Combining Market, Lapse, and Expense Stresses

Why Combined Stresses Need a Joint Projection

A market fall, increased lapses, and higher expenses can occur in the same projection. They do not affect every cashflow in the same way: a market fall can increase a maturity guarantee, while lapses can reduce the number of policies reaching maturity. At the same time, a lower fund can reduce a percentage-based surrender charge, and expense inflation can increase costs on both lapsed and continuing policies. Adding three separately calculated impacts can therefore give the wrong answer.

The method is to specify one stressed scenario, project its cashflows consistently, and then compare its result with the baseline and with isolated stresses. The example below is hypothetical. Its capital measure is an explicitly defined comparison of reserves, not a regulatory capital formula.

Model and assumptions: Consider one hypothetical one-year equity-linked policy. At the start of the year its fund account is $A_0=\$1{,}000$. A fixed charge $C_0=\$40$ is deducted, leaving $A_0-C_0=\$960$ invested. The fund earns a deterministic annual market return $r$, so its year-end value is $F_1=960(1+r)$. At year-end, the policy lapses with probability $q$; otherwise it reaches maturity. A lapse pays $98\%$ of $F_1$, so the insurer receives a surrender charge of $2\%F_1$. A policy reaching maturity receives the fund value plus any guarantee top-up $G=\max(1000-F_1,0)$. There is a fixed issue expense $e_0$, a maintenance expense $e_m$ for each policy that persists, and a lapse expense $e_l$ for each policy that lapses. Lapse is assumed independent of the deterministic market return. All profit and reserve calculations use a valuation interest rate of $0\%$, so no discounting is needed. The reserve is valued after the initial charge and issue expense have occurred; it excludes the fund-value benefit and its matching fund assets, and covers only the future net guarantee and expense cashflows. If lapse behavior depends on market performance, a single independent lapse probability $q$ is not adequate.

Projection Measures and Their Relationships

For this model, the expected future net outgo is the expected guarantee top-up plus expected future expenses, less expected surrender-charge income. Thus the prospective supplemental reserve is

$$ R=(1-q)G+(1-q)e_m+qe_l-q(0.02F_1). $$

This is a net reserve for the specified cashflows and assumptions. It is not the gross maturity benefit: the fund value is paid from the linked assets, and the reserve above is for the guarantee and other specified net outgo. The issue expense is excluded because it has already occurred at the reserve valuation time.

On the same assumptions, one-year expected profit is the initial charge less issue expense, less the reserve for future net outgo:

$$ P=C_0-e_0-R. $$

This identity follows by grouping the profit cashflows by timing. The charge and issue expense have already occurred, giving $C_0-e_0$. The remaining expected cashflows are future surrender-charge income minus guarantee and operating outgo, which equal $-R$ by the reserve definition. Since the valuation interest rate is $0\%$, there is no further accumulation or discounting between the dates.

The linked fund value $F_1$ is the asset balance allocated to this policy in this simplified projection. It is useful to track it alongside the guarantee reserve and profit, but it is not itself the shareholder profit or the supplemental reserve. In other product designs, the term asset share may refer to an accumulated allocation of assets under a specified convention; the convention must be stated before comparing that measure with a reserve.

Worked Example: Establishing the Baseline and Isolated Results

Method: Calculate the fund value first, then apply the same cashflow definitions to each scenario. This avoids changing the guarantee or surrender-charge rule when a stress is introduced. Assume the baseline return is $r=5\%$, lapse probability $q=10\%$, and expenses $e_0=\$30$, $e_m=\$10$, $e_l=\$5$.

The baseline fund is $F_1=960(1.05)=\$1{,}008$, so $G=\max(1000-1008,0)=\$0$. The surrender charge on a lapse is $0.02(1008)=\$20.16$. Therefore

R=0.90(0)+0.90(10)+0.10(5)-0.10(20.16) =9+0.50-2.016=\$7.484.

The initial margin is $40-30=\$10$, so baseline profit is $P=10-7.484=\$2.516$.

Now calculate each stress in isolation, holding the other assumptions at baseline.

