Discussion Papers In Economics
And BusinessSelling out or going public? A real options signaling
approach
Michi NISHIHARA
Discussion Paper 16-28

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANSelling out or going public? A real options signaling
approach
Michi NISHIHARA
Discussion Paper 16-28

November 2016

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANSelling out or going public? A real options signaling
approach∗
Michi NISHIHARA†

Abstract
We examine a dynamic model in which a ﬁrm chooses between selling out and going
public under asymmetric information. We show that information asymmetry tends tochange the ﬁrm’s policy from selling out to IPO. More precisely, a separating equilibrium
can arise in which the good ﬁrm goes public while the bad ﬁrm follows the ﬁrst-best sales
policy because the good ﬁrm signals to market investors by doing an IPO. In order toseparate itself from the bad ﬁrm, the good ﬁrm can choose an IPO timing that is earlier
than the ﬁrst-best IPO timing. This result is consistent with the empirical evidence thatless proﬁtable ﬁrms tend to sell out to a large ﬁrm rather than going public.
JEL Classifications Code: G31; G34; D82.
Keywords: IPO; signaling; real options; asymmetric information; acquisition; earnout.

∗

This version: 15 November, 2016. This work was supported by the JSPS KAKENHI (Grant number

JP26350424, JP26285071) and the JSPS Postdoctoral Fellowships for Research Abroad. The author would
like to thank Haejun Jeon, Paulo Pereira, and the participants of the 20th EBES Conference and the 19th
Czech-Japan Seminar on Data Analysis and Decision Making under Uncertainty for helpful comments.
†
Corresponding Author. Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka,
Osaka 560-0043, Japan, E-mail: nishihara@econ.osaka-u.ac.jp, Phone: 81-6-6850-5242, Fax: 81-6-6850-5277

1Introduction

In recent years, the number of initial public oﬀerings (IPOs) has remained small because
many ﬁrms have chosen selling out to a large ﬁrm rather than going public as an independent ﬁrm (see Gao, Ritter, and Zhu (2013)). Although there have been numerous
studies on IPOs, especially on their relation to asymmetric information (for a review, see
Ritter and Welch (2002)), few works focus on the alternative. This paper is the ﬁrst work
to examine a ﬁrm’s dynamic choice between selling out and going public. In particular,
we reveal how asymmetric information regarding a ﬁrm’s growth opportunity aﬀects theﬁrm’s optimal decision.
This paper builds on the dynamic IPO model of Bustamante (2012). An entrepreneurof a private ﬁrm has an opportunity to make a growth investment along with an IPO.
There are two types of ﬁrms with good and bad growth opportunities. The entrepreneur
knows the ﬁrm’s type, while outside investors, who price the IPO, do not observe thetype. The ﬁrm only signals its type to outside investors thorough the IPO timing. In
addition to this setup of Bustamante (2012), we assume that the ﬁrm has an option tosell out to a large ﬁrm rather than going public. Large ﬁrms, which bid for the private
ﬁrm, do not observe the ﬁrm’s type. Following the empirical evidence of Gao, Ritter, andZhu (2013), we suppose that a bidder can expand the business more eﬃciently becauseof economies of scope and scale so that the entrepreneur always chooses selling out undersymmetric information.
In the model, we show that one of three types of equilibria occurs depending the parameter values: both types of ﬁrms sell out at the same time (hereafter, pooling equilibriumof sales), both types of ﬁrms go public at the same time (hereafter, pooling equilibrium ofIPOs), or the good ﬁrm goes public earlier and the bad ﬁrm sells out later on (hereafter,
separating equilibrium). In particular, we show that the separating equilibrium is likely
to occur under stronger asymmetric information and lower volatility.
We add to previous results about IPOs in Bustamante (2012) and Banerjee, Gucbilmez,
and Pawlina (2016), by providing implications about the alternative to IPOs. Indeed, theseparating equilibrium result predicts that worse ﬁrms tend to sell out. This result is
consistent with the empirical evidence. Gao, Ritter, and Zhu (2013) showed that lessproﬁtable ﬁrms tend to sell out rather than going public. Pagano, Panetta, and Zingales(1998) also showed that more proﬁtable ﬁrms are more likely to go public.
We also show that contingent payments in acquisitions can decrease ineﬃciency due toasymmetric information and lead the good ﬁrm to sell out in the separating equilibrium.
This result can account for empirical ﬁndings that contingent contracts are more likely
to be used in acquiring private and younger ﬁrms with strong asymmetric information(e.g., Kohers and Ang (2000), Datar, Frankel, and Wolfson (2002), Ragozzino and Reuer(2009)).
The remainder of this paper is organized as follows. Section 2 introduces the setup.

