SOA Advanced Long-Term Actuarial Mathematics Examination Practice Test
ALTAM practice addresses advanced long-term actuarial mathematics, especially life-contingent cash flows and insurance liabilities. The available questions use models of survival and death, benefit and premium timing, and present values to examine how assumptions translate into actuarial values.
Exam overview
About the SOA Advanced Long-Term Actuarial Mathematics Examination Exam
Problems call for translating policy terms into present value random variables and applying valuation principles consistently. Relevant calculations include premiums under the equivalence principle, variance-loaded premiums, prospective reserves after a specified duration, and values of term, endowment, and annuity benefits. Mortality concepts in the question bank include complete future lifetime, survival probabilities, the force of mortality, and the uniform distribution of deaths assumption within a year of age. Other items test discounting and how changes in interest rates affect liability values. Solving these problems requires tracking when premiums, expenses, and benefits are paid; distinguishing survival benefits from death benefits; and conditioning future cash flows on the policyholder being alive at the valuation date. These skills connect actuarial notation and probability models to the valuation of long-term insurance cash flows.
Sample Practice Questions
Review a selection from this exam's published US Practice Tests practice bank.
A life insurance policy has an annual gross premium G, an annual expense e paid at each premium date while the policy is in force, and a death benefit with present value random variable Z. Under the equivalence principle, which equation applies if premiums and expenses have the same payment timing and duration?
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Correct answer: AThe insurer's net premium income per premium date after the stated expense is G-e. Equating the expected present value of that stream to the expected present value of benefits gives the equation in A. The annuity factor must reflect the contract's actual premium duration and timing.
A liability has a fixed positive payment at time t > 0. The valuation interest rate is reduced while the payment amount and timing remain unchanged. What happens to its present value?
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Correct answer: AFor a fixed payment C > 0, present value is C/(1+i)^t. For t > 0 this value increases when i decreases, so the liability's present value rises.
A random present value Z has mean m and variance v. Under a variance loading principle with loading parameter α > 0, the premium is defined as E[Z] + α Var(Z). What premium does the principle produce?
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Correct answer: ABy definition, the variance loading premium adds α times the variance to the expected present value. Substituting the stated mean and variance gives m + αv.
A 10-year level term insurance has a death benefit payable at the end of the year of death and returns no benefit on survival. At the end of year 4, the insured is alive and the policy remains in force. Which period should be considered when calculating the prospective reserve at that time?
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Correct answer: BA prospective reserve at duration 4 compares the expected present value of benefits and expenses after duration 4 with the expected present value of future premiums after duration 4. The remaining coverage term is 6 years; past cash flows are not included.
Under the UDD assumption within each year of age, a life aged x has one-year death probability q_x. What is the probability that death occurs within t years, for 0≤t≤1?
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Correct answer: AUniform distribution of deaths over the year means the conditional probability of death by fraction t of the year is t times the full-year death probability. Thus {}_tq_x=tq_x under UDD.
A 5-year endowment insurance on a life aged x pays 1 at the end of the year of death if death occurs within 5 years, or 1 at time 5 if the life survives. Which is the present value random variable at annual effective interest rate i?
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Correct answer: AThe contract combines term insurance with a pure endowment. Death during the term produces payment at the end of the year of death, at time K_x+1; survival to time 5 produces payment then. Discounting each possible payment gives option A.
Under the standard annual effective interest convention, the discount factor for a payment due one year from now is v. Which relationship between v and i is correct?
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Correct answer: BA unit payment one year from now has present value v, and accumulating that present value for one year must produce 1. Hence v(1+i)=1, or v=1/(1+i).
How to use this practice test
The free practice session draws up to 20 random questions from the published ALTAM bank. Work each problem by writing the cash-flow timeline and identifying the mortality or interest assumption before calculating. Results appear after the session, allowing you to check answers and identify concepts to revisit.
Preparation tips
- For each policy question, mark premium, expense, and benefit payment dates before writing its present value random variable.
- Practice prospective reserve calculations by conditioning on survival to the valuation date and valuing only future benefits, premiums, and expenses.
- Translate between complete future lifetime, survival probabilities, and the force of mortality; under UDD, derive the within-year death probability from q_x.
- Compare benefit structures such as term insurance, endowment insurance, and life annuities, noting how death and survival determine payment timing.
- Check interest-rate effects by expressing a fixed future payment as a discounted value and confirming how a lower valuation rate changes that value.