Chapter 1
Descriptive Statistics for
Financial Data
Updated: February 3, 2015
In this chapter we use graphical and numerical descriptive statistics tostudy the distribution and dependence properties of daily and monthly asset
returns on a number of representative assets. The purpose of this chapter
is to introduce the techniques of exploratory data analysis for financial timeseries and to document a set of stylized facts for monthly and daily asset
returns that will be used in later chapters to motivate probability models for
asset returns.
The R packages used in this Chapter are corrplot, PerformanceAnalytics, tseries and zoo. Make sure these packages are installed and loaded
before running the R examples.

1.1

Univariate Descriptive Statistics

Let { } denote a univariate time series of asset returns (simple or continuously compounded). Throughout this chapter we will assume that { } is acovariance stationary and ergodic stochastic process such that[ ] =  independent of var( ) =  2 independent of cov(  − ) =   independent of corr(  − ) =    2 =  independent of 
1

2CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA
In addition, we will assume that each  is identically distributed with unknown pdf  ()An observed sample of size  of historical asset returns { }=1 is assumed
to be a realization from the stochastic process { } for  = 1      That is,{ }=1 = {1 = 1       =  }The goal of exploratory data analysis is to use the observed sample { }=1 tolearn about the unknown pdf  () as well as the time dependence propertiesof { }

1.1.1

Example Data

We illustrate the descriptive statistical analysis using daily and monthly adjusted closing prices on Microsoft stock and the S&P 500 index over theperiod January 1, 1998 and May 31, 2012. 1 These data are obtained fromfinance.yahoo.com. We first use the daily and monthly data to illustrate
descriptive statistical analysis and to establish a number of stylized facts
about the distribution and time dependence in daily and monthly returns.
Example 1 Getting daily and monthly adjusted closing price data from Yahoo! in R
As described in chapter 1, historical data on asset prices from finance.yahoo.com
can be downloaded and loaded into R automatically in a number of ways.
Here we use the get.hist.quote() function from the tseries package to
get daily adjusted closing prices and end-of-month adjusted closing prices on
Microsoft stock (ticker symbol msft) and the S&P 500 index (ticker symbol^gspc):2> msftPrices = get.hist.quote(instrument="msft", start="1998-01-01",+
end="2012-05-31", quote="AdjClose",1

An adjusted closing price is adjusted for dividend payments and stock splits. Any
dividend payment received between closing dates are added to the close price. If a stock
split occurs between the closing dates then the all past prices are divided by the split ratio.
2
The ticker symbol ^gspc refers to the actual S&P 500 index, which is not a tradable
security. There are several mutual funds (e.g., Vanguard’s S&P 500 fund with ticker
VFINF) and exchange traded funds (e.g., State Street’s SPDR S&P 500 ETF with ticker
SPY) which track the S&P 500 index that are investable.

1.1 UNIVARIATE DESCRIPTIVE STATISTICS

3+
provider="yahoo", origin="1970-01-01",+
compression="m", retclass="zoo")> sp500Prices = get.hist.quote(instrument="^gspc", start="1998-01-01",+
end="2012-05-31", quote="AdjClose",+
provider="yahoo", origin="1970-01-01",+
compression="m", retclass="zoo")> msftDailyPrices = get.hist.quote(instrument="msft", start="1998-01-01",+
end="2012-05-31", quote="AdjClose",+
provider="yahoo", origin="1970-01-01",+
compression="d", retclass="zoo")> sp500DailyPrices = get.hist.quote(instrument="^gspc", start="1998-01-01",+
end="2012-05-31", quote="AdjClose",+
provider="yahoo", origin="1970-01-01",+
compression="d", retclass="zoo")> class(msftPrices)[1] "zoo"> colnames(msftPrices)[1] "AdjClose"> start(msftPrices)[1] "1998-01-02"> end(msftPrices)[1] "2012-05-01"> head(msftPrices, n=3)
AdjClose
1998-01-02
13.53
1998-02-02
15.37
1998-03-02
16.24> head(msftDailyPrices, n=3)
AdjClose
1998-01-02
11.89
1998-01-05
11.83
1998-01-06
11.89

The objects msftPrices, sp500Prices, msftDailyPrices, and sp500DailyPrices
are of class "zoo" and each have a column called AdjClose containing theend-of-month adjusted closing prices. Notice, however, that the dates asso-

4CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA
ciated with the monthly closing prices are beginning-of-month dates.3 It will
be helpful for our analysis to change the column names in each object, andto change the class of the date index for the monthly prices to "yearmon"> colnames(msftPrices) = colnames(msftDailyPrices) = "MSFT"> colnames(sp500Prices) = colnames(sp500DailyPrices) = "SP500"> index(msftPrices) = as.yearmon(index(msftPrices))> index(sp500Prices) = as.yearmon(index(sp500Prices))
It will also be convenient to create merged "zoo" objects containing both
the Microsoft and S&P500 prices> msftSp500Prices = merge(msftPrices, sp500Prices)> msftSp500DailyPrices = merge(msftDailyPrices, sp500DailyPrices)> head(msftSp500Prices, n=3)
MSFT SP500
Jan 1998 13.53 980.3
Feb 1998 15.37 1049.3
Mar 1998 16.24 1101.8> head(msftSp500DailyPrices, n=3)
MSFT SP500
1998-01-02 11.89 975.0
1998-01-05 11.83 977.1
1998-01-06 11.89 966.6
We create "zoo" objects containing simple returns using the PerformanceAnalytics function Return.calculate()> msftRetS = Return.calculate(msftPrices, method="simple")> msftDailyRetS = Return.calculate(msftDailyPrices, method="simple")> sp500RetS = Return.calculate(sp500Prices, method="simple")> sp500DailyRetS = Return.calculate(sp500DailyPrices, method="simple")> msftSp500RetS = Return.calculate(msftSp500Prices, method="simple")> msftSp500DailyRetS = Return.calculate(msftSp500DailyPrices, method="simple")
We remove the first NA value of each object to avoid problems that someR functions have when missing values are encountered
3

When retrieving monthly data from Yahoo!, the full set of data contains the open,
high, low, close, adjusted close, and volume for the month. The convention in Yahoo! isto report the date associated with the open price for the month.

1.1 UNIVARIATE DESCRIPTIVE STATISTICS

5> msftRetS = msftRetS[-1]> msftDailyRetS = msftDailyRetS[-1]> sp500RetS = sp500RetS[-1]> sp500DailyRetS = sp500DailyRetS[-1]> msftSp500RetS = msftSp500RetS[-1]> msftSp500DailyRetS = msftSp500DailyRetS[-1]We also create "zoo" objects containing continuously compounded (cc) returns> msftRetC = log(1 + msftRetS)> sp500RetC = log(1 + sp500RetS)> MSFTsp500RetC = merge(msftRetC, sp500RetC)
¥

1.1.2

Time Plots

A natural graphical descriptive statistic for time series data is a time plot.
This is simply a line plot with the time series data on the y-axis and thetime index on the x-axis. Time plots are useful for quickly visualizing many
features of the time series data.
Example 2 Time plots of monthly prices and returns.
A two-panel plot showing the monthly prices is given in Figure 1.1, and iscreated using the plot method for "zoo" objects:> plot(msftSp500Prices, main="", lwd=2, col="blue")
The prices exhibit random walk like behavior (no tendency to revert to a time
independent mean) and appear to be non-stationary. Both prices show twolarge boom-bust periods associated with the dot-com period of the late 1990s
and the run-up to the financial crisis of 2008. Notice the strong common trend
behavior of the two price series.
A time plot for the monthly returns is created using:> my.panel <- function(...) {+lines(...)+
abline(h=0)+ }> plot(msftSp500RetS, main="", panel=my.panel, lwd=2, col="blue")

