The profit calculation assumes that you borrow at a fixed interest rate to finance investments. An alternative way to borrow is to short-sell stock. What complications would arise in calculating profit if you financed a $1000 S&R index investment by shorting stock, rather than by borrowing $1000?
If you borrow money from a bank to buy a $1000 S&R index, your borrowing cost is known at the time of borrowing. Suppose the annual effective risk free interest rate is \( r \). If you borrow $1000 at \( t=0 \), then at \( T \) you just pay the bank \( 1000(1+r)^T \). You know that your borrowing cost is fixed in advance.
In contrast, if you short-sell \( n \) number of stocks and use the short sale proceeds to buy a $1000 S&R index, you owe the brokerage firm \( n \) number of stocks. If you want to close your short position at time \( T \), you need to buy \( n \) stocks at \( T \). The cost of n stocks at \( T \) is \( nS_T \) , where \( S_T \) is the price of stocks per share at \( T \). Since \( S_T \) is not known in advance, if you use short selling to finance your purchase of a $1000 S&R index, your borrowing cost \( nS_T \) cannot be known in advance. This brings additional risk to your position. As such, you can’t determine your profit.
If you short sell the stocks, then when you need to close your position in the future at time \( T \) you would need to buy back \( n \) stock at a price of \( nS_T \) where \( S_T \) is the stock price at time \( T \) but \( S_T \) is not known in advance and introduces risk to the position and makes it difficult to calculate profit.