Context
You have an initial amount of money, \( V_{\text{initial}} \), divided equally between:
- Account A in Country A using Currency A.
- Account B in Country B using Currency B.
Due to fluctuations in exchange rates, the value of Account B (converted to Currency A) may differ from Account A. Each transaction, we calculate the difference in value:
\( x = \text{Value of Account B in Currency A} - \text{Value of Account A} \)
To rebalance, we transfer funds between accounts. The bank charges a fee of percentage \( k \) (where \( 0 < k < 1 \)) on the amount of money converted.
Rebalancing Formulas
The amount to transfer, \( z \), is calculated as:
- If \( x > 0 \), transfer from Account B to Account A:
\( z = \dfrac{x}{2 + k} \)
- If \( x < 0 \), transfer from Account A to Account B:
\( z = \dfrac{-x}{2 + k} \)
Final Formulas
After \( n \) transactions:
Total Fees:
\( \text{Total Fees} = \dfrac{k (m + p)}{2 + k} \)
Total Combined Account Value:
\( V_{\text{after } n \text{ transactions}} = V_{\text{initial}} + (m - p) - \dfrac{k (m + p)}{2 + k} \)
Where:
- \( m = \) Sum of \( x \) values when \( x > 0 \).
- \( p = \) Sum of absolute values of \( x \) when \( x < 0 \).
Probability Calculation
The probability that the total combined Account value after \( n \) transactions is greater than the initial value is given by:
\( \text{Probability} = \Pr\left( m (1 - k) > p \right) \)
This means that for the total combined account value to increase, the net gain from positive exchange differences (after accounting for fees) must exceed the losses from negative differences.
From the condition \( m(1 - k) > p \), this inequality shows that even with transaction fees, if the adjusted gains outpace the losses, the overall value increases.