Actuarial Formulas

Mathematical foundations and formulas used throughout Lactuca.

This page is a concise mathematical reference for each actuarial quantity Lactuca computes. Sections are ordered from foundational concepts to derived products. Each section documents the formula in force, the required hypothesis or mode, and any input constraints — including the constraints whose violation raises a ValueError.

For narrative explanations and runnable code examples see Life Annuities, Life Insurances, and Calculation Modes. For a full list of errors see Error Reference.

Section	Key symbols
Survival and mortality functions	\(l_x\), \(d_x\), \({}_tp_x\), \({}_tq_x\), \(L_x\), \(T_x\)
Select-ultimate notation	\(q_{[x]+d}\), select period
Fractional age interpolation	UDD ("linear"), CFM ("exponential")
Commutation functions	\(D_x\), \(N_x\), \(S_x\), \(C_x\), \(M_x\), \(R_x\), \(L_x\), \(L_{x+s}\), \(T_x\), \(T_{x+s}\)
Life annuities	\(\ddot{a}_x\), temporary, deferred, \(m\)-thly, immediate, continuous
Life insurances	\(A_x\), term, endowment, pure endowment, deferred, continuous
Annuity–insurance relationship	\(\ddot{a}_x = (1 - A_x)/d\)
Life expectancy	\(\mathring{e}_x\), \(e_x\)
Interest rate conversions	\(i^{(m)}\), \(\delta\), \(d^{(m)}\)
Growth rates (escalating benefits)	\(v_g = (1+g)/(1+i)\)
Multiple decrements	\({}_tq_x^{(j)}\), \({}_tp_x^{(\tau)}\)
Joint life and last survivor	\(\ddot{a}_{xy}\), \(\ddot{a}_{\overline{xy}}\), \(A_{xy}\)
Formulas by calculation mode	mode-specific formulas
References	—

Survival and mortality functions

Life table functions

The fundamental survival function \(l_x\) represents the number of lives at exact age \(x\) from an initial cohort \(l_0\).

\[ l_x = l_0 \cdot \prod_{k=0}^{x-1} (1 - q_k) \]

Survival probability

The probability that a life aged \(x\) survives to age \(x+t\):

\[ {}_t p_x = \frac{l_{x+t}}{l_x} \]

Mortality probability

The probability that a life aged \(x\) dies within \(t\) years:

\[ {}_t q_x = 1 - {}_t p_x = \frac{l_x - l_{x+t}}{l_x} \]

Deferred mortality

The probability of surviving \(u\) years and then dying within the next \(t\) years:

\[ {}_{u|t} q_x = {}_u p_x \cdot {}_t q_{x+u} \]

Survivors and deaths

The number of survivors in a cohort at aged \(x\) is the life table function \(l_x\). Deaths between consecutive integer ages are:

\[ d_x = l_x - l_{x+1} \]

The central exposed-to-risk (person-years lived in \([x, x+1)\)) under UDD is \(L_x = l_x - d_x/2\). See the Lx commutation function definition below.

Select-ultimate notation

For select-ultimate tables, mortality depends on both attained age \(x\) and duration \(d\) since underwriting. The conventional actuarial notation for the rate at duration \(d\) is \(q_{[x]+d}\), read as “the rate for a life selected at age \(x\) and now \(d\) years into the select period”.

In Lactuca the minimum valid integer duration is given by TableSource.start_duration:

• start_duration = 1 — standard convention (e.g. DAV 2004 R, most international tables); duration=1 is the freshly-underwritten rate \(q_{[x]}\).

• start_duration = 0 — CMI/UK convention (e.g. AM92/AF92); duration=0 is the freshly-underwritten rate \(q_{[x]+0}\).

