Jing YU And Bin XU

er ch s ha pre e ap has rspe e of oundings. The synergy multiplication coefficient is introduced into the evaluation o study s the e ection, evaluation approach, contemporary option-based asset evaluation the perspective of real optionswith reference to the analyses proposed Economic Modelling 28 (2011) 1587–1594 Contents lists available at ScienceDirect Economic M e ls approach, and the integrationof the above-mentioned approaches. It is obvious that the pricing of the target enterprise of M&A is a critically hot topic that has remained unsolved due to the uncertain surround- ings of M&A. The existing evaluation approaches can be classified in three categories: (1) the traditional NPV-based approaches, such as the Rappaport model (Rappaport, 1998), Weston formula method (Weston and Thomas, 1992),and others; (2) models based on the perspective of real options (Black and Scholes, 1973; Lambrecht, 2003; McDonald and Siegel, 1984; Merton, 1977; Smith et al., 1995); and (3) models based on the game perspective (Burkart, 1995; Farrell and by Dixit (Dixit and Pindyck, 1994). We then investigate how to price the target enterprise by using the famous Rubinstein offer–counter- offer theorem, which is improved for adaptation to the assumed stochastic surroundings of M&A. The price formula of target enterprise of M&A is then developed and some corresponding simulations show the interpretation power of price formula to reality. The rest of this paper is organized as follows. Section 2 discusses how to evaluate the target enterprise of M&A from the perspective of real options. How to price the target enterprise of M&A is investigated in detail in Section 3. Some simulations are drawn to show how well Shapiro, 1990; Grossman and Hart, 1980; Hir Though extant literatures have investigated t enterprises from the integration perspective ⁎ Corresponding author. Tel.: +86 10 62288613. E-mail addresses: yujing3721@163.com (J. Yu), zhuo (B. Xu). 0264-9993/$ – see front matter © 2011 Elsevier B.V. Al doi:10.1016/j.econmod.2011.02.034 and so on. In addition, eveloped to reveal the value (NPV)-based asset solution can be found in two critical factors: how to evaluate the target enterprise of M&A and how to price the target enterprise of M&A. This paper discusses the evaluation of the target enterprise of M&A from many innovative approaches have been d essence of M&A such as classical net present 1. Introduction There are many extant literatures t (M&A), in which the research include factor analyses, target enterprise sel surroundings. The price formula is further discussed on condition that the operating cost is more than or less than profit flow, which is assumed to follow geometric Brownian motion process. The numeric simulations show that the proposed formulas in this paper can perfectly well reflect the realistic practice of M&A. © 2011 Elsevier B.V. All rights reserved. merger and acquisition valuation model, impact approaches, simply using the famous Rubinstein offer–counteroffer theorem to attain the price of the target enterprise has caused imperfectly dynamic analyses process and the corresponding unprac- tical price result in the distortion of M&A reality. The key to the shleifer and Ping, 1990). he pricing of M&A target of the above-mentioned the proposed pr Section 5 conclu the future are pr 2. Evaluating th Generally spe composed of tw hongxubin@hotmail.com l rights reserved. in theorem, which is improved to fit for the stochastic Game theory Merger and acquisition model to reflect the synergy management process of M&A, and the equilibrium price formula is proposed by applying the famous offer–counteroffer Rubinste Real option perspective to price the target enterpris theory under stochastic surr The game analyses to price the target ent the perspective of real options under sto Jing Yu a,c, Bin Xu b,⁎ a School of International Auditing,Nanjing Audit University,211815,Nanjing,China b School of