Non-hierarchical signalling: two-stage financing game

Transcription

1 MPRA Munich Personal RePEc Archive Non-hierarchical signalling: two-stage financing game Anton Miglo and Nikolay Zenkevich 2005 Online at MPRA Paper No. 1264, posted 28. Decemer 2006

2 Non-hierarchical signalling: two-stage nancing game Anton Miglo y, and Nikolay Zenkevich z y University of Guelph, Department of Economics, Guelph, Ontario, Canada, N1G 2W1, tel. (519) , ext , amiglo@uoguelph.ca. z Saint-Petersurg State University, Faculty of Applied Mathematics and Control Processes, Universitetskii prospekt 35, Petergof, Saint-Petersurg, Russia , tel. +7 (812) , Zenkevich@isdgrus.ru Astract The literature analyzing games where some players have private information aout their "types" is usually ased on the duality of "good" and "ad" types (GB approach), where "good" type denotes the type with etter quality. In contrast, this paper analyzes a signalling game without types hierarchy. Di erent types have the same average qualities ut di erent pro les of quality over time which are their private information. We apply this idea to analyze a nancing-investment game where rms insiders have private information aout the rm s pro t pro le over time. If transporting cash etween period is costless equilirium is pooling with up-front equity nancing. Otherwise equilirium is either pooling with det when the economy is stagnating, or separating when the economy is growing (some rms issue det and some rms issue shares). This provides new theoretical results that cannot e explained y the standard GB models and which are consistent with some nancial market phenomena. Keywords: Asymmetric information, Non-hierarchical signalling, Financing, Det-equity choice, Equilirium re nements, Intuitive criterion, Mispricing 1

3 1 Introduction The literature analyzing games where some players have private information aout their "types" is usually ased on the duality of "good" and "ad" types (GB approach), where "good" type denotes the type with etter quality. Depending on the context, the quality could mean the quality of produced good, the aility to work etc. Typically in such a game, the "good" type tries to signal its type to uninformed players y sending the messages which cannot e mimicked y the "ad" type. 1 In contrast, this paper analyzes a signalling game without types hierarchy. Di erent types have the same average qualities ut di erent performances pro les over time which are their private information. Hence a "good" type in the eginning may ecome "ad" in the end or "ad" in the eginning may ecome "good" in the end. We apply this idea to analyze a nancing-investment game where rms insiders have private information aout the rm s pro t pro le over time. 2 More speci cally, we analyze a situation where a rm s initial shareholders have to raise funds for nancing an investment project. There is no internal funds availale and therefore the nancing should e external. The cost of investment is known to the shareholders and to the potential investors while the expected pro t is the shareholders private information. Such a situation in a static context (one period) was well studied in the literature. The equilirium is typically pooling where all rms issue det which survive usual equilirium re nements and which minimizes mispricing (undervaluation) for a "good" type, i. e. for a rm with high expected pro t. 3 We thus consider a two-period situation. In each period there is an investment and a pro t. As was noted previously we suppose that di erent types have the same average pro t ut di erent pro t pro les over time which are their private information. Also, we assume that managers have the choice etween issuing det or equity. The solution of the game we otained shares with the standard models the existence of pooling equilirium with det. However, in our game a separating equilirium (which is e cient y de nition) may exist as well. Which equilirium prevails depends on the initial distriution of types in the economy. To provide asic ideas aout how the private information aout rms pro t pro le over time can a ect nancing choice let us suppose that there are only two types of rms. One is performance-improving (I) and have an increasing expected pro t, while others are stagnant (S) and have a atter or decreasing expected pro t. In such an environment, prices can e a ected y the lemon e ect in oth periods. 4 Intuitively, I would seem to have an informational advantage in the rst period: ecause of lower pro ts in this period, this type of rm can capitalize on the adverse selection prolem. On the other hand, in the second period the informational advantage passes to S. We show that I and S face very di erent incentives regarding nancial decisions. The point is that, generally speaking, det has a shorter maturity than equity, which has y de nition in nite maturity. Thus, the price of rst-period equity is type-independent due to the two-period maturity of equity (contrary to the one-period of det) and to the fact that oth types face the same total 1 For a review of signalling games see, for instance, Fudenerg and Tirole (1990) or Petrosjan and Zenkevich (1996). 2 In the similar spirit, some researchers assume that rms have the same average pro t ut di erent parameteres of risk which are their private information. For example, in the second part of Brennan and Kraus (1986) cash ows are ordered y mean-preserving spread condition. It is shown that optimal securities are neither convex nor concave in this case. In Brick, Frierman and Kim (1998) rms pro ts have the same average value ut di erent variances. The authors do otain some results aout rm dividend policy. 3 See, for instance, Nachman and Noe (1994). 4 We use the term "lemon" prolem to descrie a situation where private information leads to underpricing for a "good" type. See Akerlo (1970) for a classical example. 2

