session on stylized properties
TRANSCRIPT
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Quantitative Applications inFinance
Distributional Properties of Returns Stylized Properties of Financial Time Series
Readings: Chapter 1, Text
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Dealing with Returns
Most financial studies involve returns, instead ofprices, of assets.
Two main reasons: First, for average investors,
return of an asset is a complete and scale-freesummary of the investment opportunity.
Second, return series are easier to handle thanprice series because the former have more
attractive statistical properties.
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Various definitions of an assetreturn
Pt= price of an asset at time index t
One-Period Simple Return:
(i) simplegross return, (1+Rt), (growth factor):
(ii) simplenet returnor simple return:
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Multiperiod Simple Return
Holding the asset for kperiods between datestkand tgives a k-period simplegrossreturn(1+Rt(k)):
k-period simplenetreturnis:
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Average Return
Actual time interval is important in comparingreturns (e.g., monthly or annual) return).
If asset was held for k periods, then the average
(e.g. annualized) simple gross return is
Average (1+Rt(k))=
(geometric mean of the k one-period simplegross returns)
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Average Net Return Rt(k)) =
Arithmetic Mean easier to compute than geometricmean and the one-period returns tend to be small. Can use a first-order Taylor expansion to approximatethe average net return Rt(k)) as:
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Continuous Compounding
Rate of interest over a unit time (from t-1 to tperiod) = R
Pt-n = initial capital; Pt = net asset value after ntime units
Discrete Compounding (m times in unit time):Pt = Pt-n*(1+(R/m))
m*n
Continuous Compounding (over unit time):P
t= P
t-n*exp(n*r
c). Thus, continuously
compounded return rc is given by
==
)ln(1
)ln(nt
tc
nt
tc
P
P
nr
P
Pnr
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Continuous Compounding
Continuous Compounding (over unit time): Pt =Pt-1*exp(rt). Thus, continuously compoundedreturn (or log of simple gross return) is
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Multiperiod Continuously CompoundedReturn
Continuously compounded multiperiod returnis sum of continuously compounded one-period
returns. Statistical properties of log returns are moretractable.
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Densities of Various Distributions
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Standard Normal pdf
0.00
0.05
0.100.15
0.20
0.25
0.30
0.35
0.40
0.45
-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0
x
p(x)
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Log normal distribution
Log-return is
normal iff gross-return is lognormal
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>win.graph(width=4.875,height=2.5,pointsize=8)>hist(logret,breaks=30,freq=FALSE,main='RIL logret')
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Shape of a Distribution
Measure of asymmetry = skewness,
Measure of Peaked-ness = kurtosis
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Skewness
Mean = Median =ModeMean < Median < Mode Mode < Median < Mean
Right-SkewedLeft-Skewed Symmetric
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Distribution with a positive skewness has along right tail
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Distribution with a negative skewness has along left tail
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Positive kurtosis (or leptokurtosis, shown by solid line)indicates that observations cluster more than those in thenormal distribution and negative kurtosis (or platykurtosis,
shown by dotted line) indicates observations cluster less.
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Higher kurtosis implies more of the variance is because of rare extremedeviations, as opposed to frequent modestly-sized deviations
UNIFORM dist
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Skewness & Kurtosis of a r.v. X
S(x) = skewness
K(x) 3 is called the excess kurtosis [K(x) = 3
for a normal Distribution]. Positive excesskurtosis means heavier tails.
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Estimates of skewness and kurtosis
Let {x1, . . . , xT} be a random sample of Xwith Tobs. Define sample mean, standard deviation,skewness and kurtosis by
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Test of skewness
Under the normality assumption, sampleskewnessand (sample kurtosis 3) aredistributed asymptotically as normal with zero
mean and variances 6/Tand 24/T . Given an asset return series {r1, . . . , rT}, to test
skewness of the returns, we consider the nullhypothesis H0: S(r)=0
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Test of Kurtosis
Under the normality assumption, (samplekurtosis 3) is distributed asymptotically asnormal with zero mean and variances 24/T .
Given an asset return series {r1, . . . , rT}, to testexcess kurtosis of the returns, we consider thenull hypothesis H0: K(r)-3=0
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Jarque Bera Test of Normality
Jarque and Bera (1987) combine two prior testsand use the test statistic
JBis asymptotically distributed as a chi-squared
random variable with 2 degrees of freedom, to testfor normality of rt .
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Modified Jarque-Bera (JB) test statistic is anothermeasure of departure from normality, based on thesample kurtosis and skewness. The test statistic is
defined as (taking into account k number of estimatedcoefficients used to create series for which normality isbeing tested)
where S= skewness, K= kurtosis, T = number ofobservations.
The JB statistic has an asymptotic 2
2-distribution
+
=
4
)3(
6
)(
22 KS
kTJBModified
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> x jarque.bera.test(x)
Jarque Bera Testdata: xX-squared = 0.2494, df = 2, p-value = 0.8828
> x jarque.bera.test(x)
Jarque Bera Testdata: xX-squared = 7.4205, df = 2, p-value = 0.02447
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Shapiro-Wilks test
Shapiro-Wilks test tests the null hypothesis that
a sample x1, ..., xncame from a normallydistributed population.
Test statistic W is square of correlation coeff fordata (x(1),z(1)), (x(2),z(2)), , (x(n),z(n)) where
x(i) = i-th ordered value from the sample andz(i) = i-th ordered value from the sample taken from
N(0,1) [ e.g., z(i) = -1( (i0.5)/n ) ]
If W-value is close to 1 then data support the null
hypothesis of normality. Why ?
How close is close ?
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If W-value is close to 1 then data support thenull hypothesis of normality. Why ?
