Practical aspects of modelling parameter uncertainty for risk capital calculation

Noname manuscript No. (will be inserted by the editor) Practical aspects of modelling parameter uncertainty for risk capital calculation David Blanco and Annegret Weng the date of receipt and acceptance should be inserted later Abstract We assume that an insurance undertaking models its risk by a random variable X=X(θ) with a fixed parameter (vector) θ. If the undertaking does not know θ, it faces parameter uncertainty (see e.g. [1,2,4,5,10]). It is well-known that neglecting parameter uncertainty can lead to an underestimation of the true risk capital requirement. In this contribution we address some practical questions. A risk capital requirement not taking into account parameter uncertainty can imply a probability of solvency significantly below the required confidence level. However, the underestimation of the confidence level depends on the distribution, the size of the sample and, in general, on the true parameters of the distribution. We determine the probability of solvency for different distributions and samples sizes. We then follow the “inversion method” introduced in [4], which is known to model an appropriate risk capital requirement respecting parameter uncertainty for a wide class of distributions and common estimation methods. We extend the idea to distribution families and estimation methods that have not been considered so far but are frequently used to model the losses of an insurance undertaking: the lognormal distribution together with the method of moments and the two-parameter gamma distribution. Experimental data demonstrate that the inversion method also succeeds for these cases in modelling a risk capital requirement that achieves the required probability of solvency in good approximation. Keywords Parameter uncertainty, parameter risk, Solvency capital requirement Schellingstr. 24 Hochschule f¨ur Technik Tel.: (0)7118926-2730 Fax: (0)711 8926 2553 E-mail: annegret.w[email protected] 2 David Blanco and Annegret Weng 1 Introduction The first pillar of the Solvency II project (cf. [11]) requires the quantification of all material risks of an insurance undertaking. Mathematically, the potential losses of the next business year can be described by a random variable X. The required risk capital is then given as the 99.5%-quantile of X. Throughout this contribution, we assume that Xis equal to X(θ0) for a fixed parameter (vector) θ0. We concentrate on the parameter risk, that means, we assume that Xis a member of a parametric distribution family F={X(θ)|θ∈ I⊆R.}and that the entity knows F, but can only estimate θ0∈I⊆Rdwith X=X(θ0) resp. FX=FX(θ0)from historical data x1, . . . , xn, drawn from the random variable X= (X1,...,Xn) where Xi∼Xand (X1,...,Xn) are independent of X. In this setting, the undertaking faces a parameter uncertainty. We demonstrate the problem of parameter uncertainty from the undertaking’s perspective by an example. Example 1 Consider a set of historical data given as a sample of size n= 10 drawn from a normally distributed loss variable X: {98.56; 105.66; 104.80; 109.04; 125.43; 108.50; 105.48; 98.07; 93.99; 107.92}. The true parameters (µ, σ) are unknown to the undertaking. It estimates the parameters of the distribution using the maximum likelihood method and finds (ˆµ, ˆσ) = (105.75,8.13). The 99.5%-quantile of the normally distributed random variable with parameters (ˆµ, ˆσ) is equal to 126.68. Is this an appropriate risk capital taking the parameter uncertainty with respect to µand σinto account? If not, how should we calculate the risk capital? If the undertaking does not know the true parameter θ0, it has to estimate the risk capital. In case the size nof the sample is small, this underestimation can not be avoided in any situation. However, Solvency II does not require to hold an adequate risk