Catastrophe insurance and asset prices in dynamic general...
Transcript of Catastrophe insurance and asset prices in dynamic general...
-
Catastrophe insurance and asset prices in dynamic general
equilibrium: some implications for catastrophe bonds
Hisashi Nakamura∗
Graduate School of Business AdministrationHitotsubashi University
July 5, 2019
Abstract
This paper provides a dynamic (in particular, continuous-time) general equilib-rium framework to investigate corporate catastrophic risk management usinginsurance and capital markets under financial frictions. It examines (1) theefficiency of catastrophe insurance in capital markets in the presence of hid-den information, hidden actions, and insurance funds’ insolvency and (2) theeffect of the insurance on equilibrium asset prices. It considers a representative-investor production economy with firms that are subject to catastrophic riskscharacterized by Poisson processes. The risks are homogeneous ex ante, buttheir stochastic intensities become heterogenous ex post. Each firm managercan observe privately the stochastic intensity of his own Poisson process ex post,and in addition, can control it ex post in a hidden way by incurring effort costs.Other than the insurance activities, insurance funds are managed safely by in-vesting only in a risk-free asset. But, with the insurance activities, the fundsare exposed to insolvency risks. In this environment, this paper shows thatpartial insurance is optimal and has a dynamic effect on the risk-free rate andmarket returns in capital markets in equilibrium. It also derives implicationsfor catastrophe bonds.
Keywords: catastrophe insurance, catastrophe bonds, interest rates, transac-tion costs, insurance funds’ insolvency, hidden information, hidden actions,dynamic general equilibrium.
JEL Classification Codes: D53, D82, G12, G22.
∗Corresponding author: Hisashi Nakamura, Graduate School of Business Administration, HitotsubashiUniversity, 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan. Phone: +81-42-580-8826. Fax: +81-42-580-8747.Email: [email protected]. I am thankful to Toru Igarashi for his contribution to the prooffor Proposition 1. This work was supported by the Japan Society for the Promotion of Science (JSPS)Grants-in-Aid for Scientific Research (B) Grant Number JP18H00874.
1
-
1 Introduction
Since the 1990s, we had observed the convergence of capital markets and (re)insurance
sectors (Cummins and Weiss, 2009), supported by a rapid growth in financial technologies
such as financial derivatives and securitization. In the early 2000s, the market economy
seemed to overcome its vulnerability to risks by containing them in deep-pocket capital
markets. In particular, in corporate risk management, catastrophe (CAT) bonds began
to provide the insurance industry with protections against catastrophic disasters (e.g.,
earthquakes, hurricanes, and typhoons). The CAT bond market had grown strongly since
then (Polacek, 2018).
However, once the global financial crisis arose, the convergence did not advance as much
as we had expected, due to financial frictions such as hidden information, hidden actions,
transaction costs, and default. We have realized again that the market economy is still
vulnerable to catastrophic disaster risks. These observations motivate a question: how do
the capital markets and the insurance sectors diversify corporate catastrophic risks in the
whole market economy when their convergence is limited under financial frictions?
The objective of this paper is to answer the question by (1) providing a framework
to study dynamically varying state prices in a dynamic (in particular, continuous-time)
general equilibrium model with financial frictions such as hidden information, hidden ac-
tions, and insurance funds’ insolvency, and (2) examining the efficiency of CAT insurance
in capital markets. This general equilibrium framework is useful for investigating the role
of insurance in capital markets. This paper also derives implications for CAT bonds.
This paper considers a representative-investor production economy with firms that
are subject to catastrophic disaster risks characterized by Poisson processes. The risks
are homogeneous ex ante, but the stochastic intensities of the Poisson processes become
heterogenous ex post. Each firm manager can observe privately the stochastic intensity of
his own firm’s Poisson process ex post, and in addition, can control it ex post in a hidden
way. The investor can invest her funds in three technologies: a risk-free asset, firms, and
an insurance technology. Call the funds invested in the insurance technology insurance
funds., denoted by I. Let W denote the investor’s wealth invested directly in the risk-free
2
-
asset and the firms. Let X denote the investor’s total wealth: X = I +W .
With regard to the investment to the insurance technology, Raised from the investor,
the insurance funds can provide each firm manager with a bilateral insurance contract,
which transfers some part of the disaster risk from the firm to the funds in exchange for
an insurance premium. For simplicity, the insurance technology faces no transaction costs
when paying indemnities.1 Other than the insurance activities, the insurance funds manage
themselves safely by investing only in the risk-free asset. The funds can be interpreted as a
safely managed risk-neutral institutional investor from a viewpoint of asset management.
However, with the insurance activities, they are exposed to insolvency risks. From the
lender’s viewpoint, she sets the insurance premium rate ex ante so as to expect zero profit
from the insurance contract. In addition, she invests in the insurance technology for not
only the funds’ profitability but also risk pooling, which is different from risk diversification
in the capital market portfolio.
The main findings of this paper are as follows. Firstly, partial insurance is optimal
when cancellation fees are appropriately built in the insurance contract. The problem of
the hidden actions is harmful. The insurance influences the players’ welfare in two ways.
On the one hand, it mitigates each firm manager’s incentive to make efforts to reduce his
stochastic intensity: i.e., he becomes lazy in a hidden-action problem under the insurance
contract. On the other hand, at the same time, the insurance improves the investor’s
welfare. As the insurance technology increases the firm managers’ ex-ante efficiency even
under the hidden-action problem, the investor can exploit the firm managers more by
lowering the firm managers’ utility up to their reservation utility, so long as the insurance
contract avoids the firm managers’ ex-post exit. The coverage ratio (i.e., the extent of the
partialness of the insurance) is determined to balance the two effects. In addition, some
contagions of the jump shocks make the problem of the hidden actions worse, decreasing
the optimal coverage ratio, under some parametric assumptions.
Secondly, the insurance has a crucial effect on equilibrium asset prices in the capital
1Due to this simplification, no deductibles are optimal in insurance design in this model,in contrast to e.g. Arrow (1971, 1974) and Raviv (1979). Instead of omitting the costs,this present paper examines the effect of the informational frictions on optimal insurancedesign. As to a survey of optimal insurance design, see e.g. Gollier (2013).
3
-
markets. Payments of insurance premiums and claims do not change the investor’s total
wealth X directly by offsetting the changes in I and W . However, the insurance activities
influence the capital markets through two channels in this model. First, the insurance
funds’ default causes a deadweight loss to the total wealth. Second, the equilibrium state
prices are influenced by which neighborhood the ratio IW is moving in. I can choose the
neighborhood, depending on where the ratio IW is moving in on average. As the insurance
funds are invested only in the risk-free asset, the ‘average’ amount of the insurance funds
restricts the riskiness of the total wealth. Specifically, the equilibrium risk-free rate is
influenced by the ‘average’ ratio of the insurance funds to the total wealth. When the ratio
is lower (higher) on average, the equilibrium risk-free rate is lower (higher), because, as
the investor invests more (less) in the risky assets on average, the market prices of diffusive
and jump risks and the risk premium are higher (lower) in equilibrium. In addition, the
contagions of the jump shocks may increase the insurance funds on average, raise the
equilibrium risk-free rate, and lower the market prices of diffusive and jump risks.
Finally, based on the model, I can argue that CAT bonds with well-designed indemnity
triggers, in which forfeiture is linked with the losses of the insurance funds, could improve
welfare. In this model, default causes a deadweight loss in the economy. The CAT bonds
could be attractive if, financing the default costs ex ante with them, the investor could
save on the deadweight loss ex post (see e.g. Polacek (2018)). However, when the prob-
ability of being triggered is very high, the CAT bonds could be so expensive as to lose
the attractiveness. In addition, the CAT bonds are not a redundant financial derivative
that can be replicated in complete markets here. By arranging the saving on the default
costs in advance, the CAT bonds could influence equilibrium state prices. This present
general equilibrium framework enables us to make clear how the financial frictions (i.e.,
the default costs (or the saving on them) as well as hidden information, hidden actions,
and transaction costs) influence equilibrium state prices (i.e., the risk-free rate and market
prices of risks) dynamically under the physical measure.
In previous literatures, there are lots of papers that examine how financial frictions
intervene disaster risk management. The seminal paper of Froot (2001) finds eight the-
oretical explanations for inefficiencies in CAT reinsurance markets by examining capital
4
-
market imperfection regarding risk-bearing capacity. Froot and O’Connell (2008) model
the equilibrium price and quantity of risk transfer between firms and financial intermedi-
aries in the presence of financing imperfections that make the intermediary capital costly.
