# Interest Rate Model

### Overview

The interest rate model plays a pivotal role not only in providing appropriate returns for lenders but also in managing liquidity risks effectively. Interest rates are dynamically adjusted based on the utilization rate of the liquidity pool, ensuring that the pool remains balanced and functional.

Each liquidity pool can adopt various interest rate models, which are implemented as separate contracts independent of the pools themselves. This design aligns with Factorial's modular architecture, allowing for greater flexibility and customization across different pools.

### Interest Rate Parameter

Each pool has distinct interest rate parameters for each asset. The default interest rate model used is the **Jump Rate Model**, which features two kink points based on the utilization ratio of the borrowed asset.

* Base Rate ($$R\_0$$) : The interest rate when utilization is 0.
* Kink point ($$U\_{low}$$, $$U\_{high}$$) : These thresholds define ranges of utilization at which the slope of the interest rate curve changes.
* Slopes ($$Slope\_{low}$$, $$Slope\_{medium}$$, $$Slope\_{high}$$) : The rate at which interest increases within each utilization range.

### Interest Rate Model

* Utilization Ratio of the target asset in the pool is determined as follows:

$$
U\_{t}= \frac {Total \ Borrowed {}} {Total \ Supply\_{}}
$$

* Borrow Rate

$$
Borrow(U\_t)= R\_{0} + Slope\_{low} \cdot \min\[U\_{t}, U\_{low}] \ + \ Slope\_{medium}\cdot \min\[\ \max\[0,U\_{t} - U\_{low}],U\_{high}-U\_{low}] \ + \ Slope\_{high}\cdot \max\[0, U\_{t}-U\_{high}]
$$

<figure><img src="/files/7PsNoU8NshoGayr60MGH" alt=""><figcaption></figcaption></figure>

* Supply Rate

$$
SupplyRate(U\_t) = Borrow(U\_t) \cdot U\_t
$$


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