Present Values of Cause-Contingent Lump Sums

Valuing a Lump Sum Triggered by a Particular Cause

A member of a decrement group may leave because of one of several competing causes, such as death, retirement or disability. If a benefit is payable only when a specified cause occurs, its present value depends on more than the probability of that cause in a single year. It also depends on the probability that the member is still in the group at the start of that year and on the time at which the payment is made.

This tutorial uses a specific timing convention: a lump sum is paid at the end of the year in which the exit occurs. Probabilities are conditional on being in the decrement group at the start of the relevant year. This convention gives a direct formula using annual multiple-decrement probabilities. It does not value a payment made at the instant of exit.

Definition: Suppose the possible causes of exit are indexed by $j$. Let $q_{x+k}^{(j)}$ be the probability that a member who is in the group at the start of year $k+1$ exits during that year because of cause $j$. Let $p_{x+k}$ be the probability of remaining in the group for that year, so that $p_{x+k}=1-\sum_j q_{x+k}^{(j)}$ when the causes are exhaustive and mutually exclusive. The probability of reaching the start of year $k+1$ is ${}_kp_x^{(\tau)}=\prod_{r=0}^{k-1}p_{x+r}$, with ${}_0p_x^{(\tau)}=1$. The superscript $(\tau)$ indicates survival against all causes of decrement.

The cause-specific probabilities are conditional probabilities for those exposed to the risk in that year. They are not, by themselves, probabilities measured across all members present at the beginning of the full projection. To obtain the latter, multiply by the probability of surviving all competing decrements in earlier years.

Present Value Formula Under the Year-End Convention

Assume a deterministic effective annual interest rate $i$, with $v=1/(1+i)$. Let $B_{k,j}$ be the lump sum paid at the end of year $k$ if the member exits in that year due to cause $j$. For a member initially in the group, the probability of that payment being triggered is ${}_{k-1}p_x^{(\tau)}q_{x+k-1}^{(j)}$. The payment occurs at time $k$, so its discount factor is $v^k$.

\operatorname{APV} = \sum_{k=1}^{n}\sum_j {}_{k-1}p_x^{(\tau)} q_{x+k-1}^{(j)} B_{k,j}v^k

If a benefit is payable only for one specified cause $c$, the sum over causes disappears. If the benefit amount is constant, $B_{k,c}=B$, it can be taken outside the sum. When the cause probabilities vary by year, calculate survival to each year using that year's total probability of remaining in the group; do not assume one constant survival probability unless the data support that assumption.

Assumptions and scope: The formula assumes mutually exclusive causes, probabilities conditional on being in the group at each year's start, a known effective annual interest rate, and benefits paid at year-end. It values the expected present value of the specified lump sums only. It does not include premiums, expenses, annuities or payments at the instant of exit.

Why the Formula Works

For each year $k$ and cause $j$, define the event $E_{k,j}$ that the member survives all decrements through the first $k-1$ years and then exits in year $k$ because of cause $j$. Conditional probability gives \[ P(E_{k,j})={}_{k-1}p_x^{(\tau)}q_{x+k-1}^{(j)}. \] The present value contributed by this event is $B_{k,j}v^k$; if the event does not occur, this particular payment contributes zero. Therefore its expected present value is its event probability multiplied by its discounted amount. Summing these expected values gives the displayed formula. The events for different years and causes are disjoint, so this sum neither omits nor double-counts a payment.

Find survival to each year.
Start with probability 1 at time 0. Multiply the probabilities of remaining in the group in each completed year.

Find the cause-specific payment probability.
Multiply survival to the start of the year by the conditional probability of the specified cause during that year.

Discount and add.
Multiply each expected payment by $v^k$ for a payment at the end of year $k$, then sum over the relevant years.

Worked Applications

The examples treat all scenarios and data as hypothetical. Unless otherwise stated, the cause probabilities in a year sum to the probability of some exit during that year, and the remainder is the probability of staying in the group.

