Descriptive Statistics for Daily
Return Data
Econ 424
Fall 2014
Eric Zivot
Update: October 21, 2014

© Eric Zivot 2006

Daily Prices on Microsoft and the S&P 500 Index: January
1998 –May 2012, n = 3,627© Eric Zivot 2006

Mean=0.0002
SD = 0.0217

Notice the periods of high and low volatility
and the extreme choppiness of the returns.
y not constant!
Volatilityy is obviously© Eric Zivot 2006

Mean = 0.0001
SD = 0.0135

Notice how volatility appears to be time dependent:
periods of high volatility persist and periods of lowpersist,.
,volatilityy p© Eric Zivot 2006

Mean = 0.002
Sd = 00.0217
0217
Skewness = -0.0452
Excess Kurtosis = 7.0412© Eric Zivot 2006

Mean = 0.0001
Sd = 00.0135
0135
Skewness = -0.1785
Excess kurtosis = 6.860© Eric Zivot 2006

SP500 returns are
more concentrated
about zero than
MSFT returns – lessrisk!
i k!

© Eric Zivot 2006

Boxplots show a large
number of outliers
illustrating the nonnon
normality of the daily
returns© Eric Zivot 2006

Daily Historical VaR on $100K invested in MSFT# Compute daily VaR using empirical quantiles> q.01 = quantile(msftDailyReturns.mat, probs=0.01)> qq.05 = qquantile(msftDailyReturns.mat,(y, pprobs=0.05))> q.01
1%-0.05854> q.055%-0.03243> VaR.01
01 = 100000
100000*(exp(q.01)
01 - 1)
1> VaR.05 = 100000*(exp(q.05) - 1)
> VaR.01
1%-5686> VaR.05
5%-3191© Eric Zivot 2006

Daily Historical VaR on $100K invested in SP500# Compute daily VaR using empirical quantiles> q.01 = quantile(sp500DailyReturns.mat, probs=0.01)> qq.05 = qquantile(sp500DailyReturns.mat,( py, pprobs=0.05))> q.01
1%-0.03727> q.055%-0.02114> VaR.01
01 = 100000
100000*(exp(q.01)
01 - 1)
1> VaR.05 = 100000*(exp(q.05) - 1)
> VaR.01
1%-3658> VaR.05
5%-2092© Eric Zivot 2006

Daily returns arehighly non
non-normal!
normal!

© Eric Zivot 2006

Gauss a White
Gaussiane noiseo se
has constant volatility –something that MSFT
does not have!

© Eric Zivot 2006

Daily returns show very little if any timedependence

Very small negative correlation between rt and rt-1.

© Eric Zivot 2006

Daily returns show very little if any timedependence

Very small negative correlation between rt and rt-1.

© Eric Zivot 2006

TAbsolute values proxy volatility; squares proxy variance© Eric Zivot 2006

Absolute values of GWN and squared values
of GWN are not auto-correlated. This is aconsequence off independence.
i dd© Eric Zivot 2006

MSFT daily returns exhibit time varying volatility or volatility
clustering – periods of high volatility are followed by periods ofhigh volatility and periods of low volatility are followed byperiods of low volatility© Eric Zivot 2006

Daily volatility for MSFT is positively autocorrelated:
there is positive time dependence in daily volatility!
Hence daily returns are not independent!!!

© Eric Zivot 2006

SP500 daily returns exhibit time varying volatility or volatility
clustering – periods of high volatility are followed by periods ofhigh volatility and periods of low volatility are followed byperiods of low volatility© Eric Zivot 2006

Daily volatility for SP500 is positively autocorrelated:
there is positive time dependence in daily volatility!
Hence daily returns are not independent!!!

© Eric Zivot 2006

Stylized Facts for Daily Returns• Returns are not normallyy distributed. Empiricalpdistributions have fatter tails than normal
distribution (more outliers)• Returns
Rare approximately
il uncorrelatedl d overtime (no serial correlation)• Returns are not independent over time
– Squared and absolute returns are positively
autoco e atedautocorrelated– Volatility appears to be serially correlated– See Engle’sg
GARCH model© Eric Zivot 2006