An overview of the model

The general model has been described in the quoted above references. In this case study, the Tuscany region has been subdivided into M ≈ 300 sub-regions, which corresponds to the number of its municipalities. For each municipality j=1,2,...,M , number and types of buildings (and therefore their vulnerability), and number of built cubic meters are available.

These represent the so-called estimate W_j of the property values or "wealth" of the municipality j. Simulated in time and space, earthquakes ω₀,...,ω_t may occur at different municipalities, inside or outside the region, have random magnitudes and, therefore, affect a random number of municipalities. From data and models in the Petrini et al. report, a catastrophe generator has been created, based on the Gütenberg – Richter law and on the attenuation characteristics of the region (for example, see Figure 1, Aniello et al. 2000, 2000/a).

This enables the generator to calculate intensities and accelerations in each municipality. Of course, the generator could be easily adapted to incorporate different kinds of distributions, non-poissonian catastrophic processes, as well as micro-zoning within a municipality.

Fig. 1: Earthquake generator²

The municipalities affected at time t are indicated by a subset

ε

_t of municipalities j, M

j=1,2,..., . Petrini’s vulnerability relations between accelerations and losses according to the type (masonry or reinforced concrete), age and maintenance of the buildings are used to

estimate the number of cubic meters of destroyed properties. The economic loss of destroyed cubic meters of a building is defined as the cost for their reconstruction. Then it is possible to be independent of contingent pricing by considering the cost of reconstruction per cubic meters to be the monetary unit. In this way the simulation of time histories for possible earthquakes in the region produces the sets of economical losses, and enables the design of an insurance

programme.³

In its early version the Italian Design of Law 2793 (1998), to reduce the impact of natural disasters on the governmental budget, included in its Article 31bis provisions for an insurance programme against all natural hazards. It was intended not to make this insurance mandatory, but to make mandatory the extension of a fire insurance policy to all natural hazards, in a way similar to the French system quoted. In addition to tax incentives for such an insurance, it stipulated a maximum exclusion layer of 25%, the creation of a pool of insurance companies with a reserve fund corresponding to the annual average government payment for compensating losses (with some forms of state guarantee to be specified further), and linking of the premium to the premium for fire policy, rather than to the risks of a specific municipality. This article was withdrawn, and later proposals are still subject of discussion.

Starting from these principles, the case study intends to demonstrate how the model analyses and offers the decision-makers different policy options.

Let us assume that an insurance company (this might be a pool of companies or the government itself acting as an insurer) covers a fraction q (q=0.75 as in assumptions all owners buy the insurance) of earthquake losses. The rest, according to the Design of Law, would be

2 Unpublished and work still in progress at IIASA by S. Baranov, B. Digas.

3 It would also be possible to determine in which way preventive retrofitting could decrease the losses:

this is easily done by a consequent decrease of the vulnerability indices in the loss model. In this way it would be possible to study the interplay between structural measures and risk-sharing for an integrated risk management approach, and to design an insurance system linked to incentives for retrofitting of the built environonment

compensated by the state. The state would also be exposed to feed the reserve funds in case of excessive losses. This would in any case allow the government to save money with respect to usually paid compensations and to use the save for prevention measures for public infrastructures and cultural heritage.

The company has an initial fund or a risk reserve R⁰, which in general is characterised by a random variable dependent on past catastrophic events. To analyze necessary R⁰, we can set different values, e.g., R⁰ =

0

. Assume that the time span consists of t =0,1...,T, for depends on the event ω_t⁴ and the type of properties in j. The analytical structure of the probability distribution of the random variable R^t is intractable, therefore, the methodology relies on Monte Carlo simulation.

Usual actuarial approaches calculate their premiums with respect to the loss expectations.

Therefore this study considered two policy options based on similar principles:

Premiums based on the average damage over all municipalities (solidarity principle, bringing less exposed locations to pay premiums equal to more severely exposed ones, as in the spirit of the proposed insurance programme)

Location-specific premiums based on average damage in the particular municipality (risk-based).

However, stochastic optimisation allows the analysis of different criteria and takes into account location specific, dependent risks. As an example, a third policy option has been considered:

Premiums calculated in a way that equalises in a fair manner the risk of instability for the insurance company and the risk of premium overpayment for exposed persons.

For Option 3 in this study, the insurer maximises his profits taking into account the risks of his insolvency under the constraint on ‘fair’ premiums. ‘Fair’ premiums are defined in the following sense. Let municipalityj face losses (damages) L^t_j. Individuals from this municipality receive a compensation L^t_jq from the company when such a loss occurs. If W_j⁰ is the initial wealth (property), then municipality- j’s wealth at time t is

t

4 In a general model it may also depend on time dependent mitigation measures or deterioration of the built environment.

It is assumed that individuals (municipalities) maximise their wealth according to the distribution of cases when ν^tj <

0

^{, where} ν^t_j =L^t_jq−π^t_j

(

ω^t

)

. Therefore, the ‘fair’ (optimal) vector of premiums π =^t

(

π₁^t

,...,

π_m^t

)

will guarantee the given level of stability for the insurer by minimising both the risk of his insolvency and the risk of overpayments for municipalities.

Thus, in a risk based or market approach, the choice of premiums reflects a certain balance between insurance demand and supply and creates additional incentives for insurance, otherwise higher premiums may decrease profits by decreasing the number of municipalities able to pay these premiums.

Im Dokument Risk Management: Modeling and Computer Applications (Seite 37-40)