Annuities Paid While Remaining in the Decrement Group

Introduction

A periodic benefit may be payable only as long as a person remains in a specified group: for example, an active employee who has not retired, withdrawn, or become disabled. The present value of that benefit depends on the chance of still belonging to the group on each payment date. It does not depend on whether the person avoided just one particular exit cause.

The central calculation is therefore a sum of discounted scheduled payments, each weighted by the probability of remaining in the group until its payment date. The timing convention matters: an annuity-immediate pays at the end of each period, whereas an annuity-due pays at the beginning. Neither type includes a payment triggered by leaving the group.

Definition: Suppose a member is initially in a group at age $x$, and exits occur through mutually exclusive decrement causes. Let ${}_tp_x^{(\tau)}$ denote the probability that the member remains in the group for at least $t$ years, meaning that no decrement cause has removed the member by time $t$. An annuity payable while the member remains in the group pays on its specified dates only if the member is still in the group at those dates.

Group Survival and Scheduled Payments

For a one-year multiple-decrement table with $r$ exhaustive exit causes, let $q_{x}^{(j)}$ be the probability of exit during the year from cause $j$, conditional on being in the group at the start of the year. Because the causes are mutually exclusive, the probability of remaining in the group for the year is

p_x^{(\tau)}=1-\sum_{j=1}^{r}q_x^{(j)}.

Across years, group survival is obtained by multiplying the conditional one-year probabilities. If the one-year group survival probabilities are $p_x^{(\tau)},p_{x+1}^{(\tau)},\ldots$, then

{}_kp_x^{(\tau)} =\prod_{h=0}^{k-1}p_{x+h}^{(\tau)}.

This product allows the decrement probabilities to vary by year. It is the probability needed for a scheduled payment at time $k$. A cause-specific probability of avoiding one exit cause is generally not a substitute: a member who has left through any other cause is also no longer eligible for the annuity.

Under annual effective interest $i$, let $v=(1+i)^{-1}$. If a fixed amount $B$ is paid at the end of each year for $n$ years, conditional on membership at each payment date, its expected present value is

B\sum_{k=1}^{n}v^k\,{}_kp_x^{(\tau)}.

For payments at the beginnings of years $0,1,\ldots,n-1$, the corresponding value is

B\sum_{k=0}^{n-1}v^k\,{}_kp_x^{(\tau)}, \qquad {}_0p_x^{(\tau)}=1.

These expressions use the survival probability at the payment date, not merely the probability at the start of the interval. They assume the payment amount is fixed and that the annuity ends after the stated number of years. For a different payment schedule or a different amount at each date, value each scheduled payment separately.

Why the Survival-Weighted Sum Works

Let $I_k$ equal 1 if the member is in the group at payment time $k$, and 0 otherwise. For a payment $B$ at that date, the present value is $Bv^kI_k$. The expected present value is the sum of the expectations of these terms. Since $E[I_k]=P(I_k=1)={}_kp_x^{(\tau)}$, the expected value of the payment at time $k$ is $Bv^k{}_kp_x^{(\tau)}$. Summing over the payment dates gives the annuity formula.

E\!\left[\sum_{k=1}^{n}Bv^kI_k\right] =\sum_{k=1}^{n}Bv^kE[I_k] =B\sum_{k=1}^{n}v^k{}_kp_x^{(\tau)}.

This argument uses only the probability of being eligible at each scheduled date and the linearity of expectation. It does not require the eligibility indicators at different dates to be independent; in fact, they are generally dependent because remaining in the group at a later date requires remaining in it at earlier dates.

Worked Example: Annual Payments in Arrears

A hypothetical group has three possible exit causes. In years 1, 2, and 3, the cause-specific one-year exit probabilities are respectively $(0.12,0.08)$, $(0.15,0.10)$, and $(0.10,0.10)$. A member initially in the group is promised $1,000 at each year-end for three years, but only if still in the group then. The effective annual interest rate is 10%. Find the expected present value.

Method. Add the exit probabilities across causes in each year to find total group survival, multiply the annual survival probabilities to get survival to each payment date, and discount each eligible payment.

The one-year group survival probabilities are $1-0.12-0.08=0.80$, $1-0.15-0.10=0.75$, and $1-0.10-0.10=0.80$. Thus survival to the three year-ends is $0.80$, $0.80(0.75)=0.60$, and $0.80(0.75)(0.80)=0.48$. With $v=1/1.10=0.9090909$, the three discounted survival weights are

v(0.80)=0.7272727,\qquad v^2(0.60)=0.4958678,\qquad v^3(0.48)=0.3606311.

The expected present value is therefore

1000(0.7272727+0.4958678+0.3606311) =1000(1.5837716) =\$1{,}583.77.

Rounded to the nearest cent, the value is $1,583.77. As a check, the survival probabilities decrease from 0.80 to 0.60 to 0.48, and the discount factors also decrease, so the successive payment contributions should decline, as they do. The method avoids treating members who leave through the second cause as if they remained eligible.

Worked Example: Annual Payments in Advance

A hypothetical benefit pays $500 immediately and again at the beginnings of years 1, 2, and 3, provided the member is in the group on each payment date. The one-year group survival probabilities are 0.80, 0.90, and 0.75. The effective annual interest rate is 5%. Calculate the expected present value.

