Abstract Half-Mack Stochastic Reserving Frank Cuypers, Simone Dalessi July 2013 We suggest a stochastic reserving method, which uses the information gained from statistical reserving methods (such as the Mack Chain Ladder procedure), to model and fit the loss development factors with loss development functions. These loss development functions straightforwardly yield the reserves and include an implicit tail factor. The choice of the proper type of loss development function allows for actuarial or underwriting judgment based on appropriate experience with the analyzed line of business and jurisdiction. Moreover, the confidence intervals of the loss development factors determine the probability distribution of the reserves, which can be used e.g. for the purpose of modeling risk, capital and solvency. 1
1 Introduction Since Thomas Mack published his seminal article on the standard error of the Chain Ladder reserves [1] a large portion of the actuarial reserving literature has gone beyond the mere calculation of best estimates and has further explored the reserves variability. Some of the proposed methods are of statistical nature and estimate a finite number of moments of the probability distribution of the reserves. Mack s method [1] falls into this category. Other methods are of stochastic nature and generate a full probability distribution of the reserves. Prominent representatives of this approach are e.g. Chain Ladder bootstraps [2] and Markov chain Monte Carlos [3]. We suggest here a stochastic method, which makes only partial use of the information statistical methods yield, but stops short of using them for calculating the variance of the reserves. For this reason and in homage to the pioneer of the field, we call it Half-Mack Stochastic Reserving. In the following Section we quickly summarize those outputs of the Mack procedure, which are useful in the context of Half-Mack Stochastic Reserving. Next we use these results to define the loss development function and describe how to generate Half-Mack reserve distributions. Finally we apply the Half-Mack Stochastic Reserving procedure to real data and compare its results with those of other statistical and stochastic methods. 2 Mack Chain Ladder The Mack Chain Ladder procedure [1] is an example of a statistical reserving method: based on fairly simple and general assumptions, it estimates analytically the mean and the variance of the probability distribution of the reserves. For the purpose of the Half-Mack Stochastic Reserving method, however, we are not interested in the moments of the reserves, but rather in those of the cumulative loss development factors (age-to-ultimate) a2u at development year t = 1... l: E [a2u t ] = l 1 E [a2a k ] (1) k=t ( l 1 V [a2u t ] = E [a2u t ] 2 V [a2a k ] E [a2u k ] 2 k=t 1 L l t k + 1 l k 1 a=1 L a k where L a t stands for the aggregate losses of accident year a and development year t, and the mean and variance of the incremental loss development factors (age-to-age) a2a are given by ) (2) E [a2a t ] = V [a2a t ] = l t a=1 La t+1 l t a=1 La t 1 l t L a t l t a=1 ( L a t+1 L a t ) E [a2a t ] Although for definiteness we focus here on the Mack Chain Ladder method, we could as well use other statistical reserving methods, like those based on generalized linear models [3]. Nevertheless, because of its widespread use and thanks to its simple analytic formulas, Mack s Chain Ladder is a privileged candidate for this exercise. (3) (4) 2/12 +41 (41) 725 32 10
3 Half-Mack Stochastic Reserving We describe here a stochastic reserving method, which uses the confidence intervals of the cumulative loss development factors (1,2), but stops short of computing the Mack Chain Ladder reserves. However, it uses the information about the variance of these loss development factors to fit a loss development function, which in turn determines the full distribution of the reserves. 