Ancillary Benefits With Waiting Periods

Why the Waiting Period Matters

An ancillary benefit may be available only after a member has completed a specified period of service or membership. A member who dies or withdraws before that period ends might never qualify. The value of the benefit therefore depends not just on its amount and payment date, but also on the chance that the member remains eligible until then.

This tutorial values a benefit that is payable at the end of a waiting period only if the member is still active at that time. It addresses the probability of reaching eligibility and the discounting of a payment at that date. It does not assign a separate payment to a member who dies or withdraws during the waiting period; such a payment must be included only if the benefit terms specify one.

Definition: Let $w$ be the waiting period. A member is eligible at time $w$ if the member has remained active throughout the period from time 0 to time $w$. If a benefit $B$ is payable at time $w$ only to an eligible member, its present value is the benefit amount multiplied by the probability of reaching eligibility and discounted to time 0. An exit before time $w$ produces no payment under this benefit definition.

Continuous-Time Competing Exits

Suppose death and withdrawal are the only ways to leave active status during the waiting period. Let $\mu_D(t)$ and $\mu_W(t)$ be their cause-specific intensities at time $t$, conditional on being active just before $t$. The total intensity of leaving active status is $\mu_D(t)+\mu_W(t)$. In this model, the probability of remaining active to time $w$ is

{}_wp_{\text{active}}=\exp\left(-\int_0^w \left[\mu_D(t)+\mu_W(t)\right]\,dt\right).

If the intensities are constant, this simplifies to $\exp[-(\mu_D+\mu_W)w]$. This probability is not the probability of avoiding just one cause: the member must avoid every cause that terminates active status.

Assume an effective annual interest rate $i$ when discounting a payment at time $w$. Its discount factor is $v^w=(1+i)^{-w}$. Alternatively, under a constant force of interest $\delta$, the discount factor is $e^{-\delta w}$. The benefit’s expected present value is the product of the benefit amount, the probability of being eligible, and the discount factor.

Result: If a fixed benefit $B$ is paid at time $w$ if and only if the member is still active at $w$, then, under the continuous-time model, $$ \operatorname{EPV}=B\,{}_wp_{\text{active}}\,e^{-\delta w} =B\exp\left(-\int_0^w [\mu_D(t)+\mu_W(t)]\,dt-\delta w\right). $$ This result assumes the benefit is paid exactly at $w$, the specified exit causes end eligibility, and there is no additional payment on an earlier exit.

Worked Example: A Five-Year Waiting Period

A hypothetical pension plan pays a $10,000 ancillary lump sum at the end of five years, but only if the member is still active then. During the waiting period, death and withdrawal intensities are constant at 0.03 and 0.07 per year, respectively. The force of interest is $\delta=0.04$. Find the expected present value at time 0, to the nearest cent.

Method. Add the cause-specific intensities to obtain the total rate of leaving active status:

\mu_D+\mu_W=0.03+0.07=0.10.

The probability of remaining active for five years is

{}_5p_{\text{active}}=e^{-0.10(5)}=e^{-0.50}=0.6065307.

The five-year discount factor is $e^{-0.04(5)}=e^{-0.20}=0.8187308$. Thus

\operatorname{EPV}=10{,}000(0.6065307)(0.8187308) =10{,}000e^{-0.70} =\$4{,}965.85.

Check. The combined exponent is $-0.50-0.20=-0.70$, consistent with multiplying the survival and discount factors. The result is less than the discounted benefit of $8,187.31 because some members leave before becoming eligible. A common mistake is to calculate only $10{,}000e^{-0.20}=\$8{,}187.31$, which discounts the benefit but ignores pre-eligibility exits.

Piecewise Rates and the Eligibility Probability

The intensities need not be constant throughout the waiting period. If they are constant within intervals, calculate the total intensity separately in each interval and add the integrated rates. For example, if the total intensity is $c_1$ for the first $a$ years and $c_2$ for the next $w-a$ years, then the probability of remaining active to $w$ is $e^{-c_1a-c_2(w-a)}$. This is a direct application of the integrated-intensity formula.

Worked Example: Intensities Change During the Waiting Period

A hypothetical benefit of $18,000 is payable at the end of a three-year waiting period, provided the member remains active. During years 0 to 2, the death and withdrawal intensities are 0.04 and 0.08 per year. During years 2 to 3, they are 0.02 and 0.08 per year. The force of interest is 0.05 throughout. Find the expected present value, to the nearest cent.

Method. First calculate the integrated total intensity in each interval. During the first two years, the total intensity is $0.04+0.08=0.12$, giving an integrated intensity of $0.12(2)=0.24$. During the final year, it is $0.02+0.08=0.10$, giving $0.10(1)=0.10$. Therefore,

{}_3p_{\text{active}}=e^{-(0.24+0.10)}=e^{-0.34}=0.7117703.

The three-year discount factor is $e^{-0.05(3)}=e^{-0.15}=0.86070798$. The value is

\operatorname{EPV}=18{,}000(e^{-0.34})(e^{-0.15}) =18{,}000e^{-0.49} =\$11{,}027.28.

Check. The combined exponent contains both survival and discounting: $-0.34-0.15=-0.49$. The result is below the discounted amount $18{,}000e^{-0.15}=\$15{,}492.74$, as it should be when eligibility is uncertain. A common mistake is to apply the final year’s rates to all three years; the integrated rates must reflect how long each set of intensities applies.

