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Economic Capital for Insurers:

Insurance Cycle and Catastrophic Risk

ABSTRACT

This paper proposes a stochastic model to study the economic capital and performance of an

insurer of natural disasters in the presence of underwriting cycle. The model included a parameter

capturing the strategy adopted by the insurer under different cycles. We use the ruin probability

and the conditional value-at-risk adjusted returns to capture the performance of the insurer. Using

historical natural catastrophic events data of the United States from 1960 to 2008, we find different

behaviours depending on the type of catastrophic event. The existence of underwriting cycle

represents a major risk for the insurer, but it can be advantageous if the insurer chooses the right

strategy at the onset. Furthermore, we show that catastrophic insurance portfolio needs to be

composed of at least 50 elements to gain from diversification.

Keywords: Economic capital, insurance cycle, catastrophic risk. JEL Codes:

VERY PRELIMINARY, PLEASE DO NOT QUOTE WITHOUT THE AUTHORS’

PERMISSION

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I. Introduction

The regulatory framework Solvency II’s objective is to better align insurance and

reinsurance companies regulatory capital with their risk posture. It is then important for insurers

and reinsurers to have a better understanding of the risks underlying their activities in order to

optimally allocate capital among their business activities. There is therefore a need for the

development of sound internal models to estimate insurers’ obligations.

The technical specification of the « Quantitative Impact Studies 4 » (QIS4, see CEIOPS

(2007)) defines the underwriting risk as the risk related to insurance contracts. This concerns

uncertainties associated to the technical results of the insurer. Unfortunately, the QIS4 does not

require any supplementary capital for underwriting cycle. Underwriting (or insurance) cycle can

generate artificial volatility for the technical results beyond the statistically determined insurance

risk (e.g., Meyers (2007)). Therefore, insurance cycle should be accounted for when developing

internal models under Solvency II regulation since the additional volatility caused by it can lead

to a larger capital level than what is actually required.

The objective of this paper is to evaluate the economic capital for casualty and property

insurers by accounting for underwriting cycle and catastrophic risk. More specifically, the paper

addresses the following research questions: (1) How to model an insurer’s risks when facing

insurance cycle? (2) What is the impact of insurance cycle on the economic capital and

performance of an insurer in the case of catastrophic risks? (3) What is the effect of portfolio

diversification in the case of natural catastrophic risk?

To respond to the above research questions, we present a stochastic model which account

for the effect of underwriting cycles. In the proposed model, the insurer can adjust its security

loading with regards to insurance cycle and its business strategy to remain competitive. The insurer

can also adjust its claims rate to reflect its overall exposure for a given business strategy. Note

however that our aim is not to study the causes of the underwriting cycle; we instead analyze the

behaviour of the insurer when the cycle occurs. We use the probability of ruin and the conditional

value-at-risk (VaR)- adjusted returns to capture the performance of the insurer. We use historical

natural catastrophic events data of the United States from 1960 to 2008 for the empirical analyses.

The application of our model to the four catastrophic events considered (earthquake,

hurricane, tornado and flood) show different patterns depending on the type of event. The existence

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of underwriting cycle represents a major risk for the insurer, but it can be advantageous if the

insurer chooses the right strategy at the onset. Furthermore, our analyses show that disaster risk

portfolio needs to be composed of at least 50 elements to obtain the diversification benefit.

The rest of the paper is structured as follows. Section 2 presents the literature review on

underwriting cycles: existing theories and different approaches to model underwriting cycles.

Section 3 presents our model with the dynamics of the insurer’s assets and liabilities and the

insurance cycle. Section 4 presents the economic capital calculation and the solvability and

performance measures of the insurer. Section 5 presents results for natural disaster (catastrophic)

risks, the impact of insurance cycle in the case of catastrophic risk using Monte Carlo simulations

and the effect of diversification on the overall portfolio risk. Section 6 concludes.

II. Literature review on insurance cycle: theories and models

2.1. Insurance cycle theories

Insurance cycle or underwriting cycle refers to periods of positive and negative returns in

insurance business over time. As pointed out by Stewart (1981), competition among insurers will

generally determine the insurance premium proposed by insurers, because it is relatively easy for

new comers to enter the insurance business and there is no patent for insurance products.

Therefore, in the insurance industry, almost all insurers offer more or less the same products. The

only way for one company to differentiate itself from the competition is to propose the same

service at the most competitive price (low price) possible. Stewart (1981) defined insurance cycle

as the seasonal fluctuation of insurance premiums and profits over time. Generally speaking,

insurance cycle can either be “soft market” or “hard market”. Hard market is characterised by

reduced supply of insurance products, lower loss coverage and higher premiums. Unlike the hard

market, soft market corresponds to abundant supply of insurance products, larger loss coverage

and lower premiums.1

                                                            1 For instance, a cycle may start when insurers impose restrictive underwriting criteria and increase their premiums following a period of severe losses. These restrictive criteria and the increase in premiums are more likely to generate more profits in the business, and thus will attract more new capital investments in the industry (entry of new actors) and lead to an increase in the underwriting capacity (supply). However, since the goal of an insurer is to underwrite the maximum number of contracts at a high profitability level, it is more likely that, in a competitive environment, the premium will tend to decrease and the underwriting criteria will become less restrictive in order to obtain new market shares. Profits will then decrease with more generous promises for claims coverage. The insurer will then increase its premium rates and a new cycle will begin again.

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Although most industries are exposed to boom and bust cycles, cycles are particularly

critical in the insurance business because they are unpredictable. Insurance cycles affect all areas

of the insurance business, except the life insurance segment. However, the intensity and duration

of the cycle may vary from one business segment to another. Cummins and Outreville (1987)

argued that insurance cycles observed in the US and other developed countries are more likely to

exist in other parts of the world because of the development of reinsurance business around the

world.

Over the past recent years, many attempts have been made to identify and quantify

insurance cycle with some times very limited dataset, and up to now, no model has proven to

perform better than others. The literature has provided many theoretical arguments to explain

insurance cycles. There is no common agreement on the causes of insurance cycles. For instance,

Fung et al. (1998) tested statistically all the available theories and found that no theory actually

can explain fully the whole insurance cycle. Three main theories seem to prevail: (1)

disequilibrium between supply and demand (e.g., Berger (1988), Cummins and Danzon (1997),

Feldblum (2001), Froot and O’Connell (1997), Gron (1994), Harrington and Danzon (1994),

Niehaus and Terry (1993), Outreville (1981), Venezian (1985), Winter (1991, 1994)); (2) external

shocks, such as interest rate changes or institutional factors or catastrophic events (e.g., Chen et

al. (1999), Cummins et al. (1991), Cummins and Outreville (1987), Doherty and Kang (1988),

Fields and Venezian (1989), Fung et al. (1998), Haley (1993), Harrington and Niehaus (1999),

Lamm-Tennant and Weiss (1997)); and (3) the impact of economic activities (e.g., Chen et al.

(1999), Cummins et al. (1991), Meier (2003), Winter (1991)).

2.2. Insurance cycle models

Underwriting or insurance cycle is a major concern in the insurance industry when

evaluating the performance of pricing methods. Indeed, a soft market can contribute to the

insolvency of an insurer when premiums are too low due to competition; while higher prices during

a hard market can push consumers to buy less insurance products. Although there is no common

ground in the literature, several researches have been undertaken to model insurance cycles. The

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main variables used in the insurance cycle models are: the insurance premiums, the security

loading,2 the loss ratio,3 the combined ratio4 and the rate on line (ROL).5

Many actuarial models have been proposed to model underwriting cycle. The literature can

be classified into three main classes of models: (1) the deterministic models which use for example

the trigonometric functions to model the dynamics of insurance cycle (Daykin et al., 1993); (2) the

econometric models, these models are based on linear regressions or time series analysis (e.g.,

Cummins and Outreville (1987), Trufin et al. (2009), Pentilkainen et al. (1989)); and (3) the

stochastic models which embedded processes capturing changes in regime (e.g., Derien (2008),

Cummins and Danzon (1997), D’Arcy et al. (1997)). In the present paper we model underwriting

cycle using a stochastic process with seasonality.

