CAPM
Dr. Víctor M. Chung
30/9/2019
library("quantmod") ## Loading required package: xts## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric## Loading required package: TTR## Version 0.4-0 included new data defaults. See ?getSymbols.library("ggfortify")## Loading required package: ggplot2activos<-c("^GSPC","IBM")
getSymbols(activos, from="1989-12-29", to="2018-12-31")## 'getSymbols' currently uses auto.assign=TRUE by default, but will
## use auto.assign=FALSE in 0.5-0. You will still be able to use
## 'loadSymbols' to automatically load data. getOption("getSymbols.env")
## and getOption("getSymbols.auto.assign") will still be checked for
## alternate defaults.
##
## This message is shown once per session and may be disabled by setting
## options("getSymbols.warning4.0"=FALSE). See ?getSymbols for details.##
## WARNING: There have been significant changes to Yahoo Finance data.
## Please see the Warning section of '?getSymbols.yahoo' for details.
##
## This message is shown once per session and may be disabled by setting
## options("getSymbols.yahoo.warning"=FALSE).## [1] "GSPC" "IBM"head(IBM)## IBM.Open IBM.High IBM.Low IBM.Close IBM.Volume IBM.Adjusted
## 1989-12-29 23.56250 23.65625 23.50000 23.53125 6549600 12.33237
## 1990-01-02 23.68750 24.53125 23.62500 24.50000 7041600 12.84008
## 1990-01-03 24.68750 24.87500 24.59375 24.71875 9464000 12.95472
## 1990-01-04 24.75000 25.09375 24.71875 25.00000 9674800 13.10213
## 1990-01-05 24.96875 25.40625 24.87500 24.93750 7570000 13.06938
## 1990-01-08 24.81250 25.21875 24.81250 25.09375 4625200 13.15126Precios de cierre mensual
ibm.mensual<-to.monthly(IBM)head(ibm.mensual)## IBM.Open IBM.High IBM.Low IBM.Close IBM.Volume IBM.Adjusted
## Dec 1989 23.56250 23.65625 23.50000 23.53125 6549600 12.33237
## Jan 1990 23.68750 25.40625 23.62500 24.65625 156385200 12.92197
## Feb 1990 24.65625 26.37500 24.37500 25.96875 133848800 13.77932
## Mar 1990 25.93750 27.34375 25.84375 26.53125 128318400 14.07778
## Apr 1990 26.28125 27.81250 26.25000 27.25000 110108400 14.45917
## May 1990 27.37500 30.53125 26.96875 30.00000 149241200 16.09662sp500.mensual<-to.monthly(GSPC)
head(sp500.mensual)## GSPC.Open GSPC.High GSPC.Low GSPC.Close GSPC.Volume GSPC.Adjusted
## Dec 1989 350.68 353.41 350.67 353.40 145940000 353.40
## Jan 1990 353.40 360.59 319.83 329.08 3793250000 329.08
## Feb 1990 329.08 336.09 322.10 331.89 2961970000 331.89
## Mar 1990 331.89 344.49 331.08 339.94 3283280000 339.94
## Apr 1990 339.94 347.30 327.76 330.80 2801220000 330.80
## May 1990 330.80 362.26 330.80 361.23 3596680000 361.23Precios ajustados mensuales
sp500.adj<-sp500.mensual[,6]
head(sp500.adj)## GSPC.Adjusted
## Dec 1989 353.40
## Jan 1990 329.08
## Feb 1990 331.89
## Mar 1990 339.94
## Apr 1990 330.80
## May 1990 361.23ibm.adj<-ibm.mensual[,6]
head(ibm.adj)## IBM.Adjusted
## Dec 1989 12.33237
## Jan 1990 12.92197
## Feb 1990 13.77932
## Mar 1990 14.07778
## Apr 1990 14.45917
## May 1990 16.09662Retornos mensuales
sp500.retorno.mensual<-monthlyReturn(sp500.adj,type="log")
head(sp500.retorno.mensual)## monthly.returns
## Dec 1989 0.000000000
## Jan 1990 -0.071299700
## Feb 1990 0.008502791
## Mar 1990 0.023965503
## Apr 1990 -0.027255210
## May 1990 0.088000978ibm.retorno.mensual<-monthlyReturn(ibm.adj,type="log")
head(ibm.retorno.mensual)## monthly.returns
## Dec 1989 0.00000000
## Jan 1990 0.04670098
## Feb 1990 0.06424019
## Mar 1990 0.02142896
## Apr 1990 0.02673073
## May 1990 0.10728064sp500.retorno.mensual<-sp500.retorno.mensual[-1,]
ibm.retorno.mensual<-ibm.retorno.mensual[-1,]head(ibm.retorno.mensual)## monthly.returns
## Jan 1990 0.04670098
## Feb 1990 0.06424019
## Mar 1990 0.02142896
## Apr 1990 0.02673073
## May 1990 0.10728064
## Jun 1990 -0.02105238Tasa libre de riesgo
Tasa de Treasury Bill a 3 meses.
