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




                              vinzjeannin@hotmail.com
                                    1
ESGF 4IFM Q1 2012
Summary of the session (est. 4.5h)

•   R Steps by Steps
•   Reminders of last session
•   The Value at Risk




                                     vinzjeannin@hotmail.com
•   OLS & Exploration




                                           2
R Step by Step
    Downloadable for free (open source)




                                          ESGF 4IFM Q1 2012
    http://www.r-project.org/




                                          vinzjeannin@hotmail.com
                                                3
Main screen




    vinzjeannin@hotmail.com   ESGF 4IFM Q1 2012
4
Menu: File / New Script




    vinzjeannin@hotmail.com   ESGF 4IFM Q1 2012
5
Step 1, upload your data


       Excel CSV file easy to import




                                                                          ESGF 4IFM Q1 2012
                      Path C:UsersvinDesktop




                                                                          vinzjeannin@hotmail.com
                                           Note: 4 columns with headers




                                                                                6
DATA<-read.csv(file="C:/Users/vin/Desktop/DataFile.csv",header=T)
Run your instruction(s)




    vinzjeannin@hotmail.com   ESGF 4IFM Q1 2012
7
You can call variables anytime you want




                                          ESGF 4IFM Q1 2012
                                          vinzjeannin@hotmail.com
                                                8
vinzjeannin@hotmail.com   ESGF 4IFM Q1 2012
9
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




                                                                                        ESGF 4IFM Q1 2012
  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

summary(DATA$SPX)           Shows a quick summary of the distribution of one variable




                                                                                        vinzjeannin@hotmail.com
               Min. 1st Qu. Median    Mean 3rd Qu.  Max.
                 86.43   95.70 100.80   99.67 103.80 107.20


                                                 min(DATA)
      Careful using the following instructions   max(DATA)
                                                                   > min(DATA)
                                                                   [1] -0.08832874
                       This will consider DATA as one variable     > max(DATA)
                                                                   [1] 123.4793

                      > sd(DATA)
                             SPX       SPXr      AMEXr       AMEX
                      4.92763551 0.01468776 6.03035318 0.01915489                       10
       Mean & SD      > mean(DATA)
                               SPX         SPXr        AMEXr         AMEX
                      9.967046e+01 1.249283e-04 1.093951e+02 5.890780e-04
Easy to show histogram




                                                                       ESGF 4IFM Q1 2012
                                                                       vinzjeannin@hotmail.com
hist(DATA$SPXr, breaks=25, main="Distribution of SPXr", ylab="Freq",   11
                      xlab="SPXr", col="blue")
Obvious Excess Kurtosis




                                                                    ESGF 4IFM Q1 2012
                                          Obvious Asymmetry




                                                                    vinzjeannin@hotmail.com
Functions doesn’t exists directly in R…

However some VNP (Very Nice Programmer) built and shared add-in


              Package Moments                                       12
Menu: Packages / Install Package(s)




                                                      ESGF 4IFM Q1 2012
                                                      vinzjeannin@hotmail.com
• Choose whatever mirror (server) you want
• Usually France (Toulouse) is very good as it’s a
  University Server with all the packages available




                                                      13
ESGF 4IFM Q1 2012
Once installed, you can load them with the
following instructions:
   require(moments)
   library(moments)




                                             vinzjeannin@hotmail.com
         New functions can now be used!




                                             14
> require(moments)
> library(moments)
> skewness(DATA)
       SPX       SPXr      AMEXr      AMEX
-0.6358029 -0.4178701 0.1876994 -0.2453693




                                              ESGF 4IFM Q1 2012
> kurtosis(DATA)
     SPX     SPXr    AMEXr     AMEX
2.411177 5.671254 2.078366 5.770583




                                              vinzjeannin@hotmail.com
Btw, you can store any result in a variable




   > Kur<-kurtosis(DATA$SPXr)
   > Kur
   [1] 5.671254


                                              15
Lost?

Call the help!   help(kurtosis)




                                                            ESGF 4IFM Q1 2012
                            Reminds you the package




                                                            vinzjeannin@hotmail.com
                                   Syntax



                                     Arguments definition

                                                            16
Let’s store a few values
                   SPMean<-mean(DATA$SPXr)
                   SPSD<-sd(DATA$SPXr)                  Package Stats

                    Build a sequence, the x axis




                                                                                            ESGF 4IFM Q1 2012
                   x<-seq(from=SPMean-4*SPSD,to=SPMean+4*SPSD,length=500)


                    Build a normal density on these x




                                                                                            vinzjeannin@hotmail.com
                    Y1<-dnorm(x,mean=SPMean,sd=SPSD)                     Package Stats

                    Display the histogram
hist(DATA$SPXr, breaks=25,main="S&P Returns / Normal                     Package graphics
Distribution",xlab="Returns",ylab="Occurences", col="blue")


                    Display on top of it the normal density

               lines(x,y1,type="l",lwd=3,col="red")                     Package graphics    17
ESGF 4IFM Q1 2012
                                           vinzjeannin@hotmail.com
                                           18

Positive Excess Kurtosis & Negative Skew
Let’s build a spread   Spd<-DATA$SPXr-DATA$AMEX




What is the mean?




                                                             ESGF 4IFM Q1 2012
      Mean is linear   ������ ������������ + ������������ = ������������ ������ + ������������(������)

                        ������ ������ − ������ = ������ ������ − ������(������)




                                                             vinzjeannin@hotmail.com
      Let’s verify


> mean(DATA$SPXr)-mean(DATA$AMEX)-mean(Spd)
[1] 0


                                                             19
What is the standard deviation?


                    Is standard deviation linear?
                                                       NO!