Insurance cycles is an important phenomenal to account for in insurance risk management.

The next section presents the proposed model to study the impact of insurance cycles on the

performance and solvency of insurers.

III. Proposed model

We propose a stochastic model to study the impact of insurance cycles on the economic

capital and solvency of an insurer. The insurer’s assets and liabilities are characterised by

stochastic dynamic processes. We also model the dynamic of the insurance cycles using a

stochastic process with seasonality. There are many theories in the literature to explain insurance

cycles, with each one having its own rationale. The use of one theory over others depends on the

research question. Our model is inspired from the theory of competition between insurers, which

                                                            2 The security loading is directly linked to the premium as it is a safety charge added to the premium to cover losses. The value of the insurance premium against the present value of expected future losses is a signal for the state of the cycle. Hard market arrives when premiums are superior to the present value of expected losses. In soft market, it is the contrary. 3 The loss ratio is the ratio of claims payments or claims to be paid divided by the total collected premiums. 4 The combined ratio is the sum of the loss ratio and expenses ratio (management fees on premiums raised). The loss ratio and the combined ratio are usually used to evaluate the performance of an insurer. Indeed, when these ratios are superior to 100%, the insurer realises a technical loss, i.e. it pays more claims than the collected premiums. The contrary holds when these ratios are inferior to 100%. 5 The rate on line (ROL) is essentially used in reinsurance to analyse the reinsurance premiums. It is obtained by dividing reinsurance premiums by the capacity (maximum amount covered). The inverse of the ratio is called redemption period. Hence, a high value of ROL means that reinsurance premiums are high and the supply of reinsurance is low, which corresponds to a hard market condition. Whereas low values of ROL means that reinsurance premiums are relatively lower than reinsurance supply, thus more loss coverage. This corresponds to a soft market condition.

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leads to price competition. We assume that insurers change their premium level in order to remain

competitive. Therefore, we first present the different business strategies, and after introduce the

dynamic of the insurance cycles and finally the processes of the insurer’s assets and liabilities.

3.1. Business strategies

The structure of the industry and the characteristics of the products will play a crucial role

on the expected profits and the possible strategies. Feldblum (2001) argued that three

considerations should be considered for the insurance industry: (i) products differentiation and

substitution of products, (ii) cost structure and entry barriers, and (iii) the loyalty of customers.

Products differentiation and substitution: When competing companies have identical products,

they become all bind by the price of the most efficient producer. Having different products from

the competition is a good mean of increasing one’s expected profits. However, it is very difficult

in the insurance industry to propose products which are fundamentally different from competition.

Indeed, it is impossible for an insurer to protect any new product or patent new products, hence,

other insurers can imitate any new or existing product. In most industries, there are substituting

products. For example, one may substitute steel with aluminium. In the case of insurance products,

costumers have almost no choice than to acquire insurances for their cars for example. The lack of

differentiation of insurance products makes it difficult to increase prices and profits beyond the

competition. At the same time, the lack of substituting products makes it easier for the industry to

increase premiums without losing clients.

Cost structure and entry barriers: The traditional barrier for a new actor to enter into any

particular productive sector is the investment cost (i.e., the minimum capital required), the time

period necessary for entry and the learning curve of the production process. These factors can have

an impact on an investor’s decision whether to enter into a new market or not. The insurance

industry has fewer barriers. Indeed, premiums are paid at the beginning of the period and the

required capital is relatively low, equipment and office supplies are not too expensive and most of

the costs are recovered with the premiums raised. However, although easy it seems to enter into

the insurance sector, it is hard to remain in it with success; because the arrival of new insurers

brings marginal risk, and losses are relatively high in the first years of operations. It therefore

requires time to obtain a profitable business. When the insurance market is profitable, the high

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entry rate of new investors decreases the expected profits; and then depending on the losses born

over time, some players will exit the market, and the game continues so forth.

Clients’ loyalty: The price of a product is a key element in the purchase decision of a consumer.

However, when the price of a product increases, the decision of a consumer to continue to buy it

or to substitute the product will depend on the opportunity cost for him. For example, people

usually have to pay service fees when they change their Internet service provider. High « change

fees » are obstacle to competition and increases profits. These obstacles do not exist in the

insurance industry. During the renewal period, consumers are free to choose among the available

insurers, and can move to an insurer who provides them with the lowest price without bearing any

additional fees. Another important point is that consumers rarely compare premiums proposed by

different insurers (Feldblum (2001)). Hence, a consumer will start the search for a new insurer

when there is a drastic increase in its insurance premium. In the long run, having higher premium

than market average will lead to losses in market shares for the insurer, which is extremely difficult

to reconquer after. However, in the short run, an insurer with a good reputation can charge a higher

premium without losing its market share.

Knowing the consumers’ behaviour and facing insurance cycle, an insurer can adopt a strategy

which better fits its business objectives. We adopt the following four different strategies described

by Boor (2004):

- Strategy 0--No underwriting cycle. In this base case strategy, we assume the existence of

no insurance cycle. There is then a common agreement on the premium. This strategy will

be used as the benchmark for the other strategies to determine the impact of insurance cycle

on the profitability and solvency of the insurer.

- Strategy 1--Riding the cycle or the strategy of maintaining market shares. Under this

strategy, the insurer tries to keep its market shares constant during the different phases of

the cycle. In that respect, the insurer proposes the lowest possible market price. Insurance

companies following this strategy should be careful to not lose too much profit; and

threaten their viability.

- Strategy 2--The countercyclical strategy or the strategy of conserving the capital. This

strategy focuses on selling high profit margin services. It implies that the insurer

underwrites less contracts during the soft market and more during the hard market. During

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the soft market, this strategy allows the insurer to maintain its capital level. The company

however needs to underwrite fair enough contracts in order to maintain the quality of its

infrastructure while waiting for the hard market. During the hard market, the company will

have enough capital to underwrite more contracts while its competitors face capital

constraint. Insurance companies following this strategy have to make sure that they have

enough clients to survive the soft market.

- Strategy 3--The mixed strategy. This strategy combines the previous two strategies, i.e.

maintaining market shares and conserving capital. For instance, companies following

strategy 2 can choose to maintain some market share during the soft market even if it is

costly to do so, and that because they want to maintain their infrastructures. Companies

following strategy 1 instead, can decide to forgo some businesses during the soft market to

remain solvent.

Figure 1 provides a summary of the implications of the business strategies on the market shares

and the insurance premiums.

Figure 1: Business strategies, insurance premium and market share

Var

iab

le

Strategy 2 Strategy 3

Con

stan

t

Strategy 0 Strategy 1

Constant Variable

The following numerical example illustrates very well the differences in outcome from the

different strategies. Indeed, suppose that the adequate average premium to insure a given risk is

$100. An insurer using strategy 1, i.e. maintaining market shares, will insure 10 policyholders at a

premium of $80 during the soft market and 10 policyholders at $120 during the hard market. An

Insurance premium

Market share

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insurer using strategy 2, i.e. conserving the capital level, will insure 8 policyholders during the soft

market and 12 policyholders during the hard market at the same premium of $100. With no

underwriting cycle (strategy 0), all else being equal, the insurer would have 10 policyholders at a

unit premium of $100 at all times. Assuming the duration of the hard market and soft market to be

identical, the three strategies yield the same average number of policyholders (10 clients) and same

average premium ($100). In the presence of underwriting cycle, the insurer has to choose between

strategy 1 and strategy 2 or a mix of the two (strategy 3). We need to check which one between

strategy 1, strategy 2 and strategy 3 performs better than strategy 0 to be able to conclude whether

the presence of underwriting cycle makes the insurer better-off or not.