Durations at or beyond start_duration + select_period fall back to the ultimate column \(q_{x+d}^{\text{ult}}\) regardless of the value passed.

\[\begin{split} q_{[x]+d} = \begin{cases} q_{[x]+d}^{\text{select}} & \text{if } d < \text{start duration} + \text{select period} \\ q_{x+d}^{\text{ult}} & \text{if } d \ge \text{start duration} + \text{select period} \end{cases} \end{split}\]

See also

For the fallback formula, code example, and select_period metadata field see Select-and-ultimate tables below.

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Fractional age interpolation

Lactuca supports two interpolation methods for \(l_{x+s}\) at fractional ages, controlled by config.lx_interpolation.

Note

config.lx_interpolation = "linear" (UDD, default) or "exponential" (CFM). Passing any other string raises ValueError. The mode affects all probability and annuity calculations at fractional ages.

UDD — Uniform Distribution of Deaths ("linear")

For \(0 \leq s < 1\):

\[ l_{x+s} = (1-s)\,l_x + s\,l_{x+1} \]

\[ {}_s p_x = 1 - s\,q_x, \qquad {}_{1/m}p_x = 1 - \frac{q_x}{m} \]

Force of mortality under UDD (increasing throughout each year of age):

\[ \mu_{x+s} = \frac{q_x}{1 - s\,q_x} \]

CFM — Constant Force of Mortality ("exponential")

For \(0 \leq s < 1\):

\[ l_{x+s} = l_x \cdot p_x^{\,s} \]

\[ {}_s p_x = p_x^{\,s} = e^{-\mu_x s}, \qquad {}_{1/m}p_x = p_x^{\,1/m} \]

Force of mortality (truly constant within \([x, x+1)\) under CFM):

\[ \mu_{x+s} = -\ln p_x \]

Commutation functions

Commutation functions simplify actuarial calculations by precomputing discounted survival values.

Note

Dx, Nx, Cx, Mx, Sx, Rx, Lx, Tx require integer ages. Passing a fractional age raises ValueError. Lx_continuous and Tx_continuous are the counterparts for fractional ages; they raise ValueError for integer ages.

Symbol	Formula	Actuarial use	Lactuca method
\(D_x\)	\(v^x l_x\)	Denominator in all commutation formulas	lt.Dx(x, ir=)
\(N_x\)	\(\sum_{k \ge x} D_k\)	\(\ddot{a}_x = N_x/D_x\)	lt.Nx(x, ir=)
\(S_x\)	\(\sum_{k \ge x} N_k\)	Increasing annuities	lt.Sx(x, ir=)
\(C_x\)	\(v^{x+f} d_x\) *	Building block for \(M_x\)	lt.Cx(x, ir=)
\(M_x\)	\(\sum_{k \ge x} C_k\)	\(A_x = M_x/D_x\)	lt.Mx(x, ir=)
\(R_x\)	\(\sum_{k \ge x} M_k\)	Increasing insurance	lt.Rx(x, ir=)
\(L_x\)	\((l_x + l_{x+1})/2\)	Person-years in \([x, x+1)\) (integer \(x\))	lt.Lx(x)
\(L_{x+s}\)	\(\int_{x+s}^{x+s+1} l_t\,\mathrm{d}t\)	Person-years at fractional age	lt.Lx_continuous(x+s, m=)
\(T_x\)	\(\sum_{k=x}^{\omega-1} L_k\)	\(\mathring{e}_x = T_x / l_x\) (integer \(x\))	lt.Tx(x)
\(T_{x+s}\)	\(\int_{x+s}^{\omega} l_t\,\mathrm{d}t\)	Future person-years at fractional age	lt.Tx_continuous(x+s, m=)

* \(f\) = mortality placement fraction: "mid" → 0.5 (default); "end" → 1.0; "beginning" → 0. See the C function definition for details.