4 pro t over the two periods. As a result, if I were to issue equity in the rst period, they would always e mimicked y S, who stand to gain in the second period y eing perceived as growing and, therefore, as expecting high pro ts in the second period. The implication is that I are at a disadvantage for equity issues in the rst period. This is the main engine driving the results of this article. While I would de nitely prefer det to equity, incentives for S depend on the macroeconomic situation or on the initial distriution of types in the economy. The idea here is that if the economy is growing there are on average more performance-improving than stagnating rms interest rates tend to e more suitale for I. In particular, rst-period interest rates would e relatively high compared to those of the second period, ecause I is considered ad in the rst period and good in the second. Given such an interest rate pro le, we show that if S plays det, it would e ene cial to creditors, ut not to the rm. This is ecause the creditors ene t in the rst period due to the high interest rates and to the fact that S does well at that period. The rest of this paper is organized as follows. The asic model and some preliminary results are presented in Section 2. Sections 3 and 4 provide the analysis of two-type and multiple-type economies respectively. The conclusion is drawn in Section 5. 2 Model. Consider a rm with two-stage investment project. In each stage t = 1; 2 an amount has to e invested. In each stage the project can either e successful (with proaility t ) or unsuccessful (with proaility 1 t ). If the former is the case the revenue R e t equals 1 and if the latter is the case the revenue equals 0. Total expected revenue over oth periods is then Since all rms have the same total discounted pro t, total revenue can e normalized to unity without loss of generality. Hence = 1 and we write 1 = and 2 = 1. Firms di er only through the parameter. We assume < 1=2 with the s restricted to the interval [; 1 ], which implies that the investment has non-negative pro taility in each period, i. e. the expected pro t is not smaller than the amount of investments in period one ( ) and in period two ( 1 ). A rm has increasing expected revenues and pro ts if < 1=2, the pro t pro le is at or declining if = 1=2 or > 1=2 respectively. If we let e distriuted according to the density f(), then the total (average) rst-period revenue is: y = Z 1 f() d Clearly, total second-period revenue is then 1 y. This means that the economy is growing (revenues and pro ts are increasing) if y < 1=2, and it is stagnant or declining if y = 1=2 or y > 1=2. The rm s shareholders are responsile for capital structure choice, investments and pro t distriution. The initial capital structure is 100% equity, with n shares outstanding. The rm maximizes wealth of initial shareholders, who we will call the entrepreneur. Let 0 and 0 denote the initial proportions of equity owned respectively y outsiders and entrepreneur and let t and t denote their proportions immediately after the nancing and investment for stage t were done. Clearly, 0 = 1, 0 = 0 and t + t = 1 for all t. There exists universal risk-neutrality in this economy. In addition, the competition among investors is perfect. Insider shareholders know the rm s type, ut the investors do 3

5 not. The distriution of types is common knowledge. To nance the rst stage, the rm may issue either det (d) or equity (e). 5 In oth cases, the rm gets amount from the market, which is immediately invested. Holding free cash ow is costly. The reasons are well-known: empire uilding or ine cient investments and acquisitions, which spread the resources under the manager s control; increasing manager compensations or direct entrenchment; etc. 6 More speci cally, we assume that any availale free cash ow disappears immediately, producing useless loss for the shareholders. 7 Knowing this the shareholders will never keep the free cash that implies that any availale rst-period pro t will e distriuted as dividend. Thus, in the second period only det nancing is possile. 8 ;9 As in the standard literature in this eld, we assume that the contract of det is enforceale at no cost. The sequence of events is illustrated in gure 1. We assume that the rm s type is revealed to initial insiders in the period 0. The investors are identical and we will call them simply the market. The market determines the prices of issued securities. Also the market oserves the rst-period capital structure choice. However, it does not oserve the pro t previously realized y the rm. t = 0 t = 1 t = 2 s s s - Firm s type is realized It is revealed to entrepreneur Entrepreneur decides whether to issue equity or det Securities are issued Market determines the prices of securities Investment is made Project yields R 1 It is distriuted to the claimholders Firm issues second-period det Market determines the prices of securities Investment is made Project yields R 2 It is distriuted to the claimholders Figure 1. The sequence of events. 5 More complicated securities are not considered here since the model s implications are all aout equitydet choice. Also, for the simplicity of exposition, we assume that only pure strategies can e played, although this is not crucial to the results. 6 Easterrook (1984), Jensen (1986). See also [7] and [14] for empirical analysis aout the signi cance of manager s agency cost in holding cash 7 We assume free cash to e any availale cash at the end of a period, which means any resources that were not invested during the period or any received pro ts that were not used for interests or dividends. 8 This is ased on Myers and Majluf s (1984) idea that in one-period setting under asymmetric information, equity is never issued. Although our environments are quite di erent, one can show that the introduction of the possiility of equity issue in the second period does not alter any results. 9 To simplify, we assume that mixed nancing (det/equity in the rst period or cash/det in the second) is not possile. The asic intuitions developed within this paper are not a ected y introducing these possiilities. It is also important to note that the model can e extended y allowing mixed strategies (in game-theoretic terms), which can e interpreted to some extent as real mixed nancing. 4

