Amihud Illiquidity Ratio (ILLIQ): Formula, Example, and Limits

The Amihud measure asks how large a security's daily price move tends to be relative to the dollars traded that day. For each eligible trading day, it divides the absolute return by dollar volume, then averages those daily ratios over a chosen period. A larger value means that, in the observed sample, a given amount of turnover coincided with larger absolute price changes. A smaller value means that daily price changes were smaller relative to turnover.

Yakov Amihud introduced this measure as a practical proxy when transaction-level quotes and trades are unavailable or historical microstructure records are limited. His 2002 paper studies the relationship between illiquidity and stock returns using U.S. stock data, and defines the daily ratio from absolute return and dollar volume (Amihud, 2002). That motivation remains useful for historical panels, but the measure compresses each day into two numbers and cannot identify the cause of the price move.

The distinction matters. A large absolute return may reflect a thin market, a major company announcement, a broad market shock, or a combination of these. The ratio is informative about realized price movement relative to turnover under a specified data design. It does not by itself show that the market would have moved by the same amount in response to a particular investor's order.

For security (i) in period (y), a common form of the measure is

\[ \mathrm{ILLIQ}_{i,y}=\frac{1}{D_{i,y}}\sum_{d=1}^{D_{i,y}}\frac{|R_{i,y,d}|}{\mathrm{DVOL}_{i,y,d}}, \]

where \(R_{i,y,d}\) is the daily return, \(\mathrm{DVOL}_{i,y,d}\) is dollar trading volume, and \(D_{i,y}\) is the number of eligible trading days in the period. Here, assume returns are expressed as decimal fractions and dollar volume in U.S. dollars. Each daily ratio then has units of return fraction per dollar, or \(1/\text{USD}\), and the average has the same units.

The return convention must be consistent. A daily simple return of \(0.01\) represents a 1% move; if the same move is entered as \(1\) in percentage points, the calculated ILLIQ is 100 times larger. Log returns and simple returns are close for small moves but differ for large moves. State which one is used, whether prices and returns are adjusted for splits and distributions, and which currency defines dollar volume.

Researchers often multiply ILLIQ by \(10^6\) or another constant to make the printed values easier to read. This changes the displayed scale, not the underlying measure. Always name the multiplier. A scaled statistic of \(0.0023\), for example, is not interpretable without knowing whether the return inputs are decimals or percentages, the volume currency, and the scaling convention.

Consider a wholly hypothetical security with these observations. Returns are decimal fractions, and volume is daily dollar turnover:

The three-day average is

\[ \mathrm{ILLIQ}=\frac{2.5+2.5+2.0}{3}\times10^{-9}=2.333\ldots\times10^{-9}\;\text{per dollar}. \]

If the reporting convention multiplies by \(10^6\), the displayed value is \(0.002333\ldots\). This is only a unit rescaling of the sample statistic. It does not predict that a hypothetical $1 million order will cause a 0.2333% return: the ratio uses whole-day turnover and return, not a controlled order or a causal impact experiment. The arithmetic is illustrative and is not market data.

This example also shows why the measure averages daily ratios rather than dividing total absolute returns by total volume. Each eligible day receives equal weight in the stated formula. A day with low turnover can contribute a high ratio even if its dollar volume is small, so inspect daily observations and not only the period average.

Within a consistently constructed sample, higher ILLIQ means that absolute daily returns were larger relative to dollar volume on average. Analysts use this as a rough ranking or proxy for illiquidity, especially in historical data sets that do not contain reliable quotes. If the universe, return convention, volume definition, period, currency, and filters match, comparisons across securities or time can be useful descriptive evidence.

The measure is scale-sensitive and data-sensitive. Changing returns from decimal fractions to percentage points multiplies the ratio by 100. Changing the currency of volume changes the numeric value unless the conversion is applied consistently. A sample containing different listing venues, trading hours, share classes, or market-cap ranges may also have different price behavior and data quality. A table should document these choices before interpreting the ranking.

Amihud's original U.S. stock sample reported a positive relation between expected illiquidity and expected stock returns over the periods studied, including cross-sectional evidence for NYSE stocks from 1964 to 1997. This is a result from that paper's sample and design, not a guarantee that a high-ILLIQ asset earns a higher return in another market or future period. The implementation shortfall guide covers realized execution costs, a separate question from a historical security-level proxy.

Kyle's market-microstructure framework describes how order flow can move prices when informed traders and liquidity providers interact (Kyle, 1985). Amihud ILLIQ is consistent with the intuition that price movement relative to trading activity can reveal something about market liquidity, but it is not Kyle's structural price-impact coefficient. It does not estimate the effect of a signed order of a chosen size, and daily returns also contain price changes unrelated to a particular order.

A bid–ask spread measures a different object: the gap between quoted buy and sell prices, or a realized trade's distance from a midpoint. Amihud and Mendelson's asset-pricing analysis treats the spread as a transaction-cost dimension (Amihud and Mendelson, 1986); ILLIQ instead uses daily absolute returns and dollar volume. The spread can be observed from quotes or estimated with a model such as the Roll estimator. A Roll spread estimator guide explains why a spread proxy derived from transaction-price covariance is not interchangeable with ILLIQ.