Market stress only: Set $r=-10\%$. Then $F_1=960(0.90)=\$864$, $G=\$136$, and the surrender charge is $0.02(864)=\$17.28$. With $q=0.10$, $R=0.90(136)+0.90(10)+0.10(5)-0.10(17.28)=122.40+9+0.50-1.728=\$130.172$. Profit is $10-130.172=-\$120.172$.
Lapse stress only: Keep $r=5\%$ and set $q=30\%$. The fund remains $\$1{,}008$, with no top-up. Then $R=0.70(10)+0.30(5)-0.30(20.16)=7+1.50-6.048=\$2.452$. Profit is $10-2.452=\$7.548$.
Expense stress only: Keep $r=5\%$, $q=10\%$, and increase each expense by $20\%$: $e_0=\$36$, $e_m=\$12$, $e_l=\$6$. Then $R=0.90(12)+0.10(6)-0.10(20.16)=10.80+0.60-2.016=\$9.384$. Profit is $40-36-9.384=-\$5.384$.

Check and common mistake: In every case, $P=C_0-e_0-R$. In particular, the lapse-only result is more profitable than baseline in this specific model because fewer policies incur maintenance expense and more surrender charges are collected. It would be a mistake to assume that a higher lapse rate must always reduce profit; the sign depends on the modeled benefits, expenses, and charges.

Combining the Stresses

Use one joint scenario: $r=-10\%$, $q=30\%$, and expenses increased by $20\%$, so $e_0=\$36$, $e_m=\$12$, and $e_l=\$6$. The initial charge remains $\$40$, and the surrender charge remains $2\%$ of the fund. Recalculate each projected cashflow under these assumptions rather than carrying forward a result from an isolated stress.

Worked Example: Calculating the Jointly Stressed Result

Method: Project the stressed fund, calculate the guarantee and surrender charge from that fund, then weight future cashflows by the stressed lapse probability. This keeps the guarantee exposure and lapse exposure on the same scenario basis.

The stressed fund is $F_1=960(0.90)=\$864$. Thus the maturity top-up is $G=1000-864=\$136$, and the surrender charge on a lapse is $0.02(864)=\$17.28$. The expected number-weighted guarantee outgo per original policy is $0.70(136)=\$95.20$, because only policies that persist reach maturity under this model. Expected maintenance expense is $0.70(12)=\$8.40$, and expected lapse expense is $0.30(6)=\$1.80$. Expected surrender-charge income is $0.30(17.28)=\$5.184$.

R=95.20+8.40+1.80-5.184=\$100.216.

The initial margin is now $40-36=\$4$, giving

P=4-100.216=-\$96.216.

The linked asset balance is $\$864$, down from the baseline $\$1{,}008$. The guarantee top-up is $\$136$ for each policy that reaches maturity, or $\$95.20$ in expectation per policy at the start of the year. These are different quantities: the first is conditional on persistence, while the second is weighted by the probability of persistence. The net reserve is $\$100.216$, because it also includes expenses and offsets surrender-charge income. Profit additionally reflects the initial charge and issue expense.

Check and common mistake: The identity $P=C_0-e_0-R$ gives $40-36-100.216=-\$96.216$, as required. A common error is to use the stressed guarantee top-up for every policy and omit the persistence weight. That would use $\$136$ instead of the expected guarantee outgo of $\$95.20$ per original policy, overstating expected guarantee outgo by $\$40.80$.

Why the Joint Effect Is Not Additive

The isolated results are useful for diagnosis, but they do not automatically add up to the joint result. In the baseline, profit is $\$2.516$. The market-only change in profit is $-120.172-2.516=-\$122.688$; the lapse-only change is $7.548-2.516=\$5.032$; and the expense-only change is $-5.384-2.516=-\$7.900$. Adding those isolated changes to baseline predicts

2.516-122.688+5.032-7.900=-\$123.040.

The actual joint profit is $-\$96.216$, which is $\$26.824$ higher than that additive prediction. The difference arises because lapse stress changes the share of policies exposed to the market-stressed guarantee; it is not just an independent profit adjustment. It also changes expected maintenance costs and surrender-charge income. The stressed expenses then apply to the stressed mix of lapses and persisting policies.

The reserve comparison makes the interaction visible from another angle. Under the market stress alone, the reserve is $\$130.172$. Under the joint stress it is $\$100.216$, even though expenses are higher. The increased lapse probability reduces the expected guarantee outgo because fewer policies persist to maturity. It also increases surrender-charge income. Those offsets more than outweigh the higher future expenses in this example. This is a feature of the stated assumptions, not a general conclusion that lapse stress is beneficial.

