Multiple Risk-Sharing: An Analysis of the Policyholders Preference - PowerPoint PPT Presentation

Multiple Risk-Sharing: An Analysis of the Policyholder’s Preference Lukas Reichel, Hato Schmeiser and Florian Schreiber Institute of Insurance Economics University of St.Gallen EGRIE, Cyprus September 20, 2016

Motivation: Insurance Demand under Default Risk 2 Bilateral insurance contract under default risk Insurance-Demand-Curve Insurance Coverage Premium increase Policyholder Insurer 1 Indemnification reduction Indemnification is subject to a default risk of the insurer Price for Insurance What is the optimal coverage level � ∗ if the insurer’s ruin probability is non-zero? Doherty & Schlesinger (1990): • Insurance demand is affected by default risk; classic Mossin-theorem does not hold anymore • Given actuarial fair premium, no unambiguous results whether over- or under-insurance is optimal Mahul & Wright (2007): • Given actuarial fair premium, over-insurance is optimal iff the insurer’s recovery rate is above a threshold Lukas Reichel EGRIE, Cyprus September 20, 2016

Motivation: Insurance Demand under Default Risk 3 Question 1 : How is the insurance demand affected if Multiple risk-sharing in, e.g., industrial insurance, reinsurance,… default risk can be diversified by multiple risk-sharing? Insurance-Demand-Curve Insurance Coverage Insurer 1 Insurer 2 ? Premium increase Policyholder Insurer 3 Insurer 4 Indemnification reduction Indemnification is subject to Pro rata sharing of risk and premium Price for Insurance a default risk of the insurers Lukas Reichel EGRIE, Cyprus September 20, 2016

Motivation: Insurance Demand under Default Risk 4 Question 2 : How is the insurance demand affected if Multiple risk-sharing in, e.g., industrial insurance, reinsurance,… the fee for a risk-management measure increases? Insurance-Demand-Curve Insurance Coverage Insurer 1 Insurer 2 ? Premium increase Policyholder Insurer 3 Insurer 4 Indemnification reduction Indemnification is subject to Pro rata sharing of risk and premium Price for Insurance a default risk of the insurers Basic Insurance Coverage Default-Free Risk-Management (multiple with default risk) Measure (RMM) Premium Fee Insurer 1 Insurer 2 Provides replacing payment if Policyholder one/several (re)insurers fail Indemnification Indemnification Insurer 3 Insurer 4 Examples : Letter of Credit, Guarantees, Credit Default Swap, Guarantee Fund, Deposits Pro rata sharing of risk and premium Lukas Reichel EGRIE, Cyprus September 20, 2016

Model I: Insurance Coverage and External RMM 5 Basic Insurance Coverage Default-Free Risk-Management (multiple with default risk) Measure (RMM) Premium Insurer 1 Fee Insurer 2 Provides replacing payment if Policyholder one/several (re)insurers fail Indemnification Indemnification Insurer 3 Insurer 4 Examples : Letter of Credit, Guarantees, Credit Default Swap, Guarantee Fund, Deposits Pro rata sharing of risk and premium Assuming that the number of insurers � is exogenously given, the policyholder decides on two variables: The quantity of insurance coverage denoted by � � (at the premium of � � � � ) • The quantity of the risk-management measure denoted by � � (at the fee of � � � � ) • The policyholder can take three positions at the total costs of � � � � � � � � � : � � � � � means a perfect hedging  the policyholder eliminates the default risk entirely � � � � � means an under-hedging  policyholder accepts a remaining default risk � � � � � means an over-hedging  policyholder bets on the default of (re)insurers Lukas Reichel EGRIE, Cyprus September 20, 2016

Model II: Wealth States and Utility 6 (1) Binary loss event: 0 with probability 1 � �, � � 0 with probability � � � (2) Multiple risk-sharing: insurer � � 1, … , � gets � � � � � of the premium and pays � � � in a loss event (3) The insurers’ defaults are assumed to be stochastically independent (4) Each insurer fails to compensate its share with probability � (ruin probability = �� , i.i.d-assumption) � (5) The insurer’s default-to-liability ratio equals 0 � � � 1 , i.e., 1 � � � � � are still paid in a default event Lukas Reichel EGRIE, Cyprus September 20, 2016

