Sensitivity Analysis For Rating Capital Models

Why Sensitivity Analysis Matters

A capital model produces a result from a set of assumptions and exposures. That result is useful, but a single baseline calculation does not show how dependent the conclusion is on those inputs. Sensitivity analysis changes selected assumptions, reruns the calculation, and measures the effect on capital adequacy. It can reveal whether a conclusion is robust or whether modest changes in growth, reserves, asset values, or catastrophe losses could alter it.

The objective is not to predict which stress will occur. It is to understand the consequences of specified changes and identify the assumptions that drive the model result. A useful analysis states what is changed, what is held fixed, which model outputs are recalculated, and how the result is compared with a threshold. All figures below use a hypothetical insurer and a deliberately simplified rating-capital model; they are illustrations, not descriptions of an actual rating agency methodology.

Definition: A one-factor sensitivity changes one input or assumption while holding the other specified inputs fixed. A combined sensitivity changes two or more inputs in one scenario. In either case, the model should be rerun for both eligible capital and required capital if the changed assumption affects both.

Measure the Change in Adequacy

Let $C$ be eligible capital and $R$ be required capital. The capital adequacy ratio is $A=C/R$, provided $R>0$. If the relevant threshold ratio is $\tau$, define threshold headroom as $H=C-\tau R$. Positive headroom means the modeled capital exceeds the amount implied by that threshold; negative headroom means it falls short. The ratio and the headroom are related but answer different questions: the ratio expresses capital relative to required capital, while headroom expresses the dollar surplus or shortfall at a specified threshold.

A=\frac{C}{R}, \qquad H=C-\tau R

Under a stress, let $C_1=C_0+\Delta C$ and $R_1=R_0+\Delta R$, where subscript 0 denotes baseline and subscript 1 denotes stressed values. A change in the ratio cannot generally be found by looking at only one of $\Delta C$ and $\Delta R$: both affect the numerator-to-denominator relationship.

Result: The exact change in the ratio is

A_1-A_0 =\frac{C_0+\Delta C}{R_0+\Delta R}-\frac{C_0}{R_0} =\frac{R_0\Delta C-C_0\Delta R}{R_0(R_0+\Delta R)}

This identity follows by putting the two ratios over a common denominator. The numerator is $(C_0+\Delta C)R_0-C_0(R_0+\Delta R)$. Expanding gives $C_0R_0+R_0\Delta C-C_0R_0-C_0\Delta R=R_0\Delta C-C_0\Delta R$, which proves the result. For small changes, the first-order approximation is $\Delta A\approx \Delta C/R_0-(C_0/R_0^2)\Delta R$. This approximation is useful for interpreting direction and relative influence, but the exact stressed ratio should be used for a material scenario. It does not apply if the stressed required capital is zero or negative, and ordinary ratio interpretation is not meaningful when required capital is nonpositive.

One-Factor Sensitivities

A one-factor test is easiest to interpret when it isolates a defined shock and records its effects on both $C$ and $R$. In practice, some assumptions affect only one side of the ratio; others affect both. For example, adverse reserve development can reduce eligible capital through a higher liability and increase reserve-related required capital through a larger exposure base. A growth assumption may increase premium-related required capital without an immediate reduction in eligible capital in a simplified static model.

Worked Example: Premium Growth Sensitivity

A hypothetical insurer has baseline eligible capital of $180 million and required capital of $150 million. Its simplified model calculates a premium risk charge as 20% of net written premium. Baseline net written premium is $200 million, so the premium charge is $40 million; all other charges total $110 million. Test a 10% increase in premium volume. Assume the increase does not change eligible capital or any other charge in this static sensitivity. The relevant threshold ratio is 1.15.

Method. Change the premium exposure, recalculate its charge, replace the baseline charge in required capital, and then recalculate the ratio and headroom.

Stressed premium is $200(1.10)=\$220$ million. Its charge is $0.20(220)=\$44$ million, an increase of $44-40=\$4$ million. Therefore stressed required capital is $150+4=\$154$ million, while eligible capital remains $180 million.

A_0=\frac{180}{150}=1.2000,\qquad A_1=\frac{180}{154}=1.1688

Baseline headroom is $180-1.15(150)=180-172.5=\$7.5$ million. Stressed headroom is $180-1.15(154)=180-177.1=\$2.9$ million. The stress reduces the ratio by $1.2000-1.1688=0.0312$ and reduces headroom by $7.5-2.9=\$4.6$ million. The check is that the $4 million increase in required capital reduces threshold headroom by $1.15(4)=\$4.6$ million, as it should.

