Sell asset to balance the portfolio

Question

Given:

• a portfolio with only two assets (A and B)
• they respective balances (Balance_A and Balance_B)

• the function $C_{AtoB}$ that converts A into B:

  • $C_{AtoB}(x, q, cr) = x(1 - cr)q$, where
    • x = the amount of Asset A to be convert
    • q = the quotation of Asset A
    • cr = the convertion rate cost

• a target ratio between the values of Asset A and Asset B named $Tr$

• a scenario where:

$$ \frac{C_{AtoB}(Balance_A, q, cr)}{Balance_B + C_{AtoB}(Balance_A, q, cr)} > Tr$$

How to calculate the value X of asset A to be converted into asset B in order to the expression below become true?

$$ \frac{C_{AtoB}(Balance_A - X, q, cr)}{Balance_B + C_{AtoB}(Balance_A - X, q, cr)} = Tr$$

After solve the equation I found this:

$$X = \frac{Tr.Balance_B}{(1 - cr).q.(Tr - 1)} + Balance_A$$

But I am getting exactly the double of what I was expecting and I am not understanding why.

Answer 1

Ok, I got my error.

Instead of:

$$ \frac{C_{AtoB}(Balance_A - X, q, cr)}{Balance_B + C_{AtoB}(Balance_A - X, q, cr)} = Tr$$

I should have written:

$$ \frac{C_{AtoB}(Balance_A - X, q, cr)}{Balance_B + \color{red}{C_{AtoB}(X, q, cr)} + C_{AtoB}(Balance_A - X, q, cr)} = Tr$$

$\color{blue}{X}$ is the amount of asset A being converted into asset B so it will be part of the portfolio but as $\color{red}{C_{AtoB}(X, q, cr)}$ after the convertion.

Consequently, after solve it, instead of having:

$$X = \frac{Tr.Balance_B}{(1 - cr).q.(Tr - 1)} + Balance_A$$

I will get this:

$$X = \color{red}{\bigg[}\frac{Tr.Balance_B}{(1 - cr).q} + \color{red}{(Tr - 1).}Balance_A\color{red}{\bigg].-1}$$