Equilibrium Abstract
Merger Activity in Industry Equilibrium
Theodosios Dimopoulos
†
Stefano Sacchetto
††
This version: December 2014
Abstract
We study a dynamic industry-equilibrium model that features mergers, entry, and exit by heterogeneous firms. We show how different sources of synergies affect merger cyclicality.
Improvements in marginal productivity between merging firms generate a procyclical motive for mergers, while reductions in fixed costs of production generate a countercyclical one. The presence of a merger market makes poorly performing firms less likely to exit the industry in recessions, and it increases the mean and variance of the cross-sectional distribution of firmlevel productivities. Consistent with the empirical evidence, we show that announcement returns for large acquirers are lower than for small acquirers, despite large acquirers’ higher
Tobin’s Q.
JEL Classifications: D21, D92, E22, E32, G34
Keywords : Mergers, Industry Equilibrium
∗
We are grateful to Fernando Anjos, Riccardo Calcagno, Brent Glover, Rick Green, Burton Hollifield, and seminar participants at the BI Norwegian Business School, the Katz School of Business, the 2013
Western Finance Association conference, the 2013 European Finance Association annual meeting, the 4 th
Advances in Macro-Finance Tepper-LAEF conference, the 2013 CEPR/Study Center Gerzensee European
Summer Symposium in Financial Markets, and the 4 th World Finance Conference for helpful comments and suggestions. All remaining errors are our responsibility.
†
Swiss Finance Institute and University of Lausanne. Address: Department of Finance, HEC Lausanne,
Theodosios.Dimopoulos@unil.ch.
††
Corresponding author.
Tepper School of Business, Carnegie Mellon University.
Address: Tepper
School of Business, 5000 Forbes Avenue, Pittsburgh, PA 15213, USA. Phone:+1-412-268-8832. E-mail: sacchetto@cmu.edu.
1.
Introduction
Mergers and acquisitions (M&A) have a significant impact on industries’ evolution. They account for a large fraction of firm turnover, facilitate the redeployment of capital to productive uses, and accelerate the transmission of new technologies among firms. For the parties involved, M&As represent complex investment decisions: Companies weigh the value of the synergies generated in a given deal against future takeover opportunities.
Figure 1 highlights two facts about aggregate merger and exit activity. First, M&As
account for a large portion of firm turnover: between 1981 and 2010, approximately 4.5% of active public firms merged in a given year, while the exit rate due to poor performance was
Second, merger activity is procyclical, which has been documented by a large body of empirical work (see, for example, Andrade, Mitchell, and Stafford, 2001, Harford, 2005), whereas exit is countercyclical (Campbell, 1998).
Our paper’s main contribution is to study the interactions between merger activity and firms’ entry and exit decisions in industry equilibrium. We develop a dynamic model of a competitive industry populated by firms with heterogeneous productivities. The model provides a rich set of new predictions about several important questions related to M&A activity: How do different synergy motives affect merger cyclicality? What determines the joint dynamics of mergers, entry, and exit? How does M&A activity affect the dynamics of firm productivity and its cross-sectional dispersion? What explains firms’ returns on acquisition expenditures?
The key features of our model of industry dynamics are the following. In each period, potential entrants choose whether to pay an entry cost and invest in productive capital.
Incumbents search for a merger partner, invest in capital, and decide whether to exit the industry. Merger synergies are generated by improvements in productivity and reductions in the fixed costs of production. Conditional on deciding to merge, the terms of exchange are decided by Nash bargaining between the two firms, and a fixed merger-implementation cost is incurred by the merged firm. The bargaining framework accounts endogenously for potential future takeover opportunities, should one of the parties refuse to accept the deal. Because
1 M&A activity and exit rates are measured by the fraction of public firms in the Center for Research
used to measure mergers and exit activity. These rates are partly affected by the fact that CRSP coverage increased substantially in 1962 to include AMEX firms and in 1972 to include NASDAQ firms.
2
firms face persistent aggregate demand shocks and idiosyncratic productivity shocks, they are heterogeneous in terms of both the current synergies and future takeover opportunities.
Finally, in each period the output price clears aggregate production in the industry.
We characterize the time-series and cross-sectional evolution of firm productivities and merger opportunities in a rational-expectations equilibrium by employing a parameterized expectations algorithm. We then calibrate the model using cross-sectional and time-series moments that describe M&A activity, entry, and exit in the data.
Our paper provides four main new results, corresponding to the questions on merger activity presented above. The first result is that the cyclicality of M&A activity depends on the sources of synergies: a reduction in the fixed cost of production, and an increase in marginal productivity. In general, cost reductions are more relevant during periods of low aggregate demand, thus inducing countercyclicality: Firms merge in bad times to reduce costs and avoid exiting. The second source of synergies is procyclical, because productivity gains are more valuable when aggregate demand is high. At the calibrated parameters, this second effect dominates and aggregate merger activity is procyclical.
Interestingly, we find that the interaction between the two synergy sources can have important effects on merger cyclicality.
In industries characterized by high fixed costs, productivity-enhancement mergers are more desirable, because such industries feature a low number of incumbents and a high equilibrium output price. This interaction has a procyclical effect on mergers, and its magnitude depends on the level of merger-implementation costs.
Second, we study the effects of merger activity on industry dynamics. We show, by means of a comparative statics analysis, that the presence of a merger market makes low productivity firms less susceptible to economic recessions. Because the option to merge is valuable, firms have stronger incentives to stay in the market, and the exit rate becomes less countercyclical. At the same time, merger opportunities produce higher entry, as new firms are born with the prospect of potential involvement in a future takeover. Mergers also affect average productivity in the industry. On one hand, they increase the productivity of merging firms, but on the other they encourage low productivity firms to enter or remain in the industry. Overall, in our calibration the first effect prevails, and merger activity increases the average productivity of incumbents.
Because mergers represent an alternative to exit for poor performers, the contemporaneous exit and merger rates are negatively related in the model, conditional on the aggregate
3
state variables. In a regression analysis of industry-level merger and exit rates for companies in the CRSP sample, we find empirical support for this prediction. In addition, we find that the merger rate is negatively related to the lagged exit rate both in the simulated and in the actual data. Our model suggests the following explanation for this result: A higher exit rate leaves fewer poor performers in the industry and a smaller scope for productivity improvements through M&As.
Third, we show how M&A activity affects the dynamics of firm productivities over the business cycle. In the absence of merger options, the average productivity in the industry is negatively correlated with aggregate demand shocks, since worse performing firms choose to enter or to remain in the industry during periods of economic expansion. We show that this pattern is largely reversed when the possibility of mergers is taken into account. Procyclical merger synergies lead to takeover waves that are positively correlated with aggregate demand shocks and to improvements in average productivity during booms.
Finally, we provide an explanation for three important empirical findings regarding bidder returns. Moeller, Schlingemann, and Stulz (2004) show that, despite large acquirers having a higher Tobin’s Q than small acquirers, the announcement returns for large acquirers are lower. Moreover, there is a positive (negative) association between the relative size of the target and small (large) acquirers’ returns. To explain these findings, Moeller, Schlingemann, and Stulz (2004) consider theories based on managerial hubris (Roll, 1996), empire building
(Jensen, 1986), and investor misvaluation (Dong, Hirshleifer, Richardson, and Teoh, 2006).
Our model suggests a common cause for these empirical regularities that does not rely on hubris, agency, or market overvaluation: Productivity improvements through acquisitions exhibit decreasing returns to scale, that is, highly productive firms find it hard to improve their productivity through acquisitions. In a regression analysis of acquirer returns using simulated data, we find that our model is consistent with the empirical findings in Moeller,
Schlingemann, and Stulz (2004).
To the best of our knowledge, this is the first paper to study a dynamic search-based model of mergers that features heterogeneous firms and aggregate uncertainty. We con-
tribute to a wide body of research on M&A activity.
First, our model relates to theoretical studies of merger synergies. Jovanovic and Rousseau (2002) propose a Q-theory of mergers, according to which takeovers are a tool to reallocate capital from underperforming firms to those with better management or productive resources. One of the main predictions of this
2
See Betton, Eckbo, and Thorburn (2008) and Gaughan (2011) for surveys of this literature.
4
theory concerns “who buys whom”: High-productivity firms should take over the assets of low-productivity firms. Rhodes-Kropf and Robinson (2008) challenge the Q-theory based on empirical evidence that contrasts this prediction. They show that, rather than “high buys low,” mergers usually happen between firms with similar market-to-book ratios, so that a better description of the merger market would be “like buys like.” To rationalize this fact,
Rhodes-Kropf and Robinson (2008) formulate a model in which merger synergies stem from the combination of complementary assets. A recent paper that builds on the idea of complementarities is Levine (2011). In his model, firms with high revenue-growth opportunities, but high operating costs, become takeover targets of firms with lower growth prospects, but higher cost efficiency. Finally, another possible source of synergies is economies of scope that allow the merged firms to lower their fixed costs of production by eliminating redundant or inefficient activities (Gomes and Livdan, 2004).
In our paper, rather than specify merger gains according to one of these theories, we use a flexible specification that incorporates as sources of synergies both productivity gains and reductions in the fixed production costs. Depending on the parameter values, productivity gains may be maximized when the productivity gap between the merging firms is large or small. In this sense, our paper is closest to David (2011), who studies an endogenous-search model of M&As among firms with heterogeneous productivities. Compared to David (2011), our model allows for time-varying idiosyncratic shocks, endogenous exit, and aggregate demand shocks that generate fluctuations over time in the cross section of firm productivities.
Our paper relates to several recent empirical findings on the sources of merger synergies.
Sheen (2014) finds that merging firms generate gains by increasing operating efficiencies and reducing costs, and this has an effect on competition through the product price. Bena and Li (2014) show that complementarities in firms’ innovation activities are an important determinant of mergers. These findings are consistent with the sources of merger gains in our model: productivity improvements that depend on asset complementarity between the merging firms, and a reduction in the costs of production.
Maksimovic and Phillips (2001) show evidence that the productivity of the acquirers’ existing assets increases in the productivity of the targets’ assets. Regarding the pricing of mergers, Ahern (2012) shows that the gains are equally split between acquirers and targets, and Li (2013) finds that the targets’ announcement returns are positively related to the posttakeover productivity gains. Importantly, the productivity gains for the acquirers’ existing assets are significant in the case of whole-firm acquisitions (Maksimovic, Phillips,
5
and Prabhala, 2011), but not significant when including partial-firm acquisitions (Schoar,
2002). Our model incorporates all these empirical facts, unlike the capital reallocation literature, which models acquisitions as a competitive market for used capital. This underscores the focus of our paper, which highlights mergers as a method of transferring intangible and organizational capital, rather than just physical capital.
Our paper contributes to the theoretical literature on industry dynamics in economies with heterogeneous firms. In a seminal paper, Hopenhayn (1992) characterizes the entry and exit levels in stationary industry equilibrium. Building on Hopenhayn’s framework, Lee and Mukoyama (2012) and Clementi and Palazzo (2010) introduce aggregate productivity
shocks and analyze the dynamics of firms’ entry and exit decisions over the business cycle.
Yang (2008) studies a model with a fixed population of heterogeneous firms that can achieve productivity gains through asset sales. The main contribution of our model to this literature is to study the joint dynamics of mergers, entry, and exit, and their effects on the crosssectional distribution of firms’ productivities over the business cycle.
We provide three methodological contributions to the literature on dynamic models with heterogeneous agents (Krusell and Smith, 1998). First, we use a nonstochastic simulation approach that tracks a continuum of firms over time, and avoids potential pitfalls in numerical accuracy throughout iterations due to a decline in the number of simulated firms because of exit and mergers. Second, we update firms’ expectations on the evolution of the endogenous aggregate states based on an exact, rather than an approximate, law of motion. Third, we account for a large number of aggregate states using a variant of the parameterized expectations algorithm. This allows us to study, among others, the effects of the variance of the cross-sectional distribution of firm productivity on optimal merger, exit and entry dynamics.
Finally, there is a vast body of work in industrial organization that studies the effects of mergers on product-market competition and derives implications for antitrust policy (see
Whinston, 2007). While this literature studies increased market power as a source of merger gains, we focus on synergies motivated by productivity improvements and cost reductions in
perfectly competitive industries.
3 Examples of general equilibrium models with heterogeneous firms are Khan and Thomas (2008), who analyze the effects of capital-adjustment costs on aggregate investment, and Gomes and Schmid (2010), who focus on the asset-pricing implications of firm heterogeneity and entry and exit activity.