2After Section 3.1 brieﬂy explains the symmetric information case, Section 3.2 shows theequilibrium results under asymmetric information. Section 4 concludes the paper.

2

Setup

Consider an all-equity and private ﬁrm which is run by an entrepreneur who is the single shareholder. The ﬁrm receives a continuous stream of cash ﬂows X(t) following ageometric Brownian motiondX(t) = µX(t)dt + σX(t)dB(t) (t > 0),

X(0) = x,where B(t) denotes the standard Brownian motion deﬁned in a probability space (Ω, F, P)
and µ, σ(> 0) and x(> 0) are constants. For convergence, we assume that µ < r, wherea positive constant r is the risk-free interest rate. Assume that x is suﬃciently low to
exclude a ﬁrm’s investment at time 0.
The ﬁrm has a growth opportunity where the entrepreneur expands the cash ﬂows to
θi X(t) by issuing a constant fraction of shares, α ∈ (0, 1), and paying the investment costI(> 0).1 The growth option potentially takes on two types: θg (i = g: good type) andθb (i = b: bad type), where θg > θb > 1. The probability of drawing the good type isq ∈ (0, 1). The entrepreneur knows the ﬁrm’s type. Then, the value of the entrepreneur
who does an IPO at time T becomes∫ ∞(1 − α)θi X(T )e−r(t−T ) (1 − α)θi X(t)dt] − I + Pi (T ) =ET [− I + Pi (T ),
r−µ
T

(1)

where Pi (T ) denotes the proceeds from the IPO.
In addition to this basic setup of Bustamante (2012), we assume that the ﬁrm has
an option to sell out instead of going public. We denote by Qi (T ) the proceeds fromsales at time T .2 In addition to the payments Qi (T ), an acquirer pays the adjustment
and investment costs associated with the acquisition and receives a continuous stream ofcash ﬂows ηθi X(t) from the ﬁrm’s business. For simplicity, we assume that η ≥ 1 andthe acquirer’s adjustment and investment costs equal I.3 This means that an acquirer
can expand the business more eﬃciently than the entrepreneur due to economies of scopeand scale. We also assume that all agents are risk-neutral and that market investors andbidders are competitive and have the same information and beliefs.4
1

Following the basic setup of Bustamante (2012), we assume that the ﬁrm issues an exogenously given α

shares in the IPO to ﬁnance the growth investment project. Refer to the appendix of Bustamante (2012) to see
how α is determined by a ﬁnancing constraint.
2
For simplicity, we assume that the entrepreneur sells all shares to the acquirer. Our results remain unchanged
even if the entrepreneur keep holdings of a lower fraction of shares than in the IPO case, which means that theeﬀect of asymmetric information is stronger in the sales case than in the IPO case.
3
Our results remain unchanged as long as an acquirer can expand the business more eﬃciently than theentrepreneur.
4
Our model applies a situation of selling out by auction rather than by negotiation. In the negotiated sales,33Model Solutions

3.1

Symmetric information

Under symmetric information, investors and bidders observe the ﬁrm’s real type. Then,
we haveand∫ ∞αθi X(T )Pi (T ) = ET [e−r(t−T ) αθi X(t)dt] =
r−µ
T

(2)

∫ ∞
ηθi X(T )−I
Qi (T ) = ET [e−r(t−T ) ηθi X(t)dt] − I =
r−µ
T

(3)

By (1)–(3) and η ≥ 1, selling out generates the higher value than doing an IPO. The
optimal sales time is the solution to the optimal stopping problem:()∫ T
ηθi X(T )∗−rt−rT
Vi (x) = sup E[
e X(t)dt + e
−I ]
r−µ
T
0
(
)
x
(ηθi − 1)X(T )−rT=+ sup E[e− I ],
r−µ
r−µ
T

(4)

where the sales time T is optimized over all stopping times. As in the standard real options
literature (e.g., Dixit and Pindyck (1994)), we can easily derive the following solution.
Proposition 1 The ﬁrm sells out. The sales trigger isx∗i =β (r − µ)I
β − 1 ηθi − 1(i = g, b)(5)

and the entrepreneur’s value at the initial time (hereafter, called the ﬁrm value) isVi∗ (x) =where β = 1/2 − µ/σ 2 +

√

x
+
r−µ

(

x
x∗i

)β

I
β−1

(i = g, b),(6)(µ/σ 2 − 1/2)2 + 2r/σ 2 (> 1).

∗∗
For later uses, we denote the IPO trigger and ﬁrm value by x∗∗i and Vi (x), which equal(5) and (6) with η = 1.

