applied statistics iv

41
Applied Statistics Vincent JEANNIN – ESGF 4IFM Q1 2012 1 [email protected] ESGF 4IFM Q1 2012

Upload: vincent-jeannin

Post on 20-Jan-2015

161 views

Category:

Economy & Finance


2 download

DESCRIPTION

Fourth Session, MSc 4th Year

TRANSCRIPT

Page 1: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

Applied StatisticsVincent JEANNIN – ESGF 4IFM

Q1 2012

1

ESG

F 4I

FM Q

1 20

12

Page 2: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

2

Summary of the session (est. 4.5h)

• Interim Exam Sum Up• Reminders of last session• Capital Asset Pricing Model• Thinking algorithmic

ESG

F 4I

FM Q

1 20

12

Page 3: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

Interim Exam Sum-Up

3

ESG

F 4I

FM Q

1 20

12

𝑦=𝑎∗ ln (𝑥 )+𝑏+𝜀

1

Page 4: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

4

ESG

F 5I

FM Q

1 20

12

Two parameters to estimate:• Intercept α• Gradient β

Minimising residuals

𝐸=∑𝑖=1

𝑛

𝜀𝑖❑2=∑

𝑖=1

𝑛

(𝑦 𝑖− (𝑎∗ ln (𝑥𝑖)+𝑏))2

When E is minimal?

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

Attention, logarithms are not additive!

Page 5: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

5

ESG

F 5I

FM Q

1 20

12

ln (𝑎+𝑏 )≠ln (𝑎)+ ln (𝑏)

ln (∑ 𝑥𝑖 )≠𝑛 ln (𝑥 )Solution?

Change the variable Z=ln(X)

𝐸=∑𝑖=1

𝑛

𝜀𝑖❑2=∑

𝑖=1

𝑛

(𝑦 𝑖− (𝑎∗ 𝑧𝑖+𝑏))2

Page 6: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

6

ESG

F 4I

FM Q

1 20

12

𝑎∗∑𝑖=1

𝑛

𝑧 𝑖+𝑛𝑏=∑𝑖=1

𝑛

𝑦 𝑖

Leads easily to the intercept

𝑎𝑛𝑧+𝑛𝑏=𝑛𝑦

𝑎𝑧+𝑏=𝑦

𝜕𝐸𝜕𝑏

𝑏=𝑦−𝑎𝑧

( 𝑦 𝑖−𝑎𝑥 𝑖−𝑏 )2=𝑦 𝑖2−2𝑎𝑧 𝑖 𝑦 𝑖−2𝑏𝑦 𝑖+𝑎

2𝑧 𝑖2+2𝑎𝑏𝑧 𝑖+𝑏

2

Page 7: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

7

ESG

F 5I

FM Q

1 20

12

𝜕𝐸𝜕𝑎

=∑𝑖=1

𝑛

−2 𝑧𝑖 𝑦 𝑖+2𝑎𝑧𝑖❑2+2𝑏𝑧 𝑖=0

y=𝑎𝑧+ 𝑦−𝑎𝑧

y − 𝑦=𝑎(𝑧 −𝑧)

𝑏=𝑦−𝑎𝑧

∑𝑖=1

𝑛

𝑧 𝑖 (𝑦 𝑖−𝑎𝑧 𝑖❑−𝑏)=0

𝜕𝐸𝜕𝑏

=∑𝑖=1

𝑛

−2 𝑦 𝑖+2𝑏+2𝑎𝑧𝑖=0

∑𝑖=1

𝑛

𝑦 𝑖−𝑏−𝑎𝑧𝑖=0

∑𝑖=1

𝑛

𝑦 𝑖− 𝑦+𝑎𝑧−𝑎𝑧 𝑖=0

∑𝑖=1

𝑛

(𝑦 𝑖− 𝑦 )−𝑎(𝑧𝑖− 𝑧)=0

∑𝑖=1

𝑛

𝑧 𝑖 (𝑦 𝑖−𝑎𝑧 𝑖❑− 𝑦+𝑎𝑧 )=0

∑𝑖=1

𝑛

𝑧 𝑖(𝑦 𝑖−𝑦−𝑎 (𝑧 𝑖❑− 𝑧 ))=0

∑𝑖=1

𝑛

𝑧 (( 𝑦 𝑖− 𝑦 )−𝑎 (𝑧𝑖−𝑧 ))=0

Page 8: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

8

ESG

F 5I

FM Q

1 20

12

∑𝑖=1

𝑛

𝑧 𝑖(𝑦 𝑖−𝑦−𝑎 (𝑧 𝑖❑− 𝑧 ))=0 ∑

𝑖=1

𝑛

𝑧 (( 𝑦 𝑖− 𝑦 )−𝑎 (𝑧𝑖−𝑧 ))=0

∑𝑖=1

𝑛

𝑧 𝑖(𝑦 𝑖−𝑦−𝑎 (𝑧 𝑖❑− 𝑧 ))=∑

𝑖=1

𝑛

𝑧 (( 𝑦 𝑖−𝑦 )−𝑎 ( 𝑧𝑖−𝑧 ))