In contrast, this present paper examine the inefficiencies that are different from Froot and
O’Connell (2008)’s: hidden information, hidden actions, and insurance funds’ default. In
addition, it does not distinguish between insurance and reinsurance, by assuming an aggre-
gate insurance technology. In light of Froot (2001)’s theoretical explanations, this paper
examines insufficient capital in insurance, illiquidity of the insurance contract, hidden in-
formation and hidden actions, and the investor’s recursive utility as non-expected utility,
while not examining reinsurer’s market power, agency costs on either the supply side or
the demand side of reinsurance, and governmental ex-post interventions.
Moreover, this paper provides a more dynamic framework using mathematical tech-
niques of continuous-time stochastic controls on the assumption of both continuous and
jump shocks in an environment in which the jump risks are difficult to diversify based on
the law of large numbers. Due to the mathematical convenience, this model is comparable
to financial engineering models based on the Black-Scholes-Merton options valuation the-
ory, which basically assumes a unique risk-neutral measure (i.e, state price) in complete
markets without financial frictions.
Coval, et al. (2009) investigate catastrophic risks of senior trenches created in structured
finance activities (called economic catastrophe bonds), which default only in the most
adverse economic states and collect high credit ratings. They construct a state contingent
pricing framework by integrating Merton (1974)’s credit structural default model into the
CAPM. Using the framework, they analyze state prices to extract economic intuitions
from asset prices, like this paper does. However, their paper focuses on default of pooled
securities while, by contrast, this present paper examines default of the insurance funds that
pool risks from firms through bilateral contracts, not through market trading of securities.
In addition, this paper assumes hidden information and hidden actions, in addition to the
default, as financial frictions.
Finken and Laux (2009) examine a structural relationship between the markets of CAT
bonds and CAT reinsurance in an environment in which there exists asymmetric informa-
5
-
tion between reinsurers regarding an insurer’s risk. They formalize the idea, argued in
the seminal paper of Froot (2001), that CAT bonds make the reinsurance market more
contested, in that CAT bonds with information-insensitive triggers reduce the fear of ad-
verse selection and improve a competitive effect on the pricing of reinsurance contracts,
even in the absence of transaction costs or moral hazard. In their model, the asymmetric
information problem is between multi (re)insurers, while, in this model, it is between the
investor and the firm managers. In addition, this paper examines the dynamic effect of the
insurance funds’ insolvency on the capital markets.
Lakdawalla and Zanjani (2012) examine the role of collateral in catastrophic risk man-
agement, as full collateralization is an attractive feature of CAT bonds. They show that
the full collateralization feature puts some limits on their issuances while CAT reinsurance
economizes on collateral through diversification, and in addition, that the CAT bonds are
useful when the division of collateral in the event of default is incontractible. In contrast,
this present model does not examine collateralization, but rather examines ex-post costly
reorganization of the insurance funds in the event of default.
The rest of this paper is organized as follows. The next section defines the environment
of this model. Section 3 gives a formal representation of optimization problems, character-
izes equilibrium, and examines the efficiency of CAT insurance in the capital markets. In
addition, it investigates the effect of the optimal insurance on the risk-free rate and mar-
ket returns in general equilibrium. Furthermore, it discusses implications for CAT bonds.
The final section concludes, mentioning several limitations of this present study and its
directions for future research.
2 Model
2.1 Basic setup
I consider a representative-investor production economy. Time is t ∈ [0, T ] on an annual
basis where T > 0 is a constant . The filtered probability space is (Ω,F ,F = {Ft}0≤t≤T , P ),
where the filtration F is generated by a one-dimensional standard Brownian motion B and
n-dimensional independent Poisson processes {Nj ; j = 1, · · · n} (n > 1), where, for each
6
-
j , the stochastic intensity of Nj is a constant λj (0 < λj < 1), defined on the probability
space. Define the compensated Poisson process Ñ0j (t) := Nj − λjt for each j.
A representative investor lives on [0, T ] and consumes a single consumption good over
time. Let C denote the consumption process. There are n types of non-atomic firms, each
of which, denoted by j ∈ {1, · · · n}, has a continuous mass. For j ∈ {1, · · · n}, the firm j’s
production process is subject to the Brownian motion B and the Poisson process Nj . The
Brownian motion B is a common shock representing a macroeconomic continuous shock,
and is public information. The Poisson process Nj is an individual, independent shock,2
representing the probability of the firm j’s suffering a catastrophic loss. Call the stochastic
intensity λj of the Poisson process Nj the type of the firm j. The n types of the firms are
uniformly distributed. Once the investor invests in a (non-atomic) firm manager either ex
ante or ex post, the manager is assigned randomly, based on the uniform distribution, to
a (non-atomic) firm j – call him the firm manager j. For convenience, I will use female
pronouns for the investor, and male ones for the firm managers.
In contrast to standard representative-investor production economies (e.g., Cox, et al.,
1985), there are two information problems in this model. First, both of the investor and the
firm managers can observe the paths of each firm’s jump process and the Brownian motion,
but neither the investor nor the firm managers can observe any stochastic intensities of
{Nj ; j = 1, · · · n} ex ante (i.e., at time 0). The firm manager j ∈ {1, · · · , n} can observe
privately his own type λj ex post when invested. Second, each firm manager can decrease
the stochastic intensity by making efforts with some effort costs over time in a hidden way
ex post after knowing his own type privately.3 Note that, although the investor cannot
observe the intensities directly, she can influence them ex post through the ex-ante designed
insurance contract.
2I will incorporate some contagions of jump shocks in Section 3.3 below.3The jump probabilities characterized by the Poisson processes are not the ones of the
natural disasters that are exogenous to the firm managers, but rather the ones of the firm’ssuffering a catastrophic loss. Thus, I am assuming that the firm managers can control thestochastic intensities.
7
-
2.2 Investment opportunities
Let X denote the process of the total wealth process owned by the investor. The investor
has initial funds X(0) = x0 > 0. Three types of investment opportunities are available to
the investor: a risk-free asset, the firms, and an insurance technology. Let I denote the
funds invested in the insurance technology – call them insurance funds.4 Let W > 0 denote
the process of the investor’s wealth invested directly in the risk-free asset and the firms.5
For notational convenience, define θ := IW and ϑ :=WX . I.e., X = W + I = (1 + θ)W and
ϑ = 11+θ .
More specifically, firstly, the return process of the risk-free asset (with its price P0) is
characterized by
dP0(t)
P0(t)= r dt.
The risk-free rate r is endogenously determined in equilibrium. I assume that the risk-free
asset, including the investment by the insurance funds (as I will define in detail below), is
in zero net supply in the whole economy.
Secondly, with regard to the investment in the firms, the investor designs a compen-
sation scheme for the firm managers’ share ex ante. As described above, the investor can
observe the paths of the firm’s production including the jumps, but cannot observe the
stochastic intensities either ex ante or ex post. As the firms look homogeneous to the
investor, she invests in the firms as a group, not in each individual firm j ∈ {1, · · · , n}.
The investor designs ex ante the firm managers’ share at the end of the contract as a
ratio to the firm value, denoted by a constant s ∈ [0, 1], and provides the firm managers
with an investment contract, characterized by s, as a take-it-or-leave-it offer. The invest-
ment contract is enforceable on the contracting firm managers, while, on the other hand,
the lender can discharge the firm managers from the investment, due to her decision of
portfolio allocation.
When entering into the contract at time 0, the firm manager is assigned to a firm, say
4As defined below, the funds can be interpreted as a safely managed, risk-neutral insti-tutional investor from a viewpoint of asset management.
5As I will define below, the investor can also invest in the risk-free asset indirectlythrough the insurance funds.
8
-
firm j, and gets an ex-ante investment Kj(0) = 1 from the investor. The return process of
the firm j’s production (j ∈ {1, · · · , n}) is characterized by
dRj(t) =dKj(t)
Kj(t−)= µ dt+ σ dB(t) + z dNj(t).
Thus, Kj denotes the firm manager’s firm value as well. According to the terms of the
contract s, sKj is the firm manager’s share. Both µ and σ are constants, are common
to the firms, and are common knowledge. The Brownian motion drives such common,
observable macroeconomic shocks.
With regard to the jump shocks, the loss z (where −1 < z < 0) is a constant, negative
proportional jump, is common to the firms, and is common knowledge.6 As 0 < λj < 1
(j ∈ {1, · · · , n}), each shock is expected to occur less than once in one year. While
the shocks are such rare events, z is so large (i.e., catastrophic) as to influence the whole
economy. Note that I am focusing on the heterogeneity of the stochastic intensities, instead
of the homogeneity of µ, σ, and z.7 While the stochastic intensities are private information,
the firms’ capital levels are public information.
