Inflation Shocks Across Expenses and Benefits

Why Inflation Scenarios Need Separate Treatment

An inflation scenario can affect a long-term product through more than one cash-flow channel. Operating costs may rise, benefits may be contractually escalated, or both may change. These effects need not have the same escalation rate, and a stress to one does not automatically imply a stress to the other. Projecting the channels separately makes it possible to explain how a scenario changes expected outgo, its present value, and the profit available from fixed premiums.

The calculations below use deterministic annual projections and a fixed effective annual discount rate. They are hypothetical illustrations, not descriptions of actual products or company experience. In a model with decrements, the cash flows used in the projection should already reflect the expected number of policies or claimants at each time. The inflation calculation then applies to the appropriate per-policy expense or benefit amount.

Definition: Let $B_0$ and $E_0$ be benefit and expense amounts at time 0. If they escalate at annual effective rates $g_B$ and $g_E$, respectively, their amounts at the end of year $t$ are $B_t=B_0(1+g_B)^t$ and $E_t=E_0(1+g_E)^t$. At effective annual discount rate $i$, the present value of projected outgo through year $n$ is $\sum_{t=1}^{n}(B_t+E_t)/(1+i)^t$, provided the amounts are payable at the ends of those years.

The escalation exponent is the number of annual increases that have occurred by the payment date. Thus, a time-0 amount is multiplied by $1+g$ once for a year-1 payment, and by $(1+g)^2$ for a year-2 payment. The discount exponent uses the same payment time when payments are at year-end. If the contract instead has payments at the beginning of a year or at another specified date, both the projected amount and its discount period must match that timing.

Comparing Reference and Stressed Projections

A reference projection uses the baseline assumptions. A stressed projection changes one or more inflation assumptions while holding other assumptions fixed, unless the scenario explicitly says otherwise. For a given discount rate, the reserve change attributable to a cash-flow change is the present value of the difference in cash flows. With fixed premium income, that increase in reserve outgo reduces projected profit by the same amount.

\Delta V = \sum_{t=1}^{n} \frac{ B_0\left[(1+g_B^{S})^t-(1+g_B^{R})^t\right] + E_0\left[(1+g_E^{S})^t-(1+g_E^{R})^t\right] }{(1+i)^t}

Here $R$ denotes the reference assumptions and $S$ the stressed assumptions. This expression assumes the same initial amounts, payment timing, discount rate, exposure projection, and term in both scenarios. If the scenario also changes interest rates, decrements, or payment timing, those effects must be incorporated separately rather than attributed to inflation alone.

Worked Example: Separating Benefit and Expense Inflation

A hypothetical two-year contract receives a single premium of $250 at time 0. At the ends of years 1 and 2, it pays benefits with time-0 amount $100 per year and incurs expenses with time-0 amount $20 per year. There are no decrements. The reference projection has zero escalation for both items; the stress uses 10% annual benefit escalation and 5% annual expense inflation. The effective annual discount rate is 10%. Calculate the outgo present value and profit in each projection, and identify the reserve increase.

Method. Project the benefit and expense amounts independently at each year-end, add them to obtain total outgo, and discount each year's amount at 10%. The premium is received at time 0, so it is not discounted.

In the reference projection, each year's benefit is $100 and each year's expense is $20. Total outgo is therefore $120 at each year-end. Its present value is

\frac{120}{1.10}+\frac{120}{1.10^2} =109.0909+99.1736 =\$208.2645.

Under stress, year-1 benefit is $100(1.10)=\$110$, and year-1 expense is $20(1.05)=\$21$. Total year-1 outgo is $131. In year 2, the benefit is $100(1.10)^2=\$121$, and the expense is $20(1.05)^2=\$22.05$. Total year-2 outgo is $143.05. Thus,

\frac{131}{1.10}+\frac{143.05}{1.10^2} =119.0909+118.2231 =\$237.3140.

The stressed outgo present value, and hence the projected reserve for these future outgo items, is $237.3140. The reserve increase is $237.3140-208.2645=\$29.0496$, using unrounded values. Profit, defined here as the time-0 premium less the present value of future outgo, is $41.7355 in the reference projection and $12.6860 under stress. The profit decrease equals the reserve increase, subject to rounding.

Check. The difference in present values is also the present value of the cash-flow increases: year 1 adds $11, and year 2 adds $23.05. Discounting those differences gives $11/1.10+23.05/1.21=\$29.0496$. This check confirms that both inflation channels have been included.

