If the loss caused by volatility exceeds 200% of its average return, Uniswap’s rebalancing will not eliminate enough volatility. In this case, it is better to hold cash.
If the loss caused by volatility is less than 66% of its average return, the cost of rebalancing through Uniswap to offset the impact of volatility will not be worth it. At this time, it is best to just hold assets.
Within this range, becoming a Uniswap LP can create revenue.
Original title: “Is it a loss or a profit to become a Uniswap LP? 》
Author: Decentralized Financial Community
This is an article to study under what circumstances it is profitable to become an LP of a trading pair on Uniswap.
The problem is a bit confusing. For example, if you have 3 Ethereum and USDT of the same value, then put the funds in the wallet to get more income, or invest in the ETH trading pool on Uniswap To earn more fixed income?
Volatility loss is a financial mathematical term that describes the compound return of large investment losses. The following is the explanation of the term’s inventor Mark Spitznagel:
Aggressive portfolio losses can undermine the long-term compound annual growth rate (CAGR). It takes a long time to recover from a much lower starting point: if you lose 50%, you need to do 100% to get back to the original state. In this case, I call this cost the conversion of the +25% average arithmetic return of the portfolio into zero CAGR (thus making the portfolio profit zero) as “volatility loss”: it is an implicit Yes, deceptive fees, investors need to pay additional costs due to the negative impact of market fluctuations.
Uniswap LP is “forced” to get rich
1 problem
On October 14, Charlie Noyes posted on Twitter a question he and Dan Robinson had been debating: What is the best fee for any Uniswap trading pair? Can this optimal fee exceed the unrebalanced portfolio to achieve “no temporary loss” or even unexpected growth?
1.1 Basic rules
The automated market maker AMM is a decentralized trading mechanism that allows users to trade on-chain assets such as USDC and ETH.
Uniswap is the most popular AMM on Ethereum. Like most AMMs, Uniswap exchanges between trading pairs by holding reserves of two assets. And the reserve amount determines the transaction price, so that the price is consistent with the market.
The person who provides liquidity for the “fund pool” is called “LP”, and LP provides liquid assets for other users to conduct transactions. LP needs to inject two assets at the same time, and assumes the transaction risk in exchange for a part of Uniswap’s income.
1.2 Problem setting
The problem is that the pool of funds provides liquidity between funds and another asset whose price fluctuates randomly. A more cruel assumption is that almost all transactions are arbitrage transactions-only when the price of AMM exceeds market levels.
In other words, every transaction will result in a loss of funds in the fund pool.
1.3 General situation
At first glance, this situation would become a costly mistake for Uniswap’s LP.
Because the buying price required by the market maker is lower than the selling price, when the asset price does not move, the market maker directly makes a profit, and their buying and selling volumes are roughly balanced. These transactions are often referred to as “uninformed” transactions because they are not related to short-term price changes.
On the other hand, market makers will lose money if they buy before the price drops or sell before they rise. Therefore, one of the most worried counterparties of market makers is the arbitrageur, who only trades when the price changes. Every transaction of the arbitrageur is a pure profit for him and a pure loss for the market maker.
Since there are no unknowing transactions in Uniswap (actually every transaction is an arbitrage transaction), the LP will obviously lose heavily.
It can even be suspected that for some potential price fluctuations, the LP as Uniswap will be stuck in every transaction.
2 Solution
If the volatility of an asset relative to its average rate of return is sufficiently high, then over time, the LP on Uniswap will return better than the HODLer, even if only arbitrage transactions enter the market.
This is due to a phenomenon called “volatility returns”: under certain conditions, by periodically rebalancing two assets, they may perform better than any static portfolio. In this case, “rebalancing” refers to the return of the holding ratio of each asset to a fixed 50/50 through a transaction.
Therefore, when they are arbitraged, the LP will pay a fee to the market to rebalance their investment portfolio. In this particular digital setting, this rebalancing is beneficial, and you can hope to do as much as possible. This means that LP should set its cost (the price exposure that determines the rebalancing) as low as possible and not zero.
This is good news for Uniswap, because it means that even when arbitrage trading dominates, low fees still make sense, which allows Uniswap to maintain competition when on-chain orders continue to increase and begin to offer smaller spreads force.
In other words, it is worth emphasizing that these results are applicable to very specific stylized digital settings, and the assumptions involved are very similar to those of the Black-Scholes option pricing model.
2.1 Comparison criteria
We evaluate them by comparing the “progressive wealth growth rates” of different strategies. These “progressive wealth growth rates” measure the speed at which they increase (or depreciate) over a long period of time.
We compare all strategies with the “non-rebalanced portfolio”. Half of the “non-rebalanced portfolio” is in the form of cash and half is in the form of holding risk assets, and remains unchanged. This means that, in the worst case, when risky assets lose their full value, the “non-rebalanced asset portfolio” will be almost entirely composed of cash, and its growth rate will be zero in the long run. On the other hand, if risky assets grow exponentially, it will soon dominate the “non-rebalancing portfolio”, so its growth rate is the same as that of risky assets.