I assume that time-t investment is at the same level as time-t capital level of the
incumbents with the same type j.8 As the amount of the investment in each firm manager
is limited to Kj(t) either ex ante or ex post, the mass of the invested firm j (as well as that
of the invested firm managers j) is adjusted to the investor’s investment size in a perfectly
elastic way for each j.9
Each firm manager j ∈ {1, · · · , n} can decrease the stochastic intensity λj into λj(1 +
αj(t)) (−1 < αj < 0) by incurring some effort costs. Call αj the firm manager j’s efforts.6The assumption of the constant, common proportional jump shocks is a simplification
for obtaining explicit (approximate) solutions.7To obtain richer results, I could extend the model to have mean-reverting drift rates,
stochastic volatilities, and stochastic, heterogenous jump sizes. The extended model wouldbe more complex numerically.
8The firm is indexed by its type j while the investor does not know λj . In other words,I am assuming that the investor can observe j and Kj , but cannot observe λj .
9To ensure the perfect elasticity, I will impose a few more assumptions about ex-postinvestment opportunities including the firm managers’ ex-post reservation utility below.
9
-
Distinguished from Ñ0j , the compensated controlled Poisson process is defined as
Ñj(t) = Nj(t)−∫ t
0λj(1 + αj(u)) du.
Note that I will give a formal representation of the effort costs shortly below. In summary,
the return processes of the firms as a group are characterized by
1
n
n∑j=1
dRj(t) =1
n
n∑j=1
((µ+ λj(1 + αj(t))z
)dt+ σ dB(t) + z dÑj(t)
). (2.1)
Thirdly, the investor can invest in an insurance technology, which collects the firms’
disaster risks from the firm managers without knowing their types and insures the disaster
losses according terms of a term insurance contract on [0, T ].10 Specifically, raised from
the investor, the insurance funds can provide each firm manager with a bilateral insurance
contract, which transfers some part of the disaster risk from the firm to the funds in
exchange for an insurance premium. Other than the insurance activities, the insurance
funds manage themselves safely by investing only in the risk-free asset. The funds can be
interpreted as the safe funds owned by an institutional investor as well. However, with the
insurance activities, they are exposed to insolvency risks. In addition, the funds can be
reinvested in a singular way11 ex post by the investor at a proportional cost.12 Note that
I will define the reinvestment in detail below.
From the lender’s viewpoint, she invests in the insurance technology for not only the
funds’ profitability but also risk pooling, which is different from risk diversification in the
capital market portfolio. She invests in the insurance funds I(0) ex ante. The contract is
written based on ex ante information, and in addition, is not tradable ex post. The investor
sets the insurance premium rate so as to expect zero profit from the insurance contract.
I assume that the insurance contract is of a proportional type here. I also assume that
10In this paper, I do not distinguish between insurance and reinsurance, as the insurancefunds are risk-neutral and face no transaction costs when paying indemnities.
11It is called singular because the reinvestment is singular with respect to Lebesguemeasure dt. That is, the reinvestment is made at a point mass t that has a nonzero mass.In other words, the reinvestment size does not depend on time duration: its process cannotbe characterized by i dt for some i > 0.
12The reinvestment cost is proportional to the size of the reinvestment.
10
-
the firm managers cannot commit themselves to the contract: they can quit it ex post.
Note that I will later consider cancellation fees to keep the policyholders from the quitting.
The contract form consists of two parts: an insurance premium rate p > 0 and a coverage
ratio 0 ≤ c ≤ 1 where c is a constant and p is characterized by a deterministic function of
time t (i.e., p = p(t)).13 Specifically, the firm manager pays the insurance premium p · c
continuously and receives the payment of the insurance claims, namely c · z, every time the
firm suffers the loss z, unless the insurance funds become insolvent. The contract (p, c) is
common to the firms.
The timing of the ex-ante designing of the insurance contract (p, c) is as follows. First,
the insurance premium rate p is designed ex ante so as to have zero per-group expected
returns at each t ∈ [0, T ] with c given. I may write p = p(t; c) as well. Hence,
1
n
n∑j=1
E[cKj(t−)
{(p+ λj(1 + αj)z
)dt+ z dÑj(t)
}]= 0 (2.2)
where E represents the expectation operator, and in addition, Et[·] := E[· | Ft]. Next, the
lender as the firm owner designs ex ante c to maximize the firm managers’ ex-ante utility
given p(t; c).
In summary, let Rcj(t) denote the excess returns of the firm j under a given insurance
contract (p, c). The process of the excess returns of the firms as a group is characterized,
modifying Eq.(2.1), by
1
n
n∑j=1
(dRcj(t)− r dt) =1
n
n∑j=1
(µ− r − cp+ λj(1 + αj(t))(1− c)z
)dt
+σ dB(t) + (1− c)z dÑj(t)
.where p is characterized by Eq.(2.2).
13In practice, when a firm invests in his covered property during the duration of aninsurance, terms of the contact are often re-designed ex post. In contrast, in this presentmodel, the re-designing is internalized in the ex-ante contract beforehand by building thetime-varying p. This could be interpreted as a continuous-time counterpart to the insurancecontract in practice. The insurance contract could take more realistic form (e.g., annualor semi-annual insurance premium rates and an excess-of-loss layer). I could apply theexplicit (approximate) results obtained in this model to such more realistic models vianumerical approximations.
11
-
The insurance funds are subject to two financial frictions. First, each firm manager
could cancel the insurance contract when he finds ex post that his type is so good (i.e., his
stochastic intensity is so low) that the contract is costly. However, the exit would mitigate
the risk-pooling effect of the insurance technology. To overcome such limited commitment
problem, I consider the situation in which, by building some sufficiently large cancellation
fees in the insurance contract, any firm manager would not have an incentive to cancel the
insurance contract ex post even when he finds that his type λj is sufficiently low.
Second, the insurance funds could become insolvent as it either has low profitability
or suffers large losses caused by (possibly multiple) catastrophic disasters. When I ≤ 0,
it causes costly default. There are two types of default: liquidation and reorganization.
δL and δR, where 0 < δL, δR < 1 are constants, represent the proportional ratios of the
default costs relative to W in the events of liquidation and reorganization, respectively.
More specifically, when I ≤ 0, the investor can choose to liquidate the insurance funds
by incurring liquidation costs δLW and to restart with (1 + θ− δL)W . On the other hand,
to avoid such liquidation, the investor can alternatively choose to reorganize the insurance
funds by reinvesting her funds from W into I in a singular way14 with reorganization costs
δRW and reinvestment costs π dξ incurred, where 0 < π < 1 is a constant and dξ represents
the singular reinvestment from W into I. The insurance funds I then increases by (1 −
π) dξ(τ). Defining χ := dξW , the investor possesses the funds (1+θ−χ−δR)W immediately
after the reorganization. I can also write the reorganization costs by (1 + δR
χ ) dξ(τ) at a
default time τ . Call also a series of the actions of the reinvestment and the reorganization
an intervention in the terminology of stochastic controls.
In this model, with regard to the event of the insurance funds’ default, I focus on the
situation in which reorganization is chosen over liquidation. To obtain such relevance of
the reorganization, I impose the following parametric assumptions. Firstly, I assume
Assumption 1 0 < δR < δL < 1.
It means that, in the event of default, the investor prefers the reorganization to the liqui-
dation so long as the lender can finance the reinvestment (i.e., 0 < χ < 1).
14Recall that it is called singular because the reinvestment is singular with respect toLebesgue measure dt. See also Footnote 11.
12
-
Secondly,
Assumption 2 π > min{δL, 21+ 1−δ
R
δL−δR}.
The first part on the right-hand side (i.e., π > δL) means that the liquidation would be
less costly than the reinvestment of all W . With this, the reinvestment can be financed:
χ < 1. On the other hand, with regard to the second part, since 21+ 1−δ
R
δL−δR< 1 due to
Assumption 1, the inequality π > 21+ 1−δ
R
δL−δRmeans that such high level of π reduces the
reinvestment, resulting in a not quite high level of reorganized insurance funds, which looks
relevant in practice. I will return to this point later.
