## Abstract

Performance analysis of the returns provided by an actively managed portfolio is increasingly viewed as a tool not only to assess manager skill and how it adds value, but also as a basis for a discussion about the alpha factors present in an active investment strategy. We review a popular form of the most traditional of return attribution approaches and demonstrate how it fails to link the portfolio’s *realized* excess returns to the manager’s *stated strategy* and neglects the effect of extraneous risk exposures. As an alternative, we provide a framework for producing performance attribution that captures a risk-adjusted factor approach, which seeks to separate and identify the particular factors that drive a manager’s investment process. We believe that a better assessment of the management process can be accomplished by measuring and separating these factors and accounting for their contribution to the portfolio’s excess returns.

Return attribution provides a tool for measuring portfolio performance relative to established objectives, commonly represented by an appropriate benchmark index. A primary purpose of the attribution is to determine the source of the returns in excess of the designated benchmark for the portfolio. However, performance analysis of the returns provided by an actively managed portfolio is increasingly viewed as a tool not only to assess manager skill and how it adds value, but as a basis for a discussion about the distinction between portfolio return and the systematic risk factors embedded in an active investment strategy.

There is significant evidence that asset returns are driven by more than a single risk factor (i.e., the Fama-French systematic risk factors). It is well established that the returns associated with such multiple sources of systematic risk are inexpensive and relatively easy to obtain by investors. The challenge posed by the apparent confusion surrounding which risk model is appropriate is complicated by fact that the sources of alpha are as varied as the strategies marketed and held to generate it. This situation not only makes the differentiation of alpha from beta a significant challenge for the investor, but makes it considerably difficult to assess specific portfolio enhancements contributed by specific active strategies.

As a result, many different models and attribution techniques have been developed with varying degrees of accuracy and levels of interpretation. The variety of attribution models available may in some cases lead to widely divergent conclusions about the effectiveness of the investment strategy being assessed. A popular form of the most traditional of return attribution approaches fails to link the portfolio’s excess returns to the manager’s *stated strategy*, and neglects the effect of extraneous risk exposures.

If the attribution model is consistent with the investment decision making process of the manager and these decisions are quantifiable, then a meaningful conversation about the sources of risk and the generation of alpha can occur between the client and the manager. This article provides a framework for producing performance attribution that captures the effects of specific strategy factors used by managers to construct an actively managed portfolio as distinct from performance that is due to extraneous market risk exposures.

The traditional approach to performance attribution, presented as a comparison in this article, is developed by Brinson, Hood, and Beebower [1986] and Brinson, Singer, and Beebower [1991]. It is briefly outlined in Appendix 1. The Brinson approach decomposes active manager returns into allocation and security selection components against the same components of a passive benchmark index. By setting the benchmark index as a passive strategy, any deviations in performance of the active manager’s portfolio from the benchmark index are accounted for by an industry sector allocation effect, a security selection effect, or an interaction of these two measures.

Accordingly, the purpose of traditional attribution is to identify and relate the differences between the performance of the portfolio and its benchmark back to the manager’s “bets” on both industry sectors and individual security choices. Superior allocation skills are exhibited when the manager overweights industry sectors that outperform the benchmark’s return and underweights those with performance that lag the benchmark. Superior security selection skills are exhibited when the manager chooses a subset of securities that outperform the sector. The results of the attribution model may show superior decisions in one, both, or none of the model components.

While a breakdown of a manager’s performance relative to the composition of sector allocation and weighting of individual securities in a benchmark provides useful information, traditional performance attribution does not tell us how the manager performed relative to his or her *stated strategy on a risk-adjusted* basis. Therefore, the method described in this article is a variation of the traditional approach but adjusts for differences in the risk characteristics of individual stocks.

To provide a more rigorous understanding of a portfolio’s performance, we demonstrate an alternative methodology based upon factor attribution (FA) that can be customized to fit a stated strategy and varying risk exposures. The FA model represents an adjustment to the returns used in the calculation of traditional value-added performance attribution due to “allocation” and “stock selection,” substituting the targeted “factor returns” of a manager’s strategy as well as other risk factors represented in the market.

