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# numpy.pv

`numpy.``pv`(rate, nper, pmt, fv=0.0, when='end')

  Compute the present value.

  Given:
    - a future value, `fv`
    - an interest `rate` compounded once per period, of which there are
    - `nper` total
    - a (fixed) payment, `pmt`, paid either
    - at the beginning (when = {‘begin’, 1}) or the end (when = {‘end’, 0}) of each period
  Return:
    the value now

  Parameters : | rate : array_like Rate of interest (per period) nper : array_like Number of compounding periods pmt : array_like Payment fv : array_like, optional Future value when : {{‘begin’, 1}, {‘end’, 0}}, {string, int}, optional When payments are due (‘begin’ (1) or ‘end’ (0))
  Returns : | out : ndarray, float Present value of a series of payments or investments.

  Notes

  The present value 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 `pv`, which is then returned.

  References

  [WRW] | 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-formula OpenDocument-formula-20090508.odt

  Examples

  What is the present value (e.g., the initial investment) of an investment that needs to total $15692.93 after 10 years of saving $100 every month? Assume the interest rate is 5% (annually) compounded monthly.

      >>> np.pv(0.05/12, 10*12, -100, 15692.93)
      -100.00067131625819

  By convention, the negative sign represents cash flow out (i.e., money not available today). Thus, to end up with $15,692.93 in 10 years saving $100 a month at 5% annual interest, one’s initial deposit should also be $100.

  If any input is array_like, `pv` returns an array of equal shape. Let’s compare different interest rates in the example above:

      >>> a = np.array((0.05, 0.04, 0.03))/12
      >>> np.pv(a, 10*12, -100, 15692.93)
      array([ -100.00067132,  -649.26771385, -1273.78633713])

  So, to end up with the same $15692.93 under the same $100 per month “savings plan,” for annual interest rates of 4% and 3%, one would need initial investments of $649.27 and $1273.79, respectively.

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