This is an article examining the circumstances under which it is profitable to be an LP on a trading pair on Uniswap.
The question is a bit tricky. For example, if you have 3 Ethereum and equal value USDT, do you get more return by just sitting in your wallet, or do you get more fixed return by investing in the ETH trading pool on Uniswap?
This article does a lot of math, but the conclusion is simple:
If volatility losses exceed 200% of its average return, Uniswap’s rebalancing won’t eliminate enough volatility, which is best done in cash.
If volatility is less than 66% of its average return, then rebalancing through Uniswap to offset volatility is not worth the cost and it is better to just hold the asset.
Within this scope, LP as Uniswap can generate revenue.
Volatility loss is a term in financial mathematics that describes the compound return on a large investment loss. Here’s how the term’s inventor, Mark Spitznagel, explains it:
Aggressive portfolio losses undermine long-term compound annual growth rates (CAGR). It takes a long time to recover from a much lower starting point: losing 50%, you need to do 100% to get back to where you were. In this case, I call the cost of converting a portfolio’s + 25% average arithmetic return into a zero CAGR (and thus making the portfolio’s profit zero) a “volatility loss” : this is an implicit, deceptive charge, the extra cost that investors pay because of the negative impact of market volatility.
1 problem
On Oct. 14, Charlie Noyes tweeted a question he and Dan Robinson have been debating: What is the best fee for any Uniswap deal? Can this optimal expense outperform an unbalanced portfolio to achieve “no temporary losses” or even above-expectation growth?
1.1 Basic Rules
Automated market maker AMM is a decentralized trading mechanism that allows users to trade assets on a chain like USDC, ETH, etc.
Uniswap is the most popular AMM on Ethereum. Uniswap, like most AMMs, converts trading pairs by holding reserves of both assets. And reserves determine trading prices, so that the price and the market to keep consistent.
The person who provides the liquidity for the “pool” is called an “LP” and an LP provides liquid assets for other users to trade. LP needed to inject both assets simultaneously, taking on the risk of the transaction in exchange for a portion of Uniswap’s earnings.
1.2 Problem Setting
The problem is that pools provide liquidity between money and another asset whose price can fluctuate randomly. A more brutal assumption is that almost all trades are carry trades – only when AMM prices exceed market levels.
In other words, each trade results in a loss of capital in the pool.
1.3 General Conditions
At first glance, this would be a costly mistake for Uniswap’s LP.
Because the bid price required by market makers is lower than the ask price, when asset prices do not move, market makers directly profit, and they receive roughly equal amounts of buy and sell. These trades are often referred to as “blind” trades because they are not linked to short-term price movements.
Market makers, on the other hand, lose money when they buy before prices fall or sell before they rise. Marketmakers, therefore, is one of the most worried about counterparty arbitrageurs, arbitrageurs only when the change in the price. Each trade is a net profit for the arbitrageur and a net loss for the market maker.
Since there are no uninformed trades in Uniswap (virtually every trade is a carry trade), it is clear that LP will lose a lot of money.
There is even a suspicion that LP, as Uniswap, will be caught on the hook in every deal for some potential price volatility.
2 Solution
If an asset is sufficiently volatile relative to its average return, an LP on Uniswap will do better over time than A HODLer, even if only the carry trade is on offer.
This is due to a phenomenon known as “volatility returns” : under certain conditions, by periodically rebalancing two assets, it is possible for them to outperform any static portfolio. In this context, “rebalancing” refers to trading back to a fixed 50/50 ratio of holdings in each asset.
So when they are arbitraged, THE LP pays a fee to the market to rebalance the portfolio for them. In this particular numerical setup, this rebalancing is beneficial and can hopefully be done as often as possible. This means that LP should set its fees (the price exposure that determines rebalancing) as low as possible and not zero.
This is good news for Uniswap as it means that low fees still make sense even when carry trades dominate, allowing Uniswap to remain competitive as on-chain orders continue to increase and Uniswap starts to offer smaller spreads.
That said, it is worth emphasizing that these results apply to very special stylized number sets, where the assumptions involved are very similar to those of the Black-Scholes option pricing model.
2.1 Comparison Criteria
We evaluated different strategies by comparing their “asymptotic wealth growth rates,” which measure how quickly they add value (or lose value) over a long period of time.
We compared all strategies with an “unrebalanced portfolio,” which is half cash and half holding risky assets, and left unchanged. This means that in the worst case, when risky assets lose all their value, the “unbalanced portfolio” will consist almost entirely of cash, with zero growth in the long run. On the other hand, if the risky asset grows exponentially, it will soon dominate the “unbalanced portfolio” and thus grow at the same rate as the risky asset.
