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<feed xmlns="http://www.w3.org/2005/Atom"><title>Raul E. Lopez Briega</title><link href="http://relopezbriega.github.io/" rel="alternate"></link><link href="/feeds/finanzas.atom.xml" rel="self"></link><id>http://relopezbriega.github.io/</id><updated>2016-09-26T00:00:00-03:00</updated><entry><title>Series de tiempo con Python</title><link href="http://relopezbriega.github.io/blog/2016/09/26/series-de-tiempo-con-python/" rel="alternate"></link><published>2016-09-26T00:00:00-03:00</published><author><name>Raul E. Lopez Briega</name></author><id>tag:relopezbriega.github.io,2016-09-26:blog/2016/09/26/series-de-tiempo-con-python/</id><summary type="html">&lt;p&gt;
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&lt;p&gt;&lt;img alt="Series de tiempo con Python" title="Series de tiempo con Python" src="http://relopezbriega.github.io/images/timeSeries.png" high=400px width=600px&gt;&lt;/p&gt;

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&lt;h2 id="Introducci&amp;#243;n"&gt;Introducci&amp;#243;n&lt;a class="anchor-link" href="#Introducci&amp;#243;n"&gt;&amp;#182;&lt;/a&gt;&lt;/h2&gt;&lt;p&gt;Los datos obtenidos a partir de observaciones recogidas a lo largo del tiempo son extremadamente comunes. En los negocios, observamos las tasas de interés de la semana, los precios de cierre de las acciones diarios, los índices de precios mensuales, las cifras de ventas anuales, y así sucesivamente. En meteorología, observamos las temperaturas máximas y mínimas diarias, los índices anuales de precipitación y de sequía, y las velocidades del viento por hora. En la agricultura, registramos las cifras anuales de producción agrícola y ganadera, la erosión del suelo, y las ventas de exportación. En las ciencias biológicas, observamos la actividad eléctrica del corazón en intervalos de milisegundos. La lista de las áreas en las que se estudian las &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt; es prácticamente interminable.&lt;/p&gt;
&lt;h2 id="&amp;#191;Qu&amp;#233;-es-una-serie-de-tiempo?"&gt;&amp;#191;Qu&amp;#233; es una serie de tiempo?&lt;a class="anchor-link" href="#&amp;#191;Qu&amp;#233;-es-una-serie-de-tiempo?"&gt;&amp;#182;&lt;/a&gt;&lt;/h2&gt;&lt;p&gt;Una &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;serie de tiempo&lt;/a&gt; o &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;serie temporal&lt;/a&gt; es una secuencia de datos, observaciones o valores, medidos en determinados momentos y ordenados cronológicamente. Los datos pueden estar espaciados a intervalos iguales o desiguales. Una vez que se captura una &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;serie de tiempo&lt;/a&gt;, a menudo se realiza un análisis sobre ella para identificar patrones en los datos, en esencia, lo que se busca es entender que suceda a medida que el tiempo va avanzando. Ser capaz de procesar datos de &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt; es una habilidad esencial en el mundo moderno. Uno de los usos más habituales de las &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt; es su análisis para predicción y pronóstico (así se hace por ejemplo con los datos climáticos, las acciones de bolsa, o las series de datos demográficos). Resulta difícil imaginar una rama de las ciencias en la que no aparezcan datos que puedan ser considerados como &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt;.&lt;/p&gt;
&lt;h3 id="&amp;#191;Qu&amp;#233;-las-hace-especiales?"&gt;&amp;#191;Qu&amp;#233; las hace especiales?&lt;a class="anchor-link" href="#&amp;#191;Qu&amp;#233;-las-hace-especiales?"&gt;&amp;#182;&lt;/a&gt;&lt;/h3&gt;&lt;p&gt;Las características que hacen a las &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt; especiales y las diferencia de, por ejemplo, un problema de &lt;a href="https://es.wikipedia.org/wiki/An%C3%A1lisis_de_la_regresi%C3%B3n"&gt;regresión&lt;/a&gt; son las siguientes:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;&lt;p&gt;Son &lt;strong&gt;&lt;em&gt;dependientes del tiempo&lt;/em&gt;&lt;/strong&gt;; por lo tanto el supuesto básico de los modelos de &lt;a href="https://es.wikipedia.org/wiki/An%C3%A1lisis_de_la_regresi%C3%B3n"&gt;regresión&lt;/a&gt; de que las observaciones son &lt;em&gt;independientes&lt;/em&gt; no se sostiene en este caso.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;&lt;p&gt;Suelen tener una &lt;strong&gt;&lt;em&gt;tendencia&lt;/em&gt;&lt;/strong&gt;; la mayoría de las &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt; suelen tener algún tipo de tendencia de estacionalidad, es decir, las variaciones propias de un período de tiempo determinado.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;&lt;p&gt;Suelen estar &lt;strong&gt;&lt;em&gt;autocorrelacionadas&lt;/em&gt;&lt;/strong&gt;; la mayoría de los procesos físicos presentan una inercia y no cambian tan rápidamente. Esto, combinado con la frecuencia del muestreo, a menudo hace que las observaciones consecutivas estén &lt;a href="https://es.wikipedia.org/wiki/Correlaci%C3%B3n"&gt;correlacionadas&lt;/a&gt;. Esta &lt;a href="https://es.wikipedia.org/wiki/Correlaci%C3%B3n"&gt;correlación&lt;/a&gt; entre observaciones consecutivas se llama &lt;em&gt;&lt;a href="https://es.wikipedia.org/wiki/Autocorrelaci%C3%B3n"&gt;autocorrelación&lt;/a&gt;&lt;/em&gt;. Cuando los datos están &lt;em&gt;&lt;a href="https://es.wikipedia.org/wiki/Autocorrelaci%C3%B3n"&gt;autocorrelacionados&lt;/a&gt;&lt;/em&gt;, la mayoría de los métodos estadísticos estándares basados en la suposición de observaciones independientes pueden arrojar resultados engañosos o incluso ser inútiles.&lt;/p&gt;
&lt;/li&gt;
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&lt;h2 id="Series-de-tiempo-estacionarias"&gt;Series de tiempo estacionarias&lt;a class="anchor-link" href="#Series-de-tiempo-estacionarias"&gt;&amp;#182;&lt;/a&gt;&lt;/h2&gt;&lt;p&gt;Un tipo muy importante de &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt; son las &lt;a href="https://es.wikipedia.org/wiki/Proceso_estacionario"&gt;series de tiempo estacionarias&lt;/a&gt;. Una &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt; se dice que es estrictamente &lt;a href="https://es.wikipedia.org/wiki/Proceso_estacionario"&gt;estacionaria&lt;/a&gt; si sus propiedades no son afectadas por los cambios a lo largo del tiempo. Es decir, que se deberían cumplir tres criterios básicos para poder considerar a una &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt; como &lt;a href="https://es.wikipedia.org/wiki/Proceso_estacionario"&gt;estacionaria&lt;/a&gt;:&lt;/p&gt;
&lt;ol&gt;
&lt;li&gt;&lt;strong&gt;La &lt;a href="https://es.wikipedia.org/wiki/Media_aritm%C3%A9tica"&gt;media&lt;/a&gt; de la &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;serie&lt;/a&gt; no debe ser una función de tiempo&lt;/strong&gt;; sino que debe ser constante. La siguiente imagen muestra una &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;serie&lt;/a&gt; que cumple con esta condición y otra que no la cumple.
&lt;img alt="Series de tiempo con Python" title="media estacionaria - no estacionaria" src="http://relopezbriega.github.io/images/Mean_stationary.png" high=300px width=500px&gt;&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;La &lt;a href="https://es.wikipedia.org/wiki/Varianza"&gt;varianza&lt;/a&gt; de la &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;serie&lt;/a&gt; no debe ser una función del tiempo&lt;/strong&gt;. El siguiente gráfico representa una &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;serie&lt;/a&gt; cuya &lt;a href="https://es.wikipedia.org/wiki/Varianza"&gt;varianza&lt;/a&gt; no esta afectada por el tiempo (es &lt;a href="https://es.wikipedia.org/wiki/Proceso_estacionario"&gt;estacionaria&lt;/a&gt;) y otra que no cumple con esa condición.
&lt;img alt="Series de tiempo con Python" title="media estacionaria - no estacionaria" src="http://relopezbriega.github.io/images/Var_stationary.png" high=300px width=500px&gt;&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;La &lt;a href="https://es.wikipedia.org/wiki/Covarianza"&gt;covarianza&lt;/a&gt; de la &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;serie&lt;/a&gt; no debe ser una función del tiempo&lt;/strong&gt;. En el gráfico de la derecha, se puede observar que la propagación de la &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;serie&lt;/a&gt; se va encogiendo a medida que aumenta el tiempo. Por lo tanto, la &lt;a href="https://es.wikipedia.org/wiki/Covarianza"&gt;covarianza&lt;/a&gt; no es constante en el tiempo para la &lt;em&gt;serie roja&lt;/em&gt;.
&lt;img alt="Series de tiempo con Python" title="media estacionaria - no estacionaria" src="http://relopezbriega.github.io/images/Cov_stationary.png" high=300px width=500px&gt;&lt;/li&gt;
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&lt;h3 id="&amp;#191;Por-qu&amp;#233;-son-importantes-las-series-de-tiempo-estacionarias?"&gt;&amp;#191;Por qu&amp;#233; son importantes las series de tiempo estacionarias?&lt;a class="anchor-link" href="#&amp;#191;Por-qu&amp;#233;-son-importantes-las-series-de-tiempo-estacionarias?"&gt;&amp;#182;&lt;/a&gt;&lt;/h3&gt;&lt;p&gt;La razón por la que estas &lt;a href="https://es.wikipedia.org/wiki/Proceso_estacionario"&gt;series&lt;/a&gt; son importantes es que la mayoría de los modelos de &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt; funcionan bajo el supuesto de que la serie es &lt;a href="https://es.wikipedia.org/wiki/Proceso_estacionario"&gt;estacionaria&lt;/a&gt;. Intuitivamente, podemos suponer que si una &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;serie&lt;/a&gt; tiene un comportamiento particular en el tiempo, hay una &lt;a href="http://relopezbriega.github.io/blog/2015/06/27/probabilidad-y-estadistica-con-python/"&gt;probabilidad&lt;/a&gt; muy alta de que se comportamiento continúe en el futuro. Además, las teorías relacionadas con las &lt;a href="https://es.wikipedia.org/wiki/Proceso_estacionario"&gt;series estacionarias&lt;/a&gt; son más maduras y más fáciles de implementar en comparación con series no estacionarias. A pesar de que el supuesto de que la &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;serie&lt;/a&gt; es &lt;a href="https://es.wikipedia.org/wiki/Proceso_estacionario"&gt;estacionaria&lt;/a&gt; se utiliza en muchos modelos, casi ninguna de las &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt; que encontramos en la práctica son &lt;a href="https://es.wikipedia.org/wiki/Proceso_estacionario"&gt;estacionarias&lt;/a&gt;. Por tal motivo la &lt;a href="http://relopezbriega.github.io/blog/2015/06/27/probabilidad-y-estadistica-con-python/"&gt;estadística&lt;/a&gt; tuvo que desarrollar varias técnicas para hacer &lt;a href="https://es.wikipedia.org/wiki/Proceso_estacionario"&gt;estacionaria&lt;/a&gt;, o lo más cercano posible a &lt;a href="https://es.wikipedia.org/wiki/Proceso_estacionario"&gt;estacionaria&lt;/a&gt;, a una &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;serie&lt;/a&gt;.&lt;/p&gt;
&lt;h2 id="Series-de-tiempo-con-Python"&gt;Series de tiempo con Python&lt;a class="anchor-link" href="#Series-de-tiempo-con-Python"&gt;&amp;#182;&lt;/a&gt;&lt;/h2&gt;&lt;p&gt;Las principales librerías que nos ofrece &lt;a href="http://python.org/"&gt;Python&lt;/a&gt; para trabajar con &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt; son:&lt;/p&gt;
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&lt;li&gt;&lt;strong&gt;&lt;a href="http://statsmodels.sourceforge.net/stable/tsa.html"&gt;Statsmodels&lt;/a&gt;&lt;/strong&gt;: Esta librería contiene muchos objetos y funciones de suma utilidad para el análisis de &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt;. Algunos de los modelos que están cubiertos por &lt;a href="http://statsmodels.sourceforge.net/stable/tsa.html"&gt;Statsmodels&lt;/a&gt; incluyen: el &lt;a href="https://es.wikipedia.org/wiki/Modelo_autorregresivo"&gt;modelo autorregresivo&lt;/a&gt; (AR); el &lt;a href="https://en.wikipedia.org/wiki/Vector_autoregression"&gt;modelo autorregresivo de vectores&lt;/a&gt; (VAR); y el &lt;a href="https://es.wikipedia.org/wiki/Modelo_autorregresivo_de_media_m%C3%B3vil"&gt;modelo autorregresivo de media móvil&lt;/a&gt; (ARMA).  También incluye funciones de &lt;a href="http://relopezbriega.github.io/blog/2015/06/27/probabilidad-y-estadistica-con-python/"&gt;estadística descriptiva&lt;/a&gt; de &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt;, como por ejemplo la &lt;em&gt;&lt;a href="https://es.wikipedia.org/wiki/Autocorrelaci%C3%B3n"&gt;autocorrelación&lt;/a&gt;&lt;/em&gt;, así como las correspondientes propiedades teóricas de &lt;a href="https://es.wikipedia.org/wiki/Modelo_autorregresivo_de_media_m%C3%B3vil"&gt;ARMA&lt;/a&gt; o procesos relacionados. Por último, también ofrece las pruebas estadísticas relacionadas y algunas funciones auxiliares muy útiles.&lt;/li&gt;
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&lt;li&gt;&lt;strong&gt;&lt;a href="http://pandas.pydata.org/"&gt;Pandas&lt;/a&gt;&lt;/strong&gt;: &lt;a href="http://pandas.pydata.org/"&gt;Pandas&lt;/a&gt; proporciona un amplio soporte para trabajar con datos de &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt;. Generalmente cuando trabajamos con &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt; realizamos un amplio abanico de tareas, como: convertir fechas, estandarizar el tiempo de acuerdo a la zona horaria, crear secuencias a determinados intervalos o frecuencias, identificar datos faltantes, desplazar las fechas hacia atrás o hacia adelante por un determinado valor, calcular resúmenes agregados de valores a medida que el tiempo cambia, etc. &lt;a href="http://pandas.pydata.org/"&gt;Pandas&lt;/a&gt; nos brinda las herramientas para poder realizar estas y muchas otras tareas en forma muy sencilla.&lt;/li&gt;
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&lt;p&gt;Veamos algunos ejemplos de como podemos manipular y analizar &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt; con la ayuda de &lt;a href="http://python.org/"&gt;Python&lt;/a&gt;. En este caso, vamos a jugar un poco con la información de los precios de las acciones de &lt;a href="https://www.weatherford.com/en/homepage"&gt;Weatherford&lt;/a&gt; (&lt;a href="https://finance.yahoo.com/quote/wft?ltr=1"&gt;WFT&lt;/a&gt;) de este año.&lt;/p&gt;
&lt;h3 id="Manipulando-la-serie-de-tiempo-con-pandas"&gt;Manipulando la serie de tiempo con pandas&lt;a class="anchor-link" href="#Manipulando-la-serie-de-tiempo-con-pandas"&gt;&amp;#182;&lt;/a&gt;&lt;/h3&gt;
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&lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# importando modulos necesarios&lt;/span&gt;
&lt;span class="kn"&gt;import&lt;/span&gt; &lt;span class="nn"&gt;numpy&lt;/span&gt; &lt;span class="kn"&gt;as&lt;/span&gt; &lt;span class="nn"&gt;np&lt;/span&gt;
&lt;span class="kn"&gt;import&lt;/span&gt; &lt;span class="nn"&gt;pandas&lt;/span&gt; &lt;span class="kn"&gt;as&lt;/span&gt; &lt;span class="nn"&gt;pd&lt;/span&gt;
&lt;span class="kn"&gt;import&lt;/span&gt; &lt;span class="nn"&gt;pandas.io.data&lt;/span&gt; &lt;span class="kn"&gt;as&lt;/span&gt; &lt;span class="nn"&gt;web&lt;/span&gt;
&lt;span class="kn"&gt;import&lt;/span&gt; &lt;span class="nn"&gt;datetime&lt;/span&gt; &lt;span class="kn"&gt;as&lt;/span&gt; &lt;span class="nn"&gt;dt&lt;/span&gt;
&lt;span class="kn"&gt;from&lt;/span&gt; &lt;span class="nn"&gt;pydataset&lt;/span&gt; &lt;span class="kn"&gt;import&lt;/span&gt; &lt;span class="n"&gt;data&lt;/span&gt;
&lt;span class="kn"&gt;import&lt;/span&gt; &lt;span class="nn"&gt;statsmodels.api&lt;/span&gt; &lt;span class="kn"&gt;as&lt;/span&gt; &lt;span class="nn"&gt;sm&lt;/span&gt; 

&lt;span class="c1"&gt;# librerías de visualizaciones&lt;/span&gt;
&lt;span class="kn"&gt;import&lt;/span&gt; &lt;span class="nn"&gt;seaborn&lt;/span&gt; &lt;span class="kn"&gt;as&lt;/span&gt; &lt;span class="nn"&gt;sns&lt;/span&gt;
&lt;span class="kn"&gt;import&lt;/span&gt; &lt;span class="nn"&gt;matplotlib.pyplot&lt;/span&gt; &lt;span class="kn"&gt;as&lt;/span&gt; &lt;span class="nn"&gt;plt&lt;/span&gt; 


&lt;span class="c1"&gt;# graficos incrustados&lt;/span&gt;
&lt;span class="o"&gt;%&lt;/span&gt;&lt;span class="k"&gt;matplotlib&lt;/span&gt; inline

&lt;span class="c1"&gt;# pandas solo 4 decimales&lt;/span&gt;
&lt;span class="n"&gt;pd&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;set_option&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;precision&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;4&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Ejemplo serie de tiempo con Pandas&lt;/span&gt;
&lt;span class="c1"&gt;# Creando una serie de tiempo de las acciones de WFT desde yahoo finance&lt;/span&gt;
&lt;span class="n"&gt;wft&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;web&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;DataReader&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s2"&gt;&amp;quot;WFT&amp;quot;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s1"&gt;&amp;#39;yahoo&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s1"&gt;&amp;#39;2016-1-1&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s1"&gt;&amp;#39;2016-9-30&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;head&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
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      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
    &lt;/tr&gt;
  &lt;/thead&gt;
  &lt;tbody&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-01-04&lt;/th&gt;
      &lt;td&gt;8.40&lt;/td&gt;
      &lt;td&gt;8.70&lt;/td&gt;
      &lt;td&gt;8.29&lt;/td&gt;
      &lt;td&gt;8.64&lt;/td&gt;
      &lt;td&gt;10719400&lt;/td&gt;
      &lt;td&gt;8.64&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-01-05&lt;/th&gt;
      &lt;td&gt;8.67&lt;/td&gt;
      &lt;td&gt;8.80&lt;/td&gt;
      &lt;td&gt;8.13&lt;/td&gt;
      &lt;td&gt;8.26&lt;/td&gt;
      &lt;td&gt;9109100&lt;/td&gt;
      &lt;td&gt;8.26&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-01-06&lt;/th&gt;
      &lt;td&gt;7.94&lt;/td&gt;
      &lt;td&gt;8.16&lt;/td&gt;
      &lt;td&gt;7.84&lt;/td&gt;
      &lt;td&gt;7.91&lt;/td&gt;
      &lt;td&gt;13203200&lt;/td&gt;
      &lt;td&gt;7.91&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-01-07&lt;/th&gt;
      &lt;td&gt;7.69&lt;/td&gt;
      &lt;td&gt;7.83&lt;/td&gt;
      &lt;td&gt;7.34&lt;/td&gt;
      &lt;td&gt;7.34&lt;/td&gt;
      &lt;td&gt;12633800&lt;/td&gt;
      &lt;td&gt;7.34&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-01-08&lt;/th&gt;
      &lt;td&gt;7.48&lt;/td&gt;
      &lt;td&gt;7.55&lt;/td&gt;
      &lt;td&gt;6.86&lt;/td&gt;
      &lt;td&gt;6.97&lt;/td&gt;
      &lt;td&gt;18547500&lt;/td&gt;
      &lt;td&gt;6.97&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[3]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# filtrando sólo del 2016-02-04 al 2016-02-18&lt;/span&gt;
&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;2016-02-04&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;2016-02-18&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;div class="output_wrapper"&gt;
&lt;div class="output"&gt;


&lt;div class="output_area"&gt;
&lt;div class="prompt output_prompt"&gt;Out[3]:&lt;/div&gt;


