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Technical University Munich

Department of Mathematics

Master Thesis

Reduction of the Ruin Probability for an

Insurance Company by Capital Transfer in

a Bipartite Network

author

Katrin Wiersdörfer

supervised by

Prof. Dr. Claudia Klüppelberg

submission date

March 12, 2019

I hereby declare that this thesis is my own work and that no other sources have been usedexcept those clearly indicated and referenced.

Munich, March 12, 2019

Katrin Wiersdörfer

Abstract

The objective of this master thesis is the computation ofthe asymptotic ruin probability under consideration of cap-ital transfer in the context of an insurance network model.In this model, objects which are exposed to catastrophicrisks influencing the operating result and capital are in-sured. The network of the insurance companies can bemodeled by a bipartite graph. Due to large policy lim-its, the risks are shared among the insurance companiesto lower the consequences. If a large loss occurs and aninsurance company cannot cover the loss, the insurancecompany must declare insolvency. In insurance pools orgroups, the insurance members or subsidiaries might beobligated to pay the outstanding loss of the insolvent in-surance company. By reciprocal capital transfer, the riskof insolvency can be influenced. How likely is insolvencyif insurance companies transfer capital? Three cases areinvestigated: In the first case, the insurance companies arecompetitors which do not transfer capital. In the secondcase, the insurance companies transfer capital among eachother if another insurance company cannot pay the loss.Moreover, it is assumed that in this case, there are notransactions costs which means that the capital transferdoes not include any exchange rates or administrative feesof the banks. In the third case, the insurance companiesshift capital like in the second case, but every transfer mayinclude transaction costs.

Zusammenfassung

Das Ziel dieser Masterarbeit ist die Berechnung der asym-ptotischen Ruinwahrscheinlichkeit unter Berücksichtigungvon Kapitaltransfer im Kontext eines Versicherungsnetz-werkmodells. In diesem Modell werden Objekte versichert,welche Großschadenereignissen mit Auswirkung auf das ope-ratives Ergebnis und Kapital eines Versicherungsunterneh-mens auslösen können. Das Netzwerk der Versicherungsun-ternehmen kann durch einen bipartiten Graphen modelliertwerden. Aufgrund großer Versicherungssummen werden dieRisiken zur Minderung der Auswirkung solcher Schadensfäl-le unter den Versicherungsunternehmen geteilt. Wenn eingroßer Schaden eintritt und ein Versicherungsunternehmenden Schaden nicht decken kann, muss das Versicherungsun-ternehmen Insolvenz anmelden. In Versicherungspools oderKonzernen können Versicherungsmitglieder oder Tochter-gesellschaften dazu verpflichtet sein, den noch ausstehen-den Schadensbetrag des insolventen Versicherungsunter-nehmens zu decken. Durch gegenseitigen Kapitaltransferkann das Risiko einer Insolvenz beeinflusst werden. Wiewahrscheinlich ist eine Insolvenz, wenn Versicherungsun-ternehmen Kapital transferieren? Drei Fälle werden unter-sucht: Im ersten Fall sind die VersicherungsunternehmenWettbewerber, welche kein Kapital transferieren. Im zwei-ten Fall überweisen sich die Versicherungsunternehmen un-tereinander Kapital, wenn ein anderes Versicherungsunter-nehmen den Schaden nicht zahlen kann. Zudem wird ange-nommen, dass es in diesem Fall keine Transaktionskostengibt, was bedeutet, dass der Kapitaltransfer keine Wechsel-kurse und administrative Kosten der Banken beinhaltet. Imdritten Fall tauschen die Versicherungsunternehmen wie imzweiten Fall Kapital aus, wobei jede Überweisung Transak-tionskosten beinhaltet.

Contents

1 Introduction 1

2 Preliminaries 5

2.1 Poisson processes 5

2.2 Regular variation 8

3 Insurance network model 14

4 Capital transfer 19

5 Ruin sets 22

6 Ruin probability 41

7 Applications 57

7.1 Individual ruin probability 58

7.2 Network ruin probability 63

7.2.1 No capital transfer 64

7.2.2 Capital transfer without transaction costs 68

7.2.3 Capital transfer with transaction costs 71

8 Conclusion 77

References 78

List of figures

Preliminaries

Poisson Processes

2.1 Example for a homogeneous Poisson process with λ = 1 62.2 Example for a compound Poisson process with λ = 1 7

Insurance model

3.1 A bipartite network with q agents and d objects 14

Ruin sets

5.1 The solvency cone S is spanned by the vectors (π(kl)ek − e l) for1 ≤ k , l ≤ q and ek for 1 ≤ k ≤ q for q = 2. 23

5.2 The solvency set S is r -increasing. 245.3 The ruin set F is r -decreasing. 245.4 The solvency set S is a convex cone as all lines joining two points in the

set lie completely within the set. 255.5 The ruin set F is not a convex cone as the line joining the points y1 and y2

lies outside of the ruin set. 25

No capital transfer

5.6 The solvency cone S = [0,∞)q and the ruin set F = ([0,∞)q)c for q = 2 26

Capital transfer without transaction costs

5.7 The solvency cone S = {y ∈ R2 : y 1 ≥ y 2} and the ruin setF = {y ∈ R2 : y 1 < y 2} for q = 2 31

Capital transfer with transaction costs

5.8 The solvency cone S = {y ∈ R2 : y = w (12)(π(12)e1 − e2) +w (21)(π(21)e2 − e1) + z1e1 + z2e2,w (12) ≥ 0,w (21) ≥ 0, z1 ≥ 0, z2 ≥ 0}and the ruin set F = Sc 32

5.9 The vectors (π(kl)e l + ek) are orthogonal to (π(kl)ek − e l) for 1 ≤ k , l ≤ q. 355.10 The solvency cone S can be represented by the intersection of two half planes. 355.11 The ruin set F =

⋃k 6=l{y ∈ R2 : yT (π(kl)e l + ek) < 0} for k , l ∈ {1, 2} 39

5.12 The negative ruin set−F =

⋃k 6=l{y ∈ R2 : yT (π(kl)e l + ek) > 0} for k , l ∈ {1, 2} 39

5.13 The negative ruin set −F can be represented by the union of two half planes. 40

Ruin probability

6.1 The boundary of the ruin set F is the boundary of the solvency cone S i.e.∂F = ∂S. 41

6.2 The boundary ∂S of the solvency cone S is a subset of the union of twostraight lines given by

⋃k 6=l{y ∈ R2 : yT (π(kl)e l + ek) = 0} for k , l ∈ {1, 2}. 42

6.3 Geometrical interpretation of the set⋃

k 6=l B(kl) for k , l ∈ {1, 2} and

a ∈ R2+\{0} 43

6.4 The set H (kl) for k = 1, l = 2, q = 2 and a ∈ R2+\{0} 43

6.5 The set⋃

p∈Q∩(1,∞) pB(kl) for k = 1, l = 2, q = 2 and a ∈ R2

+\{0} 436.6 K δ

r ={y ∈ R2\{0} : || y

||y ||2 + r||r ||2 ||2 < δ

}for the Euclidean norm 45

6.7 The ruin set F 556.8 The ruin set −F 556.9 The set b −F 556.10 The set vAl + b −F 556.11 The set {y ∈ R2

+\{0} : y ∈ vAl + b −F} 55

Applications

Individual ruin probability

7.1 Individual ruin probability 587.2 A bipartite network with 1 agent and 1 object 587.3 Illustration of the ruin set F = (−∞, 0) 597.4 A bipartite network with 1 agent and d objects 61

Network ruin probability

No capital transfer

7.5 A bipartite network with q agents and d objects without capital transfer 64


7.6 A bipartite network with q agents and d objects with capital transfer andwithout transaction costs 68


7.7 A bipartite network with q agents and d objects with capital transfer andtransaction costs 72

List of tables

Applications

7.1 The positive safety loading lj , 1 ≤ j ≤ d for different distributions 57

Individual ruin probability

7.2 Solution (7.3) for different distributions 607.3 Solution (7.10) for different distributions 63

Network ruin probability

No capital transfer

7.4 Solution (7.15) for different distributions 66





Notation

Symbols Description

N(t) Poisson process at time t ≥ 0λ intensity rate of the Poisson processPoi Poisson distributionX (k) kth claimsT (k) claim time of the kth claimsW (k) inter claim time between the (k − 1)th and kth claimsC (t) compound Poisson process at time t ≥ 0FX distribution function of XFX right tail of the distribution function of XF IX integrated tail distribution function of X

d= same distributionµ expectation of a random variableL slowly varying functionα index of regular variationRV (α) class of regularly varying variables with index α∼ f (x) ∼ g(x), x →∞ means limx→∞

f (x)g(x)

= 1

Γ Gamma functionE state spaceE Borel σ-field of Ev→ vague convergence|| · || ||x || norm of xµ intensity measureµ intensity measure µ ◦ A−1

MRV (α,µ) class of multivariate regularly varying variables with indexα and intensity measure µ

Ai agent i for i ∈ {1, ... , q}Oj object j for j ∈ {1, ... , d}A weighted q × d adjacency matrixA∗ realization of the weighted adjacency matrix AA set of all possible realizations A∗ of the weighted adjacency

matrix AW weights which indicate the proportion of the objects’ losses

that are paid by the agentsdegA(i) number of objects O that are insured by agent Ai

degO(j) number of agents A insuring object Oj

V (t) claims process at time t ≥ 0c premium ratel positive safety loadingρ relative safety loading

Symbols Description

E (t) exposure at time t ≥ 0u initial capitalb normed initial capitalR(t) reserves at time t ≥ 0Π capital transfer matrixF ruin setF set of ruin sets FS solvency coneS set of solvency cones SMc complement of the set M∂M boundary of the set MM closure of the set MM◦ interior of the set Me standard basis vectorsθ angle (radian)γ angle (degree)ψ ruin probability

1 | Introduction

Ruin theory deals with the model-based risk assessment of an insurance company’s position.The idea is to analyze the stochastics of the claims and to compute the probability thatthe risk capital and the premiums are sufficient to cover the claims’ losses. Policyholderspay premiums to the insurance companies to ensure that the insurance company can coverthe loss in case of an insured event. If many policyholders request insurance payments,the insurance’s liabilities might exceed the insurance’s capital. Then, the insurance com-pany is called insolvent. This is usually very improbable as insurance companies have manypolicyholders and if some get money, others continuously pay their premiums. Moreover, pri-mary insurance companies can reduce their risks by ceding claims to reinsurance companies.Then, the premium income is diminished, but the excess claims over a certain contractuallyagreed threshold is covered by the reinsurance company. Since 2000, the life insurance com-pany Mannheimer Leben was the only insurance company becoming insolvent in Germany.However, the monetary amount of losses for catastrophic events has been increased in thelast years. The reasons include the growth of prosperity and population (rapid trends ofurbanization), the climate change (extreme weather conditions) or the increase of terrorism(cf. Nguyen 2007, p. 12). Catastrophic events lead to many individual losses which affectvarious insurance contracts and contracting parties. For small catastrophic risks, it is usuallysufficient that primary insurance companies are reinsured. However, the insurance’s liabilityfor events such as large terrorist attacks, aviation or nuclear risks is huge. Based on theexamples of the terrorist attacks in New York in 2001 or hurricane Katrina in 2005, theconsequences for insurance companies have became apparent. Some risks are too high sothat insurance companies cannot bear the risk alone. An internal solution within a groupcould be the joint coverage of severe and dangerous risk by the subsidiaries. In case of higherpolicy limits, insurance companies form an insurance pool which is a collective pool ofseveral insurance companies (primary insurance and reinsurance companies) ensuring thecoverage of high catastrophic risks. Due to risk sharing, the risk of declaring insolvency issmaller than for a single insurance company or subsidiary. To categorize the level of losses, ithas been established to use separate modelings for basic claim losses and large claim losses.Basic claim losses are characterized by a high loss frequency and on average a small amountof loss. The risk for insurance companies is small. Large claim losses like catastrophicrisks are losses with a low frequency and on average a large amount of loss which exceedsa predefined threshold. Compared to basic claim losses, it is often difficult to accuratelymodel large claims as relevant historic data is lacking due to the low loss frequency. Eventhough only a few insurance companies declared insolvency in the past, the computationof the ruin probability is important to allocate suitable risk reserves. This is important toensure the solvency of the insurers to pay claims on the contracts that they issue.

In 1903, the Swedish actuary Filip Lundberg examined the Poisson process as a modelfor the claim number process N . He is known as one of the pioneers of the mathematicalrisk theory. Based on his results, Harald Cramér introduced the compound Poisson processrepresenting the total claim amount in the 1930s. Lundberg’s and Cramér’s main contribu-tions led to the Cramér-Lundberg model which is capable to describe the basic dynamicsof a homogeneous insurance model. This model assumes that the claim number process isindependent of the claim sizes. The central objective is to investigate the ruin probability

1

1 | Introduction

which is considered as a univariate risk. Univariate risk models only describe the distributionof a single insurance portfolio. The ruin probability is calculated as the probability of therisk reserve process becoming negative at some point in time. For this simple model, theruin probability can be determined by closed form solutions e.g. by the Pollaczek-Khinchineformula. However, insurance portfolios are more complicated in reality and the univariaterisk model may be found too restrictive to accurately describe the complexity of the reservesof an insurance company. In reality, insurance companies typically have several insuranceportfolios which are often dependent. For instance, large insurance groups operate in differ-ent countries around the world and offer various types of insurance coverage e.g. fire, car,accident, third-party liability, storm insurances and so forth. Two examples for dependenciescould be a car accident or hurricanes. A car accident may cause a claim for the motor andhealth insurance, whereas hurricanes might cause losses in different countries. To modelthis appropriately, multivariate risk models are necessary. Multivariate risk models assess therelationships of two or more insurance portfolios and capture dependent structures betweenthem. Unlike univariate risk models, multivariate ruin probabilities can have various inter-pretations. A possible trigger for ruin could be that the reserve processes of all insuranceportfolios are simultaneously below zero. This assumption is less realistic. A more appropri-ate choice is that ruin occurs as soon as one single insurance portfolio has negative reserves.Another assumption could be that ruin occurs when the sum of the portfolio reserves isnegative. There has been a number of researchers who have examined multivariate modelsin their papers in the recent years.

Hult and Lindskog consider in their paper Heavy-tailed insurance portfolios: buffer capi-tal and ruin probabilities (2006) a single insurance company with d business lines facinglarge claims with regularly varying distribution. Each business line is assumed to be exposedto catastrophic risks as for instance earthquakes, storms, floods or terrorist attacks. Thismeans that one single large claim or one simultaneous occurrence of large claims have asignificant effect on the solvency of the insurance company. To cover the unexpected losses,insurance companies must charge their policyholders high premiums and hold a sufficientamount of capital. As holding large buffer capital can be unprofitable for an insurancecompany, Hult and Lindskog propose that insurance companies have mutual agreements offinancial support. In case of insolvency, capital may be transferred between the business linesto reduce the ruin probability. Compared to many other researchers who compute the ruinprobability as the probability of negative reserves, Hult and Lindskog use ruin sets basedon cones. The ruin probability is defined as the probability that the multivariate reserveprocess of the insurance company first hits the ruin set. In case of capital transfer, the ruinprobability is defined as the situation when negative reserves in one or multiple business linescannot be balanced.

Kley, Klüppelberg, and Reinert introduce in their paper Systemic risk in a large claimsinsurance market with bipartite graph structure (2016) an insurance network model repre-sented by a bipartite graph of agent-object relationships. A bipartite graph is characterizedby two sets (agents and objects) which are connected by an edge indicating the insured pro-portion of the objects. The agents represent the insurance companies whereas the objectsare portfolios of risks which cause a severe loss if a triggering event happens. Due to thelarge losses, the risks are shared by the agents such that the objects’ losses are jointlycovered. Compared to Hult and Lindskog whose model only considers a single insurancecompany, this model examines the network structure of several insurance companies. Kley,Klüppelberg, and Reinert show that the dependence on the network structure is an impor-

2

1 | Introduction

tant factor for the asymptotic behavior. The effect of the network structure on measures forsystemic risk is investigated by Kley, Klüppelberg, and Reinert in the paper Conditional riskmeasures in a bipartite market structure (2018). Using Pareto-tailed losses and multivari-ate regular variation, they obtain asymptotic results for conditional risk measures. Behme,Klüppelberg, and Reinert continued researching and studied in their paper Hitting probabil-ities for compound Poisson processes in a bipartite network (2018) ruin probabilities of theaggregated light-tailed risk process by means of a network Pollaczek-Khintchine formula. Inthe paper Financial risk measures for a network of individual agents holding portfolios oflight-tailed object (2018), Klüppelberg and Seifert quantify risk measures as the Value atRisk and Expected Shortfall and derive results for conditional risk measures as the Condi-tional Expected Shortfall. They compare the stochastic behavior of portfolio risks under thelight-tail assumption with heavy-tailed settings.

In this master thesis, Hult’s and Lindkog’s idea of applying capital transfer in an insur-ance model is implemented into the insurance network model of Klüppelberg et al. By thebipartite network structure as developed by Klüppelberg et al., the risk sharing of insurancepools or groups can be modeled. It is assumed that the insurance companies only haveportfolios of catastrophic risks. By mutual agreements within a group or between poolmembers, the insurance companies might be obligated to overtake the outstanding amountof loss of an insolvent insurance company. In this case, the insurance companies transfercapital among each other. An example for an insurance pool structured by a network andusing capital transfer is the pharma pool. Catastrophes caused by medical products as forinstance the Contergan scandal have shown that compensation claims can increase to anenormous height. Due to the high amount of losses, insurance companies do not insurethe risk alone. Especially, the risk of outstanding insurance claims is high. The insurancecompanies underwrite risks up to an amount of 6 million Euro per medical product. If theamount of loss exceeds this threshold, the loss is shared proportional to the risk that therespective insurance company brings in the pool. The medical products are categorized inthree risk groups: over-the-counter products (low risk), pharmacy-only products (mediumrisk) and prescription products (high risk) (cf. Ehling 2011, p. 280). Dependent on the riskgroup, the insurance companies’ proportions on the loss amount are higher or lower. If aninsurance company cannot cover the loss, the other members of the pool commit themselvesto increase their payment amounts. The members provide guarantees for cedents in case ofinsolvency (cf. Ehling 2011, p. 285).The main objective of this master thesis is the computation of the asymptotic ruin probabilitywhen the initial capital increases to infinity. For the analysis, three cases are distinguished:

1) No capital transfer

2) Capital transfer without transaction costs

3) Capital transfer with transaction costs

It is assumed that the claims are modeled by jumps such that ruin occurs in case of alarge jump. The losses of each portfolio occur at the same time. A central considerationis the modeling of the network structure and the investigation of the capital transfer. Thecomputations and proofs are mainly based on the paper Heavy-tailed insurance portfolios:buffer capital and ruin probabilities (2006) by Hult and Lindskog and Systemic risk in a largeclaims insurance market with bipartite graph structure (2016) by Kley, Klüppelberg, andReinert. Whereas Behme, Klüppelberg, and Reinert use the network Pollaczek-Khintchine

3

1 | Introduction

formula under the assumption of light-tailed claims for their ruin computations, the ruinprobability in this master thesis is based on ruin sets assuming regularly varying claims.Compared to Hult and Lindskog who neglect any network effects, the ruin probability takesthe dependence structure of the whole market into account. This master thesis is structuredas follows.

Initially, chapter 2 introduces the preliminaries representing some important results inthe risk theory. As the behavior of insurance portfolios can be modeled by Poisson pro-cesses, the homogeneous and the compound Poisson process are explained. Moreover, sincecatastrophic risks can be expressed by regularly varying distributions, the concept of regularvariation is defined.

In chapter 3, the framework of the insurance network model is described. A bipar-tite graph models the dependency between the insurance companies and portfolios. By aweighted adjacency matrix, the losses of the objects are linked to the agents. The claimsprocess for the objects are given by a multivariate compound Poisson process. Moreover,the exposure, the initial capital and the reserve of the agents are defined.

Chapter 4 is dedicated to the capital transfer among the insurance companies. Therules of the capital transfer are defined for the three cases listed on the previous page.

The ruin sets are investigated in chapter 5. Differentiated according to the three cases, itis defined when the insurance company is solvent and when it is classified as insolvent. Fortwo agents, the sets are graphically illustrated.

Chapter 6 deals with the computation of the asymptotic ruin probability. Based on theruin sets, the ruin probability is computed with respect to a fixed and a random weightedadjacency matrix. This means that on the one hand the agents are deterministically linkedwith the number objects, while on the other hand the agents choose their objects randomly.

Chapter 7 provides explicit applications of the proven asymptotic ruin probability. First,only one insurance company is considered, subsequently, a network of several insurancecompanies is investigated.

Finally, in chapter 8 a conclusion is drawn.

The references of the definitions, theorems, propositions and remarks are indicated in theright corner of the framed boxes. If a reference contains the abbreviation cf. i.e. compare,the result is based on this reference, but the wording and notation is adjusted to the overallcontext. If no reference is contained, the result is concluded by me. If not other mentioned,the explanations, figures and computations are my own work.

4

2 | Preliminaries

In this chapter, important fundaments are explained which will be needed in the followingchapters. Since the claim number process is a homogeneous Poisson process and the claimsprocess is a compound Poisson process, Poisson processes are explained at the beginningin section 2.1. In section 2.2, regular variation, which characterizes the distribution ofthe claims, is introduced. This chapter is based on the books and papers by McNeil et al.(2015), Gatto (2014), Resnick (2007), Basrak (2000), Embrechts et al. (1997), Mikosch(2009), Resnick and Willekens (1991), Resnick (1987), Hult et al. (2005), Hult and Lindskog(2005), Basrak et al. (2002). The proofs of the results can be found in the respective books.

2.1 Poisson processes

Poisson processes indicate how many random events occur within a certain time interval.There are many different types of Poisson processes. This master thesis concentrates on ho-mogeneous Poisson processes and compound Poisson processes. Poisson processesare special counting processes which are initially defined. Counting processes count thenumber of random events before the time t ≥ 0.

