# Mack's model is a Glm !

Which actuary does not know about Mack’s model ? Due to Mack (1991), this model is fairly simple. Suppose you have a triangle.

Ok seeing the origin dates of claims, thoose data are old. But who cares ? Let’s denote $$C_{i,j}$$ the value of this triangle from the $$i$$’th row and the $$j$$’th collumn.

The mack model consist of a stochastic model that enframe the classical chain-ladder estimator, denoted here by the vector $$\mathbf{\hat{f}}$$, defined $$\forall j \in \{1,...,n\}$$ by :

$\hat{f}_j = \frac{\sum\limits_{i=1}^{n-j+1} C_{i,j+1}}{\sum\limits_{i=1}^{n-j+1} C_{i,j}}$

Then we estimate future claims by setting, $$\forall i,j \,|\, i+j>n$$, $$C_{i,j+1} = \hat{f_j} C_{i,j}$$. That’s chain-ladder estimation right ?

Mack then found stochastic hypohtesis that make thoose estimator better in a certain sense (in the MSE-sense) :

• $$(H1)$$ : The lines are independants
• $$(H2)$$ : $$\exists\, \mathbf{f} \text{ such that } \forall i,j,\, \mathbb{E}(C_{i,j+1}\,|\, C_{i,1},...,C_{i,j}) = f_j C_{i,j}$$
• $$(H3)$$ : $$\exists\, \mathbf{\sigma} \text{ such that } \forall i,j,\, \mathbb{V}(C_{i,j+1}\,|\, C_{i,1},...,C_{i,j}) = \sigma_j^2 C_{i,j}$$

First, lets look a little closer at theese hypothesis. First, we should note the conditionning of $$C_{i,j+1}$$ on the past of the origin year.

Suppose that, following England and Verrall (2002), we reformulate the model in terme of individual developpements factors. Set :

$F_{i,j} = \frac{C_{i,j+1}}{C_{i,j}} \text{ and } w_{i,j} = C_{i,j}$

The 2 last hypohtesis can then be reformulated in the following way (just dividing by $$w_{i,j}$$).

• $$(H2*)$$ : $$\exists\, \mathbf{f} \text{ such that } \forall i,j,\, \mathbb{E}(F_{i,j}\,|\, w_{i,j}) = f_j$$
• $$(H3*)$$ : $$\exists\, \mathbf{\sigma} \text{ such that } \forall i,j,\, \mathbb{V}(F_{i,j}\,|\, w_{i,j}) = \frac{\sigma_j^2}{w_{i,j}}$$

Note that i assume that the information given by the past is well sumarised by $$w_ {i,j}$$, which is indeed the case under those hypohtesis.

Now that we reformulate the hypohtesis, you’l say “What about the GLM you promised ?”. Well i’m comming to it.

Look again at the formula for the estimator of the chain-ladder developpement factors. DO you see the weighted mean of $$F$$’s, weigthed by $$C$$’s ? Well you should ! Indeed, since we posed $$w_{i,j} = C_{i,j}$$,

$\hat{f}_j = \frac{\sum\limits_{i=1}^{n-j+1} F_{i,j}\, w_{i,j}}{\sum\limits_{i=1}^{n-j+1} w_{i,j}}$

Now let’s focuse on the mack estimators for the variance parameters $$\mathbf{\sigma}$$ :

$\hat{\sigma}_j = \frac{1}{n-j} \sum\limits_{i=1}^{n-j+1} w_{i,j} (F_{i,j} - \hat{f}_j)^2$

they look a lot like a simple unbiaised variance estimation, for the estimation of $$F$$’s by $$\hat{f}$$’s, with case weights $$w$$’s, right ? Well, although the normalisation is not right (should be $$\frac{1}{n-j-1}$$), they do.

Ok now that i gave you some hints, let’s get on it. To estimate the first ordre, i.e $$f$$’s, we just take a weighted average of observation $$F$$’s, with weights $$w$$’s, by column.

