NEW RISK-BASED CAPITAL STANDARDS IN THE EUROPEAN UNION: A PROPOSAL BASED ON EMPIRICAL DATA

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1 C Risk Management and Insurance Review, 2004, Vol. 7, No. 1, NEW RISK-BASED CAPITAL STANDARDS IN THE EUROPEAN UNION: A PROPOSAL BASED ON EMPIRICAL DATA Hato Schmeiser ABSTRACT In response to criticism concerning the current solvency system, the European Commission is developing new rules for insurance companies operating in the member states of the European Union (EU). Under this so-called Solvency II concept, an insurer is allowed to verify its solvency by using an internal risk management model previously approved by the regulatory authority. In this article we develop such an internal risk management approach for propertyliability insurers that is based on dynamic financial analysis (DFA). The proposed concept uses a simulation technique and models the central risk factors from the investment and underwriting areas of an insurance company. On the basis of the data provided by a German insurer, the ruin probabilities under different scenarios and varying planning horizons are calculated. INTRODUCTION The current European Union (EU) rules governing the solvency of insurance companies essentially base the required minimum equity capital on the volume of insurance business the companies write (Farny, 1997). Thus, no attempt is made to identify or quantify the central risks borne by insurance companies. 1 In response to criticism of this approach, the European Commission is currently drafting the Solvency II project, which develops Hato Schmeiser is Chair for Insurance Management, West fälische Wilhelms Universität Münster, Germany; phone: ; fax: ; hato.schmeiser@ wiwi.uni-muenster.de. The author would like to thank Michael R. Powers for helpful comments on an earlier draft of this article. In addition the author would like to thank the Gesamtverband der Deutschen Versicherungswirtschaft (German Insurance Association), Berlin, for its help in collecting the data used in this article and for its financial support. 1 This can be demonstrated by means of a simplified example. Let us take two nonlife insurance companies. Company 1 generates premium income totaling 450 million and faces normal distributed claims with a mean of 400 million and a standard deviation of 40 million; none of the claims have been reinsured, and the equity capital totals 100 million. Company 2 shows the same figures, albeit with a standard deviation for the claims of 80 million. Under the current EU rules, both companies are considered solvent (equity capital (= 100 million) > required solvency margin ( 93 million)), even though the differences measured, e.g., by the underwriting ruin probability (company percent; company percent) are considerable. 41

2 42 RISK MANAGEMENT AND INSURANCE REVIEW a totally new approach to solvency regulation (European Commission, 2002). In analogy to the ideas discussed, and in part implemented in the European banking sector (Basle Committee on Banking Supervision, 2001), a two-stage approach is being considered: An evaluation model used by an internationally recognized rating agency serves as the basic concept (Stage I). Instead of the Stage I basic concept, insurance companies can opt to use internal risk management models (Stage II). Such internal risk management models use quantification approaches whose validity is generally recognized and that are used regularly by insurers already in the context of in-house risk management. Before an internal model can be used as the external measure of a company s solvency, regulator approval is required. Should the proposed two-stage model in fact be implemented, insurance companies will be able to use a single, well-developed model for solving a number of different tasks all at the same time. Apart from providing external proof of solvency and, in Germany, meeting the demands of the KonTraG, 2 such models could be used for in-house risk and performance management. In the following, we focus on the Stage II concept from the viewpoint of insurance regulators. In this context arise two important questions: What would a consistent and feasible Stage II model of an insurance company look like? How could certain scenarios (e.g., reduction of premium income, dangerous correlations) be defined to test the insurer s solvency? The primary goal of this article is not to discuss these questions theoretically, but to show a practical implementation of the Stage II concept by using empirical data from a German property-liability insurance company. The approach originates in dynamic financial analysis (DFA) (Lowe and Stanard, 1997; Hodes, Feldblum, and Neghaiwi, 1999; Kaufmann, Gadmer, and Klett, 2001). RUIN PROBABILITY VERSUS EXPECTED POLICYHOLDER DEFICIT An insurance company s solvency is typically measured by using one of two approaches: the probability of ruin approach or the expected policyholder deficit (EPD) ratio approach (Butsic, 1994; Barth, 2000). The ruin probability approach, which is founded in actuarial risk theory, quantifies the probability of an insurer s liabilities exceeding its assets in any given period. 3 If U 1 denotes the insurer s equity capital at the end of the period, then the ruin probability ψ(1) in a one-period context can be written as: ψ(1) = Pr{U 1 < 0}. (1) 2 Gesetz zur Kontrolle und Transparenz im Unternehmensbereich (Law on Control and Transparency in Business). 3 Conversely, by setting a maximum permitted ruin probability, it is possible to determine the minimum amount of equity capital that is needed at the beginning of the planning horizon.