Definition 2.1.1. (Counting process) (McNeil et al., 2015, Definition 13.24 )

A stochastic process N = (N(t))t≥0 is a counting process if its sample paths areright continuous with left limits existing and if there exists a sequence of randomvariables T (0),T (1),T (2), ... tending almost surely to ∞ such that

N(t) =∞∑k=1

1{T (k)≤t}.

Every realisation of counting processes is a step function. For t = 0, it holds that N(0) = 0so that the counting process has not yet counted any events at the initial time point. Thehomogeneous Poisson processes are particularly characterized by a constant intensity. Thisprocess changes its state without changing its intensity. The process jumps at most onceper time point.

Definition 2.1.2. (Homogeneous Poisson process) (McNeil et al., 2015, Definition 13.25)

A stochastic process N = (N(t))t≥0 is called homogeneous Poisson process withintensity rate λ > 0 if the following properties hold:

a) N is a counting process.

b) N(0) = 0 a.s.

c) N has independent increments: For 0 = t0 < t1 < · · · < tn, the incrementsN(ti−1, ti ] := N(ti)− N(ti−1), i = 1, ... , n are independent.

d) N has stationary increments: For fixed s, N(t, t + s]d= N(s).

5

2 | Preliminaries

e) N is Poisson distributed: For each t > 0, N(t)d= Poi(λt).

Explanation:

• Independent increments:The number of incurred losses at time t − 1 does not influence the number of eventsat time t. This excludes possible chain reactions.

• Stationary Poisson distributed increments:The distribution of (N(t))t≥0 only depends on the length of the interval, but not onits position. This characteristic is called stationary increments.

Example: The probability of car accidents in the summer is identicalcompared to the winter.

Counterexample: The probability of car accidents in the winter is higher due towinterly road conditions.

• Poisson distributed:The number of incurred losses in a given time interval (t, t + s] is Poisson distributedwith λ((t + s)− t) = λs. This also explains the name of the process.

Figure 2.1: Example for a homogeneous Poisson process with λ = 1

Definition 2.1.3. (Claim times) (Gatto, 2014, Definition 3.27)

The claim times for a Poisson process (N(t))t≥0 are given by

T (k) = inf{t ≥ 0|N(t) ≥ k}, k = 0, 1, ... ,

where T (0) = 0.

For the time between the (k − 1)th and the kth incurred event, there exists the definitionof the inter claim times.

6

2 | Preliminaries

Definition 2.1.4. (Inter claim times) (Gatto, 2014, Definition 3.28)

For a homogeneous Poisson process (N(t))t≥0 with claim times (T (k))k∈N, the timeintervals between the events are given by

W (k) = T (k)− T (k − 1) for k = 1, 2, ... .

Proposition 2.1.5. (Characterization of the homogeneous Poisson process)(McNeil et al., 2015, Theorem 13.27)

For the process N = (N(t))t≥0 with N(0) = 0, the following is equivalent:

1) N is a homogeneous Poisson process with intensity rate λ > 0.

2) The inter claim times (W (k))k≥1 are i.i.d. exponential random variables withparameter λ and E[W (1)] = 1

λ.

A compound Poisson process is a jump process corresponding to a compound Poissondistribution. The jump sizes are randomly given by X (1),X (2), ... .

Definition 2.1.6. (Compound Poisson process) (Gatto, 2014, Definition 3.23)

Let (N(t))t≥0 be a counting process and X (1),X (2), ... i.i.d. random variables whichare independent of the counting process N . Then,

C (t) =

N(t)∑k=0

X (k)

with X (0) = 0 is called compound Poisson process.

Figure 2.2: Example for a compound Poisson process with λ = 1

7

2 | Preliminaries

2.2 Regular variation

The class of regularly varying random variables describes the behavior of distributionfunctions evaluated close to infinity. They are realistic distributions for catastrophic eventssuch as earthquakes, big storms, floods or terrorist attacks because they give more weightto large values. In chapter 3, it is assumed that the distribution of the claims are regularlyvarying. In the following,

FX (x) = P(X ≤ x), x ∈ R

describes the distribution function of X and

FX (x) = 1− FX (x), x ∈ R (2.1)

its right tail. First, the univariate notion of regular variation is considered and then, themultivariate definition is introduced.

Univariate regular variation

Univariate regularly varying functions are functions which behave asymptotically likepower functions, whereas slowly varying functions decay slower than any polynomial.

Definition 2.2.1. (Univariate regularly and slowly varying) (Resnick, 2007, Definition 2.1)

A measurable function f : R+ 7→ R+ is univariate regularly varying at ∞ withindex α ∈ R (written f ∈ RV (α)) if for x > 0,

limu→∞

f (ux)

f (u)= x−α.

Here, α is called index of regular variation.If α = 0, f is called slowly varying so that

limu→∞

f (ux)

f (u)= 1.

Remark 2.2.2 (Basrak, 2000, p. 24)

In probability applications, a distribution function FX has a univariate regularly varyingtail (written FX ∈ RV (α)) if for x > 0, it holds that

limu→∞

FX (ux)

FX (u)= lim

u→∞

P(X > xu)

P(X > u)= x−α

for some α > 0. It is common to say that X is univariate regularly varying (X ∈ RV (α)).

The following theorem states that integrals of regularly varying functions are again regularlyvarying. It shows that the integral behaves asymptotically like a polynomial function whenconsidering the tail. By means of this theorem, it is possible to show that the Loggammadistribution is regularly varying.

8

2 | Preliminaries

Theorem 2.2.3. (Karamata’s theorem) (Embrechts et al., 1997, Theorem A3.6)

Let L be slowly varying and locally bounded in [x0,∞) for some x0 > 0. Then,

a) for α > −1, it holds that∫ x

x0

tαL(t)dt ∼ (α + 1)−1xα+1L(x), x →∞.

b) for α < −1, it holds that∫ ∞x

tαL(t)dt ∼ −(α + 1)−1xα+1L(x), x →∞.

Note: (∼)For two functions f and g , f (x) ∼ g(x), x →∞ means

limx→∞

f (x)

g(x)= 1.

Examples

1) Pareto: F (x) = Kx−α x ≥ 1, K > 0, α > 0

P(X > xu)

P(X > u)=

K (xu)−α

Ku−α= x−α

Thus, the index of regular variation is α.

2) Burr: F (x) =

(κ

κ + xτ

)αx > 0, κ > 0, α > 0, τ > 0

P(X > xu)

P(X > u)=

( κκ+(xu)τ

)α

( κκ+uτ

)α=( κ + uτ

κ + (xu)τ

)α=( κ

κ + (xu)τ+

uτ

κ + (xu)τ

)αIf f is continuous at b and lim

x→∞g(x) = b, then lim

x→∞f (g(x)) = f (b) = f ( lim

x→∞g(x)).

It follows that( κ

κ + (xu)τ+

uτ

κ + (xu)τ

)α=( κ

κ + (xu)τ︸︷︷︸u→∞→ 0

+1

κuτ

+ xτ︸︷︷︸u→∞→ x−τ

)α u→∞→ x−τα.

Thus, the index of regular variation is τα.

3) Loggamma: f (x) =αβ

Γ(β)(ln(x))β−1x−α−1 x > 1, β > 0,α > 0, Γ(β) =

∫ ∞0

xβ−1e−xdx

L(x) = ln(x) is slowly varying as

P(X > xu)

P(X > u)=

ln(xu)

ln(u)=

ln(x) + ln(u)

ln(u)=

ln(x)

ln(u)︸︷︷︸u→∞→ 0

+1u→∞→ 1.

9

2 | Preliminaries

Moreover, since L(x) is locally bounded in [x0,∞) with x0 > 1 and α > 0, it followsby the Karamata’s theorem (Theorem 2.2.3) that∫ x

x0

αβ

Γ(β)(ln(t))β−1t−α−1dt =

αβ

Γ(β)

∫ x

x0

(ln(t))β−1t−α−1dt

∼ αβ−1

Γ(β)(ln(x))β−1x−α, x →∞.

Consequently, the Loggamma distribution has the tail

F (x) ∼ αβ−1

Γ(β)

(ln(x)

)β−1x−α, x →∞.

The Loggamma distribution is regularly varying since

P(X > xu)

P(X > u)=

αβ−1

Γ(β)(ln(xu))β−1(xu)−α

αβ−1

Γ(β)(ln(u))β−1u−α

=

(ln(x)

ln(u)+ 1

)β−1

x−αu→∞→ x−α.

Thus, the index of regular variation is α.

The following proposition and lemma are useful properties of regularly varying functions.

Proposition 2.2.4. (Moments of univariate regularly varying random variables)(Mikosch, 2009, p. 100)

Suppose that X is a regularly varying non-negative random variable with index α > 0.Then, it holds that

E[X β]{ <∞ if β < α

=∞ if β > α

i.e. moments below order α are finite and moments above α are infinite.

Lemma 2.2.5. (Potter bounds) (Resnick and Willekens, 1991, Lemma 2.2)

Let X be a non-negative random variable with distribution FX such that 1 − FX isregularly varying with index α > 0. If ε > 0 is given, there exist constants x0 = x0(ε)such that for any c > 0

1− FX ( xc

)

1− FX (x)≤{

(1 + ε)cα+ε if c ≥ 1, xc≥ x0

(1 + ε)cα−ε if c < 1, x ≥ x0.

Multivariate regular variation

The following chapters will deal with dependencies and regular variation. Since an in-surance company typically has multiple insurance portfolios, it is useful to build a modelwhere claims are multivariate distributed. Therefore, the definition of multivariate regularvariation is introduced.

To understand the definition of multivariate regular variation, the concept of vague con-vergence is explained first.

10

2 | Preliminaries

Definition 2.2.6. (Radon measure) (cf. Resnick 1987, chapter 3.4)

Let E be a locally compact space with a countable basis and denote E = B(E ) itsBorel σ-field. A measure µ on (E , E) is called Radon measure if the measure isfinite on relatively compact sets, i.e. µ(A) < ∞ for every compact set A ⊂ E . Theset of all Radon measures on E is denoted by M+(E ).

Note: (c.f. Resnick 1987, Chapter 3.1)

• A locally compact space with a countable basis is a Hausdorff space which has forevery x ∈ E a compact neighborhood. There exist open Gn, n ≥ 1 such that everyopen G can be written as G =

⋃a∈I Ga for I which is a finite or countable index set.

• The Borel σ-field E of the subsets of E is generated by the open sets.

Definition 2.2.7. (Vague convergence) (Resnick, 2007, p. 49)

Let (E , E) be the state space as described in Definition 2.2.6 and let µn ∈ M+(E ) forn ≥ 0. Then, µn converges vaguely to µ0, written µn

v→ µ0, if for every continuousfunction f : E 7→ R+ with compact support

µn(f ) :=

∫E

f (x)µn(dx)→ µ0(f ) :=

∫E

f (x)µ0(dx), n→∞.

For equivalent definitions, see Proposition 3.12 in Resnick (1987).

Multivariate regular variation is typically defined on the compactified space Rd whereR = R ∪ {−∞,∞}. The reason why ∞ is included is that it solves the problem of therequirement of relatively compact sets. After the compactification at infinity, it is convenientto uncompactify the set by removing 0. This is called one point uncompactification.The idea is to take uM → ∞ as u → ∞ for all M ∈ E . For further details, see section6.1.3 in Resnick (2007).

For multivariate regular variation on Rd , the topology on Rd\{0} is chosen such thatB(Rd\{0}) and B(Rd) coincide on Rd\{0}. Moreover, M ∈ B(Rd\{0}) is relatively com-pact in Rd\{0} if and only if M ∩Rd is bounded away from 0 in Rd i.e. 0 /∈ M ∩ Rd . Thisis referred to p. 5 in Hult et al. (2005).

Definition 2.2.8. (Multivariate regularly varying) (cf. Basrak 2000, Definition 2.1.6)

A random vector X ∈ Rd is called multivariate regularly varying if the followingholds:Let µ be a Radon measure on (E , E) = (Rd\{0},B(Rd\{0})) with µ(Rd\Rd) = 0.Suppose that a relatively compact Borel set D ∈ E exists and µ(∂uD) = 0 for u in adense subset U ⊂ (0,∞). Then, it holds that

P(X ∈ u · )

P(X ∈ uD)v→ µ(·), u →∞.

The measure µ is called intensity measure of X .

11

2 | Preliminaries

Consider the intensity measure µ with

D = {x ∈ Rd : ||x || > 1} = {||x || > 1} (2.2)

where || · || is an arbitrary norm on Rd . Then, it holds that

P(X ∈ u · )

P(||X || > u)v→ µ(·), u →∞. (2.3)

There exists some α > 0 such that the measure µ is homogeneous of order −α

µ(uM) = u−αµ(M)

for u > 0 and every M ∈ E satisfying µ(∂M) = 0.v→ denotes vague convergence. If X is multivariate regularly varying, then X ∈MRV (α,µ).

More equivalent definitions can be found in Theorem 6.1 in Resnick (2007). Multivariateregular variation on (E , E) = (Rd

+\{0},B(Rd+\{0}) is also defined in Definition 2.1 in Kley

et al. (2016).

Remark 2.2.9 (cf. Hult and Lindskog 2005, Remark 2.1)

The condition (2.3) in Definition 2.2.8 is equivalent to

P(X ∈ uM)

P(||X || > u)→ µ(M), u →∞ (2.4)

for every setM ∈ B(Rd) bounded away from 0 (i.e. 0 /∈ M (closure ofM)) with µ(∂M) = 0.

The following proposition is considered as a multivariate extension of Breiman’s result.

Proposition 2.2.10. Multivariate regularly varying with random matrix A(Basrak et al., 2002, Proposition 5.1 )

Assume that X ∈ MRV (α,µ) in the sense of (2.4). Let A be a random q×d matrixsuch that 0 < E[||A||β] <∞ for some matrix norm || · || and β > α. Then, it holdsthat

P(AX ∈ · )

P(||X || > x)v→ E[µ ◦ A−1(·)] = E[µ({x ∈ Rd\{0} : Ax ∈ · })]

where A−1 is the inverse image of A.

The next proposition can be derived from Proposition 2.2.10.

Proposition 2.2.11. Multivariate regularly varying with fixed matrix A

Assume that X ∈ MRV (α,µ) in the sense of (2.4). Let A be a fixed q × d matrix.Then, for some matrix norm || · ||, it holds that

P(AX ∈ · )

P(||X || > x)v→ µ ◦ A−1(·) = µ({x ∈ Rd\{0} : Ax ∈ · })

where A−1 is the inverse image of A.

12

2 | Preliminaries

Claims are usually positive such that extreme events can be referred to spaces as (0,∞). Inexceptional cases, incurred claims could also be negative. Then, the outstanding liabilitiesat the end of the period exceed the sum of the actual claims paid in the period. These casesare neglected and it is assumed that claims are always positive. For multivariate extremes,a convenient state space for the intensity measure µ is

E = Rd+\{0} = [0,∞]d\{0}. (2.5)

The extremal dependence information of a multivariate regularly varying vector X is encodedin this intensity measure µ.According to p. 8 in Kley et al. (2016), two cases can be differentiated:

• If µ only has mass on the axes {sej ∈ Rd+\{0} : s > 0}, then this corresponds to

independence.

• If µ only has mass on the line {s1 ∈ Rd+\{0} : s > 0} with 1 = (1, ... , 1)T ∈ Rd

+,then this corresponds to full dependence.

13

3 | Insurance network model

In this chapter, the insurance network model is explained. It is mainly based on the pa-pers by Kley et al. (2016) and Behme et al. (2018) where further explanations can be found.

The insurance model is based on a bipartite network. It consists of

• q agents denoted by Ai for i = 1, ... , q and

• d objects denoted by Oj for j = 1, ... , d .

The number of agents and objects is finite. Throughout, the index i is referred to theagents, whereas the index j represents the objects.The agents can be primary insurance or reinsurance companies. The objects are portfoliosof catastrophic risks. They cause a serve loss, affect many people simultaneously and aretriggered by one unforeseeable common event. Catastrophic events can be structured inthree categories (Nguyen, 2007, p. 7-9):

1) Natural catastrophes are catastrophes which are caused by the forces of nature.They usually result in many individual losses which affect different insurance policiesand contracting parties.

2) Man-made or technical catastrophes are large claim events which are related tohuman activities.

3) Terrorism risks are triggered by intentional human acting.

Concrete examples of natural catastrophes for the objects in figure 3.1 can be

• O1: A portfolio of household insurances in a hurricane region (e.g. in the Caribbean)

• O2: A portfolio of household insurances in an earthquake region (e.g. in Indonesia)

• O3: A portfolio of agricultural insurances as a prevention for droughts or windstorms.

Weighted adjacency matrix

The agents and objects are connected by edges which are visualized in the figure :

A1 A2 A3 A4 A5 . . . Aq

O1 O2 O3. . . Od

Figure 3.1: A bipartite network with q agents and d objects

14


If an agent insures an object, there will be an edge between the agent and the object writtenas 1{i ∼ j}. Then, the object is at least partly insured. In figure 3.1, the agent A1 insuresparts of the risks of the objects O1 and O3, agent A2 bears parts of the costs caused byobject O1, agent A3 is responsible for the risks of the objects O1 and O3 and so forth. It isalso possible that some objects may not be insured. This is a realistic assumption as somerisks are not envisioned or the damage potential is too high for the insurance company. Anexample are portfolios of elemental loss insurances for floods. The links between the agentsand the objects are encoded in a q × d weighted adjacency matrix

A = (Aij)

i=1,...,qj=1,...,d where Ai

j = 1{i ∼ j}W ij . (3.1)

The random variables W ij ≥ 0 are weights which depend on the network. They indicate

the proportion of the loss caused by object Oj that is paid by agent Ai . The weights havevalues in [0, 1] such that

0 ≤q∑

i=1

Aij =

q∑i=1

1{i ∼ j}W ij ≤ 1 for all j = 1, ... , d .

The sum determines the total proportion of object Oj which is insured by all agents. Thenumber of objects O that are insured by agent Ai is denoted by

degA(i) =d∑

j=1

1{i ∼ j} for all i = 1, ... , q.

The number of agents A insuring object Oj is given by

degO(j) =

q∑i=1

1{i ∼ j} for all j = 1, ... , d .

If degO(j) = 0 for some j ∈ {1, ... , d}, then no agent insures object Oj and the entryAij = 0 ∀ i = 1, ... , q. For figure 3.1, it holds that

degA(1) = 2 degA(2) = 1 degA(3) = 2 degA(4) = 4 degA(5) = 1 ... degA(q) = 2

anddegO(1) = 3 degO(2) = 1 degO(3) = 4 ... degO(d) = 2.

The set of all possible realisations A∗ ∈ Rq×d of the weighted adjacency matrix A is denotedas the set A.

Homogeneous system (Behme et al., 2018, Example 2.1)

In a homogeneous system, the weighted adjacency matrix A is given by

Aij =

1{i ∼ j}deg(j)

where0

0is interpreted as 0,

i.e. every object is equally shared by all agents that are linked to it.

Complete network (Behme et al., 2018, p. 6)

In a complete network, all agents are linked to all objects and vice versa such that

1{i ∼ j} = 1 for all i ∈ {1, ... , q} and j ∈ {1, ... , d}.

15


Combining a homogeneous system with a complete network, the loss proportions are equalfor each agent such that

Aij =

1

qfor all i ∈ {1, ... , q} and j ∈ {1, ... , d} with q 6= 0. (3.2)

Claims process

An insurance claim is a request to an insurance company for coverage of a loss. Theinsurance company validates the claim and once it is approved they reimburse the loss tothe insured. In this model, it is assumed that the claims are large, but as catastrophic eventsare rare events, they occur with low frequency.

The claims process for the objects Oj , j = 1, ... , d is a d-dimensional compound Poissonprocess (cf. Definition 2.1.6) V = (V1, ... ,Vd)T with the components

Vj(t) =

N(t)∑k=1

Xj(k)− cjt, t ≥ 0 and j = 1, ... , d . (3.3)

N = (N(t))t≥0 is a homogeneous Poisson process with intensity rate λ > 0 counting thenumber of claims of the objects Oj (cf. Definition 2.1.2). The claims are jumps of sizeXj(k) for j = 1, ... , d and k ≥ 1 which are positive i.i.d. random variables having meanE[Xj(1)] = µj < ∞. Xj(k) is the size of the kth claim of object Oj . FXj

(·) = P(Xj ∈ · )is the distribution function of the claims Xj satisfying FXj

(0) = 0. Assuming that the risksare primarily catastrophic risks, the distribution function FXj

has a regularly varying tailwith index α > 0 (written FXj

∈ RV (α), see Definition 2.2.2). The claims vector X ismultivariate regularly varying (X (1) ∈ MRV (α,µ), see Definition 2.2.8). The claims of theobjects Oj for j = 1, ... , d jump at the same claim times (T (k))k∈N (cf. Definition 2.1.3).This means that a hurricane (O1), an earthquake (O2) and a windstorm (O3) occur at thesame time. It is an idealistic assumption to simplify computations. The i.i.d assumptionof the claims X is appropriate to assume as the objects are of different types and could bespread over different geographic regions. The claim caused by a hurricane in the Caribbeanis independent of the claim arising from an earthquake in Indonesia.The premium rate is denoted by a constant cj > 0 such that the premium income canbe described by cjt for object Oj , j = 1, ... , d and t ≥ 0. It is a deterministic linear functionof the time t.

The positive safety loading is defined by

lj := cjE[W (k)]− E[Xj(k)] for j = 1, ... , d and k ≥ 1. (3.4)

Based on the result of Proposition 2.1.5, the inter claim times (W (k))k≥1 are i.i.d expo-nential random variables with parameter λ > 0 having mean

E[W (1)] =1

λ.

