Benchmark Rates for XL Reinsurance Revisited: Model Comparison for the Swiss MTPL Market

Transcription

1 Benchmark Rates for XL Reinsurance Revisited: Model Comparison for the Swiss MTPL Market W. Hürlimann 1 Abstract. We consider the dynamic stable benchmark rate model introduced in Verlaak et al. (005), Section 6. It is based on the assumption that the Pareto benchmark claim size remains stable over the years and the overall sum of generalized Pearson residuals of order one is minimized. A very interesting and useful economic property of this particular method is the fact that it allows one to assess the percentage changes of XL reinsurance market premiums from one year to the other, which allows for a judgement about the cyclic transition from soft to hard markets and vice versa. In the present note, the sensitivity of this benchmark model with respect to various claim size distributions is analysed. For the Swiss MTPL Market for excess-of-loss reinsurance, it is shown that the Pareto distribution has the best overall fit over the whole period While the log-gamma and the Pareto lognormal are very close to the Pareto model, the Burr and Bowers models yield higher benchmark rates in the higher excess layers. Keywords: Pareto, Log-gamma, Pareto lognormal, Burr, Bowers, excess-of-loss reinsurance, Motor Third Party Liability Introduction In the reinsurance market, the different companies competes in order to offer excess-of-loss contracts at the best possible price given their own insurance portfolio structure and the varying market conditions, as for example the cyclic transition from soft to hard markets and vice versa. Therefore, an important problem faced by (re)insurance brokers consists to summarize market rates for excess-ofloss (XL) reinsurance for different layer programs by different ceding companies into a single benchmark market model. A general approach as well as a practical analysis of the Belgium and Swiss Markets is found in Verlaak et al. (005). In this fundamental work, market benchmark rates for the excess-of-loss reinsurance of a Motor Third Party Liability insurance portfolio are expressed in percentage of the expected premium income available to cover the whole risks of the portfolio and based on various models of the claim size. It is natural to suppose that the market prices of excessof-loss contracts are based on a single claim size over all 1 IRIS integrated risk management ag, Bederstrasse 1, P.O. BOX CH-807 Zürich, werner.huerlimann@irisunified.com underwriting years. The motivation for this simplifying assumption stems from the wish to summarize the changes in the benchmark rates over the years using a single figure (percentage change of the parameter in the later formula (.6)). This economic property allows for a judgement about the cyclic transition from soft to hard markets and vice versa. Expressing the benchmark rate for a ceding company in percent of the expected premium income available to cover the whole risks of the considered portfolio, the benchmark rate associated to a specific excess-of-loss contract will depend on three unknown quantities, namely the discount rate, the loading factor and the expected number of claims, which may vary from year to year. In the present work the previous study for the Swiss Market is up-dated on the basis of an extended number of 109 data points from the years 001 to 005. Furthermore, the sensitivity of this benchmark model with respect to various claim size distributions is analysed. The cyclic transition from soft to hard markets is again observed and has lasted on the Swiss MTPL XL reinsurance market over the whole insurance period from 001 to 005. The paper is organized as follows. Section recalls the dynamic stable benchmark rate model with a fixed Pareto claim size distribution. Section 3 considers four alternative claim size distributions, which might compete with the Pareto claim size model. Finally, Section 4 summarizes the results of the