Quantmod Tutorials

Loading Packages

Setting Up Your Workspace

The TTR package is used to construct “Technical Trading Rules”. The TTR package can perform more sophisticated calculations and is worth exploring. The dygraphs library is a wrapper for a fast, open source JavaScript charting library. It is one of the htmlwidgets that makes R charting more dynamic and part of an html file instead of a static image. Lastly, the lubridate package is used for easy date manipulation

library(rvest)
library(pbapply)
library(TTR)
library(dygraphs)
library(lubridate)
library(tidyquant)
library(timetk)
pacman::p_load(dygraphs,DT,quantmod)

---

quantmod

The quantmod package for R is designed to assist the quantitative trader in the development, testing, and deployment of statistically based trading models.

Getting data

getSymbols("YHOO",src="google",from="2017-12-01", to="2018-01-01")%>%get() # from google finance 

##            YHOO.Open YHOO.High YHOO.Low YHOO.Close YHOO.Volume
## 2017-12-01     69.80     71.25    69.47      70.10    19712943
## 2017-12-04     71.06     71.34    68.16      68.31    15533597
## 2017-12-05     67.10     69.38    66.62      68.13    11963099
## 2017-12-06     67.78     69.62    67.36      69.41    11646582
## 2017-12-07     69.75     70.64    69.48      70.22     7084564
## 2017-12-08     71.00     71.63    70.82      71.25     7021125
## 2017-12-11     71.70     72.09    71.15      71.53     7359312
## 2017-12-12     71.07     71.15    69.55      69.77    11296188
## 2017-12-13     69.80     71.76    69.69      71.24    17321301
## 2017-12-14     70.60     70.62    68.84      69.23    17077065
## 2017-12-15     69.35     70.54    68.66      70.29    19517623
## 2017-12-18     70.54     70.74    69.95      70.14     6776982
## 2017-12-19     69.70     70.36    69.38      70.21    14654994
## 2017-12-20     70.60     71.05    70.17      70.90    12888149
## 2017-12-21     71.00     72.17    70.88      71.59     9682733
## 2017-12-22     71.42     71.87    71.22      71.58    10979165
## 2017-12-26     70.94     71.39    69.63      69.86     8542802
## 2017-12-27     69.77     70.49    69.69      70.06     6345124
## 2017-12-28     70.12     70.32    69.51      69.82     7556877
## 2017-12-29     69.79     70.13    69.43      69.85     6613070

getSymbols("GOOG",src="yahoo") # from yahoo finance 

## [1] "GOOG"

getSymbols("DEXJPUS",src="FRED") # FX rates from FRED 

## [1] "DEXJPUS"

 # Specify lookup parameters, and save for future sessions. 

setSymbolLookup(YHOO='google',GOOG='yahoo') 
setSymbolLookup(DEXJPUS='FRED') 
setSymbolLookup(XPTUSD=list(name="XPT/USD",src="oanda")) 
saveSymbolLookup(file="mysymbols.rda") 
# new sessions call loadSymbolLookup(file="mysymbols.rda") 
 
getSymbols(c("YHOO","GOOG","DEXJPUS","XPTUSD"))

## [1] "YHOO"    "GOOG"    "DEXJPUS" "XPTUSD"

Charting with quantmod

# Specify lookup parameters, and save for future sessions. 
getSymbols("AAPL",src="yahoo", from="2017-01-01")

## [1] "AAPL"

barChart(to.monthly(AAPL),up.col='white',dn.col='blue') 

Fig. 30

chartSeries(to.weekly(AAPL),up.col='white',dn.col='blue')

Fig. 30

 # Add multi-coloring and change background to white 
candleChart(AAPL,multi.col=TRUE,theme="white")

Fig. 30

#Non-OHLC and Volume series are handled automatically 
getSymbols("XPT/USD",src="oanda") 

## [1] "XPTUSD"

chartSeries(XPTUSD,name="Platinum (.oz) in $USD") 

Fig. 30

Platinum, now weekly with custom color candles using the quantmod function to.weekly

chartSeries(to.weekly(XPTUSD),up.col='white',dn.col='blue')

Fig. 30

Technical analysis charting tools

Technical analysis studies from package TTR to the above chart.

pacman::p_load(TTR)
 
getSymbols("AAPL") 

## [1] "AAPL"

chartSeries(to.weekly(AAPL)) 

Fig. 30

addMACD() 

Fig. 30

addBBands() 

Fig. 30

# Create a quantmod object for use in 
# in later model fitting. Note there is 
# no need to load the data before hand. 
 
setSymbolLookup(SPY='yahoo',VXN=list(name='^VIX',src='yahoo')) 
 
mm <- specifyModel(Next(OpCl(SPY)) ~ OpCl(SPY) + Cl(VIX))
 
modelData(mm)%>%head()

##            Next.OpCl.SPY      OpCl.SPY  Cl.VIX
## 2014-12-04  0.0006254149  0.0005782548 28447.7
## 2014-12-05 -0.0043851339  0.0006254149 26056.5
## 2014-12-08  0.0102755104 -0.0043851339 23582.8
## 2014-12-09 -0.0133553492  0.0102755104 21274.0
## 2014-12-10  0.0015204875 -0.0133553492 19295.0
## 2014-12-11 -0.0086360048  0.0015204875 17728.3

mm is now a quantmod object holding the model formula and data structure implying the next (Next) period’s open to close of the S&P 500 ETF (OpCl(SPY)) is modelled as a fucntion of the current period open to close and the current close of the VIX (Cl(VIX)).

The call to modelData extracts the relevant data set, with transforms magically applied. You can take the data and do with it as you’d like. A more direct function to accomplish the same end is buildData.

getSymbols("GS",from="2018-02-01") #Goldman OHLC from yahoo 

## [1] "GS"

is.OHLC(GS) # does the data contain at least OHL and C? 

## [1] TRUE

has.Vo(GS) # how about volume? 

## [1] TRUE

Op(GS) # just the Open column please. 

##            GS.Open
## 2018-02-01  266.15
## 2018-02-02  270.00
## 2018-02-05  257.51
## 2018-02-06  244.21
## 2018-02-07  255.90
## 2018-02-08  257.93
## 2018-02-09  249.62
## 2018-02-12  251.15
## 2018-02-13  252.01
## 2018-02-14  255.00
## 2018-02-15  265.76
## 2018-02-16  265.99
## 2018-02-20  266.89
## 2018-02-21  263.92
## 2018-02-22  264.40
## 2018-02-23  263.00

seriesHi(GS) # where and what was the high point

##            GS.Open GS.High GS.Low GS.Close GS.Volume GS.Adjusted
## 2018-02-01  266.15  272.59    266   272.23   3388200      272.23

OpCl(GS) #daily percent change open to close 

##                 OpCl.GS
## 2018-02-01  0.022844325
## 2018-02-02 -0.036888856
## 2018-02-05 -0.032620126
## 2018-02-06  0.059334198
## 2018-02-07  0.004689379
## 2018-02-08 -0.044895853
## 2018-02-09 -0.001281917
## 2018-02-12  0.008003225
## 2018-02-13  0.013967716
## 2018-02-14  0.029725439
## 2018-02-15  0.007224499
## 2018-02-16  0.006128069
## 2018-02-20 -0.007493724
## 2018-02-21 -0.001970366
## 2018-02-22 -0.011232984
## 2018-02-23  0.014334559

OpOp(GS) #one period open to open change 

##                  OpOp.GS
## 2018-02-01            NA
## 2018-02-02  0.0144655498
## 2018-02-05 -0.0462592222
## 2018-02-06 -0.0516484893
## 2018-02-07  0.0478685830
## 2018-02-08  0.0079327825
## 2018-02-09 -0.0322180368
## 2018-02-12  0.0061293127
## 2018-02-13  0.0034242525
## 2018-02-14  0.0118646286
## 2018-02-15  0.0421961176
## 2018-02-16  0.0008653672
## 2018-02-20  0.0033836800
## 2018-02-21 -0.0111281870
## 2018-02-22  0.0018186609
## 2018-02-23 -0.0052949850

HiCl(GS) #the percent change from high to close

##                  HiCl.GS
## 2018-02-01 -0.0013206097
## 2018-02-02 -0.0430910432
## 2018-02-05 -0.0538928509
## 2018-02-06 -0.0001931903
## 2018-02-07 -0.0185150058
## 2018-02-08 -0.0480330514
## 2018-02-09 -0.0094170740
## 2018-02-12 -0.0116728092
## 2018-02-13 -0.0033931163
## 2018-02-14 -0.0004948575
## 2018-02-15 -0.0011940560
## 2018-02-16 -0.0061646019
## 2018-02-20 -0.0147293692
## 2018-02-21 -0.0244444667
## 2018-02-22 -0.0183614219
## 2018-02-23 -0.0007866581

• Lag: What was the previous value in the series
• Next: What is the next value in the series
• Delt: Compute the change (delta) from two prices

Lag(Cl(GS)) #One period lag of the close 

##             Lag.1
## 2018-02-01     NA
## 2018-02-02 272.23
## 2018-02-05 260.04
## 2018-02-06 249.11
## 2018-02-07 258.70
## 2018-02-08 257.10
## 2018-02-09 246.35
## 2018-02-12 249.30
## 2018-02-13 253.16
## 2018-02-14 255.53
## 2018-02-15 262.58
## 2018-02-16 267.68
## 2018-02-20 267.62
## 2018-02-21 264.89
## 2018-02-22 263.40
## 2018-02-23 261.43

Lag(Cl(GS),c(1,3,5)) #One, three, and five period lags 

##             Lag.1  Lag.3  Lag.5
## 2018-02-01     NA     NA     NA
## 2018-02-02 272.23     NA     NA
## 2018-02-05 260.04     NA     NA
## 2018-02-06 249.11 272.23     NA
## 2018-02-07 258.70 260.04     NA
## 2018-02-08 257.10 249.11 272.23
## 2018-02-09 246.35 258.70 260.04
## 2018-02-12 249.30 257.10 249.11
## 2018-02-13 253.16 246.35 258.70
## 2018-02-14 255.53 249.30 257.10
## 2018-02-15 262.58 253.16 246.35
## 2018-02-16 267.68 255.53 249.30
## 2018-02-20 267.62 262.58 253.16
## 2018-02-21 264.89 267.68 255.53
## 2018-02-22 263.40 267.62 262.58
## 2018-02-23 261.43 264.89 267.68

Next(OpCl(GS)) #The next periods open to close - today! 

##                    Next
## 2018-02-01 -0.036888856
## 2018-02-02 -0.032620126
## 2018-02-05  0.059334198
## 2018-02-06  0.004689379
## 2018-02-07 -0.044895853
## 2018-02-08 -0.001281917
## 2018-02-09  0.008003225
## 2018-02-12  0.013967716
## 2018-02-13  0.029725439
## 2018-02-14  0.007224499
## 2018-02-15  0.006128069
## 2018-02-16 -0.007493724
## 2018-02-20 -0.001970366
## 2018-02-21 -0.011232984
## 2018-02-22  0.014334559
## 2018-02-23           NA

# Open to close one-day, two-day and three-day lags 
Delt(Op(GS),Cl(GS),k=1:3)

##            Delt.1.arithmetic Delt.2.arithmetic Delt.3.arithmetic
## 2018-02-01                NA                NA                NA
## 2018-02-02      -0.022956923                NA                NA
## 2018-02-05      -0.077370367      -0.064024022                NA
## 2018-02-06       0.004621187      -0.041851807     -0.0279916670
## 2018-02-07       0.052782436      -0.001592187     -0.0477777556
## 2018-02-08      -0.037319219       0.008762946     -0.0433381366
## 2018-02-09      -0.033458652      -0.025791290      0.0208427004
## 2018-02-12       0.014181592      -0.018493348     -0.0107072687
## 2018-02-13       0.017439797       0.023676004     -0.0093048271
## 2018-02-14       0.041942749       0.045510624      0.0519188857
## 2018-02-15       0.049725463       0.062180066      0.0658172383
## 2018-02-16       0.006998739       0.049490176      0.0619419877
## 2018-02-20      -0.004135400      -0.003273611      0.0387843725
## 2018-02-21      -0.013076626      -0.009737193     -0.0088802525
## 2018-02-22      -0.009434752      -0.020457948     -0.0171434910
## 2018-02-23       0.008963673       0.010798635     -0.0004497208

