Chapter 2: Individual Security Returns

Total Return and Price Return

Total return includes both capital gains and the reinvestment of these intermediate cash flows
Price return is a percentage chang in price versus the capital gains or dollar amount
Total return is the sum of
- price return
- reinvestment of intermediate cash flows

2.1: Price Returns

In this section, we will use IBM stock as the example because they pay dividends. We can compare the results for price returns and total returns

Step 1: Import IBM Data from Yahoo Finance

At the start of each chapter, load the quantmod and xts packages
Using techniques from Chapter 1, read-in the IBM Yahoo.csv file

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(xts)
data.IBM <- read.csv("IBM Yahoo.csv", header=TRUE)
date <- as.Date(data.IBM$Date,format="%Y-%m-%d")
data.IBM <- cbind(date,data.IBM[,-1])
data.IBM <- data.IBM[order(data.IBM$date),]
data.IBM <- xts(data.IBM[,2:7],order.by=data.IBM[,1])
names(data.IBM) <-
  paste(c("IBM.Open","IBM.High","IBM.Low",
  "IBM.Close","IBM.Volume","IBM.Adjusted"))
data.IBM[c(1:3,nrow(data.IBM)),]

##            IBM.Open IBM.High IBM.Low IBM.Close IBM.Volume IBM.Adjusted
## 2010-12-31   146.73   147.07  145.96    146.76    2969800     125.8031
## 2011-01-03   147.21   148.20  147.14    147.48    4603800     126.4203
## 2011-01-04   147.56   148.22  146.64    147.64    5060100     126.5574
## 2013-12-31   186.49   187.79  186.30    187.57    3619700     169.4835

Step 2: Subset the data to only include closing price

The price return is calculated off the closing price. In data.IBM, this is IBM.Close or column 4. Subset the data to only include IBM closing price

IBM.prc.ret <- data.IBM[,4]
IBM.prc.ret[c(1:3,nrow(IBM.prc.ret))]

##            IBM.Close
## 2010-12-31    146.76
## 2011-01-03    147.48
## 2011-01-04    147.64
## 2013-12-31    187.57

Step 3: Calculate IBM’s Price Return

IBM.prc.ret$IBM.prc.ret <- Delt(IBM.prc.ret$IBM.Close)
IBM.prc.ret[c(1:3,nrow(IBM.prc.ret)),]

##            IBM.Close IBM.prc.ret
## 2010-12-31    146.76          NA
## 2011-01-03    147.48 0.004905976
## 2011-01-04    147.64 0.001084913
## 2013-12-31    187.57 0.006222858

Step 4: Clean up data object

Since the first observation is NA, we can delete this observation in IBM.prc.ret
Since we are dealing with returns, we do not need IBM.Clsoe and the first column can also be deleted

options(digits=3)
IBM.prc.ret <- IBM.prc.ret[-1,2]
IBM.prc.ret[c(1:3,nrow(IBM.prc.ret)),]

##            IBM.prc.ret
## 2011-01-03     0.00491
## 2011-01-04     0.00108
## 2011-01-05    -0.00400
## 2013-12-31     0.00622

options(digits=7)

2.2: Total Returns

The return to investors from holding shares of stock are not liited to changes in the price of the security (there are also dividends)
Total return equals the sum of both the change in price of the shares and any income generated from the dividends and the reinvestment of those dividends
For a daily total return calculation, the dividend yield is zero or non-ex dividend dates
Yahoo Finance gives us an adjusted closing prie varialbe, which in data.IBM, is the 6th column (labeled IBM.Adjusted)
We will now calculate IBM’s total returns from January 2011 to December 2013

data.IBM[715:720,]

##            IBM.Open IBM.High IBM.Low IBM.Close IBM.Volume IBM.Adjusted
## 2013-11-01   179.81   180.34  178.88    179.23    3644500     161.0826
## 2013-11-04   179.90   180.80  179.34    180.27    3483300     162.0173
## 2013-11-05   179.54   179.80  177.71    177.85    6096800     159.8423
## 2013-11-06   177.91   179.75  177.78    179.19    4560700     161.9115
## 2013-11-07   179.60   181.39  179.60    180.00    5219500     162.6434
## 2013-11-08   178.83   180.08  177.35    179.99    6275000     162.6344

We now turn to calculating IBM’s daily total return

Step 1: Import Adjusted Closing Price Data

Because we already called in the IBM.data, we only need to check that t is still in the R memory. If it is, keep the adjusted close price or column 6 in data.IBM

