Lesson 14 of 25 · 7 min

The forced-sale multiple, derived from scratch

Four lines of algebra produce a number fixed by the threshold alone and knowable before anything fails.

The multiple is worth deriving rather than memorising, because the derivation shows why it depends on nothing except the threshold — not on the size of the account, not on how far prices fell, and not on which holdings are sold.

Let B be the balance, V the collateral value, t the threshold as a decimal and S the amount of pledged collateral sold, with the whole proceeds applied to the balance. After the sale the balance is B − S and the collateral value is V − S. Requiring the ratio to equal the threshold gives (B − S) = t × (V − S). Expanding: B − S = tV − tS. Collecting the S terms on one side: B − tV = S − tS. Factorising: B − tV = S × (1 − t). So S = (B − tV) ÷ (1 − t).

S = (B − t × V) ÷ (1 − t), and the multiple S ÷ (B − t × V) = 1 ÷ (1 − t)

S: Pledged collateral sold, with all proceeds applied to the balance.
B: The balance, including accrued interest.
V: Collateral value on the basis the agreement names.
t: The maintenance threshold as a decimal.
B − t × V: The shortfall. Everything to the left of the division sign is the gap itself.
1 − t: The share of each sold dollar that actually closes the gap. Smaller as the threshold rises.

Run it once with awkward numbers. Collateral value $1,315,000, threshold 58 percent, balance $803,140. The permitted balance is $762,700 and the shortfall is $40,440. The multiple is 1 ÷ 0.42 = 2.381, so the sale is $40,440 ÷ 0.42 = $96,285.71. Verify it directly: the balance becomes $706,854.29, the collateral becomes $1,218,714.29, and $1,218,714.29 × 0.58 = $706,854.29 exactly.

Two properties follow immediately. The multiple is knowable the day the agreement is signed, since it needs only t. And it is not a penalty imposed by anybody: it is the arithmetic consequence of paying down a ratio with the asset that forms its denominator.

Key terms

Forced-sale multiple: Dollars of pledged collateral that must be sold per dollar of shortfall. Equal to one divided by one minus the threshold.
Effective cure per dollar: The share of each sold dollar that narrows the gap. It is one minus the threshold, and nothing else affects it.
Threshold dependence: The fact that the multiple varies with the threshold alone, so it can be computed before any breach occurs.

Worked example

The derivation checked against itself

Collateral value $1,315,000, threshold 58 percent, balance $803,140.

Permitted balance: $762,700
Shortfall: $40,440
Multiple: 2.381×
Sale required: $96,285.71
Balance and collateral after: $706,854.29 / $1,218,714.29
Ratio after: 58.00%

A $40,440 gap required a $96,285.71 sale, and the check closes to the cent. The 2.381 came from the threshold and nowhere else.

Check yourself

Collateral value $500,000, threshold 84 percent, balance $434,000. What is the shortfall, the multiple, and the sale required?

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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.