Asset Prices under Uncertainty in Relationships

Jordi Mondria

University of Toronto

Liyan Yang

University of Toronto

February 2021

Abstract

We propose a model to study firm relationships that endogenously determine the

correlation structure of asset cash flows. Forming a relationship makes firms face the

following trade-off in their valuations: On the one hand, collaboration generates an

additional payoff component with a positive mean. On the other hand, a relationship

makes the firms’ cash flows more correlated, which lowers the investor’s diversification

benefit. We use our model to investigate the incentives of firms to form a relationship and

to disclose their relationship to the general public. We show that disclosing relationship

information can have real consequences on cash flows through affecting firm relationship

at both the intensive and the extensive margins.

Keywords: Firm relationships, asset prices, disclosure, matching quality, collaboration

intensity

1

1 Introduction

Traditional theories on portfolio choice and asset pricing assume that the correlation structure

of asset payoffs is exogenously given and known to investors. In this paper, we examine

how the payoff correlation structure is endogenously generated and study the incentives and

implications of firms to form and disclose such correlations. Since asset payoffs are cash flows

generated by production, in principle, any payoff correlation structure must trace back to the

production process. In this paper, we open the black box of the production process from a

particular perspective, namely, firm relationships.

In our setting, two firms can form a relationship to collaborate on productions, which in

turn determines the correlation between the cash flows of the two firms. The production out-

puts of the relationship depend on the interaction of two elements, which we label respectively

as “matching quality” and “collaboration intensity.” If two firms form a good (bad) match,

their collaboration will generally lead to more (less) cash flows. For a given matching quality,

more intensive collaboration between the two firms generates more cash flows to both firms

and makes their cash flows more correlated. The valuation of both firms is determined by

a representative investor who forms her portfolio in a competitive asset market. Having a

relationship has two effects on the firms’ valuation. First, the cash flows have an additional

payoff component with a positive mean (i.e., firms on average collaborate on positive NPV

projects). Second, the cash flows of the two firms become more correlated. From the investor’s

point of view, there is a benefit captured by the increase in the mean of asset payoffs that

increases asset prices, but there is also a cost captured by the increase in the variance of the

asset payoffs and the decrease in the ability to diversify her portfolio as cash flows are now

correlated, which decreases asset prices.

We use our framework to investigate the formation and implications of firm relationship

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in three concrete applications. In the first application, we study the choice of the optimal

intensity level of collaboration between the two firms when investors may be uncertain about

their matching quality (or about whether the two firms have a relationship with each other

at all). Specifically, we assume that firms can choose the mean of asset payoffs at zero cost

when there is a relationship. In the absence matching quality uncertainty, firms always choose

the highest possible level for the mean of their asset payoffs. In contrast, in the presence of

matching quality uncertainty, firms optimally choose to limit their collaboration intensity.

Thus, uncertainty about relationships will affect real decisions of the firm by influencing the

intensive margin of a relationship.

In the second application, we analyze the optimal disclosure policy about matching quality.

Unlike the existing studies on disclosure, disclosure in our setting is about disclosing the

existence of a common component in the asset payoffs of the two firms, which generates the

correlation structure between the two firms. It is not about disclosing the realization of the

fundamentals as in most research on disclosure. When firms are allowed to reveal the extent

of their relationships at no cost, we find that firms in relationships with very large or small

benefits with respect to the cost of forming a relationship will choose to fully disclose their

relationships, while firms with benefits at an intermediate level relative to the costs will choose

not to disclose any information about their relationship.

In the final application, we examine the choice of firms to form relationships. Unlike

the previous results, we do not take the relationship as given but the decision to form a

relationship is endogenous. Firms will choose to form a relationship when the expected asset

price of forming a relationship is higher than the asset price when there is no relationship. We

assume the decision is made before the firm is aware whether the relationship will be a good

or bad match. We find that when the benefits of forming a relationship are very large relative

3

to the costs, firms choose to form a relationship with disclosure. Instead, when the benefits

are at intermediate levels relative to the costs, firms form a relationship without disclosure.

Finally, when the relative benefits are low, then firms choose not to form a relationship.

In this final application, we also analyze the implications of policies that force manda-

tory disclosure of relationships. We find that mandatory disclosure may prevent relationship

formation. Specifically, we find that when firms are forced to disclose their relationships,

some relationships that would have been formed, when the benefits of forming a relationship

relative to the costs are at an intermediate range, will not be formed at all. Thus, mandatory

disclosure will have an effect on the extensive margin of relationships. Our paper provides

the specific trade-offs on Regulation SFAS No. 13, in which firms must report separately

information about an operating segment that represents more than 10 percent of sales rev-

enue. Disclosure improves the intensive margin of relationships, but at the cost of affecting

the extensive margin of relationships.

Related Literature Our paper is related to four strands of literature.

Implications of disclosure on market quality. One line of research studies the implica-

tions of financial disclosure and argues that both the amount (Tang (2014)) and the type

of the information disclosed (Goldstein & Yang (2019)) are crucial determinants of market

quality (see Verrecchia (2001), and, more recently, Goldstein & Yang (2017) for a survey).

