Applied Statistics II

ESGF 4IFM Q1 2012
Applied Statistics
Vincent JEANNIN – ESGF 4IFM
Q1 2012

vinzjeannin@hotmail.com
1

ESGF 4IFM Q1 2012
Summary of the session (est. 4.5h)

• R Steps by Steps
• Reminders of last session
• The Value at Risk

• OLS & Exploration

2

R Step by Step
Downloadable for free (open source)

ESGF 4IFM Q1 2012
http://www.r-project.org/

3

Main screen

vinzjeannin@hotmail.com ESGF 4IFM Q1 2012
4

Menu: File / New Script

5

Step 1, upload your data

Excel CSV file easy to import

ESGF 4IFM Q1 2012
Path C:UsersvinDesktop

Note: 4 columns with headers

6
DATA<-read.csv(file="C:/Users/vin/Desktop/DataFile.csv",header=T)

Run your instruction(s)

7

You can call variables anytime you want

ESGF 4IFM Q1 2012
8

9

summary(DATA) Shows a quick summary of the distribution of all variables

SPX SPXr AMEXr AMEX
Min. : 86.43 Min. :-0.0666344 Min. : 97.6 Min. :-0.0883287
1st Qu.: 95.70 1st Qu.:-0.0069082 1st Qu.:104.7 1st Qu.:-0.0094580
Median :100.79 Median : 0.0010016 Median :108.8 Median : 0.0013007

ESGF 4IFM Q1 2012
Mean : 99.67 Mean : 0.0001249 Mean :109.4 Mean : 0.0005891
3rd Qu.:103.75 3rd Qu.: 0.0075235 3rd Qu.:114.1 3rd Qu.: 0.0102923
Max. :107.21 Max. : 0.0474068 Max. :123.5 Max. : 0.0710967

summary(DATA$SPX) Shows a quick summary of the distribution of one variable

Min. 1st Qu. Median Mean 3rd Qu. Max.
86.43 95.70 100.80 99.67 103.80 107.20

min(DATA)
Careful using the following instructions max(DATA)
> min(DATA)
[1] -0.08832874
This will consider DATA as one variable > max(DATA)
[1] 123.4793

> sd(DATA)
SPX SPXr AMEXr AMEX
4.92763551 0.01468776 6.03035318 0.01915489 10
Mean & SD > mean(DATA)
SPX SPXr AMEXr AMEX
9.967046e+01 1.249283e-04 1.093951e+02 5.890780e-04

Easy to show histogram

ESGF 4IFM Q1 2012
hist(DATA$SPXr, breaks=25, main="Distribution of SPXr", ylab="Freq", 11
xlab="SPXr", col="blue")

Obvious Excess Kurtosis

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Obvious Asymmetry

Functions doesn’t exists directly in R…

However some VNP (Very Nice Programmer) built and shared add-in

Package Moments 12

Menu: Packages / Install Package(s)

ESGF 4IFM Q1 2012
• Choose whatever mirror (server) you want
• Usually France (Toulouse) is very good as it’s a
University Server with all the packages available

13

ESGF 4IFM Q1 2012
Once installed, you can load them with the
following instructions:
require(moments)
library(moments)

New functions can now be used!

14

> require(moments)
> library(moments)
> skewness(DATA)
SPX SPXr AMEXr AMEX
-0.6358029 -0.4178701 0.1876994 -0.2453693

ESGF 4IFM Q1 2012
> kurtosis(DATA)
SPX SPXr AMEXr AMEX
2.411177 5.671254 2.078366 5.770583

Btw, you can store any result in a variable

> Kur<-kurtosis(DATA$SPXr)
> Kur
[1] 5.671254

15

Lost?

Call the help! help(kurtosis)

ESGF 4IFM Q1 2012
Reminds you the package

Syntax

Arguments definition

16

Let’s store a few values
SPMean<-mean(DATA$SPXr)
SPSD<-sd(DATA$SPXr) Package Stats

Build a sequence, the x axis

ESGF 4IFM Q1 2012
x<-seq(from=SPMean-4*SPSD,to=SPMean+4*SPSD,length=500)

Build a normal density on these x

Y1<-dnorm(x,mean=SPMean,sd=SPSD) Package Stats

Display the histogram
hist(DATA$SPXr, breaks=25,main="S&P Returns / Normal Package graphics
Distribution",xlab="Returns",ylab="Occurences", col="blue")

Display on top of it the normal density

lines(x,y1,type="l",lwd=3,col="red") Package graphics 17

ESGF 4IFM Q1 2012
18

Positive Excess Kurtosis & Negative Skew

Let’s build a spread Spd<-DATA$SPXr-DATA$AMEX

What is the mean?

