Retrospective Reserves Under Gross Premium Funding

Why Gross Premiums Require an Expense Adjustment

A retrospective reserve measures the accumulated value of past cash flows for policies still in force. When a policyholder pays a gross premium, that payment is not all available to fund future benefits: the insurer also incurs expenses. A correct retrospective calculation therefore tracks the premium income, expenses, and insurance benefits separately, with each cash flow accumulated from its payment date to the valuation date.

The reserve is calculated for a specified cohort and duration. For a homogeneous cohort, divide the accumulated cohort fund by the number of policies remaining in force. The result is the retrospective reserve per survivor; it is not the cohort fund itself.

Definition: A gross-premium retrospective reserve at time $k$ is the accumulated value at time $k$ of gross premiums received through time $k$, less the accumulated value of expenses and insurance benefits paid through time $k$, divided by the number of policies in force at time $k$. The calculation uses the actual gross premium and the actual expense cash flows; it does not treat the expense portion of a premium as a separate asset.

For the annual model used here, premiums and expenses are paid at the beginning of each policy year, and death benefits are paid at the end of the year of death. The effective annual interest rate is $i$, and $v=1/(1+i)$. Let $N_j$ be the number of policies in force at time $j$, $D_j$ the number of deaths during year $j+1$, $P_j$ the gross premium per policy at time $j$, $E_j$ the expense per policy at that time, and $B_j$ the benefit per death at the end of the year. Assume no lapses or other cash flows unless stated.

The Annual Cohort-Fund Recursion

Let $F_j$ denote the cohort fund at time $j$, immediately after the previous year’s benefits have been paid and before premiums and expenses at time $j$. At issue, take $F_0=0$, unless an initial fund is explicitly given. After the year’s beginning-of-year premium and expense cash flows, interest accrues; then end-of-year benefits are paid.

F_{j+1}=\bigl(F_j+N_j(P_j-E_j)\bigr)(1+i)-D_jB_j, \qquad {}_jV^{\mathrm{ret,gross}}=\frac{F_j}{N_j}.

Here ${}_jV^{\mathrm{ret,gross}}$ denotes the retrospective reserve per policy in force at time $j$. The subscript identifies the valuation duration, and the superscript indicates that the fund reflects gross premiums and expenses. The recurrence requires $N_j>0$ to define a per-survivor reserve.

Timing condition: The factor $1+i$ applies to both the fund already held at time $j$ and the net cash flow $N_j(P_j-E_j)$, because both are available immediately after the beginning-of-year transactions. Benefits $D_jB_j$ are subtracted after interest because they are paid at year-end. If a premium, expense, or benefit is paid at a different time, its accumulation factor must change accordingly.

Why the Recursion Works

The recurrence follows directly from a one-year balance of cash flows. At time $j$, the cohort has fund $F_j$. It receives $N_jP_j$ in premiums and pays $N_jE_j$ in expenses at the same time, leaving $F_j+N_j(P_j-E_j)$ to earn interest. At the end of the year this amount has grown by $1+i$. The insurer then pays $D_jB_j$ in death benefits. The remaining amount is the fund $F_{j+1}$ carried forward.

Applying the recurrence repeatedly also gives a cumulative cash-flow expression. A beginning-of-year cash flow in year $j+1$ earns interest for $k-j$ years by time $k$; a year-end benefit from that year earns interest for $k-j-1$ years.

F_k = \sum_{j=0}^{k-1}N_j(P_j-E_j)(1+i)^{k-j} - \sum_{j=0}^{k-1}D_jB_j(1+i)^{k-j-1}.

This expression is another way to organize the same calculation, not a different reserve definition. It makes the timing visible: premiums and expenses are combined as a net beginning-of-year cash flow, while benefits are accumulated from year-end. The per-survivor reserve remains $F_k/N_k$.

