# What is the missing financial value?

Solves the time value of money equation for any one of N, I/Y, PV, PMT and FV, with payments and compounding 1 to 365 times a year, at the end or the start of each period.

- Page: https://www.acalculator.org/finance/financial-calculator
- JSON spec: https://www.acalculator.org/finance/financial-calculator.json
- Version: 9f986a745472

## Default answer

Example with the default inputs (Compute PMT, N (number of periods) 360, I/Y (yearly interest rate) 6%, PV (present value) $200,000.00, FV (future value) $0.00, P/Y (payments a year) 12, C/Y (compounding a year) 12, Payments at the End (END)): N = 360, I/Y = 6%, PV = $200,000.00, PMT = -$1,199.10, FV = $0.00.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| solve | Compute | Which of the five values to work out, like the CPT key; type the other four. |
| n | N (number of periods) | The number of payment periods, for example 360 for 30 years of monthly payments. |
| 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 is negative. |
| pmt | PMT (payment) | The payment made each period, with the same sign rule: money you pay is negative. |
| fv | FV (future value) | The amount after the last period, with the same sign rule. 0 for a loan paid off. |
| 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, for example 360 for 30 years of monthly payments. |
| 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 is negative. |
| pmt | PMT (payment) | The payment made each period, with the same sign rule: money you pay is negative. |
| fv | FV (future value) | The amount after the last period, with the same sign rule. 0 for a loan paid off. |
| 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 = pmt, n = 10, rate = 8%, pv = $10,000.00, fv = $0.00, py = 12, cy = 12, due = end gives pmt = -$1,037.03, totalPmt = -$10,370.32, interest = -$370.32. Source: Microsoft Excel PMT function, example 1: ($1,037.03) (https://support.microsoft.com/en-us/office/pmt-function-0214da64-9a63-4996-bc20-214433fa6441).
2. solve = fv, n = 10, rate = 6%, pv = -$500.00, pmt = -$200.00, py = 12, cy = 12, due = begin gives fv = $2,581.40. Source: Microsoft Excel FV function, example 1: $2,581.40 (https://support.microsoft.com/en-us/office/fv-function-2eef9f44-a084-4c61-bdd8-4fe4bb1b71b3).
3. solve = n, rate = 12%, pv = -$1,000.00, pmt = -$100.00, fv = $10,000.00, py = 12, cy = 12, due = begin gives n = 59.673866. Source: Microsoft Excel NPER function, example 1: 59.6738657 (https://support.microsoft.com/en-us/office/nper-function-240535b5-6653-4d2d-bfcf-b6a38151d815).
4. solve = rate, n = 48, pv = $8,000.00, pmt = -$200.00, fv = $0.00, py = 12, cy = 12, due = end gives rate = 9.241767%. Source: Microsoft Excel RATE function example: 9.24% a year (https://support.microsoft.com/en-us/office/rate-function-9f665657-4a7e-4bb7-a030-83fc59e748ce).
5. solve = pmt, n = 300, rate = 5%, pv = $300,000.00, fv = $0.00, py = 12, cy = 2, due = end gives pmt = -$1,744.81, periodRate = 0.412392%, ear = 5.0625%. Source: Texas Instruments BA II PLUS Guidebook, TVM worksheet with C/Y ≠ P/Y (https://education.ti.com/html/eguides/financials/pdfs/EN/BA-II-PLUS_EN.pdf).

## FAQ

### How do I use this financial calculator?

Pick the value to compute, as with the CPT key, and type the other four. For a $200,000 mortgage at 6% for 30 years, compute PMT with N = 360, I/Y = 6, PV = 200,000 and FV = 0, and keep P/Y and C/Y at 12: the payment is −$1,199.10 a month. An empty PV, PMT or FV counts as 0.

### Why is my payment negative?

The calculator follows the cash-flow sign rule of financial calculators and spreadsheets: money you receive is positive and money you pay is negative. A loan you take out is +$200,000 (you receive it) and the payments are −$1,199.10 (you pay them). A deposit into savings is negative and the balance you take out later is positive. If PV, PMT and FV all have the same sign, no interest rate can make them balance.

### What are P/Y and C/Y?

P/Y is the number of payments a year (12 for monthly payments) and C/Y is the number of times a year interest compounds. I/Y is always a yearly rate; the calculator turns it into a rate per payment period, i = (1 + I/Y ÷ C/Y)^(C/Y ÷ P/Y) − 1. When C/Y equals P/Y this is simply I/Y ÷ P/Y: 6% a year paid monthly is 0.5% a month. Canadian fixed-rate mortgages compound twice a year (C/Y = 2) but are paid monthly (P/Y = 12).

### What is the difference between END and BGN?

END means each payment is made at the end of its period (an ordinary annuity: loans, most savings plans). BGN means each payment is made at the start (an annuity due: rent, leases, insurance premiums). A BGN payment earns or costs one more period of interest, so the payment part is multiplied by (1 + i).

### Why does it say no interest rate fits?

The cash flows must change sign at least once for an interest rate to balance them: for example a positive loan and negative payments, or negative deposits and a positive balance at the end. It also says so when the rate would be below −100% or above 1,000% a year, which it does not search.

### Can there be two interest rates?

Yes, when the cash flows change sign twice, for example money received (PV), then payments made (PMT), then a large amount received at the end (FV). Two rates can then balance the equation, and the calculator shows both. With one change of sign there is only one rate.

### Is this the same as a BA II Plus or an HP 12C?

It uses the same time value of money equation and the same sign rule as the TVM keys of the Texas Instruments BA II Plus and the PV, FV, PMT, NPER and RATE functions of Excel, so it gives the same answers for the same inputs. It does not do the other worksheets of those calculators (cash flows, depreciation, statistics).

## Sources

- Texas Instruments, BA II PLUS Guidebook: the Time-Value-of-Money worksheet (N, I/Y, PV, PMT, FV, P/Y, C/Y, BGN/END) and its formulas. https://education.ti.com/html/eguides/financials/pdfs/EN/BA-II-PLUS_EN.pdf
- Microsoft, Excel PMT, FV, NPER and RATE functions, with worked examples. https://support.microsoft.com/en-us/office/pmt-function-0214da64-9a63-4996-bc20-214433fa6441
- S. A. Broverman (2017), Mathematics of Investment and Credit, 7th edition, chapters 1 and 2: equivalent rates, annuities immediate and annuities due.