Since Xj(k) are i.i.d. random variables for j = 1, ... , d and k ≥ 1 and it holds thatE[Xj(1)] = µj > 0, it follows for (3.4) that

lj = cjE[W (k)]− E[Xj(k)]

= cjE[W (1)]− E[Xj(1)] (identically distributed)

=cjλ− µj . (3.5)

16


Ifl ∈ (0,∞)d ⇔ cj

λ− µj > 0 ∀ j = 1, ... , d , (3.6)

each object Oj satisfies the net profit condition. On average, the insurance companyreceives more premium income than it must pay for the claims.

• If the net profit condition fails, then an eventual ruin is a.s.

• If the net profit condition holds, then the ruin probability is a number in (0, 1) and itsbehavior depends on the properties of the distribution function FXj

.

Thus, the net profit condition guarantees the viability of the insurance company.In the literature, it is common to define the relative safety loading

ρj :=cjE[W (1)]− E[Xj(1)]

E[Xj(1)]=

cjλ− µj

µj=

cjλµj− 1 > 0 for j = 1, ... , d

instead of the positive safety loading which is based on a per unit claim. The net profitcondition is satisfied if

ρj > 0 ∀ j = 1, ... , d . (3.7)

If l or ρ is small, the agent has low reserve resources and the risk of ruin is high. If l orρ is large, the agent has good profit margins, but the premiums are not attractive for itspotential clients.

Exposure

The vector of exposures of the q agents is a q-dimensional vector E = (E 1, ... ,E q)T

represented byE (t) = AV (t) for t ≥ 0. (3.8)

The components

E i(t) =d∑

j=1

AijVj(t) =

d∑j=1

Aij

( N(t)∑k=1

Xj(k)− cjt

)for t ≥ 0 and i = 1, ... , q

are not independent due to the matrix A.

Initial capital

The vectoru = (u1, ... , uq)T > 0 (3.9)

is the initial capital of the q agents. It is usually supposed that u is very large which willbe indicated in chapter 6 by taking the limit ||u|| → ∞. Throughout the paper, the normedinitial capital is denoted by

b :=u

||u||. (3.10)

Each component bi for i = 1, ... , q can be interpreted as the fraction of the initial capitalof agent Ai in relation to the aggregated total initial capital of all agents.

17


Reserves

The reserves are a q-dimensional vector R = (R1, ... ,Rq)T given by

R(t) = u − E (t) for t ≥ 0. (3.11)

Thus, the components are

R i(t) = ui − E i(t) = ui −d∑

j=1

AijVj(t) for t ≥ 0 and i = 1, ... , q. (3.12)

The initial risk reserve is R(0) = u > 0.

18

4 | Capital transfer

Capital transfer can be useful to reduce the ruin probability in an insurance networkdescribed in chapter 3: If a large loss occurs and some agents (e.g. members of an insurancepool or subsidiaries) cannot cover the loss by capital, then the agents can transfer capitalamong each other to cover the loss together.In this section, three cases will be considered:




Transaction costs are costs which arise for the conclusion and transaction of contracts forinternationally operating insurance companies. They include currency exchange rates andadministrative fees which banks usually charge as a margin on the exchange rate.As described in chapter 3, it is assumed that q agents Ai for i = 1, ... , q act in the market.Each agent Ai has the initial capital ui (3.9) for i = 1, ... , q. The evolution of the agents’capital depends on the claims process V (t) (3.3). At every time t ≥ 0, the agents’ capitalis represented by the reserves R(t) (3.11). If the agents are obligated to transfer capital,they will exchange the capital given by R(t).The rules for the capital transfer can be defined by a fixed matrix Π:

Definition 4.0.1. (Capital transfer) (Hult and Lindskog, 2006, p. 5)

The capital transfer is defined by a quadratic q × q matrix Π = (π(kl))k,l∈{1,...,q}.The entry π(kl) means that π(kl) units of currency k are needed to get one unit ofcurrency l for k , l ∈ {1, ... , q}. It is assumed that Π satisfies the following conditions:

i) π(kl) > 0 for k , l ∈ {1, ... , q},

ii) π(kk) = 1 for k ∈ {1, ... , q},

iii) π(kl) ≤ π(km)π(ml) for k , l ,m ∈ {1, ... , q}.

In a more financial context, Definition 4.0.1 can also be found in Schachermayer (2004). Itis assumed that the capital matrix Π does not depend on the time t.The conditions i), ii) and iii) have the following interpretations:

i) Capital transfers are positive. If they were negative or zero, this would imply thatmoney is received for free.

ii) Getting one unit of currency k equals one unit of the same currency.

iii) This condition means that the agents cannot gain anything by transferring from k tom and then from m to l instead of transferring directly from k to l (Hult and Lindskog,2006, p. 5). For example, doing exchange chains of EUR to JPY and subsequentlyJPY to CHF is not better than a direct exchange of EUR to CHF.

19



In this case, the agents are competitors and contractually independent. Even if the agentsinsure the same object, they are not liable in the case that one agent cannot cover the loss.


If capital is transferred without transaction costs, then π(kl) = 1 for k , l ∈ {1, ... , q}i.e.

Π =

π(11) π(12) · · · π(1l) · · · π(1q)

π(21) π(22) · · · π(2l) · · · π(2q)

...... . . . ... . . . ...

π(k1) π(k2) · · · π(kl) · · · π(kq)

...... . . . ... . . . ...

π(q1) π(q2) · · · π(ql) · · · π(qq)

=

1 1 · · · 1 · · · 11 1 · · · 1 · · · 1...

... . . . ... . . . ...1 1 · · · 1 · · · 1...

... . . . ... . . . ...1 1 · · · 1 · · · 1

. (4.1)

The agents are located

a) either in the same currency market i.e. the agents’ capital can be exchanged 1:1

b) or the agents are located in different markets, but quote all gains and incurring lossesin a common reference currency which is sometimes called numeraire in finance.

The capital transfer is free of charge.


If capital is transferred with transaction costs, then

Π =

π(11) π(12) · · · π(1l) · · · π(1q)

π(21) π(22) · · · π(2l) · · · π(2q)

...... . . . ... . . . ...

π(k1) π(k2) · · · π(kl) · · · π(kq)

...... . . . ... . . . ...

π(q1) π(q2) · · · π(ql) · · · π(qq)

=

1 π(12) · · · π(1l) · · · π(1q)

π(21) 1 · · · π(2l) · · · π(2q)

...... . . . ... . . . ...

π(k1) π(k2) · · · π(kl) · · · π(kq)

...... . . . ... . . . ...

π(q1) π(q2) · · · π(ql) · · · 1

.

(4.2)Case 3 differs to the previous two cases due to one of the following reasons:

a) Some agents are located in different currency markets (currency exchange rate) andthe capital transfer costs money (administrative fees).

b) Some agents are located in different currency markets, but the capital transfer doesnot include additional transaction costs. In this case,

π(kl) =1

π(lk)for k , l ∈ {1, ... , q}

as it only includes the currency exchange rates.

20


Example to a):

Consider an insurance group consisting of three agents:

• A1: Insurance in Germany (currency Euro (EUR))

• A2: Insurance in Japan (currency Japanese Yen (JPY))

• A3: Insurance in Switzerland (currency Swiss franc (CHF)).

The agents are located in different geographical areas acting in different currency markets.A1 is responsible for the European business, A2 for the Asian business and A3 for the Swissbusiness. If a loss cannot be covered by one of the agents Ai for i = 1, 2, 3, assume thatcapital is transferred among the agents. Imagine that a severe catastrophe is triggered attime t and a large claim arises which is insured by agent A1, however its reserve R1(t) isnot large enough to pay the loss. Avoiding insolvency, the agents A2 and A3 can help bytransferring capital. Due to the different currencies and some transaction costs (adminis-trative costs), it is not possible to simply add the reserves R i(t) for i = 1, 2, 3. Let thetransaction costs (including exchange costs) be given by the 3× 3 matrix

Π =

1 0.009 0.9130 1 1131.16 0.01 1

.

In chapter 5, it will be explained that the solvency set is spanned by the vectors{0.009−10

,

0.90−1

,

−1130

0

,

−10

1.16

,

0−1

0.01

,

0113−1

,

100

,

010

,

001

}.

The first vector can be interpreted that 0.009 units of currency 1 (EUR) can be exchangedfor 1 unit of currency 2 (JPY).

21

5 | Ruin sets

In this section, the ruin set, which is denoted by F , is modeled. As explained in chapter3, the reserves are the difference of the initial capital and the excess of the premiums overthe claim amounts. In the classical sense, a set is viewed as a ruin set if, at some time,the reserves are negative and the insurance company is ruined. There are no interactionsamong the agents (case 1). In this insurance network model, the agents transfer capital(case 2 and 3). The rules of capital transfer were introduced in chapter 4. Since the ruinprobability will be investigated in the following chapters, it is important to first determinethe ruin sets under the consideration of capital transfer. Similarly to the previous chapter,the following three cases are differentiated:




Since the insurance network model consists of q agents with reserves R(t) ∈ Rq (3.12) fort ≥ 0, the ruin set F ⊂ Rq. The ruin set can be represented as complements of certaincones which are called solvency cones S.

Definition 5.0.1. (Representation of the solvency cones) (cf. Schachermayer 2004, Definition 1.2)

If the agents do not transfer capital, the solvency cone S is defined by

S ={y ∈ Rq : y =

q∑k=1

zkek , zk ≥ 0 ∀ k ∈ {1, ... , q}}

= Rq+. (5.1)

For a given capital matrix Π = (π(kl))k,l∈{1,...,q}, the solvency cone S is defined by

S ={y ∈ Rq : y =

q∑k,l=1k 6=l

w (kl)(π(kl)ek − e l) +

q∑k=1

zkek ,

w (kl) ≥ 0, zk ≥ 0 ∀ k , l ∈ {1, ... , q}}

. (5.2)

It is a convex cone in Rq which is spanned by the vectors (π(kl)ek−e l) for 1 ≤ k , l ≤ qand ek for 1 ≤ k ≤ q. e1, ... , eq ∈ Rq denote the standard basis vectors so thatek = (0, ... , 1, ... , 0)T for k ∈ {1, ... , q}.

Note:

The solvency cone given in (5.1) is a special case of (5.2) by choosing w (kl) = 0 for allk , l ∈ {1, ... , q} with k 6= l .

Explanation of the representation of S in (5.2)

In (5.2), zkek for k ∈ {1, ... , q} denotes the amount of money of agent Ak quoted in

22

5 | Ruin sets

the currency k . If zkek < 0, agent Ak cannot cover its losses and needs help from theother agents Al with l ∈ {1, ... , q} and k 6= l . Recall that π(kl) means π(kl) units ofcurrency k are needed to get one unit of currency l for k , l ∈ {1, ... , q} with k 6= l (cf.Definition 4.0.1). Thus, π(kl)ek − e l for k , l ∈ {1, ... , q} with k 6= l denotes that agentAl transfers one unit of currency l to agent Ak who gets π(kl) units of currency k for thisone unit. The amount of monetary units which is transferred between the two agents Ak

and Al is determined by the parameters w (kl). It holds k 6= l as agent Ak cannot transfercapital with themselves. According to the condition ii) in Definition 4.0.1, it holds thatπ(kk) = 1 for k ∈ {1, ... , q}. Thus, π(kk)ek − ek = ek − ek = 0 for k ∈ {1, ... , q}. If the

amount of loss given by y ∈ Rq can be composed byq∑

k,l=1k 6=l

w (kl)(π(kl)ek − e l) +

q∑k=1

zkek

with w (kl) ≥ 0, zk ≥ 0 ∀ k , l ∈ {1, ... , q}, the loss can be covered.

In figure 5.1, it can be seen that the solvency cone S is spanned by the vectors (π(kl)ek−e l)for 1 ≤ k , l ≤ q and the standard basis vectors ek for 1 ≤ k ≤ q (see example in chapter4). In order to visualize the solvency cone S in (5.2), think of the following examples in R2:

• Assume that z1 = 0 and z2 = 0. If w (12) = 0 and w (21) < 0, this corresponds to avalue on the red dashed line. If w (21) = 0 and w (12) < 0, this corresponds to a valueon the blue dashed line.

• Assume that w (12) = 0 and w (21) > 0. If z1 = 0 and z2 < 0, this corresponds to avalue on the blue solid arrow. If z2 = 0 and z1 < 0, this corresponds to a value onthe blue dashed arrow.

The values on the red dashed line, blue dashed line, blue solid arrow and blues dashed arroware not contained in the solvency cone S.

−∞ +∞

+∞

−∞

0

π(21)e2 − e1

π(12)e1 − e2

S

Figure 5.1: The solvency cone S in (5.2) is spanned by the vectors (π(kl)ek − e l) for 1 ≤ k , l ≤ qand ek for 1 ≤ k ≤ q for q = 2.

If a position of the agents is in the solvency cone, the agents will be solvent. This meansthat the agents have enough capital to cover the loss.If the position is not in the solvency cone, it will be in the ruin set and the agents will beclassified as insolvent. It holds that

F = Sc . (5.3)

23

5 | Ruin sets

The solvency cone S includes all non-negative positions as well as all positions of the agentswhich can be exchanged by capital transfer into positions with non-negative entries. Thus,it holds that

Rq+ ⊂ S. (5.4)

In this insurance network model, the set of the solvency cones S can be defined by a family

S = {S ⊂ Rq : S closed, r -increasing,S convex, 0 ∈ S}

with r ∈ [0,∞)q. Due to (5.3), the set of the ruin sets F in the insurance network modelis given by a family

F = {F ⊂ Rq : F open, r -decreasing, F c convex, 0 ∈ ∂F}

where ∂F is the boundary of F .

Remark 5.0.2 (r -increasing, r -decreasing) (cf. Hult et al. 2005, Remark 3.1)

A set M ⊂ Rq is called r -increasing if m ∈ M and r ∈ [0,∞)q imply m+rt ∈ M for t ≥ 0.The solvency cone S is r -increasing as r ∈ Rq

+ ⊂ S (5.4) and y ∈ S imply y + rt ∈ S fort ≥ 0 (cf. figure 5.2).A set M ⊂ Rq is called r -decreasing if m ∈ M and r ∈ [0,∞)q imply m − rt ∈ M fort ≥ 0. In order to show that the ruin set F is r -decreasing, let r ∈ Rq

+ ⊂ S (5.4) andS ⊂ −Sc = −F (5.3). Since r ∈ −F , it follows that −r ∈ F . With y ∈ F , it resultsy − rt ∈ F for t ≥ 0 (cf. figure 5.3).

−∞ +∞

+∞

−∞

0

y

y + rt

S

Figure 5.2: The solvency set S is r -increasing.

−∞ +∞

+∞

−∞

0y

y − rt

F

Figure 5.3: The ruin set F is r -decreasing.

Remark 5.0.3 (cone) (cf. Bagirov et al. 2014, Definition 2.6)

A set M ⊂ Rq is a cone if λm ∈ M for all m ∈ M and λ ≥ 0. Moreover, if M is convex,then it is called a convex cone.Cones are sets containing all rays passing through the sets’ points emanating from the origin.Choosing λ = 0, 0 ∈ M i.e. a cone always contains the origin.Since 0 ∈ S, the solvency set S is a cone. Moreover, S is a convex cone (cf. figure 5.4).Since 0 /∈ F (only 0 ∈ ∂F), F is not a cone. However, by taking the union, F ∪ {0}q isa cone. It is possible to define F as a cone with respect to the multiplication by positivescalars. This means that for y ∈ F , it follows uy ∈ F for every u > 0. The ruin set Fis not a convex cone as F is not convex i.e. λ1y1 + λ2y2 /∈ F for all y1, y2 ∈ F and λ1,λ2 ≥ 0 (cf. figure 5.5).

24

5 | Ruin sets

−∞ +∞

+∞

−∞

0

◦ y1

◦ y2

S

Figure 5.4: The solvency set S is a con-vex cone as all lines joining twopoints in the set lie completelywithin the set.

−∞ +∞

+∞

−∞

0

◦

◦

y1

y2

F

Figure 5.5: The ruin set F is not a con-vex cone as the line joining thepoints y1 and y2 lies outside ofthe ruin set.

Note:

In the figures 5.1, 5.2, 5.3, 5.4 and 5.5, the ruin set F and the solvency cone S are modeledaccording to an example of case 3. The cases 1 and 2 are only specific cases of case 3. Allmentioned arguments also hold for both other cases.


If the agents do not transfer capital, then by Definition 5.0.1, the solvency cone S asin (5.1) is given by

S ={y ∈ Rq : y =

q∑k=1

zkek , zk ≥ 0 ∀ k ∈ {1, ... , q}}

. (5.5)

The solvency cone S is spanned by the standard basis vectors ek for k ∈ {1, ... , q}.As F = Sc (5.3), the ruin set F can be represented by

F ={y ∈ Rq : y =

q∑k=1

zkek , ∃ zk < 0 for some k ∈ {1, ... , q}}

. (5.6)

This means that the agents Ai with i = 1, ... , q are solvent as long as their respectivereserves R i(t) are positive for t ≥ 0. Thus, it holds that

R(t) ∈ S ⇔ R(t) =

R1(t)...

Rq(t)

> 0 componentwise for t ≥ 0.

Ruin occurs at time t ≥ 0 if at least one agent Ai with i = 1, ... , q cannot cover its loss.Hence, for t ≥ 0, it follows that

R(t) ∈ F ⇔ ∃i = 1, ... , q : R i(t) = ui − E i(t) = ui −d∑

j=1

Aij

( N(t)∑k=1

Xj(k)− cjt

)< 0.

25

5 | Ruin sets

The formulations

S = Rq+ = [0,∞)q F = Sc = Rq\Rq

+ = ([0,∞)q)c (5.7)

are equivalent to (5.5) and (5.6).

Geometrical interpretation of (5.7) for q = 2

If the insurance network model consists of two agents A1 and A2, then the solvency coneis given by

S = [0,∞)2 = [0,∞)× [0,∞) (green)

and the ruin set byF = ([0,∞)2)c = R2\[0,∞)2 (red).

The solvency cone S is spanned by the standard basis vectors

e1 =

(10

)(right arrow :→) and e2 =

(01

)(up arrow :↑).

−∞ +∞

+∞

−∞

0 e1

e2

S

F

Figure 5.6: The solvency cone S = [0,∞)q and the ruin set F = ([0,∞)q)c for q = 2


If capital is transferred without transaction costs, then by (4.1)

π(kl) = 1 ∀ k , l ∈ {1, ... , q}. (5.8)

Inserting (5.8) in (5.2), the solvency cone is given by

S ={y ∈ Rq : y =

q∑k,l=1k 6=l

w (kl)(ek − e l) +

q∑k=1

zkek ,w (kl) ≥ 0, zk ≥ 0 ∀ k , l ∈ {1, ... , q}}

.

(5.9)

26

5 | Ruin sets

The solvency cone S is spanned by the vectors (ek − e l) for 1 ≤ k , l ≤ q and the standardbasis vectors ek for 1 ≤ k ≤ q. Since for all k , l ∈ {1, ... , q} with k 6= l , w (kl) ≥ 0 andw (lk) ≥ 0 hold, it follows that

w (kl)(ek − e l) + w (lk)(e l − ek) = (w (kl) − w (lk)︸︷︷︸∈R

)(ek − e l),

the solvency cone in (5.9) can also be described by

S ={y ∈ Rq : y =

q∑k=1

q∑l>kk 6=l

w (kl)(ek − e l) +

q∑k=1

zkek , zk ≥ 0 ∀ k , l ∈ {1, ... , q}}

.

The difference to (5.9) is that the parameters w (kl) for k , l ∈ {1, ... , q} and k 6= l can bechosen positively or negatively. As F = Sc (5.3), the ruin set F is given by

F ={y ∈ Rq : y =

q∑k,l=1k 6=l

w (kl)(ek − e l) +

q∑k=1

zkek ,

∃ zk < 0 for some k ∈ {1, ... , q}}

.

Ruin occurs if the negative positions of the agents cannot be canceled by transferring capitalfrom other agents.

Lemma 5.0.4.

The solvency cone S in (5.9) has the equivalent representation

S ={y ∈ Rq : yT1 ≥ 0

}={y ∈ Rq : 〈y , 1〉 ≥ 0

}(5.10)

where 1 = (1, ... , 1)T ∈ Rq and 〈·, ·〉 denotes the standard scalar product.

Proof

S ⊆{

y ∈ Rq : yT1 ≥ 0}

Let y ∈ S such that

y =

q∑k,l=1k 6=l

w (kl)(ek − e l) +

q∑k=1

zkek with w (kl) ≥ 0, zk ≥ 0 ∀ k , l ∈ {1, ... , q}. (5.11)

If y given by (5.11) fulfills yT1 ≥ 0, it will prove that S ⊆{y ∈ Rq : yT1 ≥ 0

}. It holds

that

yT1 = 〈y , 1〉

=⟨ q∑

k,l=1k 6=l

w (kl)(ek − e l) +

q∑k=1

zkek , 1⟩

=⟨ q∑

k,l=1k 6=l

w (kl)(ek − e l), 1⟩

+⟨ q∑

k=1

zkek , 1⟩

=

q∑k,l=1k 6=l

w (kl)⟨

(ek − e l), 1⟩

+

q∑k=1

zk⟨ek , 1

⟩. (5.12)

27

5 | Ruin sets

Since k 6= l , the standard basis vectors do not have the same entry 1 in the same componentfor all k , l ∈ {1, ... , q}. Therefore, it holds for all k , l ∈ {1, ... , q} with k 6= l that⟨

(ek − e l), 1⟩

= 1− 1 = 0.

Moreover, since the kth component of the standard basis vector ek equals 1 and the othercomponents are 0, it holds that ⟨

ek , 1⟩

= 1.

Thus, (5.12) leads toq∑

k,l=1k 6=l

w (kl)⟨

(ek − e l), 1⟩

︸︷︷︸=0

+

q∑k=1

zk⟨ek , 1

⟩︸︷︷︸

=1

=

q∑k=1

zk . (5.13)

Since zk ≥ 0 for all k ∈ {1, ... , q}, it follows for (5.13) thatq∑

k=1

zk ≥ 0.

Hence, S ⊆{y ∈ Rq : yT1 ≥ 0

}holds.

