Scenario Valuation of Future Discretionary Benefits

Valuing Benefits Across Experience Scenarios

A future discretionary benefit, such as a terminal bonus, may depend on how experience develops before the payment date. A strong experience scenario could support a larger bonus, while the same scenario could also affect the probability that a policy remains in force. Valuing the benefit therefore requires attention to both the amount paid in each scenario and the probability of reaching the payment date in that scenario.

The central calculation is an expected present value: identify the possible scenarios, calculate the contingent payment in each one, weight it by the probability of that scenario and any other conditions for payment, and discount it to the valuation date. The method is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: Suppose experience falls into one of a finite set of mutually exclusive, exhaustive scenarios $s=1,\ldots,m$, with probabilities $p_s=P(S=s)$. Let $B_t(s)$ be the discretionary benefit payable at time $t$ in scenario $s$, and let $r_t(s)$ be the conditional probability that the benefit is payable at time $t$, given that scenario. If the effective annual interest rate is $i$, the expected present value is $$ \operatorname{EPV}=\sum_{s=1}^{m}p_s\sum_t v^t B_t(s)r_t(s), \qquad v=\frac{1}{1+i}. $$ The factor $r_t(s)$ is 1 when payment is certain in scenario $s$. If the benefit is payable only while the policy is in force, $r_t(s)$ is the probability the policy is in force at time $t$, conditional on that scenario.

This expression assumes the scenario probabilities describe the experience uncertainty relevant to the valuation, the specified payments occur at the stated times, and the valuation interest rate $i$ is fixed and effective annually. In-force probabilities must be conditional on scenarios if the scenario affects persistency. If the problem instead states that persistency is independent of scenario, the same in-force probability applies in every scenario.

For an annual effective rate, a payment at the end of year $t$ is discounted by $v^t=(1+i)^{-t}$. A payment at the end of year 1 is not discounted for two years merely because the projection also includes a year-2 payment. Each contingent amount receives the discount factor for its own payment date.

Why Scenario Weighting Comes Before the Final Sum

Let $I_t$ be an indicator equal to 1 if the benefit is payable at time $t$, and 0 otherwise. The random present value at that date is $v^t B_t(S)I_t$. Conditional on scenario $s$, its expected value is $v^tB_t(s)P(I_t=1\mid S=s)$. Averaging those conditional values over the scenarios gives:

E[v^t B_t(S)I_t] =\sum_s P(S=s)E[v^t B_t(S)I_t\mid S=s] =\sum_s p_s v^t B_t(s)r_t(s).

Summing over payment dates gives the definition above. This is the law of total expectation applied to discounted contingent payments. It explains why a benefit amount and its payment probability should generally be paired within each scenario before averaging across scenarios. Separately averaging the amounts and probabilities and multiplying those averages is only valid under suitable independence conditions.

Worked Example: A Terminal Bonus in Two Scenarios

A hypothetical participating policy pays a terminal bonus at the end of year 2 if the policy is in force. The effective annual interest rate is 5%, and the policy is assumed to be in force with probability 1 at payment. Scenario A has probability 0.30 and produces a bonus of $300; Scenario B has probability 0.70 and produces a bonus of $900. Find the expected present value.

Method. Since payment is certain once a scenario occurs, use $r_2(s)=1$. First calculate the scenario-weighted bonus:

$$ E[B_2]=0.30(300)+0.70(900)=90+630=720. $$

Both scenario payments occur at the end of year 2, so the discount factor is $v^2=1/(1.05)^2=1/1.1025$. Thus:

\operatorname{EPV}=\frac{720}{1.1025}=653.061224\ldots

Rounded to the nearest cent, the expected present value is $653.06. The calculation avoids treating a scenario-weighted undiscounted amount as though it were paid at the valuation date; it is a year-2 amount.

When Persistency Depends on the Scenario

A benefit can be linked to the same experience that affects survival or persistency. In that case, a scenario with a larger bonus may have a different probability of reaching the payment date. The correct valuation uses the scenario-specific conditional payment probability, rather than one average persistency assumption applied indiscriminately.

Worked Example: Scenario-Dependent Bonus and Mortality

A hypothetical policy pays a discretionary benefit at the end of year 2 only if the policyholder is alive and the policy remains in force. The annual effective interest rate is 4%. Scenario A has probability 0.40, a year-2 benefit of $600, and an annual mortality probability of 2% in each of the two years. Scenario B has probability 0.60, a benefit of $1,200, and annual mortality probability 5% in each year. There is no lapse. Conditional on the scenario, mortality is independent between years. Find the expected present value.

Step 1: Find the conditional probability of payment. In Scenario A, the probability of surviving both years is $0.98^2=0.9604$. In Scenario B, it is $0.95^2=0.9025$. With no lapse, these are also the probabilities that the policy is in force at the payment date.

Step 2: Calculate the probability-weighted expected payment at year 2.

$$ 0.40(600)(0.9604)+0.60(1{,}200)(0.9025) =230.496+649.800 =880.296. $$

Step 3: Discount two years. The year-2 discount factor is $1/(1.04)^2=1/1.0816$, so:

\operatorname{EPV}=\frac{880.296}{1.0816} =813.883\ldots

Rounded to the nearest cent, the expected present value is $813.88.

