Allocation of Risk Capital to Contracts in Catastrophe Reinsurance

(1)

Allocation of Risk Capital to Contracts in Catastrophe Reinsurance

R A S M U S H A N S É N

Master of Science Thesis Stockholm, Sweden 2013

(2)

(3)

Allocation of Risk Capital to Contracts in Catastrophe Reinsurance

R A S M U S H A N S É N

Master’s Thesis in Mathematical Statistics (30 ECTS credits) Master Programme in Mathematics (120 credits) Royal Institute of Technology year 2013

Supervisor at KTH was Filip Lindskog Examiner was Filip Lindskog

TRITA-MAT-E 2013:48 ISRN-KTH/MAT/E--13/48-SE

Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

(4)

(5)

Abstract

This thesis is the result of a project aimed at developing a tool for allocation of risk capital in catastrophe excess-of-loss reinsurance. Allocation of risk capital is an important tool for measuring portfolio performance and opti- mizing the capital requirement. Here, two allocation rules are described and analyzed, Euler allocation and Capital layer allocation. The rules are applied to two different portfolios. The main conclusions is that the two methods can be used together to get a better picture of how the dependence structure between the contracts affect the portfolio result. It is also illustrated how the RORAC of one of the portfolios can be increased by 1 % using the outcome from the analyses.

Keywords: Reinsurance, Allocation of risk capital, RORAC, Euler allocation, Capital layer allocation

(6)

(7)

Allokering av riskkapital

mellan katastrofåterförsäkringskontrakt

Sammanfattning

Den här uppsatsen är resultatet av ett projekt som syftat till att utveckla en mjukvara för allokering av riskkapital inom icke-proportionell katastrofåterförsäkring. Allokering av riskkapital är ett viktigt verktyg för att mäta hur väl portföljen presterar samt en grund för portföljoptimering. I den här uppsatsen un- dersöks två metoder för att allokera riskkapital, Euler-allokering och Capital layer allokering. Metoderna appliceras på två olika portföljer. Slutsatserna är att de två metoderna kan användas tillsammans för att ge en helhetsbild av beroendet mellan kon- trakten centralt i fördelningsfunktionen samt ute i svansen. Det visas också hur portföljens RORAC kan höjas med 1 % genom att använda resultaten i uppsatsen.

(8)

(9)

Acknowledgements

I would like to thank everyone at the Reinsurance department of Läns- försäkringar AB for letting me get access to data and for valuable support along the way.

Stockholm, September 2013 Rasmus Hansén

(10)

(11)

Introduction

1.1 Background

A reinsurance firm works the same way as an insurance firm but instead of selling insurance to e.g. a private person or a company the reinsurance company sells insurance to other insurance companies. Reinsurance is thus a vehicle for spreading the insurance risk between different actors in the insurance industry. One important reason for the existence of reinsurance companies is the management of extreme events. An insurance company gathers risks from different policyholders and gains from diversification ben- efits. But the policies are not completely independent since natural hazards and alike may affect many policies at the same time. With the aid of the reinsurance industry the outcomes of these events can be shared between different actors.

Both insurance and reinsurance companies must hold capital in order to guarantee that they can fulfill their policies. To protect the customers, minimum levels of the capital stock for insurance and reinsurance companies are determined through regulation. The European Union is about to imple- ment new rules and guidelines for capital management called Solvency-II. In short, Solvency-II states that the capital requirement should equal the 99.5

%-percentile of the deviation from the expected outcome during a one year term. ¹

1In simple terms, if the deterministic premium income of the insurance company is P and the expected annual loss is E[S], then the capital requirement for the firm is VaR0.995((P − E[S]) − S) since P − E[S] is the expected annual outcome. In reality these calculations becomes more complex since other risks than the underwriting risk, e.q. market risks and counterparty risks must be included in the calculations. Also the insurance portfolio may be divided in many different lines-of-businesses with some inter- dependence.

(14)

Locking up capital inflicts capital costs on the company. Thus it is relevant to share these costs between the divisions and individual policies of the company in a fair manner. Dividing the costs between the business units may give the management board insight in how the different parts contributes to the total capital need of the company and ultimately how to optimize the portfolio. For the division of the capital requirement, capital allocation can be used.

The analysis of catastrophe reinsurance agreements and portfolios has long been somewhat of a mix between statistical analysis and craft work because of the lack of historical data - extreme events occur seldom per se. However, lately a number of vendors have developed tools for catastrophe modeling.

These tools are used to simulate catastrophes as a complement to historical experience. Therefore, it is relevant to investigate how these tools can be used to aid the capital and portfolio management of catastrophe reinsurance.

1.2 Purpose

The work behind this thesis has aimed at developing a software tool that can be used to allocate capital to individual contracts in a catastrophe reinsurance portfolio based on a specific catastrophe modeling tool. This thesis describes the allocation methods used in the software as well as an analysis performed on two different catastrophe reinsurance portfolios for which modeling data were available.

1.3 Outline

The outline of the thesis is as follows. First, the principles of reinsurance and especially non-proportional catastrophe reinsurance are explained together with some actuarial methods. Second, the main problem behind capital allocation is presented as well as two different methods to solve it in a general setting. Third, the loss simulation process based on the modeling tools is briefly described. Finally, the results and conclusions are presented.

(15)

Chapter 2

Reinsurance

In this chapter the fundamentals of reinsurance and in particular non-proportional catastrophe reinsurance is described in the first two sections. The third and last section described the collective model which is an intuitive way of modeling insurance portfolios. The collective model is used in the analysis part of this thesis.

2.1 Classifications of reinsurance contracts

Reinsurance is a vehicle used to spread the risk between different companies.

The agreement is entered between a reinsurance firm that accepts to cover some of the risks of a regular insurance company, called the cedent. The reasons for an insurance company to reinsure itself can be to increase its underwriting capacity for single risks and thus be able to take on risks which it would not otherwise be able to accept, to smooth the result between different time periods or to decrease its level of risk capital needed for catastrophic events. Reinsurance agreements are categorized in roughy three dimensions.

