1 Análisis del Desempeño de las Administradoras de Fondos para el Retiro: Un Estudio Bootstrap Estacionario Analysis of the Performance of Mexican Pension Funds: New Evidence from a Stationary Bootstrap Application Autores: Gerardo Zúñiga Villaseñor Arnulfo Rodríguez Hernández Pedro Nahúm Rodríguez Hernández RESEÑA 1. Objetivo El objetivo del presente trabajo es evaluar el desempeño de las Administradoras de Fondos para el Retiro (AFORES) en México desde una perspectiva de rendimientos de inversión. Se busca medir el valor que agregan las administradoras de fondos para el retiro al portafolio de inversión mediante la estimación de la alfa de Jensen en un modelo de valuación tipo APT (Arbitrage Pricing Theory). Asimismo, este trabajo busca determinar si el desempeño estimado puede distinguirse del factor suerte. 2. Planteamiento Los sistema de cuentas individuales, como el sistema establecido hace casi 1 años mediante la reforma a la Ley del Seguro Social de 1997, tienen como característica que el monto potencial de la pensión que un trabajador puede obtener es función directa tanto de las características específicas de cada trabajador (salario, edad, tasa de cotización, vida laboral, etc.), como del valor que las AFORES puedan agregar al fondo de retiro. El valor que las AFORES pueden agregar es, a su vez, una función de los rendimientos que obtienen de la inversión de los fondos de pensiones y de las comisiones que cobran por sus servicios. Por lo tanto, es de gran relevancia estudiar el desempeño de las administradoras de fondos para el retiro en relación a cierto parámetro (benchmark) e identificar si este desempeño se puede separar del factor suerte. En la medida en que se puedan comparar las medidas de desempeño relativas a rendimientos netos de comisiones con un
2 benchmark en particular, podrá determinarse la importancia de incentivos que busquen maximizar los rendimientos netos que reciben los trabajadores. 3. Hipótesis En este trabajo identificaremos el valor que agregan los administradores de fondos para el retiro al portafolio de inversión en relación a un benchmark que consiste en una canasta de bonos gubernamentales. Estudios previos han identificado que ciertos administradores de fondos de inversión tienen habilidades superiores para agregar valor al portafolio mientras que otros estudios no han encontrado ninguna habilidad superior. Este trabajo estima la existencia o no de habilidades superiores y si los resultados obtenidos pueden distinguirse del factor suerte. 4. Metodología La metodología utilizada en el presente trabajo se divide en dos partes. La primera parte consiste en la estimación de un modelo APT (por sus siglas en inglés, Arbitrage Pricing Theory) que explica los rendimientos obtenidos por las AFORES (tanto brutos como netos de comisiones) mediante la incorporación de variables económicas fundamentales, como cambios inesperados en la tasa de inflación, cambios en los precios de la mezcla mexicana de petróleo de exportación, y cambios en los rendimientos del Bono del Tesoro de Estados Unidos a 1 años (1-year Treasury Bond). La segunda parte del trabajo introduce un método de remuestreo (bootstrap) estacionario que permite obtener una distribución aleatoria de la variable de desempeño (el alfa de Jensen) y evaluar si el desempeño estimado en la primera etapa mediante la estimación del modelo APT puede distinguirse del factor suerte. Ambas etapas de la estimación se realizan tanto para los rendimientos brutos como para los rendimientos netos de comisiones. La incorporación del modelo bootstrap estacionario constituye una contribución a la literatura ya que hasta donde los autores tienen conocimiento, no se ha llevado a cabo una evaluación del desempeño de los administradores de fondos para el retiro en México que permita separar el desempeño del factor suerte.
