Performance Attribution and the Fundamental Law

Transcription

1 Financial Analysts Journal Volume 6 umber , CFA Institute Performance Attribution and the Fundamental Law Roger Clarke, Harindra de Silva, CFA, and Steven Thorley, CFA The reported study operationalized the fundamental law of active management in the context of a factor-based performance attribution system. The system incorporates factor payoffs in the linear regression framework that many portfolio managers and external reviewers use to judge what is being rewarded in the market. The study indicates that parameters of the fundamental law can be used to approximate and interpret the results of the regression-based performance attribution system. The procedure is illustrated by the use of security holdings, returns, and factor exposure data for two portfolios benchmarked to the S&P 500 Index for April 995 to March Factor-based performance attribution systems are often used to evaluate the sources of benchmark-relative returns in actively managed portfolios. Some of the relative return can be attributed to marketwide factor exposures that differ from the benchmark, such as beta, company size, and company sector membership, and the realized payoffs to those factors. Relative performance not captured by these marketwide factor exposures is generally attributed to security selection. However, the information content of the security-ranking system is typically measured by an information coefficient or the performance of stocks grouped within quantile rankings, with little attempt to relate the security-ranking system to its actual basis point contribution to performance. In this article, we show how a regression-based attribution system can be extended to decompose the active return associated with stock selection into the information content of the rankings and constraintinduced noise. We use the mathematics of the fundamental law of active management in Grinold (989) with our extension (see Clarke, De Silva, and Thorley 2002). The extension shows that, in addition to the forecasting power of the ranking system, performance is influenced by how well the manager can structure the portfolio to capture the most attractive securities. The relationship between the security rankings and actual over- and underweight Roger Clarke is chairman of Analytic Investors, Inc., Los Angeles. Harindra de Silva, CFA, is president of Analytic Investors, Inc., Los Angeles. Steven Thorley, CFA, is the H. Taylor Peery Professor of Finance at the Marriott School, Brigham Young University, Provo, Utah. positions in the portfolio is measured by the transfer coefficient. Many managers find that their ranking system works well in a given period but is only loosely related to the actual performance of the managed portfolio. The extension of the fundamental law demonstrates that the lower the transfer coefficient, the more noise in the active return. The procedures we discuss here allow the contribution from the security rankings to be separated from the noise component and give the manager insight into the determinants of portfolio performance. To illustrate the attribution procedure and test the accuracy of the fundamental law, we use data on two portfolios benchmarked to the S&P 500 Index for 08 months. We examine the performance attribution results for both a long-only and a long short portfolio constructed on the basis of the same signal to illustrate the implementation efficiency advantages of long short strategies. We then use the 08 monthly time-series observations to test two key predictions of the fundamental law an ex ante or expectational relationship for the information ratio and an ex post relationship describing the sources of realized variance in active returns. The Fundamental Law of Active Management The fundamental law of active management provides an important strategic framework for active portfolio management. The law states that a portfolio s ratio of expected active return to active risk, the information ratio, is a function of security return forecasting skill, implementation efficiency, and breadth of application. Skill is measured by the ability to correctly rank stocks by forecasted return , CFA Institute Copyright 2005, CFA Institute. Reproduced and republished from Financial Analysts Journal with permission from CFA Institute. All rights reserved.

2 Implementation efficiency refers to the transfer of return forecasts into actual security positions after satisfying constraints. Breadth of application refers to the number of independent security investment decisions. We will argue that for uncorrelated residual security returns, a practical measure of breadth is the number of investable securities. A more detailed analysis of breadth is given in Buckle (2004). Goodwin (998) and Thomas (2000) provided helpful intuitive support for the fundamental law as a conceptual framework. Here, we suggest that the law can be used directly in performance attribution. Mathematically, the ex ante form of the fundamental law states that the managed portfolio s expected information ratio can be expressed as ( ) = Performance Attribution and the Fundamental Law ERA IC TC, () 2 R TC TC D σ A ρa, r + ρc, r σa, A where E(R A ) = expected active return, σ A = ex ante active risk of the managed portfolio IC = expected information coefficient TC = transfer coefficient = number of securities in the investor s forecast universe In accordance with prior discussions of active management, we use the word relative to mean the simple difference between the return of the managed portfolio and the benchmark return. The word active refers to the portion of the relative return that the manager is trying to forecast by using the security-ranking system (i.e., the relative return net of unforecasted marketwide factors). According to Equation, the expected active reward-to-risk ratio is a product of three parameters two simple statistical correlations and the sample size. First, the information coefficient, IC, is the expected cross-sectional correlation between forecasted risk-adjusted security returns and realized risk-adjusted security returns, a common quantitative measure of forecasting skill. Second, the transfer coefficient, TC, is the cross-sectional correlation between forecasted risk-adjusted security returns and risk-weighted security exposures. The exposure or active weight on a security is the difference between its weight in the managed portfolio and its weight in the benchmark. In an ideal world of no portfolio constraints or other implementation issues, the value of TC in an optimal portfolio would be.0. In practice, TC values rarely exceed 0.8 and can be as low as 0.2. Third, under certain independence assumptions, is the number of securities that are considered as potential holdings in the portfolio. The fundamental law indicates that the information ratio increases as investor forecasting skill increases, as portfolio construction better reflects the security rankings, and as the ranking system is applied to more choices. The fundamental law in Equation is an ex ante relationship; expected added value is based on the expected information coefficient, which measures the forecasting skill of the manager. The other parameter values in Equation, TC and, are nonrandom and known ex ante. Clarke et al. also derived an ex post application of the fundamental law that decomposes realized active return into two terms with a common multiplier: ( ) (2) where ρ a,r, ρ c,r, and D are new ex post fundamental law parameters. The ex post or realized active return of the portfolio, R A, is a function of the ex ante parameters used in the ex ante fundamental law in Equation and the three additional ex post parameters. First, the realized information coefficient, ρ a,r, is the cross-sectional correlation coefficient of forecasted risk-adjusted active security returns, a i /σ i, and realized risk-adjusted active security returns, r i /σ i, in a given period. Subscript i is a number between and, which is the investable universe of securities, and σ i is the risk estimate of the active return for security i. Although the expected value of the realized information coefficient, IC, must be positive to justify active portfolio management, the ex post correlation, ρ a,r, will vary from period to period and may be negative if the forecasting process performs poorly. Second, the noise coefficient associated with portfolio constraints, ρ c,r, is the cross-sectional correlation between a risk-weighted security-specific weight measure, c i σ i, and risk-adjusted realized active security returns, r i /σ i. The term c i measures the optimal active weight not taken when constructing portfolio positions. In theory, the expected value of the constraint noise coefficient is 0; that is, E(ρ c,r ) = 0. The third ex post parameter, D, is the realized cross-sectional dispersion in security returns beyond the forecast implicit in the ex ante active risk parameter, σ A. Realized dispersion is calculated as the cross-sectional standard deviation of riskadjusted active security returns, r i /σ i. Because the realized active security returns are standardized by estimated risk, the expected value of the security return dispersion is ; that is, E(D) =.0. September/October