First, the contracts can be on treaty or facultative form where a treaty covers all risks in a predefined portfolio, e.g. the property portfolio, and a facultative contract covers only one risk. In the second dimension, reinsurance contracts are characterized as either proportional or non-proportional. In a proportional contract the reinsurer accepts to cover all defined risks (wether it is a treaty or a facultative contract) proportionally with perhaps an upper limit. In a non-proportional agreement the reinsurer covers only losses in a specified interval of the incurred loss to the insurer. Third, the contracts can be categorized as either per risk or per event (catastrophe). A per risk contract covers individual losses to risks whereas a per event contract covers the accumulation of losses from many risks affected by the same event, e.g.

a severe storm or earthquake. Since this thesis is limited to non-proportional

(16)

catastrophe reinsurance, this type of contract is presented more elaborated below. For a more thorough discussion of the other forms of reinsurance see e.g. Gustafsson (2000).

2.2 Non-proportional catastrophe reinsurance

In a non-proportional catastrophe reinsurance agreement the reinsurer accepts to pay a share λ of the amount of all catastrophe losses affecting a layer l xs r, where l is the limit or the maximum amount that will accrue to the layer and r is the retention which the cedent covers. Usually many layers are placed on top of each other to better spread the risk. Figure 2.2 illustrates illustrates a reinsurance structure in which the cedent keeps all accumulated event losses that does not exceed r₁. For losses exceeding r₁ Reinsurer A and Reinsurer B shares the loss amount in the interval [r₁, r1 + l1]. Rein- surer A takes a portion λ_A = 0.66 of these losses and Reinsurer B takes λ_B = 0.33. The top layer covers losses exceeding r₂ = r₁+ l₁ and is limited by l₂, thus covering losses in the interval [r₁+ l1, r1+ l1+ l2]. Now, a mathematical representation of the reinsurer’s loss is derived. This derivation is based mostly on the authors experience from working with reinsurance. For a further presentation see e.g. Antal (2003).

Let X be the from-ground-up loss amount, i.e. the loss to the cedent’s reinsured portfolio.¹ The loss to a reinsurance company with share λ in a layer l xs r is

Y = λ ·







0 ifX < r

X − r ifr < X < r + l l ifr + l < X or more compactly

Y = λ(min(X, r + l) − min(X, r)). (2.1)

The premium for a non-proportional reinsurance agreement is often stated as a share of the limit. This ratios is defined as rate-on-line (ROL). Thus, a layer with limit l paying premium P has ROL = P/l. Or if the ROL is given the premium for the agreement is P = ROL · l

1The from-ground-up loss amount is net of other reinsurance. For example it is common that the cedent has a proportional agreement covering the same portfolio as the non- proportional catastrophe agreement. In this case the from-ground-up is the total loss amount to the cedent’s portfolio reduced by recoveries from the proportional protection.

(17)

As a recovery is being made under the reinsurance agreement the cover is exhausted. Given the setting above, assume that the cedent incur a loss X = r + x. Then the cedent recovers min(x, l) under the agreement and the limit is reduced by the same amount. This means that the next time the cedent incurs a loss it only has reinsured coverage for l − min(x, l) xs r.

To reinstate the coverage a non-proportional reinsurance agreement often in- cludes a reinstatement clause. The reinstatement clause specifies how many times the coverage can be reinstated and what the cedent must pay for such extra coverage. The additional payment is often expressed as a share of the original premium in proportion to the amount recovered to the limit.

To see how a reinstatement clause affects the aggregated loss S to a reinsurer participating in a cover as the one above with rate-on-line ROL, assume that the coverage has k reinstatements for an additional premium of q/100

%. Assume further that the cedent incurs N loss events with the from- ground-up loss amounts X_i, i = 1, 2, . . . N . Then the aggregate loss with reinstatements but without additional reinstatement premium is

S⁰ = min

N

X

i=1

λ(min(Xi, r + l) − min(Xi, r)), (k + 1)l

!

. (2.2)

Equation (2.2) simply states that the aggregate loss amount is limited by (k + 1)l which makes sense since the agreement has 1 original coverage and k reinstatements. The aggregated loss, S, when a reinstatement premium is being paid is

S = S⁰− min S⁰ l , k

· q · ROL (2.3)

where the second term represents the reinstatement premium for min

S⁰ l , k

reinstatements.

It is possible to derive distributions of the layer loss Y in equation (2.1) given the distribution of X by using mixture principles, see e.g. Johansson (2013, pp. 2-3). The distribution of S is harder to derive from X. Thus numerical methods must be applied. The next section aims at presenting a common method of modeling insurance losses and some properties which are used later on.

(18)

2.3 The Collective Model

The collective model is a common and intuitive way of modeling insurance portfolios, see Johansson (2013, chap. 5). Let N ≥ 0 be an integer valued random variable representing the total number of claims and let X_i ≥ 0, i = 1, 2, . . . be the random claim amounts for each claim. Then the aggregate claim amount S can be written as

S =

N

X

i=1

X_i. (2.4)

If N = 0 the summation in (2.4) is defined to be 0. Usually, N is assumed to follow a distribution from the Panjer class². In this thesis we will exclusively be working with the Poisson distribution and the following discussion will thus be limited to this case, even though similar results holds for the other distributions in the Panjer class as well.

When N is assumed to be P o(λ)−distributed and X_i, i = 1, 2, ... are iid with cdf F (x) and independent of N , S in (2.4) is said to follow a compound poisson process, here denoted by CoP o(λ, F ). To derive some properties of the compound poisson distribution it is a good idea to work with generating functions. The moment generating function, M_S(t), of S is given by

M_S(t)= Eh Eh

e^P^Nⁱ⁼¹^Xⁱ|Nii

= E

"_N Y

i=1

Ee^X¹

#

= EM_X(t)^N

= PN(MX(t)) (2.5)

where P_N(t) is the probability generating function of N and MX(t) is the moment generating function of the iid X_i-variables. In the case of N being P o(λ)-distributed, P_N(t) = exp(λ(t − 1)) and equation (2.5) is

M_S(t) = expλ(M_X(t) − 1) . (2.6) Intuitively, if you add two independent compound poisson processes you expect to get a new compound poisson process with a higher intensity. This is

2The Panjer class consists of the poisson, negative binomial, logarithmic and binomial distributions. These distributions share the following recursive relation for the probabilities pn= P (N = n): pn= (a + b/n)pn−1for some a and b such that a + b ≥ 0.

