6.8.2 Illiquidity Measures

Basic Prerequisites: Econometrics & Time Series and Spread Decomposition & Roll Model.

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1. Intuition & Practical Objective

Before you can price liquidity risk you have to measure illiquidity - and every choice of measure changes the answer. This page is the measurement toolkit: what each proxy actually captures, how to compute it from the data you realistically have, and what each one gets wrong.

The objective is a working command of four proxies, from the most data-hungry to the most robust:

1. Bid–ask spread - the direct cost of a round trip, but needs quotes/transaction data.
2. Roll estimator - the spread inferred from the negative autocovariance that bid-ask bounce induces in prices; needs only prices.
3. Amihud ILLIQ - price impact per dollar of volume, $|r|/\text{VOLD}$; needs only daily returns and volume, which is why it became the workhorse.
4. Turnover / Amivest - how much volume passes through; a capacity/activeness proxy.

The unifying idea: illiquidity is the response of price to order flow (Kyle’s price-impact concept). Spreads and ILLIQ are two operationalizations of that same response - one in quote space, one in return/volume space.

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2. Mathematical Ground Truth & Derivations

Bid–ask spread. Relative quoted spread is $S=(a-b)/m$ with mid $m=(a+b)/2$. The effective half-spread uses the trade price $p$ against the mid: $c=|p-m|$. Round-trip cost ≈ $2c$.

Roll (1984) estimator. If trades bounce between bid and ask, successive transaction-price changes carry negative autocovariance whose magnitude is the half-spread squared:

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

Amihud (2002) ILLIQ. The average absolute price change per dollar of volume:

$${\text{ILLIQ}}_{iy}=\frac{1}{D_{iy}}\sum _{t=1}^{D_{iy}}\frac{|R_{iyt}|}{{\text{VOLD}}_{iyt}}.$$

It is a Kyle-lambda-like price-impact measure built from daily data - the slope of price response to order flow, averaged over days. More illiquid ⇒ larger ILLIQ.

Amivest / turnover. Liquidity ratio $=\frac{{\sum }_t{\text{VOLD}}_t}{{\sum }_t|R_t|}$ (the inverse of a volume-weighted ILLIQ); turnover $=\text{volume}/\text{shares outstanding}$ measures how much of the float actually trades.

The economic content. ILLIQ dominates empirical work because (Amihud 2002) it is computable for long histories in most markets, where spread data does not exist. Its cost: it mixes the level of price impact with idiosyncratic return volatility (a volatile stock has high ILLIQ even if deeply liquid).

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3. Computational Implementation - the measures in numbers

Experiment 1 - Amihud ILLIQ across market cap. A small cap (big moves, tiny volume) vs a mega cap (small moves, huge volume):

The same absolute price response per unit of volume is ~850× steeper in the small cap - its price is far more sensitive to order flow, which is precisely the “illiquidity” the asset-pricing literature prices.

Experiment 2 - Roll spread from a bid-ask bounce. Build a price path that trades at bid/ask around a slowly-moving mid, then recover the spread from the autocovariance alone:

With fundamental moves small relative to the spread, Roll recovers the true spread (0.035 vs 0.04; 0.399 vs 0.40). The negative autocovariance is the fingerprint of the bounce - see Spread Decomposition & Roll for the full model and its breakdown when autocovariance is positive.

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4. Failure Modes & First-Principles Breakdowns

• ILLIQ confounds volatility with illiquidity. $|R|/\text{VOLD}$ is high for a volatile but deep stock; it is a noisy proxy for true price impact. Scale matters - values differ by orders of magnitude across cap sizes, so compare within, not across, groups.
• Zero-volume days are undefined. ILLIQ drops days with no volume; for infrequently traded names that discards most of the sample and biases the average.
• Roll needs negative autocovariance. When price changes are positively autocorrelated (trends, momentum), $\sqrt{-(\text{positive})}$ is undefined and the estimator clamps to zero - wrongly declaring “infinite liquidity.” See 05 · Failure Modes.
• Spreads miss depth. A tight quoted spread is worthless if you cannot trade size at it; spread measures capture the cost of a small trade, ILLIQ/Kyle capture the price impact of order flow.

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5. References

• Amihud (2002). Illiquidity and stock returns. JFM 5, 31–56
• Roll (1984). A simple implicit measure of the effective bid-ask spread. Journal of Finance 39, 1127–1139
• Amihud, Mendelson & Pedersen (2013). Market Liquidity
• Hasbrouck. Market Microstructure: Foundations

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6. Connected Graph Bridges

• Next: 03 · Liquidity as a Priced Factor
• Back: Index Hub · 01 · From Zero
• Sibling: Spread Decomposition & Roll Model · Market Impact & Depth (Kyle)