                                                                                         ESGF 4IFM Q1 2012
                     VAR ������������ + ������������ = ������2 ������������������ ������ + ������2 ������ ������ + 2������������������������������(������, ������)


> (var(DATA$SPXr)+var(DATA$AMEX)-2*cov(DATA$SPXr,DATA$AMEX))^0.5




                                                                                         vinzjeannin@hotmail.com
[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                    20
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")




                                                                              ESGF 4IFM Q1 2012
points(sd(Portf),mean(Portf),col="green")




                                                                              vinzjeannin@hotmail.com
                                                                              21
The efficient frontier




     vinzjeannin@hotmail.com   ESGF 4IFM Q1 2012
22
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")




                                                                             ESGF 4IFM Q1 2012
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")




                                                                             vinzjeannin@hotmail.com
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")
                                                                             23
plot(DATA$AMEX,DATA$SPXr)
abline(lm(DATA$AMEX~DATA$SPXr), col="blue")




                                              ESGF 4IFM Q1 2012
                                              vinzjeannin@hotmail.com
                                              24
LM stands for Linear Models


> lm(DATA$AMEX~DATA$SPXr)




                                                       ESGF 4IFM Q1 2012
Call:
lm(formula = DATA$AMEX ~ DATA$SPXr)

Coefficients:
(Intercept)      DATA$SPXr
  0.0004505      1.1096287




                                                       vinzjeannin@hotmail.com
 ������ = 1.1096������ + 0.04%


Will be used later for linear regression and hedging


                                                       25
Do you remember what is the most platykurtic distribution in the nature?



                                  Toss         Head = Success = 1 / Tail = Failure = 0




                                                                                         ESGF 4IFM Q1 2012
                               100 toss… Else memory issue…

> require(moments)
Loading required package: moments
> library(moments)




                                                                                         vinzjeannin@hotmail.com
> 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
                                                                                         26
                     Let’s test the fairness
Density of a binomial distribution

                                     ������ + 1 ! ℎ
             ������ ������ ������ = ℎ, ������ = ������ =         ������ (1 − ������)������
                                      ℎ! ������!




                                                             ESGF 4IFM Q1 2012
               Let’s plot this density with

                 ℎ = 52
                 ������ = 48




                                                             vinzjeannin@hotmail.com
                 ������ = 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")
                                                             27
If the probability between 45% and 55% is significant we’ll accept the fairness




                                                                                  ESGF 4IFM Q1 2012
                                                                                  vinzjeannin@hotmail.com
                                                                                  28

                What do you think?
What is the problem with this coin?

                                 Obvious fake! Assuming the probability of head is 0.7

                                 Toss it! Head = Success = 1 / Tail = Failure = 0




                                                                                         ESGF 4IFM Q1 2012
                                100 toss

> require(moments)
Loading required package: moments
> library(moments)




                                                                                         vinzjeannin@hotmail.com
> 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
                                                                                         29

                  Let’s test the fairness (assuming you don’t know it’s a trick)
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




                                                                                   ESGF 4IFM Q1 2012
plot(r,y,type="l",col="red",main="Probability density or r given 72
head out 100 flips")




                                                                                   vinzjeannin@hotmail.com
                                                            Trick coin!



                                                                                   30
Reminders of last session




                                   ESGF 4IFM Q1 2012
Normal Standard Distribution

           Snapshot, 4 moments:




                                   vinzjeannin@hotmail.com
           Mean                0
           SD                  1
           Skewness            0
           Kurtosis            3




                                   31
������ ������ ≤ ������ = 0.5                                       ������ ������ − ������ ≤ ������ ≤ ������ + ������ = 0.682
                       ������ ������ ≤ −������ + ������ = 0.159
������ ������ ≤ −1.645 ∗ ������ + ������ = 0.05                ������ ������ − 2 ∗ ������ ≤ ������ ≤ ������ + 2 ∗ ������   = 0.954
                        ������ ������ ≤ −2 ∗ ������ + ������ = 0.023




                                                                                             ESGF 4IFM Q1 2012
������ ������ ≤ −2.326 ∗ ������ + ������ = 0.01                ������ ������ − 3 ∗ ������ ≤ ������ ≤ ������ + 3 ∗ ������ = 0.996
                        ������ ������ ≤ −3 ∗ ������ + ������ = 0.001




                                                                                             vinzjeannin@hotmail.com
                                                                                             32
Notation   ������(������, ������)
                         1           (������−������)2
                                    −
Density    ������ ������ =                ������ 2������2
                        2������������ 2




                                                              ESGF 4IFM Q1 2012
                                            ������
CDF        ������ ������ ≤ ������ = ������ ������ =                  ������ ������ ������������
                                           −∞




                                                              vinzjeannin@hotmail.com
                                                              33
Let be X~N(1,1.5)
      Find:
        ������ ������ ≤ 4.75




                                ESGF 4IFM Q1 2012
                       4.75−1
������ ������ ≤ 4.75 =P ������ ≤
                         1.5

     With Y~N(0,1)
      P ������ ≤ 2.5 =?




                                vinzjeannin@hotmail.com
          Use the table!


      P ������ ≤ −2.5 =0.0062

      P ������ ≤ 2.5 =0.9938


                                34

     P ������ ≤ 4.75 =0.9938
QQ Plot




                     >qqnorm(FCOJ$V1)




                                        ESGF 4IFM Q1 2012
                     >qqline(FCOJ$V1)




                                        vinzjeannin@hotmail.com
Fat Tail




                                        35
Geometric Brownian Motion

             Based on Stochastic Differential Equation             ������������������ = ������������������ ������������ + ������������������ ������������

                   B&S




                                                                                                                     ESGF 4IFM Q1 2012
                                                                                                                     vinzjeannin@hotmail.com
             Discrete form ������������������ = ������������������ ������������ + ������������������ ������������������   with ������~N(0,1)

                   CRR                                                      S������ ������������ = ������������������ + 1 − ������ ������������
                                                                              ������ ������������ = ������������ + 1 − ������ ������
                                                                                            ������ ������������ − ������
������ = ������ ������    ������                                                                      ������ =
                                                                                             ������ − ������
    1                                                 BV= OpUp ∗ p + OpDown ∗ 1 − p                     ∗ ������ −������������   36
������ = = ������ −������        ������
    ������
Greeks Approximation – Taylor Development



                      1
������������ = ������ + ∆ ∗ ������������ + ∗ ������ ∗ ������������ 2




                                                                                    ESGF 4IFM Q1 2012
                      2

                                   1
                                  + ∗ ������������������������������ ∗ ������������ 3
                                   6




                                                                                    vinzjeannin@hotmail.com
                                                      1
                                                 +      ∗ ������������������������������4������ℎ ∗ ������������ 4
                                                     24




                                                                       etc…


                                                                                    37
The Value at Risk
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




                                                                                       ESGF 4IFM Q1 2012
the distribution




                                                                                       vinzjeannin@hotmail.com
3 Methods

          • Historical
          • Parametrical
          • Monte-Carlo
                                                                                       38

              For now, we’ll focus on VaR on one linear asset… FCOJ is back!
Historical VaR


•   No assumption about the distribution
•   Easy to implement and calculate




                                                                                    ESGF 4IFM Q1 2012
•   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




                                                                                    vinzjeannin@hotmail.com
                 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
                                                                                    39

          Works as well on weekly, monthly, quarterly series
Historical VaR




                                                            ESGF 4IFM Q1 2012
            Can be worked as well with prices variations
            instead of returns but it’s going to be price
            sensitive! So careful to the bias.