IBM.ret <- data.IBM[,6]
IBM.ret[c(1:3,nrow(IBM.ret)),]

##            IBM.Adjusted
## 2010-12-31     125.8031
## 2011-01-03     126.4203
## 2011-01-04     126.5574
## 2013-12-31     169.4835

Step 2: Calculate Total Return

When we apply the Delt command on the adjusted close price we calcualte the total return
Total return equals price return on non ex-dividend days and the total return is higher than the price return on ex-dividend days

IBM.ret$IBM.tot.ret=Delt(IBM.ret$IBM.Adjusted)
IBM.ret[c(1:3,nrow(IBM.ret)),]

##            IBM.Adjusted IBM.tot.ret
## 2010-12-31     125.8031          NA
## 2011-01-03     126.4203 0.004905977
## 2011-01-04     126.5574 0.001084913
## 2013-12-31     169.4835 0.006222858

Step 3: Clean up the data

To output the data, show the total return coumn and limit the number of decimals on display using the digits=3 option

options(digits=3)
IBM.ret <- IBM.ret[,2]
IBM.ret[c(1:3,nrow(IBM.ret)),]

##            IBM.tot.ret
## 2010-12-31          NA
## 2011-01-03     0.00491
## 2011-01-04     0.00108
## 2013-12-31     0.00622

options(digits=7)

2.3: Logarithmic Total Returns

The returns using the Delt command are called arithmetic returns, or simple returns
Logartithmic returns are used a lot in various areas of finance
Calculate the logarithmic total return for IBM stock from January 2011 to December 2013

Step 1: Import Adjusted Closing Price Data

Because we are calculating logarithmic total returns, we need the adjusted closing price data as our base data source

IBM.log.ret <- data.IBM[,6]
IBM.log.ret[c(1:3,nrow(IBM.log.ret)),]

##            IBM.Adjusted
## 2010-12-31     125.8031
## 2011-01-03     126.4203
## 2011-01-04     126.5574
## 2013-12-31     169.4835

Step 2: Calculate Log Returns

Use a combination of the diff and log commands to calculate logarithmic returns

IBM.log.ret$IBM.log.ret <- diff(log(IBM.log.ret$IBM.Adjusted))
IBM.log.ret[c(1:3,nrow(IBM.log.ret)),]

##            IBM.Adjusted IBM.log.ret
## 2010-12-31     125.8031          NA
## 2011-01-03     126.4203 0.004893982
## 2011-01-04     126.5574 0.001084325
## 2013-12-31     169.4835 0.006203576

Step 3: Clean up the data

Clean up IBM.log.ret to delete the first coumn and keep only the logarithmic return series

options(digits=3)
IBM.log.ret <- IBM.log.ret[,2]
IBM.log.ret[c(1:3,nrow(IBM.log.ret)),]

##            IBM.log.ret
## 2010-12-31          NA
## 2011-01-03     0.00489
## 2011-01-04     0.00108
## 2013-12-31     0.00620

options(digits=7)

Compare Log Returns with Artihmetic Returns

Combine the two total return calculations using the cbind command
Use the na.rm=TRUE option so that R still calculates a max and a min even though there is an NA in the data

options(digits=3,scipen=100)
tot.rets <- cbind(IBM.ret,IBM.log.ret)
tot.rets[c(1:3,nrow(tot.rets)),]

##            IBM.tot.ret IBM.log.ret
## 2010-12-31          NA          NA
## 2011-01-03     0.00491     0.00489
## 2011-01-04     0.00108     0.00108
## 2013-12-31     0.00622     0.00620

max(abs(tot.rets$IBM.tot.ret-tot.rets$IBM.log.ret),na.rm=TRUE)

## [1] 0.00363

min(abs(tot.rets$IBM.tot.ret-tot.rets$IBM.log.ret),na.rm=TRUE)

## [1] 0.00000000118

options(digits=7,scipen=0)

We used scipen=100 to increase the threshold before R converts the output into scientific notation
Once done, revert the scipen option back to zero so that the output of subsequent analyses will revert back to the default display options

2.4: Cumulating Multi-day Returns

To fully capture the effects of being able to reinvest dividends, we should calculate daily returns and string them together for longer periods
The returns of that stock going forward determines whether the reinvested dividend earned a positive or negative return
To string together multiple days of arithmetic returns, we have to take the product of the daily gross returns

Step 1: Import Data and Calculate Arithmetic Returns

Call IBM.ret to see that we have the correct data still available in R memory

IBM.acum <- IBM.ret
IBM.acum[c(1:3,nrow(IBM.acum)),]