Discount factor

\[ v = \frac{1}{1+i} \]

D function

\[ D_x = v^x \cdot l_x \]

N function (sum of D)

\[ N_x = \sum_{k=x}^{\omega} D_k = D_x + D_{x+1} + \cdots + D_{\omega} \]

S function (sum of N)

\[ S_x = \sum_{k=x}^{\omega} N_k \]

C function (deaths)

\[ C_x = v^{x+f} \cdot d_x = v^{x+f} \cdot (l_x - l_{x+1}) \]

where \(f\) is the mortality placement fraction controlled by config.mortality_placement: "beginning" → \(f = 0\); "mid" → \(f = 0.5\) (default); "end" → \(f = 1\). The classical actuarial formula \(C_x = v^{x+1} d_x\) (end-of-year) corresponds to "end". The commutation identity \(A_x = M_x / D_x\) holds exactly for any placement: \(M_x = \sum C_k\) uses the same offset \(f\) throughout, so different placements yield internally consistent but numerically distinct insurance values (representing payment timing at \(k+f\), not approximations of each other).

M function (sum of C)

\[ M_x = \sum_{k=x}^{\omega} C_k \]

R function (sum of M)

\[ R_x = \sum_{k=x}^{\omega} M_k \]

Lx function (person-years lived)

At integer ages (uses lt.Lx(x)):

\[ L_x = \int_0^1 l_{x+t}\,\mathrm{d}t \approx \frac{l_x + l_{x+1}}{2} = l_x - \frac{d_x}{2} \]

Exact under UDD. Requires integer age; fractional age raises ValueError.

At fractional ages (uses lt.Lx_continuous(x+s, m=)):

\[ L_{x+s} = \int_{x+s}^{x+s+1} l_t\,\mathrm{d}t \]

Evaluated via the trapezoidal rule with \(m+1\) equally-spaced nodes (default \(m=12\)). Requires a strictly fractional age; integer age raises ValueError.

Tx function (total future person-years)

At integer ages (uses lt.Tx(x)):

\[ T_x = \sum_{k=x}^{\omega-1} L_k \]

Used to compute life expectancy: \(\mathring{e}_x = T_x / l_x\). Requires integer age.

At fractional ages (uses lt.Tx_continuous(x+s, m=)):

\[ T_{x+s} = \int_{x+s}^{\omega} l_t\,\mathrm{d}t \]

Evaluated via the trapezoidal rule (default \(m=12\)). Requires a strictly fractional age; integer age raises ValueError. Used internally by ex_continuous.

Life annuities

Whole life annuity due

Present value of annual payments of 1 at the beginning of each year while the annuitant survives:

\[ \ddot{a}_x = \sum_{k=0}^{\omega - x} v^k \cdot {}_k p_x = \frac{N_x}{D_x} \]

Temporary annuity due

Present value for \(n\) years:

\[ \ddot{a}_{x:\overline{n}|} = \sum_{k=0}^{n-1} v^k \cdot {}_k p_x = \frac{N_x - N_{x+n}}{D_x} \]

Deferred annuity

Annuity starting after \(d\) years:

\[ {}_{d|}\ddot{a}_x = v^d \cdot {}_d p_x \cdot \ddot{a}_{x+d} = \frac{N_{x+d}}{D_x} \]

Deferred temporary annuity

Annuity of at most \(n\) payments starting after a \(d\)-year deferral period:

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

Equivalently, using the pure endowment:

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

In Lactuca, pass d= and n= together: lt.äx(x, n=n, d=d, ir=ir).

m-thly annuity due

Payments \(m\) times per year:

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

Using Woolhouse approximation (first order):

\[ \ddot{a}_x^{(m)} \approx \ddot{a}_x - \frac{m-1}{2m} \]

Immediate annuity

Payments at the end of each period (\(m = 1\), whole-life):

\[ a_x = \ddot{a}_x - 1 \]

For \(m\)-thly frequency the relationship generalises to:

\[ a_x^{(m)} = \ddot{a}_x^{(m)} - \frac{1}{m} \]

\[ a_{x:\overline{n}|}^{(m)} = \ddot{a}_{x:\overline{n}|}^{(m)} - \frac{1}{m}\,(1 - {}_n E_x) \]

These follow directly from Lactuca’s payment-grid definition: the first immediate payment falls at \(1/m\) rather than \(0\), shifting every payment by \(1/m\) relative to the annuity-due.

