Worked Projection of a Unit-Linked Death Benefit

Projecting a Unit-Linked Death Benefit

A unit-linked policy’s death benefit may depend on an investment fund that changes as premiums are allocated, investment returns are credited, and charges are deducted. If the contract promises at least a fixed sum assured, the benefit at death is not necessarily the fund alone: the insurer pays whichever is greater. A correct projection must therefore get both the fund calculation and the benefit comparison right.

This tutorial uses a deterministic fund projection: the investment return and cash flows are specified, so each projected fund value is calculated directly. Mortality probabilities can then be applied to the projected benefit amounts to find expected claims. The projection does not imply that actual investment returns or the timing of death are known in advance.

Definition: Let $F_t$ be the fund value at the end of period $t$, immediately after that period’s specified fund charge. Let $P_t$ be the premium paid at the beginning of period $t$, $a_t$ the proportion allocated to the fund, $j_t$ the effective investment return credited over the period, and $C_t$ a charge deducted from the fund at the end of the period. Under these timing assumptions, $$ F_t=(F_{t-1}+a_tP_t)(1+j_t)-C_t. $$ If death at the end of period $t$ produces a benefit equal to the greater of the fund and a fixed sum assured $S$, then $$ B_t=\max(F_t,S). $$

The recurrence assumes that the premium is paid before investment growth, only the allocated part of the premium enters the fund, and the stated charge is deducted after growth. If a contract specifies a different order, the recurrence must be changed to match it. For example, a charge deducted before growth would itself earn a return only if the contract says so. The examples below assume no lapse, no additional cash flows, and no fund deductions other than those stated.

Why the Fund Recurrence Has This Form

The recurrence follows directly from the period’s cash-flow sequence. The opening fund $F_{t-1}$ is combined with the allocated premium $a_tP_t$. This amount earns the period’s investment return, so it is multiplied by $1+j_t$. The end-of-period charge $C_t$ is then subtracted. Thus the value immediately after the charge is the result shown in the recurrence.

Add the allocated premium.
The fund immediately after the premium is $F_{t-1}+a_tP_t$, not $F_{t-1}+P_t$, unless the full premium is allocated.

Credit investment growth.
With effective return $j_t$ for the period, the fund before the end-period charge is $(F_{t-1}+a_tP_t)(1+j_t)$.

Deduct the stated charge, then compare benefits.
Subtract $C_t$ to obtain $F_t$, and use $\max(F_t,S)$ if the death benefit is the greater of the fund and sum assured.

The benefit can also be separated into two components: the fund value and any insurer-funded top-up. The top-up is the amount by which the sum assured exceeds the fund, if it does:

B_t=F_t+\max(S-F_t,0).

This identity is useful for checking a projection. If $F_t$ exceeds $S$, the top-up is zero and the benefit is the fund. If $F_t$ is below $S$, the top-up fills the difference and the total benefit is exactly $S$.

One Period: Allocation, Growth, Charge, and Benefit

Worked Example: Finding the Year-End Fund and Guarantee Top-Up

A hypothetical policy has an opening fund of $10,000. A $2,000 premium is paid at the beginning of the year, and 90% is allocated to the fund. The fund earns an effective annual return of 5%. A $300 charge is deducted at year-end. The death benefit at year-end is the greater of the fund and a $15,000 sum assured. Find the projected fund, the death benefit, and the insurer-funded top-up if death occurs at year-end.

Method. Apply the cash flows in their stated order: allocate the premium, credit the annual return, then deduct the year-end charge.

The allocated premium is $0.90(2{,}000)=\$1{,}800$. The fund immediately after the premium is therefore $10{,}000+1{,}800=\$11{,}800$. After the 5% return, it is $11{,}800(1.05)=\$12{,}390$. After the charge, the year-end fund is $12{,}390-300=\$12{,}090$.

F_1=(10{,}000+0.90(2{,}000))(1.05)-300 =12{,}090.

The benefit is $\max(12{,}090,15{,}000)=\$15{,}000$. The top-up is $15{,}000-12{,}090=\$2{,}910$, which checks because $12{,}090+2{,}910=\$15{,}000$.

Common mistake avoided: Adding the full $2,000 premium to the fund would ignore the 10% not allocated. The correct addition is $1,800, and the resulting fund—not the opening fund or pre-charge fund—is compared with the sum assured.

