# What is the proper way to calculate returns for Pair Trading?

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I am assuming that I don't need to use margin account to short here:

What is a standard way to calculate return for pairs trading strategy? For example, I bought 100 dollars worth of a loser (L) and shorted 100 dollars worth of winner (W), and when their prices converged I sold L for 110 dollars worth, and paid 90 dollars for W. At the end I made dollars 20, but what is my return? Normally I would divide profit by investment, but here my net investment is zero, so I can't do that. It seems reasonable to count each of my 100 dollars positions as separate investments, in that case I would have:

$$\mbox{return} = \frac{20}{100 + 100} * 100 \% = 10 \%$$

Is the above right? If not what is the proper method?

If I need to use a margin account:

Under Regulation T, it is mandatory for short trades that 150% of the value of the position at the time the short is created be held in a margin account. This 150% is comprised of the full value of the short (100%), plus an additional margin requirement of 50% or half the value of the position. (The margin requirement for a long position is also 50%.) For example, if you were to short a stock and the position had a value of \$20,000, you would be required to have the \$20,000 that came from the short sale plus an additional \$10,000, for a total of$30,000, in the account to meet the requirements of Regulation T

Investopedia

Then I would need to put 100 * 1.5 = 150 in the margin account for shorting a winner, and buy 100 dollars worth a loser, so my return is: $$\mbox{return} = \frac{20}{100 + 150} * 100\% = 8\%$$

I think the above is right, but a bit unsure.

Thank You

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You had to put-up money for the margin account, right? –  chrisaycock Mar 10 '13 at 20:37
@chrisaycock, thanks for the pointer, I think thinking about margin account explicitly actually makes things easier. I updated my a question, I think I am calculating the return correctly now, is that right? –  Akavall Mar 10 '13 at 22:46
It differs on which market you're investing, notice that Regulation T only applies in america. Other countries will have similar rules though. However, you're still not calculating it correctly. Notice that if you need a collateral of 150% of the value of the shorted stock, you'll still get cash from selling it. So the cash thats taken from your pocket is only 50% of the value of the stock. Also, I think you can use some percentage of the long position as collateral, but ofc this depends on what kind of player you are. –  Good Guy Mike Mar 11 '13 at 17:18
@GoodGuyMike, I got it. Thanks a lot! –  Akavall Mar 11 '13 at 19:25

It depends on which return you precisely attempt to measure:

• Gross Return on trade: Forget about margin or not margin, it does not matter. When you evaluate the performance of a single position you look at the notional to which you exposed yourself to. So if you bought a stock worth 100 dollars and later on sell it for 110 dollars then you generated a return of 10 dollars or 10% of exposed notional. It does not matter at this point whether you borrowed the funds from someone or not in calculation of gross return on your trade.

• Net return on trade: If you want to calculate the total return on your investment then you may have to factor in the costs of funding as well.

• Return on capital (employed): In this case you need to look at how much of your capital did you lock up/ margin in order to trade such position. Here, margins come into play. Obviously, your returns on leveraged investments will be much more pronounced when measured as return on capital rather than return on the actual notional traded, but your risk will also be magnified.

Now, a pair trade is really not much different in terms of return attribution from 2 separate trades. Thus all you do is you aggregate the returns (after you define precisely what return type you attempt to evaluate). Make sure you understand that risk is NOT aggregated because it does not linearly scale. But as this question is just about returns you should simply add return profiles of each leg together.

In your example if you longed 100 dollars and you shorted 100 dollars on another name then your total notional traded is 200 dollars. You generated an aggregate of 20 dollars and hence your gross return on the trade is 10%. Your net return depends on how much you paid in funding and borrow charges. Your return on capital depends on how much you leveraged your position. I can't answer it without knowing your leverage rate on this particular trade. But its pretty straight forward for you to derive.

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