Notation and Glossary

This page is the authoritative notation reference for the Lactuca documentation and API. It maps every symbol, method name, and parameter to its actuarial definition. For derivations and worked formulas consult Actuarial Formulas; for configuration options see Configuration.

Life table symbols

Symbol	Lactuca name	Description
\(x\)	x	Exact age (years)
\(\omega\)	omega	Terminal age: last age index stored in the table (effective upper bound may decrease as w after modifications)
\(l_x\)	lx	Number of lives at age \(x\) (from radix \(l_0\))
\(q_x\)	qx	Probability of death in \([x, x+1)\)
\(p_x\)	px	Probability of survival in \([x, x+1)\); \(p_x = 1 - q_x\)
\(d_x\)	dx	Expected deaths in \([x, x+1)\); \(d_x = l_x - l_{x+1}\)
\(L_x\)	Lx	Person-years lived in \([x, x+1)\)
\(T_x\)	Tx	Total person-years lived from age \(x\) (integer ages)
\(T_{x+s}\)	Tx_continuous(x, m=)	Total person-years lived from fractional age \(x+s\), \(0 < s < 1\); numerical integration with \(m\) sub-intervals/year
\(\mathring{e}_x\)	ex	Complete life expectancy at integer age \(x\); \(\mathring{e}_x = T_x / l_x\)
\(e_x\)	ex_curtate	Curtate life expectancy at integer age \(x\); \(e_x = \sum_{k=1}^{\omega-x} {}_k p_x\)
\(\mathring{e}_{x+s}\)	ex_continuous(x, m=)	Complete life expectancy at fractional age \(x+s\), \(0 < s < 1\) (numerical integration)
\({}_t p_x\)	tpx	Probability of surviving \(t\) years from age \(x\)
\({}_t q_x\)	tqx	Probability of dying within \(t\) years from age \(x\)

Decrements

Symbol	Lactuca column	Table class	Meaning
\(q_x\)	qx_m, qx_f, qx_u	LifeTable	Annual mortality rate
\(i_x\)	ix_m, ix_f, ix_u	DisabilityTable	Annual disability incidence rate
\(o_x\)	ox_m, ox_f, ox_u	ExitTable	Annual exit/withdrawal rate

Interest rate symbols

Symbol	Lactuca	Description
\(i\)	InterestRate(i) or table.interest_rate = i	Annual effective interest rate
\(v\)	ir.vn(n) computes \(v^n = (1+i)^{-n}\)	Discount factor; \(v = 1/(1+i)\)
\(\delta\)	ir.delta(t=)	Force of interest; \(\delta = \ln(1+i)\) for constant rates (ir.delta()). For piecewise term structures, provide t (ir.delta(t))
\(i^{(m)}\)	ir.i_m(m, t=)	Nominal rate convertible \(m\) times per year; \(i^{(m)} = m\left[(1+i)^{1/m} - 1\right]\)
\(d^{(m)}\)	ir.d_m(m, t=)	Nominal discount rate convertible \(m\) times per year; \(d^{(m)} = m\left[1 - v^{1/m}\right]\)

Multi-scenario rate containers

Term	Lactuca meaning
Active scenario	Name of the sub-curve (base, stress, …) used by vn(), ax(), factor(), etc. on this call
Reference semantics	LifeTable.interest_rate and ir= / gr= share the container object; the active scenario is not snapshotted at construction
copy()	Deep copy that isolates active_scenario and sub-curves from the parent container

Commutation functions

Symbol	Lactuca method	Formula
\(D_x\)	Dx	\(v^x \cdot l_x\)
\(N_x\)	Nx	\(\sum_{k=x}^{\omega} D_k\)
\(S_x\)	Sx	\(\sum_{k=x}^{\omega} N_k\)
\(C_x\)	Cx	\(v^{x+f} \cdot d_x\) where \(f\) follows config.mortality_placement (0, 0.5, or 1; default mid → \(f=0.5\))
\(M_x\)	Mx	\(\sum_{k=x}^{\omega} C_k\)
\(R_x\)	Rx	\(\sum_{k=x}^{\omega} M_k\)
\(L_x\)	Lx	\(\int_0^1 l_{x+t}\,dt \approx (l_x + l_{x+1})/2\) (integer ages)
\(L_{x+s}\)	Lx_continuous(x, m=)	Person-years lived in \([x+s,\, x+s+1)\) at fractional age; numerical integration with \(m\) sub-intervals/year

Note

\(L_x\) and \(T_x\) are life-table demographic quantities rather than discounted commutation functions, but Lactuca exposes them alongside \(D_x\), \(N_x\), etc. as LifeTable methods. Their continuous variants Lx_continuous and Tx_continuous support fractional starting ages via numerical integration.

