4.6.3 Liquidation Cost & Liquidity-Adjusted VaR (L-VaR)

Basic Prerequisites: VaR & Expected Shortfall and 02 · Market vs Funding Liquidity.

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1. Intuition & Practical Objective

Value at Risk is built on a lie of convenience: it assumes you can liquidate the whole position in zero time at the mid price. Neither clause is true. This page removes them, one at a time, and rebuilds VaR into a liquidity-adjusted VaR (L-VaR):

$$\text{L-VaR}=\underset{\text{price risk over the horizon}}{\underbrace{{\text{VaR}}_{\text{market}}}}+\underset{\text{spread you cross}}{\underbrace{{\text{LC}}_{\text{exogenous}}}}+\underset{\text{price you move by selling}}{\underbrace{{\text{LC}}_{\text{impact}}}}.$$

The objective is the decomposition itself: know which component dominates, because the fix differs. If spread dominates, trade smaller/patiently; if impact dominates, the position is simply too large for the market and must be split over days.

Three ideas carry the page:

1. Two liquidation costs, two mechanisms. The exogenous cost is the spread and its variability - independent of your size. The endogenous cost is your own market impact - it grows with the square of your size, because both the number of shares and the price concession per share rise with $Q$.
2. The liquidation horizon is a risk multiplier. Liquidating over $T$ days means you are exposed for $T$ days, so market risk scales like $\sqrt{T}$ - but the schedule of forced execution adds cost that scales like $T$ in the worst (linear-impact) case. This is exactly why Basel’s market-risk horizon is 10 days, and why the FRTB introduces liquidity horizons that can be far longer for illiquid risk factors.
3. L-VaR is a coherent upgrade of VaR, not a different animal. It keeps the quantile (so backtesting machinery survives) and adds a deterministic liquidation cost - the Bangia–Diebold–Schuermann–Stroughair (1999) construction.

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2. Mathematical Ground Truth & Derivations

2.1 Exogenous liquidation cost (spread). Selling realises the half-spread plus a worsening of the spread itself in the tail. With relative spread $s$ and spread volatility ${\sigma }_s$, the $\alpha$-confidence liquidation cost on value $V$ is

$${\text{LC}}_{\text{exog}}=\frac{1}{2}V\left(S+z_{\alpha }{\sigma }_S\right).$$

The first term is the mechanical cost of crossing a normal quarter-spread; the second prices the realistic possibility that when you are forced to sell, the spread is wider than usual - the cost and the tail are correlated. This is the pillar’s established L-VaR form (see the flat liquidity note for the same formula in ${\sum }_i{P}_i{Q}_i$ form).

2.2 Endogenous liquidation cost (market impact). With linear price impact $\Delta p=\lambda q$ (Foucault eq. 2.8) and depth $D=1/\lambda$ shares to move the price by one unit, liquidating $Q$ shares moves the price by $Q/D$. The volume-weighted execution price sits at the mid of the impact, so the shortfall per share is $\frac{1}{2}Q/D$ and

$$\boxed{{\text{LC}}_{\text{impact}}=\frac{Q^2}{2D}=\frac{\lambda Q^2}{2}}$$

The $Q^2$ is the whole story: doubling the position quadruples the impact cost. Equivalently, in return terms the impact drag is $\frac{\lambda Q}{2}$ per unit of value traded.

Reality check on the exponent. Real impact is closer to square-root (Almgren–Chriss 2000 temporary impact; Bouchaud et al. 2009 “How Markets Slowly Digest Changes in Supply and Demand”: impact $\propto \sigma (Q/V{)}^{1/2}$). The linear model is the conservative-for-reasoning baseline used in risk-adjusted measures because (i) it is closed-form and (ii) at the sizes at which L-VaR matters, the quadratic overstates less than the linear-per-share model understates. Square-root impact is treated in 06 · Advanced Extensions and in the Pillar 6 microstructure refs.

2.3 The liquidation horizon. If the order may not exceed a fraction $\alpha$ of average daily volume,

$$T_{\text{liq}}=\frac{Q}{\text{ADV}\cdot \alpha }(\text{days}).$$

The horizon-matched market VaR is then ${\mathrm{VaR}}_T=z_{\alpha }{\sigma }_{\text{daily}}\sqrt{T_{\text{liq}}}V$ under i.i.d. scaling (Hull §22.4) - and the L-VaR must be evaluated at that horizon, not at 1 day. The FRTB makes this explicit: each risk factor gets a liquidity horizon (10/20/40/60/120 days) and the ES is computed on the scaled shock $\sigma \sqrt{LH/10}$.

