Including Expenses in Retrospective Reserves

Why Expenses Change a Retrospective Reserve

A retrospective reserve measures the accumulated past cash flows associated with policies that remain in force at a valuation date. A premium is not entirely available to fund benefits when the insurer also incurs expenses. An acquisition expense may be incurred at issue, while renewal expenses may be incurred when later premiums are collected. The reserve calculation must put each expense at its actual time and apply survival weights only when the cash flow depends on the policy still being in force.

The central bookkeeping distinction is between a gross premium, the amount collected from the policyholder, and the cash flow remaining after expenses. On a net-premium basis, expenses are excluded and the premium is treated as available for benefits. On a gross-premium basis, expenses are explicitly subtracted from gross premiums. The two bases should not be mixed.

Definition: For a cohort of policies issued at age $x$, a retrospective reserve at duration $t$ is the cohort's accumulated net cash flow through time $t$, divided by the expected number of policies from the original cohort still in force at time $t$. If premiums are collected at issue and at anniversaries while the policy is in force, acquisition expenses occur at issue, and renewal expenses occur with later premiums, each cash flow is weighted and accumulated according to its timing.

Cash-Flow Timing and the Cohort Formula

Assume an annual discrete model with effective annual interest rate $i$. A gross premium $P_0$ and acquisition expense $e_0$ occur at issue. For $k=1,\ldots,t-1$, a renewal premium $P_k$ and renewal expense $e_k$ are due at time $k$ only if the policy is still in force. A death benefit $b_{k+1}$ is paid at the end of policy year $k+1$ if death occurs in that year. Expenses in this formula are expenses paid by the insurer; they therefore reduce the accumulated cash available for the reserve.

Write ${}_kp_x$ for the probability a life aged $x$ survives $k$ years, and ${}_1q_{x+k}$ for the probability of death in the following year conditional on survival to age $x+k$. Under the stated timing assumptions, the retrospective reserve per policy in force at time $t$ is

{}_tV_x^R = \frac{ (P_0-e_0)(1+i)^t +\displaystyle\sum_{k=1}^{t-1}{}_kp_x(P_k-e_k)(1+i)^{t-k} -\displaystyle\sum_{k=0}^{t-1}{}_kp_x\,{}_1q_{x+k}\,b_{k+1}(1+i)^{t-k-1} }{{}_tp_x}.

The first term is the net cash flow at issue, accumulated to time $t$. A renewal cash flow at time $k$ is expected to occur for the fraction ${}_kp_x$ of the original cohort that reaches that anniversary, and it accumulates for $t-k$ years. A benefit paid at the end of year $k+1$ is expected for the fraction ${}_kp_x{}_1q_{x+k}$ of the original cohort that dies in that year. It accumulates from time $k+1$ to time $t$, or $t-k-1$ years. Finally, the denominator converts the expected accumulated amount per original policy into an amount per policy surviving to time $t$.

This expression assumes the policyholder pays each renewal premium and the insurer incurs its corresponding renewal expense only when the policy is in force at the premium date. A premium due at a different time, an expense payable on death, or an expense paid at a different point in the year requires a cash-flow term with the appropriate timing and probability weight. Do not apply this particular timing formula unchanged in those cases.

Set the cohort and valuation date.
Start with one policy at issue and identify the duration $t$ at which the reserve is measured.

Record net premium cash flows.
At issue use gross premium less acquisition expense. At each later premium date use gross premium less renewal expense, weighted by survival to that date.

Subtract expected benefits.
For each year, weight the benefit by the probability of death in that year and accumulate it from its payment date.

Condition on survival at the valuation date.
Divide the accumulated cohort amount by ${}_tp_x$, not by the survival probability for some earlier cash-flow date.

Worked Applications

Worked Example: Correct Survival Weights for an Earlier Renewal

A hypothetical two-year contract has no interest, so $i=0$. The gross premium is $90 at issue and $100 at time 1. The acquisition expense is $15 and the renewal expense is $8. The year-1 death probability is ${}_1q_x=0.20$; conditional on surviving year 1, the year-2 death probability is ${}_1q_{x+1}=0.25$. The death benefits are $120 in year 1 and $160 in year 2. Calculate the retrospective reserve at time 2, per policy then in force.