Periodic Charges and Repeated Premiums

For monthly projections, the same logic applies one month at a time. Use the return for a month, not the annual effective return, and apply the charge at the timing specified. If each month has the same allocated premium, return, and charge, the recurrence can be repeated for each month; it is not necessary to combine the cash flows into an annual calculation.

Worked Example: Three Monthly Periods with a Fund Charge

A hypothetical policy starts with a fund of $3,000. At the beginning of each of the next three months, the policyholder pays a $200 premium, of which 90% is allocated to the fund. The fund earns 1% per month, and a $10 charge is deducted at the end of each month. Find the fund after month 3 and the month-3 death benefit if the sum assured is $3,500. Assume the monthly rate is an effective rate for each monthly period.

Method. In each month, add the allocated premium of $0.90(200)=\$180$, multiply by $1.01$, and subtract $10. Retain full precision until the final reported amount.

For month 1, the fund is $(3{,}000+180)(1.01)-10=3{,}211.80-10=\$3{,}201.80$.

For month 2, the opening fund is $3,201.80. Thus the fund is $(3{,}201.80+180)(1.01)-10=3{,}415.618-10=\$3{,}405.618$.

For month 3, the fund is $(3{,}405.618+180)(1.01)-10=3{,}621.47418-10=\$3{,}611.47418$. To the nearest cent, the projected fund is $3,611.47.

\begin{aligned} F_1&=(3{,}000+180)(1.01)-10=3{,}201.80,\\ F_2&=(3{,}201.80+180)(1.01)-10=3{,}405.618,\\ F_3&=(3{,}405.618+180)(1.01)-10=3{,}611.47418. \end{aligned}

The month-3 death benefit is $\max(3{,}611.47418,3{,}500)=\$3{,}611.47$, to the nearest cent. The fund exceeds the sum assured, so there is no guarantee top-up.

Common mistake avoided: The $10 charge is taken after the monthly return, as specified. Deducting it before growth would produce a different projection. Also, applying 1% just once to the opening fund would omit two monthly periods of premium, growth, and charges.

Projecting Benefits Across Years and Applying Mortality

A deterministic fund projection gives the benefit in each period conditional on the policy being in force at the start of that period. To calculate expected claims from issue, multiply each period’s benefit by the probability that death occurs in that period, measured from issue. This is not generally the same as multiplying every benefit by its conditional death probability alone: later-period claims also require survival through the earlier periods.

Suppose the conditional probability of death during year $t$, given survival to the start of that year, is $q_t$. The probability from issue of death in year $t$ is the probability of surviving the preceding years times $q_t$. If ${}_{t-1}p_x$ denotes the probability of surviving $t-1$ years from age $x$, this is ${}_{t-1}p_xq_t$, where $q_t$ is the conditional one-year death probability at the start of year $t$. The expected undiscounted claims are the sum of benefit amounts weighted by these issue-date death probabilities.

\operatorname{E}[\text{claims}] =\sum_{t=1}^{n} {}_{t-1}p_x q_t B_t, \qquad {}_0p_x=1.

This calculation assumes the projected benefits are paid at the stated period-end amounts, the given mortality probabilities apply to the in-force policy, and there are no other decrements or claim adjustments. It is an undiscounted expected amount; if a present value is required, each period’s expected claim must also be discounted using the specified interest basis.

Worked Example: Three-Year Projection and Expected Claims

A hypothetical policy has no opening fund. At the beginning of each year, a $1,000 premium is paid and 90% is allocated. The fund earns an effective annual return of 5%, and a $100 charge is deducted at each year-end. The death benefit at the end of each year is the greater of the fund and a $2,000 sum assured. Conditional probabilities of death during years 1, 2, and 3, given survival to the start of the respective year, are 0.02, 0.03, and 0.04. Find the year-end funds, death benefits, and total expected undiscounted claims from issue.

Method. First project the fund through all three years. Then find the death benefit in each year. Finally, weight each benefit by the probability of reaching that year alive and dying during it.

The allocated premium each year is $0.90(1{,}000)=\$900$. Starting with $F_0=0$, the annual projections are:

Year 1: $F_1=(0+900)(1.05)-100=945-100=\$845$.

Year 2: $F_2=(845+900)(1.05)-100=1{,}832.25-100=\$1{,}732.25$.

