Acquisition Costs and Initial Asset-Share Deficits

Why Acquisition Costs Matter

A policy can receive a premium at issue and still begin its investment projection with a deficit. This happens when the premium is smaller than the commission and other expenses incurred to acquire the policy. The size of that shortfall, and the time it takes for later cash inflows to offset it, depend on the expense amounts, the premium pattern, the investment returns and, importantly, the timing convention.

This tutorial considers a hypothetical per-policy asset-share projection. It isolates acquisition costs and initial deficits; other cash flows are excluded unless specified. The calculation is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus.

Definition: An acquisition cost is an expense incurred to issue or sell a policy, such as commission or an issue expense. An initial asset-share deficit occurs when cumulative allocated cash flows, after applying investment returns, produce a negative asset share. In a per-policy ledger, the negative value is a measure of the amount not yet recovered by the projected inflows; it does not imply that a policyholder has a negative account balance.

State the Cash-Flow Timing First

For the calculations below, each year’s premium and expenses are paid at the beginning of that policy year. The remaining balance earns that year’s effective investment return. Let $AS_t$ denote the asset share at the end of policy year $t$, and let $j_t$ be the effective investment return during year $t$. If $P_t$ is the premium received and $E_t$ is the expense paid at the beginning of the year, then

AS_0=0,\qquad AS_t=(AS_{t-1}+P_t-E_t)(1+j_t).

For an acquisition-cost projection, $E_1$ includes the issue costs. If the later years contain only premiums and investment returns, set $E_t=0$ for $t\geq 2$. The recurrence is a signed calculation: if $AS_{t-1}+P_t-E_t$ is negative, applying a positive return makes the deficit larger in magnitude.

Acquisition expenses often have more than one component. For example, if a commission is a proportion $c$ of the issue premium and the other issue expense is a fixed amount $K$, then the total acquisition cost is $E_1=cP_1+K$. The net amount available to invest at issue is $P_1-E_1$. The commission percentage applies to the premium specified in the problem, not automatically to the premium after expenses.

Worked Example: Separating Commission and Fixed Issue Expense

A hypothetical policy receives a single premium of $900 at issue. The commission is 30% of that premium, and other acquisition expenses are $510. Both the premium and expenses are paid at the start of year 1. The effective investment return is 5% in year 1 and 4% in year 2. A second premium of $900 is paid at the start of year 2, with no further expenses. Calculate the asset share at the end of each year.

Method. Calculate the commission on the stated issue premium, combine it with the fixed expense, and then apply the beginning-of-year recurrence.

The year 1 commission is $0.30(900)=270$. Total acquisition expenses are $270+510=780$, so the issue-date net cash flow is $900-780=120$. After the 5% return,

$$ AS_1=(900-780)(1.05)=120(1.05)=126. $$

At the start of year 2, the existing $126 asset share and the new $900 premium are invested together. The year 2 return is 4%, so

AS_2=(126+900)(1.04)=1{,}026(1.04)=1{,}067.04.

The results are $AS_1=\$126.00$ and $AS_2=\$1{,}067.04$. A useful check is that the year 1 net cash flow is positive, so a positive return should leave a positive year-end asset share. This method avoids treating the commission as a percentage of the premium remaining after expenses.

How a Deficit Accumulates and Recovers

When acquisition expenses exceed the issue premium, the issue-date net cash flow is negative. The deficit must be carried forward using the same investment-return convention as a positive balance. If later premiums are received, they first offset the carried deficit in the signed ledger; any remaining positive amount then earns the applicable return.

The recurrence also shows why recovery depends on both cash flows and investment returns. A premium can make the balance less negative, or turn it positive, before the year’s return is applied. If the balance is negative after that premium is added, a positive return increases the amount of the deficit. If it is positive, a positive return increases the asset share.

Worked Example: A Deficit Recovered by the Next Premium

A hypothetical annual-premium policy receives $1,000 at issue. The commission is 35% of the premium and other acquisition expenses are $900. The effective return is 4% in year 1. A $1,000 renewal premium is received at the beginning of year 2, when the effective return is 5%; assume there are no year 2 expenses. Find both year-end asset shares and identify when the deficit is recovered.

Method. Find the issue-date net cash flow first, then carry the signed asset share into the next year before adding the renewal premium.

The issue commission is $0.35(1{,}000)=350$. Total acquisition expenses are $350+900=1{,}250$, which exceed the issue premium by $250. Thus,

AS_1=(1{,}000-1{,}250)(1.04)=(-250)(1.04)=-260.

At the beginning of year 2, the $1,000 premium offsets the $260 deficit, leaving $ -260+1{,}000=740$ to invest. Applying the 5% return gives

AS_2=(-260+1{,}000)(1.05)=740(1.05)=777.

The year-end values are $AS_1=-\$260.00$ and $AS_2=\$777.00$. The deficit is present at the end of year 1 but has been recovered by the beginning of year 2; the year 2 investment return then increases the positive balance. The check is that the year 2 premium exceeds the carried deficit, leaving $740 before investment.

Why the Recurrence Holds

The recurrence follows from a direct cash-flow balance at the beginning of each year. Start with the previous year’s end balance, add the premium received, and subtract the expense paid. This gives the amount invested during the year. An effective annual return $j_t$ multiplies that amount by $1+j_t$ over the year.