2.4 The combined measure.

$${\mathrm{LVaR}}_{\alpha }={\mathrm{VaR}}_{\text{market}}(\sigma ,T_{\text{liq}})+\frac{1}{2}V\left(S+z_{\alpha }{\sigma }_S\right)+\frac{Q^2}{2D}.$$

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3. Computational Implementation - VaR, L-VaR, and the horizon

We decompose a $10M position’s 1-day 99% VaR into its three liquidity components, then show how each scales with the liquidation horizon. Stdlib only.

Interpretation: the impact cost ($25,000, 25 bp) exceeds the exogenous spread cost (15.8 bp) even for this ~$10M order - and total cost grows as $Q^2$, so a 5× larger order carries 5× the per-unit impact drag and ~25× the total impact cost. The horizon row shows the other multiplier: mis-specifying the horizon (1 day instead of 10) understates the market-risk term by $\sqrt{10}\approx 3.16$.

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4. Failure Modes & First-Principles Breakdowns

1. Ignoring the horizon, then compounding it. Two separate errors live here: using a 1-day VaR for a position that takes 10 days to unwind (understates market risk by $\sqrt{10}$), and omitting the liquidation cost entirely. The two do not cancel; they add.
2. Assuming the spread is constant. ${\text{LC}}_{\text{exog}}=\frac{1}{2}V(s+z_{\alpha }{\sigma }_s)$ prices the correlation between “I am forced to sell” and “the spread is wide.” Setting ${\sigma }_s=0$ removes exactly the part that mattered in 2008.
3. Linear impact at explosive sizes. ${\text{LC}}_{\text{impact}}=\lambda Q^2/2$ understates for very large orders only if real impact is super-linear (it is not - it is sub-linear, square-root), so the quadratic model overstates for very large $Q$ and can be conservative; but the linear slope $\lambda$ is itself underestimated in calm periods, so the two errors run in opposite directions. Calibrate $\lambda$ from stress-period data, not calm data.
4. Level, not path, thinking. L-VaR prices the cost of a liquidation but treats the liquidation as one-shot. Patient execution (Almgren–Chriss optimal schedules) trades impact against timing risk; the realised cost can be above or below the static L-VaR depending on the path. See Optimal Execution.
5. Non-coherence sneaks back in. Adding a size-dependent cost $Q^2/2D$ (convex in $Q$) makes L-VaR super-additive in positions - the opposite of the subadditivity failure of plain VaR, and a reminder that “liquidity-adjusted” changes the aggregation properties that risk governance relies on.
6. Currency of the horizon. ADV and $D$ are themselves state-dependent; in a crisis ADV collapses and $D$ shrinks simultaneously, so $T_{\text{liq}}$ lengthens and ${\text{LC}}_{\text{impact}}$ rises - the two horizon terms move together (see 05 · Failure Modes).

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5. References

• Bangia, A., Diebold, F., Schuermann, T. & Stroughair, J. - Modeling Liquidity Risk, With Implications for Traditional Market Risk Measurement and Management (1999/2000)
• Almgren, R. & Chriss, N. - Optimal Execution of Portfolio Transactions, J. Risk 3(2):5–39 (2000)
• Bouchaud, J.-P., Farmer, J.D. & Lillo, F. - How Markets Slowly Digest Changes in Supply and Demand (2009)
• Foucault, Pagano & Röell - Market Liquidity (2013)
• Hull - Risk Management and Financial Institutions, liquidity-risk chapter (liquidity trading risk and L-VaR) and Options, Futures and Other Derivatives
• BCBS - FRTB: Minimum Capital Requirements for Market Risk (2019, d457)

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6. Connected Graph Bridges

• Back: 02 · Market vs Funding Liquidity · Index Hub
• Forward: 04 · Margin & Funding Spirals · 05 · Failure Modes
• Measure base: VaR & Expected Shortfall · Parametric, Historical & Monte Carlo VaR
• Execution bridge: Optimal Execution & Almgren–Chriss · Transaction Costs & Turnover Constraints