Una Anatomía De Las Estrategias Comerciales

Una Anatomía de las Estrategias de Negociación: Evidencia de China Usando un enfoque de series de tiempo cruzadas, evaluamos los beneficios de las estrategias de impulso e identificamos las fuentes de beneficios en el mercado de valores de China. Las estrategias Momentum generan retornos significativos y negativos en el mercado de acciones A en horizontes de inversión en un mes y en nueve meses. En el mercado de acciones B, las estrategias de impulso producen rendimientos significativos y negativos en y por encima de doce meses. El análisis de descomposición determina que los rendimientos negativos se atribuyen predominantemente a la rentabilidad de la serie temporal de los rendimientos de las acciones. Aunque las estrategias de impulso generan retornos positivos y significativos durante el período posterior a que China abriera su mercado de acciones B, una vez restringido en el extranjero, a inversores individuales nacionales, la importancia relativa de la previsibilidad de las series temporales y la variación transversal no cambia. Si experimenta problemas al descargar un archivo, compruebe si tiene la aplicación adecuada para verla primero. En caso de problemas adicionales, lea la página de ayuda de IDEAS. Tenga en cuenta que estos archivos no están en el sitio IDEAS. Por favor sea paciente ya que los archivos pueden ser grandes. Como el acceso a este documento está restringido, es posible que desee buscar una versión diferente en Investigación relacionada (más adelante) o buscar una versión diferente de la misma. Artículo proporcionado por M. E. Sharpe, Inc. en su revista Emerging Markets Finance and Trade. 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Regístrese para acceder al resto del documento. En este artículo usamos un único marco unificador para analizar las fuentes de protones a un amplio espectro de estrategias de negociación basadas en el retorno implementadas en la literatura. Mostramos que menos de 50 de las 120 estrategias implementadas en el artículo producen apoyos estadísticamente significativos y, incondicionalmente, las estrategias de impulso y contraria tienen igual probabilidad de tener éxito. Sin embargo, cuando condicionamos el horizonte de retorno (corto, medio o largo) de la estrategia, o el período de tiempo durante el cual se implementa, surgen dos patrones. Una estrategia de impulso suele ser protable en el horizonte medio (de 3 a 12 meses), mientras que una estrategia contraria establece redes de apoyo estadísticamente significativas en horizontes largos, pero sólo durante el subperíodo 1926 de 1947. Más importante aún, nuestros resultados muestran que la variación transversal de los rendimientos medios de los valores individuales incluidos en estas estrategias juega un papel importante en su proba - bilidad. La variación transversal puede potencialmente dar cuenta de la proba - bilidad de las estrategias de momentum y también es responsable de la aten - ción. Litekin, Narasimhan Jegadeesh, Charles Jones, Roni Michaely, Steve Manaster, Vasant Naik, Sheridan Titman, Jamie Zender y participantes del seminario en la Universidad Estatal de Arizona, la Universidad Cornell, la Universidad Estatal de Michigan, la Universidad de Arizona, la Universidad de Columbia Británica. Carolina del Norte, Universidad de Notre Dame, Universidad de Utah, Conferencia de Finanzas de Invierno en la Universidad de Utah, Reuniones de la American Finance Association, San Francisco y Seminario de Finanzas de Verano en la Universidad de Tel Aviv para comentarios y sugerencias útiles. Estamos especialmente agradecidos a un árbitro anónimo ya Bob Korajczyk y Ravi Jagannathan por ayudarnos a enfocarnos en los principales temas tratados en este artículo, a Michael Cooper por su invaluable ayuda de investigación ya Sonja Dodenbier por ayudar a preparar este manuscrito. La financiación parcial para este proyecto es proporcionada por la Kenan-Flagler Business School, la Universidad de Carolina del Norte y la Universidad de Michigan Business School. Dirección de la correspondencia a Gautam Kaul, Departamento de Finanzas, Universidad de Michigan Business School, Ann Arbor, MI 48109-1234. La revisión de los estudios financieros Fall 1998 Vol. En el caso de la Sociedad de Estudios Financieros, la Sociedad de Estudios Financieros / Rev. Las estrategias de negociación que al parecer superaron el mercado se remontan al inicio de la negociación de activos financieros. Varios practicantes y académicos en la era de pre-mercado-ecuación (es decir, antes de 1960) creían que los patrones predecibles en los retornos de las acciones podrían conducir a promesas anormales a las estrategias comerciales. De hecho, Keynes (1936) resumió sucintamente las opiniones de muchos al afirmar que la mayoría de las decisiones de los inversores sólo pueden tomarse como resultado de los espíritus animales. En los últimos años ha habido un resurgimiento dramático del interés académico en la predictibilidad de los rendimientos de los activos basados ​​en su historia pasada. Un número cada vez mayor de investigadores sostienen que los patrones de series temporales en los retornos se deben a las ineficiencias del mercado y, en consecuencia, pueden ser consistentemente traducidos en prótesis anormales.1 En términos generales, estos artículos analizan dos estrategias diametralmente opuestas en filosofía y ejecución: La estrategia contraria que se basa en las reversiones de precios y la estrategia de impulso basada en la continuación de los precios (o el impulso en los precios de los activos). Hasta hace poco tiempo se ha hecho relativamente más hincapié en las estrategias contraria, pero hay cada vez más evidencia de que las continuaciones de precios resultan en constantes anormales consistentes en estrategias de impulso. Uno de los aspectos más desconcertantes de esta literatura es que estas dos estrategias diametralmente opuestas parecen funcionar simultáneamente, aunque para diferentes horizontes de inversión. Específicamente, las estrategias contrarias son aparentemente protables para los intervalos a corto plazo (semanal, mensual) ya largo plazo (3 a 5 años, o más), mientras que la estrategia momentánea es protable a mediano plazo (3 a 12- Mes) períodos de tenencia. En este artículo intentamos determinar las fuentes de los beneficios esperados de toda la clase de estrategias de negociación que se basan en la información contenida en los rendimientos anteriores de los valores individuales. La fuerza de nuestro análisis es que utilizamos un marco único, que se basa en los análisis de Lehmann (1990) y Lo y MacKinlay (1990), para descomponer los protones de todas las estrategias, contraria o momentum ya corto plazo a largo plazo. Esta descomposición es importante debido a que las estrategias de negociación basadas en estos patrones, incluyen entre otros, Alexander (1961, 1964), Cootner (1964), Fama 1963, 1970), Fama y Blume (1966), Levy (1967), Van Horne y Parker (1967), James (1968) y Jensen y Bennington (1970). Conrad y Kaul (1988, 1989), Fama y French (1988), Lo y MacKinlay (1988), Porterba y Summers (1988), Campbell, Grossman y Wang (1988) son algunos de los numerosos artículos recientes que tratan sobre la previsibilidad del retorno. 1993), Richardson (1993), Boudoukh, Richardson y Whitelaw (1994), Conrad, Hameed y Niden (1994) y Jones (1994). DeBondt y Thaler (1985), Chan (1988), Sweeney (1988), Jegadeesh (1990), Lehmann (1990), Lo y MacKinlay (1990), entre los artículos recientes que documentan la aparente proba - bilidad de las estrategias de negociación basadas en dicha previsibilidad ), Levch y Thomas (1991), Brock, Lakonishok y LeBaron (1992), Chopra, Lakonishok y Ritter (1992), Allen y Karjalainen (1993), Jegadeesh y Titman (1993, 1995a) y Asness (1994). Kaul (1997) proporciona una revisión de las metodologías empíricas utilizadas para descubrir la previsibilidad de retorno. El rendimiento de los valores en el pasado tiene dos componentes: uno que resulta de la previsibilidad de las series temporales en los retornos de seguridad y otro que surge como consecuencia de la variación transversal de los rendimientos medios de los valores que componen la cartera. La mayoría de las estrategias de comercio basadas en el retorno implementadas en la literatura se basan exclusivamente en la existencia de patrones de series de tiempo en los retornos. Específicamente, todas estas estrategias se basan en la premisa de que los precios de las acciones no siguen los paseos aleatorios. Sin embargo, los propósitos reales a las estrategias de negociación implementadas sobre la base del rendimiento pasado contienen un componente transversal que surgiría incluso si los precios de las acciones son completamente impredecibles y siguen los paseos aleatorios. Consideremos, por ejemplo, una estrategia de impulso. La compra repetida de ganadores a partir del producto de la venta de perdedores será, en promedio, equivalente a la compra de valores de alta media de la venta de valores bajas. En consecuencia, mientras exista una dispersión transversal en los retornos medios del universo de valores, una estrategia de impulso será protable. Por el contrario, una estrategia contraria será inprograble en promedio, incluso en un mundo donde los precios de las acciones siguen los paseos aleatorios. Es importante determinar las fuentes de la aparente protabilidad de las estrategias de negociación debido a (i) la hipótesis explícita en la literatura de que los patrones de series temporales en los precios de las acciones constituyen la única base de las estrategias comerciales basadas en el retorno, y (ii) La falta de previsibilidad en los rendimientos de las acciones es vista por algunos como sinónimo de efciencia del mercado, véase Fama (1970, 1991). Implementamos y analizamos un amplio espectro de estrategias de negociación durante el período 19261989 y durante los subperíodos dentro, utilizando toda la muestra de valores NYSE / AMEX disponibles. Específicamente analizamos ocho estrategias básicas con períodos de espera que oscilan entre 1 semana y 36 meses. De hecho, 55 de las 120 estrategias de negociación implementadas utilizando todos los valores de NYSE / AMEX generan signos estadísticamente significativos. Las probabilidades incondicionales de éxito de las estrategias de impulso y contraria son aproximadamente iguales: de las 55 estrategias estadísticamente protables, 30 son impulso, mientras que 25 son estrategias contraria. Más importante aún, cuando ex post condición en el horizonte de retorno de la estrategia y / o el subperíodo durante el cual se