S ⊇{

y ∈ Rq : yT1 ≥ 0}

Let y ∈ {y ∈ Rq : yT1 ≥ 0} such that

yT1 =

q∑k=1

y k ≥ 0 (5.14)

for (1, ... , 1)T ∈ Rq. If the vector y given by (5.14) has the representation

y =

q∑k,l=1k 6=l

w (kl)(ek − e l) +

q∑k=1

zkek with w (kl) ≥ 0, zk ≥ 0 ∀ k , l ∈ {1, ... , q}, (5.15)

it will prove that S ⊇ {y ∈ Rq : yT1 ≥ 0}. The following two cases will be considered:

1) y ∈ Rq+

If y ∈ Rq+, choose zk := y k ≥ 0 and w (kl) := 0 for all k , l ∈ {1, ... , q} with k 6= l .

2) y ∈ Rq

If y ∈ Rq\Rq+, y also has negative components. Therefore, it is not possible to

represent y only by the second sum of the solvency cone S such that y 6=∑q

k=1 zkek

with zk ≥ 0 for all k ∈ {1, ... , q}. To show that y has the representation of S, let

y = y+ − y−

such that y+ ∈ Rq+ and y− ∈ Rq

+. This representation splits the positive and negativecomponents of the vector y in two separate vectors y+ and y−. For each componentk with k ∈ {1, ... , q}, it holds that

y k = y k+ − y k

− =

{y k

+ if y k ≥ 0y k− else.

28

5 | Ruin sets

By (5.14), it follows that

q∑k=1

y k+ − y k

− ≥ 0⇔q∑

k=1

y k+ −

q∑l=1

y l− ≥ 0. (5.16)

The negative components of the vector y can only be represented by the first sum ofthe solvency cone S which is given by

q∑k,l=1k 6=l

w (kl)(ek − e l).

Choose

w (kl)

= 0 if y k

+ = 0= 0 if y l

− = 0> 0 else

for k , l ∈ {1, ... , q} with k 6= l . Then, for l ∈ {1, ... , q}, let

y l− :=

{ ∑qk=1 w

(kl)1{y k+ > 0} if y l

− > 00 else

and

y− :=

q∑l=1

(( q∑k=1k 6=l

w (kl)1{y k+ > 0}

)e l1{y l

− > 0})

.

Moreover, choose

zk{

= 0 if y k+ = 0

> 0 else

for k ∈ {1, ... , q}. Then, for k ∈ {1, ... , q}, let

y k+ :=

{zk +

∑ql=1 w

(kl)1{y l− > 0} if y k

+ > 00 else

and

y+ :=

q∑k=1

((zk +

q∑l=1k 6=l

w (kl)1{y l− > 0}

)ek1{y k

+ > 0})

.

Thus, the vectors y+ and y− fulfill (5.16) because

q∑k=1

y k+ −

q∑l=1

y l−

=

q∑k=1

((zk +

q∑l=1k 6=l

w (kl)1{y l− > 0}

)1{y k

+ > 0})

−q∑

l=1

(( q∑k=1k 6=l

w (kl)1{y k+ > 0}

)1{y l

− > 0})

=

q∑k=1

zk1{y k+ > 0} ≥ 0.

29

5 | Ruin sets

Moreover, since

y =y+ − y−

=

q∑k=1

((zk +

q∑l=1k 6=l

w (kl)1{y l− > 0}

)ek1{y k

+ > 0})

−q∑

l=1

(( q∑k=1k 6=l

w (kl)1{y k+ > 0}

)e l1{y l

− > 0})

=

q∑k=1

zkek1{y k+ > 0}+

q∑k,l=1k 6=l

(w (kl)ek1{y l

− > 0}1{y k+ > 0}

− w (kl)e l1{y k+ > 0}1{y l

− > 0})

=

q∑k,l=1k 6=l

w (kl)(ek − e l)1{y l− > 0}1{y k

+ > 0}+

q∑k=1

zkek1{y k+ > 0},

the vector y can be represented by (5.15) and S ⊇{y ∈ Rq : yT1 ≥ 0

}.

Remark 5.0.5

Since F = Sc (5.3), it follows by Lemma 5.0.4 that the ruin set F is given by

F ={y ∈ Rq : yT1 < 0

}={y ∈ Rq : 〈y , 1〉 < 0

}(5.17)

where 1 = (1, ... , 1)T ∈ Rq. It shows that there is not enough capital to cover the totalloss if the aggregated positions of all agents are negative.

Remark 5.0.6

The ruin set given in Remark 5.0.5 has the equivalent representation

F ={y ∈ Rq :

q∑k=1

max(y k , 0) < −q∑

k=1

min(y k , 0)}

.

This follows by rearranging (5.17) to

F ={y ∈ Rq : yT1 < 0

}={y ∈ Rq :

q∑k=1

y k < 0}

={y ∈ Rq :

q∑k=1

(max(y k , 0) + min(y k , 0)

)< 0}

={y ∈ Rq :

q∑k=1

max(y k , 0) +

q∑k=1

min(y k , 0) < 0}

={y ∈ Rq :

q∑k=1

max(y k , 0)︸︷︷︸(1)

< −q∑

k=1

min(y k , 0)︸︷︷︸(2)

}. (5.18)

30

5 | Ruin sets

The left-hand side (1) of (5.18) sums up all agents’ positions which have positive reservesR i(t) for i = 1, ... , q (3.12) at time t ≥ 0. These agents are solvent at time t. Theright-hand side (2) of (5.18) sums up all agents’ positions which have negative reservesR i(t) for i = 1, ... , q (3.12) at time t ≥ 0. These are the agents who cannot cover theirlosses. Ruin is triggered (R(t) ∈ F) if the agents’ positive reserves (1) do not exceed theagents’ negative reserves (2).

Geometrical interpretation of (5.10) and (5.17) for q = 2

If the insurance network model consists of two agents A1 and A2, then the solvency coneis given by

S ={y ∈ R2 : yT1 ≥ 0

}={y ∈ R2 : y 1 ≥ y 2

}(green)

and the ruin set by

F ={y ∈ R2 : yT1 < 0

}={y ∈ R2 : y 1 < y 2

}(red).

The ray{y ∈ R2 : y 1 = y 2

}separates the two sets S and F into two half planes.

−∞ +∞

+∞

−∞

0

e2 − e1

e1 − e2

S

F

Figure 5.7: The solvency cone S = {y ∈ R2 : y1 ≥ y2} and the ruin set F = {y ∈ R2 : y1 < y2}for q = 2


If capital is transferred with transaction costs, then according to Definition 5.0.1, the sol-vency cone is given by

S ={y ∈ Rq : y =

q∑k,l=1k 6=l

w (kl)(π(kl)ek−e l)+

q∑k=1

zkek ,w (kl) ≥ 0, zk ≥ 0 ∀ k , l ∈ {1, ... , q}}

(5.19)

31

5 | Ruin sets

for

Π =

1 π(12) · · · π(1l) · · · π(1q)

π(21) 1 · · · π(2l) · · · π(2q)

...... . . . ... . . . ...

π(k1) π(k2) · · · π(kl) · · · π(kq)

...... . . . ... . . . ...

π(q1) π(q2) · · · π(ql) · · · 1

described in (4.2). The solvency cone S is spanned by the vectors (π(kl)ek − e l) for1 ≤ k , l ≤ q and the standard basis vectors ek for 1 ≤ k ≤ q. Since in this case, thesolvency cone has a complex representation, only two dimensions will be considered.

Geometrical interpretation of (5.19) for q = 2

If the insurance network model consists of two agents A1 and A2, the solvency cone isgiven by

S ={y ∈ R2 : y =

2∑k,l=1k 6=l

w (kl)(π(kl)ek − e l) +2∑

k=1

zkek ,w (kl) ≥ 0, zk ≥ 0 ∀ k , l ∈ {1, 2}}

={y ∈ R2 : y = w (12)(π(12)e1 − e2) + w (21)(π(21)e2 − e1) + z1e1 + z2e2,

w (12) ≥ 0,w (21) ≥ 0, z1 ≥ 0, z2 ≥ 0}

(green)

(5.20)

and the ruin set byF = Sc . (red) (5.21)

The solvency cone S given in (5.20) is spanned by the vectors π(12)e1− e2 and π(21)e2− e1

and the standard basis vectors e1 and e2.

−∞ +∞

+∞

−∞

0

π(21)e2 − e1

π(12)e1 − e2

S

F

Figure 5.8: The solvency cone S in (5.20) and the ruin set F in (5.21) for q = 2

32

5 | Ruin sets

Lemma 5.0.7.

If π(12) 6= 1

π(21), the solvency cone S ⊂ R2 in (5.20) is given by

S ={y ∈ R2 : y =

2∑k,l=1k 6=l

w (kl)(π(kl)ek − e l),w (kl) ≥ 0 ∀ k , l ∈ {1, 2}}

={y ∈ R2 : y = w (12)(π(12)e1 − e2) + w (21)(π(21)e2 − e1),w (12) ≥ 0,w (21) ≥ 0

}.

Proof

In R2, the two vectors π(12)e1 − e2 and π(21)e2 − e1 are linearly independent if and only if

det

(π(12) −1−1 π(21)

)= π(12)π(21) − 1 6= 0.

This results to π(12) 6= 1π(21) . Under the assumption that π(12) 6= 1

π(21) , it follows by conditioniii) of the Definition 4.0.1 that π(12)π(21) > 1. The standard basis vector e1 ∈ R2 is linearlydependent if

w (12)

(π(12)

−1

)+ w (21)

(−1π(21)

)=

(10

)(5.22)

for w (12) ≥ 0 and w (21) ≥ 0. Solving (5.22) leads to

1) w (12)π(12) − w (21) = 1⇔ w (12) =1 + w (21)

π(12)(5.23)

2) − w (12) + w (21)π(21) = 0⇔ w (12) = w (21)π(21). (5.24)

It follows for the parameter w (21) that

1 + w (21)

π(12)= w (21)π(21) ⇔ w (21)

(π(21) − 1

π(12)

)=

1

π(12)⇔ w (21) =

1

π(12)π(21) − 1≥ 0

since π(12)π(21) > 1. Since w (21) ≥ 0, it follows that w (12) ≥ 0 in (5.23) and (5.24). Thus,the standard basis vector e1 can be represented as a non-negative linear combination of thetwo vectors π(12)e1 − e2 and π(21)e2 − e1 if π(12) 6= 1

π(21) .Similarly, the standard basis vector e2 ∈ R2 is linearly dependent if

w (12)

(π(12)

−1

)+ w (21)

(−1π(21)

)=

(01

)(5.25)

for w (12) ≥ 0 and w (21) ≥ 0. Solving (5.25) leads to

1) w (12)π(12) − w (21) = 0⇔ w (21) = w (12)π(12) (5.26)

2) − w (12) + w (21)π(21) = 1⇔ w (21) =1 + w (12)

π(21). (5.27)

It follows for the parameter w (12) that

w (12)π(12) =1 + w (12)

π(21)⇔ w (12)

(π(12) − 1

π(21)

)=

1

π(21)⇔ w (12) =

1

π(12)π(21) − 1≥ 0

33

5 | Ruin sets

since π(12)π(21) > 1. Since w (12) ≥ 0, it follows that w (21) ≥ 0 in (5.26) and (5.27). Thus,the standard basis vector e2 can be represented as a non-negative linear combination of thetwo vectors π(12)e1 − e2 and π(21)e2 − e1 if π(12) 6= 1

π(21) .

Another possibility to describe the solvency cone S in (5.20) is

S =⋂k 6=l

{y ∈ R2 : yT (π(kl)e l + ek) ≥ 0} for k , l ∈ {1, 2} (5.28)

which will be proven in Lemma 5.0.8. Before proving the equality in (5.28), the set will begeometrically explained.

Which set does (5.28) represent?

Let ~y and ~z be two arbitrary vectors in Rq. 〈·, ·〉 denotes the standard scalar product and|| · ||2 is the Euclidean norm determining the length of the vectors ~y and ~z . It holds that

cos(θ) =〈~y ,~z〉||~y ||2||~z ||2

=~yT~z

||~y ||2||~z ||2

where θ ∈ (0, π]. γ = 360◦

2πθ ∈ (0, 180◦] describes the angle between the two vectors. The

two vectors ~y and ~z

• have an acute angle i.e. θ ∈ (0, π2

)⇔ γ ∈ (0, 90◦) if

0 < cos(θ) =~yT~z

||~y ||2||~z ||2⇔ ~yT~z > 0.

• are orthogonal i.e. θ = π2⇔ γ = 90◦ (written ~y ⊥ ~z) if

0 = cos(θ) =~yT~z

||~y ||2||~z ||2⇔ ~yT~z = 0. (5.29)

• have an obtuse angle i.e. θ ∈ (π2

, π]⇔ γ ∈ (90◦, 180◦] if

0 > cos(θ) =~yT~z

||~y ||2||~z ||2⇔ ~yT~z < 0.

The solvency cone S given by (5.19) is spanned by the vectors (π(kl)ek−e l) for 1 ≤ k , l ≤ qand the standard basis vectors ek for 1 ≤ k ≤ q. The vectors which are orthogonal to(π(kl)ek − e l) are (π(kl)e l + ek) for 1 ≤ k , l ≤ q and k 6= l leading to

(π(kl)ek − e l)T (π(kl)e l + ek) = 0 ∀ k , l ∈ {1, ... , q}. (5.30)

These are the normal vectors. For q = 2, this means that

(π(12)e1 − e2)T (π(12)e2 + e1) = 0 ·

and

(π(21)e2 − e1)T (π(21)e1 + e2) = 0. ·

34

5 | Ruin sets

−∞ +∞

+∞

−∞

0

π(21)e2 − e1

π(12)e1 − e2

π(12)e2 + e1

π(21)e1 + e2

S

F

Figure 5.9: The vectors (π(kl)e l + ek) are orthogonal to (π(kl)ek − e l) for 1 ≤ k , l ≤ q.

Thus, every vector y ∈ R2 for which

yT (π(12)e2 + e1) ≥ 0 ··

and

yT (π(21)e1 + e2) ≥ 0··

holds, has an angle γ that is equal or smaller than 90◦. The solvency cone S can bedescribed as the intersection of both sets

S =⋂k 6=l

{y ∈ R2 : yT (π(kl)e l + ek) ≥ 0

}for k , l ∈ {1, 2}.

−∞ +∞

+∞

−∞

0

S

Figure 5.10: The solvency cone S can be represented by the intersection of two half planes.

In the following lemma, (5.28) will be proven.

35

5 | Ruin sets

Lemma 5.0.8.

The solvency cone S in (5.20) has the equivalent representation

S =⋂k 6=l

{y ∈ R2 : yT (π(kl)e l + ek) ≥ 0

}for k , l ∈ {1, 2}.

Proof

By Definition 5.0.1, the solvency cone S for q = 2 is defined by

S ={y ∈ R2 : y =

2∑k,l=1k 6=l

w (kl)(π(kl)ek − e l) +2∑

k=1

zkek ,w (kl) ≥ 0, zk ≥ 0 ∀ k , l ∈ {1, 2}}

={y ∈ R2 : y = w (12)(π(12)e1 − e2) + w (21)(π(21)e2 − e1) + z1e1 + z2e2,

w (12) ≥ 0,w (21) ≥ 0, z1 ≥ 0, z2 ≥ 0}

=

{(y 1

y 2

)∈ R2 :

(y 1

y 2

)= w (12)

(π(12)

−1

)+ w (21)

(−1π(21)

)+ z1

(10

)+ z2

(01

),

w (12) ≥ 0,w (21) ≥ 0, z1 ≥ 0, z2 ≥ 0

}

=

{(y 1

y 2

)∈ R2 :

(y 1

y 2

)=

(w (12)π(12) − w (21) + z1

−w (12) + w (21)π(21) + z2

),

w (12) ≥ 0,w (21) ≥ 0, z1 ≥ 0, z2 ≥ 0

}.

It will be proven that this representation is equivalent to⋂k 6=l

{y ∈ R2 : yT (π(kl)e l + ek) ≥ 0

}for k , l ∈ {1, 2}

={y ∈ R2 : yT (π(12)e2 + e1) ≥ 0

}∩{y ∈ R2 : yT (π(21)e1 + e2) ≥ 0

}={y ∈ R2 : y 1 + π(12)y 2 ≥ 0

}∩{y ∈ R2 : π(21)y 1 + y 2 ≥ 0

}.

S ⊆⋂

k 6=l

{y ∈ R2 : yT(π(kl)el + ek) ≥ 0

}for k, l ∈ {1, 2}

Let y ∈ S such that

y =

(y 1

y 2

)=

(w (12)π(12) − w (21) + z1

−w (12) + w (21)π(21) + z2

)for w (12) ≥ 0,w (21) ≥ 0, z1 ≥ 0, z2 ≥ 0.

(5.31)To prove that y ∈

⋂k 6=l

{y ∈ R2 : yT (π(kl)e l + ek) ≥ 0

}for k , l ∈ {1, 2}, verify that

y ∈{y ∈ R2 : y 1 + π(12)y 2 ≥ 0

}︸︷︷︸(a)

and y ∈{y ∈ R2 : π(21)y 1 + y 2 ≥ 0

}︸︷︷︸(b)

. (5.32)

36

5 | Ruin sets

(a) For the vector y given by (5.31), it follows that

y 1 + π(12)y 2

= w (12)π(12) − w (21) + z1 + π(12)(−w (12) + w (21)π(21) + z2)

= w (12)π(12) − w (21) + z1 − w (12)π(12) + w (21)π(12)π(21) + π(12)z2

= w (21)(π(12)π(21) − 1) + z1 + π(12)z2.

The condition i) of the capital transfer Definition 4.0.1 leads to π(12) > 0. Moreover,due to condition iii), it holds that π(12)π(21) ≥ π(11) = 1. Since w (21) ≥ 0, z1 ≥0, z2 ≥ 0, it follows that

w (21)︸︷︷︸≥0

(π(12)π(21)︸︷︷︸≥1

−1︸︷︷︸≥0

) + z1︸︷︷︸≥0

+π(12)︸︷︷︸≥0

z2︸︷︷︸≥0

≥ 0.

Thus, (a) in (5.32) is satisfied.

(b) For the vector y given by (5.31), it follows that

π(21)y 1 + y 2

= π(21)(w (12)π(12) − w (21) + z1)− w (12) + w (21)π(21) + z2

= w (12)π(21)π(12) − w (21)π(21) + z1π(21) − w (12) + w (21)π(21) + z2

= w (12)(π(21)π(12) − 1) + z1π(21) + z2.

The condition i) of the capital transfer Definition 4.0.1 leads to π(21) > 0. Moreover,due to condition iii), it holds that π(21)π(12) ≥ π(22) = 1. Since w (12) ≥ 0, z1 ≥ 0,z2 ≥ 0 , it follows that

w (12)︸︷︷︸≥0

(π(21)π(12)︸︷︷︸≥1

−1︸︷︷︸≥0

) + z1︸︷︷︸≥0

π(21)︸︷︷︸≥0

+ z2︸︷︷︸≥0

≥ 0.

Thus, (b) in (5.32) is satisfied.

By (a) and (b), it results S ⊆⋂

k 6=l

{y ∈ R2 : yT (π(kl)e l + ek) ≥ 0

}for k , l ∈ {1, 2}.

S ⊇⋂

k 6=l

{y ∈ R2 : yT(π(kl)el + ek) ≥ 0

}for k, l ∈ {1, 2}

Let y ∈⋂

k 6=l

{y ∈ R2 : yT (π(kl)e l + ek) ≥ 0

}for k , l ∈ {1, 2} such that y ∈ R2 fulfills

y 1 + π(12)y 2 ≥ 0 and π(21)y 1 + y 2 ≥ 0. (5.33)

Assume that y /∈ S. Since S is convex and nonempty (0 ∈ S), there exists by the hyperplaneseparation theorem a hyperplane

H :={z ∈ R2 : aT z = 0

}(5.34)

which separates the vector y ∈ R2 and the set S. The vector a ∈ R2 is the normal vector tothe hyperplane H . As explained in (5.30), π(12)e2 + e1 is the normal vector to π(12)e1 − e2

37

5 | Ruin sets

and π(21)e1 + e2 is the normal vector to π(21)e2 − e1. The normal vector of the separatinghyperplane has to be a convex combination of π(12)e2 + e1 and π(21)e1 + e2 such that

a : = β1(π(12)e2 + e1) + β2(π(21)e1 + e2)

=

(β1 + β2π(21)

β1π(12) + β2

)where β1 ≥ 0 and β2 ≥ 0. Then, the hyperplane H in (5.34) is given by

H :=

{z ∈ R2 :

(β1 + β2π(21)

β1π(12) + β2

)T

z = 0

}.

For the vector y which fulfills (5.33), it follows that

aTy =

(β1 + β2π(21)

β1π(12) + β2

)T

y

= (β1 + β2π(21))y 1 + (β1π(12) + β2)y 2

= β1︸︷︷︸≥0

(y 1 + π(12)y 2)︸︷︷︸≥0

+ β2︸︷︷︸≥0

(π(21)y 1 + y 2)︸︷︷︸≥0

≥ 0.