3 NEW RISK-BASED CAPITAL STANDARDS IN THE EUROPEAN UNION 43 The ruin probability approach suffers from several shortcomings (Butsic, 1994; Powers, 1995). In particular, it has been criticized for not considering the severity of an insolvency. Thus, an EPD approach has been recommended (Butsic, 1994; Wang, 1998). The EPD approach calculates the expected costs of the ruin, i.e., the probability of ruin is multiplied by the expected cost of that ruin should it actually occur. Hence, we get in a one-period context: EPD = E[Max{0 U 1, 0}] = Pr{U 1 < 0} E[0 U 1 U 1 < 0]. (2) Given a fixed value for EPD, Equation (2) illustrates that there is a substitutional relationship between the ruin probability and the expected cost of ruin, assuming that a ruin occurs. As can be seen from a simple example, this is not entirely without problems. 4 To set comparable standards for insurance companies of different sizes, the EPD is not considered directly; instead, an EPD ratio is formed. An EPD ratio is typically defined as the EPD divided by the expected value of the liabilities (Butsic, 1994; Barth, 2000). As Barth (2000) has shown on the basis of an empirical study, setting an EPD ratio in this way typically means that a relatively higher probability of ruin is sufficient for large insurance companies. 5 This might be problematic because large insurers pose a greater threat to market stability as the collapse of a single large insurer results in significant indirect costs (Barth, 2000). The problems associated with the two approaches to measuring risk do not, in our opinion, make it possible for one particular method to be considered fundamentally superior to the other. It is more important to clarify what conclusions can or cannot be drawn with the use of any particular risk measure. In this article, we use the ruin probability approach. Although the ruin probability is, in general, a useful tool for regulators, it ignores the time value of money (Powers, 1995). In other words, the ruin probability concept does not take into account that ceteris paribus insolvencies occurring in the near future generally cause much higher concern than insolvencies occurring in the distant future. If the regulators want to address this point, the expected discounted cost of insolvency (EDCI) concept, a generalization of the ruin probability approach, should be used (Powers, 1995; see also Gerber and Shiu, 1998). RUIN PROBABILITY: SOME THEORETICAL REMARKS Let us assume that U denotes the equity capital and G represents the insurance company s gains. From this the following basic relation can be obtained: U t = U t 1 + G t (3) for some time period t T. 4 For example, an insurance company with a ruin probability of 1 percent and an expected cost of 1,000,000 assuming that a ruin occurs has approximately the same EPD as an insurance company with a ruin probability of 99 percent and an expected cost of 10,101 should the ruin actually occur (see Barth, 2000, pp ). 5 Fixing a certain EPD ratio for a large insurer (i.e., one with high expected liabilities) results in a high EPD value being set. Compared with a small company, this typically results in relatively greater values for Pr{U 1 < 0} and E[ 0 U 1 U 1 < 0] being permitted (see Equation (2)).

4 44 RISK MANAGEMENT AND INSURANCE REVIEW If N stands for the first occurrence of ruin, we can write: N = inf {t : U t < 0}. (4) Hence in the multi-period case, the probability of ruin can generally be defined as (Heilmann, 1988, p. 247): ψ T = Pr {N < }. (5) Depending on the chosen set T of time point t, one can distinguish the following four cases (Bühlmann, 1996, p. 134; Straub, 1997, p. 36): Case T1 : Finite planning horizon and discrete time parameter (only a countable number of time points in the planning interval are of interest). Case T2 :Infinite planning horizon and discrete time parameter. Case T3 : Finite planning horizon and continuous time parameter (all points of time in the planning interval are of interest). Case T4 :Infinite planning horizon and continuous time parameter. For the different versions T 1, T 2, T 3, and T 4, the following inequalities hold for the probability of ruin (Bühlmann, 1996, p. 134): and ψ T1 ψ T2 ψ T4, (6) ψ T1 ψ T3 ψ T4. (7) In general, calculating ruin probabilities in a closed mathematical formula is possible only in a few cases under very restrictive assumptions (Straub, 1997, p. 37). Hence to determine the ruin probability, it is generally necessary to use suitable approximation procedures, which can be roughly divided into analytical (i.e., the normal power approximation) or numerical methods (i.e., various simulation techniques). 6 In the following, we concentrate on case T 1 discrete model with finite planning horizon and employ a Latin-Hypercube simulation (McKay, Conover, and Beckman, 1979). 7 A strong argument in favor of using simulation techniques is the opportunity to easily take into account correlations between differently distributed random variables. A SIMULATION MODEL BASED ON EMPIRICAL DATA Let us now look at the simulation results for a German property-liability insurance company. To protect the company s anonymity, the volume of the data has been 6 See Daykin, Pentikäinen, and Pesonen (1994), pp In general, with a given number of iterations, the approximation from the Latin-Hypercube technique is clearly superior to the Monte Carlo simulation.

