Swiss part3

Arthur CHARPENTIER, Life insurance, and actuarial models, with R

Actuarial Science with
3. nonlife insurance (a quick overview)

Arthur Charpentier

joint work with Christophe Dutang & Vincent Goulet
and Markus Gesmann’s ChainLadder package

Meielisalp 2012 Conference, June

6th R/Rmetrics Meielisalp Workshop & Summer School
on Computational Finance and Financial Engineering

1


Some (standard) references

Wüthrich, M. & Merz, M. (2008)
Stochastic Claims Reserving Methods in Insurance
Wiley Finance Series

Frees, J.E. (2007)
Regression Modeling with Actuarial
and Financial Applications
Cambridge University Press

2


Lexis diagram in insurance
Lexis diagrams have been designed to visualize dynamics of life among several
individuals, but can be used also to follow claims’life dynamics, from the
occurrence until closure,

in life insurance in nonlife insurance

3


but usually we do not work on continuous time individual observations
(individuals or claims) : we summarized information per year occurrence until
closure,


4


individual lives or claims can also be followed looking at diagonals, occurrence
until closure,occurrence until closure,occurrence until closure,occurrence until
closure,


5


and usually, in nonlife insurance, instead of looking at (calendar) time, we follow
observations per year of birth, or year of occurrence occurrence until
closure,occurrence until closure,occurrence until closure,occurrence until closure,


6


and ﬁnally, recall that in standard models in nonlife insurance, we look at the
transposed triangle occurrence until closure,occurrence until closure,occurrence
until closure,occurrence until closure,


7


note that whatever the way we look at triangles, there are still three dimensions,
year of occurrence or birth, age or development and calendar time,calendar time
calendar time

8


and in both cases, we want to answer a prediction question...calendar time
calendar time calendar time calendar time calendar time calendar time calendar
time calendar timer time calendar time

9


Loss development triangle in nonlife insurance
Let PAID denote a triangle of cumulated payments, over time

> PAID
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 3209 4372 4411 4428 4435 4456
[2,] 3367 4659 4696 4720 4730 NA
[3,] 3871 5345 5398 5420 NA NA
[4,] 4239 5917 6020 NA NA NA
[5,] 4929 6794 NA NA NA NA
[6,] 5217 NA NA NA NA NA

– row i the accident year
– column j the development year
– diagonal i + j the calendar year (for incremental payments)
– Yi,j are incremental payments
– Ci,j are cumulated payments, per accident year, Ci,j = Yi,0 + Yi,1 + · · · + Yi,j

10


Incurred But Not Reported - IBNR
At time n, all claims occurred before have not necessarily been (fully) reported,
or settled, so claims manager have to predict the total amount of payments. Let
Hn denote the information available at time n (the upper part of the triangle)

Hn = {(Yi,j ), i + j ≤ n} = {(Ci,j ), i + j ≤ n}.

For accident year i, we want to get a best estimate of the total amount of
payment, i.e.
(n−i)
Ci,∞ = lim E[Ci,j |Hn ] = E[Ci,∞ |Hn ]
j→∞

and the diﬀerence with the payment done as at year i will be the required reserve
(n−i)
Ri = Ci,∞ − Ci,n−i .

One interesting quantity can be the (ultimate) uncertainty,
(n−i)
Var[Ci,∞ |Hn ] or Var[Ci,∞ ]

11


One year uncertainty in Solvency II
In Solvency II, it is required to study the CDR (claims development result)
deﬁned as
(n−i+1) (n−i)
∆n = Ci,∞
i − Ci,∞ = CDRi (n),

Note that E[∆n |Fi,n−i ] = 0, but insurers have to estimate (and set a reserve for
i
that uncertainty) Var[∆n |Fi,n−i ].
i

12


Chain Ladder
The most popular model is the Chain Ladder methodology, assuming that

Ci,j+1 = λj · Ci,j for all i, j = 1, · · · , n.

A natural estimator for λj is
n−j
i=1 Ci,j+1
λj = n−j
for all j = 1, · · · , n − 1.
i=1 Ci,j

Then it is natural to predict the non-observed part of the triangle using

Ci,j = λn+1−i · · · λj−1 · Ci,n+1−i .

> k <- 1
> sum(PAID[1:(nl-k),k+1])/sum(PAID[1:(nl-k),k])
[1] 1.380933

13


Chain Ladder
Those transition factors can be obtained using a simple loop

> LAMBDA <- rep(NA,nc-1)
> for(k in 1:(nc-1)){
+ LAMBDA[k]=(sum(PAID[1:(nl-k),k+1])/sum(PAID[1:(nl-k),k]))}
> LAMBDA
[1] 1.380933 1.011433 1.004343 1.001858 1.004735

It is possible to rewrite Ci,j+1 = λj · Ci,j for all i, j = 1, · · · , n, as follows

Ci,j = γj · Ci,∞ or Yi,j = ϕj · Ci,∞ .