It is worth noting that two assets can share the same “asymptotic wealth growth rate”, but their performance is also very different. For example, if the growth rate of risky assets is zero, the value of Uniswap, which enjoys zero commission, will always be lower than the “non-rebalancing portfolio”, but since neither of them is expected to compound growth or loss over time, Wealth growth rates will all be zero.
2.2 Volatility resistance
The effect of volatility resistance on 50% loss / 75% gain
To understand these results, we must first understand the concept of volatility resistance. Assuming that the price of our risky assets either falls by 75% or rises by 50% each year, the probability of both occurrences is equal.
In any given year, if we invest $100, the “expected value” is 50/2+175/2=$112.5. If you just buy and hold, the investment portfolio is expected to increase by 12.5% every year-this seems to be a good deal.
Unfortunately, in the real world, our profits cannot be realized. If we buy and hold this combination, we will eventually lose everything. This is because, as time goes by, increasing wealth will bring huge losses.
If you lose 50% in the first year and increase by 75% in the second year, the ending balance of the second year will only be 50%*175%=87.5%. Similarly, if the first year gains 75% and the second year loses 50%, the ending balance of the second year will still be 175%*50%=87.5%. Over time, the internal rate of return under the law of large numbers will be an annualized -12.5%, which will inevitably go bankrupt.
2.3 What’s going on?
You may find the above conclusions very strange or even wrong.
In fact, the expected value is a theoretical quantity used to measure what happens when we “simultaneously” copy a given “gambling” behavior. But in fact, each “gambling” is carried out in sequence, and the result will be formed over time.
Bring in the numbers. When we gamble over and over again according to the “-50%/+75%” win rate, and reinvest our funds each time, the expectation value will increase significantly. This is mainly because there are only a few paths. Can be completely correct, thus bringing astronomical returns. But as time goes by, these paths account for smaller and smaller proportions of all possible paths, and we actually see that the probability of one of them being realized also shrinks to zero.
2.4 The value of rebalancing
Faced with the impact of volatility, even if the expected value may be a positive investment decision, it is necessary to retain some funds. In this way, when there is a problem, losses can be reduced, which will bring compound benefits in the long run.
When the price rises, part of the position is closed to lock in profits to prevent the price from falling again. When prices fall, it is sometimes necessary to buy at a low price to obtain the expected future returns.
In some cases, the best strategy is to constantly adjust the investment portfolio so that a fixed proportion of wealth is invested in each position, such as half cash and half risk assets. But this is not always the best balance. Generally speaking, the more risky assets you want in your portfolio, the higher the rate of return relative to its volatility.
The benefits of rebalancing long-term wealth growth can be huge and can mean the difference between profitability and bankruptcy. Even if the price of each rebalancing transaction is unfavorable and causes instantaneous losses, the result is the same.
2.5 Alchemy
Asset growth rate starting from 0%
In the above setting, rebalancing more frequently with minimal cost will benefit LP. Therefore, the fee needs to be set to >0% to reduce the price volatility to trigger rebalancing. But when the cost happens to be =0%, all the benefits of rebalancing will disappear, and there is a high probability that LP will have worse returns than holding a non-rebalancing portfolio.
Uniswap uses the “constant product” unchanged, which means that in the absence of fees, each transaction must maintain the product of the reserve balance unchanged. This article is expressed as Rα Rβ=C, although readers who are already familiar with Uniswap may be more accustomed to x*y=k.
However, it turns out that this C must be increased in number in order for rebalancing to provide us with wealth growth. In the case of free, C will remain unchanged, and there will be no engine for wealth growth.
The non-0% fee implemented in Uniswap or the previous setting can ensure that C increases with every transaction. The increase in C over time means that the reserve balance is not only growing, but also maintaining balance, which provides benefits.
3 Mathematics
In summary, it is now possible to answer the questions raised by Charlie Noyes accurately. To repeat, they are concerned about the wealth growth rate of AMMs such as Uniswap. This AMM charges a fee of 1−γ percentage to form a market between cash and an asset, and the price of this asset follows a geometric Brownian motion The form changes with parameters μ (offset) and σ (volatility).
3.1 Growth rate of LP assets
3.2 Optimal fees and excess returns
If and only if μ>0 and At that time, becoming an LP has more benefits than a non-rebalanced portfolio holding half of the cash and half of the tokens.
In this case, LPs should set their expenses as low as possible instead of 0%, and their asset growth rate will be approximately μ/2-σ²/8.
3.3 Explanation
Since “Geometric Brownian Motion” simulates compound growth, they will also be affected by volatility resistance. Mathematically, GBM’s asset growth rate can be expressed as -σ²/2:
G=μ-σ²/2
Which means in the range Within, the LP on Uniswap corresponds to the asset growth rate of -μ<G<μ/3.
This result shows that rebalancing can offset part of the impact of the volatility of underlying assets.
On the other hand, if the average return without volatility is positive:
- If the loss caused by volatility exceeds 200% of its average return, Uniswap’s rebalancing will not eliminate enough volatility. In this case, it is better to hold cash.
- If the loss caused by volatility is less than 66% of its average return, the cost of rebalancing through Uniswap to offset the impact of volatility will not be worth it. At this time, it is best to just hold assets.
- Within this range, becoming a Uniswap LP can create revenue.