∑𝑖=1

𝑛

𝑧 𝑖(𝑦 𝑖−𝑦−𝑎 (𝑧 𝑖❑− 𝑧 ))−∑

𝑖=1

𝑛

𝑧 (( 𝑦 𝑖− 𝑦 )−𝑎 (𝑧𝑖−𝑧 ))=0

∑𝑖=1

𝑛

(𝑧¿¿ 𝑖−𝑧)(𝑦 𝑖−𝑦−𝑎 (𝑧 𝑖❑− 𝑧 ))=0¿

𝑎=∑𝑖=1

𝑛

(𝑧¿¿ 𝑖− 𝑧)(𝑦 𝑖− 𝑦 )

∑𝑖=1

𝑛

(𝑧 ¿¿ 𝑖−𝑧)2¿¿

Finally…

We have

and

Page 9: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

9

ESG

F 5I

FM Q

1 20

12

𝑎=∑𝑖=1

𝑛

(𝑧¿¿ 𝑖− 𝑧)(𝑦 𝑖− 𝑦 )

∑𝑖=1

𝑛

(𝑧 ¿¿ 𝑖−𝑧)2¿¿ 𝑏=𝑦−𝑎𝑧

Don’t forget…

Z=ln(X)

Page 10: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

10

ESG

F 5I

FM Q

1 20

12

No forecast possible (one particular stock against the market)

Hedging is linear…

Accept or reject the regression?

Check correlation and R Squared

Check the normality of residuals

Page 11: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

11

ESG

F 5I

FM Q

1 20

12

Page 12: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

12

ESG

F 5I

FM Q

1 20

12

)-))2

𝑃 (𝑋 ≤−2.44 )=0.0073

𝑃 (𝑋 ≤2.44 )=0.9927

𝑃 (𝑋 ≥2.44 )=0.0073

𝑃 (𝑋 ≥−2.4 4 )=9927

N(0,1)

N(-1,2)

𝑌=𝑋+12

𝑃 (𝑋 ≤−2.44 )=𝑃 (𝑌 ≤−0.72 )

𝑃 (𝑋 ≤−2.44 )=0.2358

Page 13: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

13

ESG

F 4I

FM Q

1 20

12

Let’s build a tree with 5 steps, with S=104.57, σ=10%, 1 year to maturity

104.57

𝑢=𝑒𝜎 √𝑡=𝑒0.1 √15=1.045736

𝑑=𝑒−𝜎 √𝑡=𝑒− 0.1√ 15=0.956264

3𝑑∗𝑢=1

109.35114.45

119.58125.05

130.77

10095.62

91.4487.44

83.62

104.57109.35

104.57109.35

100

114.45119.58

10095.62 91.44

Page 14: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

14

ESG

F 4I

FM Q

1 20

12

130.77

83.62

109.35

100

119.58

91.44

20

19.58

5

5

0

0

Last node value

• Pay off capped to 20• Pay off between 100 inclusive and 109.35

inclusive: 5.00

𝑝=𝑒𝑟 𝑡−𝑑𝑢−𝑑

=0.741553

Page 15: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

15

ESG

F 4I

FM Q

1 20

12

BV=

Final Value 12.50

Page 16: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

16

ESG

F 4I

FM Q

1 20

12𝑑𝐶=𝐶+∆∗𝑑𝑆+12∗𝛾∗𝑑𝑆2

4

+16∗𝑆𝑝𝑒𝑒𝑑∗𝑑𝑆3

+124

∗𝐺𝑟𝑒𝑒𝑘4 h𝑡 ∗𝑑𝑆4

What is the new price of the Call (initial price $8.00) if S moves up $2.5 with delta=0.5525 and a gamma of 0.0222, volatility moves up 1.75 point with a 0.8422 Vega, r moves up 1.2 basis point with Rho=178.5448 and placing you 3 days after with a final Theta of -0.9723?

10.73

Page 17: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

17

ESG

F 4I

FM Q

1 20

12

5

Random walk! Past series has no importance! Trial s Independents!

Page 18: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

18

ESG

F 4I

FM Q

1 20

12

++…+

More than one explanatory variables

Reminder of the last session

Multiple regression

Extension

APT

+ “Pure” factors

R-Square is very often very poor

Page 19: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

19

ESG

F 4I

FM Q

1 20

12

Ratio Investment / GDP , World Bank, developing countries

Let’s discuss…

• Corruption: current corruption• CorruptionPrediction: future corruption• School: level of education• GDP: GDP• Distortion: how badly policies are run

Page 20: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

20

ESG

F 4I

FM Q

1 20

12

• General to specific: this starts off with a comprehensive model, including all the likely explanatory variables, then simplifies it.

• Specific to general: this begins with a simple model that is easy to understand, then explanatory variables are added to improve the model’s explanatory power.

How to find the right model?

Be logic

Have the best R-Squared

Not over complicate

Page 21: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

21

ESG

F 4I

FM Q

1 20

12

3 steps

Identify

Fit

Forecast

𝑂𝑏𝑠=𝑀𝑜𝑑𝑒𝑙+𝜀 with being a white noise What is a model?

