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: hisashi.nakamura@r.hit-u.ac.jp. 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.

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