Life annuity symbols

Symbol	Lactuca method	Description
\(a_x\)	ax	Whole-life immediate annuity
\(\ddot{a}_x\)	äx	Whole-life annuity-due
\(a_{x:\overline{n}\vert}\)	ax(n=n)	Temporary (term) immediate annuity
\(\ddot{a}_{x:\overline{n}\vert}\)	äx(n=n)	Temporary annuity-due
\({}_{d\vert}\ddot{a}_{x:\overline{n}\vert}\)	äx(d=d, n=n)	Deferred temporary annuity-due
\(\ddot{a}_{x:\overline{n}\vert}^{(m)}\)	äx(m=m)	Annuity-due with frequency \(m\)
\(a_{xy}\)	axy	Joint-life immediate annuity (two lives; pays while both survive)
\(\ddot{a}_{xy}\)	äxy	Joint-life annuity-due (two lives; pays while both survive)
\(a_{xyz}\)	axyz	Three-life joint immediate annuity (all three survive)
\(\ddot{a}_{xyz}\)	äxyz	Three-life joint annuity-due (all three survive)
\(a_{x_1 x_2 \cdots x_n}\)	ajoint	Joint-life immediate annuity (\(n\) lives; pays while all survive)
\(\ddot{a}_{x_1 x_2 \cdots x_n}\)	äjoint	Joint-life annuity-due (\(n\) lives; pays while all survive)

Life insurance symbols

Symbol	Lactuca method	Description
\(A_x\)	Ax	Whole-life insurance
\(A_{x:\overline{n}\vert}^1\)	Ax(n=n)	Term insurance
\({}_n E_x\)	nEx	Pure endowment
\(A_{xy}\)	Axy	First-death insurance (two lives)
\({}_n E_{xy}\)	nExy	Joint pure endowment (two lives; both must survive \(n\) years)
\(A_{xyz}\)	Axyz	First-death insurance (three lives)
\({}_n E_{xyz}\)	nExyz	Joint pure endowment (three lives; all must survive \(n\) years)
\({}_n E_{x_1 \cdots x_n}\)	nEjoint	Joint pure endowment (\(n\) lives; all must survive \(n\) years; N-life, scalar or batch)
\(A_{x_1 \cdots x_n}\)	Afirst	First-death insurance (\(n\) lives)

Calculation parameters

Most parameters are shared by annuity (ax, äx, axy, ajoint, …) and insurance (Ax, Axy, Afirst, nEx) functions. Annuity-only parameters are noted.

Parameter	Type	Meaning
x	float	Entry age in years
n	float or None	Term / duration in years; None = whole life
m	int	Payment frequency per year. Valid values: 1 (annual), 2 (semi-annual), 3 (every 4 months), 4 (quarterly), 6 (bi-monthly), 12 (monthly), 14 (Spanish “14 pagas”, approximate), 24 (twice-monthly), 26 (biweekly), 52 (weekly), 365 (daily). See Calculation Modes for details and caveats on m=14.
d	float	Deferment period in years
ts	float	Temporal shift from policy origin (years already elapsed)
ir	float, InterestRate, or None	Interest rate for discounting; if None, uses the table’s interest_rate attribute
gr	float, GrowthRate, or None	Benefit growth rate per year
cashflow_times	Sequence[float] or None	Custom payment times in years since origin; only on ax, axy, axyz, ajoint, and insurance methods (Ax, Axy, …). Requires calculation_mode='discrete_precision'; n must be None or 0. For annuity-immediate methods (ax, axy, axyz, ajoint), m must be 1 when cashflow_times is provided. Insurance methods (Ax, Axy, …) may use m > 1 if payment times align with the 1/m grid (misalignment emits a warning, not an error).
cashflow_amounts	Sequence[float] or None	Custom per-payment amounts on the standard schedule (m, n); disables gr. Supported on due paths (äx, äxy, äjoint, …) and immediate paths (ax, axy, ajoint, …). Mutually exclusive with using cashflow_times for timing overrides. Requires calculation_mode='discrete_precision'.
return_flows	bool	If True, return detailed cash-flow breakdown

Payment timing is determined by the method name: äx/äjoint = annuity-due (prepayable); ax/ajoint = annuity-immediate (postpayable).

