Asset-Share Sensitivity to Expense Deviations

Why Expense Deviations Matter

An asset share reflects premiums and other inflows, less expenses and benefits, together with investment accumulation. If actual acquisition or renewal expenses differ from the assumptions used in a baseline projection, the asset share changes. The size of that change depends not only on the amount of the deviation but also on when the expense is paid: an earlier overrun reduces the fund for longer and therefore also reduces the investment income it could have earned.

This tutorial isolates expense experience. The baseline and comparison projections use the same premiums, benefits, decrements, and investment returns; only expenses differ. This makes it possible to separate the immediate cash-flow effect from the subsequent accumulation effect. It does not measure the effect of an investment-return deviation, which requires changing the return assumptions as well.

Definition: An expense deviation at a payment date is actual expense minus the corresponding expense assumed in the baseline projection. A positive deviation is an expense overrun; a negative deviation is an expense saving. The asset-share sensitivity to expenses at a later date is the difference between the asset share under the deviating expenses and the baseline asset share, with all other projection inputs held fixed.

Projection Convention and Sensitivity Formula

First use a clear annual timing convention. Let $A_k$ be the baseline asset share at the start of year $k$. Suppose the relevant cash flows, including expenses, are taken at the start of the year, and the resulting fund earns the effective annual return $i_k$ over that year. A simplified baseline recurrence is

A_{k+1}=(A_k+P_k-E_k-B_k)(1+i_k),

where $P_k$ is premium income, $E_k$ is expense, and $B_k$ is benefit outgo under this stated timing convention. The expense-only comparison uses the same $P_k$, $B_k$, and $i_k$, but replaces $E_k$ by $E_k+\Delta E_k$, where $\Delta E_k$ is the expense deviation at the start of year $k$. Let $D_k$ be comparison asset share minus baseline asset share at time $k$. Subtracting the two recurrences gives

D_{k+1}=(D_k-\Delta E_k)(1+i_k).

Thus, a positive expense deviation immediately lowers the fund; a negative deviation raises it. The difference is then accumulated at the return earned after the payment. Iterating the recurrence from $D_0=0$ gives the sensitivity at time $n$:

D_n=-\sum_{k=0}^{n-1}\Delta E_k\prod_{j=k}^{n-1}(1+i_j).

This formula assumes that each expense deviation is paid at the start of its indicated year, that the return factors apply after that payment, and that the deviations do not change other cash flows or the return earned. If expenses are paid at year-end instead, the current year's return does not accumulate the deviation. The payment dates must therefore be matched to the projection convention rather than inferred from the formula alone.

Interpretation: The immediate cash-flow effect of a deviation is $-\Delta E_k$. Its later asset-share effect also includes the returns that would have been earned on that amount from its payment date to the measurement date. An expense saving has the reverse sign: it adds to the asset share and can itself earn returns.

Why the Formula Holds

The formula follows by comparing the projections year by year. At the start of year $k$, the difference between funds immediately after the cash flows is $D_k-\Delta E_k$: the existing difference is $D_k$, and the comparison projection pays $\Delta E_k$ more in expenses. Both funds then earn the same factor $1+i_k$, so their difference at the next year is $(D_k-\Delta E_k)(1+i_k)$. Repeated substitution gives each deviation its own accumulation factors from the year it is paid through year $n-1$:

\begin{aligned} D_1&=-\Delta E_0(1+i_0),\\ D_2&=-\Delta E_0(1+i_0)(1+i_1)-\Delta E_1(1+i_1),\\ D_n&=-\sum_{k=0}^{n-1}\Delta E_k\prod_{j=k}^{n-1}(1+i_j). \end{aligned}

Each term is the same idea: an expense deviation first changes the fund by its negative, then the resulting difference accumulates over the returns earned after that expense date. The derivation also explains why applying the full-period return to every expense is usually wrong: later expenses have fewer periods in which to accumulate.

Worked Example: Acquisition Expense Overrun

A hypothetical policy has an acquisition expense at issue that is $120 higher than the baseline assumption. The comparison is made two years later. The annual net investment return credited in year 1 is 5% and in year 2 is 4%. All other cash flows and assumptions are identical. Find the asset-share difference at the end of year 2.

Method. Treat the overrun as a positive expense deviation of $120 at time 0. Under the stated start-of-year convention, it reduces the asset share immediately by $120 and is accumulated through both years.

$$ D_2=-120(1.05)(1.04). $$

First, $1.05\times1.04=1.092$. Therefore $D_2=-120\times1.092=-\$131.04$. The acquisition overrun reduces the later asset share by $131.04: $120 is the immediate expense effect, and the additional $11.04 is the return not earned on that $120.

Check. The negative sign is appropriate because actual expenses exceed baseline expenses. The absolute reduction exceeds $120 because the difference would have earned investment returns. This calculation avoids the mistake of reporting only the issue-date cash-flow reduction.

Worked Example: Renewal Deviations in Different Years

A hypothetical insurer projects three annual renewal expenses, each paid at the start of its year. Relative to baseline, the expense deviations at times 0, 1, and 2 are respectively +$20, -$10, and +$30. The annual net returns in those years are 4%, 5%, and 3%. Calculate the difference in asset share at time 3.

