Q1 2012

ESGF 4IFM Q1 2012

Vincent JEANNIN – ESGF 4IFM

vinzjeannin@hotmail.com

Applied Statistics

1
Interim Exam Sum Up
Reminders of last session
Capital Asset Pricing Model
Thinking algorithmic

ESGF 4IFM Q1 2012

•
•
•
•...
ESGF 4IFM Q1 2012

1

vinzjeannin@hotmail.com

Interim Exam Sum-Up

3
When E is minimal?
When partial derivatives i.r.w. a and b are 0

Attention, logarithms are not additive!

vinzjeannin@hot...
Change the variable
Z=ln(X)
vinzjeannin@hotmail.com

ESGF 5IFM Q1 2012

Solution?

5
vinzjeannin@hotmail.com

ESGF 4IFM Q1 2012

Leads easily to the intercept

6
7

vinzjeannin@hotmail.com

ESGF 5IFM Q1 2012
vinzjeannin@hotmail.com

ESGF 5IFM Q1 2012

We have
and

Finally…
8
Z=ln(X)
vinzjeannin@hotmail.com

ESGF 5IFM Q1 2012

Don’t forget…

9
Accept or reject the regression?

vinzjeannin@hotmail.com

Hedging is linear…

ESGF 5IFM Q1 2012

No forecast possible (on...
11

vinzjeannin@hotmail.com

ESGF 5IFM Q1 2012
N(-1,2)
vinzjeannin@hotmail.com

ESGF 5IFM Q1 2012

2
N(0,1)

12
Let’s build a tree with 5 steps, with S=104.57, σ=10%, 1 year to maturity

125.05

119.58
114.45
109.35
104.57

100

104.5...
130.77
119.58

20
19.58

109.35 5
100

5

91.44 0
83.62

vinzjeannin@hotmail.com

• Pay off capped to 20
• Pay off between...
vinzjeannin@hotmail.com

Final Value 12.50

15

ESGF 4IFM Q1 2012
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, volati...
Random walk! Past series has no importance! Trial s Independents!

vinzjeannin@hotmail.com

ESGF 4IFM Q1 2012

5

17
Reminder of the last session
ESGF 4IFM Q1 2012

Multiple regression

vinzjeannin@hotmail.com

More than one explanatory va...
•
•
•
•
•

Corruption: current corruption
CorruptionPrediction: future corruption
School: level of education
GDP: GDP
Dist...
Be logic

vinzjeannin@hotmail.com

• General to specific: this starts off with a comprehensive
model, including all the li...
3 steps

Identify
Fit

Forecast

vinzjeannin@hotmail.com

ESGF 4IFM Q1 2012

What is a model?

21
Trend

Seasonality

Residual
vinzjeannin@hotmail.com

ESGF 4IFM Q1 2012

3 components

22
ESGF 4IFM Q1 2012
vinzjeannin@hotmail.com

Variation (price or percentage is a differentiation)
23
Series with stationarit...
On the values

On the residuals

vinzjeannin@hotmail.com

Most cases you will find autocorrelation

ESGF 4IFM Q1 2012

Onc...
Parameters of the model
White noise

vinzjeannin@hotmail.com

ESGF 5IFM Q1 2012

Auto Regressive model

AR(n)
25
f ( x)

n!
x! ( n

x

p (1

p)

(n x)

x )!

Large sample: Normal Distribution

N np ,

np (1

p)

n is the size of the sa...
Estimate a proportion

Normal approximation

Standardisation possible

Normal approximation works only if

vinzjeannin@hot...
Easy solve!

vinzjeannin@hotmail.com

ESGF 4IFM Q1 2012

Let’s look for p with a 95% confidence interval

28
vinzjeannin@hotmail.com

95% confidence interval

ESGF 4IFM Q1 2012

52 Heads out of 100 toss…

29
Mean estimation

vinzjeannin@hotmail.com

Student’s Statistic

ESGF 4IFM Q1 2012

Mean has a Student’s distribution

Degre...
SD:

DF:
S:

ESGF 4IFM Q1 2012
vinzjeannin@hotmail.com

IPO Premiums
IPO1 / 12%
IPO2 / 15%
IPO3 / 13%
IPO4 / 18%
IPO5 / 20...
vinzjeannin@hotmail.com

ESGF 4IFM Q1 2012

Is Martingale safe?

