4.6 Liquidity Risk & Funding

Basic Prerequisites: VaR & Expected Shortfall and Parametric, Historical & Monte Carlo VaR. (these are the folder-level prerequisites; page 01 states its own, smaller, entry requirements. The [VaR & ES] prerequisite is needed from page 03 onward, the [Parametric/Historical/MC VaR] one only where the pages price a liquidation horizon.)

1. Intuition & Practical Objective

A firm can be solvent on paper - assets exceed liabilities at mark-to-market - and still die in 48 hours. Liquidity risk is the gap between what a position is worth on the screen and what you actually receive when you must sell it now, with borrowed money that can be recalled.

This folder separates the two things everyone conflates:

• Market liquidity - the cost and speed of trading an asset. Its price is the bid–ask spread, its depth is the price impact of an order, its recovery is resiliency (Hasbrouck Ch 1.2; Foucault Ch 2).
• Funding liquidity - the ease of financing a position with cash or collateralised borrowing (repo). Its price is the haircut/margin, its limit is the margin constraint $P\le N/m$.

They are two sides of one feedback loop. When prices fall, the same losses (i) widen spreads and thin depth (market liquidity ↓) and (ii) raise margins and haircuts (funding liquidity ↓). The two effects feed each other - the liquidity spiral of Brunnermeier & Pedersen (2009), more precisely a margin spiral (margin ↑ → forced sale → price ↓ → margin ↑) riding on a loss spiral (loss ↑ → equity ↓ → forced sale → loss ↑).

This folder is the hub: it (a) gives the fast formula lookup below, and (b) routes to six sub-pages that walk from raw intuition through the two-liquidities distinction, the liquidation-cost/L-VaR machinery, the margin recursion, the failure modes, and the systemic extensions.

The one-sentence essence. “Standard risk measures assume you can liquidate at the mid in zero time; liquidity risk is the price you pay - in spread, in market impact, and in margin - for the fact that you cannot, and the danger is that these three costs increase together exactly when you need them.”

2. Mathematical Ground Truth & Derivations

Quick-Reference Lookup (job #1). Notation: $V$ position value; $S=a-b$ absolute quoted spread, $s=(a-b)/m$ relative spread (Foucault eq. 2.1); ${\sigma }_s$ spread volatility; $z_{\alpha }={\Phi }^{-1}(\alpha )$; $Q$ shares to liquidate; $D$ market depth (shares to move price by 1 currency unit); $N$ equity; $m$ margin/haircut; $\lambda$ price-impact coefficient (Kyle/Foucault).

The margin recursion (Brunnermeier–Pedersen 2009). A leveraged trader with equity $N_t$, position $P_t$, and margin $m_t=N_t^{\text{req}}/P_t$ is forced to sell whenever the constraint binds. One stress round is:

Two multiplicative forces hit the constraint at once: $N_t$ falls in the numerator (loss spiral) and $m_{t+1}$ rises in the denominator (margin spiral). Neither alone closes the firm; together they do.

Regulatory definitions (BCBS). Funding-liquidity risk is regulated in two ratios:

LCR is a 30-day survival test (BCBS 2013, d238); NSFR is a one-year structural funding test (BCBS 2014, d295). Both are funding-liquidity rules - they do not constrain the market liquidity of the assets, which is the other half of the spiral.

3. Computational Implementation - the formula engine

Standard library only. This reproduces every verified number above (all six experiments were executed and reproduced exactly; the full runs live on the sub-pages).

The point of the check. The two liquidation costs together add $40,816 - $8.77\%$ - to a VaR that a standard risk engine would report as the whole story. On an illiquid book the impact term dominates and can exceed the market-risk term outright.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

1. Ignoring the liquidation horizon. VaR assumes a zero-time exit at the mid; real liquidation takes $T_{\text{liq}}=Q/(\text{ADV}\cdot \alpha )$ days, during which the market keeps moving against you. The two errors compound: you scale risk by $\sqrt{T}$ but the cost of the forced sale by $T$ (§5).
2. Fire-sale externality. A fund’s private liquidation cost uses its own order size; when a crowd liquidates the same asset, each bears the others’ impact. Private cost understates social cost - the exact mechanism behind crowded-unwind contagion and the reason liquidity is a system-wide, not firm-level, risk (§5).
3. Funding vs market liquidity interaction. Margin depends on volatility, and volatility is caused by forced selling. Treating the haircut as an exogenous constant breaks precisely when it matters - in the crisis (§4).

5. References

• Brunnermeier, Markus K. & Pedersen, Lasse Heje - Market Liquidity and Funding Liquidity, Review of Financial Studies 22(6):2201–2238 (2
• Brunnermeier, Markus K. - Deciphering the Liquidity and Credit Crunch 2007–2008, Journal of Economic Perspectives 23(1):77–100 (2
• Foucault, Thierry, Pagano, Marco & Röell, Ailsa - Market Liquidity: Theory, Evidence and Policy (Oxford, 2013)
• Hasbrouck, Joel - Empirical Market Microstructure (Oxford, 2007)
• Hull, John C. - Risk Management and Financial Institutions (Wiley), liquidity-risk chapter (liquidity trading risk, L-VaR, the Basel LCR/NSFR). Risk-engineering companion to this folder.
• Amihud, Yakov - Illiquidity and Stock Returns: Cross-Section and Time-Series Effects, J. Financial Markets 5(1):31–56 (2
• Acharya, Viral & Pedersen, Lasse Heje - Asset Pricing with Liquidity Risk, J. Financial Economics 77(2):375–410 (2005). Liquidity as a priced factor
• [REG] BCBS - Basel III: The Liquidity Coverage Ratio and Liquidity Risk Monitoring Tools (2013, BIS d238) and Basel III: The Net Stable Funding Ratio (2014, BIS d295). The regulatory definitions of funding-liquidity risk.

6. Connected Graph Bridges

• Foundational base: Probability & Measure Theory · VaR & Expected Shortfall
• Sub-pages (in-folder): 01 From Zero · 02 Market vs Funding · 03 Liquidation Cost & L-VaR · 04 Margin & Funding Spirals · 05 Failure Modes · 06 Advanced Extensions
• Sibling topics: VaR & Expected Shortfall · Parametric, Historical & Monte Carlo VaR · Stress Testing & Scenario Analysis · Credit Risk & the Merton Model
• Execution bridge: Optimal Execution & Almgren–Chriss · Transaction Costs & Turnover Constraints

Beginner: start at 01 · Practitioner: start at 05