# What is the missing TVM value?

Solves the time value of money (TVM) equation for the number of periods, the interest rate, the present value, the payment or the future value.

- Page: https://www.acalculator.org/finance/tvm-calculator
- JSON spec: https://www.acalculator.org/finance/tvm-calculator.json
- Version: 2586099c1e17

## Default answer

Example with the default inputs (Compute FV, N (number of periods) 120, I/Y (yearly interest rate) 7%, PV (present value) -$10,000.00, PMT (payment) -$500.00, P/Y (payments a year) 12, C/Y (compounding a year) 12, Payments at the End (END)): N = 120, I/Y = 7%, PV = -$10,000.00, PMT = -$500.00, FV = $106,639.02.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| solve | Compute | Which of the five values to work out; type the other four. |
| n | N (number of periods) | The number of payment periods. |
| rate | I/Y (yearly interest rate) | The nominal yearly interest rate in percent, compounded C/Y times a year. |
| pv | PV (present value) | The amount at the start. Money you receive is positive; money you pay or deposit is negative. |
| pmt | PMT (payment) | The payment each period, with the same sign rule. 0 for a single sum. |
| fv | FV (future value) | The amount after the last period, with the same sign rule. |
| py | P/Y (payments a year) | How many payments are made in a year: 12 for monthly. |
| cy | C/Y (compounding a year) | How many times a year interest compounds. Usually the same as P/Y. |
| due | Payments at the | The end of each period (END, an ordinary annuity) or the start of each period (BGN, an annuity due). |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| n | N (number of periods) | The number of payment periods. |
| rate | I/Y (yearly interest rate) | The nominal yearly interest rate in percent, compounded C/Y times a year. |
| pv | PV (present value) | The amount at the start. Money you receive is positive; money you pay or deposit is negative. |
| pmt | PMT (payment) | The payment each period, with the same sign rule. 0 for a single sum. |
| fv | FV (future value) | The amount after the last period, with the same sign rule. |
| periodRate | Rate per period | The interest rate for one payment period, i = (1 + I/Y ÷ C/Y)^(C/Y ÷ P/Y) − 1. |
| ear | Effective yearly rate | The rate over a whole year once interest compounds: (1 + i)^(P/Y) − 1. |
| totalPmt | Total of payments | N × PMT, with the payment’s sign. |
| interest | Net interest | PV + N × PMT + FV: interest you earn (positive) or pay (negative) over the whole time, from the signs of the cash flows. |

## Method

PV × (1 + i)^N + PMT × (1 + i × t) × ((1 + i)^N − 1) ÷ i + FV = 0, with i = (1 + I/Y ÷ (100 × C/Y))^(C/Y ÷ P/Y) − 1 and t = 1 for payments at the start of each period (BGN), 0 at the end (END). Money received is positive, money paid is negative.

## Assumptions

- The interest rate is the same in every period, and every payment is the same size.
- The yearly rate is nominal, compounded as many times a year as set; payments are made as many times a year as set.
- No fees or taxes are included. This is an estimate for planning, not financial advice.

## Worked examples

1. solve = fv, n = 12, rate = 12%, pv = $0.00, pmt = -$1,000.00, py = 12, cy = 12, due = end gives fv = $12,682.50, totalPmt = -$12,000.00, interest = $682.50. Source: Microsoft Excel FV function, example 2: $12,682.50 (https://support.microsoft.com/en-us/office/fv-function-2eef9f44-a084-4c61-bdd8-4fe4bb1b71b3).
2. solve = fv, n = 35, rate = 11%, pv = $0.00, pmt = -$2,000.00, py = 12, cy = 12, due = begin gives fv = $82,846.25. Source: Microsoft Excel FV function, example 3: $82,846.25 (https://support.microsoft.com/en-us/office/fv-function-2eef9f44-a084-4c61-bdd8-4fe4bb1b71b3).
3. solve = pv, n = 240, rate = 8%, pmt = $500.00, fv = $0.00, py = 12, cy = 12, due = end gives pv = -$59,777.15. Source: Microsoft Excel PV function example: ($59,777.15) (https://support.microsoft.com/en-us/office/pv-function-23879d31-0e02-4321-be01-da16e8168cbd).
4. solve = pmt, n = 216, rate = 6%, pv = $0.00, fv = $50,000.00, py = 12, cy = 12, due = end gives pmt = -$129.08. Source: Microsoft Excel PMT function, example 3: ($129.08) (https://support.microsoft.com/en-us/office/pmt-function-0214da64-9a63-4996-bc20-214433fa6441).
5. solve = fv, n = 10, rate = 0%, pv = -$1,000.00, pmt = -$100.00, py = 12, cy = 12, due = end gives fv = $2,000.00, interest = $0.00. Source: The 0% case of the TVM equation, PV + PMT × N + FV = 0 (Microsoft Excel PV function, https://support.microsoft.com/en-us/office/pv-function-23879d31-0e02-4321-be01-da16e8168cbd).

## FAQ

### What is the time value of money?

The time value of money is the idea that a dollar today is worth more than a dollar later, because today’s dollar can earn interest. $10,000 today at 7% a year, compounded monthly, is worth $20,096.61 in 10 years, and $20,096.61 in 10 years is worth $10,000 today at that rate. Every savings plan, loan and annuity is the same equation read in a different direction.

### What do N, I/Y, PV, PMT and FV stand for?

N is the number of periods, I/Y the interest rate per year, PV the present value (the amount at the start), PMT the payment each period and FV the future value (the amount at the end). They are the five time value of money keys on a financial calculator. Know any four and the fifth follows.

### What is the TVM formula?

PV × (1 + i)^N + PMT × ((1 + i)^N − 1) ÷ i + FV = 0, where i is the interest rate per period. For payments at the start of each period (an annuity due) the payment part is multiplied by (1 + i). At a 0% rate the formula is PV + PMT × N + FV = 0.

### Why must PV and FV have opposite signs?

The equation balances money you pay against money you receive. A deposit today is money you pay (negative), and the balance you take out later is money you receive (positive). If every amount had the same sign, no interest rate could balance them, and the calculator says there is no answer.

### How do I work out a monthly savings plan?

Set P/Y and C/Y to 12, type N in months, the yearly rate as I/Y, what you have now as a negative PV, and your monthly deposit as a negative PMT, then compute FV. $10,000 now plus $500 a month for 10 years at 7% a year grows to $106,639.02.

### What if the rate compounds differently from the payments?

Set C/Y to the compounding periods and P/Y to the payments a year. The rate per payment period becomes (1 + I/Y ÷ C/Y)^(C/Y ÷ P/Y) − 1. For example, 6% compounded daily (C/Y = 365) with monthly payments (P/Y = 12) is 0.501% a month.

## Sources

- Texas Instruments, BA II PLUS Guidebook: the Time-Value-of-Money worksheet and its formulas. https://education.ti.com/html/eguides/financials/pdfs/EN/BA-II-PLUS_EN.pdf
- Microsoft, Excel FV, PV and PMT functions, with worked examples. https://support.microsoft.com/en-us/office/fv-function-2eef9f44-a084-4c61-bdd8-4fe4bb1b71b3
- S. A. Broverman (2017), Mathematics of Investment and Credit, 7th edition, chapters 1 and 2.