> (GAMMA <- rev(cumprod(rev(1/LAMBDA))))
[1] 0.7081910 0.9779643 0.9891449 0.9934411 0.9952873
> (PHI <- c(GAMMA[1],diff(GAMMA)))
[1] 0.708191033 0.269773306 0.011180591 0.004296183 0.001846141

14


Chain Ladder
> barplot(PHI,names=1:5)
0.6
0.4
0.2
0.0

1 2 3 4 5

0 1 2 3 4 n
λj 1,38093 1,01143 1,00434 1,00186 1,00474 1,0000
γj 70,819% 97,796% 98,914% 99,344% 99,529% 100,000%
ϕj 70,819% 26,977% 1,118% 0,430% 0,185% 0,000%

15


Chain Ladder
Note that we can write
n−j
Ci,j Ci,j+1
λj = ωi,j · λi,j où ωi,j = n−j
et λi,j = .
i=1 i=1 Ci,j Ci,j
i.e.
> k <- 1
> weighted.mean(x=PAID[,k+1]/PAID[,k],w=PAID[,k],na.rm=TRUE)
[1] 1.380933

An alternative is to write that weighted mean as the solution of a weighted least
squares problem, i.e.
> lm(PAID[,k+1]~0+PAID[,k],weights=1/PAID[,k])$coefficients
PAID[, k]
1.380933

The main interest of that expression is that is remains valid when triangle
contains NA’s
16


> LAMBDA <- rep(NA,nc-1)
> for(k in 1:(nc-1)){
+ LAMBDA[k]=lm(PAID[,k+1]~0+PAID[,k],
+ weights=1/PAID[,k])$coefficients}
> LAMBDA
[1] 1.380933 1.011433 1.004343 1.001858 1.004735

The idea here is to iterate, by column,
> TRIANGLE <- PAID
> for(i in 1:(nc-1)){
+ TRIANGLE[(nl-i+1):(nl),i+1]=LAMBDA[i]*TRIANGLE[(nl-i+1):(nl),i]}
> TRIANGLE
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 3209 4372.000 4411.000 4428.000 4435.000 4456.000
[2,] 3367 4659.000 4696.000 4720.000 4730.000 4752.397
[3,] 3871 5345.000 5398.000 5420.000 5430.072 5455.784
[4,] 4239 5917.000 6020.000 6046.147 6057.383 6086.065
[5,] 4929 6794.000 6871.672 6901.518 6914.344 6947.084
[6,] 5217 7204.327 7286.691 7318.339 7331.939 7366.656

> tail.factor <- 1

17


> chargeultime <- TRIANGLE[,nc]*tail.factor
> paiements <- diag(TRIANGLE[,nc:1])
> (RESERVES <- chargeultime-paiements)
[1] 0.00000 22.39684 35.78388 66.06466 153.08358 2149.65640

Here sum(RESERVES) is equal to 2426.985, which is the best estimate of the
amount of required reserves,

> DIAG <- diag(TRIANGLE[,nc:1])
> PRODUIT <- c(1,rev(LAMBDA))
> sum((cumprod(PRODUIT)-1)*DIAG)
[1] 2426.985

It is possible to write a function Chainladder() which return the total amount
of reserves

> Chainladder<-function(TR,f=1){
+ nc <- ncol(TR); nl <- nrow(TR)
+ L <- rep(NA,nc-1)
+ for(k in 1:(nc-1)){

18


+ L[k]=lm(TR[,k+1]~0+ TR[,k],
+ weights=1/TR[,k])$coefficients}
+ TRc <- TR
+ for(i in 1:(nc-1)){
+ TRc[(nl-i+1):(nl),i+1]=L[i]* TRc[(nl-i+1):(nl),i]}
+ C <- TRc[,nc]*f
+ R <- (cumprod(c(1,rev(L)))-1)*diag(TR[,nc:1])
+ return(list(charge=C,reserves=R,restot=sum(R)))
+ }
> Chainladder(PAID)
$charge
[1] 4456.000 4752.397 5455.784 6086.065 6947.084 7366.656

$reserves
[1] 0.00000 22.39684 35.78388 66.06466 153.08358 2149.65640

$restot
[1] 2426.985

19


A stochastic model for paiments
Assume is that (Ci,j )j≥0 is a Markovian process and that there are λ = (λj ) and
2
σ = (σj ) such that

i,j+1 |Hi+j ) = E(Ci,j+1 |Ci,j ) = λj · Ci,j
 E(C
 Var(Ci,j+1 |Hi+j ) = Var(Ci,j+1 |Ci,j ) = σ 2 · Ci,j
j

Under those assumptions,

E(Ci,j+k |Hi+j ) = E(Ci,j+k |Ci,j ) = λj · λj+1 · · · λj+k−1 Ci,j

Mack (1993) assumes moreover that (Ci,j )j=1,...,n and (Ci ,j )j=1,...,n are
independent for all i = i .
Remark : It is also possible to write

Ci,j+1 = λj Ci,j + σj Ci,j · εi,j ,

where residuals (εi,j ) are i.i.d., centred, with unit variance.

