Cash Dividends in Participating Policy Values

Introduction

A cash dividend contributes to a policy’s value only when its payment is due and the policy satisfies the contract’s eligibility conditions. Those conditions might require that the insured be alive, that the policy remain in force, or both. The valuation therefore has two separate parts: discount the amount from its scheduled payment time, and weight it by the probability that it will be payable.

This tutorial values projected cash dividends during a policy term. The dividends are hypothetical projected amounts for valuation; they are not a promise that a future discretionary declaration will occur. The calculation treats each projected amount and its payment conditions explicitly. It does not combine cash dividends with claim-time benefits.

Definition: Suppose a policy has projected cash dividends $D_j$ payable at times $t_j$, where $t_j$ is measured in years from the valuation date. Let $I_j$ equal 1 if the policy satisfies the stated eligibility conditions at the payment time and 0 otherwise. At effective annual interest rate $i$, the expected present value of the dividends is $$ \operatorname{EPV}=\sum_j v^{t_j}E[D_j I_j], \qquad v=\frac{1}{1+i}. $$ If the amount $D_j$ is fixed and eligibility is simply a yes-or-no condition, this becomes $\sum_j v^{t_j}D_jP(I_j=1)$.

Payment Time and Eligibility

The scheduled payment time determines the discount factor. Eligibility is assessed at the time specified by the contract. For example, a dividend payable at the end of year 2 is discounted for two years, not one year. If payment requires the policy to be active at that date, the relevant probability is the probability that it is active then—not merely the probability that it was active at the start of the year.

For a dividend payable at time $t$ only if the insured is alive and the policy is in force, the eligibility probability is the joint probability of those conditions. It is not automatically the product of two unconditional probabilities. A product can be used when its factors correctly express the required conditional structure, such as lapse probability conditional on survival.

When a policy has several possible payment dates, value each date separately and add the results. This follows from the linearity of expectation: the present value is a sum of discounted payment indicators, so its expectation is the sum of their expectations. No independence between eligibility at different dates is needed for this addition.

E\left[\sum_j v^{t_j}D_jI_j\right] =\sum_j v^{t_j}E[D_jI_j].

If the dividend amount depends on the policy’s state at the payment time, use the expected amount conditional on each eligible state. For example, if different eligible states produce different amounts, sum each amount multiplied by the probability of its corresponding state. The fixed-amount formula is a special case.

Annual Survival and In-Force Conditions

A common annual model specifies a one-year death probability and a lapse probability conditional on survival. If the dividend is payable at the end of the year only when the insured survives and the policy has not lapsed, then the one-year eligibility probability is

P(\text{eligible at year end}) =P(\text{survive})P(\text{no lapse}\mid\text{survive}).

If the same conditional probabilities apply in each year, the probability of being eligible at the end of year $k$ is the product of the one-year eligibility probabilities. This multiplication uses the stated sequential conditional model; it does not assume that death and lapse are independent events. If probabilities vary by year, multiply the appropriate year-specific factors instead.

Worked Example: Dividends Conditional on Survival

A hypothetical participating policy is projected to pay $100 at the end of year 1, $80 at the end of year 2, and $60 at the end of year 3. Each dividend is payable only if the insured is alive at that year end. Assume a 25% effective annual interest rate and an independent annual death probability of 20% in each year. There is no lapse. Calculate the expected present value.

Method. Calculate the survival probability to each payment date, then discount each expected payment to time 0. The one-year survival probability is $1-0.20=0.80$. With the stated annual model, survival probabilities to the three dates are

{}_1p=0.80,\qquad {}_2p=0.80^2=0.64,\qquad {}_3p=0.80^3=0.512.