Another line of research studies voluntary and mandatory disclosures. The reason to disclose

information voluntarily is to reduce information asymmetries between investor and firms (

Diamond & Verrecchia (1991), Easley & O’hara (2004)) or reducing uncertainty about future

payoffs (Barry & Brown (1985), Coles & Loewenstein (1988), Cheynel (2013)). Fishman &

Hagerty (2003) show that customer’s sophistication level is of great importance to determine

the benefit of mandatory disclosure. Unlike these prior studies, where disclosure is related to

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the realization of private information, our paper focuses on disclosure of customer-supplier

relationships and its implications on asset prices. In addition to that, our paper contributes

to this literature by showing that full disclosure of relationships may not always be optimal

for the welfare of investors.

Optimal information disclosure. Another line of research studies optimal disclosure when

there is competition (Wagenhofer (1990), Ellis et al. (2012), Yang (2017)). Among these

papers, our project is closely related to Ellis et al. (2012), who studied optimal disclosure on

customer-supplier relationships when there is a benefit of reducing information asymmetries at

the cost of revealing proprietary information to competitors. In contrast to Ellis et al. (2012),

where customer-supplier relationships are exogenous, our paper endogenizes this decision by

considering two opposing effects on the cash flows when firms choose to form a relationship.

Network formation. A growing literature studies endogenous network formation, in which

the aim of forming such network could be acquiring information (Herskovic & Ramos (2018),

Galeotti & Goyal (2010)), or forming input-ouput relationships (Oberfield (2018), Lim (2018),

Taschereau-Dumouchel (2017), Tintelnot et al. (2018), Acemoglu & Azar (2017))1. In contrast

to these papers, disclosure policy is at the core of our analysis. Furthermore, our paper

contributes to this literature by showing that disclosure policies can affect customer-supplier

relationships. More specifically, mandatory disclosure may prevent relationship formation.

Uncertainty about economic linkages. Finally, our paper is also related to empirical studies

that show the connection between news and economic linkages. Even though any operating

segment that represents more than 10 percent of sales revenue has to be disclosed, firms still

behave strategically for the remaining operating segments. In fact, the findings of Scherbina

& Schlusche (2015) and Schwenkler & Zheng (2019) support this idea that news stories have

1See also Bernard & Moxnes (2018) and Carvalho & Tahbaz-Salehi (2018) for recent literature reviews onproduction networks in international trade and macroeconomics, respectively

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information about economic linkages apart from the disclosed information. Moreover, Cohen

& Frazzini (2008) investigate the asset pricing implications of news that reveal information

about economic linkages and show that investor’s limited attention prevents the use of all

available information, and hence leads to return predictability. The asset pricing implications

of our framework are aligned with the empirical evidence documented by Cohen & Frazzini

(2008).

2 Conceptual Framework

In this section, we build a framework to study the effects of uncertainty of a relationship

between two firms on asset prices.

2.1 Model Description

Consider two symmetric firms: A and B. We normalize the number of shares of each firm

to 1. The cash flows of each firm have two components and are given by FA = VA + ∆

and FB = VB + ∆. The components VA and VB are firm-specific and normally distributed:

VA ∼ N(V , σ2V ) and VB ∼ N(V , σ2

V ), where V is the mean and σ2V is the variance of the

firm-specific cash flows. The component ∆ is common between the two firms and reflects the

cash flow correlation driven by the relationship between both firms. This second component

is given by

∆ = ρδ, with δ ∼ N(δ, σ2

δ

)and ρ =

ρh with probability π,

ρl with probability 1− π,(1)

where ρh > ρl ≥ 0 and δ ≥ 0. The random variables ρ, δ, VA and VB are mutually independent.

6

In this setup, a relationship between the two firms determines their payoff correlation

through two elements, ρ and δ. First, a relationship may turn out to be a good match with ρh

or a bad match with ρl. We refer to ρ as “matching quality.” Second, given matching quality,

if the two firms engage in intensive collaboration, say, collaborating across multiple product

lines, their payoff correlation tends to be strong. We interpret δ, or more specifically, its mean

δ, as “collaboration intensity.”

A special case of (1) is π = ρl = 0, which implies that ∆ = 0. This degenerate setting

corresponds to a benchmark setting with independent cash flows (no relationship between the

the two firms). Another interesting but less specialized case is ρl = 0 but π > 0. In this

case, the uncertainty about matching quality ρ can be reinterpreted as uncertainty about the

existence of a relationship between firms: the market is ex ante uncertain about whether two

firms are related, but once the market becomes certain that there is a relationship between

the two firms, then it is known that the matching quality of the two firms is ρh.

There exists a representative investor, whose preference is −e−γW , where γ > 0 is the

absolute risk aversion coefficient and W is the final wealth. The investor is initially endowed

with W0 wealth and chooses the asset holdings that maximize her preferences subject to the

following budget constraint

W = W0R + qA(FA −RPA) + qB(FB −RPB), (2)

where R is the risk-free interest rate, qA and qB are the asset holdings of the risky assets, PA

and PB are asset prices (price of the riskless asset is normalized to 1).

In the following two subsections, we examine the pricing implications of firm relationships.

In Subsection 2.2, we assume that the matching quality is known to the representative investor.

In Subsection 2.3, we assume that the matching quality is random. The results in these two

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subsections cover various possible settings and form the basis for our later applications.

