Mathematical Principle Analysis: How does Curve founder's new project Yield Basis reduce uncompensated losses to 0?
In fact, when people hold tokens, especially when they plan to hold them for a long time, they still hope to have a place where they can deposit and withdraw them at any time, manage their single currency, and earn interest based on the currency. However, impermanent loss and the need to add two tokens have always been the biggest obstacles preventing users and institutions from adding LPs. Therefore, is it possible to express the relationship between LP's impermanent loss and normal single currency price fluctuations through a formula? Then adjust the LP according to the formula so that the LP value fluctuates with the spot value, thereby reducing the impermanent loss to 0? Curve founder Michael Egorov's new project, Yield Basis, addresses this problem. He discovered that if the change in spot value is P, then when tokens are added to the LP, the change in LP value is √p (square root of p). (Related article: " Curve founder's new project is about to launch on mainnet: How to earn Bitcoin and avoid impermanent loss using Yield Basis? ") Readers who have learned second-grade mathematics should know that √p * √p = P. As long as the value of LP is doubled, the changes in LP value can be anchored to the spot price. How do you double your investment? That’s right, leverage! This is the compounding leverage strategy, one of the core principles of yieldbasis. This strategy leverages user deposits to collateralize loans on the backend, achieving 2x compound leverage. This allows LP value fluctuations to be anchored to the spot market, eliminating impermanent loss. The problem is that when LP is added, it is still represented by two tokens. For example, in the BTC-USDT trading pair, although in yieldbasis, users only need to deposit a single currency into the agreement, and complex steps such as lending are automatically completed by the back-end smart contract, what should be done if the token price fluctuates and the LP debt deviates? This requires the help of a third party, an arbitrageur. This is another core principle of yieldbasis: Virtual Pool, Rebalancing AMM, and Flash Loans. Through these functions, third-party arbitrageurs help users balance their LP debts. It's important to note that YieldBasis reserves a portion of its total revenue to incentivize arbitrageurs and maintain system balance. Therefore, LPs do not suffer losses during the arbitrage process. YieldBasis backtested historical data from 2019 to 2024. During the 2021 bull market, the APR peaked at 60%. During relatively quiet market conditions, the APR was around 9-10%. During this period, the price risk exposure of the BTC/USD liquidity pool was similar to that of holding BTC alone. The project will issue a separate token. According to The Block, it has already raised $5 million at a token valuation of $50 million, and the round of financing was oversubscribed 15 times. The TVL contributed to yieldbasis will eventually be added to Curve. Essentially, yieldbasis adds liquidity to Curve by addressing impermanent loss. Impact on Curve, The direct impact is mainly reflected in: 1) Increasing Curve TVL and pool depth, expected to bring more trading volume and fee income; 2) Rebalancing generates additional transactions, increasing Curve revenue; 3) Increase the demand for crvUSD and generate minting income. ———————————————————————————————————— In terms of product implementation, Yieldbasis abstracts the complexity of the mathematical principles behind it. All users need to do is deposit a single currency. According to the information disclosed so far, only BTC is supported in the early stage. https://x.com/yieldbasis/status/1962636033641849063 As mentioned earlier, the essence of the yieldbasis principle is to use leveraged lending to transform the mathematical curve of LP value fluctuations (√p) into P. This allows LP value fluctuations to track spot market fluctuations, reducing uncompensated losses to zero. Concepts such as lending, virtual pools, rebalancing AMMs, and arbitrage all serve this purpose. In actual backend operations, yieldbasis will automatically implement a compound leverage strategy, pledging user assets, lending crvUSD, and then forming LPs to maintain a 2x compound leverage. Through virtual pools, rebalancing AMMs, and flash loans, arbitrageurs are allowed to participate, and the leverage ratio of user positions is maintained at 2x, thereby eliminating LP's impermanent loss and allowing LP value fluctuations to be anchored to token fluctuations. The yieldbasis product process is as follows: 1/ Users deposit BTC (or ETH) and receive ybBTC (or ybETH) as a voucher. (The operation required by the user has actually been completed at this step.) After that, everything is automatically operated by the yieldbasis system. 2/ Leveraged lending Use the user's deposited BTC (or ETH) as collateral to borrow an equivalent amount of crvUSD. Deposit BTC (or ETH) and borrowed crvUSD into the Curve liquidity pool, maintaining 2x leverage (debt is always half of the LP value). Regarding the implementation of 2x leverage, although the white paper does not explain it in detail, official documents suggest that: The key lies in the particularity of LP Token: 1) LP Token itself contains 50% stablecoins, which is lower risk as collateral than a single asset; 2) The system may set special collateral parameters for LP Tokens: 3) Achieving a collateralization ratio close to 100% through dedicated CDPs; 3/ Automatic rebalancing to cope with price fluctuations For the specific implementation process, please refer to the white paper. The mathematical calculations involved are really too complicated. But in general, Automatically maintained by rebalancing AMMs and arbitrageurs: 1) When BTC rises: arbitrageurs help the system borrow more crvUSD, increasing LP; 2) When BTC falls: Arbitrageurs help the system redeem some LPs and repay debts; 3) The arbitrageur makes a small profit and the system returns to 2x leverage. ———————————————————————————————————— Okay, finally, I'll try to explain the mathematical foundations of yieldbasis, because it's truly fascinating. Of course, I recommend reading it before bed for excellent results. The mathematical core of yieldbasis is, pLP =√p. In summary, in classic AMMs, liquidity prices follow the relationship pLP = √p. By applying compound leverage of L=2, the price performance can be transformed from √p to p, which makes the leveraged LP position price performance the same as a single asset (such as BTC). Explain, AMM constant product formula x * y = k, where x = the amount of stablecoin (e.g. USD) in the pool y = the amount of crypto assets (e.g. BTC) in the pool k = constant Assuming the BTC price is p (denominated in USD), then x = p * y In fact, the two tokens of the LP group are equal in value, 50/50. Therefore, the total value of the LP can be expressed as py squared, that is, k = py². So, y = √(k/p) (√ is not a check sign, it’s a square root. Think back to high school math, oh no) x = p · y = p · √(k/p) = √(p*k) Total LP value = x + p * y = √(p·k) + p·√(k/p) = √(p·k) + √(p²·k/p) = √(p·k) + √(p·k) = 2√(p·k) Then, assuming that the asset price is at t0 at the initial moment and then changes to t1, then LP initial total value = 2√(p₀·k) LP total value after change = 2√(p₁·k) Change ratio = LP total value after change / LP initial total value = 2√(p₁·k) / 2√(p₀·k) = √(p₁/p₀) Assuming that at the initial time t0, the asset price is 1 unit, then the rate of change in value = √(p₁/1) = √p₁ This is the origin of pLP = √p, pLP is the relative change in LP value, and √p is this value. That is, when the BTC price quadruples, the LP value only doubles by √4 = 2. This is the root cause of impermanent loss. This is why we need to add 2x leverage. (√p)² = p. This means that after adding 2x leverage, the change in LP becomes P, the spot price, eliminating impermanent loss. You can now focus on collecting transaction fees. 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