                                                            vinzjeannin@hotmail.com
• 1% Percentile in term of price movement is -$8.11 cents
• 5% Percentile in term of price movement is -$4.14 cents




                                                            40
Parametric VaR


• Easy to implement and calculate
• Assumes a particular shape of the distribution




                                                                            ESGF 4IFM Q1 2012
• Not really sensitive to fat tails


             FCOJ Mean Return: 0.1364%




                                                                            vinzjeannin@hotmail.com
             FCOJ SD: 2.1664%

   We already know:

        ������ ������ ≤ −1.645 ∗ ������ + ������ = 0.05
        ������ ������ ≤ −2.326 ∗ ������ + ������ = 0.01
     Then:

         ������ ������ ≤ −3.43% = 0.05                     VaR 95% (-$5.15 cents)   41
         ������ ������ ≤ −4.90% = 0.01                     VaR 99% (-$7.35 cents)
Parametric VaR


Very often you assume anyway a 0 mean, therefore:




                                                                               ESGF 4IFM Q1 2012
          ������ ������ ≤ −3.57% = 0.05                       VaR 95% (-$5.36 cents)
          ������ ������ ≤ −5.04% = 0.01                       VaR 99% (-$8.10 cents)




                                                                               vinzjeannin@hotmail.com
       Lower values than the historical VaR


       Problem with leptokurtic distributions, impact of fat tails isn’t
       strong on the method

                                                                               42
Monte Carlo VaR




                                                                          ESGF 4IFM Q1 2012
• Most efficient method when asset aren’t linear
• Tough to implement
• Assumes a particular shape of the distribution




                                                                          vinzjeannin@hotmail.com
     Based on an assumption of a price process (for example GBM)

     Great number of random simulations on the price process to build a
     distribution and outline the VaR




                                                                          43
Monte Carlo VaR


      Let’s simulate 10,000 GBM, 252 steps and store the final result




                                                                               ESGF 4IFM Q1 2012
library(sde)
require(sde)
FCOJ<-
read.csv(file="C:/Users/Vinz/Desktop/FCOJStats.csv",head=FALSE,sep=",")

Drift<-mean(FCOJ$V1)




                                                                               vinzjeannin@hotmail.com
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)

     Don’t be fooled by the 252, we’re still making a daily simulation: what   44
     to change in the code to make it yearly?
Monte Carlo VaR



                > quantile(Final, 0.05)
                    5%




                                                    ESGF 4IFM Q1 2012
                144.93
                > quantile(Final, 0.01)
                       1%
                142.7941




                                                    vinzjeannin@hotmail.com
• 95% Daily VaR is -$5.07 cents
• 99% Daily VaR is -$7.21 cents




                 Let’s take off the drift
                                                    45
Monte Carlo VaR



                > quantile(Final, 0.05)
                      5%




                                                    ESGF 4IFM Q1 2012
                144.7583
                > quantile(Final, 0.01)
                      1%
                142.6412




                                                    vinzjeannin@hotmail.com
• 95% Daily VaR is -$5.35 cents
• 99% Daily VaR is -$7.36 cents




                                                    46
Which is the best?
                                                                        Comparison




                          vinzjeannin@hotmail.com   ESGF 4IFM Q1 2012
47
Going forward on the VaR




                                                ESGF 4IFM Q1 2012
All method give different but coherent values

Easy? Yes but…

 • We’ve involved one asset only




                                                vinzjeannin@hotmail.com
 • We’ve involved a linear asset




What about an option?

What about 2 assets?

                                                48
Going forward on the VaR


Portfolio scale: what to look at to calculate the VaR?




                                                                                     ESGF 4IFM Q1 2012
               Big question, is the VaR additive?




                                                                                     vinzjeannin@hotmail.com
                            NO!
                 Keywords for the future: covariance, correlation, diversification

                                                                                     49
Going forward on the VaR


Options: what to look at to calculate the VaR?




                                                                               ESGF 4IFM Q1 2012
            4 risk factors:
            • Underlying price
            • Interest rate
            • Volatility




                                                                               vinzjeannin@hotmail.com
            • 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?         50

                 Might be a bit more complicated than expected…
OLS & Exploration
         OLS: Ordinary Least Square




                                                                           ESGF 5IFM Q1 2012
         Linear regression model
         Minimize the sum of the square vertical distances
         between the observations and the linear
         approximation




                                                                           vinzjeannin@hotmail.com
                                                  ������ = ������ ������ = ������������ + ������

                                                    Residual ε




                                                                           51
Two parameters to estimate:
   • Intercept α
   • Slope β




                                                                     ESGF 5IFM Q1 2012
Minimising residuals


           ������                ������

   ������ =          ������������ 2 =          ������������ − ������������������ + ������   2




                                                                     vinzjeannin@hotmail.com
          ������=1              ������=1




          When E is minimal?



                     When partial derivatives i.r.w. a and b are 0
                                                                     52
������                ������                                                 ������

������ =            ������������ 2 =             ������������ − ������������������ + ������               2   =          ������������ − ������������������ − ������      2

         ������=1              ������=1                                               ������=1


                    Quick high school reminder if necessary…




                                                                                                                                        ESGF 5IFM Q1 2012
       ������������ − ������������������ − ������        2    = ������������ 2 − 2������������������ ������������ − 2������������������ + ������ 2 ������������ 2 + 2������������������������ + ������2



               ������                                                                                    ������
������������                                                                                  ������������




                                                                                                                                        vinzjeannin@hotmail.com
     =               −2������������ ������������ + 2������������������ 2 + 2������������������ = 0                                 =               −2������������ + 2������ + 2������������������ = 0
������������                                                                                  ������������
              ������=1                                                                                  ������=1

 ������                                                                                     ������

