insurance

Life insurance - Computes present values of insurance benefits

MIT License. Copyright (c) 2022-2023 Terence Lim

class actuarialmath.insurance.Insurance(udd: bool = True, **kwargs)[source]

Bases: Fractional

Compute expected present values of life insurance

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

Pure endowment: t_E_x

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

Examples:

>>> life = Insurance().set_survival(mu=lambda x,t: .02*t).set_interest(i=.03)
>>> var = life.E_x(0, t=15, moment=life.VARIANCE, endowment=10000)
A_x(x: int, s: int = 0, t: int = -999, u: int = 0, benefit: ~typing.Callable = <function Insurance.<lambda>>, endowment: float = 0.0, moment: int = 1, discrete: bool = True) → float[source]

Numerically compute EPV of insurance from basic survival functions

Parameters:

  • x – age of selection

  • s – years after selection

  • u – year deferred

  • t – term of insurance

  • benefit – benefit as a function of age and year

  • endowment – amount of endowment for endowment insurance

  • moment – compute first or second moment

  • discrete – benefit paid yearend (True) or moment of death (False)

Examples:

>>> life = Insurance().set_interest(delta=.05)        >>>                   .set_survival(mu=lambda x,s: .03)
>>> benefit = lambda x,t: math.exp(0.04 * t)
>>> A = life.A_x(0, benefit=benefit)
>>> print(A)   # 0.75
>>> A2 = life.A_x(0, moment=2, benefit=benefit)
>>> print(A2)  #0.60
static insurance_variance(A2: float, A1: float, b: float = 1) → float[source]

Compute variance of insurance given moments and benefit

Parameters:

  • A2 – second moment of insurance r.v.

  • A1 – first moment of insurance r.v.

  • b – benefit amount

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

Whole life insurance: A_x

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)

Examples:

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

Term life insurance: A_x:t^1

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)

Examples:

>>> life.term_insurance(x=0, t=30)
deferred_insurance(x: int, s: int = 0, u: int = 0, t: int = -999, b: int = 1, moment: int = 1, discrete: bool = True) → float[source]

Deferred insurance n|_A_x:t^1 = discounted term or whole life

Parameters:

  • x – age of selection

  • s – years after selection

  • u – year deferred

  • 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)

Examples:

>>> life.deferred_insurance(x=0, u=10, t=20)
endowment_insurance(x: int, s: int = 0, t: int = 1, b: int = 1, endowment: int = -1, moment: int = 1, discrete: bool = True) → float[source]

Endowment insurance: A_x^1:t = term insurance + pure endowment

Parameters:

  • x – age of selection

  • s – years after selection

  • t – term of insurance

  • b – amount of benefit

  • endowment – amount of endowment paid at end of term if survive

  • moment – compute first or second moment

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

Examples:

>>> life.endowment_insurance(x=0, t=10)
increasing_insurance(x: int, s: int = 0, t: int = -999, b: int = 1, discrete: bool = True) → float[source]

Increasing life insurance: (IA)_x

Parameters:

  • x – age of selection

  • s – years after selection

  • t – term of insurance

  • b – amount of benefit in first year

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

Examples:

>>> life.increasing_insurance(x=0, t=10)
decreasing_insurance(x, s: int = 0, t: int = 1, b: int = 1, discrete: bool = True) → float[source]

Decreasing life insurance: (DA)_x

Parameters:

  • x – age of selection

  • s – years after selection

  • t – term of insurance

  • b – amount of benefit in first year

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

Examples:

>>> life.decreasing_insurance(x=0, t=10)
Z_x(x, s: int = 0, t: float = 0.0, discrete: bool = True)[source]

EPV of year t insurance death benefit for life aged [x]+s: b_x[s]+s(t)

Parameters:

  • x – age of selection

  • s – years after selection

  • t – year of benefit

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

Z_t(x: int, prob: float, discrete: bool = True) → float[source]

T_x, given the prob of the PV of life insurance, i.e. r.v. Z(t)

Parameters:

  • x – age initially insured

  • prob – desired probability threshold

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

Returns:

T s.t. S(t)==prob; if discrete, return K=floor(T) s.t. S(K)>=prob

Examples

>>> t = life.Z_t(x=20, prob=0.8, discrete=True)
Z_from_t(t: float, discrete: bool = True) → float[source]

PV of annual or continuous insurance payment Z(t) at t=T_x

Parameters:

  • t – year of death

  • b – benefit paid at t

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

Examples

>>> t = life.Z_t(x=20, prob=0.8, discrete=True)
>>> print(Z, life.Z_from_t(t, discrete=True))
Z_from_prob(x: int, prob: float, discrete: bool = True) → float[source]

Percentile of annual or continuous WL insurance PV r.v. Z given probability

Parameters:

  • x – age initially insured

  • prob – threshold for probability of survival

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

Examples

>>> Z = life.Z_from_prob(x=20, prob=0.8, discrete=True)
Z_to_t(Z: float) → float[source]

T_x s.t. PV of continuous WL insurance payment is Z

Parameters:

Z – Present value of benefit paid

Examples

>>> t = life.Z_t(x=20, prob=0.8, discrete=False)
>>> Z = life.Z_from_prob(x=20, prob=0.8, discrete=False)
>>> print(t, life.Z_to_t(Z))
Z_to_prob(x: int, Z: float) → float[source]

Probability that continuous WL insurance PV r.v. is no more than Z

Parameters:

  • x – age initially insured

  • Z – present value of benefit paid

Examples

>>> Z = life.Z_from_prob(x=20, prob=0.8, discrete=False)
>>> print(life.Z_to_prob(x=20, Z=Z))
Z_plot(x: int, s: int = 0, stop: int = 0, benefit: ~typing.Callable = <function Insurance.<lambda>>, T: float | None = None, discrete: bool = True, ax: ~typing.Any = None, dual: bool = False, title: str | None = None, color: str = 'r', alpha: float = 0.3) → float | None[source]

Plot of PV of insurance r.v. Z vs t

Parameters:

  • x – age of selection

  • s – years after selection

  • stop – time to end plot

  • benefit – benefit as a function of age and time

  • T – specific time to annotate

  • insurance (discrete discrete or continuous) –

  • ax – figure object to plot in

  • dual – whether to plot survival function on secondary axis

  • color – color of primary plot

  • alpha – transparency of plot area

  • title – title of plot