The Threshold For Staked EOS Collateral: A Robust Framework

The Equilibrium framework’s main features are its EOSDT-to-USD parity protection and its ability to maintain a high level of confidence in its responsiveness to negative market events. In this in-depth analysis, we explain the steps to collateral asset adjustment. Read the full version of the article with a description of the mechanisms supporting the collateralization ratio, and some insights on Equilibrium user behavior based on the data modeled from the DAI-ETH and EOSDT-EOS use cases here.

EOS holdings have to be staked for three consecutive days before granting their holder the right to vote for an EOS block producer. Equilibrium lets a portion of the EOS collateral be staked as long as the value of the remaining liquid portion (post-liquidation if necessary) can cover at least 170% of the global EOSDT position. At all times, and particularly in extreme events, Equilibrium must ensure that there is enough EOS available for liquidation when necessary, defending the critical 170% collateralization ratio.

The following steps describe a statistical approach to determining the portion of EOS collateral that can be staked.

STEP 1: Adjust the collateral asset returns for stochasticity.

Position holders are naturally risk averse. This explains why the actual collateralization ratio (like the DAI-ETH case) is significantly above the critical 170% ratio. The collateralization ratio furthermore has a natural resistance level (most likely around 250%), that is explained by a holder’s aversion to liquidation. Position holders have a tendency to increase their collateral in periods of higher volatility to avoid auto-liquidation.

In this first step, we are looking at the significant negative jumps no matter the market’s regime or volatility state. In other words, we’re looking for negative shocks or surprises in EOS volatility and returns. These shocks are the only movements that can break the natural resistance level without warning. Position holders and bad debt keepers wouldn’t have any time to replenish their collateral position or sell it at a reasonable market price. Extreme events like these drive Equilibrium’s insolvency risk.

In order to eliminate the effects of stochastic volatility (time-varying or regime-dependent volatility), the first step is to scale each daily return by a measure of lagged volatility. Applying the suggested measure by Jurek, the chosen volatility measure is the three-month lagged standard deviation. The resulting log-return to volatility ratios is a series of z-scores.

STEP 2: define the empirical crash distribution.

In order to isolate the collateral’s crash risk exposure, we have to define a lower threshold below which z-scores are considered “crash” material. Given the limited historical data on EOS, we set the threshold at -2.0 for our exercise.

At a z-score of -2.0 and a standard deviation level of 5% (corresponding to an annualized volatility of 200%), the daily negative log-return equivalent is -10%.

At a z-score of -6, the daily log-return equivalent is -30%.

If no portion of the collateral is staked, the 41% (=70%/170%) required haircut and an extra 20–30% posted by risk-averse EOSDT generators should largely compensate for extreme adverse market movements (like an EOS collateral drop of 60%). In these conditions, staking a portion of EOS collateral can be a reasonable option. But how reasonable exactly?

Figure [1] outlines the selected crash observations.

Figure 1: EOS/USD Z-scores computed with the daily log-return divided by 3-months lagged standard deviation

The conditional crash distribution is obtained by transforming the z-scores into losses. The current volatility level is used with the distribution of z-scores to simulate a series of potential losses:

STEP 3: fit a Pareto distribution to the crash risk distribution.

The Generalized Pareto Distribution (GPD) is a classic alternative to the normal distribution to better fit the probability distribution of extreme events. It has two parameters: the scale parameter and the shape parameter. The maximum likelihood method produces point estimates of these parameters, as well as a standard error that we use to determine a 99% confidence interval.

The worst z-score observed is -3.36, computed with a -15.75% negative return scaled by a lagged three-month standard deviation at 4.68%. It is not the worst return recorded — that distinction belongs to a -30% return recorded on Sep 4th, 2017, with a standard deviation of 10.40%. -3.36 is the worst negative shock.

The worst z-score (worst shock) combined with the worst volatility level observed sets a potential negative return at -34.96%!

Furthermore, the worst z-score observed is not the worst z-score of the population of log-returns. The observed z-scores constitute a sample of the population. We mitigate the sample bias using the 99% confidence interval.

The confidence interval on the GPD parameters implies a confidence interval on the quantile of the worst z-score observed. In this interval, the lowest quantile candidate for the -3.36 worst z-score is only at 0.655. In other words, if we have a strong negative return that belongs to the crash distribution (z-score < -2), then the conditional probability that its corresponding z-score exceeds the -3.36 level is 34.5%, regardless of the current volatility level. If the three-month lagged standard deviation is at 10% with a shock level at -3.36, then the equivalent negative return is -33.6%.

Figure 2: EOS/USD Crash distribution (in red) fit to a Pareto distribution.