capital in any case but only in 99.5% of the cases. More precisely, to apply article 101 of the Solvency II regulation guidelines [11] we need to model the risk capital requirement SCR such that it will not be exceeded by the loss Xof the next business year with probability 99.5% - taking into account the randomness of Xand the randomness of the historical sample X1,...,Xn(cf. [4], Definition 1). In this article, we investigate the practical aspects of parameter uncertainty. In Section 2 we recall the definition of the probability of solvency (see also [5], Section 2.1). It measures the underestimation of the risk capital requirement. We explicitly state the probability of solvency for different confidence levels, different distributions and different sample sizes. This helps to assess the impact of the parameter risk in practice. The inversion method introduced in [4] is an approach to model the risk capital requirement taking parameter uncertainty into account (cf. Subsection Practical aspects of parameter uncertainty 3 2.3. If Fis a transformed location-scale family and the parameter θ0is estimated using either the maximum likelihood method, percentile matching or the Bayesian estimate, the inversion method has been proven to lead to a risk capital meeting the required confidence level. The same holds if Fis locationscale and the parameter (vector) is estimated using the method of moments. However, not all parametric distribution families used in practice are transformed location-scale and in some situations estimation methods different from the ones mentioned above a more popular. In this contribution, we apply the inversion method to the lognormal distribution (which is not a location-scale family) in the case where the estimation method is the method of moments. Moreover, we consider the two-parameter Gamma distribution (which is not a transformed location-scale family) together with the method of moments and the maximum likelihood method. Our experimental data demonstrate that the inversion method leads to a modelled risk capital achieving the required confidence level in good approximation. Notation (cf. [4]): Throughout the article all random variables are printed in bold. We define ζand ξas uniformly distributed random variables on [0; 1]n resp. [0; 1]. By ζand ξwe denote fixed realizations of these random variables. Let I⊆Rdbe a set of parameters and let {X(θ)|θ∈I}and {FX(θ)|θ∈I} be the corresponding set of random variables resp. the set of corresponding distribution functions. We assume that the inverse F−1 X(θ)of the cumulative distribution function of Xexists. We define the function X: [0; 1] ×I→Rby X(ξ;θ) := F−1 X(θ)(ξ) and use X(ξ;θ) to denote the random variable X(θ). 2 The probability of solvency and the inversion method 2.1 The probability of solvency We recall the approach chosen in [4], Section 2. Given the historical data (x1, . . . , xn) drawn from the random variable X= (X1,...,Xn), whose cumulative distribution function FX=FX(θ)is known except for the true, but unknown parameter θ0∈I⊆Rd, we assume that the undertaking determines its risk capital by the following two step procedure: 1. Using some method M, the undertaking generates a probability distribution P=P(x1, . . . , xn;M) for the parameter θsim depending on the sample (x1, . . . , xn). 2. The modelled risk Yis defined by Y(ˆ θ) := X(ξ,θsim), where ξis a [0;1]-uniformly distributed random variable and the modelled risk capital requirement with confidence level αis set to SCR(α;x1, . . . , xn;M) = F−1 Y(ˆ θ)(α).(1) 4 David Blanco and Annegret Weng We define the probability of solvency given a method Mresp. the probability distribution P: Definition 1 Let Mbe a method to generate a probability distribution P= P(ˆ θ;M) for the simulated parameter θsim. The probability P(X≤SCR(α;X1,...,Xn;M)) (2) is called the probability of solvency for α∈[0; 1]. We say that Mresp. θsim are appropriate with respect to αif the probability of solvency for αequals the required confidence level α. A method Mresp. a probability distribution θsim is called appropriate if it is appropriate for every α∈(0; 1). 