Summary
There is nearly $15 billion worth of tokenized U.S. Treasury funds on the Ethereum blockchain, and if you wanted to meet every holder of two of the largest projects, Circle's USYC and BlackRock's BUIDL you could do it in a medium-sized conference room. There are fewer than a hundred of them. This is sort of the dream of tokenization, and sort of its opposite: real assets, real size, onchain, but doing more or less nothing.
Tokenized stocks had the reverse problem. Lots of holders, not much size, and little trading. Then, over the past month, Robinhood launched a layer-2 (L2) chain, and the situation changed rather abruptly.
Decentralized exchange volume on the chain exploded to nearly $5 billion a day, vaulting it to No. 3 in DEX volume behind Solana and Ethereum and making it, by a wide margin, the top chain for real-world asset (RWA) DEX volume. Liquidity providers (LPs) in the automated market makers (AMMs) behind those swaps earned APYs in the hundreds to thousands of percent over the course of several weeks.
The timing couldn't be better. In mid-September, the SEC released proposed language for its innovation exemption, opening a path for fully tokenized stocks (not just the "mostly" tokenized versions on Robinhood Chain) to trade on AMMs, legally (at least for the next two years).
With the regulators nodding and smiling, and real demand finally showing up, one question matters: are AMMs actually a good place to trade stocks?
The launch-month frenzy of Robinhood’s L2 has settled, but savvy traders can still earn APYs in the hundreds of percent depositing tokenized stocks into AMM pools. Are those equilibrium rates, or a flash in the pan that will fade as more tokenized stocks arrive to compete for the same fees? And if stocks do belong in AMMs, how should they go in? Every stock in your portfolio individually, only part of your portfolio, or tokenize your portfolio as a whole and deposit that into an AMM instead?
This paper works through the math of those various options: what drives returns as pools scale to many assets, and what does and doesn't make sense to do with tokenized assets once they're onchain.
We find that tokenizing every asset in an index or portfolio and running them inside an AMM is likely a bad idea. Using the S&P 500 as a test case, a 500-asset S&P AMM pool would have surrendered roughly 3% relative to holding year to date and required approximately 45x annual turnover at a five basis-point swap fee merely to break even, more than 3x the SPY turnover in TradFi (16x).
The conclusion is not that tokenized assets do not belong in DeFi, but that most stocks do not belong indiscriminately in passive AMMs. Broad exposure is better expressed through an index token or a tokenized portfolio deposited into an AMM; single stocks are generally better lent or held; and AMMs should be reserved for structurally linked, low-dispersion assets.
The most important factor in choosing an AMM pool's asset mix is dispersion: how far the assets inside the pool move apart. Skew, kurtosis, and other features of the return distribution refine the estimate of LP returns, particularly when a few extreme winners or losers dominate the pool.
Key takeaways
The S&P 500 makes a terrible liquidity pool. An AMM holding all 500 constituents lost roughly 3% relative to simply holding them over the past year, and it would have needed about 45x annual turnover at a five-basis-point fee just to break even.
Dispersion is the main determinant of impermanent loss. How far a pool's assets drift apart explains ~90% of impermanent loss (IL) in many-asset pools, and skew and kurtosis account for the majority of the remainder.
Pool the index, not the stocks. Depositing a single index token instead of its constituents can cut breakeven turnover by over 60%. Beyond that, AMMs are best reserved for assets with a durable reason to stay close, such as dual share classes, mature-industry pairs, and perhaps short-dated Treasuries.
Introduction: The Three-Semi Problem
Somewhere on Robinhood's new blockchain there is a Curve liquidity pool that holds three semiconductor stocks and no dollars.
Just NVDA, AMD, and SNDK, pooled against one another, quoting prices to anyone who wants to swap one chipmaker for another at four in the morning on a Sunday. No fiat leg. Someone looked at three of the twitchiest stocks in the market and decided they should trade against each other, automatically, forever.
It was me. I was the first depositor in those pools. This is either the future of finance or an elaborate way to donate money to arbitrageurs, and I decided to find the answer with some arithmetic.
The question that gets you there is not the one everyone in digital assets spent the last few years shouting about. Whether a stock can be a token, that is settled. It can. It's an ERC-20 standard token with 18 decimals of precision and (in this case as a non-issuer sponsored token) no voting rights. The question now is what you do with the thing once it's in your wallet. Where should it be deployed onchain? What should it be paired with? What is the opportunity cost of tokenized stocks onchain and what are the trade-offs?
The whole promise of decentralized finance (DeFi) was an open system for the people: all the plumbing the banks provided for a fee, suddenly yours for the price of gas (the onchain kind, not at the pump). Great. So, what are you supposed to do with it?
Broadly, there are two primary ways to use tokenized stocks in DeFi right now. The first choice is to lend it out. Someone wants to short NVDA, or borrow against it, or needs the token to settle a trade, and they'll pay to borrow yours. This is not exotic; it is the oldest business your broker never told you about. When you hold shares in a brokerage account, your broker is quietly lending them out and pocketing the spread. You're the one holding the asset. They're the one getting paid to lend it (most of the time).
Onchain, that spread is yours. You deposit the token, someone borrows it, you collect interest. The catch is that "someone" is a smart contract and a pile of collateral, rather than a brokerage with a balance sheet, regulatory margin requirements, and a clearinghouse behind it, but we'll get into the details of what that trade actually costs later.
The lending option is well understood and mostly boring. One can reason about the risks rather easily.
We are primarily focused here on the second option for tokenized securities in DeFi, and that is deployed in DeFi-native automated market makers (AMMs) such as Curve, Uniswap, and YieldSpace (this last one will make sense later).
When depositing assets into an AMM, you don't lend the token to a trader. You are the trader, standing there quoting a price to anyone who wants to buy or sell, collecting a fee on every swap. Except you're not standing there, because you're asleep, because it's four in the morning on a Sunday. You've handed the job to the AMM: a little formula that holds your assets, quotes a price off them, and takes the other side of every trade that comes in, forever, without asking you anything.
