MSF AFTS KP 2014 Lectures 2 Stud

8/10/2019 MSF AFTS KP 2014 Lectures 2 Stud

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Krzysztof Piontek

Department of Financial Investments and Risk Management

1

Analysis of Financial Time Series

statistics - revision


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Course Requirements

The basic knowledge of:

financial markets,

financial instruments,

risk management,portfolio management,

mathematics,

statistics,

probability calculus,

econometrics.


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Two main tasks in financial data modeling:

- Modeling dynamics (econometrics)

- Modeling statistical distributions (statistics)

the order of the observations over the time


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Analysis of financial time seriesis main part of the

discipline called financial econometrics

Two main objectives of financial econometrics:

1) Verification of different models developed in theory of finance

(financial economics) by using financial time series models

2) Analysis of financial data to identify main characteristics

(for example risk characteristics)


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Time series:

A sequence of random variables measuring certain quantity of

interest over time

Financial Time Series:

collection of a financial measurement over time.

Example: log return r

t

Data: {r

1

, r

2

, ..., r

T

} (t data pints)

Futures prices or returns are random variables since we do not

know their values until we observe theirs.

A random variable can be thought of as an unknown value that may

change every time it is inspected.


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Name, Date, Open, High, Low, Close, Volume

KGHM, 20090904, 83.45, 85.70, 83.00, 83.00, 661561

KGHM, 20090907, 84.50, 85.95, 84.40, 85.95, 389379

KGHM, 20090908, 87.00, 89.60, 87.00, 88.90, 1251472

KGHM, 20090909, 87.70, 89.30, 87.70, 88.90, 701582

KGHM, 20090910, 89.90, 90.40, 86.15, 88.00, 970480

KGHM, 20090911, 88.00, 88.00, 85.00, 85.20, 865165KGHM, 20090914, 84.00, 84.65, 82.50, 83.60, 401915

KGHM, 20090915, 84.60, 85.40, 82.90, 84.00, 495518

KGHM, 20090916, 85.00, 86.95, 84.95, 86.95, 722730

KGHM, 20090917, 88.30, 88.85, 86.10, 87.00, 454126

KGHM, 20090918, 86.00, 86.90, 84.20, 84.60, 1359126

KGHM, 20090921, 85.10, 85.80, 84.00, 84.00, 448032KGHM, 20090922, 85.50, 90.50, 85.50, 90.50, 2367387

KGHM, 20090923, 90.50, 92.40, 89.40, 91.00, 1273626

KGHM, 20090924, 89.50, 92.25, 88.00, 88.00, 950405

KGHM, 20090925, 88.00, 88.45, 85.20, 86.50, 1388713


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Main index of Warsaw Stock Exchange - WIG

A graph of daily levels


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Characteristics of financial time series

usually we think about

returns


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Rates of return

(clusteringof variance)

Quantile-quantile plot

(normal quantiles)

Autocorrelation of squared returns

(lags)

Autocorrelation of returns

(lags)


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1 0,2264b

Skewness: 30.10.9519.08.04

2200 observations

Histogram of COMPUTERLAND rates of return vs. normal density

Compute the intervals x + s, x + 2s, x + 3s and compare the percentage of

data in these intervals to the Empirical Rule (68%, 95%, 99.7%)


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Statistics

short revision

Descriptive Statistics

Provides numerical and

graphic procedures to

summarize theinformation of the data in

a clear and

understandable way

Inferential Statistics Provides

procedures to

draw inferences

about a populationfrom a sample


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Numerical Data Properties

Central Tendency

(Location)

Variation

(Dispersion)

Shape


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Bell-shaped

Bimodal

(Multimodal)

Uniform

(Unimodal)

Right-skewed Left-skewed

Truncated

a histogramis a graphical display of

tabulated frequencies, shown as bars.


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Percentiles and Quartiles

The pth percentile is the value of a givendistribution such that p% of the

distribution is less than or equal to that

value.


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Quantile

Some quantiles have special names:

The 2-quantile is called the median

The 4-quantiles are called quartilesThe 10-quantiles are called deciles

The 100-quantiles are called percentiles

Quantiles provide information about the shape of data as well as its

location and spread.


