Suppose are is timeseries A stock returns either simple or continuously compounded
Rt is a weekly stationary
e ergodic process ttTej decaywith tie
Var Rt angry 2
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y to learn about the pdf trek
oh independent
Also assume realization
endrendent
each Ry is identically distributed with pdf frfr
We observe a sample size of T historied returns this is a
of the random process
Goal Use hrs re
over er years a monthly returns
Daily simple retain
Monthly simple return
compounded return Nfo n Nfe i Ncen
continuously
a t of o T E F G o
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t Cor x xz
aag var re t Varley
her Xin t Cor Xuxa
Var x ere ox y
Vor Xi 1 0 ‘s
area a very
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Empirical CDF example
Empirical CDF Quantile
if Semple size
then É ol IP X E o
Application
Historical
initial investment STATISTICAL CHARACTERISTICS OF SAMPLE
multiply by
Mean M E x X is Variance oh mercy E
dentin variable
captures the spreed captures the symmetry
Elegy captures the thickness
normed dit Excess kurtoni
E Y Kurt 3
the tails 3
we don’t know the distribution of the RV est these quantities from a sample
sample size
Sample mean
ÉÉXt oh5a IÉXe
Tarrance slammer
sample kurtosis
TWO WAYS OF Use
DEALING WITH robust
OUTLIERS statistics
instead f sd
Inter quartile
25 quantile
251quanth MY
a throw Detection
Find outliers Outlier
Common rule ofthumb define outlier as any obsenchin that falls more than 3 sd away from the mean
point being more than Bsd away from near E 0.26 1
I ed are already
p Pollen if
outlier are present then ween wrong
Use quartiles to detect
data point x suchthat Doit FI e x
thub nudeeate outer
in theright tail D.ptIOD
Extreme outher in the right tail aim sobythis replace
IQRCX C y 15IAR in the left tail
757quartile gaffer
any data rite x Moderate outliers in the left
date point a
extreme outliers
any data pointX x c go 3 Box
AILA NFunction
We can ask for correlation between
own history
stock’s return and autocovariance
sample autocorrelation acf function in R J
var function in R
F Xej T Ireturn t another stock I s
covariance
Standard Deviations fg in R cov function
right aligned Overaleconclusion
leftaligned
Thereis some evidence Monthly returns serene to be closeto normally distributed
there is some excesskurtry Daily returns have significant excess kurtosis
Stock nature
that volatility changes over time
3 Returns on individual stocks have higher than returns on indexes or portfolios
be positively time
volatility Csd
correlated
Returns on different stocks Cang comeletion a o.nu
I Monthy Daily
uncorrelated over
returns do exhibit negative correlation once one day
is no cross leg
correlation between shek returns
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