Method. Use the cohort accumulation formula. Since $i=0$, all accumulation factors are 1. First calculate survival: ${}_1p_x=1-0.20=0.80$, and ${}_2p_x=0.80(1-0.25)=0.80(0.75)=0.60$.

The net cash flow at issue is $90-15=75$. The time-1 net renewal cash flow is $100-8=92$, and it occurs for $0.80$ of the original cohort. Its expected cohort contribution is $0.80(92)=73.60$.

Expected year-1 benefits per original policy are $0.20(120)=24$. Expected year-2 benefits per original policy are $0.80(0.25)(160)=32$. Thus the accumulated net cash flow per original policy at time 2 is $75+73.60-24-32=92.60$. Divide by the proportion surviving to time 2:

{}_2V_x^R=\frac{92.60}{0.60}=154.333\ldots

Rounded to the nearest cent, the reserve is $154.33. Check: multiplying $154.333… by the time-2 survival proportion $0.60$ returns the cohort amount $92.60.

Common mistake avoided. The time-1 renewal cash flow is weighted by ${}_1p_x=0.80$, because it is due at time 1. Weighting it by ${}_2p_x=0.60$ incorrectly requires survival through the following year as well. That wrong weight would give a cohort amount of $75+0.60(92)-24-32=74.20$, and an incorrect reserve of $74.20/0.60=$123.67$.

Worked Example: Expenses With Interest and Unequal Benefits

A hypothetical two-year policy has an effective annual interest rate $i=0.05$. The gross premium is $200 at issue and $120 at time 1. Acquisition expense is $20, and renewal expense is $10. The year-1 death probability is $0.10$, and the conditional year-2 death probability is $0.20$. Death benefits are $300 in year 1 and $400 in year 2. Find the retrospective reserve at time 2.

Method. Net premium cash flows are $200-20=180$ at issue and $120-10=110$ at time 1. The survival probabilities are ${}_1p_x=0.90$ and ${}_2p_x=0.90(0.80)=0.72$.

Accumulate each term to time 2. The issue cash flow contributes $180(1.05)^2=180(1.1025)=198.45$. The time-1 renewal cash flow contributes $0.90(110)(1.05)=103.95$. The year-1 expected benefit is paid at time 1, so its accumulated amount is $0.10(300)(1.05)=31.50$. The expected year-2 benefit, paid at time 2, is $0.90(0.20)(400)=72.00$.

The accumulated net cash flow per original policy is $198.45+103.95-31.50-72.00=198.90$. Therefore,

{}_2V_x^R=\frac{198.90}{0.72}=$276.25.

Check: $0.72(276.25)=$198.90$, matching the accumulated cohort amount. The important timing check is that the year-1 benefit accumulates one year, whereas the year-2 benefit accumulates zero years.

Worked Example: Gross Premium Cash Flows Versus Net Premium Cash Flows

Consider a hypothetical two-year contract with $i=0$, year-1 death probability $0.20$, and conditional year-2 death probability $0.25$. It pays a $90 death benefit at the end of either year. Gross premiums of $30 are due at issue and time 1 while the policy is in force. Acquisition expense is $10, and renewal expense is $5. Find the time-2 retrospective reserve on a gross-premium basis, then compare it with the result using an expense-free level net premium.

Gross-premium basis. The net cash flow at issue is $30-10=20$. The time-1 net cash flow is also $30-5=25$, and $0.80$ of the original cohort reaches that premium date. Expected benefits are $0.20(90)=18$ in year 1 and $0.80(0.25)(90)=18$ in year 2. Thus the accumulated cohort amount is $20+0.80(25)-18-18=4$. Since ${}_2p_x=0.80(0.75)=0.60$, the reserve is $4/0.60=$6.67$, rounded to the nearest cent.

Expense-free net-premium basis. The expected present value of benefits at issue is $18+18=36$, because $i=0$. The expected number of premiums is $1+0.80=1.80$, so the level net premium is $36/1.80=$20$. Treating this net premium as available for benefits, the accumulated cohort amount at time 2 is $20+0.80(20)-18-18=0$; the corresponding retrospective reserve is $0/0.60=$0.00$.

