DRAFT. Half-Mack Stochastic Reserving. Frank Cuypers, Simone Dalessi. July 2013
|
|
- Dwight Elliott
- 5 years ago
- Views:
Transcription
1 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
2 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/ (41)
3 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/ (41)
4 inverse cumulativ ve loss development factors 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/ (41)
5 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/ (41)
6 probability chi^2 Half-Mack Stochastic Reserving 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' 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/ (41)
7 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 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) (14) 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/ (41)
8 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 commercial auto medical malpractice workers compensation general liability product liability 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 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/ (41)
9 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 probability 5 Half-Mack Szwejk 5 AF T probability '000'000 16'000'000 17'000'000 18'000'000 reserves 19'000'000 1'500'000 20'000'000 1'600' '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 DR '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 probability 7 Half-Mack Szwejk '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 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/ (41)
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 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/ (41)
11 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/ (41)
12 References [1] Thomas Mack. Distribution-Free Calculation of the Standard Error of Chain Ladder Reserve Estimates. ASTIN Bulletin 23: , [2] Peter England and Robert Verral. Analytic and Bootstrap Estimates of Prediction Errors in Claims Reserving. Insurance: Mathematics and Economics 25: , [3] Mario V. Wüthrich and Michael Merz. Stochastic Claims Reserving Methods in Insurance. Wiley Finance, [4] Rameshwar D. Gupta and Debasis Kundu. Generalized Exponential Distributions. Austral. & New Zealand J. Statist. 41: , [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, [6] Glenn G. Meyers and Peng Shi. Loss Reserving Data pulled from NAIC schedule P. [7] Markus Knecht and Stefan Küttel. The Czeledin Distribution Function. XXXIV ASTIN Colloquium, [8] Frank Cuypers and Simone Dalessi. The Švejk Distribution Function. ASTIN Colloquium, / (41)
Statistical Modeling Techniques for Reserve Ranges: A Simulation Approach
Statistical Modeling Techniques for Reserve Ranges: A Simulation Approach by Chandu C. Patel, FCAS, MAAA KPMG Peat Marwick LLP Alfred Raws III, ACAS, FSA, MAAA KPMG Peat Marwick LLP STATISTICAL MODELING
More informationModelling the Claims Development Result for Solvency Purposes
Modelling the Claims Development Result for Solvency Purposes Mario V Wüthrich ETH Zurich Financial and Actuarial Mathematics Vienna University of Technology October 6, 2009 wwwmathethzch/ wueth c 2009
More informationPrediction Uncertainty in the Chain-Ladder Reserving Method
Prediction Uncertainty in the Chain-Ladder Reserving Method Mario V. Wüthrich RiskLab, ETH Zurich joint work with Michael Merz (University of Hamburg) Insights, May 8, 2015 Institute of Actuaries of Australia
More informationWindow Width Selection for L 2 Adjusted Quantile Regression
Window Width Selection for L 2 Adjusted Quantile Regression Yoonsuh Jung, The Ohio State University Steven N. MacEachern, The Ohio State University Yoonkyung Lee, The Ohio State University Technical Report
More informationReserve Risk Modelling: Theoretical and Practical Aspects
Reserve Risk Modelling: Theoretical and Practical Aspects Peter England PhD ERM and Financial Modelling Seminar EMB and The Israeli Association of Actuaries Tel-Aviv Stock Exchange, December 2009 2008-2009
More informationIncorporating Model Error into the Actuary s Estimate of Uncertainty
Incorporating Model Error into the Actuary s Estimate of Uncertainty Abstract Current approaches to measuring uncertainty in an unpaid claim estimate often focus on parameter risk and process risk but
More informationDouble Chain Ladder and Bornhutter-Ferguson
Double Chain Ladder and Bornhutter-Ferguson María Dolores Martínez Miranda University of Granada, Spain mmiranda@ugr.es Jens Perch Nielsen Cass Business School, City University, London, U.K. Jens.Nielsen.1@city.ac.uk,
More information[D7] PROBABILITY DISTRIBUTION OF OUTSTANDING LIABILITY FROM INDIVIDUAL PAYMENTS DATA Contributed by T S Wright
Faculty and Institute of Actuaries Claims Reserving Manual v.2 (09/1997) Section D7 [D7] PROBABILITY DISTRIBUTION OF OUTSTANDING LIABILITY FROM INDIVIDUAL PAYMENTS DATA Contributed by T S Wright 1. Introduction
More informationClark. Outside of a few technical sections, this is a very process-oriented paper. Practice problems are key!
