Actuarial statistics basics

In short

Actuarial statistics applies probability, survival analysis and compound interest to the random financial consequences of death, sickness, accident, retirement and general insurance claims. Its core objects are the life table (an age-wise record of survivors and deaths), the interest function, and loss distributions for claim amounts and claim counts. It appears in B.Sc Statistics applied-statistics papers, in Institute of Actuaries of India (IAI) CS1/CS2/CM1 syllabi and in ISS/UGC-NET questions, and is the working mathematics of LIC, SBI Life, HDFC Life, GIC Re, New India Assurance and actuarial consulting firms. Indian insurers compute statutory reserves and premium rates under IRDAI regulations, using Indian Assured Lives Mortality tables published by IAI.

Explained

Survival models and the lifetime random variable

• Let T(x) be the future lifetime of a person aged x (a continuous non-negative random variable); K(x) = integer part of T(x) is the curtate future lifetime (complete years lived).
• Survival function: S(t) = P(T > t) for a newborn; for age x, tpx = P(T(x) > t) = S(x + t)/S(x). tqx = 1 − tpx is the probability of death within t years.
• Force of mortality (hazard rate): μx = −(d/dx) ln S(x) = f(x)/S(x) — instantaneous death rate per unit time at exact age x; units are per year.
• Relation: tpx = exp(−∫₀ᵗ μ(x+s) ds). Under a constant force μ over a year, px = e^−μ.
• Fractional-age assumptions for interpolating between integer ages: Uniform Distribution of Deaths (UDD) — s·qx for 0 ≤ s ≤ 1; constant force; Balducci (hyperbolic) — 1−sq(x+s) = (1−s)qx.
• Parametric laws: De Moivre μx = 1/(ω − x); Gompertz μx = B·c^x; Makeham μx = A + B·c^x (an age-independent accident term A plus a senescent term); Weibull μx = k·x^n.

The life table

• A life table is a tabulation, usually for integer ages 0 to ω, of a hypothetical cohort starting with a radix l₀ (commonly 100,000 or 10,00,000).
• Columns: x (age), lx (survivors at exact age x), dx = lx − l(x+1) (deaths in the year), qx = dx/lx (mortality rate), px = 1 − qx, Lx (person-years lived in the year), Tx (person-years above age x), ex (expectation of life).
• Relations: lx+1 = lx·px; tpx = l(x+t)/lx; t|qx = (l(x+t) − l(x+t+1))/lx (deferred probability of dying in the year after t years).
• Central death rate: mx = dx/Lx, where Lx ≈ (lx + l(x+1))/2 under UDD; Lx = ∫₀¹ l(x+t) dt exactly.
• Curtate expectation of life: ex = (Σ_{t≥1} l(x+t))/lx. Complete expectation: e̊x = ∫₀^∞ tpx dt ≈ ex + 0.5 under UDD.
• Types: aggregate versus select tables (select tables allow for underwriting — a newly selected life has lighter mortality for a select period, written q[x]+t); cohort (generation) versus current (period) tables; single-decrement versus multiple-decrement (death, withdrawal, disability, retirement).
• Indian tables: Indian Assured Lives Mortality (IALM) tables, e.g. IALM 2012-14, built from insured-lives data by IAI and the basis for valuation; general-population mortality comes from the SRS Abridged Life Tables of the Office of the Registrar General of India. Insured lives show lighter mortality than the general population (selection effect).

Interest, discounting and life annuities

• Effective annual rate i; discount factor v = 1/(1+i); nominal rate i^m convertible m-thly; force of interest δ = ln(1+i); accumulation (1+i)ⁿ = e^δn.
• Actuarial present value (APV) = E[present value of payments], taking expectation over the lifetime random variable. This "equivalence principle" is the basis of every premium and reserve.
• Whole life annuity-due: äx = Σ_{t≥0} v^t · tpx (₹1 at the start of each year while alive). Immediate: ax = äx − 1.
• Temporary annuity-due: äx:n = Σ_{t=0}^{n−1} v^t · tpx. Deferred: n|äx = vⁿ · npx · ä(x+n).
• Continuous annuity: āx = ∫₀^∞ v^t · tpx dt ≈ äx − 0.5.
• Insurance APVs: whole life Ax = Σ_{t≥0} v^t+1 · tpx · q(x+t); term A¹x:n (death within n years); pure endowment Ax:n¹ = vⁿ·npx; endowment assurance Ax:n = A¹x:n + Ax:n¹.
• Key identity: Ax = 1 − d·äx, where d = iv is the rate of discount; similarly Ax:n = 1 − d·äx:n.
• Commutation functions (for hand calculation from a table): Dx = v^x·lx; Nx = Σ_{t≥0} D(x+t); Cx = v^x+1·dx; Mx = Σ_{t≥0} C(x+t); Sx = Σ Nx. Then äx = Nx/Dx, Ax = Mx/Dx, A¹x:n = (Mx − M(x+n))/Dx, Ax:n¹ = D(x+n)/Dx.