30
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SP500

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6CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA

1998

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2006

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2012

Index

Figure 1.1: End-of-month closing prices on Microsoft stock and the S&P 500
index.
and is given in Figure 1.2. The horizontal line at zero in each panel is createdusing the custom panel function my.panel() passed to plot(). In contrast to
prices, returns show clear mean-reverting behavior and the common monthly
mean values look to be very close to zero. Hence, the common mean value
assumption of covariance stationarity looks to be satisfied. However, thevolatility (i.e., fluctuation of returns about the mean) of both series appears
to change over time. Both series show higher volatility over the periods
1998 - 2003 and 2008 - 2012 than over the period 2003 - 2008. This is anindication of possible non-stationarity in volatility.4 Also, the coincidence ofhigh and low volatility periods across assets suggests a common driver to thetime varying behavior of volatility. There does not appear to be any visualevidence of systematic time dependence in the returns. Later on we will seethat the estimated autocorrelations are very close to zero. The returns for
4

The retuns can still be convariance stationary and exhibit time varying conditional
volatility.

7

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1.1 UNIVARIATE DESCRIPTIVE STATISTICS

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Index

Figure 1.2: Monthly continuously compounded returns on Microsoft stockand the S&P 500 index.
Microsoft and the S&P 500 tend to go up and down together suggesting a
positive correlation. ¥Example 3 Plotting returns on the same graph
In Figure 1.2, the volatility of the returns on Microsoft and the S&P 500
looks to be similar but this is illusory. The y-axis scale for Microsoft is muchlarger than the scale for the S&P 500 index and so the volatility of Microsoft
returns is actually much larger than the volatility of the S&P 500 returns.
Figure 1.3 shows both returns series on the same time plot created using> plot(msftSp500RetS, plot.type="single", main="",+col = c("red", "blue"), lty=c("dashed", "solid"),+
lwd=2, ylab="Returns")> abline(h=0)> legend(x="bottomright", legend=colnames(msftSp500RetS),

0.0
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Returns0.2

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8CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATAMSFT
S P 500
1999

2001

2003

2005

2007

2009

2011

Index

Figure 1.3: Monthly continuously compounded returns for Microsoft and
S&P 500 index on the same graph.
++

lty=c("dashed", "solid"), lwd=2,
col=c("red","blue"))

Now the higher volatility of Microsoft returns, especially before 2003, isclearly visible. However, after 2008 the volatilities of the two series lookquite similar. In general, the lower volatility of the S&P 500 index represents risk reduction due to holding a large diversified portfolio. ¥Example 4 Comparing simple and continuously compounded returns
Figure 1.4 compares the simple and cc monthly returns for Microsoft
created using> retDiff = msftRetS - msftRetC> dataToPlot = merge(msftRetS, msftRetC, retDiff)> plot(dataToPlot, plot.type="multiple", main="",

9

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retDiff

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msftRetC

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1.1 UNIVARIATE DESCRIPTIVE STATISTICS

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2003

2004

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2006

2007

2008

2009

2010

2011

2012

Index

Figure 1.4: Monthly simple and cc returns on Microsoft. Top panel: simple
returns; middle panel: cc returns; bottom panel: diﬀerence between simple
and cc returns.
++

panel = my.panel,
lwd=2, col=c("black", "blue", "red"))

The top panel shows the simple returns, the middle panel shows the cc returns, and the bottom panel shows the diﬀerence between the simple and cc
returns. Qualitatively, the simple and cc returns look almost identical. The
main diﬀerences occur when the simple returns are large in absolute value(e.g., mid 2000, early 2001 and early 2008). In this case the diﬀerence between the simple and cc returns can be fairly large (e.g., as large as 0.08 in
mid 2000). When the simple return is large and positive, the cc return is notquite as large; when the simple return is large and negative, the cc return isa larger negative number.
Example 5 Plotting Daily Returns

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10CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA

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Figure 1.5: Daily returns on Microsoft and the S&P 500 index.
Figure 1.5 shows the daily simple returns on Microsoft and the S&P 500
index created with> plot(msftSp500DailyRetS, main="",+panel=my.panel, col=c("black", "blue"))
Compared with the monthly returns, the daily returns are closer to zeroand the magnitude of the fluctuations (volatility) is smaller. However, theclustering of periods of high and low volatility is more pronounced in the daily
returns. As with the monthly returns, the volatility of the daily returns on
Microsoft is larger than the volatility of the S&P 500 returns. Also, thedaily returns show some large and small “spikes” that represent unusuallylarge (in absolute value) daily movements (e.g., Microsoft up 20% in one day
and down 15% on another day). The monthly returns do not exhibit suchextreme movements relative to typical volatility.

1.1 UNIVARIATE DESCRIPTIVE STATISTICS

11

Equity Curves
To directly compare the investment performance of two or more assets, plot
the simple multi-period cumulative returns of each asset on the same graph.
This type of graph, sometimes called an equity curve, shows how a one dollar investment amount in each asset grows over time. Better preforming
assets have higher equity curves. For simple returns, the k-period returns
−1Qare  () =(1 + − ) and represent the growth of one dollar invested
=0

for  periods. For continuously compounded returns, the k-period returns
−1P
−  However, this cc -period return must be converted to aare  () ==0

simple -period return, using  () = exp ( ()) − 1 to properly represent
the growth of one dollar invested for  periods.
Example 6 Equity curves for Microsoft and S&P 500 monthly returns
To create the equity curves for Microsoft and the S&P 500 index based on
simple returns use:
> equityCurveMsft = cumprod(1 + msftRetS)> equityCurveSp500 = cumprod(1 + sp500RetS)> dataToPlot = merge(equityCurveMsft, equityCurveSp500)> plot(dataToPlot, plot.type="single", ylab="Cumulative Returns",+
col=c("black", "blue"), lwd=2)> legend(x="topright", legend=c("MSFT", "SP500"),+
col=c("black", "blue"), lwd=2)

The R function cumprod() creates the cumulative products needed for theequity curves. Figure 1.6 shows that a one dollar investment in Microsoft
dominated a one dollar investment in the S&P 500 index over the given
period. In particular, $1 invested in Microsoft grew to about $2.10 (overabout 12 years) whereas $1 invested in the S&P 500 index only grew toabout $1.40. Notice the huge increases and decreases in value of Microsoft
during the dot-com bubble and bust over the period 1998 - 2001. ¥Drawdowns
To be completed

12CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA

2.0
1.0

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S P500

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2009

2011

Ind e xFigure 1.6: Monthly cumulative continuously componded returns on Microsoft and the S&P 500 index.

