Life contingencies: assurances and annuities

In short

Life contingencies values cash flows whose timing depends on whether a life is alive or dead, by taking the expected present value (EPV) of the random payment stream. Assurances pay a sum on death (or on survival to maturity); annuities pay while the life survives. Every life insurance premium, every reserve and every pension liability in India is built from these two sets of EPVs, so the topic is the core of university life-contingencies papers, of the Institute of Actuaries of India (IAI) subject CM1 (formerly CT5) and of the IFoA's CM1, and of actuarial-analyst interviews at LIC, SBI Life, HDFC Life, ICICI Prudential, Max Life, Tata AIA, GIC Re, Milliman, WTW, Mercer, Aon, Swiss Re and Deloitte/EY actuarial teams. Examiners test it as symbol manipulation (prove a relation), as life-table arithmetic (compute an EPV from q_x and i) and as approximation (monthly annuities, continuous assurances).

Explained

Building blocks and notation

• Survival probabilities come from the life table: tp_x = l_{x+t}/l_x, tq_x = 1 − tp_x, and the deferred mortality probability t|q_x = tp_x · q_{x+t} = (d_{x+t})/l_x.
• Discounting uses v = 1/(1+i); the force of interest δ = ln(1+i); the rate of discount d = i/(1+i) = 1 − v.
• A random present value is written Z (for assurances) or Y (for annuities). An EPV is E[Z]: the probability-weighted discounted value, not a certain amount.
• Convention: a bar (Ā, ā) means payable continuously / at the moment of death; no bar means annually at the end of the year (assurances) or the standard annuity timing; a superscript (m) means m times a year.
• The equivalence principle (priced in the premium topic) uses these EPVs; here the task is only to construct and evaluate them.

Assurance (death and survival benefit) functions

• Whole life assurance A_x: ₹1 at the end of the year of death, whenever it occurs. A_x = Σ_{t=0}^{∞} v^{t+1} · t|q_x.
• Term (temporary) assurance A¹_{x:n}: ₹1 only if death occurs within n years. A¹_{x:n} = Σ_{t=0}^{n−1} v^{t+1} · t|q_x.
• Pure endowment nE_x = A_{x:n}¹ = vⁿ · np_x: ₹1 only if the life survives n years. This is the "survival discount factor" used everywhere.
• Endowment assurance A_{x:n} = A¹_{x:n} + nE_x: ₹1 on death within n years or on survival to n. Basis of LIC-style traditional savings policies.
• Deferred assurance n|A_x = nE_x · A_{x+n}: cover starts only after n years.
• Increasing assurance (IA)¹_{x:n} = Σ_{t=0}^{n−1} (t+1) v^{t+1} · t|q_x — benefit ₹1, ₹2, ₹3 … used for simple-bonus and indexed covers.
• Continuous (immediate) assurance Ā_x = ∫₀^∞ v^t · tp_x · μ_{x+t} dt — paid at the exact moment of death; the realistic model, since claims are not paid on 31 December.
• Second-moment functions ²A_x, ²Ā_x: the same formulas evaluated at force 2δ (i.e. rate (1+i)²−1). Needed for variances.

Annuity functions

• Whole life annuity-due ä_x = Σ_{t=0}^{∞} v^t · tp_x — ₹1 at the start of each year while alive (first payment certain, hence the leading 1).
• Whole life annuity-immediate a_x = Σ_{t=1}^{∞} v^t · tp_x = ä_x − 1 — ₹1 at each year end, only if then alive.
• Temporary annuity-due ä_{x:n} = Σ_{t=0}^{n−1} v^t · tp_x; annuity-immediate a_{x:n} = ä_{x:n} − 1 + nE_x.
• Deferred annuity n|ä_x = nE_x · ä_{x+n} = ä_x − ä_{x:n} — the standard pension model: contribute for n years, draw thereafter.
• Continuous annuity ā_x = ∫₀^∞ v^t · tp_x dt; mthly annuity ä^m_x pays 1/m, m times a year, in advance.
• Guaranteed annuity ä_{x:\overline{n}|} (annuity-certain for n years then life contingent) = ä_n (certain) + n|ä_x — the "annuity with return of purchase price / 5-, 10-, 15-year guarantee" sold in India.
• Increasing annuity (Iä)_x = Σ_{t=0}^{∞} (t+1) v^t · tp_x — escalating pensions.
• Complete expectation of life e̊_x = ā_x evaluated at i = 0; so an annuity value is a "discounted expectation of lifetime".