5 NEW RISK-BASED CAPITAL STANDARDS IN THE EUROPEAN UNION 45 TABLE 1 Input Data for the Underwriting Business Distribution Parameter 1 Parameter 2 Mean Standard Deviation π 1 1,346,880,000 S 1 Extreme value 1,144,082,000 39,190,000 1,166,702,860 50,263,128 CS 1 Normal 90,000,000 18,000, ,000,000 18,000,000 E 1 309,000,000 modified; however, the insurer s underlying risk structure remains unaffected. To get results from a risk management tool such as the one considered below, two main steps must be taken: development of a consistent model and obtaining input data. In this article, we concentrate on developing a simulation model 8 and presenting some main results. Hence, we do not focus on the statistical methods used to obtain the input data for the model. First we will refer to Equation (3). If U t denotes the equity capital (in t), G t the gain, Z t the underwriting result, I t the investment result, and OR t the other results, then one can obtain: U t = U t 1 + G t = U t 1 + Z t + I t + OR t ; t = 1, 2,...,. (8) Let us first consider a one-period case (e.g., one year). With = 1, the underwriting result Z 1 can be derived as follows: Z 1 = π 1 S 1 CS 1 E 1. (9) Thereby π 1 represents the net premium income, S 1 the (whole account) net claims 9 (including incurred, but not yet settled, claims), CS 1 the loss development for prior-year claims, and E 1 the underwriting expenses in. Table 1 shows the data examined for the underwriting business. The investment income I t results from the product of C t 1 (=average volume of capital investment based on market prices) and r t 1,t (=rate of return) minus the expenses from the investment area (IE t ). Hence, for the one-period case we get: I 1 = C 0 r 0,1 IE 1. (10) 8 All subsequent simulation examples have been calculated on the basis of a Latin-Hypercube simulation with 400,000 iterations. To ensure that the results could be compared accurately with one another, we used the same sequence of random numbers for all simulations. 9 We focus on a presentation of the whole account claim distribution S 1. Taking the existing correlations into account, S 1 is simulated on the basis of the insurer s seven different lines of business.

6 46 RISK MANAGEMENT AND INSURANCE REVIEW TABLE 2 Input Data for Various Asset Classes Held j Type Relevant Index Distribution E[r j ] SD[r j ] 1 Real Estate (Germany) DIX Normal 4.2% 3.0% 2 Money Market Normal 3.3% 1.0% 3 Stocks (Germany) DAX Normal 13.2% 19.3% 4 Stocks (Europe) MSCI-Europe Normal 12.3% 15.5% 5 Stocks (World) MSCI-World Normal 11.2% 11.7% 6 Bonds (Germany) REXP Normal 6.9% 5.7% 7 Affiliated Enterprises Normal 7.3% 11.3% 8 Mortgages (Germany) 3.0% 0.0% TABLE 3 Correlations Between Different Rates of Return of the Asset Classes j The rate of return r 0,1 can be obtained from the returns R T = [ r 1,..., r z ]ofthezdifferent asset classes examined. 10 If we now let w T = [ w 1,..., w z ] represent the proportion of individual asset classes as part of the total capital investment C 0, the following relations are obtained: r 0,1 = w T R; w T = 1. (11) 1 The data compiled in Tables 2 and 3 form the basis for the different asset classes and their correlations. The average volume of investment capital based on market prices (C 0 )is 4,038,277,600, the expenses from the investment area (IE 1 )are 19,651,000. We will operate with a deterministic value of 29,924,000 for the other results OR 1 (e.g., the results from noninsurance business). 10 The superscript T indicates the transpose of a vector.