The discount factor is $v=1/1.25=0.80$, so the discount factors at years 1, 2, and 3 are $0.80$, $0.80^2=0.64$, and $0.80^3=0.512$, respectively. The three contributions are therefore

\begin{aligned} \text{Year 1: }&100(0.80)(0.80)=64.00000,\\ \text{Year 2: }&80(0.64)(0.64)=32.76800,\\ \text{Year 3: }&60(0.512)(0.512)=15.72864. \end{aligned}

Adding them gives $64.00000+32.76800+15.72864=112.49664$. Rounded to the nearest cent, the expected present value is $112.50.

Check. Each contribution is less than its undiscounted dividend because both survival weighting and discounting reduce its value. The calculation uses the full time to each payment.

Combining Survival and Lapse

When a dividend requires both survival and the policy remaining in force, use the probability of satisfying both conditions. If lapse is specified conditional on survival, the annual eligibility probability is

(1-q)(1-\lambda),

where $q$ is the annual death probability and $\lambda$ is the annual lapse probability conditional on survival. If the same probabilities apply in each year, eligibility through year $k$ has probability $\bigl((1-q)(1-\lambda)\bigr)^k$. The lapse factor is conditional for a reason: in this model, lapse is considered among policies that have survived the year.

Worked Example: Correctly Applying Conditional Lapse

A hypothetical policy pays $50 and $75 at the ends of years 1 and 2, respectively, only if the insured is alive and the policy remains in force. Each year, the death probability is 0.10; conditional on survival, the lapse probability is 0.10. Assume these probabilities are the same in both years and the effective annual interest rate is 5%. Find the expected present value.

Method. First calculate the probability of being eligible through one year. Survival probability is $1-0.10=0.90$, and the probability of no lapse conditional on survival is also $0.90$. Thus

P(\text{eligible at year 1})=0.90(0.90)=0.81.

For eligibility at the end of year 2, the same conditional structure applies in each year, so $P(\text{eligible at year 2})=0.81^2=0.6561$. Also, $v=1/1.05$, giving $v^2=1/1.1025$. The present-value contributions are

\begin{aligned} \text{Year 1: }&\frac{50(0.81)}{1.05} =\frac{40.50}{1.05}=38.5714286,\\ \text{Year 2: }&\frac{75(0.6561)}{1.1025} =\frac{49.2075}{1.1025}=44.6326531. \end{aligned}

The total is $38.5714286+44.6326531=83.2040817$. Rounded to the nearest cent, the expected present value is $83.20.

Check. The year 2 eligibility probability is smaller than the year 1 probability, and its dividend is discounted for two years. Both effects are reflected in the second contribution.

Continuous-Time Eligibility and Non-Integer Payment Dates

Payment dates need not be whole years. Under a constant force of mortality $\mu$, the probability of survival for $t$ years is $e^{-\mu t}$. If lapse is also modeled by a constant force $\lambda$, and the competing exit rates are specified so that the probability of remaining alive and in force for $t$ years is $e^{-(\mu+\lambda)t}$, use that joint eligibility probability. With force of interest $\delta$, discount over $t$ years by $e^{-\delta t}$.

This continuous-time formula requires the stated constant-force model over the relevant period. It should not be substituted for a discrete annual probability model unless the assumptions justify the conversion. In particular, identify whether the given interest assumption is an effective annual rate or a force of interest. An effective annual rate $i$ uses $v^t=(1+i)^{-t}$; a force $\delta$ uses $e^{-\delta t}$.

Worked Example: Two Non-Integer Dividend Dates

A hypothetical contract projects dividends of $20 at time $t=0.5$, $40 at $t=1.5$, and $100 at $t=2$, all measured in years. Each is payable only if the policy remains alive and in force at the payment time. Assume a constant force of mortality of 0.03, a constant lapse force of 0.02, and a force of interest of 0.05 per year. Calculate the expected present value.

