Expected Event-Contingent Cash Flows

Introduction

A policy-year may end with a claim, a surrender, or continued cover, and the amount paid depends on which event occurs. To calculate expected outgo, the actuary must combine two distinct pieces of information: how likely each event is and how much is paid if it happens. If payments occur on different dates, their timing must also remain visible in the calculation.

This tutorial develops that calculation for a hypothetical policy. It assumes that event probabilities are specified for a policy in force at the start of the policy-year. The methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: An event-contingent payment is a payment made only if a specified event occurs. For events $E_1,\ldots,E_n$ with associated payments $b_1,\ldots,b_n$, the expected payment is the probability-weighted sum of the amounts. If payment dates differ, calculate and retain a separate expected amount for each date before combining them.

From Event Probabilities to Expected Outgo

Begin by specifying the population to which the probabilities apply. In the examples below, each probability is conditional on the policy being in force at the beginning of the policy-year. If the events are the possible year-end outcomes, they are assumed to be mutually exclusive and exhaustive: exactly one outcome occurs. Their probabilities must therefore sum to one.

Suppose the possible outcomes are a claim, a lapse, and survival without either event. If the claim benefit is paid only on a claim and the surrender payment only on lapse, the expected outgo is the sum of the claim benefit multiplied by the claim probability and the surrender payment multiplied by the lapse probability. The survival outcome contributes zero to this particular outgo calculation, but it still matters when checking that the probabilities form a complete set of outcomes.

E[Y] = \sum_{j=1}^{n} P(E_j)b_j

Here $Y$ is the payment for the policy-year under the stated payment convention. The formula applies when the amounts $b_j$ are the payments associated with the events and the probabilities refer to the same starting population and period. It does not mean that every event has the same payment, nor does it itself specify when payment is made. For payments at different times, use a separate amount for each time or discount each event-contingent payment to a common valuation date.

Why Probability Weighting Works

The expected-value formula follows directly by representing each possible payment with an indicator. Let $I_j$ equal 1 if event $E_j$ occurs and 0 otherwise. The total payment is $Y=\sum_j b_j I_j$. Taking expectations and using linearity of expectation gives the probability-weighted sum.

Y = \sum_{j=1}^{n} b_j I_j \qquad\Longrightarrow\qquad E[Y] = \sum_{j=1}^{n} b_j E[I_j] = \sum_{j=1}^{n} b_j P(E_j)

This argument does not require independence. In the common case where exactly one of several mutually exclusive outcomes occurs, the formula is especially easy to interpret: each possible payment contributes its amount multiplied by the chance of the outcome that triggers it. If an outcome can trigger multiple payments, include each payment in $Y$; the same linearity argument still applies.

Timing: Expected Amount Is Not Always Present Value

An expected dollar amount and its present value are not interchangeable. If $i$ is the effective annual interest rate and a payment is made at time $t$ years after the valuation date, its discount factor is $v^t=(1+i)^{-t}$. Thus a payment $b_j$ made at time $t_j$ contributes $P(E_j)b_jv^{t_j}$ to expected present value at the start of the year.

E[\text{present value at time }0] = \sum_{j=1}^{n} P(E_j)b_jv^{t_j}, \qquad v=\frac{1}{1+i}

This calculation assumes a deterministic effective annual interest rate and that $t_j$ is measured in years from the valuation date. If all payments are made at the end of the year, use $t_j=1$. A payment made at midyear uses $t_j=0.5$, not $t_j=1$. If the task asks only for the expected undiscounted outgo during the year, do not discount it; state clearly which quantity is being calculated.

Define the starting group and period.
For example, take one policy in force at the beginning of the policy-year and specify that event probabilities are conditional on that status.

List the outcomes, probabilities, amounts, and dates.
Check that probabilities for a complete set of mutually exclusive outcomes sum to one.

Weight each payment by its event probability.
For present value, also apply the discount factor for the payment date.

Aggregate by date and check the result.
Confirm that no event payment was omitted or counted under an outcome that does not trigger it.

Worked Applications

Worked Example: Year-End Claim and Surrender Outgo

A hypothetical insurer models one policy in force at the start of a policy-year. The mutually exclusive outcomes are: a claim with probability 0.020, lapse with probability 0.080, and survival without either event with probability 0.900. A claim pays $50,000 at year-end; lapse pays $4,000 at year-end; survival pays nothing. Calculate the expected undiscounted year-end outgo and its present value at the start of the year using an effective annual interest rate of 4%.

Method. Weight each year-end payment by the probability of the event that triggers it. First check completeness: $0.020+0.080+0.900=1.000$.

The expected claim payment is $0.020\times \$50{,}000=\$1{,}000$. The expected lapse payment is $0.080\times \$4{,}000=\$320$. The survival outcome contributes $\$0$.

E[\text{year-end outgo}] = \$1{,}000+\$320 = \$1{,}320

Both payments are made at time 1, so the present value is $v$ times the expected year-end outgo. At $i=0.04$, $v=1/1.04=0.96153846$.

E[\text{present value}] = \$1{,}320\left(\frac{1}{1.04}\right) = \$1{,}269.2308 \approx \boxed{\$1{,}269.23}

The check is that the present value is less than the year-end amount because the interest rate is positive. A common mistake is to multiply the benefit by the probability of avoiding the claim: that would give $0.980\times \$50{,}000=\$49{,}000$, which is not expected claim outgo. The claim is paid when the claim occurs, so its correct weight is 0.020.

Worked Example: Claim and Lapse Payments at Different Times

For a hypothetical policy in force at the start of the year, a claim occurs with probability 0.020 and pays $20,000 at time $t=0.25$. Lapse occurs with probability 0.050 and pays a $3,000 surrender amount at $t=0.5$. With probability 0.930, neither event occurs and no payment is made. Assume an effective annual interest rate of 4%. Calculate the expected undiscounted total payment and the expected present value at the start of the year.