20


From that expression, one might think of weigthed least squares techniques, to
estimation transition factors, i.e. given j, solve
n−j
1 2
min (Ci,j+1 − λj Ci,j ) .
i=1
Ci,j

Remark : The interpretation of the ﬁrst assumption is that points C·,j+1 versus
C·,j should be on a straight line that goes through 0.

> par(mfrow = c(1, 2))
> j=1
> plot(PAID[,j],PAID[,j+1],pch=19,cex=1.5)
> abline(lm(PAID[,j+1]~0+PAID[,j],weights=1/PAID[,j]))
> j=2
> plot(PAID[,j],PAID[,j+1],pch=19,cex=1.5)
> abline(lm(PAID[,j+1]~0+PAID[,j],weights=1/PAID[,j]))

21


6000
q q

6500

5500
6000

q
q
PAID[, j + 1]

PAID[, j + 1]
5500

q

5000
5000

q

q

4500
4500

q q

3500 4000 4500 5000 4500 5000 5500 6000 6500

PAID[, j] PAID[, j]

Ci,j+1 − λj Ci,j
Then, a natural idea is to use weighted residuals), εi,j = .
Ci,j

22


> j=1
> RESIDUALS <- (PAID[,j+1]-LAMBDA[j]*PAID[,j])/sqrt(PAID[,j])

Based on those residuals, it is then possible to estimate the volatility parameter
n−j−1 2
2 1 Ci,j+1 − λj Ci,j
σj =
n−j−1 i=0
Ci,j
or equivalenlty
n−j−1 2
2 1 Ci,j+1
σj = − λj · Ci,j
n−j−1 i=0
Ci,j
> lambda <- PAID[,2:nc]/PAID[,1:(nc-1)]
> SIGMA <- rep(NA,nc-1)
> for(i in 1:(nc-1)){
+ D <- PAID[,i]*(lambda[,i]-t(rep(LAMBDA[i],nc)))^2
+ SIGMA[i] <- 1/(nc-i-1)*sum(D[,1:(nc-i)])}
> SIGMA[nc-1] <- min(SIGMA[(nc-3):(nc-2)])
> (SIGMA=sqrt(SIGMA))
[1] 0.72485777 0.32036422 0.04587297 0.02570564 0.02570564

23


Tail factor, and closure of claim ﬁles
Assume that there exists λ∞ > 1 such that
Ci,∞ = Ci,n × λ∞ .
A standard technique to extrapolate λi is to consider an exponential
extrapolation (i.e. linear extrapolation on log(λk − 1)’s), then set
λ∞ = λk .
k≥n

> logL <- log(LAMBDA-1)
> tps <- 1:(nc-1)
> modele <- lm(logL~tps)
> plot(tps,logL,xlim=c(1,20),ylim=c(-30,0))
> abline(modele)
> tpsP <- seq(6,1000)
> logP <- predict(modele,newdata=data.frame(tps=tpsP))
> points(tpsP,logP ,pch=0)
> (facteur <- prod(exp(logP)+1))
[1] 1.000707

24


I.e. 0.07% should be added on top of estimated ﬁnal charge

> chargeultime <- TRIANGLE[,nc]*facteur
> paiements <- diag(TRIANGLE[,nc:1])
> (RESERVES <- chargeultime-paiements)
[1] 3.148948 25.755248 39.639346 70.365538 157.992918 2154.862234
> sum(RESERVES)
[1] 2451.764
0

q

q
q q
q
−10
logL

−20
−30

5 10 15 20

tps

25


The ChainLadder package
> library(ChainLadder)
> MackChainLadder(PAID)
MackChainLadder(Triangle = PAID)

Latest Dev.To.Date Ultimate IBNR Mack.S.E CV(IBNR)
1 4,456 1.000 4,456 0.0 0.000 NaN
2 4,730 0.995 4,752 22.4 0.639 0.0285
3 5,420 0.993 5,456 35.8 2.503 0.0699
4 6,020 0.989 6,086 66.1 5.046 0.0764
5 6,794 0.978 6,947 153.1 31.332 0.2047
6 5,217 0.708 7,367 2,149.7 68.449 0.0318

In this ﬁrst part, we have outputs per accident year, namely reserves Ri in the
IBNR, column, e.g. 2,149.7, for present year, and msep(Ri ) given by Mack S.E.
e.g. 68.449 for present year.