&lt;div class="output_html rendered_html output_subarea output_execute_result"&gt;
&lt;div&gt;
&lt;table border="1" class="dataframe"&gt;
  &lt;thead&gt;
    &lt;tr style="text-align: right;"&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;Open&lt;/th&gt;
      &lt;th&gt;High&lt;/th&gt;
      &lt;th&gt;Low&lt;/th&gt;
      &lt;th&gt;Close&lt;/th&gt;
      &lt;th&gt;Volume&lt;/th&gt;
      &lt;th&gt;Adj Close&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;Date&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
    &lt;/tr&gt;
  &lt;/thead&gt;
  &lt;tbody&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-04&lt;/th&gt;
      &lt;td&gt;7.10&lt;/td&gt;
      &lt;td&gt;7.82&lt;/td&gt;
      &lt;td&gt;6.99&lt;/td&gt;
      &lt;td&gt;7.39&lt;/td&gt;
      &lt;td&gt;34474500&lt;/td&gt;
      &lt;td&gt;7.39&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-05&lt;/th&gt;
      &lt;td&gt;7.37&lt;/td&gt;
      &lt;td&gt;7.52&lt;/td&gt;
      &lt;td&gt;6.87&lt;/td&gt;
      &lt;td&gt;6.94&lt;/td&gt;
      &lt;td&gt;27775700&lt;/td&gt;
      &lt;td&gt;6.94&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-08&lt;/th&gt;
      &lt;td&gt;6.68&lt;/td&gt;
      &lt;td&gt;6.79&lt;/td&gt;
      &lt;td&gt;6.41&lt;/td&gt;
      &lt;td&gt;6.74&lt;/td&gt;
      &lt;td&gt;17611300&lt;/td&gt;
      &lt;td&gt;6.74&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-09&lt;/th&gt;
      &lt;td&gt;6.60&lt;/td&gt;
      &lt;td&gt;6.72&lt;/td&gt;
      &lt;td&gt;6.07&lt;/td&gt;
      &lt;td&gt;6.34&lt;/td&gt;
      &lt;td&gt;13741100&lt;/td&gt;
      &lt;td&gt;6.34&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-10&lt;/th&gt;
      &lt;td&gt;6.28&lt;/td&gt;
      &lt;td&gt;6.59&lt;/td&gt;
      &lt;td&gt;6.11&lt;/td&gt;
      &lt;td&gt;6.24&lt;/td&gt;
      &lt;td&gt;8623900&lt;/td&gt;
      &lt;td&gt;6.24&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-11&lt;/th&gt;
      &lt;td&gt;6.02&lt;/td&gt;
      &lt;td&gt;6.27&lt;/td&gt;
      &lt;td&gt;5.74&lt;/td&gt;
      &lt;td&gt;6.06&lt;/td&gt;
      &lt;td&gt;17133900&lt;/td&gt;
      &lt;td&gt;6.06&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-12&lt;/th&gt;
      &lt;td&gt;6.14&lt;/td&gt;
      &lt;td&gt;6.66&lt;/td&gt;
      &lt;td&gt;6.06&lt;/td&gt;
      &lt;td&gt;6.47&lt;/td&gt;
      &lt;td&gt;13498600&lt;/td&gt;
      &lt;td&gt;6.47&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-16&lt;/th&gt;
      &lt;td&gt;6.66&lt;/td&gt;
      &lt;td&gt;6.74&lt;/td&gt;
      &lt;td&gt;6.33&lt;/td&gt;
      &lt;td&gt;6.62&lt;/td&gt;
      &lt;td&gt;11453500&lt;/td&gt;
      &lt;td&gt;6.62&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-17&lt;/th&gt;
      &lt;td&gt;6.70&lt;/td&gt;
      &lt;td&gt;7.13&lt;/td&gt;
      &lt;td&gt;6.55&lt;/td&gt;
      &lt;td&gt;6.72&lt;/td&gt;
      &lt;td&gt;29061300&lt;/td&gt;
      &lt;td&gt;6.72&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-18&lt;/th&gt;
      &lt;td&gt;6.95&lt;/td&gt;
      &lt;td&gt;6.96&lt;/td&gt;
      &lt;td&gt;6.22&lt;/td&gt;
      &lt;td&gt;6.51&lt;/td&gt;
      &lt;td&gt;13587900&lt;/td&gt;
      &lt;td&gt;6.51&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[4]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# valores al 2016-02-16&lt;/span&gt;
&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;loc&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;2016-2-16&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;div class="output_wrapper"&gt;
&lt;div class="output"&gt;


&lt;div class="output_area"&gt;
&lt;div class="prompt output_prompt"&gt;Out[4]:&lt;/div&gt;



&lt;div class="output_text output_subarea output_execute_result"&gt;
&lt;pre class="ipynb"&gt;Open         6.6600e+00
High         6.7400e+00
Low          6.3300e+00
Close        6.6200e+00
Volume       1.1454e+07
Adj Close    6.6200e+00
Name: 2016-02-16 00:00:00, dtype: float64&lt;/pre&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[5]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# valor de la columna Adj Close al 2016-2-16&lt;/span&gt;
&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Adj Close&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;][&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;2016-2-16&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;div class="output_wrapper"&gt;
&lt;div class="output"&gt;


&lt;div class="output_area"&gt;
&lt;div class="prompt output_prompt"&gt;Out[5]:&lt;/div&gt;



&lt;div class="output_text output_subarea output_execute_result"&gt;
&lt;pre class="ipynb"&gt;6.6200000000000001&lt;/pre&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[6]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# filtrando todo febrero de 2016&lt;/span&gt;
&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;2016-2&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;div class="output_wrapper"&gt;
&lt;div class="output"&gt;


&lt;div class="output_area"&gt;
&lt;div class="prompt output_prompt"&gt;Out[6]:&lt;/div&gt;


&lt;div class="output_html rendered_html output_subarea output_execute_result"&gt;
&lt;div&gt;
&lt;table border="1" class="dataframe"&gt;
  &lt;thead&gt;
    &lt;tr style="text-align: right;"&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;Open&lt;/th&gt;
      &lt;th&gt;High&lt;/th&gt;
      &lt;th&gt;Low&lt;/th&gt;
      &lt;th&gt;Close&lt;/th&gt;
      &lt;th&gt;Volume&lt;/th&gt;
      &lt;th&gt;Adj Close&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;Date&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
    &lt;/tr&gt;
  &lt;/thead&gt;
  &lt;tbody&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-01&lt;/th&gt;
      &lt;td&gt;6.51&lt;/td&gt;
      &lt;td&gt;6.67&lt;/td&gt;
      &lt;td&gt;5.90&lt;/td&gt;
      &lt;td&gt;6.33&lt;/td&gt;
      &lt;td&gt;36665900&lt;/td&gt;
      &lt;td&gt;6.33&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-02&lt;/th&gt;
      &lt;td&gt;6.12&lt;/td&gt;
      &lt;td&gt;6.16&lt;/td&gt;
      &lt;td&gt;5.82&lt;/td&gt;
      &lt;td&gt;5.97&lt;/td&gt;
      &lt;td&gt;21091100&lt;/td&gt;
      &lt;td&gt;5.97&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-03&lt;/th&gt;
      &lt;td&gt;6.04&lt;/td&gt;
      &lt;td&gt;6.40&lt;/td&gt;
      &lt;td&gt;5.60&lt;/td&gt;
      &lt;td&gt;6.27&lt;/td&gt;
      &lt;td&gt;24870400&lt;/td&gt;
      &lt;td&gt;6.27&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-04&lt;/th&gt;
      &lt;td&gt;7.10&lt;/td&gt;
      &lt;td&gt;7.82&lt;/td&gt;
      &lt;td&gt;6.99&lt;/td&gt;
      &lt;td&gt;7.39&lt;/td&gt;
      &lt;td&gt;34474500&lt;/td&gt;
      &lt;td&gt;7.39&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-05&lt;/th&gt;
      &lt;td&gt;7.37&lt;/td&gt;
      &lt;td&gt;7.52&lt;/td&gt;
      &lt;td&gt;6.87&lt;/td&gt;
      &lt;td&gt;6.94&lt;/td&gt;
      &lt;td&gt;27775700&lt;/td&gt;
      &lt;td&gt;6.94&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-08&lt;/th&gt;
      &lt;td&gt;6.68&lt;/td&gt;
      &lt;td&gt;6.79&lt;/td&gt;
      &lt;td&gt;6.41&lt;/td&gt;
      &lt;td&gt;6.74&lt;/td&gt;
      &lt;td&gt;17611300&lt;/td&gt;
      &lt;td&gt;6.74&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-09&lt;/th&gt;
      &lt;td&gt;6.60&lt;/td&gt;
      &lt;td&gt;6.72&lt;/td&gt;
      &lt;td&gt;6.07&lt;/td&gt;
      &lt;td&gt;6.34&lt;/td&gt;
      &lt;td&gt;13741100&lt;/td&gt;
      &lt;td&gt;6.34&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-10&lt;/th&gt;
      &lt;td&gt;6.28&lt;/td&gt;
      &lt;td&gt;6.59&lt;/td&gt;
      &lt;td&gt;6.11&lt;/td&gt;
      &lt;td&gt;6.24&lt;/td&gt;
      &lt;td&gt;8623900&lt;/td&gt;
      &lt;td&gt;6.24&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-11&lt;/th&gt;
      &lt;td&gt;6.02&lt;/td&gt;
      &lt;td&gt;6.27&lt;/td&gt;
      &lt;td&gt;5.74&lt;/td&gt;
      &lt;td&gt;6.06&lt;/td&gt;
      &lt;td&gt;17133900&lt;/td&gt;
      &lt;td&gt;6.06&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-12&lt;/th&gt;
      &lt;td&gt;6.14&lt;/td&gt;
      &lt;td&gt;6.66&lt;/td&gt;
      &lt;td&gt;6.06&lt;/td&gt;
      &lt;td&gt;6.47&lt;/td&gt;
      &lt;td&gt;13498600&lt;/td&gt;
      &lt;td&gt;6.47&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-16&lt;/th&gt;
      &lt;td&gt;6.66&lt;/td&gt;
      &lt;td&gt;6.74&lt;/td&gt;
      &lt;td&gt;6.33&lt;/td&gt;
      &lt;td&gt;6.62&lt;/td&gt;
      &lt;td&gt;11453500&lt;/td&gt;
      &lt;td&gt;6.62&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-17&lt;/th&gt;
      &lt;td&gt;6.70&lt;/td&gt;
      &lt;td&gt;7.13&lt;/td&gt;
      &lt;td&gt;6.55&lt;/td&gt;
      &lt;td&gt;6.72&lt;/td&gt;
      &lt;td&gt;29061300&lt;/td&gt;
      &lt;td&gt;6.72&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-18&lt;/th&gt;
      &lt;td&gt;6.95&lt;/td&gt;
      &lt;td&gt;6.96&lt;/td&gt;
      &lt;td&gt;6.22&lt;/td&gt;
      &lt;td&gt;6.51&lt;/td&gt;
      &lt;td&gt;13587900&lt;/td&gt;
      &lt;td&gt;6.51&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-19&lt;/th&gt;
      &lt;td&gt;6.47&lt;/td&gt;
      &lt;td&gt;6.51&lt;/td&gt;
      &lt;td&gt;5.97&lt;/td&gt;
      &lt;td&gt;6.20&lt;/td&gt;
      &lt;td&gt;14541500&lt;/td&gt;
      &lt;td&gt;6.20&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-22&lt;/th&gt;
      &lt;td&gt;6.20&lt;/td&gt;
      &lt;td&gt;6.92&lt;/td&gt;
      &lt;td&gt;6.20&lt;/td&gt;
      &lt;td&gt;6.75&lt;/td&gt;
      &lt;td&gt;11878300&lt;/td&gt;
      &lt;td&gt;6.75&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-23&lt;/th&gt;
      &lt;td&gt;6.61&lt;/td&gt;
      &lt;td&gt;6.68&lt;/td&gt;
      &lt;td&gt;6.07&lt;/td&gt;
      &lt;td&gt;6.12&lt;/td&gt;
      &lt;td&gt;9486500&lt;/td&gt;
      &lt;td&gt;6.12&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-24&lt;/th&gt;
      &lt;td&gt;5.93&lt;/td&gt;
      &lt;td&gt;6.14&lt;/td&gt;
      &lt;td&gt;5.77&lt;/td&gt;
      &lt;td&gt;6.08&lt;/td&gt;
      &lt;td&gt;8333800&lt;/td&gt;
      &lt;td&gt;6.08&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-25&lt;/th&gt;
      &lt;td&gt;6.07&lt;/td&gt;
      &lt;td&gt;6.13&lt;/td&gt;
      &lt;td&gt;5.68&lt;/td&gt;
      &lt;td&gt;5.93&lt;/td&gt;
      &lt;td&gt;8972000&lt;/td&gt;
      &lt;td&gt;5.93&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-26&lt;/th&gt;
      &lt;td&gt;6.13&lt;/td&gt;
      &lt;td&gt;6.55&lt;/td&gt;
      &lt;td&gt;6.07&lt;/td&gt;
      &lt;td&gt;6.43&lt;/td&gt;
      &lt;td&gt;12288100&lt;/td&gt;
      &lt;td&gt;6.43&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-29&lt;/th&gt;
      &lt;td&gt;6.43&lt;/td&gt;
      &lt;td&gt;6.62&lt;/td&gt;
      &lt;td&gt;6.33&lt;/td&gt;
      &lt;td&gt;6.40&lt;/td&gt;
      &lt;td&gt;14120300&lt;/td&gt;
      &lt;td&gt;6.40&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[7]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Valores al cierre de cada mes.&lt;/span&gt;
&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;asfreq&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;M&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;method&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;ffill&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;div class="output_wrapper"&gt;
&lt;div class="output"&gt;


&lt;div class="output_area"&gt;
&lt;div class="prompt output_prompt"&gt;Out[7]:&lt;/div&gt;


&lt;div class="output_html rendered_html output_subarea output_execute_result"&gt;
&lt;div&gt;
&lt;table border="1" class="dataframe"&gt;
  &lt;thead&gt;
    &lt;tr style="text-align: right;"&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;Open&lt;/th&gt;
      &lt;th&gt;High&lt;/th&gt;
      &lt;th&gt;Low&lt;/th&gt;
      &lt;th&gt;Close&lt;/th&gt;
      &lt;th&gt;Volume&lt;/th&gt;
      &lt;th&gt;Adj Close&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;Date&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
    &lt;/tr&gt;
  &lt;/thead&gt;
  &lt;tbody&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-01-31&lt;/th&gt;
      &lt;td&gt;6.26&lt;/td&gt;
      &lt;td&gt;6.77&lt;/td&gt;
      &lt;td&gt;6.20&lt;/td&gt;
      &lt;td&gt;6.74&lt;/td&gt;
      &lt;td&gt;17661000&lt;/td&gt;
      &lt;td&gt;6.74&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-29&lt;/th&gt;
      &lt;td&gt;6.43&lt;/td&gt;
      &lt;td&gt;6.62&lt;/td&gt;
      &lt;td&gt;6.33&lt;/td&gt;
      &lt;td&gt;6.40&lt;/td&gt;
      &lt;td&gt;14120300&lt;/td&gt;
      &lt;td&gt;6.40&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-03-31&lt;/th&gt;
      &lt;td&gt;7.62&lt;/td&gt;
      &lt;td&gt;7.86&lt;/td&gt;
      &lt;td&gt;7.55&lt;/td&gt;
      &lt;td&gt;7.78&lt;/td&gt;
      &lt;td&gt;13224600&lt;/td&gt;
      &lt;td&gt;7.78&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-04-30&lt;/th&gt;
      &lt;td&gt;8.10&lt;/td&gt;
      &lt;td&gt;8.34&lt;/td&gt;
      &lt;td&gt;7.88&lt;/td&gt;
      &lt;td&gt;8.13&lt;/td&gt;
      &lt;td&gt;21137000&lt;/td&gt;
      &lt;td&gt;8.13&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-05-31&lt;/th&gt;
      &lt;td&gt;5.61&lt;/td&gt;
      &lt;td&gt;5.74&lt;/td&gt;
      &lt;td&gt;5.55&lt;/td&gt;
      &lt;td&gt;5.61&lt;/td&gt;
      &lt;td&gt;8481400&lt;/td&gt;
      &lt;td&gt;5.61&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-06-30&lt;/th&gt;
      &lt;td&gt;5.50&lt;/td&gt;
      &lt;td&gt;5.58&lt;/td&gt;
      &lt;td&gt;5.36&lt;/td&gt;
      &lt;td&gt;5.55&lt;/td&gt;
      &lt;td&gt;14896000&lt;/td&gt;
      &lt;td&gt;5.55&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-07-31&lt;/th&gt;
      &lt;td&gt;5.64&lt;/td&gt;
      &lt;td&gt;5.81&lt;/td&gt;
      &lt;td&gt;5.59&lt;/td&gt;
      &lt;td&gt;5.68&lt;/td&gt;
      &lt;td&gt;19153500&lt;/td&gt;
      &lt;td&gt;5.68&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-08-31&lt;/th&gt;
      &lt;td&gt;5.48&lt;/td&gt;
      &lt;td&gt;5.59&lt;/td&gt;
      &lt;td&gt;5.37&lt;/td&gt;
      &lt;td&gt;5.47&lt;/td&gt;
      &lt;td&gt;11293500&lt;/td&gt;
      &lt;td&gt;5.47&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[8]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Valores al cierre de cada mes (días laborales).&lt;/span&gt;
&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;asfreq&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;BM&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
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&lt;/div&gt;

&lt;div class="output_wrapper"&gt;
&lt;div class="output"&gt;


&lt;div class="output_area"&gt;
&lt;div class="prompt output_prompt"&gt;Out[8]:&lt;/div&gt;


&lt;div class="output_html rendered_html output_subarea output_execute_result"&gt;
&lt;div&gt;
&lt;table border="1" class="dataframe"&gt;
  &lt;thead&gt;
    &lt;tr style="text-align: right;"&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;Open&lt;/th&gt;
      &lt;th&gt;High&lt;/th&gt;
      &lt;th&gt;Low&lt;/th&gt;
      &lt;th&gt;Close&lt;/th&gt;
      &lt;th&gt;Volume&lt;/th&gt;
      &lt;th&gt;Adj Close&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;Date&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
    &lt;/tr&gt;
  &lt;/thead&gt;
  &lt;tbody&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-01-29&lt;/th&gt;
      &lt;td&gt;6.26&lt;/td&gt;
      &lt;td&gt;6.77&lt;/td&gt;
      &lt;td&gt;6.20&lt;/td&gt;
      &lt;td&gt;6.74&lt;/td&gt;
      &lt;td&gt;17661000&lt;/td&gt;
      &lt;td&gt;6.74&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-02-29&lt;/th&gt;
      &lt;td&gt;6.43&lt;/td&gt;
      &lt;td&gt;6.62&lt;/td&gt;
      &lt;td&gt;6.33&lt;/td&gt;
      &lt;td&gt;6.40&lt;/td&gt;
      &lt;td&gt;14120300&lt;/td&gt;
      &lt;td&gt;6.40&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-03-31&lt;/th&gt;
      &lt;td&gt;7.62&lt;/td&gt;
      &lt;td&gt;7.86&lt;/td&gt;
      &lt;td&gt;7.55&lt;/td&gt;
      &lt;td&gt;7.78&lt;/td&gt;
      &lt;td&gt;13224600&lt;/td&gt;
      &lt;td&gt;7.78&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-04-29&lt;/th&gt;
      &lt;td&gt;8.10&lt;/td&gt;
      &lt;td&gt;8.34&lt;/td&gt;
      &lt;td&gt;7.88&lt;/td&gt;
      &lt;td&gt;8.13&lt;/td&gt;
      &lt;td&gt;21137000&lt;/td&gt;
      &lt;td&gt;8.13&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-05-31&lt;/th&gt;
      &lt;td&gt;5.61&lt;/td&gt;
      &lt;td&gt;5.74&lt;/td&gt;
      &lt;td&gt;5.55&lt;/td&gt;
      &lt;td&gt;5.61&lt;/td&gt;
      &lt;td&gt;8481400&lt;/td&gt;
      &lt;td&gt;5.61&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-06-30&lt;/th&gt;
      &lt;td&gt;5.50&lt;/td&gt;
      &lt;td&gt;5.58&lt;/td&gt;
      &lt;td&gt;5.36&lt;/td&gt;
      &lt;td&gt;5.55&lt;/td&gt;
      &lt;td&gt;14896000&lt;/td&gt;
      &lt;td&gt;5.55&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-07-29&lt;/th&gt;
      &lt;td&gt;5.64&lt;/td&gt;
      &lt;td&gt;5.81&lt;/td&gt;
      &lt;td&gt;5.59&lt;/td&gt;
      &lt;td&gt;5.68&lt;/td&gt;
      &lt;td&gt;19153500&lt;/td&gt;
      &lt;td&gt;5.68&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-08-31&lt;/th&gt;
      &lt;td&gt;5.48&lt;/td&gt;
      &lt;td&gt;5.59&lt;/td&gt;
      &lt;td&gt;5.37&lt;/td&gt;
      &lt;td&gt;5.47&lt;/td&gt;
      &lt;td&gt;11293500&lt;/td&gt;
      &lt;td&gt;5.47&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;/div&gt;
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&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[9]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# valores al cierre de cada trimestre&lt;/span&gt;
&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;asfreq&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;BQ&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
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&lt;div class="output_wrapper"&gt;
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&lt;div class="prompt output_prompt"&gt;Out[9]:&lt;/div&gt;


&lt;div class="output_html rendered_html output_subarea output_execute_result"&gt;
&lt;div&gt;
&lt;table border="1" class="dataframe"&gt;
  &lt;thead&gt;
    &lt;tr style="text-align: right;"&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;Open&lt;/th&gt;
      &lt;th&gt;High&lt;/th&gt;
      &lt;th&gt;Low&lt;/th&gt;
      &lt;th&gt;Close&lt;/th&gt;
      &lt;th&gt;Volume&lt;/th&gt;
      &lt;th&gt;Adj Close&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;Date&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
    &lt;/tr&gt;
  &lt;/thead&gt;
  &lt;tbody&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-03-31&lt;/th&gt;
      &lt;td&gt;7.62&lt;/td&gt;
      &lt;td&gt;7.86&lt;/td&gt;
      &lt;td&gt;7.55&lt;/td&gt;
      &lt;td&gt;7.78&lt;/td&gt;
      &lt;td&gt;13224600&lt;/td&gt;
      &lt;td&gt;7.78&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2016-06-30&lt;/th&gt;
      &lt;td&gt;5.50&lt;/td&gt;
      &lt;td&gt;5.58&lt;/td&gt;
      &lt;td&gt;5.36&lt;/td&gt;
      &lt;td&gt;5.55&lt;/td&gt;
      &lt;td&gt;14896000&lt;/td&gt;
      &lt;td&gt;5.55&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;
&lt;/div&gt;
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&lt;div class="text_cell_render border-box-sizing rendered_html"&gt;
&lt;h3 id="Desplazando-los-valores-de-la-serie"&gt;Desplazando los valores de la serie&lt;a class="anchor-link" href="#Desplazando-los-valores-de-la-serie"&gt;&amp;#182;&lt;/a&gt;&lt;/h3&gt;&lt;p&gt;Una operación común en los datos de &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt; es desplazar los valores hacia atrás y adelante en el tiempo, como por ejemplo para calcular el cambio porcentual de una muestra a otra. En &lt;a href="http://pandas.pydata.org/"&gt;Pandas&lt;/a&gt; podemos utilizar el método &lt;code&gt;.shift()&lt;/code&gt;.&lt;/p&gt;

