Daily Archives: 31/03/2013

2013, ACT2040, Hiver, IBNR

Chain Ladder, avec R

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Un billet rapide pour mettre en ligne des parties du code tapé en cours, mercredi dernier. On avait commencé par convertir la feuille du classeur excel en un fichier texte, pour faciliter la lecture,

> setwd("C:\\Users\\savsalledecours\\Desktop")
> triangle=read.table("exACT2040.csv",header=TRUE,sep=";")	
> triangle
  ANNEE   X0   X1   X2   X3   X4   X5
1  2000 3209 4372 4411 4428 4435 4456
2  2001 3367 4659 4696 4720 4730   NA
3  2002 3871 5345 5398 5420   NA   NA
4  2003 4239 5917 6020   NA   NA   NA
5  2004 4929 6794   NA   NA   NA   NA
6  2005 5217   NA   NA   NA   NA   NA

L’idée – quand on importe un triangle – est de récupérer une base sous la forme précédente, avec des valeurs manquantes dans la partie inférieure du triangle (on verra l’intérêt quand on fait une régression). On avait ensuite calculé les facteurs de transition, et en même temps complété le triangle,

> T=triangle[,2:7]
> rownames(T)=triangle$ANNEE
> T2=T
> n=ncol(T)
> L=rep(NA,n-1)
> for(j in 1:(n-1)){
+ L[j]=sum(T[1:(n-j),j+1])/sum(T[1:(n-j),j])
+ T2[(n-j+1):n,j+1]=L[j]*T2[(n-j+1):n,j]
+ }

Les facteurs de transition sont ici,

> L
[1] 1.380933 1.011433 1.004343 1.001858 1.004735

et le triangle complété

> T2
       X0       X1       X2       X3       X4       X5
2000 3209 4372.000 4411.000 4428.000 4435.000 4456.000
2001 3367 4659.000 4696.000 4720.000 4730.000 4752.397
2002 3871 5345.000 5398.000 5420.000 5430.072 5455.784
2003 4239 5917.000 6020.000 6046.147 6057.383 6086.065
2004 4929 6794.000 6871.672 6901.518 6914.344 6947.084
2005 5217 7204.327 7286.691 7318.339 7331.939 7366.656

Le montant de provision est ici en faisant la différence entre la charge ultime (dans la dernière colonne) et les derniers paiements observés (sur la seconde diagonale)

> CU=T2[,n]
> Pat=diag(as.matrix(T2[,n:1]))
> Ri=CU-Pat
> R=sum(Ri)

soit, numériquement

> R
[1] 2426.985

On avait alors vu que l’on pouvait calculer un tail factor, en supposant une décroissance exponentielle des facteurs de transition, et on rajoutait alors une colonne correspondant au montant ultime, par année d’accident,

> logL=log(L-1)
> t=1:5
> b=data.frame(logL,t)
> reg=lm(logL~t,data=b)
> logLp=predict(reg,newdata=data.frame(t=6:100))
> Lp=exp(logLp)+1
> Linf=prod(Lp)
> T3=T2
> T3$Xinf=T3$X5*Linf

On a ici

> T3
       X0       X1       X2       X3       X4       X5     Xinf
2000 3209 4372.000 4411.000 4428.000 4435.000 4456.000 4459.149
2001 3367 4659.000 4696.000 4720.000 4730.000 4752.397 4755.755
2002 3871 5345.000 5398.000 5420.000 5430.072 5455.784 5459.639
2003 4239 5917.000 6020.000 6046.147 6057.383 6086.065 6090.366
2004 4929 6794.000 6871.672 6901.518 6914.344 6947.084 6951.993
2005 5217 7204.327 7286.691 7318.339 7331.939 7366.656 7371.862

(je laisse reprendre le code pour calculer le montant de provisions). Enfin, on avait montré comment utiliser une régression pondérée, pour calculer les facteurs de transition,

> T4=as.matrix(T$X0,n,1)
> for(j in 1:(n-1)){
+ Y=T[,j+1]
+ X=T[,j]
+ base=data.frame(X,Y)
+ reg=lm(Y~0+X,weights=1/X)
+ T4=cbind(T4,
+ predict(reg,
+ newdata=data.frame(X=T4[,j]
+ )))
+ }