(19)

exactly what the next proposition shows.

Theorem 2.1. Let S₁ ∼ CoP o(λ₁, F1) and S2 ∼ CoP o(λ₂, F2). Define S = S₁ + S₂, then S ∼ CoP o(λ, F ) where λ = λ₁+ λ₂ and F is the cdf of the mixture variable

X =

(X₁ with probability^λ_λ¹ X₂ with probability^λ_λ² where X_i has cdf F_i.

Intuitively the stated theorem makes sense. If you add two compound poisson processes you expect the intensity of events to be the sum of the two intensities. Given that an event occurs, it must be from either the first or the second process. The probability that it is from the first process is simply the relative intensity of the first process compared with the combined process.

A formal proof can be completed using the theory of generating functions, under the assumption that they exist.

Proof. Moment generating function of independent random variables M_X+Y(t) = MX(t)MY(t) Moment generating function of compound poisson MS(t) = P_N(M_X(t))

Thus

MS(t) = PN1(MX1(t))PN2(MX2(t))

= exp{λ₁(M_X₁(t) − 1)} exp{λ₂(M_X₂(t) − 1)}

= expn

(λ₁+ λ₂)

λ₁

λ₁+ λ₂M_X₁(t) + λ₂

λ₁+ λ₂M_X₂(t) − 1

o

This is recognized as the moment generating function of a compound poisson process with intensity λ₁+ λ₂ and with event outcomes from a distribution with the moment generating function

M (t) =

λ₁ λ1+ λ2

M_X₁(t) + λ₂ λ1+ λ2

M_X₂(t)

.

To complete the proof we need to show that this moment generating function corresponds to the random variable X in theorem 2.1. But this follows simply from

(20)

M_X(t) = Ee^Xt

= Ee^Xt|X = X₁ P (X = X₁) + Ee^Xt|X = X₂ P (X = X₂)

= Ee^X¹^t P (X = X₁) + Ee^X²^t P (X = X₂)

= M (t)

The following corollary shows that we can add and subtract processes that do not result in any losses without altering the initial process. This is not completely intuitive, what if we add such a process with an extremely high intensity. Intuitively the compound process will take the value zero very often. But this is compensated for by the increased intensity of the combined process.

Corollary 2.2. Let S₁ ∼ CoP o(λ₁, F1) and S2 ∼ CoP o(λ₂, F2) where F2

is the cdf of a degenerate random variable in 0. Define S = S₁+ S₂, then S= S^d ₁.

Proof. From theorem 2.1 we know that S ∼ CoP o(λ, F ) where λ = λ₁+ λ2

and F is the distribution function of

X =

(X₁ with probability^λ_λ¹ 0 with probability^λ_λ² . The moment generating function of X is

MX(t) = Ee^Xt = E e^X¹^t P (X = X₁) + Ee^0t P (X = X₂) =

= MX1(t)λ1

λ +λ2

λ. Thus,

MS(t) = exp

λ λ₁

λMX1(t) +λ2

λ − 1

= exp{λ1MX1(t) + λ2− λ₁− λ₂}

= exp{λ₁(M_X₁(t) − 1)}

= M_S₁(t)

(21)

2.4 Pricing and Loading

Pricing of reinsurance contracts follows the same methodology as pricing of most insurance policies. Let S be a random variable representing the annual loss of the contract. By the methods in the previous sections it is possible to derive the distribution of S. The pure premium is defined as E[S]. Thus, the pure premium can be thought of as the fair price of the contract. However, the realized outcome of S may deviate from E[S] due to the variance of the underlying distribution as well as parameter uncertainties. This risk is priced by adding a loading to the pure premium. Let h be the loading, then the price of the agreement is

P = E[S] + h. (2.7)

(22)

Figure 2.1: A simple non-proportional reinsurance structure. The cedent keeps all accumulated losses not exceeding r₁. Thereafter Reinsurer A and B shares the losses in the layer l₁ xs r₁, i.e. they cover the loss amount in the interval [r₁, r1+ l1]. Reinsurer A, B and C then shares all loss amounts in the layer l₂ xs r₂ = r₁+ l₁, i.e. they covers the loss amount in the interval [r1+ l1, r1+ l1+ l2].

(23)

Chapter 3

Capital Allocation Rules

This chapter summarizes the field of capital allocation from a mathematical point of view. More precisely, two methods are presented which highlight different aspects of the allocation problem and the purpose to conduct it.

The first two sections present risk measures, the allocation problem and some measures that can be used to benchmark the portfolio. The following two sections present two solutions to the allocation problem. The first is the Euler allocation rule which is probably the most common in the literature.

The second is the Capital layer allocation rule which is not as elaborated in the literature but it has an intuitive background which makes it interesting.

Finally, the last section presents some important aspects of and differences between the two allocation rules. Especially the let section discusses negative allocations and the allocations under different dependence structures between the contracts.

3.1 Risk measures and Allocation rules

Formally a risk measure ρ is a function from the set of random variables to the real line. If the random variable X represents the change in discounted wealth during a predefined time interval, then ρ(X) may be interpreted as the amount of capital that must be held to compensate for the risk associated with having a negative wealth development. In this theses the discount factor is assumed to be one for simplicity.

As was mentioned in the introduction, the new Solvency-II regime requires companies to hold surplus capital equal to the 99.5 %-percentile of the negative deviation from the expected annual outcome, i.e. VaR_0.995(X), where X is the annual result. Therefore, this thesis will work with value-at-risk at one year as the risk measure. As will be shown later in this chapter, there

(24)

is no easy way to carry out the allocation based on sample data when using VaR as the risk measure. Therefore, expected shortfall (ESα) will be considered parallel to VaR in this theory chapter and the analysis in the following chapter will be based solely on ES.

