TLDR
 The researchers formalize a model to analyze undercollateralized DeFi lending platforms and understand the risks of onchain leverage.
 They identify strategies to mitigate socalled impermanent loss, arbitrage loss, and loan liquidations.
 They also conduct an analysis of 9 months of leverage traders’ positions on Alpha Homorra protocol and discovered that while lenders are profitable, 72.35% of borrowers suffered from a negative APY (without governance incentives), and stablecoins pools attract higher leverage multipliers.
Core Research Question
What are the risks related to undercollateralized lending platforms and how can they be mitigated?
Citation
Z. Wang, K. Qin, D. V. Minh, and A. Gervais, “Speculative Multipliers on DeFi: Quantifying OnChain Leverage Risks,” p. 24. https://fc22.ifca.ai/preproceedings/71.pdf
Background
 Leverage Trading: Leverage is the practice of taking on debt to borrow assets for a subsequent financial operation amplifying trading risks in pursuit of higher returns. The degree of amplification is determined by the leverage multiplier (ratio of the total assets to the collateral that a trader holds). The leverage multiplier can be adjusted by providing or removing ad hoc collateral from the leverage position.  Summarized from the publication.
 Lending protocol: A multiasset management pool that allows capitalproviding entities to earn interest on their capital as well as capitaltaking entities to trade with a multiple of the capital they hold. Essentially, three actors interact with a lending pool: Lenders, Borrowers as well as Liquidators. Summarized from the publication.
 Liquidation: If the value of debt collateral in a lending system declines below a custom threshold, then the debt position may be opened for liquidation. A liquidation is then an event in which a liquidator repays outstanding debts of a position and, in return, receives a portion of the collateral of the position as a reward. Summarized from the publication.
 Automated Market Maker (AMM): Decentralized exchanges that implement liquidity pools of trading pairs rather than a centralized orderbook. SCRF glossary.
 Constant Product Markets: Automated market makers that use a formula of the form xy=k where k is constant to relate price to reserve balances, e.g. Uniswap. SCRF glossary.*
 Impermanent/divergence loss: Liquidity providers (LP) deposit their assets to a liquidity pool. If the value of the LP tokens drops below simply holding the assets, there exists an impermanent loss (IL). LP may see their accumulative asset value decline when the tokens in a liquidity pool diverge in price from each other. If the token values revert to the price ratio at the time of the LP deposit, the IL is reverted (an IL is therefore only realized if exiting the pool). Summarized from the publication.
 Arbitrage: Arbitrage is the process of profiting by selling/buying assets among multiple markets, leveraging price differences. From the publication.
 Alpha Homorra: A yield farming product in DeFi. Documentation could be found here: V1, V2.
Summary
 Almost half of the DeFi Total Value Locked (TVL) is dedicated to lending protocols (44%). However, the most used lending protocols provide capitalinefficient loans as they are overcollateralized (MakerDAO, Compound, Aave, etc.).
 More recent platforms such as Alpha Homora (AH) provide undercollateralized loans (i.e. onchain leverage) but to date there has been no academic study exploring and formalizing such architectures.
 This study provides a generic and formal model for onchain lending platforms with a leverage > 2x. Then they conduct an analytical and empirical study on various aspects.
 Authors show that with reasonable system settings, an onchain lending system can achieve leverage of up to 91.9x (instead of the current maximum factor of 8x in AH).
 Over a 9month period of study on AHv1 and v2 data, lenders are always profitable (positive APY) while 72.35% of the leveragetaking borrower are not (negative APY when ignoring governance token incentives).
 Borrower losses are mainly caused by three factors/risks: Impermanent Loss (IL), asset arbitrage, and collateral liquidation. The authors found that of 10,430 AH’s positions, 1,139 suffered from IL, 149 were susceptible to arbitrage, and 270 suffered from liquidation.
 Borrowers can mitigate the risk of IL by using a leverage superior to 4x and executing a margin trade to swap borrowed to collateralized assets before supplying into AMMs (while still being liquidatable).
Method

The authors started with a formalization of an onchain leverage platform before using it to conduct analytical studies on how to quantify and mitigate various inherent risks. Then they performed data analysis on the onchain data extracted from AH platform usage during a 9month period to analyze users’ behaviors.

