Purpose and Projection Framework
An expense stress and an investment-return stress affect a long-term product through different mechanisms. Higher expenses reduce cash available to invest and may increase the prospective reserve if the changed expense assumption is recognized in the valuation basis. Lower investment returns reduce the accumulation of assets. A useful stress case keeps those mechanisms separate before combining them.
The calculations below use a hypothetical two-year endowment contract, projected per policy. It pays a premium of $1,000 at the start of each year, has expenses at the same times as the premiums, and pays a $2,200 benefit at the end of year 2. There is no mortality, lapse, tax, or other contingent cash flow in this deliberately simplified example. The annual effective valuation interest rate is 4%. The base expenses are $100 at issue and $40 at the start of year 2.
Let \(P_t\) and \(E_t\) denote the premium and expense at the start of year \(t\), and let \(B_{t+1}\) denote the benefit paid at the end of that year. Let \(V_t\) be the prospective reserve immediately before the premium and expense at time \(t\). Set the terminal reserve \(V_2=0\). The realized annual effective investment return in year \(t\) is \(j_t\), which need not equal the valuation rate \(i\).
Keeping the reserve basis fixed while changing actual experience measures the effect of experience deviating from the basis. Recalculating reserves under stressed assumptions instead measures a liability revaluation. Those are related but distinct results; adding both without identifying the valuation convention can count the same assumption change twice.
Reserve Recursion and Profit Interpretation
For a one-year interval, the prospective reserve at time \(t\) is the current expense less the current premium, plus the present value of the end-of-year benefit and next reserve. Thus, at valuation rate \(i\),
This equation follows directly by valuing the contract’s future net outgo at time \(t\). Rearranging gives \(V_t+P_t-E_t=(B_{t+1}+V_{t+1})/(1+i)\). If the realized return equals the valuation rate, substituting this expression into the profit formula gives
Zero profit under matching assumptions is a useful check on the projection. If actual return or expense differs from the valuation basis, the profit formula shows where the difference enters. In particular, an expense paid at the start of a year also changes the amount invested during that year.
Work backward from the terminal reserve using the prospective reserve recursion and the valuation rate.
Change the relevant actual expense or realized return, keeping other assumptions unchanged.
Use the opening reserve, cash flows, realized return, benefit, and closing reserve for each year.
Project the joint scenario directly, then compare it with the separate results while allowing for timing interactions.
Worked Example: Establishing the Base Projection
Method. Work backward from \(V_2=0\), using a 4% effective annual valuation rate and the base premiums, expenses, and benefit. This calculation also provides a check that the matching-return base projection produces zero annual profit.
At time 1, the second $1,000 premium and $40 expense occur immediately, and the $2,200 benefit is due at time 2. The reserve immediately before those time-1 cash flows is
At time 0, the corresponding reserve is
With the realized return equal to 4% in year 1, the accumulated amount after the time-0 premium and expense is \((210.946746+1{,}000-100)(1.04)=1{,}155.384615\). Subtracting the time-1 reserve gives \(\Pi_1=0\). In year 2, the amount accumulated after the time-1 premium and expense is \((1{,}155.384615+1{,}000-40)(1.04)=2{,}200\). After paying the benefit, \(\Pi_2=0\). The reserves and profits are therefore internally consistent with the stated basis.
Check. Both year-end profits are zero because the valuation rate and realized returns match, and the expenses used in the reserve recursion are the expenses projected to occur. A common mistake is to omit the current expense from the reserve recursion even though \(V_t\) is defined immediately before that expense.
Expense Stress: Cash-Flow Loss and Reserve Revaluation
Now suppose actual expenses are $130 at issue and $60 at the start of year 2. First hold the base reserves fixed and retain a 4% realized return. This isolates the profit impact of actual expenses exceeding the base assumptions. The starting reserve remains \(V_0=210.946746\), and the closing year-1 reserve remains \(V_1=1{,}155.384615\).
Worked Example: Higher Expenses with Base Reserves
Method. Apply the actual stressed expenses in the profit formula while retaining the base valuation reserves and a 4% realized return.
In year 1, the extra $30 expense is paid at the start of the year. The amount available to accumulate is \(210.946746+1{,}000-130=1{,}080.946746\). Thus,
The loss is $31.20 rather than $30 because the $30 expense shortfall also loses a year of investment return. In year 2, the amount available after the $60 expense is \(1{,}155.384615+1{,}000-60=2{,}095.384615\). Therefore,
The year-1 loss accumulates for one more year, so the total at time 2 is \(-31.20(1.04)-20.80=-53.248\), or \(-$53.25\) to the nearest cent. The period profits are reported at different times; simply adding \(-$31.20\) and \(-$20.80\) would not put them on a common date.
Reserve revaluation check. If the higher expense assumption is instead recognized in the prospective reserves, the time-1 reserve increases by the extra $20 renewal expense: \(V_1'=1{,}155.384615+20=1{,}175.384615\). The time-0 reserve also includes the extra $30 issue expense and the present value of the extra $20 renewal expense:
These are reserve increases of $20 at time 1 and $49.23 at time 0, not the same measure as the experience profits calculated using base reserves. A common mistake is to treat the reserve increase as an additional expense cash flow. It is a change in the measured liability; the expense itself is the cash flow.
