Building Pension Values from Multiple-Decrement Tables

Why Decrement Tables Must Be Converted Before Valuation

A multiple-decrement table records exits from active membership by cause. Its entries are not, by themselves, probabilities that a person starting at the beginning of the valuation period will receive a particular benefit in a later year. To obtain those probabilities, first account for the chance that the member remains active long enough to reach that year. Then attach the appropriate benefit to each cause and discount it according to when it is paid.

This tutorial develops that conversion and uses it to construct an expected present value (EPV) of hypothetical pension and exit benefits. The key discipline is to keep three quantities distinct: the probability of being active at a year’s start, the probability of a particular exit during that year conditional on being active, and the probability of that exit measured from the initial valuation date.

Definition: In a multiple-decrement table, let $q_{j,k}$ be the probability that an active member exits during year $k+1$ because of cause $j$, conditional on being active at the start of that year. The causes are mutually exclusive, so the probability of remaining active through that year is $p_k=1-\sum_j q_{j,k}$. If $l_k$ members are active at the start of the year, the number exiting from cause $j$ is $d_{j,k}=l_kq_{j,k}$, and the number active at the next year’s start is $l_{k+1}=l_kp_k$.

The starting point may be a table of conditional rates $q_{j,k}$, or a table of active lives and decrement counts $l_k$ and $d_{j,k}$. With counts, the conditional rate is $q_{j,k}=d_{j,k}/l_k$, provided the decrement counts and active-member exposure refer to the same year and population. The relative frequency $d_{j,k}/l_0$, by contrast, is the probability of that exit in year $k+1$ for a member drawn from the initial group, when the table represents the relevant assumed population.

From Conditional Rates to Cause-Specific Weights

Suppose the member is active at time 0. The probability of remaining active until the start of year $k+1$ is the product of the annual active-survival probabilities for all preceding years. Multiplying that probability by the conditional rate for cause $j$ gives the probability of exiting from cause $j$ in year $k+1$, measured from time 0.

{}_kp^{\mathrm{active}}=\prod_{r=0}^{k-1}\left(1-\sum_j q_{j,r}\right), \qquad w_{j,k}={}_kp^{\mathrm{active}}q_{j,k}.

Here $w_{j,k}$ is the cause-specific payment weight for exits during year $k+1$. If a cause-specific benefit $B_{j,k}$ is paid at the end of that year, and the effective annual interest rate is $i$, its contribution to the EPV is $v^{k+1}w_{j,k}B_{j,k}$, where $v=1/(1+i)$. Summing across causes and years gives the EPV of such end-of-year payments.

\operatorname{EPV} = \sum_{k}\sum_j v^{k+1}w_{j,k}B_{j,k} = \sum_{k}\sum_j v^{k+1}{}_kp^{\mathrm{active}}q_{j,k}B_{j,k}.

This expression assumes that the listed payments are made at the end of the year of exit. If a benefit is payable at another time, its discount factor must reflect that timing. For a retirement pension valued as an annuity at the retirement date, replace the retirement lump sum $B_{R,k}$ with the pension’s value at that date, and then discount that value back to time 0.

Timing condition: If retirement occurs at the end of year $k+1$ and an $n$-year certain annuity-due begins immediately at retirement, with annual amount $C_k$, its value at retirement is $C_k\ddot{a}_{\overline{n}|}$. Its time-0 EPV contribution is $v^{k+1}w_{R,k}C_k\ddot{a}_{\overline{n}|}$. This assumes the annuity payments are certain for $n$ years and the same effective annual rate $i$ is used for valuation.

Why the Weights Have This Form

The cause-specific weight follows directly from conditional probability. To exit from cause $j$ in year $k+1$, a member must first have avoided every decrement in the earlier years and then experience cause $j$ during the year in question. Those are successive conditions, so their probabilities multiply.

Survive the earlier years as an active member.
In year $r+1$, the conditional probability of staying active is $1-\sum_j q_{j,r}$. The probability of reaching the start of year $k+1$ is the product of these quantities for $r=0,\ldots,k-1$.

Apply the current year’s cause rate.
Conditional on reaching the start of year $k+1$, the probability of cause $j$ during that year is $q_{j,k}$. Multiplying gives $w_{j,k}={}_kp^{\mathrm{active}}q_{j,k}$.

Attach and discount the benefit.
Multiply the cause-specific probability by the benefit payable under that cause. Discount from the actual payment date, not automatically from the start or midpoint of the year.

A useful check is that, for each year, the weights for all causes sum to the probability of leaving active membership during that year. They must not sum to more than the probability of reaching the year. If every possible exit cause is included, the active-survival probability plus all cumulative cause-of-exit probabilities through a given year should equal 1.

