Lesson 20 of 25 · 8 min

Reverse stress testing: starting from the failure and working back

Fix the breach and solve for the inputs, and the answer is not a scenario but a curve.

A forward stress test picks a decline and computes what it does. A reverse stress test fixes the outcome — the breach — and solves for every combination of conditions that produces it. The output is not a scenario. It is a frontier, and the frontier says more than any point on it.

Take a facility with $2,340,000 of collateral at a 32 percent haircut, a balance of $1,104,000, and a maintenance threshold of 85 percent of lending value. The lending value is $1,591,200 and the current ratio is 69.38 percent.

Write d for a price decline and h for the haircut in force. Lending value after both is the starting collateral multiplied by (1 − d) and by (1 − h), and the breach is the point at which the balance equals the threshold times that figure. Everything known collects on one side.

(1 − d) × (1 − h) = B ÷ (τ × V₀) = 1,104,000 ÷ (0.85 × 2,340,000) = 0.555053

d: The decline in collateral market value, as a decimal.
h: The haircut in force after any revision, as a decimal.
B: The balance owed, $1,104,000 here. Accrued interest belongs in it.
τ: The maintenance threshold as a fraction of lending value, 0.85 here.
V₀: Collateral market value today, $2,340,000.

One constant defines the whole frontier, and it rearranges to d = 1 − 0.555053 ÷ (1 − h). At the current 32 percent haircut, 1 − 0.555053 ÷ 0.68 = 18.37 percent. At 35 percent, 14.61 percent. At 37 percent, 11.90 percent. At 40 percent, 7.49 percent. At 42 percent, 4.30 percent. At 44.49 percent the required decline is zero: the haircut revision alone breaches the facility with the market exactly where it is today.

The slope of the frontier is the exchange rate between the two variables, and it is not 1 to 1. Differentiating gives 0.555053 ÷ (1 − h)², which is 1.20 at the current haircut: one point of haircut widening consumes as much headroom as 1.20 points of price decline. At a 40 percent haircut the rate has risen to 1.54, because the same point is now taken from a smaller remaining base.

The answer to what breaks this facility is not 18 percent. It is a curve, and its steepest end is governed by the variable that does not appear on any statement.

Key terms

Reverse stress test: Fixing the failure and solving backwards for the conditions that cause it, rather than stipulating conditions and computing forwards.
Breach frontier: The full set of combinations that exactly reach the threshold. Every point on it is a breach and every point inside it is not.
Substitution rate: How many points of price decline one point of haircut widening is worth. It rises as the haircut widens, because the base it is taken from shrinks.

Worked example

Solving the frontier

Collateral $2,340,000, haircut 32 percent, balance $1,104,000, threshold 85 percent of lending value. The constant is 0.555053.

Haircut unchanged at 32 percent: Decline of 18.37%
Haircut at 37 percent: Decline of 11.90%
Haircut at 40 percent: Decline of 7.49%
Haircut at 42 percent: Decline of 4.30%
Haircut at 44.49 percent: Decline of 0%
Substitution rate at the current haircut: 1.20 points per point

Asking what a 30 percent fall would do never surfaces the row that matters: a 12.49 point haircut revision breaches this facility on a flat market.

Check yourself

Collateral $2,340,000, haircut 32 percent, balance $1,104,000, threshold 85 percent of lending value. If the haircut is revised to 40 percent, what price decline now breaches the facility, and what was it before?

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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.