&lt;/div&gt;
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&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[10]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# desplazando el 1 dia el valor de cierre&lt;/span&gt;
&lt;span class="n"&gt;desplazado&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Adj Close&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;shift&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;desplazado&lt;/span&gt;&lt;span class="p"&gt;[:&lt;/span&gt;&lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

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&lt;div class="prompt output_prompt"&gt;Out[10]:&lt;/div&gt;



&lt;div class="output_text output_subarea output_execute_result"&gt;
&lt;pre class="ipynb"&gt;Date
2016-01-04     NaN
2016-01-05    8.64
2016-01-06    8.26
2016-01-07    7.91
2016-01-08    7.34
Name: Adj Close, dtype: float64&lt;/pre&gt;
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&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[11]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# calculando el porcentaje de variación del día.&lt;/span&gt;
&lt;span class="n"&gt;variacion_diaria&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Adj Close&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="o"&gt;/&lt;/span&gt; &lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Adj Close&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;shift&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; &lt;span class="o"&gt;-&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;
&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;var_diaria&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;variacion_diaria&lt;/span&gt;
&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;var_diaria&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;][:&lt;/span&gt;&lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
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&lt;div class="prompt output_prompt"&gt;Out[11]:&lt;/div&gt;



&lt;div class="output_text output_subarea output_execute_result"&gt;
&lt;pre class="ipynb"&gt;Date
2016-01-04       NaN
2016-01-05   -0.0440
2016-01-06   -0.0424
2016-01-07   -0.0721
2016-01-08   -0.0504
Name: var_diaria, dtype: float64&lt;/pre&gt;
&lt;/div&gt;

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&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[12]:&lt;/div&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# mismo resultado utilizando pct_change()&lt;/span&gt;
&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Adj Close&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;pct_change&lt;/span&gt;&lt;span class="p"&gt;()[:&lt;/span&gt;&lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
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&lt;div class="prompt output_prompt"&gt;Out[12]:&lt;/div&gt;



&lt;div class="output_text output_subarea output_execute_result"&gt;
&lt;pre class="ipynb"&gt;Date
2016-01-04       NaN
2016-01-05   -0.0440
2016-01-06   -0.0424
2016-01-07   -0.0721
2016-01-08   -0.0504
Name: Adj Close, dtype: float64&lt;/pre&gt;
&lt;/div&gt;

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&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[13]:&lt;/div&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# calculando rendimiento acumulado diario&lt;/span&gt;
&lt;span class="n"&gt;rendimiento_diario&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt; &lt;span class="o"&gt;+&lt;/span&gt; &lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Adj Close&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;pct_change&lt;/span&gt;&lt;span class="p"&gt;())&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;cumprod&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;rend_diario&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;rendimiento_diario&lt;/span&gt;
&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;rend_diario&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;][:&lt;/span&gt;&lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
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&lt;div class="prompt output_prompt"&gt;Out[13]:&lt;/div&gt;



&lt;div class="output_text output_subarea output_execute_result"&gt;
&lt;pre class="ipynb"&gt;Date
2016-01-04       NaN
2016-01-05    0.9560
2016-01-06    0.9155
2016-01-07    0.8495
2016-01-08    0.8067
Name: rend_diario, dtype: float64&lt;/pre&gt;
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&lt;h3 id="Visualizando-las-series-de-tiempo"&gt;Visualizando las series de tiempo&lt;a class="anchor-link" href="#Visualizando-las-series-de-tiempo"&gt;&amp;#182;&lt;/a&gt;&lt;/h3&gt;&lt;p&gt;Una operación fundamental para entender el comportamiento de una &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;serie de tiempo&lt;/a&gt; y poder determinar si se trata de una serie &lt;a href="https://es.wikipedia.org/wiki/Proceso_estacionario"&gt;estacionaria&lt;/a&gt; o no; es realizar gráficos de la misma. En &lt;a href="http://pandas.pydata.org/"&gt;Pandas&lt;/a&gt; esto lo podemos realizar en forma muy sencilla con el método &lt;code&gt;.plot()&lt;/code&gt;.&lt;/p&gt;

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&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[14]:&lt;/div&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# graficando Adj Close&lt;/span&gt;
&lt;span class="n"&gt;plot&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Adj Close&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;plot&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;figsize&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;10&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;8&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
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"
&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[15]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Aplicando el filtro Hodrick-Prescott para separar en tendencia y &lt;/span&gt;
&lt;span class="c1"&gt;# componente ciclico.&lt;/span&gt;
&lt;span class="n"&gt;wft_ciclo&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;wft_tend&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;sm&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;tsa&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;filters&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;hpfilter&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Adj Close&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;tend&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;wft_tend&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[16]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# graficando la variacion del precio real con la tendencia.&lt;/span&gt;
&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Adj Close&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s1"&gt;&amp;#39;tend&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]]&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;plot&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;figsize&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;10&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;8&lt;/span&gt;&lt;span class="p"&gt;),&lt;/span&gt; &lt;span class="n"&gt;fontsize&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="mi"&gt;12&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;span class="n"&gt;legend&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;legend&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
&lt;span class="n"&gt;legend&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;prop&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;set_size&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;14&lt;/span&gt;&lt;span class="p"&gt;);&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;div class="output_wrapper"&gt;
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"
&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[17]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# graficando rendimiento diario&lt;/span&gt;
&lt;span class="n"&gt;plot&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;var_diaria&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;plot&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;figsize&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;10&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;8&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;div class="output_wrapper"&gt;
&lt;div class="output"&gt;


&lt;div class="output_area"&gt;
&lt;div class="prompt"&gt;&lt;/div&gt;



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&lt;h3 id="Promedios-m&amp;#243;viles-y-descomposici&amp;#243;n"&gt;Promedios m&amp;#243;viles y descomposici&amp;#243;n&lt;a class="anchor-link" href="#Promedios-m&amp;#243;viles-y-descomposici&amp;#243;n"&gt;&amp;#182;&lt;/a&gt;&lt;/h3&gt;&lt;p&gt;&lt;a href="http://pandas.pydata.org/"&gt;Pandas&lt;/a&gt; también nos ofrece una serie de funciones para calcular estadísticas móviles, en ellas la función estadística se calcula sobre una ventana de datos representados por un determinado período de tiempo y luego se desplaza la ventana de datos por un intervalo especificado, calculando continuamente la estadística, siempre y cuando la ventana este dentro de las fechas de la &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;serie de tiempo&lt;/a&gt;. El ejemplo más utilizado es el de &lt;a href="https://es.wikipedia.org/wiki/Media_m%C3%B3vil"&gt;media móvil&lt;/a&gt;, que se usa comúnmente en el análisis de &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt; financieras para suavizar las fluctuaciones a corto plazo y poner de relieve las tendencias a largo plazo en los datos.&lt;/p&gt;
&lt;p&gt;Otra técnica interesante que podemos intentar también es la descomposición. Esta es una técnica que trata de descomponer una &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;serie de tiempo&lt;/a&gt; en su tendencia, su estacionalidad y sus factores residuales. &lt;a href="http://statsmodels.sourceforge.net/stable/tsa.html"&gt;Statsmodels&lt;/a&gt; viene con una función de descomposición que nos facilita en sobremanera el trabajo. Veamos unos ejemplos.&lt;/p&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Calculando promedios móviles cada 5 días&lt;/span&gt;
&lt;span class="n"&gt;wft_ma&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;pd&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;rolling_mean&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Adj Close&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;prod_mov&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;wft_ma&lt;/span&gt;
&lt;span class="n"&gt;plot&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Adj Close&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s1"&gt;&amp;#39;prod_mov&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]]&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;plot&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;figsize&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;10&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;8&lt;/span&gt;&lt;span class="p"&gt;),&lt;/span&gt; &lt;span class="n"&gt;fontsize&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="mi"&gt;12&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

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"
&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[19]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Ejemplo de descomposición de serie de tiempo&lt;/span&gt;
&lt;span class="n"&gt;descomposicion&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;sm&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;tsa&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;seasonal_decompose&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Adj Close&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt;
                                                  &lt;span class="n"&gt;model&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;additive&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;freq&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="mi"&gt;30&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;  
&lt;span class="n"&gt;fig&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;descomposicion&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;plot&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;div class="output_wrapper"&gt;
&lt;div class="output"&gt;


&lt;div class="output_area"&gt;
&lt;div class="prompt"&gt;&lt;/div&gt;



&lt;div class="output_png output_subarea "&gt;
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&lt;h3 id="Pronosticando-la-serie-con-ARIMA"&gt;Pronosticando la serie con ARIMA&lt;a class="anchor-link" href="#Pronosticando-la-serie-con-ARIMA"&gt;&amp;#182;&lt;/a&gt;&lt;/h3&gt;&lt;p&gt;Como podemos observar en los gráficos que realizamos anteriormente, el comportamiento de la &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;serie de tiempo&lt;/a&gt; con la que estamos trabajando parece ser totalmente aleatorio y las medidas móviles que calculamos tampoco parecen ser de mucha utilidad para acercar la serie a un comportamiento &lt;a href="https://es.wikipedia.org/wiki/Proceso_estacionario"&gt;estacionario&lt;/a&gt;. De todas formas podemos intentar aplicar un modelo &lt;a href="https://es.wikipedia.org/wiki/Modelo_autorregresivo_integrado_de_media_m%C3%B3vil"&gt;ARIMA&lt;/a&gt; sobre la serie y ver que tan bien nos va con el pronostico del modelo. El modelo &lt;a href="https://es.wikipedia.org/wiki/Modelo_autorregresivo_integrado_de_media_m%C3%B3vil"&gt;ARIMA&lt;/a&gt; es similar a una &lt;a href="https://es.wikipedia.org/wiki/An%C3%A1lisis_de_la_regresi%C3%B3n"&gt;regresión estadística&lt;/a&gt; pero aplicando los conceptos de las &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt;; por tanto, los pronósticos del modelo vienen explicadas por los datos del pasado y no por variables independientes.&lt;/p&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Modelo ARIMA sobre el valor de cierre de la acción.&lt;/span&gt;
&lt;span class="n"&gt;modelo&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;sm&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;tsa&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;ARIMA&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Adj Close&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;iloc&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;:],&lt;/span&gt; &lt;span class="n"&gt;order&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;  
&lt;span class="n"&gt;resultados&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;modelo&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;fit&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;disp&lt;/span&gt;&lt;span class="o"&gt;=-&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;  
&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;pronostico&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;resultados&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;fittedvalues&lt;/span&gt;  
&lt;span class="n"&gt;plot&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Adj Close&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s1"&gt;&amp;#39;pronostico&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]]&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;plot&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;figsize&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;10&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;8&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;  
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&lt;p&gt;Aquí el modelo parece ser bastante efectivo, las líneas en el gráfico son muy similares. Pero para armar el modelo hemos utilizado el valor de cierre de la acción, y la variación de precio en el día a día es muy pequeña en comparación al precio absoluto. Lo que realmente nos interesa predecir es la variación diaria del precio de la acción, por lo tanto deberíamos armar el modelo utilizando la columna de variación diaria que calculamos previamente.&lt;/p&gt;

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&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[21]:&lt;/div&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# modelo ARIMA sobre variación diaria&lt;/span&gt;
&lt;span class="n"&gt;modelo&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;sm&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;tsa&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;ARIMA&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;var_diaria&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;iloc&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;:],&lt;/span&gt; &lt;span class="n"&gt;order&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;  
&lt;span class="n"&gt;resultados&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;modelo&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;fit&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;disp&lt;/span&gt;&lt;span class="o"&gt;=-&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;  
&lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;pronostico&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;resultados&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;fittedvalues&lt;/span&gt;  
&lt;span class="n"&gt;plot&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;wft&lt;/span&gt;&lt;span class="p"&gt;[[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;var_diaria&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s1"&gt;&amp;#39;pronostico&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]]&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;plot&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;figsize&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;10&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;8&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt; 
&lt;/pre&gt;&lt;/div&gt;

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&lt;p&gt;En este gráfico podemos ver que es bastante obvio que el pronóstico esta muy lejos. Nuestro modelo predice variaciones muy pequeñas en comparación con lo que ocurre en la realidad del día a día. Este era un resultado esperado ya que solo aplicamos un modelo sencillo de &lt;a href="https://es.wikipedia.org/wiki/Modelo_autorregresivo_integrado_de_media_m%C3%B3vil"&gt;promedios móviles&lt;/a&gt; a una serie no &lt;a href="https://es.wikipedia.org/wiki/Proceso_estacionario"&gt;estacionaria&lt;/a&gt;; después de todo, si fuera tan fácil predecir el movimiento del mercado, todos seríamos millonarios!. No hay suficiente información en los días anteriores para poder predecir con exactitud lo que va a pasar al día siguiente.&lt;/p&gt;

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&lt;p&gt;Aquí concluye este paseo por el mundo de las &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt;; como vimos, &lt;a href="http://pandas.pydata.org/"&gt;Pandas&lt;/a&gt; y &lt;a href="http://statsmodels.sourceforge.net/stable/tsa.html"&gt;Statsmodels&lt;/a&gt; pueden ser de mucha ayuda para trabajar con ellas. Espero que lo hayan encontrado útil.&lt;/p&gt;
&lt;p&gt;Saludos!&lt;/p&gt;
&lt;p&gt;&lt;em&gt;Este post fue escrito utilizando IPython notebook. Pueden descargar este &lt;a href="https://github.com/relopezbriega/relopezbriega.github.io/blob/master/downloads/TimeSeriePy.ipynb"&gt;notebook&lt;/a&gt; o ver su version estática en &lt;a href="http://nbviewer.ipython.org/github/relopezbriega/relopezbriega.github.io/blob/master/downloads/TimeSeriePy.ipynb"&gt;nbviewer&lt;/a&gt;.&lt;/em&gt;&lt;/p&gt;