Subsetting by Time and Date? – xts Makes It Easy

GS['2018'] #returns all Goldman's 2018 OHLC 

##            GS.Open GS.High GS.Low GS.Close GS.Volume GS.Adjusted
## 2018-02-01  266.15  272.59 266.00   272.23   3388200      272.23
## 2018-02-02  270.00  271.75 259.35   260.04   5913900      260.04
## 2018-02-05  257.51  263.30 241.50   249.11   6528500      249.11
## 2018-02-06  244.21  258.75 242.11   258.70   6739900      258.70
## 2018-02-07  255.90  261.95 255.66   257.10   4125000      257.10
## 2018-02-08  257.93  258.78 246.05   246.35   4166200      246.35
## 2018-02-09  249.62  251.67 239.29   249.30   5978600      249.30
## 2018-02-12  251.15  256.15 248.82   253.16   3692500      253.16
## 2018-02-13  252.01  256.40 250.58   255.53   2700500      255.53
## 2018-02-14  255.00  262.71 254.48   262.58   3793800      262.58
## 2018-02-15  265.76  268.00 262.91   267.68   3523100      267.68
## 2018-02-16  265.99  269.28 265.90   267.62   2602500      267.62
## 2018-02-20  266.89  268.85 263.31   264.89   2618400      264.89
## 2018-02-21  263.92  270.00 263.13   263.40   3017900      263.40
## 2018-02-22  264.40  266.32 260.72   261.43   2678300      261.43
## 2018-02-23  263.00  266.98 262.40   266.77   2477900      266.77

GS['2018'] #now just 2008 

##            GS.Open GS.High GS.Low GS.Close GS.Volume GS.Adjusted
## 2018-02-01  266.15  272.59 266.00   272.23   3388200      272.23
## 2018-02-02  270.00  271.75 259.35   260.04   5913900      260.04
## 2018-02-05  257.51  263.30 241.50   249.11   6528500      249.11
## 2018-02-06  244.21  258.75 242.11   258.70   6739900      258.70
## 2018-02-07  255.90  261.95 255.66   257.10   4125000      257.10
## 2018-02-08  257.93  258.78 246.05   246.35   4166200      246.35
## 2018-02-09  249.62  251.67 239.29   249.30   5978600      249.30
## 2018-02-12  251.15  256.15 248.82   253.16   3692500      253.16
## 2018-02-13  252.01  256.40 250.58   255.53   2700500      255.53
## 2018-02-14  255.00  262.71 254.48   262.58   3793800      262.58
## 2018-02-15  265.76  268.00 262.91   267.68   3523100      267.68
## 2018-02-16  265.99  269.28 265.90   267.62   2602500      267.62
## 2018-02-20  266.89  268.85 263.31   264.89   2618400      264.89
## 2018-02-21  263.92  270.00 263.13   263.40   3017900      263.40
## 2018-02-22  264.40  266.32 260.72   261.43   2678300      261.43
## 2018-02-23  263.00  266.98 262.40   266.77   2477900      266.77

GS['2018-01'] #now just January of 2008 

##      GS.Open GS.High GS.Low GS.Close GS.Volume GS.Adjusted

GS['2017-06::2018-01-12'] #Jun of 17 through Jan 12 of 18

##      GS.Open GS.High GS.Low GS.Close GS.Volume GS.Adjusted

GS['::'] # everything in GS 

##            GS.Open GS.High GS.Low GS.Close GS.Volume GS.Adjusted
## 2018-02-01  266.15  272.59 266.00   272.23   3388200      272.23
## 2018-02-02  270.00  271.75 259.35   260.04   5913900      260.04
## 2018-02-05  257.51  263.30 241.50   249.11   6528500      249.11
## 2018-02-06  244.21  258.75 242.11   258.70   6739900      258.70
## 2018-02-07  255.90  261.95 255.66   257.10   4125000      257.10
## 2018-02-08  257.93  258.78 246.05   246.35   4166200      246.35
## 2018-02-09  249.62  251.67 239.29   249.30   5978600      249.30
## 2018-02-12  251.15  256.15 248.82   253.16   3692500      253.16
## 2018-02-13  252.01  256.40 250.58   255.53   2700500      255.53
## 2018-02-14  255.00  262.71 254.48   262.58   3793800      262.58
## 2018-02-15  265.76  268.00 262.91   267.68   3523100      267.68
## 2018-02-16  265.99  269.28 265.90   267.62   2602500      267.62
## 2018-02-20  266.89  268.85 263.31   264.89   2618400      264.89
## 2018-02-21  263.92  270.00 263.13   263.40   3017900      263.40
## 2018-02-22  264.40  266.32 260.72   261.43   2678300      261.43
## 2018-02-23  263.00  266.98 262.40   266.77   2477900      266.77

GS['2018::'] # everything in GS, from 2018 onward 

##            GS.Open GS.High GS.Low GS.Close GS.Volume GS.Adjusted
## 2018-02-01  266.15  272.59 266.00   272.23   3388200      272.23
## 2018-02-02  270.00  271.75 259.35   260.04   5913900      260.04
## 2018-02-05  257.51  263.30 241.50   249.11   6528500      249.11
## 2018-02-06  244.21  258.75 242.11   258.70   6739900      258.70
## 2018-02-07  255.90  261.95 255.66   257.10   4125000      257.10
## 2018-02-08  257.93  258.78 246.05   246.35   4166200      246.35
## 2018-02-09  249.62  251.67 239.29   249.30   5978600      249.30
## 2018-02-12  251.15  256.15 248.82   253.16   3692500      253.16
## 2018-02-13  252.01  256.40 250.58   255.53   2700500      255.53
## 2018-02-14  255.00  262.71 254.48   262.58   3793800      262.58
## 2018-02-15  265.76  268.00 262.91   267.68   3523100      267.68
## 2018-02-16  265.99  269.28 265.90   267.62   2602500      267.62
## 2018-02-20  266.89  268.85 263.31   264.89   2618400      264.89
## 2018-02-21  263.92  270.00 263.13   263.40   3017900      263.40
## 2018-02-22  264.40  266.32 260.72   261.43   2678300      261.43
## 2018-02-23  263.00  266.98 262.40   266.77   2477900      266.77

non.contiguous <- c('2017-01','2007-02','2007-12') 
GS[non.contiguous]

##      GS.Open GS.High GS.Low GS.Close GS.Volume GS.Adjusted

The Last 3 Days of The First 2 Weeks

Another common problem when trying to subset time series data often involves looking at a particular time period. Often the last n-periods may be desired when charting the price of a security, or when constructing a model for trading or analysis. [Note: these functions are now in the standalone package xts - which quantmod requires]

To facilitate this ‘time-based’ subsetting, one can use the functions first and last. Essentially extending the concept of head and tail, one can now use character strings to describe the part of the data to be returned. As is probably expected by now - an example may help to clarify.

last(GS) #returns the last obs. 

## [1] 266.77

#last(GS, '3 months') 

last(first(GS, '2 weeks'), '3 days')

## [1] 266.15

# let's try something a bit cooler. 
#last(GS, '3 weeks') 

#last(GS, '-2 weeks') # all except the last 2 weeks 

Aggregating to a different time scale

Often, and especially with higher frequency data, it is necessary to aggregate data into lower frequency terms. For example, take daily data - OHLC or a standard time series - and convert it to weekly or monthly OHLC data.

With xts it is as simple as to.weekly or to.monthly. In fact, it is currently possible to take everything from minute data all the way up to quarterly data and convert it into something lower frequency. Minute data can become 5 or 10 minute data (to.minutes5 and to.minutes10, respectively), which can in turn be turned into hourly or daily data. Daily data can become weekly, monthly, or even yearly. All carried out in compiled code, and all blazingly fast, yes blazingly fast - convert 2 months of 1-minute bars into 3-minute bars in less than 0.1 seconds and anything lower in half that time. A full year of minute bars in less than a second on a moderately fast computer.

Is your data weekly, daily, or hourly? A call to periodicity will provide the answer; a call to nweeks will tell you the number of weeks as well.

periodicity(GS) 

## Daily periodicity from 2018-02-01 to 2018-02-23

unclass(periodicity(GS)) 

## $difftime
## Time difference of 86400 days
## 
## $frequency
## [1] 86400
## 
## $start
## [1] "2018-02-01"
## 
## $end
## [1] "2018-02-23"
## 
## $units
## [1] "days"
## 
## $scale
## [1] "daily"
## 
## $label
## [1] "day"

to.weekly(GS) 

##            GS.Open GS.High GS.Low GS.Close GS.Volume GS.Adjusted
## 2018-02-02  266.15  272.59 259.35   260.04   9302100      260.04
## 2018-02-09  257.51  263.30 239.29   249.30  27538200      249.30
## 2018-02-16  251.15  269.28 248.82   267.62  16312400      267.62
## 2018-02-23  266.89  270.00 260.72   266.77  10792500      266.77

to.monthly(GS) 

##          GS.Open GS.High GS.Low GS.Close GS.Volume GS.Adjusted
## Feb 2018  266.15  272.59 239.29   266.77  63945200      266.77

#periodicity(to.monthly(GS)) 

ndays(GS); nweeks(GS); nyears(GS) 

## [1] 16

## [1] 4

## [1] 1

# Let's try some non-OHLC to start 
getFX("USD/EUR") 

## [1] "USDEUR"

periodicity(USDEUR) 

## Daily periodicity from 2017-08-31 to 2018-02-25

to.weekly(USDEUR) 

##            USDEUR.Open USDEUR.High USDEUR.Low USDEUR.Close
## 2017-09-03    0.841538    0.843034   0.841190     0.841720
## 2017-09-10    0.840512    0.840512   0.830135     0.831980
## 2017-09-17    0.834033    0.840856   0.834033     0.837547
## 2017-09-24    0.837044    0.839532   0.834508     0.839094
## 2017-10-01    0.841394    0.850608   0.841394     0.846481
## 2017-10-08    0.850892    0.853762   0.850040     0.852244
## 2017-10-15    0.851759    0.851759   0.843702     0.846069
## 2017-10-22    0.847410    0.849895   0.845750     0.848862
## 2017-10-29    0.850896    0.861444   0.848574     0.861444
## 2017-11-05    0.860084    0.861384   0.858084     0.860658
## 2017-11-12    0.861710    0.863012   0.857192     0.857226
## 2017-11-19    0.857868    0.857868   0.846988     0.847846
## 2017-11-26    0.850961    0.852056   0.838040     0.838040
## 2017-12-03    0.838817    0.843773   0.838817     0.840976
## 2017-12-10    0.843260    0.850480   0.843260     0.849527
## 2017-12-17    0.848514    0.851020   0.846602     0.851020
## 2017-12-24    0.848798    0.848798   0.842464     0.843826
## 2017-12-31    0.842984    0.842984   0.834758     0.834758
## 2018-01-07    0.833110    0.833110   0.829714     0.831254
## 2018-01-14    0.833770    0.837326   0.819653     0.819653
## 2018-01-21    0.816692    0.818451   0.816481     0.818240
## 2018-01-28    0.816724    0.816724   0.804242     0.804498
## 2018-02-04    0.807013    0.807013   0.801417     0.802910
## 2018-02-11    0.804272    0.816290   0.804272     0.816245
## 2018-02-18    0.814634    0.814634   0.801250     0.805784
## 2018-02-25    0.805853    0.813392   0.805853     0.813392

periodicity(to.weekly(USDEUR))

## Weekly periodicity from 2017-09-03 to 2018-02-25

Apply by Period

It may be useful to identify endpoints in your data by date with the function endpoints. You can use those endpoints (or ones generated automatically) with the functions in the period.apply family. Quickly calculate periodic minimums, maximums, sums, and products - as well as general applys (with the periodic slant) with a few simple functions.

endpoints(GS,on="months") 

## [1]  0 16

# find the maximum closing price each week 
apply.weekly(GS,FUN=function(x) { max(Cl(x)) } ) 

##              [,1]
## 2018-02-02 272.23
## 2018-02-09 258.70
## 2018-02-16 267.68
## 2018-02-23 266.77

# the same thing - only more general 
period.apply(GS,endpoints(GS,on='weeks'), 
 FUN=function(x) { max(Cl(x)) } ) 

##              [,1]
## 2018-02-02 272.23
## 2018-02-09 258.70
## 2018-02-16 267.68
## 2018-02-23 266.77

 # same thing - only 50x faster! 
as.numeric(period.max(Cl(GS),endpoints(GS,on='weeks')))

## [1] 272.23 258.70 267.68 266.77

Of course, additional wrappers exist to quickly apply arbitrary functions over daily, monthly, quarterly and annual time periods as well. There are also Fortran-based routines for period.min, period.sum, and period.prod, in addition to the period.max function.

