Deferred Benefits Triggered by a Specified Cause

Introduction

A benefit tied to a particular decrement cause may be payable only if the covered person first remains in the decrement group through a specified waiting period. For example, a hypothetical contract might pay a lump sum for withdrawal due to a defined cause, but only when that withdrawal occurs after two complete years of participation. Valuing the benefit requires two distinct probabilities: survival in the group during the deferral period, and exit for the specified cause afterward.

The key is to keep the timing and conditioning explicit. A cause-specific exit probability for one year is conditional on being in the group at the start of that year; it is not, by itself, the probability that a person initially in the group will receive a deferred benefit.

Definition: In a discrete multiple-decrement model, let $q_k^{(j)}$ be the probability that a member of the decrement group at the start of year $k$ exits during that year from cause $j$. Let $p_k$ be the probability of remaining in the group through year $k$, conditional on being in the group at its start. Thus, if the listed causes are exhaustive, $p_k=1-\sum_j q_k^{(j)}$. A benefit deferred for $n$ complete years is not paid for exits in years $1,\ldots,n$; it may be paid for a specified-cause exit in a later year.

Probability of a Deferred Cause-Specific Payment

Suppose the benefit is payable at the end of the year of exit. Starting in year $k$, the probability of a cause-$j$ exit in that year is $q_k^{(j)}$. To receive a benefit deferred for $n$ complete years and then exit for cause $j$ in year $k>n$, the member must also have remained in the group through years $1,\ldots,k-1$.

P(\text{cause }j\text{ exit in year }k) = \left(\prod_{r=1}^{k-1}p_r\right)q_k^{(j)},\qquad k>n.

The product ends at $k-1$, not $k$: the member must survive in the group until the start of the exit year, but then exits during that year. The deferral condition is enforced by including only years $k>n$.

Result: If $B_k$ is the benefit payable at the end of year $k$ for an eligible cause-$j$ exit, and the effective annual interest rate is $i$, the expected present value at time 0 is

\operatorname{EPV}_0 = \sum_{k=n+1}^{N} v^k B_k \left(\prod_{r=1}^{k-1}p_r\right)q_k^{(j)}, \qquad v=\frac{1}{1+i}.

This expression assumes annual decrement probabilities, a benefit paid at the end of the exit year, and no payment for an exit during the deferral period. The probabilities can vary from year to year. If a contract instead pays immediately at exit, uses a different payment date, or defines a different eligibility rule, the payment timing or included events must be changed accordingly.

Why the Formula Holds

For a specified year $k$, the required event is a sequence: remain in the group through each of the first $k-1$ years, then exit from cause $j$ during year $k$. Conditional on reaching each year, the probability of continuing is $p_r$, and the conditional probability of the specified exit in year $k$ is $q_k^{(j)}$. Multiplying these successive conditional probabilities gives the probability of the full sequence.

Reach the exit year.
The probability of remaining in the group through the first $k-1$ years is $\prod_{r=1}^{k-1}p_r$.

Exit for the specified cause.
Given membership at the start of year $k$, the cause-specific exit probability is $q_k^{(j)}$. Multiplication gives the unconditional probability of that eligible exit.

Discount the payment.
A benefit paid at the end of year $k$ has present value $v^kB_k$. Multiplying this by the event probability and summing over eligible years gives the EPV.

There is also a useful conditional form. Let ${}_np=\prod_{r=1}^{n}p_r$, the probability of remaining in the group through the deferral period. Conditional on being in the group at time $n$, value the later payments at time $n$, and then multiply by ${}_np$ and discount $n$ years to return to time 0. This is the same calculation, expressed with a different valuation date.

\operatorname{EPV}_0 = v^n\,{}_np \sum_{k=n+1}^{N} v^{k-n}B_k \left(\prod_{r=n+1}^{k-1}p_r\right)q_k^{(j)}.

An empty product, as occurs when $k=n+1$, is defined to equal 1. Thus, for an exit in the first year after deferral, no additional post-deferral survival factor is required before that exit-year probability.

Worked Valuations

Worked Example: One-Year Deferral With Two Eligible Exit Years

A hypothetical contract pays $10,000 for a cause-$c$ exit in year 2 and $15,000 for a cause-$c$ exit in year 3. No benefit is payable for an exit in year 1. The annual effective interest rate is 4%. The probabilities are $p_1=0.90$, $q_2^{(c)}=0.04$, $p_2=0.86$, and $q_3^{(c)}=0.06$. Find the EPV at time 0. Assume the listed probabilities are conditional on membership at the start of the relevant year and benefits are paid at year-end.

Method. For each eligible year, multiply the benefit by the probability of surviving in the group to the start of the exit year and then exiting for cause $c$. Discount the year-end payment to time 0.

For year 2, the probability of the eligible exit is $0.90(0.04)=0.036$. The expected year-2 payment is $$10{,}000(0.036)=\$360$, so its present value is \[ \frac{\$360}{1.04^2}=\$332.8402367. \]

For year 3, the probability of the eligible exit is $0.90(0.86)(0.06)=0.04644$. The expected year-3 payment is $$15{,}000(0.04644)=\$696.60$, and its present value is \[ \frac{\$696.60}{1.04^3}\approx \$619.2749. \]

Adding the unrounded present values gives approximately $$332.8402+\$619.2749=\$952.1151$. The EPV is $952.12, rounded to the nearest cent.

