Payment Timing for Participating Benefits

Why Payment Timing Must Be Specified

A participating policy may provide a death benefit, a surrender value, a maturity benefit, and dividends or bonuses. Knowing the amount of each benefit is not enough to describe its cash flow. The valuation date also depends on the contract: a death benefit may be payable at the time of death, while a dividend may be payable only on a policy anniversary. A maturity benefit is normally payable on the stated maturity date if the policy remains in force then.

The distinction affects both the probability assigned to a payment and its time. A policyholder who dies during the second policy year generates a claim at the actual time of death under an immediate-payment clause, not automatically at the second anniversary. Conversely, an annual dividend condition may require the policyholder to survive and remain in force through an anniversary. This tutorial focuses on translating those conditions into event dates and probabilities; it does not calculate present values.

Definition: Let $T$ be the time from policy issue to the first event that ends or changes the policy, such as death or surrender. A payment is immediate if its date is the event time $T$. A payment is anniversary-based if it is due only on a specified policy anniversary, subject to the contract’s conditions at that date. A maturity payment is due at the fixed maturity time $n$ if the policy reaches maturity in force.

For a policy with annual periods, policy year $k$ is the interval from time $k-1$ to time $k$. Under an annual mortality assumption, $q_k$ is the probability of death during that interval, conditional on being alive at its start; $p_k=1-q_k$. Thus, if there is no other exit, the probability of surviving to anniversary $k$ is the product of the preceding conditional survival probabilities. The notation ${}_kp_x$ and ${}_kq_x$ expresses survival and death probabilities over $k$ years for a life aged $x$.

Translate Conditions Before Calculating

For each benefit, record four items: the event that triggers it, any survival or in-force condition, the payment date, and the probability of meeting those conditions. In particular, separate a claim amount from the time at which the amount is paid. If a contract says “payable immediately on death,” the benefit date follows the death time even if the probability model only identifies death as occurring within a policy year.

Name the trigger.
Identify death, surrender, survival to an anniversary, or survival to maturity. State whether triggers can occur between anniversaries.

Apply the conditions in order.
For example, an anniversary dividend might require survival to the date and continued in-force status after any surrender election specified by the contract.

Assign the payment date.
Use the event time for an immediate claim or the specified anniversary for a scheduled payment. Do not replace one timing convention with the other.

Calculate the relevant probability.
Use conditional probabilities successively, or use the appropriate event-time distribution when risks operate continuously.

Worked Example: Death Claims and an Anniversary Dividend

A hypothetical two-year participating endowment has a death benefit of $22,000 payable immediately on death, a maturity benefit of $24,000 payable at time 2 if the policy is in force, and a $300 cash dividend payable at time 1 only if the policyholder is alive then. There is no surrender. Conditional annual death probabilities are $q_1=0.10$ and $q_2=0.20$. Use an effective annual interest rate $i=4\%$; no discounting is required here. The mortality model is discrete by policy year, so it gives the probability of death in each year, not a more detailed distribution of death times within that year.

Method. First calculate the probabilities of death in each year and survival to each anniversary. Then match each payment to its trigger and timing condition.

The probability of death in year 1 is $q_1=0.10$. The probability of surviving the first year is $p_1=1-0.10=0.90$, so the probability of death in year 2 is $p_1q_2=0.90(0.20)=0.18$. The probability of survival to maturity is $p_1p_2=0.90(0.80)=0.72$.

P(\text{death in year 1})=0.10,\quad P(\text{death in year 2})=0.18,\quad P(\text{survive to time 2})=0.72.

The $22,000 death benefit is paid at the actual death time for either death outcome. The annual mortality assumption identifies the relevant year and probability, but does not justify assigning every death in that year to its end. The $300 dividend is due at time 1 for the $0.90$ of policyholders who survive to that anniversary. The $24,000 maturity benefit is due at time 2 for the $0.72$ who survive both years.

As a probability check, the mutually exclusive terminal outcomes sum to $0.10+0.18+0.72=1.00$. The dividend is not a terminal outcome: a policyholder receiving it at time 1 may subsequently die or survive to maturity. The common mistake is to treat the dividend as an alternative to later death or maturity benefits.

Continuous-Time Competing Events

When death and surrender can occur at any time, it is useful to model them as competing risks. Suppose that, while the policy is in force, the force of death is constant at $\mu_D$ and the force of surrender is constant at $\mu_S$, with independent event mechanisms conditional on remaining in force. The first exit time then has total force $\mu=\mu_D+\mu_S$. Death and surrender payments are made immediately at their respective event times; if neither occurs before maturity $n$, the maturity payment is made at $n$.