In summary, the process of W is characterized by
dW (t) = W (t−)
r dt+ ψ µ− r − pc
+ 1n∑n
j=1(1− c)λj(1 + αj)z)
dt+σ dB(t) + 1n
∑nj=1(1− c)z dÑj(t)
(2.3)
−C(t) dt− (1 + δR
χ) dξ(t)
where ψ (0 < ψ < 1) represents the ratio of the investment in the firms relative to W . On
the other hand, the process of the insurance funds I is characterized by
dI(t) = I(t−)r dt+W (t−)ψ1
n
n∑j=1
c{(p+ λj(1 + αj)z
)dt+ z dÑj(t)
}+(
1− π)
dξ(t). (2.4)
Thus, from Eqs.(2.3)(2.4), the process of the total wealth X is characterized by
dX(t) = X(t−)
r dt+ ψ̃(µ− r + 1n
∑nj=1 λj(1 + αj)z)
)dt
+σ dB(t) + 1n∑n
j=1 z dÑj(t)
−C(t) dt−(π +
δR
χ
)dξ(t) (2.5)
where ψ̃ := ψWX .
13
-
2.3 Players’ controls and performance criteria
The firm manager j (for each j) receives utility of his terminal share sKj(T ) at T . The
terminal utility form is of a logarithm type: a log(sKj(T )
)for the type j where a > 0 is
a constant. In addition, he suffers the effort costs to control the stochastic intensity after
observing his own type λj during the production process. The effort costs, denoted by Γ,
are defined as: for the type j,
Γ(αj) = e−βTaE
[λj
∫ T0
((1 + αj(u)
)log(
1 + αj(u))− αj(u)
)du]
where a constant β > 0 represents the instantaneous rate of time preference. Strictly
speaking, Γ is not a relative entropy – a measure of statistical discrimination between λj
and λj(1 + αj) –, in that the expectation is taken under the physical measure. However,
it is equal to the relative entropy in equilibrium, because the optimal αj is a constant as I
will show later.
The firm manager values his performance based on expected utility. The firm manager’s
ex-ante expected utility (net of the effort costs), denoted by U , is written as follows: for
some {αj ; j = 1, · · · , n}, c, p and s,
U(0) = ae−βT1
n
n∑j=1
{E[
log(sKj(T )
)]− Γ(αj)
}.
The firm managers control their own efforts {αj ; j = 1, · · · , n}. In addition, they are given
ex-ante reservation utility e−βT log κ at time 0 where κ > 0 is a constant. This forms the
participation constraint on the investment contract. Note that I will define their ex-post
reservation utility below.
On the other hand, the investor receives a Duffle and Epstein (1992) type stochastic
differential utility, denoted by V , as follows: for some C, φ,dξ, {αj ; j = 1, · · · , n}, c, p
and s,
V (0) = E[ ∫ T
0f(C(u), V (u)) du+ e−βT
X(T )1−γ
1− γ
]
14
-
where f(C, v) := β(1−γ)v[logC − 11−γ log((1− γ)v)
]. β > 0 is the common instantaneous
rate of time preference. A constant γ > 0 represents the coefficient of (comparative) risk
aversion. The elasticity of intertemporal substitution is one here. That is, γ − 1 represent
the investor’s preference for early resolution of uncertainty.
The lender’s controls are summarized as follows. At the ex-ante stage, first, she designs
the compensation scheme s in the investment contract so as to reduce the firm managers’
utility up to their reservation utility, by providing a take-it-or-leave-it offer to the firm
managers. Second, under the investment contract s, she sets the insurance premium rate
p(t; c) so as to hold the no expected profit condition (2.2) with c given. Third, she controls
the coverage ratio c to maximize the ex-ante homogeneous firm managers’ utility with s
and p(t, c) given. Next, at the ex-post stage, the lender controls her own consumption C,
the firm investment ratio ψ (0 < ψ < 1), and the reinvestment dξ. Note that I will define
below the firm managers’ ex-post reservation utility to ensure perfectly elastic supply of
firms.
3 Equilibrium
3.1 Optimization
3.1.1 The firm managers’ optimization problem
When invested at time 0, the firm manager j possesses one unit of invested funds Kj(0) = 1
for each j. For a compensation scheme s and an insurance contract (p, c), each firm
manager’s ex-ante utility process that is optimized with regard to the efforts {αj ; j =
1, · · · , n}, denoted by L, is written as follows:
L(0) := sup{αj ;j=1,··· ,n}
U(0)
= sup{αj ;j=1,··· ,n}
ae−βT1
n
n∑j=1
{E[
log(sKj(T ))]− Γ(αj)
}s.t.
dKj(t)
Kj(t−)=
(µ− pc+ (1− c)zλj(1 + αj(t))
)dt+ σ dB(t) + (1− c)z dÑj(t).
15
-
For notational convenience, define
λ̄ :=n∑j=1
λjn,
VAR(λ) :=1
n
n∑j=1
(λj − λ̄)2
which represent the mean and the variance, respectively, of {λj ; j = 1, · · · , n} under the
uniform distribution. Solving the firm managers’ optimization problem with regard to the
efforts given (s, p, c),
Proposition 1 When no firm managers are expected to quit the insurance contract ex
post,
L(0) = ae−βT(
log s+(µ+ λ̄(1− c)z − σ
2
2
)T − c
∫ T0p(u) du
)
for (s, c, p). Moreover, the optimal effort αj = (1− c)z for j ∈ {0, · · · , n}.
Proof: Let K0j denote the firm j’s value in the case of no effort αj = 0 where K0j (0) =
Kj(0) = 1. Defining Yj(t) := E[elog(sK
0j (T ))
]= E
[sK0j (T )
],
log Yj(T ) =(µ− σ
2
2+ λj log(1 + (1− c)z)
)T − c
∫ T0p(u) du
+σB(T ) + log(1 + (1− c)z)Ñ0j (T ).
In addition, since Yj(t) is a martingale,Yj(t)Yj(0)
= E(σB + (1 + (1 − c)z)Ñ0j
)t
where E
represents the stochastic exponential. Thus,
Yj(T ) = Yj(0) exp
σB −12σ
2T + log(1 + (1− c)z)Ñ0j (T )
−λj(
(1− c)z − log(1 + (1− c)z))T
where Yj(0) = exp
{log(sKj(0)) +
(µ+ λj(1− c)z
)T − c
∫ T0p(u) du
}= exp
{log s+
(µ+ λj(1− c)z
)T − c
∫ T0p(u) du
}.
16
-
On the other hand, when the firm manager j makes the efforts αj over time,
E[
log(sKj(T ))]
= log Yj(0) +
−λjzT − 12σ2T+∫ T
0 λj(1 + αj(u)) log(
1 + (1− c)z)
du
.
Accordingly, solving supαj ae−βT
(E[
log(sKj(T ))]− Γ(αj)
),
ae−βT(
log(1 + (1− c)z)− λj log(1 + αj) + 1− 1))
= 0
I.e., αj = (1− c)z.
Hence, supαj ae−βT
(E[
log(sKj(T ))]− Γ(αj)
)= log Yj(0)− ae−βT 12σ
2T . �
For notational convenience, define
Πj(t; c) :=E[Kj(t)]∑nj=1 E[Kj(t)]
=exp{λj(1 + (1− c))(1− c)zt}∑nj=1 exp{λj(1 + (1− c)z)(1− c)zt}
for j, c, t, which represents a twisted probability of λj(t; c) (j = 1, · · · , n) at each j, t given
c. Note that 0 < Πj(t; c) < 1 for j, t, c, and in addition, that∑n
j=1 Πj(t; c) = 1 for t, c. Let
Π(t; c) := {Πj(t; c); j = 1, · · · , n} for t, c denote the twisted probability distribution at t
given c. As E[Kj(t)] is evolving over time for each j, the impact of λj on the risk pooling
becomes diverse over the types {j | 1, 2, · · · , n} ex post. Π(t; c) stands for the distribution
of the diverse impact at t given c. Define also
λ̄Π(c)(t) :=n∑j=1
λjΠj(t; c),
VARΠ(c)(λ)(t) :=
n∑j=1
(λj − λ̄)2Πj(t; c),
SDΠ(c)(λ)(t) :=(
VARΠ(c)(λ)(t)) 1
2,
SKEWΠ(c)(λ)(t) :=
∑nj=1
(λj − λ̄Π(c)(t)
)3Πj(t; c)(
SDΠ(c)(λ))3 ,
which represent the mean, the variance, the standard deviation, and the skewness, respec-
tively, of {λj ; j = 1, · · · , n} under the twisted probability distribution Π(c, t) at t given c.
