại bỏ các giới hạn đó. Ví dụ có lời giải đã đối chiếu với nguồn độc lập: 8. Ví dụ, “Future value of 1,000 a year for 10 years at 5%” được kiểm tra bằng Future value annuity factor table, 5% and 10 periods: 12.5779; Python decimal 12,577.8925355….

### Phương pháp này lấy từ đâu?

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).

## Nguồn

- [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)

_Chỉ dùng để lập kế hoạch. Bên cho vay, cơ quan thuế và thị trường áp dụng cách làm tròn, phí và quy tắc riêng; hãy xác nhận số liệu với họ trước khi cam kết._