# Aave — Credit Risk Analysis. Part 1

# Introduction

In the decentralized finance (DeFi) space, the Aave Protocol stands out as one of the liquidity market protocols. As participants engage in lending and borrowing activities through Aave, comprehending and evaluating the associated credit risk within the decentralized financial landscape becomes paramount.

The report examines the credit risk dynamics of the Aave Protocol, utilizing structural models to gauge the probability of default. Within the structural model’s framework, a default event is defined by a critical imbalance wherein liabilities surpass the value of assets. By delving into the interplay between assets and liabilities, these models offer valuable insights into the underlying mechanisms driving default events, thereby contributing to a more comprehensive risk assessment. The emphasis on structural models allows for an intuitive examination of default events, providing insights into the inherent dynamics of credit risk within the decentralized financial ecosystem.

The report begins by establishing the foundation for Aave Protocol analysis, defining crucial variables for understanding financial intricacies. It covers assets, liabilities, collateralization ratio, and capital efficiency. Also, it explains financial metrics like Value-at-risk (VaR) and Conditional Value-at-risk (CVaR). The subsequent section describes structural models used to analyze the Aave Protocol’s credit risk, focusing on insolvency modeling from an asset values perspective. Two key models, Merton and Black-Cox, are detailed. The Merton model calculates the default distance, and the Black-Cox model addresses the default timing limitations of the Merton model. The report then introduces a crucial metric — Expected Default Frequency (EDF). The analysis shifts to the explanation of first and second-order statistics of asset estimation, serving as fundamental inputs for credit risk modeling. Finally, the report provides a comprehensive analysis of Aave V3 Ethereum, the largest network, providing insights into network dynamics.

# Definitions of Key Metrics

The section provides definitions of crucial variables central to the Aave Protocol analysis. Comprehensive descriptions are integral to understanding the financial intricacies of the protocol and are instrumental in assessing the risk associated with its operations. Readers will have a solid foundation for the forthcoming analysis by having these variables defined in detail.

## Assets

Assets in the protocol represent the total amount of different tokens supplied by users, enabled as collateral, and the fees collected over time. They play a crucial role in securing loans by offering the necessary backing for the protocol and facilitating the underwriting process of loans. Daily on-chain data is gathered for each user separately, offering a granular perspective for in-depth analysis. The dynamic nature of assets, subject to fluctuations based on user and market activities, adds a layer of complexity that is integral to comprehending the evolving risk landscape associated with the Aave Protocol.

## Liabilities

Liabilities in the Aave Protocol consist of loans issued by the protocol, collateralized by user-supplied assets. In contrast to traditional financial systems, Aave’s loans are perpetual, allowing a position to remain open indefinitely and accumulate interest. Therefore, the loan closure occurs whenever a borrower repays both the borrowed amount and the accrued fees. This perpetual nature introduces distinctive risk factors, necessitating robust risk management strategies within the protocol. Similar to assets, daily data is collected from the chain, providing a comprehensive basis for analysis. Understanding the perpetual nature of Aave’s loans and the associated risk factors is crucial for evaluating the protocol’s financial health and sustainability.

## Collateralization Ratio

The collateralization ratio is an essential parameter that assesses the overall protocol financial health and the risk associated with individual borrowing positions. The ratio is computed by dividing the total value of the collateral by the total amount borrowed against that collateral asset. In contrast to a smaller ratio, which denotes an elevated risk level, a larger collateralization ratio suggests a safer situation for the protocol. In the event of borrower default, a high collateralization ratio ensures that the protocol is sufficiently protected from potential losses. This assurance stems from having a sufficient buffer to cover outstanding loans. The stability and solvency of the protocol depend on this safeguard. However, while a high collateralization ratio ensures stability and solvency, it comes with a trade-off in terms of capital efficiency. The protocol’s overall capital utilization is decreased when borrowers are required to lock more collateral than they are borrowing. As a result, it restricts users’ ability to borrow money and hinders the protocol expansion. In essence, striking a balance between a secure collateralization ratio and optimal capital efficiency is pivotal for maintaining the protocol’s stability, protecting against default risks, and promoting sustainable growth.

