"""Commutation functions and actuarial notations
The ``LifeTable`` Space provides
commutation functions and actuarial notations, such as
:math:`D_{x}` and :math:`\\require{enclose}{}_{f|}\\overline{A}_{x}`.
Mortality tables are read from *input.xlsx* into an `ExcelRange`_ object.
The `ExcelRange`_ object is bound to a Reference, :attr:`MortalityTable`.
This Space is included in:
* :mod:`simplelife`
* :mod:`nestedlife`
* :mod:`ifrs17sim`
* :mod:`solvency2`
.. _ExcelRange:
https://docs.modelx.io/en/latest/reference/dataclient.html#excelrange
.. rubric:: Parameters
``LifeTable`` Space is parameterized with :attr:`Sex`,
:attr:`IntRate` and :attr:`TableID`::
>>> simplelife.LifeTable.parameters
('Sex', 'IntRate', 'TableID')
Each ItemSpace represents commutations functions actuarial notations
for a combination of :attr:`Sex`, :attr:`IntRate` and :attr:`TableID`.
For example, ``LifeTable['M', 0.03, 1]`` contains commutation functions
and actuarial notations for Male, the interest rate of 3%, mortality table 1.
Attributes:
Sex(:obj:`str`): 'M' or 'F' to indicate male or female column in the mortality table.
IntRate(:obj:`float`): The constant interest rate for discounting.
TableID(:obj:`int`): The identifier of the mortality table
.. rubric:: References
Attributes:
MortalityTable: `ExcelRange`_ object holding mortality tables.
The data is read from *MortalityTables* range in *input.xlsx*.
Example:
An example of ``LifeTable`` in the :mod:`simplelife` model::
>>> simplelife.LifeTable['M', 0.03, 1].AnnDuenx(40, 10)
8.725179890621531
External Links:
* `International actuarial notation by F.S.Perryman <https://www.casact.org/pubs/proceed/proceed49/49123.pdf>`_
* `Actuarial notations on Wikipedia <https://en.wikipedia.org/wiki/Actuarial_notation>`_
"""
from modelx.serialize.jsonvalues import *
_formula = lambda Sex, IntRate, TableID: None
_bases = []
_allow_none = None
_spaces = []
# ---------------------------------------------------------------------------
# Cells
[docs]def AnnDuenx(x, n, k=1, f=0):
""" The present value of an annuity-due.
.. math::
\\require{enclose}{}_{f|}\\ddot{a}_{x:\\enclose{actuarial}{n}}^{(k)}
Args:
x(int): age
n(int): length of annuity payments in years
k(int, optional): number of split payments in a year
f(int, optional): waiting period in years
"""
if Dx(x) == 0:
return 0
result = (Nx(x+f) - Nx(x+f+n)) / Dx(x)
if k > 1:
return result - (k-1) / (2*k) * (1 - Dx(x+f+n) / Dx(x))
else:
return result
[docs]def AnnDuex(x, k, f=0):
"""The present value of a lifetime annuity due.
Args:
x(int): age
k(int, optional): number of split payments in a year
f(int, optional): waiting period in years
"""
if Dx(x) == 0:
return 0
result = (Nx(x+f)) / Dx(x)
if k > 1:
return result - (k-1) / (2*k)
else:
return result
[docs]def Ax(x, f=0):
"""The present value of a lifetime assurance on a person at age ``x``
payable immediately upon death, optionally with an waiting period of ``f`` years.
.. math::
\\require{enclose}{}_{f|}\\overline{A}_{x}
"""
if Dx(x) == 0:
return 0
else:
return Mx(x+f) / Dx(x)
[docs]def Axn(x, n, f=0):
"""The present value of an assurance on a person at age ``x`` payable
immediately upon death, optionally with an waiting period of ``f`` years.
.. math::
\\require{enclose}{}_{f|}\\overline{A}^{1}_{x:\\enclose{actuarial}{n}}
"""
if Dx(x) == 0:
return 0
else:
return (Mx(x+f) - Mx(x+f+n)) / Dx(x)
[docs]def Cx(x):
"""The commutation column :math:`\\overline{C_x}`.
"""
return dx(x) * disc**(x+1/2)
[docs]def Dx(x):
"""The commutation column :math:`D_{x} = l_{x}v^{x}`.
"""
return lx(x) * disc ** x
[docs]def Exn(x, n):
""" The value of an endowment on a person at age ``x``
payable after n years
.. math::
{}_{n}E_x
"""
if Dx(x) == 0:
return 0
else:
return Dx(x+n) / Dx(x)
[docs]def Mx(x):
"""The commutation column :math:`M_x`."""
if x >= 110:
return Dx(x)
else:
return Mx(x+1) + Cx(x)
[docs]def Nx(x):
"""The commutation column :math:`N_x`."""
if x >= 110: # TODO: Get the last age from the table
return Dx(x)
else:
return Nx(x+1) + Dx(x)
[docs]def disc():
"""The discount factor :math:`v = 1/(1 + i)`."""
return 1 / (1 + IntRate)
[docs]def dx(x):
"""The number of persons who die between ages ``x`` and ``x+1``"""
return lx(x) * qx(x)
[docs]def lx(x):
"""The number of persons remaining at age ``x``. """
if x == 0:
return 100000
else:
return lx(x-1) - dx(x-1)
[docs]def qx(x):
"""Probability that a person at age ``x`` will die in one year."""
return MortalityTable[TableID, Sex, x]
# ---------------------------------------------------------------------------
# References
Sex = "M"
IntRate = 0.01
TableID = 1
MortalityTable = ("Pickle", 2325070426248)