The targeted factor returns that we refer to are the key determinants of performance as defined by a manager’s strategy. For example, a manager characterized as “Value” may isolate particular measures of value such as low P/E or low P/B as the main determinants of the alpha generation process in the investment strategy. Using factor attribution, these measures of value can be isolated within the manager’s portfolio and evaluated in relation to the benchmark index to provide an assessment of how the manager’s strategy fared. In addition to the factors defined by the manager’s strategy, adjustments for exposures to exogenous influences of the general market can also be isolated and taken into account.

To illustrate the differences between the traditional and factor attribution approaches, we have applied the performance attribution methodologies described above to an actively managed equity portfolio for a four-year period, 1998 to 2001. The management strategy is based upon a process that utilizes three stock selection factors used by the manager to capture security mispricing that can be attributed to specific “behavioral” anomalies. The three stock selection factors are denoted X, Y, and Z. The manager’s strategy has a stated preference for Factor X, relative to Factors Y and Z. Positive alpha expectations are associated with Factor X and negative alpha expectations are correlated with Factor Z. Therefore, stocks with high loadings on X are overweighted in the portfolio at the expense of weightings in stocks with high loadings on Z. Stocks without a high loading on Factor X or Z are included in the Y grouping by default; the management strategy is indifferent to these stocks.

In addition, there are several endogenous risk factors used to adjust for general market conditions: price momentum (M), market capitalization or size (S), and market risk (beta, B). The basic FA analysis proceeds in three steps: 1) estimate returns to the risk factors via cross-sectional regression; 2) determine the active factor and risk exposures in the active portfolio; and 3) determine the returns to the all active exposures. The first step of the process requires the identification of the attribution universe and is defined as the benchmark holdings, plus any non-benchmark securities contained in the manager’s portfolio. Subsequently, a cross-sectional ordinary least squares regression is applied to the attribution universe to calculate a coefficient for each of the representative factors as the set of independent variables and described by the manager’s decision-making process. Coefficients are produced by regressing the monthly observations of total return for each stock in the universe upon the set of factor variables. The regression model used in the decomposition of active returns is presented in Appendix 2.

The second step is to produce weighted returns for each security in the managed portfolio. The weighted returns are determined relative to their active exposure in the portfolio to each of the manager’s specified factors, as well as to any exogenous risk control factors. The active returns to these factors are calculated as given in Appendix 2. Finally, the sum of the returns due to all active exposures plus the return due to specific risk are calculated and related to the actual active return to the managed portfolio.

## ATTRIBUTION OF THE ACTUAL PERFORMANCE RECORD: THE TRADITIONAL APPROACH VS. FACTOR ATTRIBUTION

For illustrative purposes, we will apply each attribution methodology to an actively managed portfolio. The manager in the following examples uses a quantitative approach described previously. It categorizes stocks within the portfolio into three distinct groups (X, Y, Z), each explained by a unique set of factors. Each stock is categorized into only one of the factor groups and is allowed no dual membership between factor groups.

### An Example of the Traditional Attribution of the Sources of Outperformance

The first example (Exhibit 1) compares the traditional approach to that of the factor attribution methodology for the month of June 2001, in which the portfolio outperformed its benchmark by 0.13% (13 basis points).

Traditional attribution reveals that the manager’s allocation to Factors X, Y, and Z have contributed a total of 54 basis points of positive performance for the month, when compared to the benchmark index. However, individual security selections within each of the factor groups, particularly Factor Z, have combined to offset much of the gains (–0.39%) generated by the allocation decision. The interaction of the allocation/selection effect reduces, the portfolio’s performance by -0.02% relative to the benchmark.

Interpretation of the results provided by the traditional attribution approach would indicate that the manager’s process correctly identified and allocated across the three factors, which combined to outperform the benchmark. However, the manager’s poor selection of securities within each factor group, particularly Factor Z, offset a large portion of the gains produced by the allocation decision.

