applied statistics ii
DESCRIPTION
Second course of Applied Statistics, MSc level in Buisiness SchoolTRANSCRIPT
Applied Statistics Vincent JEANNIN – ESGF 4IFM
Q1 2012
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Summary of the session (est. 4.5h) • R Steps by Steps • Reminders of last session • The Value at Risk • OLS & Exploration
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R Step by Step
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http://www.r-project.org/
Downloadable for free (open source)
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Main screen
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Menu: File / New Script
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Step 1, upload your data
Excel CSV file easy to import
Path C:\Users\vin\Desktop
DATA<-read.csv(file="C:/Users/vin/Desktop/DataFile.csv",header=T)
Note: 4 columns with headers
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Run your instruction(s)
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You can call variables anytime you want
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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
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
Min. 1st Qu. Median Mean 3rd Qu. Max.
86.43 95.70 100.80 99.67 103.80 107.20
summary(DATA$SPX) Shows a quick summary of the distribution of one variable
Careful using the following instructions min(DATA)
max(DATA)
This will consider DATA as one variable
> min(DATA)
[1] -0.08832874
> max(DATA)
[1] 123.4793
> sd(DATA)
SPX SPXr AMEXr AMEX
4.92763551 0.01468776 6.03035318 0.01915489
> mean(DATA)
SPX SPXr AMEXr AMEX
9.967046e+01 1.249283e-04 1.093951e+02 5.890780e-04
Mean & SD
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Easy to show histogram
hist(DATA$SPXr, breaks=25, main="Distribution of SPXr", ylab="Freq",
xlab="SPXr", col="blue")
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Obvious Excess Kurtosis Obvious Asymmetry
Functions doesn’t exists directly in R…
However some VNP (Very Nice Programmer) built and shared add-in
Package Moments
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Menu: Packages / Install Package(s)
• Choose whatever mirror (server) you want • Usually France (Toulouse) is very good as it’s a
University Server with all the packages available
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require(moments)
library(moments)
Once installed, you can load them with the following instructions:
New functions can now be used!
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> require(moments)
> library(moments)
> skewness(DATA)
SPX SPXr AMEXr AMEX
-0.6358029 -0.4178701 0.1876994 -0.2453693
> 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$SPXr)
> Kur
[1] 5.671254
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Lost?
Call the help! help(kurtosis)
Reminds you the package
Syntax
Arguments definition
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Let’s store a few values
x<-seq(from=SPMean-4*SPSD,to=SPMean+4*SPSD,length=500)
Build a sequence, the x axis
SPMean<-mean(DATA$SPXr)
SPSD<-sd(DATA$SPXr) Package Stats
Build a normal density on these x
Y1<-dnorm(x,mean=SPMean,sd=SPSD) Package Stats
hist(DATA$SPXr, breaks=25,main="S&P Returns / Normal
Distribution",xlab="Returns",ylab="Occurences", col="blue")
Display the histogram
Display on top of it the normal density
lines(x,y1,type="l",lwd=3,col="red")
Package graphics
Package graphics
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Positive Excess Kurtosis & Negative Skew
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Let’s build a spread Spd<-DATA$SPXr-DATA$AMEX
What is the mean?
Mean is linear 𝐸 𝑎𝑋 + 𝑏𝑌 = 𝑎𝐸 𝑋 + 𝑏𝐸(𝑌)
𝐸 𝑋 − 𝑌 = 𝐸 𝑋 − 𝐸(𝑌)
> mean(DATA$SPXr)-mean(DATA$AMEX)-mean(Spd)
[1] 0
Let’s verify
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What is the standard deviation?
