# How to compute portfolio returns when constructing a dollar-neutral portfolio

I am trying to wrap my head around this statement:

dollar-neutral portfolios are built: dollar amounts of both long and short positions are equal. Furthermore, it is also true at the stock level: each position, long or short, may be normalized to one dollar. the naive 1/ N portfolio strategy, out-of-sample, is not inefficient.

I want to be able to implement, that is write code so to compute the daily return of such a portfolio (and then compute other statistics on daily returns). The context is: I am trying to apply statistical arbitrage strategy which performs k long operations and k short operations with a certain frequency, in my case daily. I want to be able to backtest this strategy and write some code that does so, but I am not sure how to do that. I do understand that we are speaking about a portfolio that rebalances daily, formed by 2*k stocks, equal weights. If the portfolio is dollar neutral => same amount of dollars is invested for both long and short. Here comes my doubt: long operations require some initial capital whereas short do not so I do not understand why the investment in needed in the short stoks?

Second question:

Furthermore, it is also true at the stock level: each position, long or short, may be normalized to one dollar.

How are the positions normalized to one dollar?

Third question: How is the return computed in such context? How do I set the number of shares for each stock that I want to trade (long or short) in a particular day knowing their prices?

Fourth question: How do I apply transaction costs for this? What is a half-turn?

In a dollar neutral portfolio $$|MVS|=MVL$$. The exact values do not matter (could be 1 dollar or 7.5 million) as long as they are equal in magnitude. (Also the initial investment does not matter, maybe you invested a long time ago at very different prices but these are current market values).