Final Workshop , Financial Econometrics II
Eduardo Velasco Reyes
13 de mayo de 2021
Intruduction to Portfolio Theory:
A portfolio is set of financial assets:
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 zoogetSymbols(Symbols = c("TSLA", "WMT") , from="2018-01-01", periodicity="monthly")## '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] "TSLA" "WMT"I merge both dataset and extract only adjusted prices
prices = merge(Ad(TSLA),Ad(WMT))I calculate historical cc returns
r = na.omit(diff(log(prices)))I calculate the Geometric average of historical returns:
# We start calculating the arithmetic average of cc returns:
avgccr =colMeans(r)
# The cc returns are ADITIVE; we can add them and taking its arithmetic average;
# But for simple historical returns (R), we CANNOT add them not take its arithmetic mean.
avgccr## TSLA.Adjusted WMT.Adjusted
## 0.053599540 0.008834128# The Geometric mean will be the Expected return of the stock:
# I get the simple return from cc return:
ER = exp(avgccr) - 1
ER## TSLA.Adjusted WMT.Adjusted
## 0.055062008 0.008873264I can calculate the expected return a portfolio as follows
If I assign 70% to TSLA and 30% to WMT
w1 = 0.7
w2 = 0.3
ERP1 = w1 * ER[1] + w2 *ER[2]
names(ERP1)=c("ExpRet Por1")
ERP1## ExpRet Por1
## 0.04120538We can also get the expected return of the portfolio using Matrix Algebra. Matrix Algebra is very effective to SUM OF PRODUCTS:
I define a matrix for the weigths:
W = c(w1,w2)
W## [1] 0.7 0.3I have 2 vectors, W and ER, wich are matrices of 1 column. I multiply both vectors:
ERP1_2 = t(W) %% ER
ERP1_2## [,1] [,2]
## [1,] 0.03925591 0.007182294Now, How can I estimate the Expected Risk of the portfolio?
This was respond by Harry Markowitz in 1953 in his Ph.D. Dissertation Finanacial Markets had existed in the US since early 1800`s
Before Markowitz, the risk measures for portfolio did not consider the correlations between pair of assets returns!!
The right way to estimate the RISK of a portfolio is applying Probability Theory to the Portfolio equiation:
Let´s start with the VARIANCE of the PORTFOLIO Equation, and at the end, take the SQUARE ROOT to get the Std. Dev or volatility of the portfolios:
\(VAR[P] = VAR[w1*R1 + w2*R2 + w3*R3 + ... + wN*R4]\)
For a portfolio of 2 assets, the result fot he Expected Portfolio Variance is:
\(VAR[P] = w1^2 * VAR(R1) + w2^2 * VAR(R2) + 2*w1*w2*COV(R1,R2)]\)
THEN THE CONCLUTION :
\(SD[P] = sqrt(VAR[P])\)
SINCE THIS PART THE CODE EVEN IF THERE ARE TOO MANY ASSETS WILL WORK
We can calculate the exp. risk of our 2-asset portfolio with matrix algebra as follows:
COV = cov(r)
COV## TSLA.Adjusted WMT.Adjusted
## TSLA.Adjusted 0.039947094 0.002796124
## WMT.Adjusted 0.002796124 0.002880296ERISKP = t(W) %*% COV %*% W
#This is the expected risk for the portfolio monthly
ERISKP## [,1]
## [1,] 0.02100767Our portfolio expected MONTHLY risk/volatility is 2.12%
EVOLT = sqrt(ERISKP)
EVOLT## [,1]
## [1,] 0.1449402Portfolio Optimization Applications
Lets calculate the Global Minimum Variance Portfolio
Install the IntroCompFinR package.
Load the package and download stock data
library(IntroCompFinR)
library(quantmod)
# I select 5 assets
tickers = c("MSFT","AAPL", "SPCE", "WMT", "FB")
getSymbols(Symbols = tickers, from="2018-01-01", periodicity="monthly")## [1] "MSFT" "AAPL" "SPCE" "WMT" "FB"Calculating cc monthly returns
# Merge the datasets into one
prices <- Ad(merge(AAPL,FB,MSFT,SPCE,WMT))
returns <- na.omit(diff(log(prices)))Now I calculate the expectef return for each assets:
