# Interest Rate and Utilisation Model

Both the interest rate recieved by depositors and paid by borrowers is derived from the two models below.

## Interest Rate Model 

#### Standard Interest Rate Model

$$
R\_t=Multiplier∗U\_t+R\_0
$$

#### Jump Rate model

$$
\begin{align\*}
R\_t​ &= Multiplier∗min(U\_t,U\_0) \\
&+ JumpMultiplier∗max(0,U\_t−U\_0) \\
&+ R\_0
\end{align\*}
$$

where:

* Rt = borrowing rate at Ut 
* Ro = base interest rate 
* Multiplier = the rate of increase in interest rate with respect to utilization 
* JumpMultiplier = the rate of increase in interest rate with respect to utilization after the kink
* Ut = current utilisation
* Uo = Kink

#### Supply Interest Rate

$$
S\_t=R\_t
∗ U\_t
∗(1−Reserve Factor\_a)
$$

where

* St = supply rate at Ut 
* ReserverFactor = percentage of the spread between the supply and borrow rates that the protocol keeps as profit

## Utilisation Model

The utilisation rate of each pool is a function of the current loaned amount and the amount available to loan out. This number is refreshed intraday. Over time, protocol use will provide data points to assess and refine the best parameters for our utilisation model.

Each asset pool will have a specific optimal utilisation rate (Uo). This is a function of market liquidity pool size, historical utilisation rate and risk buffering for sudden large-sum withdrawals within the given market pools.

Utilisation rate of market pool 'x' is calculated as:

$$
U\_x = \cfrac {B\_x} {C\_x + B\_x - R\_x}
$$

​where:

* Ux = Utilisation rate of market pool x
* Bx = Borrowings of market pool x
* Cx = Total liquid assets in market pool x
* Rx= Reserves of market pool x


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