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When calculating the long-short excess returns for a portfolio. Do I have to first calculate the excess returns of the long and short leg and then add them together or first calculate the average long short return and then subtract the risk free rate?

So either

  1. Calculate average returns for both the long and short portfolio.
  2. Calculate excess returns for the long and short portfolio separately.
  3. Calculate the long short return by adding them together.

OR

  1. Calculate average returns for both the long and short portfolio.
  2. Calculate the average long short portfolio return
  3. Calculate the excess return for the long short portfolio.

Many thanks in Advance!

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    $\begingroup$ After thinking about it for a while. It wouldn't make sense to first calculate the excess return for the Long an Short leg separately as we would simply remove the rf later on: (Long - rf) - (Short - rf) = Long - Short $\endgroup$ – thxclown Mar 11 at 16:06
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You're right, there is no difference between the long-short (LS) portfolio between two returns or two excess returns, the risk-free rate cancels out.

But there is an economic reason why we consider returns, excess returns and long-short returns. A simple raw return does not tell you much as you need to incorporate how much it cost you to obtain that performance. If a stock returns 1% but the risk-free rate is 2%, then your real return -1% after considering that you first need to borrow $1 to invest into the stock. So, the raw stock return really doesn't tell you much if you don't subtract the cost of having this return.

The long-short portfolio is different. Here, you gain $1 from selling one portfolio and invest this \$1 into the long portfolio. Hence, people refer to a long-short portfolio as ``zero-cost portfolio'' because you don't need to borrow \$1 at the risk-free rate (the funding cost is covered by the selling of the short portfolio). Equivalently, you can of course gain \$0.5 from selling one portfolio and investing 50 cent in the long portfolio. It doesn't matter.

An example is when you run a simple time series regressions using the Fama-French factors. You never regress the raw returns on the factors, you always first subtract the risk-free rate because this is the true performance an investor would have by investing in this particular portfolio. Similarly, that's why Fama & French subtract the risk-free rate from the market portfolio ... one needs to borrow \$1 to be able to obtain the market return. The other factors, SMB and HML (1993) or CMA, RMW (2015) or UMD (1997) etc. are all long-short portfolios and hence do not include the risk-free rate as they have zero funding cost.

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