Associate of the Society of Actuaries (ASA/FSA) ALTAM — Advanced Long-Term Actuarial Mathematics Syllabus

ALTAM — Advanced Long-Term Actuarial Mathematics syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for ALTAM — Advanced Long-Term Actuarial Mathematics in Associate of the Society of Actuaries (ASA/FSA), not a summary of it.

1. 1

   Multi-State and Multiple-Decrement Models

   3 topics

   • Markov Multi-State Models
     • Transition intensities and probabilities
     • Kolmogorov forward equations
     • Numerical (Euler) solution methods
   • Multiple-Decrement Theory
     • Associated single-decrement tables
     • Dependent vs. independent rates
   • Premiums and Reserves in Multi-State Models
     • Thiele's differential equation
     • State-dependent benefits
2. 2

   Pension Mathematics and Retirement Benefits

   3 topics

   • Pension Plan Design
     • Defined benefit vs. defined contribution
     • Salary scales and replacement ratios
   • Pension Funding and Valuation
     • Actuarial cost methods
     • Normal cost and actuarial liability
     • Service tables (decrements)
   • Retiree Health and Other Postemployment Benefits
     • OPEB valuation basics
     • Trend assumptions for retiree medical
3. 3

   Profit Testing and Universal Life

   3 topics

   • Profit Measures
     • Profit vectors and signatures
     • Net present value and internal rate of return of profits
   • Universal Life Insurance
     • Account value mechanics
     • Cost of insurance and crediting rates
   • Participating and Non-Traditional Products
     • Dividends and bonus structures
4. 4

   Health, Disability, and Long-Term Care Insurance

   3 topics

   • Disability Income Models
     • Sickness-death-recovery multi-state structure
     • Benefit period and elimination period
   • Long-Term Care Insurance
     • Activities of daily living triggers
     • Continuance and morbidity assumptions
   • Critical Illness and Health Reserves
     • Active life reserves
     • Disabled life reserves

ALTAM — Advanced Long-Term Actuarial Mathematics flashcards for Associate of the Society of Actuaries (ASA/FSA)

21 of 60 cards from the ALTAM — Advanced Long-Term Actuarial Mathematics deck — real questions with worked answers.

1. In a Markov multi-state model, what is the defining "Markov property" for the state process?

   The future evolution depends only on the current state, not on the history of how it was reached. Formally, for the present state $i$ at time $x$, transition probabilities to future states are independent of states occupied before time $x$.

2. Define the transition probability $\,_{t}p_x^{ij}$ in a multi-state model.

   $\,_{t}p_x^{ij}=\Pr[\,$ in state $j$ at age $x+t \mid$ in state $i$ at age $x\,]$, allowing any path between $i$ and $j$ during the interval.

3. Define the occupancy (sojourn) probability $\,_{t}\bar{p}_x^{ii}$ and how it differs from $\,_{t}p_x^{ii}$.

   $\,_{t}\bar{p}_x^{ii}$ is the probability of remaining continuously in state $i$ for the whole interval $[x,x+t]$ (never leaving). $\,_{t}p_x^{ii}$ allows leaving and returning, so $\,_{t}\bar{p}_x^{ii}\le\,_{t}p_x^{ii}$.

4. Write the Kolmogorov forward differential equation for $\,_{t}p_x^{ij}$ in a multi-state model.

   $$\frac{d}{dt}\,_{t}p_x^{ij}=\sum_{k\neq j}\left(\,_{t}p_x^{ik}\,\mu_{x+t}^{kj}-\,_{t}p_x^{ij}\,\mu_{x+t}^{jk}\right)$$ representing inflows to $j$ minus outflows from $j$.

5. Give the formula for the occupancy probability $\,_{t}\bar{p}_x^{ii}$ in terms of total transition intensity out of state $i$.

   $$\,_{t}\bar{p}_x^{ii}=\exp\!\left(-\int_0^{t}\sum_{j\neq i}\mu_{x+s}^{ij}\,ds\right)$$

6. In a multi-state model, what is the total force of transition out of state $i$, $\mu_{x}^{i\bullet}$?

   $\mu_x^{i\bullet}=\sum_{j\neq i}\mu_x^{ij}$, the sum of all transition intensities leaving state $i$.

7. What is the key structural feature of a multiple-decrement model expressed as a multi-state model?

   A single starting (active) state with several absorbing decrement states and no return transitions; once a life leaves the active state it cannot come back.