The two mentioned risk measures appears to be defined somewhat differently in respect to the parameter α in different sources. This thesis follows the definitions of Rau-Bredow (2003, equations 4 and 6) which are stated below¹: Definition 3.1. Let X be a random variable such that E[X] < ∞. Then the value-at-risk at level α (VaR_α) and the expected shortfall at level α (ES_α) are defined as

VaRα(X)=inf{q|P (−X ≤ q) ≥ α} (3.1a)

ES_α(X)=E−X1_{X≤−VaR_α_(X)} 1 − α

−VaRα(X) · (α − P (X > −VaRα(X)))

1 − α . (3.1b)

Observe that when X represents the positive change in future wealth, VaR_α is the α-quantile of the loss L = −X. In this sense, VaR_α(X) is the upper bound of the loss with a probability greater than or equal to α. Thus, it may be interpreted as the minimum amount of capital that must be held to have a positive result with a probability greater than or equal to α.²

The definition of ES_α, equation (3.1b), is used since it is valid even when X is discrete. Since the analysis of this thesis is based on simulated data it will deal with discrete empirical distributions. In the continuous case, however, the second term in equation (3.1b) vanishes since P (X > −VaR_α(X) = P (L < VaRα(X)) = P (L ≤ VaRα(X)) = α. Equation (3.1b) is thus reduced to

1The definitions are equivalent to the definitions used by Tasche (2008, equations (3.6) and (3.10b)) in the continuous case.

2Another definition of VaR is given by Hult et al. (2012, section 6.2). They define VaRα(X) as VaRα(X) = inf{q|P (qR0+ X < 0) ≤ α} where R0 is the risk-free rate. If X is continuous and L is defined as the discounted loss L = −_R^X

0 this definition can be reformulated as VaRα(X) = inf{q|P (L ≤ q) ≥ 1 − α}. Thus VaRα is interpreted as the (1−α)-quantile of the discounted loss. This definition is however equivalent to (3.1a) since it is just a matter of how the parameter α and the change in wealth X are defined. If X in (3.1a) is defined as the discounted loss then VaRα in (3.1a) corresponds to VaR1−α in the alternative definition by Hult et al. (2012).

(25)

ESα(X)=E−X1_{X≤−VaR_α_(X)}

1 − α = E [−X|X ≤ −VaRα(X)] =

=E [L|L ≥ VaRα(X)]

ES_αis thus interpreted as the expected loss given that the loss exceeds VaR_α with a correction term in the discrete setting. Thus, one important difference between VaR_α and ES_α is that the former only captures the behavior of the cdf in a single point whereas the later captures the whole tail.

Now that the connection between risk measures and risk capital is explained we are in a position to define the allocation problem. Let X₁, X₂, . . . , X_nbe random variables representing the future profits of n sub-portfolios/contracts constituting a portfolio with profit X =P X_i. Further let the risk capital of the entire portfolio be ρ(X) = ρ. Then, following van Gulick et al. (2012, Definition 2.1), the allocation problem is the problem of allocating the capital in a fair manner between the contracts and can be represented by the pair ({X₁, X₂, . . . , X_n}, ρ). Next we formally define the concepts capital allocation and capital allocation rule.

Definition 3.2. (i) The capital allocation corresponding to an allocation problem ({X₁, X₂, . . . , X_n}, ρ) is a vector a ∈ R^n×1 such that

n

X

i=1

ai = ρ

n

X

i=1

Xi

!

. (3.2)

(ii) Given a set of allocation problems R a capital allocation rule for R⁰ ⊆ R is a function A : R⁰ → R^n×1 such that A(R) is a capital allocation for any

R ∈ R⁰.

The two definitions may seem identical but they differ in the sense that (ii) defines a rule that guarantees that it results in a capital allocation for all allocation problems ({X₁, X₂, . . . , X_n}, ρ) in a certain set. For example we will later derive an allocation rule that is only valid for the set of problems with continuously differentiable risk measures ρ(X). The property that the allocation fulfillsPn

i=1a_i = ρ (Pn

i=1X_i) is important since it guarantees that the whole amount of risk capital is allocated. One intuitive way of allocating capital that does not necessarily fulfill this requirement is the marginal allocation rule in the following example.

Example 3.1 (The marginal allocation rule). Let the allocation be given by a^marg such that a^marg_i = ρ(X) − ρ(X − X_i), i.e. the difference between

(26)

the risk of the portfolio including the contract and the portfolio excluding the contract. If we have the simple portfolio represented by table 3.1 and choose ρ(·) to be VaR_0.9(·), then we get

ρ(X) = 9.6 since the cdf of the loss is

FL(l) =











0.0, l < −10 0.4, −10 ≤ l < −6 0.5, −6 ≤ l < 8.5 0.7, 8.5 ≤ l < 9.6 1.0, 9.6 ≤ l

and P (L ≤ 8.5) = 0.7 < 0.9 < 1.0 = P (L ≤ 9.6). In a similar way one can evaluate the following risk measures:

ρ(X − X1)= 7.6 ρ(X − X2)= 10.5 ρ(X − X₃)= 10.6.

Thus the corresponding allocations are

a^marg₁ =9.6 − 7.6 = 2.0 a^marg₂ =9.6 − 10.5 = −0.9 a^marg₃ =9.6 − 10.6 = −1.0

and clearly a^marg₁ + a^marg₂ + a^marg₃ 6= ρ(X). Similar results are obtained if ES0.9 is used as risk measure.

3.2 Capital, pricing and loading

Consider again a portfolio consisting of n contracts with X_i, i = 1, 2, . . . , n representing the profits of each portfolio and X =Pn

i=1Xi representing the profit of the portfolio. Then, if each contract is priced according to equation (2.7) we get

(27)

Change in Wealth (annual profit)

Contract 1 Contract 2 Contract 3 Portfolio

Prob X1 X2 X3 X

0.1 -5.0 -10.0 21.0 6.0

0.2 -2.5 2.0 -8.0 -8.5

0.3 -2.0 -8.6 1.0 -9.6

0.4 4.0 8.0 -2.0 10.0

Table 3.1: The change in wealth of a simple insurance portfolio with three contracts under four different states of the world with specified probabilities.