Is standard deviation linear? NO! VAR 𝑎𝑋 + 𝑏𝑌 = 𝑎2𝑉𝐴𝑅 𝑋 + 𝑏2𝐸 𝑌 + 2𝑎𝑏𝐶𝑂𝑉(𝑋, 𝑌)
> (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
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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")
points(sd(Portf),mean(Portf),col="green")
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The efficient frontier
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points(sd(0.1*DATA$SPXr+0.9*DATA$AMEX),mean(0.1*DATA$SPXr+0.9*DATA$AMEX),c
ol="green")
points(sd(0.2*DATA$SPXr+0.8*DATA$AMEX),mean(0.2*DATA$SPXr+0.8*DATA$AMEX),c
ol="green")
points(sd(0.3*DATA$SPXr+0.7*DATA$AMEX),mean(0.3*DATA$SPXr+0.7*DATA$AMEX),c
ol="green")
points(sd(0.4*DATA$SPXr+0.6*DATA$AMEX),mean(0.4*DATA$SPXr+0.6*DATA$AMEX),c
ol="green")
points(sd(0.6*DATA$SPXr+0.4*DATA$AMEX),mean(0.6*DATA$SPXr+0.4*DATA$AMEX),c
ol="green")
points(sd(0.7*DATA$SPXr+0.3*DATA$AMEX),mean(0.7*DATA$SPXr+0.3*DATA$AMEX),c
ol="green")
points(sd(0.8*DATA$SPXr+0.2*DATA$AMEX),mean(0.8*DATA$SPXr+0.2*DATA$AMEX),c
ol="green")
points(sd(0.9*DATA$SPXr+0.1*DATA$AMEX),mean(0.9*DATA$SPXr+0.1*DATA$AMEX),c
ol="green")
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plot(DATA$AMEX,DATA$SPXr)
abline(lm(DATA$AMEX~DATA$SPXr), col="blue")
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LM stands for Linear Models
> lm(DATA$AMEX~DATA$SPXr)
Call:
lm(formula = DATA$AMEX ~ DATA$SPXr)
Coefficients:
(Intercept) DATA$SPXr
0.0004505 1.1096287
𝑦 = 1.1096𝑥 + 0.04%
Will be used later for linear regression and hedging
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Do you remember what is the most platykurtic distribution in the nature?
Toss Head = Success = 1 / Tail = Failure = 0
> 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
Let’s test the fairness
100 toss… Else memory issue…
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𝑓 𝑟 𝐻 = ℎ, 𝑇 = 𝑡 =𝑁 + 1 !
ℎ! 𝑡!𝑟ℎ(1 − 𝑟)𝑡
Density of a binomial distribution
Let’s plot this density with
ℎ = 52
𝑡 = 48
𝑁 = 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")
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If the probability between 45% and 55% is significant we’ll accept the fairness
What do you think?
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What is the problem with this coin?
Toss it! Head = Success = 1 / Tail = Failure = 0
> 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
Let’s test the fairness (assuming you don’t know it’s a trick)
100 toss
Obvious fake! Assuming the probability of head is 0.7
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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
plot(r,y,type="l",col="red",main="Probability density or r given 72
head out 100 flips")
Trick coin!
Reminders of last session
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Snapshot, 4 moments:
Mean
SD
Skewness
Kurtosis
0
1
0
3
Normal Standard Distribution
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𝑃 𝑋 ≤ 𝜇 = 𝑃 𝑋 ≤ −𝜎 + 𝜇
𝑃 𝑋 ≤ −2 ∗ 𝜎 + 𝜇
𝑃 𝑋 ≤ −3 ∗ 𝜎 + 𝜇
𝑃 𝜇 − 𝜎 ≤ 𝑋 ≤ 𝜇 + 𝜎
𝑃 𝜇 − 2 ∗ 𝜎 ≤ 𝑋 ≤ 𝜇 + 2 ∗ 𝜎
𝑃 𝜇 − 3 ∗ 𝜎 ≤ 𝑋 ≤ 𝜇 + 3 ∗ 𝜎
𝑃 𝑋 ≤ −1.645 ∗ 𝜎 + 𝜇
𝑃 𝑋 ≤ −2.326 ∗ 𝜎 + 𝜇
0.5
= 0.05
= 0.01
= 0.159
= 0.023
= 0.001
= 0.682
= 0.954
= 0.996
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𝑓 𝑥 =1
2𝜋𝜎2𝑒−(𝑥−𝜇)2
2𝜎2 Density
𝑁(𝜇, 𝜎) Notation
𝑃 𝑋 ≤ 𝑥 = 𝜙 𝑥 = 𝑓 𝑥 𝑑𝑥𝑥
−∞
CDF
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Let be X~N(1,1.5) Find:
𝑃 𝑋 ≤ 4.75
𝑃 𝑋 ≤ 4.75 =P 𝑌 ≤4.75−1
1.5
With Y~N(0,1)
P 𝑌 ≤ 2.5 =?
Use the table!