Accountancy,Central University of Finance and Economics,100081,Beijing,China c Research Center of Fictitious Economy and Data Science,CAS,100190,Beijing,China a b s t r a c ta r t i c l e i n f o Article history: Accepted 10 February 2011 Keywords: Decision analysis Asset pricing Although several approache (M&A) such as classical net pricing processes in all thes target enterprise of M&A perspective, real options pe j ourna l homepage: www. prise of merger and acquisition based on astic surroundings ve been developed to evaluate the target enterprise of merger and acquisition sent value (NPV) evaluation model and real options techniques, the logic of proaches is still faulty. The classical approach of NPV perspective to price the been replaced by integration of contemporary perspectives such as NPV ctive, game perspective, and so on. In this paper, the dynamic analyses model M&A is developed from the perspective of real options integrated with game odelling ev ie r.com/ locate /ecmod ice formulas match the reality of M&A in Section 4. des the paper and some possible research direction in oposed. e target enterprise of M&A aking, in this paper, the value of target enterprise is o parts from the perspective of real options: the delling 28 (2011) 1587–1594 intrinsic value and its corresponding implied value. The former can be calculated based on the future forecasted cash flow in its sustainable operating life cycle, which covers the whole operating process of target enterprise after M&A, while the latter can be measured by its corresponding real option values. Hereafter, the value of the target enterprise is equal to the summation of the above two mentioned parts without considering the synergy effect of M&A, which is introduced into the above summation model for a time. The following parts in this section demonstrate how to develop in detail the evaluation formula according to the above-proposed idea. 2.1. Evaluating the intrinsic value of target enterprise of M&A For convenience, it is assumed that there are only two players in the M&A process: the merger or acquiring enterprise and the merged or acquired enterprise. They can be denoted as A and T respectively. Dixit (1989) (Dixit and Pindyck, 1994) proposed an innovative idea to measure the intrinsic value of enterprise and its corresponding implied value of real options, which allows us to measure the value of the above two mentioned parts. For convenience, the main content of deducing process in literature (Dixit, 1989; Dixit and Pindyck, 1994) is compactly introduced in this section. Assume P to denote the future profit flow of the target enterprise, which follows geometric Brownian motion process (Cox and Miller, 1965; Dixit, 1991; Dixit and Pindyck, 1994; Hull, 1989; Merton, 1971) dP = αPdt + σPdz ð1Þ where, the profit flow P increases with a velocity of α, which also denotes the drifting coefficient, σ denotes fluctuation rate, and dz denotes increment of Weiner stochastic process dz = ffiffiffiffiffi dt p . It is apparent that if the foreseeable future profit flow P is discounted by μ, the intrinsic value of T can be formulated as V = P= μ−αð Þ. We can then conclude that the intrinsic value V also follows geometric Brownian motion process with the same drifting coefficient α and the same fluctuation rate σ. Based on Capital Asset Pricing Model (CAPM) formulation, we have the following formula: μ = r + ϕσρpm ð2Þ where δ=μ−α, the symbol r denotes the discount rate without risk, ρpm denotes the correlation coefficient between the market portfolio and the asset of tracking profit flow P, and ϕ denotes risk rate. As V = P= μ−αð Þ is forever positive, the inequality μNαmust be assumed. Without the loss of generality, we assume that the intrinsic value V of T is only the function of profit flow P with its operating cost C. Thus, the increment of profit flow P is exactly formulated as π(P)=max(P−C,0). Based on the ITO lemma together with the feature of risk