3.1 The Loss Development Function When plotting the inverse cumulative loss development factors (1) of an aggregate loss triangle as a function of time (typically the development years), we obtain a series of points that tends to saturate towards an horizontal asymptote at. We call a curve that reproduces this time dependence of the loss development pattern a loss development function. Such loss development functions F (t) describe in continuous time t how aggregate losses (paid or incurred) L a t of any accident year a tend to develop over the discrete development years t and smoothly converge towards their ultimates: L a = a2u t L a t = La t F (t) Most lines of business display a characteristic loss development function, whose shape can vary from one jurisdiction to another. In Figure 1 we display some typical generic behaviors of loss development functions. In general an experienced actuary or underwriter knows what shape the loss development pattern (paid or incurred) of a particular business assumes, and she will accordingly choose the appropriate family of loss development functions that should best reproduce the behavior of the inverse cumulative loss development factors. If l is the number of observed accident years and development years of the observed aggregate loss triangle, we develop the last observed (diagonal) losses L a l a+1 of accident years a = 1... l according to the value of the continuous loss development function at that particular point in time F (l a + 1), and sum over all accident years to obtain the reserves R = = l ( L a L a l a+1) a=1 l a=1 L a 1 F (l a + 1) l a+1 F (l a + 1) (5) (6) (Strictly speaking, if the L a are the paid losses, then R are the total reserves, whereas if they are the incurred losses (paid losses + case reserves), then R are the IBNR.) This procedures introduces in a natural way a tail factor, which further develops even the last observed loss of the first accident year L 1 l. For the sake of definiteness and without claiming to be exhaustive, we shall focus in our application to real data in Section 4 on the following exponential family of loss development functions [4]: 3/12 +41 (41) 725 32 10
inverse cumulativ ve loss development factors 1 8 6 time saturation CH GTPL DE MTPL FR décénale Figure 1: Generic patterns of loss development functions F (t) = 1/a2u t (5). The blue curve is typical for French Décénale business (construction insurance) and reflects the long reporting delay. The green curve is typical for German Motor business and reflects the market s tendency to overreserve. The red curve is typical for Swiss Liability business. By definition a loss development function saturates on an asymptote at. F exp (t) = ( ( 1 exp t τ )) α (7) λ where t is the continuous time of the development years, and τ, λ and α are respectively the location, scale and shape parameters. This exponential family of loss development functions turns out to yield excellent fits in the numerical examples of Section 4. In general it matches well the behaviors of the French Décénale and Swiss Liability businesses depicted in Figure 1. However, it cannot reproduce over-reserving patterns like the one observed in the German Motor market. 3.2 Fitting the Loss Development Function We fit the parameters of the chosen inverse loss development function F (t) 1 (5) to the observed loss development factors E [a2u t ] (1) according to their statistical relevance V [a2u t ] (2) by means of the χ 2 estimator [5] χ 2 = l 2 t=1 ( F (t) 1 E [a2u t ] ) 2 V [a2u t ] (8) which depends explicitly on the parameters of the chosen loss development function (In our example (7) these are τ, λ and α.) and depends implicitly on the reserves R (6). The sum runs from 1 to 4/12 +41 (41) 725 32 10
l 2 because a loss triangle with l accident s and development years has only l 1 loss development factors, of which the last one s variance cannot be determined in the context of the Mack Chain Ladder procedure. By minimizing this χ 2 estimator with respect to the reserves one obtains the best estimate reserves R dχ 2 dr = 0 (9) χ 2 ( R) = χ 2 min (10) The goodness of the fit (9) is given by the value taken by the χ 2 estimator (8) at its minimum, divided by the number of degrees of freedom: χ 2 min dof where dof = # degrees of freedom (11) = # observations # parameters = l 2 # parameters (In our example (7) this number of degrees of freedom is thus dof = l 2 3.) In general a goodness of fit exceeding 1.5 is indicative of a poor fit and requires rejecting the hypothesis that the chosen loss development function F (t) correctly reproduces the temporal behavior of the losses. If the chosen loss development function yields an acceptable best fit, values of its parameters in the neighborhood of the best fit parameters may also yield a statistically acceptable fit. The more their χ 2 estimator (8) exceeds the minimum (10), the less likely these parameters assume the true values. Anticipating on the results of Section 4, where we apply the half-mack procedure to real Medical Malpractice data, we plot in the upper graph of Figure 2 the values the χ 2 estimator (8) takes for a large random sample of different