Deferred Life Contingencies#

A deferred life contingency is one in which the payment stream — whether an annuity or an insurance benefit — does not start immediately but only after a waiting period of \(d\) years measured from the reference age \(x\). The parameter d controls this waiting period uniformly across all life annuity and life insurance methods in Lactuca.

Note

This page covers d in isolation (the default case where the valuation date coincides with the reference age, i.e. ts=0). When combining deferment with an off-anniversary time shift ts, the interaction rule \(d_{\text{eff}} = \max(d - ts,\; 0)\) applies; see Prospective Reserves and the ts Parameter for a full treatment.

The deferment parameter#

d is the number of years that must elapse — from age \(x\) — before the first payment is made. Internally, every payment time is shifted by \(d\):

\[ t_j = \frac{j + p_{\text{timing}}}{m} + d \]

where \(j\) counts payments from zero, \(m\) is the annual payment frequency, and \(p_{\text{timing}}\) equals 0 for annuity-due (payments at start of each period) and 1 for annuity-immediate (payments at end of each period).

Key properties:

  • Default: d=0 — payments start immediately.

  • Type: non-negative float; sub-year values such as d=0.5 are fully supported. A negative value raises a ValueError.

  • Scope: d is accepted by all life annuity methods (äx, ax, äxy, axy, äxyz, axyz), all life insurance methods (Ax, Axy, Axyz), and the financial annuity methods on InterestRate (ä, a). Pure endowment methods (nEx, nExy, nExyz) do not accept d; see Interaction with pure endowments.

Standard formulas#

Deferred whole-life annuities#

Annuity-due (prepayable, äx):

\[ {}_{d|}\ddot{a}_x^{(m)} = \frac{1}{m} \sum_{j=0}^{\infty} v^{j/m\,+\,d} \cdot {}_{j/m\,+\,d}p_x = \nEx{d}{x} \cdot \ddot{a}_{x+d}^{(m)} \]

The factorisation uses the chain rule \({}_{j/m+d}p_x = \px{d}{x} \cdot {}_{j/m}p_{x+d}\) and separates \(v^{j/m+d} = v^d \cdot v^{j/m}\):

\[ \frac{1}{m}\sum_{j=0}^{\infty} v^{j/m+d}\cdot {}_{j/m+d}p_x = v^d\,\px{d}{x}\cdot\frac{1}{m}\sum_{j=0}^{\infty} v^{j/m}\cdot {}_{j/m}p_{x+d} = \nEx{d}{x}\cdot\ddot{a}_{x+d}^{(m)} \]

Annuity-immediate (postpayable, ax):

\[ {}_{d|}a_x^{(m)} = \frac{1}{m} \sum_{j=1}^{\infty} v^{j/m\,+\,d} \cdot {}_{j/m\,+\,d}p_x = \nEx{d}{x} \cdot a_{x+d}^{(m)} \]

The factorisation via the pure endowment \(\nEx{d}{x} = v^d \cdot \px{d}{x}\) is exact for both forms: the pure endowment discounts survival to age \(x + d\), and from that age a standard whole-life annuity applies. The two forms are related by

\[ {}_{d|}a_x^{(m)} = {}_{d|}\ddot{a}_x^{(m)} - \frac{1}{m}\,\nEx{d}{x} \]

which is the classical due-to-immediate conversion shifted to the deferred origin.

Deferred temporary annuities#

Annuity-due (prepayable, äx with n=):

\[ {}_{d|}\ddot{a}_{x:\overline{n}|}^{(m)} = \ddot{a}_{x:\overline{d+n}|}^{(m)} - \ddot{a}_{x:\overline{d}|}^{(m)} = \nEx{d}{x} \cdot \ddot{a}_{x+d:\overline{n}|}^{(m)} \]

Annuity-immediate (postpayable, ax with n=):

\[ {}_{d|}a_{x:\overline{n}|}^{(m)} = a_{x:\overline{d+n}|}^{(m)} - a_{x:\overline{d}|}^{(m)} = \nEx{d}{x} \cdot a_{x+d:\overline{n}|}^{(m)} \]

In both cases pass d= and n= together: lt.äx(x, n=n, d=d) / lt.ax(x, n=n, d=d).

The due-to-immediate relation for the deferred temporary case generalises to:

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

The first pure-endowment term accounts for the payment at the start of the deferment period (present in the due form, absent in the immediate) and the second for the extra payment at the end of term \(d+n\) (present in the immediate, absent in the due).