       −������������ ������������ + ������������������ 2 + ������������������ = 0                                                     −������������ + ������ + ������������������ = 0
������=1                                                                                   ������=1
         ������                          ������             ������                                         ������                     ������

������ ∗           ������������ 2 + ������ ∗              ������������ =          ������������ ������������                   ������ ∗           ������������ + ������������ =          ������������
       ������=1                      ������=1              ������=1                                       ������=1                   ������=1
                                                                                                                                        53
������������
       Leads easily to the intercept
������������
                       ������                     ������

               ������ ∗          ������������ + ������������ =          ������������
                      ������=1                   ������=1




                                                            ESGF 5IFM Q1 2012
               ������������������ + ������������ = ������������


              ������������ + ������ = ������




                                                            vinzjeannin@hotmail.com
              ������ = ������ − ������������


       The regression line is going through (������ , ������)


       The distance of this point to the line is 0 indeed

                                                            54
������ = ������ − ������������               y = ������������ + ������ − ������������

                                      y − ������ = ������(������ − ������ )




                                                                                                            ESGF 5IFM Q1 2012
          ������                                                             ������
������������                                                          ������������
     =          −2������������ ������������ + 2������������������ 2 + 2������������������ = 0              =           −2������������ + 2������ + 2������������������ = 0
������������                                                          ������������
         ������=1                                                           ������=1

  ������                                                            ������




                                                                                                            vinzjeannin@hotmail.com
        ������������ ������������ − ������������������ − ������ = 0                                   ������������ − ������ − ������������������ = 0
 ������=1                                                          ������=1
  ������
                                                                ������
        ������������ ������������ − ������������������ − ������ + ������������ = 0
 ������=1
                                                                      ������������ − ������ + ������������ − ������������������ = 0
                                                               ������=1
   ������                                                            ������

         ������������ (������������ − ������ − ������ ������������ − ������ ) = 0                         (������������ − ������) − ������(������������ − ������ ) = 0
  ������=1                                                         ������=1
                                                                 ������                                         55
                                                                      ������ ( ������������ − ������ − ������ ������������ − ������ ) = 0
                                                               ������=1
We have
 ������                                                           ������

       ������������ (������������ − ������ − ������ ������������ − ������ ) = 0       and               ������ ( ������������ − ������ − ������ ������������ − ������ ) = 0
������=1                                                       ������=1




                                                                                                            ESGF 5IFM Q1 2012
                   ������                                          ������

                         ������������ (������������ − ������ − ������ ������������ − ������ ) =          ������ ( ������������ − ������ − ������ ������������ − ������ )
                  ������=1                                        ������=1


                   ������                                          ������




                                                                                                            vinzjeannin@hotmail.com
                         ������������ (������������ − ������ − ������ ������������ − ������ ) −          ������ ������������ − ������ − ������ ������������ − ������       =0
                  ������=1                                        ������=1

                   ������

                         (������������ −������ )(������������ − ������ − ������ ������������ − ������ ) = 0
                  ������=1


                                                 Finally…

                                            ������
                                            ������=1(������������ −������ )(������������ −    ������)                                   56
                                   ������ =         ������               2
                                                ������=1(������������ −������ )
������                                       Covariance
       ������=1(������������ − ������ )(������������ −   ������)
������ =        ������               2
            ������=1(������������ − ������ )                    Variance




                                                             ESGF 5IFM Q1 2012
                                              ������������������������������
                                       ������ =
                                                ������2������




                                                             vinzjeannin@hotmail.com
                                       ������ = ������ − ������������




       You can use Excel function INTERCEPT and SLOPE


                                                             57
Calculate the Variances and Covariance of X{1,2,3,3,1,2} and Y{2,3,1,1,3,2}




                                                                              ESGF 5IFM Q1 2012
                                                                              vinzjeannin@hotmail.com
                                                                              58

      You can use Excel function VAR.P, COVARIANCE.P and STDEV.P
Let’s asses the quality of the regression

Let’s calculate the correlation coefficient (aka Pearson Product-Moment
Correlation Coefficient – PPMCC):




                                                                          ESGF 5IFM Q1 2012
                    ������������������������������
             ������ =                       Value between -1 and 1
                     ������������ ������������


               ������ = 1




                                                                          vinzjeannin@hotmail.com
                                        Perfect dependence


               ������ ~0                    No dependence




    Give an idea of the dispersion of the scatterplot
                                                                          59

    You can use Excel function CORREL
Poor quality
                    R=0.62
                                                                            R=0.96

                                                             High quality




                         vinzjeannin@hotmail.com   ESGF 5IFM Q1 2012
60
What is good quality?




                                                                           ESGF 5IFM Q1 2012
      Slightly discretionary…




                                                                           vinzjeannin@hotmail.com
If
             3
     ������ ≥      = 0.8666 …
            2
            It’s largely admitted as the threshold for acceptable / poor




                                                                           61
The regression itself introduces a bias


                  Let’s introduce the coefficient of determination R-Squared




                                                                                 ESGF 5IFM Q1 2012
Total Dispersion = Dispersion Regression + Dispersion Residual




                                                                                 vinzjeannin@hotmail.com
                               2                     2                   2
                   ������������ − ������       =   ������������ − ������������       +   ������������ − ������




                           Dispersion Regression
                ������2 =
                              Total Dispersion

   In other words the part of the total dispersion explained by the regression   62


     You can use Excel function RSQ
In a simple linear regression with intercept ������2 = ������ 2




                                                                         ESGF 5IFM Q1 2012
Is a good correlation coefficient and a good coefficient of
determination enough to accept the regression?




                                                                         vinzjeannin@hotmail.com
  Not necessarily!




  Residuals need to have no effect, in other word to be a white noise!

                                                                         63
vinzjeannin@hotmail.com   ESGF 5IFM Q1 2012
64
Don’t get fooled by numbers!




                                                                   ESGF 5IFM Q1 2012
    For every dataset of the Quarter

                         ������ = 9
                         ������ = 7.5




                                                                   vinzjeannin@hotmail.com
                         ������ = 3 + 0.5������
                         ������ = 0.82
                         ������2 = 0.67




         Can you say at this stage which regression is the best?
                                                                   65

Certainly not those on the right you need a LINEAR dependence
ESGF 5IFM Q1 2012
Is any linear regression useless?