Definition of the formal model of an onchain leverage platform
 This table summarize most notations defined in the model
Notation Definition LV Leverage Platform P_{id} = (C,B) Debt Position x X x amount of cryptocurrency X DebtRatio_t(P_{id}) debt ratio LM_t(P_{id}) leverage multiplier Loss^{IL} impermanent loss Loss^{AR} arbitrage loss Loss^{LQ} liquidation loss Coll_t(P^c_{id}) amount of collateral cryptocurrency Borr_t(P^B_{id}) amount of borrowing cryptocurrency p^{B \rarr C}_t price of B in unit C at time t Credit (B, C) how much credit a position gains when collateralizing 1 C and borrowing 1 B m the initial leverage multiplier when opening a position Return^{IL,Mg}_{cp} the return from impermanent loss and margin trading Return^{Mg}_{cp} the return from margin trading without impermanent loss LS liquidation spread, which determines the rewards for a liquidator after repaying the debt  This formula represents the debt ratio of a position (liquidatable iff > 100), with Credit(B,C) fixed by the leverage platform LV.
\text{DebtRatio}_t(P_{id}) = \frac{Borr_t(P^B_{id})}{Coll_t(P^C_{id})} \cdot \text Credit(B,C) \cdot p^{B\rarr C}_t \cdot 100 \%
Formalization of over/undercollateralization of a borrowed position P_id.
\text{A position } P_{id} \text{ is overcollateralized, if } Coll_t(P^c_{id}) > Borr_t(P^B_{id}) \cdot p^{B \rarr C}_t \text{, and undercollateralized otherwise.} 
Leverage Multiplier as the maximum factor of leverage for the position.
LM_t(P_{id}) = \frac{Borr_t(P^B_{id} \cdot p^{B \rarr C}_t) + Coll_t(P^C_{id})}{Coll_t(P^C_{id})}

Impermanent Loss (IL):
 Impermanent Loss definitions (generic definition on the left and for constant product AMMs on the right) with g^C_{t_0}(p_0) + g^B_{t_0}(p_0) the being assets supplied by the borrower and g^C_{t_{0+∆}}(p) + g^B_{t_{0+∆}}(p) redeemable assets a time t_{0+∆}.
Loss^{IL} = \frac{g^C_{t_0+\Delta}(p) \cdot 1 + g^B_{t_0+\Delta}(p) \cdot p}{g^C_{t_0}(p_0) \cdot 1 + g^B_{t_0+\Delta}(P) \cdot p}  1Loss^{IL}_{cp} = \frac {2 \sqrt{\frac{p}{p_0}}}{1+\frac{p}{p_0}}  1 Resulting return from impermanent loss and margin trading for a constant product AMM
Return^{{IL},{Mg}}_{cp} = m(\sqrt{\frac{p}{p_0}}\frac{p}{p_0}) + \frac{p}{p_0}  1 Resulting return from margin trading may outweigh the IL If and only if leverage m satisfies m > \frac{1\frac{p}{p_0}}{\sqrt{\frac{p}{p_0}}\frac{p}{p_0}}. Indeed, as the borrower shorts his asset borrowed for the collateralized asset when swapping to equilibrate assets amounts, the decline of price p can result in a positive financial return.
Return^{Mg}_{cp} = Return^{IL,Mg}_{cp}  Loss^{IL}_{cp} = m(\sqrt{\frac{p}{p_0}}\frac{p}{p_0}) + \frac{p}{p_0}  \frac{2\sqrt{\frac{p}{p_0}}}{1+\frac{p}{p_0}} 
Arbitrage Loss
 Generic formula for arbitrage loss (or arbitrageur profit) and arbitrage risk in constant product AMMs, with δy2 the amount swapped.
Loss^{AR} = \delta_{y2} \cdot (\frac{p^{B\rarr C}_t}{p^{B\rarr C}_{t+\delta}}1)BLoss^{AR}_{cp} = \delta_{y2} \cdot (\frac{x_t+\delta x}{x_t}1)B = \frac{(\frac{m}{2}1)c\cdot\delta y_2}{x_t}B 
Collateral Liquidation
 A position P_id is liquitable if and only if the debt is unhealthy due to a price change.
DebtRatio_{t+\Delta}(P_id) > 100\% With \hat{p}^{B\rarr C}_l the liquidation threshold and m the leverage multiplier
DebtRatio_{t+\Delta}(P_id) \leq 1 \iff \frac{\hat{p}^{B \rarr C}_l}{p_{t_0}} \leq \frac{1}{Credit(B,C) \cdot (m1)} Financial loss due to liquidation, with Liquidation Spread LS fixed by the leverage platform LV.
Loss^{LQ} = \frac{Borr_t(P^B_id)\cdot LS \cdot c_l \cdot p^{B \rarr C}_t}{Coll_t(P^C_{id})} = (m1) \cdot LS \cdot c_l \cdot \frac{p^{B \rarr C}_t}{p_{t_0}} 
Note that financial return for AH borrowers comes from 1) supplying a pair of assets into an AMM liquidity pool and 2) stake received LP tokens into liquidity mining pools for extra profits. However, for the sake of simplicity, the model considers those 2 incomes as a “black box”.