free portfolio (Dixit, 1991; 1992; Dixit and Pindyck, 1994; Pindyck, 1991), the following equation can be deduced: 1 2 σ2P2V ″ Pð Þ + r−δð ÞV′ Pð Þ−rV Pð Þ + π Pð Þ = 0: ð3Þ Thus, the solution of Eq. (3) can be obtained on condition that PbC and PNC, respectively. Details on the solving process can be seen from the literature (Dixit and Pindyck, 1994). If PbC, the solution can be formulated as: V Pð Þ = K1Pβ1 + K2Pβ2 ; ð4Þ and if PNC, the solution can also be formulated as V Pð Þ = B1Pβ1 + B2Pβ2 + P−C ð5Þ 1588 J. Yu, B. Xu / Economic Mo δ r where, β1 = 1 2 − r−δ σ2 + ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r−δ σ2 −1 2 � �2 + 2r σ2 s N 1 ð6:1Þ β2 = 1 2 − r−δ σ2 − ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r−δ σ2 −1 2 � �2 + 2r σ2 s b 0: ð6:2Þ The coefficients K1,K2 and B1,B2 are undetermined, which can be solved based on boundary conditions, and their general solutions can be expressed as V Pð Þ = K1P β1 ; P b C B2P β2 + P δ −C r ; P N C 8>< >: ð7Þ where, K1 = C1−β1 β1−β2 β2 r −β2−1 δ � � B2 = C1−β2 β1−β2 β1 r −β1−1 δ � � ð8Þ which is clearly K1N0, B2N0 and rNβ1(r−δ), rNβ2(r−δ). 2.2. Evaluating the implied value of target enterprise of M&A Measuring the optionvalueof target enterprise is investigated in this subsection. We can directly reference the formulation from the literatures (Dixit, 1991; 1992; Merton, 1973; 1977; 1971; Pindyck, 1991). F Pð Þ = A1Pβ1 + A2Pβ2 ð9Þ From literatures (Dixit, 1989; 1991; 1992; Dixit and Pindyck, 1994), both values match the condition and the smooth paste condition at the optimum critical point P*, which can be expressed as follows: F P�ð Þ = V P�ð Þ−I F′ P�ð Þ = V′ P�ð Þ ð10Þ where A1,A2 are undetermined coefficients, I denotes initial investment cost. Substituting Eqs. (7) and (9) into Eq. (10), we have following: A1 P �ð Þβ1 = B2 P�ð Þβ2 + P� δ −C r −I β1A1 P �ð Þβ1−1 = β2B2 P�ð Þβ2−1 + 1 δ : ð11Þ It is apparent that the coefficient B2 can be known from Eq. (8); therefore, the coefficients A1 and P* can be solved. Eq. (11) at the critical point P* can be transferred to β1−β2ð ÞB2 P� � �β2 + β1−1ð ÞP� δ − β1 C r + I � � = 0: ð12Þ Unfortunately, it is difficult to acquire the analytic expression of solutions P*,A1 except for some special coefficients β1,β2. 1589J. Yu, B. Xu / Economic Modelling 28 (2011) 1587–1594 2.3. Measuring the synergy effect of M&A From Subsections 2.1 and 2.2, the value G(P) of target enterprise can be formulated as G Pð Þ = V Pð Þ + F Pð Þ = ( F Pð Þ + K1Pβ1 ; P b C F Pð Þ + B2Pβ2 + P δ −C r ; P N C = A1P β1 + K1P β1 ; P b C A1P β1 + B2P β2 + P δ −C r ; P N C : 8>< >: ð13Þ It is obviously improper not to consider the synergy effect of M&A in Eq. (13), which is improved to reflect the synergy effect. In fact, the synergy coefficient λ(t) is a function of time t (tN0), which covers the sustainable operating life cycle of the target enterprise. Generally speaking, the positive synergy effect of M&A can be visually expressed as 1+1N2, which means λ(t)N1; whereas the negative synergy effect can also be visually expressed as 1+1b2, which means 0bλ(t)b1. It is obvious that the phenomena λ(t)≤0 would not occur in real practice. Based on the practice of M&A, the synergy multiplication λ(t) is less than 1 at the beginning of M&A because of the management entrenchment existing in the process of M&A, and the positive synergy effect λ(t)N1 occurs after M&A due to entrenchment lessening with the progress of M&A. The function λ(t)=λ ln(t+1) (λN0, tN0) can obviously reflect the feature of the process of M&A owing to the function chart trend changing from small to big as time goes on. Hence, it is reasonable to be selected as the synergy effective function. Thus, the formulation (13) can be transformed to the following: G Pð Þ = λ tð Þ V Pð Þ + F Pð Þð Þ = λ ln t + 1ð Þ V Pð Þ + F Pð Þð Þ: ð14Þ Many types of functions can be selected as the synergy effect function as long as they can adequately approximate the synergy effective process of M&A