_financial.py

"""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).

Functions support the :class:`decimal.Decimal` type unless
otherwise stated.
"""
from __future__ import absolute_import, division, print_function

from decimal import Decimal

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
    --------
    >>> import numpy as np
    >>> import numpy_financial as npf

    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?

    >>> npf.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
    >>> npf.fv(a, 10*12, -100, -100)
    array([ 15692.92889434,  16569.87435405,  17509.44688102]) # may vary

    """
    when = _convert_when(when)
    rate, nper, pmt, pv, when = np.broadcast_arrays(rate, nper, pmt, pv, when)

    fv_array = np.empty_like(rate)
    zero = rate == 0
    nonzero = ~zero

    fv_array[zero] = -(pv[zero] + pmt[zero] * nper[zero])

    rate_nonzero = rate[nonzero]
    temp = (1 + rate_nonzero)**nper[nonzero]
    fv_array[nonzero] = (
        - pv[nonzero] * temp
        - pmt[nonzero] * (1 + rate_nonzero * when[nonzero]) / rate_nonzero
        * (temp - 1)
    )

    if np.ndim(fv_array) == 0:
        # Follow the ufunc convention of returning scalars for scalar
        # and 0d array inputs.
        return fv_array.item(0)
    return fv_array


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
    --------
    >>> import numpy_financial as npf

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

    >>> npf.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.array, [rate, nper, pv, fv, when])
    temp = (1 + rate)**nper
    mask = (rate == 0)
    masked_rate = np.where(mask, 1, rate)
    fact = np.where(mask != 0, nper,
                    (1 + masked_rate*when)*(temp - 1)/masked_rate)
    return -(fv + pv*temp) / fact


def nper(rate, pmt, pv, fv=0, when='end'):
    """
    Compute the number of periodic payments.

    :class:`decimal.Decimal` type is not supported.

    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
    --------
    >>> import numpy as np
    >>> import numpy_financial as npf

    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(np.round(npf.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.

    >>> npf.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 = np.broadcast_arrays(rate, pmt, pv, fv, when)
    nper_array = np.empty_like(rate, dtype=np.float64)

    zero = rate == 0
    nonzero = ~zero

    with np.errstate(divide='ignore'):
        # Infinite numbers of payments are okay, so ignore the
        # potential divide by zero.
        nper_array[zero] = -(fv[zero] + pv[zero]) / pmt[zero]

    nonzero_rate = rate[nonzero]
    z = pmt[nonzero] * (1 + nonzero_rate * when[nonzero]) / nonzero_rate
    nper_array[nonzero] = (
        np.log((-fv[nonzero] + z) / (pv[nonzero] + z))
        / np.log(1 + nonzero_rate)
    )

    return nper_array


def _value_like(arr, value):
    entry = arr.item(0)
    if isinstance(entry, Decimal):
        return Decimal(value)
    else:
        return np.array(value, dtype=arr.dtype).item(0)


def ipmt(rate, per, nper, pv, fv=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
    --------
    >>> import numpy as np
    >>> import numpy_financial as npf

    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 = npf.ipmt(0.0824/12, per, 1*12, principal)
    >>> ppmt = npf.ppmt(0.0824/12, per, 1*12, principal)

    Each element of the sum of the 'ipmt' and 'ppmt' arrays should equal
    'pmt'.

    >>> pmt = npf.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_array = np.array(_rbl(rate, per, total_pmt, pv, when) * rate)

    # Payments start at the first period, so payments before that
    # don't make any sense.
    ipmt_array[per < 1] = _value_like(ipmt_array, np.nan)
    # If payments occur at the beginning of a period and this is the
    # first period, then no interest has accrued.
    per1_and_begin = (when == 1) & (per == 1)
    ipmt_array[per1_and_begin] = _value_like(ipmt_array, 0)
    # If paying at the beginning we need to discount by one period.
    per_gt_1_and_begin = (when == 1) & (per > 1)
    ipmt_array[per_gt_1_and_begin] = (
        ipmt_array[per_gt_1_and_begin] / (1 + rate[per_gt_1_and_begin])
    )

    if np.ndim(ipmt_array) == 0:
        # Follow the ufunc convention of returning scalars for scalar
        # and 0d array inputs.
        return ipmt_array.item(0)
    return ipmt_array


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, 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, 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
    --------
    >>> import numpy as np
    >>> import numpy_financial as npf

    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.