3.2

Asymmetric Information

Under asymmetric information, investors and bidders do not know the ﬁrm’s real type.
The entrepreneur cannot directly prove the ﬁrm’s type to them and can signal the typeonly through the choice between selling out and going public and its timing.5 Note thatthe bad ﬁrm may have an incentive to imitate the good type and receive Pg (T ) and Qg (T )in (2) and (3). Following Bustamante (2012), we derive the least-cost equilibrium for theunlike in our setup, through the time-consuming negotiation process with a particular ﬁrm, the seller and theacquirer ﬁll gaps in information and prices.
5
Following the basic model of Bustamante (2012), we omit signaling by underpricing in the IPO. Bustamante(2012) also showed that signaling by the IPO timing tends to be less costly than signaling by underpricing.

4good ﬁrm that pays all costs due to asymmetric information. We ﬁrst derive the least-cost
separating equilibrium, in which the good ﬁrm maximizes the ﬁrm value conditional on
separating the bad ﬁrm. As usual, we restrict our attention to threshold policies expressedas Tg = {t ≥ 0 | X(t) ≥ xg }.
When the good ﬁrm do an IPO, separates the bad type, and receives Pg (T ) deﬁnedby (2), the good ﬁrm value becomes
)
(
∫ Tg
∫ ∞
αθg X(Tg )
−rt
−rt
−rTg
−I ]
E[
e X(t)dt +e (1 − α)θg X(t)dt + e
r−µ0Tg( )β ()(θg − 1)xg
x
x
=
+
−I ,
r−µ
xg
r−µ

(7)

where Tg and xg denote the IPO time and trigger, respectively. The good ﬁrm maximizes(7) under the incentive compatibility condition (ICC):( )β ()(θg − 1)xgxxss
Vg (x) = max+−Ixg ≥x r − µxr−µ( g )β ()((1 − α)θb + αθg − 1)xg
x
x
s.t.
+
− I ≤ Vb∗ (x).
r−µ
xg
r−µ

(8)

In (8), the left-hand side of the ICC corresponds to the value of the bad ﬁrm that imitates
the good-type IPO timing and receives Pg (T ) deﬁned by (2), whereas the right-hand sideof the ICC corresponds to the value of the bad ﬁrm that follows the policy in Proposition
1. Then, under the ICC, the bad ﬁrm has no incentive to imitate the good type.
When the good ﬁrm sells out, separates the bad type, and receives Qg (T ) deﬁned by(3), the good ﬁrm value becomes()( )β ()∫ Tg
ηθg X(Tg )(ηθg − 1)xg
x
x
−rt
−rTg
e X(t)dt + e−I ]=+− I , (9)
E[
r−µ
r−µ
xg
r−µ
0
where Tg and xg denote the sales time and trigger, respectively. As in the IPO case, thegood ﬁrm value maximizes (9) under the ICC:( )β ()(ηθg − 1)xgxxsVg (x) = max+−Ixg ≥x r − µxr−µ( g )β ()(ηθg − 1)xg
x
x
s.t.
+
− I ≤ Vb∗ (x),
r−µ
xg
r−µ

(10)

where the left-hand side of the ICC corresponds to the value of the bad ﬁrm that imitates
the good-type sales timing and receives Qg (T ) deﬁned by (3), whereas the right-hand sideof the ICC corresponds to the value of the bad ﬁrm that follows the policy in Proposition
1. By taking the higher value of (8) and (10), we have the following solution.6
Lemma 1 Separating equilibrium:
Case (I): η ≥ θg /θb . The good ﬁrm sells out. The good ﬁrm’s sales trigger and value are
xsg and Vgs (x) = Vb∗ (x), where xsg ∈ (0, x∗g ) is the solution to ηθg xsg /(r − µ) − I = Vb∗ (xsg ).
6

We can easily show that the solution satisﬁes the ICC for the good ﬁrm, i.e., the good ﬁrm has no incentive

to imitate the bad type.

