Survival models and life tables

In short

A survival model describes the random future lifetime of a life aged x, usually through the survival probability tpx or the force of mortality μx. A life table is that model tabulated: a column of survivors lx from which every probability, expectation and (later) premium is read off. It is the foundation of every life insurance, annuity, pension and health valuation, so it is examined in university B.Sc Actuarial Science papers and in Institute of Actuaries of India (IAI) subjects CS2 (Risk Modelling and Survival Analysis) and CM1 (Actuarial Mathematics), and in UGC-NET/ISS statistics papers. In India the working tables are the Indian Assured Lives Mortality (IALM) table for insured lives, LIC a(96-98) ultimate for annuitants, and the SRS abridged life tables published by the Office of the Registrar General of India for the general population.

Explained

The random future lifetime and the four basic functions

• T(x) = future lifetime of a life aged exactly x, a continuous non-negative random variable; T(0) = X is age at death.
• Survival function: S(t) = Pr[T(0) > t], decreasing, S(0)=1, S(∞)=0. For a life aged x, tpx = Pr[T(x) > t] = S(x+t)/S(x).
• Distribution function: tqx = 1 − tpx = probability of dying within t years. With t omitted, px and qx mean the one-year probabilities.
• Density: f(t) = tpx · μ(x+t) — the probability of surviving t years and then dying "immediately".
• Force of mortality (hazard rate): μx = −(1/S(x))·dS(x)/dx = −d/dx ln lx. It is an instantaneous *rate* per year, not a probability, so it can exceed 1; qx ≈ μ(x+½) for small mortality.
• Curtate future lifetime: K(x) = ⌊T(x)⌋, the number of *complete* years lived; Pr[K = k] = kpx · q(x+k) = k|qx.
• Deferred probability: t|uqx = tpx − t+upx = tpx · uq(x+t) — survives t years, dies in the next u.
• Consistency (multiplicative) rule: t+upx = tpx · up(x+t). All life-table work reduces to this one identity.
• ω (omega): the limiting age, beyond which lx = 0. IALM-type insured-lives tables run to about age 120; many textbook tables stop at 100.

The life table and its columns

• lx — expected number alive at exact age x out of an arbitrary radix l₀ (usually 100,000 or 10,000,000). Only *ratios* of lx matter.
• dx = lx − l(x+1) = deaths between x and x+1. qx = dx/lx, px = l(x+1)/lx.
• Building a table: choose radix → l(x+1) = lx(1 − qx), dx = lx·qx, repeated to ω. Reading a table: tpx = l(x+t)/lx, tqx = (lx − l(x+t))/lx, t|qx = d(x+t)/lx.
• Lx = ∫₀¹ l(x+t) dt = person-years lived between ages x and x+1. Under a linear lx (UDD), Lx ≈ ½(lx + l(x+1)) = lx − ½dx.
• Tx = ∫₀^∞ l(x+t) dt = Σ_{y≥x} Ly = total future person-years lived by the lx survivors.
• Complete expectation of life e°x = Tx/lx; curtate expectation ex = Σ_{k≥1} kpx = (Σ_{y>x} ly)/lx; approximately e°x ≈ ex + ½.
• Central rate of mortality mx = dx/Lx — the rate used in demography and in graduation; mx ≈ qx/(1 − ½qx) under UDD. Distinguish it from the *initial* rate qx (denominator = lives at start) and from μx.
• a(x) = average fraction of the year of age lived by those dying in it; a(x) = ½ except at age 0 and the very old ages, where deaths cluster early (infant a(0) ≈ 0.1–0.3 in Indian SRS tables).
• Types: *complete* (single years of age) vs *abridged* (5-year groups, as SRS publishes); *period/current* (one calendar period's rates applied to a synthetic cohort) vs *cohort/generation* (an actual birth cohort followed through, needing projection); *aggregate* vs *select*; single-decrement (death only) vs multiple-decrement (death, withdrawal, retirement — covered in the pension note).