    >>> npf.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
    >>> npf.pv(a, 10*12, -100, 15692.93)
    array([ -100.00067132,  -649.26771385, -1273.78633713]) # may vary

    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
    fact = np.where(rate == 0, nper, (1+rate*when)*(temp-1)/rate)
    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):
    # Evaluate g(r_n)/g'(r_n), where g =
    # fv + pv*(1+rate)**nper + pmt*(1+rate*when)/rate * ((1+rate)**nper - 1)
    t1 = (r+1)**n
    t2 = (r+1)**(n-1)
    g = y + t1*x + p*(t1 - 1) * (r*w + 1) / r
    gp = (n*t2*x
          - p*(t1 - 1) * (r*w + 1) / (r**2)
          + n*p*t2 * (r*w + 1) / r
          + p*(t1 - 1) * w/r)
    return g / gp


# 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=None, tol=None, 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 : Number, optional
        Starting guess for solving the rate of interest, default 0.1
    tol : Number, optional
        Required tolerance for the solution, default 1e-6
    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)
    default_type = Decimal if isinstance(pmt, Decimal) else float

    # Handle casting defaults to Decimal if/when pmt is a Decimal and
    # guess and/or tol are not given default values
    if guess is None:
        guess = default_type('0.1')

    if tol is None:
        tol = default_type('1e-6')

    (nper, pmt, pv, fv, when) = map(np.asarray, [nper, pmt, pv, fv, when])

    rn = guess
    iterator = 0
    close = False
    while (iterator < maxiter) and not np.all(close):
        rnp1 = rn - _g_div_gp(rn, nper, pmt, pv, fv, when)
        diff = abs(rnp1-rn)
        close = diff < tol
        iterator += 1
        rn = rnp1

    if not np.all(close):
        if np.isscalar(rn):
            return default_type(np.nan)
        else:
            # Return nan's in array of the same shape as rn
            # where the solution is not close to tol.
            rn[~close] = np.nan
    return rn

def irr_guess(values, axis=-1):
    """
    Return a guess for an Internal Rate of Return (IRR).

    A continuously compounded IRR r is defined as a solution to the equation:
    log(sum_{k=0}^{n-1} values[k]^+ exp(-r k)) - log(sum_{k=0}^{n-1} values[k]^- exp(-r k)) = 0
    Newton's method can be used to find and r to solve this equation.

    This function implements the first step of Newton's method, starting from r_0 = 0.
    """
    negative_part = -np.minimum(values, 0.)
    positive_part = np.maximum(values, 0.)
    negative_npv = np.sum(negative_part, axis=axis)
    positive_npv = np.sum(positive_part, axis=axis)
    times = np.arange(values.shape[axis])
    negative_duration = np.sum(negative_part * times, axis=axis)/negative_npv
    positive_duration = np.sum(positive_part * times, axis=axis)/positive_npv
    delta_t = positive_duration - negative_duration
    return (positive_npv/negative_npv)**(1/delta_t) - 1

def irr(values, guess=None, tol=1e-12, maxiter=100):
    """
    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.

    :class:`decimal.Decimal` type is not supported.

    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.
    guess : float, optional
        Initial guess of the IRR for the iterative solver. If no guess is
        given an heuristic is used to estimate the guess through the ratio of
        positive to negative cash lows
    tol : float, optional
        Required tolerance to accept solution. Default is 1e-12.
    maxiter : int, optional
        Maximum iterations to perform in finding a solution. Default is 100.