Relations between assurances and annuities (the most examined derivations)

• From 1 = d·ä_∞ logic: Z + d·Y = 1 pathwise for a whole life policy and a whole life annuity-due, giving A_x = 1 − d·ä_x; continuously, Ā_x = 1 − δ·ā_x; temporarily, A_{x:n} = 1 − d·ä_{x:n} (note: the endowment assurance, not the term assurance).
• Because Z = v^{K+1} and Y = ä_{K+1} are linear in each other, Var(Z) = d² · Var(Y) — one variance gives the other.
• Recursions (ideal for spreadsheets and for exam "backward" questions):
  • A_x = v q_x + v p_x A_{x+1}
  • ä_x = 1 + v p_x ä_{x+1}
  • nE_x = v p_x · (n−1)E_{x+1}
• Premium conversion in reverse: ä_x = (1 − A_x)/d, so a table of A_x is enough to get every annuity value at the same interest rate.
• Approximations linking discrete and continuous/mthly values:
  • Uniform distribution of deaths (UDD) over each year: Ā_x ≈ (i/δ) A_x; A^m_x ≈ (i/i^m) A_x.
  • Claims acceleration: Ā_x ≈ (1+i)^{0.5} A_x (death on average mid-year) — simpler and numerically very close to UDD.
  • Woolhouse (3-term): ä^m_x ≈ ä_x − (m−1)/(2m) − (m²−1)/(12m²)·(δ + μ_x); the common exam shortcut is ä¹²_x ≈ ä_x − 11/24. For a temporary annuity, ä^m_{x:n} ≈ ä_{x:n} − ((m−1)/2m)(1 − nE_x).

Commutation functions and tabulated bases

• Defined on a life table at a fixed rate of interest: D_x = v^x l_x; N_x = Σ_{y≥x} D_y; M_x = Σ_{y≥x} C_y where C_x = v^{x+1} d_x; R_x = Σ_{y≥x} M_y; S_x = Σ_{y≥x} N_y.
• Then A_x = M_x/D_x; ä_x = N_x/D_x; a_x = N_{x+1}/D_x; nE_x = D_{x+n}/D_x; A¹_{x:n} = (M_x − M_{x+n})/D_x; A_{x:n} = (M_x − M_{x+n} + D_{x+n})/D_x; ä_{x:n} = (N_x − N_{x+n})/D_x; (IA)¹_{x:n} = (R_x − R_{x+n} − n·M_{x+n})/D_x.
• Advantage: any EPV becomes one division — fast, exam-friendly, still printed in Indian textbook tables (for example whole-life and endowment values on LIC (1996-98) Ultimate or IALM 2012-14 at 6%).
• Limitation: locked to one interest rate and one mortality table, cannot handle select mortality, changing interest or stochastic models — modern work uses spreadsheets/R/Python recursions instead.

Joint lives, variability and the Indian framework

• Joint life status (xy) fails on the first death: A_xy, ä_xy with tp_xy = tp_x · tp_y under independence. Last-survivor status fails on the second death: ä_{\overline{xy}} = ä_x + ä_y − ä_xy and A_{\overline{xy}} = A_x + A_y − A_xy.
• Joint-life annuities underlie "joint life last survivor" annuity options (NPS and LIC immediate annuities) and spouse/widow pensions.
• Variance matters: for a single whole life policy the standard deviation of the present value is typically larger than the mean, because the payment is all-or-nothing. For n independent identical policies the EPV scales by n but the standard deviation only by √n — the mathematical statement of pooling, and a standard interview question.
• Indian practice: life offices are regulated by IRDAI; the Appointed Actuary must be a Fellow of the Institute of Actuaries of India and follow the IAI's Actuarial Practice Standards and IRDAI's valuation and product regulations. Mortality bases commonly referenced are the IAI's Indian Assured Lives Mortality (2012-14) for assured lives and the Indian individual annuitants' table (2012-15 investigation) for annuities, with company-specific adjustments; annuity valuation also allows for future mortality improvement. Pension annuities bought with NPS corpus are governed by PFRDA; IRDAI's standard immediate annuity product is Saral Pension.
• Tax and GST (as at November 2026): deduction for life insurance premium and the exemption of policy proceeds (subject to premium thresholds — aggregate ULIP premium above ₹2.5 lakh, and above ₹5 lakh for other life policies issued on or after 1 April 2023) now sit in the Income-tax Act, 2025, which replaced the 1961 Act from 1 April 2026 and renumbered every section — quote the provision by name rather than an old "80C/10(10D)" number. Since the GST rate reform of 22 September 2025 the slabs are 5%/18%/40% and premiums on individual life and health insurance policies are GST-exempt (group/corporate treatment differs). India's move to an Ind AS 117 (IFRS 17) insurance-contracts standard has been in progress; the implementation date has shifted more than once, so do not quote one as fixed.

