# 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.

Interactive version: https://www.calcopenly.com/finance/time-value-of-money-calculator
Subject: Finance calculators

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.

## Inputs

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

## Results

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

## Formula

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

## Worked examples

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

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

## Questions

### 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.

### How accurate is the time value of money calculator?

Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 9 worked examples whose answers come from independent sources; for example, “Future value of 10,000 at 6% for 10 years” is checked against 10000 × 1.06^10 = 17908.4769654… (Python decimal).

### Where does the method come from?

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

## Sources

- [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)

_Note: financial information, not professional advice._