Given that the insurer, depending on the strategy adopted, may underpriced during the soft

market and overpriced during the hard market, they are several questions to be answered:

- For an insurer willing to maintain its market shares (strategy 1), will it be possible to

generate enough profit to compensate for the technical losses and remain solvent?

- Knowing that insurance is based on pooling risks, for an insurer adopting the capital

conservation strategy (strategy 2), will it be possible for the insurer to have enough

policyholders during the soft market to generate enough profits?

- Which of these strategies is the most profitable and reliable?

3.2. Modeling insurance cycles

We extended the model of Jones and Ren (2006) for the underwriting cycle dynamic. Jones

and Ren (2006) assumed the security loading θI(t) and the claim rate λI(t) to follow deterministic

and periodic cyclical dynamics and use the trigonometric functions with periodicity 2π. The length

of the underwriting cycle is therefore 2π years (approximately 6 years). We assume a continuous

price dynamic, instead of having a process with jumps with respect to cyclical changes such as

D’Arcy et al. (1997). Moreover, Asmussen and Rolski (1991) showed that when the number of

states of nature increases, the Markov model converges toward a deterministic and periodic

continuous process.

Security loading

An insurer can adjust its premium during the underwriting cycle by changing its security

loading. We consider the modified version of the model described by Jones and Ren (2006) by

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assuming that the intensity of the cycle varies with time. The security loading coefficient is then

as follows:

( ) ( ) sin( )I t t I t , (1)

where I represents the initial state of the underwriting cycle, which takes value 0 for immature

hard market, π/2 for mature hard market, π for immature soft market and 3π/2 for mature soft

market. t represents the time. α(t) captures the intensity of the cycle. Note that the cycle intensity

is constant in the model described by Jones and Ren (2006), whereas in our modified version of

the model, it varies with the time parameter t. This is more realistic as the intensity of the

underwriting cycle is not stable over time. δ designates the business strategy adopted by the

insurer. When δ = 0, the insurer adopts a capital conservation strategy (strategy 2). When δ = 1,

the insurer adopts a strategy to maintain its market share (strategy 1). The insurer will use a mixed

strategy when 0 < δ < 1; in our case, we use δ = 0.5 for the mixed strategy (strategy 3). θ is the

gross security loading parameter and represents the average security loading for a complete cycle.

Assuming α(t) constant, θ is obtained as the average I as follows:

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2

0

)cos(2

)sin(2

1tIdttII .

Strategies 0 and 2 have a fix security loading rate θ, while strategies 1 and 3 have variable security

loading rates.

The first part of the cycle represents the hard market and the second part the soft market.

Hence, any initial state of the cycle (I) comprises between 0 and π corresponds to the hard market

and any initial state between π and 2π corresponds to the soft market.

Figure 2 shows the security loading over a 15 years period for different initial states of the

cycle. An insurer following strategy 2 (capital conservation) will have a constant security loading

rate over the duration of the cycle. An insurers following strategy 1 (maintaining market shares)

or strategy 3 (mixed) will have a variable security loading rate, with higher rates (over-pricing)

during hard markets, and lower rates (under-pricing) during soft markets.

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Figure 2: Evolution of the security loading rate for different initial states of the

underwriting cycle (as a function of time)

We use the following arbitrary parameters values: θ = 0.2; α = 0.5; δ = 0, 0.5 and 1. The graphs show the evolution of the security loading rate as a function of time for 15 years. The left upper graph assumes that the cycle starts in immature hard market and the right upper graph assumes mature hard market. The bottom left graph supposes the process starts in immature soft market and the bottom right assumes mature soft market as initial state. For each graph, there are four strategies represented: strategy 0 (no underwriting cycle), strategy 1 (maintain market share), strategy 2 (conserve capital) and strategy 3 (mixed strategy).

Claims rate

Again, following Jones and Ren (2006), we assume the claims rate to have the following

modified dynamic:

( ) (1 ( ) (1 )cos( ))I t t I t , (2)

where λ represents the gross claim rate, i.e. the average claim rate for a complete cycle, β captures

the loyalty of customers. It measures the sensitivity of policyholders to premium increase.

Strategies 0 and 1 have a fixed claim rate λ and strategies 2 and 3 have variable claim rates. Here

also the intensity of the underwriting cycle α(t) is time dependent, contrary to Jones and Ren

(2006), where it is assumed constant.

Figure 3 gives the claim rate over a period of 15 years. The security loading and the claim

rate are linked. Increasing the security loading rate will increase the premium, thus have a

decreasing effect on the market share, hence leading to less claims. Therefore, an insurer adopting

a strategy to maintain its market share (strategy 1) will have a fixed claim rate, while an insurer

0 5 10 15−0.5

0

0.5

1Immature Hard Market

Strategy 3Strategy 1Strategy 0,2

0 5 10 15−0.5

0

0.5

1Mature Hard Market

Strategy 3Strategy 1Strategy 0,2

0 5 10 15−0.5

0

0.5

1Immature Soft Market

Strategy 3Strategy 1Strategy 0,2

0 5 10 15−0.5

0

0.5

1Mature Soft Market

Strategy 3Strategy 1Strategy 0,2

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with a capital conservation strategy (strategy 2) or using a mixed strategy (strategy 3), will have

variable claims rates (i.e. variable market shares), all else being equal.

Figure 3: Evolution of the claims rate for different initial states of the underwriting cycles

(as a function of time)

We use the following arbitrary parameters values: λ = 1; α = 0.5; β = 1; δ = 0, 0.5 and 1. The graphs show the evolution of the claims rate as a function of time for 15 years. The left upper graph assumes that the cycle starts in immature hard market and the right upper graph assumes mature hard market. The bottom left graph supposes the process starts in immature soft market and the bottom right assumes mature soft market as initial state. For each graph, there are four strategies represented: strategy 0 (no underwriting cycle), strategy 1 (maintain market share), strategy 2 (conserve capital) and strategy 3 (mixed strategy).

3.3. Dynamics of the insurer’s assets and liabilities

We use the processes proposed by Cummins (1988) to model the insurer’s assets and

liabilities. We denote by SI(t) the total liability of the insurer at time t; its dynamic is as follows:

I S I I I S I SdS t S t t S t dt S t dW t , (3)

where γI(t) is the amount of new claims at time t with process:

))((

1

)(tN

iiI

I

Xt

, (4)

0 5 10 150.5

1

1.5Immature Hard Market

Strategy 0,1Strategy 2Strategy 3

0 5 10 150.5

1

1.5Mature Hard Market

Strategy 0,1Strategy 2Strategy 3

0 5 10 150.5

1

1.5Immature Soft Market

Strategy 0,1Strategy 2Strategy 3

0 5 10 150.5

1

1.5Mature Soft Market

Strategy 0,1Strategy 2Strategy 3

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with N(λI(t)) the loss frequency with intensity λI(t) and severity of claims X. The distribution of

these variables will be provided later in the calibration section. μS is the inflation rate of claims

and is the instantaneous variance of claims per unit of time. WS(t) is a Wiener process. The

total liability SI(t) increases with inflation and the arrival of new claims, and decreases when claims

are paid (SI).

We denote by AI(t) the value of the insurer’s total asset at time t with dynamic given as

follows:

I A I I I A I AdA t A t P t S t dt A t dW t , (5)

where PI(t) is the premium loading at time t and its process is:

1I I IP t t E t , (6)

with μA the instantaneous rate of return on the assets, the instantaneous variance of assets’

returns. WA(t) is a Wiener process non correlated with the Wiener process WS(t). The insurer’s

total asset AI(t) is assumed to grow with its investments and the premium raised, and decrease with

the claims payments.

