Modeling Cross-Risk Contagion and Feedback

Why Cross-Risk Feedback Matters

A stress scenario can do more than produce separate losses in separate risk categories. A market shock may increase collateral calls; meeting those calls may require asset sales, which deepen market losses. A counterparty failure may reduce recoveries and also make other counterparties less willing to provide support. Policyholders may respond to perceived financial strain by surrendering policies, creating cash needs that intensify the original strain.

These are transmission and feedback mechanisms. Modeling them explicitly helps distinguish a direct shock from the effects it triggers. The objective here is to represent and solve such mechanisms within a defined scenario. It is not to assume that every risk moves together, or to combine all risk-category losses into one measure.

Definition: A cross-risk transmission is a specified relationship through which a shock or stress measure in one risk channel changes a loss, cash need, or behavior in another channel. A feedback loop occurs when a transmitted effect then changes the measure that originally drove the transmission. A fixed point is a set of stress values that satisfies all the scenario equations at the same time.

A useful scenario model names each variable, gives it a unit, states each relationship and its assumptions, and checks the solution against any thresholds or caps. The coefficients are scenario sensitivities: they describe the assumed response within the model, not universal causal laws.

Representing Feedback With Equations

For a simple two-channel linear scenario, let $x$ and $y$ be nonnegative stress amounts, measured in the same stated unit. Suppose $x=b_x+a y$ and $y=b_y+c x$, where $b_x$ and $b_y$ are direct scenario shocks and $a,c$ are nonnegative transmission coefficients. Substitution gives $x=b_x+a(b_y+cx)$, so $(1-ac)x=b_x+ab_y$.

x=\frac{b_x+ab_y}{1-ac}, \qquad y=b_y+cx

This solution applies when $ac\ne 1$, the linear relationships remain valid at the solution, and the result is consistent with the scenario’s nonnegativity and any specified constraints. In particular, if $ac<1$, the feedback does not grow without bound in the repeated-transmission interpretation. A value $ac\ge 1$ is a warning: the simple linear feedback model does not provide a finite, stable amplification for positive direct shocks. A model with caps, changing sensitivities, or another mechanism may be needed.

The algebra also explains why it is incorrect to stop after applying the direct shock once. Substituting repeatedly gives $x=b_x+ab_y+acx$: the stress in $x$ includes the effect of $y$, which itself includes a further effect of $x$. Solving the simultaneous equations accounts for all rounds of transmission at once.

Modeling principle: Write one equation for each modeled stress measure, with its direct shock and its incoming transmission effects. Solve the equations simultaneously, then check that the solution satisfies the model’s assumptions, thresholds, and limits.

Worked Examples

Worked Example: Collateral Calls and Forced Sales

In a hypothetical scenario, let $M$ be market loss in millions of dollars and $Q$ be the amount of forced sales, also in millions of dollars. The direct market shock is $5.00$. Each million dollars of forced sales adds $0.30$ million dollars of market loss, so $M=5.00+0.30Q$. The collateral call is $C(M)=2.00+0.20M$ million dollars. The insurer has $2.50$ million dollars available to meet the call; any excess call requires forced sales. Find the simultaneous stress values.

Method. The sales amount is determined by the excess call, but it cannot be negative: $Q=\max(C(M)-2.50,0)$. First test the active branch, on which the call exceeds available resources and $Q=C(M)-2.50$. Substituting the call formula gives $Q=2.00+0.20M-2.50=0.20M-0.50$.

Substitute this expression into the market-loss equation: $M=5.00+0.30(0.20M-0.50)=5.00+0.06M-0.15=4.85+0.06M$. Therefore $0.94M=4.85$, and $M=4.85/0.94=5.159574\ldots$ million dollars. Then $Q=0.20(5.159574\ldots)-0.50=1.031915\ldots-0.50=0.531915\ldots$ million dollars.

Check the branch. The call is $C(M)=2.00+0.20(5.159574\ldots)=3.031915\ldots$ million dollars. This exceeds the $2.50$ million available by $0.531915\ldots$ million dollars, exactly the calculated $Q$. Thus the active-branch solution is consistent. The market loss also checks: $5.00+0.30(0.531915\ldots)=5.159574\ldots$.

Rounded only at the end to four decimal places, $M=\$5.1596$ million and $Q=\$0.5319$ million. The loop gain is $0.30(0.20)=0.06$, below one. Common mistake avoided: setting $Q=0$ gives $M=5.00$, but then the call is $3.00$, above available resources, so $Q$ cannot be zero.

Worked Example: Credit and Market Stress Reinforce Each Other

For a hypothetical counterparty scenario, let $x$ be credit loss and $y$ be market loss, both in millions of dollars. The direct credit shock is $4.00$, and each million dollars of market loss adds $0.25$ million dollars of credit loss through reduced recoveries. The direct market shock is $1.00$, and each million dollars of credit loss adds $0.40$ million dollars of market loss through repricing. Find the simultaneous values.

Method. Represent the scenario as $x=4.00+0.25y$ and $y=1.00+0.40x$. Substitute the second equation into the first: $x=4.00+0.25(1.00+0.40x)=4.25+0.10x$. Thus $0.90x=4.25$, so $x=4.25/0.90=4.722222\ldots$ million dollars. Then $y=1.00+0.40(4.722222\ldots)=1.00+1.888889\ldots=2.888889\ldots$ million dollars.