implementa, aparecen dos patrones que son consistentes con la literatura sobre las estrategias de negociación basadas en retornos ver, p. DeBondt y Thaler (1985) y Jegadeesh y Titman (1993). La estrategia de impulso por lo general suele ser positiva, y frecuentemente estadísticamente signi fi cativa, en los horizontes medianos, excepto durante el subperíodo 19261947, mientras que una estrategia contraria tiene éxito en horizontes largos, aunque los protones a estas estrategias son estadísticamente significativos sólo durante el subperíodo 19261947. Una descomposición empírica de los protones de las estrategias sugiere que la variación transversal de los rendimientos medios de los valores individuales incluidos en la estrategia es un determinante importante de su proba - bilidad. Específicamente, no podemos rechazar la hipótesis de que la variación seccional de la muestra en los rendimientos medios puede explicar la proba - bilidad de las estrategias de impulso. La dispersión transversal en los retornos medios parece ser también responsable de la escasez de estrategias contraria estadísticamente protables. Aunque constantemente observamos reversiones significativas de precios en prácticamente todos los horizontes, las protestas que emanan de estas inversiones son neutralizadas por las pérdidas debido a la gran variación transversal de los retornos medios. Consecuentemente, los apoyos netos estadísticamente significativos a estrategias contrarias se observan solamente en el inusual subperíodo 19261947. Es importante señalar que nuestra descomposición de los préstamos comerciales se basa en el supuesto de estacionaridad media de los rendimientos de los valores individuales durante el período en que se implementan las estrategias. Además, los retornos medios se estiman para una amplia sección transversal de rms con un conjunto nito de observaciones de series de tiempo, lo que resultará en una exageración de la importancia de la variación transversal en los retornos medios. Para calibrar la robustez de nuestra descomposición empírica de los protones de las estrategias de negociación, realizamos simulaciones de bootstrap y Monte Carlo de las estrategias de mediano plazo (3 a 12 meses) en las que intentamos eliminar las propiedades de las series de tiempo de los rendimientos de seguridad, Sus incondicionales características transversales. Los resultados de las simulaciones son consistentes con la hipótesis de que las promesas de las estrategias de impulso se deben en gran medida a la variación transversal de los retornos medios. Nuestros experimentos de Monte Carlo también sugieren que nuestros resultados son robustos a la exclusión de retornos medios extremos en la muestra. Finalmente, presentamos algunas estimaciones alternativas de la importancia relativa de la variación transversal en los retornos medios en la generación de los protones de las estrategias de negociación. Incluso las estimaciones más conservadoras sugieren que la variación transversal en los retornos medios es un determinante no trivial de la proba - bilidad de las estrategias comerciales. Claramente, diferentes especificaciones de los retornos esperados de los valores individuales podrían alterar nuestras conclusiones. Además, los comerciantes pueden ver la variación de la sección transversal en los retornos medios como una fuente de prótesis anormales. No se intenta analizar las fuentes, racionales o irracionales, de la variación transversal en los retornos medios, es decir, no intentamos explicar las diferencias transversales en los rendimientos medios usando un modelo de tasación de activos. Nuestro objetivo es determinar la importancia relativa de las propiedades de los rendimientos de los activos en la determinación de la proba - bilidad de las estrategias de negociación. Creemos que nuestro análisis y resultados deben ser de interés tanto para los comerciantes técnicos como para los productores de modelos de tasación de activos. La Sección 1 contiene una descripción de las estrategias de negociación implementadas en este artículo y su proba - bilidad cuando se aplican a los valores de NYSE / AMEX durante varios períodos de tiempo. En las Secciones 2 y 3 presentamos un análisis detallado de la descomposición de los protones de las estrategias. La sección 4 contiene un breve resumen y nuestras conclusiones. 492 Una Anatomía de las Estrategias de Negocio 1. La Protabilidad de las Estrategias de Negociación Consideramos un conjunto de estrategias comerciales que explícitamente imitan o capturan la esencia de las estrategias implementadas con anterioridad. Específicamente, considere comprar o vender acciones en el tiempo t 1 en base a su desempeño desde el tiempo t 2 al t 1, donde el período abarca cualquier intervalo de tiempo nite. Además, suponga que el rendimiento de una acción se determina en relación con el rendimiento promedio de todas las acciones que se utilizan en la estrategia de negociación. En consecuencia, si se incluye todo el universo de activos en la estrategia, entonces cada rendimiento de las acciones se mide en relación con el rendimiento de la cartera de mercado de igual ponderación, Rmt. Por último, supongamos que 1 denote la fracción de la cartera de estrategias de negociación dedicada a la seguridad, véase Lehmann (1990) y Lo y MacKinlay (1990), es decir, wit 1 (k) 1 Rt 1 (k) Rmt 1 (k) N (1) donde Rt 1 (k) es la rentabilidad de la garantía i en el tiempo t 1, i 1. N. Rmt 1 (k) es la rentabilidad de la cartera de todos los valores con ponderación igual, yk es la duración del tiempo - intervalo . Dado que los pesos se basan enteramente en la información en el tiempo t 1, wit 1 tiene un subíndice de t 1. La expresión de los pesos en la ecuación (1) recoge sucintamente la filosofía de todas las estrategias de negociación basadas en el retorno. En primer lugar, el signo positivo o negativo que precede a la expresión del lado derecho refleja las creencias de los inversores (instituciones), es decir, si el inversionista cree en las continuaciones o reversiones de los precios (y por lo tanto recomienda y / o sigue un impulso o una estrategia contraria) . En segundo lugar, es importante señalar que, independientemente de si una estrategia es contraria o momentum, la premisa es que su éxito se basa en el comportamiento de la serie temporal de los precios de los activos. Específicamente, se supone que el rendimiento pasado de los valores en relación con algún punto de referencia (por ejemplo, el rendimiento medio de la cartera de todos los valores) es informativo sobre innovaciones futuras en los precios de los valores. Esto es totalmente contrario, por ejemplo, al modelo de los precios de las acciones, que implica que los cambios en los precios de las acciones son completamente impredecibles (ver sección 2 para más detalles). En tercer lugar, los pesos en dólares en la ecuación (1), es decir, w1t 1 (k). W N t 1 (k) conducen a una cartera de arbitraje (coste cero) por construcción N wit 1 (k) 0 k. (2a) i 1 y la inversión en dólares larga (o corta) viene dada por 1 (k) 1 2 N wit 1 (k). Cuarto, puesto que los pesos de la ecuación (1) son proporcionales al valor absoluto de las desviaciones de la rentabilidad de una garantía por la rentabilidad de una cartera de igual ponderación de Todos los valores, capturan la creencia general de que los movimientos extremos de los precios son seguidos por movimientos extremos, ver, por ejemplo DeBondt y Thaler (1985), Lehmann (1990), Lo y MacKinlay (1990), y Jegadeesh y Titman (1993). Finalmente, y lo que es más importante, las ponderaciones de la ecuación (1) nos permiten descomponer convenientemente los beneficios de las estrategias de negociación, véase Lehmann (1990) y Lo y MacKinlay (1990), independientemente de la naturaleza inherente de la estrategia Es una estrategia contraria o de impulso). Esto, a su vez, nos permite determinar la importancia relativa de los diferentes componentes.2 Las protuberancias realizadas en el tiempo t. T (k), a las estrategias de negociación implícitas por los pesos en la ecuación (1) están dadas por Nt (k) wit 1 (k) Rit (k). (3) i 1 Dado que todas las estrategias consideradas en este artículo (y típicamente en la literatura) son estrategias de costo cero, sólo los prots en dólares (y no los retornos) se dened como en la ecuación (3). Y si los mercados no tienen fricción, los pesos se pueden escalar arbitrariamente para obtener cualquier nivel de prótesis. Por lo tanto, vamos a confiar en gran medida en el signo y la significación estadística de los promedios de la serie temporal de la t (k) s que es, se examina si los protés esperados son estadísticamente signi cativamente positivas (o negativas). La Tabla 1 contiene protados medios / esperados para estrategias de negociación implementadas durante diferentes períodos de tiempo y para diferentes períodos de tenencia (es decir, diferentes k). Consideramos períodos de tiempo: 19621989 19261989 y tres subperíodos de igual tamaño dentro del período 19261989 (enero de 1926, abril de 1947, mayo de 1947, agosto de 1968, septiembre de 1968, diciembre de 1989). Primero implementamos las estrategias para el periodo 19621989 porque se corresponde con el periodo de tiempo usado en varios estudios anteriores, ver, p. Lehmann (1990), Lo y MacKinlay (1990), y Jegadeesh y Titman (1993). El período 1926-1989 se utiliza (para todos, excepto el período de tenencia semanal), ya que cubre un intervalo de tiempo mucho más largo, y este intervalo (y los subperíodos dentro de él) proporcionan una comprobación de robustez de la potencial proba - bilidad de las estrategias comerciales. Utilizamos ocho períodos de tenencia k diferentes. Donde k varía de 1 semana 2 Jegadeesh y Titman (1993) usan una variante de esta estrategia en la que los valores se clasifican en función de su desempeño pasado y luego se combinan en 10 carteras que se mantienen durante un período de tiempo específico. Una cartera de arbitraje también se forma comprando a los mejores y vendiendo los peores resultados. Observan que la correlación entre los retornos a la estrategia utilizada en este artículo y su trabajo es 0.95 sin embargo, los prots de su estrategia no pueden ser fácilmente descompuesto. Seguimos el esquema de ponderación implícito en la ecuación (1), especialmente porque la descomposición de las protuberancias es fundamental para este artículo. Sin embargo, los pesos de la ecuación (1) conservan la misma filosofía que la otra estrategia basada en el retorno. 