Method. Include the payment at time 0 with certainty, then use cumulative group survival to weight the later payments. The cumulative probabilities at times 1, 2, and 3 are $0.80$, $0.80(0.90)=0.72$, and $0.80(0.90)(0.75)=0.54$.

Here $v=1/1.05=0.9523810$, $v^2=0.9070295$, and $v^3=0.8638376$. The discounted survival weights are

1,\qquad v(0.80)=0.7619048,\qquad v^2(0.72)=0.6530612,\qquad v^3(0.54)=0.4664723.

Thus

500(1+0.7619048+0.6530612+0.4664723) =500(2.8814383) =\$1{,}440.72.

The answer, rounded to the nearest cent, is $1,440.72. The time-0 payment is not discounted and is not multiplied by a survival probability less than 1 because the member is initially in the group. Including it only once is also a useful check on the annuity-due timing.

Fractional Payment Dates

Payments need not be annual. If $m$ equal installments are paid per year, an annuity-immediate over $n$ years pays at times $j/m$, for $j=1,\ldots,mn$. If the installment amount is $C$, its expected present value is

C\sum_{j=1}^{mn}v^{j/m}\,{}_{j/m}p_x^{(\tau)}.

The fractional-year group survival probabilities must be consistent with the interpolation assumption used for the multiple-decrement model. For instance, if the total force of decrement is constant at $\lambda$ within each year, then survival for a fraction $t$ of a year within that interval is $e^{-\lambda t}$. A different interpolation assumption can give different fractional survival. Likewise, $v^t$ is the discount factor at time $t$ when annual effective interest is used with fractional powers. Do not apply a fractional survival formula unless its underlying assumption is specified.

Worked Example: Quarterly Installments Under Constant Forces

A hypothetical one-year annuity pays $250 at the end of each quarter if the member is still in the group. The two competing causes have constant forces of decrement 0.06 and 0.04 per year throughout the year. The effective annual interest rate is $i=e^{0.10}-1$, so the force of interest is $\delta=0.10$. Find the expected present value.

Method. Under constant competing forces, the total force of decrement is $0.06+0.04=0.10$. Group survival to time $t$ is therefore $e^{-0.10t}$. The discount factor is also exponential, $e^{-0.10t}$, because $\delta=0.10$. At time $j/4$, the product of survival and discount is $e^{-0.20(j/4)}=e^{-0.05j}$.

The four survival-discount weights are

e^{-0.05}=0.9512294,\quad e^{-0.10}=0.9048374,\quad e^{-0.15}=0.8607080,\quad e^{-0.20}=0.8187308.

Their sum is $3.5355056$, giving

250(3.5355056)=\$883.88.

Rounded to the nearest cent, the expected present value is $883.88. As a check, each quarter's weight is smaller than the preceding quarter's weight, since both the probability of remaining in the group and the discount factor decrease with time. The constant-force assumption is what justifies using these fractional-year survival probabilities.

Common Mistakes and Practical Checks

A frequent error is to use a probability of avoiding only one cause. In the first worked example, suppose the calculation incorrectly treated cause 1 as the only cause that could end eligibility. The resulting year-by-year “survival” factors would be $0.88$, $0.85$, and $0.90$, giving cumulative values $0.88$, $0.748$, and $0.6732$. The incorrect present value would be

1000\left(\frac{0.88}{1.10} +\frac{0.748}{1.10^2} +\frac{0.6732}{1.10^3}\right) \approx \$1{,}923.97,

rather than $1,583.77. The error counts people who have already exited through cause 2 as eligible. For an annuity payable while remaining in the group, combine all relevant exit causes when finding group survival.

Other checks help identify timing errors:

For an annuity-immediate, the first scheduled payment is at the end of the first period, so it is discounted and weighted by survival to that date.
For an annuity-due, a payment at time 0 is certain for a member initially in the group and is not discounted.
At every later payment date, use cumulative survival through all intervals up to that date.
For fractional dates, state and apply the interpolation or force assumption; do not treat fractional survival as automatically determined by annual probabilities.
A scheduled payment is included only at its stated date if the member is eligible then; it is not an additional payment made at exit.

These valuations are useful wherever a contract or benefit is conditional on continued status. The main modelling decision is to identify precisely which exits end eligibility; the main calculation is then to value each scheduled payment using the probability of having experienced none of those exits by its payment date. This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the payment dates and use group survival to weight each scheduled payment.

A hypothetical annuity pays $1,000 at the ends of years 1 and 2 while the member remains in the group. The annual group survival probabilities are 0.90 and 0.80, and the effective annual interest rate is 10%. Find the expected present value. (Answer: $1,413.22)
A benefit pays $200 at times 0, 1, and 2 if the member remains in the group. The annual group survival probabilities are 0.85 and 0.80, and interest is 0%. Find the expected present value. (Answer: $506.00)
A one-year hypothetical annuity pays $100 at each quarter-end. The total force of decrement is constant at 0.20 per year and interest is 0%. Find its expected present value, rounded to the nearest cent. (Answer: $353.55)
A hypothetical benefit pays $600 at the beginnings of years 0 and 1 if the member is in the group at the payment date. The first-year group survival probability is 0.90, and the effective annual interest rate is 5%. Find the expected present value. (Answer: $1,114.29)

What You'll Learn