Mortality Improvement (MI)#

What is mortality improvement?#

Mortality Improvement (MI) is the systematic reduction of mortality rates over time caused by advances in medicine, nutrition, sanitation, and living standards. For a given age \(x\), the mortality rate observed in year \(t\) is lower than the rate observed in a base year \(t_0\):

\[ q_{x,t} < q_{x,t_0} \qquad \text{for } t > t_0 \]

This improvement is measured by the MI factor \(\lambda_x\) (or \(AA_x\), depending on the projection formula), which quantifies the annual rate of reduction at each age.

In Lactuca, MI factors are stored in the .ltk file alongside the base mortality rates. All columns whose names start with mi_ carry MI data:

Column name

Content

mi_m, mi_f

Constant MI factor by age and sex (scalar per age)

mi_m_YYYY, mi_f_YYYY

MI factor for calendar year YYYY by age and sex

The cohort parameter (birth year) is the only projection input required by the public API. A person born in year \(c\) reaches age \(x\) in calendar year \(c + x\), so the relevant mortality rate is the one that applied in that specific year — not the base-year rate. Given a cohort, Lactuca evaluates each diagonal \(t = \text{cohort} + x\) automatically.

Projection formulas#

Lactuca supports four projection formulas. The formula used by a table is stored in its .ltk metadata field generational_formula_type and is selected automatically at load time. The symbol used for the MI factor varies by publishing body. Lactuca stores all MI factors under mi_m / mi_f columns regardless of formula, but the interpretation differs depending on the formula type:

Symbol

Interpretation

Formula

\(\lambda_x\)

Continuous force of mortality improvement

exponential_improvement

\(AA_x\)

Discrete annual reduction rate (SOA convention, e.g. Scale AA)

discrete_improvement, projected_improvement

generic \(mi_x\)

Absolute annual improvement (units: mortality points per year)

linear_improvement

These are not interchangeable notations for the same quantity. A value of 0.02 in an exponential_improvement column means \(e^{-0.02(t-t_0)} \approx 0.980\) per year; the same value in a discrete_improvement column means \((1-0.02)^{t-t_0} = 0.98^{t-t_0}\) — numerically close but conceptually distinct parametric families. Other publishing conventions (\(F(x)\) in UK CMI tables; factores de mejora in Spanish and Chilean regulations) map to one of these Lactuca formula types depending on the underlying mathematics used.

exponential_improvement#

The most common formula in European and Latin-American regulation:

\[ q_{x,t} = q_{x,t_0} \cdot e^{-\lambda_x \,(t - t_0)} \]

where \(\lambda_x = \text{MI}_x\) is the annual force of improvement at age \(x\).

Bundled tables — constant MI columns (mi_m, mi_f): PER2020_Ind_*, PER2020_Col_*, DAV2004R_Agg_1o, DAV2004R_SelUlt_1o. Bundled tables — year-indexed columns (mi_m_YYYY, mi_f_YYYY): DAV2004R_Agg_2o, DAV2004R_SelUlt_2o.

linear_improvement#

A simpler linear reduction (flat extrapolation risk for large \(t - t_0\)):

\[ q_{x,t} = q_{x,t_0} - mi_x \,(t - t_0) \]

where \(mi_x\) is the absolute annual improvement in mortality points at age \(x\).

Warning

This formula can produce negative rates if improvement factors are large or the projection horizon is long. No automatic floor is applied — a ValueError is raised if any projected probability falls outside \([0, 1]\). Verify that \(mi_x \cdot (t - t_0) \leq q_{x,t_0}\) for all ages and intended cohorts before using this formula in production.

Note

No bundled tables currently use this formula. It is available for custom tables built with lactuca.tables.builder.TableBuilder; see Building Custom Table Files (.ltk).

discrete_improvement#

Each year the rate is multiplied by \((1 - AA_x)\), accumulating via a power:

\[ q_{x,t} = q_{x,t_0} \cdot (1 - AA_x)^{\,t - t_0} \]

where \(AA_x\) is the annual percentage reduction factor at age \(x\).

Bundled tables: GAM94_AA (SOA 1994 Group Annuity Mortality, Scale AA).

projected_improvement#

A cumulative discrete product applied year by year along the cohort diagonal: the path through the age × calendar-year improvement matrix traced by a person born in year \(c\), reaching age \(x\) in calendar year \(c + x\). The annual factor \(AA_{x,t}\) may vary both by age and by calendar year:

\[ q_{x,\,\text{cohort}+x} = q_{x,t_0} \cdot \prod_{t=t_0+1}^{\text{cohort}+x} \bigl(1 - AA_{x,t}\bigr) \]

Bundled tables (year-indexed columns, mi_m_YYYY / mi_f_YYYY): CB_H_2020, MI_H_2020, RV_M_2020, B_M_2020, MI_M_2020 (Chile CMF NCG 305/2023).

from lactuca import LifeTable

# Chilean CMF NCG 305/2023 male table — projected_improvement with annual grid
cb_1960 = LifeTable("CB_H_2020", "m", cohort=1960)
cb_1985 = LifeTable("CB_H_2020", "m", cohort=1985)

# Cohort 1960 reaches age 65 in 2025; cohort 1985 reaches it in 2050
print(f"q65 (cohort 1960): {cb_1960.qx(65):.6f}")
print(f"q65 (cohort 1985): {cb_1985.qx(65):.6f}")   # lower — 25 more years of improvement

Constant vs. year-indexed MI factors#

Constant MI factors#

The MI factor is a single value per age and sex, independent of calendar year. Stored as mi_m and mi_f columns. Used when the published table provides one improvement vector without year breakdowns.

