Scenario-Weighted Values of Equity-Linked Benefits

Why Weight Values Across Scenarios?

For an equity-linked policy, a benefit may depend on a fund value that is uncertain at the valuation date. A deterministic investment scenario specifies one possible path for that fund. If several such paths are assigned scenario weights, the actuarial present value can be calculated by valuing the benefit under each path and combining those values using the weights.

The order matters. First project the fund under each scenario, then apply the contract’s benefit formula, then account for the probability and timing of payment, and finally discount at the valuation interest rate. Scenario investment returns determine the benefit amount; they do not, by themselves, determine the discount rate.

Definition: Suppose there are $m$ deterministic investment scenarios. Scenario $j$ has weight $w_j$, with $w_j\geq 0$ and $\sum_{j=1}^{m}w_j=1$. If $Z_j$ is the present value of the policy benefit under scenario $j$, the scenario-weighted actuarial present value is $\sum_{j=1}^{m}w_jZ_j$. When the weights represent probabilities, this is the expected present value over the stated scenario set.

The weights must be interpreted according to the valuation specification. In this tutorial they are probability weights for mutually exclusive scenarios that cover all possibilities in the model. A scenario weight is not a benefit amount, a mortality probability, or an interest rate. If the specified weights do not sum to one, they cannot be treated as probabilities without further instructions.

Calculate Within Each Scenario, Then Combine

Let $B_j(t)$ denote the amount payable at time $t$ under scenario $j$, if the payment is due. For a deterministic payment at time $n$, and an effective annual valuation interest rate $i$, the present value under scenario $j$ is $v^nB_j(n)$, where $v=1/(1+i)$. Its scenario-weighted value is therefore:

\operatorname{APV}=\sum_{j=1}^{m} w_jv^nB_j(n).

When payment depends on death or survival, the scenario-specific present value must also include the probabilities of the relevant payment events. For example, in a two-year term with death benefits at the end of each year and a maturity benefit at the end of year two, let $q_1$ be the first-year death probability and $q_2$ the second-year death probability conditional on surviving year one. If the benefit amounts under scenario $j$ are $D_{j,1}$, $D_{j,2}$, and $M_j$, then:

Z_j = v q_1D_{j,1} + v^2\,{}_1p_x q_2D_{j,2} + v^2\,{}_1p_x{}_1p_{x+1}M_j.

Here the probabilities are assumed to be the same across investment scenarios. The final value is $\sum_j w_jZ_j$. If a valuation specification instead gives mortality probabilities that vary by scenario, use the corresponding probabilities inside each $Z_j$ before applying the scenario weights.

Key distinction: A scenario return changes the projected fund and therefore may change the benefit. The valuation rate $i$ discounts the resulting payment to the valuation date. Use the same valuation basis in every scenario unless the problem explicitly specifies a different basis.

Why the Weighted-Value Formula Holds

The formula follows from conditioning on which scenario occurs. Let $J$ identify the scenario, with $P(J=j)=w_j$. Let $Z$ be the present value of the benefit, including any death or survival contingency. Under scenario $j$, its conditional expected value is $E[Z\mid J=j]=Z_j$, where $Z_j$ already averages over the relevant policy payment events.

Partition by scenario.
The scenarios are mutually exclusive and exhaustive, so the events $J=1,\ldots,J=m$ partition all outcomes in the model.

Apply total expectation.
By conditioning on this partition, $E[Z]=\sum_{j=1}^{m}P(J=j)E[Z\mid J=j]$.

Insert the scenario weights and values.
Since $P(J=j)=w_j$ and $E[Z\mid J=j]=Z_j$, the result is $E[Z]=\sum_{j=1}^{m}w_jZ_j$.

This argument also explains why a guarantee must be applied before scenario values are averaged. A nonlinear payoff function $g$ generally satisfies $\sum_jw_jg(F_j)\ne g(\sum_jw_jF_j)$. In particular, averaging fund values first can erase the effect of a floor, cap, or participation rule.

Worked Example: A One-Year Maturity Guarantee

A hypothetical policy has a fund of $10,000 at the start of the year. At the end of the year, the fund earns one of three deterministic returns: $-10\%$, $5\%$, or $20\%$, with weights $0.25$, $0.50$, and $0.25$, respectively. The maturity benefit is the greater of the fund and $10,500. The effective annual valuation interest rate is $4\%$. Calculate the actuarial present value.

Method. Project the fund in each scenario, apply the guarantee to each projected value, and then weight and discount the resulting benefits.

