Sensitivity Analysis For Formula Capital Drivers

Why Sensitivity Analysis Matters

A formula-based capital result is an output of many inputs: exposure amounts, prescribed factors, and covariance assumptions, among others. Knowing the total capital figure is not enough to explain what drives it. Sensitivity analysis measures how that figure responds when an input changes, helping an actuary identify influential assumptions, test proposed scenarios, and explain the result to decision-makers.

The central discipline is to distinguish a one-factor sensitivity, which changes one input while holding the others fixed, from a multi-factor scenario, which changes several inputs together. One-factor results help isolate drivers; a multi-factor result evaluates the combined effect. They answer different questions, and a collection of isolated sensitivities should not be mistaken for a jointly calculated scenario.

Definition: A sensitivity is the change in a formula output associated with a specified change in one or more inputs. A local sensitivity is a derivative evaluated at a particular set of inputs. A finite-change sensitivity is the difference between the formula output at a stated scenario and the output at the original inputs.

This tutorial uses a covariance aggregation formula to illustrate the method. It is a simplified model for analysis, not a substitute for the applicable regulatory instructions. The approach is aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Set Up the Formula Before Measuring Sensitivity

Suppose a formula aggregates \(n\) nonnegative risk charges \(x_1,\ldots,x_n\) using a fixed correlation matrix \(R\). The matrix is symmetric, has diagonal entries of 1, and its off-diagonal entries \(\rho_{ij}\) represent the prescribed pairwise correlations. Under this model, aggregate capital is

K=\sqrt{\mathbf{x}^{\mathsf T}R\mathbf{x}} =\sqrt{\sum_{i=1}^n x_i^2+2\sum_{i<j}\rho_{ij}x_ix_j}.

This formula assumes that the aggregation rule is exactly the stated quadratic form and that the inputs are within its applicable range. In particular, \(R\) must be a valid correlation matrix, and the quantity under the square root must be nonnegative. If the prescribed formula includes a floor, cap, tax adjustment, or other piecewise rule, calculate the output using that rule; the smooth sensitivities below apply only within a region where the same formula branch is active.

A risk charge is often itself calculated from an exposure base. For a simple linear illustration, let \(x_i=c_iE_i\), where \(E_i\) is an exposure amount and \(c_i\) is its charge factor. A change in \(E_i\) changes \(x_i\), which then affects aggregate capital through the aggregation formula. Keeping these two stages separate helps identify whether a result is driven by the size of the exposure, the charge factor, or the aggregation.

Deriving Local Sensitivities

For \(K>0\), differentiate the quadratic-form expression with respect to one charge \(x_i\), while holding the other charges and the correlation matrix fixed. Because each cross-product involving \(x_i\) appears in the symmetric quadratic form, its derivative is represented by the \(i\)-th component of \(2R\mathbf{x}\). Differentiating the square root then gives

\frac{\partial K}{\partial x_i} =\frac{(R\mathbf{x})_i}{K} =\frac{x_i+\sum_{j\ne i}\rho_{ij}x_j}{K}.

The numerator includes the charge’s own contribution and its covariance-weighted contributions with the other charges. Thus the effect of increasing one charge depends not only on its size but also on its relationships with the other charges. If \(x_i=c_iE_i\), with \(c_i\) fixed, the chain rule gives

\frac{\partial K}{\partial E_i} =c_i\frac{\partial K}{\partial x_i} =c_i\frac{x_i+\sum_{j\ne i}\rho_{ij}x_j}{K}.

For one pairwise correlation \(\rho_{ij}\), holding all charges and other correlations fixed, the corresponding derivative is

\frac{\partial K}{\partial \rho_{ij}}=\frac{x_ix_j}{K}, \qquad i<j.

These derivatives describe local change per unit of input, not the exact effect of a large change. For a finite change, recompute the formula at the new inputs. A derivative-based estimate is useful as a diagnostic, but it does not replace an exact scenario calculation when the formula is nonlinear.

Worked Example: Sensitivity to a Premium Exposure Base

A hypothetical insurer has a premium risk charge of 10 million dollars, calculated as 12.5% of an 80 million dollar premium base. Its reserve risk charge is 20 million dollars, and their prescribed correlation is 0.25. Calculate aggregate capital, then determine the exact effect of a 10% increase in the premium base, with the factor, reserve charge, and correlation held fixed. Use millions of dollars throughout.