Common mistake avoided: Adding $4 million to required capital but leaving the ratio at 1.20 confuses the baseline result with the stressed result. The denominator must be updated before the ratio is recalculated.

The size of a ratio change depends on the starting point as well as on the model response. When comparing sensitivities, report the input shock alongside the output change. A 10% premium increase and a severe catastrophe scenario are not automatically comparable tests simply because each is labeled a “stress.”

Reserve Development and Combined Scenarios

Reserve development can affect the balance sheet and the required-capital calculation through distinct channels. To avoid counting an effect twice, identify whether the modeled change represents a claim liability increase, a change in the risk exposure used for a capital charge, or both. The assumptions should also specify whether the development is already reflected in eligible capital before applying the stress.

Worked Example: Adverse Development With Growth

Use the same baseline insurer, with $C_0=\$180$ million and $R_0=\$150$ million. Net reserves are $250 million and the reserve risk charge is 12% of that amount, or $30 million. Net written premium is $200 million and the premium charge is 20%, or $40 million. Other charges total $80 million, so the three components sum to the stated $150 million. Consider a combined scenario: net premium grows 10%, and reserves develop adversely by 8% of the original reserve amount. Assume the reserve development increases liabilities dollar for dollar, reducing eligible capital, and that the reserve charge is recalculated on the increased reserve base. No other model inputs change. The threshold is 1.15.

Method. Calculate each exposure change from its stated base, then apply the balance-sheet effect and the two charge changes separately.

The reserve increase is $0.08(250)=\$20$ million. It reduces eligible capital to $180-20=\$160$ million. The new reserve base is $250+20=\$270$ million, so the reserve charge becomes $0.12(270)=\$32.4$ million, an increase of $2.4 million.

Premium after growth is $200(1.10)=\$220$ million. Its charge becomes $0.20(220)=\$44$ million, an increase of $4 million. Required capital is therefore $150+2.4+4=\$156.4$ million. The stressed ratio is $160/156.4=1.0230$, rounded to four decimal places. Stressed threshold headroom is $160-1.15(156.4)=160-179.86=-\$19.86$ million.

The baseline ratio was $180/150=1.2000$, so the combined scenario reduces it by approximately 0.1770. The calculation checks against the component totals: the original $30 million reserve charge becomes $32.4 million, the original $40 million premium charge becomes $44 million, and the unchanged $80 million of other charges gives $32.4+44+80=156.4$. The reserve loss is reflected once in eligible capital; it is not added a second time as a separate charge in this specified model.

Common mistake avoided: Applying the 8% development to the new reserve base after premium growth, or applying the increase to the already-stressed reserve amount, would use a different assumption from the one stated. Here the development is explicitly 8% of the original $250 million.

A combined result need not equal the sum of the separate ratio changes. Even if model charges add linearly, the adequacy ratio is a quotient, and a combined stress may change both its numerator and denominator. If the model itself has nonlinear rules, caps, floors, or interaction terms, the combined scenario must be run through those rules rather than assembled by adding isolated results.

Asset Values and Catastrophe Losses

Asset and catastrophe stresses illustrate why the direction of a required-capital change should not be assumed. A fall in asset values can reduce eligible capital. If the same decline also reduces the exposure base used for an asset charge, the required charge may fall even as the ratio worsens. A catastrophe event can reduce capital through an incurred loss and may also lead the model to assign a higher prospective catastrophe charge. Those are different effects: the first is a loss already borne; the second represents modeled future exposure after the event.

Worked Example: Joint Asset and Catastrophe Stress

A hypothetical insurer starts with eligible capital of $180 million and required capital of $150 million, including a $35 million asset charge and a $25 million catastrophe charge. In a simplified model, the asset charge is 10% of a $350 million asset exposure. Apply an asset-value decline of $12 million, so the exposure becomes $338 million and the loss reduces eligible capital dollar for dollar. Also apply a hypothetical catastrophe event that reduces eligible capital by $15 million. After that event, assume the prospective catastrophe charge increases from $25 million to $30 million because the modeled remaining exposure is more severe. The asset charge is recalculated on the reduced asset base. Treat the event loss and prospective charge as separate effects, with no other changes.