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&lt;/p&gt;</summary><category term="python"></category><category term="programacion"></category><category term="analisis de datos"></category><category term="estadistica"></category><category term="visualizaciones"></category><category term="series de tiempo"></category><category term="finanzas"></category></entry><entry><title>El número $e$. El número de las finanzas</title><link href="http://relopezbriega.github.io/blog/2015/12/13/el-numero-e-el-numero-de-las-finanzas/" rel="alternate"></link><published>2015-12-13T00:00:00-03:00</published><author><name>Raul E. Lopez Briega</name></author><id>tag:relopezbriega.github.io,2015-12-13:blog/2015/12/13/el-numero-e-el-numero-de-las-finanzas/</id><summary type="html">&lt;p&gt;
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&lt;p&gt;&lt;img alt="El número e" title="El número e" src="http://relopezbriega.github.io/images/number_e.png
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&lt;h2 id="Introducci&amp;#243;n"&gt;Introducci&amp;#243;n&lt;a class="anchor-link" href="#Introducci&amp;#243;n"&gt;&amp;#182;&lt;/a&gt;&lt;/h2&gt;&lt;p&gt;El artículo de hoy se lo voy a dedicar a uno los números más fascinantes de las &lt;a href="https://es.wikipedia.org/wiki/Matem%C3%A1ticas"&gt;Matemáticas&lt;/a&gt;, el número &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt;. Este número es de suma importancia, ya que lo podemos encontrar en una gran variedad de fenómenos, desde Física y Biología, hasta Finanzas, Arte y Música. Así como el famoso &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_%CF%80"&gt;$\pi$&lt;/a&gt;, gobierna el círculo, &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt; gobierna el &lt;a href="http://relopezbriega.github.io/blog/2015/12/02/introduccion-al-calculo-con-python/"&gt;Cálculo&lt;/a&gt; y los &lt;a href="https://es.wikipedia.org/wiki/Logaritmo"&gt;Logaritmos&lt;/a&gt;.&lt;/p&gt;
&lt;h2 id="&amp;#191;Qu&amp;#233;-es-el-n&amp;#250;mero-$e$?"&gt;&amp;#191;Qu&amp;#233; es el n&amp;#250;mero $e$?&lt;a class="anchor-link" href="#&amp;#191;Qu&amp;#233;-es-el-n&amp;#250;mero-$e$?"&gt;&amp;#182;&lt;/a&gt;&lt;/h2&gt;&lt;p&gt;El número &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt; es una de las más importantes &lt;em&gt;constantes matemáticas&lt;/em&gt;, pertenece al grupo de los &lt;a href="http://es.wikipedia.org/wiki/N%C3%BAmero_irracional"&gt;números irracionales&lt;/a&gt;, es decir, que el mismo no puede ser expresado como una &lt;a href="http://es.wikipedia.org/wiki/Fracci%C3%B3n"&gt;fracción&lt;/a&gt; de dos &lt;a href="http://es.wikipedia.org/wiki/Enteros"&gt;números enteros&lt;/a&gt; y su expansión decimal es infinita. Asimismo, también es un &lt;a href="http://es.wikipedia.org/wiki/N%C3%BAmero_trascendente"&gt;número trascendental&lt;/a&gt;, lo que quiere decir que tampoco puede ser expresado algebraicamente. Sus primeros dígitos son:&lt;/p&gt;
$$e = 2.7182818284590452353602874713527\dots$$&lt;p&gt;Otra cosa que hace a &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt; sumamente interesante, es que es la base del &lt;a href="https://es.wikipedia.org/wiki/Logaritmo_natural"&gt;Logaritmo natural&lt;/a&gt;, es decir que si $\ln(x) = y$, entonces $e^y = x$. También podemos encontrar a &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt;, en la &lt;a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_exponencial"&gt;función exponencial&lt;/a&gt;, $e^x$, la cual es la inversa de la función del &lt;a href="https://es.wikipedia.org/wiki/Logaritmo_natural"&gt;Logaritmo natural&lt;/a&gt;, y tiene la particularidad de que la &lt;a href="https://es.wikipedia.org/wiki/Derivada"&gt;derivada&lt;/a&gt; de $e^x$, es la misma función $e^x$. Esta característica, es lo que que hace a &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt; fundamental para el &lt;a href="http://relopezbriega.github.io/blog/2015/12/02/introduccion-al-calculo-con-python/"&gt;Cálculo&lt;/a&gt;.&lt;/p&gt;
&lt;h2 id="Historia-del-n&amp;#250;mero-$e$"&gt;Historia del n&amp;#250;mero $e$&lt;a class="anchor-link" href="#Historia-del-n&amp;#250;mero-$e$"&gt;&amp;#182;&lt;/a&gt;&lt;/h2&gt;&lt;p&gt;Para poder entender más en profundidad la naturaleza del número &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt;, debemos recorrer un poco de su historia, que a diferencia de $\pi$, que tiene una historia milenaria, es relativamente reciente.&lt;/p&gt;
&lt;h3 id="Invenci&amp;#243;n-del-Logaritmo"&gt;Invenci&amp;#243;n del Logaritmo&lt;a class="anchor-link" href="#Invenci&amp;#243;n-del-Logaritmo"&gt;&amp;#182;&lt;/a&gt;&lt;/h3&gt;&lt;p&gt;La historia del número &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt; comienza en el siglo XVII, cuando &lt;a href="https://es.wikipedia.org/wiki/John_Napier"&gt;John Napier&lt;/a&gt; inventó el concepto de &lt;a href="https://es.wikipedia.org/wiki/Logaritmo"&gt;Logaritmo&lt;/a&gt;. Pocas veces en la historia de la ciencia, un concepto matemático fue recibido con tanto entusiasmo por la comunidad científica, como fue el caso del &lt;a href="https://es.wikipedia.org/wiki/Logaritmo"&gt;Logaritmo&lt;/a&gt;. La idea básica detrás de este concepto abstracto es la siguiente: Si uno pudiera escribir cualquier número positivo como la potencia de otro número fijo (llamado base), entonces la multiplicación y la división de números pasaría a ser equivalente a sumar o restar sus potencias; lo que simplificaría y facilitaría los cálculos. Para entenderlo mejor, veamos un ejemplo, supongamos que tenemos la siguiente tabla de potencias de 2.&lt;/p&gt;
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&lt;thead&gt;&lt;tr&gt;
&lt;th style="text-align:center"&gt;$n$&lt;/th&gt;
&lt;th style="text-align:center"&gt;-3&lt;/th&gt;
&lt;th style="text-align:center"&gt;-2&lt;/th&gt;
&lt;th style="text-align:center"&gt;-1&lt;/th&gt;
&lt;th style="text-align:center"&gt;0&lt;/th&gt;
&lt;th style="text-align:center"&gt;1&lt;/th&gt;
&lt;th style="text-align:center"&gt;2&lt;/th&gt;
&lt;th style="text-align:center"&gt;3&lt;/th&gt;
&lt;th style="text-align:center"&gt;4&lt;/th&gt;
&lt;th style="text-align:center"&gt;5&lt;/th&gt;
&lt;th style="text-align:center"&gt;6&lt;/th&gt;
&lt;th style="text-align:center"&gt;7&lt;/th&gt;
&lt;th style="text-align:center"&gt;8&lt;/th&gt;
&lt;th style="text-align:center"&gt;9&lt;/th&gt;
&lt;th style="text-align:center"&gt;10&lt;/th&gt;
&lt;th style="text-align:center"&gt;11&lt;/th&gt;
&lt;th style="text-align:center"&gt;12&lt;/th&gt;
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&lt;td style="text-align:center"&gt;$2^n$&lt;/td&gt;
&lt;td style="text-align:center"&gt;1/8&lt;/td&gt;
&lt;td style="text-align:center"&gt;1/4&lt;/td&gt;
&lt;td style="text-align:center"&gt;1/2&lt;/td&gt;
&lt;td style="text-align:center"&gt;1&lt;/td&gt;
&lt;td style="text-align:center"&gt;2&lt;/td&gt;
&lt;td style="text-align:center"&gt;4&lt;/td&gt;
&lt;td style="text-align:center"&gt;8&lt;/td&gt;
&lt;td style="text-align:center"&gt;16&lt;/td&gt;
&lt;td style="text-align:center"&gt;32&lt;/td&gt;
&lt;td style="text-align:center"&gt;64&lt;/td&gt;
&lt;td style="text-align:center"&gt;128&lt;/td&gt;
&lt;td style="text-align:center"&gt;256&lt;/td&gt;
&lt;td style="text-align:center"&gt;512&lt;/td&gt;
&lt;td style="text-align:center"&gt;1024&lt;/td&gt;
&lt;td style="text-align:center"&gt;2048&lt;/td&gt;
&lt;td style="text-align:center"&gt;4096&lt;/td&gt;
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&lt;p&gt;Ahora si quisiéramos saber el resultado de multiplicar 32 por 64; simplemente deberíamos buscar las potencias de estos dos números en la tabla anterior, 5 para 32 y 6 para 64, luego sumamos estos dos números, los que nos da como resultado 11 y por último si vamos a buscar el resultado de elevar 2 a la 11 potencia en la misma tabla, encontramos el resultado de nuestra multiplicación, que es 2048. Utilizando una simple tabla de &lt;a href="https://es.wikipedia.org/wiki/Logaritmo"&gt;Logaritmos&lt;/a&gt;, se podían resolver multiplicaciones y divisiones complejas en segundos, algo muy útil en el siglo XVII, dónde no existían las calculadoras!.&lt;/p&gt;
&lt;p&gt;Si bien &lt;a href="https://es.wikipedia.org/wiki/John_Napier"&gt;John Napier&lt;/a&gt; desarrollo el concepto de &lt;a href="https://es.wikipedia.org/wiki/Logaritmo"&gt;Logaritmo&lt;/a&gt;, el mismo difiere un poco del concepto moderno que tenemos hoy en día; y tampoco fue el responsable de notar el poder &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt; como base de los mismos. Tendríamos que esperar algunos años más hasta que &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt; se apoderara por completo de los &lt;a href="https://es.wikipedia.org/wiki/Logaritmo"&gt;Logaritmos&lt;/a&gt;, como la base más natural para ellos. Veamos el camino que hubo que recorrer.&lt;/p&gt;
&lt;h3 id="El-&amp;#225;rea-de-la-hip&amp;#233;rbola"&gt;El &amp;#225;rea de la hip&amp;#233;rbola&lt;a class="anchor-link" href="#El-&amp;#225;rea-de-la-hip&amp;#233;rbola"&gt;&amp;#182;&lt;/a&gt;&lt;/h3&gt;&lt;p&gt;Un problema muy común en &lt;a href="https://es.wikipedia.org/wiki/Matem%C3%A1ticas"&gt;Matemáticas&lt;/a&gt;, es el de encontrar el &lt;a href="https://es.wikipedia.org/wiki/%C3%81rea"&gt;área&lt;/a&gt; de una figura plana, este proceso se conoce con el nombre de &lt;em&gt;cuadratura&lt;/em&gt; o &lt;a href="https://es.wikipedia.org/wiki/Integraci%C3%B3n"&gt;integración&lt;/a&gt;. Desde que los matemáticos comenzaron a trabajar en estos problemas, una de las figuras que más ha resistido obstinadamente todos los intentos de cuadratura era la &lt;a href="https://es.wikipedia.org/wiki/Hip%C3%A9rbola"&gt;hipérbola&lt;/a&gt;, la cual esta definida por la función $f(x) = \frac{1}{x}$.&lt;/p&gt;
&lt;p&gt;Un matemático que se interesó por un tema muy cercano a este, también en el siglo XVII, fue &lt;a href="https://es.wikipedia.org/wiki/Pierre_de_Fermat"&gt;Pierre de Fermat&lt;/a&gt;. &lt;a href="https://es.wikipedia.org/wiki/Pierre_de_Fermat"&gt;Fermat&lt;/a&gt; se interesó en la &lt;em&gt;cuadratura&lt;/em&gt; de curvas cuya función general tiene la forma $f(x) = x^n$, dónde $n$ es un &lt;a href="http://es.wikipedia.org/wiki/Enteros"&gt;número entero&lt;/a&gt;. Estas curvas son llamadas generalmente &lt;a href="https://es.wikipedia.org/wiki/Par%C3%A1bola_(matem%C3%A1tica)"&gt;parábolas&lt;/a&gt;. Utilizando el &lt;a href="https://es.wikipedia.org/wiki/M%C3%A9todo_exhaustivo"&gt;método de exhaución&lt;/a&gt;, el cuál explique en mi &lt;a href="http://relopezbriega.github.io/blog/2015/12/02/introduccion-al-calculo-con-python/"&gt;artículo anterior&lt;/a&gt;, &lt;a href="https://es.wikipedia.org/wiki/Pierre_de_Fermat"&gt;Fermat&lt;/a&gt; fue capaz de llegar a una fórmula general para encontrar el área de esta familia de curvas. La fórmula a la que arribó fue la siguiente:&lt;/p&gt;
$$A = \frac{a^{n + 1}}{n + 1}$$&lt;p&gt;Como podemos ver, esta no es ni más ni menos que la regla de &lt;a href="https://es.wikipedia.org/wiki/Integraci%C3%B3n"&gt;integración&lt;/a&gt; para funciones de grado $n$.&lt;/p&gt;
$$\int x^{n} \ dx = \frac{x^{n + 1}}{n + 1}$$&lt;p&gt;Debemos recordar, sin embargo, que el trabajo de &lt;a href="https://es.wikipedia.org/wiki/Pierre_de_Fermat"&gt;Fermat&lt;/a&gt; fue hecho alrededor de 1640, aproximadamente unos 30 años antes de que &lt;a href="https://es.wikipedia.org/wiki/Isaac_Newton"&gt;Newton&lt;/a&gt; y &lt;a href="https://es.wikipedia.org/wiki/Gottfried_Leibniz"&gt;Leibniz&lt;/a&gt; establecieran esta fórmula como parte del &lt;a href="http://relopezbriega.github.io/blog/2015/12/02/introduccion-al-calculo-con-python/"&gt;cálculo integral&lt;/a&gt;. A pesar de que &lt;a href="https://es.wikipedia.org/wiki/Pierre_de_Fermat"&gt;Fermat&lt;/a&gt; pudo probar que su fórmula podía ser aplicada tanto para los casos de $n$ siendo un &lt;a href="http://es.wikipedia.org/wiki/Enteros"&gt;entero&lt;/a&gt; positivo como negativo; hubo un caso que se le siguió escapando, el caso de $n = -1$, es decir el caso del &lt;a href="https://es.wikipedia.org/wiki/%C3%81rea"&gt;área&lt;/a&gt; de la &lt;a href="https://es.wikipedia.org/wiki/Hip%C3%A9rbola"&gt;hipérbola&lt;/a&gt;, $f(x) = \frac{1}{x}$; ya que este caso hace al denominador de su formula ($n + 1$) igual a 0. A pesar de todos sus esfuerzos, &lt;a href="https://es.wikipedia.org/wiki/Pierre_de_Fermat"&gt;Fermat&lt;/a&gt; nunca pudo llegar a resolver este caso.&lt;/p&gt;
&lt;p&gt;La luz recién llegaría en el año 1661, cuando &lt;a href="https://es.wikipedia.org/wiki/Christiaan_Huygens"&gt;Christiaan Huygens&lt;/a&gt; se diera cuenta de que para el caso de $n = -1$, todos los rectángulos utilizados en la aproximación del &lt;a href="https://es.wikipedia.org/wiki/%C3%81rea"&gt;área&lt;/a&gt; bajo la &lt;a href="https://es.wikipedia.org/wiki/Hip%C3%A9rbola"&gt;hipérbola&lt;/a&gt;, tenían la misma &lt;a href="https://es.wikipedia.org/wiki/%C3%81rea"&gt;área&lt;/a&gt;, es decir que a medida que la distancia crece geométricamente desde 0, las &lt;a href="https://es.wikipedia.org/wiki/%C3%81rea"&gt;área&lt;/a&gt; se incrementan en la misma proporción. Lo que implica que la relación entre el &lt;a href="https://es.wikipedia.org/wiki/%C3%81rea"&gt;área&lt;/a&gt; y la distancia, es &lt;a href="https://es.wikipedia.org/wiki/Logaritmo"&gt;logarítmica&lt;/a&gt;!. Por tanto para resolver el problema del &lt;a href="https://es.wikipedia.org/wiki/%C3%81rea"&gt;área&lt;/a&gt; de la &lt;a href="https://es.wikipedia.org/wiki/Hip%C3%A9rbola"&gt;hipérbola&lt;/a&gt;, $y = \frac{1}{x}$, simplemente deberíamos aplicar la función &lt;a href="https://es.wikipedia.org/wiki/Logaritmo"&gt;logarítmica&lt;/a&gt; sobre $x$; el único problema restante, era encontrar la &lt;em&gt;base&lt;/em&gt; de esa función &lt;a href="https://es.wikipedia.org/wiki/Logaritmo"&gt;logarítmica&lt;/a&gt;, que como ya se pueden ir imaginando, no es ni nada más ni nada menos que el número &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt;.&lt;/p&gt;
&lt;p&gt;&lt;img alt="Relación entre la hipérbola y los logaritmos" title="Relación entre la hipérbola y los logaritmos" src="http://relopezbriega.github.io/images/hiperbola-logaritmo.png
" width="400px" height="300px"&gt;&lt;/p&gt;
&lt;p&gt;Es decir, que entonces para resolver el &lt;a href="https://es.wikipedia.org/wiki/%C3%81rea"&gt;área&lt;/a&gt; bajo la &lt;a href="https://es.wikipedia.org/wiki/Hip%C3%A9rbola"&gt;hipérbola&lt;/a&gt;, debemos calcular el &lt;a href="https://es.wikipedia.org/wiki/Logaritmo_natural"&gt;Logaritmo natural&lt;/a&gt; de $x$. Lo que llevado a la notación actual del &lt;a href="http://relopezbriega.github.io/blog/2015/12/02/introduccion-al-calculo-con-python/"&gt;cálculo integral&lt;/a&gt; equivale a decir que:&lt;/p&gt;
$$\int \left(\frac{1}{x}\right) \ dx = \ln (x)$$&lt;p&gt;Desde este momento en adelante, &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt; comenzaría a convertirse en el amo y señor de los &lt;a href="https://es.wikipedia.org/wiki/Logaritmo"&gt;Logaritmos&lt;/a&gt;. Pero no se iba a quedar solo con eso, ya que aún quedaban por descubrirse otras importantes propiedades de este misterioso número.&lt;/p&gt;
&lt;h3 id="Euler"&gt;Euler&lt;a class="anchor-link" href="#Euler"&gt;&amp;#182;&lt;/a&gt;&lt;/h3&gt;&lt;p&gt;Sin dudas, uno de los más grandes matemáticos de todos los tiempos fue &lt;a href="https://es.wikipedia.org/wiki/Leonhard_Euler"&gt;Leonhard Euler&lt;/a&gt;. Prácticamente no dejó una rama de las &lt;a href="https://es.wikipedia.org/wiki/Matem%C3%A1ticas"&gt;Matemáticas&lt;/a&gt; sin tocar, dejando su marca en campos tan diversos como el &lt;a href="https://es.wikipedia.org/wiki/An%C3%A1lisis_matem%C3%A1tico"&gt;análisis matemático&lt;/a&gt;, la &lt;a href="https://es.wikipedia.org/wiki/Teor%C3%ADa_de_n%C3%BAmeros"&gt;teoría de números&lt;/a&gt;, la &lt;a href="https://es.wikipedia.org/wiki/Mec%C3%A1nica"&gt;mecánica&lt;/a&gt; y la &lt;a href="https://es.wikipedia.org/wiki/Hidrodin%C3%A1mica"&gt;hidrodinámica&lt;/a&gt;, la &lt;a href="https://es.wikipedia.org/wiki/Cartograf%C3%ADa"&gt;cartografía&lt;/a&gt;, la &lt;a href="https://es.wikipedia.org/wiki/Topolog%C3%ADa"&gt;topología&lt;/a&gt;, la &lt;a href="https://es.wikipedia.org/wiki/%C3%93ptica"&gt;óptica&lt;/a&gt; y la &lt;a href="https://es.wikipedia.org/wiki/Astronom%C3%ADa"&gt;astronomía&lt;/a&gt;. A él también le debemos muchos de los símbolos matemáticos que usamos hoy en día, como ser, $i, \ f(x), \ \pi$ y el mismo símbolo del número &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt;. La más influyente de sus numerosas obras fue su &lt;em&gt;Introductio in analysin infinitorum&lt;/em&gt;, una obra en dos volúmenes publicados en 1748 y considerada como la base de &lt;a href="https://es.wikipedia.org/wiki/An%C3%A1lisis_matem%C3%A1tico"&gt;análisis matemático&lt;/a&gt; moderno. En esta obra, &lt;a href="https://es.wikipedia.org/wiki/Leonhard_Euler"&gt;Euler&lt;/a&gt; convierte a la &lt;em&gt;&lt;a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_matem%C3%A1tica"&gt;función&lt;/a&gt;&lt;/em&gt; en el concepto central del &lt;a href="https://es.wikipedia.org/wiki/An%C3%A1lisis_matem%C3%A1tico"&gt;análisis&lt;/a&gt;. Su definición de &lt;em&gt;&lt;a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_matem%C3%A1tica"&gt;función&lt;/a&gt;&lt;/em&gt; es la que se suele utilizar hoy en día tanto en &lt;a href="https://es.wikipedia.org/wiki/Matem%C3%A1ticas"&gt;Matemáticas&lt;/a&gt; aplicadas, como en &lt;a href="https://es.wikipedia.org/wiki/F%C3%ADsica"&gt;Física&lt;/a&gt;.&lt;/p&gt;
&lt;p&gt;En el &lt;em&gt;Introductio&lt;/em&gt;, &lt;a href="https://es.wikipedia.org/wiki/Leonhard_Euler"&gt;Euler&lt;/a&gt;, hace notar por primera vez, el rol central que tiene el número &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt;, y más precisamente la &lt;a href="https://es.wikipedia.org/wiki/Funci%C3%B3n_exponencial"&gt;función exponencial&lt;/a&gt; $e^x$, y  su función inversa, la &lt;a href="https://es.wikipedia.org/wiki/Logaritmo"&gt;función logarítmica&lt;/a&gt; $\ln x$ para el &lt;a href="http://relopezbriega.github.io/blog/2015/12/02/introduccion-al-calculo-con-python/"&gt;Cálculo&lt;/a&gt;. &lt;a href="https://es.wikipedia.org/wiki/Leonhard_Euler"&gt;Euler&lt;/a&gt; puso a éstas dos últimas funciones en iguales condiciones, al darles definiciones independientes a cada una de ellas. Como ser:&lt;/p&gt;
$$e^x = \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^n
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\ln x = \lim_{n \to \infty} n(x^{1/n} - 1)$$&lt;p&gt;Finalmente, a &lt;a href="https://es.wikipedia.org/wiki/Leonhard_Euler"&gt;Euler&lt;/a&gt; también le debemos una de las &lt;a href="https://es.wikipedia.org/wiki/Ecuaci%C3%B3n"&gt;ecuaciones&lt;/a&gt; más famosas e increíbles de todas las &lt;a href="https://es.wikipedia.org/wiki/Matem%C3%A1ticas"&gt;Matemáticas&lt;/a&gt;. La ecuación:&lt;/p&gt;
$$e^{i\pi} + 1 = 0$$&lt;p&gt;En una sola fórmula, &lt;a href="https://es.wikipedia.org/wiki/Leonhard_Euler"&gt;Euler&lt;/a&gt; logró conectar a las cinco más importantes &lt;em&gt;constantes&lt;/em&gt; de las &lt;a href="https://es.wikipedia.org/wiki/Matem%C3%A1ticas"&gt;Matemáticas&lt;/a&gt;, y también tres de las más importantes &lt;em&gt;operaciones&lt;/em&gt; matemáticas (&lt;a href="https://es.wikipedia.org/wiki/Adici%C3%B3n_(matem%C3%A1ticas)"&gt;adición&lt;/a&gt;, &lt;a href="https://es.wikipedia.org/wiki/Multiplicaci%C3%B3n"&gt;multiplicación&lt;/a&gt; y &lt;a href="https://es.wikipedia.org/wiki/Exponenciaci%C3%B3n"&gt;exponenciación&lt;/a&gt;). Las cinco &lt;em&gt;constantes&lt;/em&gt; simbolizan las cuatro principales ramas de las &lt;a href="https://es.wikipedia.org/wiki/Matem%C3%A1ticas"&gt;Matemáticas&lt;/a&gt; clásicas: la &lt;a href="https://es.wikipedia.org/wiki/Aritm%C3%A9tica"&gt;Aritmética&lt;/a&gt;, representada por el 0 y el 1; el &lt;a href="https://es.wikipedia.org/wiki/%C3%81lgebra"&gt;Álgebra&lt;/a&gt; por $i$; la &lt;a href="https://es.wikipedia.org/wiki/Geometr%C3%ADa"&gt;Geometría&lt;/a&gt;, por $\pi$; y el &lt;a href="https://es.wikipedia.org/wiki/An%C3%A1lisis_matem%C3%A1tico"&gt;Análisis&lt;/a&gt;, representado por &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt;.&lt;/p&gt;
&lt;p&gt;A partir de la obra de &lt;a href="https://es.wikipedia.org/wiki/Leonhard_Euler"&gt;Euler&lt;/a&gt;, el número &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt;, se convirtió también en amo y señor del &lt;a href="http://relopezbriega.github.io/blog/2015/12/02/introduccion-al-calculo-con-python/"&gt;Cálculo&lt;/a&gt;; gobernando así, tanto a los &lt;a href="https://es.wikipedia.org/wiki/Logaritmo"&gt;Logaritmos&lt;/a&gt; como al &lt;a href="https://es.wikipedia.org/wiki/An%C3%A1lisis_matem%C3%A1tico"&gt;Análisis&lt;/a&gt;.&lt;/p&gt;
&lt;h2 id="Calculando-a-$e$"&gt;Calculando a $e$&lt;a class="anchor-link" href="#Calculando-a-$e$"&gt;&amp;#182;&lt;/a&gt;&lt;/h2&gt;&lt;p&gt;Existen muchas maneras de calcular al número &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt;, por ejemplo una forma de calcularlo es aplicando el siguiente &lt;a href="https://es.wikipedia.org/wiki/L%C3%ADmite_matem%C3%A1tico"&gt;límite&lt;/a&gt;:&lt;/p&gt;
$$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$&lt;p&gt;A medida que $n$ se va haciendo cada vez más grande, vamos ganando precisión en el cálculo de los decimales de &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt;.&lt;/p&gt;

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&lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="o"&gt;%&lt;/span&gt;&lt;span class="k"&gt;matplotlib&lt;/span&gt; inline

&lt;span class="kn"&gt;import&lt;/span&gt; &lt;span class="nn"&gt;matplotlib.pyplot&lt;/span&gt; &lt;span class="kn"&gt;as&lt;/span&gt; &lt;span class="nn"&gt;plt&lt;/span&gt;
&lt;span class="kn"&gt;import&lt;/span&gt; &lt;span class="nn"&gt;numpy&lt;/span&gt; &lt;span class="kn"&gt;as&lt;/span&gt; &lt;span class="nn"&gt;np&lt;/span&gt;
&lt;span class="kn"&gt;import&lt;/span&gt; &lt;span class="nn"&gt;pandas&lt;/span&gt; &lt;span class="kn"&gt;as&lt;/span&gt; &lt;span class="nn"&gt;pd&lt;/span&gt;

&lt;span class="k"&gt;def&lt;/span&gt; &lt;span class="nf"&gt;f&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;n&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
    &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt; &lt;span class="o"&gt;+&lt;/span&gt; &lt;span class="mf"&gt;1.0&lt;/span&gt; &lt;span class="o"&gt;/&lt;/span&gt; &lt;span class="n"&gt;n&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="o"&gt;**&lt;/span&gt;&lt;span class="n"&gt;n&lt;/span&gt;