32
How many portfolio can be built?
How to chose the weights?

Using Variance/Covariance Matrix to select the portfolio

Opti...
vinzjeannin@hotmail.com

ESGF 5IFM Q1 2012

Infinite number of long only portfolios

34
vinzjeannin@hotmail.com

ESGF 5IFM Q1 2012

Would you buy just Air Liquide?

35
vinzjeannin@hotmail.com

ESGF 5IFM Q1 2012

You’d only invest on the so called Efficient Frontier

36
vinzjeannin@hotmail.com

ESGF 5IFM Q1 2012

For a particular return, you take the lowest risk

37
For a particular risk, y...
vinzjeannin@hotmail.com

ESGF 5IFM Q1 2012

Unless there’s a risk free rate

38
Straight forward, mean is linear, weighted average

ESGF 5IFM Q1 2012
vinzjeannin@hotmail.com

For a particular combinatio...
We already know

ESGF 5IFM Q1 2012
vinzjeannin@hotmail.com

For a particular combination you need to calculate the varianc...
No linear formula to select the good one

Need a computer and algorithms

vinzjeannin@hotmail.com

Millions of portfolio

...
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Applied Statistics IV

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Fourth Session, MSc 4th Year

Published in: Economy & Finance
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  • As we have b, we can replace it in the equation of the regression line
  • Rf will be used in the backward inductionDrift is cancelled through risk neutral probability
  • Rf will be used in the backward inductionDrift is cancelled through risk neutral probability
  • Rf will be used in the backward inductionDrift is cancelled through risk neutral probability
  • V(X/n)=1/n^2 * np(1-p)
  • 0.52+- 1.96*0.04669
  • S/SQRT(6)*2.571 +- Actual mean
  • Applied Statistics IV

