Expense Shocks and Expense Reserves

Why Expense Shocks Need a Timing Convention

An expense assumption affects profit in two ways that should not be confused. It affects the cash flow when the expense is paid, and, if the reserve basis is changed, it affects the amount held for future obligations. A higher maintenance expense can therefore reduce current profit and increase a prospective reserve. A higher acquisition expense is usually paid at issue, so its direct effect is concentrated at issue rather than spread across later policy years.

The calculations below use an annual model. At the start of policy year $t$, an in-force policy receives premium $P_t$ and incurs expense $e_t$. Interest at effective annual rate $i$ is earned during the year. At the end of the year, a death benefit $B_{t+1}$ is paid if death occurs; otherwise the policy continues with reserve $V_{t+1}$. Let $q_t$ be the probability of death during the year and $p_t=1-q_t$ the probability of survival. The reserve $V_t$ is measured immediately before the start-of-year premium and expense.

Definition: Under this timing convention, the prospective reserve $V_t$ is the expected present value, at time $t$, of future benefits and expenses less future premiums, conditional on the policy being in force at time $t$. A maintenance expense paid at the start of year $t$ is included in $V_t$; a benefit and continuation reserve are valued at the end of the year.

For a one-year step, when the reserve and cash flows use the same assumptions, the prospective reserve satisfies

V_t=v\left(q_tB_{t+1}+p_tV_{t+1}\right)-P_t+e_t, \qquad v=\frac{1}{1+i}.

This relation is a local valuation equation, not a claim that every actual experience outcome will match the assumptions. If actual expenses differ from those used to set the reserve, the difference emerges in profit.

How a Maintenance Expense Change Moves the Reserve

Suppose the valuation assumptions are changed by $\Delta e_t$ in the current year and $\Delta e_{t+1}$ in the following year, while premiums, benefits, mortality, and interest are unchanged. Subtracting the original reserve equation from the revised one gives

\Delta V_t=\Delta e_t+v p_t\Delta V_{t+1}.

The current expense change enters dollar for dollar because it is paid at time $t$. A future reserve change contributes only if the policy survives, and it is discounted back one year. For a single future expense increase, therefore, the reserve change is not simply the sum of the two expense increases.

Fix the cash-flow timing.
Identify whether the expense is paid at issue, at the start of a policy year, or at its end.

Revalue future obligations.
Use the changed expense assumption in the prospective reserve and include survival probabilities and discounting for later years.

Separate valuation from experience.
For actual profit, compare actual expense cash flows with the reserve basis. Do not treat a reserve recalculation as though it were itself an expense payment.

Worked Example: Revaluing a Two-Year Maintenance Expense

A hypothetical two-year term policy has annual premium $P_t=\$100$, death benefit $\$1{,}000$ at each year-end, and no maturity benefit. The conditional probability of death is $q_t=0.10$ in each year, so $p_t=0.90$. The effective annual interest rate is $i=5\%$, and $V_2=0$. Maintenance expense of $\$10$ is paid at the start of each policy year. Recalculate the reserves if the assumed maintenance expense rises to $\$15$ in each year.

Method. Work backward from the terminal reserve using the prospective reserve recursion. This calculation avoids adding a future expense increase without allowing for survival and discounting.

The discount factor is $v=1/1.05=0.95238095$. In the second year, the base reserve is

V_1=\frac{0.10(1{,}000)+0.90(0)}{1.05}-100+10 =95.238095-90 =\$5.238095.

With expense $\$15$, the second-year reserve becomes

V_1^{\mathrm{new}}=95.238095-100+15 =\$10.238095.

At issue, the base reserve is

V_0=\frac{100+0.90(5.238095)}{1.05}-100+10 =\frac{104.714286}{1.05}-90 =\$9.727891.

Using the revised second-year reserve and the revised first-year expense gives

V_0^{\mathrm{new}} =\frac{100+0.90(10.238095)}{1.05}-100+15 =\frac{109.214286}{1.05}-85 =\$19.013605.