parameter multiplets. Many of these multiplets yield similar a amount of the reserves (6), with albeit different likelihoods, hence the cloud of points. In all generality the envelope of this cloud yields via the χ 2 statistic the confidence intervals of the reserves around their best estimate R (10): those values of the reserves, which yield a χ 2 estimator (8) that takes values χ 2 (R) χ 2 min + χ 2 1[q] (12) lie within a confidence interval of size q, where χ 2 1[q] is the value taken by a χ 2 distribution of one degree of freedom at its quantile q. As depicted by the dotted lines in the upper graph of Figure 2 one obtains by inverting Equation (12) two values of the reserves, which bound the q confidence interval from above and below the best estimate reserve R: R q R R + q (13) 5/12 +41 (41) 725 32 10
probability chi^2 Half-Mack Stochastic Reserving 10 9 8 7 6 5 4 3 2 1 Best Fit chi^2 envelope 0 1'000'000 1'100'000 1'200'000 1'300'000 1'400'000 1'500'000 1'600'000 1'700'000 9 8 7 6 5 3 1 chi^2 1'000'000 1'100'000 1'200'000 1'300'000 1'400'000 1'500'000 1'600'000 1'700'000 reserve Figure 2: Half-Mack reserves distribution of the US Medical Malpractice market [6]. The upper graph outlines the minimum χ 2 values as a function of the reserves, whereas the lower graph depicts the resulting probability distribution of the reserves. The dotted red lines illustrate the correspondence between the confidence intervals (upper graph) and the probabilities (lower graph). The cloud of dots in the upper graph represents different multiplets of parameters yielding the same reserves with lesser likelihoods. 6/12 +41 (41) 725 32 10
inverse cumulative loss development factors Half-Mack Stochastic Reserving These confidence intervals determine the shape of the reserves probability distribution, as depicted by the dotted lines connecting the upper and lower graphs of Figure 2: P [ R R q ± ] = 1 ± q 2 3.3 Sampling the Loss Development Function The analytic χ 2 procedure yields an elegant closed form (12 14) for the distribution of the reserves. The price to pay for this elegance is that it implicitly assumes the observed loss development factors are normally and independently distributed around their means E [a2u t ] (1) with the variance V [a2u t ] (2). Although this is a robust assumption much used in practice, there may be some situations where it is desirable to depart from it. One can e.g. determine the shape of the reserves distribution by sampling the loss development functions stochastically, according to any other probability distribution of the cumulative loss development factors, whose first two moments are estimated with e.g. the Mack Chain Ladder procedure (1,2). If the loss development functions are sampled according with normally distributed loss development factors, the so numerically generated reserves distribution converges towards the analytical χ 2 results (12 14). 9 8 7 6 5 3 1 (14) 1 2 3 4 5 6 7 8 9 development years Figure 3: Inverse cumulative loss development factors and their confidence intervals (1,2) of the US Medical Malpractice market [6]. The continuous colored curves depict 3 typical loss development functions (7) sampled with Gaussian loss development factors. A possible alternative to the normal behavior is to assume the loss development factors obey a lognormal distribution, which guarantees their positive definiteness. Other alternatives generating fatter tails include the Czeledin distribution [7] or the Szwejk distribution [8]. 7/12 +41 (41) 725 32 10
Having chosen for each inverse cumulative loss development factor a probability distribution whose moments match those given by the Mack Chain Ladder procedure (or any other statistical reserving method), it is straightforward to bootstrap the reserves by sampling these distributions. Each realization generates a different sequence of loss development factors, to which we adjust (e.g. with an Ordinary Least Squares fit) a loss development function, which in turn uniquely determines the reserves associated with this realization (6). In this way a sufficiently large sample of independent realizations generates a full distribution of the reserves. Anticipating on the results of Section 4, where we apply the half-mack procedure to real Medical Malpractice data, we display in Figure 3 how 3 (out of 1 000) exponential loss development functions (7), osculate the observed data within their confidence intervals. 