Method. Recalculate each affected charge, apply each realized balance-sheet loss once, and combine the resulting stressed capital and required capital.

The asset charge changes from $0.10(350)=\$35$ million to $0.10(338)=\$33.8$ million, a decrease of $1.2 million. The two losses reduce eligible capital to $180-12-15=\$153$ million. Required capital changes by $-1.2$ million for assets and $+5$ million for catastrophe, so it is $150-1.2+5=\$153.8$ million.

A_1=\frac{153}{153.8}=0.9948,\qquad H_1=153-1.15(153.8)=-23.87\text{ million}

The ratio is below 1.00, and threshold headroom is negative $23.87 million. As a check, the asset charge is $33.8$, the catastrophe charge is $30$, and the unchanged charges total $150-35-25=90$; thus stressed required capital is $33.8+30+90=153.8$ million, consistent with the direct adjustment.

Common mistake avoided: It would be incorrect to subtract the catastrophe loss from eligible capital and also add that same $15 million as a catastrophe charge unless the model explicitly defines both as separate effects. In this example the additional charge is $5 million, not $15 million, and represents a changed prospective requirement.

Interpret the Driver, Not Just the Number

A useful sensitivity summary reports at least the stress definition, stressed eligible capital, stressed required capital, the resulting ratio, and the change in threshold headroom. A ranked list of ratio changes can help direct review, but it does not establish which assumption is intrinsically most important unless stress sizes are made comparable. Consider both the magnitude of the input perturbation and the model’s response per unit of that perturbation.

For example, a reserve scenario may produce a larger ratio decline than a growth scenario because it reduces eligible capital as well as increasing required capital. That does not prove reserves are always the dominant driver: the comparison depends on the selected development and growth levels, the company’s starting balance sheet, and the model’s charge rules. Test alternative stress sizes when necessary, and distinguish an input’s direct effect from interactions with other assumptions.

State the base and the shock. Specify whether a percentage is applied to the original or already-stressed amount.
Recalculate both sides. Determine whether the assumption changes eligible capital, required capital, or both.
Show absolute and relative measures. The dollar headroom and the adequacy ratio provide complementary information.
Separate event loss from prospective risk. Do not count the same loss twice under different labels.
Use comparable stresses for ranking. A ranking based on arbitrary scenario sizes can be misleading.

Key takeaway: A rating-capital sensitivity is a controlled rerun of the model, not merely a percentage adjustment to the final ratio. Trace each changed assumption through eligible capital and required capital, then interpret the stressed ratio and threshold headroom together.

Check Your Understanding

For each problem, state the stressed adequacy ratio and, where a threshold is supplied, the stressed headroom. Use the same definitions of $A$ and $H$ given above.

A hypothetical insurer has $C=\$240$ million and $R=\$200$ million. Premium growth increases required capital by 5%, with no change in eligible capital. Find the stressed ratio and headroom at threshold 1.10. (Answer: $R_1=\$210$ million, $A_1=1.1429$, $H_1=\$9$ million.)
A reserve stress reduces eligible capital by $8 million and increases required capital by $4 million from a baseline $C=\$120$ million and $R=\$100$ million. Find the stressed ratio. (Answer: $112/104=1.0769$.)
An asset shock reduces eligible capital by $18 million and reduces required capital by $3 million from $C=\$300$ million and $R=\$250$ million. Find the stressed ratio and headroom at threshold 1.10. (Answer: $C_1=\$282$ million, $R_1=\$247$ million, $A_1=1.1417$, $H_1=\$10.3$ million.)
A catastrophe scenario reduces eligible capital by $12 million and increases required capital by $8 million from $C=\$200$ million and $R=\$160$ million. Find the stressed ratio and headroom at threshold 1.10. (Answer: $C_1=\$188$ million, $R_1=\$168$ million, $A_1=1.1190$, $H_1=\$3.2$ million.)
A combined scenario reduces eligible capital by $25 million. A growth stress increases required capital by $20 million, while an asset exposure change reduces it by $5 million. From baseline $C=\$500$ million and $R=\$400$ million, find the stressed ratio and headroom at threshold 1.15. (Answer: $C_1=\$475$ million, $R_1=\$415$ million, $A_1=1.1446$, $H_1=-\$2.25$ million.)

What You'll Learn