Amihud Illiquidity

Most good liquidity measures need tick-by-tick data — every quote, every trade. Amihud's illiquidity measure is beloved because it needs almost nothing: just the daily return and the daily dollar volume, numbers you can pull for any stock going back decades. Yet it captures the thing that matters most about liquidity — how much the price moves when people trade — and it lines up remarkably well with the sophisticated impact models built from high-frequency data.

The intuition is one sentence. In a liquid stock, a huge amount of dollar volume barely nudges the price, so the ratio of "price move" to "dollars traded" is tiny. In an illiquid stock, a modest amount of trading whips the price around, so that ratio is large. Amihud just averages that ratio over time.

The formula

For stock $i$ over $D$ trading days, the Amihud illiquidity measure is

$$\text{ILLIQ}_i = \frac{1}{D}\sum_{d=1}^{D} \frac{|R_{i,d}|}{\text{DVOL}_{i,d}} .$$

Here $R_{i,d}$ is the stock's return on day $d$ (so $|R_{i,d}|$ is how far it moved, up or down), and $\text{DVOL}_{i,d}$ is the dollar volume traded that day (price times shares). The ratio $|R|/\text{DVOL}$ is the price move per dollar of trading on a single day; averaging over $D$ days smooths out the noise. A big ILLIQ means "expensive to move" — illiquid. A small ILLIQ means "hard to budge" — liquid.

Worked example

Take a small-cap stock over one week. The raw numbers:

Day	Return $R_d$	$|R_d|$	Dollar volume (\$m)	$|R_d|/\text{DVOL}$ (per \$m)
Mon	+1.2%	0.012	3.0	0.00400
Tue	−0.8%	0.008	2.0	0.00400
Wed	+2.0%	0.020	2.5	0.00800
Thu	−0.5%	0.005	5.0	0.00100
Fri	+1.5%	0.015	3.0	0.00500

Average the last column:

$$\text{ILLIQ} = \frac{0.004 + 0.004 + 0.008 + 0.001 + 0.005}{5} = \frac{0.022}{5} = 0.0044 .$$

So on a typical day this stock moves about 0.44% for every $1 million traded. Now compare a large-cap that trades $500 million a day and moves the same 1% — its ratio is $0.01/500 = 0.00002$ per $m, more than two hundred times more liquid. That enormous spread is why Amihud illiquidity is one of the sharpest ways to sort a universe from most to least tradeable, and why it feeds directly into a The Liquidity Factor.

Why it earns a risk premium

Amihud's original point was not just measurement — it was pricing. Illiquid stocks are expensive and risky to get out of, especially in a crisis when everyone heads for the exit at once. Investors demand compensation for holding them, so high-ILLIQ stocks earn higher average returns over time. This "illiquidity premium" is a cousin of the size and value premia, and it partly explains why small, thinly traded names historically outperformed: some of that return is just payment for bearing liquidity risk, not free alpha. When you build a The Liquidity Factor, ILLIQ is often the raw ingredient.

Where it fits

Amihud illiquidity is the low-frequency workhorse of the liquidity toolkit. It approximates the price impact that Market Impact models and the The Kyle Model describe from first principles, it complements the Roll spread (which measures the bid-ask cost rather than the impact cost), and it is a core input to liquidity-based factors and to sizing rules that cap how much of a name you can hold. When someone asks "how liquid is this stock and roughly what will it cost me to move it?", ILLIQ is the number to reach for first.