Worked Example: A Two-Year Benefit for One Cause

A hypothetical group has two causes of exit, A and B. In year 1, $q_x^{(A)}=0.10$ and $q_x^{(B)}=0.05$. In year 2, conditional on reaching that year, $q_{x+1}^{(A)}=0.08$ and $q_{x+1}^{(B)}=0.12$. A benefit of $20,000 is paid at year-end for a B exit in either year. The effective annual interest rate is $i=5\%$. Find the APV per member initially in the group.

Method. Value the B benefit year by year. The year-1 probability of a B exit is $0.05$, since every initial member is exposed at the start of year 1. The probability of remaining after year 1 is \[ p_x=1-0.10-0.05=0.85. \] Thus the probability of a B exit in year 2, measured across the initial group, is $0.85(0.12)=0.102$.

The discount factor is $v=1/1.05$, so $v^2=1/1.1025$. The year-1 present value contribution is \[ 0.05(20{,}000)v=\frac{1{,}000}{1.05}=\$952.38095\ldots. \] The year-2 contribution is \[ 0.85(0.12)(20{,}000)v^2 =\frac{2{,}040}{1.1025} =\$1{,}850.34014\ldots. \] Adding the contributions gives $952.38095\ldots+1{,}850.34014\ldots=\$2{,}802.72109\ldots$. Rounded to the nearest cent, the APV is $2,802.72.

Check. The probabilities of a B exit across the initial group are $0.05$ in year 1 and $0.102$ in year 2. Their sum, $0.152$, is less than 1, as it should be; some members exit by A or remain beyond year 2. The discounting is also consistent with the payment dates: one year and two years after the valuation date.

Common mistake avoided: $0.12$ is conditional on reaching the start of year 2, not a probability for every initial member. Using $0.12$ directly for year 2 would overstate that year's contribution.

Worked Example: A Changing Benefit Amount by Year

A hypothetical product pays a lump sum only for cause C, at the end of the year of exit. The effective annual interest rate is $i=25\%$. In year 1 the cause probabilities are $q^{(C)}=0.04$ and $q^{(D)}=0.11$. In year 2 they are $0.06$ for C and $0.14$ for D. In year 3 they are $0.10$ for C and $0.15$ for D. The C benefit is $10,000 in year 1, $12,000 in year 2 and $15,000 in year 3. Find the APV per initial member.

Method. First find the annual probabilities of remaining in the group: $p_1=1-0.04-0.11=0.85$, $p_2=1-0.06-0.14=0.80$, and $p_3=1-0.10-0.15=0.75$. Survival to the start of years 1, 2 and 3 is therefore $1$, $0.85$ and $0.85(0.80)=0.68$, respectively. Since $i=25\%$, $v=0.80$.

For year 1, the contribution is $1(0.04)(10{,}000)(0.80)=\$320.00$. For year 2, it is $0.85(0.06)(12{,}000)(0.80)^2=\$391.68$. For year 3, it is $0.68(0.10)(15{,}000)(0.80)^3=\$522.24$. The total is $320.00+391.68+522.24=\$1{,}233.92$.

Check. The probability of a year-3 C exit across the original group is $0.68(0.10)=0.068$. Its undiscounted expected payment is therefore $0.068(15{,}000)=\$1{,}020$, and discounting three years gives $1{,}020(0.80)^3=\$522.24$, matching the year-3 calculation.

Common mistake avoided: Applying the year-2 survival probability of $0.85$ to the year-3 benefit would ignore exits in year 2. The correct year-3 survival probability includes both completed years: $0.85(0.80)=0.68$.

Worked Example: Valuing Benefits From Decrement Counts

A hypothetical decrement study begins with 1,000 members. In year 1, 50 exit by cause R and 50 by other causes, leaving 900. In year 2, among those 900, 90 exit by R and 90 by other causes, leaving 720. In year 3, among those 720, 72 exit by R. A $10,000 benefit is paid at year-end for each R exit. The effective annual interest rate is $i=5\%$. Find the APV per initial member for R benefits paid in the first three years.