                                                                              vinzjeannin@hotmail.com
               Think what you could do to the series



               Polynomial transformation, log transformation,…


                                                                              66
                       Else, non linear regressions, but it’s another story
First application on financial market


     S&P / AmEx in 2011




                                        ESGF 5IFM Q1 2012
                                        vinzjeannin@hotmail.com
                                        67
������������������������������������������,������&������
                         ������ =                      = 0.8501
                               ������������������������������ ������������&������


                              ������2 = ������ 2 = 0.7227




                                                                              ESGF 5IFM Q1 2012
    Oups :-o
    Is Excel wrong?




                                                                              vinzjeannin@hotmail.com
               R-Squared has different calculation methods




Let’s accept the following regression then as the quality seems pretty good

                      ������������������������������ = 0.06% + 1.1046 ∗ ������������&������

                                                                              68
How to use this?




                                                                          ESGF 5IFM Q1 2012
     • Forecasting?              Not really…
                                 Both are random variables




                                                                          vinzjeannin@hotmail.com
     • Hedging?                  Yes but basis risk
                                 Yes but careful to the residuals…


               In theory, what is the daily result of the hedge?     ������


Let’s have a try!

                                                                          69
Hedging $1.0M of AmEx Stocks with $1.1046M of S&P




                                                        ESGF 5IFM Q1 2012
                                                        vinzjeannin@hotmail.com
 It would have been too easy… Great differences… Why?


            Sensitivity to the size of the sample
                                                        70
            Heteroscedasticity         Basis Risk
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




                                                                                      ESGF 5IFM Q1 2012
But attention!
    We are looking for causation, not correlation!

    Causation implies correlation




                                                                                      vinzjeannin@hotmail.com
    Reciprocity is not true!




    DON’T BE FOOLED BY PRETTY NUMBERS
                                                                                      71
                       Let prove this…
ESGF 5IFM Q1 2012
                                 vinzjeannin@hotmail.com
Perfect linear dependence

Excellent R-Squared
                                 72
Residuals are a white noise

      What’s the problem then?
ESGF 5IFM Q1 2012
                                                          vinzjeannin@hotmail.com
Do you really think fresh lemon reduces car fatalities?
                                                          73
vinzjeannin@hotmail.com   ESGF 5IFM Q1 2012
74
Conclusion

