Expense-Level Shocks

Why Expense-Level Shocks Need Careful Exposure Bases

An expense stress changes the assumed level of one or more costs. The arithmetic may look straightforward, but the result depends on applying each change to the right exposure. An additional $10 per policy-year is multiplied by policy-years, while an additional $10 per claim is multiplied by expected claims. A charge set as a percentage of premium is based on premium, not on policy count or claim count.

This tutorial treats a level shock as a change to an expense amount or rate for a specified period. It does not assume that costs grow automatically from year to year. In a projection, distinguish the shock’s size from its effective date, identify the relevant exposure in each period, and then account for payment timing. These steps are useful when assessing the effect on projected profit or net present value (NPV), including in hypothetical solvency scenarios.

Definition: An expense-level shock is a change to an expense assumption, such as a fixed amount per policy, an amount per claim, or a proportion of premium. Its projected cost is determined by applying the changed assumption to the corresponding exposure over the period in which the shock applies. Acquisition expenses are applied to new policies; renewal expenses to the policies covered by the relevant renewal expense; claim expenses to claims; and premium-based expenses to premium.

Set Up the Projection Before Applying a Shock

For a period-by-period projection, record the number of new policies, the number of policies subject to renewal expenses, expected claims, and premium. These exposure measures need not be identical. For example, a policy can generate premium and a renewal expense in a year without generating a claim. Expected claim counts can be fractional because they are expected values across a portfolio.

For clarity, use the following notation in this tutorial: $M_t$ is the number of new policies at time $t$; $N_t$ is the number of policies subject to renewal expense in period $t$; $Q_t$ is the expected number of claims in period $t$; and $G_t$ is the premium subject to a percentage expense. Let $c^A_t$ be acquisition expense per new policy, $c^R_t$ renewal expense per applicable policy, $c^C_t$ expense per claim, and $\alpha_t$ the percentage of premium charged as an expense.

E_t = M_t c^A_t + N_t c^R_t + Q_t c^C_t + \alpha_t G_t

Here $E_t$ is total projected expense for the period. This expression assumes each expense component is charged on the stated basis and that the projection counts the relevant exposures consistently. If a product has more detailed expense rules—for example, a charge only on premium actually collected—use those rules to define the exposure rather than substituting a different base.

When premiums and benefits are otherwise held fixed, additional expense reduces projected profit dollar for dollar. A shock may also affect policyholder behavior or the amount of business written, but those are not automatic consequences of changing an expense assumption. If such effects are part of the scenario, update the affected projection assumptions and exposures rather than treating the direct expense change as the whole result.

Deriving the Direct Profit and NPV Impact

Suppose the expense assumptions change by $\Delta c^A_t$, $\Delta c^R_t$, $\Delta c^C_t$, and $\Delta\alpha_t$. If the exposure counts and premium are held fixed, the increase in expense in period $t$ is the sum of each assumption change times its own exposure.

\Delta E_t = M_t\Delta c^A_t +N_t\Delta c^R_t +Q_t\Delta c^C_t +G_t\Delta\alpha_t

This follows by subtracting the original expense from the stressed expense. For example, the renewal component changes from $N_t c^R_t$ to $N_t(c^R_t+\Delta c^R_t)$; their difference is $N_t\Delta c^R_t$. The same subtraction gives the acquisition and claim terms. The premium-based component changes from $\alpha_tG_t$ to $(\alpha_t+\Delta\alpha_t)G_t$, so its increase is $G_t\Delta\alpha_t$. Adding the component differences gives the formula.

If a positive expense increase reduces profit, the direct profit change is $-\Delta E_t$. For an effective annual interest rate $i$, define $v=1/(1+i)$. If period-$t$ expenses are paid at the end of year $t$, and acquisition expenses at time 0 are paid immediately, the present value of the direct expense increase is:

\Delta\operatorname{PV}(E) = M_0\Delta c^A_0 + \sum_{t=1}^{n}v^t \left( M_t\Delta c^A_t +N_t\Delta c^R_t +Q_t\Delta c^C_t +G_t\Delta\alpha_t \right)

A positive value here is an increase in the present value of expenses and, with other cash flows unchanged, an equal decrease in NPV. This timing convention is an assumption, not a universal payment rule. If the projected cash flows occur at different times, discount them at their actual times. Likewise, do not discount an immediate time-0 expense.

Worked Examples

Worked Example: A Simultaneous Shock to Four Expense Bases

A hypothetical portfolio has 100 new policies at time 0. It has 100 policies in force in year 1 and 90 in year 2. Annual premium is $1,000 per in-force policy, and expected benefit outgo is $20,000 in year 1 and $30,000 in year 2. Acquisition expense is $200 per new policy at time 0. Renewal expense is $80 per policy-year, claim-handling expense is $150 per claim, and a further expense is 5% of premium. Expected claims are 2 in year 1 and 3 in year 2. Calculate the baseline and stressed NPV at an effective annual interest rate of 5%. The stress increases acquisition expense by 20%, renewal expense by 25%, and claim-handling expense by 10%, and raises the premium-expense rate from 5% to 6%. All year-end cash flows are paid at year end.