This is where it gets interesting, because becoming the market sounds like the good kind of passive income but is actually a financial derivative of sorts, with surprising variables that affect the outcomes.
That pool of three chipmakers I seeded? That's the second option. Let's figure out what I signed up for.
Start with what an AMM actually is, because "provide liquidity" makes it sound like you're doing everyone a favor and collecting rent for it. You're not just collecting rent. You're taking a position that is materially different than just holding the asset. Here are the basic mechanics.
You put in two assets, say NVDA and AMD (don’t worry, the third is coming later), and the pool quotes a price between them off a fixed formula. Someone wants to swap NVDA for AMD. They trade against your pool, the formula moves the price a little to account for the trade size relative to the pool’s liquidity, and you collect a fee for providing the liquidity facilitating the swap. Do this a few million times and the fees add up.
That's the pitch, and it works. Over $330 million has been accrued by Curve AMM depositors to date, and nearly $5 billion by Uniswap AMM depositors.
The problem is what the formula does to your assets while it's collecting those fees. It has one rule, and the rule is naïve on purpose: keep the pool balanced. So when NVDA rallies and AMD doesn't, the pool sells your NVDA and buys AMD, because the rule says “rebalance,” and it keeps doing it the whole way up. You wanted to be long the winner. The formula spent the entire rally selling it to buy the underperforming asset. By the NVDA top, you're holding less of what went up and more of what didn't, which is precisely the opposite of what you'd have done if you'd just sat there.
This is the trade-off you make in exchange for fees, and the payoff structure has a name relative to the underlying assets: negative gamma. You're short convexity. It doesn't matter which direction things move. Every time the two assets pull apart, the pool trims the winner to buy the loser, and you come out behind simply holding them. The further they diverge, the more the winner is sold, and the worse it hurts compared to just holding.
You are, structurally, selling strength and buying weakness, on autopilot, collecting swap fees along the way.
A real market maker would never do this, because a real market maker gets to (is forced to) think more critically: quote tight when it likes the trade, wide when it doesn’t, hedge somewhere else, walk away from a name that's running, manage inventory. The AMM has none of those responsibilities (or capabilities). It quotes the same curve for every trade in every market condition. No discretion, no memory.
Now, how much does this naïve quoting approach cost you? It depends on how much the prices of the tokens in the pool diverge from each other, but it goes by the cheerfully optimistic name “Impermanent loss.” You compare the value of a pool to just having held the two tokens, at two points in time. It's called impermanent because if the prices wander apart and then wander back to where they started, the gap closes and the “loss” evaporates. On paper, it was never realized. Fun marketing by early DeFi proponents; the loss is “impermanent” because you haven’t exited the pool, and the price could always change (however unlikely).
Which explains why every big AMM you've heard of is stuffed with stablecoins. Pair USDC with USDT, two things that are supposed to be worth a dollar and usually are, and they never really diverge. Nothing to rebalance, no price divergence, nothing for the arbitrageur to skim, and the fees add to LPs’ bottom line like they were supposed to. It's a beautiful business, great return profile for assets that revert to the mean in price ratio, as stablecoins should. No impermanent loss, just fees, right up until you swap one of those dollars for something that can actually move apart in price on a long-term basis. Which is, of course, exactly what happens when both legs are stocks, or a stablecoin issuer picks the wrong place to custody their dollars..
With that as the backdrop, let’s get to the arithmetic. And let’s not limit it to my three chipmakers. We’ll come back to those, and whether I should be embarrassed, but let’s go bigger. Much bigger. Let’s do it on the biggest, most boring, most diversified basket anyone would actually want to tokenize: the S&P 500. We’re not looking at the existing SPY-USD pools in DeFi. We’re going to do the more interesting thought experiment: put all the 500 S&P constituents into one AMM pool and look deeply into the economics.
Take the S&P, drop all 500 names into one giant AMM pool, and back-cast its performance over different time frames. The AMM does what an AMM does, selling the winners into strength and buying the losers, and we compare that pool with simply holding the same stocks. That gap between the value of the AMM pool after rebalancing and holding the portfolio static is the impermanent loss. That impermanent loss is what you get in exchange for swap fees, instead of maintaining a static (periodically rebalanced, if we are being precise) portfolio.
One important note: This paper models IL using Balancer-style constant-weight AMM, which is slightly different from a Uniswap v2 constant product (x*y=k) AMM. The main difference is that constant-product AMMs take assets in equal proportion, while constant-weight AMMs take assets in uneven proportions. This point matters here because the S&P500 is a market cap-weighted index, where a stock’s weighting is proportional to its market cap relative to the other members, rather than an equal weight index, where each stock gets equal weight in the index construction.
An AMM LP collects fees to offset impermanent loss. So the real question isn't "how much does the pool lose?" It's "how much volume does it have to trade to break even?" Every swap costs a fee; do enough swaps, and the fees cover the drag from the negative gamma. So, we solve for turnover: given a fee level, how many times over does the pool have to fully change hands each year just for the LP to claw back to flat versus having held the basket? Put a ceiling on the fee while we're at it: five basis points, the cost of redeeming these tokens through Robinhood at the time of writing. That’s as high as a fee can go before DeFi traders just skip swapping with the pool and mint/redeem the token from the tokenizer instead (unless there is a hurdle to tokenization beyond cost, which there are for most of the current tokenized stocks, creating or redeeming them is restricted to whitelisted parties).
That breakeven turnover is one of the most important variables for selecting assets for an AMM pool. It's a single, honest number that says: here's how hard this pool has to work to justify its existence. Hold a diversified basket of American stocks in an AMM for a year, and the pool has to churn its entire value, and then some, just to break even against having done nothing. But how much “and then some,” and how can we forecast what turnover we would need for any particular combination of assets? That's where the math gets interesting, and where the details start to tell us a story of where tokenized securities will sit in DeFi in the future.
