part 1 financial variables

8/11/2019 Part 1 Financial Variables

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ETF5930 Financial econometrics

Part 1: Some financial variables, their time series and distributions

As an expert in Finance, you are interested in quantities such as:

The value of an equity (share, unit etc.)

The return on an equity over a particular time period

The dividend paid on a share

Dividend yields

The value of an index: stock market indices such as the Dow Jones or the S&P 500

Interest rates

Exchange rates, etc.

In Financial Econometrics, we think of these quantities as statistical variables, and we study the

distributions of these variables, and test for evidence of relationships between them. We consider

their behaviour over time, and also the cross-sectional distribution.

The distributions

Cross-sectional distributions

We can study the cross-sectional distributions: for example, consider the variable return on a share

and consider all companies included in the S&P 500, and the return on a share in each of these

companies between close on 3 February 2014 and 4 February 2014. So the variable return on a

share takes different values on different companiesobserved at the same time. Here we can graph

a histogram of the returns and identify whether a particular company is for example in the top 10%

of returns.

(Data fromhttp://www.asxallordinaries.com/)

Linear regression can be used to study the relationship between two or more cross-sectional

variables. However, in finance we are frequently concerned with variables that vary across time.

0

50

100

150

200

Numberof

companies

Distribution of Proportional change in price from COB 3 Feb 2014 to COB 4 Feb 2014,

All Ordinaries
http://www.asxallordinaries.com/http://www.asxallordinaries.com/http://www.asxallordinaries.com/http://www.asxallordinaries.com/


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Variation across time

We can study for a particular company how the price of a share for example varies across time.

Then the price of a share is regarded as a random variable. It takes different values at different

points in time and might be denoted by tY where truns from 1 to T.(Here we may be considering

the values tick by tick, hourly, daily, annually.) We can graph the distribution of prices, throwing

away the information about what time each price occurred. Or we can graph the price of the share

against the time variable. We can also study the relationship between two or more variables that

vary across time.

Dynamical models

And eventually we can consider dynamical modelsof the variation across time. We will start

discussing this in about Week 6 and much of the second half of the semester will be devoted to

dynamic models.

Index numbers

It is often of interest to compare the behaviour over time of a single equity with the behaviour of the

market as a whole. Index numbers measuring the overall movement of the market are commonly

used and can be tracked and studied in the same way as a single equity.

The following graphs were obtained from data on the yahoo website

(http://au.finance.yahoo.com/q/hp?s=%5EAORD etc.)

Note that the two mining shares have fairly similar behaviour over time.

Question: Can you tell from this graph which of the mining shares is performing better? Why or why

not?

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3040

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60

70

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90

100

Sha

reprice($)

Time series for share prices of several

Australian companies

NAB Fairfax RIO BHP
http://au.finance.yahoo.com/q/hp?s=%5EAORDhttp://au.finance.yahoo.com/q/hp?s=%5EAORDhttp://au.finance.yahoo.com/q/hp?s=%5EAORDhttp://au.finance.yahoo.com/q/hp?s=%5EAORD


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Next we graph the All Ordinaries index, showing the behaviour of the market overall.

Notice that all the example shares exhibit some of the variation shown by the index, but sometimes

other things are going on as well. For example compare Fairfax with the index:

In weeks 3-5 we will discuss the relationship of individual share prices to the market.

The next graph is of an American index: the time series for the American S&P 500 from 1957 to the

present:

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2000

3000

4000

5000

6000

7000

8000

Closing value of the ASX

All Ordinaries index, 1988 - 2013

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3

4

5

Shareprice($)

Fairfax


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Time series graphs of equities and indexes are often given in terms of the log(Price) or log(index)instead of the price itself. For example if we graph the log of the S&P 500 we get:

An increase in the logarithm by a fixed amount at any point corresponds to the sameproportionate

increase in the index. For example an increase of 50% in the index corresponds to an increase of

approximately 0.4 in the logarithm of the index. And a fall of one third corresponds to a decrease of

approximately 0.4. This means that catastrophic falls for example that happened a long time ago are

not made to look negligible relative to recent events just because the base value was lower then.

As another example, we show a time series of the Australian All Ordinaries index, and of the log of

the index.

