Empirical Bayes Credibility Models for Economic Catastrophic Losses by Regions

. Catastrophic events affect various regions of the world with increasing frequency and intensity. The number of catastrophic events and the amount of economic losses is varying in different regions of the world. Part of these losses is covered by insurance. Catastrophe events in last years are associated with increases in premiums for some lines of business. The article focus on estimating the amount of net premiums that would be needed to cover the total or insured catastrophic losses in different regions of the world using empirical credibility models.


Introduction
Based on Sigma Swiss Re terminology [19] the term natural catastrophe refers to an event caused by natural forces.Such an event generally results in a large number of individual losses involving many insurance policies.An event is classified as a catastrophe and included in the sigma database when insured claims, total losses or the number of casualties exceed certain thresholds, detailed in Table1.Source: Sigma Swiss Re, No 1/2016, [19].
The occurrences of catastrophic events are becoming more frequent (Fig. 1) and also grow indemnity of insurance and reinsurance companies at these events (Fig. 2).

Fig. 1. Number of catastrophes events 1970-2015
By sigma [19] there were 198 natural catastrophes in 2015, and 155 man-made disaster events.The estimated total economic losses from natural catastrophes and man-made disasters across the world were USD 92 billion in 2015.This is less than the USD 113 billion total loss in 2014, and is well below the inflationadjusted average of the previous 10 years (USD 192 billion).
The insurance industry covered almost USD 37 billion -less than half -of the total losses from natural and man-made disasters in 2015, and well below the inflationadjusted previous 10-year annual average of USD 62 billion.Natural catastrophes resulted in claims of USD 28 billion, the lowest since 2009 and again much lower than the previous 10-year inflation-adjusted annual average (USD 55 billion).Large man-made disasters led to claims of USD 9 billion, up from USD 7 billion in 2014.
Fig. 2 shows also the difference between insured and total losses over time period 1970-2015, termed the insurance protection or funding gap.It is the amount of financial loss generated by catastrophes not covered by insurance.In this figure it is evident growing trend both total and insured catastrophe losses in time period 1970-2015, determined by 10-years moving averages.The rate of growth of total losses has outpaced the growth of insured losses.Catastrophic events affect various regions of the world with increasing frequency and intensity.Large catastrophic events can be caused by natural phenomena or are caused by man.
The number of catastrophic events is also varying in different regions of the world.The catastrophic events by Sigma Swiss Re are monitored for seven regions -North America, Latin America, Europe, Africa, Asia, Oceania/Australia and Seas/space.Number of catastrophe events by regions in 2009-2015 present Table 2 and Fig. 3.The worldwide insurance industry has been rocked by the increasing catastrophes in recent years and increased demand for catastrophe cover (e.g., per occurrence excess of loss reinsurance), leading to a capacity shortage in catastrophe reinsurance.Catastrophe events in last years are associated with increases in premiums for some lines of business.Source: Own processing by Sigma Swiss Re, 2010-2016 Regional differences can be observed not only in the number of catastrophic events, but also in the amount of economic losses.Part of these losses is covered by insurance.The share of insured losses on total losses again varies by region.The collected premiums used to cover the insured losses not only in the region of insured, but if necessary, in other regions of the world.It is caused by the globalization of the insurance and reinsurance market.The article will therefore focus on estimating the amount of net premiums that would be needed to cover the total catastrophic losses in each region.For this purpose we will use empirical credibility models.