8. Distinguish between the dependent (multiple-decrement) probability $\,_{t}q_x^{(j)}$ and the independent (associated single-decrement) probability $\,_{t}q_x^{\prime(j)}$.

   $\,_{t}q_x^{(j)}$ is the probability of decrement $j$ in the presence of all other decrements (competing risks). $\,_{t}q_x^{\prime(j)}$ is the probability of decrement $j$ if it were the only decrement acting. Since intensities are equal, $\,_{t}q_x^{(j)}\le\,_{t}q_x^{\prime(j)}$.

9. State the relationship between the total survival probability and the associated single-decrement survival probabilities.

   $$\,_{t}p_x^{(\tau)}=\prod_{j}\,_{t}p_x^{\prime(j)}=\exp\!\left(-\int_0^{t}\sum_j\mu_{x+s}^{(j)}\,ds\right)$$

10. Give the integral expression for the multiple-decrement probability $\,_{t}q_x^{(j)}$ in terms of intensities.

    $$\,_{t}q_x^{(j)}=\int_0^{t}\,_{s}p_x^{(\tau)}\,\mu_{x+s}^{(j)}\,ds$$

11. Under the constant-force (over a year of age) assumption in a multiple-decrement table, express $q_x^{(j)}$ in terms of $q_x^{(\tau)}$.

    $$q_x^{(j)}=\frac{\mu^{(j)}}{\mu^{(\tau)}}\,q_x^{(\tau)}$$ where the force ratio $\mu^{(j)}/\mu^{(\tau)}$ is constant over the year of age.

12. Under the uniform-distribution-of-decrements (UDD) in the multiple-decrement table, give the formula linking $q_x^{\prime(j)}$ to dependent rates.

    $$q_x^{\prime(j)}=\left(p_x^{(\tau)}\right)^{\,q_x^{(j)}/q_x^{(\tau)}}$$

13. State Thiele's differential equation for the policy value $\,_t V^{(i)}$ in state $i$ of a multi-state model.

    $$\frac{d}{dt}\,_tV^{(i)}=\delta_t\,_tV^{(i)}-B_t^{(i)}-\sum_{j\neq i}\mu_{x+t}^{ij}\left(S_t^{(ij)}+\,_tV^{(j)}-\,_tV^{(i)}\right)$$ where $B^{(i)}$ is the rate of benefit/premium in state $i$ and $S^{(ij)}$ is the lump sum on transition $i\to j$.

14. How is the EPV of a continuous benefit of rate $1$ payable while in state $j$, starting from state $i$, written?

    $$\bar{a}_x^{ij}=\int_0^{\infty}e^{-\delta t}\,_{t}p_x^{ij}\,dt$$

15. How is the EPV of a unit lump sum paid on each transition from state $i$ to state $k$ (starting in state $i_0$) expressed?

    $$\bar{A}_x^{i_0,\,ik}=\int_0^{\infty}e^{-\delta t}\,_{t}p_x^{i_0 i}\,\mu_{x+t}^{ik}\,dt$$

16. In multi-state premium calculation, state the equivalence principle for setting a level premium.

    Set the premium so that at issue (time 0, in the starting state) the EPV of premiums equals the EPV of benefits, i.e. $\,_0V=0$ (or equivalently EPV benefits − EPV premiums $=0$).

17. What distinguishes a defined benefit (DB) pension plan from a defined contribution (DC) plan in terms of risk bearing?

    In DB the employer bears investment and longevity risk and the benefit is formula-driven (e.g., based on salary and service). In DC the employee bears investment risk; the benefit equals the accumulated account balance from defined contributions.

18. Define the "final salary" (final average) DB accrual benefit formula.

    Annual pension $=\alpha\times N\times \bar{S}_{\text{final}}$, where $\alpha$ is the accrual rate per year of service, $N$ is years of service, and $\bar{S}_{\text{final}}$ is the final (or final-average) salary.

19. What is a "career average" (CARE) DB benefit, and how does it differ from final salary?

    CARE accrues each year a benefit based on that year's salary (often revalued/indexed), then sums across all service years. Unlike final-salary, it does not depend solely on salary near retirement, so it is less sensitive to late-career raises.

20. Define the replacement ratio in pension design.

    The replacement ratio is the retirement income (pension plus other sources) divided by pre-retirement (final) salary; it measures how well the pension maintains the member's standard of living.

21. What is the salary scale function $s_y/s_x$ used for in pension valuation?

    It projects future salaries: if current salary at age $x$ is $S$, projected salary at age $y$ is $S\cdot s_y/s_x$. The ratio captures merit, promotion, and inflation increases.

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