# Why are factor models so popular for risk analysis of portfolios?

As titled, my question consists on asking for why in the most of academic papers one almost always finds that when you try to model asset returns, one needs to adjust for risk factors before analyzing asset portfolio returns. Someone can explain why is it important and how to do that?

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I was really confused by your usage of the word "risk-adjusted". In fact you talk about factor (risk) models. A risk-adjusted return is something different: en.wikipedia.org/wiki/Risk-adjusted_return_on_capital Please change the title. –  Richard May 8 at 6:27
Agree with @Richard. The risk adjusted return is Sharpe Ratio for comparing portfolios. Factor model is to itemize the components of portfolio return for various purposes. –  user12348 May 8 at 19:12
Any suggestion on what would be the proper title for the question? I'm confused about the topic (this is the reason for which I asked for that here) and I may be in wrong again. Can you suggest some more correct title, please? –  Quantopic May 8 at 23:20
You could use the title "Why are factor models so popular for risk analysis of portfolios?" or something in the way. –  Richard May 9 at 7:12
Thanks for the hint @Richard :) –  Quantopic May 9 at 10:28

There are a few reasons to use factor models.

Most importantly, stocks tend to move together. Stated alternately, the first principal component of the securities in a domestic market tends to explain a large share of the variance. If you're concerned with multiple securities (as in portfolio optimization), then you have to account for this or you will estimate too large a diversification effect. More sophisticated factor models also try to explain things like why do small cap stocks tend to outperform large cap stocks. Why do stocks with low P/Bs tend to outperform stocks with high P/Bs. And so on.

Dimensionality is also a major reason. For $N$ assets, a covariance matrix has $N$ variance terms and $N(N-1)/2$ correlation terms. A factor model with $F$ factors has $N$ idiosyncratic variance terms (assuming you are making that assumption for the errors), $F$ factor variances, and $NF$ betas. So long as $2F<N(N-1)/(N+1)$, you're estimating fewer parameters with a factor model.

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Portfolio returns are analyzed to account for risk factors only to determine what the risk factor contributed to the returns, was it the underlying assets or the skill of the portfolio manager. Fama French model explains the returns in terms of principal component such SMB and HML besides the market related returns from CAPM. These links have more detais likes of which you may have already read Fama and French Three Factor Model, and this Factor Analysis example. What ever is left over is either alpha (skill) or stochastic ( luck) return.

Another important reason for using risk-adjusted returns is to disentangle "skill" from "risk-taking". Think of a equation for a fund's performance like: $r_{i,t}-r_f=\alpha_i+\epsilon_{i,t}$ where $\alpha_i$ gives you the average excess return of fund $i$. Alpha is often interpreted as measure of a managers' skill in timing the market and selecting securities. If one fund simply takes illiquid assets than on average he will gain a greater alpha. But this is simply because he receives the additional risk premia for holding illiquid securities and not because of the fund managers skill. Therefore one needs additional risk factors to "control" for the additional risk premia. And this is what "risk-adjusting" means.