Xi= E[Si] + hi− S_i =⇒ X =

n

X

i=1

Xi = E[S] + h − S (3.3)

where S = P S_i and h = P h_i. The risk measure evaluated for X, ρ(X) is sometimes referred to as risk-adjusted-capital (RAC) since it corresponds to the amount of capital that must be added to the portfolio to compensate for the risk. A measure of how well the portfolio as well as a single contract performs is the return on risk-adjusted-capital (RORAC) defined as

RORAC= E[X]

ρ(X) RORACi= E[Xi]

ai

for the entire portfolio and the individual contracts respectively. If X and Xi are substituted for their counterparts in equation (3.3) we get

RORAC= h

ρ(X) (3.4a)

RORACi= hi

ai

(3.4b)

Another performance indicator when working with capital management is the diversification index defined as

DI= ρ(X)

Pn

i=1ρ(Xi) (3.5a)

DI_i= a_i

ρ(X_i) (3.5b)

(28)

for the portfolio respectively the contracts. The diversification index is thus a measure of how much a contract gains from the diversification of the portfolio. In particular, DI_i = 1 implies that the contract gains nothing from joining the portfolio. As is shown in section 3.5.2 this happens when the loss-distributions of the contracts are comonotonic and one of the proposed allocation rules is used.

We are now continuing by specifying two allocation rules and their advan- tages.

3.3 The Euler Allocation Rule

The most common allocation rule in the academic literature is the Euler allocation rule³ (cf. Tasche (2008) or Denault (2001)). The basic setting assumes that you have a portfolio consisting of sub-portfolios or as in this case reinsurance contracts. It is also assumed that the risk measure is continuously differentiable and that it is possible to (at least approximately) continuously change the degree of participation in each contract. The degree of participation is represented by u_i ∈ R and is not to be confused with the parameter λ representing the share in a certain layer in equation (2.1).

Instead u_i is interpreted as factors representing deviations from the current portfolio. In this sense u_i = 1 for i = 1, 2, . . . , n represents the current portfolio, i.e. X = Pn

i=1Xi. Further, u_i = 1.2 for i = 1, 2, . . . , n represents a portfolio where the participation is increased by 20 % in each contract as compared with status quo.

With the participation levels introduced, the risk measure of the portfolio can be expressed using the cost function defined by (Denault, 2001, eq. (1) section 4.3) as

r(u) = ρ

n

X

i=1

uiXi

! .

Especially we note that ρ(X) = r(1_n) where 1n is the vector of n ones. We are now in a position to define the euler allocation rule.

Definition 3.3. If r(u) is continuously differentiable⁴ then the Euler allocation rule is given by

3For the interested reader this rule is also referred to as the Coherent allocation rule or the Aumann-Shapley allocation rule in the literature.

4r(u) being continuously differentiable is equivalent to ρ(X + hXi) being continuously differentiable in h. The concept of derivatives of risk measures is presented in section 3.3.1.

(29)

a^euler_i = ∂r

∂ui

(1_n), i = 1, 2, . . . , n (3.6)

Observe that the euler allocation rule can be thought of as a limiting case of the marginal allocation rule since

a^euler_i = ∂r

∂ui

(1n) = ∂ρ(X + hXi)

∂h h=0

follows from the definition of the cost function r. Instead of looking at the contribution when the whole contract is added as is the case in the marginal allocation, the euler allocation looks at the marginal effect when an infinitesimal share of the contract is added. We will later derive expressions of the derivatives of the common risk measures VaR and ES.

The name of the rule stems from a nice property that follows from Euler’s theorem about positive homogeneous functions. A function f : Rⁿ⊃ U → R is said to be positive homogeneous of degree τ if ∀h > 0 and ∀u ∈ U it holds that f (hu) = h^τf (u). We are only going to be working with positive homogeneous functions of degree 1 and in this case euler’s theorem states the following.

Theorem 3.1 (Euler’s theorem for positive homogeneous functions of degree 1). Let U ⊂ Rⁿ be an open set and f (u) : U → R be a continuously differentiable function. f is 1-homogeneous if and only if it holds that

f (u) =

n

X

i=1

ui

∂f (u)

∂ui

= u^T∇f (u). (3.7)

Proof. For the purpose of this thesis, the if-part of the theorem is of no inter- est and is therefore not proved here. To prove the only if-part we introduce the function g(λ) : [0, ∞) → R by

g(λ) = f (λu) − λf (u).

If f is assumed to be positve homogeneous then, by definition, g(λ) = 0 ∀λ and thus g⁰(λ) = 0 ∀λ. But

g⁰(λ) = u^T∇f (λu) − f (u)

(30)

and since g⁰(λ) = 0 ∀λ it must be true for λ = 1 and equation (3.7) is fulfilled.

The following corollary guarantees that the euler allocation rule satisfies equation (3.2) whenever ρ(·) is a positive homogeneous risk measure.

Corollary 3.2. If ρ(·) is a positive homogeneous risk measure, then the cost function r(·) is positive homogeneous of degree 1 and the euler allocation rule guarantees that all capital is being allocated.

Proof. We only need to prove that if ρ(·) is positive homogeneous, then r(u) is positive homogeneous of degree 1. But this follows trivially since

r(λu) = ρ

n

X

i=1

λu_iX_i

!

= λρ

n

X

i=1

u_iX_i

!

= λr(u).

