What is the difference between T_x and K_x in FAM lifetime notation?

T_x is the complete future lifetime of a life age x. K_x is the curtate future lifetime, the integer number of complete future years lived.

If the force of mortality is constant at 0.02 for five years, calculate the five-year survival probability.

The survival probability is exp(-0.02(5)) = exp(-0.10) = 0.9048. Steps: Use survival under a force of mortality. Integrate the constant force over five years. Evaluate exp(-0.10).

For a fully discrete whole life insurance, A_x = 0.32 and a-double-dot_x = 11.5. Find the annual net premium for unit benefit.

Using the equivalence principle, P_x = A_x / a-double-dot_x = 0.32 / 11.5 = 0.02783. Steps: Identify the benefit APV. Identify the premium annuity-due APV. Divide benefit APV by premium APV.

At duration 5, A_{x+5} = 0.42, a-double-dot_{x+5} = 8.0, and the annual net premium is 0.03. Calculate the net premium policy value.

The net premium policy value is A_{x+5} - P a-double-dot_{x+5} = 0.42 - 0.03(8.0) = 0.18. Steps: Use future benefit APV minus future premium APV. Substitute the attained-age values. Subtract 0.24 from 0.42.

Exam guide

FAM Long-Term Models

FAM long-term models turn survival, mortality, interest, and benefit timing into present value random variables, premiums, and policy values.

Credential side

SOA

Primary intent

FAM long-term models

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Quick Answer

The long-term side of FAM is about contingent payment models. A benefit is paid only if a life status or retirement condition occurs, so the random variable must include both timing and discounting.

The fastest way to avoid formula confusion is to write the present value random variable before writing its expectation. FAM tests that translation repeatedly.

Coverages And Retirement Programs

FAM expects candidates to identify life insurance, health insurance, annuities, defined benefit plans, defined contribution plans, and insurable interest. This is the conceptual entry point before the formulas start.

Do not skip this block because its weight is smaller. It supplies the product language that later premium and policy-value questions assume.

Mortality Models

Mortality models include survival functions, life tables, force of mortality, complete future lifetime, curtate future lifetime, select and ultimate notation, and fractional age assumptions.

The FAM notation note says the exam may use standard symbols such as force of mortality, number of lives at age x, complete future lifetime, curtate future lifetime, and selected-life notation. Candidates need those symbols to feel ordinary before doing timed problems.

Survival and force relationship

{}_tp_x=\exp\left(-\int_0^t \mu_{x+s}\,ds\right)

Curtate future lifetime

K_x=\lfloor T_x\rfloor

Present Value Random Variables

A present value random variable combines amount, timing, and discounting. For fully discrete whole life insurance with unit benefit, the random variable pays at the end of the year of death. For annuities, it pays while the life is alive at the payment dates.

The relationships between insurance and annuity expected values matter because they let candidates check work without recomputing every sum from scratch.

Fully discrete whole life insurance PV

Z=v^{K_x+1}

Whole life APV relation

A_x+d\ddot a_x=1

Premiums

The equivalence principle sets expected present value of premiums equal to expected present value of benefits and expenses under the stated basis. FAM also includes portfolio percentile premium and premiums for a stated expected present value of profit.

For net premiums, expenses are zero. For gross premiums, expenses must be included if the question specifies them. The notation note states that expenses are zero unless the question says otherwise.

Fully discrete whole life net premium

P_x=\frac{A_x}{\ddot a_x}

Policy Values

On FAM, policy value is the expected value of the future loss random variable. The notation note deliberately uses policy value language where older exams and practice sometimes used reserve language.

At duration t, the policy value looks forward from that time. That means the issue-age premium may still appear, but the mortality and annuity values are evaluated at attained age or with the stated selected-life status.

Net premium policy value shape

{}_tV=A_{x+t}-P\ddot a_{x+t}

Practice

Original Source-Backed Practice

4 questions built from syllabus outcomes and released-exam patterns. The prompts and answers are original, so they train the skill without copying official exam text.