Method. First calculate the probability-weighted amounts without discounting. Then discount each expected amount using its own payment date. The event probabilities sum to $0.020+0.050+0.930=1.000$.

The expected claim payment is $0.020\times \$20{,}000=\$400$. The expected surrender payment is $0.050\times \$3{,}000=\$150$. Therefore the expected undiscounted total is $\$400+\$150=\$550$.

For the present value, the discount factors are $v^{0.25}=1.04^{-0.25}=0.99024274$ and $v^{0.5}=1.04^{-0.5}=0.98058068$. The separate discounted contributions are:

\$400(0.99024274)=\$396.0971, \qquad \$150(0.98058068)=\$147.0871

Adding these gives $ \$396.0971+\$147.0871=\$543.1842$, or $\boxed{\$543.18}$ to the nearest cent. The result is below the undiscounted expected amount of $550, as both payments occur after time 0.

A timing trap is to discount the surrender payment for a full year. That incorrect calculation gives $\$400v^{0.25}+\$150v=\$396.0971+\$144.2308=\$540.3279$. The correction is to use the stated midyear date: the surrender contribution is $\$150v^{0.5}=\$147.0871$, not $\$150v$.

Worked Example: Several Outgo Types and Two Payment Dates

A hypothetical long-term policy is in force at the beginning of the year. The mutually exclusive outcomes are death with probability 0.015, disability claim with probability 0.025, lapse with probability 0.100, and persistence without these events with probability 0.860. Death pays $40,000 at year-end; a disability claim pays $8,000 at year-end; lapse pays a $600 refund at midyear; persistence pays nothing. At an effective annual interest rate of 4%, calculate the expected payment by date, the total undiscounted expected outgo, and its present value at the start of the year.

Method. Group the event-weighted payments by date. The probabilities are complete because $0.015+0.025+0.100+0.860=1.000$.

At year-end, the expected death payment is $0.015\times \$40{,}000=\$600$, and the expected disability payment is $0.025\times \$8{,}000=\$200$. The total expected year-end payment is $\$600+\$200=\$800$. At midyear, the expected lapse refund is $0.100\times \$600=\$60$.

The expected undiscounted outgo is $\$800+\$60=\$860$. For present value, the year-end amount is discounted by $v=1/1.04=0.96153846$, and the midyear amount by $v^{0.5}=0.98058068$.

\$800v = \$800(0.96153846)=\$769.2308

\$60v^{0.5}=\$60(0.98058068)=\$58.8348

The total present value is $\$769.2308+\$58.8348=\$828.0656$, or $\boxed{\$828.07}$ to the nearest cent. It is also below the undiscounted expected outgo of $860. The date-by-date calculation avoids treating the midyear refund as if it were paid at year-end.

Common Errors and Checks

The most reliable way to prevent errors is to show the probability, amount, and payment time together for every possible payment. In particular, distinguish the probability of an event from the probability of its complement, and distinguish an expected amount from an expected present value.

Using the wrong probability. A claim payment is weighted by the probability of claim, not by the probability of survival or of no claim.
Mixing conditional and unconditional probabilities. If an event probability is stated conditional on a policy being in force at the start of the year, apply it to that starting in-force policy. Do not multiply by an additional probability of being in force unless the calculation explicitly begins with a broader population.
Forgetting an event-contingent payment. A lapse refund belongs to the lapse outcome, not to every policy that survives or to every policy in the starting group.
Using one payment date for all cash flows. A claim at year-end and a surrender payment at midyear have different discount factors, even if both occur in the same policy-year.
Discounting when the requested measure is undiscounted. State whether the result is a nominal expected payment during the year or a present value at the beginning of the year.

Key takeaway: For each event-contingent payment, identify the event, its probability, the payment amount, and the payment date. Weight by the event probability first; if a present value is required, discount each weighted payment using its own date.

Check Your Understanding

In each problem, probabilities are conditional on a hypothetical policy being in force at the beginning of the year. Give monetary answers to the nearest cent where needed.

A year-end claim of $60,000 occurs with probability 0.012; lapse with probability 0.080 produces a year-end payment of $500; the remaining outcome has no payment. At an effective annual rate of 5%, find the expected undiscounted outgo and its time-0 present value. (Answer: expected outgo $760; present value $723.81)
A $25,000 claim is paid at year-end with probability 0.020. A $2,000 surrender payment is made at midyear with probability 0.060. Otherwise no payment is made. At an effective annual rate of 4%, find the expected present value at time 0. (Answer: $598.44)
A $100,000 death benefit is paid at year-end with probability 0.010, and a $2,500 lapse refund is paid at midyear with probability 0.040; otherwise no payment is made. At an effective annual rate of 5%, calculate the two expected payments by date and their combined present value. (Answer: year-end expected payment $1,000; midyear expected payment $100; present value $1,049.97)
A policy has a claim probability of 0.030 with a $12,000 payment at $t=0.25$, and a lapse probability of 0.020 with a $4,000 payment at $t=0.75$. The remaining probability produces no payment. At an effective annual rate of 4%, calculate expected undiscounted outgo and expected present value. (Answer: expected outgo $440; present value $434.17)
Death with probability 0.012 pays $75,000 at year-end; disability with probability 0.018 pays $10,000 at year-end; lapse with probability 0.080 pays a $900 refund at midyear; otherwise there is no payment. At an effective annual rate of 4%, calculate the probability of the no-payment outcome and the time-0 present value of expected outgo. (Answer: no-payment probability 0.890; present value $1,109.06)

What You'll Learn