# Overcollateralized Loans

### Core Concepts: Liquidation Risk Elimination and General Conditions

#### Introduction

Overcollateralized loans have become an aspect of the ever changing decentralized finance (DeFi) landscape. These financial tools allow users to deposit assets as collateral in order to borrow assets. While they provide a means of obtaining loans without intermediaries it is essential to comprehend the mathematical foundations behind them. This section delves into the concepts with a focus on the mechanisms that aim to minimize liquidation risk and the overall requirements, for overcollateralized loans.

## Liquidation Risk Elimination Overview

Liquidation risk refers to the possibility of the borrowers assets being sold off if their value drops below a threshold in relation to the borrowed amount. Ensuring that this risk is minimized or eliminated is crucial, for maintaining the reliability and trustworthiness of DeFi platforms. To achieve this various mathematical and functional strategies are utilized.

**Dynamic Collateral Adjustment (DCA)**

**Mathematical Model**

**Inputs**

Loan Amount: The initial amount borrowed.

New Volatility: The updated volatility rate.

Old Volatility: The previous volatility rate.

**Output**

Additional Collateral: The extra collateral needed to balance the loan.

**Code Example**

**Insurance Pools**

**Insurance Pools**

**Mathematical Model**

The insurance premiums can be dynamically adjusted using the formula:

**Inputs**

Loan Amount: The initial amount borrowed.

Risk Score: A metric to quantify the borrower's risk.

Volatility: The volatility of the collateral.

**Output**

Premium: The insurance premium.

**Code Example**

## Dynamic Mathematical Model for Liquidation Risk Elimination

### Mathematical Model

#### Inputs:

$LA$: Loan Amount

$CV$: Current Collateral Value

$LR$: Liquidation Ratio (often set at 1.028 or 102.8%)

$P$: Market Price of the asset used as collateral

#### Outputs:

$LTV$: Loan to Value ratio

$L$: Liquidation Point

#### Equations:

The liquidation risk is eliminated if $CV$ remains above $L$, given the current market price $P$.

**Code Example:**

## Mathematical Model for Insurance Pools

### Mathematical Model for Insurance Premiums

#### Inputs:

$R$: Risk Profile of the Borrower

$V$: Volatility of the Collateral

$I$: Initial Insurance Premium rate

#### Outputs:

$P$: Premium for Insurance

#### Equation:

**Code Example:**

### Risk Profiles and Dynamic Adjustments

The level of risk `R`

is continuously modified based on factors, including the behavior of borrowers market conditions and the performance of collateral. This adaptive adjustment enables insurance pricing to be more responsive creating an ecosystem.

## General Conditions for Overcollateralized Loans

**Loan-to-Value (LTV) Ratio**

**Mathematical Model**

**Inputs**

Loan Amount: The initial amount borrowed.

Collateral Amount: The total amount of collateral deposited.

**Output**

LTV Ratio: The Loan-to-Value ratio.

**Code Example**

**Interest Rates**

While the initial interest rate is set at 4.5%, the formula to calculate the interest after $nn$ years is:

**Code Example**

### Loan Amount Calculation

#### Mathematical Model

The loan amount is determined by the collateral amount and the asset's volatility.

**Inputs**

Collateral: The total amount of assets deposited as collateral.

Volatility: The expected fluctuation in the asset price.

**Output**

Loan Amount: The maximum amount that can be borrowed.

**Code Example**

### Interest Rates

#### Mathematical Model

The interest rate is capped at a maximum of 4.5%, and the loan term is restricted to a maximum of one year.

**Inputs**

Loan Amount: The amount borrowed.

**Output**

Interest: The maximum amount of interest accrued, capped at 4.5% for a loan duration of 1 year.

**Code Example**

### Repayment

#### Mathematical Model

Each repayment $x$ affects the stop-loss limit.

**Inputs**

Old Stop-Loss: Previous stop-loss limit.

$x$: The amount repaid.

**Output**

New Stop-Loss: Updated stop-loss limit.

**Code Example**

### Liquidation

#### Mathematical Model

**Inputs**

Loan Amount: The amount borrowed.

**Output**

Liquidation Value: The value at which the loan will be liquidated.

**Code Example**

### Stop-Loss

#### Mathematical Model

The stop-loss is set at 2.8% above the loan value to mitigate default risk.

**Inputs**

Loan Amount: The amount borrowed.

**Output**

Stop-Loss: The asset value at which automatic liquidation occurs if the collateral falls below this level.

**Code Example**

## Dynamic Insurance Pools

Insurance pools that are dynamic play a role in decentralized finance (DeFi) systems by providing loans that are overcollateralized. These pools serve as a safety net protecting against defaults or unexpected events. In this section we will delve into the aspects that make these pools effective tools, for mitigating risks.

### Insurance Pool Size and Coverage Ratio

#### Mathematical Model

The size of the insurance pool ($I$) is often determined by a Coverage Ratio ($CR$) that specifies what percentage of the total loan value is covered by the pool. The formula to calculate the ideal size of the insurance pool is:

**Inputs**

$CR$: Coverage Ratio, a constant usually ranging from 0.1 to 1.