Method. First calculate the aggregate charge from the covariance formula. Then change only the premium base and recompute. This avoids treating a percentage change in the exposure as the same percentage change in aggregate capital.

At the original inputs, \(x_1=0.125(80)=10\) and \(x_2=20\). Therefore,

$$ K^2=10^2+20^2+2(0.25)(10)(20)=100+400+100=600, $$

so \(K=\sqrt{600}=24.4949\) million dollars before final rounding.

A 10% increase in the premium base gives \(80(1.10)=88\) million dollars. The new premium charge is \(0.125(88)=11\) million dollars. Holding the other inputs fixed,

K_{\mathrm{new}}^2=11^2+20^2+2(0.25)(11)(20) =121+400+110=631,

so \(K_{\mathrm{new}}=\sqrt{631}=25.1197\) million dollars. The exact increase is \(25.1197-24.4949=0.6248\) million dollars. Rounded to four decimals in millions, the increase is 0.6248 million dollars.

As a local check, the derivative with respect to the premium charge at the original point is \((10+0.25(20))/24.4949=15/24.4949=0.6124\). The charge rises by 1 million dollars, so the derivative predicts an increase of about 0.6124 million dollars. The exact increase is somewhat larger because the derivative is local and the formula is curved. The common mistake is to report a 10% increase in aggregate capital just because the exposure base rose by 10%; the calculation shows that the aggregate increase is much smaller.

One-Factor and Multi-Factor Scenario Analysis

A useful one-factor analysis records the baseline, changes exactly one input, and recomputes the output. For a multi-factor scenario, change all specified inputs in one formula calculation. The following sequence makes the calculation auditable:

Specify the output and model.
Write the capital formula, state its active rules, and identify the units used.

Record the baseline inputs.
List each exposure, charge, factor, and covariance input needed for the calculation.

Define the change precisely.
State whether a factor or correlation changes by a percentage or by a specified number of percentage points.

Recalculate and compare.
Apply the formula to the changed inputs, then report the dollar change and, where useful, the percentage change from baseline.

For a multi-factor scenario, the total change generally cannot be inferred by adding isolated one-factor changes. Changes can interact through products such as \(\rho_{ij}x_ix_j\), and the square root makes the output nonlinear. If a driver ranking is required, state whether it is based on local derivatives, exact one-factor scenarios of a specified size, or another defined comparison. Otherwise, the ranking may depend on the size and direction of the changes chosen.

Worked Example: Combining Exposure and Correlation Changes

A hypothetical capital model has three charges of 6, 8, and 5 million dollars. The correlations are \(\rho_{12}=0.20\), \(\rho_{13}=0.10\), and \(\rho_{23}=0.30\). In a scenario, the first charge increases by 10%, the second decreases by 5%, the third is unchanged, and \(\rho_{23}\) increases to 0.40. The other correlations do not change. Calculate the baseline and scenario capital and their difference.

Method. Apply the full quadratic formula to the baseline, then apply it again with all scenario changes made together. This avoids adding separate sensitivities as if the inputs were independent.

The baseline squared capital is

K_0^2=6^2+8^2+5^2 +2\bigl(0.20(6)(8)+0.10(6)(5)+0.30(8)(5)\bigr) =125+2(9.6+3+12)=174.2.

Thus \(K_0=\sqrt{174.2}=13.1985\) million dollars. The scenario charges are \(6(1.10)=6.6\), \(8(0.95)=7.6\), and 5 million dollars. The scenario squared capital is

K_1^2=6.6^2+7.6^2+5^2 +2\bigl(0.20(6.6)(7.6)+0.10(6.6)(5)+0.40(7.6)(5)\bigr)

$$ =126.32+2(10.032+3.3+15.2)=126.32+57.064=183.384. $$

Therefore \(K_1=\sqrt{183.384}=13.5419\) million dollars, and the change is \(13.5419-13.1985=0.3434\) million dollars. Rounded to four decimals in millions, the scenario increases capital by 0.3434 million dollars. The second charge’s decrease does not guarantee that capital will fall: the first charge and the \(\rho_{23}\) change also affect the result. The common mistake is to evaluate just one changed input, or to combine individual percentage changes without recomputing the covariance terms.