                                                        R




                         VaR

                OLS
                                  Normal Distribution




     vinzjeannin@hotmail.com   ESGF 4IFM Q1 2012
75

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Applied Statistics II

  • 1. ESGF 4IFM Q1 2012 Applied Statistics Vincent JEANNIN – ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com 1
  • 2. ESGF 4IFM Q1 2012 Summary of the session (est. 4.5h) • R Steps by Steps • Reminders of last session • The Value at Risk vinzjeannin@hotmail.com • OLS & Exploration 2
  • 3. R Step by Step Downloadable for free (open source) ESGF 4IFM Q1 2012 http://www.r-project.org/ vinzjeannin@hotmail.com 3
  • 4. Main screen vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 4
  • 5. Menu: File / New Script vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 5
  • 6. Step 1, upload your data Excel CSV file easy to import ESGF 4IFM Q1 2012 Path C:UsersvinDesktop vinzjeannin@hotmail.com Note: 4 columns with headers 6 DATA<-read.csv(file="C:/Users/vin/Desktop/DataFile.csv",header=T)
  • 7. Run your instruction(s) vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 7
  • 8. You can call variables anytime you want ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com 8
  • 9. vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 9
  • 10. 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 ESGF 4IFM Q1 2012 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 summary(DATA$SPX) Shows a quick summary of the distribution of one variable vinzjeannin@hotmail.com Min. 1st Qu. Median Mean 3rd Qu. Max. 86.43 95.70 100.80 99.67 103.80 107.20 min(DATA) Careful using the following instructions max(DATA) > min(DATA) [1] -0.08832874 This will consider DATA as one variable > max(DATA) [1] 123.4793 > sd(DATA) SPX SPXr AMEXr AMEX 4.92763551 0.01468776 6.03035318 0.01915489 10 Mean & SD > mean(DATA) SPX SPXr AMEXr AMEX 9.967046e+01 1.249283e-04 1.093951e+02 5.890780e-04
  • 11. Easy to show histogram ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com hist(DATA$SPXr, breaks=25, main="Distribution of SPXr", ylab="Freq", 11 xlab="SPXr", col="blue")
  • 12. Obvious Excess Kurtosis ESGF 4IFM Q1 2012 Obvious Asymmetry vinzjeannin@hotmail.com Functions doesn’t exists directly in R… However some VNP (Very Nice Programmer) built and shared add-in Package Moments 12
  • 13. Menu: Packages / Install Package(s) ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com • Choose whatever mirror (server) you want • Usually France (Toulouse) is very good as it’s a University Server with all the packages available 13
  • 14. ESGF 4IFM Q1 2012 Once installed, you can load them with the following instructions: require(moments) library(moments) vinzjeannin@hotmail.com New functions can now be used! 14
  • 15. > require(moments) > library(moments) > skewness(DATA) SPX SPXr AMEXr AMEX -0.6358029 -0.4178701 0.1876994 -0.2453693 ESGF 4IFM Q1 2012 > kurtosis(DATA) SPX SPXr AMEXr AMEX 2.411177 5.671254 2.078366 5.770583 vinzjeannin@hotmail.com Btw, you can store any result in a variable > Kur<-kurtosis(DATA$SPXr) > Kur [1] 5.671254 15
  • 16. Lost? Call the help! help(kurtosis) ESGF 4IFM Q1 2012 Reminds you the package vinzjeannin@hotmail.com Syntax Arguments definition 16
  • 17. Let’s store a few values SPMean<-mean(DATA$SPXr) SPSD<-sd(DATA$SPXr) Package Stats Build a sequence, the x axis ESGF 4IFM Q1 2012 x<-seq(from=SPMean-4*SPSD,to=SPMean+4*SPSD,length=500) Build a normal density on these x vinzjeannin@hotmail.com Y1<-dnorm(x,mean=SPMean,sd=SPSD) Package Stats Display the histogram hist(DATA$SPXr, breaks=25,main="S&P Returns / Normal Package graphics Distribution",xlab="Returns",ylab="Occurences", col="blue") Display on top of it the normal density lines(x,y1,type="l",lwd=3,col="red") Package graphics 17
  • 18. ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com 18 Positive Excess Kurtosis & Negative Skew
  • 19. Let’s build a spread Spd<-DATA$SPXr-DATA$AMEX What is the mean? ESGF 4IFM Q1 2012 Mean is linear ������ ������������ + ������������ = ������������ ������ + ������������(������) ������ ������ − ������ = ������ ������ − ������(������) vinzjeannin@hotmail.com Let’s verify > mean(DATA$SPXr)-mean(DATA$AMEX)-mean(Spd) [1] 0 19
  • 20. What is the standard deviation? Is standard deviation linear? NO! ESGF 4IFM Q1 2012 VAR ������������ + ������������ = ������2 ������������������ ������ + ������2 ������ ������ + 2������������������������������(������, ������) > (var(DATA$SPXr)+var(DATA$AMEX)-2*cov(DATA$SPXr,DATA$AMEX))^0.5 vinzjeannin@hotmail.com [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 20
  • 22. The efficient frontier vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 22
  • 23. 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") ESGF 4IFM Q1 2012 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") vinzjeannin@hotmail.com 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") 23
  • 24. plot(DATA$AMEX,DATA$SPXr) abline(lm(DATA$AMEX~DATA$SPXr), col="blue") ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com 24
  • 25. LM stands for Linear Models > lm(DATA$AMEX~DATA$SPXr) ESGF 4IFM Q1 2012 Call: lm(formula = DATA$AMEX ~ DATA$SPXr) Coefficients: (Intercept) DATA$SPXr 0.0004505 1.1096287 vinzjeannin@hotmail.com ������ = 1.1096������ + 0.04% Will be used later for linear regression and hedging 25
  • 26. Do you remember what is the most platykurtic distribution in the nature? Toss Head = Success = 1 / Tail = Failure = 0 ESGF 4IFM Q1 2012 100 toss… Else memory issue… > require(moments) Loading required package: moments > library(moments) vinzjeannin@hotmail.com > 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 26 Let’s test the fairness
  • 27. Density of a binomial distribution ������ + 1 ! ℎ ������ ������ ������ = ℎ, ������ = ������ = ������ (1 − ������)������ ℎ! ������! ESGF 4IFM Q1 2012 Let’s plot this density with ℎ = 52 ������ = 48 vinzjeannin@hotmail.com ������ = 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") 27
  • 28. If the probability between 45% and 55% is significant we’ll accept the fairness ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com 28 What do you think?
  • 29. What is the problem with this coin? Obvious fake! Assuming the probability of head is 0.7 Toss it! Head = Success = 1 / Tail = Failure = 0 ESGF 4IFM Q1 2012 100 toss > require(moments) Loading required package: moments > library(moments) vinzjeannin@hotmail.com > 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 29 Let’s test the fairness (assuming you don’t know it’s a trick)
  • 30. 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 ESGF 4IFM Q1 2012 plot(r,y,type="l",col="red",main="Probability density or r given 72 head out 100 flips") vinzjeannin@hotmail.com Trick coin! 30
  • 31. Reminders of last session ESGF 4IFM Q1 2012 Normal Standard Distribution Snapshot, 4 moments: vinzjeannin@hotmail.com Mean 0 SD 1 Skewness 0 Kurtosis 3 31