5

Case (II): η < θg /θb . The good ﬁrm goes public. The good ﬁrm’s IPO trigger and
ss
∗∗
∗ ∗∗
value are xss
g and Vg (x) as follows. If ((1 − α)θb + αθg )xg /(r − µ) − I > Vb (xg ) holds,∗∗xssg ∈ (x, xg ) is the solution to((1 − α)θb + αθg )xssg
− I = Vb∗ (xss
g ),
r−µ

(11)

and
Vgss (x) =

x
+
r−µ

(

x
xss
g

)β (

(θg − 1)xss
g
−I
r−µ

)

= Vb∗ (x) +

(

x
xss
g

)β

(1 − α)(θg − θb )
r−µ

∗∗
ss
∗∗
Otherwise, xss
g = xg and Vg (x) = Vg (x). In each case, the bad ﬁrm sells out, and its

sales trigger and value are x∗b (x) and Vb∗ (x).
Proof. In Case (I), problem (10) dominates (8). We can easily derive the trivial solution
xsg to problem (10). In Case (II), problem (8) dominates (10). Note that x∗∗
g is the solution
to problem (8) without the ICC. If x∗∗
g does not satisfy the ICC, we obtain the solution
xss
g deﬁned by (11) because the objective function in (8) monotonically increases withxg ∈ (x, x∗∗g ]. From the triggers, we can easily calculate the ﬁrm values. The belief of investors and bidders is as follows: For sales at a trigger in (x, xsg ] andan IPO at a trigger in (x, xssg ], the ﬁrm is good at probability one; otherwise, the ﬁrm
is bad at probability one. For η ≥ θg /θb , the result is trivial. For η < θg /θb , both thepolicy and the value depend on the ﬁrm’s type. The good ﬁrm signals to outsiders by
doing an IPO although the ﬁrm is better oﬀ selling out under symmetric information. In
the binding case, as in Bustamante (2012), the good ﬁrm accelerates the IPO timing to
avoid imitation, while in the non-binding case the choice of an IPO is enough to separate
the bad ﬁrm.
Next, we derive the least-cost pooling equilibrium for the good ﬁrm. Consider thesales case, in which a bidder pays η θ̄X(T )/(r − µ) − I, where θ̄ = qθg + (1 − q)θb . The
sales time is the solution to the optimal stopping problem:()∫ Tη θ̄X(T )p−rt−rTV (x) = sup E[
e X(t)dt + e− I ].
r−µT0
In a standard manner, we can easily derive the following solution.
Lemma 2 Pooling equilibrium of sales:
The ﬁrm sells out. The sales trigger isxp =and the ﬁrm value isV p (x) =β (r − µ)I
β − 1 η θ̄ − 1( x )β I
x
+
,
r−µ
xp
β−1

regardless of its type.

6

The belief of outsiders is as follows: For selling out at the trigger xp , the ﬁrm is good at
probability q and bad at probability 1 − q; otherwise (oﬀ the equilibrium path), the ﬁrm is
bad type at probability one. It is easy to see x∗g < xp < x∗b and Vb∗ (x) < V p (x) < Vg∗ (x).
We now consider the IPO case, in which investors pay αθ̄X(T )/(r − µ). The IPO timeis the solution to the optimal stopping problem:∫ T

Vgpp (x) = sup E[
T
0

e

−rt

∫ ∞
X(t)dt +−rte(1 − α)θg X(t)dt + e

−rT

T

(

)
αθ̄X(T )−I ]r−µ

In a standard manner, we can easily derive the following solution.
Lemma 3 Pooling equilibrium of IPOs:
The ﬁrm goes public. The IPO trigger isxpp =β(r − µ)I,
β − 1 (1 − α)θg + αθ̄ − 1

regardless of its type. The good ﬁrm value isVgpp (x) =( x )β I
x
+
r−µ
xpp
β−1

and the bad ﬁrm value is7x+Vbpp (x) =r−µ( x )β ( ((1 − α)θ + αθ̄ − 1)xpp
b

r−µ

xpp

)
−I .

The belief of outsiders is as follows: For an IPO at the trigger xpp , the ﬁrm is good at
probability q and bad at probability 1 − q; otherwise (oﬀ the equilibrium path), the ﬁrm
pp < x∗∗ and V ∗∗ (x) < V pp (x) < V ∗∗ (x).
is bad type at probability one. We have x∗∗
g
g <x
g
b
b

By comparing Vgs (x), Vgss (x), V p (x), and Vgpp (x) in Lemmas 1–3, we obtain the leastcost equilibrium for the good type.8
Proposition 2 The least-cost equilibrium:
Case (I): η ≥ θg /θb . The pooling equilibrium of sales is chosen.
Case (II-i): (1 − α)θg /θ̄ + α ≤ η < θg /θb . If V p (x) ≥ Vgss (x) holds, the pooling equilibriumof sales is chosen. Otherwise, the separating equilibrium is chosen.
Case (II-ii): 1 ≤ η < (1 − α)θg /θ̄ + α. If Vgpp (x) ≥ Vgss (x) holds, the pooling equilibriumof IPOs is chosen. Otherwise, the separating equilibrium is chosen.