2.2 Neglecting parameter uncertainty The most common way of generating a probability distribution for the simulated parameter θsim is to set θsim ≡ˆ θ. This approach ignores the parameter uncertainty. For Example 1 in Section 1, it yields SCR(α;x1, . . . , xn;without) := F−1 X(ˆµ,ˆσ)(99.5%) = 126.68. We determine the probability of solvency P(X≤SCR(α;X1,...,Xn;without)) where both Xand X1,...,Xnare random using a Monte-Carlo simulation for different confidence levels, different distributions and estimation methods for the case where the sample size nis equal to n(see Table 1 on p. 5). Note that this extends Table 1 in [5]. The results for sample sizes n= 20, 50 and 100 are given in the appendix. Remark 1 In all cases considered the probability of solvency is significantly lower than the required confidence level. The 99.5%-quantile can be interpreted as the event which occurs at most once in 1 out of 200 years. For the normal distribution together with the maximum likelihood estimation insolvency would actually be expected in less than 46 years, for the two-parameter Gamma distribution it is even more likely. For Example 1 in Section 1 we conclude that a risk capital of 126.68 is not sufficient to cover the 200-year event. The figures are only slightly better for n= 20 (see Table 8 to Table 9 in the appendix). For n= 20 almost all distributions have a probability of solvency of less than 99% for the required confidence level of 99.5%. Only for samples of size n= 50 the probability of solvency lies above 99%. Practical aspects of parameter uncertainty 5 Distribution Estimation 95% 99% 99.5%method True parameter Gamma ML 91.21% 96.40% 97.40% k= 0.5, β= 1 ML 91.41% 96.70% 97.70% k= 2, β= 1 MM 90.42% 95.70% 96.79% k= 0.5, β= 1 MM 91.59% 96.79% 97.76% k= 2, β= 1 Normal/ ML* 91.45% 96.77% 97.76%Lognormal (two parameter) MM 91.41% 96.74% 97.74% µ= 0.1, σ= 0.1 Normal/ MM 91.41% 96.74% 97.74% Lognormal µ= 1, σ= 0.1 MM 89.37% 95.17% 96.44% µ= 1, σ= 1 Exponential ML* 92.75% 97.74% 98.58% (one parameter) Pareto ML* 91.40% 96.99% 98.02% (two parameter) Table 1 Solvency probabilities in the case that the risk capital is calculated without taking parameter uncertainty into account for n= 10 for different confidence levels, different continuous distributions and different methods of estimation, ML=maximum likelihood, MM=method of moments. For the distributions and estimation methods with * the probability of solvency can be proven to be independent of the chosen parameter (see [5]). The figures have been determined using a Monte-Carlo simulation with 10.000.000 realizations of Xand 10.000.000 different samples {x1,...,xn}of size nto determining SCR(α;{x1,...,xn};M). 2.3 Description of the inversion method The inversion method proposed by Fr¨ohlich and Weng (see [4]) leads to an appropriate probability distribution θinv sim in the sense of Definition 1 for transformed location-scale families together with the maximum likelihood method, the percentile matching or the Bayesian estimation method with a certain prior distribution and location-scale families together with the method of moments. In particular, it works for the normal distribution. In the case of Example 1 in Section 1, the inversion method yields an appropriate risk capital requirement of 175.70, an increase by 38% compared to the risk capital requirement not taking parameter uncertainty into account. Given the random variable Xwith fixed, but unknown parameter θ, the inversion method consists of two steps: 1. The historical data (x1, . . . , xn) are realizations of the independent, identically distributed random variables X1,...,Xn, such