Datos de las pólizas Distribuciones discretas

datos<-c(rep(1,6),rep(2,180),rep(3,462),rep(4,594),rep(5,741),
         rep(6,831),rep(7,701),rep(8,229),rep(9,80),rep(10,74),rep(11,67)
         ,rep(12,57),rep(13,50),rep(14,43),rep(15,34),rep(16,27),
         rep(17,19),rep(18,11),rep(19,2),rep(20,0))

la Esperanza y varianza observadas son:

esperanza_obs<-mean(datos)
varianza_obs<-var(datos)
esperanza_obs
## [1] 5.986692
varianza_obs
## [1] 7.708404
x<-c(1:20)
probas<-c(6/4208, 180/4208, 462/4208, 594/4208, 741/4208, 831/4208, 
          701/4208, 229/4208, 80/4208, 74/4208, 67/4208, 57/4208, 
          50/4208, 43/4208, 34/4208, 27/4208, 19/4208, 11/4208, 2/4208, 0)

El VaR y TVaR observados al 0.95 son

var_observado<-quantile(datos, 0.95)
tvar_observado<- sum(x[c(13:20)]*probas[(c(13:20))])/(1-0.95)
var_observado
## 95% 
##  12
tvar_observado
## [1] 13.0846
hist(datos)
abline(v=esperanza_obs, col="blue")
text(esperanza_obs,0, "esperanza obs", col="blue")
abline(v=var_observado, col="red")
text(var_observado,0, "VaR obs", col="red")
abline(v=tvar_observado, col="yellow")
text(tvar_observado,0, "TVaR obs", col="yellow")

Ajuste Binomial negativa

require(fitdistrplus)
## Loading required package: fitdistrplus
## Loading required package: MASS
## Loading required package: survival
## Loading required package: npsurv
## Loading required package: lsei
binom_neg<-fitdist(datos,distr="nbinom", method=c("mle"),start = NULL )

La esperanza y varianza teóricas son

mu_nbinom<-binom_neg$estimate[2]
p<-binom_neg$estimate[1]/(mu_nbinom+binom_neg$estimate[1])
var_nbinom<-binom_neg$estimate[1]*(1-p)/(p*p)
mu_nbinom
##       mu 
## 5.986313
var_nbinom
##     size 
## 7.309887

El VaR teórico

VaR_Nbinom<-quantile(binom_neg, 0.95)
VaR_Nbinom
## Estimated quantiles for each specified probability (non-censored data)
##          p=0.95
## estimate     11
denscomp(binom_neg)
abline(v=mu_nbinom, col="blue")
text(mu_nbinom,0, "esperanza nbinom", col="blue")
abline(v=11, col="red")
text(11,0, "VaR nbinom", col="red")

Ajuste Poisson

pois<-fitdist(datos,distr="pois", method=c("mle"),start = NULL )

La esperanza y varianza teórica en la distribución poisson, es el parámetro lambda que es:

lambda<-pois$estimate
lambda
##   lambda 
## 5.986692

El VaR teórico de la distribución Poisson

VaR_Poisson<-quantile(pois, 0.95)
VaR_Poisson
## Estimated quantiles for each specified probability (non-censored data)
##          p=0.95
## estimate     10
j<-10-2
s<-c()
for(i in 0:j){
s[i+1]=lambda^i/factorial(i)
}
s
## [1]  1.000000  5.986692 17.920241 35.760987 53.522504 64.084550 63.942410
## [8] 54.686217 40.923692
TVaR_Poisson<-lambda*exp(-lambda)*(exp(lambda)-sum(s))/0.05
TVaR_Poisson
##   lambda 
## 18.12667
denscomp(pois)
abline(v=lambda, col="blue")
text(lambda,0, "esperanza Poisson", col="blue")
abline(v=10, col="red")
text(10,0, "VaR Poisson", col="red")

Distribuciones continuas