1 Log normal claim models with common shock s

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Log normal claim models with common

shocks

Greg Taylor

School of Risk and Actuarial Studies

University of New South Wales

UNSW Sydney, NSW 2052

AUSTRALIA

October 2022

Abstract. This paper is concerned with modelling multiple claim arrays that are subject to one

or more common shocks. It uses a structure that involves very general forms both idiosyncratic

and common shock components of cell means. The dependencies between arrays, or between

cells within an array, generated by the shocks are also of very general form. All of this appears

in the prior literature, where the idiosyncratic and shock components are additive. This has

created the awkwardness of unbalanced shocks. The present paper rectifies this by defining

these components as multiplicative. Observations in individuals cells of claim arrays are

assumed log normal (later log Tweedie) in order to accommodate the multiplicativity.

Conveniently, the log normal case reduced parameter estimation to linear regression, yielding

closed form solution of location parameters, and even of dispersion parameters in some cases.

Keywords: common shock, log normal, log Tweedie, loss reserving.

1. Introduction

Common shock models were introduced to the actuarial literature by Lindskog and McNeil

(2003). These provide a means creating simple models of data sets that contain dependencies

between observations.

They have been used in a variety of actuarial settings. For example, in claim modelling, Meyers

(2007) applied the concept within the context of collective risk theory. Other fields of

application include capital modelling (Furman and Landsman, 2010) and mortality modelling

(Alai, Landsman and Sherris, 2013, 2016).

Avanzi, Taylor, Vu and Wong (2016) applied a common shock model to claim triangles,

creating dependencies between triangles. In this model, cells were assumed Tweedie

distributed, and the authors generated a multivariate Tweedie distribution to describe the

triangles, including the shock.

Avanzi, Taylor and Wong (2018) established a very general framework for the introduction of

common shocks into claim triangles. This included dependencies between corresponding cells

of different triangles, corresponding accident, development or calendar periods, or

combinations of these; as well as more general dependencies.

Many models of claim triangles are multiplicative, especially those resembling the chain

ladder, with multiplicative row and column effects. When an additive common shock is added,

it can cause an awkward feature whereby the proportion of claim experience contributed by the

shock can vary widely over a triangle.

This is the “unbalanced” feature discussed by Avanzi, Taylor, Vu and Wong (2021). Those

authors increased balance by modifying the shock to assume different magnitudes in different

development periods. Subsequently, Taylor and Vu (2022) produced “auto-balanced” models,

in which the awkward imbalance is totally eliminated.

Essentially, the difficulties of balance arise from the assumed structure of cell means, which

contain a multiplicative chain-ladder-like term supplemented additively by a common shock.

3 
 
This can be simplified by subjecting the claim experience in each cell to a log transform, thus 
producing an additive model, and supplementing this with an additive shock. 
This would usually mean that the individual cell observations are the exponential of some 
distribution with an asymptotically exponential tail, such as log normal. This will be suitable 
for some data sets, and not others. In any case, log normal claim models are no stranger to the 
actuarial literature. See, for example, Taylor (2000), Wüthrich and Merz (2008), Shi, Basu and 
Meyers (2012).  
The objective of the present paper is to adopt this log normal or similar structure, but apply it 
within  the  very  general  common  shock  framework  of  Avanzi,  Taylor  and  Wong  (2018), 
mentioned  above.  The  benefits  of  doing  so  are,  first,  that  balance  of  the  shock  is  easily 
achieved; second, that parameter estimation is reduced to linear regression in the log normal 
case; and, third, that even variance estimation can sometimes be achieved in closed form. 
The paper is concerned with claim triangles, and the forecast of loss reserves based on them. It 
proceeds as follows. After some notational and mathematical preliminaries in Section 2, both 
additive and multiplicative common shocks are described in Section 3. The full detail of a log 
normal common shock model, including model structure, parameter estimation and forecast, is 
discussed in Section 4, and Section 5 notes how these models may be generalized from log 
normal to log Tweedie. Section 6 discusses the special cases of these models in which the 
idiosyncratic (non-common-shock)  component  of  each  cell  assumes a  multiplicative  chain 
ladder form, and Section  7  provides a numerical example. Section  8 winds up  with  some 
concluding remarks. 
 
