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Loan Concepts In Depth

Market Parameters

Each crvUSD market has the following parameters which affect all loans and change automatically due to market forces:

  • Base Price: The base price is the upper price limit of band number 0. Borrow rate increases the base price over time.
  • Oracle Price: The oracle price is the current price of the collateral as determined by the oracle. The oracle price is used to calculate the collateral's value and the loan's health.
  • Borrow Rate: The borrow rate is the annual interest rate charged on the loan. This rate is variable and can change based on market conditions. The borrow rate is expressed as a percentage. For example, a borrow rate of 7.62% means that the user will be charged 7.62% interest on the loan's outstanding debt. See here for how it's calculated.

Each market also has the following parameters which only change if the CurveDAO votes them to change:

  • A: The amplification parameter A is used to calculate the density of liquidity and band width, as well as the maximum LTV of a market.
  • Loan Discount: The loan discount defines how much the collateral is discounted when taking a loan, it is directly related to the maximum LTV of each crvUSD market. See here for more information.
  • Liquidation Discount: The liquidation discount is used to discount the collateral when calculating the health of the loan. See the health section for more information.
  • Sigma: Sigma changes how quickly rates increase and decrease when crvUSD depegs. With a higher sigma interest rates will increase slower when crvUSD depegs. See here for more information.

LLAMMA and Liquidations

LLAMMA (Lending-Liquidating AMM Algorithm) is a fully functional two-token AMM containing the collateral token and crvUSD, which is responsible for the liquidation mechanism. For more detailed documentation, please refer to the technical docs.

When creating a new loan, the put-up collateral will be deposited into a specified number of bands across the AMM. Unlike regular liquidation, which has a single liquidation price, LLAMMA has multiple liquidation ranges (represented by the bands) and continuously liquidates the collateral if needed. All bands have lower and upper price limits, each representing a "small liquidation range." The user's total liquidation range is represented by the upper price of the highest band to the lower price of the lowest band.

A loan only enters soft-liquidation mode once the price of the collateral asset is within a band. If the price is outside the bands, there is no need to partially liquidate and therefore not in soft-liquidation.

The AMM works in a way that the collateral price within the AMM and the "regular price" are treated a bit differently. If the price falls into a band, prices are adjusted in a way that external arbitrageurs are incentivized to sell the collateral token and buy crvUSD in the band. So, if the price is within a band, the user's collateral will be sold for crvUSD, meaning the user's collateral is now a combination of both tokens. This happens for each individual band the user has liquidity deposited into.

This liquidation process does not only happen when prices fall but also when they rise again. If the collateral in a band has been fully converted into crvUSD and the collateral price rises again, the earlier sold-off collateral will be bought up again.

In short: External traders will soft-liquidate a users collateral when the collateral token's price is falling and de-liquidate it again when prices rise again.

Losses in Soft-Liquidation

Positions in soft-liquidation / de-liquidation are suffering losses due to the selling and buying of collateral. If the position is not in soft-liquidation, no losses occur. These losses decrease the health of the loan. Once a user's health is at 0%, the user's position may face a hard-liquidation, which closes the loan.

Hard Liquidations

Hard liquidations occur when the health of a loan falls below 0%, allowing a liquidator to liquidate the loan. Anyone can act as a liquidator and liquidate eligible loans, but this is typically done by searchers.

When a liquidator initiates the process, the following occurs within a single transaction, using a market with WETH collateral and crvUSD debt as an example:

  1. Any collateral which has been swapped to crvUSD in soft liquidation is transferred to Curve and removed from the user.
  2. The remaining crvUSD debt is repaid to Curve by the liquidator.
  3. The liquidator receives the remaining WETH collateral as a reward, which is normally more than the amount repaid.

This process is illustrated in the image below:

Hard Liquidation Hard Liquidation

Bad Debt

Bad debt occurs when a loan is not profitable to liquidate. This could happen for many reasons, including gas prices being higher than the profit from a liquidation, a sequencer being down on an L2, or simply no one searching for profitable liquidations in a new market. It looks like the following:

Bad Debt Bad Debt

In this example no rational liquidator will begin the liquidation process because they will lose value by doing so.

crvUSD is only minted on Ethereum and uses high-quality assets with strong liquidity to mitigate the risk of bad debt. Due to these precautions, bad debt is not expected to occur within the crvUSD minting system. However, bad debt can and has occurred within specific Curve Lending markets, as they are permissionless and do not affect the integrity of the crvUSD stablecoin.