The S&P 500 Portfolio: Dispersion Is King
The year we're looking at was a good year for the S&P, up 11.66%, comfortably in the 60th percentile of year-to-date returns since its inception, the kind of year that makes everyone feel good for being allocated. If there were ever a stretch where being long a basket of American stocks was easy money, this was it.
And yet, if you had held all the constituents of the S&P in an AMM pool instead, you would have lost ~3% of your upside compared to just holding the portfolio, more than a quarter of the year’s returns. To compensate for that loss, the pool holding that same basket would have had to turn over ~45x its entire value, at a full five basis points a swap, just to break even against having done nothing at all. Not to profit, to break even. The S&P turns over about 16x per year in TradFi, and while transaction velocity is higher onchain than in traditional markets, that is quite the gap to overcome.
What produced the 3% downside gap? Not the return of the S&P itself. The pool does not particularly care whether the index finished up 20%, down 20%, or flat. It cares how differently the stocks inside it got there. That spread among the constituent returns is dispersion, and for a multi-asset AMM it is the main variable behind impermanent loss.
The reason comes down to arithmetic versus geometric returns. Comparing holding a portfolio of assets with holding those same assets in an AMM is effectively comparing their arithmetic return with their geometric return. The gap between the two grows with dispersion. In other words, it isn't the average return of the assets that drives the impermanent loss, but the variance around that average.
You can see the intuition in the simplest case. If every stock in the pool returns the same 20%, the arithmetic and geometric returns are basically the same, and there is very little for the AMM to rebalance. But if half the stocks are up 40% while the other half are flat, the geometric result is lower. The AMM realizes that difference between geometric and arithmetic returns as a cost because it is continually selling the stocks that are outperforming and buying the ones that are lagging. The AMM pool’s value is the weighted geometric mean of constituent returns; buy-and-hold yields their weighted arithmetic mean.
In more technical terms, the arithmetic-geometric gap is approximately proportional to the cross-sectional variance of returns.
So dispersion is the name of the game for LP-ing, but “how far did they spread” is a single number, and a single number is a blunt instrument. But before getting into the finer points of the distribution, there are three obvious reasons you might distrust that conclusion. Maybe this was just a particularly weird year. Maybe our replication of the index is doing something artificial. Or maybe negative gamma of the AMM position is beneficial in down markets.
Objection 1: A banner year skews the math?
Yeah, but this was a weird year. Fair guess, maybe the analysis caught the pool at some freak starting point and lost all the value immediately because Situational Awareness thought it had more skill and less hubris running back the Three Arrows Capital playbook but better, without getting liquidated.
So we ran it back across the prior years too, and while this year is particularly bad from the perspective of the AMM’s LPs, the prior years weren’t good either. The negative gamma drag isn't a one-off. It shows up in good years for the stock market and bad years, big years and quiet years. Over one-year, two-year, three-year, and five-year windows, we see material amounts of impermanent loss, ranging from roughly 6% to 20%.
The important point is that these results are not being driven by the direction of the index. They are being driven by what happens to the constituents relative to one another. The dispersion of returns within an AMM pool is the primary driver of impermanent loss when pooling more than two assets.
Objection 2: But everyone rebalances the index
Another reasonable criticism is, is there an issue with replication of the index and its quarterly rebalancing and changing membership?
We do not need to reproduce every committee date, float adjustment, or constituent swap to see what a reset can and cannot do (but we did it anyway). A reset will adjust the weights in the AMM that have drifted during the rebalancing; it does not reverse the winner-selling already done, and between resets arbitrage still pushes the pool around, deviating from the index.
In the five-year test, IL is 18.4% with frozen weights, 18.9% with annual resets, 18.1% with monthly resets, and 20.2% with quarterly resets. Only fixed portfolio composition avoids the IL. Mechanically, AMMs cannot support a fixed portfolio composition and enable trading; it is one or the other.
We also ran the pool with actual quarterly reconstitution, resetting its membership and target weights to match the index. The reset mainly brings the pool back into line with the index. Whether it modestly raises or lowers subsequent IL depends on timing; there is no consistent structural reduction. Look at the drag by vintage, measured as days since the last reset, and you see the same shape accrue inside every quarter, over and over. Rebalancing the portfolio just relocates the weights; it does not negate the negative gamma.
Objection 3: Maybe AMMs are better in a down market
Another common misconception is that the negative gamma can be helpful during downturns, so AMMs might be a reasonable place to park your assets in such times. The general idea is that selling the winners and buying the losers helps one average into the underperformers during a downturn and rebound more.
While it is a naïve simplification of the mechanics, it has a sprinkle of truth, but not for the reason one would expect. In a market panic it is a common refrain that correlations go to one on the way down, but that is just a tongue-in-cheek way of saying everything is being sold. It doesn’t mean they are all being sold in the same proportion.
Correlation is reductive. it hides the details of dispersion of returns, and importantly the relative dispersion. Down markets can have more correlated moves but still enough relative dispersion to hurt the LP. The S&P constituents on average experience 3.17% IL in down windows versus 2.82% in up windows, with dispersion rising from 23.3% to 25.6%. IL is reduced if dispersion is reduced, irrespective of the portfolio return.
So yes, the market becomes more correlated on the way down. But that does not mean the stocks are moving together closely enough to materially reduce IL. Stocks can fall together and still fall by very different amounts. And for an AMM, those relative differences are the part that matters.
Three reasonable objections, one answer
Three swings, three misses, and each one brings us to the same conclusion: the driver of impermanent loss isn't the timing, the rebalancing method, or the market conditions. It’s the dispersion of the assets inside the pool.