getSymbols('TB3MS',src='FRED',from="1990-01-01", to="2018-12-31")## [1] "TB3MS"head(TB3MS)## TB3MS
## 1934-01-01 0.72
## 1934-02-01 0.62
## 1934-03-01 0.24
## 1934-04-01 0.15
## 1934-05-01 0.16
## 1934-06-01 0.15TB3MS_2<-TB3MS["1990/2018"]
head(TB3MS_2)## TB3MS
## 1990-01-01 7.64
## 1990-02-01 7.74
## 1990-03-01 7.90
## 1990-04-01 7.77
## 1990-05-01 7.74
## 1990-06-01 7.73tail(TB3MS_2)## TB3MS
## 2018-07-01 1.96
## 2018-08-01 2.03
## 2018-09-01 2.13
## 2018-10-01 2.25
## 2018-11-01 2.33
## 2018-12-01 2.37TB3MS_2<-to.monthly(TB3MS_2)
TB3MS.m<- (1+TB3MS_2[,1]/100)^(1/12)-1
head(TB3MS.m)## TB3MS_2.Open
## Jan 1990 0.006154037
## Feb 1990 0.006231899
## Mar 1990 0.006356340
## Apr 1990 0.006255245
## May 1990 0.006231899
## Jun 1990 0.006224116autoplot(sp500.retorno.mensual)
autoplot(ibm.retorno.mensual)
autoplot(TB3MS.m)
dim(ibm.retorno.mensual)## [1] 348 1dim(sp500.retorno.mensual)## [1] 348 1dim(TB3MS.m)## [1] 348 1Cálculo del beta usando la covarinza
\[ \beta_i=\frac{Cov(R_i,R_m)}{var(R_m)} \]
CAPM1<-cov(ibm.retorno.mensual,sp500.retorno.mensual)/var(sp500.retorno.mensual)
CAPM1## monthly.returns
## monthly.returns 1.012649Cálculo del beta usando la regresión
ri_rf <-ibm.retorno.mensual-TB3MS.m
rm_rf <-sp500.retorno.mensual-TB3MS.m
regresion<-lm(ri_rf ~ rm_rf)
summary(regresion)##
## Call:
## lm(formula = ri_rf ~ rm_rf)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.314431 -0.031437 -0.000612 0.033169 0.240931
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0006333 0.0034683 0.183 0.855
## rm_rf 1.0078179 0.0834378 12.079 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.06449 on 346 degrees of freedom
## Multiple R-squared: 0.2966, Adjusted R-squared: 0.2946
## F-statistic: 145.9 on 1 and 346 DF, p-value: < 2.2e-16Modelo de Mercado
\[ r_i=\alpha+\beta r_m \]
regresion2 <- lm(ibm.retorno.mensual ~ sp500.retorno.mensual)
summary(regresion2)##
## Call:
## lm(formula = ibm.retorno.mensual ~ sp500.retorno.mensual)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.314456 -0.031391 -0.000494 0.033228 0.240778
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.0005888 0.0034883 0.169 0.866
## sp500.retorno.mensual 1.0126494 0.0834588 12.134 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.06449 on 346 degrees of freedom
## Multiple R-squared: 0.2985, Adjusted R-squared: 0.2965
## F-statistic: 147.2 on 1 and 346 DF, p-value: < 2.2e-16Cálculo del beta y alfa usando PerformanceAnalytics
El paquete PerformanceAnalytics tiene la función CAPM.beta(Ra, Rb, Rf = 0) para calcular el valor beta y CAPM.alpha(Ra, Rb, Rf = 0) para calcular el valor alfa.
Donde:
- Ra: Es el retorno esperado del activo.
- Rb: Es el retorno esperado del mercado.
- Rf: Es la tasa libre de riesgo.
library("PerformanceAnalytics")##
## Attaching package: 'PerformanceAnalytics'## The following object is masked from 'package:graphics':
##
## legendCAPM.beta(ibm.retorno.mensual,sp500.retorno.mensual,TB3MS.m)## [1] 1.007818CAPM.alpha(ibm.retorno.mensual,sp500.retorno.mensual,TB3MS.m)## [1] 0.0006333284