Under the assumption that y /∈ S and since

y ∈ H≥ :=

{z ∈ R2 :

(β1 + β2π(21)

β1π(12) + β2

)T

z ≥ 0

},

it has to follow that

x ∈ H≤ :=

{z ∈ R2 :

(β1 + β2π(21)

β1π(12) + β2

)T

z ≤ 0

}for every x ∈ S. (5.35)

For every x ∈ S given by (x1

x2

)=

(w (12)π(12) − w (21) + z1

−w (12) + w (21)π(21) + z2

)with w (12) ≥ 0, w (21) ≥ 0, z1 ≥ 0 and z2 ≥ 0, it follows that(β1 + β2π(21)

β1π(12) + β2

)T (w (12)π(12) − w (21) + z1

−w (12) + w (21)π(21) + z2

)= (β1 + β2π(21))(w (12)π(12) − w (21) + z1) + (β1π(12) + β2)(−w (12) + w (21)π(21) + z2)

= β1w (12)π(12) − β1w (21) + β1z1 + β2w (12)π(21)π(12) − β2w (21)π(21) + β2z1π(21)

− β1w (12)π(12) + β1w (21)π(12)π(21) + β1z2π(12) − β2w (12) + β2w (21)π(21) + β2z2

= β1w (21)(π(12)π(21) − 1)+β2w (12)(π(21)π(12) − 1)+β1z1 +β2z1π(21) +β1z2π(12) +β2z2.(5.36)

The condition i) of the capital transfer Definition 4.0.1 leads to π(12) > 0 and π(21) > 0.Due to condition iii), it holds that π(12)π(21) ≥ π(11) = 1 and π(21)π(12) ≥ π(22) = 1. Since

38

5 | Ruin sets

w (12) ≥ 0, w (21) ≥ 0, z1 ≥ 0, z2 ≥ 0, (5.36) leads to

β1︸︷︷︸≥0

w (21)︸︷︷︸≥0

(π(12)π(21)︸︷︷︸≥1

−1︸︷︷︸≥0

) + β2︸︷︷︸≥0

w (12)︸︷︷︸≥0

(π(21)π(12)︸︷︷︸≥1

−1︸︷︷︸≥0

) + β1︸︷︷︸≥0

z1︸︷︷︸≥0

+ β2︸︷︷︸≥0

z1︸︷︷︸≥0

π(21)︸︷︷︸≥0

+ β1︸︷︷︸≥0

z2︸︷︷︸≥0

π(12)︸︷︷︸≥0

+ β2︸︷︷︸≥0

z2︸︷︷︸≥0

≥ 0.

This is a contradiction to (5.35). It results S ⊇⋂

k 6=l

{y ∈ R2 : yT (π(kl)e l + ek) ≥ 0

}for

k , l ∈ {1, 2}.

Remark 5.0.9

Since F = Sc (5.3), it follows by Lemma 5.0.8 that the ruin set F is given by

F =

(⋂k 6=l

{y ∈ R2 : yT (π(kl)e l + ek) ≥ 0

})c

for k , l ∈ {1, 2}.

By De Morgan’s laws, it follows that

F =⋃k 6=l

{y ∈ R2 : yT (π(kl)e l + ek) < 0

}for k , l ∈ {1, 2}. (5.37)

The ruin set F , which visualized in figure (5.11), can be represented as the union of twohalf planes.

Remark 5.0.10

The negative ruin set F is given by

−F = −⋃k 6=l

{y ∈ R2 : yT (π(kl)e l + ek) < 0

}=⋃k 6=l

{y ∈ R2 : yT (π(kl)e l + ek) > 0} for k , l ∈ {1, 2} (5.38)

which is visualized in figure 5.12. It can be represented as the union of two half planes.

−∞ +∞

+∞

−∞

0

F

Figure 5.11: The ruin set F given by (5.37)

−∞ +∞

+∞

−∞

0

−F

Figure 5.12: The negative ruin set −F given by(5.38)

39

5 | Ruin sets

Note that the boundary in the figures is not contained in the sets. The representationin (5.38) can be reasoned according to the same argumentation as for (5.28). Every vectory ∈ Rq for which

yT (π(kl)e l + ek) > 0 ∀ k , l ∈ {1, ... , q}, k 6= l

holds, has an angle γ that is smaller than 90◦. For q = 2, the negative ruin set −F can berepresented by the union of

yT (π(12)e2 + e1) > 0 ··

and

yT (π(21)e1 + e2) > 0··

for y ∈ R2.

−∞ +∞

+∞

−∞

0

Figure 5.13: The negative ruin set −F can be represented by the union of two half planes.

40

6 | Ruin probability

The ruin probability has become a standard risk measure to assess regulatory capital.It represents an important indicator of functional quality for an insurance company. It isdefined as the probability that the company runs out of money at some point. Running outof money is usually defined as the probability that the risk reserves become negative. Inthis chapter, the ruin probability is regarded with respect to a ruin set F which is definedin chapter 5. The objective is to consider the probability that the risk reserve process R ofthe company with initial capital u ∈ Rq

+\{0} and normed initial capital b ∈ Rq+\{0} hits

the ruin set F = Rq\S, namely

ψb−F(u) := P(R(t) ∈ F for some t ≥ 0). (6.1)

Throughout, || · || is an arbitrary norm in Rd or Rq. First, some auxiliary results (Lemma6.0.1, Theorem 6.0.2, Remark 6.0.3) will be presented which are referred to the papers byHult and Lindskog (2006) and Hult et al. (2005). These results are necessary to proveLemma 6.0.4, Lemma 6.0.5, Lemma 6.0.6, Theorem 6.0.7 and Proposition 6.0.9.

Lemma 6.0.1. (Hult and Lindskog, 2006, Lemma 7)

Let the solvency cone S be given as in Definition 5.0.1, the ruin set F = Rq\S andµ defined by (2.4). If a ∈ Rq\{0}, then µ(∂(a − F)) = 0 where ∂(a − F) is theboundary of a −F .

Proof

Recall that the boundary of a set in Rq is the boundary of its complement. The solvencycone S has the representation as in Definition 5.0.1. Since F = Sc (5.3), it holds that

∂F = ∂(Sc) = ∂S. (6.2)

−∞ +∞

+∞

−∞

0

∂F = ∂S

S

F

Figure 6.1: The boundary of the ruin set F is the boundary of the solvency cone S i.e. ∂F = ∂S.

41


Therefore, for a ∈ Rq\{0}, (6.2) leads to

∂(a −F) = a − ∂F = a − ∂S (6.3)

and for the boundaries ∂F and ∂S, it holds that

∂F = ∂S ⊂⋃k 6=l

{y ∈ Rq : yT (π(kl)e l + ek) = 0

}for 1 ≤ k , l ≤ q. (6.4)

The boundary of S is a subset of unions of lower-dimensional hyperplanes in Rq\{0}.Referring to (5.29) in chapter 5, the two vectors y and (π(kl)e l + ek)1≤k,l≤q are orthogonal.In figure 6.2, the set

⋃k 6=l

{y ∈ R2 : yT (π(kl)e l + ek) = 0

}for k , l ∈ {1, 2} is visualized by

the two dashed lines.

−∞ +∞

+∞

−∞

0

π(21)e1 + e2π(12)e2 + e1

∂F = ∂S

−∂F = −∂S

S

Figure 6.2: The boundary ∂S of the solvency cone S is a subset of the union of two straight linesgiven by

⋃k 6=l{y ∈ R2 : yT (π(kl)e l + ek) = 0} for k , l ∈ {1, 2}.

It follows that

∂(a −F)(6.3)= a − ∂S

(6.4)⊂⋃k 6=l

{y ∈ Rq : (a − y)T (π(kl)e l + ek) = 0

}=⋃k 6=l

{y ∈ Rq : (−y)T (π(kl)e l + ek) = −aT (π(kl)e l + ek)

}=⋃k 6=l

{y ∈ Rq : yT (π(kl)e l + ek) = alπ(kl) + ak

}=:⋃k 6=l

B (kl) for 1 ≤ k , l ≤ q. (6.5)

The set in (6.4) is shifted by the vector a ∈ Rq\{0}. In figure 6.3, the set⋃

k 6=l B(kl) for

k , l ∈ {1, 2} is visualized by the two dashed lines assuming a ∈ R2+\{0}.

42


−∞ +∞

+∞

−∞

0

π(21)e2 + e1π(12)e1 + e2

a

a

a − ∂F = a − ∂S

Figure 6.3: Geometrical interpretation of the set⋃

k 6=l B(kl) for k, l ∈ {1, 2} and a ∈ R2

+\{0}

Open, closed and compact sets in Rq are Borel measurable. Thus, ∂(a − F) and B (kl)

are measurable. Moreover, countable unions of measurable sets are measurable. Hence,⋃k 6=l B

(kl) for 1 ≤ k , l ≤ q is measurable. If two arbitrary sets M1 and M2 are measurablesets with M1 ⊆ M2, it implies that µ(M1) ≤ µ(M2). Therefore, it holds with (6.5) that

µ(∂(a −F)) ≤ µ(⋃

k 6=l

B (kl))

for 1 ≤ k , l ≤ q. (6.6)

Define

H (kl) :={y ∈ Rq : yT (π(kl)e l + ek) ≥ alπ(kl) + ak

}for 1 ≤ k , l ≤ q with k 6= l . (6.7)

Let Q be the countable set of rational numbers. Since for 1 ≤ k , l ≤ q, it holds that

(light blue) H (kl) ⊇⋃

p∈Q∩(1,∞)

pB (kl), (dark blue)

it follows by monotonicity that

µ(H (kl)

)≥ µ

( ⋃p∈Q∩(1,∞)

pB (kl))

. (6.8)

−∞ +∞

+∞

−∞

0

π(21)e1 + e2

a

Figure 6.4: The set H(kl) in (6.7) for k = 1,l = 2, q = 2 and a ∈ R2

+\{0}

−∞ +∞

+∞

−∞

0

π(21)e1 + e2

a

Figure 6.5: The set⋃

p∈Q∩(1,∞) pB(kl) in

(6.8) for k = 1, l = 2, q = 2and a ∈ R2

+\{0}

43


Then, it holds that

µ(H (kl)

)(6.8)≥ µ

( ⋃p∈Q∩(1,∞)

pB (kl))σ−additivity

=∑

p∈Q∩(1,∞)

µ(pB (kl)

) homogeneity= µ

(B (kl)

) ∑p∈Q∩(1,∞)

p−α

(6.9)for α > 0. As H (kl) for 1 ≤ k , l ≤ q and k 6= l is bounded and closed, it is a compact set inRq\{0}. By the Definition 2.2.6 of a Radon measure, it follows that µ(H (kl)) <∞. Thus,for 1 ≤ k , l ≤ q, it holds that

µ(H (kl)

)∈ (0,∞). (6.10)

Choose α > 0 fixed. Let In := ( α√n, α√n + 1) 6= ∅ for n ≥ 1. Since Q is dense in R, it

holds that pn ∈ Q ∩ In. Thus, for α > 0, it follows that∑p∈Q∩(1,∞)

p−αsubset≥

∑n≥1

p−αn

monotonicity≥

∑n≥1

(α√n + 1

)−α=∑n≥1

1

n + 1=∞. (6.11)

Therefore, (6.9) leads to

µ(H (kl)

)︸︷︷︸∈(0,∞) by (6.10)

≥ µ(B (kl)

) ∑p∈Q∩(1,∞)

p−α︸︷︷︸=∞ by (6.11)

which results in µ(B (kl)) = 0.Hence, it holds for 1 ≤ k , l ≤ q that

µ(∂(a −F))(6.6)≤ µ

(⋃k 6=l

B (kl))σ−additivity

=∑k 6=l

µ(B (kl)

)︸︷︷︸=0

= 0

which implies thatµ(∂(a −F)) = 0.

The result of Theorem 6.0.2 is important to prove the asymptotic behavior in Theorem6.0.7. It should be achieved that ruin can only occur due to large jumps of the randomwalk. By defining a set K δ

r , it can be avoided that the random walk hits the ruin set bysimply drifting in the direction r ∈ Rq\{0}. Let δ > 0 such that the set

K δr :=

{y ∈ Rq\{0} : || y

||y ||+

r

||r |||| < δ

}(6.12)

satisfies µ((∂K δr )\{0}) = 0. Here, δ > 0 can be chosen arbitrarily small.

Example:

Consider d = 1 and assume that r > 0 and δ ∈ (0, 1). Then, the set K δr in (6.12) is given

by

K δr =

{y ∈ R\{0} : | y

|y |+

r

|r || < δ

}={y ∈ R\{0} : | y

|y |+ 1| < δ

}= (−∞, 0).

44


−∞ +∞

+∞

−∞

0

y||y||2

r||r||2

Kδr

x

fixed value

Figure 6.6: K δr =

{y ∈ R2\{0} : || y

||y ||2 + r||r ||2 ||2 < δ

}in (6.12) for the Euclidean norm

In the following Theorem 6.0.2, the parameter u ∈ R+\{0} in Theorem 3.1 by Hult et al.(2005) is replaced by the normed vector ||u|| with u ∈ Rq

+\{0} due to the multivariatesetting. Hult and Lindskog only consider one insurance company such that the initial cap-ital u is univariate, but in this insurance network model q agents are regarded such thatu ∈ Rq

+\{0}. Moreover, the Theorem 6.0.2 is defined on Rq instead on Rd . The proof canbe found in Hult et al. (2005) on p. 17.

Theorem 6.0.2. (Hult et al., 2005, Theorem 3.1)

Let Y (k) = (Y 1(k), ... , Y q(k))T ∈ Rq for k ≥ 1 be i.i.d. random vectors and assumethat Y (1) ∈ MRV (α,µ) for some α > 1 and µ as defined in (2.4). Moreover, itholds that E[Y (1)] = 0. Define

ψM(u) = P( n∑

k=1

Y (k)− rn ∈ ||u||M for some n ≥ 1)

where r ∈ Rq\{0} and M ∈ B(Rq\K δr ) such that M is bounded away from 0. Then,

µ∗(M◦) ≤ lim inf||u||→∞

ψM(u)

||u||P(||Y || > ||u||)≤ lim sup||u||→∞

ψM(u)

||u||P(||Y || > ||u||)≤ µ∗(M)

where M◦ is the interior, M the closure of the set M ,

µ∗(M) :=

∫ ∞0

µ({y ∈ Rq\{0} : y ∈ rv + Mr})dv (6.13)

andMr = {m + rt : m ∈ M , t ≥ 0}.

If a set M is r -increasing (cf. Remark 5.0.2), then Mr = M .

45


Explanation of the appearance of the set Mr :

Due to the assumption E[Y (1)] = 0, the process Y (1) fluctuates around zero. For large||u||, the multivariate random walk

n∑k=1

Y (k)− rn, n ≥ 1 (6.14)

starting in 0 hits a set ||u||M for some n ≥ 1 by taking a large jump y ∈ Rq\{0} to the set.First, the process (6.14) drifts in the direction −r . Assume that at time v , a large jumpoccurs with size y . Then, the process jumps to the point y − rn and then continues to driftin the direction −r . The set ||u||M is hit at time t ≥ v if the jump y has the form

y = rv + m + rt

for some m ∈ M . Thus, it holds that

y ∈ rv + {m + rt : m ∈ M , t ≥ 0} = rv + Mr .

Remark 6.0.3 (Hult et al., 2005, Remark 3.2)

If µ(vr + ∂M) = 0 for almost all v ∈ R+, then µ∗(M◦) = µ∗(M) where µ∗(M) is given by(6.13). By Theorem 6.0.2, it follows that

lim||u||→∞

ψM(u)

||u||P(||Y || > ||u||)= µ∗(M).

In the following, this theory is applied to prove results for this insurance network model.

Lemma 6.0.4.

Let the claims process V = (V1, ... ,Vd)T be given by Vj(t) =∑N(t)

k=1 Xj(k) − cjtas in (3.3) for t ≥ 0 and j = 1, ... , d . Moreover, let the claims vector X (k) =(X1(k), ... ,Xd(k))T for k ≥ 1 be i.i.d. random vectors with X (1) ∈ MRV (α,µ) forα > 1 and µ as defined in (2.4). Let (R(t))t≥0 be the reserve process as in (3.11) andF the ruin set as in chapter 5. Suppose that lj := cjE[W (1)] − E[Xj(1)] ∈ (0,∞)for j = 1, ... , d denotes the positive safety loading as in (3.5). Then, for everyweighted adjacency matrix A ∈ Rq×d

+ and normed initial capital b ∈ Rq+\{0}, the

ruin probability in (6.1) is given by

ψb−F(u) : = P(R(t) ∈ F for some t ≥ 0

)= P

( n∑k=1

Y (k)− nAl ∈ ||u||(b −F) for some n ≥ 1)

where

Y (k) =

Y 1(k)...

Y q(k)

= A

Y1(k)...

Yd(k)

= A

X1(k)− c1W (k)− E[X1(k)− c1W (k)]...

Xd(k)− cdW (k)− E[Xd(k)− cdW (k)]

.

46


Proof

Let

V (t) =

V1(t)...

Vd(t)

=

∑N(t)

k=1 X1(k)− c1t...∑N(t)

k=1 Xd(k)− cdt

(6.15)

be the multivariate claims process for the objects Oj for j = 1, ... , d (3.3). Suppose thatthe claims processes for all objects jump at the same times. Then, ruin can only occur at aclaim time T (n) for some n ≥ 1 when at least one agent Ai for some i ∈ {1, ... , q} cannotcover its loss. n denotes the nth jump triggering the ruin. Thus,

V (T (n)) =

V1(T (n))...

Vd(T (n))

=

∑n

k=1 X1(k)− c1T (n)...∑n

k=1 Xd(k)− cdT (n)

=

∑n

k=1

(X1(k)− c1W (k)

)...∑n

k=1

(Xd(k)− cdW (k)

)

(6.16)refers to the claims process at time T (n). W (k) is exponentially distributed as describedin Proposition 2.1.5 with mean E[W (1)] = 1

λ<∞. Let

E (T (n)) =

E 1(T (n))...

E q(T (n))

= A

V1(T (n))...

Vd(T (n))

(6.17)

be the exposure (3.8) at time T (n). The expectation of (6.17) given by

E[E (T (n))] =

E[E 1(T (n))]...

E[E q(T (n))]

= A

E[∑n

k=1

(X1(k)− c1W (k)

)]

...E[∑n

k=1

(Xd(k)− cdW (k)

)]

= A

nE[X1(1)− c1W (1)]...

nE[Xd(1)− cdW (1)]

= nA

−l1...−ld

= −nAl (6.18)

with positive safety loading lj = cjE[W (1)]−E[Xj(1)] for j ∈ {1, ... , d} (3.5) is the expectedloss for agent Ai , i = 1, ... , q. Furthermore, define

Z (T (n)) := E (T (n))− E[E (T (n))]

= A

∑n

k=1

(X1(k)− c1W (k)

)− E[X1(k)− c1W (k)]

...∑nk=1

(Xd(k)− cdW (k)

)− E[Xd(k)− cdW (k)]

47


=: A

∑n

k=1 Y1(k)...∑n

k=1 Yd(k)

(6.19)

= An∑

k=1

Y (k)

=n∑

k=1

AY (k)

=:n∑

k=1

Y (k). (6.20)

The ruin probability with claims processes (V (t))t≥0 (6.15) and V (T (n)) (6.16), exposureE (T (n)) (6.17), E[E (T (n))] (6.18) and Z (T (n)) (6.20) is given by

ψb−F(u) : = P(R(t) ∈ F for some t ≥ 0)

= P(u − E (t) ∈ F for some t ≥ 0)

= P(u − AV (t) ∈ F for some t ≥ 0)

= P(AV (t) ∈ u −F for some t ≥ 0)

= P(AV (T (n)) ∈ u −F for some n ≥ 1)

= P(E (T (n)) ∈ u −F for some n ≥ 1)

= P(E (T (n))− E[E (T (n))]︸︷︷︸:=Z(T (n))

+E[E (T (n))]︸︷︷︸=−nAl (6.18)

∈ u −F for some n ≥ 1)

= P(Z (T (n))− nAl ∈ u −F for some n ≥ 1)

= P(Z (T (n))− nAl ∈ ||u||( u

||u||︸︷︷︸=:b

−F) for some n ≥ 1)

(||u||F = F by Remark 5.0.3)

= P(Z (T (n))− nAl ∈ ||u||(b −F) for some n ≥ 1)

= P( n∑

k=1

AY (k)− nAl ∈ ||u||(b −F) for some n ≥ 1)

= P( n∑

k=1


.

Lemma 6.0.5.

If X (k) = (X1(k), ... ,Xd(k))T ∈ Rd+ for k ≥ 1 are i.i.d. random vectors and

X (1) ∈ MRV (α,µ) for α > 1 and µ as defined in (2.4), then Y (k) ∈ Rd defined by

Y (k) =

Y1(k)...

Yd(k)

=

X1(k)− c1W (k)− E[X1(k)− c1W (k)]...


are i.i.d. random vectors and Y (1) ∈ MRV (α,µ).

48


Proof

Since X (k) for k ≥ 1 are i.i.d. random vectors, Y (k) for k ≥ 1 are also i.i.d. randomvectors. Therefore, it is enough to show that Y (1) ∈ Rd is multivariate regularly varying(Y (1) ∈ MRV (α,µ)). As defined in (6.19), Y (1) is given by

Y (1) = X (1)− cW (1)− E[X (1)− cW (1)]︸︷︷︸=−l (see (3.5))

(6.21)

where X ∈ Rd+, W ∈ R+, c ∈ Rd

+ and l ∈ Rd+. Let M be the set consisting of rectangles

with m = (m1, ... , md)T ∈ Rd\{0} defined by

M := (m1,∞)× · · · × (md ,∞)

which is a generating system in Rd\{0}. According to the Definition 2.2.8 and Remark2.2.9, Y (1) in (6.21) is multivariate regularly varying if

lim||u||→∞

P(Y (1) ∈ ||u||M)

P(||Y (1)|| > ||u||)= µ(M). (6.22)

The multivariate regular variation of Y (1) in (6.22) is shown in two steps given by

lim||u||→∞

P(Y (1) ∈ ||u||M)

P(||X (1)|| > ||u||)︸︷︷︸i)

P(||X (1)|| > ||u||)P(||Y (1)|| > ||u||)︸︷︷︸

ii)

= µ(M). (6.23)

i) lim||u||→∞

P(Y (1) ∈ ||u||M)

P(||X (1)|| > ||u||)= µ(M)

For the numerator of i), it follows with (6.21) that

P(Y (1) ∈ ||u||M) = P(X (1)− cW (1) + l ∈ ||u||M)

= P(X (1)− cW (1) ∈ ||u||M − l)

= E[P(X (1)− cW (1) ∈ ||u||M − l |W (1))]

=

∫ ∞0

P(X (1) ∈ ||u||M − l + cv)λe−λvdv . (6.24)

Inserting (6.24) into i) leads to

lim||u||→∞

P(Y (1) ∈ ||u||M)

P(||X (1)|| > ||u||)= lim||u||→∞

∫∞0

P(X (1) ∈ ||u||M − l + cv)λe−λvdv

P(||X (1)|| > ||u||)

= lim||u||→∞

∫ ∞0

P(X (1) ∈ ||u||M − l + cv)

P(||X (1)|| > ||u||)λe−λvdv

= lim||u||→∞

∫ ∞0

P(X (1) ∈ ||u||(M + cv−l||u|| ))

P(||X (1)|| > ||u||)λe−λvdv .