1.1.3

Descriptive Statistics for the Distribution of Returns

In this section, we consider graphical and numerical descriptive statistics forthe unknown marginal pdf,  () of returns. Recall, we assume that theobserved sample { }=1 is a realization from a covariance stationary andergodic time series { } where each  is a continuous random variable with
common pdf  () The goal is to use { }=1 to describe properties of  ()We study returns and not prices because prices are non-stationary. Sample descriptive statistics are only meaningful for covariance stationary andergodic time series.
Histograms
A histogram of returns is a graphical summary used to describe the generalshape of the unknown pdf  () It is constructed as follows. Order returns
from smallest to largest. Divide the range of observed values into  equally1.1 UNIVARIATE DESCRIPTIVE STATISTICS

13

sized bins. Show the number or fraction of observations in each bin using abar chart.
Example 7 Histograms for the daily and monthly returns on Microsoft andthe S&P 500 index
Figure 1.7 shows the histograms of the daily and monthly returns on Microsoft stock and the S&P 500 index created using the R function hist():> par(mfrow=c(2,2))>
hist(msftRetS, main="", col="cornflowerblue")>hist(msftDailyRetS, main="", col="cornflowerblue")>hist(sp500RetS, main="", col="cornflowerblue")>hist(sp500DailyRetS, main="", col="cornflowerblue")> par(mfrow=c(1,1))
All histograms have a bell-shape like the normal distribution. The histogramsof daily returns are centered around zero and those for the monthly returnare centered around values slightly larger than zero. The bulk of the daily(monthly) returns for Microsoft and the S&P 500 are between -5% and 5%(-20% and 20%) and -3% and 3% (-10% and 10%), respectively. The histogram for the S&P 500 monthly returns is slightly skewed left (long lefttail) due to more large negative returns than large positive returns whereas
the histograms for the other returns are roughly symmetric.
When comparing two or more return distributions, it is useful to use thesame bins for each histogram. Figure ?? shows the histograms for Microsoft
and S&P 500 returns using the same 15 bins, created with the R code:> msftHist = hist(msftRetS, plot=FALSE, breaks=15)> par(mfrow=c(2,2))>
hist(msftRetS, main="", col="cornflowerblue")>hist(msftDailyRetS, main="", col="cornflowerblue",+
breaks=msftHist$breaks)>hist(sp500RetS, main="", col="cornflowerblue",+
breaks=msftHist$breaks)>hist(sp500DailyRetS, main="", col="cornflowerblue",+
breaks=msftHist$breaks)> par(mfrow=c(1,1))

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14CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA

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sp500Daily RetS

Figure 1.7: Histograms of monthly continuously compounded returns on Microsoft stock and S&P 500 index.
Using the same bins for all histograms allows us to see more clearly that the
distribution of monthly returns is more spread out than the distributions fordaily returns, and that the distribution of S&P 500 returns is more tightly
concentrated around zero than the distribution of Microsoft returns. ¥Example 8 Are Microsoft returns normally distributed? A first look.
The shape of the histogram for Microsoft returns suggests that a normal
distribution might be a good candidate for the unknown distribution of Microsoft returns. To investigate this conjecture, we simulate random returns
from a normal distribution with mean and standard deviation calibrated to
the Microsoft daily and monthly returns using:> set.seed(123)> gwnDaily = rnorm(length(msftDailyRetS), mean=mean(msftDailyRetS),

15

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1.1 UNIVARIATE DESCRIPTIVE STATISTICS

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Figure 1.8: Histograms for Microsoft and S&P 500 returns using the samebins.
+
sd=sd(msftDailyRetS))> gwnDaily = zoo(gwnDaily, index(msftDailyRetS))> gwnMonthly = rnorm(length(msftRetS), mean=mean(msftRetS),+
sd=sd(msftRetS))> gwnMonthly = zoo(gwnMonthly, index(msftRetS))
Figure 1.9 shows the Microsoft monthly returns together with the simulated
normal returns created using> par(mfrow=c(2,2))>plot(msftRetS, main="Monthly Returns on MSFT",+
lwd=2, col="blue", ylim=c(-0.4, 0.4))>
abline(h=0)

16CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA>
plot(gwnMonthly, main="Simulated Normal Returns",+
lwd=2, col="blue", ylim=c(-0.4, 0.4))>
abline(h=0)>
hist(msftRetS, main="", col="cornflowerblue",+
xlab="returns")>
hist(gwnMonthly, main="", col="cornflowerblue",+
xlab="returns", breaks=msftHist$breaks)> par(mfrow=c(1,1))
The simulated normal returns shares many of the same features as the Microsoft returns: both fluctuate randomly about zero. However, there aresome important diﬀerences. In particular, the volatility of Microsoft returns
appears to change over time (large before 2003, small between 2003 and2008, and large again after 2008) whereas the simulated returns has constantvolatility. Additionally, the distribution of Microsoft returns has fatter tails(more extreme large and small returns) than the simulated normal returns.
Apart from these features, the simulated normal returns look remarkably likethe Microsoft monthly returns.
Figure 1.10 shows the Microsoft daily returns together with the simulated
normal returns. The daily returns look much less like GWN than the monthly
returns. Here the constant volatility of simulated GWN does not matchthe volatility patterns of the Microsoft daily returns, and the tails of thehistogram for the Microsoft returns are “fatter” than the tails of the GWN
histogram. ¥Smoothed histogram
Histograms give a good visual representation of the data distribution. The
shape of the histogram, however, depends on the number of bins used. Witha small number of bins, the histogram often appears blocky and fine details of
the distribution are not revealed. With a large number of bins, the histogram
might have many bins with very few observations. The hist() function in R
smartly chooses the number of bins so that the resulting histogram typically
looks good.
The main drawback of the histogram as descriptive statistic for the underlying pdf of the data is that it is discontinuous. If it is believed that theunderlying pdf is continuous, it is desirable to have a continuous graphicalsummary of the pdf. The smoothed histogram achieves this goal. Given a

17

1.1 UNIVARIATE DESCRIPTIVE STATISTICS
Simulated Normal Returns

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Figure 1.9: Comparison of Microsoft monthly returns with simulated normal
returns with the same mean and standard deviation as the Microsoft returns.
sample of data { }=1 the R function density() computes a smoothed estimate of the underlying pdf at each point  in the bins of the histogram using
the formula
¶
µ

X
1

−


ˆ () =  =1
where (·) is a continuous smoothing function (typically a standard normal
distribution) and  is a bandwidth (or bin-width) parameter that determines
the width of the bin around  in which the smoothing takes place. The
resulting pdf estimate ˆ () is a two-sided weighted average of the histogram
values around 
Example 9 Smoothed histogram for Microsoft monthly returns

18CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA

0.05
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Figure 1.10: Comparison of Microsoft daily returns with simulated normal
returns with the same mean and standard deviation as the Microsoft returns.
Figure 1.11 shows the histogram of Microsoft returns overlaid with the smoothed
histogram created using> MSFT.density = density(msftRetS)> hist(msftRetS, main="", xlab="Microsoft Monthly Returns",+
col="cornflowerblue", probability=T, ylim=c(0,5))> points(MSFT.density,type="l", col="orange", lwd=2)
In Figure 1.11, the histogram is normalized (using the argument probability=TRUE),
so that its total area is equal to one. The smoothed density estimate transforms the blocky shape of the histogram into a smooth continuous graph.
¥

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1.1 UNIVARIATE DESCRIPTIVE STATISTICS

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Figure 1.11: Histogram and smoothed density estimate for the monthly returns on Microsoft.
Empirical CDF
Recall, the CDF of a random variable  is the function  () = Pr( ≤ )
The empirical CDF of a data sample { }=1 is the function that counts thefraction of observations less than or equal to  :1(# ≤ )number of  values ≤ =
sample size̂ () =Example 10 Empirical CDF for daily and monthly returns on Microsoft
To be completed

20CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA
Empirical quantiles/percentiles
Recall, for  ∈ (0 1) the  × 100% quantile of the distribution of a continuous random variable  with CDF  is the point  such that  ( ) =Pr( ≤  ) =  Accordingly, the  × 100% empirical quantile (or 100 × 
percentile) of a data sample { }=1 is the data value ̂ such that  · 100%of the data are less than or equal to ̂  Empirical quantiles can be easily
determined by ordering the data from smallest to largest giving the orderedsample (also know as order statistics)
(1)  (2)  · · ·  ( ) The empirical quantile ̂ is the order statistic closest to  ×  5
The empirical quartiles are the empirical quantiles for  = 025 05 and075 respectively. The second empirical quartile ̂50 is called the sample
median and is the data point such that half of the data is less than or equal
to its value. The interquartile range (IQR) is the diﬀerence between the 3rd
and 1st quartile
IQR = ̂75 − ̂25 
and shows the size of the middle of the data distribution.
Example 11 Empirical quantiles of the Microsoft and S&P 500 monthly
returns
The R function quantile() computes empirical quantiles for a single data
series. By default, quantile() returns the empirical quartiles as well as theminimum and maximum values:> quantile(msftRetS)
0%
25%
50%
75%
100%
-0.343450 -0.048561 0.008377 0.057303 0.407878> quantile(sp500RetS)
0%
25%
50%
75%
100%
-0.16942 -0.02094 0.00809 0.03532 0.10772
5

There is no unique way to determine the empirical quantile from a sample of size 
for all values of . The R function quantile() can compute empirical quantile using oneof seven diﬀerent definitions.

