"""Some simple financial calculations patterned after spreadsheet computations. There is some complexity in each function so that the functions behave like ufuncs with broadcasting and being able to be called with scalars or arrays (or other sequences). """ from __future__ import division, absolute_import, print_function import numpy as np __all__ = ['fv', 'pmt', 'nper', 'ipmt', 'ppmt', 'pv', 'rate', 'irr', 'npv', 'mirr'] _when_to_num = {'end':0, 'begin':1, 'e':0, 'b':1, 0:0, 1:1, 'beginning':1, 'start':1, 'finish':0} def _convert_when(when): #Test to see if when has already been converted to ndarray #This will happen if one function calls another, for example ppmt if isinstance(when, np.ndarray): return when try: return _when_to_num[when] except (KeyError, TypeError): return [_when_to_num[x] for x in when] def fv(rate, nper, pmt, pv, when='end'): """ Compute the future value. Given: * a present value, pv * 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 at the end of the nper periods Parameters ---------- rate : scalar or array_like of shape(M, ) Rate of interest as decimal (not per cent) per period nper : scalar or array_like of shape(M, ) Number of compounding periods pmt : scalar or array_like of shape(M, ) Payment pv : scalar or array_like of shape(M, ) Present value when : {{'begin', 1}, {'end', 0}}, {string, int}, optional When payments are due ('begin' (1) or 'end' (0)). Defaults to {'end', 0}. Returns ------- out : ndarray Future values. If all input is scalar, returns a scalar float. If any input is array_like, returns future values for each input element. If multiple inputs are array_like, they all must have the same shape. Notes ----- The future 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 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 future value after 10 years of saving $100 now, with an additional monthly savings of$100. Assume the interest rate is 5% (annually) compounded monthly? >>> np.fv(0.05/12, 10*12, -100, -100) 15692.928894335748 By convention, the negative sign represents cash flow out (i.e. money not available today). Thus, saving $100 a month at 5% annual interest leads to$15,692.93 available to spend in 10 years. If any input is array_like, returns an array of equal shape. Let's compare different interest rates from the example above. >>> a = np.array((0.05, 0.06, 0.07))/12 >>> np.fv(a, 10*12, -100, -100) array([ 15692.92889434, 16569.87435405, 17509.44688102]) """ when = _convert_when(when) (rate, nper, pmt, pv, when) = map(np.asarray, [rate, nper, pmt, pv, when]) temp = (1+rate)**nper miter = np.broadcast(rate, nper, pmt, pv, when) zer = np.zeros(miter.shape) fact = np.where(rate == zer, nper + zer, (1 + rate*when)*(temp - 1)/rate + zer) return -(pv*temp + pmt*fact) def pmt(rate, nper, pv, fv=0, when='end'): """ 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)) Returns ------- 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 ---------- .. [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-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. """ when = _convert_when(when) (rate, nper, pv, fv, when) = map(np.asarray, [rate, nper, pv, fv, when]) temp = (1+rate)**nper miter = np.broadcast(rate, nper, pv, fv, when) zer = np.zeros(miter.shape) fact = np.where(rate == zer, nper + zer, (1 + rate*when)*(temp - 1)/rate + zer) return -(fv + pv*temp) / fact def nper(rate, pmt, pv, fv=0, when='end'): """ Compute the number of periodic payments. Parameters ---------- rate : array_like Rate of interest (per period) pmt : array_like Payment pv : array_like Present value fv : array_like, optional Future value when : {{'begin', 1}, {'end', 0}}, {string, int}, optional When payments are due ('begin' (1) or 'end' (0)) Notes ----- The number of periods nper is computed by solving the equation:: fv + pv*(1+rate)**nper + pmt*(1+rate*when)/rate*((1+rate)**nper-1) = 0 but if rate = 0 then:: fv + pv + pmt*nper = 0 Examples -------- If you only had$150/month to pay towards the loan, how long would it take to pay-off a loan of $8,000 at 7% annual interest? >>> print round(np.nper(0.07/12, -150, 8000), 5) 64.07335 So, over 64 months would be required to pay off the loan. The same analysis could be done with several different interest rates and/or payments and/or total amounts to produce an entire table. >>> np.nper(*(np.ogrid[0.07/12: 0.08/12: 0.01/12, ... -150 : -99 : 50 , ... 8000 : 9001 : 1000])) array([[[ 64.07334877, 74.06368256], [ 108.07548412, 127.99022654]], [[ 66.12443902, 76.87897353], [ 114.70165583, 137.90124779]]]) """ when = _convert_when(when) (rate, pmt, pv, fv, when) = map(np.asarray, [rate, pmt, pv, fv, when]) use_zero_rate = False with np.errstate(divide="raise"): try: z = pmt*(1.0+rate*when)/rate except FloatingPointError: use_zero_rate = True if use_zero_rate: return (-fv + pv) / (pmt + 0.0) else: A = -(fv + pv)/(pmt+0.0) B = np.log((-fv+z) / (pv+z))/np.log(1.0+rate) miter = np.broadcast(rate, pmt, pv, fv, when) zer = np.zeros(miter.shape) return np.where(rate == zer, A + zer, B + zer) + 0.0 def ipmt(rate, per, nper, pv, fv=0.0, when='end'): """ Compute the interest portion of a payment. Parameters ---------- rate : scalar or array_like of shape(M, ) Rate of interest as decimal (not per cent) per period per : scalar or array_like of shape(M, ) Interest paid against the loan changes during the life or the loan. The per is the payment period to calculate the interest amount. nper : scalar or array_like of shape(M, ) Number of compounding periods pv : scalar or array_like of shape(M, ) Present value fv : scalar or array_like of shape(M, ), optional Future value when : {{'begin', 1}, {'end', 0}}, {string, int}, optional When payments are due ('begin' (1) or 'end' (0)). Defaults to {'end', 0}. Returns ------- out : ndarray Interest portion of payment. If all input is scalar, returns a scalar float. If any input is array_like, returns interest payment for each input element. If multiple inputs are array_like, they all must have the same shape. See Also -------- ppmt, pmt, pv Notes ----- The total payment is made up of payment against principal plus interest. pmt = ppmt + ipmt Examples -------- What is the amortization schedule for a 1 year loan of$2500 at 8.24% interest per year compounded monthly? >>> principal = 2500.00 The 'per' variable represents the periods of the loan. Remember that financial equations start the period count at 1! >>> per = np.arange(1*12) + 1 >>> ipmt = np.ipmt(0.0824/12, per, 1*12, principal) >>> ppmt = np.ppmt(0.0824/12, per, 1*12, principal) Each element of the sum of the 'ipmt' and 'ppmt' arrays should equal 'pmt'. >>> pmt = np.pmt(0.0824/12, 1*12, principal) >>> np.allclose(ipmt + ppmt, pmt) True >>> fmt = '{0:2d} {1:8.2f} {2:8.2f} {3:8.2f}' >>> for payment in per: ... index = payment - 1 ... principal = principal + ppmt[index] ... print fmt.format(payment, ppmt[index], ipmt[index], principal) 1 -200.58 -17.17 2299.42 2 -201.96 -15.79 2097.46 3 -203.35 -14.40 1894.11 4 -204.74 -13.01 1689.37 5 -206.15 -11.60 1483.22 6 -207.56 -10.18 1275.66 7 -208.99 -8.76 1066.67 8 -210.42 -7.32 856.25 9 -211.87 -5.88 644.38 10 -213.32 -4.42 431.05 11 -214.79 -2.96 216.26 12 -216.26 -1.49 -0.00 >>> interestpd = np.sum(ipmt) >>> np.round(interestpd, 2) -112.98 """ when = _convert_when(when) rate, per, nper, pv, fv, when = np.broadcast_arrays(rate, per, nper, pv, fv, when) total_pmt = pmt(rate, nper, pv, fv, when) ipmt = _rbl(rate, per, total_pmt, pv, when)*rate try: ipmt = np.where(when == 1, ipmt/(1 + rate), ipmt) ipmt = np.where(np.logical_and(when == 1, per == 1), 0.0, ipmt) except IndexError: pass return ipmt def _rbl(rate, per, pmt, pv, when): """ This function is here to simply have a different name for the 'fv' function to not interfere with the 'fv' keyword argument within the 'ipmt' function. It is the 'remaining balance on loan' which might be useful as it's own function, but is easily calculated with the 'fv' function. """ return fv(rate, (per - 1), pmt, pv, when) def ppmt(rate, per, nper, pv, fv=0.0, when='end'): """ Compute the payment against loan principal. Parameters ---------- rate : array_like Rate of interest (per period) per : array_like, int Amount paid against the loan changes. The per is the period of interest. nper : array_like Number of compounding periods pv : array_like Present value fv : array_like, optional Future value when : {{'begin', 1}, {'end', 0}}, {string, int} When payments are due ('begin' (1) or 'end' (0)) See Also -------- pmt, pv, ipmt """ total = pmt(rate, nper, pv, fv, when) return total - ipmt(rate, per, nper, pv, fv, when) def 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. """ when = _convert_when(when) (rate, nper, pmt, fv, when) = map(np.asarray, [rate, nper, pmt, fv, when]) temp = (1+rate)**nper miter = np.broadcast(rate, nper, pmt, fv, when) zer = np.zeros(miter.shape) fact = np.where(rate == zer, nper+zer, (1+rate*when)*(temp-1)/rate+zer) return -(fv + pmt*fact)/temp # Computed with Sage # (y + (r + 1)^n*x + p*((r + 1)^n - 1)*(r*w + 1)/r)/(n*(r + 1)^(n - 1)*x - # p*((r + 1)^n - 1)*(r*w + 1)/r^2 + n*p*(r + 1)^(n - 1)*(r*w + 1)/r + # p*((r + 1)^n - 1)*w/r) def _g_div_gp(r, n, p, x, y, w): t1 = (r+1)**n t2 = (r+1)**(n-1) return ((y + t1*x + p*(t1 - 1)*(r*w + 1)/r) / (n*t2*x - p*(t1 - 1)*(r*w + 1)/(r**2) + n*p*t2*(r*w + 1)/r + p*(t1 - 1)*w/r)) # Use Newton's iteration until the change is less than 1e-6 # for all values or a maximum of 100 iterations is reached. # Newton's rule is # r_{n+1} = r_{n} - g(r_n)/g'(r_n) # where # g(r) is the formula # g'(r) is the derivative with respect to r. def rate(nper, pmt, pv, fv, when='end', guess=0.10, tol=1e-6, maxiter=100): """ Compute the rate of interest per period. Parameters ---------- nper : array_like Number of compounding periods pmt : array_like Payment pv : array_like Present value fv : array_like Future value when : {{'begin', 1}, {'end', 0}}, {string, int}, optional When payments are due ('begin' (1) or 'end' (0)) guess : float, optional Starting guess for solving the rate of interest tol : float, optional Required tolerance for the solution maxiter : int, optional Maximum iterations in finding the solution Notes ----- The rate of interest is computed by iteratively solving the (non-linear) equation:: fv + pv*(1+rate)**nper + pmt*(1+rate*when)/rate * ((1+rate)**nper - 1) = 0 for rate. References ---------- 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 """ when = _convert_when(when) (nper, pmt, pv, fv, when) = map(np.asarray, [nper, pmt, pv, fv, when]) rn = guess iter = 0 close = False while (iter < maxiter) and not close: rnp1 = rn - _g_div_gp(rn, nper, pmt, pv, fv, when) diff = abs(rnp1-rn) close = np.all(diff < tol) iter += 1 rn = rnp1 if not close: # Return nan's in array of the same shape as rn return np.nan + rn else: return rn def irr(values): """ Return the Internal Rate of Return (IRR). This is the "average" periodically compounded rate of return that gives a net present value of 0.0; for a more complete explanation, see Notes below. Parameters ---------- values : array_like, shape(N,) Input cash flows per time period. By convention, net "deposits" are negative and net "withdrawals" are positive. Thus, for example, at least the first element of values, which represents the initial investment, will typically be negative. Returns ------- out : float Internal Rate of Return for periodic input values. Notes ----- The IRR is perhaps best understood through an example (illustrated using np.irr in the Examples section below). Suppose one invests 100 units and then makes the following withdrawals at regular (fixed) intervals: 39, 59, 55, 20. Assuming the ending value is 0, one's 100 unit investment yields 173 units; however, due to the combination of compounding and the periodic withdrawals, the "average" rate of return is neither simply 0.73/4 nor (1.73)^0.25-1. Rather, it is the solution (for :math:r) of the equation: .. math:: -100 + \\frac{39}{1+r} + \\frac{59}{(1+r)^2} + \\frac{55}{(1+r)^3} + \\frac{20}{(1+r)^4} = 0 In general, for values :math:= [v_0, v_1, ... v_M], irr is the solution of the equation: [G]_ .. math:: \\sum_{t=0}^M{\\frac{v_t}{(1+irr)^{t}}} = 0 References ---------- .. [G] L. J. Gitman, "Principles of Managerial Finance, Brief," 3rd ed., Addison-Wesley, 2003, pg. 348. Examples -------- >>> round(irr([-100, 39, 59, 55, 20]), 5) 0.28095 >>> round(irr([-100, 0, 0, 74]), 5) -0.0955 >>> round(irr([-100, 100, 0, -7]), 5) -0.0833 >>> round(irr([-100, 100, 0, 7]), 5) 0.06206 >>> round(irr([-5, 10.5, 1, -8, 1]), 5) 0.0886 (Compare with the Example given for numpy.lib.financial.npv) """ res = np.roots(values[::-1]) mask = (res.imag == 0) & (res.real > 0) if res.size == 0: return np.nan res = res[mask].real # NPV(rate) = 0 can have more than one solution so we return # only the solution closest to zero. rate = 1.0/res - 1 rate = rate.item(np.argmin(np.abs(rate))) return rate def npv(rate, values): """ Returns the NPV (Net Present Value) of a cash flow series. Parameters ---------- rate : scalar The discount rate. values : array_like, shape(M, ) The values of the time series of cash flows. The (fixed) time interval between cash flow "events" must be the same as that for which rate is given (i.e., if rate is per year, then precisely a year is understood to elapse between each cash flow event). By convention, investments or "deposits" are negative, income or "withdrawals" are positive; values must begin with the initial investment, thus values[0] will typically be negative. Returns ------- out : float The NPV of the input cash flow series values at the discount rate. Notes ----- Returns the result of: [G]_ .. math :: \\sum_{t=0}^{M-1}{\\frac{values_t}{(1+rate)^{t}}} References ---------- .. [G] L. J. Gitman, "Principles of Managerial Finance, Brief," 3rd ed., Addison-Wesley, 2003, pg. 346. Examples -------- >>> np.npv(0.281,[-100, 39, 59, 55, 20]) -0.0084785916384548798 (Compare with the Example given for numpy.lib.financial.irr) """ values = np.asarray(values) return (values / (1+rate)**np.arange(0, len(values))).sum(axis=0) def mirr(values, finance_rate, reinvest_rate): """ Modified internal rate of return. Parameters ---------- values : array_like Cash flows (must contain at least one positive and one negative value) or nan is returned. The first value is considered a sunk cost at time zero. finance_rate : scalar Interest rate paid on the cash flows reinvest_rate : scalar Interest rate received on the cash flows upon reinvestment Returns ------- out : float Modified internal rate of return """ values = np.asarray(values, dtype=np.double) n = values.size pos = values > 0 neg = values < 0 if not (pos.any() and neg.any()): return np.nan numer = np.abs(npv(reinvest_rate, values*pos)) denom = np.abs(npv(finance_rate, values*neg)) return (numer/denom)**(1.0/(n - 1))*(1 + reinvest_rate) - 1