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# Interest Rate Model

### Interest Rate Flow

Unlike the traditional financial market lending/borrowing process, ELYFI allows investors and borrowers to invest and take collateral-based loans without use of a third-party intermediary. Interest rates are determined by algorithms, and investors and borrowers can trade via these algorithms without negotiating interest rates. Interest accrued to borrowers and investors is the same, and the flow for this can be formulated as follows:
$B(t)*BR(t)=L(t)*LR(t)$
• $B(t)$
: Total Amount Borrowed
• $BR(t)$
: Overall Average Interest Rate on Borrowed Amount
• $L(t)$
: Total Investment
• $LR(t)$
: Overall Average Interest Rate on Investment

### Utilization Ratio

When borrowing demand increases, borrowing interest and Money Pool ROI increase, suppressing excessive borrowing demand and inducing investors to supply liquidity. Therefore, ELYFI's interest rates are influenced by the Money Pool utilization ratio (U). The Money Pool utilization ratio is a variable representing the current borrowing and investment status of the Money Pool, and is defined as follows:
$U(t)=\frac{B(t)}{L(t)}=\frac{B(t)}{L_L(t)+L_A(t)}$
• $B(t)$
: Total Amount of Loan
• $L(t)$
: Total Liquidity
• $L_L(t)$
: Liquidity Directly Supplied to the Money Pool
• $L_A(t)$
: Liquidity Supplied through Investment in Asset-Backed Bonds

### Kinked Borrowing Rates Model

There are various types of interest rate models that can minimize Money Pool risks. ELYFI uses the “kinked rates model” that applies the Compound and AAVE interest models.
$if \hspace{1mm} U < U_{optimal}: \hspace{1cm} BR = k_ + \frac{U(t)}{U_{optimal}} (m-k)$
$if \hspace{1mm} U \geq U_{optimal}: \hspace{1cm} BR = k + m + \frac{U(t)-U_{optimal}}{1-U_{optimal}}(n-m-k)$
• $BR$
: Interest Rate on the Borrowed Amount
• $k$
: Base Interest Rate
• $m$
: Interest rate when the Money Pool utilization ratio is optimal
• $n$
: Interest rate when the Money Pool utilization ratio is 1
The kinked rates model adjusts interest rates when the Money Pool utilization ratio exceeds the optimal ratio and approaches 1 through two different straight line graphs so that interest rates on borrowings can increase steeply. This reduces Money Pool risks, which increase rapidly as the Money Pool utilization ratio approaches 1.

### Model Parameters

Interest rates on borrowings can be classified as interest rates of crypto asset-backed loans
$BR_{C}$
and interest rates of real asset-backed loans
$BR_{R}$
. Crypto asset-backed loans are subject to variable interest rates that are adjusted according to Money Pool circumstances, and fixed interest rates are applied to real asset-backed loans.
In the case of variable interest rates, the stability of the Money Pool can be easily ensured because interest rates are adjusted according to its circumstances even if loans occur. However, in the case of fixed interest rates, once a loan is created, the fixed interest rate does not change for a specified period. In traditional financial markets, where liquidity is very high and accumulated data is sufficient, it is relatively easy to control loans even when generated at a fixed rate. However, ELYFI is in a different situation from that of traditional financial markets. Therefore, measures that are applied to ELYFI have to be different from those used in conventional financial markets because of the lack of liquidity and data.
In other words, each variable is set to lower the ratio of real asset-backed loans until sufficient liquidity is secured and data to identify indicators to minimize Money Pool risks are secured. By setting fixed interest rates higher than variable interest rates, the supply and demand of borrowings at fixed/variable interest rates can be controlled in the protocol.
Below are the values ​​determined by simulation, which can be changed through discussions of the Governance and as a result of feedback gained as data is accumulated in the future.
 Fixed Rate ​$U_{optimal}$ ​$k_R$ ​$m_R$​ ​$n_R$​ Dai 80% 4% 12% 90% Ether 70% 6% 12% 100%
 Variable Rate ​$U_{optimal}$ ​$k_C$ ​$m_C$​ ​$n_C$​ Dai 80% 0% 6% 90% Ether 70% 0% 8% 100%
In the case of real-asset-backed loans, the upper limit of interest rates on borrowings is determined by the location of collateral, and the region and regulations/laws in which the owner is located. Therefore, depending on the collateral service providers that handle contracts and the associated terms and conditions, there may be situations in which it is not possible to borrow real asset-backed loans.

### Average Real Asset Collateralized Borrowing Rate

When User “
$x$
” borrows “
$m$
” of real asset-backed cryptoassets at the point in time “
$t$
” , the average interest rate of the total real asset-backed loan is as follows:
$\bar{BR}_R=\frac{\bar{BR}_R(t-1)B_R(t-1)+\bar{BR}_R(t)m}{B_R(t-1)+m}$

### Overall Borrowing Rate

Overall borrowing rate calculated based on the real asset-backed loan interest rate and crypto asset-backed loan interest rate is as follows:
$BR(t)=\bar{BR}_R(t)SB_R(t)+BR_C(t)SB_C(t)$
• $BR(t)$
: Total Amount of Borrowing
• $SB_R(t)=\frac{B_R(t)}{L(t)}$
• $\bar{BR}_R(t)$
: Overall Average Interest Rate on the Real Asset-Backed Borrowing
• $SB_C(t)=\frac{B_C(t)}{L(t)}$
• $BR_C(t)$
: Interest Rate on Crypto Asset-Backed Borrowing

### Liquidity

The total liquidity supplied to the Money Pool is divided into the liquidity supplied from the investment in the Money Pool and the liquidity supplied from the investment in the “A” token. It is defined as follows:
$L(t)=L_L(t)+L_A(t)$
Liquidity providers can supply liquidity to the Money Pool by depositing cryptoassets directly into the Money Pool or through investment in ABTokens. Interest rates on loans are determined based on the liquidity supply, and rewards are provided to liquidity providers through loan interest accrued.

### AToken Lending Rate

Users who supply liquidity through investment in asset-backed bonds can receive interest until the relevant bonds expire. Interest rates are applied in the same way as with collateral-based loans of suitable asset-backed bonds.
$AToken~APY=ALR(i)=BR_R(t)$
Since fixed interest rates are applied to real asset-backed borrowing, users who purchase securitized asset-backed bonds can receive profits at a fixed interest rate. In this case, interest is calculated in block units, and is expressed as follows:
$rewardPerBlock=\frac{I(i)*ALR(i)}{T_{years}}*secondPerBlock$
• $I(i)$
: Value of i-th Real Asset Collateral
• $ALR(i)$
: Interest Rate of i-th Real Asset-Backed Loan
• $T_{years}$
: Second in 1 Year (Uni)
• $secondPerBlock$
: Time Per Block

#### Example

A lender makes a loan against real assets with 100,000 Dai collateral value at an annual interest rate of 12%. In this case, the interest that Alice, the ABToken investor, can receive when investing in it is as follows:
• Investment Amount: 20,000 dai
• $rewardPerBlock=\frac{100,000*0.12}{31536000}*13=4.946*10^-3$
• $rewardPerBlock=\frac{20,000}{100,000}*4.946*10^-3=9.893*10^-4~Dai$

### LToken Lending Rate

When liquidity is supplied to the Money Pool, ROI is calculated based on the demand for loans and the interest accruing to them in the Money Pool. There is a reserve factor to maintain the ecosystem and distribute the profits to the Governance participants.
$LR(t)=BR(t)*U(t)*(1-F)$