Applied Statistics IV

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Information about Applied Statistics IV

Published on February 27, 2014

Author: vinzjeannnin



Fourth Session, MSc 4th Year

Q1 2012 ESGF 4IFM Q1 2012 Vincent JEANNIN – ESGF 4IFM Applied Statistics 1

Interim Exam Sum Up Reminders of last session Capital Asset Pricing Model Thinking algorithmic ESGF 4IFM Q1 2012 • • • • Summary of the session (est. 4.5h) 2

ESGF 4IFM Q1 2012 1 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! Minimising residuals ESGF 5IFM Q1 2012 Two parameters to estimate: • Intercept α • Gradient β 4

Change the variable Z=ln(X) ESGF 5IFM Q1 2012 Solution? 5 ESGF 4IFM Q1 2012 Leads easily to the intercept 6

7 ESGF 5IFM Q1 2012 ESGF 5IFM Q1 2012 We have and Finally… 8

Z=ln(X) ESGF 5IFM Q1 2012 Don’t forget… 9

Accept or reject the regression? 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 ESGF 5IFM Q1 2012

N(-1,2) 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 125.05 119.58 114.45 109.35 104.57 100 104.57 95.62 114.45 109.35 109.35 104.57 100 91.44 119.58 95.62 87.44 100 91.44 83.62 130.77 ESGF 4IFM Q1 2012 3 13

130.77 119.58 20 19.58 109.35 5 100 5 91.44 0 83.62 • 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 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, 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 ESGF 4IFM Q1 2012 4 16

Random walk! Past series has no importance! Trial s Independents! ESGF 4IFM Q1 2012 5 17

Reminder of the last session ESGF 4IFM Q1 2012 Multiple regression More than one explanatory variables R-Square is very often very poor Extension APT 18 “Pure” factors

• • • • • Corruption: current corruption CorruptionPrediction: future corruption School: level of education GDP: GDP Distortion: how badly policies are run Let’s discuss… ESGF 4IFM Q1 2012 Ratio Investment / GDP , World Bank, developing countries 19

Be logic • 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

3 steps Identify Fit Forecast ESGF 4IFM Q1 2012 What is a model? 21

Trend Seasonality Residual ESGF 4IFM Q1 2012 3 components 22

ESGF 4IFM Q1 2012 Variation (price or percentage is a differentiation) 23 Series with stationarity much easier to modelise

On the values On the residuals Most cases you will find autocorrelation ESGF 4IFM Q1 2012 Once the series is stationary, look for autoccorrelation 24

Parameters of the model White noise 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 sample, x, the number individuals with the particular characteristic Small sample: Binomial Distribution ESGF 4IFM Q1 2012 Estimations 26

Estimate a proportion Normal approximation Standardisation possible Normal approximation works only if ESGF 4IFM Q1 2012 Binomial Distribution 27

Easy solve! ESGF 4IFM Q1 2012 Let’s look for p with a 95% confidence interval 28 95% confidence interval ESGF 4IFM Q1 2012 52 Heads out of 100 toss… 29

Mean estimation Student’s Statistic ESGF 4IFM Q1 2012 Mean has a Student’s distribution Degree of freedom 30 n-1

SD: DF: S: ESGF 4IFM Q1 2012 IPO Premiums IPO1 / 12% IPO2 / 15% IPO3 / 13% IPO4 / 18% IPO5 / 20% IPO6 / 5% t: 31 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 Optimisation of either the risk or the return 5 stocks available ESGF 5IFM Q1 2012 Capital Asset Pricing Model 33 ESGF 5IFM Q1 2012 Infinite number of long only portfolios 34 ESGF 5IFM Q1 2012 Would you buy just Air Liquide? 35 ESGF 5IFM Q1 2012 You’d only invest on the so called Efficient Frontier 36 ESGF 5IFM Q1 2012 For a particular return, you take the lowest risk 37 For a particular risk, you take the highest return ESGF 5IFM Q1 2012 Unless there’s a risk free rate 38

Straight forward, mean is linear, weighted average ESGF 5IFM Q1 2012 For a particular combination you need to calculate the expected return 39

We already know ESGF 5IFM Q1 2012 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

No linear formula to select the good one Need a computer and algorithms Millions of portfolio ESGF 5IFM Q1 2012 Thinking Algorithmic 41

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