6.8.4 Liquidity Risk & Crises

Basic Prerequisites: 03 · Liquidity as a Priced Factor and Liquidity Risk & Funding (Pillar 4).

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1. Intuition & Practical Objective

The single most dangerous fact about liquidity is that it is common and pro-cyclical: illiquidity rises across all assets in bad times, so you cannot diversify it away by holding many illiquid assets - they all become illiquid together. This page makes that precise in three ways:

1. Commonality - a large fraction of a stock’s illiquidity moves with market illiquidity (Chordia, Roll & Subrahmanyam 2000).
2. The liquidity-adjusted CAPM - Acharya & Pedersen (2005) show that expected return depends not just on the asset’s own illiquidity but on three covariances: with market illiquidity, of its return with market illiquidity, and of its illiquidity with the market return.
3. Crises & flight to quality - when these covariances spike (2007–08, 2020), the premium jumps and liquidity provision vanishes; investors flee to Treasuries and the spread between liquid and illiquid assets explodes.

The objective: understand why liquidity risk cannot be hedged by diversification, and be able to estimate the three betas that capture the risk.

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2. Mathematical Ground Truth & Derivations

Commonality in liquidity. Regress stock $i$‘s illiquidity innovation on the market’s:

$$\Delta {\text{ILLIQ}}_{i,t}={\alpha }_i+{\beta }_i\Delta {\text{ILLIQ}}_{M,t}+{\epsilon }_{i,t}.$$

The $R^2$ (and average ${\beta }_i$) measures how much of each stock’s liquidity variation is a systematic market-wide shock. Chordia et al. find a large common component; the practical consequence is that liquidity risk is largely undiversifiable.

The Acharya–Pedersen liquidity-adjusted CAPM. Define $r^i$ = return, $c^i$ = relative illiquidity cost, $r^M$ = market return, $c^M$ = market illiquidity. The required excess return is

$$E_t(r_{t+1}^i)=r^f+E_t(c_{t+1}^i)+{\lambda }_t\left[\frac{{\mathrm{cov}}_t(r^i,r^M)}{\mathrm{var}(r^M-c^M)}+\frac{{\mathrm{cov}}_t(c^i,c^M)}{\mathrm{var}(r^M-c^M)}-\frac{{\mathrm{cov}}_t(r^i,c^M)}{\mathrm{var}(r^M-c^M)}-\frac{{\mathrm{cov}}_t(c^i,r^M)}{\mathrm{var}(r^M-c^M)}\right],$$

where the four betas share the same denominator (the variance of the net market return $r^M-c^M$). The three liquidity-risk betas:

• ${\beta }_1=\mathrm{cov}(c^i,c^M)/\mathrm{var}(r^M-c^M)$ - commonality: asset illiquid when the market is illiquid ⇒ higher required return (positive premium).
• ${\beta }_2=\mathrm{cov}(r^i,c^M)/\mathrm{var}(r^M-c^M)$ - return-vs-liquidity (the PS channel): return high when market illiquid ⇒ lower required return (negative premium).
• ${\beta }_3=\mathrm{cov}(c^i,r^M)/\mathrm{var}(r^M-c^M)$ - down-market liquidity: asset stays liquid when the market falls ⇒ lower required return (negative premium).

Crises. In a crisis, $c^M$ spikes and its covariation with both $r^M$ and asset illiquidities increases, so the priced covariances rise precisely when risk aversion is highest - the liquidity premium becomes large exactly when you need to be compensated most. This is the asset-pricing reflection of the margin spiral.

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3. Computational Implementation - the three liquidity-risk betas

Simulate a market where illiquidity rises in down markets, and a stock that shares market illiquidity (positive commonality) yet stays liquid when the market falls. Estimate all three betas and confirm the signs the theory predicts. Stdlib only.

Read the signs through the AP convention (the price of risk on ${\beta }_2,{\beta }_3$ is negative, so each term enters with a minus sign): a negative ${\beta }_2$ means the stock’s return is low exactly when the market is illiquid, and a negative ${\beta }_3$ means it becomes illiquid when the market falls. Both are bad risks and raise the required return. Here ${\beta }_1=+0.0043$, ${\beta }_2=-0.0440$, ${\beta }_3=-0.0468$, so with the minus signs the three liquidity terms are $+0.0043-(-0.0440)-(-0.0468)=+0.0951$ - all three push required return the same way (up), which is what the simulated data (a pro-cyclical return, $r_i=0.9r^M$, and illiquidity that shares the market’s commonality, $c_i=1.0c^M$) actually builds in.

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4. Failure Modes & First-Principles Breakdowns

• Commonality defeats diversification. Holding many illiquid assets does not diversify liquidity risk because their illiquidities all load on the same market factor - you are long a single bet.
• Betas estimated in calm times mislead. If you estimate ${\beta }_2,{\beta }_3$ from a low-volatility sample, you miss the crisis-state covariances that are the whole point. This is a regime-dependence failure: the risk is in the tail, not the average.
• Flight to quality is a correlated bet. “Safe” assets (Treasuries) see their liquidity improve and their price rise exactly when everything else dries up - so the flight itself is what makes the illiquid side so costly.
• The funding link. The covariance spikes in a crisis are driven by the margin spiral (leverage → forced sales → more illiquidity) - the asset-pricing betas here are the shadow of the funding constraint’s failure.

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5. References

• Acharya & Pedersen (2005). Asset pricing with liquidity risk. JFE 77(2), 375–410. The full model, Proposition 1, and the three betas; empirical premia for each channel.
• Chordia, Roll & Subrahmanyam (2000). Commonality in liquidity. JFE 56, 3–28. The empirical fact of a market-wide liquidity factor.
• Pástor & Stambaugh (2003). Liquidity risk and expected stock returns. JPE 111(3)
• Bao, Pan & Wang (2011). The illiquidity of corporate bonds. Journal of Finance 66(3)
• Amihud, Mendelson & Pedersen (2013). Market Liquidity

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6. Connected Graph Bridges

• Next: 05 · Failure Modes & Practice · 06 · Advanced Extensions
• Back: 03 · Liquidity as a Priced Factor · Index Hub
• Funding spiral: Liquidity Risk & Funding · Margin & Funding Spirals