    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
    --------
    >>> import numpy_financial as npf

    >>> round(npf.irr([-100, 39, 59, 55, 20]), 5)
    0.28095
    >>> round(npf.irr([-100, 0, 0, 74]), 5)
    -0.0955
    >>> round(npf.irr([-100, 100, 0, -7]), 5)
    -0.0833
    >>> round(npf.irr([-100, 100, 0, 7]), 5)
    0.06206
    >>> round(npf.irr([-5, 10.5, 1, -8, 1]), 5)
    0.0886

    """
    values = np.atleast_1d(values)
    if values.ndim != 1:
        raise ValueError("Cashflows must be a rank-1 array")

    # If all values are of the same sign no solution exists
    # we don't perform any further calculations and exit early
    same_sign = np.all(values > 0) if values[0] > 0 else np.all(values < 0)
    if same_sign:
        return np.nan

    # If no value is passed for `guess`, then make a heuristic estimate
    if guess is None:
        guess = irr_guess(values)

    # We aim to solve eirr such that NPV is exactly zero. This can be framed as
    # simply finding the closest root of a polynomial to a given initial guess
    # as follows:
    #           V0           V1           V2           V3
    # NPV = ---------- + ---------- + ---------- + ---------- + ... = 0
    #       (1+eirr)^0   (1+eirr)^1   (1+eirr)^2   (1+eirr)^3
    #
    # by letting g = (1+eirr), we substitute to get
    #
    # NPV = V0 * 1/g^0   + V1 * 1/g^1   +  V2 * 1/x^2  +  V3 * 1/g^3  + ... = 0
    #
    # Multiplying by g^N this becomes
    #
    # V0 * g^N   + V1 * g^{N-1}   +  V2 * g^{N-2}  +  V3 * g^{N-3}  + ... = 0
    #
    # which we solve using Newton-Raphson and then reverse out the solution
    # as eirr = g - 1 (if we are close enough to a solution)
    npv_ = np.polynomial.Polynomial(values[::-1])
    d_npv = npv_.deriv()
    g = 1 + guess

    for _ in range(maxiter):
        delta = npv_(g) / d_npv(g)
        if abs(delta) < tol:
            return g - 1
        g -= delta

    return np.nan


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`.

    Warnings
    --------
    ``npv`` considers a series of cashflows starting in the present (t = 0).
    NPV can also be defined with a series of future cashflows, paid at the
    end, rather than the start, of each period. If future cashflows are used,
    the first cashflow `values[0]` must be zeroed and added to the net
    present value of the future cashflows. This is demonstrated in the
    examples.

    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
    --------
    >>> import numpy as np
    >>> import numpy_financial as npf

    Consider a potential project with an initial investment of $40 000 and
    projected cashflows of $5 000, $8 000, $12 000 and $30 000 at the end of
    each period discounted at a rate of 8% per period. To find the project's
    net present value:

    >>> rate, cashflows = 0.08, [-40_000, 5_000, 8_000, 12_000, 30_000]
    >>> npf.npv(rate, cashflows).round(5)
    3065.22267

    It may be preferable to split the projected cashflow into an initial
    investment and expected future cashflows. In this case, the value of
    the initial cashflow is zero and the initial investment is later added
    to the future cashflows net present value:

    >>> initial_cashflow = cashflows[0]
    >>> cashflows[0] = 0
    >>> np.round(npf.npv(rate, cashflows) + initial_cashflow, 5)
    3065.22267

    """
    values = np.atleast_2d(values)
    timestep_array = np.arange(0, values.shape[1])
    npv = (values / (1 + rate) ** timestep_array).sum(axis=1)
    try:
        # If size of array is one, return scalar
        return npv.item()
    except ValueError:
        # Otherwise, return entire array
        return npv


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)
    n = values.size

    # Without this explicit cast the 1/(n - 1) computation below
    # becomes a float, which causes TypeError when using Decimal
    # values.
    if isinstance(finance_rate, Decimal):
        n = Decimal(n)

    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/(n - 1))*(1 + reinvest_rate) - 1