In this section, we discuss applications of multivariate models in an insurance context. We illustrate such model evaluations based on variance-covariance, risk factor, and copula approaches. In the subsections, we summarise the au- thor’s scientific contribution to the topic of numerical model evaluation. In Sec- tion 4.1, the AEP algorithm is illustrated. The GAEP algorithm, a generalization of AEP, is introduced in Section 4.2. In Section 4.3, we give a method for the nu- merical approximation of the hierarchical risk aggregation model introduced in Section 2.1.
Suppose we have selected a specific multivariate model for the risks of the insurance company of interest according to the procedures outlined in Sec- tion 2. Furthermore, assume the model has been calibrated as illustrated in Section 3. The following list gives possible applications, most of them very rele- vant for the business of an insurance company.
• In many cases, the most important model application isrisk aggregation. That is, the calculation of the distribution of the sum of all risks, or of some non-linear aggregation functional. Diversification effects can usually also be assessed through the risk aggregation methodology.
• Strongly reliant on risk aggregation is the calculation ofsolvency capital requirements, which is often measured by applying a risk measure to the sum of all risks. For instance, such requirements are mandated by the European Solvency II framework or the Swiss Solvency Test.
• An involved multivariate risk model can also cover market risk factors, such as interest rates, currencies, and inflation. Knowing the sensitivity of single and aggregated insurance risks with respect to these market risks allows forasset liability management, risk hedging, duration matching, and calculation of replicating portfolios. The investment strategy should take these sensitivities into account in order to manage risk exposures. • Knowing the joint distribution of single risks and the aggregated risk al-
lows forcapital allocation, i.e. the splitting of risk capital to single risks
36 CHAPTER 4. MODEL EVALUATION
or risk classes. Through capital allocation and associated hurdle rates, portfolios can be steered towards profitability, see Besson et al. (2008). A related profitability measurement concept is RoRaC (Return on Risk ad- justed Capital).
• Having a precise multivariate model for insurance risks allows to identify the risks which constitute the major threats for the profitability or even existence of a reinsurance company. Throughreinsurance optimization
or insurance linked securities such threats can be mitigated.
We now illustrate how the tasks listed above can be solved for variance- covariance, risk factor, and copula approaches.
The conclusions that can be drawn from a variance-covariance approach
are inherently limited to mean, variance and covariance of linear combinations of the risksX1, . . . ,Xd. For instance, the characteristics of the aggregate riskS=
X1+ ··· +Xd are given by E[S]=µ1+ ··· +µd and var(S)= d X i=1 d X j=1 cov(Xi,Xj)= d X i=1 d X j=1 σiσjρi,j.
As the model assumptions are restricted to first and second moments, no fur- ther properties ofScan be deduced. Therefore, the only sensible approach to determine risk capital in this model is to set it proportional to variance or stan- dard deviation of S. Capital can be allocated to the single risks by calculating the single risk’s contribution to the aggregate in terms of the covariance
cov(Xi,S)= d X j=1 cov(Xi,Xj)= d X j=1 Σi,j.
Portfolio optimization can be done by minimizing the variance under constraints on the portfolio decomposition. Finding optimal reinsurance strategies is anal- ogous if only proportional reinsurance contracts are considered.
In a risk factor model, Monte Carlo simulations are usually employed for model evaluation purposes. Suppose the distribution of theYj andǫi, as well
as the parametersµi,ri,j andσ2i are known, or estimated as illustrated in Sec-
tion 3. A sample of the random vector (X1, . . . ,Xd) can now easily be created by
simulating the risk factorsYj and the residualsǫi. If there is sufficient amount
of observations, one may also draw fromYj andǫi through bootstrapping. The
empirical distribution obtained through the simulations can now be used to es- timate functionals of (X1, . . . ,Xd).