17
-
Using the results of Proposition 1, from Eq.(2.2),
Corollary 1 In the environment of Proposition 1, for c,
p = −z(
1 + (1− c)z)λ̄Π(c)(t). (3.1)
With regard to the time effect,
∂p
∂t= −z2(1 + (1− c)z)2(1− c) < 0.
That is, the insurance premium rate is decreasing in time: it is front-loading. This is
because
dλ̄Π(c)(t)
dt=(
1 + (1− c)z)
(1− c)zVARΠ(c)(λ)(t) < 0 ∀ t, c, (3.2)
i.e., the twisted mean λ̄Π(c)(t) is decreasing in t. In other words, the impact of the lower
intensities on the risk pooling is expected to be larger, as the firms with the lower intensities
are expected to have higher Kj , driven by the time-varying E[Kj ] for all j. Note that I will
examine a relationship between the coverage ratio c and the insurance premium rate p in
the proof for Corollary 2 below.
Next, plugging Eq.(3.1) into the optimal L(0) solved in Proposition 1, the first-order
condition for the firm manager’s utility L(0) (given s) with respect to the coverage ratio c
is
− 1T
∫ T0p(t; c) dt− zλ̄− c 1
T
∫ T0
∂p(t; c)
∂cdt ≤ 0
for 0 ≤ c ≤ 1. Let the left-hand side of the first-order condition be denoted by
g(c) := − 1T
∫ T0p(t; c) dt− zλ̄− c 1
T
∫ T0
∂p(t; c)
∂cdt. (3.3)
Corollary 2 In the environment of Proposition 1, partial insurance is optimal (i.e., 0 <
c < 1). In addition, if −12 < z < 0, then there is a unique solution 0 < c < 1.
18
-
Proof: First, I look at g(0) (i.e. g in the case of no insurance c = 0). From Eq.(3.3),
g(0) =1
T
∫ T0z(1 + z)λ̄Π(0)(t) dt− zλ̄
> (−z)(λ̄− 1
T
∫ T0λ̄Π(0)(t) dt
).
Sincedλ̄Π(c)(t)
dt =(
1 + (1− c)z)
(1− c)zVARΠ(c)(λ)(t) < 0 ∀ t, c from Eq.(3.2),
g(0) > (−z)(λ̄− 1
T
∫ T0λ̄Π(0)(0) dt
)= 0. (∵ λ̄Π(0)(0) = λ̄). (3.4)
That is, c > 0, i.e., insurance is optimal.
Next, I look at g(1) (i.e., g in the case of full insurance c = 1). Noting that VARΠ(1)(λ)(t) =
VAR(λ) for all t, from Eq.(3.3) again,
g(1) = −z2(λ̄+
T
2VAR(λ)
)< 0. (3.5)
As g(c) is continuous in c, there exists some solution c to hold g(c) = 0 for 0 < c < 1.
Hence, partial insurance is optimal.
Furthermore, I examine the function form of g(t; c) more specifically. From Corollary 1,
∂p(t; c)
∂c= z2
(λ̄Π(c) + t
(1 + (1− c)z
)(1 + 2(1− c)z
)VARΠ(c)(λ)(t)
). (3.6)
If −12 < z < 0,∂p(t;c)∂c > 0 (∵ 1 + 2(1− c)z > 0). In addition,
∂2p(t; c)
∂c2= (−z)3tVARΠ(c)(λ)(t)
(1 + 2(1− c)z
)· 1 + 3
(1 + 2(1− c)z
)+t(
1 + (1− c)z)(
1 + 2(1− c)z)2
SDΠ(c)(λ)(t)SKEWΠ(c)(λ)(t)
.Since Π(t; c) is right-skewed (with no skewness only when c = 1), SKEWΠ(c)(λ)(t) ≥ 0
with the equality only when c = 1. If −12 < z < 0,∂ 2p(t:c)∂c2
> 0 (∵ 1 + 2(1 − c)z > 0).
19
-
Thus, if −12 < z < 0,
dg(c)
dc= −2 1
T
∫ T0
∂p(t; c)
∂cdt− c 1
T
∫ T0
∂ 2p(t, c)
∂c2dt < 0.
That is, if −12 < z < 0, g is monotonically decreasing in c ∈ [0, 1]. Because of Eqs.(3.4)(3.5),
the desired result is obtained. �
With regard to the first half of the corollary, the optimality of the partialness is resulting
from the firm managers’ hidden laziness (the lower effort αj = (1 − c)z, i.e., the higher
stochastic intensity λj(1 + (1− c)z)) under the protection of the insurance. The problem
of the hidden actions is mitigated by coinsurance, but still remains harmful.
On the other hand, with regard to the second half of Corollary 2, the condition
−12 < z < 0 means that the jump shock is not so severe as half destruction. Accord-
ingly, the second half means that, when the jump risk is lighter than half destruction,
there exists uniquely the optimal partial insurance coverage ratio. Note, however, that
the condition −12 < z < 0 is just a sufficient condition. From the proof, I can argue
that, even if the condition −12 < z < 0 does not hold, either small T , high (twisted)
mean λ̄Π(c)(·), low (twisted) variance (standard deviation) VARΠ(c)(λ)(·), or low (twisted)
skewness SKEWΠ(c)(λ)(·) may result in a unique solution of the partial insurance.
Furthermore, due to the firm manager’s participation constraint,
Corollary 3 In the environment of Proposition 1, the compensation scheme s is
s = κ1a exp
{−(µ+ λ̄(1− c)z − σ
2
2
)T +
∫ T0p(t) dt
}
where p(t) is defined in Eq.(3.1).
Finally, as the insurance contract is not enforceable, some firm manager with a low
stochastic intensity may have an incentive to quit the contract when the insurance premium
rate p, which is derived ex ante based on the average type λ̄ =∑n
j=1λjn , turns out to
be costly ex post for him. As the true type is unobservable to the investor, all the firm
managers stay in the contract if cancellation fees are so high that the firm manager with the
lowest type λL := minj∈{1,··· ,n}{λj} does not have an incentive to quit, i.e., if there exists
20
-
a cancellation fee rate 0 ≤ ζ(t) < 1 at time t such that, for k := Kj(t), the cancellation
fees are represented by ζ(t)k satisfying
ae−β(T−t)(
log s+ log(
(1− ζ(t))k)
+(µ+ λLz −
σ2
2
)(T − t)
)= ae−β(T−t)
(log s+ log k +
(µ+ λL(1− c)z −
σ2
2
)(T − t)−
∫ T−t0
p(u) du).
Thus,
Corollary 4 No firm managers quit the insurance contract over time if the following can-
cellation fee rate ζ(t) relative to k := Kj(t) at time t for j is built in it:
ζ(t) = max
{0, 1− exp
{−c(−z)
∫ T−t0
((1 + (1− c)z)λ̄Π(c)(u)− λL
)du
}}.
When λL is so low as to hold λL <(
1 + (1− c)z)λ̄Π(c)(t), the cancellation fee rate ζ(t)
is positive. Otherwise, any firm manager stays in the contract, even without cancellation
fees (i.e., when ζ(t) = 0).
In the following, I build the cancellation fee rate ζ(t) in the insurance contract. Thus,
no firm managers quit it ex post. Accordingly, the problem of the hidden information is
harmless in this model, while, on the other hand, the problem of the hidden actions remains
harmful although it is mitigated by coinsurance.
3.1.2 The investor’s optimization problem
Taking the firm managers’ optimization as given, the investor’s optimization problem is
formulated as follows:
J(0) = sup{C,ψ̃,dξ}
E
∫ T0f(C(u), V (u)) du+ e−βT
((1− s)W (T ) + I(T )
)1−γ1− γ
subject to the wealth constraint Eqs.(2.3)(2.4)(2.5) and the firm managers’ participation
constraint characterized in Corollary 3:
a
(log s+
(µ+ λ̄(1− c)z − σ
2
2
)T − c
∫ T0p(t) dt
)= κ
21
-
where the firm managers’ optimal efforts αj = (1− c)z for each j ∈ {0, · · · , n}.
In addition, to ensure the perfectly elastic supply of the firms, I need to assume well-
defined ex-post reservation utility of potential entrants. Let e−β(T−t) log κ(t) denote the
ex-post reservation utility at t and kj = Kj(t) It is characterized as follows:
e−β(T−t) log κ(t) = ae−β(T−t)1
n
n∑j=0
log(skj) +(µ+ λ̄(1− c)z − σ22
)(T − t)
−c∫ Tt p(u) du
.I.e., κ(t) = κ exp
a 1n∑nj=1 log kj −
(µ+ λ̄(1− c)z − σ22
)t
+c∫ t
0 p(u) du
.
