Expense Gains and Losses

Why Expense Experience Needs a Clear Comparison

An insurer prices and values a hypothetical participating policy using assumptions about the expenses needed to issue and administer it. Actual expenses may differ. If the actual expense outgo is lower than the amount allowed for on the same basis, the difference is an expense gain; if actual outgo is higher, it is an expense loss. The direction is easy to reverse unless the comparison is stated explicitly.

Expense experience can include acquisition costs, ongoing maintenance costs, and claim-handling costs. Some costs are fixed over a period, while others vary with the number of new policies, policy-years, or claims handled. A sound calculation matches each assumed cost with its actual counterpart and uses the relevant exposure count.

Definition: For a specified period and a consistent set of exposures, the expense gain is assumed expense outgo minus actual expense outgo. A positive result is a gain; a negative result is an expense loss. Thus, for a cost category, $G_{\mathrm{exp}}=E_{\mathrm{assumed}}-E_{\mathrm{actual}}$.

This convention measures the difference in expense outgo only. It does not by itself measure total surplus, determine a dividend, or include investment or other experience. Those questions require additional components and a defined allocation basis.

Match Expense Types to Their Exposure Bases

A fixed expense is stated as a total amount for the period, rather than as a cost per unit of exposure. A variable expense is expressed per relevant unit. Common bases include a cost per new policy issued, per policy-year in force, or per claim processed. A per-policy cost is not automatically a per-policy-year cost: the first may be incurred once, while the second recurs with time in force.

Expense basis	Typical assumed amount	Exposure used to calculate the amount
Fixed expense	Total dollars for the period	None; use the stated period total
Acquisition expense	Dollars per new policy	Number of policies issued
Maintenance expense	Dollars per policy-year	Policy-years in the period
Claim-handling expense	Dollars per claim	Number of claims handled

For each variable category, multiply the rate by its exposure count to obtain a total expense. Compare totals on the same basis: the assumed rate and actual rate must apply to the same exposure definition and period. For example, if the actual maintenance experience is measured over 400 policy-years, calculate the assumed maintenance amount for those same 400 policy-years when isolating the expense-rate result.

Deriving the Expense-Gain Formula

Suppose a cost category has an assumed fixed amount $F_{\mathrm{assumed}}$, an actual fixed amount $F_{\mathrm{actual}}$, and a variable cost per exposure unit. Let $n$ be the number of matching exposure units, with assumed rate $r_{\mathrm{assumed}}$ and actual rate $r_{\mathrm{actual}}$. Assume both rates apply to the same type of exposure in the same period, and there are no timing or scope differences between the compared totals.

E_{\mathrm{assumed}}=F_{\mathrm{assumed}}+n r_{\mathrm{assumed}}, \qquad E_{\mathrm{actual}}=F_{\mathrm{actual}}+n r_{\mathrm{actual}}.

By definition, gain is assumed outgo minus actual outgo. Substituting the two expense totals and grouping the fixed and variable differences gives the result:

\begin{aligned} G_{\mathrm{exp}} &=E_{\mathrm{assumed}}-E_{\mathrm{actual}}\\ &=(F_{\mathrm{assumed}}-F_{\mathrm{actual}}) +n(r_{\mathrm{assumed}}-r_{\mathrm{actual}}). \end{aligned}

This derivation explains why a favorable result in one component can be offset by an unfavorable result in another. It also shows why a fixed-cost difference is not multiplied by the policy count, while a per-unit difference is. If there are several expense categories, calculate each on its proper basis and add their gains and losses, provided they refer to the same period and scope.

Important condition: The formula compares assumed and actual outgo for matching exposures. If the question instead asks for a difference between actual total outgo and a forecast based on a different number of policies, separate the exposure-volume effect from the expense-rate effect before interpreting the result. Do not silently apply a per-policy rate to a policy-year count, or vice versa.

Worked Calculations

Worked Example: Fixed Administration Cost and Maintenance Cost

A hypothetical insurer compares one year's administration expenses. The assumed fixed expense is $7,200 and the actual fixed expense is $6,950. Maintenance expense is assumed to be $15 per policy-year and is actually $15.60 per policy-year. There are 500 policy-years in the period. Calculate the combined expense result. Assume the fixed amounts cover the same year and the policy-year count is the exposure for both maintenance rates.

Method. Calculate the fixed-cost difference directly, then calculate the variable-cost difference using 500 policy-years. Add the results using gain = assumed expense minus actual expense.

For fixed administration, assumed minus actual is $7{,}200-6{,}950=\$250$, a $250 gain.

The assumed maintenance total is $500\times \$15=\$7{,}500$. The actual maintenance total is $500\times \$15.60=\$7{,}800$. Therefore the maintenance result is $\$7{,}500-\$7{,}800=-\$300$, a $300 loss.

Combining the components gives $\$250-\$300=-\$50$. The result is a $50 expense loss.

Check. Total assumed expense is $\$7{,}200+\$7{,}500=\$14{,}700$. Total actual expense is $\$6{,}950+\$7{,}800=\$14{,}750$. Assumed minus actual is $-\$50$, confirming the result. The common mistake avoided here is reporting the favorable fixed-cost difference alone while overlooking the larger maintenance overrun.

