Problem: select and weight 4 constituents of an index to hedge a particular stock

I would appreciate any feedback on my approach, here is I would go about it:

  1. Create a new index of four constituents based on the correlation/co-integration of their time series as well as their risk similarities
  2. Optimize weights of each constituent through the least squares method
  3. Estimate optimal hedge ratio by calculating the beta of the stock being hedged relative to our new index and infer $ amount for each stock Thanks!
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    $\begingroup$ I imagine the firm must be aware of or have accounted for the possibility, but IMO feels a little lame submitting 'your' solution after getting input from several others in this way, particularly in the context of a take home as part of an interview evaluation. $\endgroup$ – Chris Feb 18 '20 at 0:26
  • $\begingroup$ @Chris I agree, I’m not even sure whether this is the type of question we want on here. On the other hand, the motivation for the question can be obscured so I whatever the outcome of the debate it will be hard to police so I didn’t take it to meta. $\endgroup$ – Bob Jansen Feb 18 '20 at 17:43
  • $\begingroup$ @BobJansen, agree. He appears to have edited to remove details about this being a question from a prop fund. THAT (intentionally obscuring for personal benefit) seems unambiguously wrong. $\endgroup$ – Chris Feb 18 '20 at 19:00
  • $\begingroup$ It improves the question though, I’ll let it be but feel free to put the issue on meta. $\endgroup$ – Bob Jansen Feb 18 '20 at 20:45

Here are a couple of the flaws I see:

  1. Your measures of correlation/cointegration are likely to depend on the period chosen for the analysis. These can be quite variable, and your hedge portfolio may not perform as well as you expect out of sample. Given that, your hedge portfolio is likely to place an emphasis on the stocks that appear closest to the alcoholic beverage firm in sample.
  2. Your method of selection is limiting the subset of stocks used to build the hedge to the subset of EuroStoxx stocks that are most like the alcoholic beverage firm. This may not be the best universe to build the overall hedge for the firm since you may wish to incorporate other companies in the hedge universe that most closely model the difference between the alcoholic beverage firm and a close firm. Hence, your overall hedge portfolio may not be the best.

For me, I would say that there are two problems here - what is the best metric for measuring the performance of the hedge relative to the stock, and how to pick the 5 candidates for the final portfolio. Both of those have a range of options.

Typically, most people would start by looking to minimize the variance of the relative returns of the stock and the hedge. In other words, they are looking to minimise $var(r_h - r_s)$. If that variance is small, then we can expect that the hedge portfolio will closely track the stock. This is akin to your proposal to minimize the correlation. If we know the covariance matrix of the EuroStoxx constituents and the alcoholic beverage firm, $\Sigma$, then we can express the portfolio variance as $(x-t)’\Sigma(x-t)$, where $x$ is the hedging portfolio, and $t$ is a portfolio consisting of a unit holding in the target alcoholic beverage company. This is classical Markowitz Mean-Variance Optimisation.

There is a huge amount of literature out there about adjusting covariance matrices for better performance. For example, people use factor models (as @mark leeds suggests), or apply shrinkage estimates to the covariance matrix (see, e.g., papers by Ledoit).

If we settle on the variance as the performance measure, it would allow us to use MIQP to build a hedge portfolio that selects the best portfolio of 5 stocks to minimise the variance of the hedge portfolio wrt the alcoholic beverage firm. Set up the quadratic optimisation with a constraint that no position is larger than 40%. You may not be able to find a MIQP Python library out there, and it may be necessary to look at heuristics to pick the subset. For instance, you could build up a hedging portfolio by finding the best single hedge in the EuroStoxx, then you find the best hedge for the remaining relative returns. Repeat this process until the hedge portfolio contains 5 stocks. This subset of 5 stocks is not likely to be statistically different from the true, optimal subset.

  • $\begingroup$ Thanks Tim for the great breakdown, very helpful $\endgroup$ – aallove Feb 18 '20 at 8:51

Hi: Assuming you can buy a risk model for the EuroStoxx 50 or estimate one, then this implies that you have all the exposures (risk and industry) for the stocks in that index.

Then, once you have that, you can construct an optimization that says

minimize risk and industry factor differences between portfolio and alcoholic beverage company subject to

A) no weight being greater than 0.4 and

B) number of positions less than or equal to 5.

The number of positions less than or equal to 5 is a difficult constraint but I bet there's some quadratic optimizer out there that can handle it.

  • $\begingroup$ Thanks Mark for your thoughts, this is an approach I had thought about but struggled to quantify all the exposures $\endgroup$ – aallove Feb 18 '20 at 8:52
  • $\begingroup$ @aalove: yes, the exposures are a problem and leads to the issue of building a risk-factor model which is a project in itself !!! Using some heuristicy approach such as those described by Tim Wilding might be your best bet. Eric Zivot decribes the construction of basic factor models in his SplusFinmetrics book but it's probably pretty messy once you get into all the gory details. Note that, since you're answering a question from someone else, you could say: "Given that we have a factor model along with the various company exposures to the factors", .... $\endgroup$ – mark leeds Feb 18 '20 at 13:06

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