Designing Single-Risk Sensitivity Tests

Why Isolate One Assumption?

A projected solvency result depends on many assumptions and exposures. If claim severity, interest rates, lapse rates, and business volume all change together, a difference from the baseline does not reveal which input drove the result. A single-risk sensitivity test changes one specified assumption or exposure at a time, holds the other inputs constant, and measures the resulting change in projected capital adequacy.

The purpose is not to claim that the changed input will occur, or to assign a probability to it. The test answers a conditional question: if this one input were different, while the other modeled inputs stayed fixed, how would the projected solvency result change? This makes sensitivities useful for identifying model responsiveness, checking capital conclusions, and explaining which assumptions matter. They are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Definition: A single-risk sensitivity test compares a baseline projection with a projection in which one selected assumption or exposure is changed and all other assumptions, model settings, and calculation methods are held constant. The test reports the effect on one or more specified solvency measures.

A sensitivity should be precisely defined. “Test higher claims” is not precise enough: the test should state whether it changes claim frequency, severity, or another input; the amount of the change; the affected business; and the projection period. It should also identify the output being compared. A useful comparison often includes available capital $K$, required capital $C$, the capital margin $M$, and the coverage ratio $K/C$, where

K=\text{eligible assets}-\text{liabilities}, \qquad M=K-C, \qquad \text{coverage ratio}=\frac{K}{C}.

These are measures of the modeled position, not interchangeable quantities. A change can reduce available capital and increase required capital at the same time. Reporting only one of the two movements can therefore obscure the total effect on the margin.

A Controlled Comparison

A sound test begins with a documented baseline: the starting balance sheet, projection horizon, business plan, model version, assumptions, and capital definition. The sensitivity then replaces exactly one input with a stated alternative. The model is rerun, and the outputs are compared on the same basis and at the same projection dates.

Specify the baseline.
Record the assumptions, exposure, projection dates, and solvency measures that define the reference result.

Choose one input.
Name the assumption or exposure to be changed, its baseline value, its test value, and the portion of the projection it affects.

Hold the rest fixed.
Keep other assumptions and model methods unchanged. If the chosen input mechanically affects another model output, allow the model to recalculate that output, but do not independently alter a second assumption.

Recalculate and compare.
Report the baseline and sensitivity values, their changes, and any relevant coverage or margin measures.

Check and explain.
Verify the direction and arithmetic of the changes, then state what the comparison indicates and what it does not establish.

The central arithmetic follows directly from the margin definition. Let $\Delta K=K_{\text{test}}-K_{\text{base}}$ and $\Delta C=C_{\text{test}}-C_{\text{base}}$. Then

\Delta M =M_{\text{test}}-M_{\text{base}} =(K_{\text{test}}-C_{\text{test}})-(K_{\text{base}}-C_{\text{base}}) =\Delta K-\Delta C.

This identity explains why the two capital movements must be signed. If available capital falls by 2.4 million, then $\Delta K=-2.4$ million. If required capital rises by 0.48 million, then $\Delta C=+0.48$ million. Therefore $\Delta M=-2.4-0.48=-2.88$ million: the margin decreases by 2.88 million.

Worked Applications

Worked Example: Higher Claim Severity

A hypothetical insurer projects year-end available capital of 30.0 million and required capital of 24.0 million. A liability component for claims is 45.0 million. Test a 8% increase in that component, interpreted as a severity change that increases the projected claim liability by 8%. The model also recalculates required capital, which rises by 1.2 million. Assets and all other liabilities are unchanged. Find the new capital measures and the change in margin.

Method. Change the claim-severity input only, apply its specified effect to the liability component, and then recalculate available capital and required capital.

Step 1: Calculate the liability increase. The increase is $0.08\times45.0=3.6$ million. The new claim liability component is $45.0+3.6=48.6$ million.

Step 2: Recalculate available capital. The 3.6 million higher liability reduces available capital by 3.6 million, so $K_{\text{test}}=30.0-3.6=26.4$ million.

Step 3: Recalculate required capital and margin. Required capital rises from 24.0 million to $C_{\text{test}}=24.0+1.2=25.2$ million. The baseline margin is $30.0-24.0=6.0$ million. The test margin is $26.4-25.2=1.2$ million.

Step 4: Check the change. Available capital changes by $\Delta K=-3.6$ million, and required capital changes by $\Delta C=+1.2$ million. Thus $\Delta M=-3.6-1.2=-4.8$ million, which agrees with $1.2-6.0=-4.8$ million. The coverage ratio changes from $30.0/24.0=1.2500$ to $26.4/25.2=1.0476$.