3 Financial Analysts Journal ote that the ex post law in Equation 2 reduces to the ex ante law in Equation under the three expectation values given previously. The approximate equality notation,, is used in Equation 2 because of simplifying assumptions in the development of the ex post fundamental law relationship (see the Technical Appendix in Clarke et al.). The key managerial perspective from the ex ante fundamental law in Equation is that the parameters combine in a simple multiplicative relationship to determine expected active return over risk. The key managerial perspective from the ex post relationship in Equation 2 is that realized active return in a given investment period can be attributed to both variations in the ex post success of the return-forecasting process and variations in the noise term. In fact, Equation 2 provides a precise decomposition of realized active return into two terms one that includes ex post signal success, ρ a,r, and one that includes ex post noise associated with portfolio constraints, ρ c,r. Specifically, Equation 2 can be split into signal and noise contributions: Signal ρ ar, TC σ D A (2a) 2 oise ρcr, TC σad. (2b) Under the assumption of intertemporal stationarity, the two stochastic correlation coefficients, ρ a,r and ρ c,r, have the same variance over time. Consequently, the relative magnitudes of the signal and noise contributions to active return are based on the TC and TC 2 multipliers. For example, if portfolio constraints and other implementation issues lead to a transfer coefficient of 0.40, then the multiplier in the noise term is 0.92, predicting that the impact of constraint-induced noise on the realized active return is substantially greater than the signal. With a transfer coefficient value of 0.40, only 6 percent (that is, TC 2 ) of the variance in realized performance over time will be associated with the success of the signal; the other 84 percent is associated with constraint-induced noise. Managers frequently experience periods when the forecasting process or signal works but the active performance of the portfolio is poor and, conversely, periods when the signal performs poorly but the active performance is good. In the absence of the perspective from the ex post fundamental law relationship in Equation 2, portfolio managers tend to attribute discrepancies between signal quality and actual portfolio performance to measurement or accounting error, when, in fact, they are simply the result of constraints imposed on the portfolio construction process. Regression-Based Performance Attribution Statistical regression analysis of security returns on factors is commonplace in both financial economics and quantitative portfolio management. For example, the influential Fama and French (992) study is based on cross-sectional regressions of U.S. stock returns on three pervasive marketwide factors market beta, size (log market capitalization), and value (book-to-market ratio). Based on Carhart (997), academic equity research often includes a fourth factor price momentum (recent historical return). Chan, Karceski, and Lakonishok (998) verified that these four factors are among the most important in explaining the cross-sectional variation in U.S. stock returns. Practitioners typically view value and momentum as exploitable signal factors, whereas they view beta and size as risk factors. Regression on identifiable or statistical factors is also central to estimations of return covariance in many risk models. In addition, quantitative portfolio managers, in a search for factors that help predict returns (i.e., generate signals), often use regression analysis to determine historical payoffs. Quantitative managers may also use regression analysis ex post to calculate payoffs to various signal or risk factors in each period. Less frequently, managers tie factor payoffs and signal performance together in a formal attribution system. In this section, we describe a regression-based performance attribution system that decomposes realized benchmark-relative returns into marketwide factor contributions, the performance of the securityranking system, and constraint-induced noise. To develop the regression-based attribution system, we assume that each of the i = to security returns, R i, is split into several pieces, including the return to general market exposure, return to other marketwide factors, and return to an active component, r i, that is the focus of the manager s forecast: where β i K i M i k ki i k= R = R β + λ F + r, (3) =security i s exposure to the general market return, R M F ki =security i s exposure to the kth other unforecasted factor λ k = the payoff to the other factor The K other factors in Equation 3 depend on the investor s forecasting process. Specifically, any marketwide factor netted out of the security s return before forecasting should be included as a , CFA Institute

