Descriptive Statistics for Financial Time Series
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1 Descriptive Statistics for Financial Time Series Econ 424/Amath 462 Summer 2014 Eric Zivot Updated: July 10, 2014
2 # Load libraries > library(tseries) > library(performanceanalytics) Data for Examples # Get adjusted closing price data from Yahoo! > MSFT.prices = get.hist.quote(instrument="msft", start=" ", + end=" ", quote="adjclose", + provider="yahoo", origin=" ", + compression="m", retclass="zoo") > SP500.prices = get.hist.quote(instrument="^gspc", start=" ", + end=" ", quote="adjclose", + provider="yahoo", origin=" ", + compression="m", retclass="zoo")
3 Price Data > plot(msftsp500.prices, p main="adjusted Closing Prices", lwd=2, col="blue")
4 Monthly Continuously Compounded Returns Q: What common features do you see?
5 MSFT and S&P 500 tend to move together S&P 500 has much lower volatility than MSFT
6 Compare Cumulative Returns > chart.cumreturns(msftsp500.rets, main=", legend.loc="topright") 3.0 MSFT SP500 Value Feb 98 Feb 00 Feb 02 Feb 04 Feb 06 Feb 08 Feb 10 Jan 12
7 Are Monthly Returns Gaussian White Noise? > set.seed(123) > gwn = rnorm(length(msft),mean=mean(msft),sd=std(msft))
8 Estimating the pdf: Histogram > hist(msft,main= main="histogram of MSFT monthly cc returns, + col= slateblue1 )
9 Note: Simulated data has the same mean and variance as the returns on Microsoft > hist(gwn,main= main="histogram of simulated Gaussian data, + col= slateblue1 )
10 > hist(sp500,main="histogram of SP500 monthly cc returns, + col= slateblue1 )
11 Note: MSFT has larger SD (volatility) than S&P 500
12 > MSFT.density = density(msft) > plot(msft.density,type="l",xlab="monthly return", + ylab="density estimate",main="smoothed t i "S th histogram for MSFT + monthly cc returns, col= orange, lwd=2)
13 > hist(msft,main="histogram and smoothed density of MSFT + monthly returns", probability=t, col= slateblue1, + ylim=c(0,5)) > points(msft.density, type="l, col= orange, lwd=2)
14 Computing quantiles > quantile(msft.ret.mat) 0% 25% 50% 75% 100% # 1% and 5% empirical quantiles > quantile(msft.ret.mat,probs=c(0.01,0.05)) 1% 5% 1% and 5% quantiles are used for Value-at- at # compare to 1% and 5% normal quantiles Risk calculations > qnorm(p=c(0.01,0.05), mean=mean(msft.ret.mat), + sd=sd(msft.ret.mat)) [1] # SP500 empirical and normal quantiles > quantile(sp500.ret.mat,probs=c(0.01,0.05)) 1% 5% > qnorm(p=c(0.01,0.05), mean=mean(sp500.ret.mat), + sd=sd(sp500.ret.mat)) sd(sp500.ret.mat)) [1]
15 Monthly VaR Using Empirical Quantiles > q.01 = quantile(msft.ret.mat, probs=0.01) > q.05 = quantile(msft.ret.mat, probs=0.05) > q.01 1% > q.05 5% # Monthly VaR on $100,000 investment > VaR.01 = *(exp(q.01) - 1) > VaR.05 = *(exp(q.05) - 1) > VaR.01 1% > VaR.05 5%
16 Summary Statistics > mean(msft.ret.mat) MSFT > var(msft.ret.mat) MSFT MSFT > sd(msft.ret.mat) MSFT > skewness(msft.ret.mat) [1] > kurtosis(msft.ret.mat) ret [1] Skewness() function is in package PerformanceAnalytics kurtosis() function is in package PerformanceAnalytics and computes excess kurtosis
17 Summary Statistics by Column > apply(msftsp500.ret.mat, 2, mean) MSFT SP Note: MSFT has a higher mean and higher SD than > apply(msftsp500.ret.mat, 2, sd) SP500 MSFT SP > apply(msftsp500.ret.mat, 2, skewness) MSFT SP > apply(msftsp500.ret.mat, 2, kurtosis) MSFT SP
