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I feel like an idiot asking this but i haven't found the answer anywhere.

I have backtestest a paris trading strategy, while calculating the returns of the strategy I run into some problems when the P&L just gets more negative. Lets take for example the data above. From 2005-02-16 to 2005-02-17 the arithmetic return is 39.13% or for the dates 2005-02-23 to 2005-02-24 the return is -16311.20% which isn't right obviously. So my question is how do I calculate the returns when I have a P&L which allows negative negative values.

2005-02-14 5010 2005-02-15 -23315 2005-02-16 -14371 2005-02-17 -19995 2005-02-18 -17064 2005-02-21 -25018 2005-02-22 736 2005-02-23 -125 2005-02-25 20264

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  • $\begingroup$ I assume these are profits/losses in dollars each day. But what is the Equity (or Capital) under management to which these P&Ls are being applied? How much money did you assume to start with and how much is it now? The Equity must remain positive at all times or you are out of business. $\endgroup$
    – nbbo2
    Commented Feb 17, 2017 at 23:16
  • $\begingroup$ @noob2 It is a self financing portfolio so I short one stock and with that money i buy the other, so in theory the starting Equity is 0. I know in real life i would need to have a margin but that's something I would like to add later to the model. No I just need the returns calculated the right way. $\endgroup$
    – Alejandro Andrade
    Commented Feb 18, 2017 at 4:10
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    $\begingroup$ In real life the starting equity cannot be zero. You must open an account with a prime broker and deposit some cash before you can start trading. (Otherwise who bears the loss if you lose money on the 1st day?). Without an assumption about starting capital the percent return is undefined. You can analyze dollar P&L if you cannot compute percent return. $\endgroup$
    – Alex C
    Commented Feb 18, 2017 at 4:31
  • $\begingroup$ The simplest assumption is that you have as much capital as the value of your longs, which is also equal to the value of your shorts. $\endgroup$
    – Alex C
    Commented Feb 18, 2017 at 4:51
  • $\begingroup$ @alejandro andrade How are you calculating the position size that is responsible for the P&L? It is not possible to do so without assuming a certain amount of starting equity. You cannot back test and derive hypothetical P&L without that. 'Self financing portfolio' is not how the real world works. Why test in a way that is not applicable? $\endgroup$
    – amdopt
    Commented Feb 18, 2017 at 21:45

2 Answers 2

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There are two ways to calculate the returns.

One way is to calculate the net asset value (NAV) of your portfolio.
For the long side the NAV is the value of your stock holdings.
For the short side the initial NAV is zero since the cash proceeds from the sale balances the liabilities of the short holdings.
The portfolio NAV is hence initially equal to the value of the long holdings.
At a future date the short NAV is equal to the initial cash proceed from the sale minus the current liability of the short position, which is the negative value of the stocks that are shorted. The portfolio NAV is hence the value of the long stocks + cash proceeds from the sales - value of the short stocks. To find the return $R(t_1,t_2)$ between dates $t_1$ and $t_2$ one takes $R(t_1,t_2) = NAV(t_2)/NAV(t_1) -1 $.

Another way is to calculate the period return (say one week) of the long stocks and the negative returns of the short stocks and average them (assuming equal weighting) giving the long/short return over the period.

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  • $\begingroup$ @RPG but by the definition you give of a NAV it is possible to have a negative one and again if the NAV(t_2) is negative and smaller the NAV(t_1) I get a positive return which isn't right. And to be honest that's the way I'm calculating the P&L so it just give's me the same problem. Thanks anyway $\endgroup$
    – Alejandro Andrade
    Commented Feb 19, 2017 at 22:18
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    $\begingroup$ @alejandro andrade If your NAV is negative then you have a liability that needs to be covered with funds such that NAV>=0. If you want to calculate returns while disregarding this, then you can use the second method I outlined. $\endgroup$
    – RRG
    Commented Feb 20, 2017 at 7:03
  • $\begingroup$ The second method may be good enough for a basic evaluation. $\endgroup$
    – nbbo2
    Commented Mar 20, 2017 at 20:30
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I worked in long-short neutral fund. For a long-short neutral strategy, typically in practice book size is assumed to be long size + abs(short size). So daily return would be daily profit / book size as defined

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    $\begingroup$ I would add that long size + abs(short size) is sometimes called "gross exposure". And long size - abs(short size) is called "net exposure". $\endgroup$
    – nbbo2
    Commented Sep 22, 2022 at 18:31

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