Gauntlet Analysis: Market Risks of Listing LP Tokens as Collateral

Risk General
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Thanks, @yaron and Risk DAO, for looking into this. Impermanent loss (loss vs. holding) and loss vs. rebalancing are undoubtedly crucial in setting risk parameters for LP tokens.

Loss vs. rebalancing (LVR) and loss vs. holding (IL) are two different frameworks to understand the AMM. The LVR framework assumes LPs are continuously rebalancing the portfolio, which better captures the profit extracted by arbitrageurs. Please refer to Tim Roughgarden’s talk at SBC '22 to better understand the LVR framework. For simplicity, we dive into loss vs. holding below, which is more intuitive to comprehend via example.

The math you outlined above is incorrect. One of the fundamental equations in DeFi is x*y=k. Under trades without removing/adding liquidity, k never changes (assuming no fees). With a 50/50 ETH/USDC pool, if ETH = 1k USDC, and the LP value is $1, then the pool has 1 ETH and 1000 USDC and:

  • x*y=k translates into 1 * 1000 = 1000
  • If LP value is $1, then this means that the user has 1/2000 ownership of the pool (as you implied in your formula above). We’d note that if it is a 50/50 portfolio and you own x%, then you own x% of each asset.
  • If ETH goes down to $250, then x (the quantity of ETH tokens in the pool) would be 2, and y (the quantity of USDC tokens in the pool) would be 500 after arbitrageurs rebalance the pool.
    • x*y=k translates into 2*500=1000. The weights remain equal (the pool is 50% ETH, 50% USDC). k has remained constant.
    • You doubled k in your math, but k should be constant under this scenario.
  • So now, the LP who had 1/2000 ownership of the pool has 1/2000 of 2 ETH and 1/2000 of 500 USDC, which translates into a total of $0.50. It would not be $0.75 that you stated above.
    • This represents a 50% loss from the LP’s original position (which was $1).
    • If the LP had instead HODLed the $1 in the form of 1/2000 ETH and 1000/2000 USDC, they would have instead had $0.625. Thus, the impermanent loss is (0.50-0.625)/0.625=-20%.
  • To conclude, impermanent loss function is not sublinear here. Of course, fees act as a counter to the loss.

Risk parameters for Aave should be tuned with a strong understanding of the impermanent loss function and loss vs. rebalancing of the LP tokens.

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