Mathematical Principle Analysis: How the new project Yield Basis by Curve's founder reduces impermanent loss to 0?

In fact, when everyone holds tokens, especially when planning to hold them for the long term, they still hope to have a place where they can deposit and withdraw at any time, manage single coins, and earn interest in the coin denomination.

However, impermanent loss and the need to add two types of tokens have always been the biggest obstacles preventing users and institutions from adding LP.

So, is it possible to express the relationship between the impermanent loss of LP and the normal price fluctuations of a single coin through a formula?

Then adjust the LP according to the formula, so that the value of the LP follows the fluctuations of the spot value, thereby reducing impermanent loss to 0?

The new project yield basis by Curve founder Michael Egorov addresses this issue. He found that if the spot value changes by P, then when adding tokens to the LP, the value of the LP changes by √p (the square root of p). (Related article: "Curve founder's new work is set to launch on the mainnet soon, how to earn Bitcoin with Yield Basis and avoid impermanent loss?")

Readers who have studied second grade mathematics should know that √p * √p = P. By simply doubling the value of LP, the fluctuations of LP's value can be anchored to the spot.

How to double it? That's right, leverage!! This is one of the core principles of yieldbasis, the Compounding Leverage strategy, which completes collateral lending on the backend with the assets deposited by users, achieving 2x compounding leverage to anchor the value changes of LP to the spot market and eliminate impermanent loss.

The issue is that LP ultimately still manifests as two types of tokens when added, such as in the BTC-USDT trading pair. Although in yieldbasis, users only need to deposit a single token into the protocol, and the more complex steps like lending are automatically handled by the backend smart contract, what should be done if the token price fluctuates and LP debt deviates?

Third-party assistance is needed, namely arbitrageurs. This is the other core principle of yieldbasis, the virtual pool (Virtual Pool), rebalancing (Rebalancing AMM), and flash loans. Through these functions, third-party arbitrageurs assist users in balancing LP debts.

It should be noted that yieldbasis reserves a portion of the total income to incentivize arbitrageurs to maintain system balance. Therefore, during the arbitrage process, LP does not incur any losses.

yieldbasis backtested 6 years of historical data from 2019 to 2024, and during the bull market in 2021, the peak APR reached 60%. Under comparatively lukewarm market conditions, the APR is about 9-10%. During this period, the price risk exposure of the BTC/USD liquidity pool is similar to holding BTC as a single coin.

The project will issue tokens separately. According to The Block, it has currently raised $5 million at a token valuation of $50 million, with this round of financing being oversubscribed by 15 times.

The TVL contributed by everyone for yieldbasis will ultimately be added to Curve. Essentially, yieldbasis increases liquidity for Curve by addressing impermanent loss.

Impact on Curve,

The direct impact is mainly reflected in,

1. Increase Curve TVL and pool depth, expected to bring more trading volume and fee revenue;

2. Rebalancing generates additional trades, increasing Curve income;

3. Increase crvUSD demand to generate minting revenue.

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In terms of specific implementation of the product,

yieldbasis abstracts the complexity of the underlying mathematical principles, and what users need to do is deposit a single coin. According to the disclosed information, initially only BTC is supported.

As mentioned earlier, the essence of the yieldbasis principle is to use leveraged borrowing to transform the mathematical curve of LP value fluctuations √p into P. This allows LP value fluctuations to align with spot fluctuations, reducing impermanent loss to 0. Concepts such as borrowing, virtual pool (Virtual Pool), rebalancing (Rebalancing AMM), and arbitrage all serve this purpose.

In actual backend operations, yieldbasis will automatically complete the ( Compounding Leverage ) strategy, collateralizing user assets, lending out crvUSD, and reconstituting LP to maintain a 2x compounding leverage.

And through the virtual pool (Virtual Pool), rebalancing (Rebalancing AMM) and flash loans, it allows arbitrageurs to participate, while maintaining the leverage ratio of user positions at 2x, thereby eliminating the impermanent loss for LPs and anchoring the value fluctuations of LPs to token fluctuations.

The product process of yieldbasis is as follows:

1/ Users deposit BTC (or ETH) single coin and receive ybBTC (or ybETH) as a certificate.

(The user's required operation has actually ended at this point.)

Next, everything will be operated automatically by the yieldbasis system.

2/ Margin lending with leverage

Use the BTC (or ETH) deposited by users as collateral to borrow an equivalent amount of crvUSD.

Deposit BTC (or ETH) along with the borrowed crvUSD into the Curve liquidity pool, maintaining 2x leverage (the debt is always half of the LP value).

Regarding the implementation of 2x leverage, although the white paper does not elaborate in detail, it is implied in the official documentation.

The key lies in the uniqueness of the LP Token:

1. LP Token itself contains 50% stablecoin, which reduces the collateral risk compared to a single asset;

2. The system may have set special collateral parameters for LP Token:

3. Achieve an almost 100% collateral rate through a specialized CDP;

3/ Automatic rebalancing to cope with price fluctuations

For the specific implementation process, please refer to the white paper, as the mathematical calculations involved are indeed too complex.

But overall,

Automatically maintained through 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 part of the LP and repay the debt;

3. Arbitrageurs gain small profits, and the system restores to 2x leverage.

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Alright, finally, I still need to explain the mathematical foundation of yieldbasis as much as possible, because this part is indeed remarkable. Of course, I recommend reading it before sleep for best results.

The core of yieldbasis mathematics is,

pLP =√p.

Overall, in classic AMM, the liquidity price follows the relationship pLP = √p. By applying a compound leverage of L=2, the price performance can be transformed from √p to p, which makes the price performance of leveraged LP positions the same as that of a single asset (such as BTC).

Explain.

AMM constant product formula x * y = k, where

x = the number of stablecoins in the pool (such as USD)

y = the number of encrypted assets (such as BTC) in the pool

k = constant

Assuming the price of BTC is p (in USD), then

x = p * y

In fact, it just means that the values of the two tokens that make up the LP are equal, 50/50. Therefore, the total value of the LP can be expressed as k = py².

y = √(k/p) ( √ is not a check mark, it's a square root, remember high school math, -害)

x = p · y = p · √)k/p( = √(p*k)

Total value of LP = x + p * y

= √(p·k) + p·√(k/p)

= √(p·k) + √(p²·k/p) = √(p·k) + √(p·k)

= 2√(p·k)

Then, assuming the asset price at the initial moment is t0, it subsequently changes to t1.

LP initial total value = 2√(p₀·k) LP total value after change = 2√(p₁·k)

Rate of change = Total value after LP change / Initial total value of LP

= 2√(p₁·k) / 2√(p₀·k)

= √(p₁/p₀)

Assuming that at the initial moment t0, the asset price is 1 unit, the rate of change in value = √(p₁/1) = √p₁

This is the source of pLP = √p, where pLP is the relative change value of LP value, and √p is this value.

That is to say, when the BTC price increases by 4 times, the LP value only increases by √4 = 2 times, which is the root of impermanent loss.

So this is also why we need to add 2x leverage, because (√p)² = p, which means that after adding 2x leverage, the change in LP becomes P, which is the spot price, thus eliminating impermanent loss. We can安心 eat the transaction fees now.

Hey, are you asleep?