I am implicitly assuming that, when the investor discharges firms from the investment due
to portfolio allocation, the discharged firm managers lose the types that they had while
being invested, and then join the group of the potential entrants. In the environment, the
potential entrant firm managers are indifferent between being invested and living outside.
As the insurance is optimal for the firm managers, the lender can enjoy the welfare
improvement due to the insurance by reducing the firm managers’ utility up to their reser-
vation utility. Assume that {C(t), ψ(t); t ∈ [0, τ ∧T ]} are Markovian controls. The lender’s
optimization problem is an impulse control problem (for the details of its mathematical
regularities, see e.g. Øksendal and Sulem (2007)).
3.2 Characterization of equilibrium: asset pricing implications
The economy is said to be in equilibrium if both the investor and the firm managers
optimize their own utility, and in addition, if the markets are cleared: ψ = 1 (i.e., ψ̃ = WX ).
I assume that there exists a C1,2([0, T ] × R) function J satisfying J(t) = J(t, x) where
x = X(t). However, any exact (analytical) solution cannot be obtained. Thus, to obtain
approximate solutions instead, I focus on the situation in which the ratio ϑ = WX is in
the neighborhood of ϑ = ϑ0 where 0 < ϑ0 < 1 is a constant. Note that I can choose any
neighborhood of WX here. Define C(t) = φX(t) – call φ a consumption rate.
Proposition 2 The optimal consumption rate is φ = β. In addition, when the ratio WX is
22
-
in the neighborhood of WX = ϑ0, the equilibrium risk-free rate r is
r = µ− ϑ0γσ2 + (1 + ϑ0z)−γz
and the value function J takes the form of log(
(1− γ)J)
= q(t) + (1− γ) log x where
q(t) =1
β
β(1− γ)(log β − 1) + (1− γ)r + 12γ(1− γ)(ϑ0)2σ2+
{(1 + ϑ0z
)1−γ(1− (1−γ)ϑ0z1+ϑ0z
)− 1}
+
(1− γ) log(1− ϑ0s)
− 1β
β(1− γ)(log β − 1) + (1− γ)r + 12γ(1− γ)(ϑ0)2σ2+
{(1 + ϑ0z
)1−γ(1− (1−γ)ϑ0z1+ϑ0z
)− 1}
e−β(T−t).
Moreover, the process of an equilibrium state price density, denoted by Λ, is characterized
by dΛΛ = −r dt− ηB dB(t)−
∑nj=1 η
Nj dÑj(t) such that
ηB = ϑ0γσ,
ηNj =1
n2
(ϑ0z
1 + ϑ0z
){(1 + ϑ0z
)1−γ− 1}
for j ∈ {1, · · · , n}.
Furthermore, under Assumptions 1 and 2, the optimal singular reinvestment dξ is dξ =
χW = δL−δRπ W in the event of default.
Proof: The Hamilton-Jacobi-Bellman equation is written as follows:
0 = β((1− γ)φ− q) + q′ + (1− γ)
r − φ+ ψ̃(µ− r + 1n
n∑j=1
zλj(1 + (1− c)z))
−12γ(1− γ)(ψ̃σ)2 + 1
n
n∑j=1
(((1 + ψ̃z)1−γ − 1
)− (1− γ)λj(1 + (1− c)z)ψ̃z
).
Due to the first-order condition and the market clearing condition, in the neighborhood of
WX = ϑ0,
φ = β,
23
-
r = µ− ϑ0γσ2 + (1 + ϑ0z)−γz.
There exists such r satisfying ψ̃ = ϑ0. Since the process of an equilibrium state price
density, denoted by Λ, is characterized by dΛΛ = −r dt − ηB dB(t) −
∑nj=1 η
Nj dÑj(t) =
fv dt+dfcfc
(see e.g. Skiadas (2008) in a Brownian motion case),
ηB = ϑ0γσ,
ηNj =1
n2
(ϑ0z
1 + ϑ0z
){(1 + ϑ0z
)1−γ− 1}
for each j.
Thus,
βq − q′ = β(1− γ)(log β − 1) + (1− γ)r + 12γ(1− γ)(ϑ0)2σ2
+
((1 + ϑ0z
)1−γ(1− (1− γ)ϑ0z
1 + ϑ0z
)− 1)
;
q(T ) = (1− γ) log(1− ϑ0s).
I.e.,
q(t) =1
β
β(1− γ)(log β − 1) + (1− γ)r + 12γ(1− γ)(ϑ0)2σ2+
{(1 + ϑ0z
)1−γ(1− (1−γ)ϑ0z1+ϑ0z
)− 1}
+
(1− γ) log(1− ϑ0s)
− 1β
β(1− γ)(log β − 1) + (1− γ)r + 12γ(1− γ)(ϑ0)2σ2+
{(1 + ϑ0z
)1−γ(1− (1−γ)ϑ0z1+ϑ0z
)− 1}
e−β(T−t).
Next, I examine the situation in which default occurs for the first time τ := arg inft{θ(t) ≤
0}. Note that θ = 1ϑ − 1 by definition. From the above equilibrium results, the process of
θ = IW without interventions is characterized by
dθ(t) = θ(t−)
(− µ+ r + (1 + 1θ(t−)) cp(t) + σ
2)
dt− σ dB(t)
− c(1−c)z2
1+(1−c)z1n2∑n
j=1 dNj(t)
.In the event of default (i.e., θ ≤ 0), due to the value-matching and smooth-pasting condi-
24
-
tion, letting w = W (τ),
J(τ, (1 + θ − δL)w) = J(τ, (1 + θ − πχ− δR)w),dJ(τ, (1 + θ − πχ− δR)w)
dχ≤ 0.
Since dJ(τ,(1+θ−πχ−δR)w)
dχ < 0, χ =δL−δRπ . Under Assumptions 1 and 2, the investor can
finance the optimal reinvestment (i.e., 0 < χ < 1). �
The insurance has a crucial effect on equilibrium asset prices in the capital markets.
Payments of insurance premiums cp and claims cz do not change the investor’s total wealth
directly by offsetting the changes in I and W . However, the insurance activities influence
the capital markets through two channels in this model. First, the insurance funds’ default
causes a deadweight loss to the total wealth. Second, the equilibrium state prices are
influenced by which neighborhood (i.e., ϑ0 (or θ0)) the ratio ϑ (or θ) is moving in. I can
choose the neighborhood ϑ0 (or θ0), depending on where the ratio ϑ (or θ) is moving in
on average. As the insurance funds are invested only in the risk-free asset, the ‘average’
amount of the insurance funds restricts the riskiness of the total wealth. Specifically, the
equilibrium risk-free rate is influenced by the ‘average’ ratio of the insurance funds to the
total wealth. When the ratio is lower (higher) on average, the equilibrium risk-free rate is
lower (higher), because, as the investor invests more (less) in the risky assets on average,
the market prices of diffusive and jump risks and the risk premium are higher (lower) in
equilibrium.
In addition, I make clear the equilibrium default behavior more specifically. Note that,
for notational convenience, the representation of the dynamics of the ratio is changed from
ϑ into θ := 1ϑ − 1. Correspondingly, θ0 :=1ϑ0− 1. As shown in the proof for Proposition 2,
the equilibrium process of θ = IW in the neighborhood of θ0 without interventions is
characterized by
dθ(t) = θ(t−)
(− µ+ σ2 + d(t)
)dt
−σ dB(t)− c(1−c)z2
1+(1−c)z1n2∑n
j=1 dÑj(t)
(3.7)
25
-
where
d(t) := r + (1 +1
θ(t−)) cp(t)− c(1− c)z
2
1 + (1− c)z1
n2
n∑j=1
λj . (3.8)
First, I examine the case in which the process of θ hits θ = 0 continuously from above,
Corollary 5 In the environment of Proposition 2, when the process θ hits θ = 0 continu-
ously from above, it jumps to
θ′ =1− π
π 1−δR
δL−δR − 1
where 0 < θ′ < 1.
On the other hand, second, I examine the case in which a jump causes the default of
the insurance funds. When a jump causes θ ≤ 0, the process jumps from θ to
θ′ =θ + (1− π) δL−δRπ1− δR − δL−δRπ
.