Theorem: Under constant competing forces $\mu_D$ and $\mu_S$, the probability of remaining in force for $t$ years is $e^{-(\mu_D+\mu_S)t}$. The probability that death is the first exit during $(a,b]$ is $e^{-(\mu_D+\mu_S)a}\dfrac{\mu_D}{\mu_D+\mu_S}\left(1-e^{-(\mu_D+\mu_S)(b-a)}\right)$. Replace $\mu_D$ by $\mu_S$ for surrender. These formulas require constant forces over the interval and do not apply unchanged if forces vary with duration or policy status.

The reason for the death formula is that a death in a short interval of length $dt$ after time $t$ requires the policy to have remained in force to $t$, which has probability $e^{-\mu t}$. Given that it is still in force, the probability of death in $(t,t+dt]$ is approximately $\mu_Ddt$. Hence the death density is $e^{-\mu t}\mu_D$. Integrating that density from $a$ to $b$ gives

\int_a^b \mu_D e^{-\mu t}\,dt =\frac{\mu_D}{\mu}\left(e^{-\mu a}-e^{-\mu b}\right) =e^{-\mu a}\frac{\mu_D}{\mu}\left(1-e^{-\mu(b-a)}\right).

The corresponding surrender probability follows by integrating $\mu_Se^{-\mu t}$. The maturity probability is the probability of no exit by $n$, namely $e^{-\mu n}$. Death, surrender, and survival to maturity are mutually exclusive outcomes, and their probabilities sum to one.

Worked Example: Immediate Death and Surrender Claims

A hypothetical two-year participating policy is subject to constant forces $\mu_D=0.02$ per year for death and $\mu_S=0.08$ per year for surrender. Death and surrender benefits are both payable immediately at the relevant event. A maturity benefit is payable at time 2 if neither event has occurred. Take $i=3\%$ effective annually; interest is not needed to calculate these event probabilities.

Method. The total exit force is $\mu=0.02+0.08=0.10$. Apply the competing-risk formula separately to each policy year. Since the event-time distributions are continuous, an exit at a precise anniversary has probability zero.

For year 1, set $a=0$, $b=1$. The death probability is $e^0(0.02/0.10)(1-e^{-0.10})=0.2(0.0951626)=0.0190325$, or 0.0190 to four decimals. The surrender probability is $e^0(0.08/0.10)(1-e^{-0.10})=0.8(0.0951626)=0.0761301$, or 0.0761.

For year 2, set $a=1$, $b=2$. The death probability is $e^{-0.10}(0.2)(1-e^{-0.10})=0.9048374(0.0190325)=0.0172213$, or 0.0172. The surrender probability is $e^{-0.10}(0.8)(1-e^{-0.10})=0.9048374(0.0761301)=0.0688853$, or 0.0689.

The probability of reaching maturity is $e^{-0.10(2)}=e^{-0.2}=0.8187308$, or 0.8187. Before rounding, the outcomes sum to $0.0190325+0.0761301+0.0172213+0.0688853+0.8187308=1.0000000$. The rounded values sum to 0.9999 because each is rounded separately.

The death and surrender benefits are paid at the event times in their respective years; neither is automatically postponed to the next anniversary. A common error is to use the death share $\mu_D/\mu=0.2$ as the probability of death over the full two years. That ratio is the share of exits attributable to death under the constant-force model, not the probability of a death before maturity. The probability must also account for whether an exit occurs, and when.

Anniversary Surrender and In-Force Benefits

Some contracts restrict surrender to policy anniversaries, even though death can occur during a policy year. In that model, the event order matters. A clear convention might be: first determine whether death occurred during the year; if the policyholder survives to the anniversary, pay any anniversary dividend to policies that continue in force; then process surrender elections. Another contract might process surrender before a dividend is due. The calculation must follow the stated order rather than assume one.

An anniversary payment can also coexist with later payments. A dividend paid at time 1 does not prevent a death claim during year 2 or a maturity payment at time 2. Consequently, probabilities of receiving payments at different dates are not necessarily probabilities of mutually exclusive outcomes. Use a terminal-outcome partition to check the claim and maturity probabilities, and separately identify eligibility for each scheduled payment.

Worked Example: Dividend Eligibility With Anniversary Surrender

A hypothetical two-year participating policy pays a $15,000 death benefit immediately on death, a $5,000 surrender value at anniversary 1 if the policyholder elects to surrender, and an $18,000 maturity benefit at time 2 if the policy remains in force. A $400 dividend is paid at anniversary 1 only to policyholders who are alive and remain in force after the surrender election. At maturity, maturity payment takes precedence over any surrender election. Conditional death probabilities are $q_1=0.02$ and $q_2=0.03$; surrender probabilities among survivors at anniversaries 1 and 2 are $r_1=0.10$ and $r_2=0.05$. Use $i=5\%$, which is not needed for these probabilities. Death is modeled by policy year; surrender occurs only at anniversaries.