Common mistake and correction. A tempting but incorrect shortcut is to multiply the average benefit by the average survival probability. The average benefit is $0.40(600)+0.60(1{,}200)=960$, and the average survival probability is $0.40(0.9604)+0.60(0.9025)=0.92566$. This shortcut gives $960(0.92566)/1.0816=821.59$, which is wrong because the benefit and survival probability are paired differently in each scenario. The correct scenario-by-scenario numerator is $880.296$, not $960(0.92566)=888.6336$.

Benefits at More Than One Payment Date

When discretionary payments may occur at several dates, value each date separately. At a given date, first account for the scenario-specific amount and probability of payment; then apply the discount factor for that date. The resulting present values can be summed because expectation is additive. No assumption of independence between payments at different dates is needed for this expected-value calculation.

Worked Example: A Two-Year Bonus Schedule Across Three Scenarios

A hypothetical contract has discretionary bonuses at the ends of years 1 and 2. The annual effective interest rate is 5%. The three mutually exclusive scenarios have probabilities 0.20, 0.50, and 0.30. The table gives the bonus and the conditional probability that the policy is in force at each payment date.

Scenario	Probability	Year-1 bonus	In force at year 1	Year-2 bonus	In force at year 2
1	0.20	$100	0.98	$500	0.95
2	0.50	$200	0.97	$700	0.92
3	0.30	$400	0.95	$900	0.88

The in-force probabilities are supplied directly as conditional probabilities for each scenario and date; do not multiply them together across dates. The task is to value the two specified contingent payments.

Step 1: Calculate the expected amount payable at the end of year 1.

$$ 0.20(100)(0.98)+0.50(200)(0.97)+0.30(400)(0.95) =19.6+97.0+114.0=230.6. $$

Discounting one year gives $230.6/1.05=219.619047\ldots$.

Step 2: Calculate the expected amount payable at the end of year 2.

$$ 0.20(500)(0.95)+0.50(700)(0.92)+0.30(900)(0.88) =95.0+322.0+237.6=654.6. $$

The year-2 present value is $654.6/(1.05)^2=654.6/1.1025=593.741496\ldots$.

Step 3: Add the present values.

\operatorname{EPV}=219.619047\ldots+593.741496\ldots =813.360544\ldots

Rounded to the nearest cent, the expected present value is $813.36. The method avoids both using the year-2 discount factor for the year-1 benefit and omitting the conditional in-force probabilities.

Checks and Practical Discipline

Before accepting a scenario valuation, confirm that the scenario probabilities sum to 1, that every payment has the correct date, and that each conditional payment probability matches the stated conditioning information. For each payment date, its scenario-weighted expected amount should be between the smallest and largest scenario-specific expected amounts for that date. With nonnegative benefits and probabilities no greater than 1, applying an in-force condition cannot increase the expected payment relative to valuing the same benefits as certain.

Keep the distinction between a scenario probability and a conditional payment probability explicit. The former weights the scenario itself. The latter describes whether payment occurs after that scenario is known. Multiplying both is appropriate when both conditions must hold, but neither should be counted twice. Likewise, use probabilities at the payment date: if annual mortality probabilities are given and mortality is independent from year to year conditional on the scenario, multiply annual survival probabilities to obtain survival to the payment date. If a survival probability to that date is already supplied, do not derive it again.

Key takeaway: For each payment date, calculate the conditional expected payment within each scenario, weight by the scenario probability, and discount at that payment date. Sum across dates only after each contingent amount has been valued on the same valuation basis.

Check Your Understanding

For each problem, use the stated payment timing and conditioning assumptions. Answers are rounded to the nearest cent.

A benefit of $800 at the end of year 3 occurs with probability 0.25; otherwise the benefit is $400. The effective annual interest rate is 4%, and payment is certain once the scenario is known. Find the expected present value. (Answer: $444.50)
A year-2 bonus is payable only if the policy remains in force. The effective annual interest rate is 3%, annual mortality is 4% in each year, and there is no lapse. Two scenarios have probabilities 0.60 and 0.40 and bonuses of $500 and $900, respectively. Mortality is independent of the scenario. Find the expected present value. (Answer: $573.34)
A year-1 bonus is payable if the policy is in force. The effective annual interest rate is 5%. Scenario A has probability 0.50, a bonus of $300, and an in-force probability of 0.99 at year 1. Scenario B has probability 0.50, a bonus of $700, and an in-force probability of 0.95. Find the expected present value. (Answer: $458.10)
A hypothetical policy pays bonuses at the ends of years 1 and 3. The effective annual interest rate is 4%, and payment is certain. Two scenarios have probabilities 0.25 and 0.75. Their year-1 bonuses are $100 and $200, and their year-3 bonuses are $500 and $1,000, respectively. Find the total expected present value. (Answer: $946.14)
A year-2 benefit is payable only if the policy is in force. The effective annual interest rate is 5%. Three scenarios have probabilities 0.20, 0.30, and 0.50; their benefits are $400, $800, and $1,000, and their conditional year-2 in-force probabilities are 0.98, 0.94, and 0.90, respectively. Find the expected present value. (Answer: $683.90)

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