Method. Calculate each expense on its own exposure basis, form each year’s profit, and discount the cash flows. Amounts are shown in thousands of dollars.

Baseline. Acquisition expense at time 0 is $100(0.200)=20.000$. Year 1 premium is $100(1.000)=100.000$; renewal expense is $100(0.080)=8.000$; claim-handling expense is $2(0.150)=0.300$; and premium-based expense is $0.05(100.000)=5.000$. Thus year 1 profit is $100.000-20.000-8.000-0.300-5.000=66.700$.

Year 2 premium is $90(1.000)=90.000$; renewal expense is $90(0.080)=7.200$; claim-handling expense is $3(0.150)=0.450$; and premium-based expense is $0.05(90.000)=4.500$. Year 2 profit is $90.000-30.000-7.200-0.450-4.500=47.850$.

Stressed projection. The new acquisition amount is $200(1.20)=\$240$ per policy, giving time-0 acquisition expense $100(0.240)=24.000$. The year 1 renewal amount is $80(1.25)=\$100$, so renewal expense is $100(0.100)=10.000$. Claim expense is $150(1.10)=\$165$ per claim, giving $2(0.165)=0.330$. Premium expense is $0.06(100.000)=6.000$. Stressed year 1 profit is $100.000-20.000-10.000-0.330-6.000=63.670$.

In year 2, renewal expense is $90(0.100)=9.000$, claim-handling expense is $3(0.165)=0.495$, and premium expense is $0.06(90.000)=5.400$. Stressed year 2 profit is $90.000-30.000-9.000-0.495-5.400=45.105$.

NPV calculation. With $v=1/1.05$, baseline NPV is $-20.000+66.700/1.05+47.850/1.05^2=86.925$ thousand dollars. Stressed NPV is $-24.000+63.670/1.05+45.105/1.05^2=77.550$ thousand dollars. The NPV decrease is $86.925-77.550=9.375$ thousand dollars. Values are rounded to three decimal places in thousands, equivalent to the nearest dollar.

Check and trap avoided. The undiscounted expense increases are $4,000 at time 0, $3,030 in year 1, and $2,745 in year 2. Discounting the latter two gives $4,000+3,030/1.05+2,745/1.05^2=\$9,375.51$, consistent with the NPV difference before rounding. A common error is to apply the 10% claim-handling shock to policies rather than claims; the correct calculation uses 2 and 3 expected claims.

Worked Example: Separating Per-Policy and Percentage-of-Premium Shocks

A hypothetical one-year portfolio has 60 policies with an annual premium of $900 each and 40 policies with an annual premium of $1,500 each. Every policy incurs a $70 renewal expense. A further expense is 3% of premium. The portfolio has 5 expected claims, each with $200 of claim-handling expense. A stress raises renewal expense by $12 per policy, raises the premium-based rate from 3% to 3.8%, and raises claim-handling expense by $30 per claim. Find total baseline and stressed expenses and the change in one-year profit, assuming premium, benefits, and policy counts do not change.

Method. First calculate total premium, then apply each expense assumption to its matching exposure. Because this is one period and no discounting is requested, the expense increase equals the one-year profit decrease.

Total premium is $60(900)+40(1,500)=54,000+60,000=\$114,000$. Baseline renewal expense is $100(70)=\$7,000$. Baseline premium-based expense is $0.03(114,000)=\$3,420$. Baseline claim-handling expense is $5(200)=\$1,000$. Total baseline expense is $7,000+3,420+1,000=\$11,420$.

Stressed renewal expense is $100(70+12)=\$8,200$. The stressed premium-based rate is $0.038$, so that expense is $0.038(114,000)=\$4,332$. Stressed claim-handling expense is $5(200+30)=\$1,150$. Total stressed expense is $8,200+4,332+1,150=\$13,682$.

The increase is $13,682-11,420=\$2,262$. Equivalently, its components are $100(12)=\$1,200$, $114,000(0.038-0.030)=\$912$, and $5(30)=\$150$; their sum is $2,262. With all other projected items fixed, one-year profit decreases by $2,262.

Check and trap avoided. The premium-based increase is 0.8 percentage points, or $0.008$, not 0.8 as a decimal. Applying it to the $114,000 premium gives $912. Applying the percentage shock to the 100 policies instead would use the wrong exposure and produce an invalid result.

Worked Example: Applying a Shock Only When It Takes Effect

A hypothetical block has 120 policies in year 1 and 100 in year 2. Premium is $1,000 per policy-year. Expected benefit outgo is $30,000 in year 1 and $25,000 in year 2. At time 0, 120 new policies incur acquisition expense of $250 each. Renewal expense is $60 per in-force policy-year; claim-handling expense is $100 per claim; and premium-based expense is 4% of premium. Expected claims are 1.2 in year 1 and 1.0 in year 2. A stress raises time-0 acquisition expense by 20%. The other changes begin in year 2 only: renewal expense increases by $10 per policy, claim-handling expense increases by 20%, and the premium-based rate rises to 4.5%. Calculate baseline and stressed NPV at an effective annual interest rate of 4%. Acquisition expense is paid at time 0; all other cash flows are paid at year end.

