Agents’ Strategic Behavior in Risk Sharing
Letting agents play risk sharing games
Michail Anthropelos
Department of Banking & Financial Management
University of Piraeus
Weekly Seminar Series, Feb. 2013
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Outline
1
Motivation & Contributions
2
Optimal Risk Sharing
3
How much Risk Should an Agent Share?
4
Nash Equilibria in Risk Sharing
5
Conclusion & Further Tasks
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Outline
1
Motivation & Contributions
2
Optimal Risk Sharing
3
How much Risk Should an Agent Share?
4
Nash Equilibria in Risk Sharing
5
Conclusion & Further Tasks
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Motivation
Risk sharing
Financial agents could reduce their risk exposures by sharing parts of their random
portfolios with other agents in a mutually beneficial way.
Two types of settings
Optimal (unconstrained) risk sharing: Agents design and trading new
financial contracts.
Partially optimal (constrained) risk sharing: Agents partially share their
risks by trading at equilibrium a given vector of financial securities.
Motivation to act strategically
→ So far the related literature ignores the fact that agents can affect the risk
sharing equilibrium by exploiting their market power.
→ Why do we care?
→ A strategic use of market power reduces the efficiency of risk sharing.
→ The use of market power is in favor of speculators and low risk averse hedgers.
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Contributions
Modelling of the best response problem in optimal risk sharing
How much risk should an agent share?
Which is the random payoff an agent should put into the sharing with the other agents.
How and at which point does the market equilibrate?
Establishment of the Nash risk sharing equilibrium and comparison with the Pareto one.
Modelling of the best response problem in constrained risk sharing setting
Which is agent’s best equilibrium price?
Given the demand of the rest of the agents, which equilibrium price maximizes his utility?
How and at which point does the market equilibrate?
Equilibrium prices when “few” agents negotiate the one-shot transaction of a given vector
of assets (under quadratic preferences this is a variation of the CAPM).
CAPM prices remain unchanged if and only if agents are homogenous.
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A Short Tour on the Related Literature
Optimal risk sharing
Finding the financial assets that span the agents’ risky payoffs and their
allocations bring agents in a Pareto optimal condition.
X Seminal works of Borch (1962, 1968) and Wilson (1968).
X Many developments on financial oriented problems e.g. Duffie and Jackson
(1989), Demange and Laroque (1995), Duffie and Rahi (1995), Rahi (1995),
Pesendorfer (1995), Athanasoulis and Shiller (2000), Acharya and Bisin
(2005), Willen (2005), Peress (2010).
X The challenging mathematical problem of the conditions on agents’ risk
preferences that guarantee the existence of the optimal risk sharing has
recently studied by a number of authors e.g. Barrieu and El Karoui (2005),
Jouini et al. (2008), Dana (2011) etc.
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A Short Tour on the Related Literature, cont’d
Equilibrium pricing of given securities
Equilibrium pricing of non-replicable assets with given maturity.
X Classical references on equilibrium pricing models e.g. Lintner (1965), Black
(1972), Rubistein (1974), Duffie (1988).
X Recently new developments that consider weaker standing assumptions
e.g. Vanden (2004) and Cvitanić et al. (2009) or incorporate more abstract
risk measures Horst et al. (2010), A. and Žitković (2010), Cvitanić et al.
(2011).
Game theory on financial sharing
Although there is a rich literature on game theoretic structure on pricing financial
assets, only recently of this theory has been applied to risk (profit) sharing issues.
X For the adverse selection problem Page and Monteiro (2003, 2007), Sung
(2005), Horst and Moreno-Bromberg (2008, 2012).
X Rahi and Zigrand (2009).
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Outline
1
Motivation & Contributions
2
Optimal Risk Sharing
3
How much Risk Should an Agent Share?
4
Nash Equilibria in Risk Sharing
5
Conclusion & Further Tasks
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The Set Up
We impose a static model where (Ω, F, P) is a standard probability space and
X ⊆ L0 (F) denotes the set of the discounted payoff of all possible financial
positions with maturity up to a given terminal time T .
We assume the existence of n financial agents with common information of
traded assets and common beliefs.
Ei ∈ X , i = 1, ..., n stands for the random payoff of the agent i, i.e., the
accumulated unhedgeable payoff of his already undertaken portfolio (also
called random endowment).
Agents’ risk preferences are modelled by utility functionals:
Ui : X → R.
Example: Ui (X ) = E[X ] − γi Var[X ].
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Utility Maximization Problems
Standing assumptions:
(i) No budget constraints.
(ii) No short selling constrains.
(iii) No further transaction costs.