Bring forward the prior balance.
At the beginning of year $t$, the previous end-year amount is $AS_{t-1}$.

Record beginning-of-year cash flows.
The amount exposed to investment is $AS_{t-1}+P_t-E_t$.

Apply the annual return.
Multiplying the invested amount by $1+j_t$ gives the year-end asset share: $AS_t=(AS_{t-1}+P_t-E_t)(1+j_t)$.

Repeated substitution gives a useful check on multi-year calculations:

AS_n=AS_0\prod_{r=1}^{n}(1+j_r) +\sum_{k=1}^{n}(P_k-E_k)\prod_{r=k}^{n}(1+j_r).

Each year’s net beginning-of-year cash flow is multiplied by the investment factors for that year and every later year in the projection. For example, a year 1 cash flow earns the year 1 return and all subsequent returns through year $n$, which explains the product beginning at $r=1$. The expression is equivalent to applying the recurrence year by year; it is not a separate timing assumption.

Worked Example: Projecting a Deficit Under Changing Returns

A hypothetical policy has an issue premium of $1,500. Its commission is 20% of the issue premium, and other acquisition expenses are $1,350. The year 1 return is 6%. A $500 premium is received at the start of each of years 2 and 3, with no further expenses. Returns are 4% in year 2 and 3% in year 3. Calculate the year-end asset share in each year.

Method. Calculate the acquisition deficit at issue and then apply the recurrence separately with each year’s specified return.

The commission is $0.20(1{,}500)=300$, so total year 1 acquisition expenses are $300+1{,}350=1{,}650$. The net cash flow at issue is $1{,}500-1{,}650=-150$. Therefore,

$$ AS_1=(-150)(1.06)=-159. $$

At the start of year 2, the $500 premium reduces the deficit to $-159+500=341$. Applying the 4% return gives

$$ AS_2=341(1.04)=354.64. $$

At the start of year 3, the $500 premium increases the amount to $354.64+500=854.64$. Applying the 3% return gives

$$ AS_3=854.64(1.03)=880.2792. $$

Rounded to the nearest cent, $AS_1=-\$159.00$, $AS_2=\$354.64$, and $AS_3=\$880.28$. The deficit is recovered during year 2, before that year’s investment return. As a check, the amount invested in year 3 is positive, so a positive return increases it from $854.64 to $880.28.

Common Timing Trap

A frequent error is to accumulate the premium for a full year and then subtract an expense that was actually paid at the beginning of the year. These operations are not interchangeable: a beginning-of-year expense is also subject to the year’s investment return in the signed cash-flow calculation.

In the previous example, applying the year 1 return to the premium and subtracting the full expense afterward would give the incorrect result

$$ 1{,}500(1.06)-1{,}650=1{,}590-1{,}650=-60. $$

The specified timing instead gives $(1{,}500-1{,}650)(1.06)=-159$. The incorrect answer understates the deficit by $99, which equals $1{,}650(0.06)$: it has failed to apply the return effect to the beginning-of-year expense. If a problem specifies that an expense is paid at year-end, it should not be accumulated for the full year. Use the stated timing rather than assuming one convention for all expenses.

Practical Checks

Before finalizing an acquisition-cost projection, verify the cash-flow basis and the sign of each balance. In particular:

Calculate percentage commission on the premium base stated in the problem, and add fixed acquisition expenses separately.
Apply each premium and expense at the specified time within the year before applying the return.
Carry negative asset shares as signed amounts; do not replace them by zero unless the problem explicitly defines a floor.
Check that a positive return increases a positive balance and makes a negative balance more negative when no intervening cash flow changes its sign.
Distinguish the year in which the deficit becomes positive from the year in which the asset share reaches a particular target amount.

Acquisition costs matter because a strong first-year premium can still fail to fund the issue outgo. Projecting the signed balance makes the initial shortfall visible and shows how later inflows and returns affect recovery. A reliable calculation is therefore a short ledger with clear timing, not just a total of premiums minus total expenses.

Check Your Understanding

For each hypothetical policy, use beginning-of-year premium and expense timing, followed by the stated annual return. Unless a question says otherwise, expenses after issue are zero.

A policy has an issue premium of $800, a commission of 20% of that premium, and other acquisition expenses of $500. The year 1 effective return is 5%. Find $AS_1$. (Answer: $147.00)
A $1,200 issue premium is subject to a 25% commission and $1,050 of other issue expenses. The year 1 return is 8%. Find the end-of-year asset share, including its sign. (Answer: $-\$162.00$)
A policy begins year 2 with an asset share of $-\$300$. It receives a $700 premium at the start of year 2, then earns 4%. Find $AS_2$ and state whether the deficit is recovered. (Answer: $416.00; yes)
A policy receives $2,000 at issue. Commission is 15% of the premium, other acquisition expenses are $1,800, and the year 1 return is 5%. At the start of year 2, it receives a $500 premium and earns 2% during year 2. Find $AS_1$ and $AS_2$. (Answer: $AS_1=-\$105.00$, $AS_2=\$402.90$)
A policy has a $1,000 issue premium, total acquisition expenses of $1,600, and a 10% year 1 return. It receives an $800 premium at the start of year 2 and earns 5% that year. Find the year-end asset share after year 2. (Answer: $147.00)

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