If X~N(, 2) and Z~N(0,1) then X(i) + Z(i)
Hence X(i) and Z(i) are supposed to be highlycorrelated if the null hypothesis is true.
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References
1. Shapiro, S. S. and Wilk, M. B. (1965). "Ananalysis of variance test for normality (completesamples)", Biometrika, 52, 3 and 4, pages 591-611.
2. Bera, Anil K., Carlos M. Jarque (1980)."Efficient tests for normality, homoscedasticityand serial independence of regressionresiduals". Economics Letters6 (3): 255259.
3. Bera, Anil K., Carlos M. Jarque (1981)."Efficient tests for normality, homoscedasticityand serial independence of regressionresiduals: Monte Carlo evidence". EconomicsLetters7 (4): 313318
http://en.wikipedia.org/w/index.php?title=Carlos_Jarque&action=edithttp://en.wikipedia.org/w/index.php?title=Carlos_Jarque&action=edithttp://en.wikipedia.org/w/index.php?title=Carlos_Jarque&action=edithttp://en.wikipedia.org/w/index.php?title=Carlos_Jarque&action=edit -
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Skewness test for IBM Return
t=(-0.0775/0.023966) = -3.2337
http://faculty.chicagobooth.edu/ruey.tsay/teaching/fts2/ (Tsay Book data)
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RIL log(price) 23 Aug 2004 -17Aug 2009
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RIL logret 23 Aug 2004 -17Aug 2009
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RIL log return data
> jarque.bera.test(logret)
Jarque Bera Test
data: logret
X-squared = 11897.81, df = 2, p-value < 2.2e-16
> # Another normality test method> shapiro.test(na.omit(logret)) # Reported on Cryer-Chan p.283
Shapiro-Wilk normality test
data: na.omit(logret)W = 0.8969, p-value < 2.2e-16
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Normal Q-Q Plot
A Q-Q plot ("Q" stands for quantile) is a graphical
method for comparing two distributions by plotting theirquantiles against each other. If the two distributionsbeing compared are similar, the points in the Q-Q plotwill approximately lie on the line y= x.
> win.graph(width=4.875,height=3,pointsize=8)> qqnorm(logret,
ylab='logret data for RIL')> qqline(logret)
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Stylized Properties of Financial
Time Series(Tsay, p.19, Sec 3.1 (p.98)
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What is a Stylized Fact?
Empirical studies on financial time series showsseemingly random variations of asset prices do
share some quite nontrivial statistical properties,across a wide range of instruments, markets andtime periods.
Such properties are called stylized empiricalfacts.
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Some Stylized Statistical Propertiesof Asset Returns
Mean of daily return series usually close to zero
Skewness of daily return is not a serious problem
Heavy tails (e.g., daily returns tend to have highexcess kurtosis)
Absence of autocorrelations in many assetreturns
Gain/loss asymmetry (one observes large draw-downs in stock prices but not equally largeupward movements)
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RELIND log return (in %) over 20 Aug 2004 to 17 Aug
2009
-30.0
-25.0
-20.0
-15.0-10.0
-5.0
0.0
5.0
10.015.0
20.0
8/23/2004
2/23/2005
8/23/2005
2/23/2006
8/23/2006
2/23/2007
8/23/2007
2/23/2008
8/23/2008
2/23/2009
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Summary
> summary(logret)Min. 1st Qu. Median Mean 3rd Qu. Max.
-29.0900 -1.1730 0.1943 0.1155 1.4750 19.1400> mean(logret)
[1] 0.1154729> var(logret)[1] 7.529715> sd(logret)[1] 2.744033> skewness(logret)[1] -1.122057> kurtosis(logret)[1] 15.00814
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Histogram
0
50
100
150
200
250
300
350
400
450
-29.5
-24.7
-19.9
-15.0
-10.2
-5.4
-0.6 4.
39.
113
.918
.7
Bin
Frequency
Mean 0.11547
Standard Error 0.07793
Median 0.19432Standard Devia 2.74403
Kurtosis 15.0736
Skewness -1.1234
Range 48.2329
Minimum -29.094Maximum 19.1391
Sum 143.186
Count 1240
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>win.graph(width=4.875,height=2.5,pointsize=8)>hist(logret,breaks=30,freq=FALSE,main='RIL logret')
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> stem(logret)The decimal point is at the |
-28 | 1-26 |
-24 |-22 |-20 |-18 | 0-16 |-14 |-12 | 74-10 |-8 | 6400-6 | 99877762088888644330-4 | 854433211100098876544433222211110-2 | 99999998887776666655554444444322222221100000099998888888877777776666+55-0 | 99999999998888888888888877777777777777666666666655555555555555555544+2930 | 00000000000000011111111111111111111111111122222222222222222222222233+3602 | 00000000000000111111111111111112222222333333333334444444444555555555+85
4 | 0001111111222333445567777778889990000122456796 | 012234589033578 | 0161
10 | 112 | 914 |16 |18 | 1
S
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Stylized Properties of VolatilitySec 3.1 (p.98)
Volatility means (conditional) variance of log-return of an underlying asset
First, there exist volatility clusters (i.e., volatility
may be high for certain time periods and low forother periods).
Second, volatility evolves over time in acontinuous manner (i.e., volatility jumps are rare).
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Stylized Properties on Volatility (contd)
Third, volatility does not diverge (i.e., volatilityvaries within some fixed range). Statisticallyspeaking, volatility is often stationary.
Fourth, volatility seems to react differently to abig price increase or a big price drop, referred toas the leverageeffect. EGARCH model was developed to capture the
asymmetry in volatility induced by big positive and
negative asset returns.