494 Anatomía de las estrategias de negociación Tabla 1 Promedio de las estrategias de negociación para diferentes horizontes y períodos Intervalo de estrategia Subperiodos (19261989) (I) 19261946 (II) 19471967 (III) 19681989 19621989 19261989 0,035 (23,30) 3 meses 0,027 (0,67) 0,165 2,42) 0,557 (2,99) 0,070 (2,91) 0,020 (0,43) 6 meses 0,360 (4,55) 0,147 (1,91) 0,204 (1,03) 0,333 (4,97) 0,273 (3,63) 9 meses 0,708 (5,81) 0,488 (5,48) 0,276 (1,37) 0,487 (5,09) 0,634 (5,44) 12 meses 0,701 (4,64) 0,198 (1,29) 0,557 (-1,44) 0,372 (3,80) 0,611 (3,70) 18 meses 0,094 (0,35) 0,761 (2,88) 2,466 (3,49) 0,117 (0,77) 0,444 (1,51) 24 meses 0,501 (0,97) 1,181 (2,98) 2,831 (2,92) 0,434 (1,62) 0,792 (1,54) 36 meses 3,304 (3,39) 4,176 (6,48) 7,727 (6,08) 0,922 (1,24) 0,873 (0,84) 1 semana Contiene prots medios a estrategias de negociación de costo cero que compran ganadores de NYSE / AMEX y venden perdedores basados ​​en su desempeño pasado en relación con el desempeño de un índice de igual ponderación de todas las acciones. El valor de los protones en dólares viene dado por t (k) iN 1 wit 1 (k) Rt (k) i 1. N. donde t (k) es el dólar prot en el tiempo t de una estrategia de negociación de k periodos, (K) N Rit 1 (k) Rmt 1 (k) y Rmt 1 (k) N i 1 Rit 1 (k), donde k 1 semana y 3, 6, 9, 12, 18, 24 y 36 meses . Los números entre paréntesis son estadísticos z que son asintóticamente N (0, 1) bajo la hipótesis nula de que los protts verdaderos son cero y son robustos a la heteroscedasticidad y la autocorrelación, y tienen en cuenta cualquier correlación cruzada en los protés realizados de las estrategias dentro de un horizonte (Corto, medio o largo horizonte) de las estrategias. Todas las estimaciones prot se multiplican por 100. a 36 meses. Por brevedad, implementamos estrategias para las cuales la duración de los períodos de evaluación de desempeño pasados ​​y los períodos futuros de tenencia son idénticos. Por ejemplo, si evaluamos el rendimiento de los valores en los últimos tres meses, el período de tenencia de la estrategia de negociación también es de 3 meses. Debido a consideraciones de disponibilidad de datos, implementamos la estrategia de comercio semanal sólo durante el período 19621989. Para minimizar los sesgos de las muestras pequeñas en los estimadores de los componentes de las estrategias de negociación (véase el Apéndice), y para aumentar la potencia de nuestras pruebas, implementamos estrategias de negociación para períodos de retención superpuestos en una frecuencia mensual (para todos k excepto k 1 semana). Específicamente, determinamos los pesos, ingenio 1 (k). En el tiempo t 1 para todos los ks diferentes basados ​​en los retornos desde el tiempo t2 a t1. Las diferentes estrategias de período de tenencia pueden por lo tanto contener diferentes conjuntos de valores. Luego calculamos las protestas realizadas en el tiempo t usando la ecuación (3) para cada k. Para evitar posibles sesgos de supervivencia, véase, p. Brown, Goetzmann y Ross (1995), no exigimos que todos los valores incluidos en una estrategia en particular en el momento t 1 también tengan precios disponibles en el momento t. Si se incluye un valor en la estrategia de un período de tiempo basado en su desempeño anterior en el período k, pero sobrevive durante menos de k periodos en el futuro de 1998, Utilice un retorno del período (kj) para calcular t (k), donde j es el período de exclusión. Las estimaciones de la Tabla 1 son los promedios de las series temporales, para cada k. De los apoyos en cada momento t. T (k). Finalmente, puesto que las estrategias de impulso de impulso (contraria) son exactamente iguales a las pérdidas de estrategias contrarianas (impulso) ver las ecuaciones (1) y (3), implementamos solamente estrategias de momento (es decir, usamos los pesos 1 con 1 N Rit 1 (K) Rmt 1 (k) k). Por consiguiente, una estimación positiva (negativa) en la Tabla 1 implica que en promedio una estrategia de impulso (contraria) es protable. La tabla 1 también contiene los estadísticos z entre paréntesis para probar la significancia estadística de las protestas (pérdidas) promedio que estas estadísticas son asintóticamente N (0, 1) bajo la hipótesis nula de que los verdaderos protones son cero. Utilizamos un método de momentos generalizado ver el procedimiento de Hansen (1982) para calcular los errores estándar. Este procedimiento tiene en cuenta las correlaciones transversales (dentro de un período de tiempo particular) en los prots realizados de múltiples estrategias dentro de las clases de mediano y largo plazo, además de proporcionar errores estándar que son robustos a la heterocedasticidad y la autocorrelación. Varias características interesantes de la proba - bilidad de las estrategias de negociación surgen de una inspección de la Tabla 1. En primer lugar, el número de estimaciones positivas y negativas de prot medio son exactamente los mismos 18 frente a 18. Por lo tanto, incondicionalmente, Exitoso (al menos basado en las 36 estrategias evaluadas en la Tabla 1). Esto es digno de mención, dado que las estrategias de impulso y contraria son (como se ha señalado en la introducción) diametralmente opuestas en la filosofía. En segundo lugar, 21 de las 36 estrategias de negociación son estadísticamente signi fi cativamente rentables, el número de estrategias contraria - mente versus momentum estadísticamente significativas es aproximadamente el mismo, 11 frente a 10, respectivamente. Tercero, una vez que condicionamos el horizonte de retorno y / o el período de tiempo, sin embargo, las similitudes entre las estrategias de comercio contraria y momentum desaparecen. Específicamente, existe una relación sistemática entre el horizonte de la estrategia y la filosofía comercial que parece funcionar. Una estrategia de impulso suele ser protable en los horizontes medianos (de 3 a 12 meses): de las 20 estrategias de mediano plazo presentadas en la Tabla 1, una estrategia de impulso es protable en 15 de los casos. Más importante aún, todas las estrategias de impulso que producen protts estadísticamente signi fi cativos son estrategias de horizonte medio. Para medir el éxito de la estrategia de impulso en el horizonte medio, probamos la significancia conjunta de las estrategias de 3 a 12 meses dentro de cada período de tiempo. No es sorprendente que haya una fuerte evidencia de que la estrategia de momento medio-horizonte es protable en todos los períodos de tiempo excepto el subperíodo 19261947: las estadísticas de chi-cuadrado para cada uno de los otros cuatro períodos de tiempo tienen p-valores de cero. Durante el subperíodo 19261947, sin embargo, una estrategia contraria tiene éxito en el horizonte medio la estadística chisquare para la significancia conjunta de las estrategias de 3 a 12 meses tiene un valor p de 0,016. Por otro lado, el éxito de las estrategias contraria depende de un subperiodo, 19261947. De las 10 estrategias contrarianas que ganan prots estadísticamente signicantes, cuatro ocurren en el período 19261947 y también son responsables de la significancia estadística De los apoyos contrarios de cuatro estrategias más en el período general 19261989. De manera más significativa, una estrategia contraria es estadísticamente protable sólo dos veces en los tres subperíodos de posguerra. Nuevamente realizamos pruebas estadísticas para la significación conjunta de las estrategias contraria a largo plazo (18 a 36 meses). La estadística chi-cuadrado para el período 1926-1989 apoya firmemente la protabilidad de la estrategia contraria a largo plazo con un valor p de cero, pero esta evidencia depende de un intervalo de tiempo, el subperíodo 19261947. Mientras que el valor p para la estadística de chi cuadrado es 0,009 para el subperíodo 19261947, es 0,193 para el subperíodo 19481968 y 0,203 para el período 1969-1989.3 Por lo tanto, la protabilidad neta de la estrategia contraria se limita a la de largo plazo y A los datos anteriores a 1947. Esta evidencia es también consistente con los resultados de Fama y French (1988) y Kim, Nelson y Startz (1991), que indican que la reversión media a largo plazo de los precios de las carteras de valores es propia de la preguerra. Finalmente, aunque una estrategia contraria es obviamente protable en el horizonte semanal en el período 1962-1989, investigaciones recientes muestran que la protabilidad de las estrategias a corto plazo puede ser espuria debido a que es generada por los sesgos de la microestructura del mercado.4 La evidencia más convincente en la Tabla 1, Está a favor de la estrategia de impulso, que brinda apoyo a los proponentes de la estrategia de impulso, tanto en Wall Street como entre académicos. Asness (1994), Grinblatt, Titman y Wermers (1994), Jegadeesh y Titman (1993, 1995a) y Levy (1967) Hendricks, Patel y Zeckhauser (1993) proporcionan pruebas relacionadas. Por ejemplo, en el estudio más reciente sobre estrategias comerciales, Jegadeesh y Titman (1993) implementan 32 diferentes estrategias de impulso de 3 a 12 meses durante el período 19621989 y cada una es protable. Grinblatt, Titman y Wermers (1994) muestran que alrededor de 77 de los 155 fondos mutuos de su muestra siguen el impulso 3 Reestimamos todos los prots medios de la Tabla 1 condicionados al comportamiento del mercado, es decir, Sobre la rentabilidad del mercado superior a la tasa libre de riesgo. Los alphas de estas regresiones son típicamente similares a los prots medios incondicionales reportados en la Tabla 1. Por ejemplo, aparte de hacer que los prots medios de la estrategia de momento de 12 meses sean estadísticamente significativos, las estimaciones promedio de los protones de todas las otras estrategias y su estadística El signicance permanece en gran parte sin cambios para el período total 19261989. Durante los subperiodos, la única diferencia notable es que las estrategias contraria de 24 y 36 meses en el subperíodo 19471967 y la estrategia de 36 meses en el subperíodo 1968-1989 producen protestas condicionales estadísticamente significativas. Este último resultado proporciona un cierto apoyo para la Ball, Kothari y Shanken (1995), que indican que los protones contra el riesgo ajustados al riesgo son más altos en relación con los protésicos crudos en el período de posguerra. 4 Los efectos de la microestructura del mercado (por ejemplo, los efectos de rebote y de inventario) presentes en las devoluciones de las transacciones pueden explicar proporciones significativas de las reversiones de precios que conducen al aparente éxito de las estrategias contraria a corto plazo (véase Jegadeesh y Titman, 1995b) y Conrad, Gultekin , Y Kaul (1997). Bessembinder y Chan (1994) y Conrad, Gultekin, y Kaul (1997) consideran que todos los apoyos restantes a estas estrategias a corto plazo desaparecen a bajos niveles de costos de transacción, incluso para los grandes inversionistas institucionales. 