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.

Course Overview

Title	Advanced Actuarial Statistics
Theory Credits	3 (3 hrs/week)
Practical Credits	1 (2 hrs/week)
Pre-requisite	Actuarial Statistics, Probability, Continuous Distributions

Program Objectives

1. To extend the introductory actuarial framework with rigorous mathematical models for future lifetime, life-insurance benefits, life annuities, premiums and reserves.
2. To prepare students for the early professional actuarial examinations (CT5 / IFoA CM1 / SOA LTAM/FAM-L).
3. To develop the analytical, recursive and computational fluency required of professional actuaries.

Learning Outcomes

After successful completion of this course, students will be able to:

1. Model future lifetime using survival functions, force of mortality and the standard mortality laws (De Moivre, Constant Force, Gompertz, Makeham, Weibull).
2. Derive actuarial present values for the full range of life-insurance benefit functions, including varying and joint-life benefits.
3. Compute life-annuity present values for whole-life, temporary, deferred, guaranteed and \(m\)thly annuities, including joint and last-survivor annuities.
4. Apply the equivalence principle to set net (benefit) premiums for any standard plan, and compute the variance of the loss-at-issue random variable.
5. Build benefit reserves prospectively, retrospectively and recursively (Thiele's equation); compute gross-premium reserves and modified reserves (FPT, Zillmer).

Theory — Five Units

Unit 1: Future Lifetime & Mortality Laws

Future-lifetime random variable \(T(x)\); curtate future lifetime \(K(x)\); force of mortality \(\mu_x\); survival function \(s(x)\); analytical laws of mortality — De Moivre, constant force, Gompertz, Makeham, Weibull; select and ultimate life tables; fractional-age assumptions — UDD, constant force, Balducci.

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Unit 2: Life-Insurance Benefits

Present-value random variables; whole-life insurance — continuous (\(\bar A_x\)) and discrete (\(A_x\)) cases; term insurance, pure endowment, endowment, deferred insurance; varying benefits — annually-increasing (\(IA\)), continuously-increasing (\(\bar{IA}\)), decreasing term (\(DA\)); recursion formulas; variance of present value; joint-life and last-survivor insurances.

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Unit 3: Life Annuities

Annuities-certain; continuous whole-life annuity \(\bar a_x\); discrete annuities-due \(\ddot a_x\) and annuities-immediate \(a_x\); temporary and deferred annuities; annuity-certain-and-life ("guaranteed"); \(m\)thly life annuities; joint-life, last-survivor and reversionary annuities; key identity \(\delta\bar a_x + \bar A_x = 1\).

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Unit 4: Net (Benefit) Premiums

Equivalence principle and loss-at-issue random variable; fully continuous premiums \(\bar P(\bar A_x)\); fully discrete premiums \(P_x\); premiums for term, endowment, pure endowment and limited-pay plans; semi-continuous and apportionable premiums; variance of loss; portfolio-percentile premiums and risk pooling.

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Unit 5: Policy Reserves

Benefit (net) reserves — prospective and retrospective formulas; whole-life reserve \({}_tV_x\) and endowment reserve \({}_tV_{x:\overline n|}\); Thiele's recursive equation; continuous-time Thiele ODE; net amount at risk; gross-premium reserves; expense models; modified reserves — full preliminary term (FPT) and Zillmer modification.

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Practical — List of Experiments (10)

1. Fitting Gompertz's law of mortality to given \(\mu_x\) data.
2. Construction of a life table from a given \(q_x\) series.
3. Computation of \(A_x\) by recursion from the limiting age.
4. Computation of \(\bar A_x\) via the UDD factor \(i/\delta\).
5. Computation of \(\ddot a_x\), \(a_x\) and \(\bar a_x\) from \(A_x\).
6. Net annual premium for a whole-life policy.
7. Reserve \({}_tV_x\) computation at multiple durations.
8. Thiele's recursion to build year-by-year reserve schedule.
9. Joint-life and last-survivor APVs from independent-lives data.
10. Gross-premium reserve with explicit expense loading.

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Text Books

1. Bowers, N.L.; Gerber, H.U.; Hickman, J.C.; Jones, D.A.; Nesbitt, C.J. (1997) — Actuarial Mathematics, 2nd Edn., Society of Actuaries.
2. Dickson, D.C.M.; Hardy, M.R.; Waters, H.R. (2013) — Actuarial Mathematics for Life Contingent Risks, 2nd Edn.
3. Promislow, S.D. (2014) — Fundamentals of Actuarial Mathematics, 3rd Edn., John Wiley.

References

1. S. R. Deshmukh (2009) — Actuarial Statistics: An Introduction Using R.
2. Klugman, S.A.; Panjer, H.H.; Willmot, G.E. (2012) — Loss Models: From Data to Decisions, 4th Edn., Wiley.
3. IFoA / IAI Core-Reading material for CM1 (Actuarial Mathematics) — current edition.

Suggested Co-Curricular Activities

1. Hands-on workshops using R packages lifecontingencies, actuar and Excel solver.
2. Visits / virtual sessions with practising actuaries from LIC, GIC and private insurers.
3. Past-paper practice from IFoA CM1, SOA FAM-L, IAI Subject CM1.
4. Mini-projects: pricing a real-world product (term, endowment, ULIP) using market mortality.
5. Seminars on emerging areas — longevity risk, Solvency II reserving, IFRS-17.

Exam Blueprint (recommended distribution)

Unit	Topic	Approx. Weightage
1	Future Lifetime & Mortality Laws	15%
2	Life-Insurance Benefits	25%
3	Life Annuities	20%
4	Net Premiums	20%
5	Policy Reserves	20%