Deferred life insurance#

A deferred life insurance pays the benefit only if death occurs after the deferment period ends:

\[ {}_{d|}A_x = \nEx{d}{x} \cdot A_{x+d} \]

The same identity holds for a finite term:

\[ {}_{d|}A^1_{x:\overline{n}|} = \nEx{d}{x} \cdot A^1_{x+d:\overline{n}|} \]

Pass d= (and optionally n=) to lt.Ax(x, d=d).

API examples#

Single-life annuities#

from lactuca import LifeTable, config

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

# 10-year deferred whole-life annuity-due:  10|äx(55)
a_def_whole_due = lt.äx(55, d=10)
print(a_def_whole_due)    # → 11.3534

# 10-year deferred whole-life annuity-immediate:  10|ax(55)
a_def_whole_imm = lt.ax(55, d=10)
print(a_def_whole_imm)    # → 10.6478

# 10-year deferred, 20-year temporary annuity-due:  10|ä55:20|
a_def_temp_due = lt.äx(55, n=20, d=10)
print(a_def_temp_due)     # → 9.5844

# 10-year deferred, 20-year temporary annuity-immediate:  10|a55:20|
a_def_temp_imm = lt.ax(55, n=20, d=10)
print(a_def_temp_imm)     # → 9.1223

# For comparison: standard whole-life annuity at age 65 (no deferment)
a_reference = lt.äx(65)
print(a_reference)        # → 16.0899

config.reset()

The pure endowment factorisation produces the same result:

from lactuca import LifeTable, config

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

# Direct: 10|äx(55)
a_direct = lt.äx(55, d=10)
print(a_direct)                                     # → 11.3534

# Equivalent via pure endowment:  10Ex(55) × äx(65)
via_endowment = round(lt.nEx(55, n=10) * lt.äx(65), 4)
print(via_endowment)                                # → 11.3534

config.reset()

Single-life insurance#

Ax accepts d and all four calculation modes honour it. The benefit is payable on death only if death occurs after the deferment period:

from lactuca import LifeTable, config

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

# Standard whole-life insurance:  Ax(55)
print(lt.Ax(55))              # → 0.424462

# 10-year deferred whole-life insurance:  10|Ax(55)
print(lt.Ax(55, d=10))        # → 0.380524

# 5-year deferred, 20-year term insurance:  5|A¹55:20|
print(lt.Ax(55, n=20, d=5))   # → 0.144705

config.reset()

Note

A deferred insurance always produces a lower present value than the non-deferred equivalent: deaths during the deferment period \([0, d)\) are excluded from the benefit, reducing the expected present value.

Fractional deferment#

d is a float and any non-negative value is valid, including sub-year periods:

from lactuca import LifeTable, config

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

# 6-month deferment, 20-year temporary:  0.5|ä60:20|
a_half = lt.äx(60, d=0.5, n=20)
print(a_half)    # → 13.8892

config.reset()

Survival at the fractional age \(x + d\) is evaluated under the configured interpolation hypothesis (UDD by default — see lx Interpolation).

Multiple payment frequencies with deferment#

d and m are fully orthogonal: the payment grid is \(t_j = j/(m) + d\) for annuity-due, regardless of frequency:

from lactuca import LifeTable, config

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

# Monthly payments, 10-year deferred whole-life:  10|ä⁽¹²⁾x(55)
a_monthly = lt.äx(55, d=10, m=12)
print(a_monthly)    # → 11.0274

config.reset()

Joint-life deferred contingencies#

All joint-life annuity and insurance methods accept d:

from lactuca import LifeTable, config

config.decimals.annuities = 4
config.decimals.insurances = 6
lt_m, lt_f = LifeTable("PASEM2020_Rel_1o", ["m", "f"], interest_rate=0.03)

# Standard joint-life annuity-due (no deferment):  äxy(60, 58)
a_joint = lt_m.äxy([60, 58], table_y=lt_f)
print(a_joint)           # → 16.7085

# 10-year deferred joint-life annuity-due:  10|äxy(60, 58)
a_joint_def = lt_m.äxy([60, 58], table_y=lt_f, d=10)
print(a_joint_def)       # → 8.2606

# 10-year deferred, 15-year temporary:  10|äxy:15|(60, 58)
a_joint_def_tmp = lt_m.äxy([60, 58], table_y=lt_f, d=10, n=15)
print(a_joint_def_tmp)   # → 6.9214

# 5-year deferred joint first-death insurance:  5|Axy(60, 58)
Axy_def = lt_m.Axy([60, 58], table_y=lt_f, d=5)
print(Axy_def)           # → 0.481055

config.reset()

For the independence assumption, last-survivor and reversionary annuity formulas, see Joint-Life Calculations.