3 5. Conclusiones Los resultados de nuestras estimaciones sugieren que los administradores de fondos de ahorro para el retiro no muestran habilidades superiores (en términos de rendimientos) en comparación con un benchmark específico (una canasta de bonos gubernamentales). Además, cuando se lleva a cabo el análisis para los rendimientos netos de comisiones nuestros resultados sugieren que se extrae valor del portafolio. Las estimaciones del modelo utilizando una técnica de remuestreo (bootstrap) estacionario que arrojan una distribución aleatoria de alfas, sugiere que el desempeño de los fondos (medido con las alfas estimadas) no se puede atribuir a la suerte. El mismo resultado se obtiene tanto para rendimientos brutos como para rendimientos netos de comisiones. Nuestros resultados deben interpretarse tomando en cuenta varios factores. En primer lugar, se han realizado cambios importantes al régimen de inversión al que están sujetas las SIEFORES y sus efectos pueden manifestarse con cierto retraso. En segundo lugar, el horizonte de planeación e inversión en un sistema de ahorro para el retiro es de largo plazo, por lo que los trabajadores aún pueden recibir valor agregado a su portafolio por encima de cierto benchmark. No obstante, también hay que resaltar la importancia de diseñar un mecanismo de incentivos que promueva la competencia en términos de rendimientos netos, ya que esta variable refleja directamente el valor que reciben los trabajadores de parte de las administradoras de fondos de ahorro para el retiro. En la medida en que las AFORES puedan desarrollar habilidades superiores de inversión (por encima del benchmark) podrán obtener mayores rendimientos netos y atraer a más clientes. Una competencia en esta dimensión traerá beneficios indiscutibles para los trabajadores.
4 Análisis del Desempeño de las Administradoras de Fondos para el Retiro: Un Estudio Bootstrap Estacionario Analysis of the Performance of Mexican Pension Funds: New Evidence from a Stationary Bootstrap Application Abstract This paper assesses the performance of Mexican pension funds (AFORES) by using an asset pricing model that includes macroeconomic factors. We apply a bootstrap statistical technique to obtain the cross-sectional distribution of performance measures ( alphas ) across all pension funds. This is done to determine whether a pension fund is good (bad) at adding (subtracting) value to (from) their customers portfolio before and after commissions charges, and whether the performance observed is explained simply by luck. Our results provide strong evidence that most funds show inferior performance relative to a particular benchmark, with even the highest-ranked fund having no significant superior performance when returns before commission charges are considered. Moreover, when returns net of commissions charges are analyzed, our results show that all funds subtract value from their customers portfolio and this performance cannot be attributed to bad luck. Our results highlight the importance of evaluating the performance of AFORES in terms of net returns on investments and of giving pension fund administrators incentives to improve in this regard. JEL Classification: C14, G11, G23 Keywords: Pension funds, performance evaluation, stationary bootstrap.
5 1. Introduction It is almost ten years since the pension system for workers of the formal private sector in Mexico was reformed from a pay-as-you-go (defined benefit) scheme into fully funded (defined contribution) individual accounts. These accounts consist of mandatory contributions from workers, employers and the government, and are administered and invested by specialized firms called AFORES (Administradoras de Fondos de Ahorro para el Retiro). One characteristic of pension systems based on individual accounts is that workers retirement income depends, almost entirely, on the accumulated funds of their accounts. These funds grow over time based on several worker-specific variables such as age, income, contribution rate, and working life. In addition, the growth of these funds also depends on the rate of return paid by pension fund managers and commissions charged for management and investment services. Therefore, it is of great relevance to assess the relative performance of AFORES regarding the returns on funds under management. In this paper, we conduct a performance assessment of Mexican AFORES portfolios by explicitly controlling for luck, without assuming an ex-ante parametric distribution from which pension funds returns are drawn. The main purpose of this study is to determine whether a pension fund is good (bad) at adding (subtracting) value to (from) their customers portfolio before and after commissions charges, or whether it is simply lucky (unlucky). We apply a stationary bootstrap technique to obtain the cross-sectional distribution of performance measures across all pension funds. We use an Arbitrage Pricing Theory (APT) model with fundamental variables found to be relevant to explaining the portfolio gross (before commissions charges) returns on those funds. 1 One of the innovations of this work is the use of 1 The underlying premise of the APT is that asset returns are generated by a set of systematic factors and that asset expected returns are linear in the factor loadings. 2