4
In a survey on empirical M&A studies, Betton, Eckbo, and Thorburn (2008) conclude that there is weak empirical support for the market-power hypothesis, whereas the evidence in favor of efficiency improvements
6
The paper proceeds as follows. Section 2 develops a model of industry dynamics with
mergers, entry, and exit; section 3 presents the results of the calibration and section 4
discusses the predictions of the model. Section 5 discusses a number of avenues for future
research and concludes.
Appendix A provides the proof of Proposition 1; Appendix B
describes the sample construction and includes the definitions of the empirical variables; and
Appendix C contains the details of the numerical procedure used in the calibration.
2.
Model
We model the dynamics of an industry populated by a continuum of firms with symmetric information. The population of firms is divided between incumbents with variable mass N and potential entrants, with fixed mass N
E
. Time is discrete and the horizon is infinite.
The timing of events within each period is described in Figure 2. Incumbents participate
in a merger market, produce and realize profits, invest to accumulate productive capital, and decide to stay or exit the industry. Potential entrants observe a signal about their own productivity and decide whether to enter the industry by paying a fixed cost.
2.1.
Production and Profits
Each firm employs capital k to produce per-period quantity q = zk
α of a homogeneous good, where α ∈ (0 , 1) is a returns-to-scale parameter and z ∈ Z = [ z, ¯ ] is an idiosyncratic productivity shock, independently distributed across firms. The firm sells the quantity produced in a competitive market with isoelastic demand at price P = sQ
−
1
η
, where Q is total output in the industry, s ∈ S = [ s, ¯ ] is an industry demand shock and η ≥ 0 is the price
We assume that z ≥ 0 and s ≥ 0. The demand and productivity shocks are mutually independent first-order Markov processes with transition probabilities
F
Z
( z
0 | z ) for z
0
, z ∈ Z and F
S
( s
0 | s ) for s
0
, s ∈ S , respectively. Throughout the paper, primes denote next-period values. Period operating profits are π ( P, z, k ) = P zk
α − c f
, where c f
≥ 0 are fixed costs of production. Firms can buy and sell capital at unit price, and capital depreciates at rate δ ∈ (0 , 1). Each firm’s investment expenditure is I = k
0 − (1 − δ ) k . Firms are risk neutral and discount the stream of future operating profits at rate r .
is stronger.
5
The results of the paper remain qualitatively unchanged if we consider an aggregate productivity rather than an aggregate demand shock.
7
2.2.
Mergers and Exit
In each period, incumbents participate in the merger market by entering a pool of firms to be matched. Specifically, each firm pays a fee C ( e ) and enters the matching pool with probability e ∈ [0 , 1). We assume that firms choose the search effort, e , simultaneously, and that C ( · ) is twice differentiable, strictly increasing, strictly convex, satisfying C (0) = 0 and lim e → 1
−
C ( e ) = ∞ . Participants in the matching pool are sorted in random pairs. Members of bargaining weights. Firms that choose to merge in a given period combine their operations in the next period. Namely, in the period in which firms i and j merge, the profit of the new merged firm M is the sum of the profits of the two separate firms. In the same period, the merged firm pays a one-off integration cost c
M
≥ 0.
Organizational changes result in a new productivity level ζ
M
( z i
, z j
), and the next-period shock, z
0
, will be drawn according to F
Z
( z
0 | ζ
M
( z i
, z j
)). The merged firm’s productivity depends on the productivities of firm i and j and is given by
ζ
M
( z i
, z j
) = z + (¯ − z ) [ λ max { ˜ i
, ˜ j
} + (1 − λ ) min { ˜ i
, ˜ j
} ]
θ
, (1) where ˜ = z − z
¯ − z
, 0 6 λ 6 1, and 0 < θ < 1. We scale the function ζ
M
( z i
, z j
) so that it does not assume values outside the closed interval [¯ ]. We provide a discussion of our specification
of merger productivity gains in subsection 2.5.
After mergers take place, firms decide whether to stay or exit the industry. Exit decisions are implemented at the end of the period, and firms that exit cannot reenter the market.
Hence, the payoff of an exiting firm consists of the period payoff and the liquidation value of capital.
2.3.
Entry
In each period, there is a pool of potential entrants with fixed mass N
E
> 0. At the beginning of the period, potential entrants observe a signal y ∈ Z regarding their productivity level that is drawn from a time-invariant distribution G ( y ), and they decide whether to pay an entry cost c
E
. Entrants are not immediately productive, because they have no capital at their disposal, but they can invest to accumulate capital for production in the next period, when their productivity is distributed as F
Z
( z
0 | y ). Moreover, they do not participate in the
8
merger market for the period in which they make their entry decision.
2.4.
Equilibrium
Denote the vector of aggregate states as x ∈ X , with transition function F
X
( x
0 | x ). As we discuss below, x is composed of the industry demand shock, the number of incumbents, the output price, and the cross-sectional cumulative distribution function (c.d.f.) of firms’ productivities at the start of the period, F ( z ). At the investment stage, merger decisions for the period have been undertaken, and the incumbents’ value function is
V ( x, z, k ) = π ( P, z, k ) + (1 − δ ) k + max { U ( x, z ) , 0 } .
(2)
In the above formula, U ( x, z ) denotes the end-of-period continuation value, and it incorporates the possibility that in the next period the firm merges. The firm decides to exit whenever this continuation value is negative:
U ( x, z ) ≡ max k
0
− k
0
1
+
1 + r
Z
Z
Z
X
V S ( x
0
, z
0
, k
0
) d F
X
( x
0
| x ) d F
Z
( z
0
| z ) < 0 , (3) where V S is the value of the firm at the start of the period. Notice that since cash flows are sunk at the investment stage, the continuation value does not depend on the current product
By assumption, in the period in which two firms agree on a merger, their operating profits remain unaffected. Therefore, the value function of the merged firm is
V
M
( x, z i
, z j, k i
, k j
) = π ( P, z i
, k i
)+ π ( P, z j
, k j
)+(1 − δ )( k i
+ k j
)+max { U ( x, ζ
M
( z i
, z j
)) , 0 }− c
M
.
(4)
Synergies are therefore independent of capital levels:
W ( x, z i
, z j
) ≡ V
M
( x, z i
, z j, k i
, k j
) − V ( x, z i
, k i
) − V ( x, z j
, k j
)
= max { U ( x, ζ
M
( z i
, z j
)) , 0 } − max { U ( x, z i
) , 0 } − max { U ( x, z j
) , 0 } − c
M
.
(5)
Two firms, i and j , matched in the market for corporate control, merge only if they produce positive synergies, that is if W ( x, z i
, z j
) ≥ 0. In that case, given the exit decision rule in
Equation 3, it is optimal for the merged firm to remain in the industry and not exit.
6
It does depend, however, on the distribution of the next-period product price.
9
Let F
M
( z | x ) denote the productivity c.d.f. of firms in the matching pool. These are the
only firms that can merge in the current period. Given the synergy function in Equation 5,
firm i ’s option value of merging is
W
∗
( x, z i
) =
Z
Z max { W ( x, z i
, z ) , 0 } d F
M
( z | x ) .
Firm i ’s expected net gains from participating in the merger market are
W ( e, x, z i
) ≡ e
2
W
∗
( x, z i
) − C ( e ) , and its optimal search effort is
(6)
(7) e
∗
( x, z i
) = argmax e ∈ [0 , 1)
W ( e, x, z i
) .
(8)
Our formulation accounts for the fact that firms only merge if synergies are positive, that synergies are split equally between the merging firms, and that search costs are sunk at the merger stage.
Bayes’ rule implies that the cross-sectional productivity densities of matched and unmatched firms are, respectively, f
M
( z | x ) = f
U
( z | x ) = e
∗
( x, z ) f ( z | x )
R
Z e ∗ ( x, z ) f ( z | x ) d z
(1 − e
∗
( x, z )) f ( z | x )
R
Z
(1 − e ∗ ( x, z )) f ( z | x ) d z
,
(9)
(10) where f ( z | x ) is the start-of-period density function of incumbents’ productivities given the aggregate state x .
The value of the firm at the start of the period, V S , reflects its stand-alone value at the end of the period, V , and the net gains from participating in the merger market, W :
V S ( x, z, k ) = V ( x, z, k ) + W ( e
∗
( x, z ) , x, z ) .
(11)
10
The stock market return for firm i around the announcement of a merger with firm j is
1
2
[ W ( x, z i
, z j
) − e
∗
( x, z i
) W
∗
( x, z i
)]
.
V S ( x, z i
, k i
)
(12)
The numerator of Equation 12 reflects the part of the synergies that accrues to firm
i , minus the takeover gains expected by the stock market before the deal announcement. The difference between these two components reflects the news to be incorporated in the firm’s stock price after the deal announcement. The denominator represents the stock market capitalization of the firm before the announcement.
Since potential entrants lack productive capital, they make their entry decision by comparing their continuation value to the cost of entry. Their value function is the solution to:
V
E
( x, y ) = max { U ( x, y ) − c
E
, 0 } .
(13)
The firms’ investment choice is independent of anticipated next-period synergies, because these are invariant to the firm’s employed capital. It is straightforward to show, then, that for an incumbent with productivity z (or an entrant with an equal productivity signal), its choice of next period capital is determined by the policy function
κ ( x, z ) =
α
R
S p ( s
0
, x ) d F
S
( s
0 | s )
R
Z z
0 d F
Z
( z
0 | z ) r + δ
1
1 − α
, (14) where p ( s
0
, x ) is the next-period equilibrium price, which we characterize in Proposition 1.
Proposition 1 Denote the incumbents’ end-of-period c.d.f. of productivity by e
( z | x ) . The next-period price is p ( s
0
, x ) = s
0
αE ( s
0 | s ) r + δ
−
α
(1 − α ) η + α
N
0
Z
Z
E ( z
0
| z )
1
1 − α d e
( z | x )
−
(1 − α ) η
(1 − α ) η + α
, and its expectation at the end of the current period is
(15)
E ( p ( s
0
, x ) | s ) = E ( s
0
| s )
(1 − α ) η
(1 − α ) η + α
"
N
0
α r + δ
α
1 − α
Z
Z
E ( z
0
| z )
1
1 − α d e
( z | x )
#
−
1 − α
(1 − α ) η + α
.
(16)
Proof
11
Notice that the next-period price in Equation 15 depends on the next-period demand
shock s
0 and the current aggregate states x . This is because the next-period aggregate output
at the start of the period, x
, as shown in Equations 22 to 29 below. Output price depends
positively on realized aggregate demand shocks and negatively on the total output supplied in the industry. The latter increases with the number of incumbents and with aggregate investment, which varies positively with expected demand and expected productivity shocks.
Four probabilities characterize the rate of firm turnover in the industry: i) The probability of entry:
P
E
( x ) =
Z
Z
1 ( U ( x, y ) > c
E
) d G ( y ) , where 1 ( .
) is the indicator function.
ii) The probability that an unmatched firm remains an incumbent:
P ( x ) =
Z
Z
1 ( U ( x, z ) > 0) d F
U
( z | x ) .
(17)
(18) iii) The probability that two matched incumbents i and j merge:
P
M
( x ) =
Z
Z 2
1 ( W ( x, z i
, z j
) > 0) d F
M
( z i
| x ) d F
M
( z j
| x ) .
(19) iv) The probability that a matched firm i remains an independent incumbent:
P
N
( x ) =
Z
Z 2
1 ( W ( x, z i
, z j
) < 0 , U ( x, z i
) > 0) d F
M
( z i
| x ) d F
M
( z j
| x ) .
(20)
The recursive competitive equilibrium is a collection of i) value functions V ( x, z, k ),
V
E
( x, y ); ii) policy functions κ ( x, z ), of-period and end-of-period productivities, bounded sequences of output prices e
{
∗
P
( x, z ); iii) cumulative distribution functions for startt
}
∞ t =0
{ F t
( z ) }
∞ t =0 and n e t
( z ) o
∞
, respectively; and v) t =0 and mass of incumbents { N t
}
∞ t =0 such that:
12
1.
V ( x, z, k ), e
∗
( x, z ), and κ ( x, z
) solve the incumbent’s problem defined by Equations 2,
2.
V
E
( x, y ) and κ ( x, y
) solve the potential entrant’s problem defined by Equations 13 and
3. Output markets clear for any period t ≥ 0:
P t +1
= s t +1
N t +1
Z
Z 2 z t +1
κ ( x t
, z t
)
α d F
Z
( z t +1
| z t
) d e
( z t
| x t
)
−
1
η
.