1.1 UNIVARIATE DESCRIPTIVE STATISTICS

21

The left (right) quantiles of the Microsoft monthly returns are smaller (larger)
than the respective quantiles for the S&P 500 index.
To compute quantiles for a specified  use the probs argument. For
example, to compute the 1% and 5% quantiles of the monthly returns use> quantile(msftRetS,probs=c(0.01,0.05))
1%5%-0.1892 -0.1371> quantile(sp500RetS,probs=c(0.01,0.05))
1%5%-0.12040 -0.08184
Here we see that 1% of the Microsoft cc returns are less than -18.9% and 5%of the returns are less than -13.7%, respectively. For the S&P 500 returns,
these values are -12.0% and -8.2%, respectively.
To compute the median and IQR values for monthly returns use the Rfunctions median() and IQR(), respectively> apply(msftSp500RetS, 2, median)
MSFT
SP500
0.008377 0.008090> apply(msftSp500RetS, 2, IQR)
MSFT
SP500
0.10586 0.05626
The median returns are similar (about 0.8% per month) but the IQR for
Microsoft is about twice as large as the IQR for the S&P 500 index. ¥Historical/Empirical VaR
Recall, the  × 100% value-at-risk (VaR) of an investment of $ is VaR =$ ×   where  is the  × 100% quantile of the probability distribution of
the investment simple rate of return  The  × 100% historical VaR (sometimes called empirical VaR or Historical Simulation VaR) of an investmentof $ is defined asVaR = $ × ̂ 
where ̂ is the empirical  quantile of a sample of simple returns { }=1 
For a sample of continuously compounded returns { }=1 with empirical quantile ̂ ,VaR
 = $ × (exp(̂ ) − 1) 

22CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA
Historical VaR is based on the distribution of the observed returns andnot on any assumed distribution for returns (e.g., the normal distribution).
Example 12 Using empirical quantiles to compute historical Value-at-Risk
Consider investing  = $100 000 in Microsoft and the S&P 500 over amonth. The 1% and 5% historical VaR values for these investments basedon the historical samples of monthly returns are:
> W = 100000> msftQuantiles = quantile(msftRetS, probs=c(0.01, 0.05))> sp500Quantiles = quantile(sp500RetS, probs=c(0.01, 0.05))> msftVaR = W*msftQuantiles> sp500VaR = W*sp500Quantiles> msftVaR
1%
5%-18923 -13711> sp500VaR
1%
5%-12040 -8184
Based on the empirical distribution of the monthly returns, a $100,000 monthly
investment in Microsoft will lose $13,711 or more with 5% probability andwill lose $18,923 or more with 1% probability. The corresponding values for
the S&P 500 are $8,184 and $12,040, respectively. The historical VaR values
for the S&P 500 are considerable smaller than those for Microsoft. In thissense, investing in Microsoft is a riskier than investing in the S&P 500 index.

1.1.4

QQ-plots

Often it is of interest to see if a given data sample could be viewed as a random
sample from a specified probability distribution. One easy and eﬀective way
to do this is to compare the empirical quantiles of a data sample to thosefrom a reference probability distribution. If the quantiles match up, thenthis provides strong evidence that the reference distribution is appropriate
for describing the distribution of the observed data. If the quantiles do notmatch up, then the observed diﬀerences between the empirical quantiles andthe reference quantiles can be used to determine a more appropriate reference

23

1.1 UNIVARIATE DESCRIPTIVE STATISTICS
SP500 Monthly Returns
0.10

MSFT Monthly Returns

0.05
0.00
-0.05

Sample Quantiles

-0.15

-0.10

0.2
-0.2

-2

-1

0

1

2

-2

-1

0

1

2

-2

-1

0

1

2

Theoretical Quantiles

GWN Daily

MSFT Daily Returns

SP500 Daily Returns

0.05

-0.15

-0.05

0.00

Sample Quantiles

0.05
-0.05

-0.02

0.02

Sample Quantiles

0.15

0.10

Theoretical Quantiles

0.06

Theoretical Quantiles-0.06

Sample Quantiles

0.0

Sample Quantiles

0.1
0.0
-0.1
-0.2

Sample Quantiles

0.2

0.4

GWN Monthly

-2

0

2

Theoretical Quantiles

-2

0

2

Theoretical Quantiles

-2

0

2

Theoretical Quantiles

Figure 1.12: Normal QQ-plots for GWN, Microsoft returns and S&P 500
returns.
distribution. It is common to use the normal distribution as the reference
distribution, but any distribution can, in principle be used.
The quantile-quantile plot (QQ-plot) gives a graphical comparison of theempirical quantiles of a data sample to those from a specified reference distribution. The QQ-plot is an xy-plot with the reference distribution quantiles
on the x-axis and the empirical quantiles on the y-axis. If the quantiles exactly match up then the QQ-plot is a straight line. If the quantiles do notmatch up, then the shape of the QQ-plot indicates which features of the data
are not captured by the reference distribution.
Example 13 Normal QQ-plots for GWN, Microsoft and S&P 500 returns
The R function qqnorm() creates a QQ-plot for a data sample using thenormal distribution as the reference distribution. Figure 1.12 shows nor-

24CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA
mal QQ-plots for the simulated GWN data, Microsoft returns and S&P 500
returns created using> par(mfrow=c(2,3))> qqnorm(gwnMonthly, main="GWN Monthly", col="cornflowerblue")> qqline(gwnMonthly)> qqnorm(msftRetS, main="MSFT Monthly Returns", col="cornflowerblue")> qqline(msftRetS)> qqnorm(sp500RetS, main="SP500 Monthly Returns", col="cornflowerblue")> qqline(sp500RetS)> qqnorm(gwnDaily, main="GWN Daily", col="cornflowerblue")> qqline(gwnDaily)> qqnorm(msftDailyRetS, main="MSFT Daily Returns", col="cornflowerblue")> qqline(msftDailyRetS)> qqnorm(sp500DailyRetS, main="SP500 Daily Returns", col="cornflowerblue")> qqline(sp500DailyRetS)> par(mfrow=c(1,1))
The normal QQ-plot for the simulated GWN data is very close to a straight
line, as it should be since the data are simulated from a normal distribution.
The qqline() function draws a straight line through the points to help determine if the quantiles match up. The normal QQ-plots for the Microsoft
and S&P 500 monthly returns are linear in the middle of the distribution but
deviate from linearity in the tails of the distribution. In the normal QQ-plot
for Microsoft monthly returns, the theoretical normal quantiles on the x-axis
are too small in both the left and right tails because the points fall below thestraight line in the left tail and fall above the straight line in the right tail.
Hence, the normal distribution does not match the empirical distribution of
Microsoft monthly returns in the extreme tails of the distribution. In otherwords, the Microsoft returns have fatter tails than the normal distribution.
For the S&P 500 monthly returns, the theoretical normal quantiles are too
small only for the left tail of the empirical distribution of returns (points fall
below the straight line in the left tail only). This reflect the long left tail(negative skewness) of the empirical distribution of S&P 500 returns. The
normal QQ-plots for the daily returns on Microsoft and the S&P 500 index
both exhibit a pronounced tilted S-shape with extreme departures from linearity in the left and right tails of the distributions. The daily returns clearly
have much fatter tails than the normal distribution. ¥1.1 UNIVARIATE DESCRIPTIVE STATISTICS