Net results would indicate that the manager outperformed the benchmark, but they provide little information about how the results were obtained. This may lead the manager to believe that the strategy was effective in outperforming on a risk-adjusted basis.

### Comparison of the Traditional with a Factor Approach to the Attribution of Outperformance

Subjecting the identical portfolio to a factor attribution approach for the same one-month period provides differing results and, subsequently, a distinctly different interpretation. Once again, the 13 basis points of superior performance are delineated into the three factors defined by the management strategy. However, in addition to the three factors characterizing the manager’s strategy, three separate and exogenous factors are provided: price momentum, size, and beta. These three factors are introduced into the attribution process to control for general market risk. As shown in Exhibit 2, applying Equation (6) from Appendix 2 to each of the six identified factors results in the attribution assignments to the six factors and produces a seventh factor: residual or firm-specific risk.

Interpretation of the factor analysis indicates that the manager’s selection of the three factors combined to produce 10 bps of underperformance, indicating that the manager’s stated strategy, as defined by the three factors, detracted from performance. Why is there such a disparity in outcomes between the two attribution approaches when explaining results in context of the factors X, Y, and Z?

Differences in the results of the two methods can be measured by comparing the exhibits in the “Risk Adjusted Contribution” column of Exhibit 2 with the numbers in the “Total” column of Exhibit 1. Reconciling the differences between these results provides an understanding of the differences between the two attribution methods’ ability to explain which factors are generating returns.

The traditional approach presents an ambiguous explanation of Factor X’s contribution to the portfolio’s total excess performance. It indicates that the manager’s allocation and selection decisions completely offset one another, while the interaction of these decisions produced a 0.09% performance shortfall. The factor approach also attributes 9 basis points of the portfolio’s negative return to Factor X and directly estimates a negative return to Factor X. The return to Factor X was -0.46% for the month. The poor performance results from combining the portfolio’s 6.5X overweighting of the factor with its relatively weak return.

Factor Y in the traditional approach provides a return of 0.14% and is responsible for the bulk of the positive effects. However, the factor approach would indicate that Factor Y’s contribution to the portfolio’s excess return is, in fact, negative! Although the factor’s return for the month was positive (+1.48%), the portfolio only held approximately 62% of factor Y’s exposure to the index. As a result, the portfolio’s underallocation to the positive performing factor resulted in a net negative performance.

Both the traditional and factor approaches assign positive performance attribution to Factor Z. Traditional attribution accounts for the factor’s superior performance by attributing the effect to allocation within the factor universe. The factor approach reveals that Factor Z, when regressed upon the portfolio returns, experienced negative (–0.45%) performance. However, the outperformance of Factor Z resulted from the portfolio holding 14.7% less of the factor relative to the benchmark.

### Controlling for Market and Extra-Market Risk Exposures

The factor attribution approach also provides additional information about the three factors defined by the manager’s strategy through recognition of several exogenous risk factors within the general market. In this example, the portfolio’s slight relative underweight (–0.24%) to the minimally positive effects of price momentum (+0.91%) produced 1 basis point of positive performance. The nonexistent effect of Size (+0.00%) as a factor in explaining excess returns contributed nothing and the portfolio’s slight underweight to Beta, a general measure of market risk upon which the regression placed a -1.37% return, added two basis points to performance. The traditional attribution approach does not recognize the separate influences of these market variables.

The final factor presented by the factor model is firm-specific risk, which is a residual of the regression equation and represents that portion of the excess returns that cannot be directly explained by factors as defined by either the strategy or the general market. It is the error term in the general form regression Equation (4) in Appendix 2. Again, the traditional attribution method does not separate the randomness of firm-specific influences from the factors defining the manager’s strategy. These effects are subsumed in the selection effect.