2.2 Asset Prices under Perfect Matching Information

We use superscript “PI” to denote the case in which an investor has perfect information about

their matching quality ρ. That is, the representative investor knows the realization of ρ. The

expectation and variance of final wealth in equation (2) for any realization of ρ are given by

E[W | ρ] = W0R + qA(V + ρδ −RP PIA ) + qB(V + ρδ −RP PI

B ),

and

V [W | ρ] = q2A(σ2

V + ρ2σ2δ ) + q2

B(σ2V + ρ2σ2

δ ) + 2qAqBρ2σ2

δ .

When the investor maximizes her preferences subject to the budget constraint, her demand

for asset A is given by

qPIA =(σ2

V + ρ2σ2δ )(V + ρδ −RP PI

A )− ρ2σ2δ (V + ρδ −RP PI

B )

γ[(σ2V + ρ2σ2

δ )2 − (ρ2σ2

δ )2]

. (3)

Similarly, one can find the demand for asset B. The demand of one asset is affected by the

demand of the other asset. We can find the asset prices under perfect information by imposing

the market clearing conditions qA = 1 and qB = 1. The price for asset j = {A,B} under

perfect information is given by

P PIj (ρ) =

V − γσ2V

R+δ

Rρ− 2γσ2

δ

Rρ2, for j = {A,B}. (4)

The first term of the price would be the price when ρ = ∆ = 0, i.e., when firms have

no relationship. The second and third terms convey the additional effects of relationships in

8

prices. Specifically, the second term captures the benefit of having a relationship, which is an

additional payoff factor. The third term, instead, captures the cost of forming a relationship:

payoffs of both assets become correlated and it is more difficult for the investor to diversify

risk. The risk aversion parameter γ affects prices in two ways. First, the weight of the variance

in the price increases with risk aversion. Second, the risk aversion parameter also affects the

weight associated with the decrease in the investor’s ability to diversify when firms form a

relationship. Under both effects, prices decrease with risk aversion.

2.3 Asset Prices under Matching Uncertainty

Facing uncertainty about matching quality, the representative investor has to compute her

expected utility without knowing ρ. For a given realization of ρ, final wealth follows a Nor-

mal distribution. If ρ = ρh, then W ∼ N(µW (h), σ2W (h)), while if ρ = ρl, then W ∼

N(µW (l), σ2W (l)), where µW (s) and σ2

W (s) for s = {l, h} are the expectation and variance of

final wealth conditional on ρ = ρs and are given by

µW (s) = E[W | ρ = ρs] = W0R + qA(V + ρsδ −RPA) + qB(V + ρsδ −RPB),

and

σ2W (s) = V [W | ρ = ρs] = q2

A(σ2V + ρ2

sσ2δ ) + q2

B(σ2V + ρ2

sσ2δ ) + 2qAqBρ

2sσ

2δ .

However, before ρ is realized, final wealth does not follow a Normal distribution and we cannot

apply standard results of the CARA-Normal framework. The distribution of final wealth is a

mixture of two Normal distributions. To solve the portfolio choice, we need to calculate the

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expected utility of final wealth by using the Law of Iterated Expectations:

EU = E[E(− exp(−γW ) | ρ)

]= −π exp

[−γ(µW (h)− γσ2

W (h)

2

)]− (1− π) exp

[−γ(µW (l)− γσ2

W (l)

2

)], (5)

If we take the first order conditions with respect to qA and qB and plug the market clearing

conditions qA = 1 and qB = 1, we can find prices for asset j = {A,B} as specified in the

following proposition:

Proposition 1 (Asset Prices under Matching Uncertainty) The price of asset j = {A,B}

is given by

Pj =V − γσ2

V

R+

+δ

R

[πρhe

2γ2ρ2hσ

2δ+2γρlδ + (1− π)ρle

2γ2ρ2l σ

2δ+2γρhδ

πe2γ2ρ2hσ

2δ+2γρlδ + (1− π)e2γ2ρ2

l σ2δ+2γρhδ

]

− 2γσ2δ

R

[πρ2

he2γ2ρ2

hσ2δ+2γρlδ + (1− π)ρ2

l e2γ2ρ2

l σ2δ+2γρhδ

πe2γ2ρ2hσ

2δ+2γρlδ + (1− π)e2γ2ρ2

l σ2δ+2γρhδ

](6)

The asset price in this economy has three distinctive terms. The first term of the price

would be the price of the asset if there was no common element ∆ in the cash flows, i.e., firms

A and B would be independent of each other. The second term captures the benefit of having

a relationship, while the third term, instead, captures the cost of forming a relationship.

In the following three sections, we use three concrete applications to illustrate how one can

use our framework to examine the formation and implications of firm relationships. Specifi-

cally, in Sections 3 and 4, we assume that the two firms have already formed a relationship,

and they decide on the collaboration intensity δ in Section 3 and on the disclosure policy

about matching quality ρ in Section 4. In Section 5, we allow firms to endogenously deter-

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mine whether to form a relationship.

3 Optimal Collaboration Intensity

In this section, we study the optimal collaboration intensity δ from a firm’s point of view.

Suppose that the firms’ objective function is to maximize their prices. Since the two firms

are symmetric, we focus on one firm. To illustrate the result most transparently, we assume

that there is no cost of the firms to change δ.

For the case without matching uncertainty, by pricing function (4), firms would choose

the largest δ possible (i.e., with unbounded support, firms would choose δ to be infinity).

