A Ratings-Based Model for Credit Events in MakerDAO

Alex Evans
July 2019

Abstract

This article introduces a ratings-based model for credit events in the MakerDAO lending system. To account for stochastic loan terms, we extend conventional ratings-based models to include two absorbing states, full repayment and delinquency. This extension enables the exploration of the long-term behaviors of individual loans and global system states. The transition rates of the continuous-time Markov Chain are estimated using on-chain data for MakerDAO from Ethereum. The model can be used in credit risk management applications and for pricing derivatives with state-dependent payoffs such as MKR.

1 Introduction

We introduce a ratings-based Markov model for credit events in MakerDAO. Unlike fixed-term loans, Maker allows borrowers to repay ("Wipe") their loans including all accrued interest ("stability fees") at any time. Loans are also subject to liquidation ("Bite") by third-party "Keepers" should the borrower fail to maintain an appropriate loan-to-value ratio (this is akin to a margin call). An additional 10 percent "liquidation fee" on outstanding debt is levied on liquidated loans. The model accounts for stochastic loan terms by including repayment as an absorbing state alongside delinquency. The long-run behavior of each loan can be determined from the transition rates between states. With additional assumptions about birth rates and loan sizes, nearly any aggregate system parameter in Maker can be forecasted.

The model lends itself to convenient statistical estimation using on-chain data and, optionally, values for external covariates. The intended application of the framework is in risk management. The model can be used to compute the expected magnitude of credit exposure for a pool of loans for given covariate values. Both expected and unexpected losses can be computed with additional data on recovery values. The exploration of fee policy choices is also available to users of the model, including forecasting the anticipated effects of stability fee changes. A key feature of the model is the flexibility of its assumptions. Throughout this article, effort is taken to highlight cases where assumptions can be adapted for analysts to refine the model for their intended application.

The remainder of this article will proceed as follows. Section 2 outlines the mechanics and assumptions of the model and examines the long-term behavior of individual loans and global system state. Section 3 discusses strategies for estimating model parameters using on-chain data. Section 4 applies the model to risk management problems using the example of stress testing liquidations. Section 5 provides a brief concluding note on a potential pathway to pricing MKR.

2 The Model

This section introduces a ratings-based Markov model for credit events in the Maker system. Ratings-based credit models allow loans to be in a variety of states, representing different credit ratings, the lowest of which is default. The most popular formulation of this model is found in Jarrow, Lando, and Turnbull (1997), JLT for short. The model presented here draws from this approach, with some important adjustments due to the idiosyncrasies of the Maker system. Individual "Draw" transactions are treated as discrete loans, each of which is modeled according to a Markov Chain on a discrete state space. Each loan is permitted to be in one of four states: Safe, Unsafe, Bitten, or Wiped. Unsafe loans are below the minimum collateralization ratio and may be liquidated by Keepers. Upon creation, each loan begins in the Safe state with transition rates to other states modeled as exponential parameters, forming a continuous-time Markov Chain.

The generator matrix depicts the instantaneous transition rates associated with the transition function, P(t), i.e. P⁰(0) = Q. This transition matrix is given by the matrix exponential

$$
egin{align*}
extbf{P}(t) = e^{t extbf{Q}} = extbf{I} + t extbf{Q} + rac{t^{2}}{2!} extbf{Q}^{2} + ...
ext{(n=0 to } ext{)}\
ext{.} ext{(1)}
ext{where } ext{P}{ij}(t) ext{ is the probability of transitioning from state i to j at t.} ext{(2)}
ext{However, for the embedded chain, } ext{P} = Q. ext{(3)} ext{Long-term dynamics are governed by } ext{lim}{t o ext{∞}} ext{P}{ij}(t) = ho{i}. ext{(4)}

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3 Risk Management Applications

Analysts can use the estimation of the loan Markov Chain to stress test the resilience of the system under different scenarios. This can be applied to managing both portfolio credit risk and fee policy for Maker.

5 Future Work: Note on Pricing

One of the most appealing features of JLT is that it can efficiently handle the valuation of credit derivatives whose payouts are state-dependent. This is particularly interesting in the case of MKR as loans have different payouts to MKR holders depending on whether they are closed in the Safe or Bitten states.

References

[1] Bomm, A. (2016). Understanding Credit Derivatives and Related Instruments. Waltham, MA: AP.
[2] Dobrow, R. (2016). Introduction to Stochastic Processes with R. Hoboken, NJ: Wiley.
[3] Jackson, C. (2019). Multi-state modelling with R: the msm package.
[4] Jarrow, R. and Lando, D. and Turnbull, S. (1997) A Markov Model for the Term Structure of Credit Risk Spreads. Review of Financial Studies, Vol. 10, No. 2.