# This is the simple return expected.
ER = exp(colMeans(returns)) -1
ER## AAPL.Adjusted FB.Adjusted MSFT.Adjusted SPCE.Adjusted WMT.Adjusted
## 0.029142449 0.014145438 0.025802634 0.023615392 0.008873264Before doing portfolio optimization, I need to calculate the VAR - COV matrix:
I can get descriptive statistics of the asset return
library(psych)
describe(returns)## vars n mean sd median trimmed mad min max range skew
## AAPL.Adjusted 1 40 0.03 0.09 0.05 0.03 0.09 -0.20 0.19 0.40 -0.42
## FB.Adjusted 2 40 0.01 0.09 0.01 0.01 0.10 -0.14 0.24 0.38 0.45
## MSFT.Adjusted 3 40 0.03 0.05 0.03 0.03 0.04 -0.08 0.13 0.21 -0.25
## SPCE.Adjusted 4 40 0.02 0.22 0.00 0.01 0.05 -0.51 0.62 1.13 0.58
## WMT.Adjusted 5 40 0.01 0.05 0.01 0.01 0.04 -0.17 0.10 0.27 -0.81
## kurtosis se
## AAPL.Adjusted -0.61 0.01
## FB.Adjusted -0.46 0.01
## MSFT.Adjusted -0.54 0.01
## SPCE.Adjusted 1.03 0.03
## WMT.Adjusted 1.10 0.01library(PerformanceAnalytics)##
## Attaching package: 'PerformanceAnalytics'## The following object is masked _by_ '.GlobalEnv':
##
## prices## The following object is masked from 'package:graphics':
##
## legendcharts.PerformanceSummary(returns,
main = "Performance of $1.00 over time",
wealth.index = TRUE)
# regresar a ver el video del porfe
COV = var(returns)
COV ## AAPL.Adjusted FB.Adjusted MSFT.Adjusted SPCE.Adjusted
## AAPL.Adjusted 0.0089127006 0.0052453049 0.0033715846 0.0006706595
## FB.Adjusted 0.0052453049 0.0082549697 0.0025286527 -0.0004790925
## MSFT.Adjusted 0.0033715846 0.0025286527 0.0025608460 0.0005204921
## SPCE.Adjusted 0.0006706595 -0.0004790925 0.0005204921 0.0462644315
## WMT.Adjusted 0.0015497142 0.0015795725 0.0007742004 -0.0004671065
## WMT.Adjusted
## AAPL.Adjusted 0.0015497142
## FB.Adjusted 0.0015795725
## MSFT.Adjusted 0.0007742004
## SPCE.Adjusted -0.0004671065
## WMT.Adjusted 0.0028802955Now I get the correlation matrix to have an idea about how each pair of asset will contribute to the portfolio risk:
CORR = cor(returns)
# Es bueno y raro encontra runa correlación negativa ya que reduce la volatilidad del portfolio.
CORR## AAPL.Adjusted FB.Adjusted MSFT.Adjusted SPCE.Adjusted
## AAPL.Adjusted 1.00000000 0.61151672 0.70572854 0.03302736
## FB.Adjusted 0.61151672 1.00000000 0.54997090 -0.02451534
## MSFT.Adjusted 0.70572854 0.54997090 1.00000000 0.04781881
## SPCE.Adjusted 0.03302736 -0.02451534 0.04781881 1.00000000
## WMT.Adjusted 0.30586406 0.32393904 0.28506451 -0.04046446
## WMT.Adjusted
## AAPL.Adjusted 0.30586406
## FB.Adjusted 0.32393904
## MSFT.Adjusted 0.28506451
## SPCE.Adjusted -0.04046446
## WMT.Adjusted 1.00000000Calculating the Global Minimum Variance Portfolio
gmv = globalMin.portfolio(ER, COV, )
# EL tipo de clase es unico,
gmv## Call:
## globalMin.portfolio(er = ER, cov.mat = COV)
##
## Portfolio expected return: 0.01755801
## Portfolio standard deviation: 0.03944491
## Portfolio weights:
## AAPL.Adjusted FB.Adjusted MSFT.Adjusted SPCE.Adjusted WMT.Adjusted
## -0.1633 -0.0004 0.6800 0.0329 0.4508Efficient Frontier
efrontier = efficient.frontier(ER,COV,nport = 100,
alpha.min = -0.5,
alpha.max = 1.5, shorts = TRUE)
plot(efrontier, plot.assets = TRUE, col= "green")
plot(efrontier, col= "green")
### The optimal/tangent portfolio
rf = 0.0
optport = tangency.portfolio(ER,COV,rf, )
optport## Call:
## tangency.portfolio(er = ER, cov.mat = COV, risk.free = rf)
##
## Portfolio expected return: 0.02586915
## Portfolio standard deviation: 0.04787893
## Portfolio weights:
## AAPL.Adjusted FB.Adjusted MSFT.Adjusted SPCE.Adjusted WMT.Adjusted
## -0.0341 -0.1720 1.0697 0.0330 0.1035opweights = getPortfolio(ER,COV,weights = optport$weights)
opweights## Call:
## getPortfolio(er = ER, cov.mat = COV, weights = optport$weights)
##
## Portfolio expected return: 0.02586915
## Portfolio standard deviation: 0.04787893
## Portfolio weights:
## AAPL.Adjusted FB.Adjusted MSFT.Adjusted SPCE.Adjusted WMT.Adjusted
## -0.0341 -0.1720 1.0697 0.0330 0.1035plot(opweights)
Capital market line:
The slope of the capital market line is the sharpe ratio
plot(efrontier)
# primero se pone el eje x
points(gmv$sd, col="green")
# ahora va el tangete portafolio
points(optport$er, col="red", pch = 16 )
sharper = (optport$er - rf) / optport$sd
# que tantos premium returns te da un punto de riesgo
abline(a=rf, b=sharper, col="blue")