Worked Calculations

Worked Example: A One-Year Gross-Premium Reserve

A hypothetical insurer issues 1,000 identical one-year policies. At time 0, each policy pays a gross premium of $120, and the insurer incurs an expense of $20 per policy. The effective annual interest rate is $i=0.04$. During the year, 100 policies result in a death benefit of $500 each; the other 900 policies remain in force at time 1. Find the retrospective reserve per survivor at time 1.

Method. Calculate the net beginning-of-year cash flow for the cohort, accumulate it for one year, subtract end-of-year benefits, and divide by the survivors.

Gross premium income at time 0 is $1{,}000(120)=\$120{,}000$. Expenses at time 0 are $1{,}000(20)=\$20{,}000$. The net amount invested is therefore $\$120{,}000-\$20{,}000=\$100{,}000$.

Accumulating the net amount for one year gives $\$100{,}000(1.04)=\$104{,}000$. The insurer pays $100(\$500)=\$50{,}000$ in benefits at time 1, leaving a cohort fund of $\$104{,}000-\$50{,}000=\$54{,}000$.

There are 900 survivors, so the reserve per survivor is $\$54{,}000/900=\$60.00$. Check: $900(\$60.00)=\$54{,}000$, the calculated cohort fund.

Common mistake avoided. Treating the gross premium as if no expenses were paid would give $(\$120{,}000(1.04)-\$50{,}000)/900=\$83.11$ per survivor. That calculation overstates the fund because it omits the $\$20{,}000$ expense outflow.

Worked Example: Two Years With Changing Premiums, Expenses, and Benefits

A hypothetical cohort begins with 500 policies. The annual effective interest rate is $i=0.02$. In year 1, the mortality rate is $q_x=0.08$, the gross premium is $70 per policy, the beginning-of-year expense is $12 per policy, and the end-of-year death benefit is $400. In year 2, the mortality rate is $q_{x+1}=0.10$, the gross premium is $90, the expense is $15, and the death benefit is $250. Find the retrospective reserve per survivor at time 2. Assume the stated cohort death counts are exact.

Method. Use the cohort recursion year by year, keeping the premium and expense cash flows separate before combining them.

In year 1, deaths are $500(0.08)=40$, so $N_1=500-40=460$. Gross premium income is $500(\$70)=\$35{,}000$, and expenses are $500(\$12)=\$6{,}000$. The net beginning-of-year amount is $\$29{,}000$, which accumulates to $\$29{,}000(1.02)=\$29{,}580$. Benefits are $40(\$400)=\$16{,}000$. Thus,

F_1=\$29{,}580-\$16{,}000=\$13{,}580.

In year 2, deaths are $460(0.10)=46$, leaving $N_2=460-46=414$ survivors. Gross premium income is $460(\$90)=\$41{,}400$, and expenses are $460(\$15)=\$6{,}900$. The net new cash flow is $\$34{,}500$. Add the existing fund: $\$13{,}580+\$34{,}500=\$48{,}080$. After interest, the amount is $\$48{,}080(1.02)=\$49{,}041.60$. Year-2 benefits are $46(\$250)=\$11{,}500$, so

F_2=\$49{,}041.60-\$11{,}500=\$37{,}541.60.

The reserve per survivor is $\$37{,}541.60/414=\$90.680193\ldots$, or $90.68 rounded to the nearest cent. Check: $414(\$90.680193\ldots)=\$37{,}541.60$. The method avoids mixing policy counts from different durations: year-2 premiums, expenses, and benefits apply to 460 policies at the start of year 2, while the final reserve is divided by 414 survivors.

Worked Example: Including an Initial Acquisition Expense

A hypothetical cohort has 200 policies at issue. The effective annual interest rate is $i=0.05$. At time 0, each policy pays a gross premium of $150 and incurs an acquisition expense of $30. During year 1, 10 policies die and each receives a $1,000 benefit. At time 1, each of the 190 survivors pays a gross premium of $160 and incurs an expense of $12. During year 2, 19 policies die and each receives a $900 benefit. Find the reserve per survivor at time 2.