497 La revisión de estudios financieros / v 11 n 3 estrategias de 1998, y al parecer con bastante éxito. Momentum es también un criterio de selección de acciones explícito para varios fondos de inversión, véase Bernard (1984) y Grinblatt y Titman (1989) .5 2. Fuentes de Protase a las Estrategias de Negociación En esta sección proveemos una descomposición de los apoyos esperados a estrategias de negociación de retorno. Siguiendo a Lehmann (1990) y Lo y MacKinlay (1990), los protones de las estrategias de negociación consideradas en la literatura (y en este artículo) pueden descomponerse directa y convenientemente tomando la expectativa de t (k) en la ecuación 3), y asumiendo nuevamente que implementamos estrategias de momento, E t (k) Cov Rmt (k), Rmt 1 (k) 1 N 1 NN Cov Rit (k), Rit 1 (k) (K) 2 (1) donde P (k) C1 (k) O1 (k) es la predictibilidad-protabilidad (K) es la media incondicional de la seguridad i para el intervalo de 1 longitud k. Y mt (k) N iN 1 i (k) es el retorno incondicional de un solo período de la cartera de mercado de igual ponderación en el momento t. Bajo el supuesto de la estacionariedad media de los rendimientos individuales de la seguridad, la descomposición anterior muestra que el total esperado de las estrategias de negociación proviene de dos fuentes distintas: la previsibilidad de la serie temporal de los rendimientos de los activos, medida por P (k) Dispersión en los rendimientos medios de los valores, denotada por 2 (k). El primer término en P (k) es el negativo de la autocovariancia de primer orden del rendimiento de la cartera de mercado de igual ponderación, indicado por C1 (k), y está casi completamente determinado por covarianzas cruzadas de las devoluciones de seguridad individuales ) El segundo término, denotado por O1 (k), es el promedio de autocovariancias de primer orden de los N valores individuales incluidos en la cartera de coste cero. Since P (k ) is entirely determined by return predictability which, in turn, forms the basis of all return-based trading strategies, we term it the predictability-protability index. Lo and MacKinlay (1990) also dene an identical protability index. However, their motivation is to deemphasize 5 For example, past research has demonstrated the abnormal protability of trading strategies that use the Value Line timeliness rankings which are based on price momentum, determined by price performance over the past 12 months see Copeland and Mayers (1982) and Stickel (1985). 498 An Anatomy of Trading Strategies the role of 2 (k ) since it has a small effect on prots to trading strategies that use weekly returns (see also Tables 2 and 4). We, on the other hand, dene P (k ) to emphasize that total expected prots to return-based trading strategies do not result entirely from time-series predictability in returns. 2.1 The random walk model Although Equation (4) provides a convenient decomposition of expected prots, we need a benchmark model for the return-generating process of nancial assets to interpret the two different potential sources of prots to trading strategies. Let us assume that all security prices follow random walks, so that returns can be depicted as Rit (k ) i (k ) it (k ) i 1. N (5) where E it (k ) 0 i. k and E it (k ) jt 1 (k ) 0 i. j. k .6 The usefulness of the random walk model in Equation (5) as a benchmark, particularly for this study, becomes obvious since trading strategies that rely on time-series predictability in returns cannot be protable by construction because Cov Rit (k ), R jt 1 (k ) 0 i. j. k .7 Equivalently, Equation (5) implies that there is no return predictability in either individual securities or across different securities, and hence the very basis of return-based trading strategies is ruled out. The model in Equation (5) also has economic appeal as a benchmark because changes in stock prices will (generally) be unpredictable in a risk-neutral world with an informationally efcient stock market see, e. g. Samuelson (1965). The most important property of the model in Equation (5), when combined with the decomposition of total expected prots in Equation (4), however, lies in the fact that it helps demonstrate that momentum (contrarian) strategies will be protable (unprotable) even if asset returns are completely unpredictable. More specically, from Equations (4) and (5) it follows that E t (k ) 2 (k ). (6) Equation (6) implies that as long as there are any cross-sectional differences in mean returns of individual securities, momentum strategies will generate prots equal to 2 (k ). Conversely, contrarian strategies will generate losses of an equal amount. Under the assumption that the mean returns of individual securities are stationary, these prots (losses) have no relation to any time-series predictability in returns. The prots in Equa6 Technically, all we need in our benchmark model is that the it s are uncorrelated but for ease of exposition, we assume a random walk model for stock prices. 7 Of course, although predictability in asset returns is a necessary condition for the success of trading strategies considered in this article, it is not a sufcient condition for abnormal gains to be reaped from these strategies. As others have pointed out, time variation in expected returns could also lead to predictability in stock returns see, e. g. Fama (1970, 1991). 499 The Review of Financial Studies / v 11 n 3 1998 Table 2 The decomposition of average prots to trading strategies Strategy interval E t (k ) P (k ) c1 (k ) o1 (k ) 2 (k ) Panel A: 19621989 0.035 1 week (23.30) 0.035 (19.95) 0.001 (18.95) 101.45 1.45 3 months 0.027 (0.67) 0.071 (1.78) 0.098 (27.22) 265.92 365.92 6 months 0.359 (4.55) 0.027 (0.34) 0.387 (31.16) 7.60 107.60 9 months 0.708 (5.81) 0.159 (1.27) 0.868 (32.75) 22.49 112.49 12 months 0.701 (4.64) 0.849 (5.44) 1.550 (35.23) 121.09 221.09 18 months 0.094 (0.35) 3.508 (12.40) 3.602 (55.41) 3,747.76 3,847.76 24 months 0.501 (0.97) 7.252 (12.61) 6.751 (51.93) 1,446.91 1,346.91 36 months 3.304 (3.39) 21.140 (17.47) 17.836 (46.08) 639.77 539.77 Panel B: 19261989 3 months 0.165 (2.42) 0.234 (3.40) 0.070 (17.95) 142.31 42.31 6 months 0.147 (1.91) 0.117 (1.53) 0.265 (18.93) 79.57 179.57 9 months 0.488 (5.48) 0.098 (1.09) 0.585 (20.10) 20.02 120.02 12 months 0.198 (1.29) 0.870 (5.32) 1.069 (20.96) 439.10 539.10 18 months 0.761 (2.88) 3.134 (11.00) 2.372 (26.06) 411.60 311.60 24 months 1.181 (2.98) 5.438 (12.36) 4.257 (27.41) 460.34 360.34 36 months 4.176 (6.48) 14.461 (29.61) 10.285 (19.18) 346.30 246.30 0.114 (13.57) 120.40 20.40 Panel C1: Subperiod I (January 1926April 1947 3 months 0.557 0.671 (2.99) (3.61) P (k ) 2 (k ) 6 months 0.204 (1.03) 0.624 (3.20) 0.420 (14.38) 305.63 205.63 9 months 0.276 (1.37) 0.668 (3.36) 0.944 (16.28) 242.19 342.19 12 months 0.557 (1.44) 2.489 (6.13) 1.932 (18.40) 446.54 346.54 18 months 2.466 (3.49) 7.033 (9.48) 4.567 (22.50) 285.16 185.16 24 months 2.831 (2.92) 11.250 (10.83) 8.419 (26.39) 397.42 297.42 36 months 7.727 (6.08) 27.882 (21.17) 20.155 (40.64) 360.84 260.84 500 An Anatomy of Trading Strategies Table 2 (continued) Strategy interval E t (k )b P (k ) c1 (k ) o1 (k ) Panel C2: Subperiod II (May 1947August 1968) 3 months 0.070 0.007 (2.91) (0.30) 2 (k ) P (k ) 2 (k ) 0.063 (10.50) 10.17 89.83 6 months 0.333 (4.97) 0.059 (0.88) 0.274 (10.15) 17.67 82.33 9 months 0.487 (5.09) 0.195 (1.73) 0.682 (9.61) 40.16 140.16 12 months 0.372 (3.80) 0.927 (5.81) 1.299 (9.41) 249.11 349.11 18 months 0.117 (0.77) 3.135 (10.02) 3.017 (10.93) 2,672.29 2,572.69 24 months 0.434 (1.62) 5.583 (10.15) 5.149 (11.39) 1,287.77 1,187.77 36 months 0.922 (1.24) 14.150 (7.99) 13.228 (11.12) 1,535.33 1,435.33 682.83 582.83 Panel C3: Subperiod III (September 1968December 1989) 3 months 0.020 0.135 0.115 (0.43) (3.00) (26.74) 6 months 0.273 (3.63) 0.171 (2.28) 0.444 (29.60) 62.84 162.84 9 months 0.634 (5.44) 0.321 (2.68) 0.955 (31.83) 50.62 150.62 12 months 0.611 (3.70) 1.041 (6.12) 1.651 (36.69) 170.44 270.44 18 months 0.444 (1.51) 3.205 (4.32) 3.649 (17.98) 721.82 821.82 24 months 0.792 (1.54) 5.854 (5.37) 6.646 (20.83) 739.50 839.50 36 months 0.873 (0.84) 19.363 (14.70) 18.490 (37.51) 2,217.74 2,117.74 This table contains the decomposition of average prots of trading strategies using NYSE/AMEX stocks. The decomposition of the average dollar prots is given by E t (k ) P (k ) 2 (k ), where the predictability-protability index is given by P (k ) C1 (k ) O1 (k ), C1 (k ) is (approximately) equal to the rst-order autocovariance of the return of the equal-weighted portfolio of all securities used in the zero-cost strategy, O1 (k ) is the average rst-order autocovariance of the returns of the N individual securities in the zero-cost portfolio, and 2 (k ) measures the cross-sectional variance of the mean returns of the N individual securities. The numbers in parentheses are z-statistics that are asymptotically N (0, 1) under the null hypothesis that the relevant parameter is zero and are robust to heteroscedasticity and autocorrelation, and account for any cross-correlation in the realized prots and the realized components of prots within a horizon class (short, medium, or long horizon) strategies. All prot estimates are multiplied by 100. All protable relative-strength strategies are shown in bold, while all protable contrarian strategies are in normal print. tion (6) are realized simply because in a world where security prices follow random walks (with drifts), following a momentum strategy amounts, on average, to buying high-mean securities using the proceeds from the sale of low-mean securities. That is, although a winner (loser) can have a high 501 The Review of Financial Studies / v 11 n 3 1998 (low) realization of a return due to either being a high - (low-) mean security or due to a high (low) current shock, on average winners (losers) will be high - (low-) mean securities. Consequently, this strategy will gain from any cross-sectional dispersion in the unconditional mean returns of the securities included in the portfolio of winners and losers. Conversely, if a contrarian strategy is followed, expected prots in Equation (6) will equal 2 (k ): contrarians will lose any cross-sectional variation in mean returns by on average selling high-mean securities and buying low-mean securities with the proceeds. These prots (losses) to trading strategies will disappear only under the assumption that all securities have identical mean returns. The random walk model provides economic content to the time-series versus cross-sectional decomposition of the expected prots