Period Returns

The last set of functions simply provide a fast and reliable way to calculate returns over calendar periods - derived from the function periodReturn. Named for what they return. A note on starting/ending date convention, all periods could be named in a variety of ways - the first of the period, the first trading time of the period, the last trading time of the period, or even the last day of the period. xts has adopted the last observation of a given period as the date to record for the larger period. There may be a point in the future where this is settable as well. This is now user settable through the indexAt argument to the underlying to.period versions to.monthly and to.quarterly. The full details can be found in the related help pages, but a quick explanation is indexAt lets one set the resulting index to the first of each period (firstof), the last of each period (lastof), the starting observation of the period (startof), the ending observation of the period (endof), the month of the period (yearmon) or the quarter of the period (yearqtr). For most classes of time-series data this defaults to yearmon for monthly observations and yearqtr for quarterly requests.

 # Quick returns - quantmod style 
 
getSymbols("SBUX",from="2018-01-01") 

## [1] "SBUX"

dailyReturn(SBUX) # returns by day 

##            daily.returns
## 2018-01-02  -0.005522002
## 2018-01-03   0.018740204
## 2018-01-04   0.003747249
## 2018-01-05   0.011539131
## 2018-01-08  -0.005032713
## 2018-01-09  -0.002191890
## 2018-01-10   0.010814464
## 2018-01-11   0.003009027
## 2018-01-12   0.006666700
## 2018-01-16   0.002648990
## 2018-01-17   0.001651238
## 2018-01-18   0.007088691
## 2018-01-19   0.002782747
## 2018-01-22   0.002448613
## 2018-01-23   0.004559502
## 2018-01-24  -0.013940623
## 2018-01-25  -0.004603041
## 2018-01-26  -0.042279059
## 2018-01-29  -0.016727056
## 2018-01-30   0.002981392
## 2018-01-31  -0.006644483
## 2018-02-01  -0.014258071
## 2018-02-02  -0.004107143
## 2018-02-05  -0.019365268
## 2018-02-06   0.016822125
## 2018-02-07  -0.020679769
## 2018-02-08  -0.012669831
## 2018-02-09   0.015064199
## 2018-02-12   0.014657365
## 2018-02-13   0.005236493
## 2018-02-14   0.005748231
## 2018-02-15   0.010537596
## 2018-02-16  -0.001767444
## 2018-02-20  -0.009029727
## 2018-02-21   0.002322619
## 2018-02-22  -0.012477648
## 2018-02-23   0.013357346

weeklyReturn(SBUX) # returns by week 

##            weekly.returns
## 2018-01-05    0.028645383
## 2018-01-12    0.013252826
## 2018-01-19    0.014238344
## 2018-01-26   -0.053378977
## 2018-02-02   -0.038282496
## 2018-02-09   -0.021337601
## 2018-02-16    0.034811248
## 2018-02-23   -0.006019848

monthlyReturn(SBUX) # returns by month, indexed by yearmon 

##            monthly.returns
## 2018-01-31     -0.01967213
## 2018-02-23     -0.01179373

# daily,weekly,monthly,quarterly, and yearly 

allReturns(SBUX) # note the plural

##                   daily       weekly     monthly   quarterly      yearly
## 2018-01-02           NA           NA          NA          NA          NA
## 2018-01-03  0.018740204           NA          NA          NA          NA
## 2018-01-04  0.003747249           NA          NA          NA          NA
## 2018-01-05  0.011539131  0.028645383          NA          NA          NA
## 2018-01-08 -0.005032713           NA          NA          NA          NA
## 2018-01-09 -0.002191890           NA          NA          NA          NA
## 2018-01-10  0.010814464           NA          NA          NA          NA
## 2018-01-11  0.003009027           NA          NA          NA          NA
## 2018-01-12  0.006666700  0.013252826          NA          NA          NA
## 2018-01-16  0.002648990           NA          NA          NA          NA
## 2018-01-17  0.001651238           NA          NA          NA          NA
## 2018-01-18  0.007088691           NA          NA          NA          NA
## 2018-01-19  0.002782747  0.014238344          NA          NA          NA
## 2018-01-22  0.002448613           NA          NA          NA          NA
## 2018-01-23  0.004559502           NA          NA          NA          NA
## 2018-01-24 -0.013940623           NA          NA          NA          NA
## 2018-01-25 -0.004603041           NA          NA          NA          NA
## 2018-01-26 -0.042279059 -0.053378977          NA          NA          NA
## 2018-01-29 -0.016727056           NA          NA          NA          NA
## 2018-01-30  0.002981392           NA          NA          NA          NA
## 2018-01-31 -0.006644483           NA -0.01967213          NA          NA
## 2018-02-01 -0.014258071           NA          NA          NA          NA
## 2018-02-02 -0.004107143 -0.038282496          NA          NA          NA
## 2018-02-05 -0.019365268           NA          NA          NA          NA
## 2018-02-06  0.016822125           NA          NA          NA          NA
## 2018-02-07 -0.020679769           NA          NA          NA          NA
## 2018-02-08 -0.012669831           NA          NA          NA          NA
## 2018-02-09  0.015064199 -0.021337601          NA          NA          NA
## 2018-02-12  0.014657365           NA          NA          NA          NA
## 2018-02-13  0.005236493           NA          NA          NA          NA
## 2018-02-14  0.005748231           NA          NA          NA          NA
## 2018-02-15  0.010537596           NA          NA          NA          NA
## 2018-02-16 -0.001767444  0.034811248          NA          NA          NA
## 2018-02-20 -0.009029727           NA          NA          NA          NA
## 2018-02-21  0.002322619           NA          NA          NA          NA
## 2018-02-22 -0.012477648           NA          NA          NA          NA
## 2018-02-23  0.013357346 -0.006019848 -0.01179373 -0.03123386 -0.03123386

library(quantmod)   # get stock prices; useful stock analysis functions
library(xts)        # working with extensible time series 
library(rvest)      # web scraping
library(tidyverse)  # ggplot2, purrr, dplyr, tidyr, readr, tibble
library(stringr)    # working with strings
library(forcats)    # working with factors
library(lubridate)  # working with dates in tibbles / data frames
library(plotly)     # Interactive plots
library(corrplot)   # Visuazlize correlation plots

We’ll use MasterCard, ticker symbol “MA”, for the example.

getSymbols("MA", from = "2008-01-01", to = "2018-01-01")

## [1] "MA"

MA %>% class()

## [1] "xts" "zoo"

MA %>% str()

## An 'xts' object on 2008-01-02/2017-12-29 containing:
##   Data: num [1:2518, 1:6] 21.8 21.5 20.5 20.2 20.1 ...
##  - attr(*, "dimnames")=List of 2
##   ..$ : NULL
##   ..$ : chr [1:6] "MA.Open" "MA.High" "MA.Low" "MA.Close" ...
##   Indexed by objects of class: [Date] TZ: UTC
##   xts Attributes:  
## List of 2
##  $ src    : chr "yahoo"
##  $ updated: POSIXct[1:1], format: "2018-02-26 18:48:27"

MA %>% head()

##            MA.Open MA.High MA.Low MA.Close MA.Volume MA.Adjusted
## 2008-01-02  21.790  22.029 21.260   21.468  20156000    17.31038
## 2008-01-03  21.456  21.505 20.513   21.212  28994000    17.10395
## 2008-01-04  20.482  20.920 19.844   20.119  38427000    16.22263
## 2008-01-07  20.171  20.199 18.744   20.051  43861000    16.16780
## 2008-01-08  20.091  20.770 19.599   19.800  33315000    15.96540
## 2008-01-09  19.700  20.021 18.965   19.658  38939000    15.97190

MA %>% 
    Ad() %>% 
    chartSeries()

Fig. 30

MA %>% 
    Cl() %>% 
   barChart()

Fig. 30

The chartSeries() function has a number of options help visualize the stock performance.

subset: Isolated the year to 2015. This takes dates and ranges in xts subset format. Here’s some tricks from StackOverflow. TA: Added Bollinger Bands, Volume, and Moving Average Convergence Divergence (MACD) plots. These can also be added after the fact with add___() functions (e.g addMACD()). theme: Changed to a white theme.

MA %>%
    chartSeries(TA='addBBands();
                    addBBands(draw="p");
                    addVo();
                    addMACD()', 
                subset='2017',
                theme="white"
                ) 

Fig. 30

The quantmod package includes functions to get daily, weekly, monthly, quarterly, and annual returns. We’ll use dailyReturn() to get a xts object of daily returns (see ?periodReturn, which is the base function). The default is an arithmetic return. Specify type = “log” if a logarithmic return is needed

MA %>%
    Ad() %>%
    dailyReturn(type = 'log') %>% 
    head() 

##            daily.returns
## 2008-01-02  0.0000000000
## 2008-01-03 -0.0119966588
## 2008-01-04 -0.0529024317
## 2008-01-07 -0.0033856334
## 2008-01-08 -0.0125973558
## 2008-01-09  0.0004071101

SIMULATING STOCK PRICES

The approach I use is similar to rfortraders.com’s Lecture 6 - Stochastic Processes and Monte Carlo. The fundamental idea is that stock returns, measured daily, weekly, monthly, …, are approximately normally distributed and uncorrelated. As a result, we can theoretically model the behavior of stock prices within a confidence interval based on the stock’s prior returns.

RETURNS ARE NORMALLY DISTRIBUTED

Applying the log-transformation, we can visually see that the daily returns are approximately normally distributed:

MA_log_returns <- MA %>%
    Ad() %>%
    dailyReturn(type = "log")

names(MA_log_returns) <- "MA.Log.Returns"

# Plot the log-returns    
MA_log_returns %>%    
    ggplot(aes(x = MA.Log.Returns)) + 
    geom_histogram(bins = 100) + 
    geom_density() +
    geom_rug(alpha = 0.5)

Fig. 30

We can examine the distribution of log returns by applying the quantile() function.

probs <- c(.005, .025, .25, .5, .75, .975, .995)
dist_log_returns <- MA_log_returns %>% 
    quantile(probs = probs, na.rm = TRUE)
dist_log_returns

##         0.5%         2.5%          25%          50%          75% 
## -0.087845080 -0.041184360 -0.007923496  0.001225754  0.009356280 
##        97.5%        99.5% 
##  0.041506811  0.075800172

The median daily log return is 0.001226 and the 95% confidence interval is between -0.0412 and 0.0415.

The mean and standard deviation of the daily log returns are:

mean_log_returns <- mean(MA_log_returns, na.rm = TRUE)
sd_log_returns <- sd(MA_log_returns, na.rm = TRUE)

To get the actual returns, we need to re-transform the log returns. Pipe the mean of the log returns (mean_log_returns) to exp():

mean_log_returns %>% exp()

## [1] 1.000861

On average, the mean daily return is 0.086% more than the previous day’s price. Doesn’t sound like much, but it compounds daily at an exponential rate.

RANDOM WALK

We can simulate random using a process called a random walk with the mean and standard deviation computed. We’ll simulate prices for 1000 trading days. Every year has approximately 252 trading days, so this simulation spans just under four years.

The script below does the following: we start by specifying the number of random walks (N), the mean (mu), and the standard deviation (sigma). The script then simulates prices by progressively calculating a new price using a random return from the normal distribution characterized by mu and sigma, and multiplying it to the previous day’s price. We then visualize the simulation.