Interpreting the Drivers

Local derivatives allow comparisons on a consistent unit basis. For example, \(\partial K/\partial E_i\) measures the approximate capital change per additional dollar of exposure when the exposure is measured in dollars. For large exposure bases, it may be clearer to report a sensitivity per million dollars. A derivative with respect to a correlation is per unit of correlation: a change from 0.25 to 0.35 is an increase of 0.10, not a 0.10% increase.

An elasticity can help compare inputs with different scales. For a positive input \(z\), define the output elasticity as \((z/K)(\partial K/\partial z)\). Locally, this is the approximate percentage change in capital for a 1% change in \(z\), with other inputs held fixed. It is not an exact finite-change result, and it is not generally the same for different starting points.

A sensitivity result is conditional on the model and assumptions used. Prescribed factors may not be freely adjustable assumptions, and an illustrative stress should not be described as a change in the regulatory requirement unless the relevant rule actually changes. Also check whether a changed input makes a floor, cap, or other piecewise provision active. At such a boundary, a single smooth derivative may not describe both directions of change; calculate the formula on each applicable side.

Worked Example: Measuring Sensitivity to a Correlation Input

A hypothetical two-risk model has fixed charges of 12 and 9 million dollars. The correlation increases from 0.25 to 0.35, with both charges unchanged. Calculate the original and new capital, the exact change, and the local derivative-based estimate at the original correlation.

Method. Recompute the covariance formula at both correlations, then compare the exact change with the local derivative estimate. This makes clear why a derivative is an approximation for a finite change.

At correlation 0.25,

$$ K_0^2=12^2+9^2+2(0.25)(12)(9)=144+81+54=279, $$

so \(K_0=\sqrt{279}=16.7033\) million dollars. The new correlation is 0.35, an absolute increase of 0.10. Thus

$$ K_1^2=144+81+2(0.35)(12)(9)=225+75.6=300.6, $$

and \(K_1=\sqrt{300.6}=17.3378\) million dollars. The exact increase is \(17.3378-16.7033=0.6345\) million dollars. The local derivative at the baseline is \((12)(9)/16.7033=6.4665\) million dollars per unit of correlation. For a 0.10 increase, it estimates \(6.4665(0.10)=0.6467\) million dollars, which is close but not exact.

The common mistake is to use the relative increase in correlation, \(0.10/0.25\), as the change in the formula’s correlation input. The formula uses the absolute change, \(0.35-0.25=0.10\). Rounded to four decimals in millions, the exact capital increase is 0.6345 million dollars.

Common Traps and a Practical Reporting Standard

Changing more than one input in a one-factor test. State what is held fixed; otherwise the reported effect cannot be attributed to a single driver.
Confusing a factor change with an exposure change. If \(x_i=c_iE_i\), calculate the new charge using the changed factor or exposure before aggregating.
Using a relative change where the formula takes an absolute input. Correlations enter as values such as 0.25; an increase from 0.25 to 0.35 is 0.10.
Adding sensitivities as if they were exact. Derivatives are local estimates. Recalculate the formula for material changes or joint scenarios.
Ignoring units and formula boundaries. Keep charges and exposures in consistent units, and check whether a cap, floor, or other rule changes the calculation.

A clear report gives the baseline output, the changed input or scenario, the recalculated output, the absolute and percentage differences when useful, and the assumptions held fixed. It should also label local derivative estimates as approximations and finite-change results as exact for the stated scenario and formula.

Check Your Understanding

For each problem, use the specified covariance formula and state monetary results in millions of dollars unless otherwise noted.

A two-charge model has charges 8 and 15 and correlation 0.20. Calculate aggregate capital. (Answer: 18.3576 million dollars)
A charge is \(0.08E\), where exposure \(E=50\) million dollars. The other charge is 9 million dollars, and their correlation is 0.40. Calculate the local derivative of aggregate capital with respect to \(E\), in millions of capital per million dollars of exposure. (Answer: 0.0542)
Two fixed charges are 10 and 10. The correlation changes from 0.10 to 0.20. What is the exact increase in aggregate capital? (Answer: 0.6595 million dollars)
Two charges are 6 and 8, with correlation 0.25. Both charges increase by 10%, with correlation unchanged. Calculate scenario capital and its increase from baseline. (Answer: Scenario capital 12.2491 million dollars; increase 1.1136 million dollars)
Two charges are 5 and 7, with correlation 0.30. The first charge falls by 20% while correlation rises to 0.40; the second charge is unchanged. Calculate the new aggregate capital. (Answer: 9.3488 million dollars)

What You'll Learn