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0.00

0.10

0.20

0.30

0.40

0.50

0.60

-5 -4 -3 -2 -1 0 1 2 3 4 5

10th

percentile=-1.2816

10 percent under curve

(shaded red)


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a Q-Q plot( Q stands for quantile) is a probability plot, a kind

of graphical method for comparing two probability

distributions, by plotting their quantiles against each other.

If the data sets agree or the observed set matches the

theoretical, the plot will be on a straight line.

If the plot is not on a straight line, then the model is a poor fit.

Quantile-quantile plots are used to determine whether two samples come from the same

distribution family. They are scatter plots of quantiles computed from each sample, with a

line drawn between the first and third quartiles. If the data falls near the line, it is

reasonable to assume that the two samples come from the same distribution. The method

is robust with respect to changes in the location and scale of either distribution.


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Normal quantile plot

-2 -1 0 1 2

-2

-1

0

1

2

Normal q-q plot

Theoretical Quantiles

SampleQuantiles

q-q plot 100 sample

observations from a normaldistribution with mean 0 andstandard deviation 1


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Normal quantile plot

-2 -1 0 1 2

40

45

50

55

Normal q-q plot of Height of our Sample Data

Theoretical Quantiles

SampleQuantiles


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Descriptive statistics

Types of Descriptive Statistics. 1. Measures of Central Tendency

2. Measures of Dispersion of Variability 3. Measures of distribution Shape.

4. Quantiles


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Measures of Central Tendency

Mean:

Sum of all measurements in the data divided by

the number of measurements.

the AVERAGE (or the EXPECTED VALUE)

Median (2nd quartile):

A number such that at most half of the

measurements are below it and at most half of themeasurements are above it.

Mode:

The most frequent measurement in the data.


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A car went a certain distance at a speed 60 kilometers perhour) and then the same distance again at a speed 40

kilometers per hour.

What was the car's average speed during the whole journey?


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A car went a certain distance at a speed 60 kilometers per

hour) and then the same distance again at a speed 40

kilometers per hour.

What was the car's average speed during the whole journey?

average speed


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A car travels for a certain amount of timeat a speed 60

km/hand then the same amount of time at a speed

40 km/h


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A car travels for a certain amount of timeat a speed 60

km/hand then the same amount of time at a speed

40 km/h

average speed..


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Measures of dispersion or variability

Measures how spread out around the center are

the data.

The simplest measure of variability is the RANGE.

This is simply the maximum value minus the

minimum value.

INTER-QUARTILE RANGE (IQR) which is defined as

the value at 75% minus the value at 25% of the

distribution. Thus the central 50 % of the

observations fall between these two values.

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Measures of dispersion or variability.

The most common measures of variability

are the STANDARD DEVIATIONand the

VARIANCE. The variance is the standard

deviation squared, or the standarddeviation is the square root of the variance.

2 2

1

1

( ) ( )1

N

iiS x x xN

2( ) ( )S x S x

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Measures of Shape

The two most common measures of shape are

SKEWNESS and KURTOSIS.

Skewness is a measure of the lack of symmetry in a

distribution, or whether the distribution is skewed

to the left or the right.

Positive skewness: Values clustered toward lower

range with a long tail extending to upper ranges.

Negative skewness: Values clustered toward

upper range with long tail extending to lower

ranges.

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Shape

Right-SkewedLeft-Skewed Symmetric

Mean = MedianMean Median Median Mean

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Skewness

Measures of asymmetry of data

Positive or right skewed: Longer right tail

Negative or left skewed: Longer left tail

2/3

1

2

1

3

21

)(

)(

Skewness

Then,ns.observatiobe,...,Let

n

i

i

n

i

i

n

xx

xxn

nxxx

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1 2

4

1

2

2

1

Let , ,... be observations. Then,

( )

Kurtosis

(

3

)

n

n

i

i

n

i

i

x x x n

n x x

x x

Kurtosismeasures the thickness of the tails.Higher values of kurtosis indicate more extreme

values or heavier tails. Negative kurtosis is thinnertails. Heavier or thinner than the normal curve.