We now consider the case with matching uncertainty and in this case, we show that the

optimal intensity δ is interior. For simplicity, we assume that ρl = 0. As we mentioned before,

in this setting uncertainty about matching quality ρ can also be interpreted as uncertainty

about the existence of firm relationship. The asset prices for j = {A,B} are just a special

case of the prices in the general framework given by equation (6):

Pj =V − γσ2

V

R+

+δ

R

[πρhe

2γ2ρ2hσ

2δ

πe2γ2ρ2hσ

2δ + (1− π)e2γρhδ

]

− 2γσ2δ

R

[πρ2

he2γ2ρ2

hσ2δ

πe2γ2ρ2hσ

2δ + (1− π)e2γρhδ

]. (7)

Firms choose their collaboration intensity δ ∈ [0,∞) to maximize their asset prices (7) at

zero cost.

An increase in δ has two effects on the price (7). There is a direct effect that increases the

price due to an increase in expected cash flows. There is also an indirect effect that decreases

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the price due to the risk associated with an increase in the asset demand of both assets caused

by the direct effect. A high asset demand adds risk to the investor’s portfolio as she believes

with probability 1 − π that firms A and B have no relationship (ρ = ρl = 0). When the

investor is uncertain about the matching quality between two firms, it is optimal for firms to

have a limited collaboration intensity, in contrast to a case in which the investor is certain

about the matching quality.

Proposition 2 (Optimal Collaboration Intensity) The optimal collaboration intensity

δ = δ∗ is uniquely determined by the solution to the following equation:

π(1− π)ρh + ρhπ2e2γρh(γρhσ

2δ−δ) + 4γ2π(1− π)ρ3

hσ2δ − 2γπ(1− π)ρ2

hδ = 0

In Figure 1, we report the optimal collaboration intensity δ∗ for several parameter speci-

fications. Specifically, we assume γ = 1 and π = 0.5, set several values for ρH , and depict δ∗

against σ2δ . We can see that the optimal collaboration intensity δ∗ is interior and increasing

in σ2δ .

4 Voluntary Relation Disclosure

In this section, we assume that the collaboration intensity δ is fixed and instead that firms

can disclose information about their matching quality ρ to capture relationship disclosure.

In principle, we can also model relationship disclosure as disclosing information about δ or

about ∆. We do not explore these alternatives in the current paper.

Before the realization of ρ, firms commit to a joint disclosure policy that will take place

once firms observe the realization of ρ. They can send a message m = {h, l} to indicate if

they are in the state with ρh or ρl. Specifically, firms can choose the following probability

12

Figure 1: Optimal Collaboration Intensity

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

10

20

30

40

50

60

α ∈ [12, 1] at zero cost:

Pr(m = h | ρ = ρh) = Pr(m = l | ρ = ρl) = α,

Pr(m = l | ρ = ρh) = Pr(m = h | ρ = ρl) = 1− α. (8)

In the limit, when α = 1 firms provide perfect disclosure of the realization of ρ. Instead,

when α = 1/2 firms provide no information about the realization of ρ. Any α in between

will generate only partial disclosure about the realization of ρ. Hence, we augment the main

conceptual framework to a model with three stages. At t = 1, before the realization of ρ,

firms can commit to a disclosure policy α. At t = 2, firms observe ρ and provide a joint

message m = {h, l} based on the disclosure policy α. At t = 3, the investor updates her

beliefs about the realization of ρ based on the message received and the disclosure policy,

decides her portfolio choice and prices are determined in equilibrium. The timeline of the

model is given by Figure 2.

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Firms commit to adisclosure policy α

t = 1

Firms observe therealization of ρ andprovide a joint mes-sage m = {h, l} basedon policy α

t = 2

The investor updatesbeliefs, chooses herportfolio, and pricesare determined

t = 3

Figure 2: Timeline with Voluntary Relation Disclosure

The model is solved using backwards induction. First, we solve for the investor’s demands

and prices are set. Second, firms commit to a disclosure policy α that maximizes their expected

prices.

The investor problem is the same derived in Section 2 for the conceptual framework with

probabilities π and 1 − π being updated based on the message received at t = 2. Let us

denote πh = Pr(ρ = ρh | m = h) and πl = Pr(ρ = ρh | m = l) as the probabilities that

ρ = ρh assigned by the investor after receiving the message m = h and m = l respectively.

In consequence, we denote 1− πh and 1− πl as the probabilities that ρ = ρl assigned by the

investor after receiving the message m = h and m = l respectively. These probabilities are

given by

πh = Pr(ρ = ρh | m = h) =Pr(m = h | ρ = ρh)Pr(ρ = ρh)

Pr(m = h)=

απ

απ + (1− α)(1− π),

πl = Pr(ρ = ρh | m = l) =Pr(m = l | ρ = ρh)Pr(ρ = ρh)

Pr(m = l)=

(1− α)π

(1− α)π + α(1− π). (9)

Therefore, there will be a price of the asset when m = h and another one when m = l.