Check both equations: $4.00+0.25(2.888889\ldots)=4.722222\ldots$, and $1.00+0.40(4.722222\ldots)=2.888889\ldots$. The loop gain is $0.25(0.40)=0.10$, so the positive direct shocks have a finite solution under this linear model. Rounded to four decimal places, $x=\$4.7222$ million and $y=\$2.8889$ million.

Common mistake avoided: using only the direct values $x=4.00$ and $y=1.00$ ignores both transmission paths. The simultaneous solution includes repeated feedback, not just one pass from each direct shock.

Worked Example: Capped Policyholder Response

In a hypothetical scenario, let $E$ be a capital-strain measure in millions of dollars and $U$ be surrender-related cash outflow in millions. Direct strain is $2.00$, and each million of outflow adds $0.50$ million of strain, so $E=2.00+0.50U$. Policyholders respond to strain according to $U=\min(1.20,\max(0,0.40(E-1.00)))$. Find the equilibrium, including a check that the response is neither zero nor capped.

Method. First solve the interior branch, where $0<0.40(E-1.00)<1.20$. On this branch, $U=0.40(E-1.00)$. Substitute $E=2.00+0.50U$: $U=0.40(2.00+0.50U-1.00)=0.40+0.20U$. Therefore $0.80U=0.40$, so $U=0.50$ million dollars. The strain is $E=2.00+0.50(0.50)=2.25$ million dollars.

Check the policyholder-response rule: $0.40(2.25-1.00)=0.40(1.25)=0.50$, which is positive and below the cap of $1.20$. Substitution into the strain equation also returns $2.25$. Rounded to four decimal places, $U=\$0.5000$ million and $E=\$2.2500$ million.

Without feedback, applying the response rule to the direct strain alone would give $U=0.40(2.00-1.00)=0.40$, not the equilibrium outflow. Common mistake avoided: using that one-pass response as the final answer fails to account for the additional strain generated by the outflow.

Thresholds, Caps, and Model Checks

Realistic scenario relationships are often not linear over their full range. A collateral call may trigger sales only after available resources are exhausted. A policyholder response may be negligible below a perceived-strain threshold, then increase, and eventually be limited by the number of policies that can lapse. Such rules are piecewise: the equations change depending on which threshold or cap is active.

Define the stress measures.
State what each variable represents, its unit, and whether it is a loss, a cash need, or a response. Do not use the same symbol for different quantities.

Specify direct shocks and transmission paths.
Write the direct component and each modeled incoming effect. Make clear what each coefficient means and which scenario assumptions support it.

Identify branches and solve.
For a threshold or cap, solve on a proposed branch first. Use simultaneous equations when a variable feeds back into another variable that determines it.

Verify the solution.
Substitute values into every equation and check each branch condition, cap, nonnegativity restriction, and unit.

The most important branch check is often simple: a proposed solution to the active-sales equations is invalid if the call does not exceed available resources. Likewise, a calculated policyholder response is not an interior solution if it lies above its cap or below zero. In these cases, solve the appropriate alternative branch and check it instead.

Common Modeling Traps

Stopping after one transmission. A one-pass calculation omits later rounds of feedback. Solve the simultaneous relationships and verify them by substitution.
Using an active branch without checking it. A formula for excess collateral calls or threshold-based behavior applies only when its condition holds. Check the condition after solving.
Mixing units. If stress amounts are in millions, the intercepts and calculated amounts must also be in millions. A coefficient such as $0.30$ must be interpreted as the change in one stated quantity per unit change in another.
Counting the same effect twice. If a direct scenario shock already includes a particular market impact, do not add that same impact again as a transmitted effect. Draw the transmission path and identify what each term includes.
Treating scenario sensitivities as universal laws. A slope is an assumption for the modeled range and scenario. It may not remain valid under a different severity, time horizon, or management response.

Cross-risk feedback is useful because it makes scenario mechanisms visible and testable: an analyst can challenge a transmission coefficient, threshold, or assumed response rather than treating amplification as an unexplained adjustment. It also provides a disciplined way to report what the model does and does not capture. These techniques are aligned with the CAS Exam 9 (Risk Management for Actuaries) syllabus.

Check Your Understanding

For each hypothetical scenario, solve the stated relationships and check any threshold or cap before accepting the result.

Two stress measures, in millions of dollars, satisfy $A=2.00+0.20B$ and $B=1.00+0.50A$. Find the simultaneous values and the loop gain. (Answer: $A=\$2.4444$ million, $B=\$2.2222$ million; loop gain $0.1000$.)
A market loss $M$, in millions, satisfies $M=4.00+0.20Q$, where $Q=\max(0,0.50(M-3.00))$. Solve the applicable branch and verify the threshold. (Answer: $M=\$4.1111$ million and $Q=\$0.5556$ million; the active branch is valid because $M>3.00$.)
Credit loss $x$ and market loss $y$, in millions, satisfy $x=6.00+0.30y$ and $y=2.00+0.20x$. Find both equilibrium losses and check each equation. (Answer: $x=\$7.0213$ million and $y=\$3.4043$ million.)
A strain measure $E$, in millions, satisfies $E=3.00+0.80U$, and surrender outflow is $U=\min(1.00,\max(0,0.60(E-2.00)))$, also in millions. Determine whether the cap binds and find the equilibrium. (Answer: $U=\$1.0000$ million and $E=\$3.8000$ million; the uncapped response would be $0.60(3.80-2.00)=1.08$, so the cap binds.)

What You'll Learn