4.6.2 Market Liquidity vs Funding Liquidity

Basic Prerequisites: 01 · From Zero and VaR & Expected Shortfall.

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1. Intuition & Practical Objective

This page makes the market-vs-funding distinction measurable. The objective: given a position, produce two numbers - how illiquid is the asset (market side) and how leveraged can I be against it (funding side) - and see that they are driven by the same underlying quantity, volatility.

The three notions of liquidity (Foucault Ch 1 §0.4) put the vocabulary in place:

Notion	Question it answers	Practical proxy
Market liquidity	How costly/fast to trade the asset?	spread, price impact, Amihud ratio, depth
Funding liquidity	How easily can I finance holding it?	haircut/margin, repo rollover, LCR/NSFR
Monetary liquidity	How much money exists in the system?	M1/M2, monetary base

Market and funding liquidity are the pair that interact; monetary liquidity is the macro backdrop (this folder stays with the first two).

The essential bridge. A lender sets the haircut $m$ from the risk of the collateral. A common rule is $m=z_{\alpha }\sigma$ - the collateral’s value-at-risk. So market volatility $\sigma$ is the common driver: when $\sigma$ rises, the asset simultaneously becomes costlier to trade and harder to finance. The two “liquidities” are not independent factors; they are two faces of $\sigma$ (Brunnermeier–Pedersen 2009).

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2. Mathematical Ground Truth & Derivations

2.1 Measuring market liquidity (Foucault Ch 2; Hasbrouck Ch 9).

• Quoted/relative spread (Foucault eq. 2.1): $S=a-b$, $s=(a-b)/m$.
• Effective half-spread (eq. 2.3): $S_e=d(p-m)$, $d=\pm 1$ the trade direction - the actual cost of a trade relative to the prevailing midquote, capturing price improvement.
• Price impact / Kyle lambda (eq. 2.8; Hasbrouck Ch 7):

$$\Delta m_t=\lambda q_t+{\epsilon }_t,\frac{1}{\lambda }=\text{market depth}.$$

Stoll (2000), cited in Foucault: $\lambda >0$ for 98% of stocks and significant for 63%; the price impact of a unit order is ~0.75% for the smallest NYSE/AMEX caps vs ~0.52% for the largest.

• Amihud illiquidity ratio (Amihud 2002; Foucault eq. 2.9):

$$I_t=\frac{|r_t|}{{\text{Vol}}_t}\text{(price move per currency traded)},$$

with the Amivest liquidity ratio its reciprocal $L_t={\text{Vol}}_t/|r_t|$ (eq. 2.10).

• Roll spread estimator (Foucault eq. 2.18; Hasbrouck Ch 3): from the negative autocovariance induced by bid–ask bounce,

$$S_R=2\sqrt{-\mathrm{cov}(\Delta p_{t+1},\Delta p_t)}.$$

• Resiliency: the speed at which these costs decay after a trade - the third leg of Hasbrouck’s depth/breadth/resiliency triad.

2.2 Measuring funding liquidity (the haircut).

A secured lender advances $P(1-m)$ against collateral worth $P$, so the borrower’s equity is $mP$ and the constraint is $P\le N/m$. Setting the haircut from the collateral’s VaR at confidence $\alpha$ over the margin period of risk (MPOR, the liquidation delay in the event of default) gives the risk-based rule

$$m=z_{\alpha }{\sigma }_{\text{MPOR}}=z_{\alpha }\sigma \sqrt{\Delta t_{\text{MPOR}}}.$$

This is why a vol spike automatically tightens funding: both $z_{\alpha }\sigma$ and the MPOR itself lengthen in stress.

2.3 The return premium: market liquidity is priced. Foucault eq. (9.6) shows the gross expected return required on an illiquid asset:

$$R\simeq r+\frac{s}{h},h=\text{expected holding period},$$

so an asset with spread $s$ must earn an extra $s/h$ per period; the Amihud–Mendelson clientele logic and the liquidity-adjusted CAPM of Acharya–Pedersen (2005) generalise this into four liquidity betas (see 06 · Advanced Extensions). Estimated cross-sectionally (Foucault Ch 9), $R_i=0.0036+0.00672{\beta }_i+0.211s_i$ - the spread term is economically large.

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3. Computational Implementation - Amihud across caps and margin from vol

Two computations: the asset-side illiquidity ratio on three cap tiers, and the funding-side maximal leverage implied by a VaR-margin rule at rising volatility. Stdlib only.

Two observations. First, the Amihud ratio separates cap tiers by roughly three orders of magnitude (the printed range spans ~1600×) - market liquidity is not a single number but an asset-specific attribute that varies violently across the universe. Second, because $m=z_{\alpha }\sigma$, an eightfold rise in volatility (1%→8%) cuts the permitted leverage eightfold (43×→5.4×). Funding liquidity is a strictly decreasing function of the same volatility that thins market liquidity. That is the mechanism the spiral builds on.

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4. Failure Modes & First-Principles Breakdowns

1. Roll’s estimator assumes things that break. Foucault Ch 2 shows the Roll spread is biased downward by unbalanced order flow (by factor $2\sqrt{\eta (1-\eta )}$), by autocorrelated orders (by $2(1-\delta )$; Choi–Salandro–Shastri estimate $\delta \approx 0.7\Rightarrow$ underestimate by ~0.6), and by informed flow; and can be swamped or sign-flipped by time-varying expected returns. Using a single Roll number as “the” spread is a category error in a trending market.
2. Amihud is a proxy, not a price. ${\mathrm{Vol}}_t$ is shares (or value) traded, not offered. A market with a quoter but no trades reports infinite illiquidity; one with a huge trade reports false liquidity. Hasbrouck notes the Amihud ratio is a better proxy for $\lambda$ than the Amivest ratio, but still a proxy.
3. Haircuts are not exogenous constants. The $m=z_{\alpha }\sigma$ rule embeds a positive feedback (vol↑ ⇒ margin↑ ⇒ forced selling ⇒ vol↑) that a static margin assumption suppresses. Treating $m$ as fixed converts a spiral into a straight line.
4. Funding liquidity is not the same as having assets. A firm can hold liquid assets (high market liquidity) and still fail on funding if it cannot roll short-term borrowing - the 2007 ABCP/SIV failure mode (Brunnermeier 2009). Market liquidity of the collateral ≠ access to cash against it.
5. Time-variation is the risk, not the level. Liquidity is U-shaped intraday and crisis-widening; the premium $s/h$ is priced because $s$ and $h$ move together in bad states - the priced risk is liquidity’s covariance with the market, not its average.

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5. References

• Foucault, Pagano & Röell - Market Liquidity (2013)
• Hasbrouck - Empirical Market Microstructure (2007)
• Amihud, Y. - Illiquidity and Stock Returns, JFM 5(1):31–56 (2002)
• Brunnermeier & Pedersen (2009)
• BCBS

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6. Connected Graph Bridges

• Back: 01 · From Zero · Index Hub
• Forward: 03 · Liquidation Cost & L-VaR · 04 · Margin & Funding Spirals
• Siblings: VaR & Expected Shortfall · Parametric, Historical & Monte Carlo VaR
• Execution bridge: Optimal Execution & Almgren–Chriss