  • 32. ������ ������ ≤ ������ = 0.5 ������ ������ − ������ ≤ ������ ≤ ������ + ������ = 0.682 ������ ������ ≤ −������ + ������ = 0.159 ������ ������ ≤ −1.645 ∗ ������ + ������ = 0.05 ������ ������ − 2 ∗ ������ ≤ ������ ≤ ������ + 2 ∗ ������ = 0.954 ������ ������ ≤ −2 ∗ ������ + ������ = 0.023 ESGF 4IFM Q1 2012 ������ ������ ≤ −2.326 ∗ ������ + ������ = 0.01 ������ ������ − 3 ∗ ������ ≤ ������ ≤ ������ + 3 ∗ ������ = 0.996 ������ ������ ≤ −3 ∗ ������ + ������ = 0.001 vinzjeannin@hotmail.com 32
  • 33. Notation ������(������, ������) 1 (������−������)2 − Density ������ ������ = ������ 2������2 2������������ 2 ESGF 4IFM Q1 2012 ������ CDF ������ ������ ≤ ������ = ������ ������ = ������ ������ ������������ −∞ vinzjeannin@hotmail.com 33
  • 34. Let be X~N(1,1.5) Find: ������ ������ ≤ 4.75 ESGF 4IFM Q1 2012 4.75−1 ������ ������ ≤ 4.75 =P ������ ≤ 1.5 With Y~N(0,1) P ������ ≤ 2.5 =? vinzjeannin@hotmail.com Use the table! P ������ ≤ −2.5 =0.0062 P ������ ≤ 2.5 =0.9938 34 P ������ ≤ 4.75 =0.9938
  • 35. QQ Plot >qqnorm(FCOJ$V1) ESGF 4IFM Q1 2012 >qqline(FCOJ$V1) vinzjeannin@hotmail.com Fat Tail 35
  • 36. Geometric Brownian Motion Based on Stochastic Differential Equation ������������������ = ������������������ ������������ + ������������������ ������������ B&S ESGF 4IFM Q1 2012 vinzjeannin@hotmail.com Discrete form ������������������ = ������������������ ������������ + ������������������ ������������������ with ������~N(0,1) CRR S������ ������������ = ������������������ + 1 − ������ ������������ ������ ������������ = ������������ + 1 − ������ ������ ������ ������������ − ������ ������ = ������ ������ ������ ������ = ������ − ������ 1 BV= OpUp ∗ p + OpDown ∗ 1 − p ∗ ������ −������������ 36 ������ = = ������ −������ ������ ������
  • 37. Greeks Approximation – Taylor Development 1 ������������ = ������ + ∆ ∗ ������������ + ∗ ������ ∗ ������������ 2 ESGF 4IFM Q1 2012 2 1 + ∗ ������������������������������ ∗ ������������ 3 6 vinzjeannin@hotmail.com 1 + ∗ ������������������������������4������ℎ ∗ ������������ 4 24 etc… 37
  • 38. The Value at Risk 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 ESGF 4IFM Q1 2012 the distribution vinzjeannin@hotmail.com 3 Methods • Historical • Parametrical • Monte-Carlo 38 For now, we’ll focus on VaR on one linear asset… FCOJ is back!
  • 39. Historical VaR • No assumption about the distribution • Easy to implement and calculate ESGF 4IFM Q1 2012 • 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 vinzjeannin@hotmail.com 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 39 Works as well on weekly, monthly, quarterly series
  • 40. Historical VaR ESGF 4IFM Q1 2012 Can be worked as well with prices variations instead of returns but it’s going to be price sensitive! So careful to the bias. vinzjeannin@hotmail.com • 1% Percentile in term of price movement is -$8.11 cents • 5% Percentile in term of price movement is -$4.14 cents 40
  • 41. Parametric VaR • Easy to implement and calculate • Assumes a particular shape of the distribution ESGF 4IFM Q1 2012 • Not really sensitive to fat tails FCOJ Mean Return: 0.1364% vinzjeannin@hotmail.com FCOJ SD: 2.1664% We already know: ������ ������ ≤ −1.645 ∗ ������ + ������ = 0.05 ������ ������ ≤ −2.326 ∗ ������ + ������ = 0.01 Then: ������ ������ ≤ −3.43% = 0.05 VaR 95% (-$5.15 cents) 41 ������ ������ ≤ −4.90% = 0.01 VaR 99% (-$7.35 cents)
  • 42. Parametric VaR Very often you assume anyway a 0 mean, therefore: ESGF 4IFM Q1 2012 ������ ������ ≤ −3.57% = 0.05 VaR 95% (-$5.36 cents) ������ ������ ≤ −5.04% = 0.01 VaR 99% (-$8.10 cents) vinzjeannin@hotmail.com Lower values than the historical VaR Problem with leptokurtic distributions, impact of fat tails isn’t strong on the method 42
  • 43. Monte Carlo VaR ESGF 4IFM Q1 2012 • Most efficient method when asset aren’t linear • Tough to implement • Assumes a particular shape of the distribution vinzjeannin@hotmail.com Based on an assumption of a price process (for example GBM) Great number of random simulations on the price process to build a distribution and outline the VaR 43
  • 44. Monte Carlo VaR Let’s simulate 10,000 GBM, 252 steps and store the final result ESGF 4IFM Q1 2012 library(sde) require(sde) FCOJ<- read.csv(file="C:/Users/Vinz/Desktop/FCOJStats.csv",head=FALSE,sep=",") Drift<-mean(FCOJ$V1) vinzjeannin@hotmail.com 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) Don’t be fooled by the 252, we’re still making a daily simulation: what 44 to change in the code to make it yearly?
  • 45. Monte Carlo VaR > quantile(Final, 0.05) 5% ESGF 4IFM Q1 2012 144.93 > quantile(Final, 0.01) 1% 142.7941 vinzjeannin@hotmail.com • 95% Daily VaR is -$5.07 cents • 99% Daily VaR is -$7.21 cents Let’s take off the drift 45
  • 46. Monte Carlo VaR > quantile(Final, 0.05) 5% ESGF 4IFM Q1 2012 144.7583 > quantile(Final, 0.01) 1% 142.6412 vinzjeannin@hotmail.com • 95% Daily VaR is -$5.35 cents • 99% Daily VaR is -$7.36 cents 46
  • 47. Which is the best? Comparison vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 47
  • 48. Going forward on the VaR ESGF 4IFM Q1 2012 All method give different but coherent values Easy? Yes but… • We’ve involved one asset only vinzjeannin@hotmail.com • We’ve involved a linear asset What about an option? What about 2 assets? 48
  • 49. Going forward on the VaR Portfolio scale: what to look at to calculate the VaR? ESGF 4IFM Q1 2012 Big question, is the VaR additive? vinzjeannin@hotmail.com NO! Keywords for the future: covariance, correlation, diversification 49
  • 50. Going forward on the VaR Options: what to look at to calculate the VaR? ESGF 4IFM Q1 2012 4 risk factors: • Underlying price • Interest rate • Volatility vinzjeannin@hotmail.com • 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? 50 Might be a bit more complicated than expected…
  • 51. OLS & Exploration OLS: Ordinary Least Square ESGF 5IFM Q1 2012 Linear regression model Minimize the sum of the square vertical distances between the observations and the linear approximation vinzjeannin@hotmail.com ������ = ������ ������ = ������������ + ������ Residual ε 51
  • 52. Two parameters to estimate: • Intercept α • Slope β ESGF 5IFM Q1 2012 Minimising residuals ������ ������ ������ = ������������ 2 = ������������ − ������������������ + ������ 2 vinzjeannin@hotmail.com ������=1 ������=1 When E is minimal? When partial derivatives i.r.w. a and b are 0 52