4 Performance Attribution and the Fundamental Law factor in Equation 3. For example, managers sometimes neutralize security returns by controlling for size and industry membership, as well as beta, before trying to forecast the remaining portion of security returns. In the original formulation of the fundamental law by Grinold and in the Clarke et al. extension, the authors defined the active security return as net of only the market factor, which was also assumed to be the benchmark. Equation 3 is more general and allows the inclusion of other marketwide factors that the investor is also not trying to forecast. The multifactor structure of the attribution system allows the contribution of both forecast and nonforecast influences to be properly identified. The relative return between a managed portfolio and its benchmark is defined as a simple difference, ΔR = R P R B, (4) where R P is the return on the managed portfolio and R B is the return on the benchmark portfolio. As always, portfolio returns are calculated as the weighted-average security returns that is, and P Pi i R = w R (5) RB = wbiri, (6) where w Pi and w Bi are sets of weights that each sum to based on the dollars invested in each security. The difference between security i s weight in the managed portfolio and its weight in the benchmark, Δw i, is w Pi w Bi. The Δw i weights are thus security over- and underweights relative to the benchmark and sum to 0. Some basic algebra carried out on these definitions gives the intuitive relationship Δ R= ΔwR i i. (7) Then, substituting the assumed multifactor structure of security returns in Equation 3 into Equation 7 produces K Δ R = R M Δ w iβi + λk Δ wf i ki + Δ wr ii. (8) k= The last term in Equation 8 is the forecasted component of security returns in the portfolio s active return that is, R A = Δwr. i i (9) For example, in the well-known one-factor structure in which the only unforecasted factor is the market return, Equation 8 reduces to ΔR = R M (β P β B ) + R A, (0) where β P and β B represent the market betas of, respectively, the actively managed portfolio and the benchmark portfolio. 2 Specifically, if the actively managed portfolio has a beta different from the benchmark s, only a portion of the total relative return is explained by the active return the investor is trying to capture. The simple one-factor case illustrates why the attribution system needs to identify marketwide factors before examining the portion of the total relative return that is the subject of the investor s forecasting process. To estimate the contributions of the marketwide factors separately from those of the active return, first, one regresses the cross-sectional realized security returns each period, R i, on the unforecasted marketwide factors, as specified in Equation 3. The regression can be simple ordinary least-squares (OLS), but preferably, it is generalized least-squares (GLS), which corrects for cross-sectional heteroscedasticity by weighting observations by the inverse of σ i. 3 The estimated regression residual term in Equation 3, r i, represents the active component of security return net of unforecasted marketwide factors. These residuals are then regressed on the forecasts to estimate the payoff to the signal that period, r i = ναi + ε i, () where ν is the signal payoff. This regression is also preferably carried out in a GLS framework with observations weighted by the inverse of σ i. Using the estimated coefficients and residuals from the regressions in Equations 3 and and distributing the i-subscripted summation across terms allows one to write the relative performance between the managed portfolio and the benchmark as K Δ R= R M ( βp βb)+ λ k ΔwF i ki k= (2) + ν Δ wiαi + Δwiε i. The insight in Equation 2 is that total relative return, ΔR, can be decomposed into marketwide factors, signal, and residual contribution terms that exactly add up. ote that ΔR on the left-hand side of Equation 2 is the total relative return. The fundamental law relationships shown in Equations and 2 describe only the active return, R A, which is given by the sum of the last two terms in Equation 2. September/October