Method. Accumulate each deviation only through the returns after its payment. The $20 overrun is accumulated for all three years; the $10 saving for the final two years; and the $30 overrun for the final year only.

$$ D_3=-20(1.04)(1.05)(1.03)-(-10)(1.05)(1.03)-30(1.03). $$

For the first deviation, $1.04\times1.05\times1.03=1.12476$, so its contribution is $-20\times1.12476=-\$22.4952$. For the saving, $1.05\times1.03=1.0815$, so its contribution is $+10\times1.0815=+\$10.815$. For the final overrun, the contribution is $-30\times1.03=-\$30.90$.

Adding the contributions gives $D_3=-22.4952+10.815-30.90=-\$42.5802$. Rounded to the nearest cent, the asset share is $42.58 lower than baseline. The immediate net expense deviation is +$40, but the asset-share reduction is greater because the early net overrun also loses accumulated returns. The saving is correctly shown as a positive contribution.

Check. The signs reflect the direction of each cash-flow change, and the number of accumulation factors reflects its payment date. A common error is to treat all three deviations as if they occurred at time 0, which would give each one too many years of accumulation.

Worked Example: Combining Acquisition and Renewal Effects

A hypothetical baseline projection produces an asset share of $5,000 at time 3. The acquisition expense at issue is $150 above baseline. Renewal expenses are each $25 above baseline at the starts of years 2 and 3—that is, at times 1 and 2. The net annual return is 4% in each year. Find the comparison asset share at time 3, and separate the immediate expense effect from the additional accumulation effect.

Method. Accumulate the acquisition deviation for three years, the time-1 renewal deviation for two years, and the time-2 deviation for one year. All three deviations are positive overruns.

$$ D_3=-150(1.04)^3-25(1.04)^2-25(1.04). $$

The return factors are $(1.04)^3=1.124864$, $(1.04)^2=1.0816$, and $1.04$. The accumulated contributions are:

Acquisition overrun: $-150\times1.124864=-\$168.7296$.
First renewal overrun: $-25\times1.0816=-\$27.04$.
Second renewal overrun: $-25\times1.04=-\$26.00$.

Thus $D_3=-168.7296-27.04-26.00=-\$221.7696$. The comparison asset share is $5,000-221.7696=\$4,778.2304$, or $4,778.23 to the nearest cent. The immediate cash-flow effect is $-150-25-25=-\$200$. The remaining $-\$21.7696$ is the additional accumulated effect of returns not earned on the expense amounts.

Check. The comparison asset share is below baseline because every deviation is an overrun. Its reduction is larger than the $200 total of the deviations because each amount was paid before the measurement date and therefore missed some investment return.

Reading and Using Expense Sensitivity

The calculation can be used to identify which expense deviations have the greatest effect at a chosen valuation date. Under positive returns, an equal overrun paid earlier generally has a larger accumulated effect than one paid later, because it is absent from the fund for more time. This does not mean that late expenses are unimportant: it means only that their accumulated effect at this particular date has fewer return periods.

Acquisition expenses and renewal expenses should be recorded at their actual projection dates, not combined into a single amount unless their timing is identical. For a comparison across many policies, the calculation must also use a consistent per-policy or aggregate basis. For example, aggregate expense deviations may reflect both the amount spent per policy and the number of policies to which the expense is allocated. The expense-only formula does not itself adjust for changes in policy counts or decrements.

This sensitivity is useful for explaining an asset-share shortfall or surplus attributable to expenses, and for testing how a revised expense assumption would affect a projection. It is not a complete forecast if expense overruns also change investment allocation, benefits, policyholder behaviour, or other cash flows. Such effects require those inputs to be changed explicitly rather than being attributed to the expense deviation alone.

Common Exam Trap

A frequent error is to subtract the total expense overrun from the final asset share and stop there. That gives the immediate cash-flow effect but omits the investment accumulation effect. A second error is to apply the full projection-period return to every expense deviation, even when some expenses occur later.

Key takeaway: For each expense deviation, record its sign and payment date. Subtract a positive overrun, add a saving, and accumulate that difference only over the return periods following its payment. Keep all non-expense assumptions fixed when interpreting the result as expense sensitivity.

Check Your Understanding

Use the start-of-year expense timing convention. Unless stated otherwise, assume the same return applies in each indicated year and round dollar answers to the nearest cent.

An acquisition expense is $80 above baseline at issue. The net annual return is 5% for three years. What is the asset-share difference at the end of year 3? (Answer: $-\$92.61$)
A renewal expense is $12 above baseline at the start of year 2, and the measurement date is the end of year 3. The annual net return is 4%. What is the asset-share difference at the measurement date? (Answer: $-\$12.98$)
Expense deviations are $-\$10$ at time 0 and $+\$20$ at time 1. Annual returns are 3% in both years. What is the asset-share difference at time 2? (Answer: $-\$9.99$)
The baseline asset share at time 2 is $1,200. An acquisition expense is $40 above baseline at time 0, and a renewal expense is $15 above baseline at time 1. The return is 5% in each year. What is the comparison asset share at time 2? (Answer: $1,140.15)

What You'll Learn