Fractional ages: getting non-integer-age probabilities

A table gives integer-age values only; three standard assumptions fill the gaps, for 0 ≤ t < 1:

• Uniform distribution of deaths (UDD): lx linear in t. tqx = t·qx, l(x+t) = lx − t·dx, and μ(x+t) = qx/(1 − t·qx) — which *rises* over the year. Most used in exams.
• Constant force (exponential): μ(x+t) = μ constant, so tpx = (px)^t and μ = −ln px.
• Balducci (hyperbolic): (1−t)q(x+t) = (1−t)·qx, giving tpx = px/(1 − (1−t)qx); the force *falls* over the year, which is unrealistic, so it is of historical/exam interest only.
• Ordering for 0<t<1: UDD gives the smallest tqx, Balducci the largest, constant force in between — the differences are tiny (4th decimal) at normal mortality levels.
• Useful approximation of the force from a table: μx ≈ ½(ln p(x−1) + ln px) in magnitude, or μx ≈ (l(x−1) − l(x+1))/(2lx).

Analytic laws of mortality

• De Moivre (uniform): μx = 1/(ω−x), tpx = (ω−x−t)/(ω−x), e°x = (ω−x)/2. Crude but makes algebra easy — a favourite in exams.
• Constant force / exponential: μx = μ, tpx = e^(−μt), e°x = 1/μ; memoryless, so unrealistic for humans.
• Gompertz (1825): μx = B·c^x — mortality rises geometrically with age; tpx = exp[−(B/ln c)·c^x(c^t − 1)]. Fits adult ages (about 30–85) remarkably well.
• Makeham (1860): μx = A + B·c^x — adds an age-independent "accident" term A; tpx = exp[−At − (B/ln c)·c^x(c^t − 1)].
• Weibull: μx = k·x^n; Gompertz–Makeham with a logistic cap and the Heligman–Pollard 8-parameter law are used for full-age-range fits, capturing infant mortality, the accident hump at 18–25 and late-age deceleration.
• Why a law at all: smoothness, a handful of parameters, easy extrapolation to very old ages where data are thin, and closed-form tpx.

From data to a usable table: exposure, estimation and graduation

1. Define the rate interval and exposure — *central exposed to risk* E^c_x = person-years lived at age x; *initial exposure* E_x = E^c_x + ½dx.
2. Crude estimates: m̂x = Dx/E^c_x (central), q̂x = Dx/E_x (initial). Under a constant force, the maximum likelihood estimator of μ is μ̂ = Dx/E^c_x, with variance ≈ μ̂²/Dx (Poisson model); the binomial model gives Var(q̂x) ≈ qx(1−qx)/E_x.
3. Graduation — smooth the crude rates: parametric (fit Gompertz–Makeham by weighted least squares/MLE), graphical, or by reference to a standard table (q^graduated = a + b·q^std, or a spline).
4. Tests of the graduation: chi-square goodness of fit, standardised deviations test, signs test, cumulative deviations test, grouping-of-signs test, serial correlations test — plus a check for over/under-graduation (smoothness via third differences).
5. Projection: mortality improvement factors or the Lee–Carter model, essential for annuities and pensions where longevity risk, not mortality risk, is the exposure.
6. Selection and heterogeneity: class selection, time selection, self-selection, adverse selection, temporary initial selection — the reason insured lives die lighter than the general population.

Select and ultimate tables

• Underwriting at entry makes recent entrants healthier, so mortality depends on age at entry x and duration r: written q[x]+r, with square brackets marking the entry age.
• After the select period (commonly 2 or 5 years) the effect wears off and rates revert to the ultimate table: for a 2-year select period, q[x]+2 = q(x+2).
• Probabilities chain exactly as before: 3p[60] = p[60]·p[60]+1·p62.
• A select table is read along the row for its duration, then down the ultimate column; l[x] values are usually *back-computed* from the ultimate lx: l[x]+1 = l(x+2)/p[x]+1, l[x] = l[x]+1 / p[x].
• Ignoring selection overstates early-duration mortality, understates early-duration premiums' profit and distorts lapse-and-reentry analysis.