Continuous annuity

\[ \bar{a}_x = \int_0^{\omega-x} v^t \cdot {}_t p_x \, dt \]

Approximation used by continuous_simplified mode (see continuous_simplified for the temporary variant):

\[ \bar{a}_x \approx \ddot{a}_x - \frac{1}{2} \]

Life insurances

Whole life insurance (discrete)

Present value of 1 payable on death, discounted at the mortality placement offset:

\[ A_x = \sum_{k=0}^{\omega-x-1} v^{k+f} \cdot {}_k p_x \cdot q_{x+k} = \frac{M_x}{D_x} \]

where \(f\) is the mortality placement fraction (see \(C_x\)): "mid" → \(f = 0.5\) (default); "end" → \(f = 1\); "beginning" → \(f = 0\). The classical textbook formula uses \(v^{k+1}\) and corresponds to "end" placement. The commutation identity \(A_x = M_x/D_x\) holds for any placement because Lactuca’s \(M_x = \sum C_k\) uses the same offset \(f\) throughout.

Term insurance

Insurance payable only if death occurs within \(n\) years:

\[ A^1_{x:\overline{n}|} = \frac{M_x - M_{x+n}}{D_x} \]

Pure endowment

Payment only if the insured survives \(n\) years:

\[ {}_n E_x = v^n \cdot {}_n p_x = \frac{D_{x+n}}{D_x} \]

Endowment insurance

Combination of term insurance and pure endowment:

\[ A_{x:\overline{n}|} = A^1_{x:\overline{n}|} + {}_n E_x \]

Continuous whole life insurance

\[ \bar{A}_x = \int_0^{\omega-x} v^t \cdot {}_t p_x \cdot \mu_{x+t} \, dt \]

Classical UDD approximation (theoretical reference; not the formula Lactuca’s continuous_simplified mode uses — see continuous_simplified):

\[ \bar{A}_x \approx \frac{i}{\delta} A_x \]

where \(\delta = \ln(1+i)\) is the force of interest.

Deferred insurance

Present value of a 1-unit benefit payable on death after a deferment period of \(d\) years:

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

For a deferred term insurance (benefit paid only if death occurs within \(n\) years after the deferment period):

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

In Lactuca: lt.Ax(x, d=d) and lt.Ax(x, d=d, n=n). See Deferred Life Contingencies for code examples across all products.

Relationship between annuities and insurances

Note

Notation clash — d has two meanings on this page. In formulas below, \(d = i/(1+i)\) is the discount rate (a fixed finance constant). Elsewhere on this page (deferred annuities, Lactuca’s d= argument) \(d\) is the deferral period in years. The two quantities are always typographically identical in standard actuarial notation; context determines which is meant.

Fundamental relationship:

\[ \ddot{a}_x = \frac{1 - A_x}{d} \]

where \(d = \frac{i}{1+i}\) is the discount rate.

For temporary annuity:

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

For continuous modes the analogous relationship uses the force of interest \(\delta = \ln(1+i)\):

\[ \bar{a}_x = \frac{1 - \bar{A}_x}{\delta} \]

Note

The discrete identities \(\ddot{a}_x = (1-A_x)/d\) and \(\ddot{a}_{x:\overline{n}|} = (1-A_{x:\overline{n}|})/d\) are classical algebraic results that hold when \(A_x\) is the end-of-year insurance (i.e., config.mortality_placement = "end"). Under Lactuca’s default "mid" placement, \(A_x\) represents a mid-year benefit and these exact equalities do not hold, though the approximation error is small in practice.