7 NEW RISK-BASED CAPITAL STANDARDS IN THE EUROPEAN UNION 47 TABLE 4 Impact of Different Asset Allocations and Different Amounts of Equity Capital on the One- Period Ruin Probability and the Corresponding S&P Rating w[1] w[2] w[3] U 0 = 1,573,759,200 ψ(1) 0.000% ψ(1) 0.000% ψ(1) 0.002% (S&P: AAA ) (S&P: AAA ) (S&P: AA ) U 0 = 1,250,000,000 ψ(1) 0.000% ψ(1) 0.001% ψ(1) 0.035% (S&P: AAA ) (S&P: AA ) (S&P: A ) U 0 = 1,000,000,000 ψ(1) 0.002% ψ(1) 0.009% ψ(1) 0.229% (S&P: AA ) (S&P: AA ) (S&P: BB ) U 0 = 750,000,000 ψ(1) 0.035% ψ(1) 0.105% ψ(1) 1.055% (S&P: A ) (S&P: BBB ) (S&P: B ) The equity capital U 0 has a market value (including the market value of the equalization reserve) of 1,573,759,200. Three different asset allocations are examined below, with structure w[1] standing for the insurer s current asset allocation: w [1] = ; w [2] = ; w [3] = Thus, all the variables on the right-hand side of Equation (8) for the one-year planning horizon have now been determined. Utilizing a simulation, we can now (approximately) calculate the distribution of U 1. From the standpoint of solvency regulation, one piece of important information is the probability for U 1 < 0 (i.e., the ruin probability ψ( ) = ψ T1 ( ) with = 1). Table 4 illustrates the impact of different asset structures and different levels of equity capital in t = 0 on the insurer s ruin probability and on the corresponding Standard & Poor s (S&P) rating. 12 For the sake of simplicity, it is assumed that there are no correlations between the stochastic variables S 1, CS 1, and r 0,1. Table 4 shows that the insurer s financial stability is very well assured even where there are relatively sharp deviations from the current asset allocation/current capital structure 11 Statutory restrictions on capital investments are not considered in the examples. 12 On the relationship between the ruin probability and the categorization of the S&P rating, see Brand and Bahr (2001), p. 8, Table 2.

8 48 RISK MANAGEMENT AND INSURANCE REVIEW (shaded cell in Table 4). Even if the company chose asset allocation w[3] (70 percent of the assets are invested in stocks) and reduced its equity capital to 750,000,000 (less than half of the current equity capital), the company would still be eligible for a Standard&Poor s B rating. Let us now extend the time period of our model to more than one year (hence: t = 1, 2,..., ). We refer to Equation (8), and add a variable D that represents tax payments and dividends: with U t = U t 1 + π t S t CS t E }{{} t + C t 1 r t 1,t IE t +OR t D t, (12) }{{} Z t } {{ } G t I t { 0.6Gt for G t > 0 D t = 0 for G t 0. (13) According to Equation (13), taxes and dividends will be paid only if profits were made in the different periods. Equation (12) also shows, that the volume of capital investment depends on the net profits made (i.e., C t = C t 1 + G t D t ). For the simplest case, let us assume stationary input factors: π t = π 1, E t = E 1, IE t = IE 1, OR t = OR 1, S t is for all t identically distributed, CS t is for all t identically distributed, and OR t is for all t identically distributed. Furthermore, assume that there are no autocorrelations and no correlations between the random variables S t, CS t, and r t 1,t. We utilize the same data used for the one-period model (i.e., data from Tables 1 3 with IE t = 19,651,000 (expenses from the investment area) and OR t = 29,924,000 (results from noninsurance business)). With the current asset allocation w[1] of the insurer, Table 5 shows the ruin probabilities ψ( ) for periods of 5, 10, or 15 years, respectively. TABLE 5 Ruin Probabilities and Corresponding S&P Ratings With a 5-, 10-, and 15-Year Planning Horizon (Asset Allocation w[1], Stationary Input Factors) ψ(5) ψ(10) ψ(15) U 0 = 1,573,759, % 0.008% 0.018% (S&P: AAA ) (S&P: AAA ) (S&P: AAA ) U 0 = 1,250,000, % 0.064% 0.111% (S&P: AAA ) (S&P: AAA ) (S&P: AAA ) U 0 = 1,000,000, % 0.295% 0.418% (S&P: AA ) (S&P: AAA ) (S&P: AAA ) U 0 = 750,000, % 1.235% 1.519% (S&P: BBB ) (S&P: A ) (S&P: A )

