# Large claims, and ratemaking

During the course, we have seen that it is natural to assume that not only the individual claims frequency can be explained by some covariates, but individual costs too. Of course, appropriate families should be considered to model the distribution of the cost $Y$, given some covariates $\boldsymbol{X}$.Here is the dataset we’ll use,

>  sinistre=read.table("http://freakonometrics.free.fr/sinistreACT2040.txt",
>  sinistres=sinistre[sinistre$garantie=="1RC",] > sinistres=sinistres[sinistres$cout>0,]
>  couts=merge(sinistres,contrat)
> tail(couts)
nocontrat    no garantie    cout exposition zone puissance agevehicule
1919   6104006 11933      1RC 5376.04       0.37    E         6           1
1920   6107355 12349      1RC   51.63       0.74    E         4           1
1921   6108364 13229      1RC 1320.00       0.74    B         9           1
1922   6109171 11567      1RC 1320.00       0.74    B        13           1
1923   6111208 14161      1RC  970.20       0.49    E        10           5
1924   6111650 14476      1RC 1940.40       0.48    E         4           0
ageconducteur bonus marque carburant densite region
1919            32    57     12         E      93     10
1920            45    57     12         E      72     10
1921            32   100     12         E      83      0
1922            56    50     12         E      93     13
1923            30    90     12         E      53      2
1924            69    50     12         E      93     13

Here, each line is a claim. Usual families to model the cost are the Gamma distribution, or the inverse Gaussian. Or the lognormal distribution (which is not in the exponential family, but one can assume that the logarithm of the cost can be modeled with a Gaussian distribution). Consider here only one covariate, e.g. the age of the car, and two different models: a Gamma one, and a lognormal one.

> age=0:20
+ data=couts)
> Pgamma <- predict(reggamma.sp,newdata=data.frame(agevehicule=age),type="response")

For the Gamma regression, it is a simple GLM, so it is not difficult. For a lognormal distribution, one should remember that the expected value of a lognormal distribution is not the exponential of the underlying Gaussian distribution. A correction should be made, here to get an unbiased estimator for the average cost,

> reglm.sp <- lm(log(cout)~agevehicule,data=baseCOUT)
> sigma <- summary(reglm.sp)$sigma > mu <- predict(reglm.sp,newdata=data.frame(agevehicule=age)) > Pln <- exp(mu+sigma^2/2) We can plot those two predictions on a single graph, > plot(age,Pgamma,xlab="",ylab="",col="red",type="b",pch=4) > lines(age,Pln,col="blue",type="b") Here it is, Observe that it is also possible to use splines, since there might be no reason for the age to appear here in a multiplicative way, Here, the two models are rather close. Nevertheless, one should remember that the Gamma model can be extremely sensitive to large claims (I mean here really large claims). On the other hand, with the log-transformation for the lognormal model, it seams that this model is less sensitive to large events. Actually, if I use the complete dataset, the regressions are the following, i.e. with a lognormal distribution, the average cost is decreasing with the age of the car, while it is increasing with a Gamma model. The main reason here is that there is one large (not to say huge) claim in the dataset, > couts[which.max(couts$cout),]
cout exposition zone puissance agevehicule ageconducteur
7842  4024601       0.22    B         9          13            19
marque carburant densite region
7842      2         E      93     24

One young driver got a $4 million claim, with a 13 year old car. This is an outliers for the Gamma regression, that clearly influences the estimation (the second largest if only one third of this one). Since there is a clear influence of large claims on the estimation of the average cost, a natural idea might be to remove those large claims. Or perhaps to see them as different from normal claims: normal claims can be explained by some covariates, but perhaps that those large claims should be shared not only within its own class, but within all the insured on the portfolio. To formalize this idea, observe that we can write $\mathbb{E}(Y|\boldsymbol{X}) = {\color{Blue} {\underbrace{\mathbb{E}(Y|\boldsymbol{X},Y\leq s)}_{A} \cdot {\underbrace{\mathbb{P}(Y\leq s|\boldsymbol{X})}_{B}}}}+{\color{Red} {{\underbrace{\mathbb{E}(Y|Y> s, \boldsymbol{X}) }_{C}}\cdot {\underbrace{\mathbb{P}(Y> s| \boldsymbol{X})}_{B}}}}$ where the blue part is associated to normal-sized claims, while large ones correspond to the red part. It is then possible to run three regressions: one on normal sized claims, one on large claims, and one on the indicator of having a large claims, given that a claim occurred. The code here is something like that: a large claim – here – is above$ 10,000 (one has a fix it)

> s= 10000
> couts$normal=(couts$cout<=s)
> mean(couts$normal) [1] 0.9818087 which represent 2% of the claims in our dataset.We can run 3 sets of regressions, with smoothed regression on the age of the car. The first one to model large claims individual costs, > indice = which(couts$cout>s)
> mean(couts$cout[indice]) [1] 34471.59 > library(splines) > regB=glm(cout~bs(agevehicule),data=couts, + subset=indice,family=Gamma(link="log")) > ypB=predict(regB,newdata=data.frame(agevehicule=age),type="response") > ypB2=mean(couts$cout[indice])

the second one to model normal claims individual costs,

> indice = which(couts$cout<=s) > mean(couts$cout[indice])
[1] 1335.878
> regA=glm(cout~bs(agevehicule),data=couts,
> ypA=predict(regA,newdata=data.frame(agevehicule=age),type="response")
> ypA2=mean(couts\$cout[indice])

