Insurance Market Effects of Risk Management Metrics Carole Bernard - PowerPoint PPT Presentation

Insurance Market Effects of Risk Management Metrics Carole Bernard (University of Waterloo) Weidong Tian (U. Waterloo �→ U. North Carolina) August 2008, Portland, ARIA meeting. Bernard Carole Insurance Market Effects of Risk Management Metrics 1/26

Insurance Market VaR regulation Methodology & Results Research Directions Outline I Optimal Risk Sharing in the Insurance Market (standard ◮ theory of optimal insurance design) II Optimal Risk Sharing in the Presence of Regulators ◮ III Methodology & Results: Model & Economic Implications ◮ IV Research Directions ◮ Bernard Carole Insurance Market Effects of Risk Management Metrics 2/26

Insurance Market VaR regulation Methodology & Results Research Directions Part I Optimal Risk Sharing in the Insurance Market (Standard theory of optimal insurance design) Bernard Carole Insurance Market Effects of Risk Management Metrics 3/26

Insurance Market VaR regulation Methodology & Results Research Directions Insurance Market Participants ✬ ✩ ✬ ✩ ✛ ✘ Policyholders Insurer ✚ ✙ ✫ ✪ ✫ ✪ Insurance Market The policyholder pays a premium P to the insurer. He has a loss X . And receives I ( X ) from the insurance company. Bernard Carole Insurance Market Effects of Risk Management Metrics 4/26

Insurance Market VaR regulation Methodology & Results Research Directions Optimal Risk Sharing ✬ ✩ Optimal Insurance ✫ Contract ✪ ✬ ✩ ✬ ✩ ✛ ✘ ❅ � ❅ � Policyholders Insurer ✚ ✙ ✫ ✪ ✫ ✪ Insurance Market The policyholder pays a premium P to the insurer. He has a loss X . And receives I ( X ) from the insurance company. Bernard Carole Insurance Market Effects of Risk Management Metrics 5/26

Insurance Market VaR regulation Methodology & Results Research Directions Insurance Contract Design Let I ( X ) be an insurance indemnity.  0 � I ( X ) � X      P = φ ( E [ I ( X )])     I ( X ) non − decreasing  with φ ′ > 0 and φ ( X ) � X . Bernard Carole Insurance Market Effects of Risk Management Metrics 6/26

Insurance Market VaR regulation Methodology & Results Research Directions Framework A one-period Model. • At the beginning of the period: • W p : Initial wealth of policyholders 0 W 0 : Initial wealth of the insurer At the end of the period: • W p W p = 0 − P − X + I ( X ) T = W 0 + P − I ( X ) − c ( I ( X )) W T where X = Loss of policyholders, c � 0 and c is increasing. U : utility of policyholders, V : utility of the insurer. • Bernard Carole Insurance Market Effects of Risk Management Metrics 7/26

Insurance Market VaR regulation Methodology & Results Research Directions Optimal Insurance Design From the policyholders ’ perspective:  0 � I ( X ) � X      U ( W p � � max E 0 − P − X + I ( X )) s . t . P = φ ( E [ I ( X )]) I     I ( X ) non − decreasing  From the insurer ’s perspective:  0 � I ( X ) � X      max E [ V ( W 0 + P − I ( X ) − c ( I ( X )))] s . t . P = φ ( E [ I ( X )]) I     I ( X ) non − decreasing  Bernard Carole Insurance Market Effects of Risk Management Metrics 8/26

Insurance Market VaR regulation Methodology & Results Research Directions Optimal Insurance Design from Policyholders’ Perspective From the policyholders ’ perspective:  0 � I ( X ) � X      U ( W p � � max 0 − P − X + I ( X )) P = φ ( E [ I ( X )]) E s . t . I     I ( X ) non − decreasing  Stop loss insurance / Deductible are optimal (Arrow (1963)). I ∗ ( X ) = max( X − d , 0) Uniqueness of the optimum a.s. Bernard Carole Insurance Market Effects of Risk Management Metrics 9/26

Insurance Market VaR regulation Methodology & Results Research Directions Optimal Insurance Design from the Insurer’s Perspective From the insurer ’s perspective:  0 � I ( X ) � X      max E [ V ( W 0 + P − I ( X ) − c ( I ( X )))] s . t . P = φ ( E [ I ( X )]) I     I ( X ) non − decreasing  Upper-limit policies are optimal : (Raviv 1979) I ∗ ( X ) = min( X , c ) Uniqueness of the optimum a.s. Bernard Carole Insurance Market Effects of Risk Management Metrics 10/26

Insurance Market VaR regulation Methodology & Results Research Directions Part II Optimal risk sharing in the Presence of Regulators (Value-at-Risk requirements) Bernard Carole Insurance Market Effects of Risk Management Metrics 11/26