26



Totals
Latest: 32,637.00
Ultimate: 35,063.99
IBNR: 2,426.99
Mack S.E.: 79.30
CV(IBNR): 0.03

Here, the total amount of reserves R is IBNR, qui vaut 2,426.99, as well as
msep(R) given by Mack S.E. i.e. 79.30. Function plot() can also be used on
the output of that function.

27



Mack Chain Ladder Results Chain ladder developments by origin period
●
7000

Forecast
●
Latest

7000
5
6000

●

●
5000

6000
4
● 4
●
4000

Amount
Value

3 3 3
6

5000
3000

5
2 2 2 2

1 1 1 1
1
2000

4

4000
3
1000

2
1
0

1 2 3 4 5 6 1 2 3 4 5 6

Origin period Development period

28



1.5

1.5
1.0

1.0
Standardised residuals

0.5

0.5
0.0

0.0
−0.5

−0.5
−1.0

−1.0
−1.5

−1.5
4500 5000 5500 6000 6500 1 2 3 4 5

Fitted Origin period

29



1.5

1.5
1.0

1.0

0.5

0.5
0.0

0.0
−0.5

−0.5
−1.0

−1.0
−1.5

−1.5
1 2 3 4 5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Calendar period Development period

30


One year uncertainty
Merz & Wüthrich (2008) studied the evolution of CDRi (n) with time (n).
They proved that
2
msepcn−1 (CDRi (n)) = Ci,∞ Γi,n + ∆i,n
where
n−1 2
2 2
σn−i+1 Cn−j+1,j σj
∆i,n = n+1
+ n+1
λ2
n−i+1 Sn−i+1 j=n−i+2
Sj λ2 Sj
j
n

and
2 n−1 2
σn−i+1 σj
Γi,n = 1+ 1+ n+1
Cn−j+1,j −1
λ2
n−i+1 Ci,n−i+1 j=n−i+2 λ2 [Sj ]2
j

Merz & Wüthrich (2008) mentioned that one can approximate the term above
as
n−1 2
2 2
σn−i+1 Cn−j+1,j σj
Γi,n ≈ + n+1
2
λn−i+1 Ci,n−i+1 j=n−i+2 Sj λ2 Cn−j+1,j
j

31


using (1 + ui ) ≈ 1 + ui , which is valid if ui is small i.e.
2
σj
<< Cn−j+1,j .
λ2
j

It is possible to use function MackMerzWuthrich()

> MackMerzWuthrich(PAID)
MSEP Mack MSEP MW app. MSEP MW ex.
1 0.0000000 0.000000 0.000000
2 0.6393379 1.424131 1.315292
3 2.5025153 2.543508 2.543508
4 5.0459004 4.476698 4.476698
5 31.3319292 30.915407 30.915407
6 68.4489667 60.832875 60.832898
tot 79.2954414 72.574735 72.572700

32


Regression and claims reserving
de Vylder (1978) suggested the following model, for incremental payments

Yi,j ∼ N (αi · βj , σ 2 ), for all i, j,

i.e. an accident year factor αi and a development factor βj It is possible to use
least squares techniques to estimate those factors
 
 
(α, β) = argmin [Yi,j − αi · βj ]2 .
 
i,j

Here normal equations are

j Yi,j · βj i Yi,j · αi
αi = et βj = 2 ,
i αi
2
βj
j

33


Regression and claims reserving
An alternative is to consider an additive model for the logarithm of incremental
payments
log Yi,j ∼ N (ai + bj , σ 2 ), for all i, j.

> ligne <- rep(1:nl, each=nc); colonne <- rep(1:nc, nl)
> INC <- PAID
> INC[,2:6] <- PAID[,2:6]-PAID[,1:5]
> Y <- as.vector(INC)
> lig <- as.factor(ligne)
> col <- as.factor(colonne)
> reg <- lm(log(Y)~col+lig)

34


> summary(reg)
Call:
lm(formula = log(Y) ~ col + lig)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 7.9471 0.1101 72.188 6.35e-15 ***
col2 0.1604 0.1109 1.447 0.17849
col3 0.2718 0.1208 2.250 0.04819 *
col4 0.5904 0.1342 4.399 0.00134 **
col5 0.5535 0.1562 3.543 0.00533 **
col6 0.6126 0.2070 2.959 0.01431 *
lig2 -0.9674 0.1109 -8.726 5.46e-06 ***
lig3 -4.2329 0.1208 -35.038 8.50e-12 ***
lig4 -5.0571 0.1342 -37.684 4.13e-12 ***
lig5 -5.9031 0.1562 -37.783 4.02e-12 ***
lig6 -4.9026 0.2070 -23.685 4.08e-10 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1753 on 10 degrees of freedom
(15 observations deleted due to missingness)

35


Multiple R-squared: 0.9975, Adjusted R-squared: 0.9949
F-statistic: 391.7 on 10 and 10 DF, p-value: 1.338e-11

An unbiaised estimator for Yi,j is then Yi,j = exp[ai + bj + σ 2 /2], i.e.