that Xi∼Xfor i= 1, . . . , n. We can write xias F−1 X(ζi), where (ζ1, . . . , ζn) is a realization of a vector ζ= (ζ1,...,ζn) of independent, uniformly distributed ran- 6 David Blanco and Annegret Weng dom variables and Fis the distribution function of X. Define the function hζ(θ) := ˆ θ(ζ, θ) where ˆ θ(ζ, θ) is the estimate of θdepending on θand the fixed historic observation (ζ1, . . . , ζn). The inversion method defines the probability distribution P(cf. Subsection 2.1) by θinv sim =θinv sim(ζ,ˆ θ) := h−1 ζ(ˆ θ) (3) where ζ= (ζ1,...,ζn) is a vector of independent, uniformly distributed random variables. 2. Let Xbe the true risk. We define the modelled risk (see Equation (1)) as Y(ˆ θ) := Xξ,h−1 ζ(ˆ θ)=F−1 X(θinv sim)(ξ).(4) From the two-step procedure above, we define the following algorithm to get realizations θinv sim of θinv sim: 1. Draw ζ= (ζ1, . . . , ζn) from ζ= (ζ1,...,ζn), where ζiare uniformly distributed on [0; 1]. 2. Solve the equation ˆ θ(ξ, ·) = ˆ θ0,(5) where ˆ θ0is the estimated parameter given the data (x1, . . . , xn), and set θinv sim equal to the solution. Depending on the distribution of Xthere are different methods to solve Equation (5): 1. In some cases, Equation (5) can be solved analytically. For example, if Xbelongs to a transformed location-scale familie F={h(µ+σZ)|µ∈ R, σ > 0}for some fixed random variable Zand if the chosen estimation method is the maximum likelihood method, then µsim = ˆµ0−ˆµ(Z1,...,Zn) ˆσ(Z1,...,Zn)·ˆσ0and σsim =ˆσ0 ˆσ(Z1,...,Zn) where ˆ θ0= (ˆµ0,ˆσ0) is the estimate of (µ, σ) for a given observation (x1, . . . , xn) and ˆµ(Z1,...,Zn) resp. ˆσ(Z1,...,Zn) are the random variables depending on the vector (Z1,...,Zn), with Zi∼Z, using the maximum likelihood method (cf. [4], Corollary 2). Example 2 The two-parameter Pareto distribution Par(β, k) with scale parameter βand shape parameter kgiven by the density function f(x) = k·βk xk+1 for x≥β. is a transformed location-scale family derived via the function h(x) = exp(x) from the generalized exponential distribution. It is very popular for modelling extreme risks. The maximum likelihood estimates for the parameters βand kgiven the sample {x1, . . . , xn}are (cf. [6], Section 5.3) ˆ β= min ixiand ˆ k=n Pln xi−ln ˆ β. Practical aspects of parameter uncertainty 7 Note that ˆ βand ˆ kare independent random variables [8]. Moreover, ˆ βis Pareto distributed with scale parameter βand shape parameter n·kand ˆ khas an inverse gamma distribution with shape parameter n−1 and scale parameter n·k. Using the inversion method we get a realization (ksim, βsim) of (ksim,βsim) by ksim =ˆ k n·F−1 Γ(n−1,1)(ζ1) and βsim =ˆ β·F−1 Par(1,n·ksim )(ζ2)−1 where F−1 Γ(n−1,1) resp. F−1 Par(1,n·ksim )is the inverse cumulative distribution function of the Gamma distribution with shape parameter n−1 and scale parameter 1 resp. of the Pareto distribution with shape parameter 1 and scale parameter n·ksimW and ζ1, ζ2are realizations of two independent, uniformly distributed random variables ζ1,ζ2. Let us consider two samples of different sizes n= 10 and n= 20 taken from [7], Exercise 13.57. The first sample is given by S1={132; 149; 476; 147; 135; 110; 176; 107; 147; 165} and the second sample is S2=S1∪{135; 117; 110; 111; 226; 108; 102; 108; 227; 102}. Using the maximum likelihood method for S1(resp. S2) we obtain ˆ β= 107 (resp. ˆ β= 102) and ˆ k= 2.5908 (resp. ˆ k= 3.0185). The table below displays the impact of the consideration of parameter uncertainty on the risk capital calculation for both samples. The risk capital with parameter uncertainty has been determined using a Monte-Carlo simulation with 1,000,000 realizations. Sample Risk capital Increase in %without with param. risk param. risk S1827.03 2,144.73 +159% S2590.07 837.86 +42% The Pareto distribution is a probability distribution with a heavy tail. Therefore, it is not surprising that the parameter risk is even more relevant than in the case of the normal distribution. Moreover, the result reflects the fact that the parameter risk declines with the size of the sample. 