## Capital Efficiency

The collateralization ratio (CR) and probability of default (PD) play a crucial role in assessing Aave’s capital efficiency. Optimal capital efficiency is indicated by a low PD and a potentially lower CR, reflecting a lower likelihood of default and enabling a higher portion of the borrowed funds to be supported by a relatively lower amount of collateral. Conversely, low capital efficiency is suggested by a high PD and potentially a higher CR, signifying a higher default risk and potentially an excess amount of collateral. Monitoring the probability of default, collateralization ratio, and respective variables allows Aave to fine-tune its capital efficiency. By keeping these metrics in check, Aave can work towards maximizing the use of its capital while maintaining a resilient lending environment. This approach ensures a delicate balance between risk mitigation and efficient capital utilization within the protocol.

## Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR)

Value-at-risk (VaR) is a financial risk metric to quantify potential financial loss within the protocol over a specified timeframe and confidence level. VaR calculates the maximum loss that could occur under normal market conditions, helping manage risk exposure. In other words, by incorporating a predefined confidence level, VaR estimates the maximum loss the protocol could experience with a certain probability (e.g., 95% confidence that losses will not exceed a specific amount). Understanding VaR is crucial for assessing the Aave Protocol’s financial resilience and ability to withstand adverse market movements.

Conditional Value at Risk (CVaR)**,** also known as Expected Shortfall, delves deeper into risk assessment than VaR. While VaR focuses on a single worst-case loss threshold, CVaR goes beyond that, providing a more comprehensive picture. It calculates the average loss the protocol can expect to experience in those tail scenarios where losses exceed the VaR threshold. By understanding CVaR, we can focus on the “tail” of the loss distribution, allowing for a more targeted approach to risk management within the Aave Protocol.

# Description of the Default Models

The primary framework that we use to perform the analysis of the Aave protocol is structural models. The structural approach to credit focuses on modeling bankruptcy from an asset values perspective — deterioration of the firm’s value. The credit default event occurs when the assets of a firm drop below a certain predefined level, typically associated with the principal value of liabilities.

## Merton Model

The first default model is based on the Merton (1974) credit risk model, calculating the distance to default, providing a measure of the distance — in asset value standard deviations — of the current market value of assets in a company from a specified default point. The basis of the model is the consideration of the equity as a call option on the underlying unobservable value of assets, with a strike price equal to the face value of the debt/liabilities. While the standard approach often involves approximating asset values and their standard deviations based on equity data, it is crucial to highlight that, in our case, the Aave assets are directly observable — the amount of collateral.

Default occurs if *Vᴀ < X* with probability *P(Vᴀ < X)*, and the bondholder will receive the recovery value *Vᴀ*. Otherwise, if *Vᴀ ≥ X*, the bondholder receives *X* and the equity holder is entitled to *Vᴀ–X*. Therefore, the default probability can be computed from the lognormal distribution of *Vᴀ* based on the first model assumption that the asset value follows a Geometric Brownian Motion process, meaning that the incremental changes of asset values are normally distributed. Based on this assumption, the probability of default is given by the normal cumulative distribution:

Note that under the risk-neutral probability measure we have, replacing *μ* with *r*,

giving the relationship

# Black-Cox Model

In the Merton model, the timing of the default is questionable. The default event is restricted to the debt maturity, independently of the evolution of the asset’s value before that. In other words, the model assumes that default cannot occur before debt maturity. Based on a knock-out barrier option model, the BC model (Black & Cox, 1976) is an extension of Merton’s model and an attempt to address this limitation. It is a first-time-passage model: the opportunity to account for the possibility of default before and at maturity. This model allows for a more informed evaluation of risk throughout the entire credit cycle, from initial issuance to maturity.

