Life Annuities#

A life annuity is a series of payments made as long as a designated life survives. It is the primary building block in individual and occupational pensions, annuity pricing, premium calculations, and prospective reserve evaluation.

In Lactuca, single-life life annuities are computed via lt.äx (annuity-due, prepayable) and lt.ax (annuity-immediate, postpayable). Both are also importable as standalone functions directly from lactuca.

Annuity-due and annuity-immediate#

Annuity-due (äx): a payment of \(1/m\) is made at the start of each sub-annual period, conditional on the life being alive. The first payment is at \(t = 0\) (or after the deferment \(d\) if specified).

Annuity-immediate (ax): a payment of \(1/m\) is made at the end of each sub-annual period, conditional on the life surviving to that point. The first payment falls at \(t = 1/m\).

The two forms are related by:

\[ \ddot{a}_{x:\overline{n}|}^{(m)} = a_{x:\overline{n}|}^{(m)} + \frac{1}{m}\!\left(1 - {}_{n}E_x\right) \]

For whole life (\(n \to \infty\), \({}_{n}E_x \to 0\)) this simplifies to \(\ddot{a}_x^{(m)} = a_x^{(m)} + \tfrac{1}{m}\).

The annuity formula#

The actuarial present value (APV) of a temporary life annuity-due at frequency \(m\) is:

\[ \ddot{a}_{x:\overline{n}|}^{(m)} = \frac{1}{m} \sum_{j=0}^{mn-1} v^{j/m} \cdot {}_{j/m}p_x \]

Each term discounts a payment of \(1/m\) payable at time \(j/m\) (start of the \(j\)-th sub-period) by the discount factor \(v^{j/m} = (1+i)^{-j/m}\) and weights it by the survival probability \({}_{j/m}p_x\) that \((x)\) reaches time \(j/m\).

For a whole-life annuity (n=None), the sum runs to the limiting age \(\omega\):

\[ \ddot{a}_x^{(m)} = \frac{1}{m} \sum_{j=0}^{m(\omega-x)-1} v^{j/m} \cdot {}_{j/m}p_x \]

The corresponding annuity-immediate formula:

\[ a_{x:\overline{n}|}^{(m)} = \frac{1}{m} \sum_{j=1}^{mn} v^{j/m} \cdot {}_{j/m}p_x \]

Core parameters#

Parameter	Type	Default	Description
`x`	`float`	(required)	Current attained age
`n`	`float \| None`	`None`	Term in years; `None` = whole life
`m`	`int`	`1`	Payment frequency per year (1, 2, 3, 4, 6, 12, 14, 24, 26, 52, 365)
`d`	`float`	`0.0`	Deferment period in years
`ts`	`float`	`0.0`	Elapsed time for reserve calculations
`ir`	`float \| InterestRate \| None`	table default	Interest rate
`gr`	`float \| GrowthRate \| None`	`None`	Benefit growth rate (escalation)

äx and ax are importable as functional-style wrappers:

from lactuca import äx, ax

See Deferred Life Contingencies for the full treatment of d, and Prospective Reserves and the ts Parameter for ts.

Equivalence with life insurances#

The annuity-due and whole-life insurance are linked through the annuity-insurance duality. At annual frequency (\(m = 1\)) with end-of-year death benefit:

\[ A_x + d\,\ddot{a}_x = 1 \qquad \Longleftrightarrow \qquad \ddot{a}_x = \frac{1 - A_x}{d} \]

where \(d = 1 - v = i/(1+i)\) is the annual effective discount rate.

For a temporary annuity and matching endowment insurance of the same term \(n\):

\[ A_{x:\overline{n}|} + d\,\ddot{a}_{x:\overline{n}|} = 1 \qquad \Longleftrightarrow \qquad \ddot{a}_{x:\overline{n}|} = \frac{1 - A_{x:\overline{n}|}}{d} \]

Note

In Lactuca, lt.Ax(x, n=n) returns only the term (death) insurance \(A^1_{x:\overline{n}|}\). The full endowment insurance is lt.Ax(x, n=n) + lt.nEx(x, n=n).