Life expectancy

Complete life expectancy

The complete (entire) expectation of life \(\mathring{e}_x\) is the expected total future lifetime of a life aged \(x\):

\[ \mathring{e}_x = \int_0^{\omega-x} {}_t p_x \, dt \]

Warning

ex(x) requires an integer age. Passing a fractional age raises ValueError.

ex(x) — integer ages only: uses the UDD discrete approximation \(T_x = \sum L_k\):

\[ T_x = \sum_{k=x}^{\omega-1} L_k \approx \sum_{k=x}^{\omega-1} \frac{l_k + l_{k+1}}{2}, \qquad \mathring{e}_x = \frac{T_x}{l_x} \]

Warning

ex_continuous(x + s, m=) requires a strictly fractional age. Passing an integer age raises ValueError. Default m = 12.

ex_continuous(x + s, m=) — fractional ages only: integrates \(l_t\) numerically (trapezoidal rule with \(m\) sub-intervals per year):

\[ T_{x+s} = \int_{x+s}^{\omega} l_t \, dt, \qquad \mathring{e}_{x+s} = \frac{T_{x+s}}{l_{x+s}} \]

Curtate life expectancy

Expected whole years of future life:

\[ e_x = \sum_{k=1}^{\omega-x} {}_k p_x \]

Under UDD the two expectations are related by:

\[ \mathring{e}_x = e_x + \tfrac{1}{2} \]

Interest rate conversions

Nominal to effective rate (m-thly)

\[ i = \left(1 + \frac{i^{(m)}}{m}\right)^m - 1 \]

Effective to m-thly nominal rate

\[ i^{(m)} = m\left[(1+i)^{1/m} - 1\right] \]

Effective to force of interest

\[ \delta = \ln(1 + i) \]

m-thly discount rate

\[ d^{(m)} = m \left[1 - (1+i)^{-1/m}\right] \]

Growth rates (escalating benefits)

For benefits increasing at rate \(g\) per annum, the present value factor becomes:

\[ \text{PV factor} = \left(\frac{1+g}{1+i}\right)^t = v_g^t \]

where \(v_g = \frac{1+g}{1+i}\) is the adjusted discount factor.

Escalating whole-life annuity-due (annual payments, \(m = 1\)):

\[ \ddot{a}_x^{(g)} = \sum_{k=0}^{\omega-x} (1+g)^k \cdot v^k \cdot {}_k p_x \]

For fractional payment frequency (\(m > 1\)), Lactuca uses an anniversary-based growth index: the \(j\)-th payment receives factor \((1+g)^{\lfloor j/m \rfloor}\), so that growth increments occur only at whole-year policy anniversaries regardless of payment density. The complete \(m\)-thly growing annuity formulas are in Growth Rate Conventions.

Multiple decrements

Note

Lactuca assumes independent decrements in all multiple-decrement calculations (DecrementTable, DisabilityTable, ExitTable). The independence formula \({}_{t}p_x^{(\tau)} = \prod_j {}_t p_x^{(j)}\) is applied throughout. The general inclusion-exclusion form below is shown for mathematical completeness only.

For independent decrements \(j = 1, 2, \ldots, n\):

\[ {}_t q_x^{(\tau)} = \sum_{j=1}^n {}_t q_x^{(j)} - \sum_{i<j} {}_t q_x^{(i,j)} + \cdots \]

Under independence:

\[ {}_t p_x^{(\tau)} = \prod_{j=1}^n {}_t p_x^{(j)} \]

Single decrement probability (cause \(j\)):

\[ {}_t q_x^{(j)} = \int_0^t {}_s p_x^{(\tau)} \cdot \mu_x^{(j)}(s) \, ds \]

Joint life and last survivor

Joint life status

Note

LifeTable assumes statistical independence of lives in all joint calculations (äxy, Axy, äxyz, Axyz, etc.). Correlated mortality is not supported.

