An Actuarial Model of Cross subsidization in Price-Regulated Insurance Markets under Moral Hazard

Transcription

1 An Actuarial Model of Cross subsidization in rice-regulated Insurance Markets under Moral Hazard ASTIN Colloquium Manchester, UK July 3-6, 2008 Andreas Milidonis, h.d. Lecturer in Finance Manchester Business School University of Manchester. Motivation 2. Literature Review 3. Theoretical Model: Outline a. The Compound Aggregate Loss rocess b. remium Structure Over Time i. Moral Hazard c. The Discrete Time Markov Chain 4. Illustrative Examples 5. Conclusion Andreas Milidonis (2008) /page 2

2 . Introduction - Motivation rice Regulation rice Ceiling. High Risk (HR) Individuals denied coverage. HR enter Assigned Risk ool (AR). AR divided among insurers in state (California Auto). Low Risk (LR) subsidize HR. Cross-subsidization triggers transition of LR individuals into AR due to inflated premiums. rocess catalyzed by moral hazard and level of regulated premium (price ceiling). Vicious Cycle over time due to rice Regulation. Andreas Milidonis (2008) /page 3. Introduction - Motivation Regulation LR into AR HR into AR LR into V LR subsidize HR Andreas Milidonis (2008) /page 4 2

3 Theoretical Work 2. Literature Taggard (JF, 98). Doherty & Garven (JF, 986) Blackmon & Zeckhauser (AER, 99). Harrington & Niehaus (JFI, 2003). Empirical Work Grabowski, Viscusi & Evans (JRI, 989) Cummins (2002). Harrington (2002). Andreas Milidonis (2008) /page 5 2a. Aggregate Loss Model Random Aggregate Loss rocess: S + t = L t + L2 t +... Lit, i = 0,, 2,... m t =, 2,... S = Random Aggregate Loss for year t L = Random Collective Losses by contract i Assumptions: Finite opulation No real inflation in claims Discount on Regulated remium Andreas Milidonis (2008) /page 6 3

4 2a. Aggregate Loss Model Compound Density Function: F ( l) = S = m n= 0 m n= 0 p n p r( S n F * n L ( l) p n = r( N = n) l i Frequency robability Mass Function: = n) Severity Density Function: l * k *( k ) FL ( l) = FL ( l 0 y) f L ( y) dy Andreas Milidonis (2008) /page 7 2b. remium Structure Before rice Regulation: All contracts charged their Actuarial Fair remium (AF) After rice Regulation - Before Surcharge: V : AF < β where β = rice Ceiling R : AF >= β remium for contract i: Excess loss for contract i: Total Excess Loss: Andreas Milidonis (2008) /page 8 XL it = L it m [, β ] it = MIN AF i ( t ) i= it TXL t = XL it 4

5 2b. remium Structure Andreas Milidonis (2008) /page 9 2b. remium Structure After rice Regulation - After Surcharge: Surcharge per contract i: τ =TXL w i 3 2 * i 3 Surcharge weight: remium for t=2: w i3 = i2 m i= i2 [( τ ), β ] i = MIN AF i i3 Andreas Milidonis (2008) /page 0 5

6 2b. remium Structure Moral Hazard Effect Andreas Milidonis (2008) /page 2c. The Discrete Time Markov Chain Independent States (V, R) Rows in Transition robability Matrix sum to M M t t [] t = [ π, π ] V [ t = 2] = M [ t = ] R X 2 X 2 2 X 2 t π i = opulation roportion in state i at time t t ij = Transition rob. from state i to j at time t Do you see similarities with Credit Risk Classification? VV RV VR RR Andreas Milidonis (2008) /page 2 6

7 2c. The Discrete Time Markov Chain Transition robability from V to R for contract i : ( R V ) = ( > βδ ) i i = = i3 ( + w TXL > βδ ) i2 FXL i i3 βδ i wi Random Aggregate Excess Loss Model Andreas Milidonis (2008) /page 3 2c. The Discrete Time Markov Chain Transition robability Matrix = VV RV VR RR 2 X 2 VV = VR = 0 = RV VR RR = F k XL βδp i R i Andreas Milidonis (2008) /page 4 7