# Parameters
N     <- 1000
mu    <- mean_log_returns
sigma <- sd_log_returns
day <- 1:N
price_init <- MA$MA.Adjusted[[nrow(MA$MA.Adjusted)]]
# Simulate prices
set.seed(386) 
price  <- c(price_init, rep(NA, N-1))
for(i in 2:N) {
    price[i] <- price[i-1] * exp(rnorm(1, mu, sigma))
}
price_sim <- cbind(day, price) %>% 
    as_tibble()
# Visualize price simulation
price_sim %>%
    ggplot(aes(day, price)) +
    geom_line() +
    ggtitle(str_c("MA: Simulated Prices for ", N," Trading Days"))

Fig. 30

MONTE CARLO SIMULATION

Monte-Carlo simulation repeatedly perform the random walk simulation process hundreds or thousands of times. We’ll perform 250 Monte Carlo simulations (M = 250) for one year of trading days simulations (N = 252)

# Parameters
N     <- 252 # Number of Stock Price Simulations
M     <- 250  # Number of Monte Carlo Simulations   
mu    <- mean_log_returns
sigma <- sd_log_returns
day <- 1:N
price_init <- MA$MA.Adjusted[[nrow(MA$MA.Adjusted)]]
# Simulate prices
set.seed(123)
monte_carlo_mat <- matrix(nrow = N, ncol = M)

monte_carlo_mat[1, ] <- price_init

for (j in 1:M) {
 
    for(i in 2:N) {
        monte_carlo_mat[i, j] <- monte_carlo_mat[i - 1, j] * exp(rnorm(1, mu, sigma))
    }
}
# Format and organize data frame
price_sim <- cbind(day, monte_carlo_mat) %>%
   timetk::tk_tbl()

nm <- str_c("Sim.", seq(1, M))
#works similarly to paste0 below
#nm <- paste0("Sim.", seq(1, M))
nm <- c("Day", nm)
names(price_sim) <- nm
price_sim <- price_sim %>%
    gather(key = "Simulation", value = "Stock.Price", -(Day))
# Visualize simulation
price_sim %>%
    ggplot(aes(x = Day, y = Stock.Price, Group = Simulation)) + 
    geom_line(alpha = 0.1) +
    ggtitle(str_c("MA: ", M, 
                  " Monte Carlo Simulations for Prices Over ", N, 
                  " Trading Days"))

Fig. 30

We can get confidence intervals for the stock price at the end of the simulation using the quantile() function.

end_stock_prices <- price_sim %>% 
    filter(Day == max(Day))
probs <- c(.005, .025, .25, .5, .75, .975, .995)
dist_end_stock_prices <- quantile(end_stock_prices$Stock.Price, probs = probs)
dist_end_stock_prices %>% round(2)

##   0.5%   2.5%    25%    50%    75%  97.5%  99.5% 
##  97.29 104.27 151.93 185.72 230.46 310.91 422.70

The 95% confidence interval is between $104.27 and $310.91, with a median (“most likely”) estimated price of $185.72.

compound annual growth rate (CAGR):

# Inputs
N_hist          <- nrow(MA) / 252
p_start_hist    <- MA$MA.Adjusted[[1]]
p_end_hist      <- MA$MA.Adjusted[[nrow(MA)]]
N_sim           <- N / 252
p_start_sim     <- p_end_hist
p_end_sim       <- dist_end_stock_prices[[4]]
# CAGR calculations
CAGR_historical <- (p_end_hist / p_start_hist) ^ (1 / N_hist) - 1
CAGR_sim        <- (p_end_sim / p_start_sim) ^ (1 / N_sim) - 1

summary(CAGR_sim)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   0.229   0.229   0.229   0.229   0.229   0.229

STOCK ANALYSIS: EXPANDED TO ALL S&P500 STOCKS

library(rvest)
# Web-scrape SP500 stock list
sp_500 <- read_html("https://en.wikipedia.org/wiki/List_of_S%26P_500_companies") %>%
    html_node("table.wikitable") %>%
    html_table() %>%
    select(`Ticker symbol`, Security, `GICS Sector`, `GICS Sub Industry`) %>%
    as_tibble()
# Format names
names(sp_500) <- sp_500 %>% 
    names() %>% 
    str_to_lower() %>% 
    make.names()
# Show results
sp_500 