5 Financial Analysts Journal Specifically, the contribution from the market in Equation 2 is the product of the estimated market return and the differential exposure: Market = R M ( β P β B ). (2a) Similarly, the contribution from each of the other unforecasted marketwide factors is Factor k = λ k ΔwF i ki, (2b) where the factor contribution is the product of the estimated payoff, λˆ k, and the net exposure to the factor, ΔwF. Intuitively, the net exposure to i ki the factor is the sum of the active weights times the security factor exposures and can be calculated ex ante. 4 The signal contribution is Signal = v Δw i α i, (2c) where v is the estimated signal payoff and Δw is the net exposure to the signal. The = i α i i final term in the attribution analysis in Equation 2 is the noise contribution oise = Δw i ε i. (2d) The noise contribution has no payoff or associated regression coefficient estimate; it simply aggregates the active weights and regression residuals from Equation. Performance attribution systems often separate the marketwide influences from the active return contribution, as in Equation 8, but do not assess the direct contribution of the investor s forecasts, as in Equation 2. Our proposed application of the generalized fundamental law allows the investment manager to connect the regressionbased payoff of the ranking system and the residual noise in Equations 2c and 2d to the ex post fundamental law relationships in Equations 2a and 2b. The empirical analysis in the next two sections will determine whether the simplifying assumptions in the mathematical proof of Equation 2 given in Clarke et al. lead to reasonable approximations in relating parameters from the fundamental law to the regression results. After verifying the approximate accuracy of the fundamental law based calculations, we will test two key implications of the law. Portfolio Results: Long Only We illustrate the attribution system by using nine years of historical data on, first, a long-only portfolio benchmarked to the S&P 500. The portfolio policy precludes securities outside the index, so the investable universe is exactly = 500 each period. In the study, the portfolio was reweighted at the beginning of each calendar month from April 995 to March 2004, or 08 successive months. The = 54,000 data observations, collected from a number of well-established financial market sources, are complete in terms of exact portfolio and benchmark weights, as well as the marketwide factor exposures, on the start date of each investment period. Mergers, acquisitions, and changes in index composition over time led to some missing realized security return observations (averaging fewer than in 500 per period), which were replaced with the cross-sectional average return for that month. We used a fairly generic security-ranking process based on an equal mix of the well-known momentum factor (prior-year return less priormonth return) and value factor (prior-month bookto-market ratio). 5 The signal was translated into security alphas, which were then put into a numerical optimizer that maximized the portfolio s expected active return each month by using the period-specific Barra covariance matrices. By specification, the optimization routine limited the total ex ante relative risk of the portfolio to 5 percent annualized (.44 percent a month) and included constraints as follows. The optimized portfolio in our example was constrained to be beta, size, nonlinear-size, and sector neutral with respect to the S&P 500 benchmark. These marketwide factors were used to neutralize the security returns before forecasts were made, so by design, the signal did not attempt to rank stocks based on these factors. The neutrality constraints are an explicit portfolio construction choice to avoid incurring relative risk from factors that are not part of the forecasting process. Of course, a manager could choose to relax the neutrality constraints, in which case, the contributions from these other factors would be larger than shown in the case study that follows. Whether large or small, however, other factor contributions would have little impact on the decomposition of the active return that is the focus of the forecast. Market-to-book (M/B) and momentum exposures were not neutralized because they are part of the forecasting process, so the investor wants exposure to these factors. Market beta (with respect to the complete U.S. equity market as forecasted by Barra) and company size (the log of market capitalization) are wellknown marketwide risk factors in the Fama and French paradigm. The nonlinear-size factor (the cube of the log market cap) is a lesser-known , CFA Institute

6 Performance Attribution and the Fundamental Law constraint that reduces the chance of barbell optimizations on the size factor (a barbell portfolio is a managed portfolio with many large-cap and smallcap securities but few in the middle range). Sector neutrality, meaning that the managed portfolio has weights in each of 3 economic sectors that match the weights of the S&P 500, is another constraint used by some managers. The marketwide factor and sector-neutrality constraints in this case study were imposed as narrow ranges. For example, the portfolio s relative market beta exposure was bounded between 0.0 and +0.0 (i.e., assuming a benchmark beta of.00, an absolute beta between 0.99 and.0). Finally, to further mirror common managerial practices, we placed an absolute magnitude of 3 percent active weight on each security and, once the portfolio was initially constructed, limited turnover in any given month to 5 percent. All of these constraints and implementation issues reduced the transfer coefficient, which measures the efficiency of signal implementation. A key aspect of the portfolio policy of this case study is the no-short-selling constraint. Less attractive securities could be excluded from the portfolio, but they could not be sold short. The turnover constraint and the no-short-selling policy are two of the most binding and thus interesting constraints in practice as well as in this case study. The signal implementation inefficiencies of long-only portfolios are well established in investment literature (see, for example, Brush 997; Jacobs, Levy, and Starer 998; Grinold and Kahn 2000). In the next section, we examine the historical record of a portfolio identical to the one described in this section except that short selling was allowed. The attribution system is best understood by examining a specific month, so in Table, we give a detailed decomposition of the return for February 2004 as a relatively recent month with interesting outcomes, some of which are typical of other months and others of which are unique. Panel A indicates that the managed portfolio return in February trailed the S&P 500 return by 43 bps. Panel B shows that the underperformance that month was not a result of poor performance in the manager s stockranking system or signal. The regression-based active return decomposition shows a signal contribution of 57 bps, associated with a fundamental law information coefficient value of This realized IC value is well above the long-term expected value of and, in the absence of constraint-induced noise, would have made February 2004 a highrelative-return month. Unfortunately, as Panel B also indicates, the regression-based residual contribution swamped the contribution of the signal. Table. February 2004 Performance Attribution: Long-Only Portfolio Measure Value A. Return Portfolio return 95 bps Less: Benchmark return 38 Relative return 43 bps B. Regression-based relative return decomposition Signal contribution 57 bps Residual contribution 09 Other factor contribution a 9 Total relative return 43 bps C. Fundamental law parameters Realized information coefficient, ρ a,r Realized constraint coefficient, ρ c,r Transfer coefficient, TC Breadth, 500 Ex ante active risk (not annualized), σ A.03% Realized active return dispersion, D D. Fundamental law calculations Estimated signal contribution (Equation 2a) 57 bps Estimated noise contribution (Equation 2b) 09 a See Table 2. The third element in the regression-based attribution system, the marketwide factors, contributed 9 bps to the portfolio s relative return. Although the active exposures to factors such as size and sector membership were constrained to be small, they were not exactly zero, which resulted in the nonzero contribution of marketwide factors. These unforecasted factors are incidental to our examination of the fundamental law s description of active return, but they do contribute to total relative return, so we have decomposed the 9 bps into various factor payoffs and exposures in Table 2. Table 2 indicates that most of the neutrality constraints were binding within the specified bounds. For example, the net market beta exposure was at its upper bound of 0.00 and the net size factor exposure was at its lower bound of These results are representative of most of the 08 months for the long-only portfolio; that is, the managed portfolio beta was generally as high as the constraints would allow and the size exposure was generally as low as the constraints would allow. In this particular case study, the maximum allowable market beta exposure was a result of the correlation of the signal factors, momentum and value, with beta whereas the minimum allowable size exposure resulted, in part, from limits on the small-cap active September/October