We claim that the main risks facing the insurer in our framework are: technical risk (when

claims are higher than expected), financial risk (losses from financial investments) and

underwriting risk (due to changes in insurance cycle). When claims are superior to total asset value,

the insurer realises a loss:

L(t) = S(t) – A(t). (7)

A positive value of L(t) then materialises a loss, and a negative value is a gain/surplus for the

insurer. A good year of activity will depend on the strategy adopted by the insurer.

IV. Economic capital and performance indicators

In the above section, we described the dynamics of the insurer’s assets and liabilities using

the model proposed by Cummins (1988). We also used the modified version of the actuarial model

of Jones and Ren (2006) to capture the underwriting cycle dynamic. We will use these dynamics

to study the impact of underwriting cycles on the economic capital and performance of the insurer.

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4.1. Economic capital and risk measures

Economic capital is the risk-based capital needed by an insurer to support unexpected

losses from its activities. It is a capital buffer beyond the capital provision used to mitigate

insolvency risk. Economic capital creates value for the insurer since it decreases the financial

distress costs, improves the credit rating and reinforces the confidence of consumers of insurance

products and regulators. It can be obtained as follows:

, (8)

where CVaR, the conditional value-at-risk, is the risk measure applied to the insurer’s unexpected

losses, r is the risk-free rate used as discount factor, and T is the period. The value at risk (VaR) is

defined as the minimum loss amount at a given confidence level over a given time horizon T.

Suppose that the loss L has a distribution function FL(x), the VaR at the confidence level ω is

defined by:

, ∈ 0,1 . (9)

The CVaR is the expected value of losses above the VaR, hence, CVaR at a confidence level ω is

defined by:

| , ∈ 0,1 . (10)

4.2. Performance measures

We use two different measures to analyse the performance of the insurer: the probability

of ruin and the risk-adjusted expected return.

Probability of ruin. Ruin or default arrives when the insurer is unable to honor its obligations. The

probability of ruin will be used to analyse the solvency of the insurer. The probability of ruin or

the default probability of the insurer is the probability that the amount of claims is superior to the

value of total asset, i.e.:

Pr Pr 0 , 0. (11)

Risk-adjusted expected return. The profit R of the insurer is the total premiums raised plus the

investments returns minus the claims payments. The expected return is thus:

0 . (12)

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The risk-adjusted performance measure (RAPM) is obtained by dividing the expected profits by

the economic capital CVaR (the risk measure) as follows:

. (13)

V. Application to natural catastrophic risks

We consider the following natural catastrophes: earthquake, hurricane, tornado and flood.

We first start by modeling these natural catastrophic risks and after we apply our above model to

an insurance company insuring these catastrophic events. We also analyse the effect of the

portfolio diversification on our results.

5.1. Modeling natural catastrophic risks

The composed Poisson distribution and the diffusion processes with jumps are the two

main models generally used in the literature to model natural catastrophic risks. For instance, Lee

and Yu (2007) and Jaimungal and Wang (2006) used a pure Poisson process with intensity

parameter λ to measure the arrival rate of natural catastrophic events in a composed Poisson model.

Wu and Chung (2010) and Lin et al. (2009) used similar model but assume the frequency of the

natural catastrophic events to follow a Poisson process with stochastic intensity. Cummins (1988)

and Vaugirard (2003a, 2003b) used a diffusion process with jumps where the jump process has a

log normal distribution. Chang and Hung (2009) modeled the magnitude of jumps using a double

exponential function.

We use a composed Poisson model because it fits well with the structure of our data. For

that, let’s consider {Lt : t ≥ 0} the total loss of the insurer due to natural catastrophic events. We

have:

( )

1

N t

t kk

L l

,

where {N(t) : t ≥ 0} represents the frequency of natural catastrophic events and follows a

homogeneous Poisson process with intensity λ; lk is the amount of loss caused by the kth

catastrophe. The random variables {lk, k = 1, 2,…, N(t)} are independent and identically

distributed. We assume that the claim amounts {lk} and the frequency of claims are independents.

The data necessary for the calibration of the model are given in Table A1. The table contains the

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number and amounts of damages caused by natural disasters such as earthquakes, hurricanes,

tornados, and floods from 1960 to 2008 in the United States.

Modeling the loss frequency

We estimate the parameters of three selected distributions: the negative binomial, the

Poisson process and the binomial distribution. Table 1 presents the estimation results with the

maximum likelihood technique. For earthquakes and hurricanes, the goodness of fit to the data

with the Pearson Chi-Square statistics for the three distributions suggest that they are all

appropriate for hurricanes, but only the Poisson process and the binomial distributions fit with

earthquakes data. For tornados and floods, none of the distributions seems to fit the data very well.

We retain the Poisson process N(t) with intensity parameter λ.

Table 1: Estimation of the parameters of the loss frequency distributions

A. Parameters’ estimates

Distribution Parameters Earthquake Hurricane Tornado Flood

Binomial Negative

r inf 44.4025 2.9426 1.9624

1 0.9727 0.5297 0.3303

Poisson Process

λ 0.5918 1.2449 2.6122 3.9796

Binomial n 3 5 9 9

p 0.1973 0.2490 0.2902 0.4422

B. Goodness of fit using the Chi-Square test

Earthquake Hurricane

Chi^2 test Poisson Bin. Nég. Binomial Poisson Bin. Nég. Binomial

(k,r) (3,1) (3,2) (3,1) (4,1) (4,2) (4,1)

χ2obs 1.72 . 0.29 2.81 2.93 2.71

Critical value (95%) 3.84 . 3.84 5.99 3.84 5.99

Validation GOOD . GOOD GOOD GOOD GOOD

Tornado Flooding

Chi^2 Test Poisson Bin. Nég. Binomial Poisson Bin. Nég. Binomial

(k,r) (4,1) (4,2) (4,1) (5,1) (5,2) (5,1)

χ2obs 18.34 10.06 33.99 70.19 71.34 207.18

Critical value (95%) 5.99 3.84 5.99 7.81 5.99 7.81

17

Validation BAD BAD BAD BAD BAD BAD

Modeling the loss severity

We estimate the parameters of two distributions for the severity of losses: the log-normal

distribution and the Pareto distribution. The parameters estimates using the maximum likelihood

method are given in Table 2, as well as the Kolmogorov-Smirnov statistics for goodness of fit.

Although the two distributions seem appropriate to capture the severity of losses, the Pareto

distribution tends to overestimate the value-at-risk (VaR), hence the economic capital. We will

therefore use the log normal distribution to simulate our losses from catastrophic events.

Table 2: Estimation of the parameters of the loss severity distributions

A. Parameters estimates

Distribution Parameters Earthquake Hurricane Tornado Flood Log normal distribution (κ,τ2)

Κ 18.73 20.28 18.74 20.77

τ2 4.68 3.11 0.83 1.22

Pareto (a, alpha) a 12 205 172 20 763 786 10 399 875 24 073 646

alpha 0.42 0.29 0.39 0.49

B. Goodness of fit using the Kolmogorov-Smirnov (K-S) test

Distribution K-S test Earthquake Hurricane Tornado Flood

Log normal distribution (κ,τ2)

D-max 0.20 0.13 0.13 0.11

Critical value (95%) 0.25 0.17 0.12 0.10

P-value 0.16 0.26 0.02 0.02

Pareto (a, alpha)

D-max 0.12 0.28 0.38 0.29

Critical value (95%) 0.25 0.17 0.12 0.10

P-value 0.76 0.00 0.00 0.00

Using the distributions and the calibrated values for the parameters, we simulate a sample

of natural catastrophic events using one million Monte Carlo simulations. The summary statistics

from the simulation sample are provided in Table 3. Earthquakes and Hurricanes have high

coefficients of variation. They are more risky than the other natural catastrophes.

