Integrated Competing-Decrement Table Workshop

Why an Integrated Calculation Matters

A competing-decrement table is often used to answer several linked questions: how many members leave for each cause, how long the group is exposed to risk, what the corresponding single-decrement tables would show, and what a benefit triggered by a particular cause is worth. These quantities use related data, but they are not interchangeable. In particular, an annual decrement probability does not by itself say that every exit occurs at year-end.

This workshop uses a hypothetical group with two causes of exit. The annual probabilities vary by age. We will first calculate actual cohort counts and exposures, then convert the probabilities to associated single-decrement probabilities, and finally value benefits paid at the time of exit. The within-year calculations assume a uniform distribution of decrements (UDD); results differ under other timing assumptions.

Definition: In a multiple-decrement model, a member can leave the group during an age interval for one of several mutually exclusive causes. Write $q_x^{(j)}$ for the probability that a member in the group at age $x$ exits from cause $j$ before age $x+1$, and $l_x$ for the number in the group at age $x$. The total decrement probability is $q_x=\sum_j q_x^{(j)}$, and the probability of remaining in the group through the interval is $p_x=1-q_x$.

A Consistent Set of Age-Specific Data

Consider a hypothetical group exposed to two causes, A and B. At age 50, the probabilities are $q_{50}^{(A)}=0.09$ and $q_{50}^{(B)}=0.19$. At age 51, they are $q_{51}^{(A)}=0.19$ and $q_{51}^{(B)}=0.09$. The causes are mutually exclusive during each interval, so the total decrement probability is the sum of the two cause probabilities. The calculations below start with $l_{50}=1{,}000$.

Under UDD, the expected number of exits during a year is spread uniformly over the year. Thus the expected time at risk during that year is the starting number minus half the expected exits. This is an expected exposure calculation, not an assertion that a fractional person literally exits.

Worked Example: Exits and Exposures at Two Ages

Using the hypothetical data above, calculate cause-specific exits, survivors, and expected exposure at ages 50 and 51. State the expected exposure in person-years.

Method. For each interval, multiply the number at risk at its start by each cause probability to obtain expected exits. Add the exits to obtain total decrements. Under UDD, subtract half the total decrements from the starting number to obtain exposure.

Age 50. The expected exits are $d_{50}^{(A)}=1{,}000(0.09)=90$ and $d_{50}^{(B)}=1{,}000(0.19)=190$. Total decrements are $90+190=280$, so $l_{51}=1{,}000-280=720$. Expected exposure is $1{,}000-\tfrac12(280)=860$ person-years.

Age 51. There are 720 expected members at the start of the interval. Therefore, $d_{51}^{(A)}=720(0.19)=136.8$ and $d_{51}^{(B)}=720(0.09)=64.8$. Total decrements are $136.8+64.8=201.6$, and $l_{52}=720-201.6=518.4$. Expected exposure is $720-\tfrac12(201.6)=619.2$ person-years.

Check. The survival proportions are $0.72$ in both years. Thus $1{,}000(0.72)(0.72)=518.4$, matching the age-52 cohort count. The exits at each age also sum to the total decrement count used in the exposure calculation. A common mistake is to use 1,000 person-years at age 50, as if all members remained exposed for the full year; UDD gives 860, because expected exits reduce time at risk.

Result: Under UDD, if $d_x=\sum_j d_x^{(j)}$ expected exits occur from a group of size $l_x$ during the year, expected exposure is $E_x=l_x-\tfrac12d_x$. This formula assumes exits are spread uniformly over the interval; it is not the appropriate exposure formula under an arbitrary within-year exit pattern.

Associated Single-Decrement Tables

An associated single-decrement table isolates one cause: it describes what the probability of decrement would be if that cause acted alone, on the specified conversion basis. Its probability $q_x'^{(j)}$ is therefore not generally equal to the observed competing-decrement probability $q_x^{(j)}$. When other causes are present, some members who could have exited from cause $j$ instead leave for another cause.

For two causes under the UDD associated-table conversion, the observed probabilities and associated probabilities satisfy $q_x^{(A)}=q_x'^{(A)}(1-\tfrac12q_x'^{(B)})$ and $q_x^{(B)}=q_x'^{(B)}(1-\tfrac12q_x'^{(A)})$. The adjustment accounts for overlap in exposure to the two causes within the year. Each associated table is calculated separately; its survivor count should not be mistaken for the actual number remaining after all causes operate together.