(6.25)

For taking the limit inside the integral, it is essential to find a dominating functionwhich is integrable in order to apply the dominated convergence theorem. It holdsthat

49


P(X (1) ∈ ||u||

(M +

cv − l

||u||

))= P

(X1(1) > ||u||

(m1 +

c1v − l1||u||

), ... ,Xd(1) > ||u||

(md +

cdv − ld||u||

))≤ P

(X1(1) > ||u||

(m1 +

c1v − l1||u||

))+ · · ·+ P

(Xd(1) > ||u||

(md +

cdv − ld||u||

))=

d∑j=1

P(Xj(1) > ||u||

(mj +

cjv − lj||u||

)). (6.26)

Since the claims Xj(1) for j = 1, ... , d are independent non-negative regularly varyingrandom variables,

P(||X (1)|| > ||u||

)= P

( d∑j=1

|Xj(1)| > ||u||)

non−negative= P

( d∑j=1

Xj(1) > ||u||)

∼d∑

j=1

P(Xj(1) > ||u||

)(6.27)

(cf. p. 278 in Feller (1971) and Lemma 1.3.1 in Embrechts et al. (1997)). By (6.26)and (6.27), it follows that

P(X (1) ∈ ||u||(M + cv−l

||u|| ))

P(||X (1)|| > ||u||)≤

d∑j=1

P(Xj(1) > ||u||(mj +

cjv−lj||u|| )

)P(Xj(1) > ||u||)

.

Since Xj(1) is regularly varying with index α > 1, it follows by application of thePotter bounds (cf. Proposition 2.2.5) that for ε ∈ (0,α− 1), it holds for the positiveclaims vector Xj(1) with j = 1, ... , d that

P(Xj(1) > ||u||(mj +

cjv−lj||u|| )

)P(Xj(1) > ||u||)

≤

(1 + ε)

(mj +

cjv−lj||u||

)−(α+ε)

if 0 < mj +cjv−lj||u|| ≤ 1

(1 + ε)(mj +

cjv−lj||u||

) <0︷︸︸︷−α+ε

if mj +cjv−lj||u|| > 1

≤

{(1 + ε)(bj)

−(α+ε) with bj ∈ (0, 1] if 0 < mj +cjv−lj||u|| ≤ 1

(1 + ε) if mj +cjv−lj||u|| > 1

≤ (1 + ε)cj

for cj ∈ (0,∞). As∫ ∞0

d∑j=1

(1 + ε)cjλe−λvdv =

d∑j=1

(1 + ε)cj

∫ ∞0

λe−λvdv︸︷︷︸=1

=d∑

j=1

(1 + ε)cj <∞,

50


the limit can be taken under the integral. Since X (1) ∈ MRV (α,µ), (6.25) results to∫ ∞0

lim||u||→∞

P(X (1) ∈ ||u||(M + cv−l

||u|| ))

P(||X (1)|| > ||u||)λe−λvdv =

∫ ∞0

µ(M)λe−λvdv

= µ(M)

∫ ∞0

λe−λvdv

= µ(M)[− e−λv

]∞0︸︷︷︸

=1

= µ(M).

Thus, it holds that

lim||u||→∞

P(Y (1) ∈ ||u||M)

P(||X (1)|| > ||u||)= µ(M). (6.28)

ii) lim||u||→∞P(||X (1)|| > ||u||)P(||Y (1)|| > ||u||)

= 1

Consider D = {x ∈ Rd : ||x || > 1} = {||x || > 1} (cf. Definition 2.2.8). Then, (6.28)leads to

lim||u||→∞

P(Y (1) ∈ ||u||D)

P(||X (1)|| > ||u||)= lim||u||→∞

P(||Y (1)|| > ||u||)P(||X (1)|| > ||u||)

= µ(D) = 1.

Hence, it holds that

lim||u||→∞

P(||X (1)|| > ||u||)P(||Y (1)|| > ||u||)

= µ−1(D) = 1. (6.29)

By i) and ii), it can be concluded for (6.23) that

lim||u||→∞

P(Y (1) ∈ ||u||M)

P(||Y (1)|| > ||u||)= µ(M).

Y (1) ∈ MRV (α,µ) for some α > 1.

Lemma 6.0.6.

Let A be the weighted adjacency matrix in Rq×d+ . If X (k) = (X1(k), ... ,Xd(k))T ∈

Rd+ for k ≥ 1 are i.i.d. random vectors and X (1) ∈ MRV (α,µ) for α > 1 and µ as

defined in (2.4), then Y (k) ∈ Rq defined by

Y (k) =

Y 1(k)...

Y q(k)

= A

Y1(k)...

Yd(k)

= A

X1(k)− c1W (k)− E[X1(k)− c1W (k)]...


are i.i.d. random vectors and Y (1) ∈ MRV (α, µ) with

µ(·) := µ ◦ A−1(·)(µ ◦ A−1(D))−1

andD := {y ∈ Rq : ||y || > 1}.

51


Proof

According to Lemma 6.0.5, Y (k) for k ≥ 1 are i.i.d. random vectors and Y (1) ∈MRV (α,µ). Therefore, Y (k) for k ≥ 1 are also i.i.d random vectors. By Proposition2.2.11, it results for a fixed matrix A ∈ Rq×d

+ that

P(Y (1) ∈ ||u|| · )

P(||Y (1)|| > ||u||)=

P(AY (1) ∈ ||u|| · )

P(||Y (1)|| > ||u||)v→ µ ◦ A−1(·), ||u|| → ∞.

Since for ||u|| → ∞, it holds that

P(Y (1) ∈ ||u|| · )

P(||X (1)|| > ||u||)=

P(Y (1) ∈ ||u|| · )

P(||X (1)|| > ||u||)P(||X (1)|| > ||u||)P(||Y (1)|| > ||u||)︸︷︷︸

→ 1 by (6.29)

=P(Y (1) ∈ ||u|| · )

P(||Y (1)|| > ||u||)v→ µ ◦ A−1(·),

it follows for D := {y ∈ Rq : ||y || > 1} that

P(Y (1) ∈ ||u||D)

P(||X (1)|| > ||u||)=

P(||Y (1)|| ∈ ||u||)P(||X (1)|| > ||u||)

→ µ ◦ A−1(D). (6.30)

Therefore, for ||u|| → ∞, it holds that

P(Y (1) ∈ ||u|| · )

P(||Y (1)|| > ||u||)=

P(Y (1) ∈ ||u|| · )

P(||X (1)|| > ||u||)P(||X (1)|| ∈ ||u||)P(||Y (1)|| > ||u||)

v→ µ ◦ A−1(·)(µ ◦ A−1(D))−1

and Y (1) is multivariate regularly varying for some α > 1 with

µ(·) := µ ◦ A−1(·)(µ ◦ A−1(D))−1.

In the next theorem, the asymptotic behavior of the ruin probability is determined.

Theorem 6.0.7. (Asymptotic ruin probability for fixed A)


k=1 Xj(k) − cjtas in (3.3) for t ≥ 0 and j = 1, ... , d . Let the claims vector X (1) ∈ MRV (α,µ)with α > 1 and µ as defined in (2.4) and F be a ruin set as in chapter 5. Supposethat lj := cjE[W (1)]− E[Xj(1)] ∈ (0,∞) for j = 1, ... , d denotes the positive safetyloading as in (3.5). Define for the normed initial capital b ∈ Rq

+\{0} (3.10) the ruinprobability ψb−F(u) as in Lemma 6.0.4. Then, for every fixed weighted adjacencymatrix A ∈ Rq×d

+ , it holds that

lim||u||→∞

ψb−F(u)

||u||P(||X || > ||u||)=

∫ ∞0

µ ◦ A−1({y ∈ Rq

+\{0} : y ∈ vAl + b −F})dv .

52


Proof

By Lemma 6.0.4, the ruin probability is defined by

ψb−F(u) = P(R(t) ∈ F for some t ≥ 0)

= P( n∑

k=1


.

Define M := b − F and r := Al . Theorem 6.0.2 and Remark 6.0.3 can be applied if thefollowing three conditions are fulfilled:

1) Y(k) = (Y1(k), ... , Yd(k)) ∈ Rd are i.i.d. random vectors andY(1) ∈ MRV(α, µ) for some α > 1

This condition is fulfilled by Lemma 6.0.6. The measure µ is given by

µ(·) := µ ◦ A−1(·)(µ ◦ A−1(D))−1.

2) E[Y(1)] = 0

For A ∈ Rq×d+ , it holds that

E[Y (1)] = E[AY (1)]

= A

E[Y1(1)]...

E[Yd(1)]

= A

E[X1(1)− c1W (1)− E[X1(1)− c1W (1)]

]...

E[Xd(1)− cdW (1)− E[Xd(1)− cdW (1)]

]

= A

E[X1(1)− c1W (1)]− E[X1(1)− c1W (1)]...

E[Xd(1)− cdW (1)]− E[Xd(1)− cdW (1)]

=

0...0

.

3) M := b− F is Al-increasing

Since l ∈ Rd+\{0} (3.6) and A ∈ Rq×d

+ , it holds that r := Al ∈ Rq+. As the ruin

set F is Al-decreasing by Remark 5.0.2, −F is Al-increasing. Therefore, b−F withb ∈ Rq

+\{0} (3.10) is Al-increasing.

Since the conditions for Theorem 6.0.2 are fulfilled by 1) and 2), it follows that

µ∗((b −F)◦) ≤ lim inf||u||→∞

ψb−F(u)

||u||P(||Y || > ||u||)≤ lim sup||u||→∞

ψb−F(u)

||u||P(||Y || > ||u||)≤ µ∗(b −F)

(6.31)

53


with

µ∗(b −F) =

∫ ∞0

µ ◦ A−1({y ∈ Rq+\{0} : y ∈ vAl + b −F})(µ ◦ A−1(D))−1dv

=(µ ◦ A−1(D)

)−1∫ ∞

0

µ ◦ A−1({y ∈ Rq+\{0} : y ∈ vAl + b −F})dv . (6.32)

Due to condition 3), the application of Remark 6.0.3 leads to

lim||u||→∞

ψb−F(u)

||u||P(||Y || > ||u||)= µ∗(b −F). (6.33)

It follows that

lim||u||→∞

ψb−F(u)

||u||P(||X || > ||u||)

= lim||u||→∞

ψb−F(u)

||u||P(||X || > ||u||)P(||X || > ||u||)P(||Y || > ||u||)

P(||Y || > ||u||)P(||X || > ||u||)︸︷︷︸

=1

= lim||u||→∞

ψb−F(u)

||u||P(||Y || > ||u||)︸︷︷︸=µ∗(b−F) by (6.33)

P(||Y || > ||u||)P(||X || > ||u||)︸︷︷︸

=µ◦A−1(D) by (6.30)

= µ∗(b −F) µ ◦ A−1(D)

(6.31)=

µ ◦ A−1(D)

µ ◦ A−1(D)

∫ ∞0

µ ◦ A−1({y ∈ Rq+\{0} : y ∈ vAl + b −F})dv

=

∫ ∞0

µ ◦ A−1({y ∈ Rq+\{0} : y ∈ vAl + b −F})dv .

Remark 6.0.8

The image measure µ = µ ◦ A−1 only considers positive vectors y ∈ Rq+\{0} such that

y ∈ vAl + b−F where F is the ruin set and v ∈ R+\{0}. This results from the positivityof the claims vector X and the adjacency matrix A. Since X ∈ MRV (α,µ), it holds that

P(X ∈ u · )

P(||X || > u)v→ µ(·) = µ({x ∈ Rd

+\{0} : x ∈ · }).

As the proof in Theorem 6.0.7 is based on the multivariate regular variation of X , themeasure µ = µ ◦ A−1 only considers positive vectors y ∈ Rd

+\{0}.

Geometrical interpretation of vAl + b−FThe following figures visualize the set vAl +b−F by means of an exemplary ruin set F , ma-trix A, vector b and l and scalar v . Here, the ruin set F ∈ F of case 3 as described in chapter5 is taken. The matrix A ∈ Rq×d

+ is the adjacency matrix (3.1) and l ∈ Rq+\{0} is the pos-

itive safety loading (3.5). Thus, Al ∈ Rq+ and with the scalar v ∈ R+ the vector vAl ∈ Rq

+

is directed towards the positive orthant. Moreover, b ∈ Rq+\{0} denotes the normed initial

capital (3.10) which is also a positive directed vector. Thus, vAl +b ∈ Rq+\{0} is a positive

vector.

54


−∞ +∞

+∞

−∞

F

Figure 6.7: F

−∞ +∞

+∞

−∞

−F

Figure 6.8: −F

−∞ +∞

+∞

−∞

b

Figure 6.9: b −F

−∞ +∞

+∞

−∞

b

vAl

Figure 6.10: vAl + b −F

As stated in Theorem 6.0.7, the following set is considered

{y ∈ Rq+\{0} : y ∈ vAl + b −F}.

−∞ +∞

+∞

−∞

b

vAl

Figure 6.11: {y ∈ R2+\{0} : y ∈ vAl + b −F}

55


Proposition 6.0.9. (Asymptotic ruin probability for random A)


k=1 Xj(k)− cjt fort ≥ 0 and suppose that X (1) ∈ MRV (α,µ) with α > 1 and let F be the ruin set asin chapter 5. Suppose that lj := cjE[W (1)]−E[Xj(1)] ∈ (0,∞) is the positive safetyloading as in (3.5). Then, for every random weighted adjacency matrix A ∈ Rq×d

+

and normed initial capital b ∈ Rq+\{0} (3.10), it holds that

lim||u||→∞

ψb−F(u)

||u||P(||X || > ||u||)= E

[ ∫ ∞0

µ◦A−1({y ∈ Rq

+\{0} : y ∈ vAl+b−F})dv

].

Proof

Theorem 6.0.7 states that for every fixed matrix A ∈ Rq×d+ , it holds that

lim||u||→∞

ψb−F(u)

||u||P(||X || > ||u||)=

∫ ∞0

µ ◦ A−1({y ∈ Rq

+\{0} : y ∈ vAl + b −F})dv . (6.34)

Let A∗ ∈ Rq×d+ be a realization of the random matrix A ∈ Rq×d

+ . Under the assumptionthat A ∈ Rq×d

+ is random and independent of V , it can be concluded that

lim||u||→∞

ψb−F(u)

||u||P(||X || > ||u||)

= lim||u||→∞

P(u − A(V (t)− ct) ∈ F for some t ≥ 0)

||u||P(||X || > ||u||)

= lim||u||→∞

E[P(u − A(V (t)− ct) ∈ F for some t ≥ 0)|A

]||u||P(||X || > ||u||)

= lim||u||→∞

∑A∗ P(u − A∗(V (t)− ct) ∈ F for some t ≥ 0)P(A = A∗)

||u||P(||X || > ||u||)

= lim||u||→∞

∑A∗

P(A = A∗)P(u − A∗(V (t)− ct) ∈ F for some t ≥ 0)

||u||P(||X || > ||u||)

=∑A∗

P(A = A∗) lim||u||→∞

P(u − A∗(V (t)− ct) ∈ F for some t ≥ 0)

||u||P(||X || > ||u||)(6.34)

=∑A∗

P(A = A∗)

∫ ∞0

µ ◦ (A∗)−1({y ∈ Rq

+\{0} : y ∈ vA∗l + b −F}

)dv

= E[ ∫ ∞

0

µ ◦ A−1({y ∈ Rq

+\{0} : y ∈ vAl + b −F})dv

]=

∫ ∞0

E[µ ◦ A−1

({y ∈ Rq

+\{0} : y ∈ vAl + b −F})]dv .

56

7 | Applications

In this chapter, the theory developed in the previous chapters is applied. For differentsituations, the asymptotic ruin probability for a fixed weighted adjacency matrix A (Theorem6.0.7)

lim||u||→∞

ψb−F(u)

||u||P(||X || > ||u||)=

∫ ∞0

µ ◦ A−1({y ∈ Rq+\{0} : y ∈ vAl + b −F})dv (7.1)

withlj := cjE[W (1)]− E[Xj(1)] ∈ (0,∞), j = 1, ... , d (7.2)

is computed. Throughout, let || · || be the 1-norm. In section 7.1, the individual ruinprobability is computed which means that only one agent in the network is considered. Insection 7.2, the ruin probability of several agents is described which is called network ruinprobability.In both sections, the asymptotic ruin probability (7.1) is evaluated for different claim dis-tributions (Pareto, Burr and Loggamma). In the following table, the positive safety loading(7.2) is given for these claim distributions of X . The index of regular variation is computedin section 2.2.

Distribution FXj(x) = 1− FXj

(x), j = 1, ... , d lj := cjE[W(1)]− E[Xj(1)]

E[Xj(1)], j = 1, ... , d =cj

λ− E[Xj(1)]

Index of regular variation for j = 1, ... , d

Pareto FXj(x) = Kjx

−α(x ≥ 1, Kj > 0, α > 0) lj =

cjλ− Kj

α

α− 1

E[Xj(1)] = Kjα

α− 1α

Burr FXj(x) =

(κj

κj + xτ

)αlj =

cjλ− Γ(α− 1

τ

)Γ(

1 +1

τ

) κ1τj

Γ(α)(x > 0, κj > 0, α > 0, τ > 0)

E[Xj(1)] = Γ(α− 1

τ

)Γ(

1 +1

τ

) κ1τj

j

Γ(α)τα

Loggamma FXj(x) ∼ αβj−1

Γ(βj)(ln(x))βj−1x−α lj =

cjλ−(

1− 1

α

)−βj(x →∞, βj > 0, α > 0)

E[Xj(1)] =

(1− 1

α

)−βjα

Table 7.1: The positive safety loading (7.2) for different distributions

57

7 | Applications

7.1 Individual ruin probability

The individual ruin probability is the ruin probability for a single agent going bankrupt.Based on the insurance network model described in chapter 3, the portfolios of only oneagent

Ai for i = 1, ... , q and |A| = 1

are considered. The dependence structure with respect to the other agents is neglected (grayshaded). On the one hand, it simplifies the computation, on the other hand, it gives theagent the information about its ruin probability independent on the number and structureof other agents. In this section, two cases are regarded which are visualized in figure 7.1:

• A bipartite network with 1 agent and 1 object (light blue)

• A bipartite network with 1 agent and d objects (dark blue)

A1 A2 A3 A4 A5 . . . Aq

O1 O2 O3. . . Od

Figure 7.1: Individual ruin probability

A bipartite network with 1 agent and 1 object (classical case)

This network describes an insurance company A1 which only insures one heavy-tailed objectO1 (figure 7.2). Thus, for the number of objects and the number of agents, it holds that

degA(1) = 1 and degO(1) = 1.

The insured object O1 could be for example an insurance portfolio of storms, earthquakesor terrorist attacks.

A1

O1

Figure 7.2: A bipartite network with 1 agent and 1 object

In the univariate case, the ruin set F is described by the interval F = (−∞, 0). If theagent’s reserve R1 ∈ F , the insurance company is insolvent and the amount of loss exceedsthe agent’s capital. If R1 /∈ F , the insurance company is solvent. The negative ruin set isgiven by −F = (0,∞).

58

7 | Applications

−∞ +∞0

F −F

Figure 7.3: Illustration of the ruin set F = (−∞, 0)

Since only one agent A1 insures one object O1, the weighted adjacency matrix A = A11 =

{1 ∼ 1}W 11 ∈ R+ and the positive safety loading l ∈ R+\{0}. Moreover, for the normed

initial capital, it holds that b = u||u|| = 1 as ||u|| = |u| = u (3.10). Thus, for the right-hand

side of (7.1), it follows for α > 1 that∫ ∞0

µ ◦ A−1({y ∈ R+\{0} : y ∈ vAl + b −F})dv

=

∫ ∞0

µ(A−1{y ∈ R+\{0} : y ∈ vAl + 1 + (0,∞)})dv

=

∫ ∞0

Aαµ({y ∈ R+\{0} : y ∈ vAl + (1,∞)})dv (µ is homogeneous)

=

∫ ∞0

Aαµ({y ∈ R+\{0} : y ∈ (vAl + 1,∞)})dv

=

∫ ∞0

Aαµ((vAl + 1){y ∈ R+\{0} : y ∈ (1,∞)})dv

= Aα∫ ∞

0

(vAl + 1)−αµ({y ∈ R+\{0} : y ∈ (1,∞)})dv (µ is homogeneous)

= Aα µ({y ∈ R+\{0} : y ∈ (1,∞)})︸︷︷︸=1−α=1

∫ ∞0

(vAl + 1)−αdv

= Aα[

(vAl + 1)

<0︷︸︸︷−α+1

Al(−α + 1)

]∞0

=Aα

Al(α− 1)

=Aα−1

l(α− 1). (7.3)

The weighted adjacency parameter A ∈ [0, 1] denotes the fraction of the object O1 that theagent A1 insures.

• If A = 0, then the agent does not insure the object.

• If A = 1, then the agent bears 100% of the claims.

• If A ∈ (0, 1), then the agent insures the claims partly. If A = 12, the agent pays

50% of the incurred loss (proportional). However, A can also be a threshold whichmeans that the agent only pays the claims which exceed a stated amount of loss(non-proportional).

In the following table, the solution (7.3) is evaluated for different claims distributions X(Pareto, Burr and Loggamma).

