The Double Parachute Model: a mathematical model for using debt-backed stable coins as collaterals

Debt-backed stable coins such as DAI, LUSD, sUSD and FRAX, are a source for passive yield in DeFi (e.g., as Curve LP, or in Yearn Vaults), and users could benefit from a highly leveraged position of such assets. If the leverage is done vs another stable coin such as USDC, then the liquidation risk for the user is considered small (for as long as the collateral stable coin retains its peg).

Hence, lending markets would benefit from providing high leverage to such users, however would risk bad debt accrual if the collateral stable coin loses the peg. Such bad debt can be mitigated by setting a proper liquidation threshold (aka collateral factor, aka LTV) that would allow the platform to properly liquidate the collateral as it is being deppeged. But in return, it would also limit the amount of leverage that a user can take.

In this blog post, we present a mathematical model to reason about the liquidation thresholds of a stable collateral asset. To the best of our knowledge, this is the first research on this matter. The framework we present assumes there already exists a mathematical model that reasons about liquidation thresholds of volatile assets, and uses that model as a black box. Hence, this new framework can be applied to any existing stress testing environment.

The model is presented for the setup of the Gearbox leverage protocol, and we focus on the scenario where USDC is borrowed against a stable coin collateral.

The Double Parachute Model

The LUSD stable coin is fairly elegant as it is backed by a single collateral, namely, ETH, and it has a built-in mechanism where users’ bad debt is socialised among all borrowers. Hence, we demonstrate our framework with the use of LUSD, but similar principles hold when analysing DAI and sUSD.

The Double Parachute Model (DPM) aims to simulate the bad debt that would result from permanent price depegging, and it ignores temporary deppegs that occur due to thin liquidity (we address those as well in the next section). In such a setup, the LUSD price is only affected by its percentage of ETH backing, and thus we can view a user position with LUSD collateral and USDC debt, as a position where the de facto collateral asset is ETH (and the debt is still USDC).

However, the user position is special as both Liquity (the protocol that operates LUSD) and the lending market (in our case Gearbox) will try to prevent the accrual of bad debt.

As illustrated in the figure below, as ETH price goes down, a first line of mitigation will be activated, and Liquity will try to prevent bad debt accrual in the LUSD system. Only if/when Liquity fails, and ETH price continues to go down, then the bad debt of the LUSD system will decrease the price of LUSD itself, and at this point the Gearbox system will kick in and will try to mitigate the bad debt on their own platform.

In the double parachute analogy, the first parachute is Liquity, and its strength is dictated by the ETH backing it currently has. The second parachute is Gearbox, and its strength originates from the configured liquidation threshold where lower threshold is a stronger protection. In particular, when the ETH to LUSD backing ratio is sufficiently high, the second parachute can be degenerated and set to 100% (minus applicable liquidation penalties and known oracle deviation).

Formal framework

Formally, we see the LUSD system as a single user with X amount ETH collateral and Y amount of LUSD debt. We stress test the LUSD system to find the expected value at risk/amount of bad debt, and this can be done by any standard stress test environment. Then the liquidation threshold for Gearbox is set such that it compensates for bad debt in the LUSD system. For example, if the expected value at risk in the LUSD system is 15% of the LUSD supply, then Gearbox would set a liquidation threshold of 85%.

We note that in normal times, the value at risk of LUSD is expected to be 0%.

Price fluctuations

Most decentralised stable coins do not have physical mechanisms that force them to be traded at exactly $1. Instead, they fluctuate around $1, with volatility that is correlated to their corresponding dex liquidity (typically Curve Finance liquidity).

Most of these stable coins are not subject to risk-free arbitrages even when traded above or below peg. However one might hope that being traded above (resp., below) would discourage buyers (sellers) and thus bounce the price back to one.

Unfortunately, our analysis shows that by examining short time periods of 1 hour, we observe that the trading volume on these assets is fairly one sided.

The graph below shows that when breaking down the Curve Finance volume for FRAX trades to 1 hour windows, then on (volume weighted) average over 90% of the volume is one sided (of course as the price remains around $1, it shows that every time it is unbalanced towards a different size).

That said, FRAX maintains a perfect peg thanks to a huge amount of Curve liquidity (over $0.5B) which is owned almost exclusively by the FRAX protocol itself.

This is not the case for LUSD, who has less one sided hourly volume, but suffers from almost permanent upwards deppeg.

Finally, sUSD is the most balanced in terms of sided trading, but still pretty much one sided.

Hence, we also take the stable vs USDC dex liquidity into account, and assume that liquidation cascades will not be mitigated by organic trading volume for the counter direction. That said, as long as the stable coin is solvent, the asset volatility will remain low, and hence, relatively few liquidations are expected.

Formal framework

To be on the safe side, we simulate a situation where all of Gearbox’s stable collateral is liquidated in a single day, without any price recovery after every liquidation.

Taxonomy

Raw assets

For raw assets such as sUSD and LUSD we run simulations according to the Double Parachute Model, and according to the price fluctuations model, and set the liquidation threshold to be the minimum among those two recommendations.

Curve LP token

Curve LP tokens, e.g., LUSD/3crv LP token, are special because their price is bounded above by USDC price (of $1). This is due to technical limitations in their price oracle.

As a result we get special cases in LP tokens such as LUSD/3crv, where, as long as solvent, LUSD is redeemable to $0.99 of ETH, and as its imperfect oracle bound the price by $1, we get that the asset is not subject to price fluctuations, and thus its dex liquidity can be ignored.

On the down side, depositing the asset into the Curve system carries additional smart contract risk. This risk can be mitigated by charging higher user fees. And in any case Curve smart contracts are buttle tested and considered to be with low risk.

Algorithmic stable coins

The FRAX stable coin is also partially backed by its FXS governance token, which is minted whenever some of the backing is lost.

In this case we can apply the DPM with FXS as collateral asset. However, as FRAX has huge protocol owned liquidity vs USDC, this liquidity would also be taken into account as FXS liquidity.

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