Total Loan Value: The sum of all active loans in the system.

**Output**

$I$: The ideal size of the insurance pool.

**Code Example**

### Premium Rate Calculation

#### Mathematical Model

The insurance premium rate ($PR$) might be calculated as a function of the Coverage Ratio ($CR$) and the overall risk profile ($R$) of the loan portfolio:

**Inputs**

$CR$: Coverage Ratio

$R$: Overall risk profile of the loan portfolio

**Output**

$PR$: Insurance premium rate

**Code Example**

### Dynamic Adjustment of Premiums

#### Mathematical Model

The insurance premium ($P$) is often dynamically adjusted based on both the insurance premium rate ($PR$) and the loan amount ($L$):

**Inputs**

$PR$: Insurance premium rate

$L$: Loan amount

**Output**

$P$: Insurance premium

**Code Example**

## Dynamic Collateral Adjustment

In DeFi systems that offer overcollateralized loans, dynamic collateral adjustment is a fundamental risk-mitigation strategy. It continuously monitors the market value of collateral and prompts borrowers to top up their collateral if the value depreciates below a certain threshold. This section offers a mathematical understanding of this dynamic mechanism.

### Basic Collateral Ratio

#### Mathematical Model

The Collateral Ratio ($CR$) is calculated using the formula:

**Inputs**

Collateral Value: The current market value of the collateral deposited.

Loan Value: The amount borrowed.

**Output**

$CR$: Collateral Ratio in percentage

**Code Example**

### Additional Collateral Needed

#### Mathematical Model

The Additional Collateral Needed ($AC$) can be calculated based on the change in volatility ($ΔV$) and the loan amount:

**Inputs**

New Volatility: The updated measure of asset price fluctuation.

Old Volatility: The previous measure of asset price fluctuation.

Loan Amount: The original loan value.

**Output**

$AC$: Additional Collateral Needed

**Code Example**

### Stop-Loss Trigger

#### Mathematical Model

The stop-loss trigger is set at 2.8% above the loan value. The Stop-Loss Value (`SL`

) is therefore:

**Inputs**

Loan Value: The amount borrowed.

**Output**

$SL$: Stop-Loss Value

**Code Example**

## Minting of "test_coins_erc20"

Minting in the context of DeFi usually involves generating new tokens as part of a collateralized loan agreement. When a user over-collateralizes a loan, stablecoins like "test_coins_erc20" can be minted and provided to the borrower. This section breaks down the math and mechanics of this process.

### Minting Ratio

#### Mathematical Model

The Minting Ratio ($MRMR$) determines how many test_coins_erc20 are minted per unit of collateral. The formula is:

**Inputs**

Loan Amount: The total loan value in the native asset.

Collateral Value: The total value of the collateral locked.

**Outputs**

$MR$: Minting Ratio

**Code Example**

### Number of test_coins_erc20 Minted

#### Mathematical Model

The number of test_coins_erc20 ($TC$) minted is calculated as follows:

**Inputs**

Loan Amount: The total loan value in the native asset.

$MRMR$: Minting Ratio

**Outputs**

$TC$: Number of test_coins_erc20 minted.

**Code Example**

### Gas Fee for Minting

#### Mathematical Model

The Gas Fee ($GF$) for minting is calculated based on the Ethereum gas price ($G$) and the gas needed for the minting transaction ($N$):

**Inputs**

$G$: Current Ethereum gas price in Gwei.

$N$: Gas needed for the minting transaction.

**Outputs**

$GF$: Gas Fee for minting.

**Code Example**

## Detailed Mechanism for Minting "test_coins_erc20"

### Mathematical Model

#### Inputs:

$C$: Collateral deposited

$LA$: Loan amount

$MR$: Minting Rate (fraction of the loan amount that is minted as stablecoins)

#### Outputs:

$M$: Number of "test_coins_erc20" minted

#### Equation:

The minting rate $MRMR$ can be defined as a constant or be made dynamic based on various economic factors such as supply, demand, and stability fees.

**Code Example:**

## Pool Management

### Mathematical Model for Pool Management

#### Inputs:

$TP$: Total Pool assets

$LAL$: Loan amount locked

$F$: Fees accrued

#### Outputs:

$FTA$: Free to allocate assets in the pool

#### Equation:

This equation gives a snapshot of the pool's status and how much capital is available for new loans or other investments.

**Code Example:**

## Staking Mechanisms for Validators

### Mathematical Model

#### Inputs:

$V$: Validator's stake

$TP$: Total Pool assets

$SR$: Staking rate (The rate at which staking rewards accrue)

$T$: Time (in some consistent unit like days)

#### Outputs:

$R$: Rewards

#### Equation:

The idea here is that the reward is proportional to the validator's stake, the total pool assets, and the time for which they've staked, adjusted by a staking rate $SRSR$.

**Code Example:**

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