59

7 | Applications

Distribution l = cE[W(1)]− E[X(1)] (table 7.1)

Aα−1

l(α− 1)

Pareto l =c

λ− K

α

α− 1Aα−1

cλ

(α− 1)− Kα

Burr l =c

λ− Γ(α− 1

τ

)Γ(

1 +1

τ

) κ1τ

Γ(α)Aτα−1(

cλ− Γ(α− 1

τ)Γ(1 + 1

τ) κ

1τ

Γ(α)

)(τα− 1)

Loggamma l =c

λ−(

1− 1

α

)−βAα−1(

cλ− (1− 1

α)−β)

(α− 1)

Table 7.2: Solution (7.3) for different distributions

The solution (7.3) is comparable to the classical univariate result (Embrechts et al., 1997,Theorem 1.3.8). The Cramér-Lundberg theorem for large claims states that

limu→∞

ψ(u)

F IX (u)

=1

ρ(7.4)

where ρ > 0 and F IX (u) =

1

µ

∫ ∞u

FX (x)dx with FX (x) as in (2.1).

For u ∈ R\{0}, the left-hand side of (7.1) can be reformulated such that

limu→∞

ψb−F(u)

u P(X > u)= lim

u→∞

ψb−F(u)

uFX (u)= lim

u→∞

ψb−F(u)

F IX (u)

F IX (u)

uFX (u)

(7.4)=

1

ρlimu→∞

F IX (u)

uFX (u). (7.5)

For the Pareto distribution FX (u) = Ku−α1{u ≥ 1}, it follows for F IX (u) that

F IX (u) =

1

µ

∫ ∞u

FX (x)dx

=1

µ

∫ ∞u

Kx−α1{x ≥ 1}dx

=1

µ

[K

−α + 1x

<0︷︸︸︷−α+1

]∞u

=1

µ

−K−α + 1

u−α+1

=1

µ

K

α− 1u−α+1. (7.6)

60

7 | Applications

Thus, it follows by (7.6) that

limu→∞

F IX (u)

uFX (u)= lim

u→∞

1µ

Kα−1

u−α+1

Ku−α+1

=1

µ(α− 1).

Then, (7.5) leads to

limu→∞

ψb−F(u)

u P(X > u)=

1

ρµ(α− 1)

(3.7)=

1

( cλµ− 1)µ(α− 1)

=1

( cλ− µ)(α− 1)

(3.5)=

1

l(α− 1). (7.7)

This result in (7.7) is equal to the Pareto solution in the table (7.2) for A = 1.

A bipartite network with 1 agent and d objects

Achieving a higher diversification, an insurance company usually insures more than oneobject. Thus, the simple model illustrated in figure 7.2 is extended to the objects

Oj j = 1, ... , d .

For the number of agents and the number of objects, it holds that

degA(1) = d and degO(j) = 1 ∀ j = 1, ... , d .

A1

O1 O2... Od

Figure 7.4: A bipartite network with 1 agent and d objects

The ruin set F is identical to the previous case i.e. F = (−∞, 0). Again, b = u||u|| = 1

(3.10) as ||u|| = |u| = u and only one agent is considered. Having an insurance portfolioof d objects, the weighted adjacency matrix A := (A1, ... ,Ad) ∈ R1×d

+ . With the positivesafety loading l := (l1, ... , ld)T ∈ Rd

+\{0}, it follows that

Al = (A1, ... ,Ad)

l1...ld

= 〈A, l〉 =d∑

j=1

Aj lj . (7.8)

61

7 | Applications

where 〈·, ·〉 denotes the standard scalar product. Thus, for the right-hand side of (7.1), itfollows for α > 1 that∫ ∞

0

µ ◦ A−1({y ∈ R+\{0} : y ∈ vAl + b −F})dv

=

∫ ∞0

µ({x ∈ Rd+\{0} : Ax ∈ vAl + 1 + (0,∞)})dv

=

∫ ∞0

µ({x ∈ Rd+\{0} : Ax ∈ vAl + (1,∞)})dv

=

∫ ∞0

µ({x ∈ Rd+\{0} : Ax ∈ (vAl + 1,∞)})dv

=

∫ ∞0

µ({x ∈ Rd+\{0} : Ax ∈ (vAl + 1,∞)})dv

=

∫ ∞0

d∑j=1

µ

({sej ∈ Rd

+\{0} : s(Aej) ∈(v

d∑j=1

Aj lj + 1,∞)})

dv (7.9)

=

∫ ∞0

d∑j=1

µ

({s ∈ R+\{0} : sAj ∈

(v

d∑j=1

Aj lj + 1,∞)})

dv

=

∫ ∞0

d∑j=1

µ

({s ∈ R+\{0} : s ∈

(v∑dj=1 Aj lj + 1

Aj,∞)})

dv

=

∫ ∞0

d∑j=1

µ

((v∑dj=1 Aj lj + 1

Aj

){s ∈ R+\{0} : s ∈ (1,∞)

})dv

=

∫ ∞0

d∑j=1

(v∑d

j=1 Aj lj + 1

Aj

)−αµ({

s ∈ R+\{0} : s ∈ (1,∞)})

dv (µ is homogeneous)

= µ({

s ∈ R+\{0} : s > 1})

︸︷︷︸=1−α=1

d∑j=1

∫ ∞0

(v∑d

j=1 Aj lj + 1

Aj

)−αdv

=d∑

j=1

∫ ∞0

(v

∑dj=1 Aj lj

Aj+

1

Aj

)−αdv

=d∑

j=1

[Aj

(−α + 1)∑d

j=1 Aj lj

(v

∑dj=1 Aj lj

Aj+

1

Aj

) <0︷︸︸︷−α+1

]∞0

=d∑

j=1

−Aj

(−α + 1)∑d

j=1 Aj lj

(1

Aj

)−α+1

=d∑

j=1

Aαj

(α− 1)∑d

j=1 Aj lj. (7.10)

Due to the independent components of the claims vector X , the measure µ only has masson the axes {sej ∈ Rd

+\{0} : s > 0} and (7.9) holds. For d = 1, the solution in (7.3) follows.

62

7 | Applications

In the following table, the solution (7.10) is evaluated for different claim distributions X(Pareto, Burr and Loggamma).

Distribution lj = cjE[W(1)]− E[Xj(1)] (table 7.1)d∑

j=1

Aαj

(α− 1)∑d

j=1 Ajlj

Pareto lj =cjλ− Kj

α

α− 1d∑

j=1

Aαj

(α− 1)∑d

j=1 Aj

( cjλ− Kj

αα−1

)Burr lj =

cjλ− Γ(α− 1

τ

)Γ(

1 +1

τ

) κ1τj

Γ(α)d∑

j=1

Aταj

(τα− 1)∑d

j=1 Aj

(cjλ− Γ(α− 1

τ

)Γ(1 + 1

τ

) κ 1τj

Γ(α)

)Loggamma lj =

cjλ−(

1− 1

α

)−βjd∑

j=1

Aαj

(α− 1)∑d

j=1 Aj

(cjλ−(1− 1

α

)−βj)Table 7.3: Solution (7.10) for different distributions

7.2 Network ruin probability

As the dependence structure to other agents is an essential factor for the size of the ruinprobability of each agent, this section will deal with the ruin probability of a network withseveral agents. The ruin probability is called network ruin probability. Compared tosection 7.1, it is assumed that q agents

Ai for i = 1, ... , q with |A| = q > 1

are in the market who insure d objects

Oj for j = 1, ... , d with |O| = d ≥ 1.

For the number of agents and the number of objects, it is assumed that

degA(i) ≥ 0 ∀ i = 1, ... , q and degO(j) ≥ 0 ∀ j = 1, ... , d .

In this section, a bipartite network with q agents and d objects is regarded. The followingthree cases are investigated:

• No capital transfer

• Capital transfer without transaction costs

• Capital transfer with transaction costs

63

7 | Applications

7.2.1 No capital transfer

A bipartite network with q agents and d objects

In this insurance network model, the agents do not transfer capital among each other.The agents are competitors who are solvent if they can cover the losses caused by theirinsured objects.

A1 A2 A3 A4 A5 . . . Aq

O1 O2 O3. . . Od

Figure 7.5: A bipartite network with q agents and d objects without capital transfer

If the agents do not transfer capital, the ruin set is given in (5.7) by

F ={y ∈ Rq : y ∈

([0,∞)q

)c}.

Let A ∈ Rq×d+ , l ∈ Rd

+\{0} such that Al can be written as

Al =

(Al)1

(Al)2

...(Al)q

=

A1

1 A12 · · · A1

j · · · A1d

A21 A2

2 · · · A2j · · · A2

d...

... . . . ... . . . ...Aq

1 Aq2 · · · Aq

j · · · Aqd

l1l2...lj...ld

=

∑d

j=1 A1j lj∑d

j=1 A2j lj

...∑dj=1 A

qj lj

. (7.11)

Then, for v ∈ R+ and b ∈ Rq+\{0}, it follows that

vAl + b −F = vAl + b −{y ∈ Rq : y ∈

([0,∞)q

)c}= vAl + b︸︷︷︸

positive

+{y ∈ Rq : y ∈

((−∞, 0]q

)c}={y ∈ Rq : y ∈

((−∞, vAl + b]

)c}=

q⋃i=1

{y ∈ Rq : y i > v(Al)i + bi

}=

q⋃i=1

{y ∈ Rq : y i − v(Al)i − bi > 0

}. (7.12)

Thus, for the right-hand side of (7.1), it follows for α > 1 and with Aij 6= 0 for all

i ∈ {1, ... , q} and j ∈ {1, ... , d} that∫ ∞0

µ ◦ A−1

({y ∈ Rq

+\{0} : y ∈ vAl + b −F})

dv

=

∫ ∞0

µ ◦ A−1

(q⋃

i=1

{y ∈ Rq

+\{0} : y i − v(Al)i − bi > 0

})dv (insert (7.12))

64

7 | Applications

=

∫ ∞0

µ

(q⋃

i=1

{x ∈ Rd

+\{0} : (Ax)i − v(Al)i − bi > 0

})dv

=

∫ ∞0

d∑j=1

µ

(q⋃

i=1

{sej ∈ Rd

+\{0} : s(Aej)i − v

d∑j=1

Aij lj − bi > 0

})dv (7.13)

=

∫ ∞0

d∑j=1

µ

(q⋃

i=1

{s ∈ R+\{0} : sAi

j − vd∑

j=1

Aij lj − bi > 0

})dv

=

∫ ∞0

d∑j=1

µ

(q⋃

i=1

{s ∈ R+\{0} : s >

v∑d

j=1 Aij lj + bi

Aij

})dv

=

∫ ∞0

d∑j=1

µ

({s ∈ R+\{0} : s > min

1≤i≤q

v∑d

j=1 Aij lj + bi

Aij

})dv

=

∫ ∞0

d∑j=1

(min

1≤i≤q

(v∑d

j=1 Aij lj + bi

Aij

))−αµ({s ∈ R+\{0} : s > 1}

)︸︷︷︸=1−α=1

dv

(µ is homogeneous)

=

∫ ∞0

d∑j=1

(min

1≤i≤q

(v∑d

j=1 Aij lj + bi

Aij

))−αdv

=

∫ ∞0

d∑j=1

(min

1≤i≤q

(v

∑dj=1 A

ij lj

Aij

+bi

Aij

))−αdv

=d∑

j=1

∫ ∞0

(min

1≤i≤q

(v

∑dj=1 A

ij lj

Aij

+bi

Aij

))−αdv . (7.14)


+\{0} : s > 0} and (7.13) holds.

Complete network and equal loss proportions

As explained in chapter 3, the weighted adjacency matrix is given by Aij =

1

qfor all

i ∈ {1, ... , q} and j ∈ {1, ... , d} with q 6= 0 (see (3.2)). Thus, (7.14) leads tod∑

j=1

∫ ∞0

(min

1≤i≤q

(v

∑dj=1

1qlj

1q

+bi

1q︸︷︷︸

positive

))−αdv

=d∑

j=1

∫ ∞0

(v

d∑j=1

lj︸︷︷︸positive

+q min1≤i≤q

(bi))−α

dv

=d∑

j=1

[1

(−α + 1)∑d

j=1 lj

(v

d∑j=1

lj + q min1≤i≤q

(bi)) <0︷︸︸︷−α+1

]∞0

=d∑

j=1

−1

(−α + 1)∑d

j=1 lj

(q min

1≤i≤q

(bi))−α+1

65

7 | Applications

=d q−α+1

(α− 1)∑d

j=1 lj

(min

1≤i≤q

(bi))−α+1

. (7.15)


Distribution lj = cjE[W(1)]− E[Xj(1)] (table 7.1)

d q−α+1

(α− 1)∑d

j=1 lj

(min

1≤i≤q

(bi))−α+1


α

α− 1d q−α+1

(α− 1)∑d

j=1cjλ− Kj

αα−1

(min

1≤i≤q

(bi))−α+1

Burr lj =cjλ− Γ(α− 1

τ

)Γ(

1 +1

τ

) κ1τj

Γ(α)d q−τα+1

(τα− 1)∑d

j=1cjλ− Γ(α− 1

τ)Γ(1 + 1

τ)κ

1τj

Γ(α)

(min

1≤i≤q

(bi))−τα+1

Loggamma lj =cjλ−(

1− 1

α

)−βjd q−α+1

(α− 1)∑d

j=1cjλ−(

1− 1α

)−βj ( min1≤i≤q

(bi))−α+1


A bipartite network with 2 agents and d objects

If only 2 agents A1 and A2 are in the market, (7.14) leads to

d∑j=1

∫ ∞0

(min

(v

∑dj=1 A

1j lj

A1j

+b1

A1j

, v

∑dj=1 A

2j lj

A2j

+b2

A2j

))−αdv . (7.16)

The intersection point v is given by

v

∑dj=1 A

1j lj

A1j

+b1

A1j

= v

∑dj=1 A

2j lj

A2j

+b2

A2j

⇔ v

∑dj=1 A

1j lj

A1j

− v

∑dj=1 A

2j lj

A2j

=b2

A2j

− b1

A1j

⇔ v

(∑dj=1 A

1j lj

A1j

−∑d

j=1 A2j lj

A2j

)=

b2

A2j

− b1

A1j

⇔ v

(A2j

∑dj=1 A

1j lj − A1

j

∑dj=1 A

2j lj

A1j A

2j

)=

A1j b

2 − A2j b

1

A1j A

2j

⇔ v =A1j b

2 − A2j b

1

A2j

∑dj=1 A

1j lj − A1

j

∑dj=1 A

2j lj

.

66

7 | Applications

Ensuring that 0 ≤ v ≤ ∞, the maximum is taken such that

v = max

(A1j b

2 − A2j b

1

A2j

∑dj=1 A

1j lj − A1

j

∑dj=1 A

2j lj

, 0

). (7.17)

Consequently, for the minimum in (7.16) holds that

min

(v

∑dj=1 A

1j lj

A1j

+b1

A1j

, v

∑dj=1 A

2j lj

A2j

+b2

A2j

)

=

v

∑dj=1 A

1j lj

A1j

+b1

A1j

if v ≤ v = max

(A1j b

2 − A2j b

1

A2j

∑dj=1 A

1j lj − A1

j

∑dj=1 A

2j lj

, 0

)v

∑dj=1 A

2j lj

A2j

+b2

A2j

if v > v = max

(A1j b

2 − A2j b

1

A2j

∑dj=1 A

1j lj − A1

j

∑dj=1 A

2j lj

, 0

).

(7.18)

To compute the minimum, the integral is split according to (7.18) such that

d∑j=1

(∫ v

0

(v

∑dj=1 A

1j lj

A1j

+b1

A1j

)−αdv︸︷︷︸

(1)

+

∫ ∞v

(v

∑dj=1 A

2j lj

A2j

+b2

A2j

)−αdv︸︷︷︸

(2)

). (7.19)

Computation of the integral (1)

∫ v

0

(v

∑dj=1 A

1j lj

A1j

+b1

A1j

)−αdv

=

[A1j

(−α + 1)∑d

j=1 A1j lj

(v

∑dj=1 A

1j lj

A1j

+b1

A1j

)−α+1]v

0

=A1j

(−α + 1)∑d

j=1 A1j lj

(v

∑dj=1 A

1j lj

A1j

+b1

A1j

)−α+1

−A1j

(−α + 1)∑d

j=1 A1j lj

(b1

A1j

)−α+1

(7.20)


∫ ∞v

(v

∑dj=1 A

2j lj

A2j

+b2

A2j

)−αdv

=

[A2j

(−α + 1)∑d

j=1 A2j lj

(v

∑dj=1 A

2j lj

A2j

+b2

A2j

) <0︷︸︸︷−α+1

]∞v

=−A2

j

(−α + 1)∑d

j=1 A2j lj

(v

∑dj=1 A

2j lj

A2j

+b2

A2j

)−α+1

. (7.21)

67

7 | Applications

Inserting v (7.17) into (7.20) and (7.21), (7.19) leads to

d∑j=1

(A1j

(−α + 1)∑d

j=1 A1j lj

(max

(A1j b

2 − A2j b

1

A2j

∑dj=1 A

1j lj − A1

j

∑dj=1 A

2j lj

, 0

)∑dj=1 A

1j lj

A1j

+b1

A1j

)−α+1

−A1j

(−α + 1)∑d

j=1 A1j lj

(b1

A1j

)−α+1

+−A2

j

(−α + 1)∑d

j=1 A2j lj

(max

(A1j b

2 − A2j b

1

A2j

∑dj=1 A

1j lj − A1

j

∑dj=1 A

2j lj

, 0

)∑dj=1 A

2j lj

A2j

+b2

A2j

)−α+1)

.

This expression can only be simplified if further assumptions for the weighted adjacencymatrix A, the positive safety loading l and the normed initial capital b are made.

7.2.2 Capital transfer without transaction costs


In this insurance network model, the agents transfer capital among each other. The agents’capital is quoted in the same currency. This is illustrated by the blue ellipse in figure 7.6.An example could be that all agents are located in the European Union having the samecurrency Euro. The vector R denotes the reserves which state the capital of the agents ateach time point t ≥ 0. If an agent cannot cover its loss and had to declare insolvency, theother agents would shift capital (dashed arrows).

A1 A2 A3 A4 A5 . . . Aq

O1 O2 O3. . . Od

R

same currency market

Figure 7.6: A bipartite network with q agents and d objects with capital transfer and withouttransaction costs

As described in chapter 5 by (5.17), the ruin set F is given by

F ={y ∈ Rq : yT1 < 0

}where 1 = (1, ... , 1)T ∈ Rq. Let A ∈ Rq×d

+ , l ∈ Rd+\{0} and b ∈ Rq

+\{0}. Further, notethat Al can be written as in (7.11). Then, for v ∈ R+, it holds that

vAl + b −F = vAl + b −{y ∈ Rq : yT1 < 0

}68

7 | Applications

= vAl + b +{y ∈ Rq : yT1 > 0

}= vAl +

{y ∈ Rq : (y − b)T1 > 0

}= vAl +

{y ∈ Rq : yT1 > bT1︸︷︷︸

=( u||u|| )

T 1= ||u||||u||=1

}={y ∈ Rq : yT1 > v(Al)T1 + 1

}. (7.22)

For the right-hand side of (7.1), it follows for α > 1 that∫ ∞0

µ ◦ A−1

({y ∈ Rq

+\{0} : y ∈ vAl + b −F})

dv

=

∫ ∞0

µ ◦ A−1

({y ∈ Rq

+\{0} : yT1 > v(Al)T1 + 1

})dv (insert (7.22))

=

∫ ∞0

µ

({x ∈ Rd

+\{0} : (Ax)T1 > v(Al)T1 + 1

})dv

=

∫ ∞0

d∑j=1

µ

({sej ∈ Rd

+\{0} : s(Aej)T1 > v

q∑i=1

d∑j=1

Aij lj + 1

})dv (7.23)

=

∫ ∞0

d∑j=1

µ

({s ∈ R+\{0} : s(Aj)

T1 > v

q∑i=1

d∑j=1

Aij lj + 1

})dv

=

∫ ∞0

d∑j=1

µ

({s ∈ R+\{0} : s

q∑i=1

Aij > v

q∑i=1

d∑j=1

Aij lj + 1

})dv

=

∫ ∞0

d∑j=1

µ

({s ∈ R+\{0} : s >

v∑q

i=1

∑dj=1 A

ij lj + 1∑q

i=1 Aij

})dv

=

∫ ∞0

d∑j=1

µ

((v∑qi=1

∑dj=1 A

ij lj + 1∑q

i=1 Aij

){s ∈ R+\{0} : s > 1

})dv

=

∫ ∞0

d∑j=1

(v∑qi=1

∑dj=1 A

ij lj + 1∑q

i=1 Aij

)−αµ({

s ∈ R+\{0} : s > 1})

dv

= µ({

s ∈ R+\{0} : s > 1})

︸︷︷︸=1−α=1

d∑j=1

∫ ∞0

(v∑q

i=1

∑dj=1 A

ij lj + 1∑q

i=1 Aij

)−αdv

=d∑

j=1

[ ∑qi=1 A

ij

(−α + 1)∑q

i=1

∑dj=1 A

ij lj

(v∑q

i=1

∑dj=1 A

ij lj + 1∑q

i=1 Aij

) <0︷︸︸︷−α+1

]∞0

=d∑

j=1

−∑q

i=1 Aij

(−α + 1)∑q

i=1

∑dj=1 A

ij lj

(1∑q

i=1 Aij

)−α+1

=d∑

j=1

(∑qi=1 A

ij

)α(α− 1)

∑qi=1

∑dj=1 A

ij lj

. (7.24)


+\{0} : s > 0} and (7.23) holds.

