#Using the getSymbols command, download MONTHLY prices (from Yahoo) for Amazon, Inc (AMZN) from Jan 2019 to Feb 2021.

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
## Registered S3 method overwritten by 'quantmod':
##   method            from
##   as.zoo.data.frame zoo
getSymbols(Symbols = "AMZN", from="2019-01-01", 
           to="2021-02-01", periodicity="monthly", src = "yahoo")
## '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.
## [1] "AMZN"

#Delete any NA values with the na.omit function:

r = na.omit(AMZN)

#Adjusted Prices

prices= AMZN$AMZN.Adjusted
names(prices) = c("AMZN")

#Calculate simple returns of AMZN stock

R =  na.omit(prices / lag(prices,n=1) - 1)

Continuously compounded (cc) returns of AMZN stock (you can create new R objects or new columns in the price objects)

r = diff(log(prices))
r = na.omit(r)

Histogram for the simple returns of Amazon

hist(R, main="Histogram of Amazon simple returns", 
     xlab="Simple returns", col="sky blue")

# INTERPRETATION OF THE HISTOGRAM THE Y AXIS REPRESENTS THE FREQUENCY, AND THE X AXIS IS THE RANGE OF THE MONTHLY RETURNS FOR THE AMAZON STOCK, THE HISTOGRAM IS SKEWED. THE MOST FREQUENT RETURNS OF AMAZON STOCK ARE BETWEEN -5% TO +5%. THE MEAN IS ARROUND 0.05 AND WE CAN SEE HOW THE RETURNS BETWEEN 15% AND 25% ARE NEGATIVE.

Is the average of AMZN CC returns significantly higher than 1%?

# H0: AMZN=1
# hα: AMZN>1

Which is the hypothesis test you need for this (Null and alternative hypothesis)?

N <- nrow(r)
se_AMZN <- sd(r) / sqrt(N) 
se_AMZN
## [1] 0.01577447

MANUAL CALCULATION t-tests #Mean of returns

mean_r_AMZN<- mean(r)

Calculate t-value

t_value_AMZN <- (mean_r_AMZN -0.01) / se_AMZN 
t_value_AMZN
## [1] 1.012978

HYPOTHESIS TEST USING t.test function

ttest_AMZN<- t.test(as.numeric(r), nu=0.01, alternative = "greater")
ttest_AMZN
## 
##  One Sample t-test
## 
## data:  as.numeric(r)
## t = 1.6469, df = 23, p-value = 0.05659
## alternative hypothesis: true mean is greater than 0
## 95 percent confidence interval:
##  -0.001056222          Inf
## sample estimates:
## mean of x 
## 0.0259792

##PART II Donwload from Yahoo Finance monthly prices and returns for Alfa(ALFAA.MX) and the IPCyC Mexican market index (^MXX) from Jan, 2016 to date Jan 2021.

getSymbols(c("ALFAA.MX", "^MXX"), from="2016-01-01", to= "2021-01-01", periodicity="monthly", src="yahoo")
## [1] "ALFAA.MX" "^MXX"
# CC RETURNS FOR ALFA 
r_ALFAA <- na.omit(diff(log(ALFAA.MX$ALFAA.MX.Adjusted)))
# IPC:
r_MXX <- na.omit(diff(log(MXX$MXX.Adjusted)))

# MERGE INTO SAME OBJECT:
all_rets <- merge(r_ALFAA, r_MXX)

# RENAME THE COLUMNS:
colnames(all_rets) <- c("ALFAA", "MXX")

#MARKET REGRESSION MODEL FOR ALFAA

reg <- lm(r_ALFAA ~ r_MXX)
sumreg<- summary(reg)
sumreg
## 
## Call:
## lm(formula = r_ALFAA ~ r_MXX)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.32170 -0.04475 -0.01576  0.02992  0.34625 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.01464    0.01254  -1.167    0.248    
## r_MXX        1.89300    0.27447   6.897 4.72e-09 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.09632 on 57 degrees of freedom
## Multiple R-squared:  0.4549, Adjusted R-squared:  0.4453 
## F-statistic: 47.57 on 1 and 57 DF,  p-value: 4.725e-09

WHAT IS THE REGRESSION EQUATION OF THIS MODEL?

#(E[Ri]−Rf)=β1(RM−Rf)

# (r_ALFAA) -0.01464 + 1.89300 =r_MXX

WHAT CAN YOU SAY ABOUT THE MARKET RISK OF THE STOCK?

# ALFAA is riskier than the market, since it has a B1 higher than 1.

DOES ALFAA SIGNIFICANTLY OFFER RETURNS OVER OR UNDER THE MARKET? JUSTIFY YOUR ANSWER WITH THE REGRESSION OUTPUT

# the returns are under the market, as we saw previously in the regression output bO is negative 0.014, aprox. -1.5% less than the market.

HOW MUCH THE VARIANCE OF THE STOCK RETURNS CANNOT BE EXPLAINED BY THE VARIANCE OF THE MARKET RETURNS?

# This value corresponds to the systematic error.