Premiums, reserves and policy values

• Net (pure) premium by the equivalence principle: APV of benefits = APV of premiums, with no allowance for expenses.
• Whole life, annual premium: Px = Ax/äx = (1/äx) − d. Endowment assurance: Px:n = Ax:n/äx:n.
• Gross (office) premium adds expenses — initial (commission, underwriting, medical), renewal, claim expenses — and loadings for contingency and profit: APV(benefits) + APV(expenses) = APV(gross premiums).
• Net premium reserve (prospective): tV = APV future benefits − APV future net premiums, at duration t. Retrospective: accumulated premiums less cost of insurance — equal to the prospective reserve under the same basis.
• Whole life: tV = A(x+t) − Px·ä(x+t) = 1 − ä(x+t)/äx (the "paid-up" form).
• Surrender value is a reduced reserve after a surrender penalty; paid-up value converts the reserve into a smaller sum assured. Under the Insurance Act, 1938 (as amended) and IRDAI product regulations, traditional life policies acquire a guaranteed surrender value after the premiums of the prescribed initial period are paid; the exact rule varies by product and regulation version, so quote the current IRDAI (Insurance Products) Regulations rather than a fixed figure.
• Indian statutory valuation is on a gross premium valuation basis under IRDAI (Actuarial Report and Abstract) requirements, certified by an Appointed Actuary; solvency margin must be at least the regulatory control level (historically 1.5 times the required solvency margin).

General insurance statistics: loss models

• Claim frequency N: Poisson (mean λ, variance λ), binomial, negative binomial (used when variance > mean, i.e. heterogeneous risks or contagion).
• Claim severity X: exponential, gamma, lognormal, Pareto, Weibull — right-skewed, heavy-tailed; Pareto and lognormal are standard for motor and liability losses.
• Aggregate loss (collective risk model): S = X₁ + … + X_N, with N independent of i.i.d. Xi. Then E[S] = E[N]·E[X] and Var(S) = E[N]·Var(X) + Var(N)·(E[X])².
• If N is Poisson(λ): compound Poisson, E[S] = λ·E[X], Var(S) = λ·E[X²].
• Policy modifications: deductible d (insurer pays max(X − d, 0)); policy limit u; proportional retention α; reinsurance (quota share, surplus, excess-of-loss, stop-loss).
• Stop-loss premium: E[(S − d)₊] = ∫_d^∞ (s − d) f_S(s) ds.
• Risk premium = E[S]; office premium adds expense, profit and a risk loading, e.g. expected-value principle (1+θ)E[S], standard-deviation principle E[S] + β·σ(S), variance principle E[S] + α·Var(S).
• Ruin theory: surplus U(t) = u + ct − S(t); adjustment coefficient R solves the Lundberg equation, and Lundberg's inequality gives ψ(u) ≤ e^−Ru for the probability of ultimate ruin.
• Credibility theory: premium = Z·(own experience) + (1 − Z)·(book rate), with Z increasing in the volume of own data; Bühlmann Z = n/(n + k), k = σ²(within)/τ²(between).
• Reserving for outstanding claims: chain-ladder on a run-off triangle of paid or incurred claims, Bornhuetter-Ferguson, average cost per claim; IBNR = incurred but not reported.

