Life contingencies¶

riskpy.life is the life-and-pensions half of actuarial science: a survival model (a life table), an interest rate, and the present-value formulas that turn the two into insurance and annuity prices, premiums and reserves.

It is pure Python — math only — so it works in the zero-dependency install:

pip install open-riskpy

from riskpy import life

sult = life.LifeTable.sult()                         # the AMLCR standard table
life.whole_life_insurance(sult, 40, i=0.05)          # A_40  = 0.121059
life.whole_life_annuity_due(sult, 40, i=0.05)        # ä_40  = 18.457757
life.whole_life_premium(sult, 40, i=0.05)            # P_40  = 0.006559

Those three numbers are the ones printed in Dickson, Hardy & Waters, Actuarial Mathematics for Life Contingent Risks, Table D.3 — which is the point of shipping the Standard Ultimate Life Table: every worked answer in that book is a test case.

Conventions¶

Ages and durations are integers. A table is a discrete survival model: q_x is the probability a life aged exactly x dies before x + 1.
i is the effective annual rate as a decimal — 0.05, not 5.
Insurances pay 1 at the end of the year of death (A_x); annuities-due pay 1 at the start of each year while alive (ä_x). Pass continuous=True for the uniform-distribution-of-deaths adjustment Ā = (i/δ)·A on insurances and Woolhouse's ā ≈ ä − ½ on annuities. Both are approximations; each docstring says how good.
A table must close: its last q_x is 1. Every identity below depends on it, so a table that does not close is rejected rather than truncated.

Life tables¶

table = life.LifeTable(qx=[0.001, 0.0012, ..., 1.0], start_age=20)
table = life.LifeTable.from_lx([100_000, 99_900, ...], start_age=20)
table = life.LifeTable.gompertz(B=2.7e-6, c=1.124)               # μ_x = B·c^x
table = life.LifeTable.makeham(A=0.00022, B=2.7e-6, c=1.124)     # μ_x = A + B·c^x
table = life.LifeTable.from_force(lambda x: 0.0002 + 2.7e-6 * 1.124 ** x)
table = life.LifeTable.sult()                                    # Makeham, ages 20–130

makeham uses the closed form for t_p_x, so its rates are exact; from_force integrates any force of mortality with Simpson's rule (error around 1e-12 for Gompertz-like laws at the default 64 steps).

A steep law closes early

Gompertz with c = 1.15 reaches q_x = 1 in floating point in the mid-nineties. The generated table closes there, with a RuntimeWarning saying so, rather than carrying thirty ages of lives no one can reach.

table.qx(40), table.px(40), table.lx(40), table.dx(40)
table.tpx(10, 40)              # 10_p_40
table.deferred_qx(10, 40)      # 10|q_40 — dies in the year after age 50
table.curtate_expectation(40)  # e_40 = 45.78 on the SULT
table.survival_curve()         # (ages, l_x) — for viz.survival
table.mortality_curve()        # (ages, q_x) — for viz.mortality

Survivorship

Interest¶

life.discount_factor(0.05)        # v = 0.952381
life.discount_rate(0.05)          # d = 0.047619
life.force_of_interest(0.05)      # δ = 0.048790
life.nominal_rate(0.05, m=12)     # i^(12)
life.annuity_certain(0.05, 10)    # 7.721735;  due=True → 8.107822

Insurances and annuities¶

Every function takes (table, x, i, …) and returns the actuarial present value per unit of benefit.

