Renewal Expenses and Per-Policy Allocation

Why Renewal Expenses Need a Clear Basis

Renewal expenses are costs incurred while a policy remains in force after issue. They may include administration, servicing, or periodic policy charges. In an asset-share projection, these costs reduce the funds accumulated for the policy. The calculation depends not only on the expense amount, but also on what the expense is based on and when it is deducted.

A fixed expense of $60 per policy is different from an expense of 3% of premium: the first is a stated dollar amount for each policy, while the second changes with the premium. An expense stated as a percentage of a benefit base is different again; its basis is the specified amount of coverage, not necessarily a benefit paid during the year. The examples below use hypothetical policies and explicit cash-flow timing. This treatment is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: A renewal expense per policy is an expense allocated to an in-force policy for a period after issue. It may be specified as (1) a fixed dollar amount per policy, (2) a percentage of a stated premium amount, or (3) a percentage of a stated benefit or other exposure base. The expense basis and the period to which it applies must be identified before calculating the asset share.

Annual Asset-Share Recurrence With Renewal Expenses

Let $A_{k-1}$ be the asset share per policy at the start of year $k$, $P_k$ the premium received at the start of that year, $E_k$ the renewal expense deducted at that time, and $j_k$ the effective investment return earned during the year. Suppose the policy remains in force, and suppose the premium and expense are both received or paid at the start of the year. With no other cash flows included, the end-of-year asset share is

A_k=(A_{k-1}+P_k-E_k)(1+j_k).

The recurrence is a per-policy projection: all amounts refer to one policy. If expenses are specified per policy, they are deducted once per policy for the stated period. If a total expense for a group is supplied instead, it must first be allocated to the policy or cohort being projected. That allocation should follow the problem’s stated basis.

Timing condition: The recurrence above applies when the premium and expense are both paid at the start of the year and the remaining balance earns the full year’s return. If either cash flow occurs later, it does not earn or lose investment return for the same period as a beginning-of-year cash flow. Use the specified timing rather than applying the recurrence mechanically.

A useful expense decomposition is

$$ E_k=F_k+c_kP_k+b_kB_k, $$

where $F_k$ is a fixed expense per policy, $c_k$ is a rate applied to the specified premium $P_k$, $b_k$ is a rate applied to a specified benefit base $B_k$, and any component not present is set to zero. The equation is a bookkeeping representation, not a claim that every product has all three expense components. The bases must be read from the assumptions: for example, a stated percentage of annual premium is not a percentage of the asset share.

Why the Recurrence Holds

The recurrence follows from accumulating the year’s cash flows in their stated order. At the start of the year, the opening asset share is $A_{k-1}$. The premium adds $P_k$, while the renewal expense removes $E_k$. Therefore the amount invested immediately after those cash flows is $A_{k-1}+P_k-E_k$. An effective annual return $j_k$ multiplies that amount by $1+j_k$, giving the end-of-year value.

\begin{aligned} \text{amount invested at start of year }k &=A_{k-1}+P_k-E_k,\\ \text{amount at end of year }k &=(A_{k-1}+P_k-E_k)(1+j_k)=A_k. \end{aligned}

This also explains why changing the timing changes the formula. If the expense is paid at year-end, for example, the premium and opening asset share earn the year’s return before the expense is deducted: under that different timing, $A_k=(A_{k-1}+P_k)(1+j_k)-E_k$. The two expressions are generally unequal.

Worked Projections

Worked Example: Fixed Expense Plus a Percentage of Premium

A hypothetical policy has a start-of-year asset share of $1,000. At the start of the year, the insurer receives an annual premium of $800 and deducts a renewal expense consisting of $60 per policy plus 3% of that year’s premium. The effective annual investment return is 5%. The policy remains in force for the year. Calculate its end-of-year asset share.

Method and expense basis. The fixed expense is $60. The premium-based component is calculated from the $800 premium, not from the asset share:

0.03(800)=24,\qquad E_1=60+24=84.

The amount invested at the start of the year is

$$ 1{,}000+800-84=1{,}716. $$

It earns 5% for the full year, so

A_1=1{,}716(1.05)=1{,}801.80.

Check. The premium-based expense is $24, the total expense is $84, and the net start-of-year amount is less than the opening asset share plus premium, as expected. The end-of-year asset share is $1,801.80.

Common mistake avoided. Omitting the percentage-based component would use an expense of only $60 and produce $(1{,}000+800-60)(1.05)=1{,}827.00$. That overstates the asset share by $25.20 because it omits the $24 expense and its $1.20 investment return for the year.

Worked Example: Fixed Renewal Expenses That Increase Over Time

A hypothetical policy begins with an asset share of $0. It receives a premium of $600 at the start of each of two successive years. The fixed renewal expense is $90 in year 1 and increases by 10% for year 2. The effective annual investment return is 2% in both years. Premiums and expenses are paid at the start of each year. Calculate the asset share at the end of year 2.