> sigma <- summary(reg)$sigma
> (INCpred <- matrix(exp(logY+sigma^2/2),nl,nc))
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 2871.209 1091.278 41.66208 18.27237 7.84125 21.32511
[2,] 3370.826 1281.170 48.91167 21.45193 9.20570 25.03588
[3,] 3767.972 1432.116 54.67438 23.97937 10.29030 27.98557
[4,] 5181.482 1969.357 75.18483 32.97495 14.15059 38.48403
[5,] 4994.082 1898.131 72.46559 31.78233 13.63880 37.09216
[6,] 5297.767 2013.554 76.87216 33.71498 14.46816 39.34771
> sum(exp(logY[is.na(Y)==TRUE]+sigma^2/2))
[1] 2481.857

36


A Poisson Regression on incremental payments
Hachemeister & Stanard (1975), Kremer (1985) and Mack (1991) observed
that a log-Poisson regression yield exactly the same amount of reserves as the
Chain Ladder technique
> CL <- glm(Y~lig+col, family=poisson)
> summary(CL)

Call:
glm(formula = Y ~ lig + col, family = poisson)

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 8.05697 0.01551 519.426 < 2e-16 ***
lig2 -0.96513 0.01359 -70.994 < 2e-16 ***
lig3 -4.14853 0.06613 -62.729 < 2e-16 ***
lig4 -5.10499 0.12632 -40.413 < 2e-16 ***
lig5 -5.94962 0.24279 -24.505 < 2e-16 ***
lig6 -5.01244 0.21877 -22.912 < 2e-16 ***
col2 0.06440 0.02090 3.081 0.00206 **

37


col3 0.20242 0.02025 9.995 < 2e-16 ***
col4 0.31175 0.01980 15.744 < 2e-16 ***
col5 0.44407 0.01933 22.971 < 2e-16 ***
col6 0.50271 0.02079 24.179 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for poisson family taken to be 1)

Null deviance: 46695.269 on 20 degrees of freedom
Residual deviance: 30.214 on 10 degrees of freedom
AIC: 209.52

Number of Fisher Scoring iterations: 4

38


Here, the prediction is

> Ypred <- predict(CL,newdata=data.frame(lig,col),type="response")
> (INCpred <- matrix(Ypred,nl,nc))
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 3155.699 1202.110 49.82071 19.14379 8.226405 21.00000
[2,] 3365.605 1282.070 53.13460 20.41717 8.773595 22.39684
[3,] 3863.737 1471.825 60.99889 23.43904 10.072147 25.71173
[4,] 4310.096 1641.858 68.04580 26.14685 11.235734 28.68208
[5,] 4919.862 1874.138 77.67250 29.84594 12.825297 32.73985
[6,] 5217.000 1987.327 82.36357 31.64850 13.599887 34.71719

which is the same as Chain-Ladder estimate

> sum(Ypred[is.na(Y)==TRUE])
[1] 2426.985

39


Oversdispersion and quasiPoisson regression
Note that usually, the Poisson model might be too conservative (in terms of
variance) since there might be overdispersion of incremental payments.
Remark : N has an overdispersed Poisson distribution with dispersion
parameter φ if X/φ ∼ P(λ/φ), i.e. E(X) = λ while Var(X) = φλ
> CL <- glm(Y~lig+col, family=quasipoisson)
> summary(CL)

Call:
glm(formula = Y ~ lig + col, family = quasipoisson)

Deviance Residuals:
Min 1Q Median 3Q Max
-2.3426 -0.4996 0.0000 0.2770 3.9355

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 8.05697 0.02769 290.995 < 2e-16 ***

40


lig2 -0.96513 0.02427 -39.772 2.41e-12 ***
lig3 -4.14853 0.11805 -35.142 8.26e-12 ***
lig4 -5.10499 0.22548 -22.641 6.36e-10 ***
lig5 -5.94962 0.43338 -13.728 8.17e-08 ***
lig6 -5.01244 0.39050 -12.836 1.55e-07 ***
col2 0.06440 0.03731 1.726 0.115054
col3 0.20242 0.03615 5.599 0.000228 ***
col4 0.31175 0.03535 8.820 4.96e-06 ***
col5 0.44407 0.03451 12.869 1.51e-07 ***
col6 0.50271 0.03711 13.546 9.28e-08 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for quasipoisson family taken to be 3.18623)

Null deviance: 46695.269 on 20 degrees of freedom
Residual deviance: 30.214 on 10 degrees of freedom
AIC: NA

Number of Fisher Scoring iterations: 4

41


Quantifying uncertainty
Recall that in this GLM framework

E(Yi,j |Fn ) = µi,j = exp[ηi,j ]

i.e.
Yi,j = µi,j = exp[ηi,j ].