Denault (2001) derives the Euler allocation based on coalitional game theory from the premise that each sub-portfolio or contract is a player with an agenda of minimizing its costs of holding capital. If a sub-portfolio or any coalition of sub-portfolios carries more costs by joining the whole portfolio than they carry on their own, then it is not preferable for the sub-portfolio or the coalition to join the portfolio. In this setting it is possible to prove that the Euler allocation allocates less capital to each contract or coalition of them than they must carry on their own. If the risk measure is positive homogeneous, convex and continuously differentiable, then the Euler allocation is the only such allocation rule, cf. Denault (2001, Theorems 6 and 7 with comment). Since VaR_α is non-convex this holds not for VaR_α. Indeed, it is not even a guarantee that a_i < ρ(Xi) when ρ is not convex, cf. Tasche (2008, equation (2.9b))

Finally, it is worth pointing out that the Euler allocation principle is the only RORAC-compatible allocation principle whenever ρ is partially continuously differentiable, cf. (Tasche, 2008, Proposition 2.1). RORAC-compatibility means that if the RORAC of an individual contract is higher (lower) than the RORAC of the entire portfolio, then increasing (decreasing) the shares in that contract, at least locally, will increase the RORAC of the entire portfolio.

Thus the Euler allocation principle can be used to optimize the portfolio and the optimum is achieved when RORAC₁ = RORAC2 = . . . = RORACn = RORAC. This is however not investigated further in this thesis.

(31)

3.3.1 Derivatives of VaR and ES

In order to calculate the Euler allocation (3.6) we must derive an expression for

∂r(1n)

∂u_i = ρ(X + hXi)

∂h .

In this section formulas for the derivates of VaR and ES are presented without proofs. The theorems and proofs can be found in Rau-Bredow (2003, Theorems 2 and 3).

Proposition 3.3. Let X and Y be continuous random variables, then

∂VaRα(X + hY )

∂h = E [−Y |X + hY = −VaRα(X + hY )]

∂ESα(X + hY )

∂h = E [−Y |X + hY ≤ −VaRα(X + hY )]

.

Only the expression for the derivative of VaR is valid in the discrete case also. The expression for the derivative of ES must be adjusted to compensate for the possibility of the parameter α being in between two values of the discrete cdf.

Proposition 3.4. Let X and Y be discrete random variables, then

∂ESα(X + hY )

∂h =E−Y 1{X+hY ≤−VaRα(X+hY )}

1 − α

−

∂VaRα

∂h (X + hY ) · (α − P (X + hY > −VaRα)) 1 − α

.

When working with empirical distribution functions, the discrete versions of the risk measures are prefered since it guarantees that the property (3.2) is fulfilled. This is because the empirical distribution function is discrete and in that case the discrete derivatives are the correct derivatives. Equation (3.2) is then fulfilled by corollary 3.2.

To conclude this section we must translate the derivatives to the allocation (3.6). This is simply done by identifying X as the profit of the whole portfolio and Y as the profit of an individual contract X_iand evaluating the derivatives in h = 0. In the discrete case this gives

(32)

a^euler_i =E [−X_i|X = −VaR_α(X)] (3.8a)

a^euler_i =E−X_i1_{X≤−VaR_α_(X)}

1 − α −

−E [−Xi|X = −VaR_α(X)] · (α − P (X > −VaRα(X)))

1 − α (3.8b)

for ρ(X) = VaR_α(X) and ρ(X) = ES_α(X) respectively. If one prefer to think in terms of losses instead of profits, i.e. in terms of L_i = −X_i and L = −X, then equations (3.8a) and (3.8b) can be written

a^euler_i =E [L_i|L = VaR_α(X)] (3.9a)

a^euler_i =EL_i1{L≥VaRα(X)}

1 − α −

−E [Li|L = VaR_α(X)] · (α − P (L < VaRα(X)))

1 − α (3.9b)

3.3.2 Euler allocation of sample data

Assume that a sample of m points of the annual losses of n contract has been retrieved. Each sample point can be represented as a vector l_i ∈ R^1×n, i = 1, 2, . . . , m as

li = (l1,i, l2,i, . . . , ln,i)

where l_j,i is the loss of contract j in sample point i. The sample points of the losses of the entire portfolio, l_port,i, are then given by

l^port_i =

n

X

j=1

lj,i, i = 1, 2, . . . , m

To use the expressions for the allocations we must derive estimators of VaRα(X), E [Li|L = VaR_α(X)], EL_i1{L≥VaR_α(X)} and P (L < VaR_α(X)).

This is done below.

A consistent sample estimate of VaR_α(X) = Q−X(α) is presented in Hult et al. (2012, section 7.4) and is reformulated here to be consistent with the definition of VaR in equation (3.1a):

(33)

VaR\α(X) = lbn(1−α)c+1,n^port (3.10) Further, an estimator of E [L_i|L = VaR_α(X)] is given by

Eˆ h

Li|L = dVaRα(X) i

= 1 u

m

X

j=1 l_j^port= dVaRα(X)

li,j (3.11)

where u is the number of sample points such that l^port_i = dVaRα(X). The estimator of EL_i1_{L≥VaR_α_(X)} is given by

Eˆh

L_i1_{{L≥ d}_VaR

α(X)}

i

= 1 m

m

X

j=1 l^port_j ≥ dVaRα(X)

l_i,j (3.12)

Finally, a consistent estimate of P (L < dVaR_α(X)) is given by

P (L < db VaRα(X)) = 1 m

m

X

i=1 l^port_i < dVaRα(X)}

1. (3.13)

When X is continuous the sample estimate (3.11) depends on only one sample point since the condition l^port_j = dVaRα(X), j = 1, 2, . . . , m is fulfilled for only one j, namely j = bn(1 − α) + 1c if the sample is ordered. Thus, VaR cannot be used with this naive approach. Tasche (2008, section 3.4) presents a solutions to this problem. The solution is not elaborated in this thesis.

Instead we will only be working with ES as the risk measure from now on.

Before introducing the next allocation principle it is worth pointing out that the prescribed approach results in a consistent estimator of the allocation (cf. Tasche (2008, p. 10)).

3.4 An alternative approach: Capital consumption and capital layer allocation

3.4.1 Capital consumtion

Some of the academic literature focus on capital consumption as an alternative method to capital allocation in insurance. Mango (2003) argues that the notion of capital allocation has a number of drawbacks. Thinking about

(34)

capital allocation instead of consumption gives an erroneous impression of the physical capital stock in an insurance company. Capital must not be physically allocated to individual contracts a priori. Instead all contracts share the same capital stock from which capital can be called upon when the contract is running an operating deficit. Additionally, capital allocation does not reflect the actual amount of capital that the individual contract may actually consume in any specific scenario. Instead it is the result of some formula depending on the risk measure and in many cases only captures the tail losses. Mango’s suggested approach is to work with a set of predefined scenarios and then calculate what he calls the expected risk adjusted net-present-value of each contract to valuate their performance.

