Lesson 19 of 25 · 7 min

Stress testing with no probability attached

Shock three variables one at a time, add the results, and you will be short by the interaction term.

Track 4 lesson 16 established the stress test as arithmetic that needs no probability. The refinement that matters in practice is that the shocks do not add, because they multiply through the same quantity.

Take a facility with $1,960,000 of collateral at a 28 percent haircut, a balance of $940,000, and a maintenance threshold of 80 percent of lending value. The lending value is $1,411,200 and the ratio is $940,000 ÷ $1,411,200 = 66.61 percent.

Three shocks, applied one at a time, none of which breaches:

Prices fall 14 percent. Lending value $1,213,632, ratio 77.45 percent, a rise of 10.84 points.
The haircut is revised from 28 to 34 percent. Lending value $1,293,600, ratio 72.67 percent, a rise of 6.06 points.
The rate rises 150 basis points for six months, adding $940,000 × 0.015 × 0.5 = $7,050 of accrual. Ratio 67.11 percent, a rise of 0.50 points.

Applied together, the collateral is $1,685,600 and the lending value is $1,685,600 × 0.66 = $1,112,496, against a balance of $947,050. The ratio is 85.13 percent, a rise of 18.52 points. The three separate rises sum to 17.40. The missing 1.12 points is the interaction: the wider haircut is being applied to an already-reduced price, and a percentage of a smaller number removes fewer dollars while removing the same proportion.

The breach that none of the single shocks produced is now real. The maximum balance at 80 percent of $1,112,496 is $889,997, so the shortfall is $57,053. Curing it by sale requires S where $947,050 − S = 0.80 × ($1,112,496 − 0.66S), giving S = $57,053 ÷ 0.472 = $120,875 — a multiple of 2.12, because each dollar of collateral sold removes only 66 cents of lending value.

Key terms

Interaction term: The part of a combined stress that is not in any of the single shocks. It appears whenever two shocks act on the same quantity multiplicatively.
Coherent scenario: A stress in which the variables move together in a way that could actually occur, rather than one variable moving while the rest are held still.
Cure multiple under a haircut: How much collateral must be sold per dollar of shortfall when the test runs on lending value. Equal to 1 ÷ (1 − threshold × advance rate), so 2.12 at a threshold of 80 percent and an advance rate of 66 percent.

Worked example

Three shocks separately and together

Collateral $1,960,000, haircut 28 percent, balance $940,000, threshold 80 percent of lending value. Starting ratio 66.61 percent.

Prices −14 percent: 77.45%
Haircut 28 to 34 percent: 72.67%
Rate +150bp for six months: 67.11%
Sum of the three rises: 17.40 points
All three together: 85.13%
Cure by sale: $120,875

Three shocks that individually breached nothing produced a breach together, and the combined figure exceeded their sum by 1.12 points.

Check yourself

A price fall of 14 percent raises a ratio by 10.84 points, a haircut revision raises it by 6.06, and six months of higher rates raises it by 0.50. Applied together the rise is 18.52 rather than 17.40. Where does the extra 1.12 points come from?

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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.