Pairs trading relies on opening a 'long' position' in one security and a 'short' position in another (presumably correlated) security. The best way to think about a long position is one first buys a security and, at a later time, sells that security. A short position is just the opposite. One first sells a security and, at a later time, buys that security.

What is the definition of 'returns? Portfolio returns is simply the gain in portfolio value from time t_0 to the current time expressed as a percent. It can be represented like this

portfolio_value = cash + sum_of_long_positions + sum_of_short_positions
returns = (portfolio_value_today - portfolio_value_t_0) / abs(portfolio_value_t_0)

This definition of 'returns' is an arithmetic return (not log return) and is valid for long positions and short positions or even no positions.

What is the definition of 'leverage? Account leverage is the sum of the absolute value of long and short positions divided by the portfolio value. It can be represented like this

portfolio_value = cash + sum_of_long_positions + sum_of_short_positions
leverage = ( sum_of_long_positions + abs(sum_of_short_positions) ) / portfolio_value

Let's see how these values are calculated for a long-short portfolio. Perhaps a strategy trading a pair of stocks ABC and XYZ.

day 1 - starting balance of $100,000  
        cash: $100,000  
        long positions: $0  
        short positions: $0  
        portfolio value: $100,000  (100000 + 0 + 0)  
        leverage: 0   ((0  + abs(0)) / 100000)

day 2 - open long and short positions  
        buy 1000 shares XYZ @ $100/share  
        sell 500 shares ABC @ $200/share  
        cash: $100,000  (100000 -100000 + 100000)  
        long positions: $100,000  
        short positions: -$100,000  
        portfolio value: $100,000  (100000 + 100000 - 100000)  
        leverage: 2  (100000 + abs(-100000)) / 100000)  
        returns: 0%  (100000 - 100000) / abs(100000) 

day 3 - longs go up $1/share shorts go down $1/share  
        cash: $100,000  
        long positions: $101,000  (1000 x $101)  
        short positions: -$99,500  (-500 x $199)  
        portfolio value: $101,500    (100000 + 101000 -99500)  
        leverage: 1.98  (101000 + abs(-99500)) / 101500)  
        returns: 1.5%  (101500 - 100000) / abs(100000) 

day 4 - close all positions  
        sell 1000 shares XYZ @ $101/share  
        buy 500 shares ABC @ $199/share 

        cash: $101,500  (100000 + 101000 - 99500)  
        long positions: $0  
        short positions: $0  
        portfolio value: $101,500    (101500 + 0 - 0)  
        leverage: 0  (0 + abs(0)) / 101500)  
        returns: 1.5%  (101500 - 100000) / abs(100000) 

A couple of things to note. First, the cash in the account is always $100,000. Second, the leverage goes to 2 after the initial trades are made however, the portfolio value remains unchanged (at $100,000). After, the stock prices change (favorably) the leverage goes down a bit and the returns go up. I'll leave it up to the reader to do the math if the stock prices changed unfavorably, but in that case the leverage would go up (above 2) and the returns would go down (negative).

Hope that all makes sense?

One last issue I'd like to address. The statement was made "no initial capital is required (in principle)". While that may be true "in principal" it's definitely not true in the real world. Moreover, the values for returns and leverage become incalculable, which suggests something is wrong and therefore isn't even really true "in principal". Let's do the exact trades as above but start with $0 in cash.

day 1 - starting balance of $0  
        cash: $0  
        long positions: $0  
        short positions: $0  
        portfolio value: $0  (0 + 0 + 0)  
        leverage: 0   ((0  + abs(0)) / 0)

day 2 - open long and short positions  
        buy 1000 shares XYZ @ $100/share  
        sell 500 shares ABC @ $200/share  
        cash: $0  (0 -100000 + 100000)  
        long positions: $100,000  
        short positions: -$100,000  
        portfolio value: $0  (0 + 100000 - 100000)  
        leverage: infinite  (100000 + abs(-100000)) / 0)  
        returns: 0  (100000 - 100000) / abs(0) 

Notice the leverage is infinite (because of the division by zero). In practice the most leverage (ie margin) a broker would extend is 2. One would need at least $100,000 in an account to long and short $100,000 worth of stock and maintain a leverage of 2. This is the case in the first example. In developing trading strategies ensure you constrain leverage. Ideally, start by constraining leverage to 1. This not only ensures the strategy is realistic but also allows for more meaningful comparisons between strategies.

Thanks for your answer! I'll think longer about it. It feels somehow unsatisfactory that depending on your starting cash you suddenly have different returns. How can you then compare 2 portfolios? What you've done above is somehow similar what I currently do when I say that I use as my comparison basis the MAX_GROSS_EXPOSURE (sum_of_long_positions + abs(sum_of_short_positions)).

Is there some sort of "textbook" where such topics are discussed in more detail? The textbooks I've seen so far do not talk much about how the situation looks like once short selling is included.

The other thing you mention is that once you start including short selling you have to work with percentage changes and cannot use log-normal any longer. My stomach feeling would tell me that it would be more "correct" to split the portfolio into one for long and one for short and then use log-normal for each one of them. Doesn't it somehow have a negative effect (for example for the "correctness" of the covariance matrix) if you work with percent changes rather than with log-normal (or even better log-student-t or similar)? A long position can never have -110% change, but your normal distribution for percent changes is not bounded at -100%. Similar for the short position, which has an upper limit for the percent change, but the normal distribution does not have such a constraint. All very confusing :)