## # A tibble: 505 x 4
##     ticker.symbol security         gics.sector     gics.sub.industry      
##     <chr>         <chr>            <chr>           <chr>                  
##   1 MMM           3M Company       Industrials     Industrial Conglomerat…
##   2 ABT           Abbott Laborato… Health Care     Health Care Equipment  
##   3 ABBV          AbbVie Inc.      Health Care     Pharmaceuticals        
##   4 ACN           Accenture plc    Information Te… IT Consulting & Other …
##   5 ATVI          Activision Bliz… Information Te… Home Entertainment Sof…
##   6 AYI           Acuity Brands I… Industrials     Electrical Components …
##   7 ADBE          Adobe Systems I… Information Te… Application Software   
##   8 AMD           Advanced Micro … Information Te… Semiconductors         
##   9 AAP           Advance Auto Pa… Consumer Discr… Automotive Retail      
##  10 AES           AES Corp         Utilities       Independent Power Prod…
##  11 AET           Aetna Inc        Health Care     Managed Health Care    
##  12 AMG           Affiliated Mana… Financials      Asset Management & Cus…
##  13 AFL           AFLAC Inc        Financials      Life & Health Insurance
##  14 A             Agilent Technol… Health Care     Health Care Equipment  
##  15 APD           Air Products & … Materials       Industrial Gases       
##  16 AKAM          Akamai Technolo… Information Te… Internet Software & Se…
##  17 ALK           Alaska Air Grou… Industrials     Airlines               
##  18 ALB           Albemarle Corp   Materials       Specialty Chemicals    
##  19 ARE           Alexandria Real… Real Estate     Office REITs           
##  20 ALXN          Alexion Pharmac… Health Care     Biotechnology          
##  21 ALGN          Align Technology Health Care     Health Care Supplies   
##  22 ALLE          Allegion         Industrials     Building Products      
##  23 AGN           Allergan, Plc    Health Care     Pharmaceuticals        
##  24 ADS           Alliance Data S… Information Te… Data Processing & Outs…
##  25 LNT           Alliant Energy … Utilities       Electric Utilities     
##  26 ALL           Allstate Corp    Financials      Property & Casualty In…
##  27 GOOGL         Alphabet Inc Cl… Information Te… Internet Software & Se…
##  28 GOOG          Alphabet Inc Cl… Information Te… Internet Software & Se…
##  29 MO            Altria Group Inc Consumer Stapl… Tobacco                
##  30 AMZN          Amazon.com Inc   Consumer Discr… Internet & Direct Mark…
##  31 AEE           Ameren Corp      Utilities       Multi-Utilities        
##  32 AAL           American Airlin… Industrials     Airlines               
##  33 AEP           American Electr… Utilities       Electric Utilities     
##  34 AXP           American Expres… Financials      Consumer Finance       
##  35 AIG           American Intern… Financials      Property & Casualty In…
##  36 AMT           American Tower … Real Estate     Specialized REITs      
##  37 AWK           American Water … Utilities       Water Utilities        
##  38 AMP           Ameriprise Fina… Financials      Asset Management & Cus…
##  39 ABC           AmerisourceBerg… Health Care     Health Care Distributo…
##  40 AME           AMETEK Inc       Industrials     Electrical Components …
##  41 AMGN          Amgen Inc        Health Care     Biotechnology          
##  42 APH           Amphenol Corp    Information Te… Electronic Components  
##  43 APC           Anadarko Petrol… Energy          Oil & Gas Exploration …
##  44 ADI           Analog Devices,… Information Te… Semiconductors         
##  45 ANDV          Andeavor         Energy          Oil & Gas Refining & M…
##  46 ANSS          ANSYS            Information Te… Application Software   
##  47 ANTM          Anthem Inc.      Health Care     Managed Health Care    
##  48 AON           Aon plc          Financials      Insurance Brokers      
##  49 AOS           A.O. Smith Corp  Industrials     Building Products      
##  50 APA           Apache Corporat… Energy          Oil & Gas Exploration …
##  51 AIV           Apartment Inves… Real Estate     Residential REITs      
##  52 AAPL          Apple Inc.       Information Te… Technology Hardware, S…
##  53 AMAT          Applied Materia… Information Te… Semiconductor Equipment
##  54 APTV          Aptiv Plc        Consumer Discr… Auto Parts & Equipment 
##  55 ADM           Archer-Daniels-… Consumer Stapl… Agricultural Products  
##  56 ARNC          Arconic Inc      Industrials     Aerospace & Defense    
##  57 AJG           Arthur J. Galla… Financials      Insurance Brokers      
##  58 AIZ           Assurant Inc     Financials      Multi-line Insurance   
##  59 T             AT&T Inc         Telecommunicat… Integrated Telecommuni…
##  60 ADSK          Autodesk Inc     Information Te… Application Software   
##  61 ADP           Automatic Data … Information Te… Internet Software & Se…
##  62 AZO           AutoZone Inc     Consumer Discr… Specialty Stores       
##  63 AVB           AvalonBay Commu… Real Estate     Residential REITs      
##  64 AVY           Avery Dennison … Materials       Paper Packaging        
##  65 BHGE          Baker Hughes, a… Energy          Oil & Gas Equipment & …
##  66 BLL           Ball Corp        Materials       Metal & Glass Containe…
##  67 BAC           Bank of America… Financials      Diversified Banks      
##  68 BK            The Bank of New… Financials      Asset Management & Cus…
##  69 BAX           Baxter Internat… Health Care     Health Care Equipment  
##  70 BBT           BB&T Corporation Financials      Regional Banks         
##  71 BDX           Becton Dickinson Health Care     Health Care Equipment  
##  72 BRK.B         Berkshire Hatha… Financials      Multi-Sector Holdings  
##  73 BBY           Best Buy Co. In… Consumer Discr… Computer & Electronics…
##  74 BIIB          Biogen Inc.      Health Care     Biotechnology          
##  75 BLK           BlackRock        Financials      Asset Management & Cus…
##  76 HRB           Block H&R        Financials      Consumer Finance       
##  77 BA            Boeing Company   Industrials     Aerospace & Defense    
##  78 BWA           BorgWarner       Consumer Discr… Auto Parts & Equipment 
##  79 BXP           Boston Properti… Real Estate     Office REITs           
##  80 BSX           Boston Scientif… Health Care     Health Care Equipment  
##  81 BHF           Brighthouse Fin… Financials      Life & Health Insurance
##  82 BMY           Bristol-Myers S… Health Care     Health Care Distributo…
##  83 AVGO          Broadcom         Information Te… Semiconductors         
##  84 BF.B          Brown-Forman Co… Consumer Stapl… Distillers & Vintners  
##  85 CHRW          C. H. Robinson … Industrials     Air Freight & Logistics
##  86 CA            CA, Inc.         Information Te… Systems Software       
##  87 COG           Cabot Oil & Gas  Energy          Oil & Gas Exploration …
##  88 CDNS          Cadence Design … Information Te… Application Software   
##  89 CPB           Campbell Soup    Consumer Stapl… Packaged Foods & Meats 
##  90 COF           Capital One Fin… Financials      Consumer Finance       
##  91 CAH           Cardinal Health… Health Care     Health Care Distributo…
##  92 CBOE          CBOE Holdings    Financials      Financial Exchanges & …
##  93 KMX           Carmax Inc       Consumer Discr… Specialty Stores       
##  94 CCL           Carnival Corp.   Consumer Discr… Hotels, Resorts & Crui…
##  95 CAT           Caterpillar Inc. Industrials     Construction Machinery…
##  96 CBG           CBRE Group       Real Estate     Real Estate Services   
##  97 CBS           CBS Corp.        Consumer Discr… Broadcasting           
##  98 CELG          Celgene Corp.    Health Care     Biotechnology          
##  99 CNC           Centene Corpora… Health Care     Managed Health Care    
## 100 CNP           CenterPoint Ene… Utilities       Multi-Utilities        
## 101 CTL           CenturyLink Inc  Telecommunicat… Integrated Telecommuni…
## 102 CERN          Cerner           Health Care     Health Care Technology 
## 103 CF            CF Industries H… Materials       Fertilizers & Agricult…
## 104 SCHW          Charles Schwab … Financials      Investment Banking & B…
## 105 CHTR          Charter Communi… Consumer Discr… Cable & Satellite      
## 106 CHK           Chesapeake Ener… Energy          Oil & Gas Exploration …
## 107 CVX           Chevron Corp.    Energy          Integrated Oil & Gas   
## 108 CMG           Chipotle Mexica… Consumer Discr… Restaurants            
## 109 CB            Chubb Limited    Financials      Property & Casualty In…
## 110 CHD           Church & Dwight  Consumer Stapl… Household Products     
## 111 CI            CIGNA Corp.      Health Care     Managed Health Care    
## 112 XEC           Cimarex Energy   Energy          Oil & Gas Exploration …
## 113 CINF          Cincinnati Fina… Financials      Property & Casualty In…
## 114 CTAS          Cintas Corporat… Industrials     Diversified Support Se…
## 115 CSCO          Cisco Systems    Information Te… Communications Equipme…
## 116 C             Citigroup Inc.   Financials      Diversified Banks      
## 117 CFG           Citizens Financ… Financials      Regional Banks         
## 118 CTXS          Citrix Systems   Information Te… Internet Software & Se…
## 119 CLX           The Clorox Comp… Consumer Stapl… Household Products     
## 120 CME           CME Group Inc.   Financials      Financial Exchanges & …
## 121 CMS           CMS Energy       Utilities       Multi-Utilities        
## 122 KO            Coca-Cola Compa… Consumer Stapl… Soft Drinks            
## 123 CTSH          Cognizant Techn… Information Te… IT Consulting & Other …
## 124 CL            Colgate-Palmoli… Consumer Stapl… Household Products     
## 125 CMCSA         Comcast Corp.    Consumer Discr… Cable & Satellite      
## 126 CMA           Comerica Inc.    Financials      Diversified Banks      
## 127 CAG           Conagra Brands   Consumer Stapl… Packaged Foods & Meats 
## 128 CXO           Concho Resources Energy          Oil & Gas Exploration …
## 129 COP           ConocoPhillips   Energy          Oil & Gas Exploration …
## 130 ED            Consolidated Ed… Utilities       Electric Utilities     
## 131 STZ           Constellation B… Consumer Stapl… Distillers & Vintners  
## 132 COO           The Cooper Comp… Health Care     Health Care Supplies   
## 133 GLW           Corning Inc.     Information Te… Electronic Components  
## 134 COST          Costco Wholesal… Consumer Stapl… Hypermarkets & Super C…
## 135 COTY          Coty, Inc        Consumer Stapl… Personal Products      
## 136 CCI           Crown Castle In… Real Estate     Specialized REITs      
## 137 CSRA          CSRA Inc.        Information Te… IT Consulting & Other …
## 138 CSX           CSX Corp.        Industrials     Railroads              
## 139 CMI           Cummins Inc.     Industrials     Industrial Machinery   
## 140 CVS           CVS Health       Consumer Stapl… Drug Retail            
## 141 DHI           D. R. Horton     Consumer Discr… Homebuilding           
## 142 DHR           Danaher Corp.    Health Care     Health Care Equipment  
## 143 DRI           Darden Restaura… Consumer Discr… Restaurants            
## 144 DVA           DaVita Inc.      Health Care     Health Care Facilities 
## 145 DE            Deere & Co.      Industrials     Agricultural & Farm Ma…
## 146 DAL           Delta Air Lines… Industrials     Airlines               
## 147 XRAY          Dentsply Sirona  Health Care     Health Care Supplies   
## 148 DVN           Devon Energy Co… Energy          Oil & Gas Exploration …
## 149 DLR           Digital Realty … Real Estate     Specialized REITs      
## 150 DFS           Discover Financ… Financials      Consumer Finance       
## 151 DISCA         Discovery Commu… Consumer Discr… Cable & Satellite      
## 152 DISCK         Discovery Commu… Consumer Discr… Cable & Satellite      
## 153 DISH          Dish Network     Consumer Discr… Cable & Satellite      
## 154 DG            Dollar General   Consumer Discr… General Merchandise St…
## 155 DLTR          Dollar Tree      Consumer Discr… General Merchandise St…
## 156 D             Dominion Energy  Utilities       Electric Utilities     
## 157 DOV           Dover Corp.      Industrials     Industrial Machinery   
## 158 DWDP          DowDuPont        Materials       Diversified Chemicals  
## 159 DPS           Dr Pepper Snapp… Consumer Stapl… Soft Drinks            
## 160 DTE           DTE Energy Co.   Utilities       Multi-Utilities        
## 161 DRE           Duke Realty Corp Real Estate     Industrial REITs       
## 162 DUK           Duke Energy      Utilities       Electric Utilities     
## 163 DXC           DXC Technology   Information Te… IT Consulting & Other …
## 164 ETFC          E*Trade          Financials      Investment Banking & B…
## 165 EMN           Eastman Chemical Materials       Diversified Chemicals  
## 166 ETN           Eaton Corporati… Industrials     Electrical Components …
## 167 EBAY          eBay Inc.        Information Te… Internet Software & Se…
## 168 ECL           Ecolab Inc.      Materials       Specialty Chemicals    
## 169 EIX           Edison Int'l     Utilities       Electric Utilities     
## 170 EW            Edwards Lifesci… Health Care     Health Care Equipment  
## 171 EA            Electronic Arts  Information Te… Home Entertainment Sof…
## 172 EMR           Emerson Electri… Industrials     Electrical Components …
## 173 ETR           Entergy Corp.    Utilities       Electric Utilities     
## 174 EVHC          Envision Health… Health Care     Health Care Services   
## 175 EOG           EOG Resources    Energy          Oil & Gas Exploration …
## 176 EQT           EQT Corporation  Energy          Oil & Gas Exploration …
## 177 EFX           Equifax Inc.     Industrials     Research & Consulting …
## 178 EQIX          Equinix          Real Estate     Specialized REITs      
## 179 EQR           Equity Resident… Real Estate     Residential REITs      
## 180 ESS           Essex Property … Real Estate     Residential REITs      
## 181 EL            Estee Lauder Co… Consumer Stapl… Personal Products      
## 182 ES            Eversource Ener… Utilities       Multi-Utilities        
## 183 RE            Everest Re Grou… Financials      Reinsurance            
## 184 EXC           Exelon Corp.     Utilities       Multi-Utilities        
## 185 EXPE          Expedia Inc.     Consumer Discr… Internet & Direct Mark…
## 186 EXPD          Expeditors Inte… Industrials     Air Freight & Logistics
## 187 ESRX          Express Scripts  Health Care     Health Care Distributo…
## 188 EXR           Extra Space Sto… Real Estate     Specialized REITs      
## 189 XOM           Exxon Mobil Cor… Energy          Integrated Oil & Gas   
## 190 FFIV          F5 Networks      Information Te… Communications Equipme…
## 191 FB            Facebook, Inc.   Information Te… Internet Software & Se…
## 192 FAST          Fastenal Co      Industrials     Building Products      
## 193 FRT           Federal Realty … Real Estate     Retail REITs           
## 194 FDX           FedEx Corporati… Industrials     Air Freight & Logistics
## 195 FIS           Fidelity Nation… Information Te… Internet Software & Se…
## 196 FITB          Fifth Third Ban… Financials      Regional Banks         
## 197 FE            FirstEnergy Corp Utilities       Electric Utilities     
## 198 FISV          Fiserv Inc       Information Te… Internet Software & Se…
## 199 FLIR          FLIR Systems     Information Te… Electronic Equipment &…
## 200 FLS           Flowserve Corpo… Industrials     Industrial Machinery   
## 201 FLR           Fluor Corp.      Industrials     Construction & Enginee…
## 202 FMC           FMC Corporation  Materials       Fertilizers & Agricult…
## 203 FL            Foot Locker Inc  Consumer Discr… Apparel Retail         
## 204 F             Ford Motor       Consumer Discr… Automobile Manufacture…
## 205 FTV           Fortive Corp     Industrials     Industrial Machinery   
## 206 FBHS          Fortune Brands … Industrials     Building Products      
## 207 BEN           Franklin Resour… Financials      Asset Management & Cus…
## 208 FCX           Freeport-McMoRa… Materials       Copper                 
## 209 GPS           Gap Inc.         Consumer Discr… Apparel Retail         
## 210 GRMN          Garmin Ltd.      Consumer Discr… Consumer Electronics   
## 211 IT            Gartner Inc      Information Te… IT Consulting & Other …
## 212 GD            General Dynamics Industrials     Aerospace & Defense    
## 213 GE            General Electric Industrials     Industrial Conglomerat…
## 214 GGP           General Growth … Real Estate     Retail REITs           
## 215 GIS           General Mills    Consumer Stapl… Packaged Foods & Meats 
## 216 GM            General Motors   Consumer Discr… Automobile Manufacture…
## 217 GPC           Genuine Parts    Consumer Discr… Specialty Stores       
## 218 GILD          Gilead Sciences  Health Care     Biotechnology          
## 219 GPN           Global Payments… Information Te… Data Processing & Outs…
## 220 GS            Goldman Sachs G… Financials      Investment Banking & B…
## 221 GT            Goodyear Tire &… Consumer Discr… Tires & Rubber         
## 222 GWW           Grainger (W.W.)… Industrials     Industrial Machinery   
## 223 HAL           Halliburton Co.  Energy          Oil & Gas Equipment & …
## 224 HBI           Hanesbrands Inc  Consumer Discr… Apparel, Accessories &…
## 225 HOG           Harley-Davidson  Consumer Discr… Motorcycle Manufacture…
## 226 HRS           Harris Corporat… Information Te… Communications Equipme…
## 227 HIG           Hartford Financ… Financials      Property & Casualty In…
## 228 HAS           Hasbro Inc.      Consumer Discr… Leisure Products       
## 229 HCA           HCA Holdings     Health Care     Health Care Facilities 
## 230 HCP           HCP Inc.         Real Estate     Health Care REITs      
## 231 HP            Helmerich & Pay… Energy          Oil & Gas Drilling     
## 232 HSIC          Henry Schein     Health Care     Health Care Distributo…
## 233 HSY           The Hershey Com… Consumer Stapl… Packaged Foods & Meats 
## 234 HES           Hess Corporation Energy          Integrated Oil & Gas   
## 235 HPE           Hewlett Packard… Information Te… Technology Hardware, S…
## 236 HLT           Hilton Worldwid… Consumer Discr… Hotels, Resorts & Crui…
## 237 HOLX          Hologic          Health Care     Health Care Equipment  
## 238 HD            Home Depot       Consumer Discr… Home Improvement Retail
## 239 HON           Honeywell Int'l… Industrials     Industrial Conglomerat…
## 240 HRL           Hormel Foods Co… Consumer Stapl… Packaged Foods & Meats 
## 241 HST           Host Hotels & R… Real Estate     Hotel & Resort REITs   
## 242 HPQ           HP Inc.          Information Te… Technology Hardware, S…
## 243 HUM           Humana Inc.      Health Care     Managed Health Care    
## 244 HBAN          Huntington Banc… Financials      Regional Banks         
## 245 HII           Huntington Inga… Industrials     Aerospace & Defense    
## 246 IDXX          IDEXX Laborator… Health Care     Health Care Equipment  
## 247 INFO          IHS Markit Ltd.  Industrials     Research & Consulting …
## 248 ITW           Illinois Tool W… Industrials     Industrial Machinery   
## 249 ILMN          Illumina Inc     Health Care     Life Sciences Tools & …
## 250 IR            Ingersoll-Rand … Industrials     Industrial Machinery   
## 251 INTC          Intel Corp.      