Fishing for Gains in the Zero Stability Pool: Logistic Regression Analysis

February 24, 2023
min read


A model is derived from historical liquidation behavior of Zero’s “parent” protocol Liquity. The model estimates the probability and size of a liquidation for a given market movement. This implicitly models user behavior in terms of choice of collateral ratios and how quickly users adjust their positions in the face of price movements. Historical BTC daily price data, beginning in 2013, is then used as input to the model. A stability pool that has 5% of the total stablecoin supply—the level of the Zero stability pool at the time of writing—is used to calculate relative yields for a stability pool investment strategy. The model estimates that the Zero stability pool at this funding level would have had an average annual APY of 165%—with a wide variation from year to year.


We are considering investment possibilities for the Sovryn Dollar (DLLR), an aggregated bitcoin-backed stablecoin that will soon be released by Sovryn. When opening a line of credit on Zero, DLLR will be the default currency. However, you will still have the option to select ZUSD. One promising option is an investment in the Zero stability pool. Behind the scenes, DLLR is converted to the Zero-native stablecoin ZUSD, which is then deposited into the stability pool. 

In our initial analysis of the stability pool as an investment vehicle, we saw that an analysis of Liquity’s historical returns yields some interesting insights into the yield potential of Zero’s stability pool. Liquity is the protocol from which Zero was forked, and it has a longer history than Zero. The mechanics and parameters of Zero are the same as Liquity. Therefore, it makes sense to analyze Liquity to get a sense of the performance of Zero. If you haven’t read that post, you should read it now before continuing with this one. If you don’t want to see equations, statistics, and plots, then go back to the TL;DR above and stop there.

In this post we look at how we might model the liquidation behavior of Liquity so that the model can be applied to any historical price data to get a sample of historical performance. The model doesn’t attempt to capture or predict actual price movements. Instead, it implicitly characterizes the risk tolerance of borrowers based on where they set and maintain their collateral ratios. It also characterizes their speed (or lack thereof) in adjusting to downward movements in price that could trigger liquidations based on actual behavior in historical downward movements. This borrower behavior manifests itself in the track record of liquidations under various price movements, which is what the model characterizes directly. The model allows us to evaluate a strategy of investing in the stability pool and immediately exchanging liquidated RBTC for more DLLR that is then reinvested in the stability pool.

Logistic Regression Model

We begin by developing a model that translates certain market movements into a probability of a liquidation. Remember that a liquidation occurs whenever the collateral value in an individual line of credit falls below 110% of the nominal loan value in USD. A study of the history of liquidations on Liquity shows that more than 95% of the total liquidation gains as a fraction of stablecoin supply occurred on only 11 out of the total number of days since Liquity was launched. Therefore, we can focus on the characteristics of these days in comparison to other days to develop the model. Evidently, certain days have characteristics that surprise low-collateralized users or catch them unprepared to adjust their lines of credit to a safer condition.

Gains are defined as the difference in the value of the liquidated collateral denominated in USD and the nominal stablecoin value used from the stability pool to liquidate the loan. Gains are expressed as a percentage of the total stablecoin supply (total debt of the system). These days and gains are shown in the chart below.

Note that 3 of those 11 days are clustered together in January 2022 so the points overlap.

Using those 11 days we studied the circumstances under which liquidations occurred. Obviously, a large decline on the day of the liquidation indicated a higher likelihood of a liquidation. Therefore, we used the drop from the previous day’s close to the low of the day under consideration as an indicator of liquidation probability and of the relative size of the liquidation. We expressed that in percentage terms so that the number would not depend on the absolute price over time. Call this value D.

We also noticed that a more sustained decline over time indicated a scenario where a liquidation was more likely. Some users evidently fail to pay attention or to react even when the decline develops over several days rather than happening overnight. So we used the percentage drop from the close two weeks earlier to the present daily low as another indicator. Call this value W.

We also looked at the total collateral ratio (TCR) of the system (total collateral value / total stablecoin supply). Surprisingly, we found that there was very little correlation between the TCR and a liquidation event. This can be explained by the fact that liquidations don’t happen to an average user. They happen to outlier users on the low end of the collateral ratio spectrum. Therefore, we didn’t use this in our model. The model captures outlier behavior, not average behavior.