Using the factor attribution process to review the manager’s performance would lead us to conclude that the particular strategy that the manager has chosen to exploit, and defined by Factors X, Y and Z, contributed negative excess returns (–10 bps) to the portfolio. However, general market factors of price momentum and beta contributed +3 bps to excess returns. The portfolio’s superior performance for the month was primarily the result of firm-specific or residual effects, indicating that issues that were unexplained by the strategy provided the excess returns.

The two different approaches lead to separate and opposite conclusions regarding the management strategy’s ability to produce excess return. Traditional attribution correctly identifies the generation of excess return. However, it does not adequately link the excess returns to its sources. As a result, the client is led to believe that the manager has indeed added value. On the other hand, the FA approach correctly attributes the excess return to exogenous market factors, indicating that the manager did not produce demonstrable alpha!

### An Example of the Traditional Attribution of the Sources of Underperformance

A second example (Exhibit 3) compares the traditional approach to that of the factor attribution methodology for the month of July 2001, in which the portfolio underperformed its benchmark by 0.04%.

Traditional attribution reveals that the manager’s *allocation* to Factors X, Y, and Z have contributed a total of 62 basis points of positive performance for the month, when compared to the benchmark index. However, individual security selection within each of the factor groups, particularly Factor Z, completely nullified the gains provided by the successful allocation decision. The interaction of the allocation and selection effects combined to produce the four basis points of the portfolio’s underperformance relative to the benchmark.

Interpretation of the results provided by the traditional attribution approach would indicate that the manager’s process correctly identified and allocated across the three factors, which combined to outperform the benchmark. However, the manager’s poor selection of securities, particularly those securities that load on Factor Z, cancelled gains produced by the allocation decision.

Application of the traditional attribution approach reveals where the manager’s difference in portfolio construction differed from that of the benchmark index. However, the approach does not clarify how the portfolio performed in relation to the manager’s stated strategy in excess of risk exposures. Net results would indicate that the manager was “treading water” with little information on whether the stated strategy was effective.

### Comparison of the Traditional with a Factor Approach to the Attribution of Underperformance

Subjecting the identical portfolio to a factor attribution approach for the same one-month period provides differing results and, subsequently, a distinctly different interpretation. Once again, an accounting of the four-basis-point performance shortfall is delineated by the three factors defined by the management strategy. In addition to those three factors, three separate and exogenous factors are used (price momentum, size, and beta) to control for general market risk.

Applying Equation (6) from Appendix 2 to each of the identified factors results in the attribution assignments to the six factors as shown in Exhibit 4 and produces a seventh explanatory factor: firm specific risk.

Interpretation of the factor analysis indicates that the manager’s selection of the three factors combined to produce 50 basis points of excess return. The FA model confirms that the portfolio performed well in relation to the manager’s stated strategy as defined by the three factors. Again, why is there a disparity in outcomes between the two attribution approaches when explaining results by Factors X, Y, and Z? The difference in the results of the two methods can be reconciled by comparing exhibits in the “Risk Adjusted Contribution” column of Exhibit 4 with the numbers in the “Total” column of Exhibit 3. The comparison provides a discriminating measure of the difference between the two attribution methods.

The traditional approach indicates that Factor X provided a weak contribution of three basis points to the portfolio’s total excess performance. Conversely, the factor approach assigns 84 basis points of excess return to Factor X, making it the most dominant factor for explaining the portfolio’s performance relative to the benchmark. Factor X’s strong performance results from combining the portfolio’s 7.5X overweighting of the factor with its relatively strong 3.9% return to that factor.

Factor Y in the traditional approach provides a return of 20 basis points and is responsible for the bulk of the positive effects. However, the factor approach would indicate that Factor Y’s contribution to the portfolio’s excess return is relatively minimal. Although the portfolio holds approximately one-half of Factor Y’s exposure to the index, the negative return of -1.41% to the factor did not impact the portfolio when compared to the impact of the remaining factors. The risk-adjusted contribution of Factor Y is only 0.09% as compared to 0.84% for Factor X and -0.43% for Factor Z.