8 3. Illustrative Examples 40% rice Ceiling (Beta) on Transition robabilities beta = 50 beta = 60 beta = 70 beta = 80 robability 30% 20% 0% 0% Year 2 Year 3 Year 4 Year 5 beta = % 28.49% 28.30% 29.40% beta = % 6.6% 3.85% 2.24% beta = % 8.2% 5.60% 3.22% beta = % 3.9%.96% 0.4% Loss Distribution: Uniform (0, 00) Andreas Milidonis (2008) /page 5 3. Illustrative Examples 00% rice Ceiling (Beta) on "R" population beta = 50 beta = 60 beta = 70 beta = 80 opulation roportion 80% 60% 40% 20% 0% Year Year 2 Year 3 Year 4 Year 5 beta = % 65.% 75.% 82.% 87.4% beta = % 52.7% 60.4% 65.9% 70.0% beta = % 39.% 44.% 47.3% 49.0% beta = % 25.6% 28.0% 29.4% 29.7% Loss Distribution: Uniform (0, 00) Andreas Milidonis (2008) /page 6 8

9 opulation roportion 00% 80% 60% 40% 20% 0% 3. Illustrative Examples Moral Hazard (h) on "R" population h=.05 h=.0 h=.25 h=.50 h=2.00 Year Year 2 Year 3 Year 4 Year 5 h= % 24.4% 26.% 27.0% 27.0% h= % 25.6% 28.0% 29.4% 29.7% h= % 29.3% 33.2% 36.9% 39.% h= % 34.6% 4.2% 48.8% 55.0% h= % 43.4% 54.3% 67.9% 78.7% Loss Distribution: Uniform (0, 00) Andreas Milidonis (2008) /page 7 opulation roportion 3. Illustrative Examples Discount (delta) on "R" population delta = 97% delta = 96% delta = 95% 60% 50% 40% 30% 20% 0% 0% Year Year 2 Year 3 Year 4 Year 5 delta = 97% 20% 28% 33% 38% 43% delta = 96% 20% 29% 35% 4% 47% delta = 95% 20% 29% 37% 44% 5% Andreas Milidonis (2008) /page 8 9

10 6. Conclusions Moral hazard in the Assigned Risk ool should be taken into account when price ceilings are set. Moral Hazard aggravates cross-subsidization. In price regulated systems, the shape of the underlying distribution has to be known and monitored over time. Andreas Milidonis (2008) /page 9 Contact Author: 7. Contact Information Andreas Milidonis, hd Lecturer in Finance Accounting & Finance Division Manchester Business School Andreas Milidonis (2008) /page 20 0

11 References Blackmon, Glenn B. Jr., and Richard Zeckhauser. "Mispriced Equity: Regulated Rates for Auto Insurance in Massachusetts." American Economic Review 8, no. 2 (99): Cummins, David J. "roperty-liability Insurance rice Deregulation: The Last Bastion?" In Deregulating roperty-liability Insurance: Restoring Competition and Increasing Market Efficiency, edited by David J. Cummins, -24. Washington, D.C: Brookings Institution ress, Doherty, Neil A., and James R. Garven. "rice Regulation in roperty-liability Insurance: A Contingent-Claims Approach." Journal of Finance 4, no. 5 (986): Grabowski, Henry, Kip W. Viscusi, and William E. Evans. "rice and Availability Tradeoffs of Automobile Insurance Rate Regulation." Journal of Risk and Insurance 56, no. 2 (989): Andreas Milidonis (2008) /page 2 References Harrington, Scott E. "Effects of rior-approval Rate Regulation in Auto Insurance." In Deregulating roperty-liability Insurance: Restoring Competition and Increasing Market Efficiency, edited by David J. Cummins, Washington D.C.: Brookings Institution ress, Harrington, Scott E., and Greg Niehaus. "Capital, Corporate Income Taxes, and Catastrophe Insurance." Journal of Financial Intermediation 2, no. 4 (2003): Taggard, Robert A. Jr. "Rate of Return Regulation and Utility Capital Structure Decisions." Journal of Finance 36, no. 2 (98): Andreas Milidonis (2008) /page 22