of return-based trading strategies. Given that all return-based trading strategies are based on time-series patterns in stock prices, an empirical implementation of the decomposition will help us determine the legitimacy of this fundamental premise of trading strategies. Note that if one were to assume that crosssectional differences in mean returns are due entirely to differences in risk characteristicsa viewpoint not uncommon even among proponents of the return-based trading strategies see, e. g. Jegadeesh and Titman (1993, 1995a) and Lehmann (1990)the empirical decomposition will help provide deeper insights into the potential efciency or inefciency of asset prices. Table 2 contains estimates of the total average prots, E t (k ), and its (k ) and 2 (k ), for all holding periods, k. and for all two components, P ve time periods, 19621989 (panel A), 19261989 (panel B), and the three subperiods (panels C1C3). The numbers in parentheses below E t (k ), P (k ), and 2 (k ) are their respective z - statistics, which are autocorrela tion and heteroscedasticity consistent and take into account cross-sectional correlations in the realized prot of all strategies among each holding-period class. The Appendix contains the exact formulae and procedures used to estimate each of the three components of total average prots. Since the empirical decomposition of the prots is critically dependent on estimates of the unconditional means of the returns of individual securities, it is important to note again that the components are estimated under the assumption that the unconditional mean return of each security is constant over the entire sample period under consideration. We estimate the unconditional means using all data in a particular time period, and calculate the components of the prots of a particular strategy in a particular period based only on the securities included in that strategy in that specic period. In addition, we conduct subperiod analyses to evaluate the effect of our strong mean stationarity assumption on our inferences the inferences remain largely unchanged. We use overlapping data to minimize small-sample biases in estimates of the components of prots to trading strategies, but we 502 An Anatomy of Trading Strategies recognize that measurement errors in in-sample mean returns could nevertheless affect our inferences (see Appendix). Consequently, we devote Section 3 entirely to empirically evaluate the extent to which measurement errors may affect our results and inferences. Clearly, since there are relatively few long-horizon (say 3-year) returns even in the 19261989 period, the decomposition results for the long-horizon strategies should be interpreted with special caution. The rst important aspect of the results in Table 2 is the signicant effect of the cross-sectional variance of mean returns, 2 (k ), on the prots of all trading strategies. Specically, the cross-sectional component of the prots is both the predominant source of prots to the momentum strategy at medium horizons, and a major source of losses to contrarian strategies at long horizons. Note that the 2 (k )s are always statistically signicantly greater than zero. To gauge the economic role of the cross-sectional dispersion in mean returns in determining the prots of the different trading strategies, con sider rst the dramatic increase in the absolute magnitude of 2 (k ) with the investment horizon in each of the ve sample periods. This nding is important to emphasize because a similar pattern would be observed in the data if security prices follow the random walk process in Equation (5) or, equivalently, even if there is no predictability in returns. Specically, given Equation (5), the expected prots from a momentum strategy applied to a trading horizon of k periods and continuously compounded returns is given by see Equation (6) E t (k ) k 2 2 (1) k 2 E t (1). (8) Equation (8) shows that the expected prots (losses) from a momentum (contrarian) strategy will increase geometrically with the holding period k because the cross-sectional dispersion of mean returns increases with the (square of the) length of the holding period (relative to the length of the base holding period). For example, given Equation (5), the cross-sectional dispersion of the means of 36-month holding period returns will be 144 times i. e. (36/3)2 times the cross-sectional dispersion of the means of 3-month holding period returns. An inspection of the estimates of 2 (k ) in Table 2 shows that they do increase dramatically with the investment horizon in each sample period. This nding suggests that the protability of momentum strategies at medium horizons may not be due to price continuations potentially induced by market inefciencies. Moreover, the lack of statistically profitable contrarian strategies may be because these strategies lose the crosssectional dispersion in means, with this loss being particularly severe at long horizons. 503 The Review of Financial Studies / v 11 n 3 1998 2.2 Momentum strategies Recall that the momentum strategy is usually protable at medium horizons. To evaluate the relative importance of the cross-sectional versus time-series sources of these prots, however, it is instructive to evaluate the percentage contributions of P (k ) and 2 (k ) to total prots, as well as the sign and statistical signicance of the P (k )s. Note that if stock prices follow random walks, the percentage contributions of 2 (k ) should be constant and equal to 100 see Equation (6). The evidence in Table 2 demonstrates the important role of the cross-sectional variation in mean returns, as opposed to time-series patterns in security prices, in determining the protability of momentum strategies. Of the 18 cases in which positive prots are observed for momentum strategies (see estimates in bold in Table 2), the percentage contributions of 2 (k ) are typically greater than 100. There are only two occasions on which the contribution of the cross-sectional dispersion in mean returns to momentum strategies is less than 100: the 3-month and the 6-month strategies in subperiod II, panel C.2. Even in these two cases, however, the contribution of 2 (k ) is over 80. An alternative way to evaluate the relative importance of the crosssectional versus time-series components of the prots of momentum strategies is to note that there are only two instances in which these strategies gain from continuations in asset prices, that is, the P (k )s are positive. These are (obviously) the same two cases mentioned above. However, an advantage of evaluating the relative contribution of P (k ) is that we can also determine the statistical signicance of any prots to trading strategies due to predictable time-series patterns in asset prices. The evidence shows that even in the two cases which benet from price continuations, the resulting prots are statistically indistinguishable from zero. Using our particular method of decomposing prots, the statistical signicance of medium-horizon momentum prots appears to emanate from the statistical signicance of the 2 (k )s. Given that this empirical decomposition is affected by measure ment errors in mean returns, however, our inferences at this stage should be treated with caution. 2.3 Contrarian strategies The importance of the cross-sectional dispersion in mean returns in determining the protability of trading strategies is again observed in cases where a contrarian strategy appears to work. Note that barring the weekly and the 3-month strategies, the 2 (k )s lead to substantial losses to contrarian strategies. For example, even in the seven long-term strategies that yield statistically signicant prots to a contrarian strategy, the losses due to cross-sectional dispersion in mean returns are larger than the net prots. The important role of 2 (k ) is also exemplied by the fact that there are statistically signicant prots due to the price reversals in stock prices, 504 An Anatomy of Trading Strategies especially at longer horizons, yet only a few strategies yield statistically signicant net contrarian prots. Specically, the P (k )s are statistically signicantly negative for all long-term (18- to 36-month) strategies. Yet only in less than half the cases (7 of the 15 long-term strategies) are the price reversals able to overwhelm the losses from the cross-sectional variance in mean returns and lead to statistically signicant net prots. All of this evidence appears to be an outcome of severe and unusual price movements during the 19261947 subperiod. 3. Robustness Tests: Some Simulations8 Our analysis, based on the decomposition of the prots of trading strategies, suggests that the main determinant of the prots of return-based trading strategies is the cross-sectional variation in mean returns. Contrary to the commonly held belief that forms the basis of return-based strategies, the evidence suggests that time-series patterns in security returns are unlikely to result in statistically signicant net prots to trading strategies. The decomposition of the trading prots in Table 2 is based, however, on two assumptions. First, the mean returns of individual securities are assumed to be constant over the period in which the trading strategies are implemented. Second, the cross-sectional distribution of the in-sample mean returns accurately measure the true cross-sectional variation in the mean returns. While we do not allow for time-varying mean returns that could potentially explain predictability in returns, we do attempt to address the potentially serious effects of measurement errors in in-sample mean returns by conducting several simulation exercises and providing additional evidence about the potential role of the cross-sectional differences in mean returns. The main purpose of the simulations is to analyze the protability of trading strategies using simulated returns that are devoid of any time-series patterns that may be present in the real data, while maintaining the crosssectional characteristics of each security. Conducting simulations of any trading strategy, however, involves a great deal of time and computer resources, since the returns of several thousand individual securities need to be simulated. Consequently, we chose to simulate the prots of medium-term trading strategies during the 19641989 period. We chose the medium-term strategies because they are usually protable in the real data we focus on the 19641989 period because momentum strategies are most protable during this subperiod. We rst implement the medium-term trading strategies on real data during the 19641989 period. The second column of Table 3 contains the av - 8 We thank Ravi Jagannathan for recommending the use of simulations as a robustness check for our empirical decomposition. 