Information Te… Semiconductors         
## 252 ICE           Intercontinenta… Financials      Financial Exchanges & …
## 253 IBM           International B… Information Te… IT Consulting & Other …
## 254 INCY          Incyte           Health Care     Biotechnology          
## 255 IP            International P… Materials       Paper Packaging        
## 256 IPG           Interpublic Gro… Consumer Discr… Advertising            
## 257 IFF           Intl Flavors & … Materials       Specialty Chemicals    
## 258 INTU          Intuit Inc.      Information Te… Internet Software & Se…
## 259 ISRG          Intuitive Surgi… Health Care     Health Care Equipment  
## 260 IVZ           Invesco Ltd.     Financials      Asset Management & Cus…
## 261 IQV           IQVIA Holdings … Health Care     Life Sciences Tools & …
## 262 IRM           Iron Mountain I… Real Estate     Specialized REITs      
## 263 JEC           Jacobs Engineer… Industrials     Construction & Enginee…
## 264 JBHT          J. B. Hunt Tran… Industrials     Trucking               
## 265 SJM           JM Smucker       Consumer Stapl… Packaged Foods & Meats 
## 266 JNJ           Johnson & Johns… Health Care     Health Care Equipment  
## 267 JCI           Johnson Control… Industrials     Building Products      
## 268 JPM           JPMorgan Chase … Financials      Diversified Banks      
## 269 JNPR          Juniper Networks Information Te… Communications Equipme…
## 270 KSU           Kansas City Sou… Industrials     Railroads              
## 271 K             Kellogg Co.      Consumer Stapl… Packaged Foods & Meats 
## 272 KEY           KeyCorp          Financials      Regional Banks         
## 273 KMB           Kimberly-Clark   Consumer Stapl… Household Products     
## 274 KIM           Kimco Realty     Real Estate     Retail REITs           
## 275 KMI           Kinder Morgan    Energy          Oil & Gas Storage & Tr…
## 276 KLAC          KLA-Tencor Corp. Information Te… Semiconductor Equipment
## 277 KSS           Kohl's Corp.     Consumer Discr… General Merchandise St…
## 278 KHC           Kraft Heinz Co   Consumer Stapl… Packaged Foods & Meats 
## 279 KR            Kroger Co.       Consumer Stapl… Food Retail            
## 280 LB            L Brands Inc.    Consumer Discr… Apparel Retail         
## 281 LLL           L-3 Communicati… Industrials     Aerospace & Defense    
## 282 LH            Laboratory Corp… Health Care     Health Care Services   
## 283 LRCX          Lam Research     Information Te… Semiconductor Equipment
## 284 LEG           Leggett & Platt  Consumer Discr… Home Furnishings       
## 285 LEN           Lennar Corp.     Consumer Discr… Homebuilding           
## 286 LUK           Leucadia Nation… Financials      Multi-Sector Holdings  
## 287 LLY           Lilly (Eli) & C… Health Care     Pharmaceuticals        
## 288 LNC           Lincoln National Financials      Multi-line Insurance   
## 289 LKQ           LKQ Corporation  Consumer Discr… Distributors           
## 290 LMT           Lockheed Martin… Industrials     Aerospace & Defense    
## 291 L             Loews Corp.      Financials      Multi-line Insurance   
## 292 LOW           Lowe's Cos.      Consumer Discr… Home Improvement Retail
## 293 LYB           LyondellBasell   Materials       Specialty Chemicals    
## 294 MTB           M&T Bank Corp.   Financials      Regional Banks         
## 295 MAC           Macerich         Real Estate     Retail REITs           
## 296 M             Macy's Inc.      Consumer Discr… Department Stores      
## 297 MRO           Marathon Oil Co… Energy          Oil & Gas Exploration …
## 298 MPC           Marathon Petrol… Energy          Oil & Gas Refining & M…
## 299 MAR           Marriott Int'l.  Consumer Discr… Hotels, Resorts & Crui…
## 300 MMC           Marsh & McLennan Financials      Insurance Brokers      
## 301 MLM           Martin Marietta… Materials       Construction Materials 
## 302 MAS           Masco Corp.      Industrials     Building Products      
## 303 MA            Mastercard Inc.  Information Te… Internet Software & Se…
## 304 MAT           Mattel Inc.      Consumer Discr… Leisure Products       
## 305 MKC           McCormick & Co.  Consumer Stapl… Packaged Foods & Meats 
## 306 MCD           McDonald's Corp. Consumer Discr… Restaurants            
## 307 MCK           McKesson Corp.   Health Care     Health Care Distributo…
## 308 MDT           Medtronic plc    Health Care     Health Care Equipment  
## 309 MRK           Merck & Co.      Health Care     Pharmaceuticals        
## 310 MET           MetLife Inc.     Financials      Life & Health Insurance
## 311 MTD           Mettler Toledo   Health Care     Life Sciences Tools & …
## 312 MGM           MGM Resorts Int… Consumer Discr… Casinos & Gaming       
## 313 KORS          Michael Kors Ho… Consumer Discr… Apparel, Accessories &…
## 314 MCHP          Microchip Techn… Information Te… Semiconductors         
## 315 MU            Micron Technolo… Information Te… Semiconductors         
## 316 MSFT          Microsoft Corp.  Information Te… Systems Software       
## 317 MAA           Mid-America Apa… Real Estate     Residential REITs      
## 318 MHK           Mohawk Industri… Consumer Discr… Home Furnishings       
## 319 TAP           Molson Coors Br… Consumer Stapl… Brewers                
## 320 MDLZ          Mondelez Intern… Consumer Stapl… Packaged Foods & Meats 
## 321 MON           Monsanto Co.     Materials       Fertilizers & Agricult…
## 322 MNST          Monster Beverage Consumer Stapl… Soft Drinks            
## 323 MCO           Moody's Corp     Financials      Financial Exchanges & …
## 324 MS            Morgan Stanley   Financials      Investment Banking & B…
## 325 MOS           The Mosaic Comp… Materials       Fertilizers & Agricult…
## 326 MSI           Motorola Soluti… Information Te… Communications Equipme…
## 327 MYL           Mylan N.V.       Health Care     Pharmaceuticals        
## 328 NDAQ          Nasdaq, Inc.     Financials      Financial Exchanges & …
## 329 NOV           National Oilwel… Energy          Oil & Gas Equipment & …
## 330 NAVI          Navient          Financials      Consumer Finance       
## 331 NTAP          NetApp           Information Te… Internet Software & Se…
## 332 NFLX          Netflix Inc.     Information Te… Internet Software & Se…
## 333 NWL           Newell Brands    Consumer Discr… Housewares & Specialti…
## 334 NFX           Newfield Explor… Energy          Oil & Gas Exploration …
## 335 NEM           Newmont Mining … Materials       Gold                   
## 336 NWSA          News Corp. Clas… Consumer Discr… Publishing             
## 337 NWS           News Corp. Clas… Consumer Discr… Publishing             
## 338 NEE           NextEra Energy   Utilities       Multi-Utilities        
## 339 NLSN          Nielsen Holdings Industrials     Research & Consulting …
## 340 NKE           Nike             Consumer Discr… Apparel, Accessories &…
## 341 NI            NiSource Inc.    Utilities       Multi-Utilities        
## 342 NBL           Noble Energy Inc Energy          Oil & Gas Exploration …
## 343 JWN           Nordstrom        Consumer Discr… Department Stores      
## 344 NSC           Norfolk Souther… Industrials     Railroads              
## 345 NTRS          Northern Trust … Financials      Asset Management & Cus…
## 346 NOC           Northrop Grumma… Industrials     Aerospace & Defense    
## 347 NCLH          Norwegian Cruis… Consumer Discr… Hotels, Resorts & Crui…
## 348 NRG           NRG Energy       Utilities       Independent Power Prod…
## 349 NUE           Nucor Corp.      Materials       Steel                  
## 350 NVDA          Nvidia Corporat… Information Te… Semiconductors         
## 351 ORLY          O'Reilly Automo… Consumer Discr… Specialty Stores       
## 352 OXY           Occidental Petr… Energy          Oil & Gas Exploration …
## 353 OMC           Omnicom Group    Consumer Discr… Advertising            
## 354 OKE           ONEOK            Energy          Oil & Gas Storage & Tr…
## 355 ORCL          Oracle Corp.     Information Te… Application Software   
## 356 PCAR          PACCAR Inc.      Industrials     Construction Machinery…
## 357 PKG           Packaging Corpo… Materials       Paper Packaging        
## 358 PH            Parker-Hannifin  Industrials     Industrial Machinery   
## 359 PDCO          Patterson Compa… Health Care     Health Care Supplies   
## 360 PAYX          Paychex Inc.     Information Te… Internet Software & Se…
## 361 PYPL          PayPal           Information Te… Data Processing & Outs…
## 362 PNR           Pentair Ltd.     Industrials     Industrial Machinery   
## 363 PBCT          People's United… Financials      Thrifts & Mortgage Fin…
## 364 PEP           PepsiCo Inc.     Consumer Stapl… Soft Drinks            
## 365 PKI           PerkinElmer      Health Care     Health Care Equipment  
## 366 PRGO          Perrigo          Health Care     Pharmaceuticals        
## 367 PFE           Pfizer Inc.      Health Care     Pharmaceuticals        
## 368 PCG           PG&E Corp.       Utilities       Multi-Utilities        
## 369 PM            Philip Morris I… Consumer Stapl… Tobacco                
## 370 PSX           Phillips 66      Energy          Oil & Gas Refining & M…
## 371 PNW           Pinnacle West C… Utilities       Multi-Utilities        
## 372 PXD           Pioneer Natural… Energy          Oil & Gas Exploration …
## 373 PNC           PNC Financial S… Financials      Regional Banks         
## 374 RL            Polo Ralph Laur… Consumer Discr… Apparel, Accessories &…
## 375 PPG           PPG Industries   Materials       Specialty Chemicals    
## 376 PPL           PPL Corp.        Utilities       Electric Utilities     
## 377 PX            Praxair Inc.     Materials       Industrial Gases       
## 378 PCLN          Priceline.com I… Consumer Discr… Internet & Direct Mark…
## 379 PFG           Principal Finan… Financials      Life & Health Insurance
## 380 PG            Procter & Gamble Consumer Stapl… Personal Products      
## 381 PGR           Progressive Cor… Financials      Property & Casualty In…
## 382 PLD           Prologis         Real Estate     Industrial REITs       
## 383 PRU           Prudential Fina… Financials      Life & Health Insurance
## 384 PEG           Public Serv. En… Utilities       Electric Utilities     
## 385 PSA           Public Storage   Real Estate     Specialized REITs      
## 386 PHM           Pulte Homes Inc. Consumer Discr… Homebuilding           
## 387 PVH           PVH Corp.        Consumer Discr… Apparel, Accessories &…
## 388 QRVO          Qorvo            Information Te… Semiconductors         
## 389 PWR           Quanta Services… Industrials     Construction & Enginee…
## 390 QCOM          QUALCOMM Inc.    Information Te… Semiconductors         
## 391 DGX           Quest Diagnosti… Health Care     Health Care Services   
## 392 RRC           Range Resources… Energy          Oil & Gas Exploration …
## 393 RJF           Raymond James F… Financials      Investment Banking & B…
## 394 RTN           Raytheon Co.     Industrials     Aerospace & Defense    
## 395 O             Realty Income C… Real Estate     Retail REITs           
## 396 RHT           Red Hat Inc.     Information Te… Systems Software       
## 397 REG           Regency Centers… Real Estate     Retail REITs           
## 398 REGN          Regeneron        Health Care     Biotechnology          
## 399 RF            Regions Financi… Financials      Regional Banks         
## 400 RSG           Republic Servic… Industrials     Environmental & Facili…
## 401 RMD           ResMed           Health Care     Health Care Equipment  
## 402 RHI           Robert Half Int… Industrials     Human Resource & Emplo…
## 403 ROK           Rockwell Automa… Industrials     Electrical Components …
## 404 COL           Rockwell Collins Industrials     Aerospace & Defense    
## 405 ROP           Roper Technolog… Industrials     Industrial Conglomerat…
## 406 ROST          Ross Stores      Consumer Discr… Apparel Retail         
## 407 RCL           Royal Caribbean… Consumer Discr… Hotels, Resorts & Crui…
## 408 CRM           Salesforce.com   Information Te… Internet Software & Se…
## 409 SBAC          SBA Communicati… Real Estate     Specialized REITs      
## 410 SCG           SCANA Corp       Utilities       Multi-Utilities        
## 411 SLB           Schlumberger Lt… Energy          Oil & Gas Equipment & …
## 412 SNI           Scripps Network… Consumer Discr… Cable & Satellite      
## 413 STX           Seagate Technol… Information Te… Technology Hardware, S…
## 414 SEE           Sealed Air       Materials       Paper Packaging        
## 415 SRE           Sempra Energy    Utilities       Multi-Utilities        
## 416 SHW           Sherwin-Williams Materials       Specialty Chemicals    
## 417 SIG           Signet Jewelers  Consumer Discr… Specialty Stores       
## 418 SPG           Simon Property … Real Estate     Retail REITs           
## 419 SWKS          Skyworks Soluti… Information Te… Semiconductors         
## 420 SLG           SL Green Realty  Real Estate     Office REITs           
## 421 SNA           Snap-On Inc.     Consumer Discr… Household Appliances   
## 422 SO            Southern Co.     Utilities       Electric Utilities     
## 423 LUV           Southwest Airli… Industrials     Airlines               
## 424 SPGI          S&P Global, Inc. Financials      Financial Exchanges & …
## 425 SWK           Stanley Black &… Consumer Discr… Household Appliances   
## 426 SBUX          Starbucks Corp.  Consumer Discr… Restaurants            
## 427 STT           State Street Co… Financials      Asset Management & Cus…
## 428 SRCL          Stericycle Inc   Industrials     Environmental & Facili…
## 429 SYK           Stryker Corp.    Health Care     Health Care Equipment  
## 430 STI           SunTrust Banks   Financials      Regional Banks         
## 431 SYMC          Symantec Corp.   Information Te… Application Software   
## 432 SYF           Synchrony Finan… Financials      Consumer Finance       
## 433 SNPS          Synopsys Inc.    Information Te… Application Software   
## 434 SYY           Sysco Corp.      Consumer Stapl… Food Distributors      
## 435 TROW          T. Rowe Price G… Financials      Asset Management & Cus…
## 436 TPR           Tapestry, Inc.   Consumer Discr… Apparel, Accessories &…
## 437 TGT           Target Corp.     Consumer Discr… General Merchandise St…
## 438 TEL           TE Connectivity… Information Te… Electronic Manufacturi…
## 439 FTI           TechnipFMC       Energy          Oil & Gas Equipment & …
## 440 TXN           Texas Instrumen… Information Te… Semiconductors         
## 441 TXT           Textron Inc.     Industrials     Aerospace & Defense    
## 442 TMO           Thermo Fisher S… Health Care     Health Care Equipment  
## 443 TIF           Tiffany & Co.    Consumer Discr… Apparel, Accessories &…
## 444 TWX           Time Warner Inc. Consumer Discr… Cable & Satellite      
## 445 TJX           TJX Companies I… Consumer Discr… Apparel Retail         
## 446 TMK           Torchmark Corp.  Financials      Life & Health Insurance
## 447 TSS           Total System Se… Information Te… Internet Software & Se…
## 448 TSCO          Tractor Supply … Consumer Discr… Specialty Stores       
## 449 TDG           TransDigm Group  Industrials     Aerospace & Defense    
## 450 TRV           The Travelers C… Financials      Property & Casualty In…
## 451 TRIP          TripAdvisor      Consumer Discr… Internet & Direct Mark…
## 452 FOXA          Twenty-First Ce… Consumer Discr… Publishing             
## 453 FOX           Twenty-First Ce… Consumer Discr… Publishing             
## 454 TSN           Tyson Foods      Consumer Stapl… Packaged Foods & Meats 
## 455 UDR           UDR Inc          Real Estate     Residential REITs      
## 456 ULTA          Ulta Beauty      Consumer Discr… Specialty Stores       
## 457 USB           U.S. Bancorp     Financials      Diversified Banks      
## 458 UAA           Under Armour Cl… Consumer Discr… Apparel, Accessories &…
## 459 UA            Under Armour Cl… Consumer Discr… Apparel, Accessories &…
## 460 UNP           Union Pacific    Industrials     Railroads              
## 461 UAL           United Continen… Industrials     Airlines               
## 462 UNH           United Health G… Health Care     Managed Health Care    
## 463 UPS           United Parcel S… Industrials     Air Freight & Logistics
## 464 URI           United Rentals,… Industrials     Trading Companies & Di…
## 465 UTX           United Technolo… Industrials     Aerospace & Defense    
## 466 UHS           Universal Healt… Health Care     Health Care Facilities 
## 467 UNM           Unum Group       Financials      Life & Health Insurance
## 468 VFC           V.F. Corp.       Consumer Discr… Apparel, Accessories &…
## 469 VLO           Valero Energy    Energy          Oil & Gas Refining & M…
## 470 VAR           Varian Medical … Health Care     Health Care Equipment  
## 471 VTR           Ventas Inc       Real Estate     Health Care REITs      
## 472 VRSN          Verisign Inc.    Information Te… Internet Software & Se…
## 473 VRSK          Verisk Analytics Industrials     Research & Consulting …
## 474 VZ            Verizon Communi… Telecommunicat… Integrated Telecommuni…
## 475 VRTX          Vertex Pharmace… Health Care     Biotechnology          
## 476 VIAB          Viacom Inc.      Consumer Discr… Cable & Satellite      
## 477 V             Visa Inc.        Information Te… Internet Software & Se…
## 478 VNO           Vornado Realty … Real Estate     Office REITs           
## 479 VMC           Vulcan Materials Materials       Construction Materials 
## 480 WMT           Wal-Mart Stores  Consumer Stapl… Hypermarkets & Super C…
## 481 WBA           Walgreens Boots… Consumer Stapl… Drug Retail            
## 482 DIS           The Walt Disney… Consumer Discr… Cable & Satellite      
## 483 WM            Waste Managemen… Industrials     Environmental & Facili…
## 484 WAT           Waters Corporat… Health Care     Health Care Distributo…
## 485 WEC           Wec Energy Grou… Utilities       Electric Utilities     
## 486 WFC           Wells Fargo      Financials      Diversified Banks      
## 487 HCN           Welltower Inc.   Real Estate     Health Care REITs      
## 488 WDC           Western Digital  Information Te… Technology Hardware, S…
## 489 WU            Western Union Co Information Te… Internet Software & Se…
## 490 WRK           WestRock Company Materials       Paper Packaging        
## 491 WY            Weyerhaeuser Co… Real Estate     Specialized REITs      
## 492 WHR           Whirlpool Corp.  Consumer Discr… Household Appliances   
## 493 WMB           Williams Cos.    Energy          Oil & Gas Storage & Tr…
## 494 WLTW          Willis Towers W… Financials      Insurance Brokers      
## 495 WYN           Wyndham Worldwi… Consumer Discr… Hotels, Resorts & Crui…
## 496 WYNN          Wynn Resorts Ltd Consumer Discr… Casinos & Gaming       
## 497 XEL           Xcel Energy Inc  Utilities       Multi-Utilities        
## 498 XRX           Xerox Corp.      Information Te… Technology Hardware, S…
## 499 XLNX          Xilinx Inc       Information Te… Semiconductors         
## 500 XL            XL Capital       Financials      Property & Casualty In…
## 501 XYL           Xylem Inc.       Industrials     Industrial Machinery   
## 502 YUM           Yum! Brands Inc  Consumer Discr… Restaurants            
## 503 ZBH           Zimmer Biomet H… Health Care     Health Care Equipment  
## 504 ZION          Zions Bancorp    Financials      Regional Banks         
## 505 ZTS           Zoetis           Health Care     Pharmaceuticals