Insurance		Annuity
`whole_life_insurance(t, x, i)`	`A_x`	`whole_life_annuity_due(t, x, i)`	`ä_x`
`term_insurance(t, x, n, i)`	`A¹_{x:n}`	`whole_life_annuity_immediate(t, x, i)`	`a_x`
`pure_endowment(t, x, n, i)`	`ₙE_x`	`temporary_annuity_due(t, x, n, i)`	`ä_{x:n}`
`endowment_insurance(t, x, n, i)`	`A_{x:n}`	`temporary_annuity_immediate(t, x, n, i)`	`a_{x:n}`
`deferred_whole_life_insurance(t, x, m, i)`	`ₘ\|A_x`	`deferred_annuity_due(t, x, m, i)`	`ₘ\|ä_x`
`increasing_insurance(t, x, i)`	`(IA)_x`	`increasing_annuity_due(t, x, i)`	`(Iä)_x`
		`mthly_annuity_due(t, x, i, m)`	`ä_x^{(m)}` (Woolhouse)

life.term_insurance(sult, 40, 20, 0.05)       # 0.014633
life.pure_endowment(sult, 40, 20, 0.05)       # 0.366630
life.endowment_insurance(sult, 40, 20, 0.05)  # the sum of the two
life.mthly_annuity_due(sult, 40, 0.05, 12)    # 17.999423

The identities you would check by hand are checked: A_x = 1 − d·ä_x and A_{x:n} = A¹_{x:n} + ₙE_x in the verification suite, and ä_{x:n} = ä_x − ₙE_x·ä_{x+n}, a_x = ä_x − 1 and the recursion A_x = v·q_x + v·p_x·A_{x+1} in the tests.

Joint lives¶

Independent lives, joint-life status (fails on the first death) and last-survivor status (fails on the second):

life.joint_life_annuity_due(sult, 40, 45, 0.05)      # ä_xy  = 17.181816
life.last_survivor_annuity_due(sult, 40, 45, 0.05)   # ä_x̄ȳ = 19.092153
life.joint_life_insurance(sult, 40, 45, 0.05)
life.last_survivor_insurance(sult, 40, 45, 0.05)

ä_xy + ä_x̄ȳ = ä_x + ä_y holds exactly, and is tested.

Commutation functions¶

c = life.Commutation(sult, i=0.05)
c.D(60), c.N(60), c.C(60), c.M(60), c.S(60), c.R(60)
c.M(60) / c.D(60)     # == whole_life_insurance(sult, 60, 0.05)

The present-value functions above are computed directly; commutation functions are here for checking against older tables and textbooks that quote them.

Premiums¶

life.net_premium(benefit_apv, annuity_apv)           # the equivalence principle
life.whole_life_premium(sult, 40, 0.05)              # 0.006559 per unit, payable for life
life.term_premium(sult, 40, 20, 0.05)
life.endowment_premium(sult, 40, 20, 0.05)           # 0.029343, payable for 20 years

life.gross_premium(sult, 40, 0.05, benefit=100_000,
                   initial_expense=500, renewal_expense=50,
                   premium_loading=0.03)               # 752.84 a year

gross_premium solves the equivalence principle for a whole-life contract with an initial expense, a level renewal expense from the second year, and a percentage-of-premium loading. The docstring states exactly which cash flows it includes; read it before quoting the number.

Reserves¶

Prospective net premium reserves, and the profile over the policy's life:

life.net_premium_reserve(sult, 40, t=10, i=0.05, kind="endowment", n=20)   # 0.380073
life.reserve_profile(sult, 40, 0.05, "endowment", n=20)   # [(0, 0.0), (1, …), …, (20, 1.0)]

An endowment's reserve is 0 at issue and exactly 1 at maturity — both ends are asserted in the tests, as is Fackler's recursion (ₜV + P)(1 + i) = q_{x+t} + p_{x+t}·ₜ₊₁V at every duration.

Reserve profile

What is not here¶

Select tables (mortality that depends on time since underwriting) and multiple decrement tables are out of scope for this module. Fractional ages are not modelled beyond the UDD and Woolhouse adjustments.

References¶

Bowers, Gerber, Hickman, Jones & Nesbitt, Actuarial Mathematics, 2nd ed.
Dickson, Hardy & Waters, Actuarial Mathematics for Life Contingent Risks, 2nd ed. — the source of the Standard Ultimate Life Table and the printed values the suite reproduces.