Method. The probability of remaining eligible for $t$ years is $e^{-(0.03+0.02)t}=e^{-0.05t}$. The discount factor is $e^{-0.05t}$. Thus the combined eligibility and discount factor is $e^{-0.10t}$. Apply the correct time to each payment:

\begin{aligned} t=0.5:\quad &20e^{-0.10(0.5)} =20e^{-0.05} =20(0.9512294)=19.024588,\\ t=1.5:\quad &40e^{-0.10(1.5)} =40e^{-0.15} =40(0.8607080)=34.428320,\\ t=2:\quad &100e^{-0.10(2)} =100e^{-0.20} =100(0.8187308)=81.873080. \end{aligned}

The sum is $19.024588+34.428320+81.873080=135.325988$. Rounded to the nearest cent, the expected present value is $135.33.

Check. The exponent includes both the eligibility force and the interest force. The payment at 1.5 years is discounted and survival-weighted for 1.5 years, not for one or two complete years.

Why the Formula Holds and Common Traps

For a fixed dividend $D_j$, the random present value at date $j$ is $v^{t_j}D_jI_j$. Since the dividend amount and discount factor are fixed, its expectation is $v^{t_j}D_jE[I_j]$. An indicator equals 1 when the event occurs and 0 otherwise, so $E[I_j]=P(I_j=1)$. Therefore that payment’s expected present value is $v^{t_j}D_jP(I_j=1)$. Adding across payment dates gives the general formula by linearity of expectation.

The derivation also clarifies the scope of the calculation. If the dividend amount is random or state-dependent, $E[D_jI_j]$ must be evaluated as a joint expectation; it cannot generally be replaced by $E[D_j]P(I_j=1)$ without an appropriate independence condition. A projected amount treated as fixed is a modeling assumption, not a statement that future declarations are certain.

A frequent exam trap is to omit the survival or in-force condition when discounting a scheduled dividend. For example, the incorrect year 1 calculation in the conditional-lapse example would be $50/1.05=\$47.62$, as though the payment were certain. The correct calculation is $50(0.81)/1.05=\$38.57$, because only 0.81 of policies are eligible at that date.

Other common errors include:

Using the wrong payment time. A payment at year 2 receives $v^2$, even if its eligibility probability has been calculated correctly.
Misreading conditional lapse. When lapse is conditional on survival, calculate survival times no-lapse-given-survival; do not treat the lapse probability as unconditional without justification.
Applying annual probabilities to a fraction of a year without a model. A non-integer duration requires an assumption such as constant forces or an explicitly supplied fractional-duration probability.
Assuming independence without need. The payment-date sum follows from linearity of expectation, even when eligibility events at different dates are dependent.

Key takeaway: For every projected cash dividend, identify its contractual payment time, amount, and eligibility event. Discount to that time, multiply by the probability of the correctly defined event, and then add the payment-date values.

Check Your Understanding

For each problem, use the stated eligibility conditions and payment dates. Round money to the nearest cent.

A hypothetical policy pays $120 at the end of year 1 only if the insured survives. The annual death probability is 0.04 and the effective annual interest rate is 3%. Find the expected present value. (Answer: $111.84)
A policy pays $25 at the end of year 1 and $40 at the end of year 2, only if it is alive and in force. In each year, survival probability is 0.90 and lapse probability conditional on survival is 0.05. The effective annual interest rate is 5%. Find the expected present value. (Answer: $46.88)
A policy pays $500 at time $t=3$ only if it remains alive and in force. Assume constant mortality and lapse forces of 0.02 and 0.03, respectively, and force of interest $\delta=0.04$. Find the expected present value. (Answer: $381.69)
A policy pays $30 at the end of year 1 and $50 at the end of year 2 if alive and in force. Each year the death probability is 0.10, and lapse probability conditional on survival is 0.20. The effective annual interest rate is 4%. Find the expected present value. (Answer: $44.73)
A policy pays $80 at $t=0.5$ and $100 at $t=1.5$, provided it is alive and in force at each payment time. Constant mortality and lapse forces are 0.02 and 0.01, and the force of interest is 0.03. Find the expected present value. (Answer: $169.03)

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