25

Example 14 Student’s t QQ-plot for Microsoft returns
The function qqPlot() from the R package car can be used to create a
QQ-plot against any reference distribution that has a corresponding quantile
function implemented in R. For example, a QQ-plot for the Microsoft returns
using a Student’s t reference distribution with 5 degrees of freedom can becreated using> library(car)> qqPlot(coredata(msftRetS), distribution="t", df=5,+
ylab="MSFT quantiles", envelope=FALSE)
In the function qqPlot(), the argument distribution="t" specifies thatthe quantiles are to be computed using the R function qt().6 Figure 1.13
shows the resulting graph. Here, with a reference distribution with fatter
tails than the normal distribution the QQ-plot for Microsoft returns is closer
to a straight line. This indicates that the Student’s t distribution with 5
degrees of freedom is a better reference distribution for Microsoft returns
than the normal distribution. ¥1.1.5

Shape Characteristics of the Empirical Distribution

Recall, for a random variable  the measures of center, spread, asymmetry
and tail thickness of the pdf are:center: = []
spread :  2 = var() = [( −  )2 ]
pspread :   = var()asymmetry : skew = [( −  )3 ] 3 
tail thickness : kurt = [( −  )4 ] 4 
The corresponding shape measures for the empirical distribution (e.g., asmeasured by the histogram) of a data sample { }=1 are the sample statis6

The coredata() function is used to extract the data as a “numeric” object because
qqPlot() does not work correctly with “zoo” objects.

0.0
-0.2

MSFT quantiles

0.2

0.4

26CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA

-4

-2

0

2

4

t quantiles

Figure 1.13: QQ-plot of Microsoft returns using Student’s t distribution with
5 degrees of freedom as the reference distribution.
tics:71X
̂ = ̄ =
 
 =1

(1.1)

̂ 2 = 2 =

(1.2)



7


1 X
( − ̄)2  − 1 =1q̂  = ̂ 2
P13
=1 ( − ̄)−1[skew =

3
P
1
4
=1 ( − ̄)−1dkurt =


(1.3)
(1.4)
(1.5)

Values with hats “b” denote sample estimates of the corresponding population quantity.
For example, the sample mean ̂ is the sample estimate of the population expected value 

1.1 UNIVARIATE DESCRIPTIVE STATISTICS

27

The sample mean, ̂  measures the center of the histogram; the sample
standard deviation, ̂  , measures the spread of the data about the mean[   measures thein the same units as the data; the sample skewness, skewd   measures the tailasymmetry of the histogram; the sample kurtosis, kurt
thickness of the histogram. The sample excess kurtosis, defined as the sample
kurtosis minus 3d  − 3[  = kurt
ekurt
(1.6)

measures the tail thickness of the data sample relative to that of a normal
distribution.
Notice that the divisor in (1.2)-(1.5) is  − 1 and not  This is called adegrees-of-freedom correction. In computing the sample variance, skewness
and kurtosis, one degree-of-freedom in the sample is used up in the computation of the sample mean so that there are eﬀectively only  − 1 observations
available to compute the statistics. 8

Example 15 Sample shape statistics for the returns on Microsoft and S&P
500
The R functions for computing (1.1) - (1.5) are mean(), var() and sd(),
respectively. There are no functions for computing (1.4) and (1.5) in base R.
The functions skewness() and kurtosis() in the PerformanceAnalytics
package compute (1.4) and the sample excess kurtosis (1.6), respectively.9
The sample statistics for the Microsoft and S&P 500 monthly returns are:> statsMat = rbind(apply(msftSp500RetS, 2, mean),+
apply(msftSp500RetS, 2, var),+
apply(msftSp500RetS, 2, sd),+
apply(msftSp500RetS, 2, skewness),+
apply(msftSp500RetS, 2, kurtosis))> rownames(statsMat) = c("Mean", "Variance", "Std Dev",+"Skewness", "Excess Kurtosis")> round(statsMat, digits=4)
MSFT
SP500
8

If there is only one observation in the sample then it is impossible to create a measureof spread in the sample. You need at least two observations to measure deviations fromthe sample average. Hence the eﬀective sample size for computing the sample variance is − 1
9
Similar functions are available in the moments package.

28CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA
Mean
0.0092 0.0028
Variance
0.0103 0.0023
Std Dev0.1015 0.0478
Skewness0.4853 -0.5631
Excess Kurtosis 1.9150 0.6545
The mean and standard deviation for Microsoft monthly returns are 0.9% and10%, respectively. Annualized,
these values are approximately 10.08% (.009√
×12) and 34.6% (.10 × 12), respectively. The corresponding monthly andannualized values for S&P 500 returns are .3% and 4.7% and 3.6% and 16.3%,
respectively. Microsoft has a higher mean and volatility than S&P 500. The
lower volatility for the S&P 500 reflects risk reduction due to diversification.
The sample skewness for Microsoft, 0.485, is slightly positive and reflects
the approximate symmetry in the histogram in Figure 1.7. The skewness
for S&P 500, however, is moderately negative at -0.563 which reflects thesomewhat long left tail of the histogram in Figure 1.7. The sample excess
kurtosis values for Microsoft and S&P 500 are 1.915 and 0.654, respectively,
and indicate that the tails of the histograms are slightly fatter than the tailsof a normal distribution.
The sample statistics for the daily returns are> statsMatDaily = rbind(apply(msftSp500DailyRetS, 2, mean),+
apply(msftSp500DailyRetS, 2, var),+
apply(msftSp500DailyRetS, 2, sd),+
apply(msftSp500DailyRetS, 2, skewness),+
apply(msftSp500DailyRetS, 2, kurtosis))> rownames(statsMatDaily) = rownames(statsMat)> round(statsMatDaily, digits=4)
MSFT SP500
Mean
0.0005 0.0002
Variance
0.0005 0.0002
Std Dev
0.0217 0.0135
Skewness
0.2485 0.0012
Excess Kurtosis 7.3824 6.9971
For a daily horizon, the sample mean for both series is zero (to three decimals). As with the monthly returns, the sample standard deviation forMicrosoft, 2.2%, is about twice as big as the sample standard deviation for1.1 UNIVARIATE DESCRIPTIVE STATISTICS

29

Figure 1.14: Illustration of an “outlier” in a data sample.
the S&P 500 index, 1.4%. Neither daily return series exhibits much skewness, but the excess kurtosis values are much larger than the corresponding
monthly values indicating that the daily returns have much fatter tails andare more non-normally distributed than the monthly return series.
The daily simple returns are very close to the daily cc returns so thesquare-root-of-time rule can be applied to the sample mean and standard
deviation of the simple returns:> 12*statsMatDaily["Mean", ]
MSFT
SP500
0.005597 0.002069> sqrt(12)*statsMatDaily["Std Dev", ]
MSFT
SP500
0.07518 0.04670
Comparing these values to the corresponding monthly sample statistics showsthat the square-root-of-time-rule underestimates both the monthly mean andstandard deviation. ¥30CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATAGWN with Outlier %∆Statistic

GWN

Mean

-0.0054 -0.0072

32%

Variance

0.00860.010116%

Std. Deviation 0.0930

0.10038%

Skewness-0.2101 -1.0034

377%

Kurtosis2.7635

159%

Median-0.0002 -0.0002

0%

IQR

0.13900%7.15300.1390

Table 1.1: Sample statistics for GWN with and without outlier.