Worked Example: Converting Both Ages to Associated Tables

Convert the age-50 and age-51 probabilities in the hypothetical group to associated single-decrement probabilities. Then construct the two associated tables from a common starting value of 1,000 at age 50.

Age 50 conversion. Test $q_{50}'^{(A)}=0.10$ and $q_{50}'^{(B)}=0.20$ in the two-cause relations: $0.10(1-\tfrac12(0.20))=0.10(0.90)=0.09$, and $0.20(1-\tfrac12(0.10))=0.20(0.95)=0.19$. Both reproduce the observed probabilities, so these are the associated probabilities at age 50.

Age 51 conversion. Test $q_{51}'^{(A)}=0.20$ and $q_{51}'^{(B)}=0.10$: $0.20(1-\tfrac12(0.10))=0.20(0.95)=0.19$, and $0.10(1-\tfrac12(0.20))=0.10(0.90)=0.09$. These reproduce the age-51 observations.

Cause A table. Starting with $l_{50}'^{(A)}=1{,}000$, the age-50 survivor count is $1{,}000(1-0.10)=900$. At age 51, it becomes $900(1-0.20)=720$. Thus the successive values are $1{,}000,\ 900,\ 720$.

Cause B table. Starting with $l_{50}'^{(B)}=1{,}000$, the age-50 survivor count is $1{,}000(1-0.20)=800$. At age 51, it becomes $800(1-0.10)=720$. Its successive values are $1{,}000,\ 800,\ 720$.

Check and interpretation. The two associated tables each end at 720, but the actual competing-decrement cohort ends at 518.4. The associated tables each remove only one cause, whereas the actual cohort experiences both. Confusing these terminal values is a common error. The conversion is checked by substituting the associated probabilities back into the two equations, not by requiring an associated table to reproduce actual combined survival.

Discounting Benefits Paid at the Time of Exit

If a benefit is paid immediately when a decrement occurs, its payment time is fractional within the year. Under UDD, conditional on a specified cause occurring in the interval, its exit time is uniform over that interval. For an effective annual interest rate $i$, write $v=(1+i)^{-1}$ and $\delta=\ln(1+i)$. The discount factor at time $t$ years is $v^t=e^{-\delta t}$.

The expected discount factor for a payment made at a uniform time $T$ in a one-year interval follows by integrating across the year:

E[v^T]=\int_0^1 e^{-\delta t}\,dt =\frac{1-e^{-\delta}}{\delta} =\frac{1-v}{\delta}.

Consequently, if $l_x$ members are at risk, cause $j$ has probability $q_x^{(j)}$, and its benefit paid at exit is $B_j$, the expected present value for that interval is $l_xq_x^{(j)}B_j(1-v)/\delta$. For a later interval beginning $k$ years after the valuation date, multiply its interval value by $v^k$. This result uses UDD for exit timing; it does not apply unchanged if benefits are paid at year-end or if the within-year decrement pattern is different.

Worked Example: Present Value of Two Cause-Contingent Benefits

For the hypothetical group, suppose Cause A pays \$12,000 immediately at exit and Cause B pays \$8,000 immediately at exit. Assume an effective annual interest rate of 5% and UDD within each year. Calculate the expected present value for the initial group of 1,000 members over ages 50 to 52, and the value per initial member.

Method. Calculate the cause-specific expected benefit for each interval, apply the expected fractional-year discount factor, and discount the second interval back one additional year.

The discount quantities are $v=1/1.05=0.95238095$, $\delta=\ln(1.05)=0.04879016$, and $c=(1-v)/\delta=0.97599687$. Here $c$ is the expected discount factor within a year under UDD.

Age 50 interval. The undiscounted expected benefits are $90(\$12{,}000)=\$1{,}080{,}000$ for Cause A and $190(\$8{,}000)=\$1{,}520{,}000$ for Cause B. Their sum is \$2,600,000. The present value is $\$2{,}600{,}000c=\$2{,}537{,}591.87$.

