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In a simple setup, let's say $(w_1,w_2)$ are weights invested in assets $1$ and $2$ with prices $S_1$, $S_2$.

I only saw discussions when $w_1 + w_2 = 1$, even if short selling is allowed.

Suppose $S_1(0) = S_2(0) = 100$. I short the first assed and long the second one. My initial investment is $0$.

$\bullet$ What are the 'weights' in this scenario?
$\bullet$ How do we compute the return? For example, the Capital Asset Pricing Model (CAPM) is built on traditional return, but it is not defined here. Even if it was defined, but very small, I believe it does not make sense to use it.
$\bullet$ How do we compute Value at Risk? Does it use absolute returns instead?

Excuse my ignorance. I appreciate if someone can point me in the right direction.

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2 Answers 2

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In accademia people often analyze portfolios where there are an equal size long and short positions, so the weights add up to zero. Such portfolios are usually referred to as "arbitrage portfolios" or "zero net investment portfolios". An example of a well known academic paper in this area is Jegadeesh and Titman (1993). (It is very well done and worth studying to understand the calculation methods used https://www.jstor.org/stable/2328882).

The method used to compute returns in this and similar papers is to compute the "return per dollar long". For example if you go long one million dollars of something and short one million dollars of something else, and today's market moves in such a way that you lose 50,000 USD, we would say that the rpdl is -0.05 or -5%. You can compute the normal statistics with such returns including VaR etc.

In real life, no broker will allow you to create such a position, you need to put up some cash to cover your losses (where is the 50,000 to cover the first day loss going to come from? The broker certainly does not want to pay for it). So for realistic analysis of an arbitrage strategy you have to decide how much cash (equity) the investor is going to put up, i.e. the degree of leverage you are willing to take. Only then would you be able to compute the actual returns an investor would make.

So there are two ways to analyze arbitrage portfolios, a simplified approach where the two weights l and s add up to zero (usually seen in academic paper), or a more realistic approach in which cash is also modeled and the three weights l, s and c again add up to 1.

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Welcome to the forum. I read these two answers where @nbbo2 was a great reference as usual:

  1. market neutral weights and cash values
  2. How to compute portfolio returns when constructing a dollar-neutral portfolio

The answers to your questions are:

  • What are the 'weights' in this scenario? - 0.5 for both. Take note that we are taking the absolute value of the % weights. This means that it should have been -0.5 (short) and +0.5 (long) such that they offset each other. However, we took |-0.5| = 0.5 (for the short position) for computation purposes of other metrics.
  • How do we compute the return? For example, the Capital Asset Pricing Model (CAPM) is built on traditional return, but it is not defined here. Even if it was defined, but very small, I believe it does not make sense to use it. - Why do you need the CAPM in this case though? I would just take the percentage returns of the individual securities (from historical time series) and compute a portfolio return together with the % weights.
  • How do we compute Value at Risk? Does it use absolute returns instead? - I would use volatility of the set of % daily returns of the portfolio (answer from the previous question). The formula for the classical VaR is given by:

\begin{equation} \text{VaR}_{95,1Y} = Z \sigma \sqrt{T} = 1.645\cdot\sigma\cdot\sqrt{252} \end{equation}

where you are computing the 95.0% percentile VaR at a 1-year horizon by "lengthening" your 1-day VaR.

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