Maximum reasonable onchain leverage
 To identify the maximum multiplier that a leverage platform can support, the authors are taking in account two liquidations conditions (1) avoiding instant liquidation and (2) having sufficient collateral to repay to incentivize liquidators, we can derive:
(1) DebtRatio_t(P_{id}) \leq 1(2) Loss^{LQ} \leq 1 With \frac{p_t}{p_{t0}} the price change/volatility
m_{max} = \frac{1}{max(LS, Credit(B,C)) \cdot max(\frac{p_t}{p_{t0}})} + 1
Results
 Analytical evaluation and how to mitigate risks:
 Impermanent Loss
 To mitigate IL risks, users can use a leverage superior to 4x and swap borrowed asset to collateralized one as in this example (which is equivalent to short borrowed to long collateral asset):
 The user has benefited from the price changed in USDT/REN due to its 6x leverage (started with 1 REN and ending with 1.6 REN after repaying his loan)
 Data (leverage > 4x) found based on this graph showing the resulting return according to price change in assets for various leverage factors (with and without IL).
 To mitigate IL risks, users can use a leverage superior to 4x and swap borrowed asset to collateralized one as in this example (which is equivalent to short borrowed to long collateral asset):
 Arbitrage loss
 As the arbitrage risk is notably dependent on the amount swapped, users can execute multiple transactions to supply assets (instead of a single one). This will reduce the arbitrage opportunity however each transaction implies a blockchain gas fee.
 Impermanent Loss
 Empirical analysis/evaluation for dataset: Authors also study data from AH (v1 and v2) during a 9month period to analyze users behaviors in onchain leverage platforms.

Stablecoins:
 The next graphs are representing the distribution of leverage multipliers according to the DEX/AMMs and type of asset pair, i.e. only stablecoins, partial (stable and nonstablecoin) or nonstablecoins:
 Both graphs show that stablecoins pools are being used with higher leverage (5.39x more than average leverage in nonstablecoin pools). Indeed the platform Curve also appears to attract higher leverage as it is specialized in only stablecoin pools, reducing risks of impermanent loss (due to their high correlation and low volatility) and lower liquidation event probability.
 The next graphs are representing the distribution of leverage multipliers according to the DEX/AMMs and type of asset pair, i.e. only stablecoins, partial (stable and nonstablecoin) or nonstablecoins:

Average multiplier:
 Dataset: 10,430 leverage positions in AH (7 081 in AHv1 and 3 349 in AHv2) across 3,800 borrowers.
 In AHv2 the average multiplier is 3.07x however 65% used less than 3x
 In AHv1 the average multiplier is 2.01x with a maximum of 3x

Position adjustment frequency and duration distributions:
 To avoid liquidations, users can adjust their borrowed positions by adding collateral
 On 2,581 closed positions in AHv2, 348 are adjusted more than once
 The higher the initial leverage, the less likely the position will be adjusted
 67.92% of positions are open for less than 2 weeks.

Borrower APY:
 On 2,581 closed positions in AHv2, 348 are adjusted more than once
 The higher the initial leverage, the less likely the position will be adjusted
 67.92% of positions are open for less than 2 weeks.

Discussion and Key Takeaways
 Risk mitigation: Impermanent Loss can be mitigated using a leverage > 4x and shorting the borrowed asset for the collateralized one. Arbitrage Loss can be avoided by splitting a single action into multiple actions (ie supplying assets in multiple transactions)
 Data analysis: Lenders are always profitable in AH, while 72.35% of the closed positions of borrowers suffer from a negative APY when ignoring governance tokens incentivization. Pools of only stablecoins attract higher leverage multipliers, and consequently the Curve pools too.
 Simulation/estimation of maximum leverage with appropriate settings: An onchain leverage platform can extend its leverage multiplier to 91.9x.
Implications and Followups
 The research could be expanded upon by exploring and qualifying other risks the user faces, conducting a similar analysis on other leverage platforms, or to an extended period of AHv2 platform data (As only a 9month data period was available, which may not represent the different financial cycles)
 Differentiating analysis based on the blockchain being used. AH is now deployed on Ethereum, Avalanche, and Fantom. It would be interesting to understand the impact of lowered transaction fees on the model such as in the mitigation strategy of arbitrage risks using multiple transactions, liquidator profits, etc.
 Conduct analysis of APYs including governance token incentives (using different data points for the token price such as price when the position was closed or ATH/ATL)
 Studying closed positions profits plus the yield received from governance token
 Studying how governance tokens affect the APY of borrowers
Applicability
 This study may be used by onchain leverage platforms to design future models/architectures and notably system parameters (for example maximum leverage multiplier and liquidation spread). Users’ behaviors are also pertinent data for platforms to better know their clients.
 This study may also be used by users of such platforms to learn how to mitigate their risks before taking any position.
 Other academic studies focusing on lending platforms or Blockchain/Miner Extractable Value may use the formal model or data analytics to assess their statement.