The traditional and factor approaches both assign negative performance attribution to Factor Z, with the difference due to an order of magnitude. Traditional attribution accounts for the factor’s underperformance by ascribing the shortfall to security selection within the factor universe. The factor approach reveals that Factor Z, when regressed upon the portfolio’s excess returns, experienced a 2.79% positive return. However, the negative contribution of Factor Z to the portfolio’s performance resulted from the portfolio holding 15.3% less of the factor relative to the benchmark.

### Controlling for Market and Extra-Market Risk Exposures

In this example, the portfolio’s relative overweight (+2.12%) to the positive effects of Price Momentum (+3.22%) produced seven basis points of positive performance. The minimal effect of Size (+0.03%) in explaining excess returns contributed nothing, despite the portfolio’s 3.01% overexposure to the factor. The portfolio’s slight overweight to Beta, a general measure of market risk, subtracted one basis point from performance. The traditional attribution approach does not recognize the separate influences of these market variables and accounts for their effects in the general form of the process. In fact, excess returns that are actually due to exogenous risk exposures are mistakenly attributed to the manager’s strategy.

The final factor presented by the factor model is firm-specific. This is the residual component of the regression equation. In Equation (6) of Appendix 2, it represents that portion of the excess returns that cannot be explained by factors in either the strategy or the general market. Again, the traditional attribution method cannot separate the randomness of firm-specific influences from the factors defining the manager’s strategy and usually accounts for these occurrences in the stock-selection effect.

A summary of the results of the factor attribution process for the month would lead us to conclude that the particular behavioral anomaly that the manager has chosen to exploit, and defined by Factors X, Y, and Z, is still contributing positive excess returns (+0.50%) to the portfolio. In addition, the general market factor of Price Momentum contributed seven basis points of influence. Finally, the 60 basis points of underperformance from firm-specific or residual effects indicate that the excess return produced by the manager’s strategy was offset by decisions that were unexplained by the strategy.

As indicated earlier, the residuals represent an opportunity to conduct further research into the explanatory variables that drive the management strategy. That type of analysis is an area of development for the FA attribution model.

## LINKING THE RETURNS OVER TIME

For the period January 1998 to May 2001, the managed portfolio produced a cumulative rate of return of 4.80% in excess of the Russell 1000 Index. The cumulative monthly performance and the attribution for this period are presented in Exhibits 5 and 6. The cumulative measures of return and return attribution were calculated using the methodology presented by Carino [1999]. The difference in accounting for the attribution effects between the three factors is presented in Exhibit 7.

Note that the difference in attribution suggests that the traditional approach understates the returns to Factor X by 3.73% and overstates the returns to Factors Y and Z by 2.86% and 3.85%, respectively. This ambiguity can be a source of confusion for the client or consultant charged with analyzing a manager’s performance in an effort to distinguish skill from random outperformance.

Traditional attribution not only may mispecify each factor’s return, it also can provide poor information regarding the strategy’s management of a particular factor through its allocation within the portfolio, relative to the benchmark. Factor attribution provides a direct connection between the preferred factors, which are overweighted by the strategy, and the less favorable factors that characterize the benchmark but carry an under-representation in the portfolio. Factor attribution can also separate the effects of common risk factors (market capitalization, beta, etc.) from a manager’s performance, which has implications for the remuneration of the portfolio manager’s services by the client. Isolating and measuring the sources of excess return within a portfolio’s total performance can provide the client a valuable tool in assuring that they are rewarding the manager for the production of alpha and not purchasing an expensive source of beta.

By linking returns, as described above, the factor attribution method can provide an understanding of the contribution to excess returns over time by the representative factors in a manager’s strategy. This information provides a better understanding of the sources of a manager’s return in the context of the stated management strategy.

For example, the same manager in the above study experienced a difficult year of performance in 2000, underperforming the benchmark by 0.78%. The performance shortfall begs the question: Is the strategy utilized by the manager still a viable one? Traditional attribution provides incomplete answers. However, the factor approach can offer assistance in this area by allocating the returns to the strategy’s relevant factors. The factor attribution approach would summarize an accounting of excess returns as shown in Exhibit 8.