2. In other cases, we have to solve Equation (5) numerically. We demonstrate this approach in Section 3 and Section 4. 8 David Blanco and Annegret Weng 3. In the case of an univariate distribution where the confidence interval of the estimate is known we can use the one-sided confidence interval to construct the distribution θinv sim. If we determine the one-sided confidence interval I(α;ˆ θ)=[B(α;ˆ θ); ∞) for the parameter θdefined by P(θ∈I(α, ˆ θ)) = α, we can use the lower bound B(·,ˆ θ) to simulate θinv sim setting θinv sim(ˆ θ) := B(ς;ˆ θ) (cf. [4], Proposition 1). Example 3 (a) Let Xbe a N(µ, 1)-distributed random variable. The (1 − α)-confidence interval for the parameter µis hˆµ−φ−1(α) √n,∞, where φis the distribution function of a N(0; 1)-distributed random variable (see e.g. [6]). Hence, µsim ∼ˆµ−φ−1(ς) √n∼ˆµ−Z √n, where ςis uniformly distributed on [0; 1] and Zis N(0; 1)-distributed. (b) The confidence interval approach is a tool that can also be applied to discrete distributions. Let Xbe a Poisson-distributed random variable with fixed but unknown parameter λ > 0. Given a sample x1, . . . , xn, set ˆ λ:= 1 nPxi. Using the normal approximation we determine an one-sided (1 −α)- confidence interval for λby ˆ λ−φ−1(α)qˆ λ n,∞, where φis the distribution function of the standard normal distribution. We set then λsim := ˆ λ−φ−1(ς)sˆ λ n, where ςis uniformly distributed on [0; 1]. 3 The lognormal distribution with the method of moments The inversion method explained in Subsection 2.3 has been proved to be appropriate for the lognormal distribution together with the maximum likelihood method (cf. [4]). This estimation method has a drawback: it is biased. For some applications like reserving we prefer an unbiased estimation method such as the method of moments. However, it is not known whether the inversion method is appropriate if we use the lognormal distribution together with the method of moments. 3.1 Application of the inversion method Lemma 1 Let x1, . . . , xnbe a sample drawn from X1,...,Xn, where Xiare independent, identically distributed random variables such that Xi∼LN(µ, σ) Practical aspects of parameter uncertainty 9 for i= 1, . . . , n. The estimates ˆµand ˆσof the parameters µand σusing the method of moments are ˆµ= ln 1 n n X i=1 xi!−ˆσ2 2and (6) ˆσ2= ln 1 n n X i=1 x2 i!−2 ln 1 n n X i=1 xi!.(7) Proof The assertion follows from E[X] = eµ+σ2/2and E[X2] = e2(µ+σ2)(see [6], Chapter 14, Section 3).  Note that setting SCR(α;X1,...,Xn;without)) := F−1 X(ˆ θ)(α), that is, ignoring parameter uncertainty, implies a probability of solvency below the required confidence level (cf. Table 1 on p. 5). We adapt the inversion method to the lognormal distribution together with the method of moments. Lemma 2 Let x1, . . . , xnbe a sample drawn from X1,...,Xn, where Diare independent, identically distributed random variables such that Di∼LN(µ, σ) for i= 1, . . . , n. Let (ˆµ, ˆσ)be the estimates of the true parameters (µ, σ) using the method of methods. A realization (µsim, σsim)of the distribution (µsim,σsim)is given as the simultaneous solution of the following two equations ˆσ2=−ln 1 n−2 ln n X i=1 F−1 LN(0,σsim)(ζi)!+ ln n X i=1 F−1 LN(0,2σsim)(ζi)!,(8) µsim = ˆµ+1 2ˆσ2−ln 1 n−ln n X i=1 F−1 LN(0,σsim)(ζi)!,(9) where ζ1, . . . , ζnare realizations of independent [0; 1]-uniformly distributed random variables ζ1,...,ζnand F−1 LN(0,σ)(·)is the inverse cumulative distribution function of a lognormal random variable with parameters µ= 0 and σ. Proof The function hζ(µ, σ) := (ˆµ, ˆσ) on p. 6 is given by hζ(θ) = ln 1 n n X i=1 F−1 LN(µ,σ)(ζi)!−ˆσ2 2, ln 1 n n X i=1 F−1 LN(µ,σ)(ζi)2 i!−2 ln 1 n n X i=1 F−1 LN(µ,σ)(ζi)i!! 1 2 . Solving for µand σyields µ= ˆµ+1 2ˆσ2−ln 1 n−ln n X i=1 F−1 LN(0,σ)(ζi)! 