**Why expected return factor models and risk factor models are different? Why expected return models and risk models use different factors?**

I was recently asked how to use factor models to create asset allocation input assumptions.

I always approached expected return and risk modeling as separate problems. Could you please point me to the literature that supports or contradicts this approach.

For example of the expected returns factor model, please see the Commonality In The Determinants Of Expected Stock Returns by R. Haugen, N. Baker (1996) (http://www.quantitativeinvestment.com/documents/common.pdf ) The up to date model performance is presented at Haugen Custom Financial Systems. (http://www.quantitativeinvestment.com/models.aspx )

For example of the risk (covariance matrix) factor model, please see MSCI Barra Equity Multi-Factor Models (http://www.msci.com/products/portfolio_management_analytics/equity_models/)

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If your main concern is a joint framework for estimation and attribution of risks, then you may want to check out this paper:

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1565134

Basically, the article argues it may be that the best model for jointly describing the securities you’re looking at is not the same as the model you use to attribute risk to different factors or industries. For instance, if I perform a PCA on all stocks in the U.S., the factors may not have an obvious risk intuition. So the question is how to link this with something that a PM might better understand, like the MSCI Barra factors. It’s rather obvious for stocks how to do this, but with non-linear payoffs for securities it is best to resort to the technique in that paper.

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There is an issue with trade optimization that if the expected returns use factors that are not in the risk model, then the optimizer will like those factors too much. This is known — in some circles — as the factor alignment problem.

This is actually less mysterious than it might seem at first (true in my case at least). Basically you want there to be no directions (no portfolios) that the variance matrix thinks are virtually riskless. This can happen if the variance matrix used in the optimization is a factor model, and it is of concern when some of those directions are used in the expected returns.

I suspect a solution is to use something like a Ledoit-Wolf shrinkage estimate for the variance in the optimization. That seems to be a good choice on other grounds as well.

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There are two main issues when using an expected returns factor model with a factor risk model, whose factors are different: portfolio construction and performance attribution.

As mentioned above, when constructing a portfolio using an optimizer where there is misalignment (i.e. differences between factors), the optimizer tends to build portfolios that load up on the orthogonal component of alpha- i.e. the optimizer loads up on the expected return factors that are not in the factor risk model because the risk model doesn’t see any systematic risk associated with this expected return factor. Of course, there usually is systematic risk associated with this expected return factor and thus risk is often under-predicted and the portfolio may be ‘off the efficient frontier’ (i.e. non-optimal risk/return tradeoff). Below is a research paper that talks in much detail regarding the issue of factor misalignment: the causes, symptoms, and cure.

http://www.axiomainc.com/newsandresearch/?p=878

Factor misalignment is also problematic from a factor-based performance attribution perspective. A factor based performance attribution is meant to help understand where risk and return came from from different factors in your risk model. When your expected returns model has different factors than your risk model, you cannot get a true idea of how your actual expected return factors have performed: the performance attribution report quantifies performance through the lens of the factors in your risk model. It would be much better to have alignment between your expected returns factors and your risk model factors, so you can get a true understanding of how your expected return factors have behaved.

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This issue is addressed in an MSCI Barra research paper that’s available online:http://www.msci.com/resources/research/articles/2008/RI_Do_Risk_Models_Eat_Alphas_April_08.pdf.

There is also a JPM paper authored by three people from Goldman Sachs that emphasizes the need for including alpha factors in a risk model:http://www.iijournals.com/doi/abs/10.3905/jpm.2007.674791.

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Please also find the full sets of the research papers and case study on the topic of misalignment from MSCI,

[1] Refining Portfolio Construction by Penalizing Residual Alpha – Empirical Examples

http://www.msci.com/resources/research_papers/refining_portfolio_construction_by_penalizing_residual_alpha_-_empirical_examples.html

[2] Refining Portfolio Construction When Alphas and Risk Factors Are Misaligned

http://www.msci.com/resources/research_papers/refining_portfolio_construction_when_alphas_and_risk_factors_are_misaligned.html

[3] Do Risk Factors Eat Alphas?

http://www.msci.com/resources/research_papers/do_risk_factors_eat_alphas.html

Hope it helps.

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Those references are interesting, thanks.

It seems like another approach would be to make an RMT-like adjustment to the non-dominant factors. Strictly speaking, this adjustment would be like setting the means and standard deviations equal (which may not be what the PM wants). Alternately, you could only set the means equal and allow the standard deviations to be different so that more volatile “alphas” would be penalized.

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From a simple Asset Pricing theory point of view, adding a risk factor to an asset pricing model, increases the R2 of the regression but it does not change the intercept (i.e. the alpha).

Imagine running a typical regression of the GM returns over a number of “alpha factors” (say, beta, size, hml, momentum and so on). Now add as a variable a basket of automakers. The R2 of the regression increases dramatically but the intercept does not change. In other words, risk factors are “non-priced” but are still useful in explaining the volatility. In finance theory this means that they are “diversifiable”, i.e. they do not carry a risk premium.

A good source for a clarification of these concept is the excellent book by John Cochrane “Asset Pricing”

http://www.amazon.com/Asset-Pricing-John-H-Cochrane/dp/0691121370/ref=sr_1_1?ie=UTF8&qid=1323275694&sr=8-1

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