18

Table 3: Summary statistics

Earthquake Hurricane Tornado Flood

Mean 830 732 348 3 748 587 348 543 425 843 7 680 294 084

Median 0 577 098 278 416 216 952 5 885 000 733

Maximum 4 019 519 784 589 2 777 940 255 121 11 444 593 465 353 934 597 119

Standard Deviation 10 882 007 305 14 668 217 263 508 938 426 7 083 790 018

Coefficient of variation 12.95 3.91 0.94 0.92

Skewness 153.01 33.14 2.12 3.09

Kurtosis 41 016 3 065 13 32

We can notice in Figure 4 below that earthquakes and hurricanes have the most fat tail loss

distributions; it is therefore more probable to pay high claims amounts when assuring these two

types of catastrophic risks. As a matter of fact, natural events with low frequency of occurrence

but with high severity of losses seem to be riskier than those more frequent with low severity of

losses.

Figure 4: Probability of loss exceedance

The graph presents the probability of loss exceedance for a sample of 1,000,000 simulated catastrophic events for each catastrophic risk. For the simulations, the loss frequency follows a Poisson process with parameters given in Table 1. The loss severity follows a log-normal distribution with parameters given in Table 2. The data used for the calibration are provided in Table A1.

0 2 4 6 8 10

x 1010

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Loss

Pro

babi

lity

of lo

ss e

xcee

danc

e

EarthquakeHurricaneTornadoFlood

19

5.2. Analyzing the impact of underwriting cycles (with constant intensity of the cycle)

In this section, we analyse the impact of underwriting cycle on our four natural catastrophic

risk events assuming the cycle intensity is constant. Taking into account the cycle and the above

models used for natural disasters, the amount of new claims at time t described in (4) becomes:

))((

1

)(tN

iiI

I

Xt

with N(λI(t)) a non-homogeneous Poisson process with intensity λI(t) and severity of claims X.

Each Xi follows a log normal process with parameters κ and τ2, i.e. log N(κ,τ2).

Recall the aim of this paper is to study the impact of insurance cycles on the economic

capital and performance of an insurer with respect to the strategy adopted and the underwriting

cycle. On the one hand, we want to analyse the impact of the underwriting cycle on the economic

capital of the insurer; and on the other hand, we wish to determine which strategy performs better.

We evaluate the performance of the three strategies (1, 2 and 3) and the simulation results are

presented in Table 4.

INSERT TABLE 4.

Impact on the insurer’s economic capital

In general, the required capital increases with the degree of losses from natural disasters.

The simulation results show that the economic capital required for a strategy during the cycle

depends on the pattern of the loss ratio. As we can see for the case of floods, the pattern of the

CVaR and the loss ratio given in Figure 5 are quite similar. The higher the loss ratio, the greater

the economic capital is. This result remains valid for other types of natural disasters. This means

that in terms of economic capital, a strategy will be better if its loss ratio is low. The evolution of

the loss ratio depends on the initial phase of the cycle and the strategy adopted.

Hence, in the case of earthquakes, hurricanes and tornados, the required economic capital is

lower for the capital conservation strategy (2) when the insurer starts in “immature hard market

(0)” or “mature soft market (3π/2)”. This strategy has the lowest loss ratio for all catastrophes

considered. If the insurer enters into “mature hard market (π/2)” or “immature soft market (π)” It

20

appears most optimal to adopt a maintaining market share strategy (1) by aligning its price to the

market price.

In the particular case of floods, the optimal strategy differs from that of other types of

disasters: The mixed strategy (3) is better when the insurer starts its activities in “immature hard

market (0)” or “immature soft market (π)”. Also, it seems more prudent to adopt strategy 1

(maintaining market share) in “mature soft market (3π/2)” and strategy 2 (capital conservation) in

“mature hard market (π/2)”. This demarcation from the other types of disasters can be explained

by its high initial claim rates, compared to other risks. This is an indication for the importance of

the frequency of claims in determining the least risky strategy.

Figure 5: Loss ratio and CVaR

Impact on the insurer’s performance

In terms of performance, return on assets and the inflation rate being constant, the

performance will depend mainly on the claims rates and the premiums collected, especially in

early periods. The performance increases with the premium and decreases with the claim rate. In

general, the most profitable strategy will be the one that maximizes the premium and minimizes

the claim rate during the first phase of cycle.

0,0000

0,5000

1,0000

1,5000

2,0000

2,5000

3,0000

3,5000

4,0000

2 1 3

Loss ratio

3π/2

π

π/2

0

 ‐

 20

 40

 60

 80

 100

 120

 140

 160

 180

2 1 3

Billion

CVaR 99%

3π/2

π

π/2

0

21

We observe that strategy 1 (maintaining market share) performs better when the insurer starts

in “immature soft market (π)” for all natural catastrophic risk events. Indeed, in “immature soft

market (π)”, the premium rate is equal to its initial level and then decreases. In this case, the

strategy of maintaining market share is the one minimizing the claim rate earlier in the period.

When the insurer begins in “mature hard market (π/2)”, the premium rate is maximal and

then decreases. In parallel, the claims rate is increasing at the beginning of the period. The best

strategy is the one that maximizes the premium rate (δ = 0) and minimizes the claims rate (δ = 1).

This cannot be done simultaneously. The results show that for less frequent natural disasters such

as hurricanes and earthquakes, it is more optimal to minimize the claims rate (δ = 1). Therefore

the strategy of maintaining market share is the best. While for more frequent disasters, the capital

conservation strategy is the best.

In “mature soft market (3π/2)” , the insurer must adopt the opposite approach by adopting a

capital conservation strategy for less frequent disasters and a strategy of maintaining market share

for the most frequent disasters such as tornados and floods.

When the insurer starts in “immature hard market (0)”, the strategy of maintaining market

share must be excluded as it maximizes the claim rate. Then, the choice will be made between the

strategy of capital conservation and the mixed strategy. In this case, the determining factor seems

to be the average loss. Indeed for disasters whose average loss is sufficiently high (e.g. hurricanes,

floods), it is appropriate to choose a capital conservation strategy. Otherwise, for disasters whose

average loss is relatively small (e.g. earthquakes, Tornados), the mixed strategy should be

preferred.

5.3. Analyzing the impact of a time-varying underwriting cycle

The amplitude of the underwriting cycles in recent years has exhibited a downward trend.

The convergence of capital and traditional reinsurance markets, financial regulations and

sophistication of pricing techniques, justify this trend. It looks like the amplitude of the

underwriting cycles will continue to decline in the coming years. We therefore re-run our

simulations with a time decreasing intensity function of the underwriting cycle. For that purpose,

22

we assume α t 1 √⁄ . Figure 6 shows the downward trend of the claims rate and the security

loading when the cycle starts in mature hard market.

Figure 6: Evolution of the claims rate and security loading with mature hard market as the

initial state.

We use the following arbitrary parameters values: θ = 0.2, λ = 1, α t 1 √⁄ , β = 1, δ = 0.5. The graphs show the evolution of the claims rate and the security loading as a function of time for 15 years. The left graph represents the claim rate and the right graph represents the security loading when the cycle starts in mature hard market.

The simulation results of the impact of the time dependent intensity cycle on the economic

capital and performance of the insurer are presented in Table 5 (for the case of hurricanes). Two

main results emerge from these simulations:

- Whatever the strategy, when varying the intensity down, the required capital decreases,

while performance increases. 