up-dated calculations and model comparison.. A dynamic stable benchmark rate model To remain self-contained it appears useful to recall the dynamic stable benchmark rate model with a fixed Pareto claim size distribution introduced in Verlaak et al. (005). In the framework of the classical collective model of risk theory, the aggregate claims of a portfolio of insurance risks are described by the random variable S = N X i i= 1 (.1) where the claim sizes X i are independent and identically distributed and independent from the random claim number N. Denote by X a random variable distributed as X i. In reinsurance, it is often appropriate to suppose that X follows a Pareto, whose survival function is given by α = x F ( x) 1 +, x > 0 θ (.) BELGIAN ACTUARIAL BULLETIN, Vol. 7, No. 1, 007

2 An excess-of-loss (XL) reinsurance program C xs D covers the amount of each claim that exceeds the deductible D up to the maximum cover of C. Let L = C + D denote the upper limit of the XL contract. Then the random amount of each claim covered by the XL treaty is ( D L) = ( X D) + ( X L) + X, (.3) and its expected value equals θ θ θ E [ X ( D, L) ] =, α > 1 (.4) D + θ L + θ Assuming that a number N of claims will occur in the insurance portfolio of a ceding company, and that a reinsurance company adds a loading l proportional to the expected value and takes into account a discount for future investment at a rate v, then the market premium for a contract C xs D has the value ( D, L) v ( 1 + ) E[ N ] E[ X ( D L) ] P = l, (.5) In a first step it is natural to suppose that the market prices of XL contracts are based on a claim size, which remains constant from year to year, but the discount rate, the loading factor and the expected number of claims may vary from year to year. Expressing the benchmark rate for a ceding company in percent of the expected premium income (EPI) available to cover the whole risks with random claim size X, the benchmark rate associated to a contract C xs D will depend on three unknown parameters, θ > 0, α > 1, where θ and α are constant over the years but depends on the year, and takes the form b( D, L,, θ, α ) = E[ X ( D, L) ] θ θ θ (.6) = D + θ L + θ The purpose of the benchmark rates is a fair market comparison of company rates at the renewal negotiations. Now, each ceding company has its own EPI and different reinsurance companies offer at renewal different rates r(d,l) for a given contract C xs D, which are defined in percent of the EPI. Given the whole set of XL rates in a given reinsurance market, the unknown parameters of the Pareto benchmark rates for renewal are estimated using nonlinear regression methods. Let (D i,l i ), i = 1,...,N, be the deductibles and upper limits for which a rate r i = r(d i,l i ) to an EPI P i is available on the reinsurance market, where an unlimited cover L i = may be possible. If b i = b(d i,l i,,θ,α) denotes the corresponding benchmark rates, then our fitting method is characterized by the following objective function, which is minimized over the set of feasible parameters. No attempt has been made for fitting the rates with other objective functions, in particular generalized Pearson residuals of any other order. Dynamic Stable Method: minimization of the overall sum of generalized Pearson residuals of order one ( t) ( t ) 5 M ( t) (,,,,,, ) min ( t) ( ri bi ) f θ α = P1 ( t). t= 1 i= 1 bi where r (t) i = r(d i,l i ) are the given M (t) (t) rates to an EPI P i in year t, and b (t) i = b(d i,l i, t,θ,α) denotes the corresponding benchmark rate. 3. Some alternative analytical claim size models The modeling and fitting of statistical distributions to loss or return data in actuarial science and finance is an important subject of wide interest. Well-known books from actuarial science include Hogg and Klugman (1988), Panjer and Willmot (199) and Klugman et al. (1998). The usefulness in both actuarial science and finance of some threeparameter tractable analytical distributions like the