Background<\/h3>\n
A mathematical concept that is important when thinking about variance matrices (that is, risk models<\/a>) is eigenvalue<\/em>.\u00a0 If the variables all have zero correlation with each other, then the eigenvalues are just the variances of each variable. Otherwise, the eigenvalues are the variances of a particular set of linear combinations of the variables.\u00a0 One of the rules to get the linear combinations is that they all have zero correlation with each other.\u00a0 It is traditional to sort the eigenvalues.<\/p>\n Factor models divide risk into systematic and idiosyncratic parts.\u00a0 The actual factor model can be thought of as the sum of a systematic matrix and an idiosyncratic matrix.\u00a0 The issue of alignment has to do with the eigenvalues of the systematic matrix.\u00a0 Specifically that many of those eigenvalues are zero.<\/p>\n Figure 1 shows eigenvalues related to a factor model in a toy universe of ten assets.<\/p>\n Figure 1: Eigenvalues for the systematic risk (blue) and the total risk (gold). It is those zero eigenvalues that cause all the fuss.<\/p>\n Models are a simplification of reality<\/a>.\u00a0 Sometimes the simplification is a blessing.\u00a0 Sometimes the simplification is a bother.<\/p>\n The factor model simplification supposes that there are portfolios that have no systematic risk, only the idiosyncratic risk from the assets comprising the portfolios.\u00a0 That is highly unlikely to be true.\u00a0 It will be more or less untrue depending on the factors in the model.<\/p>\n An optimizer is going to like too much the portfolios with no systematic risk — you get return, you take little risk.<\/p>\n Consider the case where the expected returns are built with factors.\u00a0 If there are factors in the alpha model that are not in the risk model, the optimizer is going to love those factors.\u00a0 It will over-emphasize the zero risk factors and under-emphasize the other factors.\u00a0 The realized utility will be less than without the distortion (assuming the expected returns really are predictive).<\/p>\n Solution 1<\/strong>: add the alpha factors that are not in the risk model to the risk model.<\/p>\n If the factors match between alpha and risk, then the zero systematic portfolios are out of the picture.<\/p>\n Problem solved? No.\u00a0 The zero eigenvalues are only out of the picture in a simplistic world.<\/p>\n Optimization is done under constraints.\u00a0 When there are binding constraints (and there will be), then the optimal portfolio will stray into the zero systematic risk region.<\/p>\n Suppose you want to go due north.\u00a0 If after you’ve gone north for a while, you come to a cliff, then you have to deviate and go either west or east to some extent.\u00a0 Similarly the optimizer will be forced to deviate from the direction of the alpha because of constraints.<\/p>\n Solution 2<\/strong>: Axioma has a method of dealing with this.\u00a0 The catch is that there is a magic number that you have to come up with.<\/p>\n As already stated, there was decidedly no consensus on the importance of this issue.<\/p>\n Some stated that while the problem exists mathematically, the practical existence is minimal — in the scheme of things this is a triviality.<\/p>\n Others claim that it can be worth several basis points per month.<\/p>\n It is possible that both opinions above are true — that it is a big issue for some funds and a trivial issue for others.\u00a0 I don’t know.\u00a0 However, it is not a problem when there is no predictive power — you can not lose what you do not have.<\/p>\n The discussion pretty much takes for granted that expected returns are created with a factor model.\u00a0 If expected returns are created some other way, then there will be no factors to align.<\/p>\n A way to avoid the whole issue is to use a variance matrix that doesn’t have portfolios with zero systematic risk.\u00a0 One such choice is Ledoit-Wolf shrinkage<\/a> towards equal correlation.<\/p>\n There is a reason that risk vendors might not be keen on this solution.\u00a0 Fund managers can get the computer code for free<\/a> and use data that they have anyway.\u00a0 Total revenue to risk vendors: zero.<\/p>\n Risk vendors are likely to come back with the argument that their risk models are more informative than a Ledoit-Wolf estimate.\u00a0 The inputs to their models are more than just prices so the output has more predictive power.<\/p>\n That’s a reasonable hypothesis.\u00a0 Being of a scientific bent, I’d like to see evidence.<\/p>\n If it is true, then I see two alternatives for optimization:<\/p>\n The Ledoit-Wolf estimate is an optimal (in a reasonable sense) combination of the sample variance matrix and the equal correlation matrix (derived from the sample variance).\u00a0 An alternative is a combination of the vendor’s factor model and the sample variance.<\/p>\n In years past there was a good reason to have zero eigenvalues because that meant that the variance matrix could be represented compactly.\u00a0 But now the size of a variance matrix is unlikely to be problematic.\u00a0 The eigenvalues of all the factors excluded from the model could be set to the mean of their eigenvalues if they were in the model.<\/p>\n What’s unclear?<\/p>\n Are there additional alternatives to solving the problem?<\/p>\n Steve Greiner<\/a> was the referee of the risk vendor panel.\u00a0 He started off the event — focused on risk — with a story.\u00a0 He wasn’t sure how well it went over, but I liked it at least.<\/p>\n Two little fish are swimming along together.\u00a0 They pass by a big fish who asks them, “How do you like the water?”\u00a0 The little fish continue to swim along.\u00a0 After a while one of them turns to the other and asks, “What’s water?”<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":" An explanation of alpha factor alignment in portfolio optimization, and a look at the spectrum of views on it. Venue FactSet recently hosted an event that included a panel of representatives from several risk model vendors.\u00a0 The first question thrown at the panel was about alpha alignment.\u00a0 The opinions varied widely.\u00a0 There was positive correlation … Continue reading ![\"\"](\"https:\/\/www.portfolioprobe.com\/wp-content\/uploads\/2012\/07\/facaleig.png\" "\\"facaleig\\"")
<\/a>From Figure 1 we can infer that the factor model has 4 factors because the first 4 systematic eigenvalues are positive, and the rest are zero.<\/p>\nOptimization<\/h3>\n
Constraints<\/h3>\n
Opinions<\/h2>\n
Not important<\/h3>\n
Important<\/h3>\n
My take<\/h3>\n
Rebuttal<\/h3>\n
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Questions<\/h2>\n
Epilogue<\/h2>\n