Lesson 13 of 25 · 6 min

Why curing by sale shrinks both sides of the ratio

Selling exactly the gap leaves a smaller gap, and the leftovers form a series that never quite ends.

The natural response to a $24,000 shortfall is to sell $24,000 of collateral, and it does not work. Watching precisely how it fails is the cleanest route into the arithmetic of the next lesson.

Set collateral value at $740,000, the threshold at 65 percent and the balance at $505,000. The permitted balance is $481,000, so the shortfall is $24,000. Sell $24,000 of pledged collateral and apply the proceeds: the balance falls to $481,000 and the collateral value falls to $716,000. The permitted balance is now $716,000 × 0.65 = $465,400, and the balance exceeds it by $15,600. The ratio is $481,000 ÷ $716,000 = 67.2 percent, still above 65.

The residual is not a coincidence and not a rounding artefact. Selling a dollar removes a dollar from the balance and 65 cents of permitted balance, so it closes 35 cents of gap and leaves 65. The residual after selling the gap is therefore the threshold times the gap: $24,000 × 0.65 = $15,600 exactly.

Repeat the mistake and watch it converge. Selling $15,600 leaves $15,600 × 0.65 = $10,140. Selling that leaves $6,591. The sales form a geometric series with ratio 0.65, and its sum is $24,000 × (1 + 0.65 + 0.65 squared + ...) = $24,000 ÷ 0.35 = $68,571.43. That total is the amount a single correctly sized sale would have been, and the fact that a series of naive attempts converges to the same figure is a good sign the arithmetic is right.

A ratio cannot be fixed by an action that shrinks its numerator and denominator by the same amount, unless the action is larger than the gap by a factor set entirely by the threshold.

Key terms

Self-defeating cure: An action that reduces the balance and the collateral together, so the ratio improves by less than the amount applied.
Residual shortfall: What remains after an undersized cure. Selling the gap exactly leaves the threshold multiplied by the gap.
Convergence: The property that repeated undersized sales approach the correct single sale rather than closing the gap in one step.

Worked example

Selling the gap, four times over

Collateral value $740,000, threshold 65 percent, balance $505,000. Each round sells exactly the current shortfall.

Shortfall: $24,000
After selling $24,000: Shortfall $15,600
After selling $15,600: Shortfall $10,140
After selling $10,140: Shortfall $6,591
Sum of all such sales: $68,571.43

Four sales and the test still fails. The sum of the infinite sequence is the single sale that would have worked at the first attempt.

Check yourself

A shortfall of $18,500 exists at a 72 percent threshold and exactly $18,500 of pledged collateral is sold. What is left of the shortfall?

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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.