mortalitylaws

Mortality Laws: Applies shortcut formulas for Beta, Uniform, Makeham and Gompertz

MIT License. Copyright 2022-2023 Terence Lim

class actuarialmath.mortalitylaws.MortalityLaws(**kwargs)[source]

Bases: Reserves

Apply shortcut formulas for special mortality laws

l_r(x: int, s: int = 0, r: float = 0.0) → float[source]

Fractional lives given special mortality law: l_[x]+s+r

Parameters:

  • x – age of selection

  • s – years after selection

  • r – fractional year after selection

p_r(x: int, s: int = 0, r: float = 0.0, t: float = 1.0) → float[source]

Fractional age survival probability given special mortality law

Parameters:

  • x – age of selection

  • s – years after selection

  • r – fractional year after selection

  • t – death within next t fractional years

q_r(x: int, s: int = 0, r: float = 0.0, t: float = 1.0, u: float = 0.0) → float[source]

Fractional age deferred mortality given special mortality law

Parameters:

  • x – age of selection

  • s – years after selection

  • r – fractional year after selection

  • u – survive u fractional years, then…

  • t – death within next t fractional years

mu_r(x: int, s: int = 0, r: float = 0.0) → float[source]

Fractional age force of mortality given special mortality law

Parameters:

  • x – age of selection

  • s – years after selection

  • r – fractional year after selection

f_r(x: int, s: int = 0, r: float = 0.0, t: float = 0.0) → float[source]

fractional age lifetime density given special mortality law

Parameters:

  • x – age of selection

  • s – years after selection

  • r – fractional year after selection

  • t – mortality at fractional year t

e_r(x: int, s: int = 0, r: float = 0.0, t: float = -999) → float[source]

Fractional age future lifetime given special mortality law

Parameters:

  • x – age of selection

  • s – years after selection

  • r – fractional year after selection

  • t – limited at t years

class actuarialmath.mortalitylaws.Beta(omega: int, alpha: float, radix: int = 100000, **kwargs)[source]

Bases: MortalityLaws

Shortcuts with beta distribution of deaths (is Uniform when alpha = 1)

Parameters:

  • omega – maximum age

  • alpha – alpha paramter of beta distribution

  • radix – assumed starting number of lives for survival function

Examples

>>> print(Beta(omega=60, alpha=1/3).mu_x(35) * 1000)
e_r(x: int, s: int = 0, t: float = -999) → float[source]

Expectation of future lifetime through fractional age: e_[x]+s:t

Parameters:

  • x – age of selection

  • s – years after selection

  • t – limited at t years

e_x(x: int, s: int = 0, n: int = -999, curtate: bool = False, moment: int = 1) → float[source]

Shortcut formula for complete expectation

Parameters:

  • x – age of selection

  • s – years after selection

  • n – death within next n fractional years

  • t – limited at t years

  • curtate – whether curtate (True) or complete (False) lifetime

  • moment – whether to compute first (1) or second (2) moment

class actuarialmath.mortalitylaws.Uniform(omega: int, **kwargs)[source]

Bases: Beta

Shortcuts with uniform distribution of deaths aka DeMoivre’s Law

Parameters:

omega – maximum age

Examples

>>> print(Uniform(95).e_x(30, t=40, curtate=False)) # 27.692
e_x(x: int, s: int = 0, t: int = -999, curtate: bool = False, moment: int = 1) → float[source]

Shortcut for expected future lifetime

Parameters:

  • x – age of selection

  • s – years after selection

  • t – limited at t years

  • curtate – whether curtate (True) or complete (False) lifetime

  • moment – whether to compute first (1) or second (2) moment

E_x(x: int, s: int = 0, t: int = -999, moment: int = 1) → float[source]

Shortcut for Pure Endowment

Parameters:

  • x – age of selection

  • s – years after selection

  • t – term of pure endowment

  • endowment – amount of pure endowment

  • moment – compute first or second moment

whole_life_insurance(x: int, s: int = 0, moment: int = 1, b: int = 1, discrete: bool = True) → float[source]

Shortcut for whole life insurance

Parameters:

  • x – age of selection

  • s – years after selection

  • b – amount of benefit

  • moment – compute first or second moment

  • discrete – benefit paid year-end (True) or moment of death (False)

term_insurance(x: int, s: int = 0, t: int = 1, b: int = 1, moment: int = 1, discrete: bool = False) → float[source]

Shortcut for term insurance

Parameters:

  • x – age of selection

  • s – years after selection

  • t – term of insurance

  • b – amount of benefit

  • moment – compute first or second moment

  • discrete – benefit paid year-end (True) or moment of death (False)

class actuarialmath.mortalitylaws.Makeham(A: float, B: float, c: float, **kwargs)[source]

Bases: MortalityLaws

Includes element in force of mortality that does not depend on age

Parameters:

  • A – parameters of Makeham distribution

  • B – parameters of Makeham distribution

  • c – parameters of Makeham distribution

Examples

>>> print(Makeham(A=0.00022, B=2.7e-6, c=1.124).mu_x(60) * 0.9803)  # 0.00316
class actuarialmath.mortalitylaws.Gompertz(B: float, c: float, **kwargs)[source]

Bases: Makeham

Is Makeham’s Law with A = 0

Parameters:

  • B – parameters of Gompertz distribution

  • c – parameters of Gompertz distribution

Examples

>>> print(Gompertz(B=0.00027, c=1.1).f_x(50, t=10)) # 0.04839