The lapply() function below loops through each column of the data set counting the length of the unique items. The result is a count of distinct values for each column.

sp_500 %>% 
    lapply(function(x) x %>% unique() %>% length()) %>%
    unlist() # show in condensed format

##     ticker.symbol          security       gics.sector gics.sub.industry 
##               505               505                11               122

We have 505 ticker symbols but only 505 securities. There are no duplicate entry in securities. Let’s inspect any securities that are duplicated. We’ll use the dplyr functions group_by(), summarize(), and filter() to find any security that appears more than once.

sp_500 %>%
    group_by(security) %>%
    summarize(count = n()) %>%
    filter(count > 1)

## # A tibble: 0 x 2
## # ... with 2 variables: security <chr>, count <int>

We know from the lapply() function that there are 505 securities that are in 122 sub industries and in 11 sectors. However, we might want to understand the distribution of securities by sector. I again use group_by() and summarise() to get counts. In the visualization, we can use forcats::fct_reorder() to organize the gics.sector by greatest frequency (count). The forcats package simplifies working with caregories in R.

sp_500 %>%
    # Summarise data by frequency
    group_by(gics.sector) %>%
    summarise(count = n()) %>%
    # Visualize 
    ggplot(aes(x = gics.sector %>% fct_reorder(count),
               y = count
               )) + 
    geom_bar(stat = "identity") +
    geom_text(aes(label = count), size = 3, nudge_y = 4, nudge_x = .1) + 
    scale_y_continuous(limits = c(0,100)) +
    ggtitle(label = "Sector Frequency Among SP500 Stocks") +
    xlab(label = "GICS Sector") +
    theme(plot.title = element_text(size = 16)) + 
    coord_flip() 

Fig. 30

GET STOCK PRICES FUNCTION

get_stock_prices() is a wrapper for quantmod::getSymbols() that takes a ticker, a return format, and any other getSymbols() arguments and returns the prices in either xts or tibble format. In this post, we’ll use the tibble format for unnesting. We set auto.assign = FALSE to return the object to the variable rather than automatically creating the variable in the Global Environment.

get_stock_prices <- function(ticker, return_format = "tibble", ...) {
    # Get stock prices
    stock_prices_xts <- getSymbols(Symbols = ticker, auto.assign = FALSE, ...)
    # Rename
    names(stock_prices_xts) <- c("Open", "High", "Low", "Close", "Volume", "Adjusted")
    # Return in xts format if tibble is not specified
    if (return_format == "tibble") {
        stock_prices <- stock_prices_xts %>%
            as_tibble() %>%
            rownames_to_column(var = "Date") %>%
            mutate(Date = ymd(Date))
    } else {
        stock_prices <- stock_prices_xts
    }
    stock_prices
}

"MA" %>%
    get_stock_prices(return_format = 'tibble')%>%head()

## # A tibble: 6 x 7
##   Date        Open  High   Low Close   Volume Adjusted
##   <date>     <dbl> <dbl> <dbl> <dbl>    <dbl>    <dbl>
## 1 2007-01-03  9.96  9.97  9.56  9.64 26289000     7.46
## 2 2007-01-04  9.69 10.2   9.53 10.1  27024000     7.83
## 3 2007-01-05 10.1  10.2   9.90 10.1  29632000     7.82
## 4 2007-01-08  9.90 10.2   9.90 10.1  16006000     7.82
## 5 2007-01-09 10.1  10.6  10.1  10.6  36952000     8.18
## 6 2007-01-10 10.6  10.6  10.3  10.5  35099000     8.17

GET LOG RETURNS FUNCTION

get_log_returns() is a wrapper for quantmod::periodReturns() that takes a stock prices in xts or tibble format, a return format, and any other periodReturns() arguments and returns the log returns in either xts or tibble format.

get_log_returns <- function(x, return_format = "tibble", period = 'daily', ...) {
    # Convert tibble to xts
    if (!is.xts(x)) {
        x <- xts(x[,-1], order.by = x$Date)
    }
    # Get log returns
    log_returns_xts <- periodReturn(x = x$Adjusted, type = 'log', period = period, ...)
    # Rename
    names(log_returns_xts) <- "Log.Returns"
    # Return in xts format if tibble is not specified
    if (return_format == "tibble") {
        log_returns <- log_returns_xts %>%
            as_tibble() %>%
            rownames_to_column(var = "Date") %>%
            mutate(Date = ymd(Date))
    } else {
        log_returns <- log_returns_xts
    }
    log_returns
}

"MA" %>%
    get_stock_prices(return_format = 'tibble')%>% 
    get_log_returns(return_format = "tibble")%>%head()  

## # A tibble: 6 x 2
##   Date       Log.Returns
##   <date>           <dbl>
## 1 2007-01-03    0       
## 2 2007-01-04    0.0479  
## 3 2007-01-05   -0.000791
## 4 2007-01-08    0.000198
## 5 2007-01-09    0.0451  
## 6 2007-01-10   -0.000954

sp_500 <- sp_500[1:50,] %>%
    mutate(
        stock.prices = map(ticker.symbol, 
                              function(.x) get_stock_prices(.x, 
                                                            return_format = "tibble",
                                                            from = "2015-01-01",
                                                            to = "2018-01-01")
                           ),
        log.returns  = map(stock.prices, 
                              function(.x) get_log_returns(.x, return_format = "tibble")),
        mean.log.returns = map_dbl(log.returns, ~ mean(.$Log.Returns)),
        sd.log.returns   = map_dbl(log.returns, ~ sd(.$Log.Returns)),
        n.trade.days = map_dbl(stock.prices, nrow)
        )  

sp_500[1:5,]

## # A tibble: 5 x 9
##   ticker.symbol security    gics.sector   gics.sub.industry  stock.prices 
##   <chr>         <chr>       <chr>         <chr>              <list>       
## 1 MMM           3M Company  Industrials   Industrial Conglo… <tibble [755…
## 2 ABT           Abbott Lab… Health Care   Health Care Equip… <tibble [755…
## 3 ABBV          AbbVie Inc. Health Care   Pharmaceuticals    <tibble [755…
## 4 ACN           Accenture … Information … IT Consulting & O… <tibble [755…
## 5 ATVI          Activision… Information … Home Entertainmen… <tibble [755…
## # ... with 4 more variables: log.returns <list>, mean.log.returns <dbl>,
## #   sd.log.returns <dbl>, n.trade.days <dbl>

sp_500 %>% 
    select(ticker.symbol, stock.prices:log.returns)

