Performance fee calculation

Fee is collected to fee_vault every time the operator sends a rebalance crank. We are currently charging 5% of profit as the performance fee. Before jumping into the formula, we define some variables:

Before rebalance:

• vault.total_amount : t1

• lp_mint.total_supply: p1

• virtual_price (value of lp token): v1 =t1/p1

After rebalance:

• vault.total_amount : t2

• lp_mint.total_supply: p1

• virtual_price: v2 =t2/p1

We charge performance fee:

$$fee = 0.1* (t2-t1)$$

Virtual price after fee:

$$v21= (t_2-fee)/p_1$$

Vault doesn't send the token directly to the treasury token account (because vault may not have enough liquidity), so vault mint more lp tokens for fee_vault. Assuming vault mints more delta lp tokens, then the new virtual price:

$$v22= t2/(p1+delta)$$

We still ensure the virtual price in (1) and (2) are the same v21=v22, so

$$(t2-fee)/p1=t2/(p1+delta)$$

Then we can calculate how many lp tokens is minted more for each rebalance crank:

$$delta = (p_1*fee)/(t_2-fee)$$

Or

$$delta=p_1*(t_2-t_1) / (9*t_2+t_1)$$