Interest rate model

The Lend & Borrow service offered by WEMIX.Fi adjusts the balance of available funds and uses a "jump interest rate model" in which interest rates increase steeply when utilization exceeds a particular level. The interest rate applied is based on the market utilization rate, with lower rates applied when loan liquidity is high, and higher rates when liquidity is low. The interest earned by depositors and the interest charged on loans are both calculated as compound interest.

WEMIX.Fi's Lend&Borrow service referenced the Jump-Rate V2 model of the Compound protocol.

The Multiplier, Kink(utilization threshold), Jump Multiplier and Minimum interest rate(Base Rate) used in the protocol's interest rate model are all constant values, the constant values defined in the Lend & Borrow service are:

Token	Multiplier	Kink	Jump Multiplier	Base Rate	Reserve Factor
WEMIX	5%	85%	800%	0	50%
WEMIX$	5%	90%	500%	0	50%
stWEMIX	5%	85%	800%	0	50%

Utilization Rate

The utilization rate is a measure of the total loan amount compared to the total loan liquidity available in the protocol and serves as an indicator of market activity. The market utilization for each asset in the protocol is calculated using the following formula:

$$U_a\space=\space Borrows_a \space /\space(Cash_a \space + \space Borrows_a)$$

• $U_a$: Utilization rate of a market
• $Borrows_a$: Total amount of asset borrowed from the market
• $Cash_a$: Total amount of available liquidity

The market utilization rate of a market provides insight into the proportion of liquidity being used for financing compared to the overall market liquidity. It shows the activity level of the market. When market utilization reaches 100%, it means that no additional financing is possible and the closer it is to 100%, the higher the demand and interest rates for financing become. For instance, with Alice and Bob each depositing $500 WEMIX, and Charlie loaned $100 WEMIX, the market utilization rate of WEMIX is calculated using the formula mentioned above.

$$\begin{align*} U_a &=Borrows_a/(Cash_a + Borrows_a)\\ &=100/(900+100)\\ &=10 \% \end{align*}$$

If Charlie repays their loan, the deposit amount will remain unchanged but the loan amount will decrease. This will result in a decrease in the market utilization rate. When the utilization rate decreases, the loan interest rate will decrease, making it more attractive to borrowers.

The deposit interest rate is tied to the loan interest rate, so a decrease in the loan interest rate will also lead to a decrease in the deposit interest rate. A lower deposit interest rate will make the asset less attractive to depositors, so they will likely move to higher-yielding pools.

To prevent utilization from reaching 100%, the protocol has established a utilization threshold. If the utilization crosses the threshold, the interest rate will increase sharply, encouraging depositors to increase their deposit amount and decreasing the loan amount. This will help the utilization rate recover to its normal level, and the interest rate will return to its fair value.

Supply interest rate

The supply interest rate is determined based on market utilization and borrowing interest rates and can be expressed by the following formula:

$$Supply\space Interest\space Rate_a = Borrowing\space Interest\space Rate_a * U_a * (1-Reserve\space Factor_a)$$

• $Borrowing\space Interest\space Rate_a$: The borrowing interest rate of the asset’s market
• $Reserve\space Factor_a$: Stored margin(50%) earned by the through deposits and loans

The market utilization rate of WEMIX can be calculated based on the deposit and loan amounts. For instance, if Alice and Bob deposit $100,000 WEMIX each, and Charlie borrows $150,000 WEMIX, the market utilization rate of WEMIX would be calculated as described in the formula mentioned above.

$$\begin{align*} U_a &=Borrows_a/(Cash_a + Borrows_a)\\ &=150,000/(100,000+100,000)\\ &=75 \% \end{align*}$$

The protocol supply interest rate calculated based on market utilization rate is:

$$\begin{align*} Supply\space Interest\space Rate_a &= Borrowing\space Interest\space Rate_a * U_a * (1-Reserve\space Factor_a) \\ &= Borrowing\space Interest\space Rate * 70 \% * (1-50 \%) \\ &= Borrowing\space Interest\space Rate * 35 \% \end{align*}$$

Borrow interest rate

The loan interest rate in the protocol is determined by the utilization rate, which is calculated based on the market's total loan amount and the total loan liquidity available.

$$Borrow\space Interest\space Rate_a = Multiplier\space *\space min(U_a ,Kink)\space + \space Jump \space Rate \space * \space max(0, U_a -Kink)\space+\space Base\space Rate$$

• $Multiplier$: a factor multiplied by the utilization
• $Jump\space Multiplier$: a factor multiplied by the utilization when it exceeds the threshold
• $Base\space Rate$: minimum interest rate
• $min(a, b)$: returns the smaller value between the two
• $max(a, b)$: returns the bigger value between the two

The utilization rate is influenced by the Base Rate, Threshold (Kink), and Jump Multiplier, with interest rates increasing steeply when the utilization rate crosses a certain Threshold.

$$\begin{align*} & if \hspace {1mm} U_a < Kink: \hspace{2mm} Borrow Interest Rate = Multiplier*U_a + Base Rate\\ & if \hspace {1mm} U_a > Kink: \hspace{2mm} Borrow Interest Rate = Multiplier*U_a + Jump Rate * (U_a - Kink)+ Base Rate \end{align*}$$

For example, if Alice and Bob deposit $100,000 WEMIX each, and Charlie loans $190,000 WEMIX, the utilization rate of the WEMIX market would be 95%. The loan interest rate, based on this utilization rate, would be calculated according to the equation.

$$\begin{align*} Borrow\space Interest\space Rate &= Multiplier*min(U_a ,Kink) + Jump\space Multiplier * max(0, U_a - Kink) + Base\space Rate \\ &= 30\% * 90\% + 109\% *(95\% -90\% ) \\ &= 0.27+0.05 \\ &= 32\% \end{align*}$$