Expense Stress in Product Profit and Capital

Why Expense Stress Must Be Traced Through the Projection

An expense overrun can affect a long-term product in several places, but those effects are not interchangeable. A contractual charge deducted from a policy account is not automatically equal to the insurer’s actual expense. A higher actual expense can reduce shareholder profit and asset share even when the policy account value is unchanged. If future expenses are also expected to be higher, the reserve may increase and available capital may fall further.

The calculations below use simplified hypothetical products to isolate these relationships. They are aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. Unless a calculation states otherwise, cash flows are annual and the effective annual interest rate is applied once over the year. The results are not a prescribed regulatory capital formula.

Definition: An expense stress is a change in the expense experience or expense assumptions used in a projection. The contractual expense charge is the amount charged to a policy under its terms; the actual expense is the insurer’s cost of administering or acquiring the business. An overrun occurs when actual expense exceeds the amount allowed for in the pricing or projection basis.

For a universal life contract, the account-value projection uses the contractual deductions and credited interest. Actual insurer expenses affect the insurer’s financial results; they do not reduce the account value a second time unless the contract specifies an additional deduction. This distinction is essential: subtracting actual expenses from the account value after already deducting a contractual charge can double-count the expense.

Worked Example: Contractual Charges and an Expense Overrun

A hypothetical universal life policy has an account value of $500 at the beginning of a year. A $100 premium is paid at the beginning of the year. The policy then receives a contractual expense charge of $10 and a cost-of-insurance deduction of $8. The account earns an effective annual rate of 4% on the amount remaining after these deductions. The insurer’s expected actual expense is $12, but the actual expense is $18. Calculate the year-end account value and the change in the insurer’s expense margin relative to expectation.

Method. Project the policy account using contractual cash flows, then compare actual expense with the expense charge for the insurer’s expense margin. This avoids treating actual expense as an extra policy deduction.

The amount in the account immediately before interest is $500+100-10-8=582$. The year-end account value is therefore $582(1.04)=\$605.28$.

The expected expense margin is the expense charge less expected expense: $\$10-\$12=-\$2$. The actual expense margin is $\$10-\$18=-\$8$. Thus, the expense overrun reduces the margin by $-\$8-(-\$2)=-\$6$ per policy for the year.

Check. The account value uses the $10 contractual charge, not the $18 actual expense. The $6 deterioration in margin equals the increase in actual expense from $12 to $18; the contractual charge did not change.

Prospective Reserves for Future Net Expenses

A prospective reserve for future expenses can be calculated as the present value of future expenses less the present value of future charges available to meet them. For a simple deterministic contract with no lapses or claims, a level annual expense $e$ and level annual charge $c$, both payable at each year-end for $n$ years, the reserve at issue is the present value of the net outgo $e-c$.

V_0=\sum_{k=1}^{n}(e-c)v^k=(e-c)a_{\overline{n}|}, \qquad v=\frac{1}{1+i}.

This expression assumes that the expense and charge occur at the same time, that the policy remains in force for all $n$ years, that the annual amounts are level, and that the discount rate is $i$. It does not allow for mortality, lapses, expense inflation, or expenses that vary by duration. If those features are present, the expected net cash flow at each time must be projected explicitly.

Derivation: At time $k$, the net expense outgo is $e-c$. A dollar due at that time has present value $(e-c)v^k$. Summing the present values for $k=1,\ldots,n$ gives $V_0=(e-c)\sum_{k=1}^{n}v^k$. By definition, $\sum_{k=1}^{n}v^k=a_{\overline{n}|}$, so $V_0=(e-c)a_{\overline{n}|}$. If $e<c$, the net cash flow is an inflow and this simplified net expense reserve is negative; it should not be interpreted as a positive liability.

A change in the future expense assumption changes the reserve even if the current year’s actual expense has already been observed. The reserve movement can therefore create an additional profit effect beyond the current-year expense variance.

Worked Example: Expense Strengthening and Profit Emergence

A hypothetical policy has two annual year-end expense payments remaining at issue. The expected expense is $12 per year and the matching annual charge is $10. At the end of year 1, the actual expense is $18, and the expense assumption for the one remaining year is strengthened to $18. The charge remains $10. Use an effective annual interest rate of 4%. Calculate the opening reserve on the original basis, the year-end reserve on the strengthened basis, and the year-1 profit arising from these expense cash flows and reserves.

Method. Value net future expenses prospectively at each valuation time. Use the reserve roll-forward: opening reserve accumulated with interest, less the year’s net expense outgo, less the closing reserve. This example assumes no premiums, claims, or other cash flows.

At issue, the expected net expense is $\$12-\$10=\$2$ in each of two years. With $v=1/1.04$, the opening reserve is $2(v+v^2)$. The discount factors are $v=0.9615385$ and $v^2=0.9245562$, so $V_0=2(0.9615385+0.9245562)=\$3.7722$, or $3.77 to the nearest cent.

At the end of year 1, one year remains. On the strengthened basis, its net expense is $\$18-\$10=\$8$, payable at the end of that remaining year. The closing reserve is $V_1=8v=8(0.9615385)=\$7.6923$, or $7.69 to the nearest cent.

The actual year-1 net expense outgo is also $\$18-\$10=\$8$. Profit at the end of the year is $V_0(1.04)-8-V_1$. Using the unrounded opening reserve, $V_0(1.04)=3.7721893(1.04)=3.9230769$, so profit is $\$3.9230769-\$8-\$7.6923077=-\$11.7692308$, or −$11.77 to the nearest cent.

Check. If the expense remained at $12 in both years, the closing reserve would be $2v=\$1.9231$, and the year-1 net outgo would be $2. Then $3.7721893(1.04)-2-1.9230769=0$. The stressed result is lower by $11.77: $6.00 from the extra current-year net outgo and $5.77 from increasing the closing reserve from $1.92 to $7.69.