Across the whole broad-index analysis, the dispersion term accounts for an average 102.6% of IL. Skew offsets 5.6%, kurtosis adds 3.9%, and the residuals a remaining 0.9%. In 2026, dispersion’s share is lower at 91.9%, with skew, kurtosis, and the residual contributing 3.6%, 3.2%, and 1.3%, respectively.
That is the useful conclusion from the S&P exercise. The index can have a good return, the portfolio can be rebalanced correctly, and the overall market can be going up or down; none of those facts tells you very much about whether an AMM pool is an appropriate place to park your assets. What matters first and foremost is how far the assets inside the pool spread apart.
But “how far apart did they spread?” turns out to be the first sentence of the story, not the last. Dispersion is a summary statistic, and the same dispersion number can hide very different return distributions. That is where the deeper analysis starts.
Dispersion Is the Main Variable, but Not the Whole Equation
Dispersion is the first-order driver. Dispersion is the width of the return distribution, and distributions have more going on than width. Two pools can post an identical dispersion and live completely different lives. As the saying goes, there are lies, damned lies, and statistics. So this section is really about the ways the dispersion number can be misleading, and what else you need to look at once you've held it fixed. How skewed is the return profile? What is its peakdness, or kurtosis? Is it broadly normal, or does it more closely match another distribution?
The most appropriate way to measure those characteristics of the return distribution are its skew and kurtosis. Across ordinary regimes, once you've fixed the dispersion, the shape of the distribution around it is a higher-order adjustment. The higher-order terms become more important as dispersion gets large. In very dispersed years, the tails and asymmetry start to matter more and more. When you get to dramatically dispersed indexes like this year’s NASDAQ, the higher-order variables start to take over the remaining IL drivers, and the image gets more muddied. By decomposing the impermanent loss equation into these additional components, we can see where impermanent loss really starts to get away from the more general conceptions.
Distribution deep dive
Dispersion, skew, and kurtosis are not separate costs bolted onto impermanent loss. They are different features of the same return distribution, all entering through the arithmetic-versus-geometric return gap. Rewriting that gap in log returns lets us separate the main dispersion term from the corrections created by skew, kurtosis, and everything beyond them.
Written in the language of mathematics, we can broadly decompose impermanent loss into the following equation:
The math is simpler than the notation makes it look. Holding the basket gives you the weighted arithmetic average of the constituent returns. The AMM gives you their weighted geometric average because it continually rebalances among them. The direction of the portfolio does not matter by itself. Add the same 10% return to every constituent and the AMM and the holding portfolio gain equally. Nothing changes relative to one another. IL appears when one stock gains 40%, another gains 10%, and another loses 20%, not because the basket went up or down, but because its constituents went different distances.
Dispersion measures those differences and gives us the first estimate of the cost. The rest of the impermanent loss approximation can be written in common English as:
Dispersion is squared and comes first, so it usually does most of the work. Skew is multiplied by dispersion cubed, and kurtosis by dispersion to the fourth power. Those terms are small when the pool is tightly dispersed but become increasingly important as the assets spread farther apart.
Skew tells us which side contains the unusual returns. Positive skew means a few large winners. That is especially expensive for an AMM because holding keeps the winner while the pool sells it throughout the run. Negative skew means most names remain together while a few suffer large losses. Given the same dispersion, negative skew can produce less relative IL than positive skew, but that only reduces the relative losses. It doesn’t eliminate them. Skew can theoretically offset dispersion, but the distribution has to be so wildly non-normal that kurtosis and the residual variables are also too large to ignore.
While a negatively skewed portfolio can help reduce IL relative to a positively skewed portfolio, it is important to note that the geometric return is strictly less than the arithmetic return of a portfolio. You will always have impermanent loss with an AMM. Multiple assets just add more dimensions to consider when forecasting.
Kurtosis tells us how much of the dispersion is concentrated in rare extremes rather than spread across ordinary observations. It does not tell us which tail those extremes occupy, so high kurtosis is not automatically good or bad. A rare runaway winner, a pair of symmetric tail events, and one severe loser can all have high kurtosis while producing different IL.
Skewness and kurtosis are intrinsically linked. Pearson’s inequality requires ordinary kurtosis to be at least one plus squared skewness; expressed using excess kurtosis, the condition is:
This gives us a range of potential skew-and-kurtosis combinations.
By holding skew or kurtosis constant and calculating the sensitivity of impermanent loss to the other variable, we can get an idea of how skew and kurtosis interact to affect impermanent loss. The general rule of thumb is to keep kurtosis low and skew and negative as possible to reduce your (relative) impermanent loss.
Dispersion therefore gives you the headline number. Skew tells you which side of the distribution is doing the damage. Kurtosis tells you how concentrated that damage is.
Example distributions
We standardized six synthetic return distributions to the same 20% log-return dispersion to illustrate the impact of the higher order variables on impermanent loss:
A normal distribution produced 1.98% IL, almost all of which came from the 1.98 percentage-point dispersion term, with skew, kurtosis, and residuals contributing less than a basis point combined to impermanent loss.
A right-skewed distribution—many ordinary names and a few large winners—produced 2.28% IL. Relative to the common dispersion term, positive skew added about 25.8 basis points of loss, kurtosis added another 3.8, and the remaining higher orders added less than one.
A left-skewed sample reversed the shape: most names finished together while a few suffered large losses. It produced only 1.75% of IL. Negative skew removed roughly 26.8 basis points from the dispersion estimate, while kurtosis added about four back. That does not make downside skew a desirable investment strategy. It merely helps to reduce the AMM LP’s losses compared to holding, not eliminate them.
A fat-tailed distribution produced 2.30% IL: the common dispersion term contributed 1.98 percentage points, skew added less than a basis point, kurtosis added about 19.8 basis points, and fifth-and-higher effects added another 11.7.
The tight-center, rare-outlier distribution was worse at 2.80% IL. In that particular sample, positive skew added 14.2 basis points, kurtosis added 56.1, and higher orders added another 11.7.