The price will be the same as the one in the conceptual framework in equation (6), but with

adjusted probabilities πm for m = h, l. Hence, for any message m = h, l, the price Pj(α;m)

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for j = {A,B} is given by

Pj(α;m) =V − γσ2

V

R+

+δ

R

[πmρhe

2γ2ρ2hσ

2δ+2γρlδ + (1− πm)ρle

2γ2ρ2l σ

2δ+2γρhδ

πme2γ2ρ2hσ

2δ+2γρlδ + (1− πm)e2γ2ρ2

l σ2δ+2γρhδ

]

− 2γσ2δ

R

[πmρ2

he2γ2ρ2

hσ2δ+2γρlδ + (1− πm)ρ2

l e2γ2ρ2

l σ2δ+2γρhδ

πme2γ2ρ2hσ

2δ+2γρlδ + (1− πm)e2γ2ρ2

l σ2δ+2γρhδ

], (10)

where πm are the posterior probabilities after receiving message m given by equation (9).

Given the asset prices for any α, firms will choose the joint optimal disclosure policy α∗ that

maximizes their expected asset prices. Since both firms are symmetric, we can just focus on

the problem of one of the firms. The expected asset prices are given by

E[Pj(α;m)] = Pj(α;m = h)Pr(m = h) + Pj(α;m = l)Pr(m = l). (11)

Without loss of generality, we assume that if a firm is indifferent between any policy α, they

will choose full disclosure (α = 1). The following proposition shows under which conditions

firms will choose to opt for a full disclosure policy or a non-disclosure policy.

Proposition 3 The optimal disclosure policy α = α∗ is a corner solution and is given by

α∗ =

12, if (ρL + ρH) < δ

γσ2δ< 2(ρL + ρH),

1, otherwise .

(12)

Firms choose to commit to a non-disclosure policy α∗ = 12

if and only if (ρL+ρH) < δγσ2δ<

2(ρL + ρH). Otherwise, they commit to a full disclosure policy α∗ = 1. Intuitively, when

the benefit of having a relationship is very high or very low δ relative to the cost σ2δ , firms

choose to commit to disclose the nature of their relationship. However, when the benefit is

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at intermediate levels relative to the cost of forming a relationship, firms prefer to commit to

a non-disclosure regime. Thus, full disclosure of a relationship is not always optimal if the

objective is to maximize the asset prices of the firm.

Corollary 1 The price of asset j = {A,B} when the optimal policy is non disclosure (α∗ = 12)

is given by equation (6). Instead, when the optimal policy is full disclosure (α∗ = 1), then

asset prices are given by equation (4) and the expected asset price is given by

E[Pj(α∗ = 1;m)] =

V − γσ2V

R+δ

R(πρH + (1− π)ρL)− 2γσ2

δ

R(πρ2

H + (1− π)ρ2L). (13)

5 Relationship Formation

In this section, we study the choice of firms to form relationships. Firms will choose to form

a relationship if the expected price of the firm when forming a relationship is higher than

the expected price of the firm under no relationship taking into account the strategic actions

of the other firm. Without loss of generality, we assume that if a firm is indifferent between

forming or not forming a relationship, the firm will choose not to form a relationship.

If firms choose not to have a relationship, then the cash flows of each firm have only one

component and are given by FA = VA and FB = VB. This corresponds to the degenerate

case of π = ρl = 0 in Section 2. We use superscript “N” to denote the case in which the two

firms have not formed a relationship. Taking prices as given, demand for asset j = {A,B} is

standard in a CARA-Normal framework and given by

qNj =V −RPN

j

γσ2V

. (14)

The demand of the investor for asset j is independent of the demand for the other asset in

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the economy. The demand depends positively on the expected excess returns and negatively

on the variance of the asset and the risk aversion of the investor. We can find prices using

the market clearing conditions: qA = 1 and qB = 1. The prices of the two firms are the same

and given by

PNj =

V − γσ2V

R, (15)

for j = {A,B}. The price is the present discounted value of expected payoffs adjusted for the

risk associated with holding the asset.

If firms choose to have a relationship, then cash flows have two components FA = VA + ∆

and FB = VB + ∆ as in the conceptual framework in Section 2. The second component ∆ is

given by equation (1) and the price for firm j = {A,B} is given by (10). Firms will compare

the expected price of forming a relationship (11) under the optimal disclosure policy α∗ from

equation (12) with the price under no relationship (15) taking into account the choice of the

other firm. We assume that once a relationship is formed, the collaboration intensity δ is

fixed. The payoff matrix is given by Figure 3.

Figure 3: Payoffs under Relationship Formation

Firm B

Relation No Relation

Firm ARelation E[PA(α∗;m)], E[PB(α∗;m)] PN

A , PNB

No Relation PNA , P

NB PN

A , PNB

5.1 Equilibrium Definitions

The timeline of the economy is now given by Figure 4.

17

Figure 4: Timeline with Relationship Formation and Voluntary Relation Disclosure

Firms choose to

form a relationship

t = 0

Firms commit to

disclosure a policy

α

t = 1

Firms observe the

realization of ρ

and provide a joint

message m = {h, l}

based on policy α

t = 2

The investor

chooses her portfo-

lio and prices are

determined

t = 3

Definition 1 An equilibrium consists of prices when there is no relationship PNA and PN

B

that satisfy the market clearing conditions and are given by (15), prices when there is a

relationship PA(α;m) and PB(α,m) that satisfy the market clearing conditions and are given

by (10), a disclosure policy α that maximizes (11) and is given by (12) and the decision to

form a relationship by firms is a Nash equilibrium of the game in Figure 3.