9 NEW RISK-BASED CAPITAL STANDARDS IN THE EUROPEAN UNION 49 TABLE 6 Ruin Probabilities and Corresponding S&P Ratings With a 5-, 10-, and 15-Year Planning Horizon (Asset Allocation w[1], Falling Premium Income) ψ(5) ψ(10) ψ(15) U 0 = 1,573,759, % 0.075% 0.638% (S&P: AAA ) (S&P: AAA ) (S&P: AA ) U 0 = 1,250,000, % 0.391% 1.847% (S&P: AAA ) (S&P: AAA ) (S&P: BBB ) U 0 = 1,000,000, % 1.260% 3.848% (S&P: AA ) (S&P: A ) (S&P: BBB ) U 0 = 750,000, % 3.449% 7.669% (S&P: BBB ) (S&P: BBB ) (S&P: BB ) Comparing Table 4 with Table 5 clearly demonstrates that the insurance company s ruin probabilities rise ceteris paribus if more time periods are considered. Because of the positive expected profits in the example, the ruin probability increases only slightly over time. Even for longer periods, the company s stability is very well assured in the current structure (shaded cells in Table 5). In Table 6, the same assumptions as used in Table 5 hold, with one exception: we assume an annual 1 percent reduction in premiums triggered, for example, by growing competition in the insurance markets. Thus, the following holds: π k+1 = 0.99 π k ; (k = 1, 2,..., 1), (14) with 1,346,880,000 remaining the initial value for π 1. Table 6 clearly illustrates that, all else being equal, even a slight decline in premiums will result in a considerable increase in the ruin probabilities (the shaded areas show the insurer s current structure). For instance, the ruin probability for the 15-year case in the current structure rises from percent (see Table 5) to percent (see Table 6). We conclude by looking at the impact of a change in the correlation structure. Empirical studies tend to show that there are significant autocorrelations between the claims of the various periods (see Cummins and Nye, 1980, and for the German insurance market, Maurer, 2000, p. 226). Moreover, it seems reasonable to assume that there is a positive correlation between the loss development for prior-year claims and the net claims in the current year. Let us now analyze three periods of observation one year, five years, and ten years. Again, we will take the circumstances upon which the calculation in Table 5 was based. 13 For the one-period case, we assume the interrelation Corr(S 1, CS 1 ) = In particular, we assume that there are neither correlations between S and r, nor between CS and r. This assumption is supported by the empirical study undertaken by Maurer (2000), p. 251.

10 50 RISK MANAGEMENT AND INSURANCE REVIEW TABLE 7 Ruin Probabilities With a 1-, 5-, and 10-Year Planning Horizon and Corresponding S&P Ratings (Asset Allocation w[1], Stationary Input Factors, Correlations Between the Random Variables) ψ(1) ψ(5) ψ(10) U 0 = 1,573,759, % 0.012% 0.998% (S&P: AAA ) (S&P: AAA ) (S&P: A ) U 0 = 1,250,000, % 0.112% 2.737% (S&P: AAA ) (S&P: AA ) (S&P: BBB ) U 0 = 1,000,000, % 0.469% 5.554% (S&P: AA ) (S&P: A ) (S&P: BB ) U 0 = 750,000, % 1.872% % (S&P: A ) (S&P: BB ) (S&P: BB ) For the multi-period case, let Corr[S k, CS k+g ] = 0.30 and Corr[CS k, CS k+g ] = 0.30 (k = 1, 2,..., 1, g = 1, 2,..., 1, k + g ). In addition, let Corr[S t, CS t+z ] = 0.30 (t = 1, 2,...,, z = 0, 1,..., 1, t + z ). Thus, e.g., for the ten-year case, we obtain for the interrelations between S t and CS t a correlation matrix with a magnitude of Table 7 sums up the results of the simulation (the shaded areas show the insurer s current structure). In comparison with Tables 4 and 5, Table 7 shows a massive increase in the ruin probability for the five- and ten-year planning horizon. For instance, the ruin probability for the ten-year case increases from percent (see Table 5) to percent (see Table 7). Furthermore, it can be seen for the five- and ten-year planning horizons that the scenario with falling premium income (Table 6) produces considerably smaller ruin probabilities than the assumed correlation structure in Table 7. In summary, the simulation model presented above offers a very flexible tool with which a whole range of different scenarios can be calculated. The various assumptions made in the examples can be disregarded successively so as to examine the impact of particular developments (e.g., from the underwriting or investment business) on the insurance company s solvency. SUMMARY The European Commission is currently developing new solvency rules governing insurance companies operating in the member states of the European Union. It has been suggested that a two-stage approach, similar to that used in the European banking industry, should be introduced. Instead of using the basic model (Stage I), an insurer could prove its solvency by using an internal risk management model previously approved by regulators (Stage II). In this article we developed a Stage II model for property-liability insurers based on DFA. This time-discrete approach uses a Latin-Hypercube simulation and models the central risk factors from the investment and underwriting areas of an insurer. The risk of