The balanced bimodal distribution was much less dramatic. Two tight return clusters produced 1.96% IL, almost identical to the normal distribution at the same dispersion. Skew contributed less than a basis point, kurtosis added a basis point, and the residual added less than a basis point as well. A basket split between two blocks can look very different on a histogram without creating much additional IL if the split is reasonably symmetric.
The practical rule is therefore not “avoid kurtosis” or “seek negative skew.” Start with dispersion, because it still explains most of the cost of LPing. Then inspect the details of the distribution and how that affected it. A few runaway winners are especially toxic to an AMM LP and can make the headline dispersion number less reliable. A concentrated downside tail may reduce relative IL at the same dispersion, but only by reducing the level of loss compared to holding, not eliminating it entirely.
That gives us a way to evaluate a pool after somebody has chosen the assets. The more useful question is how to choose them so that these distribution problems are less likely to appear in the first place.
How to Build a Portfolio Around This
So dispersion is the enemy and tight co-movement is your friend when you are providing liquidity to an AMM. What do you actually do with that information? As the industry tokenizes more assets, does anything actually make sense to tokenize en masse for usage in AMMs?
Put to one side conventional investing’s most reliable intuitions: more names mean more diversification, and more diversification means less risk. In traditional portfolio theory, adding names reduces idiosyncratic risk to the extent that their firm-specific shocks are imperfectly correlated (and reflected in price). Any one stock’s over- or underperformance has less influence on the return of the portfolio as a whole.
An AMM cares about a different quantity. It continually trades those relative moves, selling the names that outrun the pool and buying those that fall behind. The idiosyncratic variation that diversification makes less detrimental to the portfolio’s aggregate return becomes a driver of impermanent loss for the LP.
That does not mean every large pool is bad or every small pool is good. It means the number of names is the wrong variable. A tightly correlated 500-name basket could be cheaper to LP than a badly matched pair. In practice, however, adding more names to a portfolio creates more opportunities for constituents to separate, depending on your view of the random (or not-so-random) walk.
You can see it in the actual baskets. Over the past twelve months, the S&P posted ~31% dispersion and (5.24%) IL. The Magnificent Seven as an AMM pool has ~17.2% dispersion and (1.45%) IL. Fewer names, tighter co-movement, a quarter of the gamma drag. A tight pool of stocks that move together beats a broad pool of things that don't, and it isn't close. Even better is one to three names that move in near-lockstep but deviate just enough to draw swap volume.
For broad-market exposure, the cleaner answer is not a constituent pool at all. If a lot of names free to wander apart is the problem, stop having a lot of names. Tokenize the index and pool that claim against dollars. This does not eliminate negative gamma: the index can still move against the dollar, and the AMM will still sell it on the way up and buy it on the way down. What disappears is the dispersion among the 500 constituent returns. Instead of continually rebalancing across 500 random walks, the pool manages one: the index against the dollar. No complicated portfolio dispersion, kurtosis, or skew considerations.
With a 500-stock constituent pool, the amount of turnover required to compensate for the dispersion of all the index constituents is high enough to be an issue for the economics. Collapse the basket into an index token, and the complicated multi-asset multi-variable problem becomes a straightforward question: can swap fees compensate the LP for the negative gamma on one relative price?
LPing the index seems to be the most straight forward route, but if we were to get more concentrated exposure, what other factors should we consider when determining which assets to pair together?
The screening criterion is not how many stocks are in the pool, and it is not which stocks you expect to outperform. It is how tightly their relative prices move and whether anyone wants to trade between them. You want tight co-movement, low relative dispersion, and a breakeven turnover that plausible volume can clear at the five-basis-point ceiling used above.
Correlation is not enough either. Two stocks can move in the same direction almost every day and still move by very different amounts. If their betas are far apart, the pool will spend the rally selling the higher-beta asset into the lower one.
With that in mind, which potential thematic pairings make the most sense?
Dual class shares
Consider Alphabet’s GOOG and GOOGL. Same company, two tickers, economically almost the same claim; they diverge only on the thin difference in voting rights and whatever technical supply-demand wrinkle moves one a hair against the other. Dispersion is nearly zero by construction, and it stays that way because there's no fundamental force that is pulling them apart with any significant might (currently). This is as close to a stablecoin pair as equities get. The price ratio between the assets should revert to the mean.
Admittedly, this is a small universe. There are only so many dual-class stocks, and only so many people who are desperate to swap GOOG for GOOGL. Low dispersion makes the impermanent loss side of the calculation attractive, but it does not manufacture volume for the assets. Still, where the flow exists, this is probably the highest-quality equity AMM pool you can build.
Mature industries and oligopolies/duopolies
The next best pairings are duopolies and tight substitutes in mature industries: Coke and Pepsi, Visa and Mastercard. They are exposed to nearly identical forces and they move together most of the time. They can diverge (one posts a bad quarter, one wins a deal), but far less than two random names in an index, so the pool doesn’t have runaway winners to trim. Many of the more mature industries have oligopolistic, naturally monopolistic, or highly concentrated and correlated groups of public companies with stock prices that move in near lockstep with each other. Pairings in these mature industries are natural complements for AMM liquidity, with low dispersion, high correlation, and low likelihood of idiosyncratic shocks.
This category suggests a more useful portfolio construction than dropping all 500 names into one giant machine. Call it the S&P 500+. Use the S&P 500 as the benchmark, leave most of the exposure in a main basket, and carve out only the assets with a defensible reason to remain tight: Visa and Mastercard, Coca-Cola and PepsiCo, perhaps a dual-class pair where somebody wants to trade one line for the other. Each carve-out sits in its own AMM sleeve. The portfolio keeps the majority of its exposure to the underlying assets while specifically targeting the pairs with the best combination of lowest expected dispersion and highest AMM turnover. By nature, there will be drift from the AMM-enhanced LP compared to just holding the underlying assets, but the math so far shows that the impermanent loss would be more than offset by trading fees at an achievable swap level. Assuming a 25x turnover with a five-basis-point swap fee, the Cola Duopoly would have accrued 2% of additional principal value over 5 years, while the Card Duopoly would have added over 6%.