4 Application to Real Data To test the practicability of the Half-Mack Stochastic Reserving method, we apply it to the US long tail loss data published by the CAS [6]. This repository includes the aggregate loss triangles for the 10 accident years 1988 to 1997 of a large number of carriers for the lines of business listed in Table 1. line of business χ 2 /dof τ λ α personal auto 0.73 0.17 1.41 1.15 commercial auto 0.21-0.91 1.53 3.92 medical malpractice 0.04 0.35 1.95 2.53 workers compensation 0.03 0.56 2.35 0.80 general liability 0.28-5.62 1.63 134.15 product liability 0.26-7.08 1.83 231.40 Table 1: US long tail lines of business [6] and their best fit parameters to the exponential loss development functions (7). The goodness of fit is indicated by the χ 2 statistic divided by the number of degrees of freedom (8 3 = 5). For the purpose of this study we aggregate for each of the 6 lines of business listed in Table 1 the paid losses of all carriers and apply the Half-Mack Stochastic Reserving with the exponential loss development functions (7). In Table 1 we verify by means of the χ 2 statistic that this choice of loss development function is justified for all 6 lines of business. We generate the reserves distribution according to 3 different samplings of the of the loss development functions, assuming the loss development factors are distributed around their means (1,2) as 1. Gaussian 2. Czeledin [7] 3. Szwejk [8] distributions. For the Gaussian case we use the analytic procedure described in Section 3.2. For the Czeledin and Szwejk cases we sample 1 000 random realizations of the loss development functions as explained in Section 3.3: each realization yields a different reserve (6) and the full sample generates the distribution of the reserves. 8/12 +41 (41) 725 32 10
Half-Mack Stochastic Reserving Mack Chain Ladder 9 Mack Chain Ladder 9 Half-Mack Gaussian Half-Mack Gaussian Half-Mack Czeledin Half-Mack Czeledin 8 Half-Mack Szwejk 7 7 6 6 probability 5 Half-Mack Szwejk 5 AF T probability 8 3 3 1 1 15'000'000 16'000'000 17'000'000 18'000'000 reserves 19'000'000 1'500'000 20'000'000 1'600'000 9 2'000'000 (b) Commercial Auto Mack Chain Ladder 9 Half-Mack Gaussian Half-Mack Czeledin 8 Half-Mack Gaussian Half-Mack Czeledin 8 Half-Mack Szwejk 7 Half-Mack Szwejk 7 6 probability probability 1'900'000 Mack Chain Ladder 5 6 5 DR 3 3 1 1 1'000'000 1'100'000 1'200'000 1'300'000 1'400'000 1'500'000 1'600'000 2'200'000 1'700'000 2'400'000 2'600'000 reserves Half-Mack Gaussian Half-Mack Czeledin Half-Mack Czeledin 8 Half-Mack Szwejk 7 6 6 probability 7 Half-Mack Szwejk 5 3 3 1 1 1'300'000 3'400'000 Mack Chain Ladder 9 Half-Mack Gaussian 5 3'200'000 Mack Chain Ladder 8 3'000'000 (d) Workers Compensation 9 2'800'000 reserves (c) Medical Malpractice probability 1'800'000 reserves (a) Personal Auto 1'700'000 1'400'000 1'500'000 1'600'000 1'700'000 1'800'000 1'900'000 2'000'000 0 reserves 200'000 400'000 600'000 800'000 1'000'000 reserves (e) General Liability (f) Product Liability Figure 4: Reserve distributions for the different lines of business listed in Table 1. 9/12 +41 (41) 725 32 10
inverse cumulative loss development factors Half-Mack Stochastic Reserving In Figure 4 we display the cumulative probability distributions of the reserves for each of the 6 lines of business listed in Table 1. In these Figures we observe how the 3 different samplings (Gaussian, Czeledin and Szwejk) compare with each other and with lognormally distributed reserves whose moments are given by the Mack Chain Ladder method [1]. The Czeledin and Szwejk distributions have each Pareto tails setting in at σ/10 from their means. We observe that Half-Mack Stochastic Reserving with Gaussian loss development factors yields in general a reserves distribution, which is similar to lognormally distributed Mack Chain Ladder reserves, with a slight trend towards lesser volatilities. In contrast, loss development factors with a Czeledin behavior systematically yield a broader reserves distribution with a higher expectation value. This comes a no surprise since the so sampled loss development factors have a fat tail towards larger age-to-ultimate factors, and hence they generate larger reserves. 