Age 51 interval. The undiscounted expected benefits are $136.8(\$12{,}000)=\$1{,}641{,}600$ for Cause A and $64.8(\$8{,}000)=\$518{,}400$ for Cause B, totaling \$2,160,000 at the start of age 51. The present value at age 50 is $\$2{,}160{,}000vc=\$2{,}007{,}764.99$.

Total and check. The group present value is $\$2{,}537{,}591.87+\$2{,}007{,}764.99=\$4{,}545{,}356.86$. Dividing by 1,000 gives $\$4{,}545.36$ per initial member, rounded to the nearest cent. As a check, per initial member the undiscounted cause-weighted amounts are $2{,}600$ in the first year and $0.72(3{,}000)=2{,}160$ at the start of the second year; applying $c$ and $vc$, respectively, gives the same result. A common mistake is to discount every exit at year-end. That would use $v$ for first-year exits, rather than the UDD factor $c$, and would value a different payment timing.

Practical Checks and Limits

A sound solution keeps the layers of the calculation distinct. Observed competing-decrement probabilities determine actual exits and cohort survival. Associated probabilities describe separate single-cause tables on the stated conversion basis. UDD supplies the fractional-year timing needed for expected exposure and for benefits paid at exit. The interest assumption supplies the discounting.

Calculate actual exits.
Multiply each cause probability by the number at risk at the start of the age interval.

Reconcile survival and exposure.
Subtract all cause exits to find survivors; under UDD, subtract half the total exits from the starting group for exposure.

Convert and build associated tables.
Use the specified associated-table relation, then apply each cause-specific associated probability to its own table.

Value the benefit with its actual timing.
For exit-time payments under UDD use the within-year integral; for year-end payments use the year-end discount factor instead.

The conversion equations and exposure formula depend on their stated assumptions. If decrement probabilities are based on a different fractional-age assumption, or if decrement causes have a specified nonuniform timing pattern, the UDD exposure and exit-time discount factor should not be applied automatically. Likewise, a payment contingent on one cause must use that cause's exit count or probability, not the total decrement probability.

Key takeaway: Reconcile each age interval before moving on: cause-specific exits must sum to total decrements, survivors must equal starting members minus those decrements, and each present value must use the payment timing specified. Associated single-decrement tables are separate comparisons, not the actual combined cohort.

Check Your Understanding

Work each problem using the assumptions stated. Dollar answers are rounded to the nearest cent where needed.

A group has 800 members at the start of an age interval. Cause A and Cause B probabilities are 0.08 and 0.12. Under UDD, find expected exits by cause, survivors, and exposure in person-years. (Answer: 64 Cause A exits, 96 Cause B exits, 640 survivors, and 720 person-years of exposure.)
For two causes under the UDD associated-table conversion, the observed probabilities are $q_x^{(A)}=0.17$ and $q_x^{(B)}=0.27$. Verify associated probabilities of 0.20 and 0.30 in the order A, B. (Answer: $q_x'^{(A)}=0.20,\ q_x'^{(B)}=0.30$, since $0.20(1-0.30/2)=0.17$ and $0.30(1-0.20/2)=0.27$.)
A one-year cause-specific exit probability is 0.06. A benefit of \$5,000 is paid immediately at exit. At an effective annual interest rate of 4%, find the expected present value per member at risk, assuming UDD and no other benefit. (Answer: $\$5{,}000(0.06)(1-v)/\ln(1.04)=\$294.19$.)
A hypothetical group has 500 members at the start of year 1. In year 1, the probabilities of Causes A and B are 0.10 and 0.10. In year 2, conditional on reaching that year, they are 0.20 and 0.10. Cause A pays \$1,000 at the end of the year of exit, and Cause B pays nothing. At an effective annual rate of 5%, find the expected present value of Cause A benefits for the group. (Answer: year-1 A exits are 50 and year-2 A exits are $400(0.20)=80$; the value is $1{,}000(50v+80v^2)=\$120{,}181.41$.)
In a two-cause UDD conversion, associated probabilities are $q_x'^{(A)}=0.15$ and $q_x'^{(B)}=0.20$. Calculate the observed competing-decrement probabilities. (Answer: $q_x^{(A)}=0.15(1-0.20/2)=0.135$ and $q_x^{(B)}=0.20(1-0.15/2)=0.185$.)

What You'll Learn