The quantitative evaluation of acopula model is usually done with Monte Carlo simulations. Suppose the random vector of interest has a distribution
4.1. THE AEP ALGORITHM 37
given by a copula model
P[X1≤x1, . . . ,Xd≤xd]=C(F1(x1), . . . ,Fd(xd)) ,
where both copula and margins are known or estimated. Realizations of (X1, . . . ,Xd)
can be generated as follows by drawing first a sample of from the copula:
¡
U1k, . . . ,Udk¢∼C, for k=1, . . . ,n. Then, a sample of (X1, . . . ,Xd) is given by
¡
X1k, . . . ,Xdk¢=¡F1−1(U1k), . . . ,Fd−1(Udk)¢, for k=1, . . . ,n,
whereFi−1is the generalised inverse ofFi. As soon as this sample is obtained,
evaluating the problems as mentioned in the beginning of this section is rela- tively easy, by using the empirical distribution of (X1, . . . ,Xd).
However, both steps of the sampling procedure for (X1, . . . ,Xd) can be nu-
merically challenging. Some parametric copula families are easy to simulate from, for instance the Gaussian copula and most Archimedean copulas, see McNeil et al. (2005). Other copula families, such as nested Archimedean cop- ulas and vine copulas require difficult numerical procedures for sampling, see Hofert (2010b) and Kurowicka and Cooke (2006). Inverting the marginal cdfs can also be non-trivial, such as in the case of stable distributions, see Hofert (2012).
4.1 The AEP algorithm
In this section, we introduce the AEP algorithm, as published in Paper A. Sup- pose we are interested in the cdf ofS=X1+ ··· +Xd, the sum of positive depen-
dent random variablesXi, at some fixed points∈[0,∞):
P[X1+ ··· +Xd≤s]. (4.1)
The AEP algorithm is designed to approximate (4.1). This algorithm does neither rely on simulations (as Monte Carlo) nor on numerically integrating a density. Instead, it approximates (4.1) through a geometric approach, using only evaluations of the joint cdf of (X1, . . . ,Xd).
The two necessary assumptions for its application are that
• all components of the random vector X = (X1, . . . ,Xd) are positive, i.e.
P[Xi>0]=1 fori=1, . . . ,d,
• the joint cdf
H(x1, . . . ,xd)=P[X1≤x1, . . . ,Xd ≤xd]
38 CHAPTER 4. MODEL EVALUATION
The first assumption can be weakened to component-wise bounded from be- low. The second assumption is satisfied, for instance, if the distribution of X
is given through a copula model where both copula and marginal distribution functions are analytic.
Here, we give an overview on the most important aspects of the AEP algo- rithm. To that end, we concentrate on the graphically intuitive two dimensional cased=2. The general cased∈Nis given in Paper A.
First, we give some definitions. Forb=(b1,b2)∈R2andh∈R,Q(b,h)⊂R2
denotes the hypercube
Q(b,h)=
(
(b1,b1+h]×(b2,b2+h] ifh>0,
(b1+h,b1]×(b2+h,b2] ifh<0.
LetVH denote the (probability) measure onR2induced by the joint cdfH
VH[(−∞,x1]×(−∞,x2]]=H(x1,x2).
The probability P[(X1,X2)∈Q(b,h)]=VH[Q(b,h)] forh>0 can easily be ex-
pressed in terms ofH:
VH[Q(b,h)]=P[X ∈(b1,b1+h]×(b2,b2+h]]
=H(b1+h,b2+h)−H(b1,b2+h)−H(b1+h,b2)+H(b1,b2). (4.2)
The caseh<0 is analogous.
LetS(b,h)⊂R2denote the simplex
S(b,h)= (© x∈R2:x1−b1>0,x2−b2>0, andP2k=1(xk−bk)≤h ª if h>0, © x∈R2:x1−b1≤0,x2−b2≤0, andP2k=1(xk−bk)>h ª if h<0.
The definitions of the setsQ(b,h) andS(b,h) are illustrated in Figure 4.1.
Recall that we intend to calculate or approximateP[X1+X2≤s]. In terms of
the probability measureVH,P[X1+X2≤s] can be written as P[X1+X2≤s]=VH[S(0,s)] .
