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

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.

Define the actuarial liability (AL) under the Projected Unit Credit (PUC) cost method.

The AL is the EPV of accrued benefits based on service to the valuation date but using projected (final) salaries: $AL=\alpha\cdot N_{\text{past}}\cdot \bar{S}_{\text{final, projected}}\cdot(\text{annuity \& discount/decrement factors})$.

Define the normal cost (service cost) under the Projected Unit Credit method.

The normal cost is the EPV of the benefit attributed to the current year of service (one additional year of accrual), using projected final salary and appropriate discount and decrement factors.

Contrast the Traditional Unit Credit (TUC) and Projected Unit Credit (PUC) cost methods.

TUC values accrued benefits using current salary (no future salary projection); PUC uses projected final salary. For a final-salary plan PUC gives a larger, smoother increasing liability; TUC understates the ultimate cost early on.

State the relationship among actuarial liability, plan assets, and the surplus/(unfunded) position.

Funded status $=$ Assets $-$ Actuarial Liability. A positive value is a surplus; a negative value is an unfunded actuarial liability (UAL) that must be amortized.

What is the gain/loss analysis in pension funding?

It decomposes the change in unfunded liability between valuations into experience gains and losses (e.g., investment return vs. assumed, salary increases, mortality, withdrawal) by comparing actual experience to assumptions.

Define the Aggregate cost method for pension funding.

It spreads the EPV of all future benefits (less current assets) over the EPV of future salaries of the group, producing a normal cost as a level percentage of payroll; it does not separately identify an actuarial liability or UAL.

What is the funding objective expressed by the basic balance equation in pension valuation?

$$\text{PV(Future Benefits)}=\text{Assets}+\text{PV(Future Normal Costs)}$$ Each funding method allocates the PVFB between past (AL) and future (normal costs).

Define OPEB and give the most common example.

OPEB = Other Postemployment Benefits: postretirement benefits other than pensions provided to retirees, most commonly retiree health/medical care (and life insurance, dental, etc.).

What is the Expected Postretirement Benefit Obligation (EPBO) for retiree health benefits?

The EPBO is the actuarial present value of all future postretirement benefits expected to be paid to current/retired employees, regardless of how much service has been rendered to date (full projected benefit).

What is the Accumulated Postretirement Benefit Obligation (APBO) and how does it relate to EPBO?

The APBO is the portion of the EPBO attributed (accrued) to service rendered to the valuation date. At the full eligibility date APBO equals EPBO; before that, APBO $\le$ EPBO.

Why is health-care trend rate a critical assumption in retiree health (OPEB) valuation?

Per-capita claims costs grow with medical inflation (trend), often faster than general inflation; small changes in the assumed trend rate compound over decades and materially change the EPBO/APBO.

What is the "per-capita claims cost by age" used in retiree health valuation, and the role of the aging factor?

It is the expected annual medical claim cost for a retiree at each age. Aging factors scale a base cost up with age to reflect that older retirees incur higher claims; combined with trend they project future costs.

In profit testing, define the profit signature.

The profit signature is the vector of expected profits emerging at the end of each policy year per policy issued, i.e., the per-policy-in-force profit vector multiplied by the probability of being in force at the start of each year.

Distinguish the profit vector from the profit signature.

The profit vector $(Pr_t)$ is the expected profit in year $t$ per policy in force at the start of that year. The profit signature $(\Sigma_t)$ multiplies each by the survival/in-force probability: $\Sigma_t=\,_{t-1}p_x\,Pr_t$ (with $\Sigma_0$ for any initial outgo).

Define the Net Present Value (NPV) of a profit signature and the role of the risk discount rate.

$$NPV=\sum_{t}\Sigma_t\,(1+r)^{-t}$$ where $r$ is the risk discount rate (hurdle rate). NPV $>0$ indicates the contract is expected to meet the required return.

Define the internal rate of return (IRR) profit measure.

The IRR is the discount rate $j$ that makes the NPV of the profit signature zero: $\sum_t \Sigma_t(1+j)^{-t}=0$. The product is acceptable if IRR exceeds the company's hurdle rate.

Define the profit margin profit measure.

Profit margin = NPV of profit signature divided by the EPV of premiums (both at the risk discount rate); it expresses profit as a percentage of premium income.

What is the discounted payback period (DPP) in profit testing?

The DPP is the first time $t$ at which the accumulated (or discounted) value of the profit signature first becomes non-negative—i.e., when the insurer recovers its initial investment including required return.

Describe the basic mechanics of a Universal Life (UL) policy account value roll-forward.

Each period: $AV_{\text{new}}=(AV_{\text{old}}+\text{Premium}-\text{Expense charge}-\text{COI charge})\times(1+i_{\text{credited}})$, where COI is the cost of insurance charge for the net amount at risk.

In Universal Life, define the cost of insurance (COI) charge.

$$\text{COI}_t=q_{x+t}^{*}\times v_q\times(\text{Death Benefit}-AV)$$ i.e., the COI rate times the net amount at risk (death benefit minus account value), discounted for the period; $q^{*}$ is the COI mortality rate.

Contrast UL Type A (Option 1) and Type B (Option 2) death benefits.

Type A pays a level death benefit (face amount), so the net amount at risk decreases as AV grows. Type B pays face amount plus account value, keeping the net amount at risk roughly level (equal to the face amount).