ESGF 4IFM Q1 2012
Mean is linear �� + �� = �� + ��(��)

�� − �� = �� − ��(��)

Let’s verify

> mean(DATA$SPXr)-mean(DATA$AMEX)-mean(Spd)
[1] 0

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What is the standard deviation?

Is standard deviation linear?
NO!

ESGF 4IFM Q1 2012
VAR �� + �� = ��2 �� + ��2 �� + 2��(��, ��)

> (var(DATA$SPXr)+var(DATA$AMEX)-2*cov(DATA$SPXr,DATA$AMEX))^0.5

[1] 0.01019212
> sd(Spd)
[1] 0.01019212

Let’s show the implication in a proper manner

Let’s create a portfolio containing half of each stocks 20

Portf<-0.5*DATA$SPXr+0.5*DATA$AMEX

plot(sd(DATA$SPXr),mean(DATA$SPXr),col="blue",ylim=c(0,0.0008),xlim=c(0.012
,0.022),ylab="Return",xlab="Vol")

points(sd(DATA$AMEX),mean(DATA$AMEX),col="red")

ESGF 4IFM Q1 2012
points(sd(Portf),mean(Portf),col="green")

21

The efficient frontier

22

points(sd(0.1*DATA$SPXr+0.9*DATA$AMEX),mean(0.1*DATA$SPXr+0.9*DATA$AMEX),c
ol="green")

ol="green")

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ol="green")

ol="green")

ol="green")

ol="green")

ol="green")

ol="green")
23

plot(DATA$AMEX,DATA$SPXr)
abline(lm(DATA$AMEX~DATA$SPXr), col="blue")

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24

LM stands for Linear Models

> lm(DATA$AMEX~DATA$SPXr)

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Call:
lm(formula = DATA$AMEX ~ DATA$SPXr)

Coefficients:
(Intercept) DATA$SPXr
0.0004505 1.1096287

�� = 1.1096�� + 0.04%

Will be used later for linear regression and hedging

25

Do you remember what is the most platykurtic distribution in the nature?

Toss Head = Success = 1 / Tail = Failure = 0

ESGF 4IFM Q1 2012
100 toss… Else memory issue…

> require(moments)
Loading required package: moments
> library(moments)

> toss<-rbinom(100,1,0.5)
> mean(toss)
[1] 0.52
> kurtosis(toss)
[1] 1.006410
> kurtosis(toss)-3
[1] -1.993590
> hist(toss, breaks=10,main="Tossing a
coin 100 times",xlab="Result of the
trial",ylab="Occurence")
> sum(toss)
[1] 52
26
Let’s test the fairness

Density of a binomial distribution

�� + 1 ! ℎ
�� = ℎ, �� = �� = �� (1 − ��)��
ℎ! ��!

ESGF 4IFM Q1 2012
Let’s plot this density with

ℎ = 52
�� = 48

�� = 100
N<-100
h<-52
t<-48
r<-seq(0,1,length=500)
y<-
(factorial(N+1)/(factorial(h)*factori
al(t)))*r^h*(1-r)^t
plot(r,y,type="l",col="red",main="Pro
bability density to have 52 head out
100 flips")
27

If the probability between 45% and 55% is significant we’ll accept the fairness

ESGF 4IFM Q1 2012
28

What do you think?

What is the problem with this coin?

Obvious fake! Assuming the probability of head is 0.7

Toss it! Head = Success = 1 / Tail = Failure = 0

ESGF 4IFM Q1 2012
100 toss

> require(moments)
Loading required package: moments
> library(moments)

> toss<-rbinom(100,1,0.7)
> mean(toss)
[1] 0.72
> kurtosis(toss)
[1] 1.960317
> kurtosis(toss)-3
[1] -1.039683
> hist(toss, breaks=10,main="Tossing a
coin 100 times",xlab="Result of the
trial",ylab="Occurence")
> sum(toss)
[1] 72
29

Let’s test the fairness (assuming you don’t know it’s a trick)

If the probability between 45% and 55% is significant we’ll accept the fairness
N<-100
h<-72
t<-28
r<-seq(0.2,0.8,length=500)
y<-(factorial(N+1)/(factorial(h)*factorial(t)))*r^h*(1-r)^t

ESGF 4IFM Q1 2012
plot(r,y,type="l",col="red",main="Probability density or r given 72
head out 100 flips")

Trick coin!

