# Interest Rate Model

Ensuring a steady pegged value for price stability

To minimize price volatility and maintain a stable pegged value, the protocol may adjust the supply of USC in circulation by charging a stability fee, which is an interest rate paid by users to hold the stablecoin.

The stability rate is typically adjusted based on market conditions and the demand for the stablecoin to ensure that the stablecoin's value remains stable (i.e. pegged to $1) over time.

To better ensure that USC remains pegged to $1, Carbon shifted from using a static interest rate model to a dynamic interest rate model. This is aimed at promoting the stability of the stablecoin over time.

## Interest Rate Theory

A stability rate for a stablecoin is the interest rate or fee charged by the stablecoin's issuer (in this case, Carbon protocol) to maintain the stablecoin's pegged value.

The lower the interest rate,

The higher the incentive to mint and sell USC;

The lower the incentive to lend or buy USC (and vice versa).

Theoretically, there are 2 scenarios in which a collateralized stablecoin like USC can lose its peg:

Under-peg: An increase in demand for redemption of USC will exert selling pressure on USC, leading to an expansion in the circulating supply of USC and in turn causing the value of the stablecoin to decrease.

Over-peg: A increase in speculative behaviour of USC (i.e. betting that the value of the stablecoin will rise in the future) will exert buying pressure on USC, contracting the circulating supply of USC and in turn causing the value of the stablecoin to increase.

## Static Interest Rate Model

Previously, following a static interest rate model, the value of USC was determined by a fixed interest rate that could be adjusted by governance.

The interest rate for minting USC was previously fixed at 1.5%.

The fixed interest rate refers to the rate the protocol sets and maintains as the official exchange rate, such that the value of USC is pegged at $1.

### Limitations

A growing number of users were observed borrowing and selling USC for extended periods of time, causing the the circulating supply of USC to expand. Consequently, USC had a tendency to be under-peg.

To bring the peg of the stablecoin back to $1, the interest rate of both newly minted and existing USC stablecoins had to be actively adjusted upwards via governance. This was to encourage the repurchasing and returning of the borrowed USC, in turn contracting the circulating supply of USC.

Maintaining the peg of USC using the Static Interest Rate Model demands constant monitoring of the stablecoin's circulating supply, which can be challenging and impractical. Yet, failure to do so may hinder the timely adjustment of interest rates, making it difficult to react appropriately to any changes in demand for USC.

## Dynamic Interest Rate Model 🆕

To improve the peg stability of USC, Carbon introduced the use of a dynamic interest rate that ensures the value of USC always remains at $1, which is vital for its utility as collateral in cross-margin trading of futures and perpetuals.

With the implementation of the Dynamic Interest Rate Model, should USC lose its peg, the interest rate of USC will be automatically adjusted upwards or downwards every 6 hours. The adjustment will be based on the Time-Weighted Average Price (TWAP) of the market price of USC.

To maintain the peg, USC follows a dynamic interest rate model. The interest rate will be applicable to all USC, whether they are already in circulation or newly minted.

### Parameters

Interest Rate Epoch Duration: The dynamic interest rate will be updated every 6 hours

Interest Rate Adjustor Coefficient = 0.25

Following the above parameters, this means that the the interest rate will be adjusted by a 0.25 multiplier of the difference between the USC price and peg of $1.00 (i.e. 0.25 * |($1 - Price of USC)|) every 6 hours.

The Adjustor Coefficient and Epoch Duration are parameters that can be adjusted via Carbon governance.

### Equation

The equation used in the Dynamic Interest Rate Model is as follows:

According to the equation above, the algorithm will automatically adjust the interest rate of USC to retain its peg:

If USC is trading above $1, the interest rate will be lowered to encourage the selling of USC. This will in turn reduce the price of USC and maintain the peg.

If USC is trading below $1, the interest rate will be raised to encourage the buying of USC. This will in turn raise the price of USC and maintain the peg.

**Example**

**Example**

Let's assume that the current interest rate of USC = 1.5%.

If the price of USC falls below the peg to an average of $0.995 over a 6-hour epoch, the interest rate will be gradually increased to promote the purchase of USC, which will raise the price of USC back up to the peg.

The interest rate for minting USC will continue to increase until a new equilibrium is reached where USC recovers its $1 peg.

In this example, USC recovers its $1 peg at Epoch 12 (72 hours/Day 3) at the new, higher equilibrium interest rate of 3%.

As this is the rate at which the new equilibrium is established, the interest rate is maintained at 3%.

## Market Forces

As illustrated above, when USC is under-peg, the new equilibrium is attained at a higher interest rate. When the stablecoin minting interest rate is *too high*, users may be discouraged from minting USC.

In response, users have the option to repurchase USC to repay their loans, thereby decreasing the mint interest rates. The interest rate for minting USC will decrease to a point where minters are willing to mint USC again. Consequently, the value of USC will rise above $1, causing it to over-peg.

On the other hand, if USC minters perceive the interest rate of 3% to be reasonable, the interest rate will remain at 3%.

**TLDR;** The dynamic interest rate model will ensure that USC retains its peg, such that the value of USC will always be maintained at $1.

## Roadmap

To peg USC to $1, the new equilibrium is reached at higher nominal interest rate. To lower the nominal interest rate to a point where they are comfortable minting USC, minters are motivated to buy back USC and return their loans, causing the circulating supply of USC to contract - and value of USC to over-peg.

To address this, in the next upgrade, the protocol can raise the inflation rate of USC. This is done by printing and selling USC, expanding the circulating supply of USC and restoring the $1 peg.

Given that Real Interest Rate = Nominal Interest - Inflation Rate, when Inflation Rate > Nominal Interest Rate, a negative real interest rate is achieved (i.e. real interest rate <0).

A negative interest rate can

Lead to an increase in demand for USC, which would help maintain its peg (when price of USC is over-peg);

Encourage users to mint and sell USC rather than simply holding them, which can help to stimulate economic activity;

Discourage users from borrowing and holding USC for speculative purposes;

Reduce volatility and promote stability in the market.

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