Under the $-10\%$ scenario, the fund is $10{,}000(1-0.10)=\$9{,}000$, so the benefit is $\max(9{,}000,10{,}500)=\$10{,}500$.

Under the $5\%$ scenario, the fund is $10{,}000(1+0.05)=\$10{,}500$, so the benefit is $\$10{,}500$.

Under the $20\%$ scenario, the fund is $10{,}000(1+0.20)=\$12{,}000$, so the benefit is $\$12{,}000$. The weighted maturity benefit is:

0.25(10{,}500)+0.50(10{,}500)+0.25(12{,}000) =2{,}625+5{,}250+3{,}000 =\$10{,}875.

The one-year discount factor is $v=1/1.04$. Thus:

\operatorname{APV} =\frac{10{,}875}{1.04} =\$10{,}456.7308\ldots =\boxed{\$10{,}456.73}.

The check is that the weighted benefit lies between the smallest and largest scenario benefits, and its discounted value is lower than the weighted benefit because the payment is one year in the future. The common mistake avoided here is applying the guarantee to the weighted-average fund: that would give $\max(10{,}500,10{,}500)=\$10{,}500$, not the correct weighted benefit of $10,875.

Combining Scenario Weights and Mortality

For contingent benefits, a scenario-specific value is not simply the discounted benefit at each possible time. Each payment must also be multiplied by the probability that it is due. The death probabilities in the next example are conditional on being alive at the start of the relevant year; the maturity benefit is payable only if the policyholder survives both years.

Worked Example: Death and Maturity Benefits Across Two Scenarios

A hypothetical two-year policy pays at the end of the year of death, if death occurs during the term. If the insured survives to the end of year two, it pays a maturity benefit instead. Under scenario A, the funds at the ends of years one and two are $9,500 and $10,000. Under scenario B, they are $12,000 and $14,000. The death benefit at either year-end is the greater of the fund and $11,000; the year-two maturity benefit follows the same rule. Scenario weights are $0.40$ for A and $0.60$ for B. The first-year death probability is $q_1=0.02$; the second-year conditional death probability is $q_2=0.03$. The effective annual valuation rate is $5\%$. Calculate the actuarial present value.

Method. For each scenario, determine the payment amounts and multiply each by the probability and discount factor for its payment event. Then weight the two scenario values.

The probabilities of the three payment events are:

P(\text{death in year 1})=0.02,\quad P(\text{death in year 2})=(1-0.02)(0.03)=0.0294,

P(\text{survive to maturity})=(1-0.02)(1-0.03)=0.9506.

The probabilities sum to $0.02+0.0294+0.9506=1$, as required. In scenario A, each relevant benefit is $\max(F,11{,}000)=\$11{,}000$. Since the year-two death and maturity payments have the same amount, the scenario value is:

Z_A =11{,}000\left(0.02v+(0.0294+0.9506)v^2\right) =11{,}000\left(\frac{0.02}{1.05}+\frac{0.98}{1.05^2}\right) =\$9{,}987.30.

In scenario B, the year-one death benefit is $12,000, while the year-two death and maturity benefits are both $14,000. Therefore:

Z_B =12{,}000(0.02)v+14{,}000(0.0294+0.9506)v^2 =\frac{240}{1.05}+\frac{13{,}720}{1.1025} =\$12{,}673.02.

The scenario-weighted value is:

\operatorname{APV} =0.40(9{,}987.30)+0.60(12{,}673.02) =\boxed{\$11{,}598.73}.

The unrounded scenario values are used in the weighted calculation. The result is between the two scenario-specific values, which provides a useful check. A common error is to multiply the second-year death probability by $q_2$ alone; this omits the required probability of surviving year one.

Apply Path-Dependent Rules Within Each Scenario

Some benefits depend on more than the fund at the payment date. For example, a benefit may credit part of a positive increase during the final year, or use a maximum fund value reached earlier. In that case, retain the required path information for each scenario. Do not replace the path with an average fund before applying the benefit rule.

Worked Example: A Capped Benefit Based on the Final-Year Increase

A hypothetical two-year contract starts with a fund of $10,000. Its maturity benefit is the least of $16,000 and the greater of $10,000 or the year-two fund plus $25\%$ of any positive increase from year one to year two. Two deterministic paths are specified: scenario A has funds of $9,000 and $11,000 at the ends of years one and two, with weight $0.30$; scenario B has funds of $12,000 and $10,500, with weight $0.40$; scenario C has funds of $10,000 and $16,000, with weight $0.30$. The effective annual valuation interest rate is $4\%$. Calculate the actuarial present value.

