Interest rate strategy 1

Interest rate strategy parameters

These are the parameters that are stored on this interest rate strategy contract.

Optimal utilization rate

This is $U_{optimal}$ in the equations below.

The optimal utilization rate creates the kink in the interest rate curve. The higher the optimal rate, the more it encourages users to borrow up to that point by providing relatively affordable rates. The lower the optimal rate, the more it discourages users from borrowing more than that point.

Typically, before the utilization rate reaches the optimal utilization rate, the interest rates stay low, so that users will be financially incentivized to utilize the liquidity more. On the other hand, beyond the optimal utilization rate, the interest rates stay high, so that users will be incentiveized to utilize less liquidity.

Base interest rate

This is $R_0$ in the equations below.

The base interest rate is the interest rate at utilization rate = 0. If a protocol decides that an asset is popular enough to be borrowed at a higher base interest rate, it will give a higher base interest rate. However, in other cases, it will stay at 0.

Variable rate slope 1

This is $R_{slope1}$ in the equations below.

This is the variable that controls the gradient of the left half of the interest rate curve up to the kink. The bigger the number is, the steeper the curve is, meaning higher interest rates.

Variable rate slope 2

This is $R_{slope2}$ in the equations below.

This is the variable that controls the gradient of the right half of the interest rate curve beyond the kink. The bigger the number is, the steeper the curve is, meaning higher interest rates.

Equations

To understand how interest rate strategy 1 works, it's necessary to define these terms:

Variable	Meaning	Domain (or Range)
$U$	Utilization rate	$U∈[0,1]$
$U_{optimal}$	Optimal utilization rate	$U_{optimal}∈[0,1]$
$R_{borrow}$	Borrowing interest rate at utilization rate $U$	$R∈[0,∞]$
$R_{slope1}$	Variable Rate Slope 1 (the slope that is going to be applied on the left side of the graph)	$R_{slope1}∈[0,∞]$
$R_{slope2}$	Variable Rate Slope 2 (the slope that is going to be applied on the right side of the graph)	$R_{slope2}∈[0,∞]$
$R_0$	Base interest rate. This adjusts the y-intercept of the interest rate curve.	$R_0∈[0,1]$

Then, depending on whether $U <= U_{optimal}$ is true, $R$ would be calculated differently, as follows:

$$U \le U_{optimal} \rArr R_{borrow} = R_0 + \frac{U}{U_{\text{optimal}}}(R_{\text{slope1}})$$

$$U > U_{optimal} \rArr R_{borrow} = R_0 + R_{\text{slope1}} + R_{\text{slope2}}\frac{U - U_{\text{optimal}}}{1 - U_{\text{optimal}}}$$

Graphically, the equation is illustrated as below (consider $R_{borrow}$ only for $0 \le U \le 1$ and $0<R_{borrow}$, and use two different functions based on $U \le U_{optimal}$):

The demo graph uses $U_{optimal}=0.65$, $R_0=0$, $R_{slope1}=0.08$, $R_{slope2}=1$.

Let's run an example calculation. Using this strategy, $U=0.5$ will give $R_{borrow} = 0.061538$, which is 6.1538%. In other words, 50% utilization rate will give 6.1538% of borrowing interest rate.

The lending interest rate is a function of borrow interest rate and reserve factor. It is calculated as $R_{lending}=R_{borrow} \times U \times (1 - \text{Reserve factor})$ where $\text{Reserve factor} \in [0, 1)$. Following the example above, let us say that the reserve factor is 0.15. Then, the lending interest rate will be calculated as:

$$R_{lending}=0.061538 \times 0.5 \times (1-0.15) = 0.02615365$$

To summarize:

$$\text{Given }U_{optimal}=0.65, R_0=0, R_{slope1}=0.08, R_{slope2}=1, \newline U=0.5 \newline \rArr R_{borrow}=0.061538 \newline \rArr R_{lending}=0.02615365$$

Now, we can display both $R_{borrow}$ and $R_{lending}$ on the same graph:

Borrowing interest rate (blue) and lending interest rate (red) displayed on the same axes (consider $0<R$ and $0 \le U \le 1$ only, and use two different functions based on $U \le U_{optimal}$).

Using the graph above, it is possible to determine borrowing interest rate and lending interest rate.