These are different bases. In the gross-premium calculation, actual gross premiums are reduced by actual expenses. In the net-premium calculation, expenses are not included and the premium is the expense-free amount determined from benefits. Subtracting the expenses from the $20 net premium as well would count expenses on a basis that already excludes them.

Deriving the Cohort Adjustment

The survival weighting and final division follow directly from cohort counts. Imagine $N$ identical policies issued at age $x$. The expected number surviving to time $k$ is $N{}_kp_x$. Therefore, a renewal net cash flow $P_k-e_k$, due only to policies in force at time $k$, has expected cohort amount $N{}_kp_x(P_k-e_k)$. Dividing by $N$ gives the renewal term in the formula. The expected number of deaths during year $k+1$ is $N{}_kp_x{}_1q_{x+k}$, which gives the corresponding benefit term after division by $N$. The issue cash flow applies to all $N$ policies, so it has no survival weight.

After all these per-original-policy terms are accumulated to time $t$, the amount is still an average across the original cohort. Only $N{}_tp_x$ policies are expected to remain. Dividing the per-original-policy amount by ${}_tp_x$ is therefore equivalent to dividing the total cohort amount by the expected surviving count $N{}_tp_x$. This also explains why an earlier renewal is weighted by survival only to its own due date, while the reserve itself is conditioned on survival to the valuation date.

Common Traps and Practical Checks

Using the valuation-date survival probability for every premium. A renewal at time $k$ is weighted by ${}_kp_x$, not ${}_tp_x$, unless $k=t$. Apply each cash flow's own timing.
Forgetting the expense sign. An insurer-paid expense reduces available funds, so it is subtracted from premium cash flow. It is not a premium inflow.
Subtracting acquisition expense every year. An acquisition expense due only at issue appears once. Renewal expenses appear at their specified renewal dates.
Accumulating a benefit for the wrong number of years. A benefit paid at the end of year $k+1$ accumulates $t-k-1$ years to time $t$. A benefit paid at the valuation date has no accumulation factor beyond 1.
Mixing premium bases. Use gross premiums less expenses on a gross basis. Use net premiums without subtracting expenses again on an expense-free basis.

A useful final check is to state what the numerator represents: it should be the accumulated net cash flow per policy originally issued. Then verify that the denominator is the probability of being in force at the valuation date. This topic is aligned with the SOA Exam FAM (Fundamentals of Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, identify the premium and expense timing before calculating. Round monetary answers to the nearest cent.

A two-year policy has $i=0$. At issue, the gross premium is $60 and acquisition expense is $12. At time 1, the gross premium is $60 and renewal expense is $6. Death probabilities are $0.25$ in year 1 and $0.20$ in year 2 conditional on survival. Benefits are $80 and $100 at the ends of years 1 and 2. Find the time-2 retrospective reserve. (Answer: $89.17)
A two-year policy has $i=0.04$. Its issue premium and acquisition expense are $150 and $30; its time-1 premium and renewal expense are $100 and $10. Conditional death probabilities are $0.10$ in each year, and the death benefit is $200 at either year-end. Find the time-2 reserve. (Answer: $216.34)
A policy has $i=0$, issue gross premium $50, acquisition expense $5, and gross premiums of $40 with renewal expense $4 at times 1 and 2 while in force. The annual death probabilities are $0.20$, $0.25$, and $0.10$, conditional on reaching each year. Benefits at the ends of years 1, 2, and 3 are $60, $80, and $100. Find the time-3 retrospective reserve. (Answer: $113.70)
A two-year policy has $i=0$, death probabilities $0.20$ in year 1 and $0.25$ in year 2 conditional on survival, and a $90 death benefit in either year. Gross premiums of $30 are due at issue and time 1; acquisition expense is $10 and renewal expense is $5. Find the time-2 reserve on a gross-premium basis, then find the level expense-free net premium and its time-2 retrospective reserve. (Answer: Gross-basis reserve $6.67; net premium $20.00; expense-free reserve $0.00)

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