## # A tibble: 50 x 3
##    ticker.symbol stock.prices       log.returns       
##    <chr>         <list>             <list>            
##  1 MMM           <tibble [755 × 7]> <tibble [755 × 2]>
##  2 ABT           <tibble [755 × 7]> <tibble [755 × 2]>
##  3 ABBV          <tibble [755 × 7]> <tibble [755 × 2]>
##  4 ACN           <tibble [755 × 7]> <tibble [755 × 2]>
##  5 ATVI          <tibble [755 × 7]> <tibble [755 × 2]>
##  6 AYI           <tibble [755 × 7]> <tibble [755 × 2]>
##  7 ADBE          <tibble [755 × 7]> <tibble [755 × 2]>
##  8 AMD           <tibble [755 × 7]> <tibble [755 × 2]>
##  9 AAP           <tibble [755 × 7]> <tibble [755 × 2]>
## 10 AES           <tibble [755 × 7]> <tibble [755 × 2]>
## 11 AET           <tibble [755 × 7]> <tibble [755 × 2]>
## 12 AMG           <tibble [755 × 7]> <tibble [755 × 2]>
## 13 AFL           <tibble [755 × 7]> <tibble [755 × 2]>
## 14 A             <tibble [755 × 7]> <tibble [755 × 2]>
## 15 APD           <tibble [755 × 7]> <tibble [755 × 2]>
## 16 AKAM          <tibble [755 × 7]> <tibble [755 × 2]>
## 17 ALK           <tibble [755 × 7]> <tibble [755 × 2]>
## 18 ALB           <tibble [755 × 7]> <tibble [755 × 2]>
## 19 ARE           <tibble [755 × 7]> <tibble [755 × 2]>
## 20 ALXN          <tibble [755 × 7]> <tibble [755 × 2]>
## 21 ALGN          <tibble [755 × 7]> <tibble [755 × 2]>
## 22 ALLE          <tibble [755 × 7]> <tibble [755 × 2]>
## 23 AGN           <tibble [755 × 7]> <tibble [755 × 2]>
## 24 ADS           <tibble [755 × 7]> <tibble [755 × 2]>
## 25 LNT           <tibble [755 × 7]> <tibble [755 × 2]>
## 26 ALL           <tibble [755 × 7]> <tibble [755 × 2]>
## 27 GOOGL         <tibble [755 × 7]> <tibble [755 × 2]>
## 28 GOOG          <tibble [755 × 7]> <tibble [755 × 2]>
## 29 MO            <tibble [755 × 7]> <tibble [755 × 2]>
## 30 AMZN          <tibble [755 × 7]> <tibble [755 × 2]>
## 31 AEE           <tibble [755 × 7]> <tibble [755 × 2]>
## 32 AAL           <tibble [755 × 7]> <tibble [755 × 2]>
## 33 AEP           <tibble [755 × 7]> <tibble [755 × 2]>
## 34 AXP           <tibble [755 × 7]> <tibble [755 × 2]>
## 35 AIG           <tibble [755 × 7]> <tibble [755 × 2]>
## 36 AMT           <tibble [755 × 7]> <tibble [755 × 2]>
## 37 AWK           <tibble [755 × 7]> <tibble [755 × 2]>
## 38 AMP           <tibble [755 × 7]> <tibble [755 × 2]>
## 39 ABC           <tibble [755 × 7]> <tibble [755 × 2]>
## 40 AME           <tibble [755 × 7]> <tibble [755 × 2]>
## 41 AMGN          <tibble [755 × 7]> <tibble [755 × 2]>
## 42 APH           <tibble [755 × 7]> <tibble [755 × 2]>
## 43 APC           <tibble [755 × 7]> <tibble [755 × 2]>
## 44 ADI           <tibble [755 × 7]> <tibble [755 × 2]>
## 45 ANDV          <tibble [755 × 7]> <tibble [755 × 2]>
## 46 ANSS          <tibble [755 × 7]> <tibble [755 × 2]>
## 47 ANTM          <tibble [755 × 7]> <tibble [755 × 2]>
## 48 AON           <tibble [755 × 7]> <tibble [755 × 2]>
## 49 AOS           <tibble [755 × 7]> <tibble [755 × 2]>
## 50 APA           <tibble [755 × 7]> <tibble [755 × 2]>

sp_500$stock.prices[[1]]%>%head() 

## # A tibble: 6 x 7
##   Date        Open  High   Low Close  Volume Adjusted
##   <date>     <dbl> <dbl> <dbl> <dbl>   <dbl>    <dbl>
## 1 2015-01-02   165   165   163   164 2116400      151
## 2 2015-01-05   163   164   160   160 3692900      148
## 3 2015-01-06   161   161   158   159 3532000      146
## 4 2015-01-07   160   160   159   160 3081300      147
## 5 2015-01-08   161   164   161   164 3142900      151
## 6 2015-01-09   164   164   161   162 2373800      149

VISUALIZING THE RESULTS WITH PLOTLY

library(plotly)
plot_ly(data   = sp_500,
        type   = "scatter",
        mode   = "markers",
        x      = ~ sd.log.returns,
        y      = ~ mean.log.returns,
        color  = ~ n.trade.days,
        colors = "Blues",
        size   = ~ n.trade.days,
        text   = ~ str_c("<em>", security, "</em><br>",
                         "Ticker: ", ticker.symbol, "<br>",
                         "Sector: ", gics.sector, "<br>",
                         "Sub Sector: ", gics.sub.industry, "<br>",
                         "No. of Trading Days: ", n.trade.days),
        marker = list(opacity = 0.8,
                      symbol = 'circle',
                      sizemode = 'diameter',
                      sizeref = 4.0,
                      line = list(width = 2, color = '#FFFFFF'))
        ) %>%
layout(title   = 'S&amp;P500 Analysis: Stock Risk vs Reward',
       xaxis   = list(title = 'Risk/Variability (StDev Log Returns)',
                    gridcolor = 'rgb(255, 255, 255)',
                    zerolinewidth = 1,
                    ticklen = 5,
                    gridwidth = 2),
       yaxis   = list(title = 'Reward/Growth (Mean Log Returns)',
                      gridcolor = 'rgb(255, 255, 255)',
                      zerolinewidth = 1,
                      ticklen = 5,
                      gridwith = 2),
       margin = list(l = 100,
                     t = 100,
                     b = 100),
       font   = list(color = '#FFFFFF'),
       paper_bgcolor = 'rgb(0, 0, 0)',
       plot_bgcolor = 'rgb(0, 0, 0)')

Fig. 30

sp_500 %>%
    filter(mean.log.returns >= 0.001,
           sd.log.returns < 0.0315) %>%
    select(ticker.symbol, mean.log.returns:n.trade.days) %>%
    arrange(mean.log.returns %>% desc())

## # A tibble: 6 x 4
##   ticker.symbol mean.log.returns sd.log.returns n.trade.days
##   <chr>                    <dbl>          <dbl>        <dbl>
## 1 ALGN                   0.00182         0.0196          755
## 2 AMZN                   0.00176         0.0177          755
## 3 ATVI                   0.00155         0.0187          755
## 4 ADBE                   0.00117         0.0147          755
## 5 AOS                    0.00111         0.0142          755
## 6 ALB                    0.00106         0.0190          755

"PCLN" %>%
    getSymbols(auto.assign = FALSE,
               from = "2007-01-01",
               to   = "2016-10-23") %>%
    chartSeries(name = "PCLN")

Fig. 30

BONUS: COMPUTING CORRELATIONS

cut the list down to the top 30 stocks with more than two years of samples (approx 500 trading days). We’ll add a rank column using the min_rank() function on mean.log.returns, and filter the data on the top 30 ranked means.

limit <- 30
sp_500_hp <- sp_500 %>%
    filter(n.trade.days > 500) %>%
    filter(sd.log.returns < 0.2) %>%
    mutate(rank = mean.log.returns %>% desc() %>% min_rank()) %>%
    filter(rank <= limit) %>%
    arrange(rank) %>%
    select(ticker.symbol, rank, mean.log.returns, sd.log.returns, log.returns)
sp_500_hp  

## # A tibble: 30 x 5
##    ticker.symbol  rank mean.log.returns sd.log.returns log.returns       
##    <chr>         <int>            <dbl>          <dbl> <list>            
##  1 ALGN              1         0.00182         0.0196  <tibble [755 × 2]>
##  2 AMD               2         0.00179         0.0412  <tibble [755 × 2]>
##  3 AMZN              3         0.00176         0.0177  <tibble [755 × 2]>
##  4 ATVI              4         0.00155         0.0187  <tibble [755 × 2]>
##  5 ADBE              5         0.00117         0.0147  <tibble [755 × 2]>
##  6 AOS               6         0.00111         0.0142  <tibble [755 × 2]>
##  7 ALB               7         0.00106         0.0190  <tibble [755 × 2]>
##  8 AET               8         0.000980        0.0155  <tibble [755 × 2]>
##  9 GOOG              9         0.000921        0.0139  <tibble [755 × 2]>
## 10 GOOGL            10         0.000911        0.0138  <tibble [755 × 2]>
## 11 ANTM             11         0.000841        0.0146  <tibble [755 × 2]>
## 12 ACN              12         0.000816        0.0117  <tibble [755 × 2]>
## 13 AWK              13         0.000786        0.0107  <tibble [755 × 2]>
## 14 ANSS             14         0.000784        0.0128  <tibble [755 × 2]>
## 15 ADI              15         0.000727        0.0162  <tibble [755 × 2]>
## 16 A                16         0.000705        0.0133  <tibble [755 × 2]>
## 17 APH              17         0.000690        0.0110  <tibble [755 × 2]>
## 18 MO               18         0.000649        0.0107  <tibble [755 × 2]>
## 19 ABBV             19         0.000649        0.0165  <tibble [755 × 2]>
## 20 ANDV             20         0.000643        0.0206  <tibble [755 × 2]>
## 21 ARE              21         0.000621        0.0118  <tibble [755 × 2]>
## 22 ALL              22         0.000604        0.0104  <tibble [755 × 2]>
## 23 MMM              23         0.000578        0.00977 <tibble [755 × 2]>
## 24 AFL              24         0.000576        0.00989 <tibble [755 × 2]>
## 25 AMT              25         0.000554        0.0120  <tibble [755 × 2]>
## 26 LNT              26         0.000519        0.0110  <tibble [755 × 2]>
## 27 ALLE             27         0.000517        0.0127  <tibble [755 × 2]>
## 28 AON              28         0.000512        0.0102  <tibble [755 × 2]>
## 29 AEE              29         0.000471        0.0107  <tibble [755 × 2]>
## 30 AME              30         0.000450        0.0118  <tibble [755 × 2]>

sp_500_hp_unnest <- sp_500_hp %>%
    select(ticker.symbol, log.returns) %>%
    unnest()
sp_500_hp_unnest%>%head()

## # A tibble: 6 x 3
##   ticker.symbol Date       Log.Returns
##   <chr>         <date>           <dbl>
## 1 ALGN          2015-01-02     0      
## 2 ALGN          2015-01-05     0.0140 
## 3 ALGN          2015-01-06     0.00804
## 4 ALGN          2015-01-07     0.0362 
## 5 ALGN          2015-01-08     0.0309 
## 6 ALGN          2015-01-09     0.0184

sp_500_hp_spread <- sp_500_hp_unnest %>%
    spread(key = ticker.symbol, value = Log.Returns) %>%
    na.omit()
sp_500_hp_spread%>%head()

## # A tibble: 6 x 31
##   Date              A     ABBV      ACN     ADBE       ADI      AEE
##   <date>        <dbl>    <dbl>    <dbl>    <dbl>     <dbl>    <dbl>
## 1 2015-01-02  0        0        0        0        0         0      
## 2 2015-01-05 -0.0189  -0.0190  -0.0170  -0.00499 -0.0184   -0.0101 
## 3 2015-01-06 -0.0157  -0.00496 -0.00724 -0.0204  -0.0238   -0.0117 
## 4 2015-01-07  0.0132   0.0396   0.0208   0.00819  0.0105    0.00775
## 5 2015-01-08  0.0295   0.0104   0.0151   0.0251   0.0175    0.00352
## 6 2015-01-09 -0.00736 -0.0277  -0.00111 -0.0149  -0.000731 -0.0113 
## # ... with 24 more variables: AET <dbl>, AFL <dbl>, ALB <dbl>, ALGN <dbl>,
## #   ALL <dbl>, ALLE <dbl>, AMD <dbl>, AME <dbl>, AMT <dbl>, AMZN <dbl>,
## #   ANDV <dbl>, ANSS <dbl>, ANTM <dbl>, AON <dbl>, AOS <dbl>, APH <dbl>,
## #   ARE <dbl>, ATVI <dbl>, AWK <dbl>, GOOG <dbl>, GOOGL <dbl>, LNT <dbl>,
## #   MMM <dbl>, MO <dbl>

sp_500_hp_cor <- sp_500_hp_spread %>%
    select(-Date) %>%
    cor() 
sp_500_hp_cor[1:6, 1:6] # show first 6 columns and rows

##              A      ABBV       ACN      ADBE       ADI       AEE
## A    1.0000000 0.4375766 0.5364386 0.5100089 0.4855185 0.2252420
## ABBV 0.4375766 1.0000000 0.2506644 0.2966285 0.2492603 0.1730033
## ACN  0.5364386 0.2506644 1.0000000 0.4950324 0.4285719 0.2534351
## ADBE 0.5100089 0.2966285 0.4950324 1.0000000 0.4531565 0.2686607
## ADI  0.4855185 0.2492603 0.4285719 0.4531565 1.0000000 0.1658043
## AEE  0.2252420 0.1730033 0.2534351 0.2686607 0.1658043 1.0000000

The correlations can be visualized with corrplot(). It can be ordered using hclust, which groups the stocks by similarity. The addrect argument adds boxes around the highly correlated groups, and by inspecting the circle colors I settled on 11 boxes.

pacman::p_load(corrplot)


sp_500_hp_cor %>%
    corrplot(order   = "hclust", 
             addrect = 10)

Fig. 30