0

200

400

600

800

1000

1200

1400

1600

1800

2000S&P 500 index 1957 - 2013

3.0

4.0

5.0

6.0

7.0

8.0

2/01/1957 11/09/1970 20/05/1984 27/01/1998 6/10/2011

ln(S&P500 index)


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0

1000

2000

3000

4000

5000

6000

7000

8000

Closing value of the ASX All Ordinaries index,

1984 - 2013

6

6.5

7

7.5

8

8.5

9

ln(ASX All Ordinaries index), daily closing value

1984 - 2013


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Prices, returns and dividends

Returns

It is already clear from the examples above that the actual value of the shares can be widely

different. (Fairfax shares are around a few dollars while BHP shares are around a few tens of dollars.)

These kinds of differences are not of importance to us. We are more interested in the rate at which

they are rising or falling, and again the logarithm will be relevant.

Consider the daily rise and fall of an equity. The simple return over one unit in time is defined to be

(Orexpressed as a percentage - )

Solving fort

P :

andt

t

tR

P

P

11

We usually consider continuously compounded returnsinstead of simple returns.

The continuous rate of return can be defined as

1

lnt

t

t

P

Pr so it is related to

tR by

The continuously compounded returnis also referred to as the log return.

(The definition of the continuously compounded rate of return is analogous to compound interest in

the limit as compounding happens at smaller and smaller intervals: details in the following box:)

1

1

t tt

t

P PR

P

1

1

100%t ttt

P PR

P

1(1 )t t tP R P

ln(1 )t tr R

Continuous compounding:

Recall the compounding of interest: If the annual interest rate is specified as r, or 100 %r , and

it is to be compounded m times per year, then after tyears, the principle 0P has grown to

0 1

mt

t

rP P

m

and as we approach continuous compounding, that is, let m go to infinity,

0 0lim 1

mt

rt

tm

rP P P e

m

where1

lim(1 ) 2.718mm

em

And of course,

( 1)

( 1)

1 0 0lim 1

m t

r t

tm

r

P P P em

so that

1

rt

t

Pe

P

, or1

ln t

t

Pr

P

So far, we are considering a pre-determined interest rate, but we wish to consider returns that

vary with time. Ift

P is the price of an asset at time t, we can definethe continuously

compounded return at time tto be:

1

lnt

t

t

P

Pr

Since the simple return is1

1

t t

t

t

P PRP

, we have tt

tR

PP

11

so that ln(1 )t tr R .


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Additivity of returns

Continuously compounded returns are additive across time.

Recall that: ln( ) ln( ) ln( )x y xy so that ln( ) ln( ) ln( ) ln( )A B A B AB C B C C

It follows that weekly continuous compounded returns can be obtained by adding daily returns:

Simple returns are additive across a portfolio. (this is a weighted sum.)

where ptR is portfolio return at time t, itR is return of individual share at time t, iw is the weight of

the ithequity in the portfolio.

Returns and pricesIf you know the time series of prices, then you know the time series of (log) returns since

1

lnt

t

t

P

Pr .

If you know the initial price and the returns at each time point, then you know the time series of

prices: 1tr

t tP e P .

Since there is a one-to-one relationship between returns and prices, we can say that prices and

returns explain the same occurrence differently: are different representations of the same data.

Different representations of the same data are common in finance. Each representation has

different statistical properties and reveals different features of the underlying phenomena. The

different representations require different approaches to the statistical analysis.

Consider first what we learn from the log returns. Consider a time series graph of the daily log

returns:

-0.250

-0.200

-0.150

-0.100

-0.050

0.000

0.050

0.100

0.150 log Returns, S&P500

3 5 51 2 4

0 1 2 3 4 0

ln ln ln ln ln lnp p pp p p

p p p p p p

pt i it

i

R w R


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The log returns appear to fluctuate about a mean. That mean value appears to be close to zero.

There are surprisingly manybig positive and negative values. For example, Black Monday October

19 1987. (In what sense surprising? See below.)

Also volatility clustering occurs: there are periods when large changes are followed by more large

variation (positive or negative), and when small changes are followed by small changes. Thus the

volatility is time-varying, and there is clustering of volatility levels.

Returns generally do not trend up or down over time. They form a stationary series. (Stationarity

will be defined more formally later.) So we can investigate the distribution of returns even when the

observations have been made over time.

Thus the returns data can also be graphed in a histogram rather than a time series. Here we just look

at the values, not when they occurred. So the following graph is rather similar in nature to the cross

sectional graph, even though the values did occur at different points in time.