STEP 4: Determine a safe staked portion of the collateral.

The probability of a crash multiplied by the conditional level of the crash gives an insurance premium against a potential crash. The conditional level of the crash should not exceed the unstaked liquid collateral. Any remaining portion could theoretically be staked.

We can fit a gamma distribution on the negative z-scores, or similarly fit a beta distribution on the crash distribution. Both distributions have very appealing properties: we can derive the cost of insuring a crash ξ from a simple closed-form formula. The cost is written as the product of the probability of observing market failure (π Q) multiplied by the exposure to the crash, which is the expected shortfall above the critical z-score level.

a and b are the parameters from our conservative beta distribution.

β a parameter that measures the loss we want to insure against such that: (1-x)β -1 is the consequent loss function where x is the loss (crash) on the underlying EOS.

Ɣ is a risk aversion parameter.

A simplified version (β=1) is given by:

To fit the beta distribution on the crash series, we enforce two constraints, just like Jurek’s article [2]:

• the median of the crash distribution should be equal to the median of the beta distribution
• the quantile of the worst observed z-score should equal to the most conservative quantile derived from step three.

The beta distribution estimation over crash returns gives the following estimates:

z-score threshold of -2.0: a=25.6, b=177.7 (figure [3]).

z-score threshold of -1.5: a=18.9, b=151.8.

z-score threshold of -1.0: a=8.34, b=91.4 (figure [4]).

Choosing Ɣ=2.5:

z-score threshold of -2.0: the resulting loss above threshold is at 13%.

z-score threshold of -1.5: the resulting loss above threshold is at 11%.

z-score threshold of -1.0: the resulting loss above threshold is at 9%.

Assuming a safe collateralization ratio of 250% and a three-month lagged standard deviation at 10%, the loss in this ratio would amount to a 47.5% drop (10% of the collateral value, and a 9% excess loss in value), which would leave us with a 202.5% collateralization ratio if it happened. This is still above the critical 170% level.

Assuming the same level in market crash every day, this exposure over three days would be 1-(1–47.5%)³ = 85.53%. Starting from a safe collateralization ratio of 250%, the loss in this ratio would amount to 213%, which would leave us with a 37% collateralization ratio if it happened. This is below the 100% parity level. In this case, a 63% cut of the 250% EOS collateral ratio must remain liquid and reserved for potential liquidation. At least 133% should furthermore be in reserves if we wish to satisfy the critical 170% level. We can then consider staking the remaining collateral for three days.

The model offers a robust and conservative methodology to determine whether a portion of the collateral can be staked. Robustness refers here to the model’s choice to use the confidence interval rather than the point estimates of the Pareto distribution. This methodology should protect us better against potentially new drawdowns, or errors in the sample of realized log-returns. But our analysis is far from complete. Questions remain over the choice of the volatility measure, the risk aversion parameter, the probability of having extreme crash returns over three consecutive days, and most importantly, the choice of the z-score threshold that should ideally be below -6. The small sample of daily data we are working with clearly has its limitations.

Figure 3: density of the 19 worst crash returns corresponding to a z-score < -2.0 — Constraint (at 50% and the lowest percentile of the max crash return obtained by fitting the Pareto distribution): Beta fit in blue.

Figure 4: density of the 64 worst crash returns corresponding to a z-score < -1.0 — Constraint (at 50% and the lowest percentile of the max crash return obtained by fitting the Pareto distribution): Beta fit in blue.

Given the numerous sources of liquidity for trading EOS today, the expected collateral to debt ratios (~250%-300%) would sit in a very high range, compared to other traditional liquid-like collateral assets.

There are always more factors to consider, like market liquidity risk and market impact. When all these measurable factors are accounted for, only the black swan factor remains. Black swans are fundamentally unpredictable and impossible to model.

EOSIO and Equilibrium can give their community the power to deal with black swans the same way they are dealt with in the real world: by adapting or changing the rules. More than a set of static rules and parameters, Equilibrium is a robust, dynamic framework where a variety of models and parameters are possible as long as the larger group of voters agree to the changes.

With this article and more to come, we are educating the Equilibrium community, instilling confidence in the framework’s mechanisms, and ultimately benefiting the community itself.

[1] CONT, EMPIRICAL PROPERTIES OF ASSET RETURNS, STYLIZED FACTS AND STATISTICAL ISSUES. October 2000.

[2] JUREK, CRASHES AND COLLATERALIZED LENDING, NATIONAL BUREAU OF ECONOMIC RESEARCH, Sep 2011, https://www.nber.org/papers/w17422.pdf

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