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# 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).
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}
eqstr = """
nper / (1 + rate*when) \ / nper \
fv + pv*(1+rate) + pmt*|-------------------|*| (1+rate) - 1 | = 0
\ rate / \ /
fv + pv + pmt * nper = 0 (when rate == 0)
where (all can be scalars or sequences)
Parameters
----------
rate :
Rate of interest (per period)
nper :
Number of compounding periods
pmt :
Payment
pv :
Present value
fv :
Future value
when :
When payments are due ('begin' (1) or 'end' (0))
"""
def _convert_when(when):
try:
return _when_to_num[when]
except KeyError:
return [_when_to_num[x] for x in when]
def fv(rate, nper, pmt, pv, when='end'):
"""future value computed by solving the equation
"""
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)
fv.__doc__ += eqstr + """
Example
--------
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?
>>> 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.
"""
def pmt(rate, nper, pv, fv=0, when='end'):
"""Payment computed by solving the equation
"""
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
pmt.__doc__ += eqstr + """
Example
-------
What would the monthly payment need to be to pay off a $200,000 loan in 15
years at an annual interest rate of 7.5%?
>>> 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.
"""
def nper(rate, pmt, pv, fv=0, when='end'):
"""Number of periods found by solving the equation
"""
when = _convert_when(when)
rate, pmt, pv, fv, when = map(np.asarray, [rate, pmt, pv, fv, when])
try:
z = pmt*(1.0+rate*when)/rate
except ZeroDivisionError:
z = 0.0
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
nper.__doc__ += eqstr + """
Example
-------
If you only had $150 to spend as payment, how long would it take to pay-off
a loan of $8,000 at 7% annual interest?
>>> nper(0.07/12, -150, 8000)
64.073348770661852
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.
>>> nper(*(ogrid[0.06/12:0.071/12:0.01/12, -200:-99:100, 6000:7001:1000]))
array([[[ 32.58497782, 38.57048452],
[ 71.51317802, 86.37179563]],
[[ 33.07413144, 39.26244268],
[ 74.06368256, 90.22989997]]])
"""
def ipmt(rate, per, nper, pv, fv=0.0, when='end'):
total = pmt(rate, nper, pv, fv, when)
# Now, compute the nth step in the amortization
raise NotImplementedError
def ppmt(rate, per, nper, pv, fv=0.0, when='end'):
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'):
"""Number of periods found by solving the equation
"""
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
pv.__doc__ += eqstr
# 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):
"""Number of periods found by solving the equation
"""
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
rate.__doc__ += eqstr
def irr(values):
"""Internal Rate of Return
This is the rate of return that gives a net present value of 0.0
npv(irr(values), values) == 0.0
"""
res = np.roots(values[::-1])
# Find the root(s) between 0 and 1
mask = (res.imag == 0) & (res.real > 0) & (res.real <= 1)
res = res[mask].real
if res.size == 0:
return np.nan
rate = 1.0/res - 1
if rate.size == 1:
rate = rate.item()
return rate
def npv(rate, values):
"""Net Present Value
sum ( values_k / (1+rate)**k, k = 1..n)
"""
values = np.asarray(values)
return (values / (1+rate)**np.arange(1,len(values)+1)).sum(axis=0)
def mirr(values, finance_rate, reinvest_rate):
"""Modified internal rate of return
Parameters
----------
values:
Cash flows (must contain at least one positive and one negative value)
or nan is returned.
finance_rate :
Interest rate paid on the cash flows
reinvest_rate :
Interest rate received on the cash flows upon reinvestment
"""
values = np.asarray(values)
pos = values > 0
neg = values < 0
if not (pos.size > 0 and neg.size > 0):
return np.nan
n = pos.size + neg.size
numer = -npv(reinvest_rate, values[pos])*((1+reinvest_rate)**n)
denom = npv(finance_rate, values[neg])*(1+finance_rate)
return (numer / denom)**(1.0/(n-1)) - 1
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