Liquidity Risk Assessment

Bid-ask spreads, market impact, Amihud illiquidity, liquidity-adjusted VaR, and the illiquidity premium.

When to Activate

• User evaluating the liquidity profile of a portfolio or individual positions
• Estimating market impact costs for large trades
• Computing liquidity-adjusted risk measures (LVaR)
• Assessing liquidity risk during stress periods
• Analyzing the illiquidity premium in asset pricing
• Designing execution strategies that minimize market impact

Core Concepts

Dimensions of Liquidity

Tightness (Bid-Ask Spread)

• Quoted spread: ask - bid
• Effective spread: 2 * |trade_price - midpoint| (captures actual execution vs. midpoint)
• Relative spread: spread / midpoint (percentage terms for cross-asset comparison)
• Typical ranges: large-cap equities 1-5bp, IG bonds 5-50bp, HY bonds 50-200bp, emerging market 100-500bp
• Spreads widen dramatically in stress: 2008 IG bond spreads went from 10bp to 200bp+

Depth (Volume)

• Average daily volume (ADV): baseline liquidity measure
• Days to liquidate = position_size / (participation_rate * ADV)
• Participation rate: typically 10-25% of ADV to avoid excessive impact
• Depth at best quote: number of shares/contracts available at the inside bid/ask
• Book depth: cumulative volume available within N ticks of best quote

Resilience

• How quickly prices and spreads recover after a large trade
• High resilience: price returns to pre-trade level quickly (deep, liquid markets)
• Low resilience: price impact persists (thin markets, information-driven flow)
• Measured by the half-life of price impact decay

Immediacy

• Cost of executing immediately vs. patiently
• Urgent execution (market orders) pays wider spreads and higher impact
• Patient execution (limit orders, TWAP/VWAP algorithms) reduces cost but adds timing risk

Liquidity Metrics

Amihud Illiquidity Ratio

• ILLIQ = (1/D) * sum(|r_d| / V_d) over D days
• Higher ILLIQ = more price impact per dollar of volume = less liquid
• Cross-sectionally comparable and widely used in academic research
• Captures the intuition that illiquid stocks move more per unit of trading volume
• Typically computed monthly using daily data

Kyle's Lambda

• From market microstructure theory: delta_p = lambda * order_flow
• Lambda measures permanent price impact per unit of signed order flow
• Estimated from regression of price changes on order imbalance
• Higher lambda = less liquid, more information in trades

Roll's Spread Estimator

• Estimated from serial covariance of returns: spread = 2 * sqrt(-cov(r_t, r_{t-1}))
• Only valid when cov is negative (bid-ask bounce dominates)
• Useful when actual bid-ask data is unavailable (e.g., daily bond data)

Pastor-Stambaugh Liquidity Measure

• Captures return reversals following high-volume days
• Signed volume predicts next-day return reversal in illiquid stocks
• Aggregate liquidity factor: systematic liquidity risk

Methodology

Market Impact Models

Linear (Permanent) Impact

• delta_p = lambda * Q, where Q = quantity traded
• Permanent: price does not revert — information content of trade
• Too simplistic for large trades — impact is concave (square root law)

Square Root Law (Empirical)

• Impact = sigma * k * sqrt(Q / V), where sigma = volatility, V = ADV, k = constant (typically 0.5-1.5)
• Well-documented empirically across equities, futures, FX
• Total impact for executing fraction f of ADV: approximately sigma * sqrt(f)
• Example: stock vol = 2%/day, trading 10% of ADV -> impact approximately 2% * sqrt(0.10) = 0.63%

Almgren-Chriss Optimal Execution

• Tradeoff between market impact (trading fast) and timing risk (trading slow)
• Temporary impact: g(v) = eta * v (proportional to trading rate)
• Permanent impact: h(v) = gamma * v (proportional to cumulative volume)
• Optimal execution trajectory minimizes: E[cost] + lambda * Var[cost]
• Risk-neutral: trade at constant rate (TWAP)
• Risk-averse: front-load trading (trade faster initially)
• Result is an optimal trading schedule: number of shares per time interval

Liquidity-Adjusted VaR (LVaR)