69

7 | Applications

Complete network

If the d objects are completely insured by the q agents, then

q∑i=1

Aij = 1 ∀ j = 1, ... , d

and it follows for (7.24) that∫ ∞0

µ ◦ A−1({y ∈ Rq+\{0} : y ∈ vAl + b −F)dv =

d

(α− 1)∑d

j=1 lj. (7.25)

In the following table, the solution (7.25) is evaluated for different claim distributions X(Pareto, Burr, Loggamma).


d

(α− 1)∑d

j=1 lj


α

α− 1d

(α− 1)∑d

j=1cjλ− Kj

αα−1


τ)Γ(1 + 1

τ)κ

1τj

Γ(α)

d

(τα− 1)∑d


τ)Γ(1 + 1

τ)κ

1τj

Γ(α)


1− 1

α

)−βjd

(α− 1)∑d

j=1cjλ− (1− 1

α)−βj


Comparison of (7.15) with (7.25) in a complete network with equal proportions

The solution in (7.25) is more general than the solution in (7.15) as the loss proportionsof each agent can be different. For equal loss proportions i.e. Ai

j = 1q, the solution is also

given by (7.25). For this case, the solutions in (7.15) and (7.25) will be compared.

Solution in (7.15):d q−α+1

(α− 1)∑d

j=1 lj

(min

1≤i≤q

(bi))−α+1

Solution in (7.25):d

(α− 1)∑d

j=1 ljAssume that

min1≤i≤q

(bi)>

1

q, (7.26)

70

7 | Applications

then it holds that

1 =

q∑i=1

1

q<

q∑i=1

bi .

Since the sum of the normed initial capital equals

q∑i=1

bi(3.10)

=

q∑i=1

ui

||u||= 1,

this is a contradiction to (7.26). Thus, this implies that

min1≤i≤q

(bi)≤ 1

q

which leads toq min

1≤i≤q

(bi)

︸︷︷︸≤ 1

q

≤ q1

q= 1.

Since for α > 1, it holds that −α + 1 < 0 andd

(α− 1)∑d

j=1 lj≥ 0, it follows that

(q min

1≤i≤q

(bi))−α+1

≥ 1

q−α+1(

min1≤i≤q

(bi))−α+1

≥ 1

d q−α+1

(α− 1)∑d

j=1 lj

(min

1≤i≤q

(bi))−α+1

︸︷︷︸(1)

≥ d

(α− 1)∑d

j=1 lj︸︷︷︸(2)

.

The left-hand side (1) is the solution if no capital is transferred given in (7.15) and theright-hand side (2) is the solution with capital transfer given in (7.25). In the case of acomplete network with equal loss proportions, the asymptotic ruin probability with capitaltransfer (without transaction costs) is lower than the asymptotic ruin probability if no capitalis transferred.

7.2.3 Capital transfer with transaction costs


In this insurance network model, the agents transfer capital, but compared to the bipar-tite network in subsection 7.2.2, the capital transfer implies additional transaction costs(exchange rates, administrative fees). This setting refers to different currency marketswhich are illustrated in figure 7.7 by the five blue ellipses. The figure visualizes that theagents A1 and A2 as well as the agents A4 and A5 are located respectively in the samecurrency market. The currencies of A1 and A2 (for example Euro), A3 (for example JPY),A4 and A5 (for example USD) are different. The vector R denotes the reserves which statethe capital of the agents at each time point t ≥ 0. If an agent cannot cover its loss andhad to declare insolvency, the other agents would shift capital (dashed arrows).

71

7 | Applications

different currency markets

A1 A2 A3 A4 A5 . . . Aq

O1 O2 O3. . . Od

R

Figure 7.7: A bipartite network with q agents and d objects with capital transfer and transactioncosts

A bipartite network with 2 agents and d objects

As derived in (5.37), the ruin set can be represented by

F =⋃k 6=l

{y ∈ R2 : yT (π(kl)e l + ek) < 0

}for k , l ∈ {1, 2}.

Let A ∈ R2×d+ , l ∈ Rd

+\{0} and b ∈ R2+\{0}. Note that Al can be written as

Al =

((Al)1

(Al)2

)=

(A1

1 A12 ... A1

j ... A1d

A21 A2

2 ... A2j ... A2

d

)

l1l2...lj...ld

=

(∑dj=1 A

1j lj∑d

j=1 A2j lj

).

For v ∈ R+ and k , l ∈ {1, 2}, it holds that

vAl + b −F = vAl + b −⋃k 6=l

{y ∈ R2 : yT (π(kl)e l + ek) < 0

}= vAl + b︸︷︷︸

positive

+⋃k 6=l

{y ∈ R2 : yT (π(kl)e l + ek) > 0

}(cf. (5.38))

=⋃k 6=l

{y ∈ R2 :

(y − (vAl + b)

)T(π(kl)e l + ek) > 0

}=⋃k 6=l

{y ∈ R2 : yT (π(kl)e l + ek)− (vAl + b)T (π(kl)e l + ek) > 0

}=⋃k 6=l

{y ∈ R2 : yT (π(kl)e l + ek) > (vAl + b)T (π(kl)e l + ek)

}=⋃k 6=l

{y ∈ R2 : yT (π(kl)e l + ek) > v(Al)T (π(kl)e l + ek) + bT (π(kl)e l + ek)

}=⋃k 6=l

{y ∈ R2 : yT (π(kl)e l + ek) > v

(π(kl)(Al)l + (Al)k

)+ π(kl)bl + bk

}.

(7.27)

72

7 | Applications

For the right-hand side of (7.1), it follows for α > 1 that∫ ∞0

µ ◦ A−1({

y ∈ R2+\{0} : y ∈ vAl + b −F

})dv

=

∫ ∞0

µ ◦ A−1(⋃

k 6=l

{y ∈ R2

+\{0} : yT (π(kl)e l + ek) > v(π(kl)(Al)l + (Al)k

)+ π(kl)bl + bk

})dv

(insert (7.27))

=

∫ ∞0

µ(⋃

k 6=l

{x ∈ Rd

+\{0} : (Ax)T (π(kl)e l + ek) > v(π(kl)(Al)l + (Al)k

)+ π(kl)bl + bk

})dv

=

∫ ∞0

d∑j=1

µ(⋃

k 6=l

{sej ∈ Rd

+\{0} : s(Aej)T (π(kl)e l + ek) > v(π(kl)

d∑j=1

Alj lj +

d∑j=1

Akj lj)+ π(kl)bl + bk

})dv

(7.28)

=

∫ ∞0

d∑j=1

µ(⋃

k 6=l

{s ∈ R+\{0} : s(Aj)

T (π(kl)e l + ek) > v(π(kl)

d∑j=1

Alj lj +

d∑j=1


})dv

=

∫ ∞0

d∑j=1

µ(⋃

k 6=l

{s ∈ R+\{0} : s(π(kl)Al

j + Akj ) > v

(π(kl)

d∑j=1

Alj lj +

d∑j=1


})dv

=

∫ ∞0

d∑j=1

µ(⋃

k 6=l

{s ∈ R+\{0} : s >

v(π(kl)∑d

j=1 Alj lj +

∑dj=1 A

kj lj) + π(kl)bl + bk

π(kl)Alj + Ak

j

})dv

=

∫ ∞0

d∑j=1

µ(⋃

k 6=l

{s ∈ R+\{0} : s > v

π(kl)∑d

j=1 Alj lj +

∑dj=1 A

kj lj

π(kl)Alj + Ak

j

+π(kl)bl + bk

π(kl)Alj + Ak

j

})dv

=

∫ ∞0

d∑j=1

µ({

s ∈ R+\{0} : s > min1≤k,l≤q

k 6=l

(vπ(kl)

∑dj=1 A

lj lj +

∑dj=1 A

kj lj

π(kl)Alj + Ak

j

+π(kl)bl + bk

π(kl)Alj + Ak

j

)})dv

(µ is homogeneous)

=

∫ ∞0

d∑j=1

(min

1≤k,l≤qk 6=l

(vπ(kl)

∑dj=1 A

lj lj +

∑dj=1 A

kj lj

π(kl)Alj + Ak

j

+π(kl)bl + bk

π(kl)Alj + Ak

j

))−αdv

=d∑

j=1

∫ ∞0

(min

1≤k,l≤qk 6=l

(vπ(kl)

∑dj=1 A

lj lj +

∑dj=1 A

kj lj

π(kl)Alj + Ak

j

+π(kl)bl + bk

π(kl)Alj + Ak

j

))−αdv (7.29)


+\{0} : s > 0} and (7.28) holds. Inserting k , l ∈ {1, 2}, (7.29) leadsto

d∑j=1

∫ ∞0

(min

(vπ(12) ∑d

j=1 A2j lj+

∑dj=1 A

1j lj

π(12)A2j +A1

j+ π(12)b2+b1

π(12)A2j +A1

j, v

π(21) ∑dj=1 A

1j lj+

∑dj=1 A

2j lj

π(21)A1j +A2

j+ π(21)b1+b2

π(21)A1j +A2

j

))−αdv . (7.30)

The intersection point v is given by

vπ(12)

∑dj=1 A

2j lj +

∑dj=1 A

1j lj

π(12)A2j + A1

j

+π(12)b2 + b1

π(12)A2j + A1

j

= vπ(21)

∑dj=1 A

1j lj +

∑dj=1 A

2j lj

π(21)A1j + A2

j

+π(21)b1 + b2

π(21)A1j + A2

j

⇔ vπ(12)

∑dj=1 A

2j lj +

∑dj=1 A

1j lj

π(12)A2j + A1

j

− vπ(21)

∑dj=1 A

1j lj +

∑dj=1 A

2j lj

π(21)A1j + A2

j

=π(21)b1 + b2

π(21)A1j + A2

j

− π(12)b2 + b1

π(12)A2j + A1

j

⇔ v

(π(12)

∑dj=1 A

2j lj +

∑dj=1 A

1j lj

π(12)A2j + A1

j

−π(21)

∑dj=1 A

1j lj +

∑dj=1 A

2j lj

π(21)A1j + A2

j

)=π(21)b1 + b2

π(21)A1j + A2

j

− π(12)b2 + b1

π(12)A2j + A1

j

⇔ v

((π(12)

∑dj=1 A

2j lj +

∑dj=1 A

1j lj)(π

(21)A1j + A2

j )− (π(21)∑d

j=1 A1j lj +

∑dj=1 A

2j lj)(π

(12)A2j + A1

j )

(π(12)A2j + A1

j )(π(21)A1

j + A2j )

)

=(π(21)b1 + b2)(π(12)A2

j + A1j )− (π(12)b2 + b1)(π(21)A1

j + A2j )

(π(21)A1j + A2

j )(π(12)A2

j + A1j )

73

7 | Applications

⇔ v =(π(21)b1 + b2)(π(12)A2

j + A1j )− (π(12)b2 + b1)(π(21)A1

j + A2j )

(π(12)∑d

j=1 A2j lj +

∑dj=1 A

1j lj)(π

(21)A1j + A2

j )− (π(21)∑d

j=1 A1j lj +

∑dj=1 A

2j lj)(π

(12)A2j + A1

j )

⇔ v =π(21)π(12)b1A2

j + b2A1j − π(12)π(21)b2A1

j − b1A2j

π(12)π(21)A1j

∑dj=1 A

2j lj + A2

j

∑dj=1 A

1j lj − π(21)π(12)A2

j

∑dj=1 A

1j lj − A1

j

∑dj=1 A

2j lj

.

Ensuring that 0 ≤ v ≤ ∞, the maximum is taken such that

v = max

(π(21)π(12)b1A2

j +b2A1j −π

(12)π(21)b2A1j −b

1A2j

π(12)π(21)A1j

∑dj=1 A

2j lj+A2

j

∑dj=1 A

1j lj−π(21)π(12)A2

j

∑dj=1 A

1j lj−A

1j

∑dj=1 A

2j lj

, 0

). (7.31)

Consequently, for the minimum in (7.30) holds that

min

(v

π(12)∑d

j=1 A2j lj+

∑dj=1 A

1j lj

π(12)A2j +A1

j+ π(12)b2+b1

π(12)A2j +A1

j, v

π(21)∑d

j=1 A1j lj+

∑dj=1 A

2j lj

π(21)A1j +A2

j+ π(21)b1+b2

π(21)A1j +A2

j

) v

π(12)∑d

j=1 A2j lj+

∑dj=1 A

1j lj

π(12)A2j +A1

j+ π(12)b2+b1

π(12)A2j +A1

jif v ≤ v

vπ(21)

∑dj=1 A

1j lj+

∑dj=1 A

2j lj

π(21)A1j +A2

j+ π(21)b1+b2

π(21)A1j +A2

jif v > v .

(7.32)

To compute the minimum, the integral is split according to (7.32) such that∫ v

0

(vπ(12)

∑dj=1 A

2j lj+

∑dj=1 A

1j lj

π(12)A2j +A1

j+ π(12)b2+b1

π(12)A2j +A1

j

)−αdv︸︷︷︸

(1)

+

∫ ∞v

(vπ(21)

∑dj=1 A

1j lj+

∑dj=1 A

2j lj

π(21)A1j +A2

j+ π(21)b1+b2

π(21)A1j +A2

j

)−αdv .︸︷︷︸

(2)


∫ v

0

(v

π(12)∑d

j=1 A2j lj+

∑dj=1 A

1j lj

π(12)A2j +A1

j+ π(12)b2+b1

π(12)A2j +A1

j

)−αdv

=

[1

−α+1

π(12)A2j +A1

j

π(12)∑d

j=1 A2j lj+

∑dj=1 A

1j lj

(v

π(12)∑d

j=1 A2j lj+

∑dj=1 A

1j lj

π(12)A2j +A1

j+ π(12)b2+b1

π(12)A2j +A1

j

)−α+1]v

0

= 1−α+1

π(12)A2j +A1

j

π(12)∑d

j=1 A2j lj+

∑dj=1 A

1j lj

(v

π(12)∑d

j=1 A2j lj+

∑dj=1 A

1j lj

π(12)A2j +A1

j+ π(12)b2+b1

π(12)A2j +A1

j

)−α+1

− 1−α+1

π(12)A2j +A1

j

π(12)∑d

j=1 A2j lj+

∑dj=1 A

1j lj

(π(12)b2+b1

π(12)A2j +A1

j

)−α+1

(7.33)


∫ ∞v

(v

π(21)∑d

j=1 A1j lj+

∑dj=1 A

2j lj

π(21)A1j +A2

j+ π(21)b1+b2

π(21)A1j +A2

j

)−αdv

=

[1

−α+1

π(21)A1j +A2

j

π(21)∑d

j=1 A1j lj+

∑dj=1 A

2j lj

(v

π(21)∑d

j=1 A1j lj+

∑dj=1 A

2j lj

π(21)A1j +A2

j+ π(21)b1+b2

π(21)A1j +A2

j

) <0︷︸︸︷−α+1

]∞v

= −1−α+1

π(21)A1j +A2

j

π(21)∑d

j=1 A1j lj+

∑dj=1 A

2j lj

(v

π(21)∑d

j=1 A1j lj+

∑dj=1 A

2j lj

π(21)A1j +A2

j+ π(21)b1+b2

π(21)A1j +A2

j

)−α+1

(7.34)

74

7 | Applications

By (7.33) and (7.34), (7.30) leads with v (7.31) to

d∑j=1

(1

−α+1

π(12)A2j +A1

j

π(12)∑d

j=1 A2j lj+

∑dj=1 A

1j lj

(v

π(12)∑d

j=1 A2j lj+

∑dj=1 A

1j lj

π(12)A2j +A1

j+ π(12)b2+b1

π(12)A2j +A1

j

)−α+1

− 1−α+1

π(12)A2j +A1

j

π(12)∑d

j=1 A2j lj+

∑dj=1 A

1j lj

(π(12)b2+b1

π(12)A2j +A1

j

)−α+1

+ −1−α+1

π(21)A1j +A2

j

π(21)∑d

j=1 A1j lj+

∑dj=1 A

2j lj

(v

π(21)∑d

j=1 A1j lj+

∑dj=1 A

2j lj

π(21)A1j +A2

j+ π(21)b1+b2

π(21)A1j +A2

j

)−α+1)

(7.35)

Computation for π(12) = π(21) = 1

For v in (7.31), it follows that

v = max

(b1A2

j +b2A1j −b

2A1j −b

1A2j

A1j

∑dj=1 A

2j lj+A2

j

∑dj=1 A

1j lj−A

2j

∑dj=1 A

1j lj−A

1j

∑dj=1 A

2j lj

, 0

)= 0.

Inserting v = 0 in (7.35), it leads to

d∑j=1

1

α− 1

A1j + A2

j∑dj=1 A

1j lj +

∑dj=1 A

2j lj

(b1 + b2

A1j + A2

j

)−α+1

. (7.36)

Since b1 + b2 = u1

||u|| + u2

||u|| = u1+u2

||u|| = 1, it holds for (7.36) that

d∑j=1

1

α− 1

(A1j + A2

j )α∑dj=1 A

1j lj +

∑dj=1 A

2j lj

. (7.37)

Comparing (7.37) with (7.24) for q = 2, the results equal.

Complete network and equal loss proportions

As explained in chapter 3, the agents A1 and A2 are linked to all objects and vice versasuch that 1{1 ∼ j} = 1 and 1{2 ∼ j} = 1 for all j ∈ {1, ... , d} (cf. (3.2)). The weightedadjacency matrix is given by A1

j = A2j = 1

2for all j ∈ {1, ... , d}. Thus, (7.29) leads to

d∑j=1

∫ ∞0

(min

1≤k,l≤2k 6=l

vπ(kl)

∑dj=1

12lj +

∑dj=1

12lj

π(kl) 12

+ 12

+π(kl)bl + bk

π(kl) 12

+ 12

)−αdv

=d∑

j=1

∫ ∞0

(min

1≤k,l≤2k 6=l

v(π(kl) 1

2+ 1

2)∑d

j=1 lj

π(kl) 12

+ 12

+π(kl)bl + bk

π(kl) 12

+ 12

)−αdv

=d∑

j=1

∫ ∞0

(min

1≤k,l≤2k 6=l

vd∑

j=1

lj + 2π(kl)bl + bk

π(kl) + 1︸︷︷︸positive

)−αdv

=d∑

j=1

∫ ∞0

(v

d∑j=1

lj︸︷︷︸positive

+2 min1≤k,l≤2

k 6=l

(π(kl)bl + bk

π(kl) + 1

))−αdv

75

7 | Applications

=d∑

j=1

[1

(−α + 1)∑d

j=1 lj

(v

d∑j=1

lj + 2 min1≤k,l≤2

k 6=l

(π(kl)bl + bk

π(kl) + 1

)) <0︷︸︸︷−α+1

]∞0

=d∑

j=1

−1

(−α + 1)∑d

j=1 lj

(2 min

1≤k,l≤2k 6=l

(π(kl)bl + bk

π(kl) + 1

))−α+1

=d 2−α+1

(α− 1)∑d

j=1 lj

(min

1≤k,l≤2k 6=l

(π(kl)bl + bk

π(kl) + 1

))−α+1

. (7.38)



d 2−α+1

(α− 1)∑d

j=1 lj

(min

1≤k,l≤2k 6=l

(π(kl)bl + bk

π(kl) + 1

))−α+1


α

α− 1d 2−α+1

(α− 1)∑d

j=1cjλ− Kj

αα−1

(min

1≤k,l≤2k 6=l

(π(kl)bl + bk

π(kl) + 1

))−α+1


τ

)Γ(

1 +1

τ

) κ1τj

Γ(α)d 2−τα+1

(τα− 1)∑d


τ)Γ(1 + 1

τ)κ

1τj

Γ(α)

(min

1≤k,l≤2k 6=l

(π(kl)bl + bk

π(kl) + 1

))−τα+1


1− 1

α

)−βjd 2−α+1

(α− 1)∑d

j=1cjλ− (1− 1

α)−βj

(min

1≤k,l≤2k 6=l

(π(kl)bl + bk

π(kl) + 1

))−α+1


76

8 | Conclusion

In this master thesis, a bipartite insurance network model with capital transfer was intro-duced. The objective was to compute the asymptotic ruin probability by considering thefollowing three cases:



3) Capital transfer with transaction costsAfter introducing the insurance network model, the capital transfer matrix was defined forthe second and third case. Based on the paper by Hult and Lindskog (2006), ruin occurswhen the corresponding risk reserve first hits the ruin set. For all three cases, the ruinsets and solvency cones were determined. Additional figures should facilitate the reader theunderstanding of the contents. The key question was:

How likely is insolvency if insurances transfer capital?

In all three cases, closed form solutions for the asymptotic ruin probability could be derived.The solutions depend on the network structure given by the weighted adjacency matrix,the (normed) initial capital, the positive safety loading, the distribution of the claims andin third case also on the entries of the capital transfer matrix. Under the assumption ofa complete network and equal loss proportions, the asymptotic ruin probability in the firstcase is higher than in the second case. This is realistic since capital transfer can reduce theruin probability.

The formula for the asymptotic ruin probability is a theoretically modeling due to the con-sideration of some conditions and assumptions. However, trying to be less idealistic leadsto more complex computations. An idealistic condition assumed in chapter 3 is the simul-taneous jump of all insurance portfolios. Compared to the real world, it would be moreadvantageous if the claims of each insurance portfolio incur at different times. Moreover,insurance companies typically do not only have portfolios which are regularly varying, butinstead they also have portfolios distributed with lighter tails. Furthermore, in case of aninsolvency of one agent, helping agents are allowed to transfer capital up to an amount oftheir complete risk capital. It is not assumed that the agents need to comply certain bufferssuch that their liability is unlimited.

Based on the results developed in chapter 6, it is possible to continue researching. Since theapplications in chapter 7 are based on the Theorem 6.0.7 for a fixed weighted adjacency ma-trix, applications using the Proposition 6.0.9 with respect to a random weighted adjacencymatrix can be investigated. For the capital transfer, further restrictions and regulations canbe defined such that only a fraction of the positive capital may be transferred among theagents. The main Theorem 6.0.7 can be used to compute the buffer capital which is theminimum capital required to reduce the ruin probability.

By means of this master thesis, the risk of insolvency for an insurance company in a networkcan be assessed. This risk assessment is important to allocate appropriate reserve capitalto the insurance portfolios. As a consequence, insolvencies of groups or insurance pools forcatastrophic risks can be reduced in the future.

77

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O. Kley, C. Klüppelberg, and G. Reinert. Systemic risk in a large claims insurance marketwith bipartite graph structure. Operations Research, 64(5):1159–1176, 2016.

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