Other mature industries such as the big banks and oil producers were highly accretive over holding the underlying; with both baskets seeing 1% growth in principal over a one-year horizon and 5% principal grown over a five-year horizon, assuming 25x turnover with 5 bps swap fee.
Thematic baskets
This is where this paper started and where the simple correlation story falls apart. NVDA, AMD, and SNDK, a tight cluster of semiconductor names, share a demand cycle, macro sensitivity, and a sufficiently convincing AI narrative to be an interesting fit. Their returns are highly correlated, but their betas are not. They can all rise together while one rises several times as much as the others, which is exactly the relative move the AMM sells.
If you squint, the pitch looks excellent: high enough correlation to contain IL and high enough volatility to generate fees. In the simulation, the second part rarely rescues the first. The names move together, but by wildly different amounts: over the past twelve months the pool would have realized ~5.8% IL and required 120x turnover at five basis points per swap merely to match holding. My entry timing was lucky. Since depositing on July 7, I have only realized ~8 basis points of IL (44 basis points annualized), while supporting 1.1x turnover via swaps (5.9x annualized). This is still less than the 8.8x turnover needed to cover the negative gamma (at five basis points per swap). This is mostly an artifact of my timing; extrapolating from the worst IL the pool has seen in the past two months (Aug. 7), my breakeven turnover skyrockets to 115x, ~15x higher than my current breakeven turnover.
My chipmakers are a bad pool for a liquidity provider to be in. That result is not a theorem about thematic baskets though, just commentary on my particular asset allocation. I made a bad choice depositing my assets, and the other pools may be bad choices too, but at least now you have the math to help you make an informed decision.
A better thematic basket might contain names with more similar factor exposures, comparable volatility, and some durable reason for their relative prices to remain close, not merely three stocks that are high in YTD returns and memetic mindshare. That could produce materially less IL than my basket, but thematic bucketing is not a science. The relative dispersion still has to survive different market regimes, and whatever low-dispersion basket remains still has to attract enough swaps to pay the LP.
When looking at these baskets, let’s remember two things: assumptions, and who they turn you into.
Do not forget correlation is not a constant, and it breaks exactly when you need it most. The naive framing "these names move together" is a fair-weather statement. In a real stress event, correlations don't hold; they lurch, and not always the way you'd hope. While in a crisis it is a common refrain that correlations go to one on the way down, that is just tongue-in-cheek way of saying everything is being sold. It doesn’t mean they are all being sold in the same proportion. For an AMM, direction is not enough. The magnitude of each move is what determines how much the pool has to rebalance.
A single earnings surprise, a delayed product, one name catching a downgrade the others dodge, or another landing the headline deal with a frontier-model lab can separate the basket very quickly. The stocks may still finish the day with highly correlated returns, but if one falls 3% and another falls 15%, the pool has plenty of dispersion to trade against. The basket still “moved together” in the ordinary sense. It just did not move together in the sense that matters to the LP.
This scenario is where the hierarchy between the pairings matters. The share-class pair survives because almost every fundamental event affects both at once, absent severe corporate events, like complicated mergers and acquisitions (or some form of getting Zucked). The duopolies are in as mature and boring industries as it gets (Visa, Mastercard, and Coke have all been mainstays of the Berkshire Hathaway portfolio until this year), with durable economic relationships, responding to many of the same forces.
The thematic basket has a much weaker tether to a structural relationship compared to those. Its correlation is an observed historical relationship and a handful of macro forces, not a mechanism that forces the price ratio back into line. Prices can react to technical flows, product cycles, financing conditions, and company-specific news that the theme does not capture.
None of which means thematic baskets are inherently bad for AMMs. Mine may simply have been a bad selection. Another basket with more comparable betas, similar volatility, and a stronger reason for its relative prices to remain close could produce much less IL. But calling something a theme does not establish any of those things. The realized dispersion still has to survive different market regimes, and the pool still has to attract enough swaps to pay for whatever dispersion remains.
For my AMD, NVDA, SNDK pool, the massive run up in SNDK means I have trimmed (0.08%) of my gains, and made up 0.05% in swap fees since depositing, down 3 basis points overall, not a large loss, but mostly due to lucky timing of my entry.
Other potential asset classes
Now if we step back for a second and stop thinking about equity names and indices, we start to get some other potentially interesting asset classes. The asset class most naturally suited to this machinery may not be equities at all. It may be short-dated U.S. Treasuries.
Short-dated government paper has most of the characteristics we have been looking for: a common issuer, a common currency, similar durations, and a known redemption value at maturity. Group the maturities closely enough and there is very little idiosyncratic dispersion for the AMM to trade against.
That does not mean Treasury prices never move. It’s just that they have a pretty well known terminal cash flow. Rate expectations, funding conditions, liquidity, and position on the curve all matter, and bills with different maturities will respond differently. The informational asymmetry that made AMMs sub-optimal for volatile stocks, informed flow that knows something the stale pool doesn't, has essentially collapsed, because nobody has private information about where a T-bill is going. Treasuries are a low-information market, with low risk of blow-up (barring the gray swan event of a U.S. government default, but that will also result in other more serious economic issues to deal with than a silly AMM pool assumption breaking).
Low dispersion still only minimizes IL. The pool must attract fee-paying flow, and the enormous volume of the offchain Treasury market will not automatically appear onchain. The current AMM design is also probably the wrong shape for the job. Treasuries trade in yield space. What matters to traders is the yield to maturity of a bond, not its price, and the majority of DeFi assets and protocols are not designed to optimize for that fact (some protocols, such as YieldSpace, have attempted to address this problem). Redemption creates another problem here, but abstracting away all the annoyances of integrating traditional infrastructure into DeFi, the mechanics of an AMM seem well suited to optimizing a short-dated treasury portfolio.