So if i set a glm with :

• Any law in the exponential family
• case weights given by $$w$$
• response variable $$F$$
• Only covariable the indicant f the column, i.e the year of developpement

I should get the same estimate, right ? Let’s try on the ABC triangle from the ChainLadder package (see Gesmann (2009) ) :

Table 1: Incured triangle – cumulative view
1 2 3 4 5 6 7 8 9 10 11
1977 153638 342050 476584 564040 624388 666792 698030 719282 735904 750344 762544
1978 178536 404948 563842 668528 739976 787966 823542 848360 871022 889022
1979 210172 469340 657728 780802 864182 920268 958764 992532 1019932
1980 211448 464930 648300 779340 858334 918566 964134 1002134
1981 219810 486114 680764 800862 888444 951194 1002194
1982 205654 458400 635906 765428 862214 944614
1983 197716 453124 647772 790100 895700
1984 239784 569026 833828 1024228
1985 326304 798048 1173448
1986 420778 1011178
1987 496200

First, the estimated developpement factor given by standard chain-ladder :

ChainLadder::MackChainLadder(ABC)$f ## [1] 2.308599 1.421098 1.199934 1.113445 1.072736 1.047559 1.034211 1.026047 ## [9] 1.020188 1.016259 1.000000 If we create a data.frame with what we need and then try a useless glm (actualy it’s even a lm there) on it : df <- ABC %>% as.data.frame %>% mutate( weight = cbind(rep(NA,nrow(ABC)),ABC %>% as.matrix) %>% .[,1:nrow(ABC)] %>% as.vector, F = value / weight, ) %>% drop_na model <- lm(data =df,formula=F~factor(dev)+0,weights=weight) df %<>% mutate(f.hat = model$fitted.values,
rez2 = weight * (F - f.hat)^2)

I also extracted the squared preason residuals.

Indeed, if we want to model $$\sigma$$’s too, we need a second model to calculate the weighted average of… Squared pearson residuals ! Which is Normla in the context of Joint modeling of Glm’s, see e.g McCullagh and Nelder (1989) .

To calculate the weighted average of squared residuals (for the dispersions parameters), we’ll just use another linear model since only the first order is here of interest. Note that we excluse the last column of the triangle, since there is only one point (and therefore no estimation of variance is possible). We also add the last valeu estimated by Mack for sake of completeness.

model2 <- df %>%
filter(dev != nrow(ABC)) %>%
{lm(data =., formula = rez2 ~ factor(dev))}