Due to (4.2), it is very easy to compute theVH-measure of squares. The idea
behind the AEP algorithm is to approximate the simplexS(0,s) by squares and
through the probability mass of these squares approximateP[X1+X2≤s].
As a first proxy toS1
1 =S(0,s), we set
Q1
4.1. THE AEP ALGORITHM 39 0 x1 x2 b1 e b2+he b2 b2+h S(b,h) Q(be,he) e b1+he b1+h e b2 e b1
Figure 4.1: An illustration of the simplex S(b,h) and the square Q(be,he) for
some b =(b1,b2),be =(be1,be2)∈R2 and h,he >0. For h,he <0, the sets would
be flipped.
The factor 2/3 might be surprising. In Paper A, this factor is shown to provide fastest convergence. The error commited by takingVH[Q11] as an approximation
ofVH[S11] can be expressed in terms of the three simplexes
S1 2 =S((0, 2/3s), 1/3s), S2 2 =S((2/3s, 0), 1/3s), S3 2 =S((2/3s, 2/3s),−1/3s). Formally, we have S(0,s)=¡Q1 1∪S21∪S22 ¢ \S3 2, (4.3)
which is illustrated in Figure 4.2. The sets S1
2,S22 and Q11 are pairwise disjoint. Also, note that S23⊂ Q11.
Thus,P[X1+X2≤s] can be written as VH[S(0,s)]=VH £ Q1 1 ¤ +VH £ S1 2 ¤ +VH £ S2 2 ¤ −VH £ S3 2 ¤ . (4.4)
Using (4.2), we set as a first approximation ofVH[S(0,s)] the value
P1=VH £ Q1 1 ¤ =H(2/3s, 2/3s)−H(0, 2/3s)−H(2/3s, 0)+H(0, 0).
The error committed by consideringP1instead ofVH[S(0,s)] can be expressed
in terms of theVH-measure of the three simplexesS2kdefined above:
VH[S(0,s)]−P1=VH £ S1 2 ¤ +VH £ S2 2 ¤ −VH £ S3 2 ¤ .
40 CHAPTER 4. MODEL EVALUATION 0 s 2/3s 2/3s s S3 2 S2 2 S1 2
Figure 4.2: An illustration of the decomposition (4.3) of the two-dimensional simplexS(0,s) into the squareQ1
1and the simplexesS21,S22, andS23.
At this point, we apply toS1
2, S22andS23 a decomposition analogous to the
one used forS1
1. That is, fork =1, 2, 3, we approximateVH[S2k] by the prob-
ability massVH[Q2k] of a squareQ2k. The only difference between the first and
the following step is that we have to keep track of whether the measure of a sim- plex has to be added to or subtracted from the next approximation. The error terms can again be expressed in terms of simplexes. This defines recursively a sequence of estimatorsPn,n∈N, which starts withP1=VH[Q11] and iteratively
adds the signed weight of the recursively defined cubes Qk
n. The decomposi-
tion (4.4) leads to three new simplexes, hence the number of simplexes triples in each iteration: 3 in the 2nd iteration, 9 in the third and so on. The sets whose probability mass defineP1,P2andP3is illustrated in Figure 4.3. These sets also
illustrate the iterative behaviour of the algorithm.
The following theorem shows that under mild conditions, the sequencePn
converges with exponential rate to the desired value P[X1+X2≤s]. Proofs of
the following results are given in Paper A.
Theorem 4.1 Suppose(X1,X2)has a density in a neighbourhood of{(x1,x2)∈R2: x1+x2=s}. Then
¯
¯Pn−P[X1+X2≤s]
¯
¯=O¡(1/3)n¢.
AlthoughPn converges with exponential rate inn, the numerical complex-
ity to computePnisO(3n), i.e. a grows exponentially. This implies thatPncon-
verges with polynomial rate in terms ofT(n), whereT(n) is the computation time required to calculatePn.