Acquisition And Renewal Expenses

Why Policy Duration Matters for Expenses

A long-term contract can generate costs at issue and again while it remains in force. The issue process may involve application handling, underwriting, and policy setup. Later policy years may involve servicing, billing, and ongoing administration. These costs need not have the same amount, and an expense assumption may change with policy duration.

For annual profit calculations, the key tasks are to assign each expense to the policy year specified by the assumption and to use the exposure for that year. An acquisition expense is associated with issuing a new policy; a renewal expense is associated with a policy continuing into a later year. If fewer policies remain in force, fewer policies generally generate renewal expenses. The annual profit effect is negative because an expense is an insurer cash outflow.

Definition: An acquisition expense is an expense assigned to issuing a new policy, commonly recorded in policy year 1 under the stated accounting basis. A renewal expense is an expense assigned to maintaining a policy in a later policy year. Let $e_t$ denote the expense per policy exposed to expense in policy year $t$. If $N_t$ policies are exposed in that year, the expected or actual total expense is $N_t e_t$, and its annual profit effect is $-N_t e_t$.

The policy-year assignment is determined by the problem's assumptions. In particular, do not infer a payment date from the words “acquisition” or “renewal.” The expense payment timing and any investment income earned or lost depend on separate timing assumptions. Here, the focus is the expense amount assigned to each policy year and its effect on that year's profit.

A convenient schedule separates the issue-year amount from later amounts. For example, $e_1$ might be an acquisition expense, while $e_2,e_3,\ldots$ are renewal expenses that vary by duration. A constant renewal assumption has $e_t=e$ in every renewal year; a duration-dependent assumption gives different values in different years or specifies a rule for them.

Exposure and the Annual Profit Effect

An expense stated per policy is multiplied by the number of policies exposed to that expense. If the expense applies to every policy in force at the start of policy year $t$, the relevant exposure is the number in force at that point. A renewal expense should not automatically be multiplied by the number of policies originally issued: some may have terminated before the renewal year.

Suppose a cohort begins with $N_0$ identical policies. Assume no lapses, a common annual death probability $q$, and independent mortality across policies. Write $p=1-q$. If a policy is exposed to expenses in year $t$ only when it survives to the start of that year, its probability of being in force then is $p^{t-1}$. The expected number exposed is therefore $N_0p^{t-1}$.

E[\text{expense in year }t]=N_0p^{t-1}e_t, \qquad \text{expected profit effect in year }t=-N_0p^{t-1}e_t.

This exposure formula uses survival alone. If the problem specifies lapses or other termination events, the probability of being in force must reflect those assumptions as well. If expense exposure is defined on a different basis—for example, policies in force at year-end—the exposure factor must also be changed accordingly.

Deriving the Expected Cohort Expense

The survival-weighted expense follows directly by counting exposed policies. For policy $j$, define $I_{j,t}=1$ if it is in force at the start of year $t$, and $I_{j,t}=0$ otherwise. Its expense in that year is $I_{j,t}e_t$. The total cohort expense is the sum of these amounts across policies.

E\!\left[\sum_{j=1}^{N_0} I_{j,t}e_t\right] =e_t\sum_{j=1}^{N_0}E[I_{j,t}] =e_t\sum_{j=1}^{N_0}P(I_{j,t}=1) =e_tN_0p^{t-1}.

The first equality uses the fact that $e_t$ is fixed under the expense assumption. The second uses linearity of expectation, so independence among policies is not needed for this expected-value calculation. The last equality uses the assumption that each policy has the same probability $p^{t-1}$ of surviving to the start of year $t$. Since the resulting amount is an outflow, its contribution to profit has the opposite sign.

Classify the expense.
Identify which amounts are assigned to new business and which are assigned to later policy durations.

Build the duration schedule.
Write the expense per exposed policy for each relevant policy year, applying any stated duration rule.

Find the exposure.
Use the number in force on the stipulated exposure basis, allowing for mortality and any other stated termination assumptions.

Calculate the profit item.
Multiply expense per exposed policy by exposure, then use a negative sign for the insurer's profit effect.