&lt;span class="n"&gt;n&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;np&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;array&lt;/span&gt;&lt;span class="p"&gt;([&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;10&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;100&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;1000&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;10000&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;100000&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;1000000&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;100000000&lt;/span&gt; &lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="n"&gt;y&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;f&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;n&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;tabla&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;pd&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;DataFrame&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nb"&gt;list&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nb"&gt;zip&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;n&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;y&lt;/span&gt;&lt;span class="p"&gt;)),&lt;/span&gt; &lt;span class="n"&gt;columns&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;n&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s1"&gt;&amp;#39;e&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="n"&gt;tabla&lt;/span&gt;
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      &lt;th&gt;0&lt;/th&gt;
      &lt;td&gt;1&lt;/td&gt;
      &lt;td&gt;2.000000&lt;/td&gt;
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      &lt;th&gt;1&lt;/th&gt;
      &lt;td&gt;5&lt;/td&gt;
      &lt;td&gt;2.488320&lt;/td&gt;
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      &lt;th&gt;2&lt;/th&gt;
      &lt;td&gt;10&lt;/td&gt;
      &lt;td&gt;2.593742&lt;/td&gt;
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      &lt;th&gt;3&lt;/th&gt;
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      &lt;td&gt;2.704814&lt;/td&gt;
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      &lt;th&gt;4&lt;/th&gt;
      &lt;td&gt;1000&lt;/td&gt;
      &lt;td&gt;2.716924&lt;/td&gt;
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      &lt;th&gt;5&lt;/th&gt;
      &lt;td&gt;10000&lt;/td&gt;
      &lt;td&gt;2.718146&lt;/td&gt;
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      &lt;th&gt;6&lt;/th&gt;
      &lt;td&gt;100000&lt;/td&gt;
      &lt;td&gt;2.718268&lt;/td&gt;
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      &lt;td&gt;1000000&lt;/td&gt;
      &lt;td&gt;2.718280&lt;/td&gt;
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      &lt;td&gt;2.718282&lt;/td&gt;
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&lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Graficando (1 + 1/n)**n&lt;/span&gt;

&lt;span class="n"&gt;n&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;np&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;arange&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;100000&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt; 
&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;figure&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;figsize&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;8&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;6&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;
&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;title&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="sa"&gt;r&lt;/span&gt;&lt;span class="s2"&gt;&amp;quot;Graficando $(1 + 1/n)^n$&amp;quot;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;xlabel&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;n&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;ylabel&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="sa"&gt;r&lt;/span&gt;&lt;span class="s2"&gt;&amp;quot;$(1 + 1/n)^n$&amp;quot;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;plot&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;n&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;f&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;n&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;
&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;axhline&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;y&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;np&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;e&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;color&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="s1"&gt;&amp;#39;r&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;label&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;$e$&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;xlim&lt;/span&gt;&lt;span class="p"&gt;([&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;100&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;legend&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;show&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
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&lt;p&gt;Como podemos ver, tanto del ejemplo numérico como del gráfico, esta definición de &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt;, tarda bastante en &lt;em&gt;&lt;a href="https://es.wikipedia.org/wiki/Serie_convergente"&gt;converger&lt;/a&gt;&lt;/em&gt; hacia el valor exacto. Necesitamos un valor bastante grande de $n$ para ganar precisión.&lt;/p&gt;
&lt;p&gt;Otra definición de &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt; que podemos utilizar para calcularlo y que &lt;em&gt;&lt;a href="https://es.wikipedia.org/wiki/Serie_convergente"&gt;converge&lt;/a&gt;&lt;/em&gt; mucho más rápido, es su definición como una serie infinita de &lt;a href="https://es.wikipedia.org/wiki/Factorial"&gt;factoriales&lt;/a&gt;.&lt;/p&gt;
$$e = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!}+ \frac{1}{3!}+ \frac{1}{4!}+ \frac{1}{5!}+ \frac{1}{6!}+ \frac{1}{7!} + \dots$$&lt;p&gt;Por último, también podríamos expresar a &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt;, utilizando &lt;a href="https://es.wikipedia.org/wiki/Fracci%C3%B3n_continua"&gt;fracciones continuas&lt;/a&gt;, del siguiente modo:&lt;/p&gt;
&lt;p&gt;&lt;img alt="e como fracción continua" title="e como fracción continua" src="https://upload.wikimedia.org/math/1/c/a/1caed284ecf91e35cc9b39512ba2ccdb.png
" width="400px" height="300px"&gt;&lt;/p&gt;
&lt;p&gt;Al igual que ocurre con el número $\pi$ se conocen millones de dígitos del número &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt;. Por ejemplo, para generar la imagen de la cabecera del artículo, yo utilicé 390 decimales de &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt;.&lt;/p&gt;
&lt;h2 id="$e$-en-las-finanzas"&gt;$e$ en las finanzas&lt;a class="anchor-link" href="#$e$-en-las-finanzas"&gt;&amp;#182;&lt;/a&gt;&lt;/h2&gt;&lt;p&gt;Como comenté a lo largo de todo el artículo, nos podemos topar con el número &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt; en infinidad de situaciones; pero una de las áreas donde más lo podemos encontrar es en la finanzas, ya que &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt; se encuentra escondido en la definición de una de las formulas más fundamentales del &lt;a href="https://es.wikipedia.org/wiki/Matem%C3%A1tica_financiera"&gt;Cálculo financiero&lt;/a&gt;, la fórmula del &lt;a href="https://es.wikipedia.org/wiki/Inter%C3%A9s_compuesto"&gt;interés compuesto&lt;/a&gt;. Por ejemplo, si repasamos la formula del &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor futuro&lt;/a&gt;:&lt;/p&gt;
$$FV = PV \left(1 + \frac{r}{n}\right)^n$$&lt;p&gt;donde $FV$ es el &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor futuro&lt;/a&gt;; $PV$ es el &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor presente&lt;/a&gt; de nuestra inversión; $r$ es la tasa de interés anual,  expresada como valor decimal; y $n$ es el número de períodos.&lt;/p&gt;
&lt;p&gt;Y si miramos detenidamente a esta &lt;a href="https://es.wikipedia.org/wiki/Ecuaci%C3%B3n"&gt;ecuación&lt;/a&gt;, podemos ver cierta similitud con la definición de &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt; que dimos más arriba.&lt;/p&gt;
$$e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$&lt;p&gt;Es más, si tomáramos el caso hipotético en el que el &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor presente&lt;/a&gt; sea igual a 1, es decir $PV = 1$ y la tasa de interés sea del 100 %, $r = 1$. Podemos ver que la fórmula del &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor futuro&lt;/a&gt; se convierte en la definición del número &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt;!.&lt;/p&gt;
&lt;p&gt;Por esta razón, al número &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt; lo vamos a encontrar en infinidad de situaciones en el &lt;a href="https://es.wikipedia.org/wiki/Matem%C3%A1tica_financiera"&gt;Cálculo financiero&lt;/a&gt;. Por ejemplo, la definición de la fórmula de la &lt;a href="https://es.wikipedia.org/wiki/Capitalizaci%C3%B3n_continua"&gt;capitalización continua&lt;/a&gt; utiliza al número &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt; explícitamente:&lt;/p&gt;
$$FV = PV \cdot e^{rt}$$&lt;p&gt;donde $FV$ es el &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor futuro&lt;/a&gt;; $PV$ es el &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor presente&lt;/a&gt; de nuestra inversión; $r$ es la tasa de interés efectiva y $t$ es el tiempo de capitalización.&lt;/p&gt;
&lt;p&gt;Éstos son sólo algunos casos, a medida que nos adentramos más en la complejidades del &lt;a href="https://es.wikipedia.org/wiki/Matem%C3%A1tica_financiera"&gt;Cálculo financiero&lt;/a&gt; vamos a ver que &lt;a href="https://es.wikipedia.org/wiki/N%C3%BAmero_e"&gt;$e$&lt;/a&gt; continua apareciendo una y otra vez; pero eso va a quedar para próximos artículos!&lt;/p&gt;

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&lt;p&gt;Aquí concluye el artículo, espero les haya gustado y hayan sentido cierta fascinación por este místico número que encontramos por todas partes!&lt;/p&gt;
&lt;p&gt;Saludos!&lt;/p&gt;
&lt;p&gt;&lt;em&gt;Este post fue escrito utilizando Jupyter notebook. Pueden descargar este &lt;a href="https://github.com/relopezbriega/relopezbriega.github.io/blob/master/downloads/numbere.ipynb"&gt;notebook&lt;/a&gt; o ver su versión estática en &lt;a href="http://nbviewer.ipython.org/github/relopezbriega/relopezbriega.github.io/blob/master/downloads/numbere.ipynb"&gt;nbviewer&lt;/a&gt;.&lt;/em&gt;&lt;/p&gt;

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&lt;/p&gt;</summary><category term="python"></category><category term="matematica"></category><category term="calculo"></category><category term="logaritmo"></category><category term="finanzas"></category></entry><entry><title>Introducción a Finanzas con Python</title><link href="http://relopezbriega.github.io/blog/2015/08/28/introduccion-a-finanzas-con-python/" rel="alternate"></link><published>2015-08-28T00:00:00-03:00</published><author><name>Raul E. Lopez Briega</name></author><id>tag:relopezbriega.github.io,2015-08-28:blog/2015/08/28/introduccion-a-finanzas-con-python/</id><summary type="html">&lt;p&gt;
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&lt;p&gt;&lt;img alt="Introducción a Finanzas" title="Introducción a Finanzas" src="http://relopezbriega.github.io/images/Finance.jpg" width="500px" height="400px"&gt;&lt;/p&gt;

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&lt;h2 id="Introducci&amp;#243;n"&gt;Introducci&amp;#243;n&lt;a class="anchor-link" href="#Introducci&amp;#243;n"&gt;&amp;#182;&lt;/a&gt;&lt;/h2&gt;&lt;p&gt;En el vertiginoso mundo actual de las &lt;a href="https://es.wikipedia.org/wiki/Finanzas"&gt;finanzas&lt;/a&gt;; dónde la velocidad, frecuencia y volumen de los datos aumentan a un ritmo considerable; la aplicación combinada de tecnología y software, junto con algoritmos avanzados y diferentes métodos para recopilar, procesar y analizar datos se ha vuelto fundamental para obtener la información necesaria para una eficiente toma de decisiones. Es dentro de este contexto,  que se viene produciendo un gran crecimento en la utilización de &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt; dentro de la &lt;a href="https://es.wikipedia.org/wiki/Servicios_financieros"&gt;industria de las finanzas&lt;/a&gt;.&lt;/p&gt;
&lt;p&gt;&lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt; se esta comenzando a utilizar ampliamente en diversos sectores de las &lt;a href="https://es.wikipedia.org/wiki/Finanzas"&gt;finanzas&lt;/a&gt;, como la banca, la gestión de inversiones, los seguros, e incluso en los bienes raíces; se utiliza principalmente para la construcción de herramientas que ayuden en la creación de &lt;a href="https://en.wikipedia.org/wiki/Financial_modeling"&gt;modelos financieros&lt;/a&gt;, &lt;a href="https://es.wikipedia.org/wiki/Gesti%C3%B3n_de_riesgos"&gt;gestión de riesgos&lt;/a&gt;, y el comercio. Incluso las grandes corporaciones financieras, como &lt;a href="https://www.bankofamerica.com/"&gt;Bank of America&lt;/a&gt; o &lt;a href="https://www.jpmorgan.com/pages/jpmorgan"&gt;JP Morgan&lt;/a&gt;, estan comenzando a utilizar &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt; para construir su infraestructura para la gestión de posiciones financieras, &lt;a href="https://es.wikipedia.org/wiki/Modelo_de_valoraci%C3%B3n_de_activos_financieros"&gt;precios de activos&lt;/a&gt;, &lt;a href="https://es.wikipedia.org/wiki/Gesti%C3%B3n_de_riesgos"&gt;gestión de riesgos&lt;/a&gt;, sistemas de comercio y &lt;a href="https://en.wikipedia.org/wiki/Algorithmic_trading"&gt;comercio algoritmico&lt;/a&gt;.&lt;/p&gt;
&lt;p&gt;Algunas de las razones que hacen de &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt; un &lt;a href="https://es.wikipedia.org/wiki/Lenguaje_de_programaci%C3%B3n"&gt;lenguaje de programación&lt;/a&gt; tan atractivo en el mundo de las finanzas son:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Su simple sintaxis&lt;/strong&gt;: &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt; es mundialmente conocido por lo fácil que resulta leerlo, muchas veces no existen casi diferencias entre seudo código y &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt;; tampoco suelen existir muchas diferencias entre expresar un &lt;a href="https://es.wikipedia.org/wiki/Algoritmo"&gt;algoritmo&lt;/a&gt; matematicamente o en &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt;. &lt;/li&gt;
&lt;/ul&gt;
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&lt;li&gt;&lt;strong&gt;Su ecosistema&lt;/strong&gt;: &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt; es mucho más que un &lt;a href="https://es.wikipedia.org/wiki/Lenguaje_de_programaci%C3%B3n"&gt;lenguaje de programación&lt;/a&gt;, es todo un ecosistema en sí mismo; ya que podemos encontrar un sinnúmero de herramientas para realizar cualquier tipo de tareas; en &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt; podemos encontrar módulos para realizar cálculos científicos, módulos para desarrollar aplicaciones webs, módulos para realizar tareas de administración de sistemas, módulos para trabajar con bases de datos; entre otros. Todos ellos muy fácilmente integrables dentro del lenguaje. La variedad del ecosistema de herramientas de &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt;, nos ofrece la posibilidad de desarrollar una solución completa a cualquier tipo de problema utilizando un solo &lt;a href="https://es.wikipedia.org/wiki/Lenguaje_de_programaci%C3%B3n"&gt;lenguaje de programación&lt;/a&gt;.&lt;/li&gt;
&lt;/ul&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Su integración&lt;/strong&gt;: Otras de las características por la que &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt; es también famoso, es por su fácil integración con otros &lt;a href="https://es.wikipedia.org/wiki/Lenguaje_de_programaci%C3%B3n"&gt;lenguajes de programación&lt;/a&gt;. Generalmente, las grandes empresas suelen tener herramientas desarrolladas en distintos &lt;a href="https://es.wikipedia.org/wiki/Lenguaje_de_programaci%C3%B3n"&gt;lenguajes de programación&lt;/a&gt;; las características dinámicas de &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt;, hacen que sea ideal para unir todos esos distintos componentes en una sola gran aplicación. &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt; puede ser enlazado fácilmente a herramientas desarrollas en &lt;a href="https://es.wikipedia.org/wiki/C_(lenguaje_de_programaci%C3%B3n)"&gt;C&lt;/a&gt;, &lt;a href="https://es.wikipedia.org/wiki/C%2B%2B"&gt;C++&lt;/a&gt; o &lt;a href="https://es.wikipedia.org/wiki/Fortran"&gt;Fortran&lt;/a&gt;.  &lt;/li&gt;
&lt;/ul&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;Eficiencia y productividad&lt;/strong&gt;: Por último, otra de las características que hacen a &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt; tan atractivo, es que con él, se pueden lograr resultados de calidad en una forma mucho más eficiente y productiva. La mayoría de sus módulos están ampliamente testeados y cuentan con el soporte de una amplia comunidad de usuarios; sus características dinámicas e interactivas lo hacen ideal para el análisis exploratorio de datos facilitando los análisis financieros. También es sabido, que la elegancia de su sintaxis hace que se necesiten mucho menos líneas de código para desarrollar un programa en &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt; que en casi cualquier otro  &lt;a href="https://es.wikipedia.org/wiki/Lenguaje_de_programaci%C3%B3n"&gt;lenguaje de programación&lt;/a&gt;. &lt;/li&gt;
&lt;/ul&gt;
&lt;h3 id="Principales-librer&amp;#237;as"&gt;Principales librer&amp;#237;as&lt;a class="anchor-link" href="#Principales-librer&amp;#237;as"&gt;&amp;#182;&lt;/a&gt;&lt;/h3&gt;&lt;p&gt;Las principales librerías que vamos a utilizar para realizar tareas de analisis financiero con &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt; son muchas de las que ya he venido hablando en anteriores artículos; principalmente:&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;&lt;a href="http://pandas.pydata.org/"&gt;Pandas&lt;/a&gt;&lt;/strong&gt;: La cual fue diseñada desde un comienzo para facilitar el análisis de datos financieros, principalmente las &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt; propias del mercado cambiario de acciones. Con las &lt;a href="https://es.wikipedia.org/wiki/Estructura_de_datos"&gt;estructuras de datos&lt;/a&gt; que nos brinda esta librería se vuelve sumamente fácil modelar y resolver problemas financieros.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;&lt;a href="http://www.numpy.org/"&gt;Numpy&lt;/a&gt;&lt;/strong&gt;: El principal modulo matemático que nos ofrece &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt;, en el no solo vamos a encontrar las siempre prácticas &lt;a href='http://es.wikipedia.org/wiki/Matriz_(matem%C3%A1ticas)' target='_blank'&gt;matrices&lt;/a&gt; que facilitan en sobremanera el manejo de información numérica; sino que también vamos a poder encontrar un gran número de funciones matemáticas.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;&lt;a href="http://matplotlib.org/"&gt;Matplotlib&lt;/a&gt;&lt;/strong&gt;: La siempre vigente librería para realizar gráficos en &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt;.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;&lt;a href="http://statsmodels.sourceforge.net/"&gt;statsmodels&lt;/a&gt;&lt;/strong&gt;: Si de estadística se trata, no hay como esta librería para realizar cualquier tipo de analisis estadístico.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;&lt;a href="http://pythonhosted.org//PuLP/"&gt;PuLP&lt;/a&gt;&lt;/strong&gt;: La cual nos permite crear modelos de &lt;a href="http://es.wikipedia.org/wiki/Programaci%C3%B3n_lineal"&gt;programación lineal&lt;/a&gt; en forma muy sencilla.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;&lt;a href="https://www.quandl.com/help/python"&gt;Quandl&lt;/a&gt;&lt;/strong&gt;: Este módulo nos permite interactuar fácilmente con la &lt;a href="https://es.wikipedia.org/wiki/Interfaz_de_programaci%C3%B3n_de_aplicaciones"&gt;API&lt;/a&gt; de &lt;a href="https://www.quandl.com/"&gt;quandl.com&lt;/a&gt; para obtener en forma muy sencilla todo tipo de información financiera.&lt;/p&gt;
&lt;/li&gt;
&lt;li&gt;&lt;p&gt;&lt;strong&gt;&lt;a href="http://www.zipline.io/"&gt;Zipline&lt;/a&gt;&lt;/strong&gt;: &lt;a href="http://www.zipline.io/"&gt;Zipline&lt;/a&gt; es una librería para el &lt;a href="https://en.wikipedia.org/wiki/Algorithmic_trading"&gt;comercio algoritmico&lt;/a&gt;; esta basada en eventos y trata de aproximarse lo más cerca posible a como operan los verdades sistemas de comercio electrónico de las principales bolsas del mundo. &lt;a href="http://www.zipline.io/"&gt;Zipline&lt;/a&gt; es una de las principales tecnologías detrás del popular sitio &lt;a href="https://www.quantopian.com/"&gt;quantopian.com&lt;/a&gt;, la comunidad web que pone a prueba distintos &lt;a href="https://es.wikipedia.org/wiki/Algoritmo"&gt;algoritmos&lt;/a&gt; de &lt;a href="https://en.wikipedia.org/wiki/Algorithmic_trading"&gt;comercio algoritmico&lt;/a&gt;.&lt;/p&gt;
&lt;/li&gt;
&lt;/ul&gt;
&lt;p&gt;Bueno, pero basta de introducciones y pasamos a describir los principales conceptos financieros y como podemos calcularlos con la ayuda de &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt;, ya que el tiempo es dinero!!.&lt;/p&gt;

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&lt;h2 id="Conceptos-b&amp;#225;sicos-de-Finanzas"&gt;Conceptos b&amp;#225;sicos de Finanzas&lt;a class="anchor-link" href="#Conceptos-b&amp;#225;sicos-de-Finanzas"&gt;&amp;#182;&lt;/a&gt;&lt;/h2&gt;&lt;p&gt;Los conceptos más básicos que podemos encontrar dentro de las finanzas son: &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor futuro&lt;/a&gt;, &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor presente&lt;/a&gt;, y la &lt;a href="https://es.wikipedia.org/wiki/Tasa_interna_de_retorno"&gt;tasa interna de retorno&lt;/a&gt;. Estos conceptos nos dicen cuanto nuestro dinero va a crecer si lo depositamos en un banco (&lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor futuro&lt;/a&gt;), cuanto vale hoy la promesa de unos pagos que recibiremos en el futuro(&lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor presente&lt;/a&gt;), y qué tasa de rendimiento podemos obtener de nuestras inversiones (&lt;a href="https://es.wikipedia.org/wiki/Tasa_interna_de_retorno"&gt;tasa interna de retorno&lt;/a&gt;). Recordemos que todos los activos financieros y toda planificación financiera siempre tiene una dimensión de tiempo; así por ejemplo si depositamos USD 100 en un banco que nos paga una tasa de interés anual de 6% , luego de un año obtendríamos un importe de USD 106.&lt;/p&gt;
&lt;h3 id="Valor-Futuro"&gt;Valor Futuro&lt;a class="anchor-link" href="#Valor-Futuro"&gt;&amp;#182;&lt;/a&gt;&lt;/h3&gt;&lt;p&gt;El &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor futuro&lt;/a&gt; o FV (por sus siglas en inglés), nos indica el valor en el futuro que tendrá el dinero depositado hoy en una cuenta bancaria que nos pague intereses. El &lt;em&gt;&lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor futuro&lt;/a&gt;&lt;/em&gt; de  USD X depositado hoy en una cuenta que paga r% de interés anual y que es dejado en la cuenta durante n años es $X * (1 + r)^n$. 
El &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor futuro&lt;/a&gt; es nuestro primer ejemplo de &lt;a href="https://es.wikipedia.org/wiki/Inter%C3%A9s_compuesto"&gt;interés compuesto&lt;/a&gt;, es decir, el principio de que podemos ganar intereses sobre los intereses.
De la definición que dimos del &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor futuro&lt;/a&gt;, podemos obtener su expresión matemática:&lt;/p&gt;
$$FV = X * ( 1 + r )^n$$&lt;p&gt;Como podemos ver, su cálculo es bastante simple. Veamos un ejemplo de como calcular el FV de un depósito de USD 1000 a 3 años y con una tasa de interés del 6% anual.&lt;/p&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# graficos embebidos&lt;/span&gt;
&lt;span class="o"&gt;%&lt;/span&gt;&lt;span class="k"&gt;matplotlib&lt;/span&gt; inline  
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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Ejemplo FV con python&lt;/span&gt;
&lt;span class="c1"&gt;# $1000 al 6% anual por 3 años.&lt;/span&gt;