Key points

• EPV - probability-weighted present value; the single idea behind every symbol in the topic.
• A_x - ₹1 at end of year of death, whole life; A_x = Σ v^{t+1} t|q_x = M_x/D_x.
• A¹_{x:n} vs A_{x:n} - term assurance (death only) versus endowment assurance (death or survival); differ by nE_x.
• nE_x = vⁿ np_x - pure endowment, the survival-discount factor; equals D_{x+n}/D_x.
• ä vs a - annuity-due pays in advance (includes a certain first payment), annuity-immediate in arrear; ä_x = a_x + 1.
• ä_{x:n} = ä_x − nE_x ä_{x+n} - a temporary annuity is a whole life annuity less a deferred one.
• A_x = 1 − d ä_x - the premium-conversion relation; continuous version Ā_x = 1 − δ ā_x.
• Var(Z) = d² Var(Y) - assurance and annuity present values are linearly related.
• ²A_x - same function at force 2δ; needed because Var(Z) = ²A_x − (A_x)².
• Recursion A_x = v q_x + v p_x A_{x+1} - lets you build a whole column from the oldest age backwards.
• Claims acceleration Ā_x ≈ (1+i)^{0.5} A_x; UDD Ā_x ≈ (i/δ) A_x — both convert year-end to moment-of-death.
• Woolhouse ä¹²_x ≈ ä_x − 11/24 - the standard monthly-annuity adjustment.
• Commutation functions D, N, M, R - turn any EPV into one division, but are tied to one interest rate and table.
• Deferred annuity n|ä_x = nE_x ä_{x+n} - the pension/retirement model.
• Joint life (xy) fails on first death; last survivor uses ä_x + ä_y − ä_xy.
• Annuity value rises as interest falls and as mortality improves; assurance value moves the opposite way with interest.
• Pooling - EPV grows with n, standard deviation only with √n, so relative risk falls as 1/√n.
• Indian bases - IRDAI/IAI framework; IALM 2012-14 for assured lives, annuitants' table with improvement for annuities.

Formulas and examples

Symbols: x = age in years (integer), n = term in years, i = annual effective interest rate, v = 1/(1+i), q_x and p_x = annual probabilities of death and survival, μ = force of mortality per year, l_x = survivors in the table. All monetary results below are in rupees for the sum assured / annuity stated.

Example 1 — 3-year term assurance, ₹1,00,000, i = 6%. Mortality: q₆₀ = 0.010, q₆₁ = 0.012, q₆₂ = 0.015.

• p₆₀ = 0.990, 2p₆₀ = 0.990 × 0.988 = 0.978120.
• Death probabilities: 0|q = 0.010; 1|q = 0.990 × 0.012 = 0.011880; 2|q = 0.978120 × 0.015 = 0.0146718.
• v = 0.943396, v² = 0.889996, v³ = 0.839619.
• EPV per ₹1 = 0.010(0.943396) + 0.011880(0.889996) + 0.0146718(0.839619) = 0.0094340 + 0.0105731 + 0.0123187 = 0.0323258.
• EPV = 0.0323258 × 1,00,000 = ₹3,232.58.

Example 2 — same life table: pure endowment, endowment assurance and annuity.