Worked Expense Calculations

Worked Example: A Cohort With Increasing Renewal Expenses

A hypothetical insurer issues 600 identical annual policies. The acquisition expense is $420 per policy in year 1. Renewal expense is $75 per exposed policy in year 2 and $90 in year 3. Each policy has an independent annual death probability of $q=0.04$, with no lapses. Expenses apply to policies in force at the start of each year. Calculate the expected expense and profit effect in each of the first three years and the cohort total. Use an effective annual interest rate of $i=0\%$; no discounting is required for these annual profit items. Expenses are deterministic per exposed policy, with mortality as specified and no additional expense distribution assumed.

Method. Calculate the expected number in force at each year's start, multiply by that year's duration-specific expense, and then assign the expense a negative profit sign.

Year 1 exposure is 600. Therefore, year 1 expense is $600(420)=\$252{,}000$, and the profit effect is $-\$252{,}000$.

The annual survival probability is $p=1-0.04=0.96$. Expected year 2 exposure is $600(0.96)=576$. Year 2 expense is $576(75)=\$43{,}200$, so its profit effect is $-\$43{,}200$.

Expected year 3 exposure is $600(0.96)^2=600(0.9216)=552.96$. Expected year 3 expense is $552.96(90)=\$49{,}766.40$, giving a profit effect of $-\$49{,}766.40$.

The expected three-year cohort expense is $\$252{,}000+\$43{,}200+\$49{,}766.40=\$344{,}966.40$. Per policy originally issued, this is $\$344{,}966.40/600=\$574.944$, or $\$574.94$ to the nearest cent. The three annual profit effects sum to $-\$344{,}966.40$, as required. The common mistake avoided is charging the year 2 and year 3 renewal expenses to all 600 original policies without adjusting the exposure for mortality.

Worked Example: Separating Issue Costs From Duration-Based Renewals

A hypothetical cohort contains 1,000 newly issued policies. The year 1 acquisition expense is $275 per policy. Renewal expenses are $50 per policy in year 2, $55 in year 3, and $60 in year 4. Annual death probability is $q=0.10$, with no lapses, and expenses apply to policies in force at the start of each year. Find each year's expected expense and the expected total expense per issued policy through year 4. Assume $i=0\%$ for the annual profit calculation. Unit expenses are deterministic; mortality is the only source of uncertainty and follows the stated annual probability.

Method. Treat the year 1 amount as an acquisition cost and the later amounts as renewals. Calculate exposure using $p=1-0.10=0.90$.

In year 1, all 1,000 policies are newly issued, so the acquisition expense is $1{,}000(275)=\$275{,}000$.

In year 2, expected exposure is $1{,}000(0.90)=900$. Expected renewal expense is $900(50)=\$45{,}000$.

In year 3, expected exposure is $1{,}000(0.90)^2=810$. Expected renewal expense is $810(55)=\$44{,}550$.

In year 4, expected exposure is $1{,}000(0.90)^3=729$. Expected renewal expense is $729(60)=\$43{,}740$.

Total expected cohort expense is $\$275{,}000+\$45{,}000+\$44{,}550+\$43{,}740=\$408{,}290$. Dividing by 1,000 original policies gives $\$408.29$ per policy issued. Each year's profit effect is the negative of that year's expense, so the four effects total $-\$408{,}290$. This calculation avoids the error of treating renewal expenses as if they were also acquisition expenses or as if every original policy persisted.

Worked Example: Comparing Two Expense Designs

A hypothetical insurer is comparing two expense schedules for otherwise identical three-year policies. Schedule A has an acquisition expense of $450, then renewal expenses of $48 in year 2 and $52 in year 3. Schedule B has an acquisition expense of $360, then renewal expenses of $85 in year 2 and $90 in year 3. There are 5,000 new policies, annual death probability is $q=0.08$, there are no lapses, and expenses apply to policies in force at the start of each year. Find the expected expense per policy issued under each schedule and the difference in expected three-year expense. Assume $i=0\%$; expenses are deterministic per exposed policy, with the stated mortality model and no other expense uncertainty.

Method. Apply the same exposure probabilities to both schedules because the mortality assumptions are identical. Here $p=0.92$, so the expected in-force proportions at the starts of years 1, 2, and 3 are $1$, $0.92$, and $(0.92)^2=0.8464$.