- The order of the strategies previously established changes because of the cycle intensity

differential effects over time. As illustration, we can notice in the case of the hurricane

that the mixed strategy is the most optimal in terms of economic capital in “immature

hard market (0)” or “mature hard market (π/2)”. The capital conservation strategy is

better in “immature soft market (π)” or “mature soft market (3π/2).”

These results show how the intensity of the underwriting cycle influences the optimal

strategy to follow. It is therefore important for the insurer to estimate this variable accurately

before adopting any strategy.

INSERT TABLE 5.

2 4 6 8 10 12 14

0.6

1.0

1.4

1.8

t

Mature hard market

2 4 6 8 10 12 14

0.47

0.49

0.51

t

secu

rlo

ad

(t) 

cla

imr(

t) 

Mature hard market

 

23

Can the cycle be advantageous?

Overall, the performance of the best strategy in the presence of the cycle is still higher than

that of the benchmark strategy 0 (no underwriting cycle).This order is not necessarily verified

when comparing strategy 0 to another strategy different from the most efficient. This means that

the existence of the cycle represents a major risk for the insurer. However the presence of the cycle

gives the insurer the opportunity to increase its performance, provided that it chooses the right

strategy. Therefore, better timing of insurance cycles seems to be beneficial for the insurer than

strategy 0 (i.e. assuming no insurance cycle) used as benchmark.

The choice of the right strategy depends on the purpose of the insurer. If the goal of the

insurer is to reduce the required capital, then it should pay attention to its loss ratio. It is possible

for the insurer to manage its activities from this indicator. In that case, the most important factor

is the frequency of claims. If the insurer has rather interest to increase its performance, the choice

of the optimal strategy will depend heavily on the frequency of the natural disaster and its average.

5.4. The effects of risk diversification

Diversification with no dependency between disaster events

Insurance is traditionally based on the principle of risk mutualisation. It consists of pooling

individual risks in a single portfolio in order to reduce total risk through portfolio diversification.

For non-catastrophic risks, it is proven that an insurer can reduce its total risk by diversifying or

by increasing the number of policyholders in the portfolio. However, for catastrophic risks, which

are very different by nature from non-catastrophic or financial risks, what will happen when we

pool the risks from different types of natural catastrophic events?

To answer this question, we use two risk measures to quantify the diversification effect:

the probability of ruin and the conditional value-at-risk (CVaR). In an ideal world, pooling risks

should reduce the value of these two risk measures, all else being equal.

Figure 7 shows the evolution of the probability of ruin when more catastrophic events of

similar nature are pooled together for diversification purpose. We observe that for almost all four

natural disaster events, the ruin probability increases when the number of portfolio increases up to

nearly 50 elements. When the number of portfolio elements reaches 50, the probability of ruin

24

starts to decrease for earthquake. For tornado and flood, the probability of ruin becomes more or

less flat after 50 portfolio elements, and for hurricane it continues to increase but at a slower pace.

Figure 7: Probability of ruin as a function of the number of policyholders

The graph presents the probability of ruin as a function of the number of portfolio elements. The data used for the calibration are provided in Table A1.

Table 6 presents the CVaR and the probability of ruin for different number of portfolio

elements for each catastrophic event. For all four natural disaster risk events, the CVaR decreases

with the number of portfolio elements, even though the probability of ruin increases. This means

that with more portfolio elements, although there is more likelihood of ruin within the portfolio,

the unexpected loss is less on average when the portfolio is more diversified.

0 20 40 60 80 1000.01

0.011

0.012

0.013

0.014

0.015

0.016

0.017

0.018

0.019

0.02

Number

Rui

n pr

obab

ility

Earthquake

Hurricane

Tornado

Flood

25

Table 6: CVaR and probability of ruin as a function of the number of policyholders

The table presents the probability of ruin and the conditional value-at-risk (CVaR) as a function of the number of portfolio elements for the four natural disaster risks. The data used for the calibration are provided in Table A1.

CVaR 95% (in $ million)

N Earthquake Hurricane Tornado Flood

1 13531.00 40673.00 2030.90 28634.00

5 9003.50 23658.00 1131.40 16103.00

10 7367.60 18077.00 933.90 13342.00

20 5950.90 14050.00 808.11 11499.00

30 5447.70 12245.00 756.18 10720.00

50 4300.50 10143.00 705.53 9967.10

100 3631.60 8270.40 655.79 9261.10

Ruin probability

1 0.0102 0.0131 0.0175 0.0166

5 0.0120 0.0133 0.0182 0.0172

10 0.0121 0.0143 0.0186 0.0177

20 0.0126 0.0143 0.0189 0.0178

30 0.0121 0.0144 0.0189 0.0185

50 0.0135 0.0149 0.0194 0.0186

100 0.0125 0.0153 0.0194 0.0188

For further analysis, we combine the four risk events in the same portfolio, i.e. a portfolio

composed of earthquake, hurricane, tornado and flood risks events. Our objective is to analyse the

impact of diversification by increasing the number of policyholders based on the probability of

ruin. From Figure 8, we observe that as the number of elements increases in the portfolio, the

probability of ruin decreases after 50 elements. It has to be noted that, in practice, insurers try also

to hedge their natural catastrophic risks using financial markets alternative risk transfer

instruments. For that purpose, they will issue insurance linked securities based on the insurance

risks in order to transfer their catastrophic risk exposure to capital market participants, and thus

create additional risk capacity. Catastrophic (cat) bonds are good examples of these types of

insurance linked securities.

26

Figure 8: Probability of ruin as a function of the number of natural catastrophic events in

the portfolio

The graph presents the probability of ruin as a function of the number of natural catastrophe events in the portfolio.

Diversification with dependency between disaster events

The above portfolio diversification analysis assumes no correlations between the natural

disasters. However from past experiences, there may exist a correlation between hurricanes and

floods occurrences, which may affect the diversification outcome. Indeed, correlation between

hurricanes and floods seem to be the most likely among the four natural disasters, since the

hurricane is usually followed by flooding. This has the effect of generating additional capital

needs.

In order to measure this positive dependence, a Gumbel copula is used instead of the standard

Pearson correlation coefficient, in order to appropriately account for the lognormal distribution

property of the losses. Indeed, Pearson correlation coefficient is relevant only in the presence of

elliptical distributions (i.e. multivariate Normal or Student distribution) or linear dependence. The

Gumbel copula is appropriate to model the dependence between these two natural disasters risks

(hurricanes and floods). This copula is a part of the Archimedean family and has the particularity

to model positive dependencies and dependencies in the tails of the distributions. It is obtained as

follows.

0 20 40 60 80 1000.012

0.0125

0.013

0.0135

0.014

0.0145

0.015

0.0155

0.016

Number

Rui

n pr

obab

ility

27

Let’s Cb be the Gumbel copula, and DH and DF, represent, respectively, the risks associated

to hurricanes and floods. The probability distribution of the pair (DH, DF) can be written as follows:

Pr , , , (14)

with FH and FF the distribution functions of DH and DF, respectively; and

, ln ln . (15)

The value of the parameter b determines the dependence structure, i.e. positive or negative.

More concretely, the dependence will be measured by the Kendal correlation coefficient. The

Kendal rate for a Gumbel copula is expressed as follows: 1 .

To estimate these parameters, we use the following 4 estimation methods: (1) the Moment-

Based Estimation (MBE); (2) the Exact Maximum Likelihood (EML); (3) the Inference for

Margins (IFM); and (4) the Canonical Maximum Likelihood (CML). The estimation results are

summarized in Table 7 below. These estimations provide a mean value of 1.4691039. This shows

a positive dependence between the two risks (hurricane and flood) in our sample. The Kendal rate

( = 0.319) highlights a strong dependence between hurricane and flood losses.