The red line shows the normal distribution with the same mean and standard deviation as this data.

Relative to the normal distribution, the returns data has a sharper central peak (more very small

values) and heavier tails (more extreme values, both positive and negative). This is the sense in

which there are surprisingly many big fluctuations.

Prices vs returns

Log returns have the following advantages.

They are unit free

They show the short-term variation well

They are statistically quite easy to handle.

However,

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-.10 -.08 -.06 -.04 -.02 .00 .02 .04 .06 .08 .10

Normal

Density

Histogram of daily log returns for S&P500, 1957 - 2013


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1 2 3

2 3

1 2 31 (1 ) (1 )

t t tt t

t t t

D D DP

R R R

E

wherePt = price at time t, Et= expected value at time t, t nD = dividend at time t + n,Rt+n=discount

rate at time tfor dividend to be paid at time t + n.

If the expected dividend is constant and equal tot

D , and if the discount rateRis assumed constant,

(using formula for sum of geometric series)

Rearranging,t

t

DR

P . Here, remember,

tP is the present value, and

tD is the dividend level, which

we have assumed constant for the purposes of the calculation.

Of course over the long term dividends are not constant, and investors are interested in the current

level of dividends relative to price. Therefore dividend yieldat time tis definedast

t

t

DYIELD

P .

(In the newspaper, what is quoted is Dividend for previous yearcurrent price

.)

Taking logarithms and rearranging, ln ln lnt t t

P D YIELD .

Here is a graph of the dividend yield for the shares of the S&P500, monthly since 1870:

Notice that there is considerable variability, some of which can be associated with well-known

events. However, it appears to wander randomly around the level 0.05.

0

0.05

0.1

0.15

1850 1900 1950 2000 2050

S&P Dividend Yield, monthly since 1870

2 3

2

1

(1 )

...1 1 1

1 11

(1 ) (1 ) (1 )

1

(1 ) 1

t t t

t

t

t

R

t

D D DP

R R R

D

R R R

D

R

D

R


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Numerical summary statistics

An analysis always starts with a graphical investigation, because the quickest way to get a feel for

the qualitative properties of a data set is by producing relevant graphical plots. However, numerical

summary statistics are needed in order to make quantitative comparisons between series in a more

systematic analysis.

The central locationof a distribution is measured numerically as mean, median or mode.

The variabilityof a distribution is measured numerically using the variance, standard

deviation or interquartile range.

The asymmetryof a distribution is measured numerically using skewness.

The heaviness of tails is measured numerically as kurtosis.

The relatednessof the movement of two time series over time is measured numerically as

covariance or correlation.

Dependence on past values and periodic tendenciesof a single time series are measured

numerically by autocovariance and autocorrelation.

The quantities calculated from the data are sample statistics. They are estimates of the population

parameters.

Sample statistic Name Population parameter

Mean

Variance

Skewness

Kurtosis

Covariance

Correlation

Autocovariance

Autocorrelation

2=( )X

X E

= ( )X E

3 3( )

XX E

1

1 T

t

t

X XT

2 2

1

1( )

1

T

X t

t

s X XT

3

1

1

1

T

t

t X

X XS

T s

4

1

1

1

T

t

t X

X XK

T s

4 4( )XX E

1

1( )( )

1

T

XY t t

t

s X X Y YT

cov( , )

[( )( )]X Y

X Y

X Y

E

cov( , )

X Y

X Y

XY

X Y

s

s s

1

1( ) ( )( )

T

t t k

t k

k X X X X T k

( )

(0)

k

( )

(0)

k

( ) [( )( )]t X t k X k x x E


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Mean, variance and covariance

The mean return is calculated from sample data as1

1 T

t

t

r rT

.

In the absence of any other information about returns, the expected return on any given day is

estimated to be r . This is the best estimate of the expected value of the return [ ]r tr E .

However, the returns vary from day to day, and the magnitude of this variation is indicated by

calculating the variance of the returns. The population value of the variance is2 2( )tr E .

The best estimate of the variance based on the data from t= 1to Tis 2 2

1

1( )

1

T

r t

t

s r rT

.

Note that the mean of a multiple of r

[ ] [ ]t twr w r E E ,

which can also be written wr rw .