The abstract problem
sup Ui (X + Ei )
X ∈X
When the set of choices is the span of given assets
Let C ∈ X k be the payoff vector of given financial assets. If their prices are given
by a vector p ∈ Rk , agent i faces the problem:
sup {Ui (Ei + a · (C − p))}.
a∈Rk
And his demand schedule is
Zi (p) = argmax {Ui (Ei + a · (C − p))}
a∈Rk
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Utility Maximization Problems
Example: M-V preferences
Under M-V preferences the agent’s demand function is given by:
E[C] − p
Zi (p) =
− Cov(C, Ei ) · Var−1 [C]
2γi
where
E[C] = (E[C1 ], ..., E[Ck ]) and
Cov(C, X ) = (Cov(C1 , X ), ..., Cov(Ck , X )), for any X ∈ L2 (F).
X Demand comes from two distinguished sources: expectation and correlation.
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Constrained Risk Sharing (no use of market power )
Optimal span and equilibrium on C
Given a vector of payoffs C ∈ X k , the (partially) optimal risk sharing problem is
finding the Pareto equilibrium transaction of C.
Namely, the pair (p∗ , A∗ ) ∈ Rk × An×k is called a price-allocation equilibrium if
Zi (p∗ ) = a∗i
for each i ∈ {1, ..., n} (that is,
Pn
i=1
Zi (p∗ ) = 0).
Here, An×k denotes the set of allocations of securities in C among the n agents.
Theorem
(General utility functionals)
Let X = L∞ (F, P) and Ui ’s be non-decreasing and satisfy the Lebesque property.
Then, there is a unique price-allocation equilibrium (p∗ , A∗ ).
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Under M-V Preferences (no use of market power )
Proposition (CAPM)
Under M-V preferences, the equilibrium prices are given by
p∗ = E[C] − 2γ Cov(C, E)
where where E =
Pn
i=1
Ei (the market portfolio) and
1
γ
=
Pn
1
i=1 γi
.
How much utility does each agent get at equilibrium?
We define the utility level at equilibrium of agent i as
vi (p∗ ) = Ui (Ei + a∗i · (C − p∗ )).
where a∗i ’s are the optimal allocations. It holds that
vi (p∗ ) = γi Cov(C, Ci∗ ) Var−1 [C] Cov(C, Ci∗ ) + U(Ei )
where, Ci∗ is the contract that she would have purchased in the Pareto optimal
risk sharing (see next slide).
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Pareto OptimalPRisk Sharing (no use of market power )
Let A = {C ∈ X n :
n
i=1
Ci = 0}.
Definition
The vector of security payoffs C∗ ∈ A is a Pareto optimal risk sharing, if for any
C ∈ A, existence of some i ∈ {1, ..., n} such that Ui (Ei + Ci ) > Ui (Ei + Ci∗ )
implies
∃j ∈ {1, ..., n}, j 6= i such that Uj (Ej + Cj ) < Uj (Ej + Cj∗ ).
X A useful characterization (see among others Demange and Laroque (1995)):
C∗ = argmax
C ∈A
n
X
Ui (Ei + Ci ).
i=1
Proposition (see also Demange and Laroque (1995) and Barrieu & ElKaroui (2004))
Let all agents have M-V preferences. The unique (up to constants) optimal risk
sharing rule C∗ is given by
Ci∗ =
where E−i =
P
j6=i
γ
γi
E−i −
γi −γ
γi Ei
Ej .
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The Measure of Risk Sharing Inefficiency
Following the related literature (see among others Acharya and Bisin (2005)), we
define:
Risk Sharing Inefficiency = Optimal Aggregate Utility − Realized Aggregate Utility
In the case where inefficiency is caused by the constrains in risk sharing,
( n
)
( n
)
X
X
Risk Sharing Inefficiency = sup
Ui (Ei + Xi ) − sup
Ui (ai · (C + Ei ))
X∈A
i=1
A∈An×k
i=1
Proposition
Under M-V preferences, all agents suffer loss of utility because of the constrains.
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Outline
1
Motivation & Contributions
2
Optimal Risk Sharing
3
How much Risk Should an Agent Share?
4
Nash Equilibria in Risk Sharing
5
Conclusion & Further Tasks
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The Best Endowment Response
The motive
Let’s take the position of agent 1. Given:
the mechanism that produces the optimal sharing securities and their prices
and
the endowments that the rest n − 1 agents are going to share,
we are asking:
Which is the random quantity agent 1 should report as his endowment?
He “should” report what maximizes his utility after the application of the risk
sharing rules.
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The Best Endowment Response, cont’d
Let E2 , E3 , ..., En be given.