P 𝑌 ≤ −2.5 =0.0062
P 𝑋 ≤ 4.75 =0.9938
P 𝑌 ≤ 2.5 =0.9938
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>qqnorm(FCOJ$V1)
>qqline(FCOJ$V1)
Fat Tail
QQ Plot
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Discrete form 𝑑𝑠𝑡 = 𝜇𝑠𝑡𝑑𝑡 + 𝜎𝑠𝑡 𝑑𝑡𝜀
Geometric Brownian Motion
Based on Stochastic Differential Equation 𝑑𝑠𝑡 = 𝜇𝑠𝑡𝑑𝑡 + 𝜎𝑠𝑡𝑊𝑡
with 𝜀~N(0,1)
B&S
CRR
𝑢 = 𝑒𝜎 𝑡
𝑑 =1
𝑢= 𝑒−𝜎 𝑡
S𝑒𝑟𝑡 = 𝑝𝑆𝑢 + 1 − 𝑝 𝑆𝑑 𝑒𝑟𝑡 = 𝑝𝑢 + 1 − 𝑝 𝑑
𝑝 =𝑒𝑟𝑡 − 𝑑
𝑢 − 𝑑
BV= OpUp ∗ p + OpDown ∗ 1 − p ∗ 𝑒−𝑟𝑡
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Greeks Approximation – Taylor Development
𝑑𝐶 = 𝐶 + ∆ ∗ 𝑑𝑆 +1
2∗ 𝛾 ∗ 𝑑𝑆2
+1
6∗ 𝑆𝑝𝑒𝑒𝑑 ∗ 𝑑𝑆3
+1
24∗ 𝐺𝑟𝑒𝑒𝑘4𝑡ℎ ∗ 𝑑𝑆4
etc…
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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 the distribution
3 Methods
• Historical • Parametrical • Monte-Carlo
For now, we’ll focus on VaR on one linear asset… FCOJ is back!
The Value at Risk
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Historical VaR
• No assumption about the distribution • Easy to implement and calculate • 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
Works as well on weekly, monthly, quarterly series
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Historical VaR
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
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Parametric VaR
• Easy to implement and calculate • Assumes a particular shape of the distribution • Not really sensitive to fat tails
FCOJ Mean Return: 0.1364%
𝑃 𝑋 ≤ −1.645 ∗ 𝜎 + 𝜇 = 0.05
𝑃 𝑋 ≤ −2.326 ∗ 𝜎 + 𝜇 = 0.01
FCOJ SD: 2.1664%
We already know:
𝑃 𝑋 ≤ −3.43% = 0.05
Then:
𝑃 𝑋 ≤ −4.90% = 0.01
VaR 95% (-$5.15 cents)
VaR 99% (-$7.35 cents)
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Parametric VaR
𝑃 𝑋 ≤ −3.57% = 0.05
𝑃 𝑋 ≤ −5.04% = 0.01
VaR 95% (-$5.36 cents)
VaR 99% (-$8.10 cents)
Very often you assume anyway a 0 mean, therefore:
Lower values than the historical VaR
Problem with leptokurtic distributions, impact of fat tails isn’t strong on the method
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Monte Carlo VaR
Based on an assumption of a price process (for example GBM)
• Most efficient method when asset aren’t linear • Tough to implement • Assumes a particular shape of the distribution
Great number of random simulations on the price process to build a distribution and outline the VaR
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Monte Carlo VaR
library(sde)
require(sde)
FCOJ<-
read.csv(file="C:/Users/Vinz/Desktop/FCOJStats.csv",head=FALSE,sep=",")
Drift<-mean(FCOJ$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)
Let’s simulate 10,000 GBM, 252 steps and store the final result
Don’t be fooled by the 252, we’re still making a daily simulation: what to change in the code to make it yearly?
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Monte Carlo VaR
> quantile(Final, 0.05)
5%
144.93
> quantile(Final, 0.01)
1%
142.7941
• 95% Daily VaR is -$5.07 cents • 99% Daily VaR is -$7.21 cents
Let’s take off the drift
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Monte Carlo VaR
> quantile(Final, 0.05)
5%
144.7583
> quantile(Final, 0.01)
1%
142.6412
• 95% Daily VaR is -$5.35 cents • 99% Daily VaR is -$7.36 cents
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Comparison
Which is the best?
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Going forward on the VaR
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?
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Going forward on the VaR
Portfolio scale: what to look at to calculate the VaR?
Big question, is the VaR additive?
NO! Keywords for the future: covariance, correlation, diversification
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Going forward on the VaR
Options: what to look at to calculate the VaR?
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?
Might be a bit more complicated than expected…
OLS & Exploration
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Linear regression model
Minimize the sum of the square vertical distances between the observations and the linear approximation
𝑦 = 𝑓 𝑥 = 𝑎𝑥 + 𝑏
Residual ε
OLS: Ordinary Least Square
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Two parameters to estimate: • Intercept α • Slope β
Minimising residuals
𝐸 = 𝜀𝑖2
𝑛
𝑖=1
= 𝑦𝑖 − 𝑎𝑥𝑖 + 𝑏 2
𝑛
𝑖=1
When E is minimal?
