This document appears to be notes from a class on applied statistics and quantitative finance. It covers several topics: - Reminders about the capital asset pricing model and thinking algorithmically - Summarizing an interim exam and minimizing residuals when estimating parameters - Discussing the binomial and normal distributions for estimations from sample data - Explaining autoregressive models and components of time series like trends and seasonality - Discussing portfolio optimization and the efficient frontier

1. Q1 2012
ESGF 4IFM Q1 2012
Vincent JEANNIN – ESGF 4IFM
vinzjeannin@hotmail.com
Applied Statistics
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2. Interim Exam Sum Up
Reminders of last session
Capital Asset Pricing Model
Thinking algorithmic
ESGF 4IFM Q1 2012
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Summary of the session (est. 4.5h)
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3. ESGF 4IFM Q1 2012
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Interim Exam Sum-Up
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4. When E is minimal?
When partial derivatives i.r.w. a and b are 0
Attention, logarithms are not additive!
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Minimising residuals
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Two parameters to estimate:
• Intercept α
• Gradient β
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5. Change the variable
Z=ln(X)
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Solution?
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6. vinzjeannin@hotmail.com
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Leads easily to the intercept
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7. 7
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8. vinzjeannin@hotmail.com
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We have
and
Finally…
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9. Z=ln(X)
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Don’t forget…
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10. Accept or reject the regression?
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Hedging is linear…
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No forecast possible (one particular stock against the market)
Check correlation and R Squared
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Check the normality of residuals

11. 11
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12. N(-1,2)
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N(0,1)
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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
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130.77
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14. 130.77
119.58
20
19.58
109.35 5
100
5
91.44 0
83.62
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• Pay off capped to 20
• Pay off between 100 inclusive and 109.35
inclusive: 5.00
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Last node value
0
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15. vinzjeannin@hotmail.com
Final Value 12.50
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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
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17. Random walk! Past series has no importance! Trial s Independents!
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18. Reminder of the last session
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Multiple regression
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More than one explanatory variables
R-Square is very often very poor
Extension
APT
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“Pure” factors

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Corruption: current corruption
CorruptionPrediction: future corruption
School: level of education
GDP: GDP
Distortion: how badly policies are run
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Let’s discuss…
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Ratio Investment / GDP , World Bank, developing countries
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20. Be logic
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• 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.
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How to find the right model?
Have the best R-Squared
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Not over complicate

21. 3 steps
Identify
Fit
Forecast
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What is a model?
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22. Trend
Seasonality
Residual
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3 components
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23. ESGF 4IFM Q1 2012
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Variation (price or percentage is a differentiation)
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Series with stationarity much easier to modelise

24. On the values
On the residuals
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Most cases you will find autocorrelation
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Once the series is stationary, look for autoccorrelation
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25. Parameters of the model
White noise
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Auto Regressive model
AR(n)
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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
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Small sample: Binomial Distribution
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Estimations
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27. Estimate a proportion
Normal approximation
Standardisation possible
Normal approximation works only if
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Binomial Distribution
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28. Easy solve!
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Let’s look for p with a 95% confidence interval
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29. vinzjeannin@hotmail.com
95% confidence interval
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52 Heads out of 100 toss…
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30. Mean estimation
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Student’s Statistic
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Mean has a Student’s distribution
Degree of freedom
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n-1

31. SD:
DF:
S:
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IPO Premiums
IPO1 / 12%
IPO2 / 15%
IPO3 / 13%
IPO4 / 18%
IPO5 / 20%
IPO6 / 5%
t:
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32. vinzjeannin@hotmail.com
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Is Martingale safe?
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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
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5 stocks available
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Capital Asset Pricing Model
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34. vinzjeannin@hotmail.com
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Infinite number of long only portfolios
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35. vinzjeannin@hotmail.com
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Would you buy just Air Liquide?
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You’d only invest on the so called Efficient Frontier
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37. vinzjeannin@hotmail.com
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For a particular return, you take the lowest risk
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For a particular risk, you take the highest return

38. vinzjeannin@hotmail.com
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Unless there’s a risk free rate
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39. Straight forward, mean is linear, weighted average
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For a particular combination you need to calculate the expected return
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40. We already know
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For a particular combination you need to calculate the variance (or SD)
Not enough, need the general case for a bigger number of assets
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41. No linear formula to select the good one
Need a computer and algorithms
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Millions of portfolio
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Thinking Algorithmic
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