We adopted a probability modeling methodology called logistic regression. Logistic regression translates a simple score value into a probability. The score is based on a weighted sum of indicators (plus an offset). Therefore, the score S was defined as

where the weights a, b, and c need to be determined from the data.

To translate the score into a probability, we used the logistic function

which is plotted below.

This is sometimes referred to as an S-curve because of its shape. The logistic function smoothly translates a linear score into a probability. The lower the score, the closer the logistic function is to a probability of 0; the higher the score, the closer the logistic function is to a probability of 1.

To apply logistic regression, we maximize the resulting likelihood function over the weights. In essence, we find the weights that make the probability of a liquidation as close to 1 as possible on the 11 days when significant liquidations occurred and as close to zero as possible on all the other days. Our study yielded the following probability model:

In reality, the probability calculated by the model for any given day is very low—even on the days when liquidations actually occurred. This makes sense because there are many similar days in terms of large declines when no liquidations occurred. The key is that the probability tends to be higher on days similar to liquidation days, and that is sufficient for our model to be useful.


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Linear Regression Liquidation Size Model

Next, we need a way to estimate the profit P of the liquidation as a percentage of the supply for a given liquidation event. Again, we use the same indicators but this time with a simple linear regression model to estimate the profit. A linear regression yielded the following model for profit percentage:

Remember that this model characterizes liquidations given certain market movements. It doesn’t attempt to estimate the actual market movements. This means we can use this model with actual market data to calculate the probability and profit of a liquidation. This characteristic is important for two reasons: 

  1. The actual data we have only covers one specific price sequence, but we would like to explore potential liquidation returns over various market periods. This model characterizes the probability and size of daily total liquidations given certain market movements. It does not try to predict how often or when those market movements will occur. So we should be able to apply the model to different historical data and see the probabilities and sizes of liquidations under those circumstances.
  1. BTC and ETH price movements are not exactly the same. BTC has a lower volatility than ETH. Therefore, we would naturally expect liquidations to occur less often in a BTC market than in an ETH market. Our model accounts for this difference. If daily drops (D) and biweekly drops (W) are not as strong—as we would expect for BTC—then our model will predict lower likelihoods of liquidation events and will impact the statistical behavior accordingly.

Monte Carlo Simulation

Since this is only a probabilistic model, we don’t get a fixed liquidation pattern for a given set of historical data. Instead, we just get a sequence of probabilities. To illustrate this, we calculated a sample of historical liquidations using the bitcoin market price over a 10-year period. For each day, we calculated the probability of a liquidation and then randomly decided whether a liquidation had occurred based on that given probability. For example, if the probability of a liquidation on a given day was calculated to be 10%, most likely that day would not record a liquidation. However, on average we would see one liquidation occur in every ten such 10% days.

One random sample of liquidations is shown below.

Using the exact same market data, we calculate the same probabilities but get a different set of random liquidation events based on those same probabilities, leading to a different pattern below.

As you can see, the results are different. For example, the first case has four liquidations in 2021; the second one has three. There are many other differences. This reflects the fact that the scoring elements we identified don’t fully distinguish liquidation days from non-liquidation days. Instead, we have to consider the situation probabilistically.

These random results illustrate the difficulty of using a probability model to calculate expected outcomes. Unfortunately, calculating expected outcomes directly from a probability model like we’ve developed can be very difficult if not impossible. Fortunately, a simple alternative exists—create a large quantity of random results and then simply average them. This is called the Monte Carlo method. A Monte Carlo simulation calculates statistics by creating random outputs over and over again based on the probabilities that have been calculated.

To calculate expected returns over a particular historical period, we conducted a Monte Carlo simulation. Using this method we calculated random patterns of liquidations over a ten-year period over and over again—100 times. Then we calculated statistics for each year. The results are summarized in the box plot below.

This plot requires some explanation as well as study. The box over each year covers 50% of all the returns that year—-from the 25th percentile to the 75th. The middle vertical level of the box (not marked) is the mean. The red line inside the box is the median value—50% of the returns are below and 50% above that line. The bars at the extremes of the dashed lines represent the range of returns. In some cases, one or two values were significantly separated from all the others. These are considered outliers and are designated by a +.