Defined capital comparison: For this illustration, define the incremental stress capital measure as the increase in supplemental reserve relative to baseline, with a floor of zero: $K=\max(0,R_{\text{scenario}}-R_{\text{baseline}})$. Baseline reserve is $\$7.484$, so joint-stress $K=100.216-7.484=\$92.732$. Market-only $K=130.172-7.484=\$122.688$. This comparison is specific to the defined one-year model; it is not a general solvency standard or a substitute for a product’s prescribed capital calculation.

A Practical Order for Integrated Calculations

A reliable calculation keeps the drivers and measures separate until the cashflows have been projected. In particular, do not treat the linked fund, guarantee top-up, net reserve, and profit as interchangeable amounts.

Set the scenario.
Record the market return, lapse assumption, expense assumptions, charge rules, and valuation interest rate. State whether lapses depend on market performance.

Project the linked account.
Apply the charge and market return using the product’s timing convention. Calculate the fund value before applying any guarantee.

Identify conditional cashflows.
Calculate maturity top-ups for persisting policies and surrender-charge income for lapsing policies. Weight each by the probability of its relevant outcome.

Calculate net reserve and profit.
Combine expected future outgo and income for the reserve, then include initial-period cashflows when calculating profit. Check the profit-reserve identity if the timing assumptions permit it.

Compare measures and scenarios.
Report the allocated fund balance, conditional and expected guarantee costs, reserve, profit, and any explicitly defined capital comparison separately. Explain interactions rather than assuming isolated impacts add.

The same discipline applies when an asset-share projection is available: state its allocation and accumulation convention, then project it under the combined market and expense assumptions. A fund balance can fall with the market even when a net reserve falls because lapses reduce guarantee exposure. Conversely, a reserve can rise while profit falls. The measures answer different questions, so a single headline number does not describe the whole stress.

Worked Example: Separating Lapse Relief from Expense Deterioration

Method: Hold the market return at the stressed $-10\%$, change lapse assumptions first, and then apply expense inflation. This identifies which part of the joint reserve movement is associated with each change. Use ordinary expenses for the intermediate lapse-only calculation.

With $F_1=\$864$, $G=\$136$, and surrender charge $\$17.28$, set $q=30\%$ but retain $e_m=\$10$ and $e_l=\$5$. The reserve is

R=0.70(136)+0.70(10)+0.30(5)-0.30(17.28) =95.20+7+1.50-5.184=\$98.516.

Relative to the market-only reserve of $\$130.172$, the higher lapse probability reduces the reserve by $\$31.656$. Now inflate expenses to $e_m=\$12$ and $e_l=\$6$, keeping the same market and lapse assumptions. Expected future expenses rise from $7+1.50=\$8.50$ to $8.40+1.80=\$10.20$, an increase of $\$1.70$. Therefore the reserve rises from $\$98.516$ to $\$100.216$, matching the joint result.

Check and common mistake: The difference between the joint reserve and the market-only reserve is $\$100.216-\$130.172=-\$29.956$. It equals the $\$31.656$ lapse-related reduction plus the $\$1.70$ increase in future expenses. The issue expense does not enter this reserve comparison because it has already occurred at the reserve valuation time; it does affect profit, reducing the initial margin from $\$10$ to $\$4$. Including the issue expense in the future reserve as well would count it twice.

Check Your Understanding

Use the model and timing assumptions stated above. Report dollar answers to the nearest cent.

Suppose $r=-5\%$, $q=20\%$, and expenses are at baseline levels. Find the fund value, guarantee top-up per persisting policy, and supplemental reserve. (Answer: fund value $\$912.00$, top-up $\$88.00$, reserve $\$75.75$.)
For the assumptions in question 1, find expected profit using the baseline initial charge and issue expense. (Answer: $-\$65.75$.)
Suppose $r=0\%$, $q=25\%$, $e_0=\$36$, $e_m=\$12$, and $e_l=\$6$. Find the supplemental reserve. (Answer: $\$35.70$.)
Using the baseline and joint-stress results in the article, calculate the defined incremental stress capital measure $K$. (Answer: $\$92.73$.)

What You'll Learn