There is also the economic opportunity cost of acting as an LP. A token can only be deployed in one place at a time, and putting it into an AMM means giving up whatever it could have earned elsewhere. A pool can beat holding and still be the wrong trade if lending the same stock would have paid more without continually selling the upside. The relevant benchmark is not zero; it is the best available risk-adjusted use of the asset. For tokenized equities, that means the pool has to beat the lend. So what does lending actually pay, and how hard must an AMM work to clear that higher bar?
The lending benchmark
Lending onchain is simple. Somebody wants to short NVDA, or borrow against it, or needs it to settle something, and they'll pay to borrow yours. You post the NVDA-backed token to a protocol, the borrower posts collateral and pays interest, and some of that interest comes back to you (barring a credit event). The pool didn’t trim your winner. If NVDA triples, you own all of the tripling, plus the lending yield on top. The AMM's entire job is to sell your winner into strength; lending's entire job is to leave it the hell alone and pay you rent while it runs.
Lending is not some crypto-native invention, either, like the near exclusive usage of perpetuals in DeFi since their conceptual invention by Robert Schiller in the 90s. Securities lending is one of the oldest, largest, and quietest businesses run by your brokerage. When you hold shares at a brokerage, it is (right now, probably) lending your stock to short-sellers and pocketing the fee. You are holding the asset and taking the directional risk, and your broker is collecting the lending spread and keeping all of it (admittedly while managing credit risk).
Moving from a traditional brokerage desk to a DeFi protocol changes who gets the spread: with traditional brokerages, you ain’t getting it; onchain, it's yours. There is no free lunch though, and in exchange for the lending yield you now also take on the credit risk that the brokerage was managing, and its clearing house was backstopping. This is a very different risk profile that isn’t for everyone, and as we have seen from some of the lending app issues of the past year, comes with very unique crypto and DeFi risk. The main drawback is the emergent complexity of lending apps as they adopt more collateral and share in the risks of those assets (see rsETH).
Reclaiming agency over where your stock portfolio is being lent, and who gets the spread from it is incidentally the mechanic retail rediscovered during the GME short squeeze of January 2021; traders calling their brokers to revoke share-lending authorization, trying to starve the shorts of borrow. Now instead of just revoking authorization, you are taking it into your own hands. Same plumbing, same spread (mostly). Onchain, you'd be the one collecting the spread and exclusively in charge of whether it is lent out or not.
So how much does this alternative pay? Right now, organic borrow yields on tokenized stocks are generally low because utilization is low. Incentivized products can advertise much sweeter rates, but that yield comes from subsidies rather than genuine borrower demand. The most active tokenized equity lending market is on Kamino with xStocks. Over half of the pools pay 0% interest to suppliers because there are no borrowers (but they do count towards users’ collateral balance), and the pool with the highest utilization, the SPYx market, pays only 11 basis points to suppliers. Whatever the sustainable rate is, that is the number the AMM has to beat.
This lets us state the hurdle honestly. Until now, we have been measuring from the beginner tees: how much turnover does the pool need to cover its impermanent loss and match holding? The real question is how much turnover it needs to cover both the impermanent loss and the lending yield the LP gave up. At the same five-basis-point fee ceiling used above, that is a strictly higher turnover number.
The pool therefore has to do two things: first, claw back everything that dispersion and negative gamma cost it; second, out-earn the yield available from lending while the investor remains fully long. For the S&P 500 constituent pool, that bar is somewhere between David Goggins-level demanding and flatly absurd. For a tightly linked sleeve, however, it’s at least worth calculating. Dual classes and mature-industry pairs do not get a free pass either; they still need enough real, fee-paying volume to clear the lend.
Now, lending isn't free money. To give DeFi its due, automatic, overcollateralized liquidation is one feature that some ratings frameworks treat as a potential credit strength relative to conventional lending (with the S&P giving LEDN’s bitcoin-backed CLO a lower stress scenario haircut due to the successful history of the liquidation mechanism). Positions can be marked continuously, margin thresholds are explicit, and collateral can be sold as soon as a threshold is breached rather than waiting on a discretionary workout process.
The benefit is conditional, though. The oracle has to be right, the collateral has to remain liquid, and liquidators have to show up before the position suffers a shortfall. A liquidation engine can be fast and still fail if the market it needs is closed or gone.
Traditional securities lending has its own failures, but it also has regulated intermediaries, collateral agents, legal agreements, and some prospect of recovery. Onchain, those protections are replaced by technical, liquidity, and credit risks that ultimately belong to the user. When it breaks, it can break in code with far fewer (or none) of the protections afforded investors in a courtroom. Some people will take that trade happily.
For an investor with conviction in a single stock, lending still begins with the structural advantage: upside plus yield, rather than fees minus negative gamma. The AMM earns its keep only in the narrow band where the assets remain close enough to contain dispersion and trade often enough to cover both the gamma drag and the forgone lending yield. That band exists. It is just much narrower than “put your stocks in a pool and earn yield” makes it sound. Mostly, it can't. Sometimes, the right basket, the right asset types, it might.
Conclusion
So, my three chipmakers. Future of finance, or an elaborate donation to arbitrageurs? Having now done the arithmetic I probably should have done before depositing: it was mostly a donation to the arbitrageurs.
The original pitch was pretty simple. Semis move together, so correlation should contain IL; semis are volatile, so trading should generate fees. High correlation, high volume, best of both.