Worked Valuations

Worked Example: Conditional Rates and a Retirement Pension

A hypothetical plan begins with one active member at time 0. During year 1, the conditional probabilities of retirement, death, and withdrawal are $0.10$, $0.02$, and $0.08$. During year 2, conditional on being active at its start, they are $0.20$, $0.03$, and $0.07$. Assume a 5% effective annual interest rate. Retirement at each year-end produces a three-year certain annuity-due of \$900 per year, beginning immediately. Death pays \$5,000 and withdrawal pays \$2,000, each at the end of its year. Find the EPV at time 0.

Method. First calculate active survival and cause-specific weights. Then value the retirement annuity at each retirement date and discount every cause-specific payment. The common error this method avoids is treating the year-2 rates as if they were probabilities from time 0.

In year 1, the probability of remaining active is $1-0.10-0.02-0.08=0.80$. Thus the year-1 weights for retirement, death, and withdrawal are $0.10$, $0.02$, and $0.08$. In year 2, only $0.80$ of the original member group reaches the start of the year. The corresponding weights from time 0 are $0.80(0.20)=0.16$, $0.80(0.03)=0.024$, and $0.80(0.07)=0.056$. Year-2 active survival is $1-0.20-0.03-0.07=0.70$, so the probability still active at time 2 is $0.80(0.70)=0.56$.

With $i=0.05$, $v=1/1.05=0.95238095$. The pension value at a retirement date is based on three certain payments beginning immediately:

\ddot{a}_{\overline{3}|}=1+v+v^2 =1+0.95238095+0.90702948 =2.85941043, \qquad 900\ddot{a}_{\overline{3}|}=\$2{,}573.47.

The retirement contribution is the retirement-date pension value multiplied by the appropriate retirement weight and discount factor:

\$2{,}573.46939\left(0.10v+0.16v^2\right) = \$2{,}573.46939(0.09523810+0.14512472) = \$618.57.

The death benefits are paid at the respective year-ends:

\$5{,}000\left(0.02v+0.024v^2\right) = \$5{,}000(0.01904762+0.02176871) = \$204.08.

The withdrawal refunds are also paid at year-end:

\$2{,}000\left(0.08v+0.056v^2\right) = \$2{,}000(0.07619048+0.05079365) = \$253.97.

Adding the three components gives an EPV of $\$618.57+\$204.08+\$253.97=\$1{,}076.62$. As a check, total exit weights are $0.20$ in year 1 and $0.24$ in year 2; the remaining probability $0.56$ is active at time 2. These probabilities sum to 1.

Worked Example: Building Weights from Decrement Counts

A hypothetical active-member table has 1,000 members at the start of year 1. In year 1, there are 120 retirements, 30 deaths, and 50 withdrawals. In year 2, there are 120 retirements, 16 deaths, and 64 withdrawals. Assume a 4% effective annual interest rate. Retirement at each year-end produces a two-year certain annuity-due, with annual amount \$1,000 for a year-1 retirement and \$1,100 for a year-2 retirement. Death pays \$4,000 and withdrawal pays \$500 at the relevant year-end. Calculate the EPV per initial member.

Method. Derive the active-member counts and conditional rates from the decrement counts, then use the counts divided by the initial 1,000 members as the cause-specific weights. This avoids using a year-2 count as though it were a probability conditional on the initial group without converting its denominator.

At the start of year 1, $l_0=1{,}000$. The first-year exits total $120+30+50=200$, leaving $l_1=800$. The year-1 conditional rates are $120/1{,}000=0.12$, $30/1{,}000=0.03$, and $50/1{,}000=0.05$. In year 2 the table’s counts are consistent with the 800 active members: the conditional rates are $120/800=0.15$, $16/800=0.02$, and $64/800=0.08$. The second-year exits total 200, leaving 600 active members.

Measured per initial member, year-1 weights are $0.12$, $0.03$, and $0.05$. Year-2 weights are the year-2 counts divided by 1,000: $0.12$, $0.016$, and $0.064$. Equivalently, each is the 0.80 probability of reaching year 2 multiplied by its year-2 conditional rate. With $v=1/1.04=0.96153846$, the two-year annuity-due factor is $1+v=1.96153846$.

The year-1 retirement pension value is $\$1{,}000(1+v)=\$1{,}961.54$, and the year-2 value is $\$1{,}100(1+v)=\$2{,}157.69$. Their time-0 contributions are:

\$1{,}000(1+v)(0.12v) + \$1{,}100(1+v)(0.12v^2) = \$226.33+\$239.39 = \$465.72.

The death-benefit contribution is $\$4{,}000(0.03v+0.016v^2)=\$174.56$. The withdrawal contribution is $\$500(0.05v+0.064v^2)=\$53.62$. Therefore the EPV per initial member is $\$465.72+\$174.56+\$53.62=\$693.90$, rounded to the nearest cent. As a check on the table conversion, the year-2 cause weights sum to $0.12+0.016+0.064=0.20$, exactly the fraction of initial members exiting in year 2.