Exogenous Spread Approach

• LVaR = VaR + 0.5 * spread * position_value
• Adds half the bid-ask spread (cost of liquidating) to standard VaR
• Simple but ignores that spreads widen in stress

Endogenous Spread Approach

• Model spread as a function of market conditions (spread widens with vol, position size)
• LVaR = VaR + liquidity_cost(position_size, stressed_spread)
• Stressed spread: use crisis-period spreads or scale current spread by stress factor

Holding Period Adjustment

• Standard VaR assumes 1-day liquidation; illiquid positions need longer
• Adjust holding period based on days to liquidate: VaR_T = VaR_1 * sqrt(T_liquidation)
• T_liquidation = position / (participation_rate * ADV)
• For a position taking 10 days to liquidate: VaR_10 = VaR_1 * sqrt(10) = 3.16 * VaR_1

FRTB Liquidity Horizons

• Regulatory framework assigns liquidity horizons by risk factor category
• 10 days: large-cap equities, G10 FX, sovereign rates
• 20 days: small/mid-cap equities, investment grade credit
• 40 days: high yield credit, EM FX
• 60 days: volatility risk factors, exotic FX
• 120 days: correlation, structured credit

Liquidity Stress Testing

1. Identify positions that become illiquid in stress (credit, EM, small-cap)
2. Apply historical stress bid-ask spreads (2008, 2020) or multiples of current spreads (3-5x)
3. Model volume decline: in crisis, trading volumes can drop 50-80% in credit markets
4. Compute liquidation timeline under stressed volumes
5. Assess margin calls: forced liquidation at worst prices compounds losses
6. Identify liquidity mismatches: illiquid assets funded by short-term borrowing

Examples

Amihud Illiquidity Comparison

Stock A (large-cap): avg |daily return| = 1.2%, avg daily volume = $500M
  ILLIQ_A = 0.012 / 500,000,000 = 2.4e-11

Stock B (small-cap): avg |daily return| = 2.5%, avg daily volume = $5M
  ILLIQ_B = 0.025 / 5,000,000 = 5.0e-9

Stock B is 208x more illiquid than Stock A by Amihud measure.
Expected price impact of $1M trade:
  Stock A: negligible (0.2bp)
  Stock B: approximately 50bp (material)

Liquidation Cost Estimation

Portfolio: $200M in corporate bonds
Average bid-ask spread (normal): 15bp
Stressed bid-ask spread (2020 COVID): 120bp
Average daily volume per bond: $2M
Participation rate: 20%

Days to liquidate (normal): $200M / (20% * $2M * 100 bonds) = 5 days
Days to liquidate (stressed, vol drops 60%): $200M / (20% * $0.8M * 100 bonds) = 12.5 days

Liquidation cost (normal): $200M * 15bp / 2 = $150K
Liquidation cost (stressed): $200M * 120bp / 2 = $1.2M + market impact
Total stressed cost including impact: approximately $2-3M (1-1.5% of portfolio)

LVaR Calculation

Portfolio: $50M, 1-day 99% VaR = $800K
Position in illiquid small-cap: $10M, days to liquidate = 8 days
Bid-ask spread (stressed): 150bp

Standard VaR: $800K
Liquidity adjustment:
  Holding period: VaR * (sqrt(8) - 1) = $800K * 1.83 = $1,464K additional
  Spread cost: $10M * 150bp / 2 = $75K
LVaR = $800K + $1,464K + $75K = $2,339K

LVaR is 2.9x standard VaR — the illiquid position triples the effective risk.

Quality Gate

• Liquidity metrics must be computed for every position, not just portfolio-level
• Amihud or equivalent illiquidity ratio must be tracked over time to detect liquidity deterioration
• Market impact estimates must use the square root law or a calibrated model, not linear assumptions
• LVaR must include both spread cost and holding period adjustment
• Liquidation timeline must be computed at both normal and stressed volume levels
• Positions exceeding 1 day of ADV at 20% participation must be flagged as liquidity risks
• Liquidity stress tests must use crisis-period spreads and volumes, not current conditions
• Portfolio liquidity profile must be reported: percentage of portfolio liquidatable in 1, 5, 10, 20 days
• Illiquid positions funded by short-term leverage must be identified and flagged
• Liquidity metrics must be refreshed at least weekly; daily during periods of market stress