  • 53. ������ ������ ������ ������ = ������������ 2 = ������������ − ������������������ + ������ 2 = ������������ − ������������������ − ������ 2 ������=1 ������=1 ������=1 Quick high school reminder if necessary… ESGF 5IFM Q1 2012 ������������ − ������������������ − ������ 2 = ������������ 2 − 2������������������ ������������ − 2������������������ + ������ 2 ������������ 2 + 2������������������������ + ������2 ������ ������ ������������ ������������ vinzjeannin@hotmail.com = −2������������ ������������ + 2������������������ 2 + 2������������������ = 0 = −2������������ + 2������ + 2������������������ = 0 ������������ ������������ ������=1 ������=1 ������ ������ −������������ ������������ + ������������������ 2 + ������������������ = 0 −������������ + ������ + ������������������ = 0 ������=1 ������=1 ������ ������ ������ ������ ������ ������ ∗ ������������ 2 + ������ ∗ ������������ = ������������ ������������ ������ ∗ ������������ + ������������ = ������������ ������=1 ������=1 ������=1 ������=1 ������=1 53
  • 54. ������������ Leads easily to the intercept ������������ ������ ������ ������ ∗ ������������ + ������������ = ������������ ������=1 ������=1 ESGF 5IFM Q1 2012 ������������������ + ������������ = ������������ ������������ + ������ = ������ vinzjeannin@hotmail.com ������ = ������ − ������������ The regression line is going through (������ , ������) The distance of this point to the line is 0 indeed 54
  • 55. ������ = ������ − ������������ y = ������������ + ������ − ������������ y − ������ = ������(������ − ������ ) ESGF 5IFM Q1 2012 ������ ������ ������������ ������������ = −2������������ ������������ + 2������������������ 2 + 2������������������ = 0 = −2������������ + 2������ + 2������������������ = 0 ������������ ������������ ������=1 ������=1 ������ ������ vinzjeannin@hotmail.com ������������ ������������ − ������������������ − ������ = 0 ������������ − ������ − ������������������ = 0 ������=1 ������=1 ������ ������ ������������ ������������ − ������������������ − ������ + ������������ = 0 ������=1 ������������ − ������ + ������������ − ������������������ = 0 ������=1 ������ ������ ������������ (������������ − ������ − ������ ������������ − ������ ) = 0 (������������ − ������) − ������(������������ − ������ ) = 0 ������=1 ������=1 ������ 55 ������ ( ������������ − ������ − ������ ������������ − ������ ) = 0 ������=1
  • 56. We have ������ ������ ������������ (������������ − ������ − ������ ������������ − ������ ) = 0 and ������ ( ������������ − ������ − ������ ������������ − ������ ) = 0 ������=1 ������=1 ESGF 5IFM Q1 2012 ������ ������ ������������ (������������ − ������ − ������ ������������ − ������ ) = ������ ( ������������ − ������ − ������ ������������ − ������ ) ������=1 ������=1 ������ ������ vinzjeannin@hotmail.com ������������ (������������ − ������ − ������ ������������ − ������ ) − ������ ������������ − ������ − ������ ������������ − ������ =0 ������=1 ������=1 ������ (������������ −������ )(������������ − ������ − ������ ������������ − ������ ) = 0 ������=1 Finally… ������ ������=1(������������ −������ )(������������ − ������) 56 ������ = ������ 2 ������=1(������������ −������ )
  • 57. ������ Covariance ������=1(������������ − ������ )(������������ − ������) ������ = ������ 2 ������=1(������������ − ������ ) Variance ESGF 5IFM Q1 2012 ������������������������������ ������ = ������2������ vinzjeannin@hotmail.com ������ = ������ − ������������ You can use Excel function INTERCEPT and SLOPE 57
  • 58. Calculate the Variances and Covariance of X{1,2,3,3,1,2} and Y{2,3,1,1,3,2} ESGF 5IFM Q1 2012 vinzjeannin@hotmail.com 58 You can use Excel function VAR.P, COVARIANCE.P and STDEV.P
  • 59. Let’s asses the quality of the regression Let’s calculate the correlation coefficient (aka Pearson Product-Moment Correlation Coefficient – PPMCC): ESGF 5IFM Q1 2012 ������������������������������ ������ = Value between -1 and 1 ������������ ������������ ������ = 1 vinzjeannin@hotmail.com Perfect dependence ������ ~0 No dependence Give an idea of the dispersion of the scatterplot 59 You can use Excel function CORREL
  • 60. Poor quality R=0.62 R=0.96 High quality vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 60
  • 61. What is good quality? ESGF 5IFM Q1 2012 Slightly discretionary… vinzjeannin@hotmail.com If 3 ������ ≥ = 0.8666 … 2 It’s largely admitted as the threshold for acceptable / poor 61
  • 62. The regression itself introduces a bias Let’s introduce the coefficient of determination R-Squared ESGF 5IFM Q1 2012 Total Dispersion = Dispersion Regression + Dispersion Residual vinzjeannin@hotmail.com 2 2 2 ������������ − ������ = ������������ − ������������ + ������������ − ������ Dispersion Regression ������2 = Total Dispersion In other words the part of the total dispersion explained by the regression 62 You can use Excel function RSQ
  • 63. In a simple linear regression with intercept ������2 = ������ 2 ESGF 5IFM Q1 2012 Is a good correlation coefficient and a good coefficient of determination enough to accept the regression? vinzjeannin@hotmail.com Not necessarily! Residuals need to have no effect, in other word to be a white noise! 63
  • 64. vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 64
  • 65. Don’t get fooled by numbers! ESGF 5IFM Q1 2012 For every dataset of the Quarter ������ = 9 ������ = 7.5 vinzjeannin@hotmail.com ������ = 3 + 0.5������ ������ = 0.82 ������2 = 0.67 Can you say at this stage which regression is the best? 65 Certainly not those on the right you need a LINEAR dependence
  • 66. ESGF 5IFM Q1 2012 Is any linear regression useless? vinzjeannin@hotmail.com Think what you could do to the series Polynomial transformation, log transformation,… 66 Else, non linear regressions, but it’s another story
  • 67. First application on financial market S&P / AmEx in 2011 ESGF 5IFM Q1 2012 vinzjeannin@hotmail.com 67
  • 68. ������������������������������������������,������&������ ������ = = 0.8501 ������������������������������ ������������&������ ������2 = ������ 2 = 0.7227 ESGF 5IFM Q1 2012 Oups :-o Is Excel wrong? vinzjeannin@hotmail.com R-Squared has different calculation methods Let’s accept the following regression then as the quality seems pretty good ������������������������������ = 0.06% + 1.1046 ∗ ������������&������ 68
  • 69. How to use this? ESGF 5IFM Q1 2012 • Forecasting? Not really… Both are random variables vinzjeannin@hotmail.com • Hedging? Yes but basis risk Yes but careful to the residuals… In theory, what is the daily result of the hedge? ������ Let’s have a try! 69
  • 70. Hedging $1.0M of AmEx Stocks with $1.1046M of S&P ESGF 5IFM Q1 2012 vinzjeannin@hotmail.com It would have been too easy… Great differences… Why? Sensitivity to the size of the sample 70 Heteroscedasticity Basis Risk
  • 71. 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 ESGF 5IFM Q1 2012 But attention! We are looking for causation, not correlation! Causation implies correlation vinzjeannin@hotmail.com Reciprocity is not true! DON’T BE FOOLED BY PRETTY NUMBERS 71 Let prove this…
  • 72. ESGF 5IFM Q1 2012 vinzjeannin@hotmail.com Perfect linear dependence Excellent R-Squared 72 Residuals are a white noise What’s the problem then?
  • 73. ESGF 5IFM Q1 2012 vinzjeannin@hotmail.com Do you really think fresh lemon reduces car fatalities? 73
  • 74. vinzjeannin@hotmail.com ESGF 5IFM Q1 2012 74
  • 75. Conclusion R VaR OLS Normal Distribution vinzjeannin@hotmail.com ESGF 4IFM Q1 2012 75