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# Liquidity and Interest Rates on ERC-20 Fungible Tokens

## Introduction

The liquidity of the the ParaSpace protocol represents the availability of capital to support borrowing amounts and redeeming nTokens and pTokens. The liquidity of the protocol can be measured via the utilization rate, which is the share of reserve that is currently borrowed for each asset.

## Interest Rate Model

Interest rates are charged to borrowers based on the Utilization Rate,
$U$
. At a high level, the interest model helps incentivize what is needed to support liquidity. When there is capital available, borrowers pay less interest to borrow; when capital is scarce, high interest rates encourage more supply and repayments on loans.
As
$U$
gets closer to 100% the liquidity risk increases. There is an inflection point before which the interest rate changes, i.e. slope or
$\frac{dR}{dU}$
, are small and after which slope starts rising sharply. This inflection point is represented as
$U_{optimal}$
.
The interest rate
$R_t$
follows the model:
$if \hspace{1mm} U < U_{optimal}: \hspace{1cm} R_t = R_0 + \frac{U_t}{U_{optimal}} R_{slope1}$
$if \hspace{1mm} U \geq U_{optimal}: \hspace{1cm} R_t = R_0 + R_{slope1} + \frac{U_t-U_{optimal}}{1-U_{optimal}}R_{slope2}$
In the borrow rate technical implementation, the calculateCompoundedInterest method relies on an approximation that mostly affects high interest rates. The resulting actual borrow rate can is:
$Actual APY = (1+Theoretical APY/secsperyear)^{secsperyear}-1$

Asset
UOptimal
r_0
Slope 1
Slope 2
Reserve Ratio
ETH
85%
5%
8%
60%
10%
stETH
85%
0%
8%
60%
10%
wstETH
85%
0%
8%
60%
10%
USDC
85%
5%
8%
60%
10%
USDT
85%
5%
8%
60%
10%
APE
85%
40%
25%
50%
10%
sAPE
85%
40%
25%
50%
10%
cAPE
85%
5%
10%
23%
10%
BLUR
85%
0%
8%
60%
10%