Opening Thoughts Outside of a few technical sections, this is a very process-oriented paper. Practice problems are key! Outline I. Introduction Objectives in creating a formal model of loss reserving:
More informationThe Leveled Chain Ladder Model. for Stochastic Loss Reserving
The Leveled Chain Ladder Model for Stochastic Loss Reserving Glenn Meyers, FCAS, MAAA, CERA, Ph.D. Abstract The popular chain ladder model forms its estimate by applying age-to-age factors to the latest
More informationarxiv: v1 [q-fin.rm] 13 Dec 2016
arxiv:1612.04126v1 [q-fin.rm] 13 Dec 2016 The hierarchical generalized linear model and the bootstrap estimator of the error of prediction of loss reserves in a non-life insurance company Alicja Wolny-Dominiak
More informationContent Added to the Updated IAA Education Syllabus
IAA EDUCATION COMMITTEE Content Added to the Updated IAA Education Syllabus Prepared by the Syllabus Review Taskforce Paul King 8 July 2015 This proposed updated Education Syllabus has been drafted by
More information**BEGINNING OF EXAMINATION** A random sample of five observations from a population is:
**BEGINNING OF EXAMINATION** 1. You are given: (i) A random sample of five observations from a population is: 0.2 0.7 0.9 1.1 1.3 (ii) You use the Kolmogorov-Smirnov test for testing the null hypothesis,
More informationObtaining Predictive Distributions for Reserves Which Incorporate Expert Opinions R. Verrall A. Estimation of Policy Liabilities
Obtaining Predictive Distributions for Reserves Which Incorporate Expert Opinions R. Verrall A. Estimation of Policy Liabilities LEARNING OBJECTIVES 5. Describe the various sources of risk and uncertainty
More informationA Multivariate Analysis of Intercompany Loss Triangles
A Multivariate Analysis of Intercompany Loss Triangles Peng Shi School of Business University of Wisconsin-Madison ASTIN Colloquium May 21-24, 2013 Peng Shi (Wisconsin School of Business) Intercompany
More informationThe Retrospective Testing of Stochastic Loss Reserve Models. Glenn Meyers, FCAS, MAAA, CERA, Ph.D. ISO Innovative Analytics. and. Peng Shi, ASA, Ph.D.
The Retrospective Testing of Stochastic Loss Reserve Models by Glenn Meyers, FCAS, MAAA, CERA, Ph.D. ISO Innovative Analytics and Peng Shi, ASA, Ph.D. Northern Illinois University Abstract Given an n x
More informationRISK ADJUSTMENT FOR LOSS RESERVING BY A COST OF CAPITAL TECHNIQUE
RISK ADJUSTMENT FOR LOSS RESERVING BY A COST OF CAPITAL TECHNIQUE B. POSTHUMA 1, E.A. CATOR, V. LOUS, AND E.W. VAN ZWET Abstract. Primarily, Solvency II concerns the amount of capital that EU insurance
More informationLinda Allen, Jacob Boudoukh and Anthony Saunders, Understanding Market, Credit and Operational Risk: The Value at Risk Approach
P1.T4. Valuation & Risk Models Linda Allen, Jacob Boudoukh and Anthony Saunders, Understanding Market, Credit and Operational Risk: The Value at Risk Approach Bionic Turtle FRM Study Notes Reading 26 By
More informationDeveloping a reserve range, from theory to practice. CAS Spring Meeting 22 May 2013 Vancouver, British Columbia
Developing a reserve range, from theory to practice CAS Spring Meeting 22 May 2013 Vancouver, British Columbia Disclaimer The views expressed by presenter(s) are not necessarily those of Ernst & Young
More informationSOCIETY OF ACTUARIES Advanced Topics in General Insurance. Exam GIADV. Date: Thursday, May 1, 2014 Time: 2:00 p.m. 4:15 p.m.