When partial derivatives i.r.w. a and b are 0
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𝐸 = 𝜀𝑖2
𝑛
𝑖=1
= 𝑦𝑖 − 𝑎𝑥𝑖 + 𝑏 2
𝑛
𝑖=1
= 𝑦𝑖 − 𝑎𝑥𝑖 − 𝑏 2
𝑛
𝑖=1
𝜕𝐸
𝜕𝑎= −2𝑥𝑖𝑦𝑖 + 2𝑎𝑥𝑖
2 + 2𝑏𝑥𝑖
𝑛
𝑖=1
= 0
𝑦𝑖 − 𝑎𝑥𝑖 − 𝑏 2 = 𝑦𝑖2 − 2𝑎𝑥𝑖𝑦𝑖 − 2𝑏𝑦𝑖 + 𝑎2𝑥𝑖
2 + 2𝑎𝑏𝑥𝑖 + 𝑏2
Quick high school reminder if necessary…
−𝑥𝑖𝑦𝑖 + 𝑎𝑥𝑖2 + 𝑏𝑥𝑖
𝑛
𝑖=1
= 0
𝑎 ∗ 𝑥𝑖2
𝑛
𝑖=1
+ 𝑏 ∗ 𝑥𝑖
𝑛
𝑖=1
= 𝑥𝑖𝑦𝑖
𝑛
𝑖=1
𝜕𝐸
𝜕𝑏= −2𝑦𝑖 + 2𝑏 + 2𝑎𝑥𝑖
𝑛
𝑖=1
= 0
−𝑦𝑖 + 𝑏 + 𝑎𝑥𝑖
𝑛
𝑖=1
= 0
𝑎 ∗ 𝑥𝑖
𝑛
𝑖=1
+ 𝑛𝑏 = 𝑦𝑖
𝑛
𝑖=1
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𝑎 ∗ 𝑥𝑖
𝑛
𝑖=1
+ 𝑛𝑏 = 𝑦𝑖
𝑛
𝑖=1
Leads easily to the intercept
𝑎𝑛𝑥 + 𝑛𝑏 = 𝑛𝑦
𝑎𝑥 + 𝑏 = 𝑦
The regression line is going through (𝑥 , 𝑦 )
The distance of this point to the line is 0 indeed
𝜕𝐸
𝜕𝑏
𝑏 = 𝑦 − 𝑎𝑥
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𝜕𝐸
𝜕𝑎= −2𝑥𝑖𝑦𝑖 + 2𝑎𝑥𝑖
2 + 2𝑏𝑥𝑖
𝑛
𝑖=1
= 0
y = 𝑎𝑥 + 𝑦 − 𝑎𝑥
y − 𝑦 = 𝑎(𝑥 − 𝑥 )
𝑏 = 𝑦 − 𝑎𝑥
𝑥𝑖 𝑦𝑖 − 𝑎𝑥𝑖 − 𝑏 = 0
𝑛
𝑖=1
𝜕𝐸
𝜕𝑏= −2𝑦𝑖 + 2𝑏 + 2𝑎𝑥𝑖 = 0
𝑛
𝑖=1
𝑦𝑖 − 𝑏 − 𝑎𝑥𝑖
𝑛
𝑖=1
= 0
𝑦𝑖 − 𝑦 + 𝑎𝑥 − 𝑎𝑥𝑖 = 0
𝑛
𝑖=1
(𝑦𝑖 − 𝑦 ) − 𝑎(𝑥𝑖 − 𝑥 )
𝑛
𝑖=1
= 0
𝑥𝑖 𝑦𝑖 − 𝑎𝑥𝑖 − 𝑦 + 𝑎𝑥 = 0
𝑛
𝑖=1
𝑥𝑖(𝑦𝑖 − 𝑦 − 𝑎 𝑥𝑖 − 𝑥 )
𝑛
𝑖=1
= 0
𝑥 ( 𝑦𝑖 − 𝑦 − 𝑎 𝑥𝑖 − 𝑥 )
𝑛
𝑖=1
= 0
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𝑥𝑖(𝑦𝑖 − 𝑦 − 𝑎 𝑥𝑖 − 𝑥 )
𝑛
𝑖=1
= 0 𝑥 ( 𝑦𝑖 − 𝑦 − 𝑎 𝑥𝑖 − 𝑥 )
𝑛
𝑖=1
= 0
𝑥𝑖(𝑦𝑖 − 𝑦 − 𝑎 𝑥𝑖 − 𝑥 )
𝑛
𝑖=1
= 𝑥 ( 𝑦𝑖 − 𝑦 − 𝑎 𝑥𝑖 − 𝑥 )
𝑛
𝑖=1
𝑥𝑖(𝑦𝑖 − 𝑦 − 𝑎 𝑥𝑖 − 𝑥 )
𝑛
𝑖=1
− 𝑥 𝑦𝑖 − 𝑦 − 𝑎 𝑥𝑖 − 𝑥
𝑛
𝑖=1
= 0
(𝑥𝑖−𝑥 )(𝑦𝑖 − 𝑦 − 𝑎 𝑥𝑖 − 𝑥 )
𝑛
𝑖=1
= 0
𝑎 = (𝑥𝑖−𝑥 )(𝑦𝑖 − 𝑦 )𝑛
𝑖=1
(𝑥𝑖−𝑥 )2 𝑛𝑖=1
Finally…
We have
and
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𝑎 = (𝑥𝑖 − 𝑥 )(𝑦𝑖 − 𝑦 )𝑛
𝑖=1
(𝑥𝑖 − 𝑥 )2𝑛𝑖=1
Covariance
Variance
𝑎 =𝐶𝑜𝑣𝑥𝑦
𝜎2𝑥
𝑏 = 𝑦 − 𝑎 𝑥
You can use Excel function INTERCEPT and SLOPE
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Calculate the Variances and Covariance of X{1,2,3,3,1,2} and Y{2,3,1,1,3,2}
You can use Excel function VAR.P, COVARIANCE.P and STDEV.P
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Let’s asses the quality of the regression
Let’s calculate the correlation coefficient (aka Pearson Product-Moment Correlation Coefficient – PPMCC):
𝑟 =𝐶𝑜𝑣𝑥𝑦
𝜎𝑥𝜎𝑦 Value between -1 and 1
𝑟 = 1 Perfect dependence
𝑟 ~0 No dependence
Give an idea of the dispersion of the scatterplot
You can use Excel function CORREL
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R=0.96
High quality
R=0.62
Poor quality
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What is good quality?
Slightly discretionary…
𝑟 ≥3
2= 0.8666…
If
It’s largely admitted as the threshold for acceptable / poor
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The regression itself introduces a bias