7 Financial Analysts Journal Table 2. February 2004 Other Factor Contribution Details Factor Payoff Exposure Contribution A. General factors Market beta bps Size (log market cap) onlinear size (size cubed) B. Sectors Basic materials bps Commercial services Consumer cyclicals Consumer noncyclicals Consumer services Energy Finance Health Industrials Technology Telecommunications Transportation Utility Total other factor contribution 9 bps ote: Total does not sum because of rounding. weights that were a consequence of the long-only constraint. In addition to the three general marketwide factors, the exact positive and negative values for net sector exposures of percent in Table 2 verify that the sector-neutrality constraints were also binding except for the Health and Utility sectors. Panel C of Table shows estimates of the active return performance based on parameters in the fundamental law. The realized constraint correlation coefficient of is consistent with the negative sign of the residual regression contribution. The absolute magnitude of the constraint correlation coefficient (as compared with the size of the information coefficient) might suggest a slightly lower impact of noise than signal in the active return in this month, but as discussed previously, the ex post fundamental law relationship has a multiplier of TC 2 on the constraint coefficient and a multiplier of TC alone on the information coefficient. The transfer coefficient for the long-only portfolio for February shown in Panel C is 0.383, and with that value, the multiplier on the constraint noise coefficient is = , which is several times larger than the IC multiplier of The result is that constraint-induced noise has a significantly larger impact on actual performance than does the ranking signal for the longonly portfolio. Grinold and others have suggested that the number of securities can only roughly approximate breadth. We set the breadth,, to an investable universe of the S&P 500, and our results suggest that the number of securities can be used in practice for breadth with reasonable precision when stocks are ranked on factor-neutralized relative returns. The equivalence of fundamental law breadth with security count is facilitated in this attribution system by the focus on factor-neutralized security returns, which have a more diagonal covariance matrix than do total security returns. 6 From Panel C of Table, the annualized ex ante active risk for the managed portfolio is 03. 2= 357. %. February was a relatively low ex ante active risk month; the annualized active risk for most months was closer to 4 percent. ote that the ex ante active risk is below the 5 percent limit imposed on ex ante relative risk by the optimization routine. The difference is a result of the small but nonzero exposures to unforecasted marketwide factors. The final fundamental law parameter shown in Panel C of Table is the realized active return dispersion. Because this standard deviation calculation uses risk-adjusted security returns, dispersion has an expected value of.0. February 2004 turned out to have a lower-than-average active return dispersion, in part because the number of trading days is lower in February than in the average month. The fundamental law calculations in Panel D of Table are key empirical results of this study. They confirm the practicality of a performance attribution system that can be interpreted in the context of the fundamental law. The signal contribution calculated directly from the fundamental law parameters in Equation 2a is % = 57 bps, which is within a bp rounding error of the regression-based attribution system contribution of 57 bps shown at the top of Table. Thus, the performance contribution from active portfolio positions can be related to the information content of the signal (realized IC) and the efficiency of portfolio construction given investor-imposed constraints (TC). The contribution from constraint noise estimated by Equation 2b is % = 09 bps, , CFA Institute