Model II: Wealth States and Utility 7 (1) Binary loss event: 0 with probability 1 � �, � � 0 with probability � � � (2) Multiple risk-sharing: insurer � � 1, … , � gets � � � � � of the premium and pays � � � in a loss event (3) The insurers’ defaults are assumed to be stochastically independent (4) Each insurer fails to compensate its share with probability � (ruin probability = �� , i.i.d-assumption) � (5) The insurer’s default-to-liability ratio equals 0 � � � 1 , i.e., 1 � � � � � are still paid in a default event State Probability Final Wealth �� � �,� ≔ � � � � � � � � � � � � � � � � � � � � � � � Loss event and � defaults � ⋮ ⋮ ⋮ � � �,� ≔ � � � � � � � � � � � � � � � � � �� � � � 1 � � ��� Loss event and � defaults � � � � � � � ⋮ ⋮ ⋮ � 1 � � � Loss event and 0 defaults �,� ≔ � � � � � � � � � � � � � � � � � 1 � � �� ≔ � � � � � � � � � � � No loss � Lukas Reichel EGRIE, Cyprus September 20, 2016

Model II: Wealth States and Utility 8 (1) Binary loss event: 0 with probability 1 � �, � � 0 with probability � � � (2) Multiple risk-sharing: insurer � � 1, … , � gets � � � � � of the premium and pays � � � in a loss event (3) The insurers’ defaults are assumed to be stochastically independent (4) Each insurer fails to compensate its share with probability � (ruin probability = �� , i.i.d-assumption) � (5) The insurer’s default-to-liability ratio equals 0 � � � 1 , i.e., 1 � � � � � are still paid in a default event State Probability Final Wealth �� � �,� ≔ � � � � � � � � � � � � � � � � � � � � � � � Loss event and � defaults � ⋮ ⋮ ⋮ � � �,� ≔ � � � � � � � � � � � � � � � � � �� � � � 1 � � ��� Loss event and � defaults � � � � � � � ⋮ ⋮ ⋮ � 1 � � � Loss event and 0 defaults �,� ≔ � � � � � � � � � � � � � � � � � 1 � � �� ≔ � � � � � � � � � � � No loss � The policyholder possesses a vNM utility function � with � � � 0 and � �� � 0 . The policyholder’s objective: � �� � � � � � � � 1 � � ��� � � max � � ,� � � � � max 1 � � � � �,� � � ,� � ��� Lukas Reichel EGRIE, Cyprus September 20, 2016

Model III: Premium/Fee Principle 9 Expected value calculus: Expected payoffs do Payoff from insurance contract: � � � � � � ��1 � ��� , not depend on � Payoff from risk-management measure: � � � � � � ��� . � � 1 � � as � ⟶ ∞ (=variance of a default-free insurance policy) But: ��� � � ↘ � � Assumed premium/fee principle: Actuarial Fair Premium/Fee x Proportional Cost Loading Premium for insurance coverage: � � � � � � � � 1 � � � � � � � � � � ��� 1 � � � �: � � � � 1 � � � , Fee for risk-management measure: � � � � � � � � 1 � � � � � � ��� 1 � � � �: � � � � 1 � � � . By assumption, � � is non-dependent on � • Assumption might be unmet (especially for large � ) due to economies of scale (Mayers & Smith, 1981) • Premium, fee and expected wealth do not depend on � •  Multiple risk-sharing has a mean-preserving effect  Under this premium principle SOC for the maximization problem max � � ,� � � � is fulfilled Lukas Reichel EGRIE, Cyprus September 20, 2016

Results I: Demand Effect of Multiple Risk-Sharing 10 At first, � � � 0 (no risk-management measure at hand) Policyholder’s utility � � � � 1 � � ��� � � � �� � � ∑ � � � 1 � � � � ��� �,� can be interpreted as Bernstein polynomial. Thereby, one can conclude: ∗ ∗ i. � ��� � � � for all � � 1 , i.e., � ��� � �,��� � � � � �,� , �→� � � � 1 � � � � lim �� � �� � � � � � � � � � � � � � � � � �1 � ��� , ii. ∗ � �,�� ∗ �→� � �,� lim � iii. ���� . � ∗ Example for actuarial fair premiums ( � � � 0 ): lim �→� � �,� � • ���� ; i.e., over-insurance is optimal ∗ , � �,� ∗ , � �,� ∗ , … monotonously increasing, too? • Utility is monotonously increasing in �; is the sequence � �,� Lukas Reichel EGRIE, Cyprus September 20, 2016