Method. Project one year at a time, escalating the fixed expense before applying the second year’s recurrence. The year 1 expense is $90. Thus

$$ A_1=(0+600-90)(1.02)=510(1.02)=520.20. $$

The year 2 fixed expense is 10% higher than the year 1 amount:

$$ E_2=90(1.10)=99. $$

The amount invested at the start of year 2 is

$$ 520.20+600-99=1{,}021.20. $$

Applying the 2% return gives

A_2=1{,}021.20(1.02)=1{,}041.624.

Rounded to the nearest cent, the end-of-year 2 asset share is $1,041.62. The intermediate value is retained without rounding until the final answer.

Check. Year 2’s expense is $99, not $90, and the year 2 opening asset share is the year 1 result, $520.20. Both points are reflected in the calculation.

Common mistake avoided. Applying the year 1 expense again in year 2 would produce $(520.20+600-90)(1.02)=1{,}050.80$, which is too high because it ignores the stated 10% escalation.

Worked Example: Expense as a Percentage of a Benefit Base

A hypothetical policy has a start-of-year asset share of $500 and receives a $1,000 premium at the start of the year. Its renewal expense is $40 per policy plus 0.2% of a stated benefit base of $20,000. The effective annual investment return is 3%. Assume the expense is deducted at the start of the year. Calculate the end-of-year asset share.

Method and basis. The benefit-base expense uses the stated $20,000 base:

$$ 0.002(20{,}000)=40. $$

Add the fixed expense to find the total renewal expense:

$$ E_1=40+40=80. $$

The amount invested at the start of the year is

$$ 500+1{,}000-80=1{,}420. $$

After the 3% effective annual return,

A_1=1{,}420(1.03)=1{,}462.60.

The end-of-year asset share is $1,462.60. This calculation treats the benefit base only as the basis for an expense assessment; it does not treat the $20,000 as a benefit paid during the year.

Common mistake avoided. Applying 0.2% to the premium would give $2 instead of $40. The problem specifies the benefit base, so using the premium as the basis would be incorrect.

Expense Bases and Checks

Before inserting an expense into the recurrence, identify its unit, basis, and timing. “Per policy per year” describes a fixed annual amount for each policy. “A percentage of premium” requires the relevant premium amount. “A percentage of benefit base” requires the specified benefit measure. These descriptions are not interchangeable, even if they all produce dollar amounts after calculation.

Expense description	Calculation	Key check
Fixed annual expense per policy	Use the stated dollar amount for each policy and period	Do not multiply by premium unless instructed
Percentage of premium	Rate times the premium amount specified	Confirm whether the rate applies to annual, modal, or another stated premium
Percentage of benefit base	Rate times the stated benefit or coverage base	Do not assume the base is a benefit paid during the year
Expense that escalates	Apply the stated increase to the prior or original expense, as specified	Check the reference amount and the year of application

A useful numerical check is that, with a positive expense and positive return, the expense reduces the amount that earns investment income. For a beginning-of-year deduction, each dollar of expense reduces the end-of-year asset share by $1+j_k$ dollars relative to a projection that omitted that expense. In the first worked example, omitting $24 therefore overstates the end-of-year value by $24(1.05)=25.20$.

Common Exam Traps

Using the wrong base. A percentage of premium is applied to the stated premium, while a percentage of benefit base is applied to the stated base. Do not choose a base merely because it is convenient.
Forgetting the per-policy unit. A fixed expense stated per policy is deducted once for each policy for the specified period. It is not automatically a single expense for an entire group.
Ignoring escalation. If a fixed expense increases, calculate the amount for each year before applying that year’s asset-share recurrence.
Applying return to the wrong balance. Under beginning-of-year timing, deduct the expense before investment accumulation. A year-end deduction follows a different formula.
Rounding too early. Keep intermediate asset-share values to full precision where practical, then round the final dollar amount to the nearest cent.

Key takeaway: First calculate each renewal expense from its stated basis; then deduct it at the stated time and apply the appropriate investment accumulation. A correct asset-share projection depends on getting both the expense basis and cash-flow timing right.

Check Your Understanding

For each problem, use the stated timing and retain intermediate values until the final answer.

A policy begins the year with an asset share of $0. It receives a $900 premium at the start of the year. The renewal expense is $75 per policy plus 4% of premium, also deducted at the start of the year. The effective annual return is 3%. Find the end-of-year asset share. (Answer: $812.67)
A policy has a start-of-year asset share of $400 and receives a $700 premium. At the start of the year, expenses of $55 per policy plus 2% of premium are deducted. The effective annual return is 4%. Find the end-of-year asset share. (Answer: $1,072.24)
A policy starts with no asset share and receives $500 at the start of each of two years. A fixed renewal expense of $80 in year 1 increases by 5% for year 2. The effective annual return is 3% in each year. Find the end-of-year 2 asset share. (Answer: $874.06)
A policy has a start-of-year asset share of $1,200 and receives a $1,000 premium. The beginning-of-year expense is $30 per policy plus 0.15% of a $24,000 benefit base. The effective annual return is 2.5%. Find the end-of-year asset share. (Answer: $2,187.35)
A policy starts with no asset share and receives a $1,500 premium at the start of the year. The expense comprises $50 per policy, 2% of premium, and 0.1% of a $30,000 benefit base. All are deducted at the start of the year, and the effective annual return is 4%. Find the end-of-year asset share. (Answer: $1,445.60)

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