Calculation modes

All four modes value the same actuarial present values under shared conventions; they are actuarially coherent but not numerically identical. See Actuarial coherence across modes in Calculation Modes.

Mode string	Type	Handling of \(m > 1\)	Notes
"discrete_precision"	Discrete	Via \(l_x\) interpolation	Default; production reference
"discrete_simplified"	Discrete	Woolhouse (annuities) / linear age interp. (insurances); hybrid m-thly tail when \(n\) fractional	Faster approximations
"continuous_precision"	Continuous	Numerical integration	Unrounded \(l_x\) integration
"continuous_simplified"	Continuous	Trapezoidal / averaging shortcuts	Fast continuous approximations

See Calculation Modes for details.

Force of mortality methods

The force_mortality_method setting controls how \(\mu_{x+t}\) is estimated from discrete \(({}_t p_x)\) values. It is active only under "continuous_precision" and "continuous_simplified" calculation modes. Set via config.force_mortality_method.

Value	Method	Best used for
"finite_difference"	Central finite differences of \(\ln({}_t p_x)\)	Default; standard actuarial tables
"spline"	Cubic spline of \(\ln({}_t p_x)\) with analytical derivative	Smooth graduated mortality curves
"kernel"	Gaussian kernel smoothing with numerical differentiation	Noisy or irregular empirical data

Other terms

Life-table fundamentals

Radix (\(l_0\))

The initial population size from which the life table is constructed. All subsequent \(l_x\) values are proportional to \(l_0\). Lactuca default: \(l_0 = 1\,000\,000\).

Limiting age (\(\omega\))

The highest age index stored in the table. Survivors at exact age \(\omega\) are \(l_\omega > 0\) with \(q_\omega = 1\) during \([\omega, \omega+1)\); \(l_x = 0\) for \(x > \omega\). Accessible as LifeTable.omega (file metadata) or LifeTable.w (effective bound after modifications).

Force of mortality (\(\mu_x\))

The instantaneous hazard rate of death; \(\mu_x = -\frac{\mathrm{d}}{\mathrm{d}x} \ln l_x\). Used in continuous calculations. The numerical approximation method is controlled by config.force_mortality_method; see the Force of mortality methods section above.

Table types and classification

Period table

A mortality table whose \(q_x\) values represent a single calendar year, with no cohort improvement applied.

Generational table

A mortality table whose \(q_x\) values are adjusted along the life-table diagonal using improvement factors for the specified birth cohort.

Cohort

Birth year of the insured, used to project age-specific mortality along the life-table diagonal in generational tables. Passed as cohort= to LifeTable.

Mortality Improvement (MI)

The systematic reduction of mortality rates over time. In Lactuca, MI parameters are stored in mi_* columns of .ltk files alongside the base rates. The projection formula is specified by the table’s generational_formula_type metadata field:

Formula	MI identifier	Expression
exponential_improvement	\(\lambda_x\)	\(q_{x,t} = q_x^{(b)} \cdot e^{-\lambda_x(t-t_0)}\)
linear_improvement	\(mi_x\)	\(q_{x,t} = q_x^{(b)} - mi_x(t-t_0)\)
discrete_improvement	\(AA_x\)	\(q_{x,t} = q_x^{(b)} \cdot (1-AA_x)^{t-t_0}\)
projected_improvement	\(AA_{x,t}\)	\(q_{x,t} = q_x^{(b)} \cdot \prod_{t'=t_0+1}^{c+x}(1-AA_{x,t'})\)

mi_m/mi_f columns hold constant-per-age factors (\(\lambda_x\), \(mi_x\), or \(AA_x\) per generational_formula_type); mi_m_YYYY/mi_f_YYYY columns hold year-indexed factors (\(AA_{x,t}\)). The only public input required is cohort= (birth year); Lactuca resolves the diagonal automatically. See Mortality Improvement (MI) for the full reference.

First-order table

A mortality table incorporating a prudential safety margin, used for pricing and regulatory reserving (Spanish: primera orden; German: 1. Ordnung). Denoted _1o in Lactuca table identifiers.

Second-order table

A mortality table based on central best-estimate mortality rates, without prudential margins (Spanish: segunda orden; German: 2. Ordnung). Denoted _2o in Lactuca table identifiers. Typical uses: experience studies, stochastic projections, asset-liability modelling.

Aggregate table

A mortality table in which \(q_x\) depends only on attained age \(x\), with no distinction by duration since underwriting. Combines pre- and post-selection experience into a single rate per age. The complement of a select-ultimate table.