Why the Present-Value Difference Formula Holds

The comparison formula follows from the linearity of present value when the discount rate and payment dates are fixed. At each time $t$, subtract the reference amount from the stressed amount, then discount that difference. Summing those discounted differences is identical to subtracting the two present-value totals.

Write each scenario's cash flow.
At time $t$, the outgo is $C_t^R=B_t^R+E_t^R$ in the reference projection and $C_t^S=B_t^S+E_t^S$ under stress.

Subtract before or after discounting.
Because both amounts are discounted by the same $(1+i)^t$, $\frac{C_t^S}{(1+i)^t}-\frac{C_t^R}{(1+i)^t}=\frac{C_t^S-C_t^R}{(1+i)^t}$.

Sum over payment dates.
Adding the identity over $t=1,\ldots,n$ gives $V^S-V^R=\sum_{t=1}^{n}(C_t^S-C_t^R)/(1+i)^t$.

This derivation depends on consistent discounting and timing. It does not say that the present value change is independent of an interest-rate stress: if $i$ changes, the discount factors also change, and the difference combines cash-flow and discounting effects. Nor does it imply that the relationship between inflation and present value is linear in the escalation rate; annual escalation is compounded.

Worked Example: Expense Inflation with a Different Baseline Trend

A hypothetical contract has a level benefit of $500 at each of the next three year-ends and expenses with time-0 amount $100 per year. The reference expense inflation rate is 10% annually; a stress increases it to 20%. Benefits do not escalate, there are no decrements, the effective annual discount rate is 10%, and a premium of $1,500 is received at time 0. Calculate both outgo present values and profits.

Method. The benefit present value is the same in both projections. Project expenses using the reference and stressed rates separately. Since benefit cash flows are unchanged, the reserve difference comes entirely from expenses.

The benefit present value in either projection is

\frac{500}{1.10}+\frac{500}{1.10^2}+\frac{500}{1.10^3} =454.5455+413.2231+375.6574 =\$1,243.4260.

Reference expenses at the ends of years 1, 2, and 3 are $100(1.10)=\$110$, $100(1.10)^2=\$121$, and $100(1.10)^3=\$133.10$. Discounting each gives $100.0000, $100.0000, and $100.0000, respectively. The reference expense present value is therefore $300.0000, and total reference outgo present value is $1,243.4260+300.0000=\$1,543.4260$.

Stressed expenses are $100(1.20)=\$120$, $100(1.20)^2=\$144$, and $100(1.20)^3=\$172.80$. Their present values are

\frac{120}{1.10}=109.0909,\qquad \frac{144}{1.10^2}=119.0083,\qquad \frac{172.80}{1.10^3}=129.8272.

The stressed expense present value is $109.0909+119.0083+129.8272=\$357.9264$, using unrounded values. Total stressed outgo present value is $1,243.4260+357.9264=\$1,601.3524$. The reserve increase is $57.9264. Profit is $1,500-1,543.4260=-\$43.4260$ in the reference projection and $1,500-1,601.3524=-\$101.3524$ under stress. The change in profit is a decrease of $57.9264, matching the increase in outgo present value.

Check. The year-1 expense increase is $10, the year-2 increase is $23, and the year-3 increase is $39.70. Their present values sum to $10/1.10+23/1.21+39.70/1.331=\$57.9264$, confirming the reserve difference.

Joint Shocks with Projected Exposure

A per-policy escalation is not necessarily the same as the projected total cash flow. If fewer policies are expected to be in force or on claim at later durations, the expected exposure must also be applied. The following example treats the projected probabilities of being in payment as exposure weights. They are held fixed across scenarios so the calculation isolates inflation effects.

Worked Example: Joint Inflation Shocks on Expected Payments

A hypothetical contract has a benefit of $250 per year while in payment and an expense of $20 per year for each expected unit of active exposure. At the ends of years 1 and 2, the expected exposure weights are 0.8 and 0.6. The reference projection has no escalation. The stress applies 10% annual benefit escalation and 5% annual expense inflation. The effective annual discount rate is 10%, and the time-0 premium is $360. Calculate the reference and stressed outgo present values, then separate the benefit and expense contributions to the change.

Method. Multiply each escalated per-unit amount by the corresponding exposure weight, add benefit and expense cash flows at each date, and discount. The exponent $t$ applies to the per-unit amount before it is weighted by exposure.

