Lesson 19 of 25 · 7 min

From one day to a liquidation window

Four and a half times the sessions gives only 2.12 times the discount, for a reason lesson 5 undermines.

A lender's exposure is not one session. It runs from the decision to sell until the cash arrives, and the discount has to cover the adverse move available over that whole span. Stretching lesson 18's figure across several sessions uses the scaling from lesson 3.

haircut = coverage multiple × daily volatility × square root of assumed sessions

haircut: The percentage of market value withheld from lending value as a buffer.
coverage multiple: How many standard deviations of adverse move the buffer is built to survive. A lender's choice, not a market figure.
daily volatility: Annualised volatility divided by 15.8745, the square root of 252 sessions.
square root of assumed sessions: The stretch applied for an exit lasting more than one session. Its square root, not its value, because variance is what adds.

Take 33.6 percent annualised volatility, so a daily figure of 2.1166 percent, and a 3.5 multiple. A two-session exit needs 3.5 × 2.1166 × 1.4142 = 10.48 percent. A nine-session exit needs 3.5 × 2.1166 × 3 = 22.22 percent. On $683,000 of market value those are $71,555.46 and $151,792.04 of withheld value.

Nine sessions is four and a half times two sessions. The haircut is 22.22 ÷ 10.48 = 2.1213 times, which is the square root of 4.5. Scaling in proportion to the days instead would give 47.14 percent — 24.92 percentage points too much, and $321,966 of withheld value against the correct $151,792.04. That is why a large position's discount is not ruinous even when its exit is long, and it is the same fact Track 7 lesson 9 used from the other direction.

Now the part that belongs in this track rather than that one. The square-root rule requires daily returns to be independent, which lesson 3 flagged and lesson 5 measured: a lag-one autocorrelation of +0.4289 in absolute returns. Magnitudes persist, so multi-session moves grow faster than the square root of the sessions. If the effective scaling exponent were 0.58 rather than 0.50 — an illustrative figure — nine sessions would scale by 3.5765 instead of 3, and the haircut would be 26.50 percent rather than 22.22 percent.

The rule that gives lenders a discount for long windows understates by about four points at nine sessions, and it understates most in clustered regimes — which are the regimes in which exits lengthen.

Key terms

Square-root-of-time scaling: Stretching a one-session move across several sessions by multiplying by the square root of their number. Correct only when sessions are independent.
Scaling exponent: The power the number of sessions is raised to. Independence gives 0.5; persistent magnitudes give something larger, so the standard rule understates.
Withheld value: The haircut expressed in currency rather than percent. Market value multiplied by the haircut.

Worked example

Two sessions against nine, on one holding

$683,000 of market value, 33.6 percent annualised volatility so 2.1166 percent daily, coverage multiple 3.5. Illustrative throughout.

One-session haircut: 7.41%
Two-session haircut: 10.48%
Nine-session haircut: 22.22%
Ratio of the two haircuts: 2.1213
What a linear rule would give at nine sessions: 47.14%
Nine sessions at a 0.58 exponent instead: 26.50%

The square root saves the borrower 24.92 points against a linear rule and costs the lender about 4.27 points against clustered reality.

Check yourself

A lender lengthens its assumed exit from two sessions to nine on a holding with 33.6 percent volatility and a 3.5 multiple. By what factor does the haircut grow, and why not 4.5?

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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.