And finally, a third one, on the probability of having a normal sized claim, given that a claim occurred

> regC=glm(normal~bs(agevehicule),data=couts,family=binomial)
> ypC=predict(regC,newdata=data.frame(agevehicule=age),type="response")
> regC2=glm(normal~1,data=couts,family=binomial)
> ypC2=predict(regC2,newdata=data.frame(agevehicule=age),type="response")

Note that we to have, each time something that can be interpreted either as $\mathbb{E}(Y|\boldsymbol{X},Y\gtrless s)$, or $\mathbb{E}(Y|Y\gtrless s)$ – i.e. no covariate is considered on the later. On the graph below, we did plot

$\mathbb{E}(Y|\boldsymbol{X}) = {\color{Blue} {\underbrace{\mathbb{E}(Y|\boldsymbol{X},Y\leq s)}_{A} \cdot {\underbrace{\mathbb{P}(Y\leq s|\boldsymbol{X})}_{B}}}}+{\color{Red} {{\underbrace{\mathbb{E}(Y|Y> s, \boldsymbol{X}) }_{C}}\cdot {\underbrace{\mathbb{P}(Y> s| \boldsymbol{X})}_{B}}}}$

where Gamma regressions – with splines – are considered for the average costs, while logistic regressions – again with splines – are considered to model probabilities.

(but careful with splines: on borders, since we do not have a lot of observations, the behavior can be… odd. And adjustments should be made to obtain an adequate level of premium).  If it is legitimate to assume that normal-sized claims can be explained by some covariates, perhaps large claims (or extremely large ones) are just purely random, i.e. not function of any covariate, at all. I.e.

$\mathbb{E}(Y|\boldsymbol{X}) = {\color{Blue} {\underbrace{\mathbb{E}(Y|\boldsymbol{X},Y\leq s)}_{A} \cdot {\underbrace{\mathbb{P}(Y\leq s|\boldsymbol{X})}_{B}}}}+{\color{Red} {{\underbrace{\mathbb{E}(Y|Y> s) }_{C'}}\cdot {\underbrace{\mathbb{P}(Y> s| \boldsymbol{X})}_{B}}}}$

To go one step further, it might also be possible to assume that not only the size of the claim (given that it is a large one) is not a function of any covariate, but perhaps neither is the probability of having an extremely large claim, too

$\mathbb{E}(Y|\boldsymbol{X}) = {\color{Blue} {\underbrace{\mathbb{E}(Y|\boldsymbol{X},Y\leq s)}_{A} \cdot {\underbrace{\mathbb{P}(Y\leq s)}_{B'}}}}+{\color{Red} {{\underbrace{\mathbb{E}(Y|Y> s) }_{C'}}\cdot {\underbrace{\mathbb{P}(Y> s)}_{B'}}}}$

From the first part, we’ve seen that the distribution considered had an impact on the prediction, and in the second, we’ve seen that the definition of large claims (and how to deal with them) also has an impact. So clearly, actuaries have some leverage when working on ratemaking…

# Too large datasets for regression ? What about subsampling….

recently, a classmate working in an insurance company told me he had too large datasets to run simple regressions (GLM, which involves optimization issues), and that they were thinking of a reward for the one who will write the best R-code (at least the fastest). My first idea was to use subsampling techniques, saying that 10 regressions on 100,000 observations can take less time than a regression on 1,000,000 observations. And perhaps provide also better results…

• Time to run a regression, as a function of the number of observations

Here, I generate a dataset as follows

and we fit

where is a spline function (just to make it as general as possible, since in insurance ratemaking, we include continuous variates that do not influence claims frequency linearly in the score). Yes, there might be also useless variables, including one of them which is strongly correlated with one that has an impact in the regression. The code to generate the dataset is simply

> n=10000
> X1=rexp(n)
> X2=sample(c("A","B","C"),size=n,replace=TRUE)
> X3=runif(n)
> Z=rmnorm(n,c(0,0),matrix(c(1,0.8,.8,1),2,2))
> X4=Z[,1]
> X5=Z[,2]
> X6=X1^2
> E=runif(n)
> lambda=.2*X5-4*dbeta(X3,2,5)+X1+
+1*(X2=="A")-2*(X2=="B")-5*(X2=="C")
> Y=rpois(n,exp(lambda))
> base=data.frame(Y,X1,X2,X3,X4,X5,X6,E)

We would like the study the time it takes to run a regression, as a function of the size (i.e. the number of lines ) of the dataset.

