Alan Xian, UNSW Sydney Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes 4th European Actuarial Journal Conference, 2018 Benjamin Avanzi, Greg Taylor, Bernard Wong and Alan Xian 11 September 2018 Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Introduction The Big Question How do you model something Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Introduction The Big Question How do you model something that you can’t actually model? Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Introduction The Big Question How do you model something that you can’t actually model? Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Introduction Presentation Outline 1 Introduction to insurance claims analysis Why is accuracy important? What improvements can be made to current methods? 2 Markov-modulated non-homogeneous Poisson processes (MMNPP) How can we use MMNPPs for claims analysis? How do we calibrate the model? 3 Simulation Case Study Is the calibration accurate? 4 Empirical Case Study Does it work on real insurance data? What insights can we gain from the model? Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Introduction Why is it important to get it right? Insights from insurance claims analysis are used as inputs for: 1 Reserving and Capital Management 2 Premium Liability Estimation 3 Pricing and Rate-making 4 Claims Management 5 Business Strategy Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Introduction How is it (usually) done in practice? Key step : Claims are aggregated and discretised into triangles (for example, by accident year i and development year j ) Figure: Aggregate Claims Modelling: Key Step Why is this approach so prevalent? Computationally cheap, flexible and tractable. Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Introduction Macro-modelling Problems However, there are issues with this approach documented in the literature: Problems resulting from small sample sizes (Renshaw [1994], Verdonck, Wouwe, and Dhaene [2009]) Problems with underlying processes and assumptions (Halliwell [2007], Taylor [2011], Taylor and McGuire [2004]) Problems with practical implementation (Kunkler [2004], Liu and Verrall [2009], Parodi [2014]) However, the key concern is the potentially material information is unnecessarily discarded due “excessive" aggregation/discretisation. Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Introduction A relationship between motor claims and rainfall? The Pearson correlation between the two lines shown below is a statistically significant 32%. 50 12 40 10 Rainfall (mm per day) 30 8 20 10 6 0 4 -10 2 -20 -30 0 1-Oct-10 16-Oct-10 31-Oct-10 15-Nov-10 30-Nov-10 15-Dec-10 30-Dec-10 3-Day Moving Average of Claim Counts (t,t+1,t+2) Rainfall Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Introduction How do we model things that are hard (or materially infeasible) to model? In our rainfall example, the impact on the granular daily claim frequencies was overdispersion and persistence . Our main idea: Model what information/drivers you can and proxy the impact of the rest ! Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Introduction Micro-level claims analysis - A delicate balancing act Insurance data is investigated at a greater level of detail so that more information can be extracted. However, there are some important trade-offs to consider: 1 Prediction Accuracy 2 Model Complexity 3 Interpretability 4 Modellability 5 Computational Feasibility 6 Materiality Is there a micro-level model for managing all of these elements? Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Markov-modulated Poisson processes A solution! - Markov-modulated Poisson processes Siméon méon Denis is Poi oisson sson Figure: A Markov-modulated Poisson process (MMPP) Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Markov-modulated Poisson processes Or perhaps more formally... Figure: The “Formal” Definition for a MMPP Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Markov-modulated Poisson processes Where are these processes used? These processes are commonly used to fit clustered or "bursty" processes where there are hidden components/drivers. Fields that use these models include 1 Natural sciences (Lu [2012], Thayakaran and Ramesh [2013a], Thayakaran and Ramesh [2013b], Langrock, Borchers, and Skaug [2013]) 2 Signals and telecommunications (Scott and Smyth [2003], Pan, Rao, Agarwal, and Gelfand [2016]) 3 Finance and economics (Nasr and Maddah [2015]) 4 Queueing, inventory and reliability theory (Arts [2017], Landon, Özekici, and Soyer [2013]) Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Markov-modulated Poisson processes MMPPs in actuarial literature - relatively undeveloped There are few papers applying MMPPs to insurance claim analysis: 1 Guillou, Loisel, and Stupfler [2013]: Applied an MMPP model to insurance data incorporating a claim severity model 2 Guillou, Loisel, and Stupfler [2015]: Extended the claims count model component to allow for seasonality Why is the literature underdeveloped? We think that there are three barriers to practical implementation that need to be addressed first... Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Markov-modulated Poisson processes Domestic motor claim numbers over time Daily claim counts over time 1 2 3 4 5 6 Years Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes Figure:
Alan Xian, UNSW Sydney Markov-modulated Poisson processes Domestic motor claim numbers over time Daily claim counts over time 1 2 3 4 5 6 Years Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes Figure:
Alan Xian, UNSW Sydney Markov-modulated Poisson processes Domestic motor claim frequencies and number of policyholders over time Daily claim counts and number of policyholders over time 1 2 3 4 5 6 Years Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Markov-modulated Poisson processes Barriers to practical implementation What extensions are required for the MMPP micro-level model to be realistic and practicably useful? 1 Flexible risk exposure Periodic (seasonality) 1 Non-periodic (number of policyholders, structural changes) 2 2 Numerical stability during the calibration process 3 Reasonable computation times for large insurance data sets Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Markov-modulated non-homogeneous Poisson processes Our Solution: a Markov-modulated non-homogeneous Poisson process! Figure: A Markov-modulated non-homogeneous Poisson process Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Markov-modulated non-homogeneous Poisson processes Questions to answer 1 How do we define the MMNPP model? i.e. How do we introduce our flexible exposure measure? 2 How do we calibrate this new model? What about issues of numerical instability? 1 Is it computationally feasible for large insurance data sets? 2 3 How would this model be implemented in practice? 4 Does it work for real-world insurance data? Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Question 1: How do we define the MMNPP model? Question 1: How do we define the MMNPP model? Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Markov-modulated non-homogeneous Poisson processes Model Notation 1 γ ❼ t ➁ is a general exposure measure (what we can model) 2 M ❼ t ➁ is the Hidden Markov chain (what we can’t model) 3 Y ❼ t ➁ is the conditional Poisson process for claim arrivals Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
Alan Xian, UNSW Sydney Markov-modulated non-homogeneous Poisson processes Again, a bit more formally... Figure: The “Formal” Definition for a MMNPP Modelling micro-level insurance claim counts using Markov-modulated non-homogeneous Poisson processes
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