6 Throughout this article, we use the concept of Perfect-Bayesian equiliria (PBE) and also verify that o -equilirium eliefs survive usual re nements like Cho and Kreps (1987) intuitive criterion and consistency (Kreps and Wilson, 1982). The intuitive criterion seems to e not very powerful in games where pooling equilirium is Pareto-e cient (see Cadsy, Frank, and Maksimovic, 1998). Fortunately, this is not the case in the present paper. In addition, note that perfect competition etween outsiders implies zero market pro t and risk-neutral valuation for any security issued. More speci cally, we assume that there are at least two investors and the competition among them is in the Bertrand style (see Cho and Kreps (1987) or Nachman and Noe (1994)). Their pricing strategies are identical and equal to the expected value of the o ered securities. Competition in the capital market therefore results in the price that yields zero net pro t to investors. From de nition of det and equity it follows that ifhdet is issued in period t with face value D then the detholders expected payo equals E min( R e i t ; D). The shareholders are residual claimants. If new equity was issued the shareholders share the pro t according to the numer of shares owned. 2.1 Perfect market. This susection provides some useful information aout enchmark pricing when the market knows the rm s type. Consider strategy e. Denote the issue of shares in period 1 y n, the price of issued shares y p 1 e and the second period det face value y D 2 e. The relations descriing the pricing and the payo s are: 1) rst-period udget constraint: = p 1 en (1) 2) market valuation of second-period det: h = E min( R e i 2 ; De) 2 (2) 3) market valuation of equity issued in the rst period (recall that n denotes the initial numer of shares): h p 1 e = R E max(0; e i R 2 D 1 e) 2 n + n + (3) n + n where E[ e R] = R. Given the identity: and using equations (1) and (2), we can transform (3) to: min(r; D) + max(0; R D) = R (4) p 1 e = R 1 + R 2 n 2 Since R 1 + R 2 = 1, we get p 1 e = 1 2 n (5) 5

7 Remark 1. p 1 e depends only on the rm s total pro t and not on its pro t pro le over time. Using equation (2) and conducting a similar exercise for strategy d (for type ), one can otain the e cient (symmetric information) face values of det (for the rst and second period respectively): D 1 d = =; D 2 d = =(1 ) (6) If < 1=2, the interest rate pro les in the case of d corresponding to type is downward sloping (and upward sloping if > 1=2, respectively). Finally, note that regardless of how the investment is nanced, the value of the rm for the entrepreneur is: V = 1 2 (7) For example if e is played then the entrepreneur s expected payo equals h nr ne max(0; e i R 2 D 1 e) 2 n + n + n + n Taken into account (3) and (5) this equals 1 nancing does not matter. 2.2 Asymmetric information. 2. As usual, in perfect market, the choice of Now consider the situation where the rm s type is its private information. Let us introduce the payo -functions. Denote y V j (; ) the entrepreneur s nal payo if the rm is of type ut is perceived as type, given the rst-period action j = e; d. The following explains why the analysis of these functions is useful. Suppose that the market eliefs oserving strategy j are characterized y a density function j () with support [; 1 ]. Lemma 1. Let the market eliefs oserving strategy j = e; d e j. The pricing is then as if the market elieves with proaility 1 that the rm is type j, where Z j = j ()d Proof. Consider j = d. Let rst-period det face value equals Dd 1. The rst-period lenders expected rst-period payo is then: R Dd 1j ()d. Risk-neutral valuation implies that it should e equal to. Thus D 1 d = R j ()d Analogously for second-period det: Dd 2 = R (1 )j ()d Lemma 1 follows from (6). Now consider j = e. For second-period det face value the reasoning is exactly as for Dd 2. Now consider the rst-period share price. Since the rm s 6