505 The Review of Financial Studies / v 11 n 3 1998 erage prots, with their heteroscedasticity - and autocorrelation-consistent z - statistics in parentheses, for the 3- to 12-month strategies. Not surprisingly, the prot estimates are virtually identical to the prots for the 19621989 sample period in Table 2, panel A. 3.1 Bootstrap results To gauge whether the cross-sectional variance in mean returns alone can generate the prots to medium-horizon strategies, we rst conduct bootstrap simulations in which the returns of individual securities are scrambled in an attempt to eliminate any time-series relations that may be present in the real data see Efron (1979). Specically, we generate a sample of 301 monthly returns for each stock in the sample by resampling with replacement from the actual monthly returns between (December) 1964 and (December) 1989. This bootstrap sample should eliminate any time-series properties in each securitys returns, while maintaining all the other characteristics. Specically, the cross-sectional distribution of the individual-security mean returns should be preserved. In re-creating the bootstrap sample, we preserve the missing observations because that helps us retain the exact sample size used in the actual trading strategy.9 All medium-term (3- to 12-month) strategies are implemented on the bootstrap sample, and this exercise is replicated on 500 bootstrapped samples. Table 3, panel A, contains the results from the bootstrap simulations. The rst column of panel A contains the average prots of the trading strategies, the second column contains the average t - statistics of the 500 replications, and the last column shows the p - values which measure the proportion of times the simulated mean returns are greater than the mean returns of the actual strategies shown in the second column of Table 3. The bootstrap results conrm the ndings of our decomposition analysis. The mean prots of the bootstrap strategies are always greater than the corresponding estimates in panel A and the p - values are large, ranging between 0.69 and 1.00. Moreover, the average t - statistics of all the medium-term bootstrap strategies are greater than 10 and each of the t values are signicant for each strategy in all the 500 replications.10 These results suggest that the cross-sectional properties of the returns observed 9 To maintain the cross-sectional correlation in the returns, in one of our bootstrap experiments we attempted to scramble entire vectors of returns. This created a substantial mismatch between the number of securities used in the actual trading strategy and the simulated strategy, since the resampling of vectors scrambled the missing values as well. The substantial reduction in the number of securities in the simulated sample rendered the simulated and the actual samples incomparable. Consequently, we chose to preserve the placement of missing values in scrambling the individual security returns and thus maintain the same set of securities in the simulations that are used in the actual strategy. 10 The average t - values in Table 3, panel A, are always substantially larger than the corresponding t - values of the prots of the actual strategies because of a lack of cross-sectional correlation in the bootstrapped sample. 506 0.3512 (4.52) 0.7199 (5.83) 0.7183 (4.63) 6 months 9 months 12 months t 1.4704 0.8411 0.3775 0.0988 p 24.20 1.00 21.21 0.88 17.53 0.69 11.19 1.00 Panel A E t (k ) t 1.5944 0.9220 0.4041 0.1026 p 20.06 1.00 17.96 0.88 15.56 0.82 10.93 1.00 Panel B E t (k ) t 1.4345 0.8093 0.3655 0.0935 p 22.44 1.00 20.34 0.80 16.90 0.60 10.92 1.00 Panel C E t (k ) t 1.1956 0.6608 0.2913 0.0721 1.00 p 21.93 1.00 18.96 0.24 14.81 0.07 8.87 Panel D E t (k ) t 0.0144 0.0127 0.0069 0.0015 0.07 0.00 0.07 0.00 0.26 0.00 1 1 R (k ). The where t (k ) is the dollar prot at time t from a k - period trading strategy, wit 1 (k ) N Rit 1 (k ) Rmt 1 (k ) and Rmt 1 (k ) N i 1 it 1 second column contains estimates of average prots of medium-term momentum strategies implemented on the real data from December 1964 to December 1985. The numbers in parentheses are z-statistics that are asymptotically N (0, 1) under the null hypothesis that true prots are zero and are robust to heteroscedasticity and autocorrelation. The table also contains results of several simulations, each with 500 replications. Panel A contains a bootstrap simulation in which we generate 1-month returns from the sample with replacement and then implement the four medium-term (3- to 12-month) momentum strategies. The panel also contains the t-statistics average of the 500 simulated t-values and the p-values, where these values denote the proportion of times the 500 simulated mean returns are greater than the sample mean prots of the actual strategy shown in the second column. Panels BE contain Monte Carlo simulations. In Panel B we show average prots, average t-values, and the p-values of implementing the trading strategies on randomly sampled 1-month individual security returns from normal distributions that have moments (means and variances) that match the monthly moments of the securities in the sample. Panels C and D contain estimates for trading strategies implemented on randomly sampled monthly returns generated from normal distributions that exclude the extreme 1 and 5, respectively, of the high - and low-mean securities. Panel E provides the average prots, average t-values, and p-values of trading strategies implemented on randomly sampled rms from normal distributions with identical means but variances that match the sample counterparts. All prot estimates are multiplied by 100. N p 0.16 0.05 Panel E E t (k ) This table contains average actual and average simulated prots to zero-cost trading strategies that buy NYSE/AMEX winners and sell losers based on their past N w (k ) Rit (k ) i 1. N. performance relative to the performance of an equal-weighted index of all stocks. The dollar prots are given by t (k ) i 1 it 1 0.0217 (0.60) E t (k ) 3 months Strategy Interval Table 3 Average prots of actual and simulated medium-term trading strategies An Anatomy of Trading Strategies 507 The Review of Financial Studies / v 11 n 3 1998 between 1964 and 1989 alone have the potential to explain the prots of momentum strategies.11 An interesting and important aspect of the bootstrap results is the relation between the average prots of the momentum strategies and their holding periods. Specically, consistent with the prediction of the random walk model, the prots increase geometrically with the holding period, k see Equation (8) and the discussion in Section 2.2. Given the mean returns for the basic monthly measurement interval (i. e. for k 1), the relation between the average prots of the 3-month versus the 6-month and 12month strategies is virtually identical to the predictions of the random walk model: starting with an average prot of 0.099 for the 3-month strategy, there is a geometric increase to 0.378, 0.841, and 1.470 for the 6-, 9-, and 12-month strategies, respectively. This is in sharp contrast for the average prots for the real strategies reported in the second column, which increase with the holding period, but less than geometrically, and eventually exhibit no change between the 9- and 12-month strategies. This behavior in turn suggests the presence of price reversals, and not momentum, in the real data. The bootstrap results appear to conrm the ndings of the empirical decomposition of the real prots presented in Table 2. Since we do not estimate any parameters of individual-security returns in the bootstrap tests, these results should be devoid of measurement errors in mean returns present in the empirical decomposition. 3.2 Monte Carlo evidence We also conduct Monte Carlo simulations in which returns of individual securities are sampled from normal, independent, and identical distributions with moments that match the moments of the securities used in the trading strategy. We conduct these experiments for two reasons: (i) to ensure that individual security returns do not contain any time-series correlations, and (ii) to check the sensitivity of the empirical decomposition in Table 2 to measurement errors in mean returns (or specically to the extreme mean returns observed in the real data). 11 Note that all estimates in Tables 2 and 3 are prots and not returns because the strategies are zeroinvestment strategies see Equation (2a). Under the null hypothesis that stock returns follow random walks, however, the prots from the actual and simulated strategies are directly comparable. This follows because the expected value of dollar investment long (or short) see Equation (2b) is the same since it depends on the unconditional means of the returns which, in turn, are the same in the actual and each of the simulated strategies. Karolyi and Kho (1993) also conduct bootstrap and Monte Carlo experiments on momentum strategies. They simulate or shufe monthly returns to examine 6-month strategies and, like Jegadeesh and Titman (1993), rank stocks on the basis of past returns and buy (sell) equal-weighted decile portfolios of the highest (lowest) return securities. Although this portfolio method differs from ours, their results also suggest that cross-sectional variation in mean returns is importantthey nd that the average prots of the simulated zero-investment strategy, though less than the actual prots, still constitute about 80 of prots in the real data. Recall that we do not implement the Jegadeesh and Titman (1993) strategy because it does not lend itself readily to the decomposition analysis that is our main focus see the discussion in Section 1. 