Bands (N)

When creating a loan, the added collateral is spread among the number of bands selected. Minimum amount is 4 bands, and the maximum amount is 50 bands.

A band essentially is a price range, with an upper and lower price limit. If the price of the collateral is within the limits of a band, that particular band is likely to be liquidated.

Note that band price ranges drift higher over time as base price increases by the borrow rate

In the illustration above, there are multiple bands with different price ranges. The light grey areas represent the collateral token, which in this example is ETH. As depicted, the bands below the collateral token's price are entirely in ETH since there is no need for liquidation, given the higher price. The dark grey areas represent crvUSD. Because the price of ETH fell within the band on the far right, the deposited collateral (ETH) is converted into crvUSD. In this instance, the band consists of both ETH and crvUSD. If the price continues to decline, all collateral in the band will be fully converted into crvUSD, and the band to the left will undergo soft-liquidation.

Remember: When prices rise again, the opposite is happening. The ETH which was converted into crvUSD earlier will be converted back into ETH again.

A band which has fully been soft-liquidated. All collateral was converted into crvUSD because the price of the collateral is below the liquidation range.

A band which currently is in soft-liquidation. It contains both, the collateral token and crvUSD.

A band which has not been liquidated yet (composition is 100% collateral token). The price of the collateral is above the liquidation range.

Band Formulae:

A controls the density of the liquidity. This is directly related to the width of the bands. Band width at any price can be estimated to be:

\[\text{bandwidth} \approx \frac{\text{price}}{\text{A}}\]

To find the exact upper price limit and lower price limits of the bands the following formulae can be used:

\[\begin{aligned} \text{upperLimit} &= \text{basePrice} * \left( \frac{A-1}{A} \right)^{n} \\ \text{lowerLimit} &= \text{basePrice} * \left( \frac{A-1}{A} \right)^{n+1}\end{aligned}\]

Where:

  • \(\text{basePrice}\): The current base price of the desired market
  • \(A\): The amplification factor of the desired market (default is 100)
  • \(n\): The Band Number, e.g., \(-\)67.

Band Calculator

Use the calculator below to simulate how bands are shaped and how liquidity density changes with different parameters. By definition the liquidity density will be 100% at band 1. Liquidity density increases as band width decreases, because the same amount of collateral will be spread over a smaller price range.

Inputs:

Loan Health

Based on a user's collateral and debt amount, the UI will display a health score and status. If the position is in soft-liquidation mode, an additional warning will be displayed. Once a loan reaches 0% health, the loan is eligible to be hard-liquidated. In a hard-liquidation, someone else can pay off a user's debt and, in exchange, receive their collateral. The loan will then be closed.

The health of a loan decreases when the loan is in soft-liquidation mode. These losses do not only occur when prices go down but also when the collateral price rises again, resulting in the de-liquidation of the user's loan. This implies that the health of a loan can decrease even though the collateral value of the position increases. If a loan is not in soft-liquidation mode, then no losses occur.

Losses are hard to quantify. There is no general rule on how big the losses are as they are dependent on various external factors such as how fast the collateral price falls or rises or how efficient the arbitrage is. But what can be said is that the losses heavily depend on the number of bands used; the more bands used, the fewer the losses. Daily losses based on current data are shown here.

The formula for health is below, this is visualized in the Health Calculator applet as well.

\[\begin{aligned} \text{health} &= \frac{s \times (1-\text{liqDiscount}) + p}{\text{debt}} - 1 \\ p &= \text{collateral} \times \text{priceAboveBands} \end{aligned}\]

Where:

  • \(\text{collateralValue}\) : the value of all collateral at the current LLAMMA prices
  • \(\text{liqDiscount}\) : the liquidation discount for the market (how much to discount the collateral value for safety during hard-liquidation).
  • \(\text{debt}\) : the debt of the user
  • \(s\) : an estimation of how much crvUSD a user would have after converting all collateral through their bands in soft-liquidation. This can be very roughly estimated as: \(\text{collateral} \times \left( \frac{\text{softLiqUpperLimit} - \text{softLiqLowerLimit}}{2} \right)\)
  • \(p\) : The value above the soft-liquidation bands. Found by multiplying the amount of collateral by how far above soft-liquidation the current price is. If user is in or below soft-liquidation, this value is 0.
  • \(\text{collateral}\) - The amount of collateral a user has, e.g., if a user has 5 wBTC, this value is 5.
  • \(\text{priceAboveBands}\) - The price difference between the oracle price and the top of the user's soft-liquidation range (upper limit of top band). This value is 0 if user is in soft-liquidation. See applet below for a visual representation.
  • \(\text{collateralPrice}\) - The price of a single unit of the collateral asset, e.g., if the collateral asset is wBTC, this value is the price of 1 wBTC.