6 survey data on changes in inflation expectations for the next twelve months that measure unanticipated changes in inflation. 2 Previous literature on the performance assessment of Mexican pension funds is found in García-Verdú (25). He finds that a two-factor model adequately captures the returns on Mexican pension funds. However, neither of those two factors is a fundamental economic variable. 3 He also concludes that no pension fund manager has superior asset-picking skills, something that we corroborate in this paper with the asset pricing model that includes macroeconomic factors and the bootstrap technique. Elton et al. (1995) include fundamental economic variables in an asset pricing model to explain the returns on a set of U.S. bond market indices. They find that the addition of fundamental economic variables leads to a considerable improvement in the explanation of cross-sectional expected returns. Kosowski et al. (25) conduct the first comprehensive examination of U.S. mutual fund performance that explicitly controls for luck and use a bootstrap statistical technique to generate the joint distribution of performance measures across all funds. Their main results indicate that a sizable minority of managers have enough asset picking skills well enough to add value to portfolios net-of-costs. To the authors knowledge, not only is this work the first performance assessment of Mexican AFORES portfolios that explicitly controls for luck, but also the first that uses fundamental economic variables to explain the returns on these funds. Our results suggest that most funds have inferior performance (negative and significant values of the Jensen s alpha) relative to a particular benchmark and even the highest-ranked fund fails to achieve a significant superior performance for returns before commission charges (the Jensen s alpha is positive but 2 Elton et al. (1981) show that treating changes in expectations as reflecting unexpected influences is consistent with both a rational expectations view of economic decision-making and a vast empirical literature. 3 Elton et al. (1995) demonstrate the importance of the inclusion of fundamental economic variables to explain bond returns. In our work this is important because before the year 25 Afores were only allowed to invest in Mexican fixed-income assets. Moreover, García-Verdú (25) mentions that performance measures are very similar when using the CAPM and a twofactor model. He suggests that this could be a result of not having a model that contains all the relevant factors in explaining pension fund returns. 3
7 not significant). Moreover, all funds subtract value from their customers portfolio when returns net of commissions charges are considered and this cannot be attributed to bad luck. Measuring the performance of pension funds in terms of net returns is of great relevance since this is a more accurate measure of the value added by AFORES to workers retirement accounts. To the extent that pension fund managers have incentives to maximize this measure, it will result in higher pensions for workers when they retire. The remainder of this paper is organized as follows. In Section 2, we describe the current pension fund system and the data. Section 3 introduces the APT model and tests if its restrictions hold. Section 4 presents the stationary bootstrap technique along with the performance assessment statistically inferred from both cross-sectional distributions of performance measures (alphas) and t-statistics. Finally, Section 5 presents our concluding remarks. 2. The Pension System of Individual Accounts in México The 1997 reform of the pension system in Mexico introduced individual accounts for every worker in the formal private sector. The Social Security Law enacted in 1997 established compulsory contributions to a retirement account from workers, employers and the federal government amounting to 6.5% of every worker base salary. In addition, the federal government contributes with 5.5% of the minimum wage to every worker s account, regardless of their level of income. 4 Individual accounts are administered by AFORES and invested by subsidiary specialized investment entities called SIEFORES (Sociedades de Inversión Especializadas en Fondos de Ahorro para el Retiro). When a worker chooses (or is assigned to) a particular AFORE the funds in the individual account are invested by that AFORE s SIEFORE. 4 This social quota is a progressive subsidy since it represents a larger proportion of a worker s salary as this salary decreases. This quota is contingent on the worker s continued participation in the system. 4