• 3p₆₀ = 0.978120 × 0.985 = 0.963448; 3E₆₀ = 0.963448 × 0.839619 = 0.808935 → pure endowment of ₹1,00,000 costs ₹80,893.50.
• Endowment assurance A₆₀:₃ = 0.0323258 + 0.808935 = 0.841261 → ₹84,126.10.
• Annuity-due ä₆₀:₃ = 1 + 0.943396(0.990) + 0.889996(0.978120) = 1 + 0.933962 + 0.870526 = 2.80449.
• Check with A_{x:n} = 1 − d ä_{x:n}: d = 0.06/1.06 = 0.0566038; 1 − 0.0566038 × 2.80449 = 1 − 0.158732 = 0.841268 — agrees with 0.841261 to rounding.
• Annuity-immediate a₆₀:₃ = 2.80449 − 1 + 0.808935 = 2.61343, i.e. ₹2,61,343 for ₹1,00,000 a year in arrear for 3 years while alive.

Example 3 — constant force: EPV and variance. μ = 0.02 per year, δ = 0.05 per year, sum assured ₹10,00,000, benefit at the moment of death.

• Ā_x = 0.02/0.07 = 0.285714 → EPV = ₹2,85,714.
• ā_x = 1/0.07 = 14.2857 years; check Ā_x = 1 − 0.05 × 14.2857 = 0.285715. ✔
• ²Ā_x = 0.02/(0.02 + 0.10) = 0.166667; Var(Z/S) = 0.166667 − 0.285714² = 0.166667 − 0.081633 = 0.085034; SD = 0.291605 → SD = ₹2,91,605, larger than the mean — a single policy is extremely variable.
• 100 independent such lives: EPV = ₹2.857 crore, SD = ₹2,91,605 × √100 = ₹29.16 lakh, i.e. only 10.2% of the mean against 102% for one policy.

Example 4 — deferred monthly pension (Woolhouse). Age 60, retires at 65, pension ₹20,000 per month in advance for life. Basis: i = 6%, 5p₆₀ = 0.92, ä₆₅ = 10.50.

• v⁵ = 0.747258; 5E₆₀ = 0.747258 × 0.92 = 0.687477.
• ä¹²₆₅ ≈ 10.50 − 11/24 = 10.50 − 0.458333 = 10.041667 (years' worth of annual payments).
• Annual pension = ₹2,40,000. EPV at 60 = 0.687477 × 10.041667 × 2,40,000 = 6.903415 × 2,40,000 = ₹16,56,820.
• If instead the pension starts immediately at 60 with ä₆₀ = 13.20: ä¹²₆₀ ≈ 12.741667 → EPV = ₹30,58,000 — the five-year deferment and survival discount nearly halves the cost.

Example 5 — commutation functions. Given at 6%: D₄₀ = 1,000, D₆₀ = 320, M₄₀ = 190, M₆₀ = 160, N₄₀ = 15,500, N₆₀ = 4,200.

• Whole life A₄₀ = 190/1,000 = 0.1900 → ₹1,00,000 cover costs ₹19,000.
• 20-year term A¹₄₀:₂₀ = (190 − 160)/1,000 = 0.0300 → ₹3,000.
• 20-year pure endowment 20E₄₀ = 320/1,000 = 0.3200; endowment assurance A₄₀:₂₀ = 0.0300 + 0.3200 = 0.3500 → ₹35,000.
• ä₄₀ = 15,500/1,000 = 15.500; ä₄₀:₂₀ = (15,500 − 4,200)/1,000 = 11.300.
• Check: 1 − d ä₄₀:₂₀ = 1 − 0.0566038 × 11.300 = 1 − 0.639623 = 0.360377 against 0.3500 — the small gap shows the data above are illustrative, not from one consistent table; in a real table the identity holds exactly, and that check is how you catch a wrong commutation value.

Where it is used

• Life insurance pricing and product design: term, endowment, money-back, whole life and ULIP mortality charges at LIC, SBI Life, HDFC Life, ICICI Prudential, Max Life, Tata AIA — every benefit EPV in the pricing model is one of the symbols above.
• Annuity and pension business: immediate and deferred annuities, IRDAI's Saral Pension, NPS annuitisation through PFRDA-empanelled insurers, and employer gratuity/pension valuations that use ä and n|ä.
• Statutory valuation and reporting: Appointed Actuary's reserve and solvency work under IRDAI regulations, embedded value and appraisal value, and the actuarial inputs to Ind AS/IFRS-17-style measurement.
• Reinsurance and risk: GIC Re, Swiss Re, Munich Re and RGA price mortality and longevity covers from A and ä values with stressed mortality.
• Consulting and analyst roles: Milliman, WTW, Mercer, Aon, Deloitte, EY and KPMG India actuarial teams; entry tests almost always include an EPV calculation from a given q_x column and interest rate.
• Exams: IAI/IFoA CM1, university B.Sc/M.Sc Actuarial Science life-contingencies papers, and quantitative-analyst screening in banks that use the same expected-discounted-cash-flow machinery.