SOCIETY OF ACTUARIES Exam GIADV Date: Thursday, May 1, 014 Time: :00 p.m. 4:15 p.m. INSTRUCTIONS TO CANDIDATES General Instructions 1. This examination has a total of 40 points. This exam consists of 8
More information2017 IAA EDUCATION SYLLABUS
2017 IAA EDUCATION SYLLABUS 1. STATISTICS Aim: To enable students to apply core statistical techniques to actuarial applications in insurance, pensions and emerging areas of actuarial practice. 1.1 RANDOM
More informationEvidence from Large Indemnity and Medical Triangles
2009 Casualty Loss Reserve Seminar Session: Workers Compensation - How Long is the Tail? Evidence from Large Indemnity and Medical Triangles Casualty Loss Reserve Seminar September 14-15, 15, 2009 Chicago,
More informationMODELLING OF INCOME AND WAGE DISTRIBUTION USING THE METHOD OF L-MOMENTS OF PARAMETER ESTIMATION
International Days of Statistics and Economics, Prague, September -3, MODELLING OF INCOME AND WAGE DISTRIBUTION USING THE METHOD OF L-MOMENTS OF PARAMETER ESTIMATION Diana Bílková Abstract Using L-moments
More informationOvernight Index Rate: Model, calibration and simulation
Research Article Overnight Index Rate: Model, calibration and simulation Olga Yashkir and Yuri Yashkir Cogent Economics & Finance (2014), 2: 936955 Page 1 of 11 Research Article Overnight Index Rate: Model,
More informationEvidence from Large Workers
Workers Compensation Loss Development Tail Evidence from Large Workers Compensation Triangles CAS Spring Meeting May 23-26, 26, 2010 San Diego, CA Schmid, Frank A. (2009) The Workers Compensation Tail
More informationWhere s the Beef Does the Mack Method produce an undernourished range of possible outcomes?
Where s the Beef Does the Mack Method produce an undernourished range of possible outcomes? Daniel Murphy, FCAS, MAAA Trinostics LLC CLRS 2009 In the GIRO Working Party s simulation analysis, actual unpaid
More informationA Convenient Way of Generating Normal Random Variables Using Generalized Exponential Distribution
A Convenient Way of Generating Normal Random Variables Using Generalized Exponential Distribution Debasis Kundu 1, Rameshwar D. Gupta 2 & Anubhav Manglick 1 Abstract In this paper we propose a very convenient
More informationAPPROACHES TO VALIDATING METHODOLOGIES AND MODELS WITH INSURANCE APPLICATIONS
APPROACHES TO VALIDATING METHODOLOGIES AND MODELS WITH INSURANCE APPLICATIONS LIN A XU, VICTOR DE LA PAN A, SHAUN WANG 2017 Advances in Predictive Analytics December 1 2, 2017 AGENDA QCRM to Certify VaR
More informationReserving Risk and Solvency II
Reserving Risk and Solvency II Peter England, PhD Partner, EMB Consultancy LLP Applied Probability & Financial Mathematics Seminar King s College London November 21 21 EMB. All rights reserved. Slide 1
More informationCase Study: Heavy-Tailed Distribution and Reinsurance Rate-making
Case Study: Heavy-Tailed Distribution and Reinsurance Rate-making May 30, 2016 The purpose of this case study is to give a brief introduction to a heavy-tailed distribution and its distinct behaviors in
More informationEE266 Homework 5 Solutions
EE, Spring 15-1 Professor S. Lall EE Homework 5 Solutions 1. A refined inventory model. In this problem we consider an inventory model that is more refined than the one you ve seen in the lectures. The
More information1. You are given the following information about a stationary AR(2) model:
Fall 2003 Society of Actuaries **BEGINNING OF EXAMINATION** 1. You are given the following information about a stationary AR(2) model: (i) ρ 1 = 05. (ii) ρ 2 = 01. Determine φ 2. (A) 0.2 (B) 0.1 (C) 0.4
More informationANALYSIS OF THE BINOMIAL METHOD
ANALYSIS OF THE BINOMIAL METHOD School of Mathematics 2013 OUTLINE 1 CONVERGENCE AND ERRORS OUTLINE 1 CONVERGENCE AND ERRORS 2 EXOTIC OPTIONS American Options Computational Effort OUTLINE 1 CONVERGENCE
More informationComputational Finance Binomial Trees Analysis
Computational Finance Binomial Trees Analysis School of Mathematics 2018 Review - Binomial Trees Developed a multistep binomial lattice which will approximate the value of a European option Extended the
More informationLikelihood-based Optimization of Threat Operation Timeline Estimation
12th International Conference on Information Fusion Seattle, WA, USA, July 6-9, 2009 Likelihood-based Optimization of Threat Operation Timeline Estimation Gregory A. Godfrey Advanced Mathematics Applications
More informationSubject CS1 Actuarial Statistics 1 Core Principles. Syllabus. for the 2019 exams. 1 June 2018
` Subject CS1 Actuarial Statistics 1 Core Principles Syllabus for the 2019 exams 1 June 2018 Copyright in this Core Reading is the property of the Institute and Faculty of Actuaries who are the sole distributors.