It is worth noting that the two assets can share the same “asymptotic wealth growth rate”, but the performance is also very different. For example, if the risky assets grow at zero, Uniswap with zero fees will always be worth less than the “unbalanced portfolio”, but since neither is expected to compound or lose money over time, both will grow at zero wealth.
2.2 Wave resistance
To understand these results, it is important to understand the concept of wave resistance. Suppose that each year our risky asset prices either fall by 75 per cent or rise by 50 per cent with equal probability.
In any given year, if we invest 100, the expected value is 50/2+175/2=100, the expected value is 50/2+175/2=100, and the expected value is 112.5. If you just buy and hold, the portfolio is expected to grow 12.5 per cent a year — which seems like a good deal.
Unfortunately, in the real world, our profits don’t materialize. If we buy and hold this portfolio, we end up losing everything. That’s because, over time, increased wealth comes at a huge cost.
If 50 per cent was lost in the first year and 75 per cent gained in the second year, the ending balance in the second year would be only 50 per cent ∗ 175% = 87.5%. Similarly, if a gain of 75% in the first year and a loss of 50% in the second year, the ending balance in the second year remains 175% ∗ 50% = 87.5%. Over time, the internal rate of return under the law of large numbers would be -12.5% annualized and would inevitably go bust.
2.3 What’s wrong?
You may find the above conclusion strange or even wrong.
In fact, expected value is a theoretical quantity that measures what would happen if we replicated a given “gambling” behavior “simultaneously.” But in reality, each “gamble” takes place in turn, and the outcome is shaped over time.
Put the numbers in. When we gamble over and over again at a “-50% /+ 75%” win rate, reinvesting our money each time, the expectation increases dramatically, mainly because only a few paths can be exactly right, resulting in astronomical returns. But over time, these paths make up a smaller and smaller proportion of all possible paths, and the odds that we’ll actually see one of them happen shrink to zero.
2.4 The value of rebalancing
In the face of volatility, it is necessary to retain some capital even if expectations may be positive. That way, when things go wrong, you can reduce your losses and compound your gains in the long run.
When prices rise, unwind part of your position to lock in profits in case prices fall again. When prices are falling, it is sometimes necessary to buy low for expected future returns.
In some cases, the optimal strategy is to constantly adjust the portfolio so that a fixed proportion of wealth is invested in each position, such as half cash and half risky assets. But that’s not always the best balance, and as a general rule, the more risky assets you want in your portfolio, the higher the return relative to their volatility.
The benefits of rebalancing long-term wealth growth could be huge, and could mean the difference between profit and bankruptcy. This is true even if each rebalancing trade is unfavorably priced and causes instant losses.
2.5 alchemy
Rebalancing more frequently, with minimal cost, in the Settings above would benefit LP. Therefore, a fee of >0% is required to reduce price volatility to trigger rebalancing. But when fees are exactly 0%, all the benefits of rebalancing disappear, and there is a high probability that LP will do worse than holding an unbalanced portfolio.
Uniswap uses “constant product” invariance, which means that in the absence of fees, the product of the reserve balance must be kept constant for each transaction. Rα Rβ=C, although readers already familiar with Uniswap may be more accustomed to x*y=k.
But it turns out that this C has to increase in quantity in order for rebalancing to provide us with an increase in wealth. In the free case, C will stay the same, there will be no engine for wealth growth.
A non-0% fee implemented in Uniswap or the previous setup ensures that C increases with each transaction. As C increases over time, it means that the reserve balance is not only growing, it is also in balance, providing income.
3 mathematics
To sum up, the question posed by Charlie Noyes can now be answered accurately. To repeat, they focus on the wealth growth rate of an AMM like Uniswap, which charges a 1−γ percentage fee to create a market between cash and an asset whose price moves in geometric Brownian motion with parameters μ (offset) and σ (volatility).
3.1 Growth rate of LP assets
3.2 Optimal expenses and excess returns
If and only if μ>0 and
In this case, LP should set their fees to the lowest possible value rather than 0%, and their asset growth rate will be about μ/2 – σ²/8.
3.3 interpretation
Since the “geometric Brownian motion” simulates compound growth, they are also subject to volatility resistance and the asset growth rate of GBM can be mathematically expressed as -σ²/2:
That means in the range
This suggests that rebalancing can offset some of the volatility of underlying assets.
On the other hand, if the average return is positive without the impact of volatility:
- If volatility losses exceed 200% of its average return, Uniswap’s rebalancing won’t eliminate enough volatility, which is best done in cash.
- If volatility is less than 66% of its average return, then rebalancing through Uniswap to offset volatility is not worth the cost and it is better to just hold the asset.
- Within this scope, LP as Uniswap can generate revenue.
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