Indian tables, regulation and practice

• IALM 2012-14 (Indian Assured Lives Mortality, Ultimate) is the standard Indian insured-lives table, constructed by the IAI's Mortality and Morbidity Investigation from LIC and industry data; it replaced IALM 2006-08, which replaced LIC (1994-96). Whether a newer investigation (post-2014 experience, including COVID-period adjustments) has been published should be verified against the IAI's current mortality investigation report before use in a valuation.
• LIC a(96-98) ultimate remains the common annuitant table in India; annuitant mortality is materially lighter than assured-lives mortality, and using an assurance table to price an annuity understates the liability.
• SRS Abridged Life Tables (Office of the Registrar General & Census Commissioner, Ministry of Home Affairs) give population e°0 and age-specific rates by sex and rural/urban; recent SRS releases put life expectancy at birth in India at roughly 70 years, female above male — quote the exact figure only from the specific SRS report year being used.
• Regulation: the Insurance Act 1938 and the IRDAI Act 1999 require every insurer to appoint an Appointed Actuary, who certifies the mortality basis; IRDAI consolidated its actuarial rules in the IRDAI (Actuarial, Finance and Investment Functions of Insurers) Regulations, 2024. Professional conduct and basis-setting are governed by the Actuaries Act, 2006 and the IAI's Actuarial Practice Standards and Guidance Notes. The valuation basis must be prudent, with margins for adverse deviation over the best-estimate table.
• Other Indian uses: LIC/EPFO/state pension valuations, gratuity and leave-encashment liabilities under Ind AS 19 / AS 15(R), and compensation for death in motor-accident and workmen's-compensation claims (where courts use statutory multiplier tables, not actuarial life tables — a frequent confusion).

Key points

• tpx - probability a life aged x survives t more years = l(x+t)/lx; the whole table is just this one ratio.
• Consistency rule - t+upx = tpx · up(x+t); never add survival probabilities, multiply them.
• qx vs μx - qx is a probability over a year (0 ≤ qx ≤ 1); μx is an instantaneous rate per year and can exceed 1.
• mx - central rate dx/Lx, denominator is person-years; qx has lives-at-start as denominator.
• dx - deaths in the year of age = lx·qx = lx − l(x+1).
• Lx - person-years in (x, x+1] ≈ lx − ½dx; Tx = Σ Ly from x upwards.
• e°x = Tx/lx (complete), ex = (Σ_{y>x} ly)/lx (curtate), and e°x ≈ ex + ½.
• Recursion - ex = px(1 + e(x+1)); e°x = ∫₀¹ tpx dt + px·e°(x+1).
• Var(K) = Σ(2k−1)kpx − ex²; Var(T) = 2∫₀^∞ t·tpx dt − (e°x)².
• UDD - deaths spread evenly in the year: tqx = t·qx, μ(x+t) = qx/(1 − t·qx).
• Constant force - tpx = (px)^t, μ = −ln px; Balducci - tpx = px/(1 − (1−t)qx), force falls, unrealistic.
• De Moivre - μx = 1/(ω−x) and e°x = (ω−x)/2; Gompertz μx = Bc^x; Makeham μx = A + Bc^x.
• Select table - q[x]+r depends on entry age and duration; after the select period it becomes the ultimate rate q(x+r).
• Period vs cohort table - a period table applies one era's rates to a synthetic cohort; a cohort table needs projection.
• Graduation - smoothing crude rates, then validated by chi-square, signs, cumulative deviations and serial correlation tests.
• Median future lifetime - the t solving tpx = 0.5; it differs from e°x because T is skewed.
• Indian standards - IALM 2012-14 for assured lives, LIC a(96-98) for annuitants, SRS abridged tables for the population.
• Selection - insured lives die lighter than the general population; mortality improvement makes annuity and pension liabilities rise.

Formulas and examples

Example 1 — reading a life table. Illustrative segment:

• 2p60 = l62/l60 = 9,780/10,000 = 0.9780; 5p60 = 9,230/10,000 = 0.9230.
• 3q60 = 1 − 9,630/10,000 = 0.0370.
• 2|q60 = d62/l60 = 150/10,000 = 0.0150 (survives 2 years, dies in the third).
• 2|3q60 = (l62 − l65)/l60 = (9,780 − 9,230)/10,000 = 0.0550.
• L60 = 10,000 − 50 = 9,950; m60 = 100/9,950 = 0.010050 against q60 = 0.010000.
• Half-year probability at age 60: UDD 0.5q60 = 0.005000; constant force 0.5p60 = 0.99^0.5 = 0.994987 so 0.5q60 = 0.005013; Balducci 0.5p60 = 0.99/(1 − 0.005) = 0.994975, 0.5q60 = 0.005025.

Example 2 — expectation, variance and median from a terminating table. l90 = 100, l91 = 70, l92 = 40, l93 = 15, l94 = 0.

