4.6.1 Liquidity Risk from Zero

Basic Prerequisites: VaR & Expected Shortfall (helpful but not required - this page needs no prior quant-finance).

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1. Intuition & Practical Objective

This page builds the why of liquidity risk with no prior finance needed. The objective is one idea, the one every beginner gets wrong:

There are two completely different ways to be unable to sell something, and they are not the same risk.

Way 1 - market liquidity. You own an asset and want to trade it. The screen shows a mid price (say $50.00), but nobody trades at the mid: there is a bid (what a buyer offers, $49.95) and an ask (what a seller wants, $50.05). If you must sell now, you hit the bid and cross the spread. If your order is big, you do not merely cross the spread - you walk the book, hitting successively worse bids, and the price you receive is worse than the screen. Market liquidity is how much worse, and how fast the price recovers afterwards.

Way 2 - funding liquidity. You borrowed money to buy the asset. The loan is collateralised: the lender holds the asset and gives you cash, keeping a haircut (margin). If the haircut is 2% you can borrow $98 against a $100 bond; if the lender raises the haircut to 25% you can borrow only $75, and you must find cash elsewhere or sell the asset to repay the loan. Funding liquidity is your access to that cash/rollover, and it can vanish overnight regardless of what the asset is worth.

Three “aha”s:

1. The spread is a tax you pay for immediacy. You are not being cheated; you are paying a liquidity supplier to take the other side right now instead of later at a better price. The cost is measurable and small in calm times - and enormous in a crisis.
2. Leverage converts a small price move into a forced sale. With $10 of your own money controlling $50 of assets (5× leverage), a 3% price fall costs you 15% of your equity - and if the margin requirement rises at the same moment, the broker can force you to sell at exactly the worst price. The broker’s margin call does not care that you think the asset is cheap.
3. Market and funding liquidity are the same fire. Falling prices (i) make the asset harder to sell and (ii) make lenders demand more margin. Each makes the other worse. That is the liquidity spiral - the subject of 04 · Margin & Funding Spirals.

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2. Mathematical Ground Truth & Derivations

The spread, precisely (Foucault eq. 2.1). With best bid $b$, best ask $a$, and midquote $m=(a+b)/2$, the quoted spread and its relative version are

$$S=a-b,s=\frac{a-b}{m}.$$

Selling at the bid realises a half-spread cost $\frac{1}{2}S$ per share - the standard measure of immediacy’s price. So the market-liquidity cost of liquidating $Q$ shares at mid price $m$ is, to first order,

$${\text{Cost}}_{\text{spread}}=\frac{1}{2}SQ=\frac{1}{2}smQ=\frac{1}{2}sV,V=mQ.$$

The margin constraint. A position of value $P$ financed with equity $N$ and a fractional margin (haircut) $m$ must satisfy, with $L=P/N$ the leverage,

$$P\le \frac{N}{m}⟺L=\frac{P}{N}\le \frac{1}{m}.$$

$m$ is the only number in this inequality that the lender sets unilaterally - and $m$ is set from risk: a common rule is $m=z_{\alpha }\sigma$ (the position’s value-at-risk), so higher volatility mechanically forces lower leverage (see 02 · Market vs Funding Liquidity).

Why “solvent but dead” is possible. Solvency is about assets vs liabilities; liquidity is about the timing of cash flows. A fund can have $P>P_{\text{debt}}$ (positive net worth) yet be unable to meet a margin call today because its assets cannot be converted to cash at their marked value without a fire-sale discount. Formally, liquidation raises cash $\frac{1}{2}sV$ less than the mark, so a firm that is barely solvent can be made insolvent by the act of meeting its own margin call.

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3. Computational Implementation - the two costs, side by side

One position, two risks. First the market-liquidity cost of selling; then funding liquidity expressed as the maximal leverage each haircut permits. Stdlib only.

Read the two halves together. A 10 bp spread costs $12,500 on a $12.5M position - annoying but survivable. But the same asset financing a book at 10× leverage loses 80% of its allowed leverage when the haircut jumps from 2% to 10%. The market-liquidity cost is a steady tax; the funding-liquidity cost is a cliff.

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4. Failure Modes & First-Principles Breakdowns

1. “I’ll sell at the price on my screen.” You cannot. You sell at the bid, and for size you sell through it. The mark-to-market P&L you report is systematically optimistic versus the price at which the position can actually be exited - the seed of L-VaR (03 · Liquidation Cost & L-VaR).
2. “My balance sheet is solvent, so I’m safe.” Solvency ≠ liquidity. A firm can be wiped out by meeting a margin call it cannot fund without fire-selling the very assets whose value it was relying on.
3. “The haircut is a constant.” Haircuts are state-contingent and rise with volatility and with the lender’s own stress. Assuming a fixed margin overstates the leverage you will actually have in a crisis - and the failure is multiplicative, because the loss spiral and the margin spiral fire together.
4. “Liquidity is an execution detail, not a risk.” It is a risk factor: illiquidity varies over time (U-shaped intraday spreads; crisis widening) and is common across assets (Hasbrouck–Seppi 2001), so it does not diversify away.

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5. References

• Foucault, Pagano & Röell - Market Liquidity (2013)
• Hasbrouck - Empirical Market Microstructure (2007)
• Brunnermeier & Pedersen - Market Liquidity and Funding Liquidity, RFS 22(6):2201–2238 (2009): the formal pairing of the two liquidities.
• Brunnermeier - Deciphering the Liquidity and Credit Crunch 2007–2008 (2009): the plain-language case study.

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6. Connected Graph Bridges

• Base: VaR & Expected Shortfall · Probability & Measure Theory
• Continue: 02 · Market vs Funding Liquidity · Index Hub
• Execution bridge: Optimal Execution & Almgren–Chriss