8 Since 1997 the number of AFORES has changed by exits, mergers and new entries. In our analysis, we only consider AFORES that have survived in the industry ever since the inception of the new pension system. The sample period is from January 21 through October 26 (58 monthly observations). We chose that sample period mainly because of the availability of consistent data on returns net of commissions published by CONSAR (available since December 21). SIEFORES are subject to an investment regime determined by the regulatory authority, CONSAR, that sets limits to the risk that each SIEFORE can undertake with its investment choices. The investment regime has been gradually modified to allow greater diversification while maintaining risk levels within certain limits. Between 22 and 24 important changes were introduced to the investment regime. These changes involved: 1) Since 1997 a minimum of 65% of instruments had to be government bonds with a maximum maturity (or revision) period of 183 days. 5 This restriction was first modified through an increase in the average maturity period to 9 days (December 21). 6 Subsequently, the regulation was based on a daily valuation of risk (a VaR measure) in effect since November ) Originally, the type of issuer was restricted to the following limits: 8 a. Federal Government (minimum 65%) b. Private (maximum 65%) c. Financial Intermediaries (maximum 1%) 5 Circular CONSAR 15-1, Diario Oficial de la Federación de Junio 3 de Circular CONSAR 15-5, Diario Oficial de la Federación de Diciembre 5 de Circular CONSAR 15-8, Diario Oficial de la Federación de Noviembre 29 de Circular CONSAR
9 These limits were removed and replaced by credit worthiness as measured by credit rating agencies with a minimum rating of A, and instruments issued by other entities were allowed (local and state governments and government-owned companies). 9 3) The use of derivatives was introduced (November 22). 1 4) Investment in stocks was allowed up to 15% of total assets as long as they tracked stock indices (May 24). 11 5) Exclusion of foreign issuers was lifted subject to a maximum share of 2% (bonds and securities) as long as these issuers were under the regulation of the technical committee of the International Organization of Securities Commissions (IOSCO ) and the European Union (May 24). 12 6) In May 24 two investment funds were created based on life-cycle considerations: a. SIEFORE Básica 1 for workers 56 years and older. This fund can only be invested in bonds (domestic and foreign (up to 2%)) and at least 51% has to have some sort of inflationary protection. 13 b. SIEFORE Básica 2 for workers under 56 years old. This fund allows investment in securities (up to 15%) and up to 2% in foreign instruments (bonds or securities). The modifications previously discussed were gradually implemented and they occurred throughout the period under analysis in this paper. Since the funds initially could only invest in 9 Circular CONSAR 15-6, Diario Oficial de la Federación de Abril 8 de Circular CONSAR Circular CONSAR 15-12, Diario Oficial de la Federación de Mayo 26 de Ibid. 13 Either be denominated in Unidades de Inversión, UDIs (units indexed to inflation) or that guarantee a return equal or greater than the variation in UDIs. 6
10 Mexican fixed-income assets (government issued bonds predominantly) our model will use a benchmark that best captures the returns on government instruments. 2.1 Data Our analysis involves the estimation of excess returns of the SIEFORES portfolios. We chose the 28-day CETES as our risk-free asset and the 12-month SIEFORES return as our AFORE portfolio return. Since January 25 we consider returns of SIEFORE Básica 2 and we will use the term SIEFORE henceforth. In the estimation of excess returns we use an asset pricing model that uses fundamental macroeconomic variables. The choice of these variables is based on both the characteristics of the Mexican economy and findings of a vast literature on the relationship between financial markets and macroeconomic variables (see Chen, Roll, and Ross (1986)). The variables used in our estimations are: 1) monthly nominal excess gross returns obtained from the historical 12-month nominal return of the SIEFORES published by CONSAR; 2) monthly nominal excess net returns obtained from historical nominal returns net of commissions published by CONSAR; 3) nominal returns on 28-day CETES calculated from its yields which are published by Banco de México; 4) inflation expectations for the next 12 months surveyed by Banco de Mexico; 5) price of oil (per barrel in dollar terms) of the Mexican export blend (Mezcla Mexicana de Exportación), available from several sources; 6) yields of the 1-year Treasury Bond published by the Federal Reserve Board; 6) an index measure of returns of a fixed-income market portfolio consisting of all government bonds (including BREMs and BPAs) published by VALMER as MEX_GUBER. One clarification note regarding the market portfolio is in order. Given the high concentration of government bonds in the composition of pension funds portfolios we used a benchmark index that included only government bonds. This is justified as long as the components and the weights included in this benchmark index include the assets that were 7