according to the real practice of M&A. While different functions can be selected as synergy effect functions, the expressions of the equilibrium price proposed next may be different. The deduced idea must be the same as our deduced analysis. In other words, the deduced process in this paper does not lose the generality of guiding the practitioners to price the target enterprise in M&A. 3. Pricing the target enterprise of M&A 3.1. Preliminary theorems The process of pricing the target enterprise of M&A can be divided into two main phases: evaluation and pricing of the target enterprise. The first phase has been accomplished in Section 2; therefore, how to price the target enterprise is the main task in this section. It is apparent that merger/acquisition enterprise and its opposite merged/ acquired enterprise can be taken as two players in one game, and the two players play an offer–counteroffer game to attain the equilibrium price of the target enterprise of M&A. The classical Rubinstein theorem can be directly applied to attain the equilibrium price under certain surroundings. Unfortunately, the stochastic surround- ings localized here cannot satisfy the preliminary condition of Rubinstein equilibrium theorem because the game target value is of uniform distribution, which is crucial to prove the Rubinstein theorem in the following proof. Thus, it is necessary to improve the Rubinstein theorem to fit the stochastic surroundings. In order to smoothly understand the proposed improved Rubin- stein theorem, the detailed proof process of classical Rubinstein theorem, which is strictly different from that of Shaked and Sutton (1984) in this section, is again proven. The following several corollaries are then deduced to get the equilibrium price of target enterprise M&A under different stochastic surroundings. Theorem 1. (Rubinstein, 1982): Under the unlimited times of alternat- ing offer–counteroffer game, the only NASH perfect equilibrium solution of sub-game can be formulated as x� = 1−θ2 1−θ1θ2 If θ1 = θ1 = θ; x � = 1 1 + θ where x* denotes the share of first mover player, 1−x* denotes that of last mover player, the symbols θ1,θ2 denote their corresponding discount rates respectively. Theorem 1 tells us how to carve up the target value by deciding their corresponding shares. In this paper, the following discloses an inarguable fact implied in this theorem, which has long been ignored intentionally or unintentionally. The fact is that the target value is of uniform distribution in the closed interval [0, 1]. To illustrate the importance of the implied fact, this study again proves the theorem to illustrate the rationality of the above-hypothesized implied fact. The improved proof process is based on that of Shaked and Sutton (1984). Proof. Assume that the target value ξ is of uniform distribution in the closed interval [0, 1], which density function can be expressed as f xð Þ = 1; x∈ 0;1½ �0; x∉ 0;1½ � : � There are two players designated as Player 1 (first-move player) and Player 2 (last-move player). Their strategy spaces can be represented as S1={(0,M)|0≤M≤1} and S2={(M, 1)|0≤M≤1}, where M denotes the maximum attainable share for Player 1. Their payoff functions can be represented as P{x≤M} and 1−P{x≤M}, which are equal to P{x≥1−M} without regard to their corresponding discount rates θ1,θ2. Otherwise, their payoffs must be multiplied by θ1,θ2 or their power, and so on, according to real discount situations. Because the alternating offer–counteroffer game is a two-player game with unlimited game time T=∞, which means there is no final phase, themathematical induction cannot be directly applied to prove the theorem. In fact any sub-game beginning from first-move Player 1 is equivalent to the game beginning from the first phase t=1; therefore, the backward induction of alternating game with limited game times can be directly employed to attain the perfect equilibrium solution of sub-game. If thefirstmover (Player1) canacquire themaximumvalueMat Phase t≥3, themaximumshare (payoff) isP{x≤M}. To thefirstmover, the share (payoff) P{x≤M} at Phase t is equivalent to the share (payoff) θ1P{x≤M} at Phase t−1. The last mover (Player 2) knows that any offering share (payoff) beingmore than θ1P{x≤M} at phase t−1 can be accepted by the first mover (Player 1). The share (payoff) of last-move Player 2 cannot be less than 1−θ1P{x≤M}, which is equivalent to θ2(1−θ1P{x≤M}) at phase t−2. Player 1 knows that any offering share (payoff) x being less than 1−θ2(1−θ1P{x≤M}) can be accepted by Player 2, so the Player 2's payoff can attain θ2(1−θ1P{x≤M}). Since the sub-game beginning from t−2 is equivalent to the sub-game beginning from t, the share (payoff) of Player 1 at t must be equal to that of Player 1 at t−2. Thus, we have following equation: P x≤Mf g = 1−θ2 1−θ1P x≤Mf gð Þ which makes it evident that P x≤Mf g = 1−θ21−θ1θ2. Obviously, P{x≤M}=M due to the uniform distribution of ξ in the closed interval [0, 1], so we can have M = 1−θ21−θ1θ2. The above proof illustrates the maximum share offering of Player 1, as well as his minimum share. Some corollaries are be deduced below. 1590 J. Yu, B. Xu / Economic Modelling 28 (2011) 1587–1594 Corollary 1. If the value ξ of target enterprise of M&A is of uniform distribution in the closed interval [a,b](bNa≥0), the game equilibrium price can be formulated as M = a + 1−θ21−θ1θ2 b−að Þ. There are two players designated as Player 1 (first-move player) and Player 2 (last-move player). Their strategy spaces can be represented as S1={(a,M)|a≤M≤b} and S2={(M,b)|a≤M≤b}, where M denotes the maximum attainable share for Player 1. Their payoff functions can be represented as P{x≤M} and 1−P{x≤M}, which are equal to P{x≥1−M} without regard to their corresponding discount rates θ1,θ2. Otherwise, their payoffs must be multiplied by θ1,θ2 or their power, and so on, according to real discount situations. Based on Theorem 1, it is evident that P x≤Mf g = 1−θ21−θ1θ2; therefore, ∫ M a 1 b−a dx = 1−θ2 1−θ1θ2 and M−a = 1−θ2 1−θ1θ2 b−að Þ, and we have M ¼ a + 1−θ2 1−θ1θ2 b−að Þ. Corollary 2. If the value ξ of target enterprise of M&A, its density function is f(x), which is of standard normal distribution N(0,1). Thus, the equilibrium price Mmust satisfy ∫ M −∞ f xð Þdx = 1−θ2 1−θ1θ2 , which means Φ Mð Þ = 1−θ2 1−θ1θ2 and M = Φ−1 1−θ2 1−θ1θ2 � � . There are two players in M&A; one is merger enterprise and the other is merged enterprise, which are represented as Player 1 (first- move player) and Player 2 (last-move player), respectively, and their strategy spaces can be represented as S1={(−∞,M)|M∈(−∞,∞)} and S2={(M,+∞)|M∈(−∞,∞)} respectively, where M denotes the maximum attainable share for Player 1. Their payoff functions can be represented as P{x≤M} and 1−P{x≤M}, respectively, which is equal to P{x≥1−M}, subject to x∈(−∞,∞) without regard to their corresponding discount rates θ1,θ2; otherwise their payoffs must be multiplied by θ1,θ2 or their power, and so on, according to real discount situations. Its proof process, which is similar to that of Corollary 1, can be omitted; thus, we have Corollary 3. Corollary 3. If the value variable ξ of target enterprise is distributed as geometric Brownian motion with drifting feature, dx=αxdt+σxdz, the equilibrium price M of target enterprise of M&A can be formulated as M = M0 exp α− 1 2 σ2 � � t + σ ffiffi t p Φ−1 1−θ2 1−θ1θ2 � �� : where x(0)=M0, the coefficients α and σ denote drifting rate and variance rate, respectively. Naturally, the expressions of the players, their strategy spaces and payoff functions are the same with that of Corollary 2. Apparently, if its density function f(x)(x∈R) is of normal distribution N(μ,σ2), there will be ∫ M −∞ f xð Þdx = 1−θ2 1−θ1θ2 , which can be normalized as ∫ M−μ σ −∞ f yð Þdy = 1−θ2 1−θ1θ2 . Order y = x−μ σ , we can have Φ x−μ σ � = 1−θ2 1−θ1θ2 ; thus, the above formulation can be simply transformed to x = μ + σΦ−1 1−θ2 1−θ1θ2 � � .