Financial annuities without mortality#

InterestRate.ä and InterestRate.a accept d with exactly the same semantics as their life-table counterparts — no mortality is involved, so survival probabilities are identically 1 and the present value is a pure discount calculation. This is useful for pricing fixed-income structures or for isolating the interest component of a life annuity:

from lactuca import InterestRate, config

config.decimals.annuities = 4
ir = InterestRate(0.03)

# 10-year annuity-due, starting immediately:
print(ir.ä(n=10))          # → 8.7861

# The same annuity deferred 5 years:
print(ir.ä(n=10, d=5))     # → 7.5789

# Annuity-immediate, 5-year deferred:
print(ir.a(n=10, d=5))     # → 7.3582

config.reset()

For a flat interest rate the scaling is exact: \(\ddot{a}(d, n) = v^d \cdot \ddot{a}(n)\), i.e.\ a 5-year deferment at 3% multiplies the undeferrred value by \(v^5 = 1.03^{-5} \approx 0.8626\), giving approximately \(0.8626 \times 8.7861 \approx 7.5789\).

Deferment and benefit growth#

When a GrowthRate is active, d shifts payment times but does not advance the growth anniversary index. The growth exponent always counts completed years from the first payment of the stream — the deferment period \([0, d)\) is not counted.

For an annual annuity-due with constant geometric growth \(g\):

Payment at \(t\)

Growth factor

\(d\)

\((1+g)^0 = 1\)

\(d+1\)

\((1+g)^1\)

\(d+2\)

\((1+g)^2\)

\(\vdots\)

\(\vdots\)

Because the growth schedule always restarts at the first payment of any call, the pure-endowment factorisation extends unchanged to growing annuities:

\[ {}_{d|}\ddot{a}_x(g) = \nEx{d}{x} \cdot \ddot{a}_{x+d}(g) \]
from lactuca import LifeTable, GrowthRate, config

config.decimals.annuities = 4
lt = LifeTable("PASEM2020_Rel_1o", "m", interest_rate=0.03)
gr = GrowthRate(0.02)

# Deferred growing annuity: growth index starts from first payment at age 65
a_deferred = lt.äx(55, d=10, gr=gr)
print(a_deferred)           # → 14.1698

# Equivalent factorisation: nEx(55, 10) × äx(65, gr=2%)
nex = lt.nEx(55, n=10)
a_factor = round(nex * lt.äx(65, gr=gr), 4)
print(a_factor)             # → 14.1698

config.reset()

Note

This is different from ts (time shift): ts=5 consumes the first 5 anniverary years from the growth schedule, while d=5 merely delays the first payment without altering the growth starting point. See Growth Rate Conventions for the complete interaction between gr, d, and ts.

Interaction with pure endowments#

Pure endowment methods (nEx, nExy, nExyz) do not accept a d parameter. This is intentional: the pure endowment \(\nEx{n}{x} = v^n \cdot \px{n}{x}\) already has a fixed observation horizon \(n\); adding deferment on top of it would simply be equivalent to changing \(n\). To value a deferred pure endowment, increase n directly:

\[ {}_d E_x = \texttt{lt.nEx}(x,\; n=d) \qquad {}_{d+n} E_x = \texttt{lt.nEx}(x,\; n=d+n) \]

The annuity factorisation \({}_{d|}\ddot{a}_x = \nEx{d}{x} \cdot \ddot{a}_{x+d}\) makes this connection explicit: nEx and äx together reproduce the deferred annuity.

Calculation modes#

d shifts the payment grid independently of the calculation mode. The table below summarises how each mode integrates the shift:

Mode

Discrete / continuous

Effect of d

"discrete_precision" (default)

Discrete exact summation

Payment grid \(t_j = j/m + d\); each \(t_j\) enters survival and discount directly

"discrete_simplified"

Woolhouse UDD approximation

Same grid shift; Woolhouse 2-term adjustment applied after

"continuous_precision"

Numerical integration

Integration starts at \(t = d\); the grid spans \([d,\, d+n]\)

"continuous_simplified"

Trapezoidal approximation

Sub-calculations shifted by \(d\); results averaged to approximate continuous integral

Both continuous modes use unrounded \(l_x\) values (full float64 precision, no intermediate rounding).

No configuration changes are needed when using d with any mode. Switch modes via config.calculation_mode; see Calculation Modes for full details.

See also#