ACTL10001 Chap.8 Contingent Payments
Contingent Payments
Contingent payment valuation combines the time value of money with uncertainty about whether a cashflow occurs. A survival benefit is discounted to the valuation date and multiplied by the probability of surviving to payment. A death benefit requires the probability of death in each eligible interval and the matching payment date.
Term insurance, pure endowments, endowment insurance and life annuities are built from these components. The chapter emphasises event trees, mutually exclusive branches and consistent conditioning so that each term can be read in words as well as symbols.
What this chapter covers
- 01
Expected present value
- 02
Survival benefits
- 03
Death benefits
- 04
Deferred death events
- 05
Pure endowments
- 06
Term insurance
- 07
Endowment decomposition
- 08
Life annuity payments
Worked example · free
A two year death benefit
- 1Payment time is one year, so discount by v = 1/1.03.
- 1The payment event has probability q45 = 0.004.
- 1EPV = 60,000 × 1.03^(-1) × 0.004 = $233.01.
Key terms
- Contingent Payment
- A cashflow that occurs only if a specified uncertain event happens.
- Expected Present Value
- The probability weighted average of discounted payments across possible outcomes.
- Pure Endowment
- A benefit paid at a fixed future date only if the insured survives to that date.
- Term Insurance
- A death benefit limited to deaths occurring within a stated term.
- Endowment Insurance
- A contract paying on death during a term or on survival to its end.
- Life Annuity
- A sequence of payments that continues only while the recipient is alive.
Contingent Payments FAQ
Why are there two multipliers in a contingent value?
The discount factor adjusts for payment timing, while the probability adjusts for whether payment occurs. They represent different features of the same contingent contractual promise.
How do I value a later death year?
Multiply survival through earlier years by death probability in the target year, then discount to the end of that year if that is the stated timing.
Why can endowment value be split?
Death during the term and survival to the end are mutually exclusive benefit branches, so their expected present values can be added at one valuation date.
Is expected present value what one person receives?
No. It is a probability weighted portfolio average under assumptions. An individual outcome may be the full benefit or no payment at all under the contract.
How do I know whether a death probability needs a survival factor first?
Ask whether the death interval begins after the valuation date. Death in a later year requires survival to that year's start and then death during the year, so the unconditional branch probability multiplies those stages.
What is the difference between expected present value and a possible payment?
A possible payment is the contractual amount on one event branch. Expected present value averages discounted branch payments using their probabilities, so it can be an amount no individual ever receives. It is a valuation building block for pricing and reserving, not a prediction that a particular policy will pay the average.
The valuation basis must accompany the result because different interest and mortality assumptions change that average. Check that every contractual outcome belongs to one branch.
Exam move
Describe every benefit in words using event, eligible interval and payment date. Draw a branch diagram for short contracts and translate each terminal branch into amount times probability times discount. Practise decomposing endowment insurance and building life annuities from repeated survival payments. Check that probabilities cover the intended events without overlap.
Interpret expected present value as a pricing building block rather than a guaranteed individual outcome. Begin with a benefit audit containing amount, event, payment time and valuation date. Draw mutually exclusive branches for survival, death in each relevant interval and any zero-payment outcome. On every terminal branch, write probability first, then the cashflow and its discount factor.
Later-year death branches require survival to the start of that year before the conditional death probability is applied. Sum branch probabilities separately from values to check that the event partition is complete. Recalculate a short contract by direct enumeration and by a compact actuarial expression; matching results confirm both notation and timing.
Practise pure endowments and term insurance alone before combining them into endowment insurance. For life annuities, list each payment date with the survival condition that applies there, noting that a due payment at issue is certain for a life known alive.
Change one assumption at a time and predict the direction before recomputing: higher interest lowers delayed values, while higher relevant survival raises survival-contingent benefits. Keep branch probabilities unrounded until the final sum. Distinguish expected present value from the amount any individual policyholder receives, since the expectation averages across possible outcomes.
Conclude every calculation by explaining which factor handles time and which handles uncertainty. If an answer lacks either factor for a genuinely contingent future payment, the valued contract is incomplete.
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