Health Calculator

Use the applet below to simulate how health works, soft-liquidation losses are given as numbers in a comma separated list, the first number is the starting band onwards. The light blue shaded areas in the bands represent the value without using the soft-liquidation discounts, while the dark blue areas are the values after discounting.

Inputs:



Health (including value above bands):

Health (not including value above bands):

The Curve UI will either show health adding value above bands or without that value based on how close to liquidation a user is. If the active band (Oracle price band) is 3 or less bands away from the user's soft liquidation bands, the UI will show the health not including value above bands. Otherwise it will show the health including the value above bands.

The health values on the Curve UI and within smart contracts will always be slightly less than the values here. Health is calculated by estimating the amount of crvUSD/debt tokens the collateral will be swapped for in each band. This takes into account how much liquidity is in each band, the more liquidity in a band the less slippage Curve estimates will occur. This slippage estimation slightly reduces a user's health.

Loan Discount

The loan_discount is used for finding the maximum LTV a user can have in a market. At the time of writing in crvUSD markets this value is a constant 9%, in Curve Lending markets this value ranges from 7% for WETH to 33% for volatile assets like UwU. Use the calculator below to see the maximum LTVs a user can have based on the loan_discount, and amplification factor A (with 4 bands, N=4). The formula is:

\[\text{maxLTV} = \left(\frac{A - 1}{A}\right)^2 \times (1 - \text{loan_discount})\]

Maximum LTV Calculator

Inputs:

Result

Maximum LTV:

Borrow Rate

In [crvUSD minting markets](The general idea is that borrow rate increases when crvUSD goes down in value and decreases when crvUSD goes up in value. Also, contracts called PegKeepers can also affect the interest rate and crvUSD peg by minting and selling crvUSD or buying and burning crvUSD.

The formula for the borrow rate is as follows:

\[\begin{aligned}r &= \text{rate0} * e^{\text{power}} \\ \text{power} &= \frac{\text{price}_\text{peg} - {\text{price}_\text{crvUSD}}}{\text{sigma}} - \frac{\text{DebtFraction}}{\text{TargetFraction}} \\ \text{DebtFraction} &= \frac{\text{PegKeeperDebt}}{\text{TotalDebt}}\end{aligned}\]

with:

  • \(r\): The interest rate.
  • \(\text{rate0}\): The rate when PegKeepers have no debt and the price of crvUSD is exactly 1.00.
  • \(\text{price}_\text{peg}\): Desired crvUSD price: 1.00
  • \(\text{price}_\text{crvUSD}\): Current crvUSD price.
  • \(\text{sigma}\): variable which can be configured by the DAO, lower value makes the interest rates increase and decrease faster as crvUSD loses and gains value respectively.
  • \(\text{DebtFraction}\): Ratio of the PegKeeper's debt to the total outstanding debt.
  • \(\text{TargetFraction}\): Target fraction.
  • \(\text{PegKeeperDebt}\): The sum of debt of all PegKeepers.
  • \(\text{TotalDebt}\): Total crvUSD debt across all markets.

A tool to experiment with the interest rate model is available here.

PegKeepers

A PegKeeper is a contract that helps stabilize the crvUSD price. PegKeepers are deployed for special Curve pools, a list of which can be found here.

PegKeepers take certain actions based on the price of crvUSD within the pools. All these actions are fully permissionless and callable by any user.

When the price of crvUSD in a pool is above 1.00, they are allowed to take on debt by minting un-collateralized crvUSD and depositing it into specific Curve pools. This increases the balance of crvUSD in the pool, which consequently decreases its price.

If a PegKeeper has taken on debt by depositing crvUSD into a pool, it is able to withdraw those deposited crvUSD from the pool again. This can be done when the price is below 1.00. By withdrawing crvUSD, its token balance will decrease and the price within the pool increases.

More on PegKeepers here