8 Performance Attribution and the Fundamental Law also within a basis point of the regression residual contribution of 09 shown in Panel B of Table. We report in Table 3 that the standard deviation over time (08 months) between fundamental law calculations and regression-based contributions for signal and noise was approximately bp a month. Table 3 gives summary statistics for the key data items from Table for April 995 to March The cumulative impact of the three constituent contribution items signal, constraint noise, and marketwide factors and cumulative total relative return for all 08 months is shown graphically in Panel A of Figure. Constraint noise should, in theory, have a long-term mean of 0 unless the combined effect of the constraints leads to security weights not taken that have a correlation with some unknown factor with a persistently positive or negative payoff. Although the 9 bp average seems large, a simple statistical significance test cannot reject the null hypothesis of a true mean of 0. 7 In addition, the visual impression from Panel A of Figure is that almost all of the negative constraint-induced noise occurred in the calendar year The 08-month sample is large enough to allow for reasonableness checks on the practical validity of the ex ante fundamental law in Equation and the ex post relationship in Equation 2. Unfortunately, the checks are compromised by the lack of stationarity in the real-world management process over time. Specifically, the transfer coefficient, shown graphically in Figure 2, is not constant from month to month despite a fixed set of constraints. Although the constraint set and implementation policies were constant, changes in the security covariance matrix and in the decay rate in the ranking signal over time led to changes in the transfer coefficient. As shown in Figure 2, the long-only TC varied about 0.4 for the first three years, rose to a high of about 0.7 in the middle of the year 2000, and then dropped back to about 0.4 at the end of the time series. Additionally, as shown in Figure 3, the ex ante monthly active risk level was not constant; it had a high of 38 bps and low of 80 bps. The nonstationarity in the risk model s covariance matrix is evidenced by changes in the proportion of total relative risk (which was limited by the optimizer to 44 bps a month) that was captured by the fundamental law s active risk parameter, σ A. Finally, the information content of the signal itself exhibits nonstationarity, in that the variability is larger than expected. Based on the 500 crosssectional sample, repeated observations over time of a fixed information coefficient would have a sampling error of only / 500 = The intertemporal IC standard deviation of reported in Table 3 suggests that the true information coefficient changes over time, a practical problem associated with the intertemporal application of the fundamental law discussed by Qian and Hua (2004). Despite the nonstationarity problems, we can perform a rough check on the two fundamental law relationships by using the time-series averages. The predicted information ratio in Equation when the Table 3. Monthly Attribution Statistics, April 995 March 2004 Measure Mean Standard Deviation Maximum Minimum A. Regression-based relative return decomposition Signal contribution 22 bps 6 bps 332 bps 46 bps Residual contribution Other factor contribution Total relative return 7 bps 87 bps 628 bps 465 bps B. Fundamental law parameters Realized information coefficient Realized constraint coefficient Transfer coefficient Ex ante active risk (monthly) Realized active return dispersion C. Fundamental law calculations Estimated signal contribution 23 bps 6 bps 335 bps 47 bps Difference from regression 5 2 Estimated noise contribution Difference from regression 2 4 ote: Totals may not sum because of rounding. September/October

9 Financial Analysts Journal Figure. Cumulative Relative Return Attribution, Monthly for April 995 March 2004 Relative Return (%) 50 A. Long-Only Portfolio /95 3/96 3/97 3/98 3/99 3/00 3/0 3/02 3/03 3/04 Relative Return (%) 50 B. Long Short Portfolio /95 3/96 3/97 3/98 3/99 3/00 3/0 3/02 3/03 3/04 Signal oise Factors Total average information coefficient is and the average transfer coefficient is is IR = IC TC = = 02., an annualized value of = Assuming that the long-term expected value of the constraint contribution is 0, we use the observed 22 bp realized average signal contribution as a proxy for the active return. Dividing by the average 4 bp active risk, we find the observed empirical information ratio of the long-only portfolio to be 22/4 = 0.9, close to the predicted value of 0.2. ote that comparing predicted information ratios with actual empirical results may be even less precise in real time because managers often overestimate the value of their forecasting process as measured by the expected information coefficient. In addition, we have observed that realized risk is often higher than the ex ante estimates based on the Barra covariance matrix. For example, the realized relative risk over the time period reported in Table 3 averaged 87 bps rather than the limit specified to the optimizer of 44 bps , CFA Institute

10 Performance Attribution and the Fundamental Law Figure 2. Long-Only and Long Short Transfer Coefficients, Monthly for April 995 March 2004 Transfer Coefficient Long Short Portfolio Long-Only Portfolio /95 3/96 3/97 3/98 3/99 3/00 3/0 3/02 3/03 3/04 Figure 3. Long-Only and Long Short Ex Ante Active Risk, Monthly for April 995 March 2004 Risk (%) Relative Risk (both portfolios).2 Long Short Portfolio Long-Only Portfolio /95 3/96 3/97 3/98 3/99 3/00 3/0 3/02 3/03 3/04 Clarke et al. suggested that one key implication of the ex post fundamental law relationship in Equation 2 is that the variance in realized active return will be proportioned as TC 2 percent and TC 2 percent into, respectively, signal and constraintinduced noise. Based on the average transfer coefficient value of from Table 3, these proportions should be = 23 percent and = 77 percent. Squaring the standard deviations of the explained signal and noise terms shown in Table 3 gives 6 2 = 3,456 bps and 30 2 = 6,900 bps, or proportions of 44 percent and 56 percent. So, although the constraint noise variance does represent a majority of active return variance, the proportions are not as dramatic as predicted by the ex post fundamental law theory. The primary reason for this discrepancy is the unequal intertemporal variation of the two stochastic correlation coefficients, ρ a,r and ρ c,r, in Equation 2. As mentioned, repeated observations of static or fixed population correlations from a 500-stock sample would result in observed standard deviations of about / 500 = The constraint coefficient standard deviation in Table 3 is, at 0.057, only slightly higher, but the observed information coefficient standard deviation, at 0.079, is substantially higher. The September/October

