Calculating returns with trading costs

Question

This perhaps is an over simplification of calculating trading returns while including trading costs. I've made some assumptions - the commission for investing and extracting an investment is 1% and 2% respectively. The commissions do not change over the trading period which in this case is 5 time steps. I've used Python code to perform the calculations.

Set of positive and negative percentage changes in price for a given asset over 5 time steps is {0.031% , 0.00121% , 0.0231% , -0.0213% , -0.0121%}.

The commission to enter an investment is 1% of the invested amount, the commission to exit an investment is 2% of the current value of the invested amount.

If I invest 1 euro in this asset, is the following correct?

1.

The final investment amount if I do not trade the investment until $t=5$ is: the final percentage change amount at $t=5$ which is 'initial invested amount' + '% change' - 'commission to enter' - 'commission to exit', therefore:

initial_investment_amt = 1

comission_in_amt = 1

comission_out_amt = 2

price_change = -.0121

return_amt = (initial_investment_amt + (price_change / 100)) - (comission_in_amt / 100) - (comission_out_amt / 100) = 0.97 which represents a loss of 1 - .97 = .03

2.

The final investment amount if I trade the investment at each time step until $t=5$ is:

initial_investment_amt = 1

comission_in_amt = 1

comission_out_amt = 2

price_change = .031

return_amt_1 = (initial_investment_amt + (price_change / 100)) - (comission_in_amt / 100) - (comission_out_amt / 100)

price_change = .00121

return_amt_2 = (return_amt_1 + (price_change / 100)) - (comission_in_amt / 100) - (comission_out_amt / 100)

price_change = .0231

return_amt_3 = (return_amt_2 + (price_change / 100)) - (comission_in_amt / 100) - (comission_out_amt / 100)

price_change = -.0213

return_amt_4 = (return_amt_3 + (price_change / 100)) - (comission_in_amt / 100) - (comission_out_amt / 100)

price_change = -.0121

return_amt_5 = (return_amt_4 + (price_change / 100)) - (comission_in_amt / 100) - (comission_out_amt / 100)

print(return_amt_1)
print(return_amt_2)
print(return_amt_3)
print(return_amt_4)
print(return_amt_5)

prints :

0.97031
0.9403220999999999
0.9105530999999999
0.8803400999999998
0.8502190999999998

which represents a loss of $1 - 0.85 = 0.15$.

Answer 1

The easiest way to think about this would be to think of these events in chronological order. Propose you invest $100 and the commission to buy is 1%.

Then the quantity of the asset you have is ((100) - (0.01 * 100)) / buy_price_of_asset.

Suppose you then sell the asset at a price of 110. The selling process becomes: (110 * quantity) - 0.02 * (110 * quantity).

Essentially, the transaction cost is proportional to the volume you are trading with.

So in your case, both answers aren't quite fully correct. Your first one is close, as you MUST make the sum of transaction costs proportional to the capital you are trading with.