Reference expected benefits are $250(0.8)=\$200$ in year 1 and $250(0.6)=\$150$ in year 2. Reference expenses are $20(0.8)=\$16$ and $20(0.6)=\$12$. Thus reference outgo is $216 and $162, with present value

\frac{216}{1.10}+\frac{162}{1.10^2} =196.3636+133.8843 =\$330.2479.

Under stress, expected benefits are $250(1.10)(0.8)=\$220$ in year 1 and $250(1.10)^2(0.6)=\$181.50$ in year 2. Expected expenses are $20(1.05)(0.8)=\$16.80$ and $20(1.05)^2(0.6)=\$13.23$. Total stressed outgo is therefore $236.80 and $194.73. Its present value is

\frac{236.80}{1.10}+\frac{194.73}{1.10^2} =215.2727+160.9339 =\$376.2066.

For attribution, the reference benefit present value is $200/1.10+150/1.21=\$305.7851$; the stressed benefit present value is $220/1.10+181.50/1.21=\$350.0000$. The benefit shock adds $44.2149. The reference expense present value is $16/1.10+12/1.21=\$24.4628$, while the stressed expense present value is $16.80/1.10+13.23/1.21=\$26.2066$. The expense shock adds $1.7438. Together the increases are $44.2149+1.7438=\$45.9587$, equal to the total outgo present-value increase $376.2066-330.2479$, using unrounded values.

Profit is $360-330.2479=\$29.7521$ in the reference projection and $360-376.2066=-\$16.2066$ under stress. The decrease in profit is $45.9587. This reconciliation relies on the premium and all non-inflation assumptions remaining unchanged.

Common Mistakes and Interpretation

A frequent error is to apply only one year's escalation to every future payment. For example, with a $100 base amount and a 5% annual rate, the incorrect year-2 amount $100(1.05)=\$105$ misses the second increase. The correct year-2 amount is $100(1.05)^2=\$110.25$. Discounting the incorrect amount cannot fix the projection error.

Other checks help keep scenario comparisons interpretable:

Keep the channels distinct. A benefit escalation assumption changes benefit amounts; an expense inflation assumption changes expenses. Do not apply one rate to both unless the scenario specifies that.
Use the right exposure. If projected expected claim counts or in-force policies vary by year, apply the appropriate exposure to the cash flow at each time.
Match inflation and discount timing. A year-end payment at time $t$ is escalated and discounted over $t$ years in the stated annual model.
State what is held fixed. If premiums, interest rates, or decrements also change, the total profit or reserve difference cannot be attributed solely to inflation.
Reconcile totals. When only outgo changes and premium income is fixed, the reserve increase and profit decrease should agree before rounding.

Inflation stresses matter because their effects accumulate over time: a higher escalation rate changes later cash flows by more than it changes the first payment. The present-value effect also depends on the discount rate and the timing of those cash flows. Reporting the benefit and expense components separately makes the projected change easier to explain and provides a direct arithmetic check.

Key takeaway: Project each benefit and expense with its own compounded escalation rate, apply the appropriate expected exposure, and discount at the payment time. Under fixed premiums and discounting, the present value increase in outgo equals the decrease in projected profit.

Check Your Understanding

For each problem, use end-of-year payments and keep the stated discount rate fixed between scenarios.

A benefit of $300 is payable at each of the next two year-ends. Its escalation rate changes from 3% to 6%, and the effective annual discount rate is 5%. What is the increase in benefit present value? (Answer: $25.63)
Expenses have a time-0 amount of $50 at each of the next two year-ends. The reference escalation rate is 0%, the stressed rate is 10%, and the discount rate is 10%. Calculate the increase in expense present value. (Answer: $13.22)
A single benefit with time-0 amount $400 is payable at the end of year 2. Its escalation rate changes from 1% to 4%; the discount rate is 5%. What is the increase in benefit present value? (Answer: $22.31)
A contract receives a $500 premium at time 0 and pays benefits of $200 and expenses of $20 at the ends of years 1 and 2. There are no decrements. At 10% interest, both inflation rates are 0% in the reference projection; under stress, benefit escalation is 5% and expense inflation is 10%. Find the change in outgo present value and the stressed profit. (Answer: Outgo present value increases by $31.32; stressed profit is $86.86.)

What You'll Learn