Worked Example: Acquisition, Maintenance, and Claim-Handling Expenses

For a hypothetical block over one year, the assumed acquisition expense is $48 per new policy and the actual amount is $45. The insurer issues 240 policies. Maintenance expense is assumed to be $20 per policy-year and is actually $21, with 600 policy-years. Claim-handling expense is assumed to be $90 per claim and is actually $86, with 35 claims. Calculate the combined expense result. Assume all amounts apply to this same one-year accounting period.

Method. Convert each unit expense to a total using its own exposure count, then subtract actual total from assumed total by category.

Acquisition: assumed $240\times \$48=\$11{,}520$; actual $240\times \$45=\$10{,}800$. The acquisition result is $\$11{,}520-\$10{,}800=\$720$ gain.

Maintenance: assumed $600\times \$20=\$12{,}000$; actual $600\times \$21=\$12{,}600$. The maintenance result is $\$12{,}000-\$12{,}600=-\$600$, a $600 loss.

Claim handling: assumed $35\times \$90=\$3{,}150$; actual $35\times \$86=\$3{,}010$. The claim-handling result is $\$3{,}150-\$3{,}010=\$140$ gain.

The combined result is $\$720-\$600+\$140=\$260$. There is a $260 expense gain.

Check. Total assumed expense is $\$11{,}520+\$12{,}000+\$3{,}150=\$26{,}670$. Total actual expense is $\$10{,}800+\$12{,}600+\$3{,}010=\$26{,}410$. Their difference is $\$260$, as calculated. The common mistake avoided is using the number of issued policies as the exposure for maintenance or claims rather than using the stated policy-years and claim count.

Worked Example: Separating Fixed and Per-Policy-Year Experience

A hypothetical insurer's annual maintenance budget has an assumed fixed component of $4,800 and an assumed variable component of $9.50 per policy-year. Actual fixed expense is $5,050, and actual variable expense is $9.20 per policy-year. The block has 700 policy-years. Find the expense gain or loss and verify it by comparing total expenses. Assume the two fixed amounts and the exposure count cover exactly the same year and block.

Method. Compare fixed amounts without multiplying by exposure. Then compare variable rates over the 700 policy-years.

The fixed result is $\$4{,}800-\$5{,}050=-\$250$, a $250 loss.

Assumed variable expense is $700\times \$9.50=\$6{,}650$. Actual variable expense is $700\times \$9.20=\$6{,}440$. The variable result is $\$6{,}650-\$6{,}440=\$210$ gain.

The combined result is $-\$250+\$210=-\$40$, so there is a $40 expense loss.

Check. Total assumed expense is $\$4{,}800+\$6{,}650=\$11{,}450$. Total actual expense is $\$5{,}050+\$6{,}440=\$11{,}490$. Assumed minus actual is $-\$40$. The common mistake avoided is multiplying the fixed-cost difference by 700; it is a period total, not a per-policy-year amount.

Common Exam Traps

Expense calculations often involve several terms with different units. A short unit check can prevent a sign error or an exposure error before it reaches the final result.

Reversing the sign. Under the convention here, compute assumed minus actual. If actual expense is higher, the result must be a loss.
Multiplying fixed costs by exposure. A stated fixed total is already a total for its period. Only a per-unit rate is multiplied by an exposure count.
Mixing exposure bases. New policies, policy-years, and claims are not interchangeable. Match each rate to its stated unit.
Combining rates before converting to totals. Acquisition, maintenance, and claim-handling rates usually have different denominators. Calculate each total separately first.
Ignoring the period or scope. Assumed and actual amounts must cover the same period and expense categories. If one figure covers a year and another covers a month, put them on a common basis before comparing.

A reliable workflow is to label every rate with its unit, calculate assumed and actual totals separately, and then subtract. This makes the sign transparent and provides a direct check: a positive total difference means assumed outgo exceeded actual outgo, while a negative difference means actual outgo exceeded assumed outgo. Expense gains and losses are one component of participating-policy experience, aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Check Your Understanding

For each problem, state whether the result is a gain or a loss using assumed expense minus actual expense. Treat each stated exposure as applying to both assumed and actual rates.

A hypothetical block has 180 policy-years. Maintenance expense is assumed to be $24 per policy-year and is actually $22 per policy-year. Find the expense result. (Answer: $360 gain)
For one year, fixed administration expense is assumed to be $9,200 and actually $9,500. Variable expense is assumed to be $10 per policy-year and is actually $9.50, over 800 policy-years. Find the combined result. (Answer: $100 gain)
Acquisition expense is assumed to be $55 per new policy and is actually $52 for 240 new policies. Maintenance expense is assumed to be $14 per policy-year and is actually $15 over 600 policy-years. Find the combined result. (Answer: $120 gain)
In the same period, fixed expense is assumed to be $2,000 and actually $1,950. Claim handling is assumed at $30 per claim and actually $32 for 75 claims. Maintenance is assumed at $11 per policy-year and actually $10.50 over 400 policy-years. Find the combined result. (Answer: $100 gain)
A hypothetical block has 720 policy-years. Its fixed annual expense is assumed to be $3,000 and is actually $3,320. Variable maintenance is assumed to be $18 per policy-year and is actually $17.25. Calculate both total assumed and actual expense and then the result. (Answer: assumed $15,960; actual $15,740; $220 gain)

What You'll Learn