Mortality analysis, graduation and regulation

• Exposed to risk: central exposed to risk E^c_x = person-years lived at age x; initial exposed to risk E_x ≈ E^c_x + (1/2)·deaths. Estimates: m̂x = Dx/E^c_x, q̂x = Dx/E_x.
• Graduation smooths crude rates: graphical, parametric (fit Gompertz-Makeham), and standard-table methods (q^g = a + b·q^s). Tests of fit: chi-square, standardised deviations, signs test, cumulative deviations, grouping of signs, serial correlation.
• Comparing mortality: Standardised Mortality Ratio SMR = actual deaths/expected deaths; direct and indirect standardisation.
• Multiple state models (Markov): alive-dead, healthy-sick-dead, with transition intensities; used for health, critical illness and income-protection products.
• Regulatory frame in India: Insurance Act, 1938; IRDAI Act, 1999; IRDAI regulations on actuarial reports, assets-liabilities-solvency, and product design; the Actuaries Act, 2006 governs the profession, and only a Fellow can sign as Appointed Actuary. Life insurance payouts receive the exemption for life-insurance proceeds under the Income-tax Act, 2025 subject to premium-to-sum-assured conditions — the conditions have been tightened in recent years, so check the current provision rather than assuming blanket exemption. Insurance premiums attract GST at the prevailing rate; the September 2025 restructuring changed insurance rates, so verify the rate currently notified.

Key points

• T(x) - future lifetime of a life aged x; all actuarial values are expectations over T(x).
• tpx / tqx - survival and death probabilities; tpx = l(x+t)/lx from a life table.
• Force of mortality μx - instantaneous death rate; tpx = exp(−∫μ).
• Life table - lx, dx, qx, px, Lx, Tx, ex built from a radix l₀; the basic actuarial data structure.
• mx vs qx - mx = dx/Lx (per person-year), qx = dx/lx (per person at start); mx > qx at adult ages.
• ex vs e̊x - curtate and complete expectation of life; e̊x ≈ ex + 0.5 under UDD.
• Select table - lighter mortality for a select period after underwriting; q[x]+t notation.
• v and δ - v = 1/(1+i), δ = ln(1+i); discounting converts future rupees to present value.
• äx - whole life annuity-due APV; ax = äx − 1; āx ≈ äx − 0.5.
• Ax = 1 − d·äx - the single most used identity, with d = i/(1+i).
• Commutation functions - Dx, Nx, Cx, Mx turn table lookups into ratios: äx = Nx/Dx, Ax = Mx/Dx.
• Equivalence principle - net premium set so APV(benefits) = APV(premiums).
• Reserve tV - prospective APV(future benefits) − APV(future premiums); equals the retrospective value on the same basis.
• Collective risk model - S = ΣXi over N claims; E[S] = E[N]E[X], Var(S) = E[N]Var(X) + Var(N)E[X]².
• Compound Poisson - Var(S) = λE[X²]; convenient because sums of independent compound Poissons are compound Poisson.
• Credibility factor Z - weight on own experience; Bühlmann Z = n/(n+k).
• Lundberg bound - ψ(u) ≤ e^−Ru; more initial surplus u means exponentially less ruin risk.
• IALM tables and Appointed Actuary - Indian insured-lives mortality basis and the statutory signing role under IRDAI.

Formulas and examples

Symbols: l = survivors, d = deaths, q/p = death/survival probability, μ = force of mortality (per year), i = effective annual interest rate, v = discount factor, δ = force of interest, A = APV of ₹1 sum assured, ä = APV of ₹1 annuity-due, P = annual net premium per ₹1 sum assured, S = aggregate claims (₹), N = claim count, X = individual claim amount (₹), λ = Poisson mean claims per year.

Example 1 — life-table probabilities and expectation. Extract (radix 1,00,000):

• 2p60 = l62/l60 = 76,636/80,000 = 0.95795; 2q60 = 0.04205.
• 4p60 = 72,628/80,000 = 0.90785.
• Probability of dying in the third year, 2|q60 = (l62 − l63)/l60 = 1,916/80,000 = 0.02395.
• Central death rate at 60: L60 = (80,000 + 78,400)/2 = 79,200, m60 = 1,600/79,200 = 0.02020.
• If the table ends at age 65 with l65 = 0, then e60 = (l61 + l62 + l63 + l64 + l65)/l60 = (78,400 + 76,636 + 74,720 + 72,628 + 0)/80,000 = 302,384/80,000 = 3.78 years, and e̊60 ≈ 4.28 years.

Example 2 — annuity, assurance and premium at i = 6%. A tiny 3-year model: a life aged 60 with q60 = 0.02, q61 = 0.0225, q62 = 0.025 (deaths assumed at year end), v = 1/1.06 = 0.943396, d = 0.06/1.06 = 0.0566038.