16 David Blanco and Annegret Weng n β k α = 95% α= 99% α= 99.5% 10 1 1 95.07% 99.04% 99.55% 2 95.00% 99.01% 99.53% 4 95.13% 99.06% 99.52% 5 1 95.05% 98.97% 99.49% 2 95.09% 99.04% 99.52% 4 95.00% 99.01% 99.52% 10 1 94.95% 99.00% 99.48% 2 94.91% 99.01% 99.53% 4 95.04% 99.02% 99.52% 20 1 1 95.01% 89.99% 99.50% 2 95.00% 99.02% 99.51% 4 95.07% 99.01% 99.56% 5 1 95.05% 99.00% 99.48% 2 94.98% 98.97% 99.49% 4 95.05% 98.98% 99.50% 10 1 95.05% 99.07% 99.55% 2 94.95% 99.03% 99.50% 4 94.87% 98.97% 99.50% Table 6 P(X≤SCR(α;X1,...,Xn;inv)) for the two-parameter gamma distribution using the inversion method and considering different sample sizes n, confidence levels α and values of the true parameters kand βtaking 100,000 samples of size nof a gammadistributed random variable Γ(k, β) and performing 10,000 realizations of the distribution of (ksim,βsim) given a fixed sample. Estimation without the with the Increase in %method consideration of consideration of parameter risk parameter risk MM 8,554.93 11,113.24 +29,90% ML 8,790.90 11,746.60 +33,62% Table 7 Required risk capital for the sample given by Equation 12 using method of moments (MM) and maximum likelihood (ML) with and without the consideration of parameter uncertainty 5 Summary and Outlook This article deals with practical aspects of parameter uncertainty in the context of risk capital calculations. For a practitioner it is first necessary to assess the impact of parameter uncertainty. In Table 1 we give the probabilities of solvency for risk capital calculations ignoring parameter uncertainty for commonly used distribution families and estimation methods. For all distributions and all estimation methods considered, the probability of solvency is significantly lower than the given confidence level. In some cases, like e.g. for the Gamma distribution and sample size n= 10, it leads to a probability of insolvency which is five times higher than required. Next we recall the inversion method introduced in [4] and explain its use for Practical aspects of parameter uncertainty 17 different distributions (see Section 2). We apply the inversion method in two relevant cases which have not been considered so far but are commonly used in practice: the lognormal distribution together with the method of moments (Section 3) and the Gamma distribution with both, the method of moments and maximum likelihood method (Section 4). For both distributions we give experimental results supporting the hypothesis that the inversion method leads to probabilities of solvency achieving the required confidence level in good approximation. Together with the results derived in [4] the inversion method has proved to be an appropriate and practical tool for modelling parameter uncertainty in risk capital calculations. In the future, the challenge lies in the consideration of aggregate distributions where the overall risk can be written as the sum of random variables resp. depends on the several random variables, but the historical data are given on a more granular level. An example is the collective risk model where the overall risk is given by S=PN i=1 Xi,Xi∼X, and where we have historical data for the number of claims (i.e. realizations of N) and for the amount of the claims (i.e. realizations of X). 6 Acknowledgements This work has been supported by the DVfVW (Deutscher Verein f¨ur Versicherungswissenschaft) by a Modul 1 Forschungsprojekt with the title “Das Parameterrisiko in Risikokapitalberechnungen f¨ur Versicherungsbest¨ande”. The experimental results have been generated using Java and Matlab programs. The calculations are quite time consuming. Therefore, we are very grateful for the opportunity to run the program on the bwGriD cluster of the Hochschule Esslingen. References 1. Bignozzi, V., Tsanakas, A. (2015). Parameter uncertainty and residual estimation risk. Forthcoming in The Journal of Risk and Insurance. DOI: 10.1111/jori.12075 2. Borowicz, J., Norman, J. (2006). The effects of parameter uncertainty in the extreme event frequency-severity model. Presented at the 28th International Congress of Actuaries, Paris. 