Definition 2 (i) A relationship equilibrium is an equilibrium where both firms A and B

choose to form a relationship. (ii) A no-relationship equilibrium is an equilibrium where at

least one of the firms A or B chooses not to form a relationship. (iii) An equilibrium is

Pareto-dominant if it has the highest expected price.

5.2 Equilibrium Characterization

The following Lemma shows that there always exists an equilibrium with no relationship.

Lemma 1 A no-relationship equilibrium always exists.

Intuitively, no firm has any incentive to deviate from a no-relationship equilibrium as

there is no benefit from an individual deviation. It is only when both firms choose to form a

relationship that a relationship is formed. For the purpose of this paper, we will focus on the

18

Pareto-optimum equilibrium. In this model, a Pareto-optimum equilibrium is the equilibrium

with the highest expected price. Hence, if there exists an equilibrium where firms choose to

form a relationship, then that equilibrium will be the Pareto-optimum equilibrium. There

will be a Pareto-optimum equilibrium with a relationship when E[Pj(α∗;m)] > PN

j . The next

proposition shows the conditions for existence of a relationship Pareto-dominant equilibrium.

Proposition 4 There exists a relationship Pareto-dominant equilibrium where firms will form

a relationship if and only if

δ

2γσ2δ>

πρ2H+(1−π)ρ2

L

πρH+(1−π)ρL, if α∗ = 1

2,

δ2γσ2

δ>

πρ2he

2γ2ρ2hσ2δ+2γρlδ+(1−π)ρ2

l e2γ2ρ2l σ

2δ+2γρhδ

πρhe2γ2ρ2

hσ2δ

+2γρlδ+(1−π)ρle2γ2ρ2

lσ2δ

+2γρhδ, if α∗ = 1,

where the optimal disclosure policy α = α∗ is given by (12).

The next corollary finds a sufficient condition for a relationship equilibrium to be Pareto-

dominant. It also finds a sufficient condition under which the no-relationship equilibrium is

Pareto-dominant.

Corollary 2 If δ > 2γρhσ2δ , then firms have a relationship Pareto-dominant equilibrium

under both a disclosure and non-disclosure policy. If δ < 2γρlσ2δ , then firms choose to have a

no-relationship Pareto-dominant equilibrium both disclosure and non-disclosure policies.

5.3 Numerical Example

Figure 5 shows the parameter range where Pareto-dominant equilibrium with relationship

formation exists. There are six parameters that determine the existence of an equilibrium

with relationship formation that are π, γ, ρH , ρL, δ and σ2δ . In the figure, we assume γ = 1

and π = 0.5, set several values for ρH and ρL, and show equilibria with relationship formation

19

Figure 5: Pareto-Dominant Equilibrium with Relationship Formation

when there are changes in δ and σ2δ . We observe that equilibria with relationship formation

can be Pareto-dominant under both a disclosure and a non-disclosure policy. Intuitively,

when the benefit δ of forming a relationship is high relative to the cost σ2δ , a relationship

20

equilibrium is Pareto-dominant. When the benefit is very high relative to the cost of forming

a relationship, firms choose to commit to disclose the nature of their relationship. However,

when the benefit is at intermediate levels relative to the cost of forming a relationship, firms

prefer to commit to a non-disclosure regime. Finally, we observe that when δ is low relative

to σ2δ , then there is a unique equilibrium where no relationships are formed.

5.4 Mandatory Disclosure

In this section, we study the implications of regulation forcing disclosure on relationship

formation. Specifically, we analyze the impact of Regulation SFAS No. 13, in which firms

must report separately information about an operating segment that represents more than

10 percent of sales revenue. The next proposition shows that the introduction of mandatory

disclosure (α = 1) may lead to destruction of previously formed relationship under non-

disclosure (α = 12).

Proposition 5 Mandatory disclosure may destroy relationship formation. For any firm j =

{A,B}, E[Pj(α = 12;m)] > PN

j > E[Pj(α = 1;m)] if and only if

πρ2h + (1− π)ρ2

l

πρh + (1− π)ρl>

δ

2γσ2δ

>πρ2

he2γ2ρ2

hσ2δ+2γρlδ + (1− π)ρ2

l e2γ2ρ2

l σ2δ+2γρhδ

πρhe2γ2ρ2hσ

2δ+2γρlδ + (1− π)ρle2γ2ρ2

l σ2δ+2γρhδ

.

When the parameter condition is satisfied, the proposition shows that moving from a

non-disclosure regime (α = 12) to a disclosure regime (α = 1) would break relationships

previously formed. Under a non-disclosure policy, a relationship would be formed because

E[Pj(α = 12;m)] > PN

j . However, once regulation forcing disclosure would be introduced,

then relationships would break because PNj > E[Pj(α = 1;m)].

Figure 6 shows the parameter range where forcing disclosure destroys the formation of

21

Figure 6: Mandatory Disclosure and Destruction of Relationships

relationships. In the figure, we assume γ = 1 and π = 0.5, set several values for ρH and ρL,

and show relationship formation destruction when there are changes in δ and σ2δ . For most

combinations of ρh and ρl, there exist a region where relationships are formed only under

22

the non-disclosure policy when δ relative to σ2δ is in an intermediate range. Thus, when the

benefit of forming a relationship is at some intermediate level relative to the cost, mandatory

disclosure may destroy relationships and will affect the extensive margin of a relationship

between two firms.