&lt;span class="c1"&gt;# importando librerías&lt;/span&gt;
&lt;span class="kn"&gt;import&lt;/span&gt; &lt;span class="nn"&gt;numpy&lt;/span&gt; &lt;span class="k"&gt;as&lt;/span&gt; &lt;span class="nn"&gt;np&lt;/span&gt;
&lt;span class="kn"&gt;import&lt;/span&gt; &lt;span class="nn"&gt;matplotlib.pyplot&lt;/span&gt; &lt;span class="k"&gt;as&lt;/span&gt; &lt;span class="nn"&gt;plt&lt;/span&gt;

&lt;span class="n"&gt;x&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mi"&gt;1000&lt;/span&gt;  &lt;span class="c1"&gt;# deposito&lt;/span&gt;
&lt;span class="n"&gt;r&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="mi"&gt;06&lt;/span&gt;    &lt;span class="c1"&gt;# tasa de interes&lt;/span&gt;
&lt;span class="n"&gt;n&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;3&lt;/span&gt;      &lt;span class="c1"&gt;# cantidad de años&lt;/span&gt;

&lt;span class="c1"&gt;# usando la funcion fv de numpy&lt;/span&gt;
&lt;span class="n"&gt;FV&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;np&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;fv&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;pv&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;rate&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;r&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;nper&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;n&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;pmt&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Controlando el resultado&lt;/span&gt;
&lt;span class="n"&gt;x&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt; &lt;span class="o"&gt;+&lt;/span&gt; &lt;span class="n"&gt;r&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="o"&gt;**&lt;/span&gt;&lt;span class="n"&gt;n&lt;/span&gt;
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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Graficando las funciones con interes de 6 y 12 % a 20 años.&lt;/span&gt;
&lt;span class="n"&gt;t&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nb"&gt;list&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nb"&gt;range&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;21&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;

&lt;span class="k"&gt;def&lt;/span&gt; &lt;span class="nf"&gt;fv6&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;num&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
    &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;np&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;fv&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;pv&lt;/span&gt;&lt;span class="o"&gt;=-&lt;/span&gt;&lt;span class="mi"&gt;1000&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;rate&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;r&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;pmt&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;nper&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;num&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;

&lt;span class="k"&gt;def&lt;/span&gt; &lt;span class="nf"&gt;fv12&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;num&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
    &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;np&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;fv&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;pv&lt;/span&gt;&lt;span class="o"&gt;=-&lt;/span&gt;&lt;span class="mi"&gt;1000&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;rate&lt;/span&gt;&lt;span class="o"&gt;=.&lt;/span&gt;&lt;span class="mi"&gt;12&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;pmt&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;nper&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;num&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;title&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s2"&gt;&amp;quot;Graficando FV 6 y 12 %&amp;quot;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;plot&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;t&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;fv6&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;t&lt;/span&gt;&lt;span class="p"&gt;),&lt;/span&gt; &lt;span class="n"&gt;label&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="s2"&gt;&amp;quot;interes 6 %&amp;quot;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;plot&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;t&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;fv12&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;t&lt;/span&gt;&lt;span class="p"&gt;),&lt;/span&gt; &lt;span class="n"&gt;label&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="s2"&gt;&amp;quot;interes 12 %&amp;quot;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;legend&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;loc&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;upper left&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;show&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
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&gt;
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&lt;p&gt;Al graficar dos funciones de &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;FV&lt;/a&gt;, una con una tasa de interes del 6% y otra con una tasa más alta del 12%, podemos ver que el &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor futuro&lt;/a&gt; suele ser bastante sensitivo a los cambios en la tasa de interes, pequeñas variaciones en ella pueden generar grandes saltos a lo largo del tiempo.&lt;/p&gt;
&lt;h4 id="Anualidades"&gt;Anualidades&lt;a class="anchor-link" href="#Anualidades"&gt;&amp;#182;&lt;/a&gt;&lt;/h4&gt;&lt;p&gt;Como podemos ver en el ejemplo anterior, la función fv de &lt;a href="http://www.numpy.org/"&gt;Numpy&lt;/a&gt; tiene varios parámetros, esto es así, porque existen otros casos en los que el cálculo del &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor futuro&lt;/a&gt; se puede volver más complicado; es aquí cuando comenzamos a hablar de &lt;em&gt;anulidades&lt;/em&gt;. La idea de las &lt;em&gt;anualidades&lt;/em&gt; es no solo quedarse con el cálculo simple de cuanto me va a rendir un solo deposito inicial a fin de un período, sino también poder calcular el &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor futuro&lt;/a&gt; de múltiples depósitos que se reinvierten a una misma tasa de interés. Supongamos por ejemplo que queremos hacer 10 depósitos anuales de USD 1000 cada uno, los cuales vamos a ir depositando al comienzo de cada año. ¿Cuál sería en este caso el &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor futuro&lt;/a&gt; de nuestra &lt;em&gt;anualidad&lt;/em&gt; luego del décimo año?.
Ayudémonos de &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt; para calcular la respuesta!&lt;/p&gt;

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&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[6]:&lt;/div&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Calculando el valor de la anualidad con 6% anual&lt;/span&gt;
&lt;span class="n"&gt;x&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mi"&gt;1000&lt;/span&gt; &lt;span class="c1"&gt;# valor de depositos&lt;/span&gt;
&lt;span class="n"&gt;r&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="mi"&gt;06&lt;/span&gt;   &lt;span class="c1"&gt;# tasa de interes&lt;/span&gt;
&lt;span class="n"&gt;n&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;10&lt;/span&gt;    &lt;span class="c1"&gt;# cantidad de años&lt;/span&gt;

&lt;span class="c1"&gt;# usando la funcion fv de numpy&lt;/span&gt;
&lt;span class="n"&gt;FV&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;np&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;fv&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;pv&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;rate&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;r&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;nper&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;n&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;pmt&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;when&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;begin&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;FV&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
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&lt;div class="prompt output_prompt"&gt;Out[6]:&lt;/div&gt;



&lt;div class="output_text output_subarea output_execute_result"&gt;
&lt;pre class="ipynb"&gt;13971.642638923764&lt;/pre&gt;
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&lt;p&gt;Aquí comenzamos con un &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor presente&lt;/a&gt;(PV) de cero, luego realizamos el primer deposito de USD 1000 al comienzo del primer año y continuamos con los sucesivos depósitos al comienzo de cada uno de los restantes años. Para poder entender mejor como funciona la función &lt;code&gt;fv&lt;/code&gt; de &lt;a href="http://www.numpy.org/"&gt;Numpy&lt;/a&gt; voy a explicar un poco más sus parámetros.&lt;/p&gt;
&lt;ul&gt;
&lt;li&gt;&lt;strong&gt;pv&lt;/strong&gt; = este parametro es el &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor presente&lt;/a&gt; de nuestra inversión o &lt;em&gt;anualidad&lt;/em&gt;; en nuestro ejemplo empezamos con un valor de cero; ya que luego vamos a ir realizando los diferentes depósitos de USD 1000.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;rate&lt;/strong&gt; = es la tasa efectiva de interés que nos rendirá la &lt;em&gt;anualidad&lt;/em&gt; por cada período.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;nper&lt;/strong&gt; = Es el número de períodos. Tener en cuenta que si aquí estamos utilizando como unidad de medida de años, nuestra tasa de interés deberá estar expresada en la misma unidad.&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;pmt&lt;/strong&gt; = El valor de los depósitos que vamos a ir invirtiendo en nuestra &lt;em&gt;anualidad&lt;/em&gt;. En nuestro caso el valor de -1000 refleja el importe que vamos a ir depositando año a año.(se expresa con signo negativo, ya que un deposito implica una salida de dinero).&lt;/li&gt;
&lt;li&gt;&lt;strong&gt;when&lt;/strong&gt; = Este parámetro nos dice cuando se van a hacer efectivos nuestros depósitos, ya que el resultado puede ser muy distinto si realizamos el deposito al comienzo(como en nuestro ejemplo) o al final de cada período.&lt;/li&gt;
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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# mismo caso pero con la diferencia de que los depositos se &lt;/span&gt;
&lt;span class="c1"&gt;# realizan al final de cada período.&lt;/span&gt;
&lt;span class="n"&gt;FV&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;np&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;fv&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;pv&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;rate&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;r&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;nper&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;n&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;pmt&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;x&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;when&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;end&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;FV&lt;/span&gt;
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&lt;p&gt;En este último ejemplo, el valor es menor por las perdidas relativas de interés que vamos teniendo al realizar los depósitos al final de cada período en lugar de al comienzo.&lt;/p&gt;

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&lt;h3 id="Valor-Presente"&gt;Valor Presente&lt;a class="anchor-link" href="#Valor-Presente"&gt;&amp;#182;&lt;/a&gt;&lt;/h3&gt;&lt;p&gt;El &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor presente&lt;/a&gt; o PV (por sus siglas en inglés), nos indica el valor que tienen hoy un pago o pagos que recibiremos en el futuro. Supongamos por ejemplo que sabemos que un tío nos va a regalar USD 1000 dentro de 3 años porque somos su sobrino favorito, si también sabemos que un banco nos pagaría un 6% de interés por los depósitos en una caja de ahorro, podríamos calcular el &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor presente&lt;/a&gt; que tendría ese pago futuro de nuestro tío en el día de hoy. La formula para calcular el &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor presente&lt;/a&gt; la podemos derivar de la que utilizamos para calcular el &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor futuro&lt;/a&gt; y se expresaría del siguiente modo:&lt;/p&gt;
$$PV =\frac{fv}{(1 + r)^n}$$&lt;p&gt;Aplicando la esta formula sobre los datos con que contamos, podríamos calcular el valor de hoy de la promesa de pago de USD 1000 de nuestro tío, los que nos daría un &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor presente&lt;/a&gt; de USD 839.62 como se desprende del siguiente cálculo.&lt;/p&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="n"&gt;fv&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;1000&lt;/span&gt;  &lt;span class="c1"&gt;# valor futuro&lt;/span&gt;
&lt;span class="n"&gt;r&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="mi"&gt;06&lt;/span&gt;   &lt;span class="c1"&gt;# tasa de interes&lt;/span&gt;
&lt;span class="n"&gt;n&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="mi"&gt;3&lt;/span&gt;     &lt;span class="c1"&gt;# cantidad de años&lt;/span&gt;

&lt;span class="n"&gt;fv&lt;/span&gt; &lt;span class="o"&gt;/&lt;/span&gt; &lt;span class="p"&gt;((&lt;/span&gt;&lt;span class="mi"&gt;1&lt;/span&gt; &lt;span class="o"&gt;+&lt;/span&gt; &lt;span class="n"&gt;r&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;&lt;span class="o"&gt;**&lt;/span&gt;&lt;span class="n"&gt;n&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# usando la funcion pv de numpy&lt;/span&gt;
&lt;span class="n"&gt;PV&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;np&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;pv&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;fv&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;fv&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;rate&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;r&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;nper&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;n&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;pmt&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;PV&lt;/span&gt;
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&lt;p&gt;Estos USD 839.62 en realidad lo que representan es que si nosotros hoy depositáramos en la caja de ahorro de un banco que nos pague 6% anual de interés el importe de USD 839.62, obtendríamos dentro de los 3 años los mismos USD 1000 que nos ofreció dar nuestro tío dentro de 3 años; o lo que es lo mismo que decir que el &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor futuro&lt;/a&gt; dentro de 3 años de USD 839.62 son los USD 1000 de nuestro querido tío.&lt;/p&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Calculando el valor futuro de los 839.62&lt;/span&gt;
&lt;span class="n"&gt;np&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;fv&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;pv&lt;/span&gt;&lt;span class="o"&gt;=-&lt;/span&gt;&lt;span class="mf"&gt;839.62&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;rate&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;r&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;nper&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;n&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;pmt&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
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&lt;h4 id="Valor-presente-y-anualidades"&gt;Valor presente y anualidades&lt;a class="anchor-link" href="#Valor-presente-y-anualidades"&gt;&amp;#182;&lt;/a&gt;&lt;/h4&gt;&lt;p&gt;Al igual que en el caso del &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor futuro&lt;/a&gt;, aquí también podemos encontrarnos con las &lt;em&gt;anualidades&lt;/em&gt;, es decir, una serie de pagos iguales que recibiremos. El &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor presente&lt;/a&gt; de una &lt;em&gt;anualidad&lt;/em&gt; nos va a decir el valor que tienen hoy los futuros pagos de la &lt;em&gt;anualidad&lt;/em&gt;. Así, por ejemplo si nuestro tío en lugar de regalarnos USD 1000 dentro de 3 años, decide darnos USD 250 al final de cada año durante 5 años; y asumiendo la misma tasa de interés que nos ofrece el banco de 6% anual. El valor presente de esta &lt;em&gt;anualidad&lt;/em&gt; sería USD 1053.09.&lt;/p&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Calculando el valor de la anualidad&lt;/span&gt;
&lt;span class="n"&gt;PV&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;np&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;pv&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;fv&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;rate&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;r&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;nper&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;pmt&lt;/span&gt;&lt;span class="o"&gt;=-&lt;/span&gt;&lt;span class="mi"&gt;250&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;when&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;end&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;PV&lt;/span&gt;
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&lt;h4 id="Eligiendo-la-tasa-de-descuento"&gt;Eligiendo la tasa de descuento&lt;a class="anchor-link" href="#Eligiendo-la-tasa-de-descuento"&gt;&amp;#182;&lt;/a&gt;&lt;/h4&gt;&lt;p&gt;Uno de los puntos sobre el que hacer más foco al calcular el &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor presente&lt;/a&gt; de un flujo de fondos futuro, es el de como elegir la tasa para descontar estos fondos, ya que la tasa que utilicemos es la pieza clave para la exactitud de nuestros cálculos. El principio básico que nos debe guiar la elegir la &lt;em&gt;tasa de descuento&lt;/em&gt; es el de tratar de elegir que sea apropiada al &lt;em&gt;riesgo&lt;/em&gt; y la duración de los flujos de fondos que estamos descontando. En el ejemplo que venimos viendo, como sabemos que nuestro tío es una persona muy solvente y de palabra, podemos considerar que no existe mucho &lt;em&gt;riesgo&lt;/em&gt; en ese flujo de fondos, por lo que utilizar la tasa de interés de una caja de ahorro parece ser un buen criterio para descontar ese flujo. En los casos de las empresas, las mismas suelen utilizar el &lt;em&gt;costo del capital&lt;/em&gt; como una &lt;em&gt;tasa de descuento&lt;/em&gt; apropiada para descontar el flujo futuro de sus inversiones o proyectos.&lt;/p&gt;

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&lt;h3 id="Valor-Presente-Neto"&gt;Valor Presente Neto&lt;a class="anchor-link" href="#Valor-Presente-Neto"&gt;&amp;#182;&lt;/a&gt;&lt;/h3&gt;&lt;p&gt;Un concepto que merece una especial mención, por su importancia dentro del mundo de las finanzas, cuando hablamos del &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor presente&lt;/a&gt;, es el de &lt;a href="https://es.wikipedia.org/wiki/Valor_actual_neto"&gt;Valor Presente Neto&lt;/a&gt;. 
Cuando estamos &lt;em&gt;descontando&lt;/em&gt; flujos de fondos futuros para traerlos a su valor actual, puede ser que éstos flujos no sean homogeneos, por lo que ya no podríamos tratarlos como una &lt;em&gt;anualidad&lt;/em&gt;, ya que los pagos son por importes distintos todos los años; para estos casos debemos utilizar el &lt;a href="https://es.wikipedia.org/wiki/Valor_actual_neto"&gt;Valor Presente Neto&lt;/a&gt;.&lt;/p&gt;
&lt;p&gt;El &lt;a href="https://es.wikipedia.org/wiki/Valor_actual_neto"&gt;Valor Presente Neto&lt;/a&gt; o NPV (por sus siglas en inglés) de una serie de flujos futuros de fondos es su igual a su &lt;a href="https://es.wikipedia.org/wiki/Valor_tiempo_del_dinero"&gt;valor presente&lt;/a&gt; menos el importe de la inversión inicial necesaria para obtener esos mismos flujos de fondos futuros. Su expresión matemática sería la siguiente:&lt;/p&gt;
$$NPV = \sum\limits_{t=1}^n \frac{V_{t}}{(1 + r)^t} - I_{0}$$&lt;p&gt;donde, $V_{t}$ representa el flujo de fondos de cada período $t$; $I_{0}$ es el valor inicial de la inversión; $r$ es la &lt;em&gt;tasa de descuento&lt;/em&gt; utilizada; y $n$ es la cantidad de períodos considerados.&lt;/p&gt;
&lt;p&gt;Volviendo al ejemplo que veníamos utilizando de nuestro generoso tío, esta vez no ofrece pagarnos USD 500 al final del primer año, USD 750 al final del segundo año, USD 1000 al final del tercer año, USD 1250 al final del cuarto año y USD 500 al final del quinto año. El &lt;a href="https://es.wikipedia.org/wiki/Valor_actual_neto"&gt;NPV&lt;/a&gt; de este flujo de fondos sería de USD 3342.56.&lt;/p&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Calculando el valor presente neto.&lt;/span&gt;
&lt;span class="n"&gt;NPV&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;np&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;npv&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;rate&lt;/span&gt;&lt;span class="o"&gt;=.&lt;/span&gt;&lt;span class="mi"&gt;06&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;values&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;500&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;750&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;1000&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;1250&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;500&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="n"&gt;NPV&lt;/span&gt;
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&lt;p&gt;El &lt;a href="https://es.wikipedia.org/wiki/Valor_actual_neto"&gt;Valor Presente Neto&lt;/a&gt; es sumamente utilizado en los análisis financieros, principalmente para evaluar inversiones o proyectos. Como regla general se considera que si el &lt;a href="https://es.wikipedia.org/wiki/Valor_actual_neto"&gt;NPV&lt;/a&gt; de un proyecto o inversión es positivo, se trata de un proyecto rentable en el que deberíamos invertir; en cambio si el &lt;a href="https://es.wikipedia.org/wiki/Valor_actual_neto"&gt;NPV&lt;/a&gt; es negativo estamos ante un mal negocio.&lt;/p&gt;
&lt;p&gt;Si por ejemplo, tendríamos que invertir hoy USD 4000 para poder obtener un flujo de fondos de USD 500 al final del primer año, USD 750 al final del segundo año, USD 1000 al final del tercer año, USD 1250 al final del cuarto año y USD 500 al final del quinto año; estaríamos haciendo un mal negocio, ya que como sabemos el valor presente de esos flujos de fondos es de USD 3342.56, un valor mucho menor a los USD 4000 que deberíamos invertir.&lt;/p&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Calculando el NPV de la inversión de 4000.&lt;/span&gt;
&lt;span class="n"&gt;NPV&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;np&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;npv&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;rate&lt;/span&gt;&lt;span class="o"&gt;=.&lt;/span&gt;&lt;span class="mi"&gt;06&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;values&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mi"&gt;4000&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;500&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;750&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;1000&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;1250&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;500&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="n"&gt;NPV&lt;/span&gt;
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&lt;p&gt;En el ejemplo podemos ver que al utilizar la función &lt;code&gt;npv&lt;/code&gt; de &lt;a href="http://www.numpy.org/"&gt;Numpy&lt;/a&gt;, el primer valor en la lista de valores que le pasamos al parámetro &lt;code&gt;values&lt;/code&gt; debe ser el monto de la inversión inicial, y como implica un desembolso de dinero, su signo es negativo.&lt;/p&gt;
&lt;p&gt;Si en lugar de tener que invertir USD 4000, tendríamos que invertir USD 3000 para obtener el mismo flujo de fondos, ya estaríamos realizando una buena inversión, con &lt;a href="https://es.wikipedia.org/wiki/Valor_actual_neto"&gt;NPV&lt;/a&gt; positivo.&lt;/p&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Calculando el NPV de la inversión de 3000.&lt;/span&gt;
&lt;span class="n"&gt;NPV&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;np&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;npv&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;rate&lt;/span&gt;&lt;span class="o"&gt;=.&lt;/span&gt;&lt;span class="mi"&gt;06&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;values&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mi"&gt;3000&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;500&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;750&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;1000&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;1250&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;500&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="n"&gt;NPV&lt;/span&gt;
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&lt;h3 id="Tasa-interna-de-Retorno"&gt;Tasa interna de Retorno&lt;a class="anchor-link" href="#Tasa-interna-de-Retorno"&gt;&amp;#182;&lt;/a&gt;&lt;/h3&gt;&lt;p&gt;La &lt;a href="https://es.wikipedia.org/wiki/Tasa_interna_de_retorno"&gt;tasa interna de retorno&lt;/a&gt; o IRR (por sus siglas en inglés) es la &lt;em&gt;tasa de descuento&lt;/em&gt; que hace que el &lt;a href="https://es.wikipedia.org/wiki/Valor_actual_neto"&gt;Valor Presente Neto&lt;/a&gt; de los flujos de fondos futuros sea igual a &lt;em&gt;cero&lt;/em&gt;; también puede ser definida como la tasa de &lt;a href="https://es.wikipedia.org/wiki/Inter%C3%A9s_compuesto"&gt;interés compuesto&lt;/a&gt; que nos paga nuestra inversión.&lt;/p&gt;
&lt;p&gt;Al igual que como sucede con el &lt;a href="https://es.wikipedia.org/wiki/Valor_actual_neto"&gt;Valor Presente Neto&lt;/a&gt;, podemos utilizar a la &lt;a href="https://es.wikipedia.org/wiki/Tasa_interna_de_retorno"&gt;tasa interna de retorno&lt;/a&gt; para tomar decisiones financieras. Aquí la regla general es que, al momento de decidir entre diferentes inversiones, deberíamos elegir aquella con una &lt;a href="https://es.wikipedia.org/wiki/Tasa_interna_de_retorno"&gt;tasa interna de retorno&lt;/a&gt; más alta; ya que es la que en menos tiempo no va a devolver nuestra inversión inicial.&lt;/p&gt;
&lt;p&gt;Veamos un ejemplo, supongamos que tenemos USD 1000 para invertir, y que podemos decidir invertir ese dinero en una compañía que nos va a pagar USD 300 al final de cada uno de los próximos cuatro años; o por otra lado, podemos invertir el dinero en una caja de ahorro de un banco que nos va a pagar 5% anual. ¿Dónde deberíamos invertir nuestro dinero?&lt;/p&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Calculando la tasa interna de retorno de la inversion en la compañía&lt;/span&gt;
&lt;span class="n"&gt;IRR&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;np&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;irr&lt;/span&gt;&lt;span class="p"&gt;([&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mi"&gt;1000&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;300&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;300&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;300&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;300&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="n"&gt;IRR&lt;/span&gt; &lt;span class="o"&gt;*&lt;/span&gt; &lt;span class="mi"&gt;100&lt;/span&gt;
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&lt;p&gt;Al calcular la &lt;a href="https://es.wikipedia.org/wiki/Tasa_interna_de_retorno"&gt;IRR&lt;/a&gt; de la inversión que podríamos hacer en la compañía, vemos que nos da un valor de 7.71%; esta tasa es más alta que la tasa del 5% que nos ofrece el banco por el deposito en su caja de ahorro, por lo que deberíamos decidir invertir nuestro dinero en la compañía en lugar de en el banco.&lt;/p&gt;
&lt;h4 id="La-IRR-graficamente"&gt;La IRR graficamente&lt;a class="anchor-link" href="#La-IRR-graficamente"&gt;&amp;#182;&lt;/a&gt;&lt;/h4&gt;&lt;p&gt;Como se desprende de su definición, la &lt;a href="https://es.wikipedia.org/wiki/Tasa_interna_de_retorno"&gt;tasa interna de retorno&lt;/a&gt; es la tasa que hace que el &lt;a href="https://es.wikipedia.org/wiki/Valor_actual_neto"&gt;NPV&lt;/a&gt; se haga cero, por lo que si nos propusiesemos graficar el &lt;a href="https://es.wikipedia.org/wiki/Valor_actual_neto"&gt;NPV&lt;/a&gt; en función de la &lt;em&gt;tasa de descuento&lt;/em&gt;, podríamos encontrar a simple vista, cual es la &lt;a href="https://es.wikipedia.org/wiki/Tasa_interna_de_retorno"&gt;IRR&lt;/a&gt; de un determinado flujo de fondos. Veamos un ejemplo, graficando el flujo de fondos con el que trabajamos anteriormente.&lt;/p&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Graficando el NPV en función de la tasa de descuento&lt;/span&gt;
&lt;span class="k"&gt;def&lt;/span&gt; &lt;span class="nf"&gt;npv_irr&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;tasas&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
    &lt;span class="n"&gt;result&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;[]&lt;/span&gt;
    &lt;span class="k"&gt;for&lt;/span&gt; &lt;span class="n"&gt;tasa&lt;/span&gt; &lt;span class="ow"&gt;in&lt;/span&gt; &lt;span class="n"&gt;tasas&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;
        &lt;span class="n"&gt;result&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;append&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;np&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;npv&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;tasa&lt;/span&gt;&lt;span class="o"&gt;/&lt;/span&gt;&lt;span class="mi"&gt;100&lt;/span&gt;&lt;span class="p"&gt;,[&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mi"&gt;1000&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;300&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;300&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;300&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;300&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt; &lt;span class="p"&gt;))&lt;/span&gt;
    &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;result&lt;/span&gt;