normal inverted gamma mixture (generalized Student t) and the symmetric double Weibull distribution has been pointed in Hürlimann (001/04). Whereas important applications are considered, the appearance of new phenomena leads to new questions. For example, what is the impact on the dynamic stable benchmark rate model for XL reinsurance when claim size models alternative to the Pareto model are used? Is it worthwhile and important to look in practice for a sound procedure to select appropriate models? The desire to be consistent with Extreme Value Theory (EVT) in excess-of-loss reinsurance pricing has been recognized at an early time and various Pareto type claim size distributions are well-known in the literature, among others the Benktander distributions (Benktander and Segerdhal (1960) and Benktander (1970)), the log-gamma and the Burr distributions (Hogg and Klugman (1984), Mack (1997), p. 365). In the present paper we consider as alternative claim size models the log-gamma, the Pareto lognormal, the Burr and the Bowers distributions. The required formulas and some comments on these models follow. Log-gamma model Traditionally, the Pareto model has been very successful and a first choice in the practice of reinsurance (see e.g. Schmitter (1978), Schmitter and Bütikofer (1997), Doerr (1980), Schmutz and Doerr (1998)). This particular model is especially useful for pricing high deductibles or/and large claims. However, this simple model often lacks of fit in the lower and medium deductibles, and as a remedy to this Mack (1997), p , suggested to use the twoparameter log-gamma model, which is a kind of Pareto model with a varying index. To generate it, one assumes that the transformed random variable Y = ln{ (X+θ) / θ } has a Gamma distribution Γ (τ,α) with parameters τ = 1/k² and α = 1/(k²µ), where µ and k are the mean and coefficient of

3 variation of the transformed claim size. Its distribution function and stop-loss transform are given by E x + θ F, α, τ ( x) = Γ α ln ; τ, x > 0, θ θ (3.1) α τ θ + D + θ,, τ θ, α, τ D [( X D) ] = θ [ 1 F ( D) ] [ 1 F ( )], where Γ ( τ ) y 1 τ 1 ; = Γ( τ ) 0 θ t y t e dt (3.) denotes the normalized incomplete Gamma function. That the Pareto model is included as special case τ = 1 is of significance in a comparative analysis for two main reasons. First, the log-gamma is a more flexible choice than the Pareto and a valuable alternative in case the Pareto does not provide an adequate fit. Second, it may turn out that the loggamma is sufficiently close to the Pareto, or even may coincide with it, in which case only the more parsimonious Pareto will be retained. Furthermore, the Pareto like behaviour of the log-gamma model can be made intuitive by noting that for τ > 1 the varying Pareto index is increasing with increasing claim size and for τ < 1 it is decreasing with increasing claim size. Pareto lognormal model In theory, several temporal stochastic phenomena can be modeled appropriately using a geometric Brownian motion (GBM), e.g. Black-Scholes option pricing, firm sizes, city sizes and individual incomes. In practice, the empirical data of such phenomena exhibits power-law behavior, which contradicts the log-normal distribution behind GBM. Recall the following recent reconciliation. A simple mechanism, which generates the power-law behavior in the tails, consists to assume that the time of observation in a GBM is itself a random variable, whose distribution is close to an exponential distribution. For example, the state of a GBM after an exponentially distributed random time with fixed initial state generates the double Pareto distribution introduced in Reed (001). A natural generalization consists to look at the state of a GBM after an exponentially distributed random time with lognormally distributed initial state. It generates a double Pareto lognormal distribution, which has been considered in Reed (003) and Reed and Jorgensen (004). In the present case study, the right-tail of the