Rounded to cents, the revised reserves are $V_1^{\mathrm{new}}=\$10.24$ and $V_0^{\mathrm{new}}=\$19.01$. The issue reserve increases by $\$9.29$, not $\$10$: the $\$5$ first-year increase is immediate, while the second $\$5$ is survival-weighted and discounted. Indeed, $5+(0.90/1.05)5=\$9.285714$, which checks the result. The common mistake avoided here is counting both expense increases as though both occurred at time zero.

Expense Deviations in the Annual Profit Equation

For an annual profit test using opening reserve $V_t$, the year-end profit conditional on being in force at the start of year $t$ is

\pi_{t+1} =(V_t+P_t-e_t^{\mathrm{actual}})(1+i) -q_tB_{t+1}-p_tV_{t+1}.

The opening reserve, premium, and expense are accumulated to year-end. Expected year-end outgo consists of the death benefit for deaths and the continuation reserve for survivors. The equation is conditional on an in-force policy at the start of the year; portfolio profit at that duration also depends on how many policies remain in force.

If reserves are based on the same assumptions as actual experience, substituting $e_t^{\mathrm{actual}}=e_t$ and the prospective reserve equation gives zero expected profit for that step. If actual expense is higher by $\Delta e_t$, while the reserve basis is held fixed, the conditional profit change is

\Delta\pi_{t+1}=-(1+i)\Delta e_t.

The extra expense is paid at the start of the year, so it also forgoes the interest that would otherwise have been earned on that amount. This formula does not apply unchanged if the expense is paid at year-end, if other assumptions also change, or if the reserve is recalculated on the shocked basis.

A useful justification is to subtract the profit equation under the original expense from the equation under the shocked expense, holding all other cash flows and reserves fixed. The only changed term is the accumulated net amount $V_t+P_t-e_t^{\mathrm{actual}}$. Increasing expense by $\Delta e_t$ reduces this amount by $\Delta e_t$; accumulation multiplies the reduction by $1+i$. Thus the profit change is exactly $-(1+i)\Delta e_t$.

Worked Example: A Maintenance Overrun in the Second Year

Use the two-year policy in the previous example, with reserves calculated on annual maintenance expense of $\$10$. In year 2, actual maintenance expense is $\$14$, while the premium remains $\$100$. The year-2 opening reserve is $V_1=\$5.238095$; $q_1=0.10$, $p_1=0.90$, $B_2=\$1{,}000$, $V_2=0$, and $i=5\%$. Find conditional year-2 profit and its effect per policy issued.

Method. Apply the annual profit equation using actual expense and the fixed reserve basis. This avoids treating the $\$4$ overrun as if it were paid at year-end.

The assets available before interest are

V_1+P_1-e_1^{\mathrm{actual}} =5.238095+100-14 =\$91.238095.

After interest, they are $91.238095(1.05)=\$95.80$. Expected year-end outgo is $0.10(1{,}000)+0.90(0)=\$100$. Therefore

\pi_2=95.80-100=-\$4.20.

With the assumed expense of $\$10$, the same calculation gives $(5.238095+100-10)(1.05)-100=\$0$, as required by the matching reserve basis. The expense overrun therefore reduces conditional profit by $\$4(1.05)=\$4.20$. If $90\%$ of policies issued are expected to reach the start of year 2, the year-2 profit effect per original policy is $0.90(-4.20)=-\$3.78$, measured at the end of year 2. The check is that both the profit equation and the direct accumulated-overrun calculation give the same conditional result.

Acquisition Expense Shocks and Duration of Profit

Acquisition expenses are commonly paid at or near issue. Their immediate effect is therefore different from a recurring maintenance expense shock. If the issue premium is received at issue, an acquisition expense $A_0$ is paid at issue, and the initial reserve $V_0$ is established immediately after those issue transactions, the initial profit is

\pi_0=P_0-A_0-V_0.

This simple issue-date equation assumes the stated reserve is the required amount after issue and that there are no other issue-date cash flows. If acquisition expenses are included differently in the valuation basis, the reserve and profit definitions must be adjusted consistently. Once an acquisition expense is paid, changing that historical cash flow does not directly change later reserves for future benefits and maintenance expenses. It can still affect issue profit and the overall profit signature.