Select-ultimate table

A mortality table in which mortality rates depend on both attained age \(x\) and duration \(d\) since underwriting while start_duration \le d < start_duration + select_period, written \(q_{[x]+d}\). Once \(d \ge \text{start\_duration} + \text{select\_period}\), rates revert to the ultimate column \(q_x^u\), which depends only on attained age.

Select period

Count of select columns in the table metadata (LifeTable.select_period). Together with start_duration, it defines the ultimate cutoff: durations at or above start_duration + select_period use ultimate rates. Accessible via LifeTable.select_period.

Duration (select offset)

Years elapsed since the underwriting date (integer \(\ge 0\)). Determines which select column is used for select-ultimate tables. Passed as duration= to LifeTable.

Start duration

Minimum valid integer duration for a select-ultimate table. Most tables use start_duration=1 (columns named qx_m_s1 …). Tables following the CMI/UK convention — such as AM92/AF92 — use start_duration=0 (columns named qx_m_s0, qx_m_s1 …). Accessible via TableSource.start_duration and LifeTable.start_duration. Passing a duration below start_duration raises ValueError.

Ultimate

The section of a select-ultimate table where \(q_x\) depends only on attained age — i.e., after the select period has expired. Equivalent to an aggregate table restricted to post-selection lives.

Fractional-age assumptions

UDD (Uniform Distribution of Deaths)

Fractional-age assumption that deaths are uniformly distributed within each year of age: \(l_{x+s} = (1-s) l_x + s \cdot l_{x+1}\). Set via config.lx_interpolation = "linear" (default).

CFM (Constant Force of Mortality)

Fractional-age assumption that \(\mu_x\) is constant within \([x, x+1)\): \(l_{x+s} = l_x \cdot p_x^s\). Set via config.lx_interpolation = "exponential".

Joint-life and last-survivor symbols

Last-survivor annuity (\(\ddot{a}_{\overline{xy}}\))

Annuity-due while at least one of two lives survives: \(\ddot{a}_{\overline{xy}} = \ddot{a}_x + \ddot{a}_y - \ddot{a}_{xy}\).

Last-survivor insurance (\(A_{\overline{xy}}\))

Benefit on the last death; derivable from single-life and first-death insurances (\(A_x\), \(A_y\), \(A_{xy}\), …).

Reversionary annuity (\(\ddot{a}_{x|y}\))

Payable on \(y\)’s death if \(x\) is still alive; no dedicated method — compose from \(\ddot{a}_x\), \(\ddot{a}_y\), \(\ddot{a}_{xy}\).

Continuous products

\(\bar{a}_x\), \(\bar{a}_{x:\overline{n}|}\)

Continuous annuity (integral of \(v^t {}_t p_x\)); computed in continuous_precision or continuous_simplified modes.

\(\bar{A}_x\), \(\bar{A}^1_{x:\overline{n}|}\)

Continuous insurance (integrand \(v^t {}_t p_x \mu_{x+t}\)); same modes.

Reserves and premiums

\({}_{t}V\) (prospective reserve)

APV of future benefits minus APV of future net premiums, both from attained age \(x+t\) with remaining term \(n-t\). See Prospective Reserves and the ts Parameter and Actuarial Formulas.

\(P\) (net level premium)

Constant premium per unit benefit such that \({}_{0}V = 0\) at issue, e.g. \(P = A^1_{x:\overline{n}|} / \ddot{a}_{x:\overline{n}|}\) for term insurance.

Actuarial building blocks

Commutation function

Precomputed discounted survival quantity (\(D_x\), \(N_x\), \(C_x\), \(M_x\), etc.) used to evaluate annuity and insurance values in closed form. See Commutation Functions.

TableKey

A hashable NamedTuple (lactuca.TableKey) that uniquely identifies a LifeTable, DisabilityTable, or ExitTable instance by its five constructor dimensions: table_name (str), sex (str), cohort (int or None), duration (int/str or None), and unisex_blend (float or None). table_name and sex are required; the remaining fields default to None. Used as the key type in dict returns from the vectorial constructor when return_dict=True.

from lactuca import TableKey
key = TableKey("PER2020_Ind_1o", "m", 1960)    # cohort=1960, duration=None, unisex_blend=None
key = TableKey("PASEM2010", "u", unisex_blend=0.55)

Note

Always look up a key using the same literal float that was passed as unisex_blend at construction time. IEEE 754 rounding means 0.5 + 0.05 \neq 0.55 at the bit level.