• Survival: p60 = 0.98, 2p60 = 0.98 × 0.9775 = 0.957950, 3p60 = 0.957950 × 0.975 = 0.934001.
• 3-year annuity-due ä60:3 = 1 + v(0.98) + v²(0.957950) = 1 + 0.924528 + 0.852571 = 2.777099.
• 3-year endowment assurance (death benefit at end of year of death, ₹1 maturity at 63):
  • Death APV: v(0.02) + v²(0.98 × 0.0225) + v³(0.957950 × 0.025) = 0.018868 + 0.019624 + 0.020108 = 0.058600.
  • Pure endowment: v³ × 0.934001 = 0.839619 × 0.934001 = 0.784206.
  • A60:3 = 0.058600 + 0.784206 = 0.842806. Check: 1 − d·ä = 1 − 0.0566038 × 2.777099 = 1 − 0.157194 = 0.842806 (the identity holds).
• Annual net premium per ₹1: P = 0.842806/2.777099 = 0.303484. For a sum assured of ₹10,00,000, annual net premium ≈ ₹3,03,484 — large because the term is only 3 years and almost all of the benefit is the maturity payment.
• Reserve after 1 year: ä61:2 = 1 + v(0.9775) = 1.922170; A61:2 = 1 − 0.0566038 × 1.922170 = 0.891198. 1V = 0.891198 − 0.303484 × 1.922170 = 0.891198 − 0.583348 = 0.307850 per ₹1, i.e. ₹3,07,850 on ₹10,00,000.

Example 3 — constant force of mortality and a whole life annuity. μ = 0.02 per year constant at all ages, i = 6%.

• px = e^−0.02 = 0.980199; tpx = e^−0.02t.
• äx = Σ_{t≥0} v^t e^−0.02t = 1/(1 − e^−δ−0.02) where δ = ln 1.06 = 0.0582689. Ratio r = e^−0.0782689 = 0.924716, so äx = 1/(1 − 0.924716) = 13.283.
• Ax = 1 − d·äx = 1 − 0.0566038 × 13.283 = 1 − 0.751666 = 0.248334.
• Annual net premium for a whole life policy of ₹5,00,000: P = 0.248334/13.283 = 0.018696 → ₹9,348 per year.

Example 4 — aggregate claims in motor insurance. A motor portfolio has claim counts Poisson with λ = 250 per year; claim sizes are lognormal with mean ₹40,000 and standard deviation ₹60,000.

• E[X] = 40,000; E[X²] = Var + mean² = 60,000² + 40,000² = 3.6×10⁹ + 1.6×10⁹ = 5.2×10⁹.
• E[S] = 250 × 40,000 = ₹1,00,00,000 (₹1 crore).
• Var(S) = λE[X²] = 250 × 5.2×10⁹ = 1.3×10¹² → σ(S) = ₹11,40,175.
• Office premium on the standard-deviation principle with β = 0.3: 1,00,00,000 + 0.3 × 11,40,175 = ₹1,03,42,053; with a 25% expected-value loading instead: ₹1,25,00,000.
• Normal approximation: the probability that claims exceed ₹1.2 crore is P(Z > (1,20,00,000 − 1,00,00,000)/11,40,175) = P(Z > 1.754) ≈ 4.0%.

Example 5 — Bühlmann credibility for a group health scheme. Own experience: 5 years, average claim cost ₹8,000 per member per year; book (manual) rate ₹6,500; within-risk variance σ² = 40,00,000 (₹² per member-year), between-risk variance τ² = 10,00,000.

• k = σ²/τ² = 40,00,000/10,00,000 = 4; Z = n/(n + k) = 5/(5 + 4) = 0.5556.
• Credibility premium = 0.5556 × 8,000 + 0.4444 × 6,500 = 4,444.8 + 2,888.6 = ₹7,333 per member per year.
• With 20 years of data: Z = 20/24 = 0.8333 → premium = 0.8333 × 8,000 + 0.1667 × 6,500 = ₹7,750 — more data pulls the rate towards own experience.

Where it is used

• Life insurers: LIC, SBI Life, HDFC Life, ICICI Prudential — product pricing, statutory valuation, embedded value, asset-liability management under IRDAI rules.
• General and health insurers and TPAs: New India Assurance, ICICI Lombard, Star Health — motor third-party and own-damage rating, health claim cost modelling, IBNR reserving.
• Reinsurance and broking: GIC Re, Munich Re India, Swiss Re, Aon, Marsh — excess-of-loss pricing, catastrophe load, retention analysis.
• Pensions and employee benefits: actuarial valuation of gratuity, leave encashment and pension liabilities for company accounts under AS 15 (Revised) / Ind AS 19, and EPFO and NPS work.
• Job roles: actuarial analyst, pricing analyst, reserving analyst, valuation actuary, risk analyst in banks (IFRS 9 / Ind AS 109 credit loss models use the same survival framework), and consulting (Mercer, WTW, Deloitte, Milliman).
• Exams and study: IAI CS1/CS2/CM1/CM2, UK IFoA papers, ISS, UGC-NET statistics, and university applied-statistics papers.

