# Interest rate model

There are a number of interest rates used to determine the interest paid on borrowings and interest received on deposits. There are also interest indexes which are used to track the accrued interest.

### Borrow interests rates

To differentiate the variable parameters from the stable ones, we are going to use the subscript $$s$$​ for stable and $$v$$​ for variable.\
The variable borrow interest rate $$i\_{vb\_t}$$ and stable borrow interest rates $$i\_{sb\_t}$$ are calculated based on the optimal utilisation ratio $$U\_{opt}$$ and optimal stable to total debt ratio $$0\_{ratio}$$​ set by the protocol.

If $$U\_t < U\_{Opt}$$

$$
i\_{vb\_{t}}=R\_{v0}+ \frac{U\_t}{U\_{Opt}}\*R\_{v1}
$$

If $$U\_t \geqslant	 U\_{opt}$$

$$
i\_{vb\_t}=R\_{v0}+R\_{v1}+\frac{U\_t-U\_{opt}}{1-U\_{opt}}\* R\_{v2}
$$

If $$U\_t \leqslant U\_{opt}$$

$$
i\_{sb\_t}=(R\_{v1}+R\_{s0})+\frac{U\_t}{U\_{Opt}}\*R\_{s1 }
$$

If $$U\_t > U\_{opt}$$

$$
i\_{sb\_t}=(R\_{v1}+R\_{s0})+R\_{s1}+ \frac{U\_t- U\_{Opt}}{1 -U\_{Opt}}\*R\_{s2}
$$

If $$Ratio > 0\_{ratio}$$, an excess is added to the stable borrow interest rate:

$$
i\_{sb\_t}+=R\_{s3}\*\frac{ratio-0\_{ratio}}{1-0\_{ratio}}
$$

The overall borrow interest rate $$i\_{b\_t}$$ factors in all variable and stable rate borrows and is calculated by taking the weighted average of the total variable and stable borrowed amounts and their respective interest rates.

$$
i\_{b\_t}=\frac{TotalVariableBorrowAmount*i\_{vb\_t}+\sum{B\_i*i\_{sb\_i}}}{TotalDebt}
$$

Where:

* $$i\_{vb\_t}$$is the variable borrow amount,
* $$B\_i$$ is the stable borrow amount,
* $$i\_{sb\_i}$$is the stable borrow rate,
* the subscript $$i$$ represents each stable borrow taken on the protocol.

### Deposit interests rates

The deposit interest rate $$i\_{d\_t}$$ is directly dependent on the overall borrow interest rate $$i\_{b\_t}$$. The borrow interest is divided between the depositors, excluding what is retained by the protocol.

$$
i\_{d\_t}= U\_t*i\_{b\_t}*(1-RR)
$$

Where:

* $$RR$$ is the retention rate i.e. the interest kept by the protocol as revenue.


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