(21)
4. For all z ∈ Z and for all t ≥ 0, the end-of-period cross-sectional distribution of firm productivities is
F t
( z ) = e t
( z | E t
= 1) Pr( E t
= 1) + e t
( z | E t
= 0) Pr( E t
= 0) , (22) where Pr( E t
= 1) =
N
E
P
E
( x t
)
N t +1 is the probability that a productivity shock observation at the end of period t corresponds to a new entrant ( E t
= 1), and Pr( E t
= 0) =
1 − Pr( E t
= 1). The c.d.f. of the new entrants’ productivity shock is
F t
( z | E t
= 1) =
G ( z )
P
E
( x t
)
1 ( U ( x, z ) > 0) , (23) and that of the incumbents is e t
( z | E t
= 0) = 1 −
Z
Z e
∗
( x, z ) d F t
( z ) e U
( z | x t
)+
Z
Z e
∗
( x, z ) d F t
( z )
"
1
2
P
M
( x t
) e M
( z | x t
) + P
N
( x t
) e N
( z | x t
)
#
1
2
P
M
( x t
) + P
N
( x t
)
, (24) where
F
U
( z | x t
) =
Z z z
1 ( U ( x, z i
) > c
E
) f
U
( z i
| x
P ( x t
) t
) d z i e M
( z | x t
) =
Z
{ ζ
M
( z i
,z j
) ≤ z }
1 ( W ( x, z i
, z j
) > 0) f
M
( z i
| x t
) f
M
( z j
P
M
( x t
)
| x t
) d z j d z i
(25)
(26)
13
F
N
( z | x t
) =
Z z
Z
1 ( W ( x, z i
, z j
) ≤ 0 , U ( x, z i
) > 0) f
M
( z i
| x t
) f
M
( z j
P
N
( x t
)
| x t
) d z j d z i z z
(27) are the end-of-period productivity cumulative distribution functions for unmatched but non exiting firms, merging firms, and matched firms who remain independent incumbents, respectively.
5. For all z
0
, z ∈ Z and for all t ≥ 0, the start-of-period cross-sectional distribution of firm productivities is
F t +1
( z
0
) =
Z
Z
F
Z
( z
0
| z ) d e t
( z ) .
(28)
6. The dynamics of the number of incumbents follows the law
N t +1
= N t
1 −
Z
Z e
∗
( x, z ) d F t
( z ) P ( x t
)+
N t
Z
Z e
∗
( x, z ) d F t
( z )( P
M
( x t
) / 2 + P
N
( x t
)) + N
E
P
E
( x t
) .
(29)
2.5.
Discussion
A number of remarks on the model’s assumptions are in order. Two potential gains are realized when firms merge. First, economies of scope are generated by a reduction in the fixed costs of production (Gomes and Livdan, 2004). While total per-period costs for the two separate firms are 2 c f
, fixed costs for the merged firm are half at c f
. These savings may come from lower administrative costs and overhead expenses or the elimination of redundant activities.
The second potential source of merger synergies is an increase in productivity. This effect is captured in our model by the function ζ
M
( z i
, z j
) in Equation 1, which assigns
a new productivity level to the merged firm given those of the two merging firms. The
specification in Equation 1 reflects the implications of two important theories of merger
gains that have been analyzed in the prior literature. The Q-theory of mergers in Jovanovic and Rousseau (2002) implies that the largest improvements occur when firms with very different productivity levels merge. This prediction is consistent with setting λ = 1 in our synergy function. In contrast, Rhodes-Kropf and Robinson (2008) develop a model of merger synergies based on complementary assets that predicts that gains are largest for firms with
14
similar levels of productivity.
In our specification, this happens for low values of λ
illustrates the effects of parameter λ on productivity gains, plotting the ζ
M function for the two extreme cases of λ = 1 and λ = 0.
The specification of the function ζ
M
( z i
, z j
) carries important implications for merger prices. In Q-theory models (e.g., Jovanovic and Rousseau, 2002, Yang, 2008), mergers real-
locate assets between firms through an intermediary market for used capital.
Accordingly, the price per unit of capital acquired is independent of the involved firms’ productivity. In our model, the transaction price per unit of capital depends on the productivity of the merging firms. Consistently, Li (2013) finds greater posttakeover productivity improvements for the target’s assets associated with higher premiums paid by acquirers and higher combined cumulative abnormal returns around the deal announcement. Moreover, in Q-theory models, the postmerger productivity of both existing and acquired capital depends solely on the buyer’s characteristics. In our model, the productivity of existing assets after the merger is positively affected by the productivity of acquired assets, consistent with the empirical findings in Maksimovic and Phillips (2001). Maksimovic, Phillips, and Prabhala (2011) show that the productivity gains for the acquirer’s existing assets are significant in the case of
The model features two important frictions related to the merger market. First, firms face a one-off fixed cost of merger implementation and integration ( c
M
). Examples of such a cost are the expenses incurred to integrate the two firms’ information technology systems, business processes, and organizational structures (Tafti, 2011), the legal and advisory fees,
and any external financing costs associated with the deal.
Second, finding a merger partner is costly. Examples of search costs are the opportunity cost of the time spent by managers to screen potential candidates, and the fees charged by investment banks to assist the bidder
7 Maksimovic, Phillips, and Prabhala (2011) find that postacquisition productivity relates positively to a proxy of complementarity between the assets of the target and those of the acquirer.
8 Typically, Q-theory models feature capital-adjustment costs, which affect the value of merger synergies.
In our model, we abstract from this determinant of merger gains, and focus on post-merger productivity improvements.
9 Schoar (2002) finds no significant productivity change in the acquirer’s existing assets when new assets are purchased through partial- or full- firm acquisitions, or with the addition of new plants. The contrast between the evidence of Schoar (2002) and that of Maksimovic, Phillips, and Prabhala (2011) highlights the importance of mergers in realizing productivity gains.
10 For example, in the merger of Mellon Financial and The Bank of New York in 2007, the estimated integration costs were 1.3 billion dollars, or 3% of the pre-announcement market capitalization of the two companies (Baldwin and Taliaferro, 2010).
15
in the takeover process and in the target’s due diligence.
We assume that the merger surplus is split according to a Nash bargaining process in which firms have equal bargaining weights. The assumption of an equal split of the surplus is consistent with the empirical results in Ahern (2012), who find that the total dollar gains
from mergers are only marginally higher for targets compared to bidders.
Finally, in our model, merger decisions are driven by the following trade-off. If two firms decide to merge in a given period, they benefit from higher productivity and lower costs, but they incur a fixed implementation cost and they lose the option to continue as stand-alone entities. Thus, firms must consider an opportunity cost represented by their stand-alone values. In our framework, a firm’s stand-alone value depends not only on its stream of profits, but also on its future merger opportunities, which are driven by the cross section of firm productivities over time. Thus, firms care about aggregate productivity conditions, as well as the output price, and this makes our model an industry equilibrium model. Other models of heterogeneous firms also highlight the importance of industry conditions for firm value, but for different reasons (e.g., the price of labor in Clementi and Palazzo, 2010, and the price of used capital in Yang, 2008).
3.
Calibration
To investigate the implications of the model for the joint dynamics of entry, exit, and mergers, we solve the model numerically and perform a calibration exercise. We first describe the parameterization of the model and the solution algorithm. We then discuss the choice of empirical moments to use as calibration targets and the resulting parameter values.
Finally, we present the calibration results and describe firm policies and industry dynamics in equilibrium.
3.1.
Parameterization and Solution Algorithm
The system of Equations 2 to 11 shows that the firm’s value function depends on the
cross-sectional distribution of firm productivity. This distribution is an infinitely dimensional object, so the need arises to approximate it for calibration purposes with a finite set of
11
Cumulative abnormal percentage returns are on average larger for the target companies (Andrade,
Mitchell, and Stafford, 2001). However, since percentage returns are affected by the relative size of bidders and targets, inference of bargaining weights based on this measure can be misleading (Ahern, 2012).
16
moments.
We choose two moments, the cross-sectional mean m and variance v of logproductivity at the start of each period. Our aggregate states are then described by the vector x ≡ [ s, P, log( N ) , m, log( v )]. Including a second-order moment represents a methodological contribution to the literature on heterogeneous-agent models, which typically characterize heterogeneity only by the first moment of the cross-sectional distribution (cf. Krusell and
Smith, 1998). This allows us to analyze how dispersion in productivities affects firms’ merger, entry and exit decisions.
We describe the dynamics of both the idiosyncratic and the aggregate shocks by means of AR(1) processes: log( z
0
) = ρ z log( z ) + σ z
ε z and log( s
0
) = ρ s log( s ) + σ s
ε s
, where ρ z
, ρ s
∈
[0 , 1), σ s
, σ z
> 0, and ε z and ε s are uncorrelated truncated standard normal white noise processes. We set the distribution of the entrants’ log-productivity signal to be equal to the ergodic distribution of the log( z ) process, i.e., log( y ) ∼ N 0 ,
σ
2 z
1 − ρ
2 z
. Finally, to specify the incumbents’ search cost, we choose
C ( e ) = c e e
1 − e
, (30) where c e
>
0 is a search cost parameter consistent with our assumptions in subsection 2.2.
Our calibration algorithm nests two loops. In the first loop, we compute the recursive competitive equilibrium at a given value of the structural parameters, simulate the economy for a number of periods, and collect a vector of simulated moments. In the second loop, the parameter space is searched until the sum of squared deviations between the set of empirical and simulated moments chosen for calibration is minimized.
To compute the competitive equilibrium, we employ the Parameterized Expectation Algorithm (PEA), treating the continuation value U ( x, z ) as the object of convergence (see
This algorithm relies on simulating a cross section of firms for many periods, and updates the value function on the basis of realized policies. In total, we simulate 500 periods and discard the first 25 percent, to ensure updating on the basis of stationary paths of the endogenous aggregate states m , log( v ) and log( N ).
To ensure convergence, we follow a nonstochastic simulation approach that traces a continuum of firms over time. We do so because a standard Monte Carlo simulation of a finite number of firms would encounter two computational problems. First, for a given value of the
12 See Algan, Allais, Den Haan, and Rendahl (2014) for a discussion of parameterized expectation methods for the solution of dynamic models with heterogeneous agents and aggregate uncertainty. See Den Haan
(1996) for an application.
17
parameters, in an iteration in which firms’ expectations are far from equilibrium, there is a high likelihood that at least in one period all simulated firms decide to exit and none to enter.
Second, the merger and exit rates are discontinuous functions of parameters, because they are based on aggregation of firm-specific discrete choices. By comparison, these issues do not arise in our nonstochastic simulation. Indeed, with a continuum rather than a finite number of firms, there is always a positive mass of firms remaining in the industry, and the entry and
exit probabilities (Equation 17 and Equation 18) are determined by distribution functions
that are smooth in parameters. This preserves a uniform level of numerical accuracy across parameters that imply different merger, entry and exit rates.
Our algorithm requires approximating the continuation value U ( x, z ) using a flexible functional form on a grid of the states x and z
. As discussed in Appendix C, we use a polynomial
approximation, and we discretize the state space for z following Tauchen (1986), by assuming that log( z ) lies on a grid of 40 Chebyshev nodes spanning the interval √ 5 σ z
1 − ρ
2 z
, √ 5 σ z
1 − ρ
2 z
We choose a large number of standard deviations of the ergodic distribution of the process
.
of log( z
) to obtain a large enough range to incorporate post-merger productivity changes.
For our endogenous aggregate states m , log( v ) and log( N ), we employ an adaptive updating rule: We gradually adjust the bounds of the state space, depending on the states’ realized
Dynamic models with heterogeneous agents are often solved using the Krusell and Smith
(1998) algorithm. This method assumes that agents follow a forecasting rule to form expectations on the future distribution of the endogenous aggregate states (typically an autoregressive law based on the first moment of the distribution); solves for the agents’ individual policies by dynamic programming given this rule; simulates an economy with a large number of agents; updates the agents’ forecasting rule by regression on the simulated data; and iterates until convergence. The algorithm we employ offers three significant advantages compared to Krusell and Smith’s (1998) algorithm. First, we do not need to specify an ap-
proximate law of motion of the endogenous aggregate states, since Equations 15–29 provide
an exact one. Second, our algorithm is faster because it nests the updating of firm’s policies and the simulation of the economy in one loop. Finally, compared to dynamic programming approaches, our method is less memory intensive because it computes the continuation value
13 We verify by Monte Carlo experiments that these bounds are never reached.
14 This bound-adjustment approach is similar to the one described by Maliar and Maliar (2003) for the standard neoclassical growth model.
18
U ( x, z ) at the realized values of the endogenous aggregate states, rather than for a wide grid of the states that include regions not visited in equilibrium.
