Analyzing the performance of an investment strategy is an integral part of the investment process. Systematic decomposition of investment performance began with a seminal study by Brinson, Hood, and Beebower (BHB) where they breakdown a portfolio’s returns into several parts that disentangled timing, stock selection, and benchmark performance. Many variations soon followed that allowed for similar decompositions of performance that helped disentangle country selection, industry selection, and miscellaneous other groupings. The goal of this type of decomposition was to help investor gauge manager skill and for managers to analyze where their competencies are.

**BHB Method**

The original BHB analysis broke down performance in terms of timing and security selection for an active portfolio measured against a benchmark. Under BHB, the benchmark group return times the active weight is considered the **asset allocation **component, the portfolio weights times the active return is considered the **stock selection** component, and the remaining component of the active return is called the **interaction** component.

**Factor Decomposition**

The fundamental process that drove the BHB analysis, and indeed performance attribution in general, is two fold: 1) separate out parts of a portfolio return that does not require skill (benchmark) and 2) analyze the performance along various dimensions (allocation vs selection). Both processes have evolved dramatically since its original inception.

With the advent of quantitative investment and factor models, performance attributions along factor dimensions also became a common occurrence. This added a great deal of complexity to the problem. We wish the measure the contributions from each factor, but simply taking the dot product of the returns and the weights won’t suffice because the factor portfolio return will include the effects from other factors as well. Instead, we make use of regression analysis to disentangle all the factor. Suppose we had a vector ** r** of asset contributions to portfolio return and a matrix

**X**of factor exposures, then we could calculate the orthogonalized factor returns according to this linear model:

where we solve for β via OLS:

Once we have the factor returns, it’s a simple step to take the dot product of the factor exposures to the factor returns to obtain the factor contributions to portfolio return.

Finally, residuals of this regression are considered to be return contributions from market noise, which on average should equal zero if the assumptions behind OLS are met or if the weights correctly accounts for the heteroskedasticity.

**Factor Decomposition and Risk
**

Because a factor risk model is also an indispensable part of the quantitative investing process, we must also take into account risk factor exposures as well as the alpha factor exposures. This helps the manager clarify and control their risk along dimensions based on investing style, sector membership, and country membership. It also helps investors see whether the returns are truly coming from alpha factors or from accidental bets of risk factors that are presumed to carry no return over the long run.

Thus we must extend or factor decomposition to include the risk factors as well:

where Ω signifies the risk model

This approach has one drawback, namely that if the alpha factors are correlated with the risk factors, then we run the risk of multicollinearity issues. In a paper via Barra, Menchero and Poduri make the suggestion to regress the risk factors on the alpha or custom factors and use the residuals in the regression. However, this method is not useful for disentangling the parts of a manager’s strategy that simply coincides with risk factors that do not carry sustainable returns.

If we wanted to be really strict and look at strategy alpha above and beyond not just the market but also in excess of the risk factors, then we may want to reverse the roles of the two types of factors and regress the alpha factors on the risk factors:

And the residuals from this regression will be used in our new factor decomposition model:

where

The factor returns from the alpha factor part of the regression now contains only the return contributions that cannot be explained by the risk factors.

**Optimization Constraints and Factor Decomposition**

One could argue that because of the constraints in the optimization process, bets along the risk factors is inevitable, because we simply cannot achieve 100% of our ideal view portfolio. The short response here is that from the investor’s point of view, it doesn’t matter what the constraints are, what matters is the final value added. But a more nuanced response should recognize that many constraints are client mandated and are often customizable. Thus it is important to be cognizant of the magnitude of their impact.

A proper treatment of constraint decomposition and it’s relationship with performance attribution requires a dedicated post. But I’ll make two suggestions. One way to get a rough sense of how much the constraints are changing the performance attribution is to either use the returns to a view portfolio (or to a theoretical unconstrained portfolio) and look at the differences in factor contributions between that portfolio’s theoretical returns and the actual portfolio returns (need to pay attention to t-costs here). Another way to do this maybe to include the “delta” portfolios associated with each constraint as additional factors in the decomposition regression.

**Conclusions**

We have progressed much since the days when all we could distinguish was market performance and portfolio performance. With BHB and its variants, we were able to breakdown performance due to timing, stock selection, industry selection, country selection, and passive benchmark positioning. As quantitative investing and factor models came into the fore, we have moved onto using regression analysis to decompose returns amongst various factors. And finally, using a factor risk model and some simple modifications, we can also strip out the return contributions from bets along risk factor dimensions that don’t carry returns.

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Hello, Chang She, Thx for the post, I like it. I have two question for the term “delta” portfolio for each constrain.

Question 1. Can you briefly explain what does it mean for a “delta” portfolio for each constrain?

Question 2. What does it mean for using the constructed delta portfolio as predictor variable in the

decomposition regression? For all the decomposition regression in your post, the dependent variable is r, the cross-sectional record of excessive return. Each row of the decomposition regression (both r and X), refers to a specific security. Thus, if an additional factor, say Z, related to the delta portfolio is included, can you briefly explain to me what does row_i(Z) refer to ?

Thanks again,

Tan

delta portfolio: the change in portfolio weights due to the constraint