30

Reminders of last session

ESGF 4IFM Q1 2012
Normal Standard Distribution

Snapshot, 4 moments:

Mean 0
SD 1
Skewness 0
Kurtosis 3

31

�� ≤ �� = 0.5 �� − �� ≤ �� ≤ �� + �� = 0.682
�� ≤ −�� + �� = 0.159
�� ≤ −1.645 ∗ �� + �� = 0.05 �� − 2 ∗ �� ≤ �� ≤ �� + 2 ∗ �� = 0.954
�� ≤ −2 ∗ �� + �� = 0.023

ESGF 4IFM Q1 2012
�� ≤ −2.326 ∗ �� + �� = 0.01 �� − 3 ∗ �� ≤ �� ≤ �� + 3 ∗ �� = 0.996
�� ≤ −3 ∗ �� + �� = 0.001

32

Notation ��(��, ��)
1 (��−��)2
−
Density �� = �� 2��2
2�� 2

ESGF 4IFM Q1 2012
��
CDF �� ≤ �� = �� = ��
−∞

33

Let be X~N(1,1.5)
Find:
�� ≤ 4.75

ESGF 4IFM Q1 2012
4.75−1
�� ≤ 4.75 =P �� ≤
1.5

With Y~N(0,1)
P �� ≤ 2.5 =?

Use the table!

P �� ≤ −2.5 =0.0062

P �� ≤ 2.5 =0.9938

34

P �� ≤ 4.75 =0.9938

QQ Plot

>qqnorm(FCOJ$V1)

ESGF 4IFM Q1 2012
>qqline(FCOJ$V1)

Fat Tail

35

Geometric Brownian Motion

Based on Stochastic Differential Equation �� = �� + ��

B&S

ESGF 4IFM Q1 2012
Discrete form �� = �� + �� with ��~N(0,1)

CRR S�� = �� + 1 − ��
�� = �� + 1 − ��
�� − ��
�� = �� =
�� − ��
1 BV= OpUp ∗ p + OpDown ∗ 1 − p ∗ �� −�� 36
�� = = �� −��
��

Greeks Approximation – Taylor Development

1
�� = �� + ∆ ∗ �� + ∗ �� ∗ �� 2

ESGF 4IFM Q1 2012
2

1
+ ∗ �� ∗ �� 3
6

1
+ ∗ ��4��ℎ ∗ �� 4
24

etc…

37

The Value at Risk
Estimate with a specific confidence interval (usually 95% or 99%) the worth loss
possible. In other words, the point is to identify a particular point on the left of

ESGF 4IFM Q1 2012
the distribution

3 Methods

• Historical
• Parametrical
• Monte-Carlo
38

For now, we’ll focus on VaR on one linear asset… FCOJ is back!

Historical VaR

• No assumption about the distribution
• Easy to implement and calculate

ESGF 4IFM Q1 2012
• Sensitive to the length of the history
• Sensitive to very extreme values

Let’s get back to our FCOJ time series, last price is $150 cents

If we work on returns, we’ve seen the use of the PERCENTILE
Excel function

• 1% Percentile is -5.22%, 99% Historical Daily VaR is -$7.83 cents
• 5% Percentile is -3.34%, 95% Historical Daily VaR is -$5.00 cents
39

Works as well on weekly, monthly, quarterly series

Historical VaR

ESGF 4IFM Q1 2012
Can be worked as well with prices variations
instead of returns but it’s going to be price
sensitive! So careful to the bias.

• 1% Percentile in term of price movement is -$8.11 cents
• 5% Percentile in term of price movement is -$4.14 cents

40

Parametric VaR

• Easy to implement and calculate
• Assumes a particular shape of the distribution

ESGF 4IFM Q1 2012
• Not really sensitive to fat tails

FCOJ Mean Return: 0.1364%

FCOJ SD: 2.1664%

We already know:

�� ≤ −1.645 ∗ �� + �� = 0.05
�� ≤ −2.326 ∗ �� + �� = 0.01
Then:

�� ≤ −3.43% = 0.05 VaR 95% (-$5.15 cents) 41
�� ≤ −4.90% = 0.01 VaR 99% (-$7.35 cents)

Parametric VaR

Very often you assume anyway a 0 mean, therefore:

ESGF 4IFM Q1 2012
�� ≤ −3.57% = 0.05 VaR 95% (-$5.36 cents)
�� ≤ −5.04% = 0.01 VaR 99% (-$8.10 cents)

Lower values than the historical VaR

Problem with leptokurtic distributions, impact of fat tails isn’t
strong on the method

