Work backward from a portfolio loss budget

Define the outcome before solving the shock

The starting portfolio value is $10 million, and the chosen loss budget is 10% of that amount. Express the budget as a positive input, but the target P/L as negative: -$10 million * 10 / 100 = -$1 million. This is an illustrative decision boundary, not a claim about a suitable risk tolerance, a regulatory requirement or the maximum loss the portfolio could experience.

The equity sleeve is $2 million with beta one. Other-position P/L is fixed at -$200,000. These inputs are separate from the default four-holding portfolio: in particular, the default equity sleeve is not being reused. The remaining exposures are represented only by their assumed aggregate P/L. Their composition, coverage and valuation method are not reconstructed or validated by the reverse calculator.

Calculate the remaining loss and equity sensitivity

The signed equity loss needed to reach the target is target P/L minus other P/L: -$1,000,000 - (-$200,000) = -$800,000. Because the other positions have already used $200,000 of the loss budget, the equity sleeve needs to supply only the remaining $800,000. Subtracting a negative number is essential; treating the other loss as positive would solve a different problem.

Each one-percentage-point equity move changes modeled P/L by equity value * beta / 100 = $20,000. Divide the required -$800,000 equity P/L by that sensitivity to obtain -40 percentage points, reported as a -40% equity shock. The sensitivity denominator is the $2 million equity sleeve, adjusted for beta and percentage units, not the $10 million total used to define the budget.

Reconcile the answer without confusing the returns

Substituting the answer back into the equation gives $2,000,000 * 1 * -0.40 = -$800,000 for equities. Adding the fixed -$200,000 produces the requested -$1,000,000. Dividing that total P/L by $10,000,000 gives -10%. A 40% equity decline therefore reaches a 10% portfolio loss under these assumptions; the percentages refer to different exposures and should not be compared as though their denominators matched.

The engine labels -40% within-range because it falls between its -60% and +60% equity-shock bounds. That is a numerical support check, not evidence that the scenario is likely, suitably severe or economically complete. The reverse output reports the target, solved shock and reconciled P/L. It does not independently certify a full holdings valuation or generate a distribution of possible losses.

Challenge the frozen other-position assumption

Holding other P/L fixed makes the equation transparent, but those positions might react to the same economic shock. As a separate sensitivity check, fixing other losses at $400,000 instead would leave $600,000 for equities and move the boundary to -30%. That calculation changes an assumption; it is not a second estimate of the original reference case. A less negative required equity move can reflect larger losses elsewhere.

Fixed beta also ignores changing exposures, options and nonlinear responses. The solver does not search across rates, spreads, currencies or feedback from selling. It does not establish the least severe joint scenario or the first operational failure. Counterparty problems, collateral demands or unavailable funding could matter before the specified market-value loss is reached. Conversely, possible management actions are not automatically credited as offsets.

Recognize mathematical and methodological limits

With zero equity value or zero beta, the engine returns no-sensitivity and no solved shock; changing that factor cannot alter P/L. A solution outside the supported shock interval is labeled outside-range rather than treated as a validated scenario. The EBA's stress-testing guidance starts reverse analysis from a predefined outcome and considers alternative paths to it. The Basel principles similarly connect reverse testing with identifying vulnerabilities; neither endorses this simplified equation as a complete institutional exercise.

Questions that turn a boundary into a review

What decision would reaching the loss budget trigger, and who owns it? Is the other-position loss independently supported under a coherent scenario? Which funding constraint could bind sooner, and how long would any response take? Preserve the five inputs and the substitution check before testing alternatives. The useful output is a conditional boundary for discussion, not a safe operating zone, probability estimate or instruction to trade.

Review checkpoints

• A 10% budget on $10 million sets a -$1 million P/L target.
• After -$200,000 elsewhere, equities must contribute -$800,000.
• At $20,000 per percentage point, the required equity shock is -40%.
• Challenge frozen losses, nonlinear exposures and funding constraints.

Source notes

1. European Banking Authority: Guidelines on institutions' stress testing (2018), reverse stress testing
2. Basel Committee: Stress testing principles (2018), reverse testing and vulnerabilities

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