But since the variance involves the square of r, we find2 2 2

wr rw . For example if you double the

whole return series, then you double the expected value, but you multiply the variance by 4.

Since the variance is a measure of the unpredictable variation in the returns, it is a measure of risk.

Now consider two assets with return series r1andr2. The covariance between the two series is given

by 1 2 1, 1 2, 2cov( , ) [( )( )]t tr r r r E and so 1 1 2 2 1 2 1 2cov( , ) cov( , )w r w r w w r r .

Also, 2cov( , ) 0a r where ais a constant.

Portfolio risk management: Mean-variance model of relationship between risk and return

Variance is a measure of riskin the sense that if a stock is purchased because of its expected future

returns, but its variance is also high, this means there is a risk that its returns will differ from their

expected value. It is well-known that this risk can be diminished by holding a diversified portfolio. So

now we consider the variance of a portfolio.

An investor usually holds a portfolio of a variety of assets. We consider how to measure the variance

of the whole portfolio. Consider the simplest example where the portfolio consists of just two assets.

Suppose the portfolio consists of two investments, investments 1 and 2 :

Mean: ][ ,11 trE ][ ,22 trE

Variance: )][( 21,121 trE )][(

22,2

22 trE 140

Covariance: )])([( 2,21,11,22,1 tt rrE

Current weight in value of portfolio:1w 2w

(The weights sum to 1. )


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The return rpon the portfolio is the weighted sum of the returns on each of the two assets:

, 1 1, 2 2.p t t tr w r w r and the expected value of the return is 1 1 2 2p w w

The variance 2p of the portfolio is given by:

2

1 1, 2 2,

1 1, 2 2, 1 1, 2 2,

2 2

1 1, 2 2, 1 2 1, 2,

2 2 2 2

1 1 2 2 1 2 1,2

var( )

var( ) var( ) 2cov( , )

var( ) var( ) 2 cov( , )

2

p t t

t t t t

t t t t

w r w r

w r w r w r w r

w r w r w w r r

w w w w

(Note: 1, 2, 2, 1,cov( , ) cov( , )t t t t r r r r in other words 1,2 2,1 .)

We can choose the weights so as to minimise the portfolio variance.

By straightforward calculus, (not required) it can be shown that the values of1w and 2w that

minimise the portfolio variance 2

p are:

2 2

2 12 1 121 22 2 2 2

1 2 1,2 1 2 1,2

;2 2

w w

Example 1

Suppose the two assets have a variance-covariance matrix:

2

1 1,2

22,1 2

0.011332 0.002380

0.002380 0.005759

Calculate the variance-minimising weights of the two assets in a portfolio.

2

2 1,2

1 2 2

1 2 1,2

0.005759 0.0023800.274

2 0.011332 0.005759 2 0.002380w

2 11 0.726w w

Note that more weight is given to the asset with the lower variance.

Example 2

If the covariance1,2

0 then2 2

2 11 22 2 2 2

1 2 1 2

,w w

so the weight of one corresponds to

the proportion of the variance of the other.

Skewness and kurtosis and (non-)normality

Approximating a distribution by the normal distribution makes analysis easier. Assumption of normal

returns is widely used in portfolio allocation models and value-at-risk calculations and in pricing

options. But financial series are often non-normal.


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Deviation from the normal distribution: Skewness

Sample skewness:

If there is a long tail to the left, Swill be large and negative andXis said to be negatively skewed, orskewed to the left.

If there is a long tail to the right, Swill be large and positive, andXis said to be positively skewed, or

skewed to the right.

For the distribution of S&P500 daily returns 1950-2011, S = 1.0479, so the distribution is

somewhat skewed to the left.

Recall that the median is the value such that half the values are greater than it and half are less.

Note that if the mean is greater than the median, then the distribution tends to be skewed to the

right. The intuition behind this is that very large values in the right tail will increase the mean

strongly, but the median cannot tell the difference between a value that is just above the middle

value, and one that is extremely large. So you can think of the large values as pulling the mean to the

right, but not affecting the median. A similar discussion holds for negative or left skew.

Deviation from the normal distribution: kurtosis

A symmetrical distribution can deviate from the normal distribution by being more or less peaked

than the normal distribution with the same variance, and/or by having heavier or lighter tails. We

have seen the case of more peaked and heavier tails in the distribution of daily log returns.

