Executive Summary

In the assignment we look at an empirical example of Harry Markowitz’s Modern Portfolio Theory. As we increase assets to our portfolio we can reduce the volatility of the portfolio’s return. This is statistical properties of covariance.

We looked atthe connection between the volatility of a company’s stock return and it’s beta, also utilizing the properties of covariance and correlation.

Writing a function that will let us automatically download return data:

# Loading Necessary libraries

library(quantmod)
library(zoo)
library(tseries)
library(dplyr)
library(knitr)
library(dygraphs)
library(ggplot2)

# Function to calculate monthly returns on a stock 

monthly_stock_returns <- function(ticker, start_year) {
  # Download the data from Yahoo finance
  
  start_year <- paste(start_year,"-01-01",collapse = "",sep="")
  symbol <- getSymbols(ticker, src = 'yahoo', from = start_year, 
                       auto.assign = FALSE, warnings = FALSE) 
  
  # Tranform it to monthly returns using the periodReturn function from quantmod
  data <- periodReturn(symbol, period = 'monthly', type = 'log')
  
  # Let's rename the column of returns to something intuitive because the column
  # name is what will eventually be displayed on the time series graph
  colnames(data) <- as.character(ticker)
  return(data)
}

Data is downloaded at a daily price rate. The log of the ratio of the adjusted closed price of each month is then used to calculate the monthly return. This function builds upon the quantmod R package.

I chose to look at the following companies:

  1. El Al Israel Airlines Ltd. (ELAL.TA) - Aviation.
  2. Clal Biotechnology Industries Ltd. (CBI.TA) - Technology.
  3. Strauss Group Ltd. (STRS.TA) - Consumer goods.
  4. The Phoenix Holdings Ltd. (PHOE.TA) - Financial services.
  5. Bezeq The Israel Telecommunication Corporation Limited (BEZQ.TA) - Telecom.
  6. Clal Insurance (CLIS.TA) - Financials services.
  7. Migdal Holdings (MGDL.TA) - Financials services.
  8. Teva Pharmaceutical Industries Limited (TEVA.TA) - Pharma
  9. Elbit Systems Ltd. (ESLT.TA) - Weapons
  10. Bank Leumi le- Israel B.M (LUMI.TA) - Financials services.

The yahoo finance ticker for the TA125 index is still TA100.

1. Downloading Data

ta125 <- monthly_stock_returns("^TA100","2012")
elal <- monthly_stock_returns("ELAL.TA","2012")
clalbio <- monthly_stock_returns("CBI.TA","2012")
strauss <- monthly_stock_returns("STRS.TA","2012")
fnx <- monthly_stock_returns("PHOE.TA","2012")
bezeq <- monthly_stock_returns("BEZQ.TA","2012")
clal <- monthly_stock_returns("CLIS.TA","2012")
migdal <- monthly_stock_returns("MGDL.TA","2012")
teva <- monthly_stock_returns("TEVA.TA","2012")
elbit <- monthly_stock_returns("ESLT.TA","2012")
leumi <- monthly_stock_returns("LUMI.TA","2012")

2. Volatility of stocks and Portfolios

vola <- c(sd(ta125),sd(elal),sd(clalbio),sd(strauss),sd(fnx),sd(bezeq),sd(clal),sd(migdal),sd(teva),sd(elbit),sd(leumi))

names(vola) <- c("TA125","elal","clalbio","strauss","fnx","bezeq","clal","migdal","teva","elbit","leumi")
kable(t(vola))
TA125 elal clalbio strauss fnx bezeq clal migdal teva elbit leumi
0.031469 0.1258715 0.1609502 0.0578711 0.0745591 0.0723128 0.0794524 0.0742777 0.0628512 0.051678 0.0571819

We will define volatility as the sample standard deviation of monthly returns. We can see that Pharma,consumer good and financial sector stocks have lower volatility than the technology and aviation sector, these would coincide with our expectations of the riskiness of these sectors. We can see that the index has the lowest volatility as expected by MPT. This makes sense with financial theory that a diversified portfolio will be less volatile than the specific underlying stocks that it is made of. Strauss is in the consumer products industry which is considered less volatile than others. Also, strauss is a big exporter and has a large and diversified portfolio of products and projects being pursued in different markets and currencies.