More informationCHAPTER II LITERATURE STUDY
CHAPTER II LITERATURE STUDY 2.1. Risk Management Monetary crisis that strike Indonesia during 1998 and 1999 has caused bad impact to numerous government s and commercial s bank. Most of those banks eventually
More informationRobust Critical Values for the Jarque-bera Test for Normality
Robust Critical Values for the Jarque-bera Test for Normality PANAGIOTIS MANTALOS Jönköping International Business School Jönköping University JIBS Working Papers No. 00-8 ROBUST CRITICAL VALUES FOR THE
More informationCan we use kernel smoothing to estimate Value at Risk and Tail Value at Risk?
Can we use kernel smoothing to estimate Value at Risk and Tail Value at Risk? Ramon Alemany, Catalina Bolancé and Montserrat Guillén Riskcenter - IREA Universitat de Barcelona http://www.ub.edu/riskcenter
More informationA Stochastic Reserving Today (Beyond Bootstrap)
A Stochastic Reserving Today (Beyond Bootstrap) Presented by Roger M. Hayne, PhD., FCAS, MAAA Casualty Loss Reserve Seminar 6-7 September 2012 Denver, CO CAS Antitrust Notice The Casualty Actuarial Society
More informationValidating the Double Chain Ladder Stochastic Claims Reserving Model
Validating the Double Chain Ladder Stochastic Claims Reserving Model Abstract Double Chain Ladder introduced by Martínez-Miranda et al. (2012) is a statistical model to predict outstanding claim reserve.
More informationGI ADV Model Solutions Fall 2016
GI ADV Model Solutions Fall 016 1. Learning Objectives: 4. The candidate will understand how to apply the fundamental techniques of reinsurance pricing. (4c) Calculate the price for a casualty per occurrence
More informationA Comprehensive, Non-Aggregated, Stochastic Approach to. Loss Development
A Comprehensive, Non-Aggregated, Stochastic Approach to Loss Development By Uri Korn Abstract In this paper, we present a stochastic loss development approach that models all the core components of the
More informationUPDATED IAA EDUCATION SYLLABUS
II. UPDATED IAA EDUCATION SYLLABUS A. Supporting Learning Areas 1. STATISTICS Aim: To enable students to apply core statistical techniques to actuarial applications in insurance, pensions and emerging
More informationInstitute of Actuaries of India Subject CT6 Statistical Methods
Institute of Actuaries of India Subject CT6 Statistical Methods For 2014 Examinations Aim The aim of the Statistical Methods subject is to provide a further grounding in mathematical and statistical techniques
More informationValue at risk might underestimate risk when risk bites. Just bootstrap it!