Table 7: Calibration of Gumbel copula

Parameters Estimation methods

MBE EML IFM CML Mean value

b 1.1733643 1.7065692 1.7091422 1.28734 1.4691039

0.1477498 0.4140290 0.4149112 0.2232044 0.3193130

Table 8 compares the CVaR of portfolios in which the four risks are assumed to be

independent and the CVaR of portfolios taking into account the dependency between hurricanes

and floods occurrences. As expected, the CVaR of the aggregate portfolio is lower than the sum

of the CVaR from the portfolio elements taken individually. On the one hand, the aggregation of

independent risks has reduced the economic capital by almost 40% on average. On the other hand,

the capital requirement (CVaR) of the aggregate portfolio with positive dependency between

hurricanes and floods is higher than the one without dependency, but remains below the sum of

the individual CVaRs. In this latter case with dependency, economic capital represents on average

28

85% of the sum of the CVaRs, hence only a 15% gain in economic capital from diversification.

The consideration of the dependence has increased significantly the capital requirement; it almost

brings back the capital close to the same level as if there had been no diversification, hence,

reducing the diversification opportunities.

Table 8: Impact of the dependency on the insurer’s economic capital

CVaR 95% (in $ million) Sum of individual CVaRs 95% (in $ million)

N Portfolio with no

dependence Total Portfolio with

dependence

1 51,588.87 72,288.99 84,868.90 5 30,022.50 42,727.87 49,895.90

10 23,663.06 33,968.85 39,720.50 20 19,123.17 27,563.01 32,308.01 30 17,178.70 24,877.08 29,168.88 50 14,873.32 21,339.62 25,116.13

100 12,953.23 18,538.07 21,818.89

29

VI. Conclusion

The goal of this paper is to study the economic capital and the solvency of an insurer in the

presence of underwriting cycle and natural catastrophic risk. Our objective was to propose a

pricing model to answer the following questions: (1) How to model an insurer’s risks when facing

insurance cycle? (2) What is the impact of insurance cycle on the performance of an insurer in the

case of catastrophic risks? (3) What is the effect of portfolio diversification in the case of natural

catastrophic risks?

To respond to these questions, we present a stochastic model which accounted for the effect of

underwriting cycles. The model included a parameter capturing the strategy adopted by the insurer

under different cycles. We use the probability of ruin and the CVaR- adjusted returns to capture

the performance of the insurer. We use historical natural catastrophic events data of the United

States from 1960 to 2008 for the empirical analyses.

The application of our model to the four catastrophic events considered (earthquake, hurricane,

tornado and flood) show different patterns depending on the type of events. A better timing of

insurance cycle seems to be beneficial for the insurer. Moreover, we show that portfolio of natural

disasters needs to be composed of at least 50 elements to obtain the diversification benefit.

There are many ways this paper can be extended in the future. Our analysis studies the

importance of insurance cycle and tests the effect of the different business strategies. A possible

extension of our model can be the determination of an optimal strategy in order to maximize the

benefit of the insurer under the underwriting cycles. Indeed, there is an endogeneity between the

underwriting cycle itself and the business line. Also, policyholders can have different behaviour

(loyalty) depending on the type of insurance. Our model is a static model, which assumes that the

insurer does not change its strategy (δ) during the evolution of the insurance cycle. A natural

extension is to redo the analysis using a dynamic model in which the insurer can modify/adjust its

strategy during the underwriting cycle.

30

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Zhu, L. (2008), Double Exponential Jump Diffusion Model for Catastrophe Bonds Pricing, Journal of Fujian University of Technology 2008-04.

33

Table A1: Insurance losses dues to natural catastrophes

This table contains data on natural disasters losses incurred by the insurance industry in the United States over the period 1960-2008.

Year Number Damage Number Damage Number Damage Number Damage1960 0 0 1 37 141 583 1 74 283 167 0 0

1961 1 33 427 425 1 37 141 584 0 0 1 37 141 584

1962 0 0 0 0 0 0 0 0

1963 0 0 1 34 666 852 1 34 666 852 1 34 666 851

1964 1 3 744 019 968 2 1 109 339 249 1 69 333 703 1 693 337 031

1965 1 86 667 129 1 3 848 020 525 3 4 069 888 373 1 693 337 030

1966 0 0 1 33 112 379 2 422 489 438 1 64 998 375

1967 0 0 1 974 975 611 2 532 986 675 1 32 499 190

1968 0 0 2 631 626 173 1 152 942 616 0 0

1969 1 404 437 254 1 4 663 428 782 1 28 888 375 1 57 776 751

1970 0 0 1 266 485 362 1 27 368 767 0 0

1971 1 2 599 968 800 1 311 996 252 1 10 399 875 0 0

1972 0 0 1 259 996 880 2 67 599 189 1 389 995 320

1973 1 27 733 756 0 0 2 398 672 841 1 3 070 523 073

1974 0 0 1 20 763 786 2 1 966 243 316 1 124 324 956

1975 2 40 400 000 1 200 000 000 1 740 000 000 4 140 000 011

1976 0 0 1 38 517 834 1 308 142 670 4 96 294 584

1977 0 0 0 0 1 125 515 510 0 806 885 243

1978 1 48 750 366 0 0 1 243 751 828 4 845 006 468

1979 0 0 2 2 451 417 817 1 415 998 098 7 1 954 819 647

1980 1 5 199 937 601 1 1 299 984 660 1 480 994 228 6 662 991 966

1981 0 0 0 0 2 236 361 917 6 1 359 079 367

1982 0 0 1 110 639 051 2 276 597 650 7 493 450 154

1983 3 108 332 972 1 1 083 329 722 1 205 832 647 5 1 752 672 770

1984 1 15 599 975 1 31 199 950 2 308 532 916 8 374 399 401

1985 0 0 5 1 483 333 146 1 190 000 000 3 2 018 621 585

1986 0 12 205 172 0 0 1 98 112 319 5 561 202 366

1987 1 661 813 369 0 0 0 0 4 567 268 866

1988 0 0 0 0 4 173 332 360 0 0

1989 1 10 226 721 209 2 3 882 687 374 1 242 667 961 0 0

1990 1 20 965 053 0 0 2 93 599 881 7 702 962 130

1991 1 52 787 495 1 236 361 917 2 173 332 073 2 338 785 416

1992 2 253 880 860 1 2 294 104 152 1 168 234 304 2 871 759 578

1993 2 53 336 899 0 0 1 74 285 375 7 2 699 892 787

1994 1 28 888 792 593 1 72 221 866 1 158 888 359 5 246 999 177

1995 0 0 3 4 959 664 952 4 265 621 047 6 455 350 367

1996 0 0 3 4 878 415 917 5 725 672 920 8 1 777 249 846

1997 0 0 1 80 666 667 6 434 666 667 6 8 805 976 800

1998 0 0 2 2 365 832 039 7 1 695 584 634 8 2 977 328 159

1999 0 0 3 5 276 577 780 3 2 283 712 284 7 1 450 313 950

2000 1 62 650 361 0 0 6 266 289 094 7 1 371 416 399

2001 1 2 418 613 651 2 6 187 902 095 9 485 052 968 7 917 138 296

2002 1 23 908 002 2 821 000 789 8 746 646 903 6 569 010 448

2003 1 350 561 483 2 73 267 350 7 1 379 668 671 8 2 358 460 801

2004 0 0 4 22 292 982 859 4 332 342 857 7 1 284 800 000

2005 0 0 4 55 074 625 214 4 667 809 924 7 1 368 789 323

2006 1 78 268 235 0 0 6 585 510 727 9 3 594 549 100

2007 0 0 1 93 011 578 6 480 919 255 6 1 008 122 387

2008 0 0 1 3 284 352 503 6 606 210 000 6 10 561 520 000

FLOODING (USA)EARTHQUAKE (USA) HURRICANE (USA) TORNADO (USA)

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Table 4: Analysis of the performance with respect to the initial state of the cycle

The parameters values are: intensity of the cycle α=0.5, clients’ fidelity β=1, mean loading rate θ=0.2, horizon T=6 years, initial asset value A0=3λE[X], asset mean return μA = 0.035, asset return volatility A = 0.10, η=0.1, initial liability S0=25000, liability mean return μS=0.045, liability return volatility S=0.15.

