The role of insurance companies in a risky economy

Transcription

1 The role of insurance companies in a risky economy EEA-ESEM Lisbon 2017

2 Motivation Example: a simple economy composed of people, each one exposed to an endowment risk distribution with 11 possible states (endowment from 0 to 10 units).

3 Motivation Example: a simple economy composed of people, each one exposed to an endowment risk distribution with 11 possible states (endowment from 0 to 10 units). Arrow and Debreu tell us that Pareto optimality is reached with security markets.

4 Motivation Example: a simple economy composed of people, each one exposed to an endowment risk distribution with 11 possible states (endowment from 0 to 10 units). Arrow and Debreu tell us that Pareto optimality is reached with security markets. The main issues are that: it requires to have a tremendous number of security markets. it requires to make public the individual state of each person in the economy.

5 Motivation Example: a simple economy composed of people, each one exposed to an endowment risk distribution with 11 possible states (endowment from 0 to 10 units). Arrow and Debreu tell us that Pareto optimality is reached with security markets. The main issues are that: it requires to have a tremendous number of security markets. it requires to make public the individual state of each person in the economy. What role is played by insurance companies?

6 Contribution I show that Pareto optimality is reached with only of these security markets if people have also access to standard insurance contracts supplied by stock insurance companies.

7 Contribution I show that Pareto optimality is reached with only of these security markets if people have also access to standard insurance contracts supplied by stock insurance companies. The presence of insurance companies allow: to reduce the required number of security markets by many orders of magnitude. to lower tremendously the required public information.

8 Contribution I show that Pareto optimality is reached with only of these security markets if people have also access to standard insurance contracts supplied by stock insurance companies. The presence of insurance companies allow: to reduce the required number of security markets by many orders of magnitude. to lower tremendously the required public information. I show this result in a static exchange economy with: multiple commodities, heterogeneous agents in terms of preferences and risk distributions, no restriction on risk dependence.

9 Literature review Kihlstrom and Pauly (1971), Ellickson and Penalva-Zuasti (1997): one commodity, heterogeneous agents, no restriction on risk dependence. they do not explain who supplies insurance contracts.

10 Literature review Kihlstrom and Pauly (1971), Ellickson and Penalva-Zuasti (1997): one commodity, heterogeneous agents, no restriction on risk dependence. they do not explain who supplies insurance contracts. Malinvaud (1973), Cass, Chichilnisky and Wu (1996) one commodity, homogeneous agents, i.i.d. risks. mutual insurance companies. Penalva-Zuasti (2001, 2008) one commodity, heterogeneous agents, i.i.d. risks. stock insurance companies.

11 Contents The model Risky economy. Spot and security markets. Insurance companies.

12 Contents The model Risky economy. Spot and security markets. Insurance companies. The equilibrium: Equilibrium. Pareto optimality. Role of insurance companies. Insurance premiums.

13 Risky economy Exchange economy with C commodities and N risk-averse heterogeneous agents: VNM utility for agent i: vi (.) : R C + R Endowment vector for agent i: ei (s i ) = e i l i (s i ), s i = 1,.., S i

14 Risky economy Exchange economy with C commodities and N risk-averse heterogeneous agents: VNM utility for agent i: vi (.) : R C + R Endowment vector for agent i: ei (s i ) = e i l i (s i ), s i = 1,.., S i Definition of an Arrow-Debreu state An Arrow-Debreu state is a full specification of the individual endowments obtained by all the agents in the economy.

15 Risky economy Exchange economy with C commodities and N risk-averse heterogeneous agents: VNM utility for agent i: vi (.) : R C + R Endowment vector for agent i: ei (s i ) = e i l i (s i ), s i = 1,.., S i Definition of an Arrow-Debreu state An Arrow-Debreu state is a full specification of the individual endowments obtained by all the agents in the economy. Definition of an Fundamental state A fundamental state is a full specification of the aggregate endowments in the economy.

16 Risky economy Arrow-Debreu states denoted: z = 1,.., Z, with probability π(z).

17 Risky economy Arrow-Debreu states denoted: z = 1,.., Z, with probability π(z). Fundamental states: t(.) : [1, Z] [1, T ] Individual states: s i (.) : [1, Z] [1, S i ]

18 Risky economy Arrow-Debreu states denoted: z = 1,.., Z, with probability π(z). Fundamental states: t(.) : [1, Z] [1, T ] Individual states: s i (.) : [1, Z] [1, S i ] With F t the set of Arrow-Debreu states in the Fundamental state t: z F t, i e i(s i (z)) = E(t(z))

19 Risky economy Arrow-Debreu states denoted: z = 1,.., Z, with probability π(z). Fundamental states: t(.) : [1, Z] [1, T ] Individual states: s i (.) : [1, Z] [1, S i ] With F t the set of Arrow-Debreu states in the Fundamental state t: z F t, i e i(s i (z)) = E(t(z)) Consumption plan of agent i in Arrow-Debreu state z denoted: x i (z).