It's also the exact story that falls apart on contact with the real math, because the semis basket sits right at the intersection of two things you do not want stacked together in an AMM: dispersion that's high enough to matter, and correlation that hides wildly divergent betas. My little pool with no external incentives gives us an (imperfect) test of the real world volume during an onchain boom. It has produced ~$80 of swaps and ~$1 in fees while bleeding 8 basis points away in negative gamma due to the run up of SNDK. My $75 would have had to facilitated ~$120 trading volume to make up what we need in swap fees, a turnover of 8.8x, 1.4 times greater than the pool’s 5.6x (annualized) turnover. The pool has not cleared the basic bar of being superior to holding at a reasonable turnover hurdle, let alone clearing the higher lending bar.
To be fair, I couldn’t have lent out those same tokens in DeFi. No lending market existed for them, and lending markets are generally less permissionless than AMM pool creation. Curve let anyone create the pool without asking Robinhood, a broker, or anyone else. Had I lent those same tokens, stayed fully long, and collected a yield instead of paying the machine to sell my winners into every rally, I would have at least not lost out on the SNDK run.
The economically superior alternative is not much use if nobody has built it, and a lending market cannot be deployed as casually as a plain AMM pool. Somebody has to set the collateral factor, liquidation threshold, borrow cap, interest-rate curve, and oracle configuration, then continue monitoring the asset’s liquidity and volatility. Get those parameters wrong and the yield uptick you got for lending comes at the cost of principal loss when a borrower goes under in an illiquid market. That availability constraint matters when choosing where (or if) to deploy onchain.
An AMM is a machine that sells your winners to buy your losers, forever, and the only questions that matter economically are how far apart the things inside it drift, and how much flow covers the cost of pulling them back together. Dispersion is the main determinant of the cost of LPing, and you can manage most of the complicated issues associated with many asset AMMs via that one number. But it does not contain everything. Once you want the finer answer, you have to ask what shape produced that dispersion. Is the return distribution symmetric, skewed toward a few winners, or dominated by fat tails and extreme names? Skew and kurtosis tell you where the first-order dispersion estimate begins to miss, and the model translates those differences into expected IL and the turnover required to cover it.
How far did the assets spread, what shape did that spread take, and did the fee flow pay for it? That’s what it boils down to.
The resulting assets that make good pairs for AMMs are unfortunately not every stock in the S&P 500. If you want broad-market exposure, tokenize the index instead of forcing constituents to rebalance against one another. If you own a single stock and want to remain long it, lend it where a credible lending market exists, and (depending on volatility) hold it where one does not. Use passive AMMs where a structural or durable economic relationship contains dispersion and actual volume clears the turnover hurdle: dual-share classes, mature-industry pairs, and thematic baskets only after considerably more underwriting than “these stocks are all involved in AI.”
My proposed S&P 500+ sits between those choices. Keep most of the index in a residual basket and carve out only the relationships that can plausibly support productive AMM sleeves. That might improve portfolio yield, but it also introduces tracking error as the AMM sleeves trade away from their benchmark weights between rebalancings.
The same screen points beyond equities to short-dated Treasuries: low dispersion, a known terminal cash flow, and a natural role in stablecoin reserves. The implementation would need to account for yields, maturities, and redemption rather than simply dropping the bills into a constant-product pool. But Treasuries may fit the machinery better than most stocks—which raises the larger question of which parts of reserve management that machinery could eventually replace.
One notable caveat. The analysis here uses the plain constant-weight AMM: the simplest passive version of the trade and the cleanest place to see its economics. Concentrated liquidity AMMs such as Uniswap V3 lets a depositor change the shape of the exposure, concentrating fee capture inside a chosen range and losses outside it, effectively levering exposure to dispersion. Proprietary AMMs go further by quoting bespoke curves, adding inventory management and complex decision trees, rebuilding some of the human market maker in code. Those designs can manage the cost more intelligently. They do not make divergence free, and at some point, a sufficiently complex proprietary AMM becomes a different product from the passive pool analyzed here.
The implications seem to be that the most natural use for AMMs is not stock-picking yieldmaxxing at all. It may be the boring, standardized plumbing around it: short-dated Treasury management and portfolio yield enhancement.
Circle already delegates significant reserve-management infrastructure to BlackRock. The question is not whether a smart contract can replace everything BlackRock provides. It cannot (yet). Reserve managers provide custody, operational controls, liquidity, and connections to financial infrastructure that have no analog in DeFi to compare cost to. The narrower question is how much of the repetitive work of buying, rolling, and providing liquidity in highly standardized short-term government paper could eventually become programmable.
That question becomes more relevant under the reserve framework created by the GENIUS Act, as stablecoin growth creates additional demand for cash and short-dated government obligations. A yield-aware AMM connected directly to mint and redemption rails might eventually return some of the economics of reserve management to the stablecoin or its holders. It might also remain easy prey for sophisticated offchain flow. Both outcomes are plausible; the current market infrastructure is not developed enough to pretend otherwise.
Stock lending raises the parallel question. Brokers and other intermediaries provide real credit, collateral, and operational services, but they also retain a spread generated by assets their customers own. Onchain lending demonstrates that at least part of that spread can flow back to the holder. Both desks, reserve management and stock lending, are the same species: enormous, redundant, low-information intermediation layers whose margin survives mostly because the infrastructure to route around them doesn’t exist yet.
I started with three chipmakers and a pool anyone could create. The framework does identify some properly good uses for tokenized assets onchain: index tokens, carefully selected AMM sleeves, lending markets, and perhaps eventually Treasury pools behind stablecoins. It also identifies where the machinery should probably be left alone.
This raises a serious question about whether AMMs might be a fit for these commoditized markets with distributions so well-tailored to their strengths (low return dispersion and information asymmetry, high trading volume). Or as other papers have surmised, will the programmatic nature of smart contracts, delayed settlement and the slightly delayed prices of blockchain trading leave AMMs prey to sophisticated offchain flow forever?
Curve lets anyone create the pool without asking permission, but the market is under no obligation to make it a good idea.