42

Monte Carlo VaR

ESGF 4IFM Q1 2012
• Most efficient method when asset aren’t linear
• Tough to implement
• Assumes a particular shape of the distribution

Based on an assumption of a price process (for example GBM)

Great number of random simulations on the price process to build a
distribution and outline the VaR

43

Monte Carlo VaR

Let’s simulate 10,000 GBM, 252 steps and store the final result

ESGF 4IFM Q1 2012
library(sde)
require(sde)
FCOJ<-
read.csv(file="C:/Users/Vinz/Desktop/FCOJStats.csv",head=FALSE,sep=",")

Drift<-mean(FCOJ$V1)

Volat<-sd(FCOJ$V1)
nbsim<-252
Spot<-150
Final<-rep(1,10000)

for(i in 1:100000){
Matr<-GBM(x=Spot,r=Drift, sigma=Volat,N=nbsim)
Final[i]<-Matr[nbsim+1]}

quantile(Final, 0.05)
quantile(Final, 0.01)

Don’t be fooled by the 252, we’re still making a daily simulation: what 44
to change in the code to make it yearly?

Monte Carlo VaR

> quantile(Final, 0.05)
5%

ESGF 4IFM Q1 2012
144.93
1%
142.7941

• 95% Daily VaR is -$5.07 cents

Let’s take off the drift
45

Monte Carlo VaR

5%

ESGF 4IFM Q1 2012
144.7583
1%
142.6412


46

Which is the best?
Comparison

47

Going forward on the VaR

ESGF 4IFM Q1 2012
All method give different but coherent values

Easy? Yes but…

• We’ve involved one asset only

• We’ve involved a linear asset

What about an option?

What about 2 assets?

48


Portfolio scale: what to look at to calculate the VaR?

ESGF 4IFM Q1 2012
Big question, is the VaR additive?

NO!
Keywords for the future: covariance, correlation, diversification

49


Options: what to look at to calculate the VaR?

ESGF 4IFM Q1 2012
4 risk factors:
• Underlying price
• Interest rate
• Volatility

• Time

4 answers:
• Delta/Gamma approximation knowing the distribution of the underlying
• Rho approximation knowing the distribution of the underlying rate
• Vega approximation knowing the distribution of implied volatility
• Theta (time decay)

Yes but,… Does the underling price/rate/volatility vary independently? 50

Might be a bit more complicated than expected…

OLS & Exploration
OLS: Ordinary Least Square

ESGF 5IFM Q1 2012
Linear regression model
Minimize the sum of the square vertical distances
between the observations and the linear
approximation

�� = �� = �� + ��

Residual ε

51

Two parameters to estimate:
• Intercept α
• Slope β

ESGF 5IFM Q1 2012
Minimising residuals

��

�� = �� 2 = �� − �� + �� 2

��=1 ��=1

When E is minimal?

When partial derivatives i.r.w. a and b are 0
52

��

�� = �� 2 = �� − �� + �� 2 = �� − �� − �� 2

��=1 ��=1 ��=1

Quick high school reminder if necessary…

ESGF 5IFM Q1 2012
�� − �� − �� 2 = �� 2 − 2�� − 2�� + �� 2 �� 2 + 2�� + ��2

��
��

= −2�� + 2�� 2 + 2�� = 0 = −2�� + 2�� + 2�� = 0
��
��=1 ��=1

��

−�� + �� 2 + �� = 0 −�� + �� + �� = 0
��=1 ��=1
��

�� ∗ �� 2 + �� ∗ �� = �� ∗ �� + �� = ��
��=1 ��=1 ��=1 ��=1 ��=1
53

��
Leads easily to the intercept
��
��

�� ∗ �� + �� = ��
��=1 ��=1

ESGF 5IFM Q1 2012
�� + �� = ��

�� + �� = ��

�� = �� − ��

The regression line is going through (�� , ��)

The distance of this point to the line is 0 indeed

54

�� = �� − �� y = �� + �� − ��

y − �� = ��(�� − �� )

ESGF 5IFM Q1 2012
��
��
= −2�� + 2�� 2 + 2�� = 0 = −2�� + 2�� + 2�� = 0
��
��=1 ��=1

��

�� − �� − �� = 0 �� − �� − �� = 0
��=1 ��=1
��
��
�� − �� − �� + �� = 0
��=1
�� − �� + �� − �� = 0
��=1
��

�� (�� − �� − �� − �� ) = 0 (�� − ��) − ��(�� − �� ) = 0
��=1 ��=1
�� 55
�� ( �� − �� − �� − �� ) = 0
��=1