508 An Anatomy of Trading Strategies In the Monte Carlo simulations we generate 1-month individual security returns from independent and identical normal distributions that have means and variances that are identical to those observed in the real data. We simulate 500 such monthly series and implement the momentum trading strategy for the 3- to 12-month intervals for each set of returns. Table 3, panel B, contains the average prots, the average t - statistics, and the p - values denoting the proportion of times the 500 simulated mean returns are greater than the corresponding sample mean returns in the second column of the table. The results of this Monte Carlo experiment are similar to the bootstrap evidence in panel A. The mean prots are all greater than those witnessed in the real data, and the p - values range between 0.82 and 1.00, suggesting that the cross-sectional characteristics of the data could generate the prots of the momentum strategies. And again, the difference between the average prots of the simulated and real strategies increases signicantly with an increase in the holding period, implying that there are reversals in the real data at least at the 9- and 12-month horizons. The Monte Carlo simulations therefore suggest that in-sample crosssectional differences in individual security returns can account for the profitability of medium-term momentum strategies. To determine the robustness of the protability of the simulated strategies to extreme mean returns observed in the data, we conduct two additional Monte Carlo experiments. In these two simulations, we exclude individual securities that have extreme means (both positive and negative) from the entire simulated samples, that is, we exclude 1 and 5, respectively, of the securities based on the magnitudes of their estimated mean returns. This has the effect of reducing the estimated cross-sectional variance of mean returns of individual securities. It also provides a means of checking the sensitivity of our results to estimation error in the mean returns, since it is possible that the extreme means of individual security returns observed in the real data are an outcome of measurement errors rather than being true extreme means. Ideally a calibration of the underlying cross-sectional distribution of mean returns should be determined by an asset pricing model. However, given the lack of success of theoretical asset pricing models like the CAPM to explain the cross-section of required returns, we do not attempt such an exercise. Our simulation analysis is similar in spirit to the work of Knez and Ready (1997), who show that the size effect can be explained by 1 of the outliers in the data. The evidence from the Monte Carlo experiments that exclude 1 and 5 of the extreme-mean securities is shown in Table 3, panels C and D, respectively. These results show that excluding 1 and 5 of the extreme-mean securities from the simulation lowers the average prots at all horizons, but it does not change the basic conclusion that the success of the momentum strategies can be accounted for by cross-sectional differences in mean re - 509 The Review of Financial Studies / v 11 n 3 1998 turns of individual securities. In panel C, none of the mean prots are less than the corresponding real numbers reported in the second column of the table, and the p - values remain large, ranging from 0.60 to 1.00. In panel D, the mean prots are about 20 and 10 lower than the real prots for the 6- and 9-month strategies, respectively, but the p - values remain relatively high at 0.07 and 0.24. For the 3- and 12-month strategies the average simulated prots are substantially higher than the corresponding real prots, with p - values of 1.00. Finally, we attempt to determine whether there are any biases inherent in the Monte Carlo simulations by simulating individual security returns that have the same variances as the real data, but have identical (zero) means and no time-series relations. We again simulate 500 series of monthly returns for all the securities in our sample and implement the medium-term momentum strategies. The results of this experiment are shown in Table 3, panel E. The prots are invariably positive due to noise, but the magnitudes of the average prots are small: 0.0015, 0.0069, 0.0127, and 0.0144 for the 3- to 12-month strategies, respectively. Also, the p - values are all close to zero. These estimates are between 0.90 and 1.71 of the corresponding Monte Carlo estimates in panel B, which reect all the in-sample cross-sectional variation in mean returns. Moreover, the average t-statistics are also small, ranging between 0.073 and 0.259. Hence, the biases in the simulations appear to have a minor effect on the inferences because the protability of trading strategies is very small if there is no cross-sectional variation in the mean returns of individual securities.12 3.3 Some additional evidence and interpretation The empirical decomposition and the simulation evidence suggest that cross-sectional differences in mean returns could play an important role in determining the protability of return-based trading strategies. In this section, we provide some additional evidence and interpretation that may shed more light on this issue. The problem with the empirical decomposition is that it is based on estimates of the mean returns of individual securities that are measured with error in nite samples (see Appendix). The small-sample bias is potentially important, especially for longer horizons because we use k - period returns 12 We conduct another set of tests to check the robustness of our ndings. Specically we sort securities based on their betas before implementing the trading strategies. The prots of strategies implemented on securities sorted by beta should reduce the cross-sectional variation in mean returns, 2 (k ), and should also simultaneously increase our ability to highlight or emphasize the role (if any) of price continuations or reversals in generating prots for trading strategies. The most important general nding for the beta-sorted strategies is that, although there is a substantial reduction in the point estimates of the cross-sectional dispersion in mean returns for most holding periods, 2 (k ) continues to have an important effect on the prots of trading strategies. The contribution of 2 (k ) is again always statistically different from zero. Moreover, as in Table 2, 2 (k ) contributes high percentages of the prots of momentum strategies and it also continues to result in large losses to contrarian strategies. 510 An Anatomy of Trading Strategies to calculate the k - period mean returns (that is, 12-month returns are used to calculate 12-month mean returns). To assess the effects of small samples on the empirical decomposition of trading prots presented in Table 2, we now provide estimates of the cross-sectional variance in mean returns for all horizons in italics by alternative estimates of the cross-sectional variance in weekly mean returns. Since the number of weekly observations are large (up to 1,434 for the 19621989 period), the effects of measurement errors in mean returns on estimates of the cross-sectional variance in mean returns should be small (see Appendix). The implied estimate of the cross-sectional variance in mean returns for different horizons are calculated using the following formula see Equation (8): 2 (k ) n 2 2 (weekly) (9) where n is the number of weeks in the holding period, k. of the trading strategy (k 3 months. 36 months). The implied cross-sectional variances in Equation (9) are obtained under the assumption that returns follow stationary processes. We use three alternative samples to calculate the weekly cross-sectional variance in mean returns used in Equation (9). The rst sample is the survivor-sample of 512 rms that have no missing weekly returns during the entire 19621989 period. The cross-sectional variance in mean returns of these rms is likely to have little bias since each rms mean return is calculated using 1,434 observations (see Appendix). The weekly estimate of the cross-sectional variance reported in Table 4 is 0.000087, which is virtually identical to the estimate of 0.00009 reported in Lo and Mackinlay (1990) based on a sample of 551 survived rms for the 19621987 period, each with 1,330 observations. Although the cross-sectional variance of the mean returns of rms in this sample is likely to be measured with reasonable accuracy, it is also likely to provide a lower bound on the cross-sectional variance of the mean returns of the rms used in our trading strategies. Firms that survive the entire sample are likely to be large rms with similar mean returns and, in any event, a real time trading strategy could not be implemented on such a set of survived rms. The second sample of rms used to calculate the weekly cross-sectional variance in mean returns is the limited sample of 2,111 rms that have transaction prices for at least half (that is, 717 weeks) of the sample period. The bias in the cross-sectional variance of the mean returns of the individual securities in this sample should also be relatively small. On the other hand, however, the true cross-sectional variance in mean returns in this sample should again be less than the sample of rms used in the trading strategies. The estimated cross-sectional variance in mean returns of this sample (see Table 4) is 0.000140, which is 60 larger than the corresponding estimate for the survived sample. Some of this increase may be due to the increased effects of measurement errors, 511 The Review of Financial Studies / v 11 n 3 1998 Table 4 Implied cross-sectional variation in mean returns for different horizons based on weekly estimates Strategy interval 1 week 3 months 6 months 9 months 12 months Survivor rms ( N 512) Limited rms ( N 2,111) All rms ( N 6,524) 0.000087 0.014703 (21,56) 0.058812 (16,40) 0.132327 (19,48) 0.235248 (34,119) 0.000140 0.023660 (34,89) 0.094640 (26,64) 0.212940 (30,77) 0.378560 (54,191) 0.000535 0.090415 (129,333) 0.361660 (100,246) 0.813735 (115,295) 1.446640 (207,732) This table contains the implied cross-sectional variation in mean returns for trading strategies of different horizons based on weekly estimates for three alternative samples. The weekly cross-sectional variation of the survivor-rm sample is based on a set of 512 rms that have no missing returns during the 19621989 period, and each individual-securitys mean return is calculated using 1,434 observations. The cross-sectional variation of the limited-rm sample is based on a set of 2,111 rms that had a minimum of 717 returns during the 19621989 period, while the estimate for the all rm sample is based on all 6,524 rms that are included in the trading strategy reported in Table 1. The implied estimates of the cross-sectional variation in mean returns are obtained as 2 (k ) n 2 2 (week), where n is the number of weeks in the holding period, k. of the trading strategy. All estimates of the cross-sectional variation in weekly mean returns are multiplied by 100. The numbers in parentheses below the implied estimates of the cross-sectional variation in mean returns are the minimum and maximum percentages of the prots of trading strategies reported in Table 1 for different time periods that can be explained by the implied estimates. but some of it is likely to be due to larger differences in the true