If agent 1 reports endowment B ∈ X he would get contract (see page 14)
C1∗ (B) =
n
γ X
γ1 − γ
Ei −
B
γ1
γ1
i=2
The level at the equilibrium would be
G1 (B; (Ei )ni=2 ) = U1 (E1 + C1∗ (B) − pr1 (B))
where pr1 (B) is the equilibrium prices that of the contracts (B, E2 , ..., En )
given by (see page 13)


Pn
E[B] − 2γ Cov B + Pi=2 Ei , B  E[E ] − 2γ Cov B + n E , E 
2
2 
γ − γ1 γ
γ
i=2 i

, , ...,
·
pr1 (B) =

..
γ1
γ1
γ1


.
Pn
E[En ] − 2γ Cov B + i=2 Ei , En
Best endowment response problem
B1∗ = argmax{G1 (B; (Ei )ni=2 )} = argmax{G1 (B; E−1 )}.
B∈X
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The Best Endowment Response, cont’d
Proposition
Let all agents have M-V preferences. The unique up to constants best endowment
response of agent i is
Bi∗ =
γi
Ei
γ +γ
| i {z }
+
A fraction of true endowment
γ2
E−i
− γ2
{z
}
|
γi2
Risk he does not possess
Remarks
The contract that agent i is going to get when he reports Bi∗ is
i
Ci∗ (Bi∗ ) = γiγ+γ
Ci∗ (i.e. a fraction of the Pareto optimal one).
The strategic behavior is more intense for low risk averse agents. Moreover,
lim Bi∗ = Ei .
γi →∞
The speculators (i.e. agents with Ei = 0) behave as hedgers in order to
improve their cash transactions.
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The Best Demand Response
We now adapt this agent’s strategic behavior in the constrained risk sharing
environments (the lines of argument are similar to Kyle (1989)).
Let agent
P i know the aggregate demand schedule of the rest of the agents,
that is j6=i Zj (p).
The
Pmarket equilibrates at some price p̃, when agent i asks the demand
− j6=i Zj (p̃) (i.e., when the sum of demand functions equals zero).
At this equilibrium, his utility is
X
X
Zj (p̃)) = Ui (Ei −
Zj (p̃) · (C − p̃))
φi (p̃;
j6=i
j6=i
Hence, his best price is the given p̂i is
p̂i = argmax{φi (p;
p∈Rk
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X
Zj (p))}
j6=i
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The Best Demand Response, cont’d
Proposition
Let all agents have M-V preferences. The best equilibrium price of agent i is given
by
γi2
γi
Ei .
p̂i = E[C] − 2γ Cov C, 2
E−i +
γi + γ
γi − γ 2
Remarks
The market equilibrates at this price vector if agent i reports the demand
function induced by the best endowment response Bi∗ (a clear connection
among constrained and unconstrained sharing ).
In fact, agent i manipulates the market equilibrium price by changing the
effective market portfolio
from
E
to
γi2
E
γi2 −γ 2 −i
+
γi
γi +γ Ei .
When agent i is a speculator (Ei = 0 and γi small), prices are higher when he
goes long and lower when he goes short.
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Outline
1
Motivation & Contributions
2
Optimal Risk Sharing
3
How much Risk Should an Agent Share?
4
Nash Equilibria in Risk Sharing
5
Conclusion & Further Tasks
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Nash Game in Risk Sharing
Let all agents follow the same strategic behavior (indicated by the best
endowment response).
Agents agree on the sharing rules (design of contracts and their pricing
mechanism) and negotiate the endowments that they are going to share.
This is a Nash-type of game among agents. We ask:
Does it have an equilibrium point?
How much efficiency is lost by use of the market power?
Which agents benefit from this sharing game?
Definition
We call a vector of random variables (B1∗ , B2∗ , ..., Bn∗ ) ∈ X n Nash risk sharing
equilibrium if for each i ∈ {1, ..., n}
Gi (Bi∗ ; (Bj∗ )j6=i ) ≥ Gi (Bi ; (Bj∗ )j6=i )
for all Bi ∈ X .
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Nash Game in Risk Sharing, cont’d
Theorem
Let all agents have M-V preferences. Then, there exists a unique (up to
constants) Nash risk sharing equilibrium and for each i ∈ {1, ..., n}:
(I) The best endowment response of agent i at Nash equilibrium is given by
Bi∗
γi
Ei +
=
γi + γ−i
γ−i
γi + γ−i
where B ∗ is the effective market portfolio, B ∗ =
2
B∗
Pn
i=1
Bi∗ , given by
Pn
E − γ i=1 Eγii
B =
Pn
2
1 − i=1 γγ 2
∗
i
(II) The risk sharing inefficiency caused by agents strategic behavior is always
positive and equal to
n
X
Ui (Ei + Ci∗ ) −
i=1
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n
X
Ui (Ei + Ci (Bi∗ )) =
n
X
i=1
γi Var[Ei − Bi∗ ] − γ Var[B ∗ − E].