We have
��

�� (�� − �� − �� − �� ) = 0 and �� ( �� − �� − �� − �� ) = 0
��=1 ��=1

ESGF 5IFM Q1 2012
��

�� (�� − �� − �� − �� ) = �� ( �� − �� − �� − �� )
��=1 ��=1

��

�� (�� − �� − �� − �� ) − �� − �� − �� − �� =0
��=1 ��=1

��

(�� −�� )(�� − �� − �� − �� ) = 0
��=1

Finally…

��
��=1(�� −�� )(�� − ��) 56
�� = �� 2
��=1(�� −�� )

�� Covariance
��=1(�� − �� )(�� − ��)
�� = �� 2
��=1(�� − �� ) Variance

ESGF 5IFM Q1 2012
��
�� =
��2��

�� = �� − ��

You can use Excel function INTERCEPT and SLOPE

57

Calculate the Variances and Covariance of X{1,2,3,3,1,2} and Y{2,3,1,1,3,2}

ESGF 5IFM Q1 2012
58

You can use Excel function VAR.P, COVARIANCE.P and STDEV.P

Let’s asses the quality of the regression

Let’s calculate the correlation coefficient (aka Pearson Product-Moment
Correlation Coefficient – PPMCC):

ESGF 5IFM Q1 2012
��
�� = Value between -1 and 1
��

�� = 1

Perfect dependence

�� ~0 No dependence

Give an idea of the dispersion of the scatterplot
59

You can use Excel function CORREL

Poor quality
R=0.62
R=0.96

High quality

60

What is good quality?

ESGF 5IFM Q1 2012
Slightly discretionary…

If
3
�� ≥ = 0.8666 …
2
It’s largely admitted as the threshold for acceptable / poor

61

The regression itself introduces a bias

Let’s introduce the coefficient of determination R-Squared

ESGF 5IFM Q1 2012
Total Dispersion = Dispersion Regression + Dispersion Residual

2 2 2
�� − �� = �� − �� + �� − ��

Dispersion Regression
��2 =
Total Dispersion

In other words the part of the total dispersion explained by the regression 62

You can use Excel function RSQ

In a simple linear regression with intercept ��2 = �� 2

ESGF 5IFM Q1 2012
Is a good correlation coefficient and a good coefficient of
determination enough to accept the regression?

Not necessarily!

Residuals need to have no effect, in other word to be a white noise!

63

64

Don’t get fooled by numbers!

ESGF 5IFM Q1 2012
For every dataset of the Quarter

�� = 9
�� = 7.5

�� = 3 + 0.5��
�� = 0.82
��2 = 0.67

Can you say at this stage which regression is the best?
65

Certainly not those on the right you need a LINEAR dependence

ESGF 5IFM Q1 2012
Is any linear regression useless?

Think what you could do to the series

Polynomial transformation, log transformation,…

66
Else, non linear regressions, but it’s another story

First application on financial market

S&P / AmEx in 2011

ESGF 5IFM Q1 2012
67

��,��&��
�� = = 0.8501
�� &��

��2 = �� 2 = 0.7227

ESGF 5IFM Q1 2012
Oups :-o
Is Excel wrong?

R-Squared has different calculation methods

Let’s accept the following regression then as the quality seems pretty good

�� = 0.06% + 1.1046 ∗ ��&��

68

How to use this?

ESGF 5IFM Q1 2012
• Forecasting? Not really…
Both are random variables

• Hedging? Yes but basis risk
Yes but careful to the residuals…

In theory, what is the daily result of the hedge? ��

Let’s have a try!

69

Hedging $1.0M of AmEx Stocks with $1.1046M of S&P

ESGF 5IFM Q1 2012
It would have been too easy… Great differences… Why?

Sensitivity to the size of the sample
70
Heteroscedasticity Basis Risk

The purpose was to see if the market as effect an effect on a particular stock

The dependence is obvious but residuals too volatile for any stable application

ESGF 5IFM Q1 2012
But attention!
We are looking for causation, not correlation!

Causation implies correlation

Reciprocity is not true!

DON’T BE FOOLED BY PRETTY NUMBERS
71
Let prove this…

ESGF 5IFM Q1 2012
Perfect linear dependence

Excellent R-Squared
72
Residuals are a white noise

What’s the problem then?

ESGF 5IFM Q1 2012
Do you really think fresh lemon reduces car fatalities?
73

74

Conclusion

R

VaR

OLS
Normal Distribution

75

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