&lt;span class="n"&gt;tasas&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nb"&gt;list&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="nb"&gt;range&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;14&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;

&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;title&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s2"&gt;&amp;quot;NPV y la tasa de descuento&amp;quot;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;plot&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;tasas&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;npv_irr&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;tasas&lt;/span&gt;&lt;span class="p"&gt;),&lt;/span&gt; &lt;span class="n"&gt;marker&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;o&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;label&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;NPV&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;axhline&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;color&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;red&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;axes&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;gca&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
&lt;span class="n"&gt;axes&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;set_ylim&lt;/span&gt;&lt;span class="p"&gt;([&lt;/span&gt;&lt;span class="o"&gt;-&lt;/span&gt;&lt;span class="mi"&gt;200&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;250&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;xticks&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;tasas&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;legend&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;loc&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;upper right&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;plt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;show&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
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&lt;p&gt;Como podemos ver en el gráfico, la función de &lt;a href="https://es.wikipedia.org/wiki/Valor_actual_neto"&gt;NPV&lt;/a&gt;, se hace cero en aproximadamente 7.71%; es decir, el valor de la &lt;a href="https://es.wikipedia.org/wiki/Tasa_interna_de_retorno"&gt;IRR&lt;/a&gt; para ese flujo de fondos.&lt;/p&gt;

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&lt;h2 id="Informaci&amp;#243;n-financiera-y-Pandas"&gt;Informaci&amp;#243;n financiera y Pandas&lt;a class="anchor-link" href="#Informaci&amp;#243;n-financiera-y-Pandas"&gt;&amp;#182;&lt;/a&gt;&lt;/h2&gt;&lt;p&gt;En las finanzas, una de las formas de datos más comunes e importantes con la que nos vamos a encontrar son las &lt;a href="https://es.wikipedia.org/wiki/Serie_temporal"&gt;series de tiempo&lt;/a&gt;; para trabajar con este tipo de información en &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt;, no existe mejor librería que &lt;a href="http://pandas.pydata.org/"&gt;Pandas&lt;/a&gt;; sus dos estructuras de datos básicas, las &lt;em&gt;Series&lt;/em&gt; y el &lt;em&gt;Dataframe&lt;/em&gt;, nos ayudan a manipular información financiera de forma muy conveniente. Además &lt;a href="http://pandas.pydata.org/"&gt;Pandas&lt;/a&gt; nos proporciona una gran batería de métodos y funciones que nos facilitan la obtención y el análisis de datos financieros. Veamos algunos ejemplos de las cosas que podemos con &lt;a href="http://pandas.pydata.org/"&gt;Pandas&lt;/a&gt;.&lt;/p&gt;

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&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[17]:&lt;/div&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Importando pandas y datetime&lt;/span&gt;
&lt;span class="kn"&gt;import&lt;/span&gt; &lt;span class="nn"&gt;pandas&lt;/span&gt; &lt;span class="k"&gt;as&lt;/span&gt; &lt;span class="nn"&gt;pd&lt;/span&gt;
&lt;span class="kn"&gt;import&lt;/span&gt; &lt;span class="nn"&gt;pandas.io.data&lt;/span&gt; &lt;span class="k"&gt;as&lt;/span&gt; &lt;span class="nn"&gt;web&lt;/span&gt;
&lt;span class="kn"&gt;import&lt;/span&gt; &lt;span class="nn"&gt;datetime&lt;/span&gt; &lt;span class="k"&gt;as&lt;/span&gt; &lt;span class="nn"&gt;dt&lt;/span&gt;

&lt;span class="c1"&gt;# Extrayendo información financiera de Yahoo! Finance&lt;/span&gt;
&lt;span class="n"&gt;inicio&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;dt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;datetime&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;2014&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;1&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;fin&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;dt&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;datetime&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;2014&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;12&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="mi"&gt;31&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;

&lt;span class="n"&gt;msft&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;web&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;DataReader&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s2"&gt;&amp;quot;MSFT&amp;quot;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s1"&gt;&amp;#39;yahoo&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;inicio&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;fin&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;  &lt;span class="c1"&gt;# información de Microsoft&lt;/span&gt;
&lt;span class="n"&gt;aapl&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;web&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;DataReader&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s2"&gt;&amp;quot;AAPL&amp;quot;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s1"&gt;&amp;#39;yahoo&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;inicio&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;fin&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;  &lt;span class="c1"&gt;# información de Apple&lt;/span&gt;
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&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[18]:&lt;/div&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="n"&gt;msft&lt;/span&gt;&lt;span class="p"&gt;[:&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;
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  &lt;thead&gt;
    &lt;tr style="text-align: right;"&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;Open&lt;/th&gt;
      &lt;th&gt;High&lt;/th&gt;
      &lt;th&gt;Low&lt;/th&gt;
      &lt;th&gt;Close&lt;/th&gt;
      &lt;th&gt;Volume&lt;/th&gt;
      &lt;th&gt;Adj Close&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;Date&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
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  &lt;/thead&gt;
  &lt;tbody&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-02&lt;/th&gt;
      &lt;td&gt;37.349998&lt;/td&gt;
      &lt;td&gt;37.400002&lt;/td&gt;
      &lt;td&gt;37.099998&lt;/td&gt;
      &lt;td&gt;37.160000&lt;/td&gt;
      &lt;td&gt;30632200&lt;/td&gt;
      &lt;td&gt;35.448341&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-03&lt;/th&gt;
      &lt;td&gt;37.200001&lt;/td&gt;
      &lt;td&gt;37.220001&lt;/td&gt;
      &lt;td&gt;36.599998&lt;/td&gt;
      &lt;td&gt;36.910000&lt;/td&gt;
      &lt;td&gt;31134800&lt;/td&gt;
      &lt;td&gt;35.209856&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-06&lt;/th&gt;
      &lt;td&gt;36.849998&lt;/td&gt;
      &lt;td&gt;36.889999&lt;/td&gt;
      &lt;td&gt;36.110001&lt;/td&gt;
      &lt;td&gt;36.130001&lt;/td&gt;
      &lt;td&gt;43603700&lt;/td&gt;
      &lt;td&gt;34.465786&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
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&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[19]:&lt;/div&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="n"&gt;aapl&lt;/span&gt;&lt;span class="p"&gt;[:&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;
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&lt;div style="max-height:1000px;max-width:1500px;overflow:auto;"&gt;
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  &lt;thead&gt;
    &lt;tr style="text-align: right;"&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;Open&lt;/th&gt;
      &lt;th&gt;High&lt;/th&gt;
      &lt;th&gt;Low&lt;/th&gt;
      &lt;th&gt;Close&lt;/th&gt;
      &lt;th&gt;Volume&lt;/th&gt;
      &lt;th&gt;Adj Close&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;Date&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
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    &lt;tr&gt;
      &lt;th&gt;2014-01-02&lt;/th&gt;
      &lt;td&gt;555.680008&lt;/td&gt;
      &lt;td&gt;557.029945&lt;/td&gt;
      &lt;td&gt;552.020004&lt;/td&gt;
      &lt;td&gt;553.129990&lt;/td&gt;
      &lt;td&gt;58671200&lt;/td&gt;
      &lt;td&gt;76.419139&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-03&lt;/th&gt;
      &lt;td&gt;552.860023&lt;/td&gt;
      &lt;td&gt;553.699989&lt;/td&gt;
      &lt;td&gt;540.430046&lt;/td&gt;
      &lt;td&gt;540.980019&lt;/td&gt;
      &lt;td&gt;98116900&lt;/td&gt;
      &lt;td&gt;74.740527&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-06&lt;/th&gt;
      &lt;td&gt;537.450005&lt;/td&gt;
      &lt;td&gt;546.800018&lt;/td&gt;
      &lt;td&gt;533.599983&lt;/td&gt;
      &lt;td&gt;543.930046&lt;/td&gt;
      &lt;td&gt;103152700&lt;/td&gt;
      &lt;td&gt;75.148096&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

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&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[20]:&lt;/div&gt;

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        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Seleccionando solo el Adj Close price de Enero 2014&lt;/span&gt;
&lt;span class="n"&gt;msft01&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;msft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;2014-01&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;][[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Adj Close&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]]&lt;/span&gt;
&lt;span class="n"&gt;aapl01&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;aapl&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;2014-01&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;][[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Adj Close&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]]&lt;/span&gt;

&lt;span class="n"&gt;msft01&lt;/span&gt;&lt;span class="p"&gt;[:&lt;/span&gt;&lt;span class="mi"&gt;3&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
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  &lt;thead&gt;
    &lt;tr style="text-align: right;"&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;Adj Close&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;Date&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
    &lt;/tr&gt;
  &lt;/thead&gt;
  &lt;tbody&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-02&lt;/th&gt;
      &lt;td&gt;35.448341&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-03&lt;/th&gt;
      &lt;td&gt;35.209856&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-06&lt;/th&gt;
      &lt;td&gt;34.465786&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
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&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[21]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="n"&gt;aapl01&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;head&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;  &lt;span class="c1"&gt;# head() nos muestra los primeros 5 registros&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
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&lt;div class="output_wrapper"&gt;
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&lt;div class="prompt output_prompt"&gt;Out[21]:&lt;/div&gt;


&lt;div class="output_html rendered_html output_subarea output_execute_result"&gt;
&lt;div style="max-height:1000px;max-width:1500px;overflow:auto;"&gt;
&lt;table border="1" class="dataframe"&gt;
  &lt;thead&gt;
    &lt;tr style="text-align: right;"&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;Adj Close&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;Date&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
    &lt;/tr&gt;
  &lt;/thead&gt;
  &lt;tbody&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-02&lt;/th&gt;
      &lt;td&gt;76.419139&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-03&lt;/th&gt;
      &lt;td&gt;74.740527&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-06&lt;/th&gt;
      &lt;td&gt;75.148096&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-07&lt;/th&gt;
      &lt;td&gt;74.610653&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-08&lt;/th&gt;
      &lt;td&gt;75.083151&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;
&lt;/div&gt;
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&lt;/div&gt;

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&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[22]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# tambien se puede seleccionar un rango de tiempo&lt;/span&gt;
&lt;span class="n"&gt;msft&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;2014-02&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;:&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;2014-02-13&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;  &lt;span class="c1"&gt;# desde el 1 al 13 de febrero&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

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&lt;div class="output_area"&gt;
&lt;div class="prompt output_prompt"&gt;Out[22]:&lt;/div&gt;


&lt;div class="output_html rendered_html output_subarea output_execute_result"&gt;
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&lt;table border="1" class="dataframe"&gt;
  &lt;thead&gt;
    &lt;tr style="text-align: right;"&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;Open&lt;/th&gt;
      &lt;th&gt;High&lt;/th&gt;
      &lt;th&gt;Low&lt;/th&gt;
      &lt;th&gt;Close&lt;/th&gt;
      &lt;th&gt;Volume&lt;/th&gt;
      &lt;th&gt;Adj Close&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;Date&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
    &lt;/tr&gt;
  &lt;/thead&gt;
  &lt;tbody&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-02-03&lt;/th&gt;
      &lt;td&gt;37.740002&lt;/td&gt;
      &lt;td&gt;37.990002&lt;/td&gt;
      &lt;td&gt;36.430000&lt;/td&gt;
      &lt;td&gt;36.480000&lt;/td&gt;
      &lt;td&gt;64063100&lt;/td&gt;
      &lt;td&gt;34.799662&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-02-04&lt;/th&gt;
      &lt;td&gt;36.970001&lt;/td&gt;
      &lt;td&gt;37.189999&lt;/td&gt;
      &lt;td&gt;36.250000&lt;/td&gt;
      &lt;td&gt;36.349998&lt;/td&gt;
      &lt;td&gt;54697900&lt;/td&gt;
      &lt;td&gt;34.675649&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-02-05&lt;/th&gt;
      &lt;td&gt;36.290001&lt;/td&gt;
      &lt;td&gt;36.470001&lt;/td&gt;
      &lt;td&gt;35.799999&lt;/td&gt;
      &lt;td&gt;35.820000&lt;/td&gt;
      &lt;td&gt;55814400&lt;/td&gt;
      &lt;td&gt;34.170063&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-02-06&lt;/th&gt;
      &lt;td&gt;35.799999&lt;/td&gt;
      &lt;td&gt;36.250000&lt;/td&gt;
      &lt;td&gt;35.689999&lt;/td&gt;
      &lt;td&gt;36.180000&lt;/td&gt;
      &lt;td&gt;35351800&lt;/td&gt;
      &lt;td&gt;34.513482&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-02-07&lt;/th&gt;
      &lt;td&gt;36.320000&lt;/td&gt;
      &lt;td&gt;36.590000&lt;/td&gt;
      &lt;td&gt;36.009998&lt;/td&gt;
      &lt;td&gt;36.560001&lt;/td&gt;
      &lt;td&gt;33260500&lt;/td&gt;
      &lt;td&gt;34.875979&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-02-10&lt;/th&gt;
      &lt;td&gt;36.630001&lt;/td&gt;
      &lt;td&gt;36.799999&lt;/td&gt;
      &lt;td&gt;36.290001&lt;/td&gt;
      &lt;td&gt;36.799999&lt;/td&gt;
      &lt;td&gt;26767000&lt;/td&gt;
      &lt;td&gt;35.104922&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-02-11&lt;/th&gt;
      &lt;td&gt;36.880001&lt;/td&gt;
      &lt;td&gt;37.259998&lt;/td&gt;
      &lt;td&gt;36.860001&lt;/td&gt;
      &lt;td&gt;37.169998&lt;/td&gt;
      &lt;td&gt;32141400&lt;/td&gt;
      &lt;td&gt;35.457878&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-02-12&lt;/th&gt;
      &lt;td&gt;37.349998&lt;/td&gt;
      &lt;td&gt;37.599998&lt;/td&gt;
      &lt;td&gt;37.299999&lt;/td&gt;
      &lt;td&gt;37.470001&lt;/td&gt;
      &lt;td&gt;27051800&lt;/td&gt;
      &lt;td&gt;35.744063&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-02-13&lt;/th&gt;
      &lt;td&gt;37.330002&lt;/td&gt;
      &lt;td&gt;37.860001&lt;/td&gt;
      &lt;td&gt;37.330002&lt;/td&gt;
      &lt;td&gt;37.610001&lt;/td&gt;
      &lt;td&gt;37635500&lt;/td&gt;
      &lt;td&gt;35.877614&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[23]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# combinando ambos resultados&lt;/span&gt;
&lt;span class="n"&gt;close&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;pd&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;concat&lt;/span&gt;&lt;span class="p"&gt;([&lt;/span&gt;&lt;span class="n"&gt;msft01&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;aapl01&lt;/span&gt;&lt;span class="p"&gt;],&lt;/span&gt; &lt;span class="n"&gt;keys&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;MSFT&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s1"&gt;&amp;#39;AAPL&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;
&lt;span class="n"&gt;close&lt;/span&gt;&lt;span class="p"&gt;[:&lt;/span&gt;&lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;
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    &lt;/div&gt;
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  &lt;thead&gt;
    &lt;tr style="text-align: right;"&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;Adj Close&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;Date&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
    &lt;/tr&gt;
  &lt;/thead&gt;
  &lt;tbody&gt;
    &lt;tr&gt;
      &lt;th rowspan="5" valign="top"&gt;MSFT&lt;/th&gt;
      &lt;th&gt;2014-01-02&lt;/th&gt;
      &lt;td&gt;35.448341&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-03&lt;/th&gt;
      &lt;td&gt;35.209856&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-06&lt;/th&gt;
      &lt;td&gt;34.465786&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-07&lt;/th&gt;
      &lt;td&gt;34.732887&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-08&lt;/th&gt;
      &lt;td&gt;34.112826&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;
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&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[24]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# seleccionando los primeros 5 registros de AAPL&lt;/span&gt;
&lt;span class="n"&gt;close&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;ix&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;AAPL&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;][:&lt;/span&gt;&lt;span class="mi"&gt;5&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;  
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&lt;div class="prompt output_prompt"&gt;Out[24]:&lt;/div&gt;


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  &lt;thead&gt;
    &lt;tr style="text-align: right;"&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;Adj Close&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;Date&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
    &lt;/tr&gt;
  &lt;/thead&gt;
  &lt;tbody&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-02&lt;/th&gt;
      &lt;td&gt;76.419139&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-03&lt;/th&gt;
      &lt;td&gt;74.740527&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-06&lt;/th&gt;
      &lt;td&gt;75.148096&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-07&lt;/th&gt;
      &lt;td&gt;74.610653&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-08&lt;/th&gt;
      &lt;td&gt;75.083151&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;
&lt;/div&gt;
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&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[25]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# insertando una nueva columna con el simbolo&lt;/span&gt;
&lt;span class="n"&gt;msft&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;insert&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s1"&gt;&amp;#39;Symbol&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s1"&gt;&amp;#39;MSFT&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;aapl&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;insert&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;0&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s1"&gt;&amp;#39;Symbol&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s1"&gt;&amp;#39;AAPL&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;