Year 3: $F_3=(1{,}732.25+900)(1.05)-100=2{,}763.8625-100=\$2{,}663.8625$.

The first two funds are below the $2,000 sum assured, so $B_1=\$2{,}000$ and $B_2=\$2{,}000$. The third fund exceeds the sum assured, so $B_3=\$2{,}663.8625$, or $2,663.86 to the nearest cent.

The issue-date probability of death in year 1 is $0.02$. For year 2, survival through year 1 is $1-0.02=0.98$, so the probability is $0.98(0.03)=0.0294$. For year 3, survival through years 1 and 2 is $0.98(1-0.03)=0.98(0.97)=0.9506$, so the probability of death in year 3 is $0.9506(0.04)=0.038024$.

The expected claims are therefore $0.02(2{,}000)+0.0294(2{,}000)+0.038024(2{,}663.8625)$. The three contributions are $40.00, $58.80, and $101.2907077, respectively. Their sum is $200.0907077, or $200.09 to the nearest cent.

\begin{aligned} \operatorname{E}[\text{claims}] &=0.02(2{,}000)+0.98(0.03)(2{,}000)\\ &\quad+0.98(0.97)(0.04)(2{,}663.8625)\\ &=40+58.8+101.2907077\\ &=200.0907077\approx \$200.09. \end{aligned}

Check. The death probabilities from issue sum to $0.02+0.0294+0.038024=0.087424$, which is less than 1 because some policyholders survive all three years. Each year’s benefit is multiplied by its own issue-date death probability, and the expected amount is positive and below the largest individual benefit, as it should be.

Common mistake avoided: Using 0.03 and 0.04 directly as issue-date probabilities would omit survival to years 2 and 3. The correct year-3 probability is $0.98(0.97)(0.04)=0.038024$, not 0.04.

Checks That Prevent Projection Errors

A reliable projection is as much about timing and interpretation as arithmetic. Before calculating, record when premiums enter the fund, what portion is allocated, when growth is credited, and when each charge is deducted. Keep the fund calculation separate from the benefit comparison: a minimum death benefit can make the claim exceed the fund, while a fund larger than the sum assured makes the fund itself the benefit.

For each period, check that the projected fund follows the specified transaction order. Then calculate $\max(F_t,S)$, not the sum $F_t+S$. If calculating expected claims, use conditional mortality only after accounting for survival to the start of the period. Do not round intermediate fund values unless the contract explicitly requires rounding at each transaction; otherwise, round the final reported money amount to cents.

Key takeaway: Project the fund using the contract’s timing, compare the resulting fund with the sum assured to determine the death benefit, and weight each period’s benefit by the probability of death in that period from issue. These are separate steps, and each has its own check.

Check Your Understanding

For each problem, follow the stated cash-flow timing and keep full precision until the final money amount.

A policy has an opening fund of $2,400. A $600 premium is paid at the beginning of the year, 92% is allocated, the fund earns 3%, and an $18 charge is deducted at year-end. The death benefit is the greater of the year-end fund and $3,200. Find the fund and benefit. (Answer: fund $3,022.56; benefit $3,200.00)
A policy starts with $1,000. For each of two months, a $100 premium is paid at the beginning, 95% is allocated, the fund earns 2% per month, and a $5 charge is deducted at month-end. Find the month-2 fund. (Answer: $1,226.04)
A projected policy has death benefits of $1,500 in year 1 and $1,800 in year 2. Conditional death probabilities are 0.01 in year 1 and 0.02 in year 2, given survival to the start of each year. Calculate expected undiscounted claims from issue. (Answer: $50.64)
A policy’s projected death benefits are $2,200, $2,200, and $2,600 in years 1, 2, and 3. Conditional death probabilities are 0.02, 0.03, and 0.05. Calculate the total expected undiscounted claims from issue. (Answer: $232.26)
A policy has an opening fund of $5,000. A $1,000 premium is paid at the beginning of the year, 90% is allocated, the fund earns 4%, and a $50 charge is deducted at year-end. The death benefit is the greater of the year-end fund and $6,000. Find the year-end fund and benefit, and state why the charge is not applied before growth under these assumptions. (Answer: fund $6,086.00; benefit $6,086.00; the stated timing deducts the charge after growth.)

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