What is the corridor requirement in Universal Life?

To qualify as life insurance for tax purposes, the death benefit must be at least a specified multiple (corridor factor) of the account value; if AV grows too large the death benefit is forced up to maintain the minimum ratio.

What is a no-lapse guarantee (secondary guarantee) in UL?

A provision keeping the policy in force even if the account value falls to zero, provided the policyholder pays at least a specified minimum premium (or a shadow account stays positive).

Define a participating (with-profits) insurance policy.

A policy under which the policyholder shares in the insurer's surplus/profits through dividends (US) or bonuses (UK), typically funded by conservative premium assumptions that generate distributable surplus.

Distinguish reversionary bonuses from terminal bonuses in participating business.

A reversionary bonus is a permanent addition to the guaranteed sum insured declared periodically (once added it cannot be removed). A terminal bonus is paid only at claim/maturity, is not guaranteed, and reflects accumulated unallocated surplus at exit.

What are the three traditional sources of surplus distributed as dividends in participating life insurance?

Mortality savings (actual vs. assumed mortality), interest/investment surplus (actual vs. assumed earnings), and expense/loading savings (actual vs. assumed expenses) — often summarized as the three-factor dividend formula.

In a disability income (DI) multi-state model, name the typical states and which transitions are allowed.

Healthy/active (H), Disabled/sick (D), and Dead (X). Allowed: H$\to$D (sickness inception), D$\to$H (recovery), H$\to$X and D$\to$X (death). It is a model with recovery, so D$\to$H is permitted.

What is the difference between an inception-annuity ("claim cost") approach and a continuous multi-state approach to DI valuation?

The inception-annuity approach values the EPV of a disability annuity at each possible time of disablement (inception rate × annuity value for that disability). The continuous multi-state approach integrates the occupied-disabled probabilities directly via $\,_{t}p_x^{HD}$.

Why does a DI benefit often depend on the duration of disability, and what model feature captures this?

Recovery and mortality rates from disability vary by how long someone has been disabled (select effect), and benefits may have elimination/off periods. A model where the disabled state's transition intensities depend on time since disablement (a semi-Markov / duration-dependent model) captures this.

Define the elimination (waiting) period and benefit (maximum) period in a disability income policy.

The elimination period is the time a claimant must be continuously disabled before benefits begin (a deductible in time). The benefit period is the maximum length of time benefits will be paid for a single disability claim.

Describe the typical state structure of a Long-Term Care (LTC) multi-state model.

Often: Healthy, then graded levels of impairment (e.g., needing help with increasing numbers of Activities of Daily Living / ADLs, or mild/severe), and Dead. Benefits trigger on reaching a defined impairment level; transitions can include deterioration and sometimes recovery, plus death from any state.

What is the standard benefit-eligibility trigger for Long-Term Care insurance?

Inability to perform a specified number (commonly 2 or more) of the six Activities of Daily Living (ADLs)—bathing, dressing, eating, toileting, transferring, continence—or severe cognitive impairment.

Contrast reimbursement and indemnity (disability-style) LTC benefit designs.

A reimbursement (expense-incurred) policy pays actual covered LTC costs up to a daily/monthly maximum. An indemnity (disability) model pays a fixed periodic amount once the trigger is met, regardless of actual expenses incurred.

Define Critical Illness (CI) insurance and its core benefit trigger.

CI insurance pays a lump sum on the first diagnosis of (or survival a set period after) one of a defined list of serious conditions (e.g., cancer, heart attack, stroke). The trigger is diagnosis/survival, not death or expense.

Distinguish "accelerated" from "standalone (additional)" critical illness benefits.

Accelerated CI pays the CI lump sum out of an existing life insurance sum assured, reducing the death benefit by the amount paid. Standalone/additional CI pays the CI benefit independently, leaving any life cover intact.

In a critical-illness model, why is a state for "diagnosed with CI but alive" important, and how is the CI lump sum valued?

Because CI pays on transition into the critical-illness state (an inception event), not on death. The EPV uses the transition intensity into the CI state: $$\bar{A}^{CI}=\int_0^{n}e^{-\delta t}\,_{t}p_x^{(\tau)}\,\mu_{x+t}^{\text{CI}}\,dt$$

What is an active life reserve vs. a disabled life reserve in health insurance?

The active (or disabled-life) reserve concept: the active life reserve holds funds for currently-healthy insureds expected to become disabled later; the disabled life reserve (DLR/claim reserve) holds the EPV of future benefit payments for lives already in claim (disabled).

How is a disabled life reserve (claim reserve) for an in-force DI/LTC claim computed?

It is the EPV of remaining benefit payments to a currently-disabled life: $$DLR=\int_0^{\infty}e^{-\delta s}\,_{s}p^{\,\overline{dd}}_{[x]+t}\,B\,ds$$ using continuation (persistency-in-disability) probabilities from the current claim duration, reflecting recovery and death.

What is "active life reserve" inadequacy risk in long-duration health products and why does it matter for LTC?

Because premiums are level but morbidity costs rise steeply with age, large active life reserves accumulate. If morbidity, lapse, or interest assumptions prove adverse, reserves may be inadequate—an issue that has driven major LTC premium increases and reserve strengthening.