1.1.6

Outliers

Figure 1.14 nicely illustrates the concept of an outlier in a data sample.
All of the points are following a nice systematic relationship except one the “outlier”. Outliers can be thought of in two ways. First, an outlier
can be the result of a data entry error. In this view, the outlier is not avalid observation and should be removed from the data sample. Second,an outlier can be a valid data point whose behavior is seemingly unlike theother data points. In this view, the outlier provides important informationand should not be removed from data sample. For financial market data,
outliers are typically extremely large or small values that could be the resultof a data entry error (e.g. price entered as 10 instead of 100) or a valid
outcome associated with some unexpected bad or good news. Outliers areproblematic for data analysis because they can greatly influence the value ofcertain sample statistics.
Example 16 Eﬀect of outliers on sample statistics
To illustrate the impact of outliers on sample statistics, the simulated monthly
GWN data is polluted by a single large negative outlier:> gwnMonthlyOutlier = gwnMonthly> gwnMonthlyOutlier[20] = -sd(gwnMonthly)*6

31

-0.4

0.0 0.2

1.1 UNIVARIATE DESCRIPTIVE STATISTICS

1999

2001

2003

2005

2007

2009

2011

40
20
0

Frequency

60

Index

-0.6

-0.4

-0.2

0.0

0.2

GWN with outlier

Figure 1.15: Monthly GWN data polluted by a single outlier.
Figure 1.15 shows the resulting data. Visually, the outlier is much smaller
than a typical negative observation and creates a pronounced asymmetry
in the histogram. Table 1.1 compares the sample statistics (1.1) - (1.5) of
the unpolluted and polluted data. All of the sample statistics are influenced
by the single outlier with the skewness and kurtosis being influenced themost. The mean decreases by 32% and the variance and standard deviation
increase by 16% and 8%, respectively. The skewness changes by 377% andthe kurtosis by 159%. Table 1.1 also shows the median and the IQR, which
are quantile-based statistics for the center and spread, respectively. Noticethat these statistics are unaﬀected by the outlier. ¥The previous example shows that the common sample statistics (1.1) (1.5) based on the sample average and deviations from the sample averagecan be greatly influenced by a single outlier, whereas quantile-based sample
statistics are not. Sample statistics that are not greatly influenced by a single
outlier are called (outlier) robust statistics.

32CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA
Defining Outliers
A commonly used rule-of-thumb defines an outlier as an observation thatis beyond the sample mean plus or minus three times the sample standard
deviation. The intuition for this rules comes from the fact that if  ∼ ( 2 ) then Pr( − 3 ≤  ≤  + 3) ≈ 099 While the intuition forthis definition is clear, the practical implementation of the rule using thesample mean, ̂ and sample standard deviation, ̂ is problematic because
these statistics can be heavily influenced by the presence of one or more
outliers (as illustrated in the previous sub-section). In particular, both ̂and ̂ become larger (in absolute value) when there are outliers and this may
cause the rule-of-thumb rule to miss identifying an outlier or to identify anon-outlier as an outlier. That is, the rule-of-thumb outlier detection rule is
not robust to outliers!
A natural fix to the above outlier detection rule-of-thumb is to use outlier
robust statistics, such as the median and the IQR, instead of the mean andstandard deviation. To this end, a commonly used definition of a moderate
outlier in the right tail of the distribution (value above the median) is a data
point  such that̂75 + 15 ·     ̂75 + 3 · (1.7)

To understand this rule, recall that the IQR is where the middle 50% of thedata lies. If the data were normally distributed then ̂75 ≈ ̂ + 067 × ̂ and ≈ 134 × ̂ Then (1.7) is approximately
̂ + 267 × ̂    ̂ + 467 × ̂
Similarly, a moderate outlier in the left tail of the distribution (value below
the median) is a data point such that̂25 − 3 ·     ̂25 − 15 · Extreme outliers are those observations  even further out in the tails of the
distribution and are defined using  ̂75 + 3 ·   ̂25 − 3 · For example, with normally distributed data an extreme outlier would be anobservation that is more than 467 standard deviations above the mean.

1.2 TIME SERIES DESCRIPTIVE STATISTICS

1.1.7

33

Box Plots

A boxplot of a data series describes the distribution using outlier robust
statistics. The basic boxplot is illustrated in Figure. The middle of the
distribution is shown by a rectangular box whose top and bottom sides are
the first and third quartiles, ̂25 and ̂75  respectively. The line roughly in the
middle of the box is the sample median, ̂5  The top whisker is a horizontal
line that represents either the largest data value or ̂75 +15·, whicheveris smaller. Similarly, the bottom whisker is located at either the smallest data
value or ̂25 − 3 ·  whichever is larger.
Example 17 Boxplots of return distributions
Box plots are a great tool for visualizing the return distribution of one
or more assets. Figure 1.16 shows the boxplots of the monthly continuously
compounded and simple returns on Microsoft and the S&P500 index computed using10> boxplot(coredata(msftRetS), coredata(msftRetC),+
coredata(sp500RetS), coredata(sp500RetC),+
names=c("msftRetS", "msftRetC",+"sp500RetS", "sp500RetC"),+
col="cornflowerblue")
The Microsoft returns have both positive and negative outliers whereas the
S&P 500 index returns only have negative outliers. The boxplots also clearlyshow that the distributions of the continuously compounded and simple returns are very similar except for the extreme tails of the distributions. ¥1.2

Time Series Descriptive Statistics

In this section, we consider graphical and numerical descriptive statistics forsummarizing the linear time dependence in a time series.
10

The coredata() function is used to extract the data from the zoo object because
boxplot() doesn’t like zoo objects.

-0.4

-0.2

0.0

0.2

0.4

34CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA

msftRetS

msftRetC

sp500RetS

sp500RetC

Figure 1.16: Boxplots of the continuously compounded and simple monthly
returns on Microsoft stock and the S&P 500 index.

1.2.1

Sample Autocovariances and Autocorrelations

For a covariance stationary time series process { }, the pautocovariances  = cov(  − ) and autocorrelations  = cov(  − ) var( )var(− ) =   2 of a describe the linear time dependences in the process. For a sample of data { }=1 , the linear time dependences are captured the sample
autocovariances and autocorrelations:X1̂  =( − ̄)(− − ̄)  = 1 2      −  + 1
 − 1 =+1

̂ =

̂ 
  = 1 2      −  + 1̂ 2

where ̂ 2 is the sample variance (1.2). The sample autocorrelation function(SACF) is a plot of ̂ vs. , and gives a graphical view of the liner time1.2 TIME SERIES DESCRIPTIVE STATISTICS

35

dependences in the observed data.
Example 18 SACF for the Microsoft and S&P 500 returns.
The R function acf() computes and plots the sample autocorrelations ̂of a time series for a specified number of lags. For example, to compute ̂for  = 1     5 for the monthly and daily returns on Microsoft use> acf(coredata(msftDailyRetS), lag.max=5, plot=FALSE)
Autocorrelations of series ‘coredata(msftDailyRetS)’, by lag
0
1
2
3
4
5
1.000 -0.044 -0.018 -0.010 -0.042 0.018> acf(coredata(msftRetS), lag.max=5, plot=FALSE)
Autocorrelations of series ‘coredata(msftRetS)’, by lag
0
1
2
3
4
5
1.000 -0.193 -0.114 0.193 -0.140 -0.023> acf(coredata(msftDailyRetS), lag.max=5, plot=FALSE)
Autocorrelations of series ‘coredata(msftDailyRetS)’, by lag
0
1
2
3
4
1.000 -0.044 -0.018 -0.010 -0.042