Check. Conditional on being in the group at time 1, the value at time 1 is $$10{,}000(0.04)/1.04+\$15{,}000(0.86)(0.06)/1.04^2\approx\$1{,}100.2219$. Multiplying by the probability $0.90$ of reaching time 1 and discounting one year gives $0.90(1/1.04)(\$1{,}100.2219)\approx\$952.1151$, the same result. The common mistake avoided is omitting the year-1 survival factor when converting a conditional value at time 1 to a value at time 0.

Worked Example: Deferral Across Two Years With No Interest

A hypothetical benefit pays $12,000 for cause-$d$ exit in year 3 and $25,000 for cause-$d$ exit in year 4. The deferral period is two complete years. The interest rate is 0%. For years 1 through 4, the relevant probabilities are $p_1=0.80$, $p_2=0.75$, $p_3=0.90$, $q_3^{(d)}=0.05$, and $q_4^{(d)}=0.04$. Find the EPV.

Method. Include years 3 and 4 only. Since interest is 0%, discount factors are 1, but survival to each eligible exit year is still required.

For year 3, the probability of cause-$d$ exit is $0.80(0.75)(0.05)=0.03$. The year-3 expected payment is $$12{,}000(0.03)=\$360$.

For year 4, the probability is $0.80(0.75)(0.90)(0.04)=0.0216$. The year-4 expected payment is $$25{,}000(0.0216)=\$540$.

Therefore, $\operatorname{EPV}_0=\$360+\$540=\$900.00$. As a check, the probability of reaching the end of the two-year deferral period is $0.80(0.75)=0.60$; conditional on that survival, the expected eligible payments are $$12{,}000(0.05)+\$25{,}000(0.90)(0.04)=\$600+\$900=\$1{,}500$. Then $0.60(\$1{,}500)=\$900$. The calculation avoids treating the two post-deferral cause probabilities as unconditional probabilities from time 0.

Worked Example: Changing Survival and Cause Probabilities With Interest

A hypothetical contract has a two-year deferral and pays $8,000 for cause-$e$ exit in year 3 and $20,000 for cause-$e$ exit in year 4. The annual effective interest rate is 3%. Assume $p_1=0.95$, $p_2=0.90$, $q_3^{(e)}=0.10$, $p_3=0.80$, and $q_4^{(e)}=0.05$. Benefits are paid at the end of the exit year. Calculate the EPV to the nearest cent.

Method. Apply the full survival product through the year before each eligible exit, then discount each payment by its year number.

The probability of a cause-$e$ exit in year 3 is $0.95(0.90)(0.10)=0.0855$. The expected payment is $$8{,}000(0.0855)=\$684$, giving present value $\$684/1.03^3=\$625.9578$, approximately.

The probability of a cause-$e$ exit in year 4 is $0.95(0.90)(0.80)(0.05)=0.0342$. The expected payment is $$20{,}000(0.0342)=\$684$, giving present value $\$684/1.03^4=\$607.7251$, approximately.

The unrounded values sum to approximately $$625.9578+\$607.7251=\$1{,}233.6829$. Therefore the EPV is $1,233.68, rounded to the nearest cent. The equal expected payments in the two years are a useful arithmetic check; their present values differ because they occur at different times.

Common Exam Trap: Applying the Deferral Rule Incorrectly

A frequent error is to use a cause-specific exit probability without multiplying by survival through the earlier years. Another is to include a payment for an exit during the deferral period, or to treat a value conditional on survival to the deferral date as though it were already a time-0 value. These errors change the set of events being valued, not merely the presentation.

For example, in the first worked example, using $0.04$ as the unconditional probability of the year-2 payment would give $$10{,}000(0.04)/1.04^2=\$369.82$, rather than the correct $$332.84$. The incorrect calculation omits the probability $p_1=0.90$ of surviving through year 1. Similarly, the year-3 probability is not just $q_3^{(c)}=0.06$; the correct probability is $p_1p_2q_3^{(c)}=0.04644$.

Key takeaway: For every eligible exit year, identify the exact exit event, multiply the conditional survival probabilities needed to reach that year by its cause-specific exit probability, and discount according to the actual payment date. A deferral changes which exit years qualify; it does not remove the need to account for survival.

Check Your Understanding

For each problem, assume the stated probabilities are conditional on membership at the start of the relevant year and benefits are paid at year-end.

A benefit is deferred for one year. It pays $5,000 for cause-$a$ exit in year 2. If $p_1=0.80$, $q_2^{(a)}=0.10$, and $i=0\%$, find the EPV. (Answer: $400.00)
A benefit is deferred for two complete years and pays $10,000 for cause-$b$ exit in year 3. If $p_1=0.90$, $p_2=0.80$, $q_3^{(b)}=0.05$, and $i=0\%$, find the EPV. (Answer: $360.00)
A one-year-deferred benefit pays $6,000 for cause-$c$ exit in year 2 and $10,000 for cause-$c$ exit in year 3. Given $p_1=0.95$, $q_2^{(c)}=0.04$, $p_2=0.90$, $q_3^{(c)}=0.03$, and $i=0\%$, find the EPV. (Answer: $484.50)
A benefit is deferred for one year. It pays $10,000 for cause-$d$ exit in year 2 and $20,000 for cause-$d$ exit in year 3. Given $p_1=0.90$, $q_2^{(d)}=0.05$, $p_2=0.80$, $q_3^{(d)}=0.04$, and $i=5\%$, find the EPV. (Answer: $905.73)
A two-year-deferred benefit pays $15,000 for cause-$e$ exit in year 3. Given $p_1=0.95$, $p_2=0.90$, $q_3^{(e)}=0.06$, and $i=4\%$, find the EPV. (Answer: $684.08)

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