3.2.
Calibration Targets
Table 1 displays the parameter values used in the calibration. One period in the simu-
lation is assumed to correspond to one year. The parameters in Panel A are comparable to
those used in prior studies that feature a similar specification of operating profits.
We set the discount rate to r = 4%, capital depreciation to δ = 15%, the curvature of the profit function to α = 0 .
75, the persistence of the productivity process to ρ z
= 0 .
66, and the standard deviation of the productivity innovations to σ z
= 0 .
11.
The nine parameters in Panel B of Table 1 are calibrated to match a set of empirical mo-
ments, which are reported in Table 2. The sources of data are Thomson Reuters’ Securities
Data Company (SDC) Platinum for information on merger deals, the Center for Research in Security Prices (CRSP) database for exit and merger rates, Compustat for financial information on targets and acquirers, and the National Income and Product Account (NIPA)
tables from the Bureau of Economic Analysis for aggregate earnings. Appendix B describes
the sample construction in detail and provides definitions of the empirical variables. We use empirical evidence from Lee and Mukoyama (2012) for the entrants’ relative total factor productivity (TFP) and from Betton, Eckbo, and Thorburn (2008) for an estimate of the average combined merger gains.
The first set of parameters concerns M&A activity. These are the cost of merger implementation, c
M
, the search cost parameter, c e
, the synergy weight of the most productive firm between the target and the acquirer, λ , and the curvature parameter of the synergy function,
θ . Because our model is highly nonlinear, it is not possible to associate each one of these parameters to a specific moment. However, the model provides guidance on which moments
are most sensitive to a change in these parameters, and can thus be used for identification.
We start by computing the average annual merger rate, measured by the fraction of firms in the CRSP population that are delisted during a year because of a merger or acquisition,
15 See Hennessy and Whited (2007), Riddick and Whited (2009), Yang (2008), and Eisfeldt and Muir
(2012). The parameter values used in these studies are typically different from those in real business-cycle models that feature labor as a production input.
16
structural parameters.
19
and its standard deviation. We then compute the autocorrelation coefficient of the merger rate for each 2-digit Standard Industry Classification (SIC) code, and take an average across
industries. As discussed in subsection 2.5, the parameter
λ governs “who buys whom”: The higher the value of λ , the higher on average will be the difference in productivity between the merged firms. To account for this effect of λ , we match the absolute difference between targets’ and bidders’ log Tobin’s Q, scaled by the acquirer sector’s yearly standard deviation of log(Q). As a fourth moment, we choose the average combined merger gains for targets and acquirers, which are set to 1.79%, an estimate of the combined cumulative abnormal returns
(CAR) around the announcement of a takeover bid from Betton, Eckbo, and Thorburn
To calibrate the fixed cost of production, c f
, we compute the average annual percentage of firms in CRSP that are delisted because of liquidation, bankruptcy, or other reasons related to poor performance. The cost of entry c
E is set to match the average relative TFP of
the entrants computed in Lee and Mukoyama (2012).
To calibrate the parameters ρ s and
σ s
, which govern the dynamics of the aggregate shock s , we match the autocorrelation and standard deviation of annual aggregate earnings growth. Finally, the price-elasticity parameter, η , affects, among other moments, the exit rate and the autocorrelation of aggregate
earnings growth (see discussion in subsection 3.3.3).
To match the empirical moments in the data, we define the variables used to obtain their
theoretical counterparts in the model in Table 3. In the calibration, we set the number of
incumbents and the number of potential entrants in the first period to one.
3.3.
Calibration Results
In this subsection, we present the numerical results of the calibration and discuss the
implications of merger activity on industry dynamics. Table 2 reports the simulated moments
generated according to the calibrated parameters in Table 1. Because of the high degree of
nonlinearities in the model, we are not able to match exactly the moments in the data, but we can replicate them reasonably well. The two largest discrepancies between actual and
17 This estimate corresponds to the value-weighted returns for bidders and targets between day − 41 and day +1 around the announcement of the deal. See Table 7 in Betton, Eckbo, and Thorburn (2008).
18 Lee and Mukoyama (2012) find that the TFP of new entering plants ranges between 0.87 and 0.96 of the incumbents’ TFP, depending on the estimate of the returns-to-scale parameter (see Table 6 in their paper).
In our model, we use as a target the value of average relative TFP that corresponds most closely to the returns-to-scale parameter that we use in our calibration.
20
simulated moments are the relative productivity of new entrants (1.045 versus 0.87 in the data) and the average merger rate (5.64% versus 4.54% in the data).
The analysis presented in the rest of the paper is based on simulating the cross-sectional distribution of firms for 500 periods, discarding the initial 25% to ensure convergence to the steady state. Given the simulated distributions, we use numerical integration to obtain the probabilities and moments of interest in each of the remaining periods, as described in
3.3.1.
Mergers, Entry, and Exit in Industry Equilibrium
To characterize the equilibrium industry dynamics over the business cycle, we start by describing the firms’ policy functions, which determine their entry, exit, and merger decisions.
Incumbents stay in the industry and potential entrants enter if their productivity exceeds the exit and entry thresholds, denoted by ζ ( x ) and ζ
E
( x ), respectively. These thresholds are
decreasing functions of the aggregate shock (Figure 4). Thus, entry is procyclical and exit
In periods of higher aggregate demand, firms with lower productivity enter the industry, whereas incumbents close up shop at lower levels of productivity.
We now turn to merger activity. In any given period, the merger rate is determined by two quantities: the search effort that firms undertake, and the synergies realized upon matching. The optimal search effort for firm i
, using Equations 6, 7, 8, and 30, is
e
∗
( x, z i
) = max
(
1 − s
2 c e
W ∗ ( x, z i
)
, 0
)
, (31) and the firm’s expected net gains from participating in the merger market are
W ( e
∗
( x, z i
) , x, z i
) = max c e
+
1
2
W
∗
( x, z i
) − p
2 c e
W ∗ ( x, z i
) , 0 .
(32)
Therefore, firms only search for a merger partner if
W
∗
( x,z i
)
2 of merging is high enough relative to the search cost.
> c e
, that is, if the option value
Consider now the synergies realized by two firms upon matching in the merger market.
Figures 5a and 5b show, from different angles, the synergy function at the calibrated pa-
19 This is consistent with existing empirical evidence (see Campbell, 1998). In our simulation, the correlations of the entry and exit rates with the log aggregate demand shock are 0.82 and -0.78, respectively.
21
rameters in Table 1, with the aggregate states set at their average levels in the simulation.
It is worth noticing that the synergies, as a function of the productivity levels of the two merging firms, are hump shaped. To understand the intuition for this result, remember that the synergies are given by the difference between the value of the merged firm and the sum
of the stand-alone values of the two merging firms (Equation 5). On the one hand, the value
of the merged firm is influenced by the productivity improvements due to the merger. In our model, these improvements are captured by the productivity function of the merged firm,
ζ
M
( z i
, z j
), which is concave in either of its arguments, and globally concave for λ < 0 .
5 and
θ < 1. This concavity reflects the fact that there are diminishing opportunities for pro-
ductivity improvements through the takeover market for high levels of firm productivity.
On the other hand, the stand-alone value of each merging firm is a convex function of its
productivity. This is a common feature among neoclassical models of the firm, 21
and it reflects the fact that a positive productivity shock not only increases future output at a given level of capital, but it also induces the firm to invest in new assets. When a merger takes place, the value of the merged firm increases in the productivity of the merging firms, but at a diminishing rate, whereas the value of the stand-alone firms increases at an accelerating rate. As a result of these two countervailing effects, the synergy function declines at high values of the merging firm’s productivity, thus giving rise to a hump shape.
Which firms search, and which merge? In Figure 6a, the solid line represents the contour
plot of productivity pairs that yield zero synergies in a given period. Firms in the interior of that boundary would merge if matched. Not all firms in that merger opportunity set, however, decide to search. The dashed line in the plot defines the search set, that is, the set of firms with positive search effort. Its square shape reflects the fact that search effort depends on own productivity ( z i
) and on the distribution of merger partners’ productivities
( F
M
), but not directly on its realization ( z j
The intersection of the merger opportunity and search sets characterizes the merger set, that is, the productivity pairs for which mergers realize. The merger set leaves out firms that search but find unsuitable matches (failed acquisitions), and firms that could have found a suitable match, but at too high a search
20 The value of the merged firm is also affected by the reduction in the fixed cost of production. At high levels of productivity, the relative importance of this cost-cutting motive declines, and it affects the shape of the synergy function minimally.
21
See Adda and Cooper (2003) for a textbook treatment of these models, and Strebulaev and Whited
(2012) for a recent survey of the applications of these models in finance.
22
See Equation 6 and Equation 31.
22
cost.
Within the search set, effort intensity varies across firms. Since synergies are hump shaped, effort intensity inherits the same property: Firms close to the boundary of the search set have relatively little incentive to search. For a given firm, the incentives to search depend on the distribution of search intensity across firms, since this distribution affects the probability of finding a suitable candidate. The more desirable merger partners search, the higher the incentive of the firm to search. To understand this externality effect, consider
Figure 6b. The solid curve shows the equilibrium search effort of the firm (Equation 8) as
a function of its own productivity. Assume, instead, that all firms in the industry exert a constant level of effort. In this case, the productivity distribution of the firms in the matching pool, F
M
, would be the same as the distribution of all incumbents, F . The dotted curve in
Figure 6b represents the optimal search effort of a particular firm under this assumption.
The solid curve lies above the dotted curve, because the firms within the equilibrium search set yield, on average, higher synergies compared to a random match across all firms in the industry.
Figure 7 shows the effect of aggregate shocks on merger synergies. Figures 7a and 7b
provide contour plots of the merger synergies (Equation 5) as a function of the productivity
of the two merging firms, z
1 and z
2
, for two different levels of the demand shock, s = 0 .
85 and s = 1 .
18, respectively. In both figures, the states m , v , and N are set to their average values in the simulation, and the thick curve represents the zero level of merger synergies.
For the region of z
1 and z
2 points where synergies are positive (the area inside the thick curve), two matched firms decide to merge. For low levels of the demand shock, the merger gains are negative for the majority of ( z
1
, z
2
) pairs, so that few M&As occur (Figure 7a).
For higher levels of the aggregate shock, synergies rise high enough to generate mutually
profitable merger deals for a wider range of firms (Figure 7b).
3.3.2.
The Effect of Merger Options on Firm Policies
To gain deeper intuition about firms’ behavior in our calibrated model, we regress the time series of the simulated entry, exit, and merger rates, and the expected future price,
defined in Equation 16, on the aggregate state variables log(
s ), log( N ), m , and log( v ). To
highlight the effect of merger options on firm policies, Table 4 shows a comparison of the
values of the estimated coefficients to those obtained by ignoring merger activity and setting
23
the merger-implementation cost to infinity.
A positive demand shock and a decline in the number of incumbents positively affect the
expected future product price (Column 4 in Table 4). Both events increase the present value
of productivity-related synergies and lead to higher matching and merger rates (Column 1).
When the economy is characterized by higher average productivity, the merger rate is low.
This can be interpreted in two ways: First, when firms are on average very productive ( m is high), they invest aggressively, thus increasing future aggregate output, lowering its price, and reducing the value of productivity-related synergies. Second, the scope of productivity improvements in a population of highly productive firms is narrow, because synergies are hump shaped. Their expected value declines with m , along with the incentives of incumbents to search for merger partners.
Table 4 shows a positive relation between the dispersion in productivity and the merger
rate (Column 1). This positive association arises because a larger spread in productivity between matched firms provides a source of potential productivity improvements, thus increasing merger synergies.
The comparison of the regression coefficients for the two simulated economies, with and
without mergers, in Table 4 allows us to examine the effect of merger options on entry and
exit activity. In the economy without mergers, the average entry rate is 8.4%, and new firms are more likely to enter the industry when demand is high, there are few competitors, and the average productivity of incumbents is low (Column 5). When firms have the option to merge, the average entry rate increases to 9.2%, as firms weigh the value of future merger
opportunities against the entry costs.
Entry decisions also become more dependent on demand shocks and less dependent on the cross-sectional mean productivity (Column 2).
This is because the value of future merger options, embedded in the stand-alone value of a new entrant, increases in s and decreases in m .
Incumbents prefer to leave the industry when there is low demand and the sector is
populated by a large number of incumbents (Column 6 in Table 4). The exit rate increases
with the dispersion of productivity and decreases with its average value. There are two effects at play behind the latter result: First, for a given exit threshold policy, higher m implies a lower exit rate. Second, higher mean productivity implies more investment and lower future price of output, which increases the exit threshold itself. At our calibrated
23
Metrick and Yasuda (2010) find that the majority of successful venture-capital-backed firms end up being acquired, which illustrates the importance of merger options for young firms.