Still, θ′ may not be positive. When θ′ ≤ 0, the lender repeats the reinvestment until the
ratio IW flips over into a positive value for the first time. Let θ{m} denote the ratio IW after
reinvesting m times given θ < 0. That is, for an integer m ≥ 1,
θ{m} = −1− ππ
δL − δR
δR + δL−δRπ
+1(
1− δR − δL−δRπ)m
(θ +
1− ππ
δL − δR
δR + δL−δRπ
).
Note that θ{m} is increasing in m. For notational convenience, define a ceiling function
dme := min{m ∈ N | θ{m} ≥ 0} where N represents the set of natural numbers {1, 2, · · · }.
Corollary 6 In the environment of Proposition 2, when a jump causes θ ≤ 0, the ratio IWrestarts from θ{k} for an integer k = dme.
Note that, when θ′ > 0, θ′ = θ{1}.
3.3 Contagion effect of jump shocks
In practice, disaster losses often spread extensively in adjacent areas and hinder the risk
pooling. The spreading property can be characterized by the contagion effect of jump
26
-
shocks. However, the above model does not capture the property as assuming the inde-
pendence of the jump shocks. In this subsection, by contrast, I assume some contagions of
the jump shocks.
Specifically, I assume that n types of the firms are located in line on a circle from 1
to n clockwise at even intervals. The distance between two different types of the firms is
measured counterclockwise (see the image on Figure 1). I also assume that the contagion
of the shocks between the two firms – say type j1 and type j2 – who are apart by k = j1−j2
counterclockwise is ρk where ρ is a constant and 0 < ρ < 1. When the type j1 suffers a
jump loss Nj1 , it causes the type j2 a jump loss Nj2 with probability ρk. For notational
Figure 1: Firm types in line on a circle
j=2
j=3
j=n
j=n-1
j=1
Distance
convenience, define
m(i) :=
i if i > 0i+ n if i ≤ 0for i ∈ Z where Z represents the set of integers. The process of Kj for each j is then
characterized by
dKj(t)
Kj(t−)=
(µ− p(t)c
)dt+ σ dB(t) + (1− c)z dNj(t)
=(µ− p(t)c+ (1− c)zλSj
)dt+ σ dB(t) + (1− c)z dÑSj (t)
27
-
where λSj :=∑n
k=1 ρk−1λm(j−k+1) and dÑ
Sj (t) = dNj(t)− λSj dt.
With regard to the optimal efforts {αj ; j = 1, · · · , n}, Γ(αj) depends only on the firm
manager’s own efforts. Now, I focus on a Nash equilibrium solution regarding the efforts
of the firm managers. Therefore, the optimal efforts are αj = (1 − c)z for all j, which is
the same result as the case of the independent jumps. In other words, the optimal efforts
cause externality, in that the efforts influence the other types’ stochastic intensities without
being internalized in the markets.
Accordingly, as compared to the case of the independent jumps in Proposition 1, λSj
replaces λj for each j. Obviously, λSj > λj . For notational convenience, using Eq.(3.3),
let c∗ and c∗∗ denote a unique c satisfying g(c) = 0 when −12 < z < 0 in the case of the
independent jumps and the contagious jumps, respectively.
Corollary 7 If −12 < z < 0 and n > −1z , then c
∗∗ < c∗.
Proof: As compared to the case of the independent jumps, λSj replaces λj in the stochastic
differential for each j here. Since λSj > λj , we examine the situation in which λj is increased
in Corollary 2. Specifically, fixing c = c∗, we examine how an increase in λj changes g(c∗).
From Eq.(3.3),
g(c∗) = − 1T
∫ T0p(t; c∗) dt− zλ̄− c∗ 1
T
∫ T0
∂p(t; c)
∂c
∣∣∣c=c∗
dt = 0.
Hence,
∂g(c∗)
∂λj= − 1
T
∫ T0
∂p(t; c∗)
∂λjdt− z 1
n− c∗ 1
T
∫ T0
∂
∂λj
(∂p(t; c)
∂c
∣∣∣c=c∗
)dt. (3.9)
With regard to the first term on the right-hand side,
∂p(t; c∗)
∂λj= −z
(1 + (1− c∗)z
).
With regard to the third term of Eq.(3.9), from Eq.(3.6),
∂p(t; c)
∂c
∣∣∣c=c∗
= z2(λ̄Π(c∗) + t
(1 + (1− c∗)z
)(1 + 2(1− c∗)z
)VARΠ(c∗)(λ)(t)
).
28
-
Thus,
∂
∂λj
(∂p(t; c)
∂c
∣∣∣c=c∗
)= z2
(1 + t
(1 + (1− c∗)z
)(1 + 2(1− c∗)z
)∂VARΠ(c∗)(λ)∂λj
)= z2 (∵ ∂VARΠ(c∗)(λ)/∂λj = 0)
Therefore,
∂g(c∗)
∂λj= z
(1 + (1− c∗)z
)− z 1
n− c∗z2
= z(
1 + 2(1− c∗)z − z − 1n
).
When −12 < z < 0 and n > −1z ,
∂g(c∗)∂λj
< 0. In addition, from the proof for Corollary 2,
when −12 < z < 0, g(c) is monotonically decreasing in c. Thus, if −12 < z < 0 and n > −
1z ,
c∗∗ < c∗ (∵ g(c∗) = 0 and ∂g(c∗)
∂λj< 0). �
In other words, when the negative jump size is not so big as half destruction, and
in addition, when the number of types of firms are large enough to hold n > −1z , the
optimal coverage ratio is lower in the case of the contagious jumps than in the case of the
independent jumps.
With regard to the effect on the asset prices, the contagions of the jumps do not
influence the state prices with θ0 (and ϑ0) given. On the other hand, they influence the
default behavior of the insurance funds through a change in θ in the neighborhood of θ0
(and ϑ0). Applying Eqs.(3.7)(3.8) to the case of the contagious jumps,
dθ(t) = θ(t−)
(− µ+ σ2 + dS(t)
)dt
−σ dB(t)− c(1−c)z2
1+(1−c)z1n2∑n
j=1 dÑSj (t)
(3.10)where
dS(t) := r + (1 +1
θ(t−)) cp(t)− c(1− c)z
2
1 + (1− c)z1
n2
n∑j=1
λSj . (3.11)
The effect of the contagions on the dynamics of θ is expected to be directly through dS(t),
defined by Eq.(3.11), in Eq.(3.10). However, the effect is uncertain generally. Further
29
-
numerical analyses would be necessary for making clear the effect strictly. Still, in the
next subsection, I will discuss the effect a little more deeply by adding several non-strict,
numerical conjectures to this environment.
3.4 Discussions: Implications for CAT bonds
I compare the present results with Froot (2001)’s theoretical explanations. Firstly, the
problem of the hidden information is harmless due to the cancellation fees built in the
insurance contract. This is consistent with Froot (2001). On the other hand, the problem of
the hidden actions is reduced by coinsurance as discussed in Froot (2001), but still distorts
the capital markets as well as the insurance premium rate and the insurance coverage ratio.
In addition, under the above-shown parametric assumptions, the contagions of the jump
shocks make the problem more harmful, decreasing the optimal coverage ratio.
Secondly, the illiquidity of the insurance contract plays an essential role in this model.
If the insurance contract were liquid in the markets, the efficiency caused by the ex-ante
contract might be reduced. In addition, the insurance funds are subject to default risks
here. The default costs and the transaction costs of the insurance funds cause financial
inefficiency. The lender reinvests and reorganizes the funds, modeled as impulsive controls,
by incurring the costs when the funds become insolvent at some random intervals.
Thirdly, with regard to the effect of the investor’s recursive utility as non-expected
utility, I can distinguish the effect of risk aversion γ from the effect of the elasticity of
intertemporal substitution, which is set at one in this model. Higher risk aversion lowers
the equilibrium risk-free rate due to higher risk premium, preferring earlier resolution of
uncertainty.
In addition to these results about CAT insurance, this paper sheds light on the effect
of the optimal insurance on the capital markets. Firstly, the behavior of the equilibrium
state prices (and so the equilibrium asset prices) is not influenced directly by the terms of
the insurance contract (i.e., the insurance premium rate p and the insurance coverage ratio
c), but by which neighborhood (i.e., ϑ0 (or θ0)) the ratio ϑ (or θ) is moving in.15 The ratio
15Note that the terms of the insurance contract influences the equilibrium welfare throughs in q.
30
-
of the insurance funds to the total wealth on average matters, as shown in Proposition 2.
Secondly, the default behavior is govern by the dynamics of θ = IW (or ϑ =1
1+θ ) in the
neighborhood of θ0 =1
1+θ0as in Eq.(3.7).