6 Conclusion

This paper develops a new conceptual framework to analyze the incentives of firms to form and

disclose relationships and its implications for asset prices. Forming a relationship generates

synergies between firms. But relationships have a cost. Relationships make the performance

of the firms correlated, reducing the ability to hedge the risk of the two firms and generating

additional risk in financial markets. We first study the trade-offs on asset prices when there

is uncertainty about the relationship between two firms. Having a relationship has two effects

on the cash flows of the firm. First, cash flows have an additional payoff component with

positive mean. Second, the cash flows of the two firms become more correlated. From the

investor’s point of view, there is a benefit captured by the increase in the mean of asset payoffs

that increases asset prices, but there is also a cost captured by the increase in the variance

of the asset payoffs and the decrease in the ability to diversify her portfolio as cash flows are

now correlated, which decreases asset prices.

We also analyze the optimal disclosure policy about a relationship. Unlike previous litera-

ture, disclosing a relationship in our framework is about disclosing the existence of a common

component in the asset payoffs of the two firms, generating a correlation structure between the

two firms. It is not about disclosing the realization of the fundamentals as in most research

on disclosure. We finally examine under which conditions firms choose to form relationships

and their collaboration intensity. Firms will form a relationship when the expected asset price

23

of forming a relationship is higher than the asset price when there is no relationship.

In recent years, there has been an increase in regulation to bring more transparency to

financial markets. Financial transparency has been a key aspect improving the stability of

our financial system. Companies now have to disclose more information about their financial

performance and operations. It is tempting to promote financial transparency as a general

principle. Transparency is generally a good feature in regulation. Investors need transparent

financial statements to make informed investment decisions. Yet transparency comes with

some trade-offs. This study provides a conceptual framework to analyze some of these trade-

offs.

24

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26

Appendix

A Proof of Proposition 1

The representative investor maximizes equation (5) with respect to qA and qB. The first order

condition with respect to qA is given by

π exp

[−γ(µW (h)− γσ2

W (h)

2

)](V + ρsδ −RPA − γqA(σ2

V + ρ2hσ

2δ )− γqBρ2

hσ2δ +

(1− π) exp

[−γ(µW (l)− γσ2

W (l)

2

)](V + ρlδ −RPA − γqA(σ2

V + ρ2l σ

2δ )− γqBρ2

l σ2δ = o.

We can get the first order condition with respect to qB in the same way. When we plug the

market clearing conditions qA = 1 and qB = 1 into both conditions, we get prices for asset

j = {A,B}:

Pj =V − γσ2

V

R+

+δ

R

[πρhe

2γ2ρ2hσ

2δ+2γρlδ + (1− π)ρle

2γ2ρ2l σ

2δ+2γρhδ

πe2γ2ρ2hσ

2δ+2γρlδ + (1− π)e2γ2ρ2

l σ2δ+2γρhδ

]

− 2γσ2δ

R

[πρ2

he2γ2ρ2

hσ2δ+2γρlδ + (1− π)ρ2

l e2γ2ρ2

l σ2δ+2γρhδ

πe2γ2ρ2hσ

2δ+2γρlδ + (1− π)e2γ2ρ2

l σ2δ+2γρhδ

].

B Proof of Proposition 2

Taking the first order condition of the asset price under uncertainty (6) with respect to δ and

re-arranging terms, we get:

π(1− π)ρh + ρhπ2e2γρh(γρhσ

2δ−δ) + 4γ2π(1− π)ρ3

hσ2δ − 2γπ(1− π)ρ2

hδ

Re2γ2ρ2hσ

2δ+2γ2ρ2

l σ2δ+2δγρh+2δγρl

((1− π)e2γ(γρ2

l σ2δ+δρh) + πe2γ(γρ2

hσ2δ+δρl)

)2 = 0

C Proof of Proposition 3

Taking the first order condition of the expected asset price (11) with respect to α, we get:

∂E[Pj(α;m)]

∂α

∣∣∣∣α= 1

2

= 0

27

The relevant expression for the sign of the second order condition∂2E[Pj(α;m)]

∂α2 for α = 12

to be

a maximum is given by

[2γσ2

δ (ρh + ρl)− δ]

(e2γ2ρ2hσ

2δ+2γρlδ − e2γ2ρ2

l σ2δ+2γρhδ) < 0,

otherwise α = 12

is a minimum. There are two ways under which the inequality above is

satisfied for α = 12

to be a maximum:

1. If both of these conditions are satisfied: δ > 2γσ2δ (ρh+ρl) and e2γ2ρ2

hσ2δ+2γρlδ > e2γ2ρ2

l σ2δ+2γρhδ.

These two conditions are satisfied if and only if γ(ρl + ρh)σ2δ > δ > 2γ(ρl + ρh)σ

2δ . This

is not a feasible condition.

2. If both of these conditions are satisfied: δ < 2γσ2δ (ρh+ρl) and e2γ2ρ2

hσ2δ+2γρlδ < e2γ2ρ2

l σ2δ+2γρhδ.

These two conditions are satisfied if and only if γ(ρl + ρh)σ2δ < δ < 2γ(ρl + ρh)σ

2δ . This

is a feasible condition.