Insurance Market VaR regulation Methodology & Results Research Directions Insurance Market Participants ✬ ✩ ✬ ✩ ✛ ✘ ✚ Policyholders ✙ Insurer ✫ ✪ ✫ ✪ Insurance Market Bernard Carole Insurance Market Effects of Risk Management Metrics 12/26

Insurance Market VaR regulation Methodology & Results Research Directions Insurance Market Participants ✬ ✩ Regulatory Constraints ✫ ✪ ✬ ✩ ❅ � ❅ � ✬ ✩ ✛ ✘ ✚ Policyholders ✙ Insurer ✫ ✪ ✫ ✪ Insurance Market Bernard Carole Insurance Market Effects of Risk Management Metrics 13/26

Insurance Market VaR regulation Methodology & Results Research Directions Objective of our Study: the Insurance Market In Europe, the “Solvency II” project will likely introduce Value-at-Risk requirements in the insurance marketplace. What is the impact of such a change on the market? We look at the economic effects of Value-at-Risk regulation imposed to insurers on optimal risk sharing in the insurance market. Bernard Carole Insurance Market Effects of Risk Management Metrics 14/26

Insurance Market VaR regulation Methodology & Results Research Directions Market without Regulators Let I ( X ) be an insurance indemnity.  0 � I ( X ) � X      P = φ ( E [ I ( X )])     I ( X ) non − decreasing  with φ ′ > 0 and φ ( X ) � X . ⇒ Insurers can sell any indemnity I to customers (no constraints from regulators.) Bernard Carole Insurance Market Effects of Risk Management Metrics 15/26

Insurance Market VaR regulation Methodology & Results Research Directions Market with Regulators Regulators aim at protecting the insurance market and customers . Thus, they want to induce companies to control their risks. Assume that regulators require companies to satisfy: Pr ( W T < K ) � α. where W T = insurer’s final wealth. W T = W 0 + P − I ( X ) − c ( I ( X )) where c is non-negative and non-decreasing. The constraint writes also as: Pr ( I ( X ) > a ) � α, Obviously this condition can’t always be satisfied. ◮ Ignoring the reinsurance market, the company is not fully free: some indemnities are TOO RISKY to be issued. The presence of regulators influences the insurance market. Bernard Carole Insurance Market Effects of Risk Management Metrics 16/26

Insurance Market VaR regulation Methodology & Results Research Directions Part III Methodology & Results Bernard Carole Insurance Market Effects of Risk Management Metrics 17/26

Insurance Market VaR regulation Methodology & Results Research Directions Methodology We compare two situations: Optimal Insurance Contracts without Regulation ◮ Optimal Insurance Contracts under VaR Constraints ◮ We analyse optimal contracts: For risk-averse policyholders to buy, ◮ For risk-averse insurers to issue. ◮ Bernard Carole Insurance Market Effects of Risk Management Metrics 18/26

Insurance Market VaR regulation Methodology & Results Research Directions Market without Regulators From the policyholders ’ perspective:  0 � I ( X ) � X      U ( W p � � max E 0 − P − X + I ( X )) s . t . P = φ ( E [ I ( X )]) I     I ( X ) non − decreasing  From the insurer ’s perspective:  0 � I ( X ) � X      max E [ V ( W 0 + P − I ( X ) − c ( I ( X )))] s . t . P = φ ( E [ I ( X )]) I     I ( X ) non − decreasing  Bernard Carole Insurance Market Effects of Risk Management Metrics 19/26

Insurance Market VaR regulation Methodology & Results Research Directions Market with Regulators From the policyholders ’ perspective:  0 � I ( X ) � X   P = φ ( E [ I ( X )])  U ( W p � � max E 0 − P − X + I ( X )) s . t . I ( X ) non − decreasing I   Pr ( W T < K ) � α  From the insurer ’s perspective:  0 � I ( X ) � X   P = φ ( E [ I ( X )])  max E [ V ( W 0 + P − I ( X ) − c ( I ( X )))] s . t . I ( X ) non − decreasing I   Pr ( W T < K ) � α  Bernard Carole Insurance Market Effects of Risk Management Metrics 20/26

Insurance Market VaR regulation Methodology & Results Research Directions Comparison of Optimal Insurance Designs for policyholders In blue —–, with regulation. In red - - - , without regulation. 6 Insured’s optimum Arrow’s optimum 5 4 Indemnity I(x) 3 2 1 0 0 1 2 3 4 5 6 7 8 d * =2 Loss x d Arrow =3 q=6 Bernard Carole Insurance Market Effects of Risk Management Metrics 21/26