Ce qui donnait la même projection que la méthode Chain Ladder

> T4
  [,1]     [,2]     [,3]     [,4]     [,5]     [,6]
1 3209 4431.414 4482.076 4501.543 4509.909 4531.263
2 3367 4649.601 4702.758 4723.184 4731.961 4754.367
3 3871 5345.591 5406.705 5430.188 5440.279 5466.039
4 4239 5853.775 5920.698 5946.414 5957.464 5985.673
5 4929 6806.619 6884.435 6914.337 6927.186 6959.986
6 5217 7204.327 7286.691 7318.339 7331.939 7366.656

La suite mercredi prochain, même si on risque d’aller très vite sur la méthode de Mack (et les calculs d’erreur quadratique moyenne pour arriver à la régression de Poisson). A suivre donc…

Computer, Time Series

Easter

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This morning, there was an interesting post entitled “why does Easter move around so much?” online on http://economist.com/blogs/economist-explains/…

In my time series classes, I keep saying that sometimes, series can exhibit seasonlity, but the seasonal effect can be quite irregular. It is the cas for river levels, where snowmelt can have a huge impact, and it is irregular. Similarly, chocolate sales (even monthly, or quarterly) depends on Easter. Because it can be either in March, or in April, the seasonal pattern is not as regular as flower sales for instance (Valentine beeing always on February 14th, as far as I remember). If we look at the word eggs on http://google.com/trends/q=eggs…, we do observe a cycle related to Easter.

The title of the article published by http://economist.com/blogs/economist-explains/… claims that there is a lot of variability on Eater’s day. Let us check ! The answer to the question “When is Easter ?” can be the following (if we want a short answer): Easter Sunday is the first Sunday after the first full moon after vernal equinox. For more details, see e.g. http://ortelius.de/east. The algorithm used to compute the date of Easter can is online, on http://smart.net/~mmontes/….

> century = year/100
> G = year % 19
> K = (century - 17)/25
> I = (century - century/4 - (century - K)/3 + 19*G + 15) % 30
> I = I - (I/28)*(1 - (I/28)*(29/(I + 1))*((21 - G)/11))
> J = (year + year/4 + I + 2 - century + century/4) % 7
> L = I - J
> EasterMonth = 3 + (L + 40)/44
> EasterDay = L + 28 - 31*(EasterMonth/4)

Actually, this algorithm can be found in some R packages. Here we use the date of Easter from AD 1000 and AD 3000,

> library(timeDate)
> E=Easter(1000:3000)
> D=as.Date(E)
> table(months(D))/2001

    april     march 
0.7651174 0.2348826

(April being before March, in the alphabetical order) If we look at the distribution of the date, it is the following, the starting point being March 1st,

> J=as.numeric(D-as.Date(paste("01/03/",1000:3000,sep=""),"%d/%m/%Y"))
> hist(J,breaks=seq(20,55),col="light green")

And if we look at the autocorrelation function, we can observe that indeed, after 19 years, there is a strong correlation (that could be seen in the algorithm given previously),

> plot(acf(J))

But in order to get a better understanding of the dynamics, we can also look at transiftion matrices. Define

> Q=quantile(J,seq(0,1,by=.25))
> Q[1]=Q[1]-1
> C=cut(J,Q)

Then, the one year transition matrix is (in %)

> k=1; n=length(C)
> B=data.frame(X1=(C[1:(n-k)]),X2=(C[(k+1):n]))
> (T=table(B$X1,B$X2))

          (20,31] (31,39] (39,46] (46,55]
  (20,31]       0       0     265     277
  (31,39]     316       0      13     182
  (39,46]     224     264       0       0
  (46,55]       1     247     211       0
> P=T/apply(T,1,sum)
> round(P*1000)/10

          (20,31] (31,39] (39,46] (46,55]
  (20,31]     0.0     0.0    48.9    51.1
  (31,39]    61.8     0.0     2.5    35.6
  (39,46]    45.9    54.1     0.0     0.0
  (46,55]     0.2    53.8    46.0     0.0

I.e. if  Easter was early in the year (say in March, in the first quartile), then very likeliy, the year after, it will be late in the year (with 50% chance in the third quartile, and 50% chance in the fourth one).