• kp90: 1p = 0.70, 2p = 0.40, 3p = 0.15, 4p = 0.
• e90 = 0.70 + 0.40 + 0.15 = 1.25 years.
• L90 = (100+70)/2 = 85, L91 = 55, L92 = 27.5, L93 = 7.5 → T90 = 175 → e°90 = 175/100 = 1.75 years, matching e90 + ½.
• E[K²] = 1(0.70) + 3(0.40) + 5(0.15) = 0.70 + 1.20 + 0.75 = 2.65 → Var(K) = 2.65 − 1.25² = 2.65 − 1.5625 = 1.0875, SD = 1.043 years.
• Median: need tp90 = 0.5, i.e. l = 50, between l91 = 70 and l92 = 40 → t = 1 + (70−50)/(70−40) = 1.667 years, below e°90 = 1.75 because T is right-skewed.
• m90 = d90/L90 = 30/85 = 0.3529 against q90 = 0.30.

Example 3 — Gompertz and Makeham. B = 3×10⁻⁵, c = 1.09, x = 50, t = 10.

• ln c = 0.086178; c⁵⁰ = e^50×0.086178 = e^4.30889 = 74.358; c¹⁰ = e^0.86178 = 2.3674.
• B/ln c = 3×10⁻⁵/0.086178 = 3.4811×10⁻⁴.
• Integrated hazard = 3.4811×10⁻⁴ × 74.358 × (2.3674 − 1) = 3.4811×10⁻⁴ × 101.670 = 0.035393.
• 10p50 = e^(−0.035393) = 0.96523, so 10q50 = 0.03477.
• Add Makeham A = 0.0005: extra hazard = 0.0005 × 10 = 0.005 → total 0.040393 → 10p50 = e^(−0.040393) = 0.96041, 10q50 = 0.03959. The constant A adds ~0.5 percentage point of 10-year mortality at this age.
• Force at 50: Gompertz μ50 = 3×10⁻⁵ × 74.358 = 0.002231 per year; Makeham 0.002731.

Example 4 — De Moivre. ω = 100, x = 40.

• e°40 = (100−40)/2 = 30 years; 20p40 = (100−40−20)/(100−40) = 40/60 = 0.6667; q40 = 1/60 = 0.016667; μ40 = 1/60 = 0.016667; 10|10q40 = (50−40)/60 = 0.16667.
• Var(T) for De Moivre = (ω−x)²/12 = 3600/12 = 300, SD = 17.32 years — a reminder that a single expectation hides enormous spread.

Example 5 — select vs ultimate, 2-year select period.

• 3p[60] = 0.994 × 0.991 × 0.987 = 0.972248.
• 3p60 = 0.990 × 0.988 × 0.987 = 0.965305.
• Difference 0.006943 — a newly underwritten life aged 60 is about 0.69 percentage points more likely to survive 3 years than an average life of the same age. Pricing a 3-year term policy off the ultimate table would overstate expected claims by roughly 25% over this period (0.034695 vs 0.027752).

Example 6 — building lx from qx (illustrative rates, not actual IALM values). Radix l50 = 100,000, q50 = 0.00301, q51 = 0.00327, q52 = 0.00356.

• d50 = 100,000 × 0.00301 = 301 → l51 = 99,699.
• d51 = 99,699 × 0.00327 = 326.1 → l52 = 99,372.9.
• d52 = 99,372.9 × 0.00356 = 353.8 → l53 = 99,019.1.
• 3p50 = 99,019.1/100,000 = 0.99019, 3q50 = 0.00981.

Where it is used

• Life insurance pricing and valuation: LIC, SBI Life, HDFC Life, ICICI Prudential, Max Life — mortality basis for term, endowment and ULIP products, certified by the Appointed Actuary under IRDAI regulations.
• Annuities and longevity products: immediate annuities, NPS annuity service providers and the Pradhan Mantri Vaya Vandana-type schemes, priced on annuitant tables with mortality-improvement projections.
• Retirement benefits: gratuity, pension and leave-encashment liabilities valued under Ind AS 19 / AS 15(R) by consulting actuaries (Mercer, Aon, WTW, Milliman, Deloitte, KPMG, and independent Fellows of the IAI).
• Reinsurance and experience analysis: actual-versus-expected (A/E) studies, mortality investigations and graduation work at GIC Re, RGA, Munich Re and SCOR India offices.
• Public health and population policy: SRS abridged life tables feed NITI Aayog and Ministry of Health indicators, NFHS analysis, and state-level life-expectancy comparisons.
• Adjacent quantitative roles: credit-risk and telecom churn teams use the same survival-analysis toolkit (hazard rates, Kaplan–Meier, Cox regression), which is why CS2 content travels well into analytics jobs.