20 Spot and security markets C spot markets: after the state of nature has been revealed. price vector: p(z).

21 Spot and security markets C spot markets: after the state of nature has been revealed. price vector: p(z). T security markets: security market t enables to get one ex-post money unit if t occurs, in exchange for z F t π(z) ex-ante money unit. quantity of securities purchased/sold by agent i denoted a i = (a i (1),.., a i (T ))

22 Insurance companies M insurance companies in competition supply insurance contracts and are owned through stock markets.

23 Insurance companies M insurance companies in competition supply insurance contracts and are owned through stock markets. An insurance contract for agent i consists in: an indemnity in state z: τ i (p(z), s i (z)) = p(z)l i (s i (z)); in exchange for a premium in any state z: α i = z π(z )p(z )l i (s i (z )).

24 Insurance companies M insurance companies in competition supply insurance contracts and are owned through stock markets. An insurance contract for agent i consists in: an indemnity in state z: τ i (p(z), s i (z)) = p(z)l i (s i (z)); in exchange for a premium in any state z: α i = z π(z )p(z )l i (s i (z )). Quantity of insurance purchased by agent i: ni.

25 Insurance companies M insurance companies in competition supply insurance contracts and are owned through stock markets. An insurance contract for agent i consists in: an indemnity in state z: τ i (p(z), s i (z)) = p(z)l i (s i (z)); in exchange for a premium in any state z: α i = z π(z )p(z )l i (s i (z )). Quantity of insurance purchased by agent i: ni. Profit of insurer k: r k (z) = j N k (α j τ j (p(z), s j (z)))n j

26 Insurance companies M insurance companies in competition supply insurance contracts and are owned through stock markets. An insurance contract for agent i consists in: an indemnity in state z: τ i (p(z), s i (z)) = p(z)l i (s i (z)); in exchange for a premium in any state z: α i = z π(z )p(z )l i (s i (z )). Quantity of insurance purchased by agent i: ni. Profit of insurer k: r k (z) = j N k (α j τ j (p(z), s j (z)))n j Share of insurer k purchased by agent i: mki.

27 Equilibrium Agent maximization problem: max π(z)v i (x i (z)) x i,n i,m ki,a i z s.t. p(z)x i (z) = p(z)e i (s i (z)) + τ i (p(z), s i (z))n i + k r k (z)m ki + a i (t(z)), z α i n i + z π(z)a i (t(z)) = 0

28 Equilibrium Agent maximization problem: max π(z)v i (x i (z)) x i,n i,m ki,a i z s.t. p(z)x i (z) = p(z)e i (s i (z)) + τ i (p(z), s i (z))n i + k r k (z)m ki + a i (t(z)), z α i n i + z π(z)a i (t(z)) = 0 Market clearing conditions (spot, security, stock): x i (z) = E(t(z)), z Z i a i (t) = 0, t T i m ki = 1, k M i

29 Pareto optimality Result In the decentralized economy with C spot markets, T security markets and insurance companies in competition, the equilibrium exists and the allocation is Pareto optimal.

30 Role of insurance companies With insurance companies, it is sufficient to have T security markets: much lower quantity of security markets. need to make public only the aggregate endowment.

31 Role of insurance companies With insurance companies, it is sufficient to have T security markets: much lower quantity of security markets. need to make public only the aggregate endowment. One way to reach her consumption plan for each agent i is: elimination of individual endowment risk with insurance contract: n i = 1. elimination of price risk exposure and participation to aggregate risk through security and insurance stock markets: a i (t(z)) = p(z)(x i (z) e i ) + α i 1 N k r k(z) and m ki = 1. N

32 Insurance premiums Insurance companies are constrained to sell fair contracts to catch policyholders on one side and shareholders on the other side.

33 Insurance premiums Insurance companies are constrained to sell fair contracts to catch policyholders on one side and shareholders on the other side. Insurance premium: α i = z π(z )p(z )l i (s i (z )) = (1 + β i )p z π(z )l i (s i (z )), in which: p = π(z )p(z ) z 1 β i = p π(z )(p(z ) p)l z π(z )l i (s i (z i (s i (z )). )) z

34 Insurance premiums Insurance companies are constrained to sell fair contracts to catch policyholders on one side and shareholders on the other side. Insurance premium: α i = z π(z )p(z )l i (s i (z )) = (1 + β i )p z π(z )l i (s i (z )), in which: p = π(z )p(z ) z 1 β i = p π(z )(p(z ) p)l z π(z )l i (s i (z i (s i (z )). )) z For an insurer, the higher the correlation between insured individual risks and the aggregate risk, the higher the expected profit.

35 Conclusion An exchange economy with multiple commodities, heterogeneous agents and no restriction on risk dependence. Only one security market per fundamental state is sufficient if there are also stock insurance companies supplying standard insurance contracts. Insurance and security markets respectively allow to deal with endowment risks and price risk. Insurance premium has an aggregate risk loading factor which is specific to each agent.