24
parameters, the first effect dominates and the exit rate is negatively associated with mean productivity.
Merger options make exit less responsive to changes in demand and the distribution
of productivity (Column 3 in Table 4). Poorly performing firms that find themselves in
adverse aggregate conditions may choose not to exit the industry, in the hope that a merger
How do entry, exit, and mergers affect the cross section of firms in the industry? Consider first the economy with no mergers. In each period, the cross-sectional distribution of incumbent firms’ productivity reflects the history of past entry and exit decisions. Since unproductive firms exit the industry, and productive entrants become incumbents, both entry and exit increase average productivity in the industry. Indeed, in the economy without mergers, the incumbent firms’ average productivity z , 1.066, exceeds the ergodic mean of the z process at the calibrated parameters, exp
0 .
5 σ
1 − ρ
2 z
2 z
= 1 .
011.
In the simulated economy with mergers, the average productivity increases even further to 1.117. To understand this result, notice that mergers have two effects on average productivity. The first is a direct, and positive, effect: Firms merge when they can achieve productivity improvements that are high enough to compensate the merger costs. The second is an indirect effect, and it operates through the selection of firms in the industry through entry and exit. Because the option to merge is valuable, the entry and exit thresholds are lower compared to the economy without mergers. This has a negative effect on average productivity in the industry. Overall, at the calibrated parameters, the direct effect of mergers on productivity dominates, and average productivity is higher than in the economy without mergers.
Merger activity also affects the dispersion of firm productivities in the industry. In any given period, the cross section of incumbent firms reflects a mixture of the distributions of merged and unmerged firms. Because these two subsamples have different means, their mixture results in a higher cross-sectional standard deviation of firm productivity in the economy with mergers (0.156) compared to the economy without mergers (0.149).
24 Hotchkiss and Mooradian (1998) provide empirical evidence of distressed firms avoiding exit through acquisitions.
25
3.3.3.
Comparative Statics Results
With the calibrated parameters in hand, we can perform a comparative statics analysis to examine how the determinants of merger synergies affect the policy functions and the cross-sectional distribution of firms in the industry.
Table 5 presents the results. To facilitate comparison, Column 1 reports the moments at
the calibrated parameters in Table 1. In Column 2, we examine the effect of increasing the
weight of the most efficient firm in the merged firm’s productivity by setting λ to 0.7. The difference in the Tobin’s Q of the merging firms rises, because higher values of λ increase the merger gains for firms with different levels of productivity. This point is illustrated in
Figure 8, which shows contour plots of the merger synergies as a function of the productivity
of the merging firms for two different values of λ .
In industries in which mergers generate lower productivity improvements (the curvature parameter θ increases, Column 3), the merger rate is lower, and future merger options are less valuable, which increases the exit rate. Overall, these industries experience lower firm turnover.
A similar effect on firm turnover is present in industries with high merger-implementation costs, c
M
, and high costs of searching for a merger partner, c e
(Columns 4 and 5, respectively).
Interestingly, an increase in either c
M or c e raises the average combined merger gains. The intuition is the following. First, c
M and c e have a direct effect on the option value of merging,
defined in Equation 6, which decreases in
c
M and increases in c e
—firms only search when expected synergies are high enough to compensate for the search costs. However, an increase in either of the two costs reduces the expected net gains from participating in the merger
market, defined in Equation 32, and therefore the stand-alone value of the firm. For
c e
, both the direct and the selectivity effects move in the direction of higher combined merger gains
c
M
, the selectivity effect dominates the direct effect on expected synergies, explaining why the combined merger gains in Column 4 are higher compared to those in Column 1.
When fixed costs of production rise (Column 6 of Table 5) firm turnover accelerates
for two main reasons: Unmatched firms have a stronger incentive to exit the industry, and matched firms are more inclined to merge and save production costs. The latter effect
dominates and the average exit rate decreases compared to Column 1. Figure 9 shows that
the decline in the exit rate is only a local effect. When the fixed costs of production are
26
either at a low level, so that synergies are small,
or at a high level, so that the incentive to exit is strong, the average exit rate is positively related to c f
.
Industries with higher costs of entry (Column 7 of Table 5) are not only characterized
by a lower average entry rate, but also feature a higher average productivity of new entrants relative to that of the incumbents. Less persistent aggregate shocks (Column 8) imply a lower autocorrelation of aggregate operating-earnings growth and a lower average exit rate, because spells of poor demand are shorter. By comparison, a higher elasticity of demand
(Column 9) also yields lower persistence in earnings growth, but it is associated with a higher exit rate because the average level of the product price is lower. Finally, more volatile aggregate shocks imply a larger fraction of firms below the exit threshold, a higher standard deviation of earnings growth, and a more volatile merger rate (Column 10).
3.3.4.
The Effect of Merger Options on Industry Dynamics
How do merger options affect the firms’ response to aggregate shocks? How do they influence the cross-sectional distribution of firm productivity over time? We show, by means of an impulse-response analysis, that merger options have a radical effect on the cyclicality of mean productivity in the industry.
We simulate a continuum of firms using the parameter values given in Table 1 for 500
periods. In the last period, we set the aggregate demand shock, s , the product price, P , the cross-sectional mean, m , and variance, v , of the productivity shocks at their average steady-
state levels. Figure 10 shows the impulse responses to a two-standard-deviation innovation
in aggregate demand (about 7%) in period 501 of the simulation, setting future innovations in s to zero. We illustrate two cases, one in which mergers among firms in the industry are
possible (solid line in the graphs of Figure 10), and one in which merger activity is absent
(the merger implementation cost c
M is set to infinity), but entry and exit are allowed (dotted lines in the graphs).
Consider first the case in which mergers do not occur. In this scenario, a positive aggre-
sequence of effects in the dynamics of the industry. The number of incumbent firms increases
25 For low c f
, the potential cost savings due to a merger are small, and productivity gains are also small because there is a low exit rate, a high number of incumbents, and a low product price.
27
by roughly 17% following a positive aggregate shock (Figure 10c). The favorable aggregate
conditions encourage incumbents and potential entrants with relatively low productivity to stay or enter the industry, respectively. As a result of this selection effect, mean produc-
tivity among incumbents declines by 6% (Figure 10d).
Since a positive aggregate shock decreases the entry and exit thresholds, the variance of productivity among incumbents and
the increase in the number of competing firms.
When mergers are possible, the value of the synergies increases with the aggregate demand shock. These synergies create procyclical productivity gains that more than compensate the countercyclical effect of entry and exit on the cross-sectional average productivity,
which increases by 2% following the aggregate shock (Figure 10d). Therefore, the empirical
prediction of our model is that productivity is more procyclical in industries with higher merger activity. The fraction of firms taken over increases by 23% and remains above its
steady-state rate for about 7 years, thus giving rise to a merger wave (Figure 10h). The
growth rate of the number of incumbent firms remains roughly similar to the case of no mergers, but this masks the effect of merger activity on entry and exit. Indeed, due to a rise in the value of future merger options, the rate of entry accelerates and the rate of exit decelerates even more strongly than in the case of no mergers. Finally, since merger options make exit policies less dependent on aggregate states, the variance of productivity responds
less to the demand shock, compared to the no-mergers case (Figure 10e).
4.
Main Results
This section presents the main results and empirical predictions of the model. First, we focus on the time-series implications of our model for industry dynamics. In particular, we examine the cyclical properties of merger activity and the joint dynamics of exit and merger rates. We then discuss the cross-sectional predictions of our model in terms of the empirical relation between acquirer returns and firm size.
26 This countercyclical effect of entry and exit on average idiosyncratic productivity aligns with the work of Clementi and Palazzo (2010).
28
4.1.
Merger Activity over the Business Cycle
The model generates a rich set of new predictions about the cyclical properties of merger activity. The two sources of synergies that we model, productivity improvements and cost
cutting, have different implications with respect to merger cyclicality. Figure 11 plots the
average merger rate (solid curve) and the correlation of the merger rate with the log aggregate demand shock s (dashed curve) as a function of the synergy curvature parameter θ (leftcolumn plots) and the fixed cost of production c f
(right-column plots). We consider two
cases: In Figure 11a, we set the fixed merger-implementation cost,
c
M
, to zero, and in
c
M at its calibrated value of 0.252.
The effect of productivity improvements on merger cyclicality varies depending on parameter values. Consider first the case of c
M
θ decreases, and the productivity improvements in mergers rise, the merger rate is higher and more procyclical, because higher demand shocks make productivity improvements more profitable. For lower levels of θ , productivity improvements represent a larger fraction of total synergies, and the incentives for firms to merge when demand is high are stronger.
For a higher level of c
M
(Figure 11b), the correlation of the merger rate with the aggregate
shock is nonmonotonic in θ . This effect is caused by the variance of the merger rate, which decreases for low values of θ , but increases for higher values.
To recover the economic intuition for this relation, we decompose the merger rate as the product of the matching
rate, defined as the average search effort by firms in the industry (see Table 3), and the
fraction of matched firms that merge (Equation 19). In Figure 12, we plot their simulated
values in each period against the log demand shock s for three values of the curvature parameter: θ = 0 .
3 (dots), θ = 0 .
4 (crosses), and θ = 0 .
54 (circles).
For the lowest value of θ , merger synergies are high, firms have strong incentives to search for a merger partner, and the matching rate is at its highest levels, positive in all periods,
intuition is that, for high values of s , firms understand that better merger opportunities are likely to arrive in the subsequent periods, because s is positively autocorrelated. Thus, the option value of waiting is high, and firms are more inclined to reject current deals. Moreover, because the option to wait is very valuable, firms’ merger decisions are strongly dependent on aggregate states and the merger rate is very volatile.
29
For the intermediate value of θ , synergies are lower. This has two consequences: Firms exert less search effort, and they become more reluctant to reject merger deals, conditional on finding a match. As a result, both the matching rate and the fraction of matched firms that merge vary less with the aggregate state, and become less volatile.
At high values of θ , positive synergies arise rarely. Therefore, the merger market is characterized by long periods of inactivity, in which firms do not search for merger partners, and few periods of bursts in the number of deals. These bursts explain why the variance of the merger rate is increasing at high values of θ . Spikes in merger activity are less likely to appear when the fixed merger-implementation cost c
M evenly spread across periods.
is low, and search effort is more
Synergies driven by cost-cutting motives are increasing in c f
. When c
M
shows that the merger rate becomes more countercyclical as c f increases. The intuition is that cost-cutting motives are strongest in recessions, when a higher fraction of firms is up against the exit threshold. For those firms, an opportunity to economize costs by merging can make the difference between remaining in the industry and exiting.
There is a second effect, driven by industry equilibrium considerations, of fixed costs of production on merger cyclicality. As c f increases, a larger fraction of firms exit, a lower fraction of firms enter, and the output price rises. The increase in price makes productivity improvements through mergers more valuable, especially when aggregate demand conditions are favorable. Thus, an increase in c f
can make mergers more procyclical. Figure 11b shows
that this is the case for low values of c f
, when merger implementation costs are present. These implementation costs make mergers between small, poorly performing firms prohibitively expensive, which explains why the procyclical effect of c f dominates. For higher values of c f
, the cost-cutting motive becomes strong enough across more firms in the industry, and merger cyclicality becomes negatively related to c f
. Thus, the cost-cutting and productivityrelated motives interact through the output price, a result that becomes apparent only in an industry equilibrium framework.
In summary, our discussion in this section generates several new predictions about the properties of merger activity over the business cycle. First, depending on the relative importance of the sources of synergies, mergers can be procyclical or countercyclical: Productivity improvements create a procyclical motive for mergers, and reductions in the fixed cost of production a countercyclical one. At our calibrated parameters, the first motive prevails and merger activity exhibits strong procyclicality: The correlation between the merger rate
30
and the log demand shock is 0.76.
Second, merger-implementation costs create periods of freezes in the takeover market, followed by bursts in merger activity. Finally, during booms, firms exert more effort in searching for a merger partner, but they also have a higher option value of waiting, so that they are more inclined to refuse a merger once they find a match.
4.2.