11 available to pension fund managers, subject to their particular investment regime. Between January 21 and the creation of SIEFORES 1 and 2 in May 24 pension fund managers could have invested up to 1% of their assets in government bonds with at least 51% invested in instruments with inflationary protection (either denominated in inflation-indexed units, UDIs, or with a guaranteed return equal or greater than the variation in the value of UDIs). There were also restrictions on the maturity period of instruments: since 1997 and until December 21 the limit was 183 days. Between this date and November 22 the limit increased to an average of 9 days and from November 22 onwards the restriction was shifted to a VaR measure. 14 After the separation of portfolios in May 24 into SIEFORES 1 and 2 we only analyze the returns of SIEFORE Básica 2. This fund could be invested up to 1% in government bonds with no restrictions regarding inflation-indexed instruments or maturity periods. In other words, all instruments that make up our benchmark index were available to all AFORES. Following Chen et. al (1986) and Elton, Gruber and Blake (1995) we introduce (unexpected changes for) fundamental economic variables to explain expected returns in the bond market. The previous studies have shown that unexpected changes in inflation and unexpected changes in economic performance (along with a measure of market returns and a measure of default risk) contribute to explain returns in the stock and bond markets. In our study, we also include a measure of (changes in) oil prices given the great importance of oil exports in the Mexican economy. Elton et al. (1995) treat changes in forecasts as reflecting unexpected influences. However, we only have unexpected changes for inflation because we have its forecasts. The assumption that unexpected changes for the other two variables could be approximated by their actual changes will be validated only if we fail to reject the random walk hypothesis. 14 Circulares CONSAR 15-1, 15-5,
12 Table I reports the variance ratios and test statistics for oil prices of the Mezcla Mexicana and the 1-year U.S. Treasury Price Index. The values reported in the main rows are the actual variance ratios, and the entries enclosed in parenthesis are the test statistics. Table I: Random walk test Variance-ratio test of the Random Walk Hypothesis for oil prices of Mezcla Mexicana and 1-year U.S. Treasury Price Index from December, 21 to October, 26. The variance-ratios are reported in the main rows, with the heteroscedasticity-robust test statistics given in parenthesis immediately below each main row. Under the random walk hypothesis, the value of the variance ratio is one and the test statistics have a standard normal distribution (asymptotically). Number q of base observations aggregated to form variance ratio Time Period Number of observations A. Oil prices of Mezcla Mexicana 12/1-1/ (.31) (1.37) (1.55) (1.5) B. 1-year U.S. Treasury Price Index 12/1-1/ (1.43) (.85) (.49) (.37) The random walk null hypothesis may not be rejected at the usual significance levels for the entire time period. Moreover, the failure to reject the null hypothesis is not due to changes in variances since the statistic is robust to heteroscedasticity. Consequently, we can assume that unexpected changes in oil prices and unexpected changes in yields can be approximated by their realized changes. 9
13 3. The Asset Pricing Model Following Chen, Roll and Ross (1986), we assume that returns are generated by a mixture of benchmark portfolios and fundamental economic variables. We adapt the return generating process in Elton et al. (1995) and write it as r it = E K [ ri ] + ij ( R jt E[ R j ]) + β γ g + η (1) k = 1 ik kt it where r i t ( i 1,..., N ) 1. is the return on pension fund at time it 2. R is the return on benchmark portfolio j at time t ; jt g kt =. In our case N = is the unexpected change (in logs) in the k th fundamental economic variable at t g1 t, g2 t and g3t time. In our case K = 3 and are the unexpected changes (in logs) in inflation for one year forward, oil prices and yields on the 1-year U.S. Treasury Bond, respectively. 4. β ij is the sensitivity of pension fund i to the innovation of the j th benchmark 5. ik portfolio; γ is the sensitivity of pension fund i to the innovation of the k th fundamental economic variable; 6. η it is the time-t return of pension fund i that is unrelated to either benchmark portfolios or fundamental economic variables; 7. Ε denotes expectation; 8. Ε[ ] = Ε[ ] = g η. k i Note that g k represents unexpected changes in a fundamental economic variable. By definition, the expected value of an unexpected change is zero or Ε [ g ] = k. In our case, we only have 1