Method. The cause-specific probabilities conditional on reaching each year are $50/1{,}000=0.05$, $90/900=0.10$, and $72/720=0.10$. The survival probabilities to the starts of years 1, 2 and 3 are $1$, $900/1{,}000=0.90$, and $720/1{,}000=0.72$. Equivalently, the year-2 survival probability is $1-0.05-0.05=0.90$, and the year-3 survival probability is $0.90(1-0.10-0.10)=0.72$.

For year 1, the probability of an R exit per initial member is $50/1{,}000=0.05$, giving a discounted contribution of $0.05(10{,}000)/1.05=\$476.19048\ldots$. For year 2, that probability is $90/1{,}000=0.09$, giving $0.09(10{,}000)/(1.05)^2=\$816.32653\ldots$. For year 3, it is $72/1{,}000=0.072$, giving $0.072(10{,}000)/(1.05)^3=\$621.96307\ldots$. The total is $\$1{,}914.48008\ldots$, or $1,914.48 per initial member to the nearest cent.

Check. The expected numbers of R exits in the three years per initial member are $0.05$, $0.09$ and $0.072$. These match the count-based proportions $50/1{,}000$, $90/1{,}000$ and $72/1{,}000$. Multiplying each proportion by the benefit and the appropriate discount factor reproduces the APV above.

Common mistake avoided: The year-2 R probability is $90/900$ conditional on reaching year 2, but the probability relevant to an initial member is $90/1{,}000$. Either use the conditional probability with survival, $0.90(0.10)$, or use the initial-cohort proportion directly; do not use both adjustments.

Exam Traps and Practical Checks

The most frequent error is to treat a conditional cause probability as though it applied to the original group in every year. For the first worked example, the incorrect year-2 contribution would be \[ 0.12(20{,}000)v^2=\frac{2{,}400}{1.1025}=\$2{,}176.87\text{ (approximately)}, \] instead of the correct contribution \[ 0.85(0.12)(20{,}000)v^2=\$1{,}850.34\text{ (approximately)}. \] The missing factor $0.85$ fails to account for members who left by another cause in year 1.

A second trap is to use $v^{k-1}$ for a payment at the end of year $k$. The end of year 1 is one year after the valuation date, so its discount factor is $v$, not 1. A third is to discount a year-3 payment by only two years. Always label the payment time before assigning its discount factor.

A useful check is to compare two ways of finding the unconditional probability of a cause-specific exit: multiply survival to the start of the year by the conditional cause probability, or divide the cause-specific exit count for that year by the original cohort size. When both are available, they should agree. Also confirm that annual probabilities for all causes do not exceed 1 and that survival decreases or stays constant over time.

These calculations are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. The central discipline is to keep three items distinct: conditional cause probability, survival to the relevant year and discounting to the payment date.

Check Your Understanding

For each hypothetical situation, use the year-end payment convention. Round dollar answers to the nearest cent.

A $5,000 lump sum is paid at the end of year 1 for cause M. The year-1 probability of M is $0.07$, and the effective annual interest rate is $4\%$. Find the APV per initial member. (Answer: $336.54)
A $10,000 benefit is paid for cause N at the end of either of two years. In year 1, $q^{(N)}=0.08$ and the probability of another cause is $0.12$. In year 2, conditional on reaching that year, $q^{(N)}=0.10$. If $i=5\%$, find the APV. (Answer: $1,487.53)
A hypothetical group has $i=10\%$. The probability of cause P is $0.05$ in year 1, when the probability of another cause is $0.05$. In year 2, the cause-P probability is $0.08$, conditional on reaching that year. A $12,000 benefit is payable for P in either year. Find the APV. (Answer: $1,168.60)
Of 500 initial members, 25 exit by cause Q in year 1 and 75 exit by other causes. Of the 400 remaining members, 40 exit by Q in year 2. A $6,000 benefit is paid at year-end for Q, and $i=5\%$. Find the two-year APV per initial member. (Answer: $721.09)
For a three-year projection, $i=25\%$. In year 1, the specified cause has probability $0.02$ and another cause has probability $0.08$. In year 2, those probabilities are $0.04$ and $0.16$; in year 3 they are $0.10$ and $0.10$. A $5,000 benefit is paid for the specified cause at year-end. Find the APV. (Answer: $379.52)

What You'll Learn