df %<>% mutate(
estimated.phi = model2$fitted.values %>% {c(.,min(.[length(.)],.[length(.)-1],.[length(.)]^2/.[length(.)-1]))} ) df ## origin dev value weight F f.hat rez2 estimated.phi ## 1 1977 2 342050 153638 2.226337 2.308599 1039.6609948 1940.0408948 ## 2 1978 2 404948 178536 2.268159 2.308599 291.9754480 1940.0408948 ## 3 1979 2 469340 210172 2.233123 2.308599 1197.2515446 1940.0408948 ## 4 1980 2 464930 211448 2.198791 2.308599 2549.5751391 1940.0408948 ## 5 1981 2 486114 219810 2.211519 2.308599 2071.5915573 1940.0408948 ## 6 1982 2 458400 205654 2.228987 2.308599 1303.4551427 1940.0408948 ## 7 1983 2 453124 197716 2.291792 2.308599 55.8462412 1940.0408948 ## 8 1984 2 569026 239784 2.373077 2.308599 996.9031237 1940.0408948 ## 9 1985 2 798048 326304 2.445719 2.308599 6135.1878014 1940.0408948 ## 10 1986 2 1011178 420778 2.403115 2.308599 3758.9619552 1940.0408948 ## 11 1977 3 476584 342050 1.393317 1.421098 263.9876431 548.0174476 ## 12 1978 3 563842 404948 1.392381 1.421098 333.9344808 548.0174476 ## 13 1979 3 657728 469340 1.401389 1.421098 182.3038561 548.0174476 ## 14 1980 3 648300 464930 1.394403 1.421098 331.3012598 548.0174476 ## 15 1981 3 680764 486114 1.400420 1.421098 207.8370561 548.0174476 ## 16 1982 3 635906 458400 1.387229 1.421098 525.8105220 548.0174476 ## 17 1983 3 647772 453124 1.429569 1.421098 32.5170215 548.0174476 ## 18 1984 3 833828 569026 1.465360 1.421098 1114.8127986 548.0174476 ## 19 1985 3 1173448 798048 1.470398 1.421098 1939.6523906 548.0174476 ## 20 1977 4 564040 476584 1.183506 1.199934 128.6170742 208.3223889 ## 21 1978 4 668528 563842 1.185665 1.199934 114.7889365 208.3223889 ## 22 1979 4 780802 657728 1.187120 1.199934 107.9955483 208.3223889 ## 23 1980 4 779340 648300 1.202129 1.199934 3.1232034 208.3223889 ## 24 1981 4 800862 680764 1.176416 1.199934 376.5043910 208.3223889 ## 25 1982 4 765428 635906 1.203681 1.199934 8.9295071 208.3223889 ## 26 1983 4 790100 647772 1.219719 1.199934 253.5814014 208.3223889 ## 27 1984 4 1024228 833828 1.228344 1.199934 673.0390490 208.3223889 ## 28 1977 5 624388 564040 1.106992 1.113445 23.4819775 95.1739102 ## 29 1978 5 739976 668528 1.106874 1.113445 28.8663512 95.1739102 ## 30 1979 5 864182 780802 1.106788 1.113445 34.6021980 95.1739102 ## 31 1980 5 858334 779340 1.101360 1.113445 113.8120512 95.1739102 ## 32 1981 5 888444 800862 1.109360 1.113445 13.3642605 95.1739102 ## 33 1982 5 862214 765428 1.126447 1.113445 129.4015811 95.1739102 ## 34 1983 5 895700 790100 1.133654 1.113445 322.6889519 95.1739102 ## 35 1977 6 666792 624388 1.067913 1.072736 14.5232459 95.4346025 ## 36 1978 6 787966 739976 1.064853 1.072736 45.9752469 95.4346025 ## 37 1979 6 920268 864182 1.064901 1.072736 53.0508012 95.4346025 ## 38 1980 6 918566 858334 1.070173 1.072736 5.6366537 95.4346025 ## 39 1981 6 951194 888444 1.070629 1.072736 3.9429272 95.4346025 ## 40 1982 6 944614 862214 1.095568 1.072736 449.4787401 95.4346025 ## 41 1977 7 698030 666792 1.046848 1.047559 0.3369445 14.7731099 ## 42 1978 7 823542 787966 1.045149 1.047559 4.5761913 14.7731099 ## 43 1979 7 958764 920268 1.041831 1.047559 30.1914463 14.7731099 ## 44 1980 7 964134 918566 1.049608 1.047559 3.8554008 14.7731099 ## 45 1981 7 1002194 951194 1.053617 1.047559 34.9055665 14.7731099 ## 46 1977 8 719282 698030 1.030446 1.034211 9.8952347 12.6617691 ## 47 1978 8 848360 823542 1.030136 1.034211 13.6760755 12.6617691 ## 48 1979 8 992532 958764 1.035220 1.034211 0.9771982 12.6617691 ## 49 1980 8 1002134 964134 1.039414 1.034211 26.0985679 12.6617691 ## 50 1977 9 735904 719282 1.023109 1.026047 6.2066637 2.9989596 ## 51 1978 9 871022 848360 1.026713 1.026047 0.3763507 2.9989596 ## 52 1979 9 1019932 992532 1.027606 1.026047 2.4138644 2.9989596 ## 53 1977 10 750344 735904 1.019622 1.020188 0.2353251 0.2170726 ## 54 1978 10 889022 871022 1.020665 1.020188 0.1988201 0.2170726 ## 55 1977 11 762544 750344 1.016259 1.016259 0.0000000 0.2170726 And there we have it ! Thoose estimates are clearly the same as Mack’s. Indeed : model4 <- MackChainLadder(ABC,est.sigma="Mack") resultat <- df %>% group_by(dev) %>% summarise(phi=mean(estimated.phi)) %>% mutate( mack = model4$sigma^2,
degree.of.freedom = max(dev) - dev, # correction for degree of freedom
phi = (phi * (degree.of.freedom+1)/(degree.of.freedom)) %>%
.[1:(length(.)-1)] %>%
{c(.,min(.[length(.)],.[length(.)-1],.[length(.)]^2/.[length(.)-1]))},
f = model$coefficients[1:(dim(ABC)[1]-1)], f.cl = model4$f[1:(dim(ABC)[1]-1)]
) %>%
select(f,f.cl,phi,mack) %>%
as.data.frame
f f.cl phi mack
2.308599 2.308599 2155.6009942 2155.6009942
1.421098 1.421098 616.5196286 616.5196286
1.199934 1.199934 238.0827301 238.0827301
1.113445 1.113445 111.0362286 111.0362286
1.072736 1.072736 114.5215230 114.5215230
1.047559 1.047559 18.4663874 18.4663874
1.034211 1.034211 16.8823588 16.8823588
1.026047 1.026047 4.4984394 4.4984394
1.020188 1.020188 0.4341453 0.4341453
1.016259 1.016259 0.0418994 0.0418994