Why subtract 3? Because a normal curve has a kurtosis of 3: if wesubtract three then it is zero and thus kurtosis is expressed subject tocomparisons with the normal curve.

excess

kurtosis

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leptokurtic

platykurtic

mesokurtic

Kurtosis > 3

excesskurtosis >0fat tails (typical feature)

Kurtosis = 3excesskurtosis = 0

Kurtosis < 3

excesskurtosis < 0

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Expected value

risk

(measured as volatility)

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Normal Distribution

2

2

1( )

21

( ) ,2

x

f x e x

A density curvedescribes the overall pattern of a distribution.The total area under the curve is always 1.

The normaldistribution is symmetric, single-peaked and bell-shaped.

Mean, Median, and mode are same for a normal distribution

A normal distribution can be described if we know their mean andstandard deviation.(If we know and , we know every thing about thenormal distribution.)

The probability density function of a normal variable with mean and

standard deviation can be expressed as,

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Normal Distribution

.

x

z

2

21

( ) ,2

z

f z e z

Standardizing and z-Scores

If x is an observation from a distribution that has mean and

standard deviation , the standardized valueof x is

A standardized value is often called a z-score. If x is normal distribution

with mean and standard deviation , then z is a standard normal

variable with mean 0 and standard deviation 1.

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Normal Distribution

Approximately 68% of the

measurements lie in the

interval to +

Approximately 95% of the


interval 2to + 2

Approximately 99.7% of the


interval 3to + 3

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Raw (crude) moment (moment about zero)

1

1x

N

k

k i

i

mN

k

km E X

For a sample of N observations the sample moment is

Moments: values (statistics, quantities) used to

characterize the probability distributions of

random variables.

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Central moment (moment about its mean)

11

1 x

N

kk i

i

mN

k

k E X E X

Sample moment

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1

2

2

31 3 2

2

42 2

2

m

mean

variance

skewness

kurtosis

Raw (crude) moment (moment about zero)

Central moment (moment about its mean)

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Outliers

Outliers are cases that have data values very

different from the data values for the majority of

cases in the data set.

Outliers are important because they can change the

results of our data analysis.

Whether we include or exclude outliers from a data

analysis depends on the reason why the case is an

outlier and the purpose of the analysis.

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The higher moment (raw or central) the more

senitive to outliers.

outlier

1

observation

1000 observations

skewness not zero but e.g. 0.4

kurtosis not 3 but e.g. 4.5

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Expected value

risk

(measured as volatility)

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The simplest model

The world is a big gaming machine (lottery).

Each day the rate of return is drown from the same

distribution which is constant over time.

(all moments of the daily return distributions areconstant over time)

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Discrete time model (simple rates of return)

tt

t

tt

t z

P

PPr

1

1

),N(~ 2tr

)1,0N(~t

z

The return is decomposed into two parts as

rt= predictable part + not predic. part

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VALUE AT RISK

Rq

np. 5 %,

1 %

R

(R)

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VALUE AT RISK

an approach to VaR estimation under the assumption

of NORMAL return distribution:

mean of the distribution of returns,

standard deviation of returns

constant dependent on the tolerance level,

e.g.:

if q = 0.05, then c = -1.65

if q = 0.01, then c = -2.33

, - constant in time

cRq

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0 t

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The Black-Scholes model of the market for an equity

makes the following explicit assumptions:

It is possible to borrow and lend cash at a known

constant risk-free interest rate.

The price follows a geometric Brownian motion with

constant drift and volatility.

There are no transaction costs.

The stock does not pay a dividend (see below for

extensions to handle dividend payments).

All securities are perfectly divisible (i.e.it is possibleto buy any fraction of a share).

There are no restrictions on short selling.

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1 2

( ) ( )r T

c S d X e N d N2

1

2

ln r

dX

TS

T

2

2

2

ln S

r TX

d

T

N(x) denotes the standard normal cumulative distribution function

the Black-Scholes formula

MSF AFTS KP 2014 Lectures 2 Stud

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