Euler Leveraged Strategy
===

## Table of Contents

[TOC]

## How to caluculate Euler health score
To avoid to be liquidated, keep $HealthScore$ greater than 1.0 .

Generally $HealthScore$ is denoted as
$$
HealthScore = {RiskAdjustedCollateral \over RiskAdjustedLilability}
$$

Here assume a user deposit a single asset and leverage its position.

Let $a^c$ denote the balance of collateral and $s^c$ denote the self-collateralized balance of collateral.
Let $f^c$ denote the collateral factor of a asset, $f^b$ denote its borrow factor and $f^s$ denote the collateral factor of a self-collateralized asset (called self-collateralized factor).

Risk-adjusted collateral is calculated as
$$
v^c = f^c[a^c -{s^c \over f^s}] + s^c \tag{1}
$$
where $f^c \leqq f^s$.

> A pdf is found on [Euler fi Discord](https://discord.com/channels/742749441697513632/780939191012818977/1001457715568197652).
Risk-adjusted collateral seems to be calculated differently. I think the formula in the pdf is wrong. [link](https://drive.google.com/file/d/1mWscpHYOu1S7nX24QLdWBzLpkpLhDYh2/view?usp=sharing)

Risk-adjusted lilability is calculated as
$$
v^b = {a^b - s^b \over f^b} + s^b \tag{2}
$$

where self-collateralized part of any asset have always 1.0 borrow factor. $s^b$ is always equal to $s^c$ ??

> In Euler self-collateralized part of any asset has always 0.95 collateral factor and 1.0 borrow factor.

In case of our leveraged strategy $a^b$ is always equal to $s^b$.
$a^c$ is `eToken.balanceOfUnderlying(strat)`.
$s^c$ is `dToken.balanceOf(strat)`.

for example,

deposits 1000 USDC and mints 9000 USDC. normal CF of USDC is 0.9

now collateral 10000 USDC and liability 9000 USDC.

so, risk adjusted collateral = (10000 - 9000/0.95) * 0.9 + 9000 * 0.95 = 9023
risk adjusted liability  = 9000*1

### Peusdocode

  function getCurrentHealthScore() public view returns (uint256) {
        IMarkets.AssetConfig memory config = EULER_MARKETS.underlyingToAssetConfig(token);
        uint256 cf = config.collateralFactor;
        uint256 balanceInUnderlying = IEToken(config.eTokenAddress).balanceOfUnderlying(address(this));
        uint256 selfAmount = dToken.balanceOf(address(this));

        require(selfAmount != 0, "strat/no-borrow");

        // selfAmountAdjusted = selfAmount * CONFIG_FACTOR_SCALE) / SELF_COLLATERAL_FACTOR;
        uint256 riskAdjustedCollateral = (cf *
            (balanceInUnderlying - (selfAmount * CONFIG_FACTOR_SCALE) / SELF_COLLATERAL_FACTOR)) /
            CONFIG_FACTOR_SCALE +
            selfAmount;
        uint256 riskAdjustedLilability = selfAmount;
        return (riskAdjustedCollateral * EXP_SCALE) / riskAdjustedLilability;
}

## Calculate amount to `mint` or `burn` to maintain a target health score.

## Minting
Let $h_{target} (\gt 1)$ denote the target health score we want to maintain.

Let $x$ denote its newly added collateral and $x^s$ denote its newly added collateral with recursive borrowing (`eToken.mint()`).

Strategy deposits its underlyings with `eToken.deposit(amount x)` and `eToken.mint(amount x^s)`.

$$
h_{target} = \frac{f^c[a^c + x + x^s -{s^c + x^s \over f^s}] + s^c + x^s}{s^c + x^s} \tag{4}
$$

Resolve the equation for $x^s$.

$$
x^s = \frac{f^c(a^c+ x ) - (h_{target} + f^c/f^s - 1)s^c}{h_{target} + f^c(1/f^s -1) -1} \tag{5}
$$

## Burning

Strategy withdraws its underlyings with `eToken.withdraw(amount x)` and `eToken.burn(amount x^s)`.

### References
[How to calculate Euler’s health score](https://hackmd.io/wYxWkRwZTpej0Zxew4DGzw)