Where you are. A full-range pool spreads its capital across every price from zero to infinity. Almost all of that range will never be visited, so almost all of the capital is doing nothing.
Look at the constant-product curve again and notice how much of it is absurd. The pool stands ready to sell bonds at four hundred and to buy them at twenty-five. Neither price will happen. Capital is sitting there anyway, reserved for prices nobody expects.
So: put it all where the trading actually is.
The idea in one paragraph
Concentrated liquidity lets a provider choose a price range and place their whole position inside it, which multiplies the depth they offer within that range by a large factor because the same capital is no longer reserved for prices outside it. The fees follow the depth, so a narrow range earns far more per unit of capital while the price stays inside it. The moment the price leaves, the position is entirely one asset and earns nothing at all, which turns a passive position into one somebody has to watch and move. The design trades capital efficiency for attention, and that is the whole of it.
What the comparison shows
The exercise puts the same million to work both ways and walks a ten-day path that starts inside a range of ninety to a hundred and ten and wanders out of it.
Concentrated into that range, the position offers about twenty times the depth. It earns accordingly, and it earns nothing at all from day six onwards, when the price climbs past the top of the range and keeps going. Even with four idle days at the end it finishes well ahead.
That “even with” is the honest framing. The concentrated position won this path. It won it while spending forty percent of the period earning nothing, which on a different path is the whole story rather than a footnote.
The position somebody has to move
The management burden is the part worth taking seriously, because it changes who can hold the position at all.
A full-range position is genuinely passive: put the money in, and its behaviour is the same in a year. A concentrated position has to be watched, and moving it means withdrawing, converting one asset into the other and redepositing, which pays fees and locks in whatever loss the move accumulated. So the return that gets advertised is a gross figure from which several rounds of that will be subtracted by anybody actually running it.
The practical result is that concentrated positions have concentrated their providers too: the returns favour whoever can automate the watching, which is not the person the passive-yield framing was aimed at.
What it means for the analogy with an FX desk
A pool of tokens that are all meant to be worth about the same thing, several dollar stablecoins say, is a special case worth naming.
Because the assets should trade near par, a provider can concentrate extremely tightly around one, offering enormous depth for a small amount of capital. The result behaves a great deal like a currency desk quoting a very tight market, except that the quoting is a formula and the inventory is a pool. It is the closest thing in this module to module 3’s FX machinery rebuilt from arithmetic, and it works well exactly as long as the assumption in its name holds.
Check yourself
1. What does concentrating liquidity buy, and what does it cost?
It buys depth: the same capital offers many times the depth inside the chosen range, and earns fees in proportion. It costs attention, because a position whose price leaves the range earns nothing until somebody moves it, and it costs a worse payoff shape through sharper impermanent loss.
2. The concentrated position in the exercise wins despite four idle days. Why is that not a general result?
Because it won on this path. The same position on a path that leaves the range on day two and never returns earns almost nothing. The comparison is a bet on where the price will spend its time, and one favourable path does not settle it.
3. Why does concentration make impermanent loss worse rather than better?
Because a narrow position is a larger effective exposure to the same move, so the loss against holding accumulates faster. If the price exits the range, the position has been fully converted into the asset that was falling and remains there. The extra fee income is compensation for that, not a bonus on top of it.
4. Why does a pool of several dollar stablecoins behave like a currency desk?
Because the assets are all meant to be worth about the same, so a provider can concentrate very tightly around par and offer great depth with little capital. The quoting is done by a formula and the inventory by a pool, and the arrangement holds precisely as long as the assets really do stay near each other.
Do this
Deploy the same capital both ways.
python3 code/concentrated.pyThe starter provides the full-range depth and the fee accounting, and leaves you concentrated_liquidity: the depth multiple for a range is the square root of the price divided by the difference between that and the square root of the lower edge. The assertions check that the multiple is several times over, that the path genuinely leaves the range, and that the concentrated position still finishes ahead.
The completed version is in solutions/concentrated.py.
What you can now do. You can compare concentrated and full-range positions on one path, say what the concentration bought and cost, and recognise the stablecoin pool as a currency desk built out of the same arithmetic. The next lesson leaves pools behind for the venues institutions actually use at size.