This approach is not considered in the analysis of this thesis but a short description as well as an example are given to simplify the understanding of the capital layer allocation principle presented later.

In the capital consumption approach each contract makes a capital call when the associated losses exceeds the premiums. The called amount is simply the difference between the losses and the premiums. The call is accompanied with a capital cost that can be calculated with a cost function. If the firm is not risk averse the cost function may simply be f (C) = a · C where C represents the capital that is being called and a is a factor that must be calibrated. On the contrary, if the firm is risk averse the parameter a may be a function of C, for example

f (C) =

(1.0 · C, if C ≤ 4 1.5 · C, if 4 < C .

To see how this approach can be used to determine the premium of a contract or evaluate and compare its performance, consider the following example which is a modification of Mango (2003, section 5). Assume that we have a catastrophe excess of loss contract with a limit of 10 millions and paying premiums of 1.9 millions. Assume further that the three different scenarios in table 3.2 can occur.

Here the cost function for the risk averse firm above is used and represented in the table by the capital call cost charge. Now we calculate the expected result of the contract, the expected capital cost and the expected risk-adjusted result of the contract. The later is the difference between the previous two.

The results are presented in table 3.3. In this example the expected risk- adjusted-result is negative which indicates that the contract is not preferable.

In this way the approach can be used to evaluate contracts a posteriori. It can also be used a priori to determine the premium by solving for the premium that results in an expected risk-adjusted-result of zero.

(35)

Capital call

Scenario Prob. Premiums Losses Profit amount cost charge cost

1 0.80 1.9 0 1.9 0 0 0

2 0.18 1.9 5 -3.1 3.1 1 3.1

3 0.02 1.9 10 -8.1 8.1 1.5 12.15

Table 3.2: The results of a contract under three different scenarios.

Expected result 0.800

Expected capital cost 0.801 Expected risk-adjusted-result -0.001

Table 3.3: The result of the capital consumption approach used on the contract represented in table 3.2

3.4.2 Capital layer allocation

One approach that lies between pure capital allocation and pure consumption as proposed by Mango is the capital layer allocation principle presented by Bodoff (2007). This approach is based on a set of simulated portfolio-wide loss scenarios or realizations of the portfolio result. The risk capital is not necessarily determined by a risk measure and can be viewed as a parameter, but this parameter can of course be calculated by a risk measure if that is preferable. The capital layer allocation principle is close to the consumption approach because it focuses on scenarios which makes capital calls from the common capital stock.

Bodoff (2007, chapter 5) presents the Capital layer allocation only in proce- dural terms for a sample. Here, I derive this process in more general terms for discrete random losses. In order to do so we let L be a discrete random variable with a finite number of outcomes representing the net loss to a portfolio consisting of N contracts. Thus L is the sum of N discrete random variables representing the net loss of each contracts but these variables are not needed yet and are introduced later.

Let l₁ ≤ l₂ ≤, ..., ≤ l_M represent the M possible outcomes of L. To allocate the risk capital ρ to these outcome we must first define a capital layer.

Definition 3.4. Let L be a discrete random variable representing the net loss of a portfolio for a predefined time horizon. Assume that l₁ ≤ l₂, ≤ . . . ≤ lM

are M possible outcomes of L and that the capital ρ is to be divided between these realizations. Any of the intervals

[0 , l₁] , [l₁, l₂] , . . . , [l_k−1, l_k] , [l_k, ρ], k = max{j = 1, 2, . . . , M |l_j < ρ}

(36)

is referred to as a capital layer. Thus the capital layers divide the interval

[0, ρ] into subintervals.

An example of the division of [0, ρ] into capital layers is illustrated in figure 3.1. The figure shows a bar plot of the realized scenarios with M = 10 and k = 8.

1 2 3 4 5 6 7 8 9 10

l1 l2 l3 l4 l5l6 l7 l8

\rho

Lee diagram

realization number

loss amount

Figure 3.1: An example of the capital layers for a situation with 10 simulated realizations. The portfolio result was simulated by simulating three contracts with exponentially distributed losses with mean 1.5. Further a Gaussian copula with correlation parameter 0.5 among each pair of contracts were used to model the dependency. ρ(X) was chosen to be ES_0.6(X).

The main idea behind the capital layer allocation principle is that each capital layer is shared by the outcomes that penetrates it. In other terms each capital layer can be consumed by the outcomes that penetrates it. The share of the layer that each outcome receives is proportional to the conditional probability that the outcome makes a call from the capital layer given that the layer is consumed. Thus we introduce p₁, p₂, . . . , p_M to represent the probabilities for each outcome to occur and we define A_ito be the capital allocated to scenario number i. Now, the conditional probability that scenario i makes a call from the first layer given that it is being called is simply pi. Thus, the scenarios are allocated an amount of l_i · p_i, i = 1, 2, . . . , M . Further, the conditional probability that scenario i makes a call from the second capital layer of size l₂− l₁ given that the second layer is consumed

(37)

is given by 0 if i = 1 and PM^pⁱ j=2pj

, i = 2, 3, . . . , M . For the third layer the conditional probabilities are given by 0 if i = 1, 2 and PM^pⁱ

j=3pj, i = 3, 4, . . . , M and so on. The capital allocated to each outcome is thus given by











A1 = l1· p₁

A₂ = l₁· p₂+ (l₂− l₁) ·PM^p² j=2pj

A₃ = l₁· p₃+ (l₂− l₁) ·PM^p³

j=2pj + (l₃− l₂) ·PM^p³ j=3pj

...

A_k+1 = l1· p_k+1+ (l2− l₁) ·P^pM^k+1

j=2pj + (l3− l₂) ·P^pM^k+1 j=3pj+ . . . + (ρ − l_k) ·PM^p^k+1

j=k+1pj

...

