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

Question

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?

Answer 1

It is true that you do not "invest your money" in a short position, but still that position has a dollar size sometimes called MVS (market value of short securities, usually negative by convention, for ex short 100 shares of ACME priced at 5/sh gives MVS =(-100)*5 = -500 dollars), and the longs have an MVL value.

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).

The returns of a dollar neutral portfolio are usually measured on a per dollar basis in academic studies. So a profit of 0.05 in a month means you made 5 cents on a position which was short 1 dollar of stocks and was long 1 dollar of other stocks at the beginning of the month.

The price of MSFT stock today is 212.83 dollars per share. To buy shares of MSFT with a market value of 1 dollar you have to buy 1/212.83 = 0.0046985 shares, to short a MVS of -1 dollars you have to short the same number of shares. (It seems like a ridiculously small number, but rememember that this is just because of the convention of normalizing to one dollar. It is only for purposes of calculation, not in real life).

In transaction costs, a "full turn" is how much it costs to buy an asset and then sell it again right away. A "half turn" is half of this, i.e. the cost of just buying the asset or the cost of just selling it. Usually in academic studies transaction costs are expressed in basis points or bps ("bips"). For example a full turn of 40 bips means you start with some cash, buy stock, sell it again and you end up with 0.40% less cash than you started with. So you have 0.996 dollars left for every dollar you started with, assuming no price change. Again, everything is done on a "per dollar" or "normalized to a dollar" basis.