# numpy.pmt¶

`numpy.``pmt`(rate, nper, pv, fv=0, when='end')[source]

Compute the payment against loan principal plus interest.

Given:
• a present value, `pv` (e.g., an amount borrowed)
• a future value, `fv` (e.g., 0)
• an interest `rate` compounded once per period, of which there are
• `nper` total
• and (optional) specification of whether payment is made at the beginning (when = {‘begin’, 1}) or the end (when = {‘end’, 0}) of each period
Return:
the (fixed) periodic payment.
Parameters: rate : array_like Rate of interest (per period) nper : array_like Number of compounding periods pv : array_like Present value fv : array_like, optional Future value (default = 0) when : {{‘begin’, 1}, {‘end’, 0}}, {string, int} When payments are due (‘begin’ (1) or ‘end’ (0)) out : ndarray Payment against loan plus interest. If all input is scalar, returns a scalar float. If any input is array_like, returns payment for each input element. If multiple inputs are array_like, they all must have the same shape.

Notes

The payment is computed by solving the equation:

```fv +
pv*(1 + rate)**nper +
pmt*(1 + rate*when)/rate*((1 + rate)**nper - 1) == 0
```

or, when `rate == 0`:

```fv + pv + pmt * nper == 0
```

for `pmt`.

Note that computing a monthly mortgage payment is only one use for this function. For example, pmt returns the periodic deposit one must make to achieve a specified future balance given an initial deposit, a fixed, periodically compounded interest rate, and the total number of periods.

References

 [WRW108109] Wheeler, D. A., E. Rathke, and R. Weir (Eds.) (2009, May). Open Document Format for Office Applications (OpenDocument)v1.2, Part 2: Recalculated Formula (OpenFormula) Format - Annotated Version, Pre-Draft 12. Organization for the Advancement of Structured Information Standards (OASIS). Billerica, MA, USA. [ODT Document]. Available: http://www.oasis-open.org/committees/documents.php ?wg_abbrev=office-formulaOpenDocument-formula-20090508.odt

Examples

What is the monthly payment needed to pay off a \$200,000 loan in 15 years at an annual interest rate of 7.5%?

```>>> np.pmt(0.075/12, 12*15, 200000)
-1854.0247200054619
```

In order to pay-off (i.e., have a future-value of 0) the \$200,000 obtained today, a monthly payment of \$1,854.02 would be required. Note that this example illustrates usage of `fv` having a default value of 0.

numpy.npv

numpy.ppmt