The Mathematics of Long/Short

Over the past few weeks we have had the pleasure of meeting with clients and friends of the firm around Australia and New Zealand. We are fortunate to have such sophisticated advisers and investors as clients. It makes our discussions detailed and insightful. One subject of interest recently was the performance drivers of, and expectations for, long/short funds. In this article I will put some mathematics and meaning around the topic.

We have known for some time that the concept of a long/short fund brings with it interest, appeal and sometimes confusion. Much of this stems from the low and variable exposure to the market that these funds carry. And it elicits pointed feedback from advisers and clients alike.

Some of it is supportive, such as: it offers protection when markets turn down! I like being able to profit from owning great businesses and selling deteriorating ones. Others are quite negative: the shorts just drag on performance; I can get better returns in equities. And still others query: how can long/short beat the market? Should it beat the market? What should be expected?

These sorts of questions and comments are all reasonable. And especially after a very turbulent and challenging final quarter of 2018 when it seems that a lot of long/short funds didn’t perform true to label. The answer as is often the case may lie in a little calculus.

Breaking down long/short

Let’s start with a few definitions. The first is Greek (and it’s the Greek that is first): alpha or “α”. Alpha is defined as the outperformance from a stock, portfolio or fund over and above the performance of the market. So, if we denote the performance or return of the market by “rM” and the return of some portfolio by “rP” then we have:

α = rP – rM, or, rP= r+ α

Now let’s think about two portfolios. One is a long portfolio of stocks that is fully invested. That is every dollar in the portfolio is deployed to buy individual stocks and there is no cash left over. The return on this portfolio can be denoted by “rL”and its outperformance over the stock market is “αL”, such that:

rL = r+ αL

The second is a short portfolio of stocks which works inversely to the long portfolio in that 100 cents in the dollar is used to short sell individual stocks. The return on the short portfolio is denoted as “rS” Here the outperformance is a little different. We define the outperformance of a short portfolio as the amount by which its return exceeds the inverse, or negative, of the market return. This is short alpha or “αS”. It is represented as:

rS = – r+ αS

This may seem odd at first. But think about a short on a stock whose price declines say 4%. The return on the short position would be +4%. The short seller makes money when the price of the stock shorted falls (remember, profitable short sellers buy low and sell high as well, just in the opposite order). So how well has the short seller done on a relative basis?

Suppose over the same period the market also fell, but by 5%. Then we would say the short seller hasn’t done so well at all. We could have just been short the market index and generated a +5% return without the effort and risk of picking an individual stock to short. So, the short seller has detracted 1 percentage point of performance compared to a passive market short strategy. Mathematically this would look like:

rS = – r+ αS

+4% = -(-5%) + αS

Therefore, α= -1%

We can further combine these two portfolios into one fund, our long/short fund. To do this we need to introduce an additional feature – gross exposure. That is, we will scale the long and the short portfolios by some percentage. The long gross exposure can be represented by “gL” and the short gross exposure by “gS”. Denote the return of our long/short fund by “rF” and we have:

rF = gL* r+ gS* rS

This simply says the return of the fund will be the return on the long portfolio multiplied by the amount of long exposure (or the percentage by which we have scaled the long portfolio), plus the return on the short portfolio multiplied by the short exposure (or scaling factor). Breaking this down further using the formulae from above:

rF = gL* (rM+ αL)  + gS* (- r+ αS)

Therefore,rF = gL* αL+ gS* αS + rM* (gL– gS)

And in words this says the return of the fund can be broken down into three components:

  1. Outperformance of the long portfolio, scaled by the gross long exposure
  2. Outperformance of the short portfolio, scaled by the short gross exposure
  3. Market return, scaled by the “net market exposure” which is the difference between long and short gross exposures (simply the amount invested to take advantage of market gains minus the amount shorted to profit from market falls)

Using numbers

Let’s put some numbers to this to bring it to life. Assume our fund is 90% gross long, 50% gross short, that we can generate 5% of alpha on the long side, 3% of alpha on the short side, and the market return is 8%. Then we have the following result for our fund:

rF = (90% * 5%) + (50% * 3%) + 8% * (90% – 50%)

= 4.5% + 1.5% + 3.2%

= 9.2%

We can see that the long outperformance added 4.5 points of return, the short outperformance added 1.5 points of return (so our total manager outperformance added 6 points of return), and the net exposure to a rising market added 3.2 points of return.

There are three points of note:

  1. The market contribution to our fund’s return was 3.2%, much less than the total market return of 8%. This is because the net effect of being long and short gave the fund a much lower 40% exposure to the performance generated by the stock market. It’s obviously a drag when markets go up, but a benefit when markets decline. Our fund is just not fully sensitive to the market. Said another way, our long/short fund has reduced market risk.
  2. Despite the positive contribution from short alpha, or the short portfolio’s outperformance of its hurdle (inverse the market), the short portfolio produced a negative absolute return. That is, the short portfolio lost absolute dollars. We see this formulaically as:

rS= – r+ αS= -(+8%) + 3% = -5%

The negative absolute short portfolio performance has been split into a (quite significant) negative market component (the drag mentioned in the note above) and a positive outperformance component (short alpha). It is often this point that can be forgotten or confusing when it comes to disaggregating long/short fund returns.

  1. The fund outperformed the market. Even though the fund only captured 3.2% of the market’s 8% gain, the value-add from the long and short portfolios by way of their ability to surpass their passive market hurdles contributed an additional 6%. This demonstrates that when (scaled and combined) alpha exceeds the drag from a lower net market exposure, the return of a long/short fund will exceed the market return.

While putting equations on a page can often compound confusion, we thought we had an obligation to advisers, clients and friends of the firm to try our best to demystify the pattern of performance of long/short funds. By laying out the mathematics of long/short fund performance we hope to move out of the clouds and towards clarity. In this way the attributes of long/short funds can be more readily and accurately assessed, and perhaps the potential for surprise results that deviate from prior expectations can be reduced.

 Christopher Demasi is a Portfolio Manager with Montaka Global Investments.
To learn more about Montaka, please call +612 7202 0100.

2 thoughts on “The Mathematics of Long/Short”

  1. As an investor that raised this question in a webinar, thank you very much for the detailed explanation, helps enormously.

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