Asset Shares and a Simplified Capital Test

An asset share tracks accumulated amounts attributed to a policy or group of policies. In a deliberately simple annual projection, if premiums enter at the beginning of the year and expenses and benefits are paid before the end-year accumulation, one form of the recursion is $AS_{t+1}=(AS_t+P_t-E_t-B_t)(1+i)$. Here $AS_t$ is the opening asset share, $P_t$ is premium, $E_t$ is actual expense, $B_t$ is benefit outgo, and $i$ is the effective annual rate. The formula assumes all listed pre-interest cash flows occur at the beginning of the accumulation period; different payment timings require different accumulation factors.

A policy account value may follow a separate contractual recursion. Comparing the asset share with the guarantee or account-value liability can show an economic shortfall in a simplified scenario. A capital requirement, however, depends on the stated capital definition. In the next example, the measure is explicitly defined as a fixed buffer plus any shortfall between the liability and available assets; it is illustrative, not a general solvency standard.

Worked Example: Correcting an Expense-Stress Asset-Share Gap

A hypothetical universal life policy has an opening account value and opening asset share of $100. A $500 premium is paid at the beginning of the year. The contractual policy deduction is $40, while the insurer’s actual expense is $50. There are no benefits during the year, and both the account balance and the asset share earn an effective annual rate of 4% after the stated deductions. Calculate the ending account value, ending asset share, and their difference. Then calculate a simplified capital measure equal to a buffer of 5% of the account-value liability plus any positive excess of that liability over the asset share.

Method. Apply the contractual deduction in the account-value projection and actual expense in the asset-share projection. Accumulate each resulting amount at 4%. This avoids substituting actual expense for the contractual charge.

The account value before interest is $\$100+\$500-\$40=\$560$. Thus, the ending account value is $\$560(1.04)=\$582.40$.

The asset share before interest is $\$100+\$500-\$50=\$550$. Thus, the ending asset share is $\$550(1.04)=\$572.00$. The account-value excess over asset share is $\$582.40-\$572.00=\$10.40$.

The 5% buffer is $0.05(\$582.40)=\$29.12$. The defined capital measure is therefore $\$29.12+\$10.40=\$39.52$.

Check. The pre-interest difference is $\$560-\$550=\$10$. After interest it is $\$10(1.04)=\$10.40$, which agrees with the difference between the two ending values. All amounts are stated to the nearest cent.

How to Structure an Expense-Stress Projection

For a multi-year case, keep the contractual account projection, actual expense experience, reserve basis, and capital definition in separate but reconcilable calculations. A useful sequence is:

Identify the expense items.
Separate contractual policy deductions, expected expenses, actual expenses, and future expense assumptions. State their amounts and timing.

Project policy values and asset shares separately.
Use contractual deductions for the policy value and actual expenses for the asset share. Apply interest only over the period and to the cash flows specified.

Recalculate the prospective reserve.
Value future net expenses on the stressed assumption. Include only the future cash flows and decrements specified in the model.

Reconcile profit and capital.
Show current expense variance and reserve movement in profit emergence. Then apply the problem’s stated capital definition to stressed assets and liabilities.

Common Exam Traps

Deducting an actual expense from the policy account without contractual support. If the contract deducts a stated charge, use that charge in the account-value recursion. Actual expense instead affects the insurer’s profit and asset share.
Ignoring future expense assumptions in the reserve. A current overrun may signal higher future expenses. If the reserve basis is strengthened, calculate the closing reserve on that basis and include its movement in profit.
Mixing cash-flow timing. A beginning-of-year premium, an end-of-year expense, and an end-of-year reserve are not interchangeable. State the timing before applying interest.
Reporting a capital amount without defining it. A liability shortfall, a risk margin, and a regulatory capital requirement are different measures. Use the measure explicitly specified in the scenario.
Subtracting rounded figures. Keep unrounded values through the projection and round the final money amounts to cents. For example, the asset-share gap in the worked example is $10(1.04)=\$10.40$, not a difference based on prematurely rounded intermediate amounts.

Key takeaway: An expense stress should be traced through distinct mechanisms: contractual charges determine policy-value deductions, actual expenses affect current profit and asset share, future expense assumptions affect prospective reserves, and the stated capital definition determines the capital result. Reconcile each step without double-counting expenses.

Check Your Understanding

For each question, state the timing and basis used before calculating.

A hypothetical universal life policy has an opening account value of $800, receives a beginning-year premium of $200, and has contractual deductions of $25 for expenses and $15 for insurance. The remaining balance earns 3% for the year. What is the ending account value? (Answer: $988.80)
A policy has an annual expense charge of $14. Expected actual expense is $16, but actual expense is $19. By how much does the expense margin change compared with expectation? (Answer: decreases by $3.00)
A policy has three annual year-end expense payments remaining. Each expected expense is $30 and each matching charge is $22. At an effective annual rate of 5%, calculate the issue reserve for these net expenses, assuming certain persistence and no other cash flows. (Answer: $21.79)
At the beginning of a year, the reserve is $20. During the year, an end-year net expense outgo of $6 occurs, and the closing reserve is $17. At an effective annual rate of 4%, calculate year-end profit using opening reserve accumulated with interest, less net outgo and closing reserve. (Answer: −$2.20)
A policy has an opening account value and asset share of $200, a beginning-year premium of $300, a contractual account deduction of $20, and actual expense of $30. Both balances earn 4% after deductions; there are no benefits. Calculate the ending account value, ending asset share, and account-value excess over asset share. (Answer: account value $499.20; asset share $488.80; excess $10.40)

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