Worked Example: Exit Counts with Different Benefits by Year

A hypothetical plan has 500 active members at the start of year 1. The year-1 decrement counts are 40 retirements, 10 deaths, and 50 withdrawals. At the start of year 2, the remaining 400 members experience 60 retirements, 8 deaths, and 32 withdrawals. Assume a 4% effective annual interest rate. A retirement pension is a two-year certain annuity-due of \$800 for a year-1 retirement or \$1,000 for a year-2 retirement. Death benefits of \$3,000 in year 1 and \$3,500 in year 2, and withdrawal benefits of \$200 in year 1 and \$250 in year 2, are paid at the end of the year of exit. Find the EPV per initial member.

Method. Convert each exit count to a proportion of the initial 500 members, value the retirement pension at its start date, and discount each year’s benefit. The error avoided is applying the year-2 counts to the initial group without recognizing that only 400 members entered year 2.

The year-1 counts leave $500-40-10-50=400$ active members. Hence year-1 weights per initial member are retirement $40/500=0.08$, death $10/500=0.02$, and withdrawal $50/500=0.10$. Year-2 weights are retirement $60/500=0.12$, death $8/500=0.016$, and withdrawal $32/500=0.064$. These equal the year-2 conditional rates $60/400=0.15$, $8/400=0.02$, and $32/400=0.08$, each multiplied by the probability $400/500=0.80$ of reaching year 2.

At 4%, $v=0.96153846$ and $\ddot{a}_{\overline{2}|}=1+v=1.96153846$. The two retirement contributions are:

\$800(1+v)(0.08v)=\$120.71, \qquad \$1{,}000(1+v)(0.12v^2)=\$217.63.

The death-benefit EPV is $\$3{,}000(0.02v)+\$3{,}500(0.016v^2)=\$57.69+\$51.78=\$109.47$. The withdrawal-benefit EPV is $\$200(0.10v)+\$250(0.064v^2)=\$19.23+\$14.79=\$34.02$. The total, rounded to cents, is $\$120.71+\$217.63+\$109.47+\$34.02=\$481.83$. Finally, year-1 exits account for $0.20$ of the initial group, year-2 exits account for $0.20$, and $0.60$ remain active after year 2, providing a probability check.

Common Mistake: Forgetting Earlier Decrements

A year-$k+1$ decrement rate is conditional on being active at the start of that year. It is not automatically the probability, from time 0, of exiting from that cause in that year. For example, if the probability of reaching year 2 is $0.80$ and the conditional death rate in year 2 is $0.03$, the probability of death in year 2 from the initial group is $0.80(0.03)=0.024$, not $0.03$. The wrong calculation overstates the year-2 death weight by ignoring prior exits.

A second timing trap is to discount a pension as though its first payment were one year after retirement when the stated pension is an annuity-due beginning immediately at retirement. At the retirement date, the correct factor for $n$ certain annual payments is $\ddot{a}_{\overline{n}|}$, not $a_{\overline{n}|}$. Conversely, an end-of-year exit lump sum should be discounted from the end of that year, using $v^{k+1}$.

Key takeaway: For each year, identify the active population at the start, apply the cause-specific rates to that population, and then express the resulting weights relative to the initial group. Attach benefits only after the weights and payment dates are clear.

Check Your Understanding

For each problem, convert the table information into the requested survival or payment weight before applying any pension value or discount factor.

A plan has 250 active members at time 0. In year 1, $q_R=0.08$, $q_D=0.02$, and $q_W=0.10$. Find the number expected to remain active and the number expected to die during the year. (Answer: 200 remain active; 5 deaths.)
For a member active at time 0, year-1 conditional rates are retirement $0.10$, death $0.02$, and withdrawal $0.08$. Year-2 conditional rates are retirement $0.20$, death $0.03$, and withdrawal $0.05$. Find the probability, from time 0, of death during year 2. (Answer: $0.80(0.03)=0.0240$.)
At 5% effective annual interest, retirement at the end of year 1 has probability $0.10$, and retirement at the end of year 2 has probability $0.16$, both measured from time 0. Each retirement produces a two-year certain annuity-due of \$1,000 per year. Find its EPV. (Answer: $\$471.48$.)
A group has 400 initial active members. There are 32 withdrawals in year 1 and 24 in year 2. Assume the table’s year-2 withdrawal count is among 300 members active at the start of that year. At 5% effective annual interest, year-end withdrawal benefits are \$500 in year 1 and \$700 in year 2. Find the EPV per initial member. (Answer: $\$76.19$.)
At 5% effective annual interest, year-1 rates for retirement, death, and withdrawal are $0.10$, $0.02$, and $0.08$; year-2 rates are $0.20$, $0.03$, and $0.07$, conditional on being active at the start of year 2. Retirement at each year-end produces a two-year certain annuity-due of \$500 per year. Death pays \$6,000 and withdrawal pays \$1,000 at the year-end. Find the total EPV per initial member. (Answer: $\$606.52$.)

What You'll Learn