Building Portfolios

# 1 - Two stocks

port1 <- 0.5*teva +0.5*fnx
sd(port1)
## [1] 0.04653915
# 2 - Four stocks

port2 <- 0.25*teva +0.25*fnx + 0.25*strauss + 0.25*elbit
sd(port2)
## [1] 0.04013974
# 3 -  Eight stocks

port3 <- 0.1*strauss+0.1*fnx+0.1*bezeq+0.1*clal+0.1*migdal+0.2*teva+0.2*elbit+0.1*leumi
sd(port3)
## [1] 0.04081349

As expected the volatility of a portfolio decreased we add more stocks (that aren’t perfectly correlated). However this is a marginally decreasing process.

3. Stock Beta

# running regressions

capm_elal <- lm(elal~ta125)
capm_clalbio <- lm(clalbio~ta125)
capm_strauss <- lm(strauss~ta125)
capm_fnx <- lm(fnx~ta125)
capm_bezeq <- lm(bezeq~ta125)
capm_clal <- lm(clal~ta125)
capm_migdal <- lm(migdal~ta125)
capm_teva <- lm(teva~ta125)
capm_elbit <- lm(elbit~ta125)
capm_leumi <- lm(leumi~ta125)


# results
betas <- cbind(coef(capm_elal),coef(capm_clalbio),coef(capm_strauss),coef(capm_fnx),coef(capm_bezeq),coef(capm_clal),coef(capm_migdal),coef(capm_teva),coef(capm_elbit),coef(capm_leumi))

colnames(betas) <- c("elal","clalbio","strauss","fnx","bezeq","clal","migdal","teva","elbit","leumi")
rownames(betas) <- c("alpha","beta")
kable(betas)
elal clalbio strauss fnx bezeq clal migdal teva elbit leumi
alpha 0.0200988 -0.0322697 0.0016162 0.0023448 -0.0044863 -0.0045795 -0.0109707 -0.0084288 0.0125850 0.0011393
beta 0.7432829 1.4468690 0.9946945 1.0229029 0.8384177 1.3375606 1.2540865 1.0010269 0.7103524 1.2551940 
dat <- data.frame(Beta=betas[2,],Volatility=vola[2:11])

ggplot(dat, aes(Volatility, Beta)) + geom_point() + ggtitle("Beta as function of Volatility")

The mathematical definition of \(\beta\) is \[\beta_{i}=\rho_{i,m}\cdot\frac{\sigma_{i}}{\sigma_{m}}\]

In other words it’s the correlation between an asset’s return and the market times the ratio of standard deviation of the returns. Therefore three things can affect the \(\beta\) - the correlations and the marginal volatilities.

As seen in the formula the beta should increase with the underlying stock’s volatility, however it is scaled by the ratio of the correlation of te index and the stock and the index’s volatility. Meaning it’s not just the volatility of the stock but it’s how correlated it is to the index which affects the magnitude of the beta.

For example Pheonix and bezeq have similar volatilites but the betas are different since the corrleation with TA125 is different. Pheonix has a stronger corrleation and therefore will have a higher beta as expected.

kable(t(vola[5:6]))
fnx bezeq
0.0745591 0.0723128 
kable(betas[,4:5])
fnx bezeq
alpha 0.0023448 -0.0044863
beta 1.0229029 0.8384177 
kable(cor(fnx,ta125))
^TA100
PHOE.TA 0.4317346 
kable(cor(bezeq,ta125))
^TA100
BEZQ.TA 0.3648618

Bibliography

  1. Jonathan Regenstein’s Blog give me source code that I took inspiration from.
  2. Class notes