# 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.

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분야: 금융 계산기

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.

## 입력

- **Solve for** (선택 항목: Future value, Present value, Rate, 시간)
- **Present value**
- **Future value**
- **Interest rate (per year)**
- **시간**
- **Compounding** (선택 항목: Yearly, Half-yearly, Quarterly, Monthly, Daily (365), Continuous)

## 결과

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

## 공식

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

## 계산 예제

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

- Solve for: Future value
- Present value: 10,000
- Interest rate (per year): 6%
- 시간: 10 years
- Compounding: Yearly
- **Future value: 17,908.48**
- 검증 출처: 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%
- 시간: 10 years
- Compounding: Monthly
- **Present value: 10,992.65**
- **Effective annual rate: 6.1678%**
- 검증 출처: 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
- 시간: 8 years
- Compounding: Monthly
- **Interest rate needed (per year): 1.1920%**
- 검증 출처: 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: 시간
- Present value: 2000
- Future value: 2200
- Interest rate (per year): 2.5%
- Compounding: Yearly
- **Time needed: 3.86 years**
- 검증 출처: 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: 시간
- Present value: 1000
- Future value: 1200
- Interest rate (per year): 2.5%
- Compounding: Monthly
- **Compounding periods needed: 87.60**
- 검증 출처: 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: 시간
- Present value: 1000
- Future value: 2000
- Interest rate (per year): 7%
- Compounding: Monthly
- **Time needed: 9.93 years**
- **Compounding periods needed: 119.17**
- 검증 출처: ln 2 / ln(1 + 0.07/12) = 119.1715 months = 9.93096 years (Python decimal)

## 자주 묻는 질문

### 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)”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: 9. 예를 들어 “Future value of 10,000 at 6% for 10 years”은 10000 × 1.06^10 = 17908.4769654… (Python decimal)와 대조해 확인합니다.

### 이 계산 방법의 출처는 무엇인가요?

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

## 출처

- [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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