Discussions on “ Demand for Repeated Insurance Contracts with Unknown Loss Probability ” Jason Yeh Chinese University of Hong Kong 2007 ARIA - Quebec City

2 Summary (1) A theoretical model to discuss buying behavior of multiple-period insurance policies – drivers not knowing accident probability (frequency) Bayesian updating Likelihood for # accident Prior belief Posterior update

3 Summary (2) Expected utility theory Drivers are utility maximizers with CARA (stationary over time & known to self) -- optimal deductible can be found -- D*>C (no insurance), D*<0 (no deductible)

4 Summary (3) Self-selection deductible (D H or D L ) for compulsory coverage Conditions opting for High and Low deductible are found Do drivers switch? Accident(s) make drivers realize they are not of LOW-risk type? No accident makes drivers realize they are not of HIGH-risk type?

5 Summary (4) Adverse selection HIGH-risk types choosing Higher coverage (D L ) LOW-risk types choosing Lower coverage (D H ) Conditions for a switch to happen are found: One accident is a necessary condition but not sufficient from D H to D L # of accidents remains zero for long enough (or risk aversion large enough) then a switch from D L to D H will occur If coverage is voluntary, just introduce another deductible option (D=C), means no insurance can be an option.

6 Comments Premium (p.7), P(D)= (C-D)(1+loading), suggesting the insurer has full information The insurer knows, but not the policyholder? On the other extreme, if the insurer has no information at all, More realistic to assume the insurer has some information from a driver ’ s previous records.

7 Comments (Cont.) Suppose insurer observes a policyholder ’ s previous t-year claim data (n 1, n 2, …, n t ), then The limited-information premium will be set as P(D)=E(N|n 1, n 2, …, n t )(C-D)(1+loading), where

8 Suggestions Continuous likelihood? The paper assumes unknown frequency, a discrete likelihood (Poisson), but fixed and known severity (C). Can other more realistic conjugate distributions, e.g., Exponential/Gamma, for the whole loss amount distribution be helpful? MCMC simulation? Would it be helpful to facilitate readers ’ understanding by offering visualized and numerical simulations?

9 Thanks! Merci! Gracias! 謝謝 !