18 Empirical CDF of Gaussian data #x(i) <= x sort(gwn) > n1 = length(gwn) > plot(sort(gwn),(1:n1)/n1,type= (1:n1)/n1 type="s",ylim=c(0,1), col= slateblue1 + main="empirical CDF of Gaussian data", ylab="#x(i) <= x")
19 Empirical CDF vs. Normal CDF for Gaussian data Normal CDF Empirical CDF CDF standardized gwn
20 Empirical CDF is pretty close to Normal CDF but hard to tell
21 Empirical CDF is pretty close to Normal CDF but hard to tell
22 QQ-plots against Normal Distribution Nice and straight! Empirical quantiles > par(mfrow=c(2,2)) don t match > qqnorm(gwn) normal > qqline(gwn) quantiles in> > qqnorm(msft.ret) ret) the tails! > qqline(msft.ret) > qqnorm(sp500.ret) > qqline(sp500.ret) > par(mfrow=c(1,1))
23
24
25 Effect of Outliers on Descriptive Statistics Outlier
26 Summary statistics of polluted data > tmp = cbind(gwn, gwn.new) > apply(tmp, 2, mean) gwn gwn.new > apply(tmp, 2, sd) gwn gwn.new > apply(tmp, 2, skewness) gwn gwn.new > apply(tmp, 2, kurtosis) gwn gwn.new Notice how sample statistics are influenced by the single outlier # outlier robust measures > apply(tmp, 2, median) gwn gwn.new > apply(tmp, 2, IQR) gwn gwn.newnew
27 Boxplot of monthly cc returns on Microsoft 0..2 outliers Largest data point at most1.5*iqr from top of box monthly cc re eturn 0.0 IQR median Smallest data point at most1.5*iqr from bottom of box > boxplot(msft,outchar=t,main= outchar=t main="boxplot of monthly cc + returns on Microsoft",ylab="monthly cc return")
28 > boxplot(gwn,msft,sp500,names=c("gwn","msft","sp500"),outchar=t, + main="comparison of return distributions", ylab="monthly cc + return")
29 Four Graph Summary par(mfrow=c(2,2)) # plot 1 hist(msft.mat,main= main="msft monthly cc returns", probability=t, ylab="cc return", col="slateblue1") # plot 2 boxplot(msft.mat,outchar=t, ylab="cc return", col="slateblue1") # plot 3 plot(msft.density,type="l",xlab="cc l,xlab return", col="slateblue1", lwd=2, ylab="density estimate", main="smoothed density") # plot 4 qqnorm(msft.mat) qqline(msft.mat) par(mfrow=c(1,1)) (
30
31 Scatterplot Cov(MSFT,SP500)= Corr(MSFT,SP500)=0.60 > plot(msft.mat,sp500.mat,main="monthly cc returns on MSFT + and SP500, pch=16, cex=1.5, col= blue ) > abline(h=mean(sp500)) # horizontal line at SP500 mean > abline(v=mean(msft)) # vertical line at MSFT mean
32 Pairwise Scatterplots > pairs(cbind(gwn,msft,sp500), col= blue, pch=16, + cex=1.5, cex.axis=1.5)
33 Sample Covariances and Correlations > var(cbind(gwn,msft.mat,sp500.mat)) gwn MSFT SP500 gwn MSFT SP > cor(cbind(gwn,msft.mat,sp500.mat)) gwn MSFT SP500 gwn MSFT SP
34 Sample ACF for S&P 500 Monthly returns are essentially uncorrelated; i.e., unpredictable!
35 Sample ACF for MSFT Small negative 1 st order autocorrelation possibly MA(1) model
36 Stylized Facts for Monthly CC Returns Returns appear to be approximately normally distributed Some noticeable negative skewness and excess kurtosis Individual asset returns have higher SD than diversified portfolios Many assets are contemporaneously correlated Assets are approximately uncorrelated over time (no serial correlation)
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