Formulas: exponential_improvement, linear_improvement, discrete_improvement.

Example tables: PER2020_Ind_1o, GAM94_AA, DAV2004R_SelUlt_1o.

Year-indexed MI factors#

The MI factor varies by calendar year: columns mi_m_YYYY, mi_f_YYYY form an annual grid. For cohort \(c\) and age \(x\), the relevant year is \(t = c + x\).

  • Lookup: the factor for year \(t\) is read from the column whose year is the smallest grid year \(\geq t\) (ceiling lookup via np.searchsorted).

  • Before the first grid year: when cohort + x is below the earliest grid year, the first available factor is used (clamped to the lower grid boundary).

  • Beyond the last grid year: the last available factor is used (clamped to the upper grid boundary).

Formulas: exponential_improvement (taxonomy sub-type b), projected_improvement (taxonomy sub-type c). See Table Taxonomy for the full classification.

Example tables: DAV2004R_Agg_2o, DAV2004R_SelUlt_2o, CB_H_2020.

Boundary behavior when cohort + x < base_year#

The base year \(t_0\) is the calibration year of the table’s published mortality rates, stored in the .ltk metadata. It is accessible via lt.table.base_year. When the cohort diagonal falls before \(t_0\), behavior depends on the formula:

Formula

Behavior when \(\text{cohort}+x < t_0\)

exponential_improvement, linear_improvement, discrete_improvement

Base rate returned unchanged — the exponent or difference is clamped to 0; no back-projection.

projected_improvement

Base rate returned unchanged — the product range is empty (cum_rf = 1).

Year-indexed (gen-c path)

First grid year’s factor used — cal_year is clamped to first_grid before the lookup.

Note

All formulas return the base-year rate unchanged when cohort + x < base_year. Projecting mortality rates backward before \(t_0\) — i.e. inflating rather than reducing \(q_x\) — has no actuarial justification: the base table already represents the best published mortality estimate at \(t_0\).

The year-indexed row behaves differently from the scalar-formula rows because there is no single exponent or difference to clamp to zero: the lookup operates on a column grid, so the natural boundary is the first available grid year rather than a numeric zero.

Improvement in practice#

The examples below show what is specific to MI: how the projected rate varies by cohort and how to inspect MI metadata on a table instance. For constructor syntax, vectorial creation, cohort-guard patterns, and bulk portfolio iteration see Using Actuarial Tables.

Note

Generational tables require cohort at construction time. Omitting it raises a ValueError immediately — the projection year \(t = \text{cohort} + x\) cannot be computed without knowing the year of birth. Conversely, passing cohort to a static (period) table also raises a ValueError.

Observing the improvement effect#

A younger cohort reaches the same age in a later calendar year and therefore benefits from more accumulated improvement:

from lactuca import LifeTable

lt_1960 = LifeTable("PER2020_Ind_1o", "m", cohort=1960)
lt_1985 = LifeTable("PER2020_Ind_1o", "m", cohort=1985)

# Cohort 1960 reaches age 65 in 2025; cohort 1985 reaches it in 2050
print(f"q65 (cohort 1960): {lt_1960.qx(65):.6f}")
print(f"q65 (cohort 1985): {lt_1985.qx(65):.6f}")   # lower — 25 more years of improvement

Checking MI metadata#

from lactuca import LifeTable

lt = LifeTable("PASEM2020_Rel_1o", "m")
print(lt.generational)                    # False — no MI factors

lt_gen = LifeTable("PER2020_Ind_1o", "m", cohort=1969)
print(lt_gen.generational)                # True
print(lt_gen.generational_formula_type)   # 'exponential_improvement'
print(lt_gen.table.base_year)             # 2012 — calibration year of PER2020 base rates
# Equivalent via the underlying TableSource:
print(lt_gen.table.generational_formula_type)

MI and the select-ultimate structure#

For select-ultimate generational tables, the improvement diagonal shifts to \(t = \text{cohort} + x + d\), where \(d\) is the select duration since underwriting:

# DAV 2004 R select-ultimate, 1st order — duration 2 (2 years since contract inception)
dav_su = LifeTable("DAV2004R_SelUlt_1o", "m", cohort=1960, duration=2)

For projected_improvement tables (e.g. Chilean CMF), Lactuca shifts the projection diagonal to effective_cohort = cohort + d so the accumulated product correctly traces \(t = \text{cohort} + x + d\). For other formulas (e.g. exponential_improvement, used by DAV2004R_SelUlt_2o), improvement is applied uniformly across all durations using \(t = \text{cohort} + x\).

Note

No bundled tables combine the select-ultimate structure with projected_improvement (taxonomy sub-type Gen (c) select-ultimate). The engine supports this combination; see Table Taxonomy for the full classification.

See also#