3. Bowman, K.O., Shenton, L.R. (1988). Properties of estimators for the gamma distribution. STATISTICS: textbooks and monographs, Vol. 89, Marcel Dekker, Inc., New York and Basel 4. Fr¨ohlich, A., Weng, A. (2015). Modelling parameter uncertainty for risk capital calculation. European Actuarial Journal, Vol. 5, No. 1, p. 79-112. 5. Gerrard R., Tsanakas, A. (2011). Failure probability under parameter uncertainty. Risk Analysis, Vol. 8, Issue 5, p. 727-744. 6. Johnson, N. L., Kotz, S., Balakrishnan, N. (1994). Continuous Univariate Distributions. Toronto: John Wiley & Sons. 7. Klugman, S., Panjer, H., Willmot, G. (2012). Loss models: from data to decisions (4th edition). New Jersey: Wiley Series in Probability and Statistics. 18 David Blanco and Annegret Weng 8. Malik, H. J. (1970). Estimation of the parameters of the Pareto distribution. Metrika, Vol. 15, Issue 1, p. 126-132. 9. Minka, T. (2002). Estimating a gamma distribution. , available under http://research. microsoft.com/en-us/um/people/minka/papers/minka-gamma.pdf (last retrieved May 01, 2016) 10. Sauler, K. (2009). Das Pr¨amienrisiko in der Schadenversicherung unter Solvency II. Ulm: Gesellschaft f. Finanzu. Aktuarwiss. 11. Solvency II directive 2009/138/EC, available under http://eur-lex.europa.eu/ LexUriServ/LexUriServ.do?uri=OJ:L:2009:335:0001:0155:en:PDF (last retrieved May 01, 2016) 12. Thom, H.C.S. (1958). A Note on the Gamma Distribution. Monthly weather review, Vol. 86, No. 4, p. 117-122 7 Appendix We give the solvency probabilities for risk capital calculations without taking parameter uncertainty into account for sample sizes n= 20, n= 50 and n= 100. n Distribution Estimation 95% 99% 99.5%method True parameter 20 Gamma ML 93.16% 97.86% 98.63% k= 0.5, β= 1 ML 93.27% 98.00% 98.76% k= 2, β= 1 MM 92.60% 97.41% 98.26% k= 0.5, β= 1 MM 93.26% 97.96% 98.72% k= 2, β= 1 Normal/ ML* 93.29% 98.03% 98.79%Lognormal (two parameter) MM 93.26% 98.02% 98.78% µ= 0.1, σ= 0.1 Normal/ MM 93.26% 98.02% 98.78% Lognormal µ= 1, σ= 0.1 MM 91.81% 97.00% 97.98% µ= 1, σ= 1 Exponential ML* 93.91% 98.42% 99.10% (one parameter) Pareto ML* 93.29% 98.14% 98.90% (two parameter) Table 8 Solvency probabilities in the case that the risk capital is calculated without taking parameter uncertainty into account for n= 20 and different confidence levels, different continuous distributions and different methods of estimation. Practical aspects of parameter uncertainty 19 n Distribution Estimation 95% 99% 99.5%method True parameter 50 Gamma ML 94.28% 98.58% 99.20% k= 0.5, β= 1 ML 94.32% 98.64% 99.25% k= 2, β= 1 MM 93.99% 98.37% 99.03% k= 0.5, β= 1 MM 94.38% 98.59% 99.21% k= 2, β= 1 Normal/ ML* 94.33% 98.65% 99.26%Lognormal (two parameter) MM 94.32% 98.64% 99.25% µ= 0.1, σ= 0.1 Normal/ MM 94.32% 98.64% 99.25% Lognormal µ= 1, σ= 0.1 MM 93.45% 98.09% 98.84% µ= 1, σ= 1 Exponential ML* 94.54% 98.79% 99.35% (one parameter) Pareto ML* 94.34% 98.69% 99.30% (two parameter) 100 Gamma ML 94.64% 98.80% 99.36% k= 0.5, β= 1 ML 94.66% 98.83% 99.38% k= 2, β= 1 MM 94.48% 98.68% 99.27% k= 0.5, β= 1 MM 94.64% 98.80% 99.36% k= 2, β= 1 Normal/ ML* 94.67% 98.83% 99.39%Lognormal (two parameter) MM 94.66% 98.83% 99.38% µ= 0.1, σ= 0.1 Normal/ MM 94.66% 98.83% 99.38% Lognormal µ= 1, σ= 0.1 MM 94.10% 98.48% 99.13% µ= 1, σ= 1 Exponential ML* 94.74% 98.89% 99.42% (one parameter) Pareto ML* 94.67% 98.85% 99.40% (two parameter) Table 9 Solvency probabilities in the case that the risk capital is calculated without taking parameter uncertainty into account for n= 50 and n= 100 and different confidence levels, different continuous distributions and different methods of estimation.