> system.time( glm(Y~bs(X1)+X2+X3+X4+
+ X5+X6+offset(log(E)),family=poisson,
+ data=base) )
utilisateur     système      écoulé
0.25        0.00        0.25

Here, the time I look at is the last one. But so far, it was rather simple, but it is not the best model I can get. Let us use a stepwise (backward) variable selection,

> system.time( step(glm(Y~bs(X1)+X2+X3+
+ X4+X5+X6+offset(log(E)),family=poisson,
+ data=base)) )
Start:  AIC=2882.1
Y ~ bs(X1) + X2 + X3 + X4 + X5 + X6 + offset(log(E))
Step:  AIC=2882.1
Y ~ bs(X1) + X2 + X3 + X4 + X5 + offset(log(E))
Df Deviance    AIC
<none>        2236.0 2882.1
- X5      1   2240.1 2884.2
- X4      1   2244.1 2888.2
- X3      1   4783.2 5427.3
- X2      2   5311.4 5953.5
- bs(X1)  3   6273.7 6913.8
utilisateur     système      écoulé
1.82        0.03        1.86

Finally, from the first regression, we have points in black (based on 200 simulated datasets), and with a stepwise procedure, we have the points in red.

i.e. it might look linear (proportional), but if it was linear, then on a log-log scale, we should have also straigh lines, with slope 1,

Actually, it looks like a convex function.

The interpretation of that convexity might lead to misinterpretation. On the graph below on the left, on a dataset two times bigger than the previous one (black point) will be less than two times longer to run, while on the right, it will be more than two timess longer,

Convexity can simply be interpreted as “too large datasets take time, and too small too…”. Which is a first step: it should be interesting, in some cases, to run several regressions on smaller datasets….

• Running 100 regressions on 100 lines, or running 1 regression on 10,000 lines ?

Here, we have datasets with =200,000 lines. The questions is how long will it take if we subdived into subsamples (of equal size), and run regressions ?

> nk=trunc(n/k)rep(1:k,each=nk); nt=nk*k
> base=data.frame(Y[1:nt],X1[1:nt],
+ X2[1:nt],X3[1:nt],X4[1:nt],X5[1:nt],
+ X6[1:nt],E[1:nt],classe)
> system.time( for(j in 1:k){
+  glm(Y~bs(X1)+X2+X3+X4+X5+
+ X6+offset(log(E)),family=poisson
+ ,data=base,subset=classe==j) })
utilisateur     système      écoulé
1.31        0.00        1.31
> system.time( for(j in 1:k){
+      step(glm(Y~bs(X1)+X2+X3+
+ X4+X5+X6+offset(log(E)),family=
+ poisson,data=base,subset=classe==j)) })
Start:  AIC=183.97
Y ~ bs(X1) + X2 + X3 + X4 + X5 + X6 + offset(log(E))

[…]

  Df Deviance    AIC
<none>        117.15 213.04
- X2      2   250.15 342.04
- X3      1   251.00 344.89
- X4      1   420.63 514.53
- bs(X1)  3   626.84 716.74
utilisateur     système      écoulé
11.97        0.03       12.31

On the graph below, we have the time (y-axis, here on a log scale) it took to run regression on samples of size , as function of (x-axis), including the time it took to run the regression on a dataset of size which is the concentration of dots on the left (i.e. =1), both on the 6 regressors – in black – and with a strepwise procedure – in red. One has to keep in mind that I did not remove the printing option in the stepwise procedure, so it might be difficult to compare the two clouds (black vs. red). Nevertheless, we clearly see that if we run regression on samples of size , when is not too large, i.e. less than 10 or 15, it is not longer than the regression on =200,000 lines.

So here we see that running 100 regressions on 2,000 lines is longer than running 1 regression on 200,000 lines… But maybe we are not comparing things that are actually comparable: what if it takes a bit longer, but we strongely improve the quality of our estimators ?

• What about the quality of the output ?

Here, we consider only one dataset, with =100,000 lines (just to make it run a bit faster). And =20 subsets. Recall that the generated dataset is from

and we fit

Here, we plot here and a confidence interval, defined as

The lightblue segment is the initial estimator, while the blue one is obtained from the stepwise procedure. The grey area represent the estimation on the overall sample, while the segments on the right are the estimators (each on samples of size ).

We can see that we have much more volatility on those estimators, but the average (horizontal doted lines) are not so bad… The true value (i.e. the one used to generate the dataset is the dotter black horizontal line).
And if we repeat that on 1,000 simulated dataset, we obtaind the following distribution for (blue line), so we have an unbiased estimator of our parameter (the verticular line being here the true value), here including a stepwise procedure,

But if we add the the red curve is the average of the the previous one being now the clear blue line in the back, we see that taking average of estimators on subsamples is not bad at all, on the contrary,

and for those who think that the stepwise procedure is a mistake, here is what we get without it,

So what we can see is that running 20 regressions can take (a little) more time (from what we’ve seen earlier) than running only one on the whole dataset…. but it provides better estimates. So the tradeoff is not that simple, and maybe running several regressions on huge datasets can be a proper alternative.