USUAL Distribution Model
Last updated
Last updated
The following document is a comprehensive review of the USUAL minting module. It follows that USUAL is minted based on parameters tied to TVL, the underlying interest rate for the collateral of USD0 and additional factors. It should further be noted the math for this paper is specifically focused on a single asset environment where Usual only accepts and mints USD0 (which can be locked as USD0++) for collateral deposited (based on accepted Usual collateral).
The minting of USUAL is completed daily and based on many different factors. The model has three layers of calculations.
We calculate the daily distribution as such given d the global distribution rate, Mt the minting rate, Supply++ the supply of USD0++ , and P the primary market price of USD0:
The distribution rate is immutable and set to 0.25 meaning we will plan to fully distribute the minting rate per the market cap of current locked assets over 4 years total. However, given the dynamic nature of TVL and Price, the real circulating supply will be dynamic and is the aggregate of the daily distribution over time.
The minting rate is a new concept that represents the supply of USUAL per LST to be emitted over four years (considering d = 0.25). It is inversely related to the growth of the LST supply St and directly related to changes in the interest rate Rt. This minting rate is capped so it does not exceed a set maximum rate κt allowing for some sensitivity to interest rate increases and LST supply reduction. Additionally, a growth control variable γt is included, allowing the DAO to accelerate or decelerate changes in the minting rate globally for all LST assets (set to 1 by default, but adjustable through governance).
Where M0 is the initial set minting rate. This is immutable and set to 10.
There are 4 factors calculated which are used to compute the minting rate. We use the supply, rate, scale and cap factors to complete this calculation (as first illustrated in "Minting of USUAL").
St is a supply factor which inversely adjusts the minting rate as supply increases. It has a ceiling of 1 such that if the supply is ever below the initial supply at launch, we use this ceiling value. On the contrary, for any supply Supply++ above the initial supply Supply++, the factor will be below one. It t0 should further be noted we include price in the calculation Pt given this model can eventually be adapted to a multi asset environment (where we would still use the initial supply and current price of USD0 for the numerator rather than the initial supply and current price of the newly added asset - this is still under research).
Rt is a rate factor which directly adjusts the minting rate based on change in the current underlying interest rate of collateral for USD0 rt (subject to the rate floor and celing rmin and P90(πt)). We use a rate floor set by the DAO to mint and distribute a minimum amount of USUAL. Further, we use the 90th percentile of the daily rates from the last 60 days as a rate ceiling to cap the rate factor increasing significantly when large sudden changes in the underlying interest rate are realized (ideally creating a smoother transition in the number of USUAL distributed before and after large rate changes).
The scale factor γt is used to modify the minted amount of USUAL by a set base scale factor γ which is set by the DAO and by a time factor τt. The base scale factor allows the DAO to modify the daily minting of USUAL to either reduce or increase the USUAL emission rate over time; this acts as either an accelerator for inflation when set below 1 (and always above 0) and as a decelerator when set above 1. If set to 1, the scale is neutral such that the inflation is as originally intended. The time factor for this is calculated using the UNIX timestamp at which the last mint occurred and the timestamp at which the current mint occurs. This is then adjusted to calculate how many days (based on using 86400 seconds per day via DF S = 86400, the distribution frequency scalar.) have elapsed between mints to adjust the distribution (given it is supposed to be daily). Since the model is meant to operate on a daily basis but may differ in terms of times between the actual mints of USUAL This function uses the last parameter values available at the time of the call for calculations. We use this to adjust and ensure the correct amount of USUAL per time is minted from one mint to the next.
Further, we set the cap factor κt based on the initial minting rate but is modified over time by the change in the underlying interest rates and the scale factor. κt is set as follows (where r0 is the initial underlying interest rate of USD0 at launch):
The research paper on the USUAL distribution model can be downloaded below.