Displayed depth measures the quantity available at quoted prices; order-flow imbalance summarizes changes in best-quote supply and demand. Neither is directly contained in a daily return-to-volume ratio. The order-flow imbalance guide covers top-of-book changes, while trade-direction classification explains why identifying buyer- versus seller-initiated trades requires its own rule. These measures can complement ILLIQ when data permit, but they have different definitions and failure modes.

Liquidity risk is also distinct from the level of illiquidity. Pástor and Stambaugh study stock-return sensitivity to aggregate liquidity fluctuations and construct a liquidity risk factor (Pástor and Stambaugh, 2003). ILLIQ alone measures neither that sensitivity nor an asset's exposure to a market-wide liquidity shock.

For an equity, daily dollar volume is commonly calculated from the value of shares traded during the session, using a documented price and share-volume convention. Vendor fields may represent consolidated volume, one exchange, regular hours, or extended trading; they should not be silently mixed. If the source reports turnover directly, verify its currency and whether it covers the same session used to compute the daily return.

The return interval and volume interval must align. A close-to-close return paired with regular-session volume includes overnight price changes in the numerator while the denominator excludes overnight trading. This may be an intentional daily design, but it is not a like-for-like intraday impact estimate. Note market holidays, halts, partial sessions, stale closes, and whether returns span a nontrading day.

Corporate actions also matter. Splits can create artificial price changes if price histories are not adjusted; distributions affect whether total or price returns are used. For futures, contract rolls and multiplier conventions require explicit treatment; for crypto markets, venue, quote currency, stablecoin conversion, and 24-hour day boundaries matter. Use comparable instrument definitions or keep separate universes.

Zero-volume or missing observations create a division-by-zero or missing-ratio problem. Do not insert an arbitrary denominator or silently turn a missing ratio into zero. Define what makes a day eligible, report the number excluded, and evaluate how exclusions affect low-activity securities. A day with a tiny positive volume can generate an extreme ratio, so quality checks should distinguish a genuine illiquid session from bad or incomplete data.

The numerator uses an absolute return, so a volatile security can have a high ILLIQ even if the same order size would encounter a similar quoted spread or book depth as before. A major earnings release may produce a large price move alongside heavy volume; the ratio does not separate news-driven repricing from an order's price impact. The measure is therefore not a pure liquidity statistic.

Daily aggregation hides intraday timing. A thin opening auction, a price jump after news, and a quiet close can all be compressed into one absolute return and one volume total. The ratio does not show how much price impact reversed, which side initiated trades, or how much liquidity was available at a particular quote. Tick sizes, price limits, stale prices, and non-synchronous trading can further distort comparisons across assets.

Extreme ratios can dominate a mean, particularly when daily volume is very small. Researchers sometimes trim or winsorize observations, but that changes the estimand and can erase genuine stress episodes. Report the raw distribution, the outlier rule, and sensitivity to plausible alternatives. A separate volatility measure can help characterize price variation, but adjusting for volatility still does not turn ILLIQ into a direct execution-cost estimate.

The formula above gives each eligible trading day equal weight after calculating its own ratio. Dividing the sum of absolute returns by the sum of volume would instead weight high-volume days more heavily and is a different statistic. Likewise, averaging security-level ratios equally differs from weighting them by market capitalization when producing a market-level series. Name the exact aggregation rule.

Period length matters. An annual average can smooth short-lived stress but hide liquidity changes; short windows can be noisy and sensitive to one low-volume day. Compare pre-specified windows and subperiods, inspect the daily series, and avoid choosing the period because it gives a preferred ranking. If the research concerns expected illiquidity, use a clear rule for mapping past observations to a future holding period without looking ahead.

Useful checks include alternative return definitions, a common currency, a consistent session, thresholds for data quality, and sensitivity to extreme observations. For a cross-sectional analysis, document survivorship, delisted securities, venue coverage, and changes in the eligible universe. If the measure enters a return test, separate the estimation sample from the evaluation period and account for the possibility that many specifications were tried.

A reproducible report should give the formula, return convention, volume currency, scaling factor, sample frequency, date range, eligible-day rule, aggregation method, data vendor or venue coverage, and treatment of outliers and corporate actions. Include the number of valid observations and a distribution or sensitivity range when a single mean would hide large variation. Explain whether the statistic describes one security, a portfolio, or a market-wide aggregate.

Use language such as “historical absolute price movement relative to dollar turnover” or “low-frequency illiquidity proxy.” Avoid labeling the result a quoted spread, a causal impact per dollar, or the cost of a particular order. For an order decision, use current quotes, depth, order size, execution data, fees, and venue-specific conditions. For a return claim, test it out of sample with realistic costs and a design that addresses data snooping.

ILLIQ is useful when its limited question matches the data: how did daily price moves compare with daily dollar turnover over this sample? Its value comes from transparent, comparable construction. It cannot reveal liquidity on its own, identify why prices moved, guarantee a liquidity premium, or establish a profitable trading rule.