14 one benchmark portfolio (MEX_GUBER) and three fundamental economic variables: inflation forecasts, oil prices and yields on the 1-year U.S. Treasury Bond. From the Arbitrage Pricing Theory (APT) of Ross (1976), equation (1) leads to the following expression for the expected return on pension fund i Ε K γ ikλk k = 1 [ r ] i = λ + β ijλ j + (2) where 1. λ is the return on the risk-free asset ( R F ) (We use 28-day CETES). j 2. λ is the market price of sensitivity to the j th benchmark portfolio; 3. λ k is the market price of sensitivity to the k th fundamental economic variable. When variables in the return-generating process are benchmark portfolios, the APT market price of risk associated with such portfolio is the portfolio s expected return minus λ. Thus [ ] λ j = Ε R j λ for j = 1,..., J. In our case J = Substituting this expression into equation (2) and recognizing that λ = RF yields: Ε K [ r ] = R + β ( E[ R ] R ) + γ λk i F ij j F k = 1 ik (3) Following Elton et. al (1995), we substitute equation (3) into equation (1) and allow over time to obtain: R F to vary r it R Ft ij K ( R jt RFt ) + ik ( λk + g kt ) + ηit = β γ k = 1 (4) 15 García-Verdú (25) uses a two-factor model which consists of two benchmark portfolios: Pip-Guber and PipG-Real. Our MEX_GUBER portfolio is similar to the Pip-Guber. Since our benchmark portfolio also contains inflation protected government bonds, we do not use another benchmark like the PipG_real. Moreover, García-Verdú (25) finds that performance measures are quite similar between the one-factor (with the Pip-Guber as the only factor) and the twofactor models. 11
15 Rearranging equation (4) yields: where r it R Ft = α + β i ij K ( R jt RFt ) + γ ik g kt + ηit k = 1 (5) α = i K k = 1 γ ik λ k (6) We estimated Equation (5) for every pension fund to explain both gross nominal excess returns and net nominal excess returns (after considering commission charges). Tables II and III show the estimation results done with OLS, whose parameters standard errors are adjusted for potential autocorrelation and heteroscedasticity by the method proposed by Newey and West (1987). 16 The results for gross returns (Table II) indicate that innovations in the benchmark portfolio covary positively and significantly with the excess gross return of seven out of eight pension funds. In addition, the sensitivity of pension funds to the innovations of oil prices is positive and significant. Net returns (Table III) on the other hand, shows that solely the innovations in the benchmark portfolio are significant determinants of pension fund performance. However, the performance measures ( alphas ) across all pension funds in Table III are negative and statistical significant. An examination of 2 R s in Tables II and III reveals that the asset pricing model explains a larger fraction of the variance of monthly excess net returns than of excess gross returns. 16 Ferson et al. (23) use the Newey-West (1987) method when they have autocorrelation in the explanatory variables and/or heteroscedasticity in the estimated regression. 12
16 Table II: Estimated coefficients obtained with OLS regressions for every pension fund excess gross returns. The following regression model is estimated on excess gross returns using monthly data from January, 22 to October, 26: K rit R Ft = α i + β ij ( R jt R Ft ) + γ ik g kt + η it. k = 1 The p-values are below the coefficients and are corrected for heteroscedasticity and autocorrelation using 2 the Newey-West estimator. R s from each regression are reported in decimal form. Fund i α i GROSS RETURNS β ij γ i1 γ i 2 γ i 3 R 2 Inbursa *** p-value * ** * Banorte *** p-value Bancomer *** p-value * ** * Principal *** *** p-value XXI * *** p-value ** ** Santander *** p-value ** *** Profuturo *** p-value ** * Banamex -.646*** *** *** p-value
17 Table III: Estimated coefficients obtained with OLS regressions for every pension fund excess net returns. The following regression model is estimated on excess net returns using monthly data from January, 22 to October, 26: ( ) r R = α + β R R + γ g + η. it Ft i ij jt Ft ik kt it k = 1 The p-values are below the coefficients and are corrected for heteroscedasticity and autocorrelation using 2 the Newey-West estimator. R s from each regression are reported in decimal form. K Fund i α i NET RETURNS β γ ij i1 γ γ 2 i 2 i3 R ** Inbursa *** p-value Bancomer *** *** p-value XXI *** 3.85 *** p-value Santander *** *** p-value Banorte *** *** p-value Profuturo *** *** p-value Banamex *** *** p-value Principal *** *** p-value We also estimated equations (4) and (5) to explain the cross-sectional variations in returns. The latter was estimated by using the iterated non-linear seemingly unrelated regressions (ITNLSUR) used in Gallant (1987). Since equation (4) is equivalent to equation (5) 14