Common mistake avoided. Applying the 8% increase to all liabilities would change more than the specified claim component. Applying it to the 45.0 million component gives the stated 3.6 million increase. It would also be incorrect to subtract the required-capital increase from the margin change as though it were a decrease; both the liability increase and the capital requirement increase reduce the margin.

Worked Example: A Lower Discount Rate

A hypothetical insurer has an asset cash flow of 105 million at the end of year 1. Its liability cash flows are 50 million at the end of year 1 and 55 million at the end of year 2. Use a single effective annual discount rate for all these cash flows. Required capital is 5 million and is held fixed for this sensitivity. Compare a 5% baseline rate with a 4% test rate. Calculate present values, available capital, and the margin. Report money in millions to four decimal places, rounding only the final results.

Method. Discount each cash flow at its stated time using $PV=CF/(1+i)^t$. The only changed input is the discount rate; required capital and the cash flows remain fixed.

At 5%: The asset present value is $105/1.05=100.0000$ million. The year-1 liability present value is $50/1.05=47.6190$ million, and the year-2 liability present value is $55/(1.05)^2=55/1.1025=49.8866$ million. Thus $PV(L)=47.6190+49.8866=97.5057$ million, using unrounded values in the sum. Available capital is $K=100.0000-97.5057=2.4943$ million, and the margin is $M=2.4943-5.0000=-2.5057$ million.

At 4%: The asset present value is $105/1.04=100.9615$ million. The year-1 liability present value is $50/1.04=48.0769$ million, and the year-2 liability present value is $55/(1.04)^2=55/1.0816=50.8506$ million. Thus $PV(L)=48.0769+50.8506=98.9275$ million, using unrounded values in the sum. Available capital is $K=100.9615-98.9275=2.0340$ million, and the margin is $M=2.0340-5.0000=-2.9660$ million.

Check and interpretation. The asset value rises by approximately $100.9615-100.0000=0.9615$ million, but the liability value rises by approximately $98.9275-97.5057=1.4218$ million. The liability increase is larger, so available capital falls by about $1.4218-0.9615=0.4603$ million. With required capital fixed, the margin also falls by about 0.4603 million, consistent with $-2.9660-(-2.5057)=-0.4603$ million using rounded displayed results.

Common mistake avoided. Discounting both liability cash flows at year 1, or using the wrong exponent for the year-2 cash flow, gives an incorrect liability value. The 4% year-2 amount is $55/(1.04)^2$, not $55/1.04$. This sensitivity does not imply that lower rates always worsen solvency: the direction depends on the relative timing and amounts of assets and liabilities.

Worked Example: A Higher Lapse Rate

Consider a hypothetical two-year block with 20,000 policies at the start of year 1. Each active policy produces a net surplus of 100 dollars in each year it remains active. Lapses occur at each year-end, and the annual lapse rate is the same in both years. For this simplified projection, year-1 policies all earn the year-1 surplus, while the expected year-2 active policy count is the opening count multiplied by one minus the lapse rate. Assume zero interest, so no discounting is needed. Projected available capital equals opening capital of 10.0 million plus these expected surpluses. Required capital is 5.0 million plus 10% of expected year-2 claims, with claims of 900 dollars per active policy. Compare lapse rates of 5% and 10%; hold all other inputs fixed.

Method. Recalculate expected year-2 policies and the specified capital outputs at each rate. The year-1 exposure is unchanged; only the lapse-rate assumption changes.

At a 5% lapse rate: Expected year-2 active policies are $20{,}000(1-0.05)=19{,}000$. Expected total policy-years are $20{,}000+19{,}000=39{,}000$. Projected surplus is $39{,}000\times100=\$3{,}900{,}000$, so available capital is $10.0+3.9=13.9$ million. Expected year-2 claims are $19{,}000\times900=\$17{,}100{,}000$. Required capital is $5.0+0.10(17.1)=6.71$ million. The margin is $13.9-6.71=7.19$ million, and the coverage ratio is $13.9/6.71=2.0715$.

At a 10% lapse rate: Expected year-2 active policies are $20{,}000(1-0.10)=18{,}000$. Expected total policy-years are $20{,}000+18{,}000=38{,}000$. Projected surplus is $38{,}000\times100=\$3{,}800{,}000$, so available capital is $10.0+3.8=13.8$ million. Expected year-2 claims are $18{,}000\times900=\$16{,}200{,}000$. Required capital is $5.0+0.10(16.2)=6.62$ million. The margin is $13.8-6.62=7.18$ million, and the coverage ratio is $13.8/6.62=2.0846$.

Check and interpretation. The higher lapse rate reduces available capital by 0.1 million and required capital by 0.09 million. Therefore $\Delta M=-0.1-(-0.09)=-0.01$ million, matching the change from 7.19 to 7.18 million. In this simplified model the margin falls slightly, while the coverage ratio increases because both the numerator and denominator change and the denominator falls proportionally more. The result depends on the stipulated surplus and capital formulas; it is not a general conclusion that higher lapses improve coverage.