7

Vbpp (x) could be lower than Vb∗ (x) for some parameter values. However, when the pooling equilibrium of

IPOs is chosen (see Proposition 2), Vbpp (x) is higher than Vb∗ (x), which means that the bad ﬁrm has no incentive
to deviate from the pooling equilibrium of IPOs.
8
As shown in Bustamante (2012), the least-cost equilibrium for the good ﬁrm is chosen by the intuitive
criterion and the Pareto-dominant criterion of alternative equilibrium reﬁnements in the Bayes-Nash equilibrium
in which both types of ﬁrms jointly have incentives to reveal their type or cluster.

7

Case (II-ii) with η = 1 corresponds to the basic result of Bustamante (2012). The separating equilibrium in Cases (II-i) and (II-ii) is consistent with the empirical evidence
about IPOs. For instance, Bustamante (2012) and Banerjee, Gucbilmez, and Pawlina(2016) showed that ﬁrms with better growth opportunities go public earlier, while Pagano,
Panetta, and Zingales (1998) showed that more proﬁtable ﬁrms are more likely to go public. Most notably, the separating equilibrium results can also account for an empiricalﬁnding about selling out. Indeed, Gao, Ritter, and Zhu (2013) showed that less proﬁtableﬁrms tend to avoid IPOs and choose selling out.
In Cases (II-i) and (II-ii), a higher q leads to a higher θ̄, which causes the pooling
equilibrium rather than the separating equilibrium. This is consistent with the resultof Bustamante (2012). Further, we explore the eﬀects of asymmetric information andvolatility on the equilibrium in numerical examples. We set the base parameter values
as r = 0.07, µ = 0.03, σ = 0.25, α = 0.5, q = 0.5, x = 0.01, θg = 2.3 and θb = 1.7. In
Figure 1 we change levels of θg − θb with θ̄ = 2 ﬁxed, while in Figure 2 we change levelsof σ. In the ﬁgures, the regions S, P S, and P I stand for the separating equilibrium, thepooling equilibrium of sales, and the pooling equilibrium of IPOs, respectively. We can see
from the ﬁgures that the separating equilibrium tends to occur under stronger asymmetric
information and lower volatility.
Our model is also related to a contingent contract, in which an acquirer ﬁrst pays
a fraction of the total consideration and later pays the remaining value contingent on
the performance (e.g., see earnouts in Lukas, Reuer, and Welling (2012)). Consider acontingent contract in which a bidder pays a fraction of the ﬁrm value, denoted by α̃, atthe acquisition timing and pays the remaining fraction 1 − α̃ contingent on the realized
cash ﬂows. For instance, suppose that α̃ is low enough to satisfy η((1 − α̃)θb + α̃θg ) ≤(1−α)θb +αθg . In such a case, the ICC in (10) is looser than the ICC in (8); hence, the goodﬁrm sells out in the separating equilibrium. As α̃ → 0, the good ﬁrm value approaches
the ﬁrst-best value Vg∗ (x) in (6). In other words, the good ﬁrm can beneﬁt from such acontingent contract because it reduces loss due to asymmetric information. This result
is consistent with the empirical ﬁndings in Kohers and Ang (2000), Datar, Frankel, andWolfson (2002), and Ragozzino and Reuer (2009). They show that contingent payments
are more likely to be used in the acquisition of private ﬁrms, especially new ventures (i.e.,
in higher asymmetric information cases) than public and established ﬁrms (i.e., in lower
asymmetric information cases).

4

Conclusion

We examined a real options signaling game in which a ﬁrm chooses between selling out
and going public. We derive three types of equilibria; one separating and two pooling
equilibria. Most interestingly, in the separating equilibrium, the good ﬁrm goes public8while the bad ﬁrm sells out. This is because the good ﬁrm signals to outsiders by choosing
an IPO. In addition, the good ﬁrm can choose an IPO timing that is earlier than the ﬁrstbest IPO timing to avoid imitation. This result is consistent with empirical ﬁndings thatless proﬁtable ﬁrms tend to choose selling out rather than going public. We can also
account for empirical ﬁndings that contingent payments are likely to be used in M&A
under stronger asymmetric information.

References
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9

1.1
1
0.9
0.8

S

θg−θb

0.7
0.6
0.5
0.4
PS
0.3

PI

0.2
0.1

1

1.02

1.04

1.06

1.08

1.1
η

1.12

1.14

1.16

1.18

1.2

Figure 1: The eﬀects of asymmetric information on the equilibrium.

0.4
PI
0.35

PS

S

σ

0.3

0.25

0.2

0.15

0.1

1

1.02

1.04

1.06

1.08

1.1
η

1.12

1.14

1.16

1.18

1.2

Figure 2: The eﬀects of volatility on the equilibrium.

10