The Black-Cox model calculates the probability of default *P(Vᴀ < X)* at any point before and at maturity, taking into account the asset value *(Vᴀ)* and the default barrier *(X)*. The probability distribution function is given by

The model employs two crucial metrics, Direct Distance-to-Default (DDD) and Image Distance-to-Default (IDD), to quantify the risk of default. DDD is analogous to the distance-to-default calculated in the Merton model, a distance of the current market value of assets from a specified default point at maturity. In other words, the DDD reflects financial resilience, with higher values suggesting a larger buffer against potential losses, indicating a reduced likelihood of default. Conversely, a smaller DDD implies an increased risk of default, highlighting vulnerability to adverse market conditions. In other words, it indicates the distance of assets to liabilities at some defined point, offering a clear and robust indication of the protocol’s default. For simplicity, we will refer to Direct Distance-to-Default as Distance-to-Default (DD) for the remainder of the analysis.

While the second term, IDD, does not directly represent any measurement, it specifically comes into play in the Black-Cox model for calculating the probability of default before maturity. IDD helps account for the period between some time and maturity, providing insights into the buffer between the assets and the default barrier. It ensures that the model incorporates the dynamic nature of the default risk over time.

# Expected default frequency (EDF)

EDF stands for expected default frequency and serves as a metric to express the probability that a company’s liabilities surpass assets during a given time frame, usually a year. In other words, EDF essentially estimates the likelihood of default. It is a crucial component of credit risk modeling since it helps to calculate the possible losses that lenders might sustain in the event of a borrower’s failure. Notably, the DeFi market, being relatively new, lacks a sufficient history of default incidents to construct a distribution, as seen in established models like Moody’s KMV, which leverages historical default data of TradFi. Therefore, in the Merton model, the Expected Default Frequency is determined through a normal distribution, where the Distance to Default is incorporated.

As already mentioned, we will utilize the normal distribution for both Distance-to-Default (DD) and Image Distance-to-Default (IDD) when calculating the Expected Default Frequency (EDF) in the context of the Black-Cox model. This approach aligns with the current practices in the DeFi market, given the relatively limited historical data on defaults, providing a pragmatic framework for assessing credit risk and estimating potential losses.

# Key Asset Parameters

Understanding the expected rate of return and volatility is crucial since these parameters play a pivotal role in credit risk modeling. These parameters serve as fundamental inputs for the Merton and Black-Cox models and play a central role in assessing the expected default frequency (EDF) within the Aave protocol. The expected rate of return and volatility estimates involve sophisticated statistical models to ensure accuracy and reliability. In our analysis, we employ the Autoregressive Integrated Moving Average (ARIMA) and Generalized Autoregressive Conditional Heteroskedasticity (GARCH) models, respectively.

The ARIMA model is utilized to estimate the expected asset return by capturing the autoregressive and moving average dependencies within the time series. This model considers how past returns influence the current return and accounts for forecast errors, resulting in a more precise estimate of the expected return.

To determine the optimal parameters for the ARIMA model, we relied on criteria such as the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) for the determination of optimal parameters (p, d, q). Following differentiation and testing with the Augmented Dickey-Fuller (ADF) test, we iterated over various combinations of parameters (p, d, q), limiting each parameter to a maximum value of 3. This approach ensured a thorough exploration of potential parameter combinations while maintaining a balance between model complexity and practicality.

Estimating the daily standard deviation involves fitting the GARCH model to the returns of Aave assets. The daily returns amount on the platform shows the swings in the volume of cryptocurrency assets deposited and withdrawn. The GARCH model is a powerful tool designed to capture the time-varying nature of asset volatility. In our analysis, we opted for the GARCH(1,1) model, representing one autoregressive and one moving average term. This choice is not arbitrary; GARCH(1,1) is widely regarded as a gold standard in financial time series analysis. Its popularity stems from its simplicity, robustness, and effectiveness in capturing the essential features of volatility dynamics.

The accuracy of these estimated parameters contributes to a more granular credit risk analysis. By incorporating the expected return and volatility into the Merton and Black-Cox models, we can effectively assess the expected default frequency, offering insights into the resilience and risk exposure of the Aave protocol over time.

Written by: Artem Aksonov, Maksym Oliinyk, Oleksandr Severhin