Using the delta method we can write
2
∂µi,j
Var(Yi,j ) ≈ · Var(ηi,j ),
∂ηi,j

which becomes, with a log link function

∂µi,j
= µi,j
∂ηi,j

42


In the case of an overdispersed Poisson model (as in Renshaw (1992)),

E [Yi,j − Yi,j ]2 ≈ φ · µi,j + µ2 · Var(ηi,j )
i,j

for the lower part of the triangle. Further,

Cov(Yi,j , Yk,l ) ≈ µi,j · µk,l · Cov(ηi,j , ηk,l ).

Since R = i+j>n Yi,j , then
 

E [R − R]2 ≈  φ · µi,j  + µF · Var(η F ) · µF ,
i+j>n

where µF and η F are restrictions of µ and η to indices i + j > n (i.e. lower part
of the triangle).

43


Remark : This expression is only asymptotic.

> p <- nl+nc-1;
> phi <- sum(residuals(CL,"pearson")^2)/(sum(is.na(Y)==FALSE)-p)
> Sig <- vcov(CL)
> X <- model.matrix(glm(Ypred~lig+col, family=quasipoisson))
> Cov.eta <- X%*%Sig%*%t(X)
> Ypred <-predict(CL,newdata=data.frame(lig,col),type="response")*(is.na(Y)==TRUE)
> se2 <- phi * sum(Ypred) + t(Ypred) %*% Cov.eta %*% Ypred
> sqrt(se2)
[,1]
[1,] 131.7726

44


Uncertainty and errors in regression models
In regression models, there are two sources in regression models
• the process error (due to statistical inference) : error on Y
• the variance error (due to the model approximation) : error on Y = Y + ε

q
60

q

q
50

q
40

q

q q
30

q
20

q
10

q

8 10 12 14 16 18 20

45


Uncertainty and errors in regression models
In regression models, there are two sources in regression models
• the process error (due to statistical inference) : error on Y
• the variance error (due to the model approximation) : error on Y = Y + ε

q
60

q

q
50

q
40

q

q q
30

q
20

q
10

q

8 10 12 14 16 18 20

46


Quantifying uncertainty using simulations
In order to estimate those quantities, consider Person’s residuals

Yi,j − Yi,j
εi,j = .
Yi,j

or, in order to have unit variance,

n Yi,j − Yi,j
εi,j = · ,
n−k
Yi,j

where k is the number of parameters in the model (i.e. 2n − 1).

47


> (residus=residuals(CL,type="pearson"))
1 2 3 4 5 6 7 8
9.49e-01 2.40e-02 1.17e-01 -1.08e+00 1.30e-01 -1.01e-13 -1.13e+00 2.77e-01
9 10 11 13 14 15 16 19
5.67e-02 8.92e-01 -2.11e-01 -1.53e+00 -2.21e+00 -1.02e+00 4.24e+00 -4.90e-01
20 21 25 26 31
7.93e-01 -2.97e-01 -4.28e-01 4.14e-01 -6.20e-15
> n=sum(is.na(Y)==FALSE)
> k=ncol(PAID)+nrow(PAID)-1
> (residus=sqrt(n/(n-k))*residus)
1 2 3 4 5 6 7 8
1.37e+00 3.49e-02 1.69e-01 -1.57e+00 1.89e-01 -1.46e-13 -1.63e+00 4.02e-01
9 10 11 13 14 15 16 19
8.22e-02 1.29e+00 -3.06e-01 -2.22e+00 -3.21e+00 -1.48e+00 6.14e+00 -7.10e-01
20 21 25 26 31
1.15e+00 -4.31e-01 -6.20e-01 6.00e-01 -8.99e-15

48


> epsilon <- rep(NA,nl*nc)
> epsilon[is.na(Y)==FALSE]=residus
> matrix(epsilon,nl,nc)
[,1] [,2] [,3] [,4] [,5] [,6]
[1,] 1.37e+00 -1.6346 -2.22 -0.710 -0.62 -8.99e-15
[2,] 3.49e-02 0.4019 -3.21 1.149 0.60 NA
[3,] 1.69e-01 0.0822 -1.48 -0.431 NA NA
[4,] -1.57e+00 1.2926 6.14 NA NA NA
[5,] 1.89e-01 -0.3059 NA NA NA NA
[6,] -1.46e-13 NA NA NA NA NA