18 with restriction (6) imposed, we compare the results of both estimations to determine whether imposing the APT restriction (6) on our four-factor model results in a statistically decrease in explanatory power. We find that the APT restriction holds. Table IV shows the results. Table IV: Likelihood ratio test statistics We compare the APT model against the non-restricted model. The dependent sample consists of 8 Afores. The sample period is from January 21 through October 26 (58 monthly observations). Null hypothesis: restrictions hold; alternative hypothesis: restrictions do not hold. Small-sample adjustment (see Gallant (1987)): Reject null when L > qf α, where F α is F statistic at α level of significance with q degrees of freedom in numerator and nm p degrees of freedom in denominator. n = 58 observations, M = 8 equations, and α =.5. Pairwise Test Models L a p b q c qf.5 L - qf.5 Reject Null? Net Returns No Gross Returns No a L = n(ln Σ 1 - ln Σ 2 ), where Σ 1 is the variance-covariance matrix of restricted residuals and Σ 2 is the variance-covariance matrix of unrestricted residuals. b p = total number of estimated parameters. c q = number of restrictions. 4. The Stationary Bootstrap Technique and Cross-Sectional distributions of performance measure (alphas) and t-statistics. In this section we apply a bootstrap statistical technique to determine whether a pension fund is good (bad) at adding (subtracting) value to (from) their customers portfolio before and after commissions charges, or whether it is simply lucky (unlucky). We use the Stationary Bootstrap method proposed by Politis and Romano (1994). This traditional resampling algorithm was used to generate cross-sectional distributions of alphas and t-statistics. 17 In building such distributions, we impose the condition that the true alphas are zero for every pension fund with a given rank. Kosowski et al. (26) impose such condition to determine whether a mutual fund is lucky (unlucky) or good (bad) at asset-picking. However, 17 As mentioned in Kosowski et al. (26), when individual fund residuals are normal, heterogeneity in risk taking across funds will not bring about non-normalities in the crosssection if one uses the t-statistic. 15
19 they use the simple bootstrap procedure which consists of residual resampling. In constructing their artificial return series, they impose the null hypothesis of zero true performance ( α i =, or, equivalently, tˆ ˆ α = ). i Formally, in our model, when strictly following Kosowski et al. (26) we would generate the following pseudo time series for excess returns ( er it b ) of each pension fund: er b it = ˆ β ij 3 b ( R jt RFt ) + ˆ γ ik g kt + ˆ ηit k = 1 (7) where ˆb η it are the residuals resampled with replacement from the vector of residuals obtained in the estimation with the real data. Since the stationary bootstrap resamples observations (not residuals) with replacement in blocks of random size, we must find an equivalent condition for imposing the null hypothesis of zero true performance. 18 Such equivalence is found when er b it i i i ij 3 b ( R jt RFt ) + ˆ γ ik g kt + ˆ ηit ˆ α = ˆ α ˆ α + ˆ β (8) k = 1 In other words, subtracting the estimated alpha from resampled observations of excess returns is equivalent to imposing the null hypothesis of alpha equal to zero for the case of residual resampling. We ranked the funds according to their alpha and t-statistic given by the estimations with the true data. Figures I and II show the cross-sectional distributions of alphas while Figures III and IV present those of t-statistics. The vertical lines in those figures indicate the estimated alphas and t-statistics from the estimations with the true data. 18 We decided to use the stationary bootstrap instead of the simple bootstrap procedure of residual resampling in order to take into account potential autocorrelation of excess returns. 16
20 Figure I: Cross-sectional distributions of alphas for nominal excess gross returns Panel A.1 Top Pension Fund Panel B.1 Bottom Pension Fund 35 Bootstrapped alpha-values Top 1 alpha-values x 1-3 Panel A.2 Top 2 Pension Fund Panel B.2 Bottom 2 Pension Fund 4 Bootstrapped alpha-values 45 Bootstrapped aplha-values Top 2 alpha-values x Bottom 2 alpha-values x 1-3 Panel A.3 Top 3 Pension Fund Bootstrapped alpha-values Panel B.3 Bottom 3 Pension Fund 5 Bootstrapped alpha-values Top 3 alpha-values x Bottom 3 alpha-values x 1-3 Panel A.4 Top 4 Pension Fund 5 Bootstrapped alpha-values Top 4 alpha-values x 1-3 Panel B.4 Bottom 4 Pension Fund 5 Bootstrapped alpha-values Bottom 4 alpha-values x