11 Financial Analysts Journal result is a higher proportion of signal contribution variation than predicted by TC 2. To investigate this phenomenon further, we examined subperiods by dividing the full nine-year period into three subperiods of three years each. 9 The long-only transfer coefficient shown in Figure 2 was highest in the middle subperiod and lowest in the first subperiod. When the average transfer coefficient within each subperiod was used, the TC 2 values for the three subperiods were 6 percent, 30 percent, and 23 percent. The actual observed proportions of signal contribution variance, calculated as described for the full sample, were 9 percent, 53 percent, and 43 percent. Thus, although the pattern of lower signal contribution to total variance for a lower transfer coefficient is clear, the actual proportion of signal contribution was higher than predicted by TC 2 in all three subperiods. The lowest discrepancy (6 percent predicted versus 9 percent observed in the first subperiod) corresponds to more equality in that subperiod for the ρ a,r and ρ c,r correlation coefficient standard deviations of, respectively, and Thus, the subperiod analysis suggests that nonstationarity in the information coefficient is the primary reason for the observed bias toward understatement of the signal contribution variance by the ex post fundamental law. Portfolio Results: Long Short We also examined a long short portfolio; we used the same signal and implemented the portfolio with all the constraints except for the long-only constraint. The results for the two portfolios are compared in Table 4. The impact of removing the long-only constraint on implementation efficiency as measured by the transfer coefficient is dramatic; the average TC value rises to for the long short portfolio; the two portfolios transfer coefficients are graphically compared in Figure 2. Although the long short portfolio was constrained to have a market beta close to the benchmark, the combined value of the long and short positions in the portfolio were not constrained to a constant value. To illustrate, Figure 4 shows the sum of the long security weights each month during the full sample period. On average, the long short portfolio had about 200 percent long weights and about 00 percent short weights for a total weight of 00 percent each month. Despite a constant ex ante relative risk constraint of 5 percent annualized, the portfolio had long short weights that varied from about 270/ 70 at the start of the period to as low as 50/ 50 in the middle. Table 4 shows the decomposition of the long short portfolio s average monthly relative return, and Panel B of Figure depicts the cumulative long short portfolio active return and constituent contributions, which can be compared with the equivalent long-only chart in Panel A. The same signal was used in both portfolios, so the realized monthly information coefficients in the portfolios Table 4. Comparison of Long-Only and Long Short Attributions, April 995 March 2004 Long-Only Portfolio Long Short Portfolio Standard Standard Measure Mean Deviation Mean Deviation A. Regression-based relative return decomposition (bps) Signal contribution 22 bps 6 bps 4 bps 82 bps Residual contribution Other factor contribution Total relative return 7 bps 87 bps 44 bps 98 bps B. Fundamental law parameters Realized information coefficient Realized constraint coefficient Transfer coefficient Ex ante active risk (monthly) Realized active return dispersion C. Fundamental law calculations Estimated signal contribution 23 bps 6 bps 42 bps 83 bps Difference from regression 2 Estimated noise contribution Difference from regression 2 ote: Totals may not sum because of rounding , CFA Institute

12 Performance Attribution and the Fundamental Law Figure 4. Sum of the Long-Only and Long Short Long Weights, Monthly for April 995 March 2004 Long Weight Long Short Portfolio.5.0 Long-Only Portfolio /95 3/96 3/97 3/98 3/99 3/00 3/0 3/02 3/03 3/04 are identical, but the higher transfer coefficient in the long short portfolio allows more of the expected value added by the signal to be transferred into performance. As in the long-only portfolio, we assumed stationarity by using the intertemporal averages as fixed parameter values in a rough check of the fundamental law relationships. The predicted monthly information ratio of the long short portfolio in Equation with these averages used is IR = IC TC = = The actual information ratio when the average signal contribution of 4 bps and the average ex ante active risk of 28 bps, both from Table 4, are used is quite close, at 4/28 = For the long short portfolio, the standard deviation of the signal contribution of 82 bps in Table 4 is much larger than the 0 bp standard deviation in the constraint noise contribution, which is consistent with the higher transfer coefficient in the long short portfolio than in the long-only portfolio. Like the long-only portfolio, the actual proportions of performance variance, however, lean more toward the signal than is predicted by the ex post fundamental law relationship. The average transfer coefficient of squared for the long short portfolio predicted that 55 percent of performance variance would come from the signal. The actual observed proportion is 75 percent. Subperiod analysis (not reported) again indicated that, like the long-only portfolio, the difference between predicted and observed proportions corresponds to the relative magnitude of the information and constraint coefficient intertemporal standard deviations. Conclusion Existing performance attribution systems have difficulty relating the success of the manager s security-ranking system to the actual signal contribution. The effectiveness of the ranking system is evaluated outside the portfolio construction process, so no mechanism relates the payoff from the rankings to their exact contribution to performance. We proposed a regression-based attribution structure that links the two, and our method relates the results from the regression-based structure to parameters from the fundamental law of active management. The empirical investigation we report indicates that the law can be used to estimate and interpret the regression-based results without substantial error while showing the investor how much of the active return is related to forecasting accuracy and how much is noise related to constraints in constructing the portfolio. For our portfolios benchmarked to the S&P 500, we found the discrepancy between signal and constraint noise contributions estimated by the regression and the fundamental law equations to be about bp a month. The time-series results of this study confirm two strategic perspectives of the extended version of the fundamental law of active management, although precise validation is difficult because of nonstationarities in the actual data. First, the fundamental law September/October