49


Generating a pseudo triangle of payments
It is then natural to generate errors εb = (εb ), to generate a pseudo triangle of
i,j
payments
b
Yi,j = Yi,j + Yi,j · εb .
i,j

> hist(residus,breaks=seq(-3.5,6.5,by=.5),col="grey",proba=TRUE)
> u <- seq(-4,5,by=.01)
> densite <- density(residus)
> lines(densite,lwd=2)
> lines(u,dnorm(u,mean(residus),sd(residus)),lty=2)
> plot(ecdf(residus),xlab="residus",ylab="Fn(residus)")
> lines(u,pnorm(u,mean(residus),sd(residus)),lty=2)
> Femp <- cumsum(densite$y)/sum(densite$y)
> lines(densite$x,Femp,lwd=2)

Bootstrap or parametric model for ε’s ?

50


Histogram of epsilon ecdf(epsilon)

1.0
q

q

q
0.25

q

0.8
q

q
0.20

q

q

Cumulative distribution
q

0.6
q
0.15
Density

q

q

q

0.4
q
0.10

q

q

q

0.2
0.05

q

q

q

q
0.00

0.0

−5 0 5 10 −5 0 5 10

epsilon espilon

51


Once triangle Y b = (Yi,j ) is generate, compute Rb , e.g. using Chain Ladder
b

technique. Variance of Rb will denote the estimation error.
b
From triangle Yi,j , prediction for the lower part of the triangle can be derived
b
Yi,j . In order to generate scenarios of future payments, generate Poisson variates
b
with mean Yi,j ,

> CLsimul1<-function(triangle){
+ rpoisson(length(triangle),lambda=triangle)}

But generating Poisson variates might be too conservative,

> (summary(CL)$dispersion)
[1] 3.19

52


Generating quasi Poisson variates
First idea : a Gamma approximation. Recall that E(Y ) = λ et Var(Y ) = φλ for
Y ∼ G(α, β) i.e αβ = λ and αβ 2 = φλ.
> rqpoisG <- function(n, lambda, phi, roundvalue = TRUE) {
+ b <- phi ; a <- lambda/phi ; r <- rgamma(n, shape = a, scale = b)
+ if(roundvalue){r<-round(r)}
+ return(r)}

Second idea : a Negative Binomial variate, with mean µ and variance µ + µ2 /k,
i.e. µ = λ and k = λ(φλ − 1)−1 , i.e.
> rqpoisBN = function(n, lambda, phi) {
+ mu <- lambda ; k <- mu/(phi * mu - 1) ; r <- rnbinom(n, mu = mu, size = k)
+ return(r)}

Then
> CLsimul2<-function(triangle,surdispersion){
+ rqpoissonG(length(triangle),lambda=triangle,phi=surdispersion)}

53


An alternative to generate future payements is use Mack (1993)’s approach, as
suggested in England & Verrall (1998).
2
From λj and σj use the following dynamics
b b b 2 b
Ci,j+1 |Ci,j ∼ N (λj Ci,j , σj Ci,j ).

Thus, from a triangle of cumulated payments triangle a vector of transition
valuers l and a vector of conditional variances s deﬁne

> CLsimul3<-function(triangle,l,s){
+ m<-nrow(triangle)
+ for(i in 2:m){
+ triangle[(m-i+2):m,i]<-rnorm(i-1,
+ mean=triangle[(m-i+2):m,i-1]*l[i-1],
+ sd=sqrt(triangle[(m-i+2):m,i-1])*s[i-1])
+ }
+ return(triangle) }

54


Then, we run several simulations to generate best estimates and possible
scenarios of payments
> ns<-20000
> set.seed(1)
> Yp <- predict(CL,type="response",newdata=base)
> Rs <- R <- rep(NA,ns)
> for(s in 1:ns){
+ serreur <- sample(residus,size=nl*nc,replace=TRUE)
+ E <- matrix(serreur,nl,nc)
+ sY <- matrix(Yp,6,6)+E*sqrt(matrix(Yp,6,6))
+ if(min(sY[is.na(Y)==FALSE])>=0){
+ sbase <- data.frame(sY=as.vector(sY),lig,col)
+ sbase$sY[is.na(Y)==TRUE]=NA
+ sreg <- glm(sY~lig+col,
+ data=sbase,family=poisson(link="log"))
+ sYp <- predict(sreg,type="response",newdata=sbase)
+ R[s] <- sum(sYp[is.na(Y)==TRUE])
+ sYpscenario <- rqpoisG(36,sYp,phi=3.18623)
+ Rs[s] <- sum(sYpscenario[is.na(Y)==TRUE])
+ }}