Page 22: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

22

ESG

F 4I

FM Q

1 20

12

3 components

Trend

Seasonality

Residual

Page 23: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

23

ESG

F 4I

FM Q

1 20

12

Variation (price or percentage is a differentiation)

Series with stationarity much easier to modelise

Page 24: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

24

ESG

F 4I

FM Q

1 20

12

Once the series is stationary, look for autoccorrelation

Most cases you will find autocorrelation

On the values

On the residuals

Page 25: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

25

ESG

F 5I

FM Q

1 20

12

𝑋 𝑡=𝑐+𝜑1𝑋 𝑡 −1+𝜑2𝑋 𝑡 −2+…+𝜑𝑛 𝑋 𝑡−𝑛+𝜀𝑡

𝜑𝑛Parameters of the model

𝜀𝑛White noise

Auto Regressive model

AR(n)

Page 26: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

26

ESG

F 4I

FM Q

1 20

12

Small sample: Binomial Distribution

Large sample: Normal Distribution

)()1()!(!

!)( xnx pp

xnx

nxf

)1(, pnpnpN

n is the size of the sample, x, the number individuals with the particular characteristic

𝐸 ( 𝑋 )=𝑛𝑝𝑉 (𝑋 )=𝑛𝑝(1−𝑝)

Estimations

Page 27: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

27

ESG

F 4I

FM Q

1 20

12

Binomial Distribution

𝐸 (𝑌 )=𝑝 𝑉 (𝑌 )=𝑝(1−𝑝)𝑛

Normal approximation

𝑌 𝑁 (𝑝 ,√𝑝 (1−𝑝 )𝑛 ) Standardisation possible

𝑌 ∗ 𝑁 (0,1 )

𝑌 ∗=𝑌 −𝑝

√𝑝 (1−𝑝 )𝑛

Normal approximation works only if

𝑛𝑝≥5 𝑛(1−𝑝)≥5

Estimate a proportion𝑌=𝑋𝑛

Page 28: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

28

ESG

F 4I

FM Q

1 20

12

𝑃 (𝑝1<𝑝<𝑝2 )=0.95Let’s look for p with a 95% confidence interval

Easy solve!

𝑃 (𝜇−1.96∗𝜎 ≤ 𝑋≤𝜇+1.96∗𝜎 )=0.95

Page 29: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

29

ESG

F 4I

FM Q

1 20

12

52 Heads out of 100 toss…

𝑌 𝑁 (0.52 ,0.04996 )

95% confidence interval

𝑝1=0.62

𝑌 𝑁 (? ,? )

𝑝2=0.42

Page 30: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

30

ESG

F 4I

FM Q

1 20

12

Student’s Statistic

S

𝑃 (𝑥− 𝑆√𝑛

∗𝑡𝛼/2<𝜇<𝑥+𝑆√𝑛

∗𝑡𝛼 /2)=0.95

Degree of freedom

n-1

Mean estimation

Mean has a Student’s distribution

Page 31: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

31

ESG

F 4I

FM Q

1 20

12

IPO PremiumsIPO1 / 12%IPO2 / 15%IPO3 / 13%IPO4 / 18%IPO5 / 20%IPO6 / 5%

SD: =4.81%

DF: =5

S: =5.27%

t: =2.571

: =19.36%

: =13.83%

: =8.30%

Page 32: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

32

ESG

F 4I

FM Q

1 20

12

Is Martingale safe?

Page 33: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

33

ESG

F 5I

FM Q

1 20

12

Capital Asset Pricing Model

Using Variance/Covariance Matrix to select the portfolio

Optimisation of either the risk or the return

5 stocks available

How many portfolio can be built?

How to chose the weights?

Page 34: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

34

ESG

F 5I

FM Q

1 20

12

Infinite number of long only portfolios

Page 35: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

35

ESG

F 5I

FM Q

1 20

12

Would you buy just Air Liquide?

Page 36: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

36

ESG

F 5I

FM Q

1 20

12

You’d only invest on the so called Efficient Frontier

Page 37: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

37

ESG

F 5I

FM Q

1 20

12

For a particular return, you take the lowest risk

For a particular risk, you take the highest return

Page 38: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

38

ESG

F 5I

FM Q

1 20

12

Unless there’s a risk free rate

Page 39: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

39

ESG

F 5I

FM Q

1 20

12

For a particular combination you need to calculate the expected return

𝐸 (𝑃 )=∑1

𝑛

𝑥𝑖 .𝐸(𝑅𝑛)

Straight forward, mean is linear, weighted average

Page 40: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

40

ESG

F 5I

FM Q

1 20

12

For a particular combination you need to calculate the variance (or SD)

We already know

VAR

Not enough, need the general case for a bigger number of assets

Page 41: Applied Statistics IV

vinz

jean

nin@

hotm

ail.c

om

41

ESG

F 5I

FM Q

1 20

12

Thinking Algorithmic

Millions of portfolio

No linear formula to select the good one

Need a computer and algorithms