Investment Stress and the Combined Scenario
An investment-return stress changes accumulation, not the stated policy cash flows. To isolate it, retain base expenses and reserves but use a 2% realized annual effective return instead of 4%. Then combine that return stress with the higher expenses. The valuation reserves remain fixed at the base values in both projections.
Worked Example: Lower Returns Separately and with Higher Expenses
Method. Calculate the profit in each period directly from the profit vector. First change only the realized return; then use both stressed expenses and the stressed return. All dollar figures are per policy.
Investment stress only. In year 1, the funds after the base premium and expense are \(210.946746+1{,}000-100=1{,}110.946746\). At 2%, they accumulate to \(1{,}110.946746(1.02)=1{,}133.165681\). Therefore,
In year 2, the amount after the base premium and expense is \(1{,}155.384615+1{,}000-40=2{,}115.384615\). It accumulates to \(2{,}157.692308\), so
Accumulated to time 2, the total is \(-22.218934(1.02)-42.307692=-64.971005\), or \(-$64.97\) to the nearest cent.
Combined expense and investment stress. In year 1, after the $130 expense, the amount invested is \(210.946746+1{,}000-130=1{,}080.946746\). At 2%, this becomes \(1{,}102.565681\). The year-1 profit is \(1{,}102.565681-1{,}155.384615=-$52.818934\). In year 2, after the $60 expense, the amount invested is \(1{,}155.384615+1{,}000-60=2{,}095.384615\). At 2%, it becomes \(2{,}137.292308\), giving a year-2 profit of \(2{,}137.292308-2{,}200=-$62.707692\).
Accumulating the first combined profit to time 2 gives \(-52.818934(1.02)-62.707692=-116.583005\), or \(-$116.58\) to the nearest cent. For comparison, the separately calculated final-date losses sum to \(-$53.248-\$64.971005=-$118.219005\). That differs from the combined result because the interaction changes how the first-period profit or loss accumulates. The joint projection, rather than an unadjusted sum of separately dated amounts, is the appropriate combined result.
Check. The combined period profits are more negative than either isolated case, as expected for this example. A common mistake is to calculate the investment effect as a percentage of the benefit alone. The return applies to the amount invested during each year, which depends on the reserve, premium, and expense timing.
Practical Checks and Exam Traps
A clear stress analysis states both its cash-flow timing and its reserve convention. In this case, premiums and expenses are paid at the start of each year, benefits at year-end, and reserve values are measured before the start-of-year cash flows. Moving an expense from the start to the end of a year changes its investment effect and may also change the prospective reserve formula.
The expense and investment stresses should first be reported separately. The expense-only projection identifies the effect of higher outgo; the investment-only projection identifies the effect of lower accumulation. The combined projection then captures the interaction between those changes. This separation helps explain whether a result comes from a cash-flow deviation, reserve revaluation, or the timing of investment income.
- Wrong timing: treating the $30 issue expense increase as a year-end payment gives a first-year reduction of only $30. With the stated start-of-year timing, its year-end effect is \(30(1.04)=$31.20\).
- Mixing bases: changing the actual return but silently changing the reserve discount rate as well combines experience loss with reserve revaluation. State explicitly whether reserves are fixed or recalculated.
- Adding profits at different dates: the first-year profit is measured at time 1 and the second-year profit at time 2. Accumulate or discount them to a common date before comparing totals.
- Assuming additivity: separately measured stress effects need not sum to the combined result when changed cash flows also change the amount exposed to investment returns.
This framework is aligned with the SOA Exam ALTAM (Advanced Long-Term Actuarial Mathematics) syllabus. In a larger product model, the same discipline applies, but the cash flows and reserves may also depend on decrements, multiple states, guarantees, and other assumptions.
Check Your Understanding
Use the stated timing in each problem. Report dollar answers to the nearest cent.
- A one-year contract has \(i=5\%\), premium $500, expense $20, and a $600 benefit at year-end. The terminal reserve is zero. Calculate the prospective reserve immediately before the premium and expense. (Answer: $91.43)
- For the contract in Question 1, suppose the realized return is 4% rather than 5%, with the same cash flows and reserve. Calculate the year-end profit. (Answer: \(-$5.71\))
- A two-year contract has \(i=5\%\), premiums of $800 at times 0 and 1, expenses of $50 and $30 at those times, and a $1,800 benefit at time 2. With \(V_2=0\), calculate the base \(V_1\) and \(V_0\). Then find the reserves if only the time-1 expense assumption increases by $15. (Answer: Base \(V_1=$944.29\), \(V_0=$149.32\); revised \(V_1=$959.29\), \(V_0=$163.61\))
- At the start of a year, the opening reserve is $300, the premium is $700, and the stressed expense is $45. A $1,000 benefit is paid at year-end, the terminal reserve is zero, and the realized return is 1%. Calculate profit. Then calculate profit if the expense were $35 and return 3%, with all other values unchanged. (Answer: Combined stress profit \(-$35.45\); lower-expense, higher-return profit \(-$6.05\))