55


Remark : In those simulations, it might be possible to generate negative
increments

> sort(residus)[1:2]
14 13
-3.21 -2.22
> sort(sqrt(Yp[is.na(Y)==FALSE]))[1:4]
25 26 19 20
2.87 2.96 4.38 4.52

But we might assume that those scenarios while yield low reserves (and thus will
not impact high losses quantiles)

> Rna <- R
> Rna[is.na(R)==TRUE]<-0
> Rsna <- Rs
> Rsna[is.na(Rs)==TRUE]<-0
> quantile(Rna,c(.75,.95,.99,.995))
75% 95% 99% 99.5%
2470 2602 2700 2729

56


> quantile(Rsna,c(.75,.95,.99,.995))
75% 95% 99% 99.5%
2496 2645 2759 2800
2900
2800
2700
2600
2500

0.75 0.80 0.85 0.90 0.95 1.00

probabilty

57


Vector R contains best scenarios, while vector Rs contains scenarios of payments.
In order to visualize their distribution, we have to remove NA values

> Rnarm <- R[is.na(R)==FALSE]
> Rsnarm <- Rs[is.na(Rs)==FALSE]

Here quantiles are slightly higher

> quantile(Rnarm,c(.75,.95,.99,.995))
75% 95% 99% 99.5%
2478 2609 2704 2733
> quantile(Rsnarm,c(.75,.95,.99,.995))
75% 95% 99% 99.5%
2507 2653 2764 2805

58


qq q qqq q qqqq
q qq q q qq
q q q qq
q q qqqqqqqqqqqqqq qqq q qq q q
qqqqqqqq q q qqq q q q q q
qqqqqqqq q q q q q q
qqqqqqqq q
qq
qq q q
Best Estimate

q qqq qqqqq
q qqqqq
qqqq
qqq
q qqqqqqqqqqqqqqqqqqqq q qq q
qqqqqqqqqqqqqqq qqq q q q
qqqqqqqqqqqqqq q qq q
qqqqqqqq qqqqqq q qq
qqqqq qqqqqq q q
qqqq q q q
qqq
qqqqqq
qq q
qq q q qqq q
qq q q

2000 2200 2400 2600 2800 3000

> plot(density(Rnarm),col="grey",main="")
> lines(density(Rsnarm),lwd=2)
> boxplot(cbind(Rnarm,Rsnarm),
+ col=c("grey","black"),horizontal=TRUE)

59


0.004
0.003
Density

0.002
0.001
0.000

2200 2400 2600 2800 3000

N = 18079 Bandwidth = 11.07

Those quantities are quite similar to the one obtained using bootChainLadder
from library(ChainLadder),

60


> BootChainLadder(PAID,20000,"od.pois")
BootChainLadder(Triangle = PAID, R = 20000, process.distr = "od.pois")

Latest Mean Ultimate Mean IBNR SD IBNR IBNR 75% IBNR 95%
1 4,456 4,456 0.0 0.0 0 0
2 4,730 4,752 22.1 12.0 28 44
3 5,420 5,455 35.3 15.3 44 63
4 6,020 6,085 65.5 19.8 77 101
5 6,794 6,946 152.4 28.6 170 203
6 5,217 7,363 2,146.2 111.6 2,216 2,337

Totals
Latest: 32,637
Mean Ultimate: 35,059
Mean IBNR: 2,422
SD IBNR: 132
Total IBNR 75%: 2,504
Total IBNR 95%: 2,651

61


To go further, loss distributions and credibility

Krugman, S.A., Panjer, H.H. & Willmot, G.E. (1998)
Loss Models
Wiley Series in Probability

Bühlmann, H. & Gisler, A. (2005)
A Course in Credibility Theory and its Applications
Springer Verlag

62


To go further, regression GLMs

Kaas, R., Goovaerts, M, Dhaene, J & Denuit, M. (2009)
Modern Actuarial Risk Theory
Springer Verlag

de Jong, P. & Heller, G.Z. (2008)
Generalized Linear Models for Insurance Data
Cambridge University Press

63


To go further, ratemaking

Denuit, M., Marechal, X., Pitrebois, S.
& Walhin, J.F. (2007)
Actuarial Modelling of Claim Counts :
Risk Classiﬁcation, Credibility & Bonus-Malus Systems
John Wiley & Sons

Ohlsson E. & Johansson, B. (2010)
Non-Life Insurance Pricing with
Generalized Linear Models
Springer Verlag

64

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