Method. Apply the path-dependent formula to each ordered pair of fund values, then weight the resulting maturity benefits and discount two years.

In scenario A, the final-year increase is $11{,}000-9{,}000=\$2{,}000$. The credited amount is $0.25(2{,}000)=\$500$, so the benefit before the cap is $\max(10{,}000,11{,}000+500)=\$11{,}500$. The cap does not bind.

In scenario B, the final-year increase is $10{,}500-12{,}000=-\$1{,}500$, so the credited positive increase is zero. The benefit is $\max(10{,}000,10{,}500)=\$10{,}500$.

In scenario C, the final-year increase is $16{,}000-10{,}000=\$6{,}000$, and the credited amount is $0.25(6{,}000)=\$1{,}500$. The uncapped benefit is $\$17{,}500$, so the cap reduces it to $\$16{,}000$.

The weighted maturity benefit is:

0.30(11{,}500)+0.40(10{,}500)+0.30(16{,}000) =3{,}450+4{,}200+4{,}800 =\$12{,}450.

Discounting for two years gives:

\operatorname{APV} =\frac{12{,}450}{1.04^2} =\frac{12{,}450}{1.0816} =\boxed{\$11{,}510.72}.

The payment is below the weighted maturity benefit, consistent with positive interest. The common mistake avoided is using only the year-two fund in the benefit formula: doing so would miss the contract’s dependence on the year-one value.

Common Traps and Practical Checks

Scenario-weighted calculations are manageable when the components are kept distinct. The following checks help identify errors before a final value is reported.

Check the weights. For probability scenarios, verify that all weights are nonnegative and sum to one.
Apply contract terms scenario by scenario. Floors, caps, and path-dependent credits can make the payoff nonlinear, so benefit at the average fund need not equal average benefit.
Use the correct payment-event probability. A later-year death probability is conditional on survival to that year. Multiply by the probability of reaching that year when calculating an unconditional payment probability.
Discount by payment time. A year-two payment uses $v^2$, even if the investment path has different returns in the two years.
Do not discount at the scenario return. The scenario return determines the projected fund; the valuation interest rate discounts the payment under the stated valuation basis.
Check units and timing. Confirm whether a benefit is paid at the start or end of a year, and whether a stated fund value is before or after any contract deduction.

This method is useful whenever benefits depend on investment outcomes but the valuation is based on a specified finite set of deterministic paths. It gives a transparent bridge from projected scenario outcomes to a single actuarial present value, while keeping investment assumptions, contract rules, payment contingencies, and discounting separate. The approach is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, apply the benefit rule within each scenario before weighting. Round dollar amounts to the nearest cent.

A one-year policy has a $10,000 fund. Its year-end fund returns are $-10\%$, $8\%$, and $20\%$, with weights $0.20$, $0.50$, and $0.30$. The maturity benefit is the greater of the fund and $10,200, and the valuation rate is $5\%$. Find the actuarial present value. (Answer: $10,514.29)
A two-year policy pays the greater of its year-two fund and $9,000 at maturity. Scenario A has year-two fund $8,500 and weight $0.60$; scenario B has year-two fund $11,000 and weight $0.40$. At an effective annual valuation rate of $4\%$, find the value. (Answer: $9,060.65)
A two-year policy pays $5,000 at the end of the year of death, or $6,000 at the end of year two if the insured survives. The conditional death probabilities are $q_1=0.10$ and $q_2=0.20$, and the effective annual valuation rate is $5\%$. Calculate the actuarial present value. (Answer: $5,210.88)
A two-year maturity benefit is the greater of $10,000 and $F_2+0.50\max(F_2-F_1,0)$. The two equally weighted fund paths $(F_1,F_2)$ are $(\$9{,}000,\$11{,}000)$ and $(\$12{,}000,\$10{,}000)$. The effective annual valuation rate is $4\%$. Find the actuarial present value. (Answer: $10,170.12)
At the end of year one, a hypothetical policy pays a death benefit if death occurred during the year; otherwise it pays a maturity benefit. Scenario A has weight $0.40$, death probability $0.02$, death benefit $12,000, and maturity benefit $10,000. Scenario B has weight $0.60$, death probability $0.05$, death benefit $15,000, and maturity benefit $11,000. The effective annual valuation rate is $5\%$. Find the actuarial present value. (Answer: $10,224.76)

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