Merger and Exit Dynamics
We now turn our attention to the relation between exit and merger activity. To study
this relation empirically, in Table 6 we perform a regression analysis of the industry-level
merger and exit rates for the companies in the CRSP sample between 1981 and 2010. The
construction of the variables is explained in Appendix B. The regression coefficients re-
ported in Column 1 of Table 6 show that the exit rate is negatively and significantly related
to the contemporaneous merger rate, using as controls the lagged merger rate and the contemporaneous average industry profitability, measured by return on assets. To account for unobserved heterogeneity at the aggregate and industry levels, we include year and industry fixed effects. The negative relation between the exit and the merger rates is robust to performing a dynamic panel data regression (Column 2) with the merger rate as a predetermined variable (Blundell and Bond, 1998).
Using the simulated data generated according to the parameters in Table 1, we perform
a regression of the exit rate on the contemporaneous and lagged merger rates, the lagged
exit rate, the aggregate state variables, and a constant. The results (Column 5 of Table 6)
show that the model is consistent with a negative relation between the merger and exit rates, as observed in the real data (Columns 2 and 3). Accordingly, seeking a potential acquirer represents a valid strategic alternative to exit for poorly performing firms in years in which the takeover market is hot. Finally, the regression coefficients for the lagged merger and exit rates have the same signs as those in the regression using the real data.
We then study the relation between the merger rate and the previous year’s exit rate.
Column 3 of Table 6 shows that the merger rate in an industry is negatively related to
the previous year’s exit rate, controlling for year and industry fixed effects and industry
27 This result is in line with the empirical findings. Maksimovic and Phillips (2001) find that the fraction of assets sold through mergers increases in periods of high aggregate real production. Eisfeldt and Rampini
(2006) find that the correlation of capital reallocation and gross domestic product over the business cycle is highly positive and significant. See also Andrade, Mitchell, and Stafford (2001), Harford (2005), and Dittmar and Dittmar (2008) for evidence of procyclical merger activity.
31
profitability. This negative relation is robust to the inclusion of the lagged acquisition rate in a dynamic panel data regression in which the lagged exit rate is a predetermined variable
(Column 4). When we perform a similar analysis in the simulated data, we find that the model can replicate the negative association between the exit and the lagged merger rate
(Column 6). The intuition for this result is that, following periods in which the exit rate is high and firms with low productivity exit the industry, mean productivity increases. As a consequence, merger activity decreases because the potential for productivity improvements
in mergers declines (cf. Table 4).
4.3.
Acquirer Returns and Firm Size
Our model provides novel insights on the determinants of stock returns around the announcement of M&A deals. One of the main points of interest in the empirical literature on this topic is the relation between acquirer announcement returns and acquirer size. Moeller,
Schlingemann, and Stulz (2004) use a sample of 12,023 completed US takeovers in the period
1980–2001, and show that smaller acquirers enjoy higher announcement returns. Furthermore, the relative size of the target, defined as the value of the M&A deal divided by the preacquisition capitalization of the acquirer, is positively associated with the returns of small acquirers and negatively with the returns of large acquirers. Finally, large acquirers have a higher average Tobin’s Q than small acquirers.
To examine whether our model provides an interpretation for these results, we simulate
an economy for 500 years using the calibrated parameters in Table 1. For the purposes of
this section, we use a direct simulation approach, that is, we simulate a panel of firms. To
maintain the correspondence with our nonstochastic simulation results in subsection 3.3,
we impose that a unit mass of firms ( N = 1) corresponds to 10,000 firms in the direct
simulation. Figure 13a plots the synergies as a function of the productivity levels of the
two matching firms in a given simulation period, where the dots show the synergies for
the realized mergers in the simulation, and Figure 13b shows the productivity pairs of the
merging firms. Notice that firms with higher productivity tend to merge with firms that are
In any given match in the merger market, we impose that the firm with the highest
28
eter values.
32
preacquisition capitalization plays the role of the acquirer. This classification follows a wellestablished empirical pattern in the data. For example, in the sample we use to compute the moments for calibration, the market capitalization of the acquirer is larger than that of
the target in 95% of the M&A deals.
To maintain comparability with Moeller, Schlingemann, and Stulz (2004), we keep the last 22 years of simulated data and construct three proxies of acquirer size, following their
Table 5: an indicator for small firms (those in the bottom yearly quartile of capitalization), the log value of its capitalization, and the log value of its assets.
Our model generates three predictions that are consistent with Moeller, Schlingemann, and Stulz’s (2004) empirical results. First, large acquirers have a higher average Tobin’s Q
(1.14) than small acquirers (1.09), as the latter are characterized by a low level of produc-
Second, a regression of acquirer returns on the proxies of firm size, the relative size of the target and the Tobin’s Q of the acquirer shows that lower returns are associated with
large acquirers across all three proxies (Columns 1, 2, and 3 in Table 7). This is because,
for higher productivity levels, the potential productivity improvements, determined in the model by the function ζ
M
, are smaller (see discussion in subsection 3.3).
Third, the model yields a positive relation between relative size and acquirer returns when
we focus on the sample of small acquirers (Column 4 in Table 7). However, this relation
becomes negative once we run the regression on the sample of large acquirers (Column 5).
In our model, firm size is highly positively correlated with productivity z , and the synergy
surface is hump-shaped (Figure 13a). When two small firms match in the merger market,
the synergies are increasing in the productivity and size of the target. In contrast, in mergers between large firms, synergies are decreasing in the productivity and size of the target.
In summary, our model provides the following explanation for the three empirical findings regarding announcement returns documented by Moeller, Schlingemann, and Stulz (2004):
Productivity improvements in M&As exhibit decreasing returns to scale, that is, highly productive firms find it increasingly difficult to improve productivity through acquisitions.
To the best of our knowledge, this is the first paper to rationalize these empirical findings in a framework with optimal dynamic firm decisions, without using arguments that rely
29
See Appendix B for a description of the data construction. We use the market capitalization of the
target and the acquirer four weeks prior to the announcement of the bid, as reported by SDC Platinum.
30
The difference in average Tobin’s Q between the two groups is statistically significant at the 1% level.
33
on managerial hubris (Roll, 1996), empire building (Jensen, 1986), or investor misvaluation
(Dong, Hirshleifer, Richardson, and Teoh, 2006).
Finally, we examine the potential of our model to explain instances of negative abnormal returns for acquirers. Moeller, Schlingemann, and Stulz (2005) show that during the period
1998–2001, the cumulative abnormal announcement returns for acquirers were on average positive, but at the same time the aggregate dollar gains across all acquirers were very negative. This discrepancy is due to a small number of megamergers that produced what
Moeller, Schlingemann, and Stulz (2005) call “wealth destruction on a massive scale.” Our model can generate negative announcement returns for acquirers, in case realized synergies
fall short of market expectations (the numerator of Equation 12 is negative). The synergy
plot in Figure 13a shows that negative returns, which correspond to the red dots on the
synergy surface, are realized for the larger merger deals. These negative returns are small, as we abstract from agency costs and behavioral motives. However, if very large firms were to merge, for example because of empire-building motives of the managers, the prediction of our model would indeed be a massive wealth destruction for the acquirer’s shareholders
5.
Conclusion
What are the effects of merger activity on industry dynamics? To answer this question, we develop an infinite-horizon model of a competitive industry populated by firms with heterogeneous productivities that make entry, exit, and merger decisions. We show that better merger opportunities induce higher entry rates and lower exit rates, and poorly performing firms become less susceptible to recessions, because the option to merge represents for them a valid alternative to exit.
We find that the correlation between M&A activity and aggregate demand shocks depends on the sources of merger synergies. Improvements in marginal productivity represent a procyclical motive for mergers, while savings in fixed production costs constitute a countercyclical one. The relative importance of these effects varies depending on the magnitude
31 In the capital-reallocation literature, Yang (2008) studies a model in which the returns for acquirers of used capital decrease with the acquirer’s size. Her result depends on the assumption that the market for used capital faces fixed adjustment costs, whereas the market for new capital faces no such costs. Instead, our result is driven by the concave relation of the merged firms’ productivity to the productivities of merging firms. In comparison to Yang (2008), our model provides new empirical predictions regarding the relation between acquirer returns and the relative size of the target.
34
of merger-implementation costs.
The model generates a rich set of new results regarding the effects of merger activity on the cross-section of firms in the industry. The presence of a merger market increases the mean and variance of the cross-sectional distribution of firm-level productivities. Moreover, while entry and exit induce the mean of the distribution of idiosyncratic productivities to be countercyclical, we show that mergers can reverse this effect and generate procyclicality.
Finally, our model provides an explanation for the negative empirical relation between acquirer returns and firm size documented in Moeller, Schlingemann, and Stulz (2004), even in the absence of agency or behavioral motives for mergers, and it is consistent with a negative association between the contemporaneous merger and exit rates, as observed in the data.
Our paper indicates a number of potential avenues for future research.
Particularly promising is the investigation of optimal financing policies when firms can be bidders or targets in a market for corporate control. While prior work on the topic employs real-options models that consider acquisitions as one-time irreversible investments (e.g., Morellec and
Zhdanov, 2008), our model allows for repeated mergers over time. Therefore, our framework is a natural starting point to study how firms finance strategic growth through mergers and acquisitions, and how capital-structure decisions affect their future opportunities in the merger market.
35
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Appendix
A.
Proof of Proposition 1
The next period industry output is q ( x ) = N
0
Z
Z 2 z
0
κ ( x, z )
α f
Z
( z
0 | z ) ˜ ( z | x ) d z
0 d z.
(33)
The output price is p ( s
0
, x ) = s
0
[ q ( x )]
−
η
1
, which, using Equation 14 and Equation 33 and
rearranging, yields p ( s
0
, x ) = s
0
[ E ( p ( s
0
, x ) | s )]
−
α
(1 − α ) η
"
N
0
α r + δ
α
1 − α
Z
Z
E ( z
0
| z )
1
1 − α d e
( z | x )
#
−
1
η
.
(34)
Taking expectations on both sides of Equation 34 conditional on
s and rearranging gives
Equation 16. Finally, we obtain Equation 15 by substituting Equation 16 in Equation 34
and collecting terms.
B.
Data
This appendix describes the sources of data and the construction of the empirical variables used in the paper.
To compute the merger and exit rates, we use monthly data from CRSP. Our sample consists of all observations between 1926 and 2010 for listed companies (share codes of 10 or 11). For each year in the sample, we compute the number of firms active in January, the fraction of these firms that are delisted during the year because of a merger (delisting codes
200 to 299), and the fraction of firms that are delisted because of liquidation (codes 400 to 490), bankruptcy (code 574), or other reasons likely to be related to poor performance
(codes 500 and 535 to 590, excluding code 573, which denotes firms that go private). These measures of merger and exit activity are similar to those in previous studies (see, for example,
Andrade, Mitchell, and Stafford, 2001, and Alimov and Mikkelson, 2012). The moments for
the merger and exit rates reported in Table 2 refer to the 1981–2010 period, in order to
be consistent with the moments computed with other data sources. Business cycle dates in
Figure 1 are from the National Bureau of Economic Research (www.nber.org).
40
For the regression analysis in Table 6, we use the Fama and French 48-industry clas-
sifications available on Ken French’s website at http://mba.tuck.dartmouth.edu/pages/ faculty/ken.french
, and the sample period is 1981–2010. We exclude utilities (code 31) and financial companies (codes 44 and higher). The variable Profitability is measured as the average ratio of operating income before depreciation (Compustat item oibdp ) to total assets
(Compustat item at ) for firms the same industry and year.
We collect financial information on takeover targets and acquirers, and details of M&A deals, from Thomson Reuters’ Securities Data Company (SDC) Platinum. We start by downloading all completed transactions involving US companies between 1981 and 2010 that are categorized as mergers (deal form M) or acquisitions of majority interest (AM).
We exclude deals for which: the bidder holds more than 50% of the target’s shares at the announcement date of the bid; the bidder is seeking to acquire less than 50% of the target shares; there is no information on the percentage of shares involved in the transaction; either the target or the acquirer is a regulated utility (SIC codes 4900 to 4949), a financial institution
(SIC codes 6000 to 6799), or a quasi-public firm (SIC codes greater than 9000); the acquirer is a foreign company; or the identity of the acquirer is not disclosed or attributable to a specific entity (e.g. “investor group”).
We match the observations in SDC Platinum with the CRSP/Compustat merged (CMM) database. To do this, we collect the list of CUSIP codes for targets and acquirers in SDC and find the corresponding list of PERMCO codes using the CRSP translate tool. From
CCM, we collect information on the following variables: Tobin’s Q, defined as the sum of market value of equity (stock price at fiscal-year end ( prccf ) multiplied by number of common shares outstanding ( cshpri ), debt in current liabilities ( dlc ), long-term debt ( dltt ) and preferred stock ( pstkl ) minus deferred taxes ( txditc ), divided by total assets ( at ); net sales ( sale ); and profitability (operating income before depreciation ( oibp ) divided by total assets). Variables are deflated to constant 2005 dollars using the GDP deflator provided by the Bureau of Economic Analysis (www.bea.gov, NIPA Table 1.1.9).