i=1
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Nash Game in Risk Sharing, cont’d
Comparison of equilibria when n = 2
Effective Market Portfolio
Purchased contract
Pareto Sharing
E
Utility gain from sharing
γ2 E2 −γ1 E1
γ
h1 +γ2
i
2 E2
γ1 Var γ1 Eγ11−γ
+γ2
Inefficiency
0
Nash Sharing
γ1 E1 +γ2 E2
2γ
1 γ2 E2 −γ1 E1
2 γ1h+γ2
i
γ1 +2γ2
2 E2
Var γ1 Eγ11−γ
4
+γ2
h
i
γ1 E1 −γ2 E2
1
Var
γ1 +γ2
2
Corollary
B ∗ = E ⇔ Agents are homogeneous.
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Nash Game in Risk Sharing, cont’d
Discussion
The effective market portfolio is B ∗ , in which there is an overexposure of
the endowments of the more risk averse agents.
Agents with sufficiently low risk aversion benefit from playing the game.
Agents with low risk aversion reduces the efficiency of the risk sharing.
Agents have motives to share their risks with agents with high risk aversion.
It is reasonable to suppose that agents would create sub-markets of
homogeneous only agents when the transferred risk is low.
It can be proved that inefficiency goes to zero as the number of agents
increases to infinity.
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The CAPM in Oligopoly Financial Markets
Definition
The pair (p̂, Ẑ) ∈ Rk × Z is a Nash equilibrium price-demand of vector of
securities C if
Pn
(i)
i=1 Ẑi (p̂i ) = 0
(the demands clear out at price p̂) and
(ii) For each i ∈ {1, ..., n}
X
X
φi (p̂;
Ẑj (p̂)) ≥ φi (p;
Ẑj (p)), for every p ∈ Rk
j6=i
j6=i
(p̂ is the best price of all agents).
Theorem
Let all agents have M-V preferences. The unique Nash equilibrium price of a
vector C is
p̂ = E[C] − 2γ Cov(C, B ∗ )
where B ∗ is the effective market portfolio in the Nash risk sharing problem.
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The CAPM in Oligopoly Financial Markets, cont’d
Discussion
As before, in Nash equilibrium the market portfolio is B ∗ instead of the true
one E.
The Nash equilibrium prices coincide with the CAPM prices if and only if
agents are homogeneous. Even in this case however, the allocation is not the
optimal one and all agents suffer loss of utility.
Which agents benefit from this price negotiation?
Agents with low risk aversion and low risk endowments (i.e., the speculators).
Price pressure that is caused by negotiation game:
p̂j − pj∗ = 2γ Cov(Cj , E − B ∗ )
In some cases, a testable quantity!
Price pressure is against hedgers.
It is reasonable to argue that markets with low risk transfer should tend to
include homogeneous only agents.
As expected, Nash equilibrium prices approach the CAPM prices when
number of agents increases.
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Outline
1
Motivation & Contributions
2
Optimal Risk Sharing
3
How much Risk Should an Agent Share?
4
Nash Equilibria in Risk Sharing
5
Conclusion & Further Tasks
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Conclusion
This work attempts to fill a gap in the risk sharing literature by incorporating into
the modelling the strategic behavior of the participating agents. We have
seen:
X An endogenous explanation of the risk sharing inefficiency and the
security mispricing that occur on markets with few agents.
X It is never optimal for an agent to declare his true risky exposure.
X The lower the agent’s risk aversion is the more different his best
endowment response from his true endowment is.
X Strategic games benefit the agents with low risk aversion.
X The prices and the effective market portfolio remain unaffected by the
market power if and only if agents are homogeneous in their risk
preferences. Even in this case however, the volume of the market is reduced.
X The pressure on the equilibrium prices caused by the use of the market
power is in favor of speculators.
X As expected, all Nash equilibria tend to Pareto ones when the number of
agents increases to infinity.
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What is next...
−→ The form of the best response endowment and the existence of the Nash
equilibrium under general utility functionals.
· Initial results on the existence and the form of the best endowment response.
−→ The formation of the risk sharing (sub)markets, when the number of agents
is endogenously derived.
· Agents can choose not only how much risk they want to share but also with
whom.
−→ Dynamic version of the Nash risk sharing equilibria.
· How does the evolution of the information change the price pressure on the
prices?
−→ How does the ambiguity affect the Nash equilibria and the induced risk
sharing?
· Different beliefs alter the risk sharing rules and hence the Nash equilibria.
Does ambiguity decrease the inefficiency?
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THANKS!
Thank you for your attention!
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