To derive further asset pricing implications, I add several non-strict, numerical conjec-
tures to the environment assumed in Proposition 2 in the case of the independent jumps.
First, one crucial point for the derivation is how ϑ0 (or θ0) is chosen here, while, by con-
trast, ϑ0 (or θ0) is exogenously given in the above. I pick out the ‘average’ of ϑ (or θ) as
ϑ0 (or θ0) roughly here. Second, I assume that the number of the insured firm types n is
so large that the jump part become negligible in Eqs.(3.7)(3.8), discounted by the square
of n. Note that the squire of n is coming from the quadratic variation of the jump terms in
the processes of the components 1W and I that the ratio θ =1W · I is composed of. Finally,
I assume −12 < z < 0.
Then the effect of the insurance on the asset prices is as follows. Recalling that ∂p∂c > 0
when −12 < z < 0, cp is lower as compared to full insurance. A low level of cp is likely to
result in a low level of θ (a high level of ϑ), as in Eqs.(3.7)(3.8): a low level of θ0 (a high
level of ϑ0) can be chosen. Accordingly, as compared to full insurance, the risk-free rate r
is low while the risk premium and the market prices of risks are high. From Eqs.(3.7)(3.8),
such low r is consistent with the low θ (the high ϑ). Noting that the partialness of the
insurance comes from the hidden actions, these results are the effect of the hidden-action
problem. On the other hand, the effect in comparison with the case of no insurance is in
the opposite direction of the effect of the hidden-action problem.
On the other hand, their effect of the contagions of the jump risks on the equilibrium
asset prices is uncertain generally, as shown in Section 3.3. However, I dig into it a little
more deeply here by adding several non-strict, numerical conjectures again to the results
obtained in Section 3.3. I consider the situation of an increase in λj around c = c∗ in the
same way as in the proof for Corollary 7. That is, I examine the effect of the increase
in λj on dS(t), defined by Eq.(3.11), in Eq.(3.10). As in the discussion of the case of the
independent shocks in the previous paragraph, I assume that the number of the insured
firm types n is so large as to make the jump part negligible in Eq.(3.11).
The effect of the increase in λj on dS(t) of Eq.(3.11) is as follows. First, with regard
31
-
to the effect on cp,
d(cp)
dλj=
dc
dλjp+ c
dp
dλj=
dc
dλjp+ c
( ∂p∂λj
∣∣∣c=c∗
+∂p
∂c
dc
dλj
). (3.12)
Recall that dcdλj < 0 and∂p∂λj
∣∣∣c=c∗
> 0 while, if −12 < z < 0,∂p∂c > 0. With regard to the
second term on the right-hand side of Eq.(3.12), the exact effect is uncertain. However, I
assume dpdλj > 0, by supposing that λSj is large enough. On the other hand, the first term
is negative. Thus, the total effect of the two terms on the right-hand side of Eq.(3.12) is
still uncertain. But, as the size of c is limited, if λSj is large enough, it is supposed to hold
d(cp)dλj
> 0. Next, with regard to − c(1−c)z2
1+(1−c)z1n2∑n
j=1 λSj in d
S(t) defined by Eq.(3.11),
∂
∂c
(c(1− c)z2
1 + (1− c)z
)=( z
1 + (1− c)z
)2((1− 2c) + (1− c)2z
).
The effect is uncertain again. However, even if the contagions increase the intensities λSj for
all j, a sufficiently large n can make the term − c(1−c)z2
1+(1−c)z1n2∑n
j=1 λSj negligible, discounted
by the square of n. In total, given r, if (i) d(cp)dλj > 0 and (ii) n is such sufficiently large, θ is
likely to be high: θ0 (ϑ0) can be set at a high (low) level. Thus, the risk-free rate r is high
while the risk premium and the market prices of risks are low in the case of the contagious
jumps. From Eqs.(3.10)(3.11), such high r is consistent with the high θ (the low ϑ).
Finally, I argue that CAT bonds with well-designed indemnity triggers, in which forfei-
ture is linked with the losses of the insurance funds, could improve welfare. In the present
model, default causes the deadweight loss in the economy. CAT bonds could be attractive
if, financing the default costs ex ante with CAT bonds, the investor could save on the loss
ex post. However, when the probability of being triggered is very high, the CAT bonds
could be too expensive. To investigate the efficiency of the CAT bonds, I should weigh the
attractiveness and the expensiveness. Moreover, I can argue that the CAT bonds are not
a redundant financial derivative that can be replicated in complete markets, in that they
could save on the default costs and influence essentially the equilibrium state price.
To sum up, in corporate risk management against catastrophic disasters, the partial
insurance is optimal in the capital markets under the financial frictions. In addition,
32
-
the optimal insurance influences the equilibrium state prices dynamically. Furthermore,
the choices of risk management methods as well change the state prices in equilibrium.
The general equilibrium framework is useful for examining such catastrophic disaster risk
management in the capital markets.
4 Conclusion
This paper provided a framework to examine corporate catastrophic disaster risks in a
dynamic general equilibrium model under financial frictions. It has showed that partial
CAT insurance is optimal and has a crucial influence on the risk-free rate and the market
returns in equilibrium. Furthermore, the choices of risk management methods as well
influence the state prices in equilibrium. The general equilibrium framework is convenient
for analyzing the role of CAT insurance in capital markets. Due to the convenience, it also
derived some implications for CAT bonds.
However, this model has several limitations. First, the technological structures were
simplified in order to make clear numerically the role of CAT insurance in capital markets.
To modify the model to deal with more realistic situations, more elaborate numerical
approximations would be necessary. Second, with regard to the CAT bonds, I ended up
with discussing the possibility of the efficiency. To make the discussions more specific, I
need to construct and solve a model of CAT bonds by extending the present model. They
are future work in this line of research.
References
[1] Arrow, K.J. (1971), Essays in the Theory of Risk Bearing. Markham Publishing,Chicago.
[2] Arrow, K.J. (1974), Optimal Insurance and Generalized Deductibles. ScandinavianActuarial Journal, 1: 1-42.
[3] Coval, J.D., Jurek, J.W., and Stafford, E. (2009), Economic Catastrophe Bonds. Amer-ican Economic Review, 99: 628-666.
[4] Cox, J.C., Ingersoll, Jr., J.E., and Ross, S.A. (1985), An Intertemporal General Equi-librium Model of Asset Prices. Econometrica, 53: 363-384.
33
-
[5] Cummins, J.D., and Weiss, M.A. (2009), Convergence of Insurance and Financial Mar-kets: Hybrid and Securitized Risk-Transfer Solutions. Journal of Risk and Insurance,76: 493-545.
[6] Duffie, D., and Epstein, L. (1992), Stochastic Differential Utility. Econometrica. 60:353-394.
[7] Finken, S., and Laux, C. (2009), Catastrophe Bonds and Reinsurance: The Com-petitive Effect of Information-Insensitive Triggers. Journal of Risk and Insurance, 76:579-605.
[8] Froot, K.A. (2001), The Market for Catastrophe Risk: A Clinical Examination. Jour-nal of Financial Economics, 60: 529-571.
[9] Froot, K.A., and O’Connell, P. (2008), On the Pricing of Intermediated Risks: Theoryand Application to Catastrophe Reinsurance. Journal of Banking and Finance 32: 69-85.
[10] Gollier, C. (2013), The Economics of Optimal Insurance Design. In G. Dionne (Ed.),Handbook of Insurance, Second Edition, Chapter 4, Springer, 107-122.
[11] Lakdawalla, D., and Zanjani, G. (2012), Catastrophe Bonds, Reinsurance, and theOptimal Collateralization of Risk Transfer. Journal of Risk and Insurance, 79: 449-476.
[12] Merton, R.C. (1974), On the Pricing of Corporate Debt: The Risk Structure of InterestRates. Journal of Finance. 29: 449-470.
[13] Øksendal, B., and Sulem, A. (2007), Applied Stochastic Control of Jump Diffusions,Springer, 2nd Edition.
[14] Polacek, A. (2018), Catastrophe Bonds: A Primer and Retrospective. Chicago FedLetter, No. 405, Federal Reserve Bank of Chicago.
[15] Raviv, A. (1979), The Design of an Optimal Insurance Policy. American EconomicReview, 69: 84-96.
[16] Skiadas, C. (2008), Dynamic Portfolio Choice and Risk Aversion. In J.R. Birge andV. Linetsky (Eds.), Handbooks in Operations Research and Management Science:Financial Engineering, Vol. 15, Chapter 19, Elsevier, 789-843.
34