11 NEW RISK-BASED CAPITAL STANDARDS IN THE EUROPEAN UNION 51 the insurance company was measured by the ruin probability. On the basis of the data provided by a German insurance company, we began by looking at a one-year planning horizon, taking different asset allocation structures into account. We then simulated 5-, 10-, and 15-year planning horizons under different scenarios (particularly with regard to premium development and the correlation structures). A major objective of the article was to serve as a link between the theoretical demands of a consistent risk management model and insurance companies need to put such a model into practice easily. REFERENCES Barth, M. M., 2000, A Comparison of Risk-Based Capital Standards Under the Expected Policyholder Deficit and the Probability of Ruin Approaches, Journal of Risk and Insurance, 67(3): Basle Committee on Banking Supervision, 2001, Overview of The New Basle Capital Accord (Basle: Bank for International Settlements). Brand, L., and R. Bahr, 2001, Ratings Performance 2000: Default, Transition, Recovery, and Spreads (New York: Standard & Poor s). Bühlmann, H., 1996, Mathematical Methods in Risk Theory, 2nd printing (Berlin: Springer). Butsic, R. P., 1994, Solvency Measurement for Property-Liability Risk-Based Capital Applications, Journal of Risk and Insurance, 61(4): Cummins, J. D., and D. J. Nye, 1980, The Stochastic Characteristics of Property-Liability Insurance Profits, Journal of Risk and Insurance, 47(1): Daykin, C. D., T. Pentikäinen, and M. Pesonen, 1994, Practical Risk Theory for Actuaries (London: Chapman & Hall). European Commission, 2002, Study into the Methodologies to Assess the Overall Financial Position of an Insurance Undertaking from the Perspective of Prudential Supervision, World Wide Web: market/en/finances/insur/ solvency2-study en.htm (Accessed on February 5, 2003). Farny, D., 1997, The American Risk Based Capital Model versus the European Model of Solvability for Property and Casualty Insurers, The Geneva Papers on Risk and Insurance, Issues and Practice, 82(1): Gerber, H. U., and E. S. Shiu, 1998, On the Time Value of Ruin, North American Actuarial Journal, 2(1): Heilmann, W.-R., 1988, Fundamentals of Risk Theory (Karlsruhe: Verlag Versicherungswirtschaft). Hodes, D. M., S. Feldblum, and A. A. Neghaiwi, 1999, The Financial Modeling of Property-Casualty Insurance Companies, North American Actuarial Journal, 3(3): Kaufmann, R., A. Gadmer, and R. Klett, 2001, Introduction to Dynamic Financial Analysis, ASTIN Bulletin, 31(1): Lowe, S. P., and J. N. Stanard, 1997, An Integrated Dynamic Financial Analysis and Decision Support System for a Property Catastrophe Reinsurer, ASTIN Bulletin, 27(2): Maurer, R., 2000, Integrierte Erfolgssteuerung in der Schadenversicherung auf der Basis von Risiko-Wert-Modellen (Karlsruhe: Verlag Versicherungswirtschaft).

12 52 RISK MANAGEMENT AND INSURANCE REVIEW McKay, M., W. Conover, and R. Beckman, 1979, A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code, Technometrics, 21(1): Powers, M. R., 1995, A Theory of Risk, Return and Solvency, Insurance: Mathematics and Economics, 17(2): Straub, E., 1997, Non-Life Insurance Mathematics (Berlin: Springer). Wang, S., 1998, An Actuarial Index on the Right-Tail Risk, North American Actuarial Journal, 2(2):