    1. 1. Q1 2012 ESGF 4IFM Q1 2012 Vincent JEANNIN – ESGF 4IFM vinzjeannin@hotmail.com Applied Statistics 1
    2. 2. Interim Exam Sum Up Reminders of last session Capital Asset Pricing Model Thinking algorithmic ESGF 4IFM Q1 2012 • • • • vinzjeannin@hotmail.com Summary of the session (est. 4.5h) 2
    3. 3. ESGF 4IFM Q1 2012 1 vinzjeannin@hotmail.com Interim Exam Sum-Up 3
    4. 4. When E is minimal? When partial derivatives i.r.w. a and b are 0 Attention, logarithms are not additive! vinzjeannin@hotmail.com Minimising residuals ESGF 5IFM Q1 2012 Two parameters to estimate: • Intercept α • Gradient β 4
    5. 5. Change the variable Z=ln(X) vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Solution? 5
    6. 6. vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 Leads easily to the intercept 6
    7. 7. 7 vinzjeannin@hotmail.com ESGF 5IFM Q1 2012
    8. 8. vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 We have and Finally… 8
    9. 9. Z=ln(X) vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Don’t forget… 9
    10. 10. Accept or reject the regression? vinzjeannin@hotmail.com Hedging is linear… ESGF 5IFM Q1 2012 No forecast possible (one particular stock against the market) Check correlation and R Squared 10 Check the normality of residuals
    11. 11. 11 vinzjeannin@hotmail.com ESGF 5IFM Q1 2012
    12. 12. N(-1,2) vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 2 N(0,1) 12
    13. 13. Let’s build a tree with 5 steps, with S=104.57, σ=10%, 1 year to maturity 125.05 119.58 114.45 109.35 104.57 100 104.57 95.62 114.45 109.35 109.35 104.57 100 91.44 119.58 95.62 87.44 100 91.44 83.62 vinzjeannin@hotmail.com 130.77 ESGF 4IFM Q1 2012 3 13
    14. 14. 130.77 119.58 20 19.58 109.35 5 100 5 91.44 0 83.62 vinzjeannin@hotmail.com • Pay off capped to 20 • Pay off between 100 inclusive and 109.35 inclusive: 5.00 ESGF 4IFM Q1 2012 Last node value 0 14
    15. 15. vinzjeannin@hotmail.com Final Value 12.50 15 ESGF 4IFM Q1 2012
    16. 16. 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 vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 4 16
    17. 17. Random walk! Past series has no importance! Trial s Independents! vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 5 17
    18. 18. Reminder of the last session ESGF 4IFM Q1 2012 Multiple regression vinzjeannin@hotmail.com More than one explanatory variables R-Square is very often very poor Extension APT 18 “Pure” factors
    19. 19. • • • • • Corruption: current corruption CorruptionPrediction: future corruption School: level of education GDP: GDP Distortion: how badly policies are run vinzjeannin@hotmail.com Let’s discuss… ESGF 4IFM Q1 2012 Ratio Investment / GDP , World Bank, developing countries 19
    20. 20. Be logic vinzjeannin@hotmail.com • 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. ESGF 4IFM Q1 2012 How to find the right model? Have the best R-Squared 20 Not over complicate
    21. 21. 3 steps Identify Fit Forecast vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 What is a model? 21
    22. 22. Trend Seasonality Residual vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 3 components 22
    23. 23. ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com Variation (price or percentage is a differentiation) 23 Series with stationarity much easier to modelise
    24. 24. On the values On the residuals vinzjeannin@hotmail.com Most cases you will find autocorrelation ESGF 4IFM Q1 2012 Once the series is stationary, look for autoccorrelation 24
    25. 25. Parameters of the model White noise vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Auto Regressive model AR(n) 25
    26. 26. f ( x) n! x! ( n x p (1 p) (n x) x )! Large sample: Normal Distribution N np , np (1 p) n is the size of the sample, x, the number individuals with the particular characteristic vinzjeannin@hotmail.com Small sample: Binomial Distribution ESGF 4IFM Q1 2012 Estimations 26
    27. 27. Estimate a proportion Normal approximation Standardisation possible Normal approximation works only if vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 Binomial Distribution 27
    28. 28. Easy solve! vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 Let’s look for p with a 95% confidence interval 28
    29. 29. vinzjeannin@hotmail.com 95% confidence interval ESGF 4IFM Q1 2012 52 Heads out of 100 toss… 29
    30. 30. Mean estimation vinzjeannin@hotmail.com Student’s Statistic ESGF 4IFM Q1 2012 Mean has a Student’s distribution Degree of freedom 30 n-1
    31. 31. SD: DF: S: ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com IPO Premiums IPO1 / 12% IPO2 / 15% IPO3 / 13% IPO4 / 18% IPO5 / 20% IPO6 / 5% t: 31
    32. 32. vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 Is Martingale safe? 32
    33. 33. How many portfolio can be built? How to chose the weights? Using Variance/Covariance Matrix to select the portfolio Optimisation of either the risk or the return vinzjeannin@hotmail.com 5 stocks available ESGF 5IFM Q1 2012 Capital Asset Pricing Model 33
    34. 34. vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Infinite number of long only portfolios 34
    35. 35. vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Would you buy just Air Liquide? 35
    36. 36. vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 You’d only invest on the so called Efficient Frontier 36
    37. 37. vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 For a particular return, you take the lowest risk 37 For a particular risk, you take the highest return
    38. 38. vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 Unless there’s a risk free rate 38
    39. 39. Straight forward, mean is linear, weighted average ESGF 5IFM Q1 2012 vinzjeannin@hotmail.com For a particular combination you need to calculate the expected return 39
    40. 40. We already know ESGF 5IFM Q1 2012 vinzjeannin@hotmail.com For a particular combination you need to calculate the variance (or SD) Not enough, need the general case for a bigger number of assets 40
    41. 41. No linear formula to select the good one Need a computer and algorithms vinzjeannin@hotmail.com Millions of portfolio ESGF 5IFM Q1 2012 Thinking Algorithmic 41
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