Actuarium

SOA ALTAMAdvanced Long-Term Actuarial Mathematics

Associateship (ASA)
3 hours·6–8 written-answer questions, 60 points·≈400 study hours

Syllabus learning objectives

Official syllabus

Paraphrased from the SOA syllabus so the study-plan builder and practice sets track the topics you will actually be examined on. Weights are the official topic ranges.

Multi-state models & multiple decrements

30%
  • Set up transition-intensity matrices for multi-state Markov models and compute occupancy and transition probabilities via Kolmogorov's forward equations.
  • Compute actuarial present values of benefits and premiums payable while in, or upon transition between, specific states.
  • Decompose a multiple-decrement table into associated single-decrement (independent) rates and vice versa under standard assumptions.
  • Apply multiple-decrement models to disability, withdrawal and mortality decrements in long-term products.

Profit testing, universal life & embedded options

30%
  • Construct a profit-testing (asset share) cash-flow projection for a traditional or universal life policy, incorporating premiums, expenses, interest, mortality and reserves.
  • Compute profit signature, net present value and internal rate of return of a product cash-flow projection.
  • Model universal life account value roll-forward including cost of insurance, expense and surrender charges.
  • Value embedded guarantees (minimum interest, GMDB/GMIB-style options) using stochastic or deterministic scenario testing.

Pension plans & retirement benefits

25%
  • Compute the actuarial liability and normal cost under standard funding methods (unit credit, entry age normal, aggregate).
  • Determine plan benefits under final-average-pay, career-average-pay and cash-balance formulas.
  • Assess the impact of decrements (withdrawal, disability, retirement, death) on pension valuation using a service table.
  • Evaluate funded status and amortization of unfunded actuarial liability over time.

Mortality improvement & longevity risk

15%
  • Apply mortality improvement scales to project future mortality rates and compute their effect on life expectancy and reserves.
  • Describe sources and drivers of longevity risk and its financial impact on annuity and pension liabilities.
  • Compute the impact of a change in mortality improvement assumption on actuarial present values.
  • Discuss hedging and risk-transfer approaches (longevity swaps, reinsurance) for longevity risk.

Lecture videos for this exam

Open the full video library →

Introduction to Contingencies- Lecture 1

H&J Online Academy · Life Contingencies

The Life Table (Contingencies: Actuarial Mathematics)

inTuition · Life Contingencies

Life Assurance Contracts Part I (Contingencies: Actuarial Mathematics)

inTuition · Life Contingencies

Whole Life and Temporary Annuities

Mike, the Mathematician · Life Contingencies

4.4. Actuarial Math: Survival Models D

Dr. Amjad Rabi · Survival Models

036. Introduction to Continuous Time Survival Analysis

Dr. Dylan Spicker · Survival Models

Overview

ALTAM is the written-answer life exam: multi-state Markov models (Kolmogorov equations, Thiele), multiple decrements, profit testing and universal life, pension funding (projected unit credit, traditional unit credit), and mortality improvement models. Candidates choose ALTAM or ASTAM for ASA.

Duration
3 hours
Questions
6–8 written-answer questions, 60 points
Style
Computer-based written answer
Passing
Scaled 6 of 10

Syllabus map

Multi-state models & multiple decrements
30%
Life Contingencies Primer
Profit testing, universal life & embedded options
30%
Life Contingencies PrimerInterest Theory and Time Value of Money
Pension plans & retirement benefits
25%
Life Contingencies Primer
Mortality improvement & longevity risk
15%
Statistics and Estimation for Actuaries

Key formulas

Kolmogorov forward   ddttpxij=∑k≠j(tpxikμx+tkj−tpxijμx+tjk)\;\dfrac{d}{dt}{}_tp_x^{ij}=\sum_{k\ne j}\left({}_tp_x^{ik}\mu^{kj}_{x+t}-{}_tp_x^{ij}\mu^{jk}_{x+t}\right)

Thiele's differential equation   ddttV(i)=δt tV(i)+Pt(i)−∑j≠iμx+tij(bt(ij)+tV(j)−tV(i))\;\dfrac{d}{dt}{}_tV^{(i)}=\delta_t\,{}_tV^{(i)}+P^{(i)}_t-\sum_{j\ne i}\mu^{ij}_{x+t}\left(b^{(ij)}_t+{}_tV^{(j)}-{}_tV^{(i)}\right)

Projected unit credit accrued liability   ALt=accrued benefit at projected final salary×r−xpx vr−x a¨r\;AL_t=\text{accrued benefit at projected final salary}\times{}_{r-x}p_x\,v^{r-x}\,\ddot a_r

Profit signature   Πt=t−1px⋅Prt\;\Pi_t={}_{t-1}p_x\cdot Pr_t, NPV =∑tΠtvrt=\sum_t\Pi_t v_r^t at hurdle rate rr

Lee–Carter   ln⁡mx,t=αx+βxκt+εx,t\;\ln m_{x,t}=\alpha_x+\beta_x\kappa_t+\varepsilon_{x,t}

Study strategy

  1. Practice writing derivations legibly under time pressure — graders reward clear structure and stated assumptions.

  2. Work every released ALTAM/LTAM written-answer question with the model solution beside you, then again alone a week later.

  3. Draw the state diagram for every multi-state question before writing an equation.

  4. For pension items, be explicit about salary scale, decrement, and discount timing.

Common traps

  • Mixing transition intensities (forces) with one-year transition probabilities.

  • Forgetting survival-weighting when converting profit vector to profit signature.

  • Using current salary instead of projected final salary under PUC.

  • Sign conventions in Thiele (premiums increase, benefits and reserve jumps decrease the reserve growth).

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