Hence, we can conclude that α = 12

is a maximum if and only if γ(ρl + ρh)σ2δ < δ <

2γ(ρl + ρh)σ2δ . Otherwise, α = 1 is a maximum since α = 1

2is a minimum and

∂E[Pj(α;m)]

∂α> 0

for any α ∈ (12, 1].

D Proof of Corollary 1

When the optimal policy is non disclosure (α∗ = 12), then the message received by the rep-

resentative investor contains no information and we are in the scenario of section 2.3, where

the asset price is given by equation (6). In this case, the price is independent of the message,

thus E[Pj(α∗ = 1

2;m)] is also given by equation (6). Instead, when the optimal policy is

full disclosure (α∗ = 1), then the message received by the representative investor contains

full information and asset prices are given by equation (4) depending on the realization of ρ.

Using equation (11), we can calculate the expected asset price as

E[Pj(α∗ = 1;m)] = πP PI

j (ρ = ρh) + (1− π)P PIj (ρ = ρl),

where we have used that Pr(m = h) = π, Pr(m = l) = 1− π, Pj(α∗ = 1;m = h) = P PI

j (ρ =

ρh) and Pj(α∗ = 1;m = l) = P PI

j (ρ = ρl).

28

E Proof of Lemma 1

From Table 3, we can see that no firm has any incentive to deviate from a no-relationship

equilibrium as there is no benefit from an individual deviation.

F Proof of Proposition 4

Under the optimal disclosure policy α∗ = 1, using equations (15) and (13), for any firm

j = {A,B}, we get that PNj < E[Pj(α

∗ = 1;m)] if and only if

(πρh + (1− π)ρl)δ > 2γσ2δ (πρ

2H + (1− π)ρ2

L).

Under the optimal disclosure policy α∗ = 12, using equations (15) and (6), for any firm

j = {A,B}, we get that PNj < E[Pj(α

∗ = 12;m)] if and only if

(πρhe2γ2ρ2

hσ2δ+2γρlδ +(1−π)ρle

2γ2ρ2l σ

2δ+2γρhδ)δ > 2γσ2

δ (πρ2he

2γ2ρ2hσ

2δ+2γρlδ +(1−π)ρ2

l e2γ2ρ2

l σ2δ+2γρhδ)

.

G Proof of Corollary 2

If we rewrite equation (13) as

E[Pj(α∗ = 1;m)] =

V − γσ2V

R+πρhR

(δ − 2γρhσ2δ ) +

(1− π)ρlR

(δ − 2γρlσ2δ ).

Since ρh > ρl and given that the price under no relationship is given by (15), then we can

conclude that PNj < E[Pj(α

∗ = 1;m)] if δ > 2γρhσ2δ and PN

j > E[Pj(α∗ = 1;m)] if δ < 2γρlσ

2δ .

If we rewrite equation (6) as

E[Pj(α∗ =

1

2;m)] =

V − γσ2V

R+

+πρhe

2γ2ρ2hσ

2δ+2γρlδ

R

(δ − 2γρhσ2δ )

πe2γ2ρ2hσ

2δ+2γρlδ + (1− π)e2γ2ρ2

l σ2δ+2γρhδ

+

+(1− π)ρle

2γ2ρ2l σ

2δ+2γρhδ

R

(δ − 2γρlσ2δ )

πe2γ2ρ2hσ

2δ+2γρlδ + (1− π)e2γ2ρ2

l σ2δ+2γρhδ

29

Since ρh > ρl and given that the price under no relationship is given by (15), then we

can conclude that PNj < E[Pj(α

∗ = 12;m)] if δ > 2γρhσ

2δ and PN

j > E[Pj(α∗ = 1

2;m)] if

δ < 2γρlσ2δ .

H Proof of Proposition 5

There are three conditions that need to be satisfied so that E[Pj(α = 12;m)] > PN

j > E[Pj(α =

1;m)] for any firm j = {A,B}.

1. For both conditions E[Pj(α = 12;m)] > PN

j and PNj > E[Pj(α = 1;m)] to be jointly

satisfied, the following inequality must be satisfied

πρ2h + (1− π)ρ2

l

πρh + (1− π)ρl>

δ

2γσ2δ

>πρ2

he2γ2ρ2

hσ2δ+2γρlδ + (1− π)ρ2

l e2γ2ρ2

l σ2δ+2γρhδ

πρhe2γ2ρ2hσ

2δ+2γρlδ + (1− π)ρle2γ2ρ2

l σ2δ+2γρhδ

.

The first inequality arises from PNj > E[Pj(α = 1;m)] and the second inequality arises

from E[Pj(α = 12;m)] > PN

j .

2. The sufficient conditions stated in Proposition 2 need to be violated. Otherwise there

exists an equilibrium with relationship formation under both a disclosure and non-

disclosure policy or alternatively there is a unique equilibrium with no relationships in

both economy. Hence, the following condition must be satisfied

ρh >δ

2γσ2δ

> ρl

3. For the upper bound of condition 1 to be larger than the lower bound, we need

(ρh − ρl)(e2γ2ρ2l σ

2δ+2γρhδ − e2γ2ρ2

hσ2δ+2γρlδ) > 0,

which is only satisfied when

(ρL + ρH) <δ

γσ2δ

.

Conditions 3 implies that the upper bound of condition 1 is always strictly higher than the

lower bound of that condition. Also conditions 2 and 3 are weaker than condition 1. Hence,

condition 1 is a necessary and sufficient condition.

30