So, ok, we reproduced Mack’s result in a new model. But under wich hypohtesis ? I see there linear models, that require generaly gaussian error, so some law hypothesis, right ?

The hypothesis we used are :

• $$(H1**)$$ : The lines are independants
• $$(H2**)$$ : $$\exists\, \mathbf{f}\text{ and }\mathbb{\sigma} \text{ such that } \forall i,j,\, F_{i,j} |\, w_{i,j} \sim \mathcal{N}(f_j, \frac{\sigma_j^2}{w_{i,j}})$$

Well, if we forget about the law assumption and only use quasi-glm (see, once again, McCullagh and Nelder (1989)), thoose are mack hypothesis.

Therefore, Mack’s model is a Quasi-glm, conditional, and weighted, with a normal-like variance assumption.

Some remarks :

• Note that we playe a little with degree of freedom : Since we are estimating the $$\sigma$$’s after the $$f$$’s, we should take into account that $$n-1$$ parameters are already estimated. This is classical in the joint modeling process.
• Furthermore, the joint modeling process gives the choice of residuals (pearson residuals or deviance contribution), but in a the normal case they are equal.
• Last but not least, the accointance for the kurtosis is not necessary here beacause of the normal variance assumption.

So what ? Ok, Mack’s model is a Glm, and what ? Well, the categorisation of this classical hypothesis set as a GLM allows several things :

• The use of standard statistiqcal tools : cook’s distance, residual analysis, etc…
• The use of standard software : R’s glm function is probably more safe than ChainLadder’s specific function.
• The simplicity of extention : The bornhutter-fergusson extension, e.g, can be easily introduced in the modelisation within a GLM framework, it’s harder within mack’s framework.

Next time, i’ll continue my analysis of Mack model by talking about one year risk and the Merz-wuthrich principles in this quasi-glm framwork.

References :

England, P. D., and R. J. Verrall. 2002. “Stochastic Claims Reserving in General Insurance.” British Actuarial Journal 8 (03): 443–518. https://doi.org/10/csqxh7.

Gesmann, Markus. 2009. “London R-User Group Meeting Claims Reserving in R the ChainLadder Package,” March, 35.

Mack, Thomas. 1991. “A Simple Parametric Model for Rating Automobile Insurance or Estimating IBNR Claims Reserves.” ASTIN Bulletin 21 (01): 93–109. https://doi.org/10.2143/AST.21.1.2005403.

McCullagh, P, and J. A Nelder. 1989. Generalisez Linear Models.

##### Oskar Laverny
###### Actuary - P.h.d

My research interests include dependences structures in high dimensions, copulas, code and actuarial sciences.