Method. Apply mortality during each year, then apply the surrender decision at the anniversary to those who survived that year. At anniversary 1, determine dividend eligibility only after applying the stated surrender condition.

The probability of death in year 1 is $q_1=0.02$. Survival to anniversary 1 is $1-q_1=0.98$. The probability of surrender at that anniversary is $0.98(0.10)=0.098$, so the probability of remaining in force after the election—and therefore receiving the dividend—is $0.98(0.90)=0.882$.

The probability of death in year 2 is the probability of being in force after anniversary 1 multiplied by the conditional death probability: $0.882(0.03)=0.02646$. The probability of surviving year 2 is $0.882(0.97)=0.85554$. Since maturity takes precedence over surrender at time 2, this full probability receives the maturity benefit; $r_2$ does not reduce it.

The terminal outcomes are death in year 1, surrender at anniversary 1, death in year 2, and maturity. Their probabilities sum to $0.02000+0.09800+0.02646+0.85554=1.00000$. The dividend is paid to the 0.882 who continue after anniversary 1. Those policyholders may later be included among the year-2 deaths or maturities, so adding 0.882 to the terminal-outcome sum would be incorrect.

The main trap is to apply $r_1$ before testing survival to anniversary 1, which would incorrectly give a surrender probability of 0.10. The correct unconditional probability is $0.98(0.10)=0.098$, because only survivors can elect to surrender.

Common Timing Traps

Moving an immediate claim to an anniversary. If a benefit is due on death, its payment date is the death time. An annual probability of death may identify only the policy year, not an exact payment date.
Ignoring the in-force condition. A dividend payable to policies in force at an anniversary is not paid to someone who died earlier or surrendered before that date.
Using a surrender probability without its condition. If the surrender rate applies only to anniversary survivors, multiply it by the probability of reaching that anniversary in force.
Counting a recurring payment as a separate terminal outcome. A dividend recipient can later receive a death or maturity benefit. Payment eligibility at one date and terminal claim outcomes answer different probability questions.
Assuming an event order that the contract does not state. If dividend eligibility depends on whether surrender is processed first or second, the contractual timing convention must be specified before calculating.

Key takeaway: For each participating benefit, state its trigger, eligibility condition, and payment date before calculating. Use conditional survival and exit probabilities in the correct order, and keep anniversary payments distinct from mutually exclusive terminal outcomes.

Check Your Understanding

For each problem, identify the applicable timing convention as well as the requested probabilities.

A hypothetical one-year policy pays $10,000 immediately on death and $12,000 at time 1 if the policyholder survives. The annual death probability is 0.04; there is no surrender. With $i=4\%$, state the death-payment date convention and calculate the death and maturity probabilities. (Answer: Death benefit at the actual death time; death probability 0.0400; maturity probability 0.9600.)
A two-year policy has annual conditional death probabilities $q_1=0.05$ and $q_2=0.10$, with no other exits. A $250 dividend is payable at time 1 if alive, and maturity is paid at time 2 if alive. With $i=3\%$, calculate dividend eligibility probability and maturity probability. (Answer: Dividend eligibility 0.9500; maturity probability $0.95(0.90)=0.8550$.)
A policy has constant competing forces $\mu_D=0.03$ and $\mu_S=0.07$ for one year; both exit benefits are immediate. With $i=2\%$, calculate the death and surrender probabilities, to four decimals. (Answer: Death $0.3(1-e^{-0.10})=0.0285$; surrender $0.7(1-e^{-0.10})=0.0666$.)
A two-year policy has $q_1=0.04$, $q_2=0.05$, and an anniversary-1 surrender probability of 0.20 among survivors. A $500 dividend is paid only to survivors who do not surrender at time 1. With $i=4\%$, calculate surrender probability at time 1, dividend eligibility probability, and year-2 death probability. (Answer: Surrender $0.96(0.20)=0.1920$; dividend eligibility $0.96(0.80)=0.7680$; year-2 death $0.768(0.05)=0.0384$.)
Under constant forces $\mu_D=0.02$ and $\mu_S=0.08$, a two-year policy pays death and surrender benefits immediately and a maturity benefit at time 2 if there is no exit. With $i=3\%$, calculate the probability of maturity and explain whether a dividend paid at time 1 to all policies then in force should be added to the mutually exclusive terminal outcome probabilities. (Answer: Maturity probability $e^{-0.10(2)}=0.8187$; no, the dividend is an intermediate payment and recipients may later have a terminal outcome.)

What You'll Learn