23 September 215 by Zhili Cao Research & Investment Strategy at risk might underestimate risk when risk bites. Just bootstrap it! Key points at Risk (VaR) is one of the most widely used statistical tools
More informationChapter 6 Forecasting Volatility using Stochastic Volatility Model
Chapter 6 Forecasting Volatility using Stochastic Volatility Model Chapter 6 Forecasting Volatility using SV Model In this chapter, the empirical performance of GARCH(1,1), GARCH-KF and SV models from
More informationMeasuring Financial Risk using Extreme Value Theory: evidence from Pakistan
Measuring Financial Risk using Extreme Value Theory: evidence from Pakistan Dr. Abdul Qayyum and Faisal Nawaz Abstract The purpose of the paper is to show some methods of extreme value theory through analysis
More informationQQ PLOT Yunsi Wang, Tyler Steele, Eva Zhang Spring 2016
QQ PLOT INTERPRETATION: Quantiles: QQ PLOT Yunsi Wang, Tyler Steele, Eva Zhang Spring 2016 The quantiles are values dividing a probability distribution into equal intervals, with every interval having
More informationA Review of Berquist and Sherman Paper: Reserving in a Changing Environment
A Review of Berquist and Sherman Paper: Reserving in a Changing Environment Abstract In the Property & Casualty development triangle are commonly used as tool in the reserving process. In the case of a
More informationUsing Fractals to Improve Currency Risk Management Strategies
Using Fractals to Improve Currency Risk Management Strategies Michael K. Lauren Operational Analysis Section Defence Technology Agency New Zealand m.lauren@dta.mil.nz Dr_Michael_Lauren@hotmail.com Abstract
More informationBack-Testing the ODP Bootstrap of the Paid Chain-Ladder Model with Actual Historical Claims Data
Back-Testing the ODP Bootstrap of the Paid Chain-Ladder Model with Actual Historical Claims Data by Jessica (Weng Kah) Leong, Shaun Wang and Han Chen ABSTRACT This paper back-tests the popular over-dispersed
More informationA Skewed Truncated Cauchy Logistic. Distribution and its Moments
International Mathematical Forum, Vol. 11, 2016, no. 20, 975-988 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2016.6791 A Skewed Truncated Cauchy Logistic Distribution and its Moments Zahra
More informationSimulation of probability distributions commonly used in hydrological frequency analysis
HYDROLOGICAL PROCESSES Hydrol. Process. 2, 5 6 (27) Published online May 26 in Wiley InterScience (www.interscience.wiley.com) DOI: 2/hyp.676 Simulation of probability distributions commonly used in hydrological
More informationstarting on 5/1/1953 up until 2/1/2017.
An Actuary s Guide to Financial Applications: Examples with EViews By William Bourgeois An actuary is a business professional who uses statistics to determine and analyze risks for companies. In this guide,
More informationREINSURANCE RATE-MAKING WITH PARAMETRIC AND NON-PARAMETRIC MODELS
REINSURANCE RATE-MAKING WITH PARAMETRIC AND NON-PARAMETRIC MODELS By Siqi Chen, Madeleine Min Jing Leong, Yuan Yuan University of Illinois at Urbana-Champaign 1. Introduction Reinsurance contract is an
More informationPricing & Risk Management of Synthetic CDOs
Pricing & Risk Management of Synthetic CDOs Jaffar Hussain* j.hussain@alahli.com September 2006 Abstract The purpose of this paper is to analyze the risks of synthetic CDO structures and their sensitivity
More informationPractical example of an Economic Scenario Generator
Practical example of an Economic Scenario Generator Martin Schenk Actuarial & Insurance Solutions SAV 7 March 2014 Agenda Introduction Deterministic vs. stochastic approach Mathematical model Application
More informationModelling catastrophic risk in international equity markets: An extreme value approach. JOHN COTTER University College Dublin
Modelling catastrophic risk in international equity markets: An extreme value approach JOHN COTTER University College Dublin Abstract: This letter uses the Block Maxima Extreme Value approach to quantify
More informationGamma Distribution Fitting
Chapter 552 Gamma Distribution Fitting Introduction This module fits the gamma probability distributions to a complete or censored set of individual or grouped data values. It outputs various statistics
More informationPresented at the 2012 SCEA/ISPA Joint Annual Conference and Training Workshop -
Applying the Pareto Principle to Distribution Assignment in Cost Risk and Uncertainty Analysis James Glenn, Computer Sciences Corporation Christian Smart, Missile Defense Agency Hetal Patel, Missile Defense
More informationBROWNIAN MOTION Antonella Basso, Martina Nardon
