The capital efficiency problem that Uniswap v3 was designed to solve
Same pool as before. ETH at $2,000.
10 ETH + 20,000 USDC = $40,000, with k = 200,000.
Now assume ETH spends the next six months in $1,800 and $2,200 — a ±10% band where realistically all trading will take place.
How much of your $40,000 actually gets used?
The calculation
At the top of the range ($2,200):
The pool contains 9.535 ETH and 20,976 USDC, having sold 0.465 ETH along the way on the way up, worth about $930.
At the bottom ($1,800):
The pool contains 10.541 ETH and 18,974 USDC, having spent $1,026 buying ETH on the way down.
Total capital ever changed hands: $1,956.
Out of $40,000.
That's 4.9%.
So what of the other $38,044?
Its sitting there, reserved for prices that never come.
The constant product formula spreads your liquidity across every available price, from zero to infinity. Some of your capital is dedicated to ETH at $50. Some to ETH at $500,000. The curve demands it, because it can never run out of either asset.
A neat mathematics and a wasteful practice.
Widening our assumption make it very much better either. Assume a $1,000 to $4,000 range — a 4x wider span than most periods of six months — and you're really using about 29% of your capital to work.
Roughly 70% remains unused.
What Uniswap v3 did about it
Two ways to use the same finding.
Same depth, less capital. You could provide the same liquidity across the $1,800–$2,200 band with roughly $1,956 freeing up $38,000 for something else.
Same capital, more fees. Deposit the whole $40,000 concentrated into that same band and you're provideing roughly 20x the depth at prevailing prices — capturing about 20x the share of trading fees.
The v3 whitepaper quotes capital efficiency improvements of up to 4,000x for very tight ranges. That's the same maths taken to its extreme: a band of ±0.1% around the current price.
The catch
That 20x multiple is what you get paid for range risk.
Your capital works twenty times harder inside $1,800–$2,200 — and does nothing at all the moment ETH trades at $2,201. Out of range means out of the fee stream entirely, while still holding full price exposure in whichever asset you got converted into.
If you want to fix that you'd close the position, crystallise the loss at that price, and re-open at a new range. Every time.
Which is a different article.
References
- There's free money sitting in every liquidity pool. So why don't you take it?
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Adams, H., Zinsmeister, N., Salem, M., Keefer, R., Robinson, D., "Uniswap v3 Core", March 2021 — uniswap.org/whitepaper-v3.pdf
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Uniswap, "Uniswap v2 Core Whitepaper" — uniswap.org/whitepaper.pdf
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Metrix Finance, "Uniswap v3/v4 Position Backtester" — metrix.finance
