Session on Stylized Properties

7/31/2019 Session on Stylized Properties

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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.

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Session on Stylized Properties