2. Framework and notation 
2.1. Notation  
The general common shock framework of Avanzi, Taylor and Wong (2018) is adopted.  A 
claim array  will be defined here as a 2-dimensional array of random variables , 
indexed by integers , with  for some fixed integers  and 
integer-valued function .  The symbol  has been used here, rather than the more natural 
, since the latter will be used later to denote an identity matrix.  For any given pair , the 
random variable  may or may not be present. 
The subscripts  typically index accident period (row) and development period (column) 
respectively, and the  represent observations on claims, commonly claim counts or amounts.  
In the special case  and , the array reduces to 
the well known claim triangle. 
Define , so that .  Observations with common  lie on the 
-th diagonal of . 
Subsequent sections will often involve the simultaneous consideration of multiple business 
segments, with one array for each segment.  A segment could be a line of business.   
It will be necessary in this case to consider a collection  of claim 
arrays, where  denotes the array for segment .  It will be assumed that all  are 

congruent, i.e. are of the same dimensions , and that they have missing observations in the

same  locations. Consequently, the order  of  is the same for all n, and will be

denoted .

The  observation of  will be denoted 

; the entire -th row of  denoted ;

and the entire -th column . It will sometimes be useful to represent the enter array 

as a vector. This will be denoted by , with components 

 arranged in some pre-defined

order. The order might conveniently be dictionary order but, in general, it is arbitrary. It must,

however, be the same for all .

The vector random variables  may be stacked to form a larger vector









(2.1)

It will be convenient to write 



, and to stack the 

 into larger vectors  and

 in the same way as was done for the 

. Now introduce the notation





 and .

It will also be useful to consider diagonals of , where the -th diagonal is defined as the

subset 

, and represents claim observations from the -th calendar

period,  denoting the calendar period in which the first accident period falls. The entire

-th diagonal of  will be denoted .

Let  be the largest value of  occurring in . Estimates of 

for  will be

forecasts. Let the set of 

 requiring forecast be denoted by . This will correspond

with some set of ordered pairs  with .

An identity matrix will be denoted by , and a column vector with all components equal to

unity denoted by 1. When the dimensions of these quantities are not immediately apparent

from the context, they will appear as subscripts to the quantities themselves. For example, 

will denote the  identity matrix, and  the -dimensional unit vector.

2.2. Mathematical preliminaries

There will be occasion to use the Kronecker matrix product. The Kronecker product of the

 matrix  and  matrix  is the  matrix

  

 

 

(2.2)

It may be shown that



(2.3)

where the upper  denotes matrix transposition.

It will also be useful to note the so-called mixed-product property, according to which



(2.4)

where, here and below, vectors and matrices are assumed of suitable dimensions for the

products and Kronecker products that appear.

Further, if  is a column vector, then



(2.5)

and



(2.6)

2.3. Probabilistic preliminaries

2.3.1. Univariate Tweedie family

The Tweedie family is a sub-family of the exponential dispersion family. The latter has two

well known representations, the additive and reproductive forms (Jorgensen, 1987). In

common with Avanzi, Taylor, Vu and Wong (2016), the present paper develops the

multivariate Tweedie family (Section 2.3.2) within the context of the additive representation,

which has the pdf



(2.7)

where  is a location parameter,  is a dispersion parameter, and  is called the

cumulant function.

This distribution has cumulant generating function

,

(2.8)

giving

,

(2.9)

.

(2.10)

The Tweedie sub-family is obtained by the selection















(2.11)

If  is distributed according to (2.7) with , then it will be denoted .

A useful alternative form of (2.11) in the case



 is

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1LognormalclaimmodelswithcommonshocksGregTaylorSchoolofRiskandActuarialStudiesUniversityofNewSouthWalesUNSWSydney,NSW2052AUSTRALIAOctober20222Abstract.Thispaperisconcernedwithmodellingmultipleclaimarraysthataresubjecttooneormorecommonshocks.Itusesastructurethatinvolvesverygeneralformsbothidiosyncrat...

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