Method. Treat the acquisition expense as a time-0 cash outflow, then apply the same annual fund balance to each policy year.

At time 0, the cohort’s premium income is $200(\$150)=\$30{,}000$, and its acquisition expense is $200(\$30)=\$6{,}000$. The net amount is $\$24{,}000$. After one year of interest it is $\$24{,}000(1.05)=\$25{,}200$. Year-1 benefits total $10(\$1{,}000)=\$10{,}000$, leaving $F_1=\$15{,}200$, with 190 survivors.

At time 1, gross premiums total $190(\$160)=\$30{,}400$, while expenses total $190(\$12)=\$2{,}280$. The net addition is $\$28{,}120$. Combining it with the existing fund gives $\$15{,}200+\$28{,}120=\$43{,}320$, which accumulates to $\$43{,}320(1.05)=\$45{,}486$. Year-2 benefits total $19(\$900)=\$17{,}100$, so $F_2=\$45{,}486-\$17{,}100=\$28{,}386$.

The number of survivors at time 2 is $190-19=171$. Therefore the reserve per survivor is $\$28{,}386/171=\$166.00$. Check: $171(\$166.00)=\$28{,}386$, exactly matching the cohort fund.

Interpreting the Result and Avoiding Traps

A retrospective reserve is a balance of past cash flows under the stated assumptions. Its value depends on the premium, expense, and benefit amounts and on when each cash flow occurs. In particular, a gross premium does not remove the need to account for expenses: gross premiums are the actual amounts collected, while expenses are separate cash outflows. Deducting an expense twice is also incorrect. Once the expense is deducted from the gross premium cash flow in a year, it should not be subtracted again from the fund.

A frequent denominator error is to divide by the initial cohort size rather than the number of policies in force at the valuation duration. The cohort fund is an aggregate amount; the reserve per policy at time $k$ is $F_k/N_k$. Another common error is to accumulate an end-of-year death benefit for an extra year, or to omit a year of interest on a beginning-of-year premium. Use the specified payment times before assigning accumulation factors.

Key takeaway: For a beginning-of-year gross premium and expense and an end-of-year benefit, add the current cohort’s net premium cash flow to the fund, accumulate by $1+i$, subtract that year’s benefits, and divide the resulting fund by the survivors at the valuation time.

Check Your Understanding

For each problem, use the stated annual timing convention: premiums and expenses are paid at the beginning of the year, and death benefits are paid at year-end.

A cohort has 300 policies. The annual effective interest rate is $i=0.03$; the gross premium is $80, the expense is $10, and the death benefit is $500 per death. If $q_x=0.04$, find the reserve per survivor at time 1. (Answer: $54.27 per survivor)
A cohort starts with 250 policies. In year 1, $q_x=0.08$, the gross premium is $110, the expense is $15, and the death benefit is $700. In year 2, $q_{x+1}=0.10$, the gross premium is $130, the expense is $18, and the death benefit is $800. If $i=0.04$, find the reserve per survivor at time 2. Assume exact cohort death counts. (Answer: $94.29 per survivor)
A cohort has 100 policies, with $i=0.05$, a year-1 gross premium of $200, an expense of $25, and a death benefit of $1,000. If $q_x=0.06$, find the time-1 reserve per survivor. (Answer: $131.65 per survivor)
A cohort starts with 100 policies. In each of two years $i=0.02$ and $q=0.10$. The year-1 gross premium, expense, and benefit are $100, $10, and $400; the year-2 amounts are $120, $15, and $500. Find the time-2 reserve per survivor. (Answer: $128.67 per survivor)
A cohort has 100 policies and $q=0.10$ in each of two years. The annual effective interest rate is $i=0.04$; the death benefit is $400 in each year, and the beginning-of-year expense is $10 per policy in each year. A level gross premium $P$ is paid at the start of each year. What value of $P$, rounded to the nearest cent, gives a retrospective reserve of $100 per survivor at time 2? (Answer: $88.61)

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