5
0.018

None of these sample autocorrelations are very big and indicate lack of linear
time dependence in the monthly and daily returns.
Sample autocorrelations can also be visualized using the function acf().
Figure 1.17 shows the SACF for the monthly and daily returns on Microsoft
and the S&P 500 created using> par(mfrow=c(2,2))>acf(coredata(msftRetS), main="msftRetS", lwd=2)>
acf(coredata(sp500RetS), main="sp500RetS", lwd=2)>
acf(coredata(msftDailyRetS), main="msftDailyRetS", lwd=2)>
acf(coredata(sp500DailyRetS), main="sp500DailyRetS", lwd=2)> par(mfrow=c(1,1))

36CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA

0.0 0.2 0.4 0.6 0.8 1.0

ACF

-0.2

ACF

sp500RetS

0.2 0.4 0.6 0.8 1.0

msftRetS

5

10

15

20

0

10

15

Lag

msftDailyRetS

sp500DailyRetS

0.8

1.0

20

ACF

0.2

0.4

0.6

0.8
0.6
0.4
0.0

0.0

0.2

ACF

5

Lag

1.0

0

0

5

10

15

20

25

30

35

0

5

10

15

Lag

20

25

30

35

Lag

Figure 1.17: SACFs for the monthly and daily returns on Microsoft and the
S&P 500 index.
None of the series show strong evidence for linear time dependence. The
monthly returns have slightly larger sample autocorrelations at low lags thanthe daily returns. The dotted lines on the SACF plots are 5% critical values
for testing the null hypothesis that the true autocorrelations  are equal tozero. If ̂ crosses the dotted line then the null hypothesis that  = 0 canbe rejected at the 5% significance level.

1.2.2Time Dependence in Volatility

The daily and monthly returns of Microsoft and the S&P 500 index do notexhibit much evidence for linear time dependence. Their returns are essentially uncorrelated over time. However, the lack of autocorrelation does notmean that returns are independent over time. There can be nonlinear time37

1.2 TIME SERIES DESCRIPTIVE STATISTICS

ACF

0.0

0.4

0.8

0 1 2 3-2

gwn

Series gwn

0

100

200

300

400

500

0

5

10

Time

15

20

25

20

25

20

25

Lag

0.8
ACF

0

0.0

0.4

6
4
2

gwn^2

8 10

Series gwn^2

0

100

200

300

400

500

0

5

10

Time

15
Lag

0.8
ACF

0.0

0.4

2.0
1.0
0.0

abs(gwn)

3.0

Series abs(gwn)

0

100

200

300

400

500

0

Time

5

10

15
Lag

Figure 1.18:
dependence. Indeed, there is strong evidence for a particular type of nonlinear time dependence in the daily returns of Microsoft and the S&P 500 index
that is related to time varying volatility. This nonlinear time dependence,
however, is less pronounced in the monthly returns.
Example 19 GWN and Time Independence
Let  ∼ GWN(0 2 ) Then  is, by construction, independent overtime. This implies that any function of  is also independent over time.
To illustrate, Figure 1.18 shows   2 and | | together with their SACFs
where  ∼  (0 1) for  = 1     500. Notice that all of the sample
autocorrelations are essentially zero which confirms the lack of linear timedependence in   2 and | | ¥

38CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA

0.05
0.00
0.08
0.04
0.008
0.004
0.000

Returns^2

0.012 0.00

abs(Returns)

0.12

-0.05

Returns

0.10

Daily Returns

2000

2005

2010

Index

Figure 1.19: Daily returns, absolute returns, and squared returns on the S&P
500 index.
Nonlinear Time Dependence in Asset Returns
Figures 1.5 and 1.2 show that the daily and monthly returns on Microsoft
and the S&P 500 index appear to exhibit periods of volatility clustering.
That is, high periods of volatility tend to be followed by periods of highvolatility and periods of low volatility appear to be followed by periods oflow volatility. In other words, the volatility of the returns appears to exhibit
some time dependence. This time dependence in volatility causes linear timedependence in the absolute value and squared value of returns.
Example 20 Nonlinear time dependence in the S&P 500 returns
To illustrate nonlinear time dependence in asset returns, Figure 1.19shows
the daily returns, absolute returns and squared returns for the S&P 500 index

39

1.2 TIME SERIES DESCRIPTIVE STATISTICS
Daily Squared Returns

0.8
0.6

ACF

0.2
0.0

0.0

5

10

15

20

25

30

35

0

10

15

20

25

30

Monthly Absolute Returns

Monthly Squared Returns

5

10
Lag

15

20

35

0.0 0.2 0.4 0.6 0.8 1.0

Lag

ACF
0

5

Lag

0.0 0.2 0.4 0.6 0.8 1.0

0

ACF

0.4

0.6
0.4
0.2

ACF

0.8

1.0

1.0

Daily Absolute Returns

0

5

10

15

20

Lag

Figure 1.20: SACFs of daily and monthly absolute and squared returns on
the S&P 500 index.

and Figure shows the SACFs for these series. Notice how the absolute andsquared returns are large (small) when the daily returns are more (less)volatile. Figure 1.20 shows the SACFs for the absolute and squared daily
and monthly returns. There is clear evidence of time dependence in the daily
absolute and squared returns while there is some evidence of time dependence
in the monthly absolute and squared returns. Since the absolute and squared
daily returns reflect the volatility of the daily returns, the SACFs show astrong positive linear time dependence in daily return volatility.

40CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA

1.3

Bivariate Descriptive Statistics

In this section, we consider graphical and numerical descriptive statistics forsummarizing two or more data series.

1.3.1

Scatterplots

The contemporaneous dependence properties between two data series { }=1
and { }=1 can be displayed graphically in a scatterplot, which is simply an
xy-plot of the bivariate data.
Example 21 Scatterplot of Microsoft and S&P 500 returns
Figure 1.21 shows the scatterplot between the Microsoft and S&P 500 returns
created using> plot(sp500RetS,msftRetS,+
main="Monthly cc returns on MSFT and SP500",+
xlab="S&P500 returns", ylab="MSFT returns",+
lwd=2, pch=16, cex=1.25, col="blue")> abline(v=mean(sp500RetS))> abline(h=mean(msftRetS))
The S&P 500 returns are put on the x-axis and the Microsoft returns on the
y-axis because the “market”, as proxied by the S&P 500, is often thought as
an independent variable driving individual asset returns. The upward sloping
orientation of the scatterplot indicates a positive linear dependence between
Microsoft and S&P 500 returns. ¥Exercise 22 Pair-wise scatterplots for multiple series
For more than two data series, the R function pairs() plots all pair-wise
scatterplots in a single plot. For example, to plot all pair-wise scatterplots
for the GWN, Microsoft returns and S&P 500 returns use:
> pairs(cbind(gwnMonthly,msftRetS,sp500RetS), col="blue",+pch=16, cex=1.25, cex.axis=1.25)
The top row of Figure 1.22 shows the scatterplots between the pairs (MSFT,
GWN) and (SP500, GWN), the second row shows the scatterplots between
the pairs (GWN, MSFT) and (SP500, MSFT), the third row shows the scatterplots between the pairs (GWN, SP500) and (MSFT, SP500). ¥41

1.3 BIVARIATE DESCRIPTIVE STATISTICS

0.0
-0.2

MSFT returns

0.2

0.4

Monthly cc returns on MSFT and SP500

-0.15

-0.10

-0.05

0.00

0.05

0.10

S&P500 returns

Figure 1.21: Scatterplot of Monthly returns on Microsoft and the S&P 500
index.

