Return on equity

Introduction

When users deposit and borrow on Euler, they earn the Supply APY on their deposits and pay the Borrow APY on their debts. Return on equity (ROE) is a measure of the overall yield they can expect to earn over the course of a year relative to the equity they have:

$$ROE = \frac{( \text{Total Deposits} \times \text{Net Supply APY}) - ( \text{Total Debt} \times \text{Net Borrow APY})}{\text{Equity}}.$$

The ROE has two different calculations depending on whether or not an account has an ordinary borrow or a multiplied position, since this affects how an account's equity is measured.

It is generally not possible to know whether an account has an ordinary borrow or a multiplied position just by looking at the account's deposits and debts.

An account with $1000 deposits and $800 debts could have deposited $1000 and borrowed $800, or they could have deposited $200 and multiplied 5× by borrowing, swapping, and re-depositing, until they have total deposits of $1000 and total debts of $800.

Since the user-interface cannot know the context in which a position was created, users are instead given the option to toggle which calculation they prefer on their position page. The two options are detailed below.

Ordinary borrow

If a user simply deposits collateral and borrows against it, their account equity is measured as the total value of their deposits. The purpose of ROE in this context is to indicate the impact of the loan on the value of their deposited assets. This calculation is most useful for working out the cost of an ordinary loan.

Their equity is:

$$\text{Equity} = \text{Total Deposits}.$$

Their ROE is:

$$ROE = \text{Net Supply APY} - \left(\frac{\text{Total Debt}}{\text{Total Deposits}}\right) \times \text{Net Borrow APY}.$$

Since a user's Loan-to-Value (LTV) is their total debt divided by their total deposits, this simplifies to:

$$ROE = \text{Net Supply APY} - \text{LTV} \times \text{Net Borrow APY}.$$

Example

Suppose a user deposits $1000 into ETH paying 3.5% Supply APY and borrows $800 USDC costing 4.5% Borrow APY.

Their equity is:

$$\text{Equity} = \$1000.$$

Their ROE is:

$$ROE = 3.5\% - \left(\frac{\$800}{\$1000} \times 4.5\%\right) = -2.25\%.$$

This results in a negative return, because they are paying more on their loan than they are earning on their deposits.

Multiplied borrow

If a user creates a multiplied position on Euler, their account equity is measured as the difference between their total deposits and their total debts. In this case, all borrowed assets are swapped back into collateral and re-deposited. The ROE calculation helps assess the impact of the position on their net asset value (NAV). This is useful for evaluating yield farming strategies or the overall cost of long/short positions.

Their equity is:

$$\text{Equity} = \text{Total Deposits} - \text{Total Debts} = \text{Net Asset Value}.$$

Their ROE is:

$$ROE = \frac{( \text{Total Deposits} \times \text{Net Supply APY}) - ( \text{Total Debt} \times \text{Net Borrow APY})}{\text{Net Asset Value}}.$$

Example

Suppose a user deposits $200 into ETH paying 3.5% Supply APY and multiplies 5× by shorting USDC, which has a 4.5% Borrow APY. Their total deposits and debts remain the same as the first example:

Total deposits = $1000
Total debts = $800

However, their equity is now:

$$\text{Equity} = \$1000 - \$800 = \$200.$$

Their ROE is now:

$$ROE = \frac{(\$1000 \times 3.5\%) - (\$800 \times 4.5\%)}{200} = -11.25\%.$$

Even though the deposits, debts, and interest rates are identical to the first example, the ROE is very different because the denominator changes.