BROWNIAN MOTION Antonella Basso, Martina Nardon basso@unive.it, mnardon@unive.it Department of Applied Mathematics University Ca Foscari Venice Brownian motion p. 1 Brownian motion Brownian motion plays
More informationThe Application of the Theory of Power Law Distributions to U.S. Wealth Accumulation INTRODUCTION DATA
The Application of the Theory of Law Distributions to U.S. Wealth Accumulation William Wilding, University of Southern Indiana Mohammed Khayum, University of Southern Indiana INTODUCTION In the recent
More informationValue at Risk and Self Similarity
Value at Risk and Self Similarity by Olaf Menkens School of Mathematical Sciences Dublin City University (DCU) St. Andrews, March 17 th, 2009 Value at Risk and Self Similarity 1 1 Introduction The concept
More informationContinuous-Time Pension-Fund Modelling
. Continuous-Time Pension-Fund Modelling Andrew J.G. Cairns Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Riccarton, Edinburgh, EH4 4AS, United Kingdom Abstract This paper
More informationXiaoli Jin and Edward W. (Jed) Frees. August 6, 2013
Xiaoli and Edward W. (Jed) Frees Department of Actuarial Science, Risk Management, and Insurance University of Wisconsin Madison August 6, 2013 1 / 20 Outline 1 2 3 4 5 6 2 / 20 for P&C Insurance Occurrence
More informationDependent Loss Reserving Using Copulas
Dependent Loss Reserving Using Copulas Peng Shi Northern Illinois University Edward W. Frees University of Wisconsin - Madison July 29, 2010 Abstract Modeling the dependence among multiple loss triangles
More informationAlternative VaR Models
Alternative VaR Models Neil Roeth, Senior Risk Developer, TFG Financial Systems. 15 th July 2015 Abstract We describe a variety of VaR models in terms of their key attributes and differences, e.g., parametric
More informationContents Utility theory and insurance The individual risk model Collective risk models
Contents There are 10 11 stars in the galaxy. That used to be a huge number. But it s only a hundred billion. It s less than the national deficit! We used to call them astronomical numbers. Now we should
More informationF UNCTIONAL R ELATIONSHIPS BETWEEN S TOCK P RICES AND CDS S PREADS
F UNCTIONAL R ELATIONSHIPS BETWEEN S TOCK P RICES AND CDS S PREADS Amelie Hüttner XAIA Investment GmbH Sonnenstraße 19, 80331 München, Germany amelie.huettner@xaia.com March 19, 014 Abstract We aim to
More informationA Markov Chain Monte Carlo Approach to Estimate the Risks of Extremely Large Insurance Claims
International Journal of Business and Economics, 007, Vol. 6, No. 3, 5-36 A Markov Chain Monte Carlo Approach to Estimate the Risks of Extremely Large Insurance Claims Wan-Kai Pang * Department of Applied
More informationjoint work with K. Antonio 1 and E.W. Frees 2 44th Actuarial Research Conference Madison, Wisconsin 30 Jul - 1 Aug 2009
joint work with K. Antonio 1 and E.W. Frees 2 44th Actuarial Research Conference Madison, Wisconsin 30 Jul - 1 Aug 2009 University of Connecticut Storrs, Connecticut 1 U. of Amsterdam 2 U. of Wisconsin
More informationStochastic Loss Reserving with Bayesian MCMC Models Revised March 31
w w w. I C A 2 0 1 4. o r g Stochastic Loss Reserving with Bayesian MCMC Models Revised March 31 Glenn Meyers FCAS, MAAA, CERA, Ph.D. April 2, 2014 The CAS Loss Reserve Database Created by Meyers and Shi
More informationIIntroduction the framework
Author: Frédéric Planchet / Marc Juillard/ Pierre-E. Thérond Extreme disturbances on the drift of anticipated mortality Application to annuity plans 2 IIntroduction the framework We consider now the global
More informationGN47: Stochastic Modelling of Economic Risks in Life Insurance
GN47: Stochastic Modelling of Economic Risks in Life Insurance Classification Recommended Practice MEMBERS ARE REMINDED THAT THEY MUST ALWAYS COMPLY WITH THE PROFESSIONAL CONDUCT STANDARDS (PCS) AND THAT
More informationStrategies for Improving the Efficiency of Monte-Carlo Methods
Strategies for Improving the Efficiency of Monte-Carlo Methods Paul J. Atzberger General comments or corrections should be sent to: paulatz@cims.nyu.edu Introduction The Monte-Carlo method is a useful
More informationINSTITUTE AND FACULTY OF ACTUARIES. Curriculum 2019 SPECIMEN EXAMINATION
INSTITUTE AND FACULTY OF ACTUARIES Curriculum 2019 SPECIMEN EXAMINATION Subject CS1A Actuarial Statistics Time allowed: Three hours and fifteen minutes INSTRUCTIONS TO THE CANDIDATE 1. Enter all the candidate
More informationSOCIETY OF ACTUARIES Advanced Topics in General Insurance. Exam GIADV. Date: Friday, April 27, 2018 Time: 2:00 p.m. 4:15 p.m.