Worked Example: An Acquisition Cost Overrun at Issue

A hypothetical single-premium contract receives $\$300$ at issue and establishes an initial reserve of $\$45$ immediately after issue. Its acquisition expense is $\$240$, paid at issue. All later benefits and maintenance expenses remain as originally assumed. Calculate issue profit before and after an acquisition expense increase of $\$18$, and state the direct effect on later policy-year profits.

Method. Apply the issue-date cash balance, keeping the stated post-issue reserve and all future assumptions fixed. This avoids spreading a one-time expense over later durations without a contractual or accounting basis for doing so.

Before the shock, issue profit is

\pi_0=300-240-45=\$15.

The revised acquisition expense is $240+18=\$258$, so revised issue profit is

\pi_0^{\mathrm{new}}=300-258-45=-\$3.

Issue profit falls by $\$18$. Because the shock is confined to an expense already paid at issue and future maintenance, benefits, and reserves are unchanged, the direct change to later policy-year profits is $\$0$. The initial profit reduction is the full $\$18$, not an annual amount.

Interpreting the Profit Pattern and Avoiding Traps

Expense shocks can produce different duration patterns even when their total dollar amounts are similar. A recurring maintenance increase creates losses in the years in which the extra expenses are paid. If the reserve basis is revised, it also raises reserves for future expenses, with the future effect weighted for survival and discounting. An acquisition increase paid at issue instead creates an immediate issue-date loss; by itself, it does not create a recurring maintenance loss.

A reserve increase is not automatically a profit increase or a profit loss of the same amount. In a profit test, a reserve movement is one component of the reserve-adjusted profit calculation. When the reserve is recalculated to match a changed expense assumption, projected profits may be redistributed across durations even though the revised projection remains internally consistent. When actual expenses differ from a fixed reserve basis, the difference emerges as experience profit or loss.

Do not omit interest on a start-of-year overrun. Under the timing used here, an extra expense $\Delta e$ changes year-end profit by $-(1+i)\Delta e$, not merely $-\Delta e$.
Do not add future expense increases without survival and discounting. A future maintenance expense contributes to the current reserve only along the path on which the policy remains in force.
Do not count the same expense change twice. Distinguish the actual cash expense from any reserve increase caused by revising the valuation assumption.
Do not treat acquisition and maintenance costs as interchangeable. Their payment dates determine the durations at which their direct cash effects appear.

Key takeaway: State the expense payment date, reserve measurement date, and whether the reserve basis changes before calculating an expense shock. Those choices determine whether the effect appears as an immediate cash loss, a prospective reserve change, or both in the profit-test projection.

Check Your Understanding

For each problem, use the timing convention and assumptions stated in the question.

A start-of-year maintenance expense rises by $\$6$. The effective annual interest rate is $4\%$, and reserves are held fixed. What is the conditional year-end profit change? (Answer: $-\$6.24$)
A reserve at time $t$ includes a current-year expense increase of $\$3$ and, if the policy survives, a $\$4$ expense increase one year later. The annual survival probability is $0.80$, and $i=5\%$. Find the reserve increase at time $t$. (Answer: $\$6.05$)
In the final year of a policy, the death probability is $0.20$, the death benefit is $\$500$, the premium is $\$70$, and the start-of-year expense is $\$8$. There is no continuation reserve. If $i=4\%$, find the prospective reserve immediately before the premium and expense. What is the reserve if the expense assumption rises to $\$11$? (Answer: $\$34.15$ initially and $\$37.15$ after the increase)
An extra $\$2$ maintenance expense is incurred at the start of policy year 3. Survival probabilities for years 1 and 2 are $0.90$ and $0.80$, respectively. The effective annual interest rate is $5\%$, and reserves are fixed. Find the expected year-end profit change per policy issued, before discounting that year-end amount to issue. (Answer: $-\$1.51$)
A contract receives $\$500$ at issue, pays acquisition expense of $\$120$, and establishes an initial reserve of $\$420$ immediately after issue. Acquisition expense rises by $\$25$, with all later cash flows and reserves unchanged. Find the revised issue profit and the direct change in subsequent policy-year profit. (Answer: revised issue profit $-\$65$; subsequent direct change $\$0$)

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