Survival of both lives:

\[ {}_t p_{xy} = {}_t p_x \cdot {}_t p_y \quad \text{(independence)} \]

Joint life annuity:

\[ \ddot{a}_{xy} = \sum_{k=0}^{\infty} v^k \cdot {}_k p_{xy} \]

Last survivor status

At least one life survives:

\[ {}_t p_{\overline{xy}} = {}_t p_x + {}_t p_y - {}_t p_{xy} \]

Last survivor annuity:

\[ \ddot{a}_{\overline{xy}} = \ddot{a}_x + \ddot{a}_y - \ddot{a}_{xy} \]

Joint life insurance and pure endowment

Joint life insurance — benefit on first death (uses config.mortality_placement offset \(f\), same convention as single-life Ax):

\[ A_{xy} = \sum_{k=0}^{\infty} v^{k+f} \cdot {}_k p_{xy} \cdot (1 - p_{x+k}\,p_{y+k}) \]

Joint pure endowment — both lives survive \(n\) years:

\[ {}_n E_{xy} = v^n \cdot {}_n p_{xy} = v^n \cdot {}_n p_x \cdot {}_n p_y \]

Three-life joint status

Under independence of three lives:

\[ {}_t p_{xyz} = {}_t p_x \cdot {}_t p_y \cdot {}_t p_z \]

Three-life joint annuity-due, insurance, and pure endowment:

\[ \ddot{a}_{xyz} = \sum_{k=0}^{\infty} v^k \cdot {}_k p_{xyz} \]

\[ A_{xyz} = \sum_{k=0}^{\infty} v^{k+f} \cdot {}_k p_{xyz} \cdot (1 - p_{x+k}\,p_{y+k}\,p_{z+k}) \]

\[ {}_n E_{xyz} = v^n \cdot {}_n p_x \cdot {}_n p_y \cdot {}_n p_z \]

Formulas by calculation mode

Lactuca provides four calculation modes (set via config.calculation_mode). The same method (äx, Ax, etc.) uses different formulae depending on the active mode. Full details in Calculation Modes.

Note

Change mode globally with config.calculation_mode = "<mode>". Allowed values: "discrete_precision" (default), "discrete_simplified", "continuous_precision", "continuous_simplified". Any other string raises ValueError. Call config.reset() to restore the default.

discrete_precision — integer ages, \(m = 1\)

Closed-form equivalents (direct summation reduces to commutation formulas for integer ages and \(m=1\)). All identities hold exactly for any value of config.mortality_placement (see C function note above):

\[\ddot{a}_{x:\overline{n}|} = \frac{N_x - N_{x+n}}{D_x}, \qquad \ddot{a}_x = \frac{N_x}{D_x}\]

\[A^1_{x:\overline{n}|} = \frac{M_x - M_{x+n}}{D_x}, \qquad A_x = \frac{M_x}{D_x}\]

\[{}_{n}E_x = \frac{D_{x+n}}{D_x}\]

discrete_precision — \(m > 1\) or fractional age

Exact summation over the payment grid, using \(\ell_x\) interpolation at each grid point.

Annuity-due:

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

Annuity-immediate:

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

Insurance (\(m\)-thly):

\[A_x^{(m)} = \sum_{k=0}^{mn-1} v^{k/m\,+\,f/m} \cdot {}_{k/m}p_x \cdot {}_{1/m}q_{x+k/m}\]

where \(f/m\) is the mortality placement micro-offset within each \(1/m\)-year sub-period (\(f \in \{0,\,0.5,\,1\}\) from config.mortality_placement; see C function).

Survival and death probabilities are obtained by interpolating \(\ell_x\) with the method set in config.lx_interpolation (see lx Interpolation).

discrete_simplified

Classical Woolhouse first-order approximation converting \(m=1\) to frequency \(m\):

\[\ddot{a}_{x:\overline{n}|}^{(m)} \approx \ddot{a}_{x:\overline{n}|} - \frac{m-1}{2m}\,(1 - {}_{n}E_x)\]

\[\ddot{a}_x^{(m)} \approx \ddot{a}_x - \frac{m-1}{2m}\]

