MA546 Long-Term Actuarial Mathematics

Course Catalog Description


This course is designed to satisfy the Validation by ASA requirement for Long-Term Actuarial Mathematics (LTAM) Exam. The course contents comprise various aspects of long- term actuarial mathematics, including, long-term insurance coverages, survival models and their estimation, present value random variables, premium calculation, reserves calculation, pension plans and retirement benefits. Discussions are also made on applying advanced probability and statistic to evaluate various long-term insurances.

Prerequisites: MA540, MA541
Cross-listing: None

Course Objective

• Understand the key features of long-term insurance coverages
• Understand key concepts concerning parametric and non-parametric (tabular) and multi-state models including single life, or multiple life, and multiple decrements
• Perform calculations on the present value random variables associated with benefits and expenses for any of the models
• Use and explain premium-calculation methodologies
• Understand reserves for insurances and annuities for models
• Understand how the previous models apply to pension plans and retirement benefits
• Catch basics of insurance and reinsurance coverages for long-term insurances
• Use basic methods to calculate premiums and reserves for long-term insurance coverages


Teaching Associate Professor Email Office
Xiaohu Li Kidde 113A

Campus Fall Spring Summer
On Campus X

More Information

Course Outcomes

    • Describe the appropriate models to be used to calculate expected present values, premiums or contributions, and reserves for each long-term coverage
    • Calculate nonparametric estimates of survival models using the Kaplan-Meier and Nelson-Aalen formulas for seriatim data and adaptations for grouped data
    • Calculate, using both seriatim and grouped data, maximum likelihood estimates of transition probabilities assuming constant transition intensity during fixed age intervals; Calculate the variances of and construct confidence intervals for the estimators
    • Apply appropriate approximation methods such as uniform distribution of deaths, constant force, Woolhouse, and Euler
    • Calculate and interpret common profit measures such as expected profit, actual profit, gain, gain by source and period, internal rate of return, profit margin, and break-even year
    • Given particular participant data, plan provisions, and valuation assumptions, apply the concerned models to defined benefit pension plans and calculate and interpret replacement ratios, accrued benefits, gain or loss, and their expected values with adjustments such as the early retirement reduction factor
    • Given particular participant data, plan provisions, and valuation assumptions, calculate and interpret the actuarial accrued liability and the normal cost for a defined benefit plan under the projected unit credit (PUC) cost method and the traditional unit credit (TUC) cost method.
    • Identify and interpret the assumptions and methods for retiree health care plans. Given particular participant data, plan provisions, and valuation assumptions, calculate and interpret the expected present value of future benefits, accumulated postretirement benefit obligation (APBO), and the normal cost or service cost for retiree health care plans.
    • Calculate and interpret the effect of changes in underlying valuation assumptions such as mortality, discrete salary increase changes, other decrements and interest on the quantities, and apply appropriate approximation methods such as uniform distribution of deaths, constant force, Woolhouse, and Euler.

Course Resources


  • Required textbook
    Actuarial Mathematics for Life Contingent Risks, 2nd Edition, 2013 Dickson, D., Hardy, M., Waters, H Cambridge University Press, ISBN: 978-1-10704-407-4.
  • Syllabus and study materials, SOA
    Exam LTAM: Long-Term Actuarial Mathematics


Grading Policies

1 Homework 45%
2 Mid-term 25%
3 Final 35%
Maximum Possible 100%

Lecture Outline

Topic Reading(s) Homework
Week 1 Long-term insurance coverages: life insurance and annuity contract, pension benefit Chapter 1 Exercises: 1.6, 1.7
Week 2 Survival models: force of mortality, curtate future lifetime Chapter 2 Exercises: 2.4, 2.5, 2.7, 2.11, 2.12, 2.13, 2.15
Week 3 Lifetime tables: fractional age, survival models for lifetime insurance policy holder, survival model selection, heterogeneity in mortality Chapter 3 Exercises 3.1, 3.3, 3.5, 3.7, 3.9
Week 4 Insurance benefit Chapter 4 Exercises: 4.3, 4.6, 4.9, 4.12, 4.15, 4.18, 4.21
Week 5 Annuities Chapter 5 Exercises: 5.2, 5.4, 5.6, 5.8, 5.10, 5.12, 5.14, 5.16, 5. 18
Week 6 Premium calculations Chapter 6 Exercise: 6.3, 6.6, 6.9, 6.12, 6.15, 6.18, 6.2
Week 7 Midterm exam Chapters 1-6
Week 8 Policy value Chapter 7 Exercises: 7.2, 7.5, 7.8, 7.11, 7.14, 7.17, 7.20
Week 9 Multistate models Chapter 8 Exercises: 8.3, 8.6, 8.8, 8. 12, 8.14, 8.8
Week 10 Joint life and last survivor benefit Chapter 9 Exercise: 9.2, 9.4, 9. 6, 9.9, 9.11, 9.14, 9.16
Week 11 Pension mathematics Chapter 10 Exercises: 10.1, 10.3, 10.5, 10.7, 10.9, 10.11, 10.13
Week 12 Emerging costs for traditional life insurance Chapter 12 Exercises: 12.2, 12.4, 12.6, 12.8, 12.10, 12.12
Week 13 Review Chapters 7-8
Week 14 Final exam All chapters