Interpretation of the results would indicate the X, Y, Z factor strategy utilized by the manager contributed only -0.07% to the overall underperformance. The risk factor of size was also inconsequential, contributing +0.02%. Exposure to price momentum and beta had the largest impacts on portfolio performance, at -0.36% and +0.37%, respectively. However, the primary culprit of the performance shortfall resided in the firm-specific aspect of the portfolio’s management. These results might prompt the manager to analyze the residuals in an attempt to determine if there is an important factor missing from the strategy or if the existing factors are correctly specified.

### The Role of Residual Risk

The residual return for each security is assumed to be uncorrelated with each of the strategy factors. The residual for one security’s returns is also assumed to be uncorrelated with that of any other security’s returns and is therefore totally idiosyncratic with respect to that security. Under those conditions, the only sources of correlation among security returns are those that occur due to the exposures of the portfolio to the stated strategy and market risk factors. Although this is a key assumption of this and any general linear factor model, it does not impose a restriction on the size or sign of the residual returns. Given that the FA model includes only the most important strategy factors as well as the extraneous risk factors, the residual returns are allowed to be positive, negative, or zero for any specific security. As a result, in this and many other factor attribution models the residual returns can be very large. Preliminary analysis of the residual returns presented in Exhibit 5 suggest a significant residual association between secondary stock selection rules and Factor X. Although further discussion of the residual effect is beyond the scope of this article, the sheer size of the residual returns shown in Exhibit 5 present an opportunity to evaluate the effectiveness of subsidiary stock selection rules employed in the strategy that are uncorrelated with the *major strategy* factor returns.

## CONCLUSION

Traditional performance attribution is useful for helping clients understand how the managed portfolio performed relative to its benchmark in the more conventional terms of stock and industry sector weighting relative to the designated benchmark. Traditional attribution supports managers’ explanations of why they held certain stock or industry concentrations. However, it does not typically provide an explanation of the factors that a manager may be exploiting to generate excess returns. We have demonstrated a risk-adjusted factor approach to portfolio attribution that seeks to separate and identify the particular factors that drive a manager’s investment process. The authors believe that a better assessment of the management process can be accomplished by measuring and separating these factors and accounting for their contribution to the portfolio’s returns.

## APPENDIX 1

### The Brinson Attribution Model and Decomposition of Active Returns

Assume the return on the active portfolio is:

where w_{i} = percent invested in each security in the active portfolio; R_{i} = return on each security included in the active portfolio;

Also assume the return on the benchmark portfolio is:

where w_{i} = percent invested in each security in the benchmark index; R_{i} = return on each security included in the benchmark index;

Note the return-generating process for the portfolio universe is the summation of the above two equations. Therefore, the difference in returns can be represented as:

Traditional performance attribution decomposes these differences to account for the effects of sector allocation and individual security selection. In addition, there is a term to capture the interaction of these two components.

## APPENDIX 2

### The Factor Attribution Model and Decomposition of Active Returns

The factor attribution begins with the same assumptions as the traditional approach outlined above.

As a result, the reader will recognize the first three steps of factor attribution process. Assume the return on the active and benchmark portfolios are as represented in Equations (1) and (2) and the difference in returns to each portfolio is represented in Equation (3).

Assume the factors explaining the active portfolio performance are represented as a function of the assignment to Factor X, Factor Y, and Factor Z classes, where the factors represent the performance generating characteristics of the managed portfolio strategy. Other risk factors are also included to control for market or extra-market influences.

where:

a = the regression coefficient measuring the return characteristics of each factor.

e

_{i}= the residual amount, unexplained by the regression formula.

Substituting Equation (4) into Equation (3) results:

Rewriting Equation (5) allows a decomposition of the total active return on the enhanced portfolio to portions due to active exposure to the various risk factors included in the model: the return due to active exposure to “Factor X,” … to “Factor Y,” … to “Factor Z,” … to various “Risk factors.” This decomposition is presented as follows.

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