Common mistake avoided. Applying the lapse rate to year-1 policies would incorrectly reduce first-year exposure. Under the stated timing, lapse occurs at year-end, so the full opening count earns the first-year surplus and only the year-2 count changes.

Interpreting and Documenting Results

A sensitivity result is meaningful only in relation to its definition and baseline. A 5% lapse-rate test, for example, is ambiguous unless the baseline rate, direction of change, affected policies, and timing are specified. Similarly, a severity test should distinguish a percentage change in severity from the same percentage change in total claims, which could also reflect frequency.

For each test, report the baseline input and test input, the outputs under both, and the absolute changes. Percentage changes can help compare measures with different scales, but they can be misleading when the baseline is close to zero or negative. In those cases, show the absolute dollar movement and explain the direction. State whether required capital was recalculated or intentionally held fixed. If it is fixed, make that modeling choice explicit.

The principal exam trap is to treat a single-risk sensitivity as if it were a combined stress. Changing two assumptions at once prevents a clean attribution to either one. Another trap is to report only the change in available capital when the test also changes required capital. Always check the signed relationship $\Delta M=\Delta K-\Delta C$, and recalculate the coverage ratio from its numerator and denominator rather than inferring its direction from the margin alone.

A sensitivity is conditional, not a forecast or a probability statement. It does not by itself establish how plausible the changed input is, nor does it describe the effect of several risks interacting. Its value is narrower and practical: it shows how the specified projection responds to one controlled change, using the stated model and assumptions.

Key takeaway: Define one input change, keep the rest of the comparison controlled, rerun the projection, and trace the result through both available and required capital. Confirm the arithmetic and explain the sensitivity as a conditional model result.

Check Your Understanding

For each problem, state the key capital effect and verify the arithmetic before checking the answer.

A hypothetical insurer has available capital of 18.0 million and required capital of 14.0 million. A single severity sensitivity increases a 25.0 million claim-liability component by 6%; assets are unchanged, and required capital rises by 0.5 million. Find test available capital, test required capital, the baseline and test margins, and the margin change. (Answer: The liability rises by $0.06\times25.0=1.5$ million; test $K=16.5$ million and $C=14.5$ million; margins are 4.0 million and 2.0 million, so $\Delta M=-2.0$ million.)
A hypothetical insurer has an asset cash flow of 84 million at year 1, liability cash flows of 40 million at year 1 and 42 million at year 2, and fixed required capital of 3 million. Using effective annual discount rates, compare 5% with 4%. Find $PV(A)$, $PV(L)$, available capital, and margin at each rate. Report millions to four decimals. (Answer: At 5%, $PV(A)=80.0000$, $PV(L)=40/1.05+42/(1.05)^2=76.1905$, $K=3.8095$, and $M=0.8095$. At 4%, $PV(A)=80.7692$, $PV(L)=40/1.04+42/(1.04)^2=77.2929$, $K=3.4763$, and $M=0.4763$.)
A model’s available capital is 22.0 million and required capital is 18.0 million at baseline. A one-factor test decreases available capital by 1.1 million and increases required capital by 0.4 million. Calculate the test margin and the change from baseline. (Answer: Baseline margin $=4.0$ million; test $K=20.9$ million, $C=18.4$ million, and $M=2.5$ million; $\Delta M=-1.5$ million, equal to $-1.1-0.4$.)
A hypothetical two-year block has 10,000 policies, a 4% annual lapse rate, and a test lapse rate of 9%. Lapses occur at year-end. Each active policy produces 80 dollars of surplus per policy-year, and opening available capital is 4.0 million. Required capital is fixed at 2.0 million. Under the simplified model in which all opening policies earn year-1 surplus and only year-2 exposure depends on lapse, calculate available capital and margin at both lapse rates. (Answer: At 4%, year-2 policies $=9{,}600$, total policy-years $=19{,}600$, surplus $=\$1{,}568{,}000$, $K=5.568$ million, and margin $=3.568$ million. At 9%, year-2 policies $=9{,}100$, total policy-years $=19{,}100$, surplus $=\$1{,}528{,}000$, $K=5.528$ million, and margin $=3.528$ million.)
A sensitivity changes claim severity and interest rates together and reports only the resulting margin. Why is this not a single-risk sensitivity, and what is the minimum redesign needed to isolate each input’s effect? (Answer: Two inputs changed, so their individual effects cannot be attributed from that comparison. Run two separate tests against the same baseline: change severity alone while holding the rate fixed, then change the rate alone while holding severity fixed; report the same capital measures for each.)

What You'll Learn