Following Rhodes-Kropf and Robinson (2008), our measure of the difference (in absolute value) between target and acquirer log(Tobin’s Q) is scaled by the standard deviation of log(Tobin’s Q) in the acquirer’s industry in the year of the merger transaction. To reduce the effect of outliers, we winsorize the top and bottom 1% of observations. Our final sample of merged SDC/CCM observations with non-missing values for target and acquirer Tobin’s
Q consists of 2,751 completed merger deals.
41
Finally, to compute the autocorrelation and standard deviation of aggregate earnings
growth in Table 2, we use the aggregate net operating surplus for non-financial domestic
corporate business provided by the Bureau of Economic Analysis (NIPA Table 1.1.14, line
8). The time-series used is at an annual frequency for the period 1972–2012, and growth is defined as the log-difference between two consecutive years.
C.
Details of the Computation Procedure
From equations 3 and 11, the continuation value of the firm is the solution to the following
fixed point problem:
U ( x, z ) = T
U
( x, z ) , (35) where
1
T
U
( x, z ) = − κ ( x, z )+
1 + r
Z
Z
Z
X
V ( x
0
, z
0
, κ ( x, z ))+ W ( e
∗
( x
0
, z
0
) , x
0
, z
0
) d F
X
( x
0
| x ) d F
Z
( z
0
| z ) .
(36)
Let Φ( x, z ) be a matrix of polynomials of states such that U ( x , z ) = Φ( x , z ) β for a set of nodes ( x , z ) = ( x
1
, z
1
, ..., x
I
, z
I
). A value function iteration method would seek a solution to min
β d (Φ( x , z ) β, b U
( x , z ; β )) where d ( ., .
) is a distance metric, and b U
( x , z ; β ) is a numerical approximation of T
U
( x , z ). The Parameterized Expectations Algorithm (PEA), instead, minimizes a distance d (Φ( x
T
( β ) , z ) β, b U
( x
T
( β ) , z ; β )), where x
T
( β ) = ( x
1
( β ) , ..x
t
( β ) ., x
T
( β ))) are aggregate states generated by simulation for T periods, in which firms assume that
U ( x, z ) = Φ( x, z ) β . Therefore, the PEA seeks a value of the collocation parameters β such that the distance between the value of the polynomial approximating U ( x, z ) and the nu-
merical approximation of the right hand side of the Bellman equation in 36 is minimal on
a set of aggregate states consistent with β . This can be implemented numerically using the iteration
β n +1
= (1 − w ) β n
+ wB n
, (37) where
B n
= min
β d (Φ( x
T
( β n
) , z ) β, b U
( x
T
( β n
) , z ; β n
)) , (38) until the sequence β n
w ∈ (0 , 1] is a dampening parameter useful for numerical stability purposes. Intuitively, given a value of β n
, firms make merger, entry and exit decisions on the basis of U ( x , z ) = Φ( x , z ) β n
. This allows us to simulate the aggregate
42
states x
T
( β n
) and compute the continuation value on the realized states b U
( x
T
( β n
) , z ; β n
).
Then, the algorithm selects the parameter vector that most closely describes this continuation values in terms of the polynomial function Φ( .
) evaluated at the realized states x
T
( β n
).
The steps of this numerical procedure are detailed in the following three subsections. All integrations are performed by Chebyshev quadrature methods.
C.1. Algorithm initiation
We specify log( z ) to lie on a grid logz
G consisting of 40 Chebyshev nodes spanning the interval √ 5 σ z
1 − ρ
2 z
, √ 5 σ z
1 − ρ
2 z
. The polynomial matrix Φ ( x , z ) includes: the tensor product of third-degree Chebyshev polynomials for the states log( s ), log( N ) and log( z ); the linear terms for states m , log( v ) and all their pairwise correlations with log( s ), log( N ) and log( z ).
We choose this anisotropic polynomial approach to minimize multi-collinearity problems in
Φ (see discussion in Judd, Maliar, Maliar, and Valero, 2013).
C.2. Iterations
For each period t = 1 , ...T
:
1. Simulate log( s t
) = ρ s log( s t
) + σ s
ε st
, where ε st
∼ N (0 , 1).
2. Find P t by linearly interpolating p ( s | x t − 1
) on s t
.
3. For each pair i , j in the grid logz
G
, compute U it
U ijt
= Φ( x
t
, ζ
M
( z i
, z j
)) β n
, and the synergies W ijt
= Φ( x t
, z i
) β n
, U jt
= Φ( x t
, z j
) β n from matching types i , j , using
,
4. Determine the distribution of matched firms f
M
. Initiate using f
M gridpoint in logz
G
:
= f t
. For each i. Compute W given f
M
ii. Compute search effort e
∗
iii. Update f
M
Iterate steps i to iii until convergence.
5. Determine the distribution of matched firms f
U
43
6. Compute the fraction of matched firms
R
Z e
∗
( x, z ) dF t
( z ).
7. Compute the fraction of matched firms that merge:
P
M
( x t
) =
Z
Z 2
1 ( W ( x t
, z
1
, z
2
) > 0) d F
M
( z
1
) d F
M
( z
2
) .
8. Compute the fraction of matched firms that do not merge and do not exit
P
N
( x t
) =
Z
Z 2
1 ( W ( x t
, z
1
, z
2
) < 0 , U ( x t
, z
1
) > 0) d F
M
( z
1
) d F
M
( z
2
) .
(39)
(40)
9. Compute the end-of-period distributions e M
( z | x t
) and e N
( z | x t
and Equation 27, respectively.
10. Determine the entry policy: i. Find the entry threshold z
E,t
∈ √ 5 σ z
1 − ρ 2 z
, √ 5 σ z
1 − ρ 2 z such that bisection and linear interpolation between gridpoints.
U ( x t
, z
E,t
) = c
E using ii. Compute the fraction of potential entrants entering:
P
E
( x t
) = 1 − G ( z
E,t
) .
(41)
iii. Compute the end-of-period distribution for new entrants using Equation 23.
11. Determine the exit policy: i. Find the exit threshold z
,t
∈ √ 5 σ z
1 − ρ 2 z
, √ 5 σ z
1 − ρ 2 z such that bisection and linear interpolation between gridpoints.
U ( x t
, z
,t
) = 0 using ii. Compute the fraction of unmatched firms that don’t exit:
P ( x t
) = 1 − F
U
( z
,t
) .
(42)
iii. Compute end-of-period distribution of unmatched firms using Equation 25.
12. Update the end-of-period distribution e t and the next start-of-period distribution F t +1
using Equation 22 and Equation 28, respectively.
44
13. Compute p t +1
( s t +1
| x t
) for each log( s t +1
) in the gridspace of log( s
14. Determine the investment policy k t +1
15. Evaluate T
U
( x t
( β ) , z ; β ). To do so: i. For each log( s t +1
) on the gridspace of log( s ) repeat steps 2 to 14 and evaluate
π t +1
, k t +2
, f
M,t +1
, f
U,t +1
, U
1 ,t +1
, U
2 ,t +1
, U
M,t +1
, and the expected gains from participating the next period takeover market, W t +1
.
ii. Evaluate T
U
( x t
( β ) , z ; β ) using numerical integration on the right hand side of
V t +1
= π t +1
+ (1 − δ ) k t +1
+ max { U
1 ,t +1
, 0 } .
(43)
C.3. Updating of grids and coefficients
To update the grid of states and the collocation coefficients, we follow the steps below: i. Discard computations on the initial 125 periods.
ii. Update the range [ m n +1
, m n +1
] for state m , so that for n = 1, m n +1
= min { m
126
, ...m
T
} , m n +2
= max { m
126
, ...m
T
} , and for n > 1, m n +1
= 0 .
9 m n
, and m n +1
= m
2
(3 −
2 exp( − n )). Therefore, asymptotically m can be at most as large as 3 times its maximum realized value in the first iteration. This step follows Maliar and Maliar (2003).
iii. Update the range [log( v ) n +1
, log( v ) n +1
] for state log( v ), so that for n = 1, log( v ) n +1 min { log( v )
126
, ...
log( v )
T
} , log( v ) n +2
= max { log( v )
126
, ..., log( v )
T
} , and for n >
=
1, log( v ) n
= 0 .
9log( v ) n
, and log( v ) n +1
= log( v )
2
(3 − 2 exp( − n )).
Therefore, asymptotically log( v ) can be at most as large as 3 times its maximum realized value in the first iteration.
iv. Update B n by regressing b U
( x t
( β n
) , z ; β n
) on Φ( x t
( β n
) , z ) for t = 126 , ..., T using regularized least squares with truncated singular value decomposition (RLS-TSVD). This method is proposed by Judd, Maliar, and Maliar (2011) to address collinearity prob-
45
lems in the columns of the data matrix
Φ( x
126
( β n
) , z )
...
.
Φ( x
T
( β n
) , z )
(44) v. Update β n +1
= wβ n
+ (1 − w ) B n
, setting the dampening parameter to w = 0 .
2.
Iterations finish when max { ( | Φ( x
T
( β n
) , z )( β n +1
− β n
)) |} is smaller than our tolerance level,
10
− 3
, where | .
| is the element-by-element absolute value operator.
46
Table 1: Parameter values . Panel A reports the set of parameters that are chosen based on standard values in the literature (cf. Hennessy and Whited, 2007, Riddick and Whited, 2009,
Yang, 2008, and Eisfeldt and Muir, 2012). Panel B displays the values of the parameters that are
calibrated to match the empirical moments in Table 2.
Panel A : Standard parameters
Symbol
α
δ
ρ z
σ r z c
M c e
λ
θ c f c
E
ρ s
σ s
η
Description
Curvature parameter of the profit function
Depreciation of capital
Persistence of idiosyncratic productivity shock
Value
0.750
0.150
0.660
Standard deviation of idiosyncratic productivity shock 0.110
Discount rate 0.040
Panel B : Calibrated parameters
Symbol Description
Fixed merger-implementation cost
Search cost parameter
Most-productive-firm synergy weight
Curvature parameter of the synergy function
Fixed cost of production
Cost of entry
Persistence of aggregate demand shock
Standard deviation of aggregate demand shock
Price elasticity of demand
Value
0 .
252
0 .
021
0 .
493
0
0
0
0
0
0 .
.
.
.
.
.
502
154
277
788
035
452
47
Table 2: Calibration targets . The table contains the empirical moments targeted in the calibration. The definition of variables and the details about sample construction are provided in
Appendix B. The merger and exit rates are computed at an annual frequency using data from the
Center for Research in Security Prices (CRSP). Information on Tobin’s Q for merged firms is from
Thomson Reuters’ SDC Platinum and the CRSP/Compustat merged database. Aggregate earnings growth is measured using the time series of aggregate net operating surplus for non-financial domestic corporate business from the Bureau of Economic Analysis. The estimates of average combined merger gains and relative total factor productivity of entrants are from Betton, Eckbo, and
Thorburn (2008) and Lee and Mukoyama (2012), respectively. The definitions of the theoretical
counterparts to the empirical moments are in Table 3.
Description of moment
Average merger rate
Standard deviation of merger rate
Autocorrelation of merger rate
Average absolute scaled difference in log Tobin’s Q of merged firms
Average combined merger gains
Average exit rate
Average relative total factor productivity of entrants
Autocorrelation of aggregate earnings growth
Standard deviation of aggregate earnings growth
Model
5.64%
0.018
3.39%
1.045
0.197
0.088
Data
4.54%
0.016
0.317
0.291
0.917
0.953
1.55% 1.79%
3.68%
0.870
0.251
0.085
48
Table 3: Definition of variables in the model.
x denotes the vector of aggregate states, z the productivity shock, k firm’s capital, N the mass of incumbents, N
E the mass of potential entrants, F ( z ) the start-of-period cross-sectional distribution of firm productivities.
V S ( x, z, k ) is
the value of the firm at the start of the period, defined by Equation 11.
W ( x, z i generated by a merger between firm i and j
W
∗
( x, z i of merging for firm i
e
∗
( x, z i
, z j
) are the synergies
) is the option value
) is the optimal search effort for firm i , defined
P
E
( x ), P ( x ), P
M
( x ), and P
N
( x
) are probabilities defined by Equations 17, 18, 19,
and 20, respectively. The remaining parameters are defined in Table 1.