The duality holds exactly for \(m = 1\) with config.mortality_placement = "end". For other modes (especially "mid", the default) and \(m > 1\), use the \(m\)-thly discount rate \(d^{(m)} = m(1 - v^{1/m})\) in place of \(d\). Small numerical discrepancies will appear because the two quantities are computed by independent engine dispatches.

Commutation function connection#

For integer ages and annual frequency (\(m = 1\)) the standard commutation identity is:

\[ \ddot{a}_x = \frac{N_x}{D_x} \]

See Commutation Functions for the algebraic derivation. The calculation engine used by äx and ax evaluates the defining sum directly — it does not use commutation functions — so it correctly handles all frequencies, fractional ages, and piecewise interest rate curves.

Actuarial contexts#

Actuarial use case	Method call	Key parameters
Annual retirement pension (life)	`lt.äx(x)`	`m=12` for monthly payments
Pension deferred until age 65	`lt.äx(x, d=65-x)`	`d = 65 − current age`
Premium-paying temporary annuity	`lt.äx(x, n=n_premium)`	—
Widow’s reversionary pension	`lt.äx(y) - lt.äxy([x, y], ...)`	see Joint-Life Calculations
Escalating pension (CPI-linked)	`lt.äx(x, gr=GrowthRate(0.02))`	see Growth Rates
Prospective reserve at time \(t\)	`lt.äx(x, n=n, ts=t)`	see Prospective Reserves and the ts Parameter

Code examples#

Whole-life and temporary annuities#

from lactuca import LifeTable, config

config.decimals.annuities = 4
lt = LifeTable("PASEM2020_Rel_1o", "m", interest_rate=0.03)

# Annual whole-life annuity-due: äx(65)
a_wl = lt.äx(65)
print(a_wl)           # → 16.0899

# 20-year temporary annuity-due: ä_{65:20|}
a_temp = lt.äx(65, n=20)
print(a_temp)         # < a_wl (truncated at n=20)

# Monthly temporary annuity-due: ä^(12)_{65:20|}
a_m12 = lt.äx(65, n=20, m=12)
print(a_m12)          # lower than a_temp: each sub-annual payment is weighted by exact fractional-age survival probability

# Annuity-immediate (postpayable) equivalent: a_{65:20|}
a_imm = lt.ax(65, n=20)
print(a_imm)          # < a_temp by approximately 1/m × (1 - nEx)

config.reset()

Due-to-immediate conversion#

from lactuca import LifeTable, config

config.decimals.annuities = 4
lt = LifeTable("PASEM2020_Rel_1o", "m", interest_rate=0.03)

n = 20
x = 65
a_due = lt.äx(x, n=n)
nEx   = lt.nEx(x, n=n)
a_imm = lt.ax(x, n=n)

# Identity: ä = a + (1/m) * (1 - nEx)  for m=1
print(round(a_due - a_imm, 6))           # ≈ 1 - nEx  (for m=1; smaller when nEx is large)
print(round(a_due - (a_imm + 1 - nEx), 6))  # ≈ 0.0

config.reset()

Annuity-insurance duality#

from lactuca import LifeTable, config

config.decimals.annuities = 6
config.decimals.insurances = 6
lt = LifeTable("PASEM2020_Rel_1o", "m", interest_rate=0.03)

i = 0.03
d_rate = i / (1 + i)    # annual discount rate d = 1 - v

ax_val = lt.äx(65)      # annuity-due, m=1
Ax_val = lt.Ax(65)      # annual whole-life insurance

# Check: Ax + d * äx ≈ 1  (exact under m=1, mortality_placement="end")
print(round(Ax_val + d_rate * ax_val, 4))   # ≈ 1.0 (small gap from default "mid" placement)

config.reset()

Prospective reserve at elapsed time `ts`#

from lactuca import LifeTable, config

config.decimals.annuities = 4
lt = LifeTable("PASEM2020_Rel_1o", "m", interest_rate=0.03)

x, n = 55, 30

# APV at issue (ts=0)
a_issue = lt.äx(x, n=n)
print(a_issue)

# Prospective reserve after 10 years:
# effective age = 65, remaining term = 20  — identical approaches:
a_ts = lt.äx(x, n=n, ts=10)        # using ts parameter
a_direct = lt.äx(65, n=20)         # calling directly at attained age

print(round(a_ts - a_direct, 8))   # → 0.0

config.reset()