# Time value of money calculator (future and present value)

> Solve for the future value, present value, interest rate or time of a lump sum from the other three, with any compounding, including continuous.

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

The time value of money links four numbers for a single sum: the present value PV, the future value FV, the annual rate r and the time t. With compounding k times a year, FV = PV × (1 + r/k)^(kt), and continuous compounding gives FV = PV × e^(rt). Given any three, the fourth follows: dividing FV by the growth factor gives the present value, a root gives the rate, and logarithms give the time.

With the defaults, 10,000 at 6% compounded yearly grows to 17,908.48 in 10 years, 1.79 times the starting amount. Discounting runs the other way: 20,000 due in 10 years is worth 10,992.65 today at 6% compounded monthly.

This covers one deposit with no further payments; for regular contributions use the compound interest or SIP calculator. The rate is a nominal yearly rate, and the effective annual rate shows what the compounding adds to it.

## Dữ liệu đầu vào

- **Solve for** (lựa chọn: Future value, Present value, Rate, Thời gian)
- **Present value**
- **Future value**
- **Interest rate (per year)**
- **Thời gian**
- **Compounding** (lựa chọn: Yearly, Half-yearly, Quarterly, Monthly, Daily (365), Continuous)

## Kết quả

- Future value — kết quả chính
- Present value
- Interest rate needed (per year)
- Time needed (years)
- Compounding periods needed
- Effective annual rate
- Growth multiple (FV ÷ PV)

## Công thức

$$
FV = PV\left(1+\tfrac{r}{k}\right)^{kt}\ \ (\text{continuous: } PV e^{rt}),\quad r = k\left[(FV/PV)^{1/(kt)} - 1\right],\quad t = \frac{\ln(FV/PV)}{k \ln(1+r/k)}
$$

## Ví dụ có lời giải

### Future value of 10,000 at 6% for 10 years

- Solve for: Future value
- Present value: 10,000
- Interest rate (per year): 6%
- Thời gian: 10 years
- Compounding: Yearly
- **Future value: 17,908.48**
- Nguồn đối chiếu: 10000 × 1.06^10 = 17908.4769654… (Python decimal)

### Present value of 20,000 in 10 years at 6% monthly

- Solve for: Present value
- Future value: 20,000
- Interest rate (per year): 6%
- Thời gian: 10 years
- Compounding: Monthly
- **Present value: 10,992.65**
- **Effective annual rate: 6.1678%**
- Nguồn đối chiếu: 20000 / 1.005^120 = 10992.6546672… (Python decimal)

### Excel RRI: 10,000 to 11,000 over 96 months

- Solve for: Rate
- Present value: 10,000
- Future value: 11,000
- Thời gian: 8 years
- Compounding: Monthly
- **Interest rate needed (per year): 1.1920%**
- Nguồn đối chiếu: Microsoft RRI documentation: =RRI(96,10000,11000) = 0.0009933 per month; × 12 = 1.19197 % (Python decimal 12·(1.1^(1/96) − 1))

### Excel PDURATION: 2,000 to 2,200 at 2.5%

- Solve for: Thời gian
- Present value: 2000
- Future value: 2200
- Interest rate (per year): 2.5%
- Compounding: Yearly
- **Time needed: 3.86 years**
- Nguồn đối chiếu: Microsoft PDURATION documentation: =PDURATION(2.5%,2000,2200) = 3.86 years; ln 1.1 / ln 1.025 = 3.85987 (Python decimal)

### Excel PDURATION: 1,000 to 1,200 at 2.5% compounded monthly

- Solve for: Thời gian
- Present value: 1000
- Future value: 1200
- Interest rate (per year): 2.5%
- Compounding: Monthly
- **Compounding periods needed: 87.60**
- Nguồn đối chiếu: Microsoft PDURATION documentation: =PDURATION(0.025/12,1000,1200) = 87.6 months; ln 1.2 / ln(1 + 0.025/12) = 87.6055 (Python decimal)

### Doubling at 7% compounded monthly

- Solve for: Thời gian
- Present value: 1000
- Future value: 2000
- Interest rate (per year): 7%
- Compounding: Monthly
- **Time needed: 9.93 years**
- **Compounding periods needed: 119.17**
- Nguồn đối chiếu: ln 2 / ln(1 + 0.07/12) = 119.1715 months = 9.93096 years (Python decimal)

## Câu hỏi

### How do you calculate the future value of a lump sum?

Multiply the present value by (1 + r/k)^(kt), where r is the annual rate, k the compounding periods a year and t the years. 10,000 at 6% compounded yearly for 10 years grows to 10,000 × 1.06^10 = 17,908.48. Excel's =FV(6%, 10, 0, -10000) returns the same amount.

### How do you calculate present value?

Divide the future amount by the growth factor: PV = FV ÷ (1 + r/k)^(kt). At 6% compounded monthly, 20,000 due in 10 years is worth 20,000 ÷ 1.005^120 = 10,992.65 today. A higher discount rate or a longer wait lowers the present value; Excel's =PV(0.5%, 120, 0, -20000) gives the same figure.

### How do you work out the interest rate needed to reach a target?

Take the growth multiple to the power 1 ÷ (number of periods) and subtract 1: r = (FV ÷ PV)^(1/t) − 1. Doubling money in 10 years takes 2^(1/10) − 1 = 7.18% a year. Microsoft's example for Excel's RRI function, 10,000 growing to 11,000 over 96 months, gives 0.0009933 a month, 1.19% a year.

### How long does it take for money to reach a target amount?

Divide the logarithm of the growth multiple by the logarithm of one plus the rate per period: n = ln(FV ÷ PV) ÷ ln(1 + r/k). Growing 2,000 to 2,200 at 2.5% a year takes ln 1.1 ÷ ln 1.025 = 3.86 years, the answer Excel's PDURATION function gives. Doubling at 7% compounded monthly takes 119.17 months, or 9.93 years.

### What is the difference between a nominal and an effective annual rate?

The nominal rate is the quoted yearly rate; the effective rate includes interest earned on interest within the year: (1 + r/k)^k − 1. A nominal 6% compounded monthly is 6.1678% effective, 6.1831% compounded daily and 6.1837% compounded continuously (e^0.06 − 1). Compare offers on the effective rate, not the nominal one.

### “Time value of money calculator (future and present value)” 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: 9. Ví dụ, “Future value of 10,000 at 6% for 10 years” được kiểm tra bằng 10000 × 1.06^10 = 17908.4769654… (Python decimal).

### Phương pháp này lấy từ đâu?

Microsoft Excel RRI function; Microsoft Excel PDURATION function; Brealey, Myers & Allen — Principles of Corporate Finance, ch. 2 (present values).

## Nguồn

- [Microsoft Excel RRI function](https://support.microsoft.com/office/rri-function-6f5822d8-7ef1-4233-944c-79e8172930f4)
- [Microsoft Excel PDURATION function](https://support.microsoft.com/office/pduration-function-44f33460-5be5-4c90-b857-22308892adaf)
- Brealey, Myers & Allen — Principles of Corporate Finance, ch. 2 (present values)

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