AM = l1· p_M + (l2− l₁) ·PM^p^M

j=2pj + (l3− l₂) ·PM^p^M j=3pj+ . . . + (ρ − lk) ·PM^p^M

j=k+1pj

(3.14)

where k is defined as in definition 3.4. The somewhat different behavior starting from index k + 1 is due to the fact that the last capital layer is [l_k, ρ]. Compare figure 3.1 where k = 8. The overshoot above ρ of scenarios 9 and 10 are not included in the allocation. Observe that this far we have only allocated capital to the portfolio-wide outcomes. The next definition says how the capital allocated to the outcomes is to be allocated to the contracts.

Definition 3.5 (Capital Layer Allocation). Let l₁ ≤ l₂ ≤, ..., ≤ l_M be the M outcomes of the net loss to a portfolio consisting of N contracts. Further let ρ be the amount of capital to be allocated and let A_j, j = 1, 2, . . . , M be the allocated capital to each outcome according to (3.17). The Capital Layer Allocation rule is given by a ∈ R^{N ×1}such that

a_i =

M

X

j=1

lij

l_jA_j (3.15)

where l_ij with double index is the loss amount to contract i in the jth

outcome.

We will now prove that the defined allocation rule satisfies equation (3.2), i.e. that all capital is being allocated to the contracts. First we need to prove the following proposition:

(38)

Proposition 3.5. Given the allocation in equation (3.14) the following holds

M

X

l=1

A_l= ρ.

Proof. The proof follows easily by collecting the terms corresponding to the same capital layers.

M

X

l=1

A_l= l₁

M

X

i=1

p_i+ l₂− l₁ PM

i=2pi M

X

i=2

p_i+ . . . + ρ − l_k PM

i=k+1pi M

X

i=k+1

p_i

= l1+ l2− l₁+ l3− l₂+ . . . + l_k−1− l_k−2+ l_k− l_k−1+ ρ − l_k=

= ρ.

Now we can easily prove that all capital is being allocated.

Theorem 3.6. Let a ∈ R^{N ×1} be a Capital layer allocation with risk capital ρ. Then

N

X

i=1

a_i = ρ.

Proof.

N

X

i=1

ai=

N

X

i=1 M

X

j=1

l_ij lj

Aj =

M

X

j=1

Aj N

X

i=1

l_ij lj

=

M

X

j=1

Aj = ρ

where the second to last inequality follows since obviouslyPN i=1

lij

lj = 1.

3.4.3 Capital layer allocation of sample data

Let (l_1,i, l2,i, . . . , ln,i) represent the sample losses to the contracts for sample point i = 1, 2, . . . , M and let l_i=Pn

j=1lj,i be the portfolio wide loss for this sample point. By approximating the probabilities p₁, p₂, . . . , p_M by the empirical distribution, each sample point has the same probability of occurring, i.e. p₁ = p2 = . . . = pM = _M¹ . In this case the expressions in (3.14) are simplified to

(39)











A₁ = _M^l¹

A₂ = _M^l¹ +^l_{M −1}²^−l¹

A₃ = _M^l¹ +^l_{M −1}²^−l¹ +^l_{M −2}³^−l² ...

A_k+1 = l₁· p_k+1+ (l₂− l₁) ·P^pM^k+1

j=2pj + (l₃− l₂) ·P^pM^k+1 j=3pj+ . . . + (ρ − l_k) ·PM^p^k+1

j=k+1pj

...

AM = l1· p_M + (l2− l₁) ·PM^p^M

j=2pj + (l3− l₂) ·PM^p^M j=3pj+ . . . + (ρ − lk) ·PM^p^M

j=k+1pj

(3.16)

Since A₂shares the first term with A₁, A₃ shares the two first terms with A₂ and so on, the expressions in (3.16) can be reduced to the following recursive formula:

(A1 = _M^l¹

A_i = A_i−1+ ^min(lⁱ^,ρ)−min(l_{M −i+1} ⁱ⁻¹^,ρ), i = 2, 3, ..., M . (3.17) The min-function in equation (3.17) is included to secure that the sum of the allocated capital to the scenarios does not exceed ρ. Let, as above, k be the index of the last loss being strictly smaller than ρ and assume that M ≥ k + 2. Then we get







A_k= A_k−1+_{M −k+1}^l^k^−l^k−1 A_k+1= A_k+_{M −k}^ρ−l^k

Ak+2= Ak+1+_{M −k−1}^ρ−ρ = Ak+1

.

In this way, the method does not care about how much the losses extend above the capital ρ. Compare figure 3.1 with k = 8. Here the ninth and tenth realizations are allocated the same amount of capital since they penetrate the same layers regardless of how much they overshoot ρ.

Is the allocation dependent on the choice of the capital layers? Since the split- ting of the interval [0, ρ] has no physical meaning it would be inconvenient if adding or removing capital layers alters the allocations to the scenarios.

Luckily this is not the case as is shown next. Assume that an extra layer [li, x], for x ≤ li+1 is included. Then the capital layers in definition 3.4 are [0, l1], . . . , [li−1, li], [li, x], [x, li+1], . . .. The intervals to the left of [li, x] are unaltered and the allocations in (3.17) only depend on the lengths of the

Allocation of Risk Capital to Contracts in Catastrophe Reinsurance

Allocation of Risk Capital to Contracts in Catastrophe Reinsurance

Allocation of Risk Capital to Contracts in Catastrophe Reinsurance

Allokering av riskkapital

mellan katastrofåterförsäkringskontrakt

Acknowledgements

Contents

Chapter 1

Introduction

1.1 Background

1.2 Purpose

1.3 Outline

Chapter 2

Reinsurance

2.1 Classifications of reinsurance contracts

2.2 Non-proportional catastrophe reinsurance

2.3 The Collective Model

2.4 Pricing and Loading

Chapter 3

Capital Allocation Rules

3.1 Risk measures and Allocation rules

3.2 Capital, pricing and loading

3.3 The Euler Allocation Rule

3.4 An alternative approach: Capital consumption and capital layer allocation