means of individual securities. The third weekly estimate of the cross-sectional variance in mean returns is 0.000535, which is for the all-rm sample of 6,524 rms used in the actual weekly trading strategy reported in Table 1. This estimate is six (four) times larger than the corresponding estimate for the survived sample (limited sample). Table 4 contains the implied cross-sectional variances in mean returns for all three samples of rms for the medium horizons (3 to 12 months). We do not report the implied estimates for the long horizons (18 to 36 months) because, even without adjusting for the cross-sectional variation in mean returns, there appear to be reversals in the long run. The numbers in parentheses below the implied cross-sectional variance in mean returns are the minimum and maximum percentages of the actual prots of trading strategies reported in Table 1 that can be explained by them. For example, the implied cross-sectional variances of 0.094640 at the 6-month horizon for the limited-rm sample can explain a minimum of 26 of the prots of the 512 An Anatomy of Trading Strategies 6-month momentum strategy in the 19621989 period, and a maximum of 64 of the prots of the 6-month trading strategy in the 19261989 period. The results in Table 4 shed some light on the relative importance of the cross-sectional variance in mean returns in determining the protability of trading strategies. Consider the most conservative estimates of the crosssectional variation in mean returns based on the survived sample. The implied cross-sectional variance can explain between 16 and 119 of medium horizon strategies implemented over the various time periods. The corresponding percentages are 26 and 191 for the limited-rms sample. Finally, the implied cross-sectional variation in mean returns for the all-rms sample is sufcient to explain the prots of all medium-horizon strategies. Hence, even if we rely solely on the most conservative estimates in Table 4, the evidence suggests that cross-sectional differences in the mean returns of securities included in trading strategies could play a nontrivial role in determining the protability of these strategies. 4. Conclusion We present an analysis of trading strategies that rely on time-series patterns in security returns. We implement the two most commonly suggested strategiesmomentum and contrarianat eight different horizons and during several different time periods. We show that less than 50 of the 120 strategies implemented in this article yield statistically signicant prots and, unconditionally, momentum and contrarian strategies are equally likely to be successful. However, there are two systematic patterns that emerge. First, the momentum strategy usually nets positive and statistically significant prots at medium horizons, except during the 19261947 subperiod. Second, the contrarian strategy is successful at long horizons, but the profits to these strategies are statistically signicant only during the 19261947 subperiod. We nd that an important determinant of the protability of trading strategies is the estimated cross-sectional dispersion in the mean returns of individual securities comprising the portfolios used to implement these strategies. This cross-sectional variance is not related to the time-series patterns in returns that form the basis of return-based trading strategies. Specically, the cross-sectional dispersion in mean returns witnessed during different time periods can potentially generate the observed prots of the most consistently protable strategy, the momentum strategy implemented at medium horizons. Our ndings, based on the empirical decomposition of prots, bootstrap and Monte Carlo simulations, and alternative estimates based on weekly returns, suggest that cross-sectional differences in mean returns play a nontrivial role in determining the protability of momentum strategies. On the other hand, although there is substantial and statistically reliable evidence of price reversals, the net prots to contrarian strategies are statis - 513 The Review of Financial Studies / v 11 n 3 1998 tically signicant primarily during one subperiod: 19261947. In all other subperiods, the consistently signicant prots from price reversals are (statistically) neutralized at least in part by the losses due to the cross-sectional dispersion in the mean returns of securities included in the strategy. These losses again appear to have no relation to time-series patterns in security returns that form the basis of trading strategies they occur because a contrarian strategy on average involves the purchase of low-mean securities from the proceeds of the sale of high-mean securities. The results of our article are clearly dependent on the assumption that the mean returns of individual securities are constant during the periods in which the trading strategies are implemented. However, our results raise the intriguing possibility that the cross-sectional variation in mean returns can simultaneously account for the prots of momentum strategies and the typical lack of success of contrarian strategies. This nding in itself may raise questions about the protability of trading strategies and the related, and more signicant, issue about the informational efciency of stock prices. Obviously, different specications of the model for unconditional required returns could affect the conclusions of our analysis. Several recent attempts at explaining the momentum effect are being made along these lines, but with mixed results see, e. g. Fama and French (1996) and Moskowitz (1997). It is also possible that more plausible models of time-varying expected returns could provide deeper insights into the potential sources of the prots of momentum strategies see, e. g. Grundy and Martin (1997) and Karolyi and Kho (1993). Appendix A.1 Estimation of the components of prots The components of total prots see Equation (4) are estimated by allowing serial covariances (both own and cross) and the cross-sectional variance of mean returns of individual securities to be time dependent. Specically, C1 (k ) T (k ) 1 C1t (k ), T (k ) 1 t (k )2 where C1t Rmt (k ) Rmt 1 (k ) 2 1 (k ) mt O1 (k ) 514 1 2 N N i 1 Rit (k ) Rit 1 (k ) it 1 (k ) 2 T (k ) 1 O1t (k ), T (k ) 1 t (k )2 An Anatomy of Trading Strategies where O1t N 1 N2 N i 1 Rit (k ) Rit 1 (k ) it()1 (k ) 2 k and 2 (k ) T (k ) 1 2 (k ), T (k ) 1 t (k )2 t where t2 (k ) 1 N N it 1 (k ) mt 1 (k )2 i 1 and T (k ) total number of overlapping returns in the sample period for a trading strategy based on holding period k. For ease of exposition, we do not have a security-related subscript on T (k ), but each security in the trading strategy will have a different number of observations. In calculating the components of the prots to trading strategies, we assume that individual security returns are mean stationary, and we calculate all sample means of security returns for each holding period k. i (k ), us ing overlapping data over the entire sample period. The t 1 subscript on it 1 (k ) and mt 1 (k ) simply denotes that these are the sample means of securities available at time t 1 to form the trading strategy portfolios see Equation (1). The only reason the mean returns of individual securities change at each portfolio formation time t 1 is because the securities included in each strategy in each period themselves change and, consequently, the mean return of the portfolio of all these securities, m (k ), also changes. Therefore, although we require mean stationarity, estimates of all components of the prots/losses of trading strategies are time dependent. The use of the entire sample period to calculate the mean returns of individual securities, as opposed to calculating the means based on a rolling sample of data up to time t 1, should reduce the estimates of the cross-sectional variance in mean returns because each mean is estimated more precisely. The assumption of mean stationarity does not appear to affect our main inferences because they are robust across different sample periods. Finally, note that the minor differences between the population parameters C1 (k ) and O1 (k ) in Equation (4) and their sample counterparts are reected above in the last element of C1t (k ) and the NN 21 factor in O1t (k ). The estimators are calculated slightly differently so that C1 (k ) depends en tirely on cross-serial covariances, while O1 (k ) depends solely on own-serial covariances see also Lo and MacKinlay (1990). 515 The Review of Financial Studies / v 11 n 3 1998 A.2 Small-sample biases in estimators of the components of prots It can be shown that since sample means are estimated with error in small samples, covariance estimators are downward biased see Fuller (1976). Consequently, the estimators of the components of total prots see Equation (4) are biased in small samples. Specically, E C1 (k ) C1 (k ) E O1 (k ) O1 (k ) and E 2 (k ) 2 mt 1 (k ). T (k ) N 2 i 1 it 1 (k ) 1 N 2 (k ) T (k ) 1 N. N i 1 2 2 it 1 (k ) mt 1 (k ) . T (k ) T (k ) where T (k ) is the number of returns of holding period k used to calculate trading prots, i2 is the population variance of an individual securitys 2 return, and m is the population variance of the return of the equal-weighted portfolio of all securities used in the trading strategy portfolio. Finally, it is important to note that all the small-sample biases noted above are derived under the null hypothesis that returns are independently and identically distributed. An interesting aspect of the above analysis is that the biases in the com ponents offset each other. Also, note that C1 (k ) and O1 (k ) are downward N 1 2 2 biased, but since N i 1 it 1 ampgt mt 1. the downward bias in O1 (k ) 1 (k ). For a momentum strategy, is greater than the downward bias in C therefore, in small samples 2 (k ) will be upward biased while the predictability-protability index, P (k ), will be downward biased by the same magnitude. This bias could be nontrivial in small samples which, in turn, could materially affect inferences about the relative importance of the sources of prots to trading strategies. Since the bias disappears as T (k ) , however, we use overlapping holding period returns at the monthly frequency for each trading strategy (except the 1-week strategy). It is important to note that the entire discussion of the small-sample biases is based on the assumption that true returns are serially uncorrelated. This assumption will result in an over - (or under-) estimate of the bias if returns are negatively (positively) serially correlated. 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