Let’s introduce the coefficient of determination R-Squared
Total Dispersion = Dispersion Regression + Dispersion Residual
Dispersion Regression
Total Dispersion 𝑅2 =
In other words the part of the total dispersion explained by the regression
𝑦𝑖 − 𝑦 2 = 𝑦𝑖 − 𝑦𝑖 2 + 𝑦𝑖 − 𝑦 2
You can use Excel function RSQ
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In a simple linear regression with intercept 𝑅2 = 𝑟2
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!
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𝑦 = 7.5
𝑥 = 9
𝑦 = 3 + 0.5𝑥
𝑟 = 0.82
𝑅2 = 0.67
Don’t get fooled by numbers!
For every dataset of the Quarter
Can you say at this stage which regression is the best?
Certainly not those on the right you need a LINEAR dependence
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Is any linear regression useless?
Think what you could do to the series
Polynomial transformation, log transformation,…
Else, non linear regressions, but it’s another story
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First application on financial market
S&P / AmEx in 2011
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𝑅𝐴𝑚𝑒𝑥 = 0.06% + 1.1046 ∗ 𝑅𝑆&𝑃
𝑟 =𝐶𝑜𝑣𝐴𝑚𝐸𝑥,𝑆&𝑃
𝜎𝐴𝑚𝐸𝑥𝜎𝑆&𝑃= 0.8501
𝑅2 = 𝑟2 = 0.7227
Oups :-o
Is Excel wrong?
R-Squared has different calculation methods
Let’s accept the following regression then as the quality seems pretty good
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How to use this?
• Forecasting? Not really… Both are random variables
• Hedging? Yes but basis risk Yes but careful to the residuals…
Let’s have a try!
In theory, what is the daily result of the hedge? 𝑎
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Hedging $1.0M of AmEx Stocks with $1.1046M of S&P
It would have been too easy… Great differences… Why?
Sensitivity to the size of the sample
Heteroscedasticity Basis Risk
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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
But attention!
We are looking for causation, not correlation!
Causation implies correlation
Reciprocity is not true!
DON’T BE FOOLED BY PRETTY NUMBERS
Let prove this…
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Perfect linear dependence
Excellent R-Squared
Residuals are a white noise
What’s the problem then?
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Do you really think fresh lemon reduces car fatalities?
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Conclusion
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R
Normal Distribution
VaR
OLS