A. Earthquake (mean intensity λ=0.59183, loss severity distribution parameters: log normal N(κ,τ2), κ=18.73, τ2=4.68)

CVaR 99% (in $ )

Strategies

Initial state (I) 0 2 1 3

0 3 611 865 833 1 696 299 188 5 645 530 846 6 319 351 576 π/2 3 611 865 833 3 611 865 833 3 020 703 564 3 329 347 396 π 3 611 865 833 15 113 535 293 1 780 892 170 5 770 949 987

3π/2 3 611 865 833 3 611 865 833 4 180 617 452 3 926 152 511

Average 3 611 865 833 6 008 391 537 3 656 936 008 4 836 450 368

CVaR-adjusted return (99%) Ruin probability

Strategies Strategies

Initial state (I) 0 2 1 3 0 2 1 3

0 0.0750 -0.4769 -0.3358 -0.0834 0.0280 0.0000 0.2870 0.1220 π/2 0.0750 0.0750 0.2620 0.1595 0.0280 0.0280 0.0140 0.0190 π 0.0750 0.1730 1.3688 0.6205 0.0280 0.1580 0.0000 0.0290

3π/2 0.0750 0.0750 -0.0596 0.0028 0.0280 0.0280 0.0520 0.0380

Average 0.0750 -0.0384 0.3088 0.1748 0.0280 0.0713 0.0883 0.0520

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B. Hurricane (mean intensity λ=1.24489, loss severity distribution parameters: log normal N(κ,τ2), κ=20.281, τ2=3.10)

CVaR 99% (in $ million)

Strategies

Initial state (I) 0 2 1 3

0 19 317 554 025 28 974 487 684 29 591 443 790 47 624 735 507

π/2 19 317 554 026 19 317 554 025 15 929 126 009 17 907 505 204

π 19 317 554 027 48 213 626 276 9 535 295 888 26 897 363 611

3π/2 19 317 554 028 19 317 554 025 22 036 075 573 20 643 355 250

Average 19 317 554 027 28 955 805 503 19 272 985 315 28 268 239 893

CVaR-adjusted return (99%) Ruin probability

Strategies Strategies

Initial state (I) 0 2 1 3 0 2 1 3

0 0.1226 0.1264 -0.2980 0.0673 0.0360 0.0670 0.3420 0.1640 π/2 0.1226 0.1226 0.3087 0.2034 0.0360 0.0360 0.0180 0.0260 π 0.1226 0.1401 1.4218 0.3705 0.0360 0.1380 0.0010 0.0390

3π/2 0.1226 0.1226 -0.0081 0.0530 0.0360 0.0360 0.0670 0.0500

Average 0.1226 0.1280 0.3561 0.1736 0.0360 0.0693 0.1070 0.0698

36

C. Tornado (mean intensity λ=2.6122449, loss severity distribution parameters: log normal N(κ,τ2), κ=18.74, τ2=0.83)

CVaR 99% (in $ million)

Strategies

Initial state (I) 0 2 1 3

0 2 475 719 133 1 772 666 778 3 073 693 833 4 154 967 332 π/2 2 475 719 134 2 475 719 133 2 409 868 655 2 479 081 393 π 2 475 719 135 3 784 694 792 1 834 831 849 3 019 277 933

3π/2 2 475 719 136 2 475 719 133 2 513 604 877 2 481 928 540

Average 2 475 719 135 2 627 199 959 2 457 999 804 3 033 813 800

CVaR-adjusted return (99%) Ruin probability (%)

Strategies Strategies

Initial state (I) 0 2 1 3 0 2 1 3

0 -0.0029 -0.0535 -0.2269 0.0058 0.0270 0.0080 0.1060 0.0950 π/2 -0.0029 -0.0029 -0.0327 -0.0173 0.0270 0.0270 0.0280 0.0280 π -0.0029 0.0531 0.3722 0.1178 0.0270 0.0610 0.0060 0.0400

3π/2 -0.0029 -0.0029 0.0256 0.0115 0.0270 0.0270 0.0290 0.0270

Average -0.0029 -0.0016 0.0346 0.0295 0.0270 0.0308 0.0423 0.0475

37

D. Flood (mean intensity λ=3.97959, loss severity distribution parameters: logn normal N(κ,τ2), κ=20.768, τ2=1.222)

CVaR 99% (in $ million)

Strategies

Initial state (I) 0 2 1 3

0 39 685 940 350 40 857 846 577 44 224 042 223 38 331 932 801 π/2 39 685 940 350 39 700 355 418 40 419 104 059 40 205 426 704 π 39 685 940 350 21 987 304 803 34 728 096 036 17 818 084 885

3π/2 39 685 940 350 39 700 355 418 38 468 801 963 38 729 007 592

Average 39 685 940 350 35 561 465 554 39 460 011 070 33 771 112 996

CVaR-adjusted return (99%) Ruin probability

Strategies Strategies

Initial state (I) 0 2 1 3 0 2 1 3

0 -0.0157 0.0052 -0.1246 -0.0922 0.0420 0.0420 0.0760 0.0450 π/2 -0.0157 -0.0157 -0.0750 -0.0455 0.0420 0.0420 0.0470 0.0440 π -0.0157 -0.1585 0.1226 -0.1885 0.0420 0.0020 0.0230 0.0000

3π/2 -0.0157 -0.0157 0.0460 0.0150 0.0420 0.0420 0.0370 0.0400

Average -0.0157 -0.0462 -0.0077 -0.0778 0.0420 0.0320 0.0458 0.0323

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Table 5: Analysis of the performance with respect to the initial state of the cycle with time-varying intensity

The parameters values are: clients’ fidelity β=1, mean loading rate θ=0.2, horizon T=6 years, initial asset value A0=3λE[X], asset mean return μA = 0.035, asset return volatility A = 0.10, η=0.1, initial liability S0=25000, liability mean return μS=0.045, liability return volatility S=0.15. The intensity of the cycle is time varying as follows: 1 √ .⁄

B. Hurricane (mean intensity λ=1.24489, loss severity distribution parameters: log normal N(κ,τ2), κ=20.281, τ2=3.10)

CVaR-adjusted return (99%)

Strategies

Initial state (I) 0 2 1 3

0 19 317 554 025 27 158 619 622 23 745 906 996 25 623 130 682 π/2 19 317 554 026 19 317 554 025 18 486 276 426 16 421 179 740 π 19 317 554 027 14 236 251 012 14 727 101 417 15 085 364 591

3π/2 19 317 554 028 14 727 101 417 20 099 502 085 19 694 799 285

Average 19 317 554 027 18 859 881 519 19 264 696 731 19 206 118 575

CVaR-adjusted return (99%) Ruin probability (%)

Strategies Strategies

Initial state (I) 0 2 1 3 0 2 1 3

0 0.1226 0.1392 -0.1161 0.0195 0.0360 0.0620 0.1020 0.0760 π/2 0.1226 0.1226 0.1879 0.1423 0.0360 0.0360 0.0310 0.0320 π 0.1226 0.0780 0.5089 0.2678 0.0360 0.0180 0.0140 0.0170

3π/2 0.1226 0.6131 0.0858 0.1039 0.0360 0.0140 0.0430 0.0380

Average 0.1226 0.2382 0.1666 0.1334 0.0360 0.0325 0.0475 0.0408