Under schedule A, the expected expense per original policy is $450+0.92(48)+0.8464(52)$. The year 2 term is $0.92(48)=44.16$. The year 3 term is $0.8464(52)=44.0128$. Thus schedule A costs $450+44.16+44.0128=\$538.1728$ per original policy, or $\$538.17$ to the nearest cent. For 5,000 policies, the expected total is $5{,}000(538.1728)=\$2{,}690{,}864$.

Under schedule B, the expected expense per original policy is $360+0.92(85)+0.8464(90)$. The year 2 term is $78.20$, and the year 3 term is $76.176$. Therefore schedule B costs $360+78.20+76.176=\$514.376$ per original policy, or $\$514.38$ to the nearest cent. The expected total for 5,000 policies is $5{,}000(514.376)=\$2{,}571{,}880$.

Schedule B reduces expected three-year expense by $\$538.1728-\$514.376=\$23.7968$ per policy issued, or $\$118{,}984$ for the cohort. If other profit components are identical, its expected expense contribution to profit is therefore higher by $\$118{,}984$ over the three years. This comparison shows why a lower acquisition cost alone is not enough to compare schedules: renewal expenses and the exposure to them also matter.

Common Mistakes and Interpretation

The expense component is only one part of an annual profit vector. In a complete profit calculation, premiums, benefits, expenses, investment items, and reserve movements are combined according to the applicable annual profit accounting identity. The calculations here isolate expenses so their classification and exposure are clear; they do not replace the other components.

Charging renewal costs to the original issue count. If renewal expense applies only to policies still in force, first calculate the later-year exposure.
Using the same expense amount at every duration. Apply the stated policy-year schedule or rule; a duration-dependent assumption must not be flattened into one average unless the problem specifically asks for an average.
Mixing up amount and sign. The expense amount is positive as an outgo, but its contribution to insurer profit is negative.
Reclassifying by payment date without an assumption. This tutorial assigns expenses to policy years as stipulated. A separate payment-timing assumption is needed to adjust cash-flow dates or investment income.
Allocating unassigned overhead without a basis. A fixed company-level expense cannot be converted into a per-policy expense unless an allocation rule or exposure basis is provided.

Key takeaway: For each policy year, identify whether the expense is acquisition or renewal, apply the expense amount for that duration to the corresponding exposure, and record the resulting outgo as a negative profit item. Duration and persistence both affect the total.

Check Your Understanding

For each problem, use the stated exposure basis and give the expected expense; the annual profit effect has the opposite sign.

A cohort of 800 policies has a $200 acquisition expense in year 1 and a $40 renewal expense in year 2. Annual death probability is 0.03, there are no lapses, and expense exposure is at the start of each year. Find the expected total expense per policy issued through year 2. Assume $i=0\%$, deterministic unit expenses, and the stated mortality model. (Answer: $238.80 per policy)
A cohort of 1,200 policies has a $150 acquisition expense and a $30 renewal expense in year 2. Annual death probability is 0.05 with no lapses, and expenses apply at the start of each year. Find the expected cohort expense through year 2. Assume $i=0\%$, deterministic unit expenses, and the stated mortality model. (Answer: $214,200)
A cohort of 500 policies has an acquisition expense of $100 in year 1, a $20 renewal expense in year 2, and a $25 renewal expense in year 3. Annual death probability is 0.10 with no lapses. Calculate the expected total expense per policy issued through year 3. Assume $i=0\%$, deterministic unit expenses, and start-of-year exposure. (Answer: $138.25 per policy)
For 1,000 policies, compare schedule X, with acquisition expense $300 and renewal expenses $40 and $50 in years 2 and 3, against schedule Y, with acquisition expense $260 and renewal expenses $70 in each of years 2 and 3. Annual death probability is 0.04, there are no lapses, and expenses apply at the start of each year. Which schedule has lower expected three-year cohort expense, and by how much? Assume $i=0\%$, deterministic unit expenses, and the stated mortality model. (Answer: Schedule X is lower by $7,232)
A cohort of 2,000 policies has an acquisition expense of $180 in year 1 and renewal expenses of $32, $36, and $40 in years 2, 3, and 4. Annual death probability is 0.05 with no lapses. Find expected expense in each year and total expected expense per policy issued through year 4. Assume $i=0\%$, deterministic unit expenses, and start-of-year exposure. (Answer: Yearly expenses $360,000, $60,800, $64,980, and $68,590; total $277.185 per policy issued)

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