Woolhouse approximation for insurances:

\[A_x^{(m)} \approx A_x + \frac{m-1}{2m}\left(A_{x+1} - A_x\right)\]

continuous_precision

Numerical integration over the unrounded \(\ell_x\) curve (raw values):

\[\bar{a}_{x:\overline{n}|} = \int_0^n v^t \cdot {}_t p_x \, dt\]

\[\bar{A}^1_{x:\overline{n}|} = \int_0^n v^t \cdot {}_t p_x \cdot \mu_{x+t} \, dt\]

The force of mortality \(\mu_{x+t}\) is derived numerically from the raw \(\ell_x\) grid. The approximation method is controlled by config.force_mortality_method; allowed values: "finite_difference" (default), "spline", "kernel". Passing any other string raises ValueError.

continuous_simplified

Arithmetic interpolation using the rounded \(\ell_x\) grid:

\[\bar{a}_x \approx \ddot{a}_x - \frac{1}{2}\]

\[\bar{a}_{x:\overline{n}|} \approx \ddot{a}_{x:\overline{n}|} - \frac{1}{2}\,(1 - {}_n E_x)\]

\[\bar{A}_{x+s} \approx \frac{\bar{A}_x + \bar{A}_{x+1}}{2}, \quad 0 < s < 1\]

Select-and-ultimate tables

Some tables are select tables: the mortality rate \(q_{[x]+k}\) depends not only on the attained age \(x + k\) but also on the time elapsed since selection \(k\) and the age at selection \(x\). Once the select period has elapsed, the rate reverts to the ultimate table which depends on attained age only. See Select-ultimate notation for the mathematical definition.

Lactuca reads the select period from the table’s select_period metadata field. The fallback to ultimate mortality is applied automatically once duration reaches start_duration + select_period years — no code change is required in the calling layer.

Example — DAV 2004R Select-Ultimate (select_period = 5):

from lactuca import LifeTable

# Load select-ultimate table at duration 1 (freshly underwritten)
dav = LifeTable("DAV2004R_SelUlt_1o", "m", duration=1)
print(dav.select_period)      # 5

# Select mortality at age [60], 2 years after selection: q_{[60]+2}
dav.duration = 2
q_sel = dav.qx(60)

# Ultimate mortality at attained age 65: q_65
dav.duration = "ult"
q_ult = dav.qx(65)

For a full treatment see Table Taxonomy.

References

Actuarial mathematics

• Bowers, N.L., et al. (1997). Actuarial Mathematics (2nd ed.). Society of Actuaries.

• Dickson, D.C.M., Hardy, M.R., Waters, H.R. (2019). Actuarial Mathematics for Life Contingent Risks (3rd ed.). Cambridge University Press.

• Promislow, S.D. (2014). Fundamentals of Actuarial Mathematics (3rd ed.). Wiley.

• Gerber, H.U. (1997). Life Insurance Mathematics (3rd ed.). Springer / Swiss Association of Actuaries.

Numerical methods and financial mathematics

• Shiu, E.S.W. (1984). Woolhouse’s formula and other summation techniques. Transactions of the Society of Actuaries, 36.

• Norberg, R. (1990). Reserve formula for annuities and insurances with the focus on multiple state models. Scandinavian Actuarial Journal.

• Press, W.H., et al. (2007). Numerical Recipes: The Art of Scientific Computing (3rd ed.). Cambridge University Press.

Multiple decrements and multiple-state models

• Haberman, S., Pitacco, E. (1999). Actuarial Models for Disability Insurance. Chapman & Hall/CRC.

• Pitacco, E., Denuit, M., Haberman, S., Olivieri, A. (2009). Modelling Longevity Dynamics for Pensions and Annuity Business. Oxford University Press.

Mortality table sources

The bundled tables are sourced from the official publications of each jurisdiction’s supervising authority or professional association. See Bundled Actuarial Tables for the complete list and provenance of each table.

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Note: All formulas assume integer ages unless fractional age interpolation (UDD/CFM) is explicitly specified.