Source document. This page reproduces the syllabus this course was written to, as published — its semesters, credits and paper numbers are that document’s, not this site’s. The course itself is studied on its own, in any order.
| Title | Actuarial Statistics |
|---|---|
| Theory Credits | 3 (3 hrs/week) |
| Practical Credits | 1 (2 hrs/week) |
Discrete, continuous and mixed probability distributions. Insurance applications, sum of random variables. Utility theory: utility functions, expected-utility criterion, types of utility function, insurance and utility theory.
Properties of premium principles and examples (Net, Expected-value, Variance, SD, Exponential, Percentile, Esscher). Individual risk models — models for individual claims, sum of independent claims, approximations and their applications.
Uncertainty of age at death, survival function, time-until-death for a person, curate future lifetime, force of mortality, life tables with examples, deterministic survivorship group, life-table characteristics, assumptions for fractional age, analytical laws of mortality (De Moivre, Gompertz, Makeham, Weibull).
Models for insurance payable at the moment of death and at the end of the year of death; whole-life, term, pure endowment, endowment and deferred insurance; varying benefits; relationships between continuous and discrete forms; net single premium.
Continuous life annuities; discrete life annuities (annuity-due, annuity-immediate); annuities with periodic m-thly payments; insurance-annuity identities; continuous and discrete premiums via the equivalence principle; gross premiums.
Open practical course material →
| Unit | Topic | Approx. Weightage |
|---|---|---|
| 1 | Insurance Applications & Utility Theory | 20 % |
| 2 | Premium Principles & Individual Risk | 20 % |
| 3 | Survival Distributions & Life Tables | 20 % |
| 4 | Life Insurance | 20 % |
| 5 | Life Annuities & Premiums | 20 % |
| Symbol | Meaning |
|---|---|
| \(s(x)\) | Survival function: \(P(X > x)\) |
| \(\mu(x)\) | Force of mortality at age \(x\) |
| \(l_x\) | Number alive at age \(x\) in the life table |
| \(d_x = l_x - l_{x+1}\) | Number dying between ages \(x\) and \(x+1\) |
| \(p_x, q_x\) | Survival / death probability for one year from age \(x\) |
| \({}_tp_x, {}_tq_x\) | t-year survival / death probability |
| \(T(x)\) | Future lifetime (continuous) of (\(x\)) |
| \(K(x) = \lfloor T(x) \rfloor\) | Curate (integer) future lifetime |
| \(e_x = E(K(x))\) | Curate expectation of life |
| \(\bar e_x = E(T(x))\) | Complete expectation of life |
| \(i, \delta\) | Effective annual / force of interest |
| \(v = (1+i)^{-1}, d\) | Discount factor / effective discount rate |
| \(\bar A_x, A_x\) | APV of whole-life insurance (continuous / discrete) |
| \(\bar A^{\,1}_{x:\overline{n}|}\), \(A^{\,1}_{x:\overline{n}|}\) | APV of n-year term insurance |
| \({}_nE_x = v^n\, {}_np_x\) | APV of n-year pure endowment |
| \(\bar A_{x:\overline{n}|}, A_{x:\overline{n}|}\) | APV of n-year endowment |
| \(\bar a_x, \ddot a_x, a_x\) | Whole-life annuity (continuous / annuity-due / annuity-immediate) |
| \(\bar a_{x:\overline{n}|}, \ddot a_{x:\overline{n}|}\) | n-year temporary annuity |
| \(\ddot a_x^{(m)}\) | m-thly whole-life annuity-due |
| \(P_x = A_x / \ddot a_x\) | Net annual premium for whole-life |
| Identity | Notes |
|---|---|
| \(s(x) = \exp\bigl(-\int_0^x \mu(t) dt\bigr)\) | Survival from force of mortality |
| \(_tp_x = s(x+t)/s(x) = l_{x+t}/l_x\) | From survival function or life table |
| \(\bar A_x = E(v^T),\;\; A_x = E(v^{K+1})\) | Whole-life APVs |
| \(\bar A_x + \delta\, \bar a_x = 1\) | Continuous identity |
| \(A_x + d\, \ddot a_x = 1\) | Discrete identity |
| \(\bar A_x = (i/\delta) A_x\) (UDD) | Continuous-discrete conversion |
| Constant force: \(\bar A_x = \mu/(\mu+\delta);\; \bar a_x = 1/(\mu+\delta)\) | Exponential lifetimes |
| \(E(S) = E(N) E(X);\; \text{Var}(S) = E(N)\text{Var}(X) + \text{Var}(N) [E(X)]^2\) | Compound model |
| Equivalence: \(P \cdot \ddot a_x = A_x\) | Net premium formula |