13 Financial Analysts Journal states that, ex ante, the expected performance, as measured by the information ratio, is a function of the expected success of the signal, the extent to which the signal is transferred into active security weights, and the breadth of application. We confirmed that this ex ante relationship generally holds and that the higher transfer coefficient of a long short portfolio results in higher realized average performance for that portfolio structure than for a long-only portfolio based on the same signal. Second, the fundamental law predicts that, ex post, portfolios with lower transfer coefficients will have lower proportions of realized performance variation attributable to the signal. We validated this prediction by comparing the results of a long-only and a long short portfolio that were built on the same signal. As predicted, the portfolio with the higher transfer coefficient had a substantially greater proportion of signal-related performance over time. The proportion of performance variation explained by the signal was greater than predicted by the squared transfer coefficient, however, because of nonstationary in the information coefficient over time. We thank Steven Sapra for technical assistance in conducting the optimizations. otes. Clarke et al. discussed the motivation for risk weighting in the correlation coefficient calculations. Specifically, the expected and realized active security returns respectively, a i and r i are divided by the individual-security active risk estimates, the σ i variables, prior to the cross-sectional correlation calculations. Similarly, active security weight parameters Δw i and c i are multiplied by the individual-security risk estimates prior to calculating the cross-sectional correlations. 2. The market beta for a portfolio, like the exposure to any other marketwide factor, is simply the weighted-average beta of the individual securities. In the special case of the one-factor structure used in the original formulations of the fundamental law, the general market and benchmark portfolios were assumed to be the same and Equation 0 was ΔR = R B (β P ) + R A. 3. An unweighted OLS regression is justified only under the simplistic assumption that active risk is the same for all securities in the investable set. 4. The interpretation of the estimated payoff, λˆ k, depends on the unit of measure for the security factor exposures. However, the contribution of a marketwide factor to portfolio performance (i.e., estimated payoff times exposure) is invariant to the unit of measure used. ote that the estimated factor payoffs are based on a multivariate regression, which will produce results that are different from those produced by univariate regressions because of nonzero cross-sectional correlations between the various marketwide factor exposures. 5. Active portfolio managers typically use signals of a more proprietary composition and/or security rankings based on some form of fundamental analysis. Our objective in this research was to illustrate the attribution system and the practical application of the fundamental law of active management, not to promote any particular security selection procedure. 6. The derivation of the ex post fundamental law theory in Clarke et al. is based on the simplifying assumption of a perfectly diagonal covariance matrix for the forecasted component of active security returns. Although this assumption never holds in practice, the empirical results in this article suggest that the S&P 500 security returns that are net of the specified set of marketwide factors are close enough to being uncorrelated that the equations hold within an acceptably small error. 7. The statistical test of a 0 population mean uses the t-statistic, calculated as the observed sample mean of 9 bps over the standard error of the mean. The standard error of the mean based on the sample standard deviation of 30 in Table 3 and the sample size of 08 is 30/ 08 = 0.4. A t-statistic of 9/0.4 = 0.9 is not statistically significant. 8. For large sample sizes, the standard error of an estimated correlation coefficient is approximately over the square root of, where is the sample size (see Judge, Hill, Griffiths, Lutkepohl, and Lee 982, p. 685). 9. Complete statistics for each of the subperiods are available on request from the authors. References Brush, John S Comparisons and Combinations of Long and Long/Short Strategies. Financial Analysts Journal, vol. 53, no. 3 (May/June):8 89. Buckle, David How to Calculate Breadth: An Evolution of the Fundamental Law of Active Portfolio Management. Journal of Asset Management, vol. 4, no. 6 (April): Carhart, Mark M On Persistence in Mutual Fund Performance. Journal of Finance, vol. 52, no. (March): Chan, Louis K.C., Jason Karceski, and Joseph Lakonishok The Risk and Return from Factors. Journal of Financial and Quantitative Analysis, vol. 33, no. 2 (June): Clarke, Roger, Harindra de Silva, and Steven Thorley Portfolio Constraints and the Fundamental Law of Active Management. Financial Analysts Journal, vol. 58, no. 5 (September/October): Fama, Eugene F., and Kenneth R. French The Cross- Section of Expected Stock Returns. Journal of Finance, vol. 47, no. 2 (June): Goodwin, Thomas H The Information Ratio. Financial Analysts Journal, vol. 54, no. 4 (July/August): Grinold, Richard C The Fundamental Law of Active Management. Journal of Portfolio Management, vol. 5, no. 3 (Spring): , CFA Institute

14 Performance Attribution and the Fundamental Law Grinold, Richard C., and Ronald. Kahn The Efficiency Gains of Long Short Investing. Financial Analysts Journal, vol. 56, no. 6 (ovember/december): Jacobs, Bruce I., Kenneth. Levy, and David Starer On the Optimality of Long Short Strategies. Financial Analysts Journal, vol. 54, no. 2 (March/April):40 5. Judge, George G., R. Carter Hill, William E. Griffiths, Helmut Lutkepohl, and Tsoung-Chao Lee Introduction to the Theory and Practice of Econometrics. 2nd ed. ew York: John Wiley & Sons. Qian, Edward, and Ronald Hua Active Risk and Information Ratio. Journal of Investment Management, vol. 2, no. 3 (Third Quarter): Thomas, Lee R Active Management. Journal of Portfolio Management, vol. 26, no. 2 (Winter): What financial mindsneedtoknow $70.00 Cloth w/cd-rom $90.00 Cloth $80.00 Cloth wiley.com September/October