1.3.2

Sample Covariance and Correlation

For two random variables  and  the direction of linear dependence iscaptured by the covariance,   = [( −  )( −  )] and the direction
and strength of linear dependence is captured by the correlation,  =       For two data series { }=1 and { }=1  the sample covariance,1 X( − ̄)( − ̄)̂  =
 − 1 =1

(1.8)

measures the direction of linear dependence, and the sample correlation,̂ =̂ ̂  ̂ (1.9)

42CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA

0.0

0.2

0.4

0.1

0.2

-0.2

0.2

0.4

-0.2

0.0

gwnMonthly-0.15

sp500RetS

-0.05

0.05

-0.2

0.0

msftRetS

-0.2

0.0

0.1

0.2

-0.15

-0.05

0.05

Figure 1.22: Pair-wise scatterplots between simulated GWN, Microsoft returns and S&P 500 returns.

measures the direction and strength of linear dependence. In (1.9), ̂  and̂  are the sample standard deviations of { }=1 and { }=1  respectively,
defined by (1.3).

When more than two data series are being analyzed, it is often convenient
to compute all pair-wise covariances and correlations at once using matrix
algebra. Recall, for a vector of  random variables X = (1       )0 withmean vector μ = (1       )0 the  ×  variance-covariance matrix is43

1.3 BIVARIATE DESCRIPTIVE STATISTICS
defined as
⎛

 21

⎞

 12 · · ·  1
⎟
⎜
⎟
⎜
2
⎜  12  2 · · ·  2 ⎟0⎟
Σ = var(X) = cov(X) = [(X − μ)(X − μ) ] = ⎜⎜ ..
.. . .
.. ⎟ 
.
⎜ .
.
. ⎟
⎠
⎝
2
 1  2 · · ·  For  data series { }=1  where  = (1       )0  the sample covariancematrix is computed using⎞⎛2̂ ̂ 12 · · · ̂ 1
⎟
⎜ 1
⎟
⎜

2
X
⎟
⎜
̂
̂
·
·
·
̂
1
12
2
2
⎟
(1.10)Σ̂ =(x − μ̂)(x − μ̂)0 = ⎜⎜ ..
.. . .
.. ⎟ 
 − 1 =1
. . ⎟
⎜ .
.
⎠
⎝
̂ 1 ̂ 2 · · · ̂ 2

where μ̂ is the  × 1 sample mean vector. Define the  ×  diagonal matrix⎛⎞
̂ 1 0 · · · 0⎜⎟⎜⎟⎜ 0 ̂ 2 · · · 0 ⎟⎜⎟D̂ = ⎜ . . .
.. ⎟ 
.
.
.
. . ⎟
⎜ . .
⎝
⎠
0 0 · · · ̂ 
Then the  ×  sample correlation matrix R̂ is computed as⎞⎛1 ̂12 · · · ̂1⎟⎜⎟⎜⎜ ̂12 1 · · · ̂2 ⎟
−1
−1
⎟
⎜
R̂ = D̂ Σ̂D̂ = ⎜ .
.. . .
.. ⎟ 
.
. . ⎟
⎜ .
.
⎠
⎝
̂1 ̂2 · · · 1(1.11)

Example 23 Sample covariance and correlation between Microsoft and S&P
500 returns

44CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA
The scatterplot of Microsoft and S&P 500 returns in Figure 1.21 suggests a
positive linear relationship in the data. We can confirm this by computing thesample covariance and correlation using the R functions cov() and cor():> cov(sp500RetS, msftRetS)[1] 0.002979> cor(sp500RetS, msftRetS)[1] 0.6139
Indeed, the sample covariance is positive and the sample correlation shows amoderately strong linear relationship.
When passed a matrix of data, the cov() and cor() functions can also
be used to compute the sample covariance and correlation matrices Σ̂ andR̂, respectively:> cov(msftSp500RetS)
MSFT
SP500
MSFT 0.010311 0.002979
SP500 0.002979 0.002284> cor(msftSp500RetS)MSFT SP500
MSFT 1.0000 0.6139
SP500 0.6139 1.0000
The function cov2cor() transforms a sample covariance matrix to a sample
correlation matrix using (1.11):> cov2cor(cov(msftSp500RetS))
MSFT SP500
MSFT 1.0000 0.6139
SP500 0.6139 1.0000¥
Example 24 Visualizing correlation matrices
The R package corrplot contains functions for visualizing correlation
matrices. This is particularly useful for summarizing the linear dependencies
among many data series. For example, Figure 1.23 shows the correlation plot
from the corrplot function corrplot.mixed() created using

45

1.3 BIVARIATE DESCRIPTIVE STATISTICS

1

0.8

gwnMonthly
0.6

0.4

0.2

-0.09

msftRetS

0

-0.2

-0.4

-0.05

0.61

sp500RetS

-0.6

-0.8

-1

Figure 1.23:> dataToPlot = merge(gwnMonthly, msftRetS, sp500RetS)> cor.mat = cor(dataToPlot)> corrplot.mixed(cor.mat, lower="number", upper="ellipse")
The color scheme shows the magnitudes of the correlations (blue for positiveand red for negative) and the orientation of the ellipses show the magnitude
and direction of the linear associations.

1.3.3

Stylized Facts for Daily and Monthly Asset Returns

A stylized fact is something that is generally true but not always. From thedata analysis of the daily and monthly returns on Microsoft and the S&P
500 index we observe a number of stylized facts that tend to be true for other46CHAPTER 1 DESCRIPTIVE STATISTICS FOR FINANCIAL DATA
individual assets and portfolios of assets. For monthly data we observe thefollowing stylized facts:
M1. Prices appear to be random walk non-stationary and returns appear to
be mostly covariance stationary. There is evidence that return volatilitychanges over time.
M2. Returns appear to approximately normally distributed. There is some
negative skewness and excess kurtosis.
M3. Assets that have high average returns tend to have high standard deviations (volatilities) and vice-versa. This is the no free lunch principle.
M4. Returns on individual assets (stocks) have higher standard deviations
than returns on diversified portfolios of assets (stocks).
M5. Returns on diﬀerent assets tend to be positively correlated. It is unusual for returns on diﬀerent assets to be negatively correlated.
M6. Asset returns are approximately uncorrelated over time. That is, there
is little evidence of linear time dependence in asset returns.
Daily returns have some features in common with monthly returns andsome not. The common features are M1 and M3-M6 above. The stylized
facts that are specific to daily returns are :D2. Returns are not normally distributed. Empirical distributions havemuch fatter tails than the normal distribution (excess kurtosis).
D7. Returns are not independent over time. Absolute and squared returns
are positively autocorrelated and the correlation dies out very slowly.
Volatility appears to be autocorrelated and, hence, predictable.
These stylized facts of daily and monthly asset returns are the main features of returns that models of assets returns should capture. A good model
is one that can explain many stylized facts of the data. A bad model does
not capture important stylized facts.

1.4

Further Reading

To be completed

1.5 PROBLEMS

1.5

47

Problems

Exercise 25 Histogram for returns with diﬀerent number of bins
Exercise 26 Smoothed density with diﬀerent bandwidth parameters
Exercise 27 Histogram overlaid with normal density
Show PerformanceAnalytics function chart.Histogram().
Exercise 28 Extracting unique covariance elements from covariance matrix
Exercise 29 Visualizing correlation with heatmaps

1.6

References

Ruppert, D. Statistics and Data Analysis for Financial Engineering. SpringerVerlag, New York.