&lt;span class="n"&gt;msft&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;head&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
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  &lt;thead&gt;
    &lt;tr style="text-align: right;"&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;Symbol&lt;/th&gt;
      &lt;th&gt;Open&lt;/th&gt;
      &lt;th&gt;High&lt;/th&gt;
      &lt;th&gt;Low&lt;/th&gt;
      &lt;th&gt;Close&lt;/th&gt;
      &lt;th&gt;Volume&lt;/th&gt;
      &lt;th&gt;Adj Close&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;Date&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
    &lt;/tr&gt;
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  &lt;tbody&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-02&lt;/th&gt;
      &lt;td&gt;MSFT&lt;/td&gt;
      &lt;td&gt;37.349998&lt;/td&gt;
      &lt;td&gt;37.400002&lt;/td&gt;
      &lt;td&gt;37.099998&lt;/td&gt;
      &lt;td&gt;37.160000&lt;/td&gt;
      &lt;td&gt;30632200&lt;/td&gt;
      &lt;td&gt;35.448341&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-03&lt;/th&gt;
      &lt;td&gt;MSFT&lt;/td&gt;
      &lt;td&gt;37.200001&lt;/td&gt;
      &lt;td&gt;37.220001&lt;/td&gt;
      &lt;td&gt;36.599998&lt;/td&gt;
      &lt;td&gt;36.910000&lt;/td&gt;
      &lt;td&gt;31134800&lt;/td&gt;
      &lt;td&gt;35.209856&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-06&lt;/th&gt;
      &lt;td&gt;MSFT&lt;/td&gt;
      &lt;td&gt;36.849998&lt;/td&gt;
      &lt;td&gt;36.889999&lt;/td&gt;
      &lt;td&gt;36.110001&lt;/td&gt;
      &lt;td&gt;36.130001&lt;/td&gt;
      &lt;td&gt;43603700&lt;/td&gt;
      &lt;td&gt;34.465786&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-07&lt;/th&gt;
      &lt;td&gt;MSFT&lt;/td&gt;
      &lt;td&gt;36.330002&lt;/td&gt;
      &lt;td&gt;36.490002&lt;/td&gt;
      &lt;td&gt;36.209999&lt;/td&gt;
      &lt;td&gt;36.410000&lt;/td&gt;
      &lt;td&gt;35802800&lt;/td&gt;
      &lt;td&gt;34.732887&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-08&lt;/th&gt;
      &lt;td&gt;MSFT&lt;/td&gt;
      &lt;td&gt;36.000000&lt;/td&gt;
      &lt;td&gt;36.139999&lt;/td&gt;
      &lt;td&gt;35.580002&lt;/td&gt;
      &lt;td&gt;35.759998&lt;/td&gt;
      &lt;td&gt;59971700&lt;/td&gt;
      &lt;td&gt;34.112826&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[26]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# concatenando toda la información y reseteando el indice&lt;/span&gt;
&lt;span class="n"&gt;combinado&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;pd&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;concat&lt;/span&gt;&lt;span class="p"&gt;([&lt;/span&gt;&lt;span class="n"&gt;msft&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;aapl&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;sort_index&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
&lt;span class="n"&gt;datos_todo&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;combinado&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;reset_index&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;

&lt;span class="n"&gt;datos_todo&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;head&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;div class="output_wrapper"&gt;
&lt;div class="output"&gt;


&lt;div class="output_area"&gt;
&lt;div class="prompt output_prompt"&gt;Out[26]:&lt;/div&gt;


&lt;div class="output_html rendered_html output_subarea output_execute_result"&gt;
&lt;div style="max-height:1000px;max-width:1500px;overflow:auto;"&gt;
&lt;table border="1" class="dataframe"&gt;
  &lt;thead&gt;
    &lt;tr style="text-align: right;"&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;Date&lt;/th&gt;
      &lt;th&gt;Symbol&lt;/th&gt;
      &lt;th&gt;Open&lt;/th&gt;
      &lt;th&gt;High&lt;/th&gt;
      &lt;th&gt;Low&lt;/th&gt;
      &lt;th&gt;Close&lt;/th&gt;
      &lt;th&gt;Volume&lt;/th&gt;
      &lt;th&gt;Adj Close&lt;/th&gt;
    &lt;/tr&gt;
  &lt;/thead&gt;
  &lt;tbody&gt;
    &lt;tr&gt;
      &lt;th&gt;0&lt;/th&gt;
      &lt;td&gt;2014-01-02&lt;/td&gt;
      &lt;td&gt;MSFT&lt;/td&gt;
      &lt;td&gt;37.349998&lt;/td&gt;
      &lt;td&gt;37.400002&lt;/td&gt;
      &lt;td&gt;37.099998&lt;/td&gt;
      &lt;td&gt;37.160000&lt;/td&gt;
      &lt;td&gt;30632200&lt;/td&gt;
      &lt;td&gt;35.448341&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;1&lt;/th&gt;
      &lt;td&gt;2014-01-02&lt;/td&gt;
      &lt;td&gt;AAPL&lt;/td&gt;
      &lt;td&gt;555.680008&lt;/td&gt;
      &lt;td&gt;557.029945&lt;/td&gt;
      &lt;td&gt;552.020004&lt;/td&gt;
      &lt;td&gt;553.129990&lt;/td&gt;
      &lt;td&gt;58671200&lt;/td&gt;
      &lt;td&gt;76.419139&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2&lt;/th&gt;
      &lt;td&gt;2014-01-03&lt;/td&gt;
      &lt;td&gt;MSFT&lt;/td&gt;
      &lt;td&gt;37.200001&lt;/td&gt;
      &lt;td&gt;37.220001&lt;/td&gt;
      &lt;td&gt;36.599998&lt;/td&gt;
      &lt;td&gt;36.910000&lt;/td&gt;
      &lt;td&gt;31134800&lt;/td&gt;
      &lt;td&gt;35.209856&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;3&lt;/th&gt;
      &lt;td&gt;2014-01-03&lt;/td&gt;
      &lt;td&gt;AAPL&lt;/td&gt;
      &lt;td&gt;552.860023&lt;/td&gt;
      &lt;td&gt;553.699989&lt;/td&gt;
      &lt;td&gt;540.430046&lt;/td&gt;
      &lt;td&gt;540.980019&lt;/td&gt;
      &lt;td&gt;98116900&lt;/td&gt;
      &lt;td&gt;74.740527&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;4&lt;/th&gt;
      &lt;td&gt;2014-01-06&lt;/td&gt;
      &lt;td&gt;MSFT&lt;/td&gt;
      &lt;td&gt;36.849998&lt;/td&gt;
      &lt;td&gt;36.889999&lt;/td&gt;
      &lt;td&gt;36.110001&lt;/td&gt;
      &lt;td&gt;36.130001&lt;/td&gt;
      &lt;td&gt;43603700&lt;/td&gt;
      &lt;td&gt;34.465786&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[27]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Armando una tabla pivot del precio de cierre&lt;/span&gt;
&lt;span class="n"&gt;pivot&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;datos_todo&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;pivot&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;index&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Date&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;columns&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Symbol&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; 
                         &lt;span class="n"&gt;values&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Adj Close&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;

&lt;span class="n"&gt;pivot&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;head&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;div class="output_wrapper"&gt;
&lt;div class="output"&gt;


&lt;div class="output_area"&gt;
&lt;div class="prompt output_prompt"&gt;Out[27]:&lt;/div&gt;


&lt;div class="output_html rendered_html output_subarea output_execute_result"&gt;
&lt;div style="max-height:1000px;max-width:1500px;overflow:auto;"&gt;
&lt;table border="1" class="dataframe"&gt;
  &lt;thead&gt;
    &lt;tr style="text-align: right;"&gt;
      &lt;th&gt;Symbol&lt;/th&gt;
      &lt;th&gt;AAPL&lt;/th&gt;
      &lt;th&gt;MSFT&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;Date&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
    &lt;/tr&gt;
  &lt;/thead&gt;
  &lt;tbody&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-02&lt;/th&gt;
      &lt;td&gt;76.419139&lt;/td&gt;
      &lt;td&gt;35.448341&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-03&lt;/th&gt;
      &lt;td&gt;74.740527&lt;/td&gt;
      &lt;td&gt;35.209856&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-06&lt;/th&gt;
      &lt;td&gt;75.148096&lt;/td&gt;
      &lt;td&gt;34.465786&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-07&lt;/th&gt;
      &lt;td&gt;74.610653&lt;/td&gt;
      &lt;td&gt;34.732887&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-08&lt;/th&gt;
      &lt;td&gt;75.083151&lt;/td&gt;
      &lt;td&gt;34.112826&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[28]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Obteniendo datos de multiples empresas&lt;/span&gt;
&lt;span class="k"&gt;def&lt;/span&gt; &lt;span class="nf"&gt;all_stocks&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;symbols&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;start&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;end&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
       &lt;span class="k"&gt;def&lt;/span&gt; &lt;span class="nf"&gt;data&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;symbols&lt;/span&gt;&lt;span class="p"&gt;):&lt;/span&gt;
           &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;pd&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;io&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;data&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;DataReader&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;symbols&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s1"&gt;&amp;#39;yahoo&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;start&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;end&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
       &lt;span class="n"&gt;datas&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="nb"&gt;map&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;data&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;symbols&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
       &lt;span class="k"&gt;return&lt;/span&gt; &lt;span class="n"&gt;pd&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;concat&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;datas&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;keys&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="n"&gt;symbols&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;names&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;symbols&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Date&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;])&lt;/span&gt;

&lt;span class="n"&gt;simbolos&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;AAPL&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;MSFT&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;GOOG&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;IBM&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;

&lt;span class="n"&gt;all_data&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;all_stocks&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;simbolos&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;inicio&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="n"&gt;fin&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;
&lt;span class="n"&gt;all_data&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;head&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;div class="output_wrapper"&gt;
&lt;div class="output"&gt;


&lt;div class="output_area"&gt;
&lt;div class="prompt output_prompt"&gt;Out[28]:&lt;/div&gt;


&lt;div class="output_html rendered_html output_subarea output_execute_result"&gt;
&lt;div style="max-height:1000px;max-width:1500px;overflow:auto;"&gt;
&lt;table border="1" class="dataframe"&gt;
  &lt;thead&gt;
    &lt;tr style="text-align: right;"&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;Open&lt;/th&gt;
      &lt;th&gt;High&lt;/th&gt;
      &lt;th&gt;Low&lt;/th&gt;
      &lt;th&gt;Close&lt;/th&gt;
      &lt;th&gt;Volume&lt;/th&gt;
      &lt;th&gt;Adj Close&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;symbols&lt;/th&gt;
      &lt;th&gt;Date&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
    &lt;/tr&gt;
  &lt;/thead&gt;
  &lt;tbody&gt;
    &lt;tr&gt;
      &lt;th rowspan="5" valign="top"&gt;AAPL&lt;/th&gt;
      &lt;th&gt;2014-01-02&lt;/th&gt;
      &lt;td&gt;555.680008&lt;/td&gt;
      &lt;td&gt;557.029945&lt;/td&gt;
      &lt;td&gt;552.020004&lt;/td&gt;
      &lt;td&gt;553.129990&lt;/td&gt;
      &lt;td&gt;58671200&lt;/td&gt;
      &lt;td&gt;76.419139&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-03&lt;/th&gt;
      &lt;td&gt;552.860023&lt;/td&gt;
      &lt;td&gt;553.699989&lt;/td&gt;
      &lt;td&gt;540.430046&lt;/td&gt;
      &lt;td&gt;540.980019&lt;/td&gt;
      &lt;td&gt;98116900&lt;/td&gt;
      &lt;td&gt;74.740527&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-06&lt;/th&gt;
      &lt;td&gt;537.450005&lt;/td&gt;
      &lt;td&gt;546.800018&lt;/td&gt;
      &lt;td&gt;533.599983&lt;/td&gt;
      &lt;td&gt;543.930046&lt;/td&gt;
      &lt;td&gt;103152700&lt;/td&gt;
      &lt;td&gt;75.148096&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-07&lt;/th&gt;
      &lt;td&gt;544.320015&lt;/td&gt;
      &lt;td&gt;545.960052&lt;/td&gt;
      &lt;td&gt;537.919975&lt;/td&gt;
      &lt;td&gt;540.039970&lt;/td&gt;
      &lt;td&gt;79302300&lt;/td&gt;
      &lt;td&gt;74.610653&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-08&lt;/th&gt;
      &lt;td&gt;538.810036&lt;/td&gt;
      &lt;td&gt;545.559990&lt;/td&gt;
      &lt;td&gt;538.689980&lt;/td&gt;
      &lt;td&gt;543.459969&lt;/td&gt;
      &lt;td&gt;64632400&lt;/td&gt;
      &lt;td&gt;75.083151&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[29]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="n"&gt;all_data&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;ix&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;GOOG&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;head&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;  &lt;span class="c1"&gt;# información de google&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;div class="output_wrapper"&gt;
&lt;div class="output"&gt;


&lt;div class="output_area"&gt;
&lt;div class="prompt output_prompt"&gt;Out[29]:&lt;/div&gt;


&lt;div class="output_html rendered_html output_subarea output_execute_result"&gt;
&lt;div style="max-height:1000px;max-width:1500px;overflow:auto;"&gt;
&lt;table border="1" class="dataframe"&gt;
  &lt;thead&gt;
    &lt;tr style="text-align: right;"&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;Open&lt;/th&gt;
      &lt;th&gt;High&lt;/th&gt;
      &lt;th&gt;Low&lt;/th&gt;
      &lt;th&gt;Close&lt;/th&gt;
      &lt;th&gt;Volume&lt;/th&gt;
      &lt;th&gt;Adj Close&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;Date&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
    &lt;/tr&gt;
  &lt;/thead&gt;
  &lt;tbody&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-03-27&lt;/th&gt;
      &lt;td&gt;568.002570&lt;/td&gt;
      &lt;td&gt;568.002570&lt;/td&gt;
      &lt;td&gt;552.922516&lt;/td&gt;
      &lt;td&gt;558.462551&lt;/td&gt;
      &lt;td&gt;13100&lt;/td&gt;
      &lt;td&gt;558.462551&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-03-28&lt;/th&gt;
      &lt;td&gt;561.202549&lt;/td&gt;
      &lt;td&gt;566.432590&lt;/td&gt;
      &lt;td&gt;558.672477&lt;/td&gt;
      &lt;td&gt;559.992504&lt;/td&gt;
      &lt;td&gt;41200&lt;/td&gt;
      &lt;td&gt;559.992504&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-03-31&lt;/th&gt;
      &lt;td&gt;566.892592&lt;/td&gt;
      &lt;td&gt;567.002574&lt;/td&gt;
      &lt;td&gt;556.932537&lt;/td&gt;
      &lt;td&gt;556.972503&lt;/td&gt;
      &lt;td&gt;10800&lt;/td&gt;
      &lt;td&gt;556.972503&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-04-01&lt;/th&gt;
      &lt;td&gt;558.712565&lt;/td&gt;
      &lt;td&gt;568.452595&lt;/td&gt;
      &lt;td&gt;558.712565&lt;/td&gt;
      &lt;td&gt;567.162558&lt;/td&gt;
      &lt;td&gt;7900&lt;/td&gt;
      &lt;td&gt;567.162558&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-04-02&lt;/th&gt;
      &lt;td&gt;599.992707&lt;/td&gt;
      &lt;td&gt;604.832763&lt;/td&gt;
      &lt;td&gt;562.192568&lt;/td&gt;
      &lt;td&gt;567.002574&lt;/td&gt;
      &lt;td&gt;147100&lt;/td&gt;
      &lt;td&gt;567.002574&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[30]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Graficando los datos.&lt;/span&gt;
&lt;span class="n"&gt;solo_cierre&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;all_data&lt;/span&gt;&lt;span class="p"&gt;[[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Adj Close&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]]&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;reset_index&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
&lt;span class="n"&gt;pivot_cierre&lt;/span&gt; &lt;span class="o"&gt;=&lt;/span&gt; &lt;span class="n"&gt;solo_cierre&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;pivot&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;Date&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s1"&gt;&amp;#39;symbols&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt; &lt;span class="s1"&gt;&amp;#39;Adj Close&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;)&lt;/span&gt;

&lt;span class="n"&gt;pivot_cierre&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;head&lt;/span&gt;&lt;span class="p"&gt;()&lt;/span&gt;
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;div class="output_wrapper"&gt;
&lt;div class="output"&gt;


&lt;div class="output_area"&gt;
&lt;div class="prompt output_prompt"&gt;Out[30]:&lt;/div&gt;


&lt;div class="output_html rendered_html output_subarea output_execute_result"&gt;
&lt;div style="max-height:1000px;max-width:1500px;overflow:auto;"&gt;
&lt;table border="1" class="dataframe"&gt;
  &lt;thead&gt;
    &lt;tr style="text-align: right;"&gt;
      &lt;th&gt;symbols&lt;/th&gt;
      &lt;th&gt;AAPL&lt;/th&gt;
      &lt;th&gt;GOOG&lt;/th&gt;
      &lt;th&gt;IBM&lt;/th&gt;
      &lt;th&gt;MSFT&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;Date&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
      &lt;th&gt;&lt;/th&gt;
    &lt;/tr&gt;
  &lt;/thead&gt;
  &lt;tbody&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-02&lt;/th&gt;
      &lt;td&gt;76.419139&lt;/td&gt;
      &lt;td&gt;NaN&lt;/td&gt;
      &lt;td&gt;177.049572&lt;/td&gt;
      &lt;td&gt;35.448341&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-03&lt;/th&gt;
      &lt;td&gt;74.740527&lt;/td&gt;
      &lt;td&gt;NaN&lt;/td&gt;
      &lt;td&gt;178.108836&lt;/td&gt;
      &lt;td&gt;35.209856&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-06&lt;/th&gt;
      &lt;td&gt;75.148096&lt;/td&gt;
      &lt;td&gt;NaN&lt;/td&gt;
      &lt;td&gt;177.498090&lt;/td&gt;
      &lt;td&gt;34.465786&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-07&lt;/th&gt;
      &lt;td&gt;74.610653&lt;/td&gt;
      &lt;td&gt;NaN&lt;/td&gt;
      &lt;td&gt;181.038515&lt;/td&gt;
      &lt;td&gt;34.732887&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
      &lt;th&gt;2014-01-08&lt;/th&gt;
      &lt;td&gt;75.083151&lt;/td&gt;
      &lt;td&gt;NaN&lt;/td&gt;
      &lt;td&gt;179.378044&lt;/td&gt;
      &lt;td&gt;34.112826&lt;/td&gt;
    &lt;/tr&gt;
  &lt;/tbody&gt;
&lt;/table&gt;
&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[31]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Graficando la información de Apple&lt;/span&gt;
&lt;span class="n"&gt;pivot_cierre&lt;/span&gt;&lt;span class="p"&gt;[&lt;/span&gt;&lt;span class="s1"&gt;&amp;#39;AAPL&amp;#39;&lt;/span&gt;&lt;span class="p"&gt;]&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;plot&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;figsize&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;12&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;8&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt;  
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;div class="output_wrapper"&gt;
&lt;div class="output"&gt;


&lt;div class="output_area"&gt;
&lt;div class="prompt output_prompt"&gt;Out[31]:&lt;/div&gt;



&lt;div class="output_text output_subarea output_execute_result"&gt;
&lt;pre class="ipynb"&gt;&amp;lt;matplotlib.axes._subplots.AxesSubplot at 0x7f5445826080&amp;gt;&lt;/pre&gt;
&lt;/div&gt;

&lt;/div&gt;

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"
&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;
&lt;div class="cell border-box-sizing code_cell rendered"&gt;
&lt;div class="input"&gt;
&lt;div class="prompt input_prompt"&gt;In&amp;nbsp;[32]:&lt;/div&gt;

&lt;div class="inner_cell"&gt;
    &lt;div class="input_area"&gt;
        &lt;div class="highlight-ipynb"&gt;&lt;pre class="ipynb"&gt;&lt;span&gt;&lt;/span&gt;&lt;span class="c1"&gt;# Graficando todos&lt;/span&gt;
&lt;span class="n"&gt;pivot_cierre&lt;/span&gt;&lt;span class="o"&gt;.&lt;/span&gt;&lt;span class="n"&gt;plot&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="n"&gt;figsize&lt;/span&gt;&lt;span class="o"&gt;=&lt;/span&gt;&lt;span class="p"&gt;(&lt;/span&gt;&lt;span class="mi"&gt;12&lt;/span&gt;&lt;span class="p"&gt;,&lt;/span&gt;&lt;span class="mi"&gt;8&lt;/span&gt;&lt;span class="p"&gt;))&lt;/span&gt; 
&lt;/pre&gt;&lt;/div&gt;

    &lt;/div&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;div class="output_wrapper"&gt;
&lt;div class="output"&gt;


&lt;div class="output_area"&gt;
&lt;div class="prompt output_prompt"&gt;Out[32]:&lt;/div&gt;



&lt;div class="output_text output_subarea output_execute_result"&gt;
&lt;pre class="ipynb"&gt;&amp;lt;matplotlib.axes._subplots.AxesSubplot at 0x7f54457194e0&amp;gt;&lt;/pre&gt;
&lt;/div&gt;

&lt;/div&gt;

&lt;div class="output_area"&gt;
&lt;div class="prompt"&gt;&lt;/div&gt;



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YII=
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&lt;p&gt;Como podemos ver &lt;a href="http://pandas.pydata.org/"&gt;Pandas&lt;/a&gt; es una librería muy versátil, con ella podemos hacer todo tipo de manipulaciones de datos, desde obtener los datos desde la web hasta realizar concatenaciones, tablas pivot o incluso realizar gráficos.&lt;/p&gt;
&lt;p&gt;Con esto termino esta introducción a finanzas con &lt;a href="https://www.python.org/"&gt;Python&lt;/a&gt;; los dejo para que se entretengan con sus propios ejemplos, a practicar!&lt;/p&gt;

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&lt;p&gt;Saludos!&lt;/p&gt;
&lt;p&gt;&lt;em&gt;Este post fue escrito utilizando IPython notebook. Pueden descargar este &lt;a href="https://github.com/relopezbriega/relopezbriega.github.io/blob/master/downloads/pyFinance.ipynb"&gt;notebook&lt;/a&gt; o ver su version estática en &lt;a href="http://nbviewer.ipython.org/github/relopezbriega/relopezbriega.github.io/blob/master/downloads/pyFinance.ipynb"&gt;nbviewer&lt;/a&gt;.&lt;/em&gt;&lt;/p&gt;

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&lt;/p&gt;</summary><category term="python"></category><category term="finanzas"></category><category term="programacion"></category><category term="analisis de datos"></category></entry></feed>