Tobin’s Q
Combined merger gains for firms i and j
Exit rate
Merger rate
Entry rate
Matching rate
V S ( x,z,k ) k
W ( x,z i
,z j
) −
1
2
( e
∗
( x,z i
) W
∗
( x,z i
)+ e
∗
V S ( x,z i
,k i
)+ V S ( x,z j
( x,z j
,k j
)
) W
∗
( x,z j
))
1 − R
Z e ∗ ( x, z ) d F ( z ) (1 − P ( x )) + R
Z e ∗ ( x, z ) d F ( z ) 1 −
P
M
2
( x )
− P
N
( x )
P
M
( x )
2
P
E
( x ) N
E
N
R
Z e
∗
( x, z ) d F ( z )
49
50
51
52
53
Figure 1: Merger and exit rates in the sample of CRSP firms, 1926-2010. Shaded areas denote periods of economic recession as defined by the National Bureau of Economic Research (NBER).
The definition of the variables and their construction are described in Appendix B.
8%
6%
4%
2%
0%
8%
1930 1940 1950 1960 1970 1980 1990 2000 2010
% firms merged (CRSP) Recession
(a) Merger rate
6%
4%
2%
0%
1930 1940 1950 1960 1970 1980 1990 2000 2010
% Firms exiting (CRSP) Recession
(b) Exit rate
54
Figure 2: Timing of events within each period for incumbents and potential entrants.
x denotes the vector of aggregate states, z the idiosyncratic productivity shock, and k firm’s capital.
Incumbents
Observe beginning of period states
( x, z, k )
Choose matching probability e in merger market
Merger decision
Investment decision
Period payoff realized
Exit decision t t + 1
Potential entrants
Observe beginning of period states
( x, y )
Entry decision pay cost c
E
Investment decision t t + 1
Figure 3: Productivity of the merged firm, ζ
M
, as a function of the productivities of the two merging firms, z
1 and z
2
. In both figures, θ =
1
2
.
(a) λ = 1
55
(b) λ = 0
Figure 4: Entry and exit productivity thresholds ( ζ
E
, ζ ) as a function of the aggregate demand shock ( s
). The data is generated according to the algorithm described in Appendix C, and the
parameter values are shown in Table 1. States
m , v , and N are set at their average levels in the simulation.
1
0.95
0.9
1
E
0.85
0.8
0.75
0.7
0.85
0.9
0.95
1 1.05
Aggregate Shock s
1.1
(a) Entry threshold ( ζ
E
)
1.15
1
0.95
0.9
1
ε
0.85
0.8
0.75
0.7
0.65
0.85
0.9
0.95
1 1.05
Aggregate Shock s
1.1
(b) Exit threshold ( ζ )
1.15
1.2
1.2
56
Figure 5: Merger synergies. The two graphs show, from different angles, the merger synergies,
defined by Equation 5, as a function of the productivity of the two merging firms,
z
1 and z
2
. For the region of ( z
1
, z
2
) points where synergies are positive, the matched firms merge. The data is
generated according to the algorithm described in Appendix C, and the parameter values are shown
s , m , v , and N are set at their average levels in the simulation.
(a) Angle 1
(b) Angle 2
57
Figure 6: Merger synergies and search effort. Figure (a) shows the merger opportunity set (area inside the solid curve), and the search set (area delimited by the dashed line) as a function of the productivity of two firms, z
1 and z
2
. The intersection of the two sets, shaded in grey, corresponds to the merger set. Figure (b) shows the search effort of a firm as a function of its own productivity z i
for two cases: at the equilibrium level defined in Equation 8 (solid curve) and under the assumption
that all other firms in the industry, not only those in the equilibrium search set, exert a constant level of effort (dotted curve). We generate the simulated data according to the algorithm described
in Appendix C. The parameter values are shown in Table 1, and the values of the aggregate states
for the period shown are s = 1 .
14, m = 0 .
1, v = 0 .
02, N = 2 .
37.
1.4
1.3
Merger set
Merger opportunity set
Search set
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.5
0.6
0.7
0.8
0.9
z
1
1 1.1
1.2
1.3
(a) Merger opportunity set and search set
1.4
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0.6
Equilibrium effort
Effort if all other firms search equally
0.8
1 1.2
z i
(b) Search effort
58
1.4
1.6
Figure 7: Merger synergies and aggregate demand. The two graphs show contour plots of the
merger synergies, defined by Equation 5, as a function of the productivity of the two merging firms,
z
1 and z
2
. Figure (a) shows the case of s = 0 .
8451 (low aggregate demand), figure (b) shows the case of s = 1 .
1833 (high aggregate demand). The thicker curve represents the locus of points for which merger synergies are zero. For the region of ( z
1
, z
2
) points where synergies are positive, the
matched firms merge. The data is generated according to the algorithm described in Appendix C,
and the parameter values are shown in Table 1. States
m , v , and N are set at their average levels in the simulation.
1.6
-0.3
-0.5
-0.2
-0.3
1.4
-0.4
-0.3
-0.2
-0.1
1.2
-0.1
-0.2
0
1
0.8
-0.1
0
0
-0.1
-0.3
0.6
-0.3
0.6
-0.2
1.6
1.4
-0.4
-0.3
1.2
-0.2
1
-0.1
-0.2
0.8
1 1.2
z
1
(a) Low aggregate demand
1.4
-0.8
0
-0.1
-0.2
-0.4
-0.3
0.1
0
1.6
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.1
0.8
0.6
-0.2
-0.1
0
0.6
0.1
0.1
0
-0.2
-0.1
0.8
1 1.2
z
1
(b) High aggregate demand
-0.3
1.4
59
-0.4
1.6
Figure 8: Comparative statics analysis of merger synergies. The two graphs show contour plots of
the merger synergies, defined by Equation 5, as a function of the productivity of the two merging
firms, z
1 and z
2
. Figure (a) shows the case of λ = 0 .
7, figure (b) shows the case of λ = 0 .
2. The
remaining parameters are set at the values shown in Table 1. The thicker curve represents the locus
of points for which merger synergies are zero. For the region of ( z
1
, z
2
) points where synergies are positive, the matched firms merge. The data is generated according to the algorithm described in
s , m , v , and N are set at their average levels in the simulation.
1.6
-0.8
-0.2
-0.1
-0.2
-0.3
-0.6
1.4
0
1.2
0.1
0
-0.1
-0.2
-0.5
-0.4
-0.3
1
0.1
0.8
0
0.1
-0.1
0
0.1
0.6
-0.1
0.6
0.8
1 z
1
(a) λ = 0 .
7
1.2
1.6
1.4
-0.9
-0.8
-0.7
-0.6
-0.5
1.2
-0.4
1
-0.3
-0.2
-0.1
0.8
-0.1
0.6
-0.3
0.6
-0.2
0
-0.6
-0.5
-0.4
-0.3
-0.2
0
-0.1
0.1
0.8
0.1
0
-0.1
-0.2
-0.3
1 z
1
(b) λ = 0 .
2
1.2
-0.4
-0.2
-0.3
1.4
-0.8
1.6
-0.4
-0.5
-0.6
-0.7
-0.7
-0.6
-0.5
1.4
-0.8
-0.9
1.6
60
Figure 9: Average merger and exit rates as a function of the fixed cost of production, c f
. The
other parameter values are shown in Table 1. The definition of the merger and exit rates according
to the model is provided in Table 3. The data is generated according to the algorithm described in
0.25
0.2
0.15
0.1
0.05
0
0.1
Exit rate
Merger rate
0.2
0.3
0.4
0.5
Fixed cost of production, c f
0.6
0.7
61
Figure 10: Impulse responses. This figure shows the impulse responses to a two-standard-deviation innovation in aggregate demand, s , for two simulated economies: with mergers (solid line in the graphs), and without mergers (dotted line in the graphs). The parameters are set to the values
shown in Table 1 for the economy with mergers. For the economy without mergers, the merger-
implementation cost, c
M
, is set to infinity. Impulse responses are computed following the procedure
described in subsection 3.3.4.
6
5
4
3
8
7
2
1
0
0 5 10
Time
15 20
(a) Aggregate shock s
8
6
4
2
0
−2
0 5
With Mergers
Without Mergers
10
Time
(b) Price
15 20
4
2
0
−2
−4
−6
With Mergers
Without Mergers
−8
0 5 10
Time
15 20
(d) Average log idiosyncratic productivity shock z
0.6
0.4
0.2
0
−0.2
−0.4
With Mergers
Without Mergers
−0.6
−0.8
0 5 10
Time
15 20
(e) St. dev. of log idiosync.
productivity shock z
0.85
0.84
0.83
0.82
0.81
0.8
0.79
0.78
0.77
0
With Mergers
Without Mergers
5 10
Time
15
(g) Exit threshold
20
20
15
10
5
0
With Mergers
Without Mergers
−5
0 5 10
Time
15
(c) Number of firms
20
1.06
0
With Mergers
Without Mergers
With Mergers
Without Mergers
25
20
15
10
5
0
−5
0 5 10
Time
15
(h) Merger rate
20
(f ) Entry threshold
62
Figure 11: Determinants of merger cyclicality. This figure shows the average merger rate and the correlation of the merger rate with the log aggregate demand shock, log( s ), as a function of the curvature parameter of the synergy function θ (two graphs in the left column) and the fixed cost of production c f
(right column). In panel (a), the fixed merger-implementation cost, c
M
, is set to zero. In panel (b), c
M equals its calibrated value, 0.252. The other parameter values are shown in
Table 1. The definition of the merger rate according to the model is provided in Table 3. The data
is generated according to the algorithm described in Appendix C.
0.2
0.19
0.18
0.17
0.16
0.15
0.14
0.3
0.8
Merger rate
Correlation of merger rate and log(s)
0.6
0.4
0.2
0
-0.2
-0.4
0.35
0.4
0.45
0.5
Curvature parameter of the synergy function 3
0.4
0.3
0.2
0.1
0
0.1
(a) c
M
= 0
Merger rate
Correlation of merger rate and log(s)
0.1
0.05
0.15
0.2
0.25
Fixed cost of production c f
0.3
0
-0.05
0.35
-0.1
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
0
0.3
1.1
Merger rate
Correlation of merger rate and log(s)
1
0.9
0.8
0.7
0.35
0.4
0.45
0.5
Curvature parameter of the synergy function 3
0.3
0.25
0.2
0.15
0.1
1.1
Merger rate
Correlation of merger rate and log(s)
1
0.9
0.8
0.7
0.6
0.05
0.6
0.5
0
0.14
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
Fixed cost of production c f
0.5
(b) c
M
= 0 .
252
63
Figure 12: Matching rate and fraction of matched firms that merge in the simulation. Figure
(a) plots the matching rate in each simulated period for three economies with different values of θ .
Figure (b) shows the fraction of matched firms that merge. The dots, crosses, and circles correspond to simulated economies with θ = 0 .
3, θ = 0 .
4, and θ = 0 .
54, respectively. The other parameter
values are shown in Table 1. The definition of the matching rate according to the model is provided
in Table 3, and the fraction of matched firms that merge is defined in Equation 19. The data is
generated according to the algorithm described in Appendix C.
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-0.2
-0.15
3 = 0.3
3 = 0.4
3 = 0.54
-0.1
-0.05
0
Log aggregate shock s
0.05
(a) Matching rate
0.1
0.15
1
0.995
0.99
0.985
0.98
0.975
3 = 0.3
3 = 0.4
3 = 0.54
0.97
-0.2
-0.15
-0.1
-0.05
0
Log aggregate shock s
0.05
0.1
(b) Fraction of matched firms that merge
0.15
64
Figure 13: Synergies and productivity pairs for realized mergers. Figure (a) shows the
merger synergies, defined by Equation 5, as a function of the productivity of the two merg-
ing firms, z
1 and z
2
, in a given simulation period. The yellow (red) dots correspond to realized mergers in the simulation that are characterized by positive (negative) announcement returns for the acquirer. Point A represents a hypothetical merger scenario between highly productive firms. Figure (b) shows the productivity pairs ( z
1
, z
2
) of the merging firms at the given simulation period. We generate the simulated data according to the
algorithm described in Appendix C and subsection 4.3. The parameter values are shown in
Table 1, and the values of the aggregate states for the period shown are
s = 1 .
14, m = 0 .
1, v = 0 .
02, N = 2 .
37.
(a) Realized merger synergies
1.15
1.1
1.05
1
0.95
0.9
0.85
0.8
0.75
0.7
45°
0.65
0.65
0.7
0.75
0.8
0.85
0.9
z
1
0.95
1
(b) Matched merger pairs
1.05
1.1
1.15
65