distribution matters and for this reason we consider only the limiting three-parameter right-tailed Pareto lognormal with claim size distribution ln 1 x θ αθ + ( ατ ) ln α x θ Fθ, α, τ ( x) = Φ + e x Φ ατ. τ τ (3.3) Using partial integration and somewhat tedious but straightforward calculations, one obtains the following formula for the stop-loss transform α 1 θ + τ ln D θ ln D ν E[ ( X D) + ] = e Φ D Φ α 1 τ τ e + Burr model 1 αθ + ( ατ ) D ( ) ln D θ Φ ατ. τ (3.4) This three-parameter distribution, which is a power transform of the Pareto distribution, has been introduced by Burr (194). Though not perfectly, it has been noted that thick-tailed insurance claims data often fit a Burr quite well (e.g. Hogg and Klugman (1984), Klugman et al. (1998), Hürlimann (001)). The distribution function and stop-loss transform are given by E B τ x F θ, α, τ ( x) = 1 1 +, (3.5) θ D τ [( ) ] ( ) ( ) 1 1 +,, 1 θ τ + ατ X D = B θ α τ β ; ; [ 1 ( )], 1 ( ) D F,, x D τ θ α τ + τ τ θ τ + 1 ατ 1 x θ Γ ( ) ( τ ) Γ( τ ) Γ ( ) ( a + b) a 1 b 1 θ, α, τ =, β x; a, b = ( 1 ). ( ) ( ) ( ) y y dy Γ α Γ a Γ b 0 (3.6) Bowers model In Hürlimann (1993/97) the following distribution is considered: 1 ( ) x θ F θ, α x = 1+ (3.7) α + ( x θ ) A risk with distribution (3.7) has finite mean θ but infinite variance, and maximizes the stop-loss transform over the space of all random variables with finite mean θ, variance α² and support (, ). This distribution function, which realizes the upper bound of the inequality of Bowers (1969) 1 E X D + α + D θ D θ (3.8) α [( ) ] ( ) ( ) has been considered first by Stoyan (1973) (see also Stoyan (1977), chapter 1). Further applications of Bowers distribution are found in Hürlimann (1995/98a/98b/99). 4. Numerical comparisons In the present Section the dynamic stable benchmark rate model is applied to the Swiss MTPL Market for XL reinsurance. Our study is based on 109 data points from the years 001 to 005 (18 for 001, 1 for 00, 4 for 003, 5 for 004 and 1 for 005). The parameters of the fitted model for the alternative claim size distributions presented in Section 3 are summarized in Table 4.1. The first 5 parameters are the characterizing constants i, i=1,...,5 over 3

4 the 5 years 001 to 005 while the next parameters are specific to the considered claim size distribution, which is assumed to be fixed over the 5 years. Table 4. displays the relative percentage changes of the parameters i over the years. These changes show the important increase in benchmark rates for the years 00 and 003 of more than 30% and for 004 and 005 of about 0%. Table 4.3 displays the sensitivity of the benchmark rate model to the choice of the claim size distribution. The fitting statistics show that the Pareto distribution has the best overall fit over the whole period The threeparameter log-gamma and the Pareto lognormal fits are very close to the Pareto fit and differences are neither significant nor material for the rates of MTPL XL reinsurance. The fitting of the individual years is best for the Pareto in the years 001, 003 and 004, while the fit is best for the Pareto lognormal in 005 and for the Bowers model in 00. Except for the year 00 the fit of the Bowers model is always the worst one. For reason of parsimony (smallest number of parameters) the standard Pareto is the preferred model. In Table 4.3 one notes considerable differences in the fitting statistics over the years. Unfortunately we do not have an explanation. This might be due to various effects like the observed changes in the parameters i over the years, the change in EPI s over the years, the change in layer structure over the years (combination of priority and cover), or the structural change of the rates (drastic increase in higher layers without change in lower layers). To discuss the impact of the fitting for a particular company, it is appropriate to use the notion of rate on line, which is the standard communication tool among brokers, which act as intermediaries between a