Annual Decrement Probabilities

A plan may instead specify annual probabilities of death and withdrawal. Suppose these probabilities are measured over each year for a member active at the start of that year, and the two causes are mutually exclusive within the year. If $q_{D,k}$ and $q_{W,k}$ are the probabilities of death and withdrawal in year $k$, the probability of remaining active through that year is $1-q_{D,k}-q_{W,k}$. For a waiting period of $n$ complete years, multiply the one-year continuation probabilities.

With annual effective interest $i$, the value of a fixed benefit $B$ payable at the end of year $n$ is

\operatorname{EPV} =Bv^n\prod_{k=1}^{n}(1-q_{D,k}-q_{W,k}).

This formula uses the stated annual decrement probabilities directly; it does not treat them as continuous intensities. It assumes the probabilities are conditional on being active at the start of each year and that eligibility is tested at the end of the $n$-year period.

Worked Example: Annual Death and Withdrawal Probabilities

A hypothetical plan pays $12,000 at the end of a four-year waiting period if a member remains active. In each year, the conditional probability of death is 0.02 and the conditional probability of withdrawal is 0.08. The annual effective interest rate is $i=0.05$. Find the expected present value to the nearest cent.

Method. In each year, the probability of remaining active is $1-0.02-0.08=0.90$. The probability of remaining active for all four years is

{}_4p_{\text{active}}=(0.90)^4=0.6561.

The four-year discount factor is

v^4=(1.05)^{-4}=\frac{1}{1.21550625}=0.8227025.

Hence

\operatorname{EPV}=12{,}000(0.6561)(0.8227025) =12{,}000\left(\frac{0.90}{1.05}\right)^4 =\$6{,}477.30.

Check. The result is less than the discounted amount $12{,}000(1.05)^{-4}=\$9{,}872.43$, reflecting the possibility of leaving before eligibility. A common mistake is to subtract the two exit probabilities sequentially as if each were conditional on avoiding the other. Here they are mutually exclusive probabilities in the same year, so the annual continuation probability is $1-0.02-0.08$.

Why the Valuation Formula Holds

The result follows by representing the benefit as a contingent payment. Let $I_w$ equal 1 if the member remains active through time $w$, and 0 otherwise. The present value of the benefit is $B e^{-\delta w}I_w$. Taking expectations gives

E[B e^{-\delta w}I_w] =B e^{-\delta w}E[I_w] =B e^{-\delta w}P(I_w=1).

Because $I_w$ is an indicator, its expectation is the probability of the event it indicates. In the continuous-time competing-risk model, the probability of remaining active solves the survival equation

\frac{d}{dt}{}_tp_{\text{active}} =-[\mu_D(t)+\mu_W(t)]{}_tp_{\text{active}}, \qquad {}_0p_{\text{active}}=1.

Separating variables and integrating from 0 to $w$ gives $\ln({}_wp_{\text{active}})=-\int_0^w[\mu_D(t)+\mu_W(t)]dt$, and exponentiation gives the formula used above. The derivation relies on the cause-specific intensities describing all exits that terminate eligibility. If another event also ends active status, its intensity must be included; if it does not end eligibility, it must not be included merely because it is an event.

Interpretation and Exam Checks

The valuation separates naturally into three elements: the benefit amount at eligibility, the chance of reaching eligibility, and the discount factor from the eligibility date to time 0. Keeping these elements visible makes the model easier to check and helps distinguish eligibility risk from interest discounting.

Check the eligibility condition. Confirm exactly which exits before $w$ prevent payment. Include every such cause in the continuation probability.
Check when the benefit is paid. The formulas here assume payment at the end of the waiting period. A different payment date requires the corresponding discount factor and payment condition.
Use the model’s stated units. Intensities are rates per unit time, so multiply by time in matching units. Annual decrement probabilities are not interchangeable with annual intensities.
Do not add an unstated exit benefit. If a member who leaves early receives a refund or other payment, it is a separate cash flow and must be valued according to its own terms.

This conditional valuation is useful when assessing the cost of ancillary plan provisions: a nominal benefit amount alone does not measure its expected cost when eligibility can be lost before the waiting period ends. These methods are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the probability of reaching eligibility before applying the appropriate discount factor.

A hypothetical benefit of $8,000 is payable at the end of six years if the member remains active. Constant death and withdrawal intensities are 0.01 and 0.05 per year, and the force of interest is 0.03. Find the expected present value. (Answer: $4,661.99)
A $5,000 benefit is payable at the end of three years if the member remains active. Each year, conditional on being active at the start of the year, death probability is 0.03 and withdrawal probability is 0.07. The annual effective interest rate is 0.04. Find the expected present value. (Answer: $3,240.39)
A $10,000 benefit is payable at the end of four years if the member remains active. The total exit intensity is 0.08 during the first year and 0.12 during the next three years. The force of interest is 0.04. Find the expected present value. (Answer: $5,488.12)
A $6,000 benefit is payable at the end of two years if the member remains active. In year 1, death and withdrawal probabilities are 0.02 and 0.05; in year 2, they are 0.03 and 0.07. The annual effective interest rate is 0.03. Find the expected present value. (Answer: $4,733.72)

What You'll Learn