Each box plot is located at the beginning of the year it summarizes. It corresponds to the BTC price from the labeled year to the beginning of the next labeled year. The BTC price plot is also included for reference.

Several observations can be made about these results:

  1. The results vary considerably. They vary considerably from year to year due to differing market conditions, but they may also vary considerably even under very similar market action. For example, our model shows that under 2022 market movements we could expect a return anywhere between 0% and 3% about 50% of the time and outside that range about 50% of the time. Our model is only able to give a probabilistic range, not a specific prediction, even with the exact same historical data.
  1. Liquidation conditions can occur in both bear and bull markets. The biggest bull market in the last ten years occurred in 2013. And yet 2013 also had liquidation conditions more than any other year—by far! This tells us that the stability pool could be a good investment even in a bull market.
  1. The returns can be quite sparse and require patience.
  1. These returns may seem small, but remember that they are a percentage of overall supply, not the amount in the stability pool. Currently, Zero has only 5% of the supply in the stability pool, so we need to multiply these estimates by 1/0.05 = 20!.
  1. Using the current stability pool size for Zero, this analysis projects that the average APY in the stability pool would have been 165%. Of course, this rate varies tremendously by year.

Assumptions and Caveats

To apply this analysis to project future gains in Zero requires at least two critical assumptions. First, it assumes that Liquity users are similar enough to Zero users that the former can serve as a model for the latter. This assumption may not be perfectly valid. Zero users, because they come from bitcoin holders, may tend to be more conservative than Liquity users, because they come from ether holders. Also, there is reason to believe that Liquity has attracted large institutional investors, whereas there is little indication that that is the case for Zero so far. 

However, one can measure overall risk tolerance by looking at the total collateral ratio of the protocol. Zero and Liquity have had somewhat similar TCRs over time. Typically, Zero has had a slightly lower TCR than Liquity, which possibly indicates a greater willingness to risk liquidation, not less. So the supposition of less risk tolerance among Zero users is not confirmed by the TCR.

Second, the analysis assumes that user behavior with respect to liquidation risk does not change over time. It is difficult to assess this assumption. Two observations are in order. 

One, Zero will continue to attract new users who have no prior experience with the protocol. These users will begin their learning and behavioral responses when they discover Zero and begin to consider it and then use it. There is little reason to think that entire batches of new users will be significantly different in their risk tolerance than previous new users, although this could certainly drift with changes in society, the bitcoin market, and the economy. 

Two, Liquity has operated for 21 months and has seen significant liquidations throughout its history. There is no obvious adaptation in user behavior during that time frame. 

One thing that will obviously change is the level of participation in the Zero stability pool. As participation goes up relative to supply, the share of the gains will be diluted over a larger pool. As stability pool participation grows from 5%, the APY multiple will decrease from 20 over time. Thus, early participants are likely to gain more than later participants as investors increasingly identify the stability pool as an attractive investment strategy and the stability pool grows relative to supply.

This strategy has some practical challenges. It assumes that liquidations are monitored and that the liquidated collateral is instantly sold for stablecoins and put back into the stability pool. In practice, a bot would be required to monitor liquidations 24/7. Also, some liquidations could potentially use up the entire stability pool supply. In that case, further liquidations would be distributed to all remaining lines of credit and the gains would not be captured by the stability pool. Finally, selling a large amount of RBTC instantly could result in significant slippage in the price received. 

The solution to this would be to sell small amounts over time and allow arbitrageurs to return the price to the external market price. Another solution would be to transfer RBTC out to BTC and sell it on an exchange. A final possibility would be to deposit the RBTC in Zero and borrow more stablecoins to deposit in the stability pool. All of these strategies would involve some market risk of holding RBTC for a time.

The bottom line is that this model gives us some perspective on what might be expected with an investment in the stability pool. Simple models like the one above can’t guarantee any kind of precise prediction of future profits. (This one doesn’t even give a precise prediction of past profits!) However, it does make a case for a promising use case of DLLR; the stability pool could be a lucrative investment over time for patient investors. And in the meantime, it allows you to hold your value in a decentralized, bitcoin-backed stablecoin while you wait for gains to accrue.

Past performance—or model performance based on past data—is no guarantee of future results.

Stay Sovryn!


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