cedent and re-insurers. The rate on line ROL to the priority D is defined by the formula ROL ( D) r = ( D, L ) L max max EPI D (4.1) where L max = 10 8 is the maximum limit offered on the Swiss MTPL market. The impact on a typical company is illustrated in Table 4.4 and in the displayed graphs of Figure 4.1. It suffices to compare the Pareto with the Burr and Bowers models. This choice is based on the facts that the Pareto is the standard model, its global fitting is best and there is nearly no difference with the log-gamma and the Pareto lognormal. Compared to the Pareto, the fits of the Burr and Bowers models produce higher rates in the higher excess layers. More precisely, the Burr has higher rates for the first priority and the last 3 priorities. The Bowers shows a crossing effect. It starts always with lower rates and ends always with higher rates. The higher rates in the tail are quite important for the Bowers model but this might not be surprising since by construction the Bowers distribution is the most dangerous distribution in the stop-loss order or increasing convex order. A better idea of the goodness of fit of the Burr and Bowers as compared to the Pareto might be obtained from residual plots, an analysis which could not be done in the present study. Finally, Figure 4. shows the dynamic change of the rate on line Pareto benchmark curve for a typical company over the years 001 to 005. It shows in particular that the cyclic transition from soft to hard markets has lasted on the Swiss MTPL XL reinsurance market over the whole insurance period from 001 to 005. ACKNOWLEDGEMENTS I am very grateful to the referee for many detailed comments and suggestions, which have contributed to enhance the content of this note. REFERENCES 1. Benktander, G. (1970). Schadenverteilung nach Grösse in der Nicht-Lebensversicherung. Bulletin of the Swiss Association of Actuaries, Benktander, G. and C.-O. Segerdhal (1960). On the analytical representation of claim distribution with special reference to excess of loss reinsurance. Transactions of the International Congress of Actuaries, Bowers, N.L. (1969). An upper bound for the net stop-loss premium. Transactions of the Society of Actuaries XIX, Burr, I.W. (194). Cumulative frequency functions. Annals of Mathematical Statistics 13, Doerr, R. (1980). Property Excess of Loss: Pareto Rating. Swiss Re publications. 6. Hogg, R. and S. Klugman (1984). Loss Distributions. John Wiley. 7. Hürlimann, W. (1993). Solvabilité et réassurance. Bulletin of the Swiss Association of Actuaries, Hürlimann, W. (1995). Predictive stop-loss premiums and Student's t-distribution. Insurance: Mathematics and Economics 16, Hürlimann, W. (1997). Fonctions extrémales et gain financier. Elemente der Mathematik 5, Hürlimann, W. (1998a). On best stop-loss bounds for bivariate sums by known marginal means, variances and correlation. Bulletin of the Swiss Association of Actuaries, Hürlimann, W. (1998b). On distribution-free safe layeradditive pricing. Insurance: Mathematics and Economics, Hürlimann, W. (1999). Non-optimality of a linear combination of proportional and non-proportional reinsurance. Insurance : Mathematics and Economics 4, Hürlimann, W. (001). Financial data analysis with two symmetric distributions. First Prize Gunnar Benktander ASTIN Award Competition 000. ASTIN Bulletin 31, Hürlimann, W. (004). Fitting bivariate cumulative returns with copulas. Computational Statistics and Data Analysis 45(), Klugman, S., Panjer, H. and G. Willmot (1998). Loss Models. From Data to Decisions. John Wiley, New York. 16. Mack, Th. (1997). Schadenversicherungsmathematik. Verlag Versicherungswirtschaft, Karlsruhe. 17. Panjer, H. and G. Willmot (199). Insurance Risk Models. Society of Actuaries. Schaumburg. 4

