vaults (ERC-4626)

https://ethereum.org/en/developers/docs/standards/tokens/erc-4626/

how many shares to mint

• a = amount to deposit
• B = balance of vault before deposit
• s = shares to mint
• T = total shares before mint

eQ: a+B / B = s+T/T

to find s (shares) we can simplify it to:

1. Cross-multiply: [ (a + B)T = (s + T)B ]

2. Distribute both sides: [ aT + BT = sB + TB ]

3. Subtract ( BT ) from both sides: [ aT = sB ]

4. Solve for ( s ): [ s = aT/B ]

s = aT/B

How much amount to withdraw

• a = amount to withdraw
• B = balance of vault before withdraw
• s = shares to burn
• T = total shares before burn

eQ: B-a/B = T-s/T

1. Cross-multiply: [ (B + a)T = (T - s)B ]

2. Distribute both sides: [ BT + aT = TB + sB ]

3. Subtract ( BT ) from both sides: [ aT = sB ]

4. Solve for ( a ): [ a = sB/T ]

a = sB/T

Example

Step-by-Step Calculation

1. Initial Shares and Deposits:

   • User 1 deposited 125 and has 125 shares.
   • User 2 deposited 200 and has 200 shares.
   • Total shares: 125 + 200 = 325 shares.
   • Total initial deposits: 125 + 200 = 325.

2. Vault Yields:

   • The vault yields 100, so the new total value of the vault is: [ 325 + 100 = 425 ]

3. Share Value:

   • The new share value is: [ \text{New share price} = \frac{\text{Total value of the vault}}{\text{Total shares}} = \frac{425}{325} \approx 1.3077 ]

4. Withdrawals:

   • Each user will withdraw an amount proportional to the shares they hold.

User 1:

• User 1 has 125 shares.
• The value of User 1's shares: [ \text{Value of User 1's shares} = 125 \times 1.3077 \approx 163.46 ]

User 2:

• User 2 has 200 shares.
• The value of User 2's shares: [ \text{Value of User 2's shares} = 200 \times 1.3077 \approx 261.54 ]

Summary

• User 1 can withdraw approximately 163.46.
• User 2 can withdraw approximately 261.54.

These values add up to the total vault value of 425, ensuring that the withdrawals are proportional to the shares held by each user.