Interview questions

• Q: What is the difference between qx and mx? A: qx = dx/lx is the probability that a life alive at exact age x dies within the year; mx = dx/Lx is the central death rate per person-year of exposure, so it uses a mid-year denominator and is slightly larger at adult ages.
• Q: State and explain the relation Ax = 1 − d·äx. A: ₹1 payable on death plus ₹d at the start of each year while alive is equivalent to ₹1 now, because the interest earned on ₹1 funds the annuity and the capital pays the claim; taking APVs gives Ax + d·äx = 1.
• Q: What is the equivalence principle? A: The net premium is fixed so that the actuarial present value of benefits equals the actuarial present value of premiums at the chosen mortality and interest basis, with no expense or profit margin.
• Q: Why do insured lives show lighter mortality than the general population? A: Underwriting selection, medical screening and the socio-economic profile of buyers; hence select tables and separate insured-lives tables such as IALM rather than SRS population tables.
• Q: Difference between curtate and complete expectation of life? A: ex counts only whole years lived after age x; e̊x includes the fraction of the final year, and under uniform deaths e̊x ≈ ex + 0.5.
• Q: When would you prefer a negative binomial to a Poisson for claim counts? A: When the observed variance exceeds the mean (overdispersion), which happens with heterogeneous risks or claim contagion; the negative binomial arises as a Poisson with a gamma-distributed mean.
• Q: Explain the collective risk model and its variance formula. A: Aggregate loss S = ΣXi over a random number N of i.i.d. claims independent of N; Var(S) = E[N]Var(X) + Var(N)(E[X])², capturing uncertainty in both the number and the size of claims.
• Q: What does the credibility factor Z mean practically? A: It is the weight given to the risk's own claims experience against the book rate; Z rises with the volume and homogeneity of the risk's data and with the spread between risks.
• Q: What is a prospective reserve and why must it equal the retrospective reserve? A: Prospective = APV of future benefits minus future premiums; retrospective = accumulated past premiums minus the cost of insurance provided. On the same mortality, interest and premium basis the equivalence principle makes them identical.
• Q: What is IBNR and how is it estimated? A: Claims incurred but not yet reported to the insurer; estimated from development patterns in a run-off triangle using chain-ladder or Bornhuetter-Ferguson methods.
• Q: What does Lundberg's inequality tell a general insurer? A: The probability of ultimate ruin falls at least exponentially with initial surplus, ψ(u) ≤ e^−Ru, so capital requirements can be linked to the adjustment coefficient R, which depends on the premium loading and claim-size distribution.
• Q: Which basis does Indian statutory life valuation use, and who signs it? A: A gross premium valuation on prudent assumptions prescribed under IRDAI's actuarial regulations, certified by the Appointed Actuary, who must be a Fellow of the Institute of Actuaries of India.
• Q: How does graduation differ from simple smoothing? A: Graduation imposes a smooth, usually parametric or standard-table-based progression on crude rates and is then tested statistically (chi-square, signs, standardised deviations) for both smoothness and fit to the data.

Common mistakes

• Confusing lx (survivors at exact age x) with Lx (person-years lived between x and x+1), and so computing e̊x wrongly.
• Using qx when the question needs tpx, or multiplying qx values instead of subtracting cumulative survival (tqx ≠ Σ qx).
• Taking Ax = v·qx-type shortcuts: forgetting that the death benefit is paid at the end of the year of death, so the exponent is t+1, not t.
• Using d = i instead of d = i/(1+i) in Ax = 1 − d·äx, or mixing annuity-due with annuity-immediate (äx = ax + 1).
• Forgetting the variance in claim numbers in Var(S) and using only E[N]·Var(X).
• Treating net and gross premium as interchangeable — expense and profit loadings make office premium materially higher, and reserves differ by basis.
• Quoting a rupee answer per ₹1 sum assured: the premium per unit must be multiplied by the sum assured before it is reported.

Related topics

Demography and vital statistics · Probability distributions and expectation · Survival analysis and censored data · Stochastic processes and Markov chains · Risk management and insurance regulation in India · Statistical inference and estimation.