5 18. Reed, W. J. (001). The Pareto, Zipf and other power laws. Economics Letters 74, Reed, W. J. (003). The Pareto law of incomes - an explanation and an extension. Physica A319, Reed, W. J. and M. Jorgensen (004). The double Paretolognormal distribution - A new parametric model for size distribution. Communications in Statistics - Theory & Methods, Vol. 33, No. 8., Schmitter, H. (1978). Quotierung von Sach- Schadenexzedenten mit Hilfe des Paretomodells. Swiss Re publications.. Schmitter, H. and P. Bütikofer (1997). Abschätzung von Risikoprämien für Sach-Schadenexzedenten mit Hilfe des Paretomodells. Schmitter, H. (1978). 3. Schmutz, M. and R. Doerr (1998). Das Paretomodell in der Sach-Rückversicherung. Swiss Re publications. 4. Stoyan, D. (1973). Bounds for the extrema of the expected value of a convex function of independent random variables. Studia Scientiarum Mathematicarum Hungarica 8, Stoyan, D. (1977). Qualitative Eigenschaften und Abschätzungen Stochastischer Modelle. Akademie-Verlag, Berlin. (English version (1983). Comparison Methods for Queues and Other Stochastic Models. J. Wiley, New York.) 6. Verlaak, R., Hürlimann, W. and J. Beirlant (005). Benchmark Rates for Excess of Loss Reinsurance Programs - A Generalised Non-Linear Quasi-Likelihood Approach. 36th International ASTIN Colloquium, September 005, Zürich. Huerlimann_Bierlant.pdf Table 4.1: Parameters of Benchmark Rate Model for various claim size distributions Models Parameters Pareto LogGamma Pareto lognormal Burr Bowers θ α τ Table 4.: Relative increase of benchmark rates for various claim size distributions Year Pareto LogGamma Pareto lognormal Burr Bowers % 37.0% 37.5% 39.% 17.5% % 33.7% 33.3% 3.% 47.% % 19.1% 19.1% 0.1% 16.0% % 0.7% 0.9%.3% 1.0% Table 4.3: Sensitivity of Benchmark Rate Model to choice of claim size distribution Fitting Statistics in units of 10^-6 Model Overall Fit Year 001 Year 00 Year 003 Year 004 Year 005 Pareto LogGamma Pareto lognormal Burr Bowers

6 Table 4.4: Rate on Line comparison of models for a typical company Year % 6.7% 6.69% 6.94% 6.17% 3.0% -8.4% %.98%.97%.75%.67% -7.4% -10.1% % 1.75% 1.75% 1.61% 1.7% -7.7% -1.3% % 1.17% 1.18% 1.10% 1.7% -6.0% 8.4% % 0.85% 0.86% 0.8% 1.03% -3.8% 0.9% % 0.47% 0.48% 0.48% 0.69% 1.5% 45.6% % 0.31% 0.31% 0.33% 0.53% 6.4% 71.1% % 0.11% 0.11% 0.14% 0.30% 1.4% 167.% Year % 9.75% 9.74% 10.% 7.67% 4.6% -1.5% % 4.3% 4.3% 4.06% 3.3% -6.0% -3.0% %.53%.54%.37%.14% -6.3% -15.4% % 1.70% 1.71% 1.6% 1.58% -4.6% -7.0% % 1.4% 1.5% 1.1% 1.8% -.4% 3.5% % 0.69% 0.69% 0.71% 0.86% 3.% 4.9% % 0.45% 0.45% 0.49% 0.66% 8.1% 46.7% % 0.16% 0.16% 0.0% 0.37%.9% 18.3% Year % 13.99% 13.94% 14.51% 1.1% 3.5% -13.6% % 6.0% 6.18% 5.76% 5.5% -7.0% -15.% % 3.63% 3.64% 3.36% 3.37% -7.3% -6.9% %.44%.45%.30%.49% -5.6%.4% % 1.78% 1.79% 1.71%.0% -3.5% 14.0% % 0.99% 0.99% 1.01% 1.36%.0% 37.6% % 0.65% 0.64% 0.69% 1.05% 7.0% 61.6% % 0.3% 0.% 0.8% 0.59% 1.5% 151.3% Year % 17.47% 17.41% 18.8% 14.74% 4.3% -15.8% % 7.74% 7.7% 7.5% 6.39% -6.% -17.4% % 4.54% 4.55% 4.3% 4.11% -6.5% -9.3% % 3.05% 3.06%.89% 3.03% -4.8% -0.3% %.%.3%.16%.46% -.7% 11.0% % 1.3% 1.4% 1.7% 1.65%.9% 33.9% % 0.81% 0.80% 0.87% 1.7% 7.9% 57.4% % 0.9% 0.8% 0.36% 0.7%.6% 145.0% Year % 1.74% 1.69% 3.03% 18.38% 5.6% -15.7% % 9.63% 9.61% 9.14% 7.96% -5.1% -17.3% % 5.64% 5.66% 5.33% 5.1% -5.4% -9.% % 3.79% 3.8% 3.65% 3.78% -3.7% -0.% %.76%.78%.7% 3.07% -1.5% 11.% % 1.53% 1.54% 1.60%.06% 4.1% 34.1% % 1.00% 1.00% 1.10% 1.59% 9.% 57.6% % 0.36% 0.35% 0.45% 0.89% 4.0% 145.0% 6

7 Figure 4.1: Graphs of Rate on Line models for a typical company ROL Comparison % 6.00% 5.00% 3.00%.00% 1.00% 0 '000'000 4'000'000 6'000'000 8'000'000 10'000'000 ROL Comparison % % 8.00% 7.00% 6.00% 5.00% 3.00%.00% 1.00% 0 '000'000 4'000'000 6'000'000 8'000'000 10'000'000 7

8 ROL Comparison % % 6.00%.00% 0 '000'000 4'000'000 6'000'000 8'000'000 10'000'000 ROL Comparison % 16.00% % % 6.00%.00% 0 '000'000 4'000'000 6'000'000 8'000'000 10'000'000 8

9 ROL Comparison % 1.00% 8.00% 0 '000'000 4'000'000 6'000'000 8'000'000 10'000'000 Figure 4.: Rate on Line Pareto Benchmark curves for a typical company Benchmark ROL % 0.10% 0 5'000'000 10'000'000 15'000'000 0'000'000 5'000'000 Bench 001 Bench 00 Bench 003 Bench 004 Bench 005 9