9 8 7 6 5 3 1 1 2 3 4 5 6 7 8 9 development years Figure 5: Inverse cumulative loss development factors and their confidence intervals (1,2) of the US Product Liability market [6]. The continuous colored curves depict 3 typical loss development functions (7) sampled with Czeledin loss development factors. loss development functions like the red one generate unrealistically larges reserves, like those observed in Figure 4f. Similarly, loss development factors with a Szwejk behavior systematically also yield a broader reserves distribution. However, the expectation value remains similar to the result obtained with Gaussian loss development factors. Again this is in line with expectations, because the so sampled loss development factors have two fat tails towards both larger and smaller age-to-ultimate factors, and hence they generate reserves, which on average remain approximately centered around the Gaussian mean. These observations are strikingly pronounced in the case of the Product Liability line of business, where the Czeledin and Szwejk samplings generate extremely broad distributions of the reserves. As depicted in Figure 5, this is because in that line of business the loss development factors 10/12 +41 (41) 725 32 10
come with peculiarly large confidence intervals, and the inverse loss development factor of the first development year is particularly small. In this case the method samples a significant amount of loss development functions with very low initial values. These in turn generate some very large reserves, which induce absurdly fat distribution tails. In this case the choice of an unconstrained exponential loss development function (7) clearly does not correctly reflect the natural behavior of the initial inverse loss development factors. Either the family (7) should be constrained such as to avoid too small values at the origin, or another family of loss development functions should be used. 5 Conclusions Half-Mack Stochastic Reserving is a straightforward technique to estimate the full probability distribution of loss reserves. It has the following advantages: It builds upon any standard deterministic aggregate multiplicative reserving technique. (Here we have focused on the Chain Ladder method.) It accounts for the statistical and systematic errors associated with the chosen reserving technique (Here we have focused on the Mack Chain Ladder confidence intervals [1].) and yields a full probability distribution of the reserves. It automatically smooths the loss development factors in a natural fashion. (Here we have focused on an exponential loss development function (7).) It automatically incorporates tail factors in a natural fashion. (Here we have focused on an exponential loss development function (7).) It allows for introducing yet unobserved fat tails into the behavior of the loss development factors. (Here we have considered Gaussian, Czeledin [7] and Szwejk [8] sampling.) It requires the user making a deliberate actuarial decision with regard to a proper loss development function. (Here we have focused on an exponential loss development function (7).) This last item is particularly important, because there are situations where the method requires a sound actuarial or underwriting judgment based on a solid knowledge of the considered line of business and applicable jurisdiction. For instance, we observed that in spite of yielding a perfectly acceptable fit to the data (Cf. Table 1.) the exponential loss development function (7) does not suitably reproduce the behavior of the claims payments of the US Product Liability line of business. 6 Acknowledgments We are very thankful to Glenn Meyers for pointing out to us the US long tail loss data repository [6]. Furthermore we are grateful to Eric Dal Moro and Joachim Schirmer for the interesting exchange of ideas and opinions. 11/12 +41 (41) 725 32 10
References [1] Thomas Mack. Distribution-Free Calculation of the Standard Error of Chain Ladder Reserve Estimates. ASTIN Bulletin 23:2 213 225, 1993. [2] Peter England and Robert Verral. Analytic and Bootstrap Estimates of Prediction Errors in Claims Reserving. Insurance: Mathematics and Economics 25:3 281 293, 1999. [3] Mario V. Wüthrich and Michael Merz. Stochastic Claims Reserving Methods in Insurance. Wiley Finance, 2008. [4] Rameshwar D. Gupta and Debasis Kundu. Generalized Exponential Distributions. Austral. & New Zealand J. Statist. 41:2 173 188, 1999. [5] William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. Numerical Recipes 3rd Edition: The Art of Scientific Computing. Cambridge University Press, 2007. [6] Glenn G. Meyers and Peng Shi. Loss Reserving Data pulled from NAIC schedule P. http://www.casact.org/research/index.cfm?fa=loss_reserves_data. [7] Markus Knecht and Stefan Küttel. The Czeledin Distribution Function. XXXIV ASTIN Colloquium, 2008. [8] Frank Cuypers and Simone Dalessi. The Švejk Distribution Function. ASTIN Colloquium, 2012. 12/12 +41 (41) 725 32 10