Interview questions

• Q: What is the difference between qx and μx? A: qx is the probability a life aged x dies within one year, bounded by 1; μx is the instantaneous force (hazard rate) per unit time at exact age x and can exceed 1. They are linked by qx = 1 − exp(−∫₀¹ μ(x+t)dt).
• Q: Why can you not add survival probabilities? A: Survival over successive periods is a chain of conditional events, so t+upx = tpx·up(x+t) — a product, not a sum.
• Q: What does the radix of a life table mean, and does its value matter? A: It is the arbitrary starting population l₀ (often 100,000). It does not matter, because every probability and expectation is a ratio of lx values.
• Q: State the relationship between e°x and ex, and why. A: e°x ≈ ex + ½, because the curtate version discards the part-year lived in the year of death, which averages about half a year under a uniform death assumption.
• Q: What is UDD and what does it imply for the force of mortality? A: Uniform distribution of deaths over each year of age makes lx linear in t, so tqx = t·qx and μ(x+t) = qx/(1 − t·qx), which increases across the year.
• Q: Why is the Balducci assumption rarely used? A: It implies a force of mortality that falls within each year of age, which contradicts observed mortality, even though it simplifies some exposure formulae.
• Q: What is a select and ultimate table, and why do insurers need one? A: Underwriting makes new entrants temporarily healthier, so rates are indexed by entry age and duration, q[x]+r, reverting to ultimate rates after the select period. Using ultimate rates for new business overstates early claims.
• Q: Difference between a period and a cohort life table? A: A period table applies one calendar period's age-specific rates to a hypothetical cohort; a cohort table follows an actual birth cohort and therefore needs projected future rates.
• Q: Which Indian table would you use for a term assurance and which for an annuity, and why not the same one? A: IALM (currently the 2012-14 investigation) for assured lives and LIC a(96-98) ultimate for annuitants; annuitant mortality is much lighter due to self-selection, so using an assurance table would understate annuity liabilities.
• Q: Why is the median future lifetime usually less than e°x at older ages? A: Because the distribution of T is right-skewed — a minority surviving well beyond the median pull the mean up.
• Q: What is graduation and how do you know it worked? A: Smoothing crude mortality rates with a parametric law, standard table or spline; validated with the chi-square test, standardised deviations, signs test, cumulative deviations, grouping of signs and serial correlation, plus a smoothness check.
• Q: Central exposed to risk versus initial exposed to risk? A: Central exposure counts actual person-years lived at each age; initial exposure adds back roughly half a year for each death (E = E^c + ½d) and pairs with the initial rate qx, while E^c pairs with mx or μ.
• Q: Which is riskier for an insurer — mortality being heavier than assumed, or lighter? A: It depends on the product: heavier mortality hurts assurances, lighter mortality (longevity risk) hurts annuities and pensions, which is why the two are often written together as a natural hedge.

Common mistakes

• Adding probabilities instead of multiplying them: writing 2p60 = p60 + p61 instead of p60 × p61.
• Confusing qx, mx and μx, or using the central rate with the initial exposure (and vice versa) in an estimate.
• Treating a force of mortality greater than 1 as an error — it is a rate, not a probability.
• Forgetting that a select table must be entered by *entry age and duration*, then followed into the ultimate column; reading q[60]+1 as q61 is wrong.
• Using e°x = ex without the ½ correction, or quoting e°x as the age at death rather than the *remaining* lifetime.
• Pricing annuities on an assured-lives table, or valuing a pension without mortality improvement, understating longevity liability.
• Computing Lx as lx + ½dx instead of lx − ½dx, and then getting Tx and e°x too high.
• Quoting a withdrawn or superseded table (LIC 1994-96, IALM 2006-08) as current, or mixing a population SRS table with insured-lives work.

Related topics

Actuarial statistics basics · Demography and vital statistics · Life contingencies: assurances and annuities · Premium calculation and policy reserves · Pension and employee benefit valuation · Multiple-state and multiple-decrement models