Method. Project the baseline and stressed cash flows separately, keeping year 1 unstressed for the delayed changes. Amounts below are in thousands of dollars.

Baseline acquisition expense is $120(0.250)=30.000$. In year 1, premium is $120(1.000)=120.000$; renewal expense is $120(0.060)=7.200$; claim-handling expense is $1.2(0.100)=0.120$; and premium-based expense is $0.04(120.000)=4.800$. Year 1 profit is $120.000-30.000-7.200-0.120-4.800=77.880$.

In year 2, premium is $100.000$, benefit outgo is $25.000$, renewal expense is $100(0.060)=6.000$, claim-handling expense is $1.0(0.100)=0.100$, and premium-based expense is $0.04(100.000)=4.000$. Year 2 profit is $100.000-25.000-6.000-0.100-4.000=64.900$. Baseline NPV is $-30.000+77.880/1.04+64.900/1.04^2=104.888$ thousand dollars.

Under the stress, acquisition expense per policy is $250(1.20)=\$300$, so time-0 acquisition expense is $120(0.300)=36.000$. Year 1 remains unchanged, with profit of 77.880. In year 2, renewal expense is $100(0.070)=7.000$, claim-handling expense is $1.0(0.120)=0.120$, and premium-based expense is $0.045(100.000)=4.500$. Stressed year 2 profit is $100.000-25.000-7.000-0.120-4.500=63.380$. Stressed NPV is $-36.000+77.880/1.04+63.380/1.04^2=97.483$ thousand dollars.

The NPV decrease is $104.888-97.483=7.405$ thousand dollars, or $7,405.33 when calculated without rounding intermediate values. A direct check is $6,000+1,000/1.04^2+20/1.04^2+500/1.04^2=\$7,405.33$: the components are the immediate acquisition increase and the three year-2 increases.

Check and trap avoided. Year 1 profit is unchanged by the renewal, claim, and premium-based shocks because they do not begin until year 2. Applying those changes in both years would overstate the impact. The acquisition increase is immediate and therefore is not discounted.

Common Errors and Practical Checks

Before accepting a stress result, check both the calculation and its interpretation. The direct expense impact should follow from the stated shock and exposure data; any change to premiums, claims, or policy counts needs an explicit assumption or a revised projection.

Using one exposure for every expense. Multiply per-policy amounts by the policies to which they apply, per-claim amounts by expected claims, and premium-based rates by the relevant premium.
Misreading a rate change. A change from 4% to 4.5% is an increase of 0.5 percentage points, or 0.005 as a decimal.
Applying a stress outside its effective period. Match each changed assumption to the time periods in which it applies; do not retrospectively apply a delayed shock.
Discounting at the wrong time. Use the cash-flow timing specified by the projection. An immediate acquisition expense is not discounted, while a year-end expense is discounted to time 0.
Mixing a direct expense effect with secondary effects. If a scenario also changes sales, lapses, or other exposures, identify and model those changes explicitly instead of assuming they are implied by the expense increase.

A useful final check is to calculate the shock’s direct expense increase in two ways when possible: subtract total baseline expenses from total stressed expenses, and add the exposure-weighted component increases. Agreement between those calculations helps catch omitted expense categories and incorrect exposure bases. A separate check of timing then confirms that the NPV impact reflects when each increase is paid.

Key takeaway: An expense shock has no single universal multiplier. Its impact depends on the expense basis, the relevant exposure, the time period in which it applies, and the payment timing. State each of these before calculating the change in projected profit or NPV.

Check Your Understanding

For each problem, apply the shock to its stated exposure and timing.

A one-year block has 80 policies. Renewal expense rises from $75 to $90 per policy, and an expense rate rises from 4% to 4.5% of annual premium. Annual premium is $1,200 per policy. Find the total expense increase. (Answer: $1,680)
A hypothetical insurer issues 150 new policies. Acquisition expense rises by 12% from $180 per policy. Find the increase in acquisition expense. (Answer: $3,240)
A year’s projection has 240 applicable policies, 7.5 expected claims, and premium of $950 per policy. Renewal expense rises by $8 per policy, claim expense rises by $20 per claim, and the premium-based expense rate rises by 0.3 percentage points. Find the total expense increase. (Answer: $2,754)
A $2,400 expense increase is paid at the end of year 2. At an effective annual interest rate of 5%, find its present value at time 0. Round to the nearest cent. (Answer: $2,176.87)
At time 0, acquisition expense rises by $15 per policy on 200 new policies. At the end of year 1, renewal expense rises by $6 on 150 policies, and an expense rate rises by 0.4 percentage points on $150,000 of premium. At an effective annual interest rate of 4%, find the present value of the total increase. (Answer: $4,442.31)

What You'll Learn