SOCIETY OF ACTUARIES Exam GIADV Date: Friday, April 27, 2018 Time: 2:00 p.m. 4:15 p.m. INSTRUCTIONS TO CANDIDATES General Instructions 1. This examination has a total of 40 points. This exam consists of
More informationLecture Slides. Elementary Statistics Twelfth Edition. by Mario F. Triola. and the Triola Statistics Series. Section 7.4-1
Lecture Slides Elementary Statistics Twelfth Edition and the Triola Statistics Series by Mario F. Triola Section 7.4-1 Chapter 7 Estimates and Sample Sizes 7-1 Review and Preview 7- Estimating a Population
More informationProxies. Glenn Meyers, FCAS, MAAA, Ph.D. Chief Actuary, ISO Innovative Analytics Presented at the ASTIN Colloquium June 4, 2009
Proxies Glenn Meyers, FCAS, MAAA, Ph.D. Chief Actuary, ISO Innovative Analytics Presented at the ASTIN Colloquium June 4, 2009 Objective Estimate Loss Liabilities with Limited Data The term proxy is used
More informationStochastic model of flow duration curves for selected rivers in Bangladesh
Climate Variability and Change Hydrological Impacts (Proceedings of the Fifth FRIEND World Conference held at Havana, Cuba, November 2006), IAHS Publ. 308, 2006. 99 Stochastic model of flow duration curves
More informationCEEAplA WP. Universidade dos Açores
WORKING PAPER SERIES S CEEAplA WP No. 01/ /2013 The Daily Returns of the Portuguese Stock Index: A Distributional Characterization Sameer Rege João C.A. Teixeira António Gomes de Menezes October 2013 Universidade
More informationBusiness Statistics 41000: Probability 3
Business Statistics 41000: Probability 3 Drew D. Creal University of Chicago, Booth School of Business February 7 and 8, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office: 404
More informationAnalysis of Methods for Loss Reserving
Project Number: JPA0601 Analysis of Methods for Loss Reserving A Major Qualifying Project Report Submitted to the faculty of the Worcester Polytechnic Institute in partial fulfillment of the requirements
More informationA Saddlepoint Approximation to Left-Tailed Hypothesis Tests of Variance for Non-normal Populations
UNF Digital Commons UNF Theses and Dissertations Student Scholarship 2016 A Saddlepoint Approximation to Left-Tailed Hypothesis Tests of Variance for Non-normal Populations Tyler L. Grimes University of
More informationKeywords Akiake Information criterion, Automobile, Bonus-Malus, Exponential family, Linear regression, Residuals, Scaled deviance. I.
