Discussions on “ Demand for Repeated Insurance Contracts with Unknown Loss Probability ”

1. 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 (DH or DL) 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 (DL) • LOW-risk types choosing Lower coverage (DH) • Conditions for a switch to happen are found: • One accident is a necessary condition but not sufficient from DH to DL • # of accidents remains zero for long enough (or risk aversion large enough) then a switch from DL to DH 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 (n1, n2, …, nt), then • The limited-information premium will be set as P(D)=E(N|n1, n2, …, nt)(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!