> For the complete documentation index, see [llms.txt](https://superp.gitbook.io/superp-litepaper/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://superp.gitbook.io/superp-litepaper/alpha-perp/alpha-perp-mechanism/interest.md).

# Interest

**1. Overview**

The  Alpha Perp system utilizes a dynamic interest rate model based on the utilization rate of borrowed assets. The mechanism adjusts borrowing costs in response to supply and demand fluctuations, ensuring capital efficiency and sustainable lending practices.

***

#### **2. Key Parameters & Definitions**

| **Field** | **Description**                                                                       |
| --------- | ------------------------------------------------------------------------------------- |
| **U**     | Current utilization rate, defined as borrowed assets / collateral assets.             |
| **Umax**  | Maximum utilization rate, representing the upper limit of the borrowable asset ratio. |
| **Us**    | Utilization threshold, affecting interest rate adjustments.                           |
| **R**     | Final interest rate.                                                                  |
| **Rt**    | Current interest rate (iterative value).                                              |
| **Rₜ₋₁**  | Previous interest rate.                                                               |
| **R0**    | Base interest rate.                                                                   |
| **r**     | Reward interest rate (iterative value).                                               |
| **P**     | Current collateral assets.                                                            |
| **P0**    | Rewarded collateral asset quantity.                                                   |
| **t**     | Block count, acting as a time factor.                                                 |
| **Rmax**  | Maximum interest rate; Rt cannot exceed this value.                                   |
| **Rmin**  | Minimum interest rate; Rt cannot fall below this value.                               |
| **v, k**  | Constants influencing interest rate adjustments.                                      |

***

#### **3. Interest Calculation Formula**

The interest rate in Alpha Perp is adjusted iteratively based on the utilization rate (U) and other factors. The formulas governing the system are as follows:

**3.1 Final Interest Rate Calculation**

<figure><img src="https://3526437504-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2Fswmb7WC15eHGwLVBawz0%2Fuploads%2FJb1qFTBWp9U84oExFu5b%2Fimage.png?alt=media&amp;token=3c50f3fb-e8ec-4848-8699-58f1dd995d91" alt=""><figcaption></figcaption></figure>

where r >= 0

**3.2 Reward Interest Calculation**

<figure><img src="https://3526437504-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2Fswmb7WC15eHGwLVBawz0%2Fuploads%2FoJvGeQ3gb9pB6uo7RFnp%2Fimage.png?alt=media&amp;token=72632bd8-5919-4075-aa30-f0b636495f05" alt=""><figcaption></figcaption></figure>

**3.3 Iterative Interest Rate Adjustments**

* **When utilization rate (U) exceeds the threshold (Us):**&#x20;

<figure><img src="https://3526437504-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2Fswmb7WC15eHGwLVBawz0%2Fuploads%2Fe64zGSX9cFfPZb9S1cSv%2Fimage.png?alt=media&amp;token=f819735e-db4d-4ced-9317-63a95329ed8c" alt=""><figcaption></figcaption></figure>

This means the interest rate increases, encouraging repayment or additional collateral deposits to rebalance the system.

* **When utilization rate (U) is below the threshold (Us):**&#x20;

<figure><img src="https://3526437504-files.gitbook.io/~/files/v0/b/gitbook-x-prod.appspot.com/o/spaces%2Fswmb7WC15eHGwLVBawz0%2Fuploads%2FIBZRUXg8YnMSR6lyr1Zw%2Fimage.png?alt=media&amp;token=fa236fb6-c15f-480c-9f96-16e73ef2172e" alt=""><figcaption></figcaption></figure>

This results in a lower interest rate, incentivizing borrowing when there is surplus liquidity.

***

#### **4. Interest Rate Constraints**

To maintain stability and prevent excessive fluctuations, interest rates are capped within predefined boundaries:

* **Rₜ cannot exceed Rmax (Maximum Interest Rate).**
* **Rₜ cannot drop below Rmin (Minimum Interest Rate).**

These constraints ensure that interest rates remain within an economically viable range, preventing extreme volatility in borrowing costs.

***

#### **5. Summary**

The Alpha Perp interest rate mechanism dynamically adjusts borrowing costs based on utilization rates, block time, and collateral conditions. By incorporating reward-based incentives and utilization-based rate adjustments, the system optimizes liquidity distribution while protecting lenders from unexpected interest rate swings.