0

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20

30

40

50

60

70

80

-.10 -.08 -.06 -.04 -.02 .00 .02 .04 .06 .08 .10

Normal

Density

Histogram of daily log returns for S&P500, 1957 - 2013

3

1

1

1

T

t

t X

X XS

T s


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Sample kurtosis is . If there are many values far from the mean,K

will be large.

If Kis large, the distribution is said to be leptokurtic; if Kis small the distribution is said to be

platykurtic.

For the normal distribution, Kurtosis = 3; for the distribution of log returns for S&P500, Kurtosis =

32.0597.

Evidence of non-normality: Quantile-quantile graph:

4

1

1

1

T

t

t X

X XK

T s


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To explain a quantile-quantile plot, consider as an example the point on the plot corresponding to

the 99th

percentile. The 99th

percentile of the standard normal distribution is 2.327, meaning that

there is a probability of 1% that a standard normal variable is greater than 2.327. The coordinates of

the little blue circle corresponding to the 90th

percentile will be (2.327,x) wherexis the value such

that 1% of the sample log returns are greater thanxand 99% are belowx.

The line represents the expected quantiles if the distribution were normal. If the daily log returns

had a normal distribution2( , )rN r s with their actual mean and variance, then a graph of the

percentiles of the returns against the percentiles of the standard normal distribution would be the

straight line shown. Since the quantile points in the left tail are below the line, this means there are

more observations than expected for the normal distribution at low values. And at the upper end,

the fact that the quantile points are above the line means that there are more large observations

than would be expected for a normal distribution (e.g. the 99th

percentile occurs at a higher value so

the 1% are further out). We conclude that the log returns have heavy tails relative to a normal

distribution.

The Jarque-Bera testuses the values of S and K to determine whether there is sufficient evidence to

conclude that a distribution is not normal. The Jarque-Bera test statistic is

22 ( 3)

6 4

T KJB S

so that if the distribution is normal,JBshould be 0.

The hypotheses are:

0

1

: The variable is normally distributed (so 0, 3)

: The variable is not normally distributed

H S K

H

If the null hypothesis is true, the test statisticJBhas a chi square distribution with 2 degrees of

freedom,2

2~ JB . Note that the critical value of2

2 cutting off an upper tail of area 1% is 9.21.

Therefore if we wish to perform the hypothesis test at the 1% level of significance, we would reject

the null hypothesis if the sample value of JB was greater than 9.21.


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Example: Consider the daily log returns of the S&P500 index. A histogram and descriptive stats of

this series was obtained using EViews.

Question:

(a) Read off the value of skewness and kurtosis from the EViews output, and hence calculate

the sample value of the statistic JB. Compare your answer with the value given by EViews.

(b)

Complete the test.

(c) How do you interpret the line Probability 0.000000?

Covariance and correlation

Investors wish to understand the relationship between the returns on the assets in their portfolio.

For example, if they tend to move in opposite directions then the combination can be less risky than

the individual holdings. Sample covariance between two time series of returns on two assets 1r and

2r is defined by

.

A positive sample covariance between returns on two assets suggests that their returns have a

tendency to move in the same direction, while a negative covariance suggests that they have a

tendency to move in opposite directions. If the covariance is approximately zero, then there is no

particular co-movement. The absolute size of covariance is not particularly informative. For this we

need to use instead correlation, which is a normalised unit-free version of the covariance.

The sample correlation coefficient is: where

Thus correlation is the covariance, divided by the two standard deviations. The correlation is unit-

free and lies between1 and 1.

Some examples of covariances and correlations along with graphs of the time series:

RIO and BHP NAB and BHP Fairfax and BHP Fairfax and NAB

Covariance 320.97 54.46 3.56 0.12

Correlation 0.97 0.71 0.33 0.03

1 2, 1 1 2 21

1

1

T

r r t t t t

ts r r r rT

1 2

1 2

,

2 2

r r

r r

s

s s

22

1

1

1i

T

r it i

i

s r rT


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A company that has negative correlation with the market as a wholeas represented by the market

index is known as a negative beta asset (the reason will be explained in future lectures) and hasthe advantage that it is a hedging instrument against the overall market.

Auto-covariance and autocorrelation will be discussed in the dynamic time series part of the unit

(Week 6 onwards).

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Shareprice($)

Time series for share prices of several Australian companies

NAB Fairfax RIO BHP

part 1 financial variables

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