Application of the Generalized Linear Models in Actuarial Framework BY MURWAN H. M. A. SIDDIG School of Mathematics, Faculty of Engineering Physical Science, The University of Manchester, Oxford Road,
More informationECON 214 Elements of Statistics for Economists
ECON 214 Elements of Statistics for Economists Session 7 The Normal Distribution Part 1 Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education
More informationChapter 2 Uncertainty Analysis and Sampling Techniques
Chapter 2 Uncertainty Analysis and Sampling Techniques The probabilistic or stochastic modeling (Fig. 2.) iterative loop in the stochastic optimization procedure (Fig..4 in Chap. ) involves:. Specifying
More informationChoice Probabilities. Logit Choice Probabilities Derivation. Choice Probabilities. Basic Econometrics in Transportation.
1/31 Choice Probabilities Basic Econometrics in Transportation Logit Models Amir Samimi Civil Engineering Department Sharif University of Technology Primary Source: Discrete Choice Methods with Simulation
More informationThe Two Sample T-test with One Variance Unknown
The Two Sample T-test with One Variance Unknown Arnab Maity Department of Statistics, Texas A&M University, College Station TX 77843-343, U.S.A. amaity@stat.tamu.edu Michael Sherman Department of Statistics,
More informationGENERATION OF STANDARD NORMAL RANDOM NUMBERS. Naveen Kumar Boiroju and M. Krishna Reddy
GENERATION OF STANDARD NORMAL RANDOM NUMBERS Naveen Kumar Boiroju and M. Krishna Reddy Department of Statistics, Osmania University, Hyderabad- 500 007, INDIA Email: nanibyrozu@gmail.com, reddymk54@gmail.com
More informationFrom Double Chain Ladder To Double GLM
University of Amsterdam MSc Stochastics and Financial Mathematics Master Thesis From Double Chain Ladder To Double GLM Author: Robert T. Steur Examiner: dr. A.J. Bert van Es Supervisors: drs. N.R. Valkenburg
More informationValue at Risk and Self Similarity
Value at Risk and Self Similarity Olaf Menkens School of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland olaf.menkens@dcu.ie January 10, 2007 Abstract The concept of Value at
More informationCOMPARATIVE ANALYSIS OF SOME DISTRIBUTIONS ON THE CAPITAL REQUIREMENT DATA FOR THE INSURANCE COMPANY
COMPARATIVE ANALYSIS OF SOME DISTRIBUTIONS ON THE CAPITAL REQUIREMENT DATA FOR THE INSURANCE COMPANY Bright O. Osu *1 and Agatha Alaekwe2 1,2 Department of Mathematics, Gregory University, Uturu, Nigeria
More informationSample Size for Assessing Agreement between Two Methods of Measurement by Bland Altman Method
Meng-Jie Lu 1 / Wei-Hua Zhong 1 / Yu-Xiu Liu 1 / Hua-Zhang Miao 1 / Yong-Chang Li 1 / Mu-Huo Ji 2 Sample Size for Assessing Agreement between Two Methods of Measurement by Bland Altman Method Abstract:
More informationHomework Problems Stat 479
Chapter 10 91. * A random sample, X1, X2,, Xn, is drawn from a distribution with a mean of 2/3 and a variance of 1/18. ˆ = (X1 + X2 + + Xn)/(n-1) is the estimator of the distribution mean θ. Find MSE(
More informationFAV i R This paper is produced mechanically as part of FAViR. See for more information.
Basic Reserving Techniques By Benedict Escoto FAV i R This paper is produced mechanically as part of FAViR. See http://www.favir.net for more information. Contents 1 Introduction 1 2 Original Data 2 3
More informationAsymptotic methods in risk management. Advances in Financial Mathematics
Asymptotic methods in risk management Peter Tankov Based on joint work with A. Gulisashvili Advances in Financial Mathematics Paris, January 7 10, 2014 Peter Tankov (Université Paris Diderot) Asymptotic
More informationA Comprehensive, Non-Aggregated, Stochastic Approach to Loss Development
A Comprehensive, Non-Aggregated, Stochastic Approach to Loss Development by Uri Korn ABSTRACT In this paper, we present a stochastic loss development approach that models all the core components of the
More informationStochastic reserving using Bayesian models can it add value?
Stochastic reserving using Bayesian models can it add value? Prepared by Francis Beens, Lynn Bui, Scott Collings, Amitoz Gill Presented to the Institute of Actuaries of Australia 17 th General Insurance
More information