##            IBM.tot.ret
## 2010-12-31          NA
## 2011-01-03 0.004905977
## 2011-01-04 0.001084913
## 2013-12-31 0.006222858

Step 2: Set first day total return value to zero

Set the return equal to zero on the day we are aking the investment (December 31, 2010)

IBM.acum[1,1]<-0
IBM.acum[c(1:3,nrow(IBM.acum)),]

##            IBM.tot.ret
## 2010-12-31 0.000000000
## 2011-01-03 0.004905977
## 2011-01-04 0.001084913
## 2013-12-31 0.006222858

Step 3: Calculate gross daily returns

Create a new variable for the gross return, which is one plus the net total return

IBM.acum$GrossRet <- 1+IBM.acum$IBM.tot.ret
IBM.acum[c(1:3,nrow(IBM.acum)),]

##            IBM.tot.ret GrossRet
## 2010-12-31 0.000000000 1.000000
## 2011-01-03 0.004905977 1.004906
## 2011-01-04 0.001084913 1.001085
## 2013-12-31 0.006222858 1.006223

Step 4: Calculate Cumulative Gross Returns

Use the cumprod command to take the cumulative product of the gross return
cumprod takes the product of all the gross returns

IBM.acum$GrossCum <- cumprod(IBM.acum$GrossRet)
IBM.acum[c(1:3,nrow(IBM.acum)),]

##            IBM.tot.ret GrossRet GrossCum
## 2010-12-31 0.000000000 1.000000 1.000000
## 2011-01-03 0.004905977 1.004906 1.004906
## 2011-01-04 0.001084913 1.001085 1.005996
## 2013-12-31 0.006222858 1.006223 1.347213

Step 5: Convert Cumulative Gross Returns to Cumulative Net Returns

Subtract one from the GrossCum to calculate the net cumulative return or NetCum
NetCum is interpreted as the percentage return from the investment date

IBM.acum$NetCum <- IBM.acum$GrossCum-1
IBM.acum[c(1:3,nrow(IBM.acum)),]

##            IBM.tot.ret GrossRet GrossCum      NetCum
## 2010-12-31 0.000000000 1.000000 1.000000 0.000000000
## 2011-01-03 0.004905977 1.004906 1.004906 0.004905977
## 2011-01-04 0.001084913 1.001085 1.005996 0.005996213
## 2013-12-31 0.006222858 1.006223 1.347213 0.347212558

2.4.2: Cumulating Logarithmic Returns

An alternative way to calculate multi-period returns is to take the sum of the daily logarithmic returns

Step 1: Import data and calculate logarithmic returns

Call IBM.logret to see that we have the correct data still available in the R memory

IBM.logcum <- IBM.log.ret
IBM.logcum[c(1:3,nrow(IBM.logcum)),]

##            IBM.log.ret
## 2010-12-31          NA
## 2011-01-03 0.004893982
## 2011-01-04 0.001084325
## 2013-12-31 0.006203576

Step 2: Set the first log return to zero

IBM.logcum[1,1]<-0
IBM.logcum[c(1:3,nrow(IBM.logcum)),]

##            IBM.log.ret
## 2010-12-31 0.000000000
## 2011-01-03 0.004893982
## 2011-01-04 0.001084325
## 2013-12-31 0.006203576

Step 3: Take the sum of all logarithmic returns during the investment period

Using the Sum command, add up the values in IBM.log.ret

logcumret=sum(IBM.logcum$IBM.log.ret)
logcumret

## [1] 0.2980377

Step 4: Convert log return back to Arithmetic Return

We need to convert the cumulative logarithmic return to a cumulative arithmetic return. We do this by using the exp command (exponential logarithmic)

cumret=exp(logcumret)-1
cumret

## [1] 0.3472126

As we can see, it takes fewer steps to calculate multi-period returns using logarithmic returns than it is using arithmetic returns

2.4.3: Comparing price return and total return

Step 1: Import Data and Calculate Price and Total Returns

We have already combined the price and total returns data from tot.rets, so we only need to call in data
Rename the data to better identify the variables as prc.ret and tot.ret

IBM.Ret <- cbind(IBM.prc.ret,IBM.ret)
names(IBM.Ret) <- c("prc.ret","tot.ret")
IBM.Ret[c(1:3,nrow(IBM.Ret)),]

##                prc.ret     tot.ret
## 2010-12-31          NA          NA
## 2011-01-03 0.004905976 0.004905977
## 2011-01-04 0.001084913 0.001084913
## 2013-12-31 0.006222858 0.006222858