Interview questions

• Q: What is the difference between a term assurance and an endowment assurance? A: Term assurance pays only on death within the term; endowment assurance also pays the sum assured on survival to maturity, so A_{x:n} = A¹_{x:n} + nE_x and it is always the dearer of the two.
• Q: Explain A_x = 1 − d·ä_x in words. A: For every policyholder the present value of ₹1 at death plus d times the present value of an annuity-due of ₹1 while alive equals exactly ₹1, whatever the actual lifetime; taking expectations of that identity gives the relation.
• Q: Why does an annuity become more expensive when interest rates fall? A: Payments far in the future are discounted less, so their present values rise — longevity-heavy liabilities are highly interest-sensitive, which is why annuity writers match with long bonds.
• Q: Which is larger, ä_x or a_x, and by how much? A: ä_x is larger by exactly 1, because the annuity-due adds a payment at time 0 that is certain.
• Q: How do you value a monthly pension if you only have annual annuity values? A: Use Woolhouse: ä¹²_x ≈ ä_x − 11/24 (three-term version adds −(143/1728)(δ + μ_x)); for a temporary annuity subtract (11/24)(1 − nE_x).
• Q: What is claims acceleration and when is it preferred to UDD? A: Ā_x ≈ (1+i)^{0.5} A_x, assuming death on average mid-year; it is preferred for its simplicity and gives answers very close to the UDD factor i/δ.
• Q: How do you compute the variance of the present value of a whole life assurance? A: Var(Z) = ²A_x − (A_x)², where ²A_x is A_x evaluated at force 2δ, i.e. at rate (1+i)² − 1.
• Q: A single whole life policy has a standard deviation bigger than its mean. Is that a problem? A: Not for a portfolio: the EPV scales with n but the standard deviation only with √n, so relative risk falls as 1/√n — this is the pooling argument behind insurance.
• Q: What are commutation functions good for, and why are they going out of use? A: They reduce any EPV to a single division from tabulated D, N, M, R columns, but they are fixed to one interest rate and one mortality table and cannot handle select mortality, varying interest or stochastic models.
• Q: How is a joint-life last-survivor annuity valued? A: ä_{\overline{xy}} = ä_x + ä_y − ä_xy, where ä_xy uses tp_x · tp_y for independent lives; it pays until both are dead, so it exceeds either single-life annuity.
• Q: Which mortality basis would you use for a term plan and which for an annuity, and why not the same? A: Assured-lives mortality (for example the IAI's Indian Assured Lives Mortality 2012-14 with company adjustments) for assurances, and a lighter annuitants' table with future mortality improvement for annuities, because annuity buyers self-select for good health — using the assured-lives table would understate annuity cost.
• Q: Why is a deferred annuity much cheaper than an immediate one at the same age? A: Payments are both discounted over the deferment and conditional on surviving it, so the cost is multiplied by nE_x = vⁿ np_x.

Common mistakes

• Using A_{x:n} where A¹_{x:n} is meant, or applying 1 − d·ä_{x:n} to a term assurance — that identity holds only for the endowment assurance.
• Mixing the two annuity timings: writing a_{x:n} = ä_{x:n} − 1 and forgetting the + nE_x term for a temporary annuity.
• Taking ²A_x as (A_x)² or as A_x at rate 2i; it is A_x evaluated at force 2δ, i.e. interest rate (1+i)² − 1.
• Discounting death benefits by v^t instead of v^{t+1} when the benefit is payable at the end of the year of death.
• Treating tp_xy = tp_x + tp_y, or confusing joint-life (first death) with last-survivor (second death) statuses.
• Applying the Woolhouse −11/24 adjustment to an annuity-immediate or to an assurance, or to a monthly annuity already given on an mthly basis.
• Pricing an annuity on an assured-lives table with no allowance for selection or future mortality improvement, which materially understates the liability.

Related topics

Survival models and life tables · Premium calculation and policy reserves · Pension and employee benefit valuation · Actuarial statistics basics · Demography and vital statistics · Compound interest and discounted cash flow (financial mathematics).