Application of Bühlmann credibility model
Credibility theory is a technique, or set of techniques, for calculating premiums for short term insurance contracts.The technique calculates a premium for a risk (in our case for a region) using two ingredients: past data from the risk itself and collateral data, i.e. data from other sources considered to be relevant, in our cases from other regions.The essential features of a credibility premium are that it is a linear function of the past data from the risk itself and that it allows for the premium to be regularly updated as more data are collected in the future [1,2,12,13,15,18].
A credibility premium represents a compromise between the two above mentioned sources of information.The credibility formula for estimation of pure premium Pc in next year is [6,7]: where Pr is estimation based on own past data in risk, or region, μ is estimation based on collateral data and Z is a number between zero and one, known as the credibility factor.Credibility factor Z is a measure of how much reliance the company is prepared to place on the own data [8,9,10].
Empirical Bayes credibility theory is the collective name for the vast literature which has developed since Bühlmann [2] and Bühlmann and Straub [3].
The form of empirical Bayes credibility premium as a result of Bühlmann credibility model has been derived in [2] as with factor credibility In empirical Bayes credibility model we use the available data (Tab.3) to estimate the quantities , and hence obtain a Bühlman type credibility premium estimate [2] for a particular risk.
Derived relations necessary for estimates are: The derivation of these relations can be found in Bühlmann [2] and Waters [18].
Table 3 and Fig. 4 give present the total economic losses (in USD billion) to each of seven regions of the world over seven years..These values contains second column in Table 5.  4) -( 6) we have estimated Therefore using (3) we have calculated the credibility factor Z which is the same for each region: The credibility premium for each of seven regions in USD billion by ( 2) is in last column of Table 6.

Application of Bühlmann -Straub credibility model
Contrary to the Bühlmann model in section 3 Bühlmann-Straub credibility model accounts for the varying annual volume of business, as is for example observed number of claims.We have applied this model on insured catastrophe losses (Table 4) and figure 5 with acceptation of the number of catastrophe events (Table 2).
Let region, for which the estimates of insurance catastrophic loses are computed, is one of N regions.Insured losses in previous n years are known for all these regions.
Let P ij is a variable, describing number of catastrophic events of the i-th observed region (i = 1,…, N; N = 7) in the j-th year (j = 1, …, n; n = 7).
Let Y ij is the size of insured catastrophe losses for each region i = 1, …, N and each year j = 1, …, n.According to [4,5,11,13,16,17,18] for the estimation of credible net insured premium for the i-th risk, the following formula is used Credibility factor for the i-th risk is calculated in form [3,18]: The rules for estimating the parameters are by [3,18]: The value of credibility factor shows the rate of reliability of own data for each region.
The estimates of parameters are the same for all the regions.Credibility factor Z i differs for each region.The higher is the value of risk rate P i , the higher is the value of credibility factor Z i .each region in the whole period, and average insured catastrophe losses per insurance events i X for each region were computed according to (12).These characteristics present Table 6.
According to (8), values of credibility factors Z i are computed for each region.It is necessary to find these values to be able to estimate credible net premium per region according to (7).Values of credibility factors and values of credible net premiums by regions contain Tab. 7.

Source: Own calculations
Estimates of credible net premiums by regions provide information how much money in the region will be needed to cover insured catastrophe losses in next year.

Conclusion
Developments of the financial consequences of natural and man-made catastrophes have a major impact on the global insurance market and forcing the insurance and reinsurance companies to seek for new approaches and ways to cover these risks [14].Raises the concern that the capacity of the world's insurance and reinsurance markets in the future will not be sufficient to cover these risks and aims to seek alternative options for their transfer.

Fig. 2 .
Source: Sigma Swiss Re, No 1/2016 Insured and uninsured losses in 1970-2015 in USD million in 2015 prices

3 .
Source: Own processing by Sigma Swiss Re, 2010-2016 Fig. Number of catastrophe events by regions for 2009-2015

Fig. 4 .
Source: Own processing by Sigma Swiss Re, 2010-2016 Times series plot of total economic losses in USD bnUsing the Bühlmann credibility model we have calculated the credibility factor and credibility premium for next year for each of the seven regions.For each of the regions we calculate means

Fig. 5 .
Source: Own processing by Sigma Swiss Re, 2010-2016 Insured catastrophe losses by regions in 2009-2015 in USD billion

Table 1 .
The sigma event selection criteria 2015

Table 3 .
Total economic losses in USD bn.

Table 4 .
Insurance losses by region in 2009-2015 in USD billion

Table 5 .
Results of Bühlmann credibility model i X Bühlmann-Straub model was applied to compute credible net premium for seven regions of the world based insured catastrophe losses in time period 2009-2015 published in Sigma Swiss Re [20] (Table4).Total amount of insured losses Y i , where 1 by

Table 6 .
Computed characteristics of the regions

Table 7 .
Credibility factor Z i and estimates of net insurance premium per region in 2016.