Step 2: Set First Returns to Zero

Set the Dec 31, 2011 price and total return to zero

IBM.Ret$prc.ret[1] <- 0
IBM.Ret$tot.ret[1] <- 0
IBM.Ret[c(1:3,nrow(IBM.Ret)),]

##                prc.ret     tot.ret
## 2010-12-31 0.000000000 0.000000000
## 2011-01-03 0.004905976 0.004905977
## 2011-01-04 0.001084913 0.001084913
## 2013-12-31 0.006222858 0.006222858

Step 3: Calculate Gross Returns

Calculate the gross price return gross.prc and gross total return gross.tot

IBM.Ret$gross.prc<-1+IBM.Ret$prc.ret
IBM.Ret$gross.tot<-1+IBM.Ret$tot.ret
IBM.Ret[c(1:3,nrow(IBM.Ret)),]

##                prc.ret     tot.ret gross.prc gross.tot
## 2010-12-31 0.000000000 0.000000000  1.000000  1.000000
## 2011-01-03 0.004905976 0.004905977  1.004906  1.004906
## 2011-01-04 0.001084913 0.001084913  1.001085  1.001085
## 2013-12-31 0.006222858 0.006222858  1.006223  1.006223

Step 4: Cumulate the Gross Returns

Calculate the cumulative price return cum.prc and cumulative total return cum.tot by using the cumprod command

IBM.Ret$cum.prc <- cumprod(IBM.Ret$gross.prc)
IBM.Ret$cum.tot <- cumprod(IBM.Ret$gross.tot)
IBM.Ret[c(1:3,nrow(IBM.Ret)),]

##                prc.ret     tot.ret gross.prc gross.tot  cum.prc  cum.tot
## 2010-12-31 0.000000000 0.000000000  1.000000  1.000000 1.000000 1.000000
## 2011-01-03 0.004905976 0.004905977  1.004906  1.004906 1.004906 1.004906
## 2011-01-04 0.001084913 0.001084913  1.001085  1.001085 1.005996 1.005996
## 2013-12-31 0.006222858 0.006222858  1.006223  1.006223 1.278073 1.347213

Step 5: Plot the Two Return Series

First plot the cum.tot variable, and then plot the cum.prc variable
Using the abline command, add a horizontal line at $1 so we can interpret whether the investment is making money or losing money

y.range <- range(IBM.Ret[,5:6])
y.range

## [1] 1.000000 1.526869

plot(IBM.Ret$cum.tot,
     type="l",
     auto.grid=FALSE,
     xlab="Date",
     ylab="Value of Investment ($)",
     ylim=y.range,
     minor.ticks=FALSE,
     main="IBM Stock Performance Based on
Total Returns and Price Returns
     December 31, 2010 - December 31, 2013")
lines(IBM.Ret$cum.prc,
      type="l",
      lty=3)
abline(h=1,col="black")
legend("topleft",
       col=c("black","black"),
       lty=c(1,3),
       c("Value Based on Total Return",
         "Value Based on Price Return"))

As the graph shows, the value of an investment based on IBM’s total return is equal to or greater than the value on an investment based on IBM’s price return

Financial Modeling Chapter 2

Carter Medeiros

4/12/2017

Chapter 2: Individual Security Returns

2.1: Price Returns

Step 1: Import IBM Data from Yahoo Finance

Step 2: Subset the data to only include closing price

Step 3: Calculate IBM’s Price Return

Step 4: Clean up data object

2.2: Total Returns

Step 1: Import Adjusted Closing Price Data

Step 2: Calculate Total Return

Step 3: Clean up the data

2.3: Logarithmic Total Returns

Step 1: Import Adjusted Closing Price Data

Step 2: Calculate Log Returns

Step 3: Clean up the data

2.4: Cumulating Multi-day Returns

Step 1: Import Data and Calculate Arithmetic Returns

Step 2: Set first day total return value to zero

Step 3: Calculate gross daily returns

Step 4: Calculate Cumulative Gross Returns

Step 5: Convert Cumulative Gross Returns to Cumulative Net Returns

2.4.2: Cumulating Logarithmic Returns

Step 1: Import data and calculate logarithmic returns

Step 2: Set the first log return to zero

Step 3: Take the sum of all logarithmic returns during the investment period

Step 4: Convert log return back to Arithmetic Return

2.4.3: Comparing price return and total return

Step 1: Import Data and Calculate Price and Total Returns

Step 2: Set First Returns to Zero

Step 3: Calculate Gross Returns

Step 4: Cumulate the Gross Returns

Step 5: Plot the Two Return Series