8 total expected pro t equals 1 and since the second-period lenders expected payo equals, the pro t of insiders plus the pro t of rst-period outsiders of rm j equals 1. Also, in the case of e, 2 o (that also shows the fraction of equity held y the rst-period outsiders in the moment of rst-period pro t distriution) equals rst-period outsider shareholders is: n (1 ) n + n n n+n. Thus, expected pro t of Risk-neutral valuation implies that the expected revenue of rst-period outsiders is. n n Thus: n+n (1 ) = and n = 1 2, which implies (ecause we know the udget constraint p 1 en = ) that p 1 e = 1 2 (8) n End proof. 10 Note that under perfect information, the rst-period share price equals 1 2 n, regardless of the issuer s type (equation (5)). The same result holds true under asymmetric information. It provides the intuition as to why growing rms prefer det to equity they cannot use their informational advantage in the rst-period playing equity ecause the price is always the same. Consider the features of functions V j (; ) for j = e; d. If the entrepreneur plays e 2 c = n n + n = n n + =p 1 e = as implied y (8). Thus Also: V e (; ) = ( + (1 )(1 )) (9) 1 V d (; ) = (1 = ) + (1 )(1 =(1 )) (10) Oviously, V j (; ) = 1 2; j = e; d, since this corresponds to complete information valuation. Oserve also that V d (; 1=2) = 1 2. The following properties are ovious: Lemma > 0 < 0: The idea ehind Lemma 2 is that since the rst-est share price in the rst period is the same for all types, the types with high ene t from their informational advantages in the second period (when they are really lemons ). On the other hand, a larger means a larger second-period interest rate, which is unpro tale. Lemma 3. V d (; ) 1 2 if and only if 1=2 or 1=2. Furthermore d(; ) ) = 1=2) (11) 10 Note that the same result holds true if one introduces the possiility for second-period outsiders to oserve rst-period pro t realization, given that market eliefs are Bayesian. 7

9 @V d (; ) < min(; 1=2) > 0 (12) d (; ) > max(; 1=2) < 0 (13) V d (; ) 1 2, , ( ) (1 ) + (1 ) 2 (1 ) 0; where is convex with roots = 1=2 and =. This proves the rst statement. The proof of (11) follows d (; To prove (12) and (13) one can check that = (1 2 ) (1 ) d(; ) = sign( 1 ( ) 2 ) 1 Now < min(; 1=2) implies 1 > ( 1 )2 while > max(; 1=2) implies 1 < ( 1 )2. End proof. Intuitively, y analogy with perfect information case, a downward sloping interest rates pro le ( 1=2) is suitale for growing rms, i. e. for rms with and not for rms with lower than average rate of growth ( > ), which are etter o with upward sloping interest rate pro le. Conversely for the case of stagnating economy (1=2 ). The intuition ehind condition (11) is the same. Now consider equation (12). If the interest rate pro le is downward sloping then, for a rm that has lower than average rate of growth, making interest rate pro le less upward sloping is pro tale. On the other hand, if the interest rate pro le is upward sloping then, for a rm with higher than average rate of growth, making the interest rate pro le deeper is unpro tale (equation (13)). Lemma 4. sign(v d (; ) V e (; )) = sign( ). Proof. Consider V d (; ) 1 ( ) V e (; ). That is: ( + (1 )(1 1 )) = ( )(1 ) (1 )(1 ) (14) The sign of the last expression depend oviously on the sign of. End proof. Figure 2 illustrates the rst parts of Lemmas 2 and 3, condition (11) and Lemma 4. 8

10 6V V d H HHHH 1 2 H HHHH 0 1=2 a V e H - 6V V e 1 2 V d - 0 1=2 Figure 2. V d (; ) and V e (; ) when: a) < 1=2; ) > 1=2. In oth cases V e (; ) is increasing in. When < 1=2 (Figure 2a), V d (; ) is downward sloping in and is upward sloping if > 1=2 (Figure 2). If the latter is the case the slope of V e (; ) is greater than that of V d (; ) meaning that the payo from the strategy det is less sensitive to adverse selection prolem as compared to equity. This is in keeping with most of the literature in this eld. Intuitively, if a rm is perceived as a less growing type than it is in reality then it will prefer det to equity. This is ecause the second-period interest rate is the same in either case, ut y playing det, the rm gains in the rst period y eing a ad rm-type. If, in contrast, a rm is perceived y the market as a less growing than in reality type, it would prefer equity. 3 Two-type economy. To generate the asic ideas, we rst consider a two-type economy. Firm I is characterized y the parameter I, rm S has parameter S where I < S. By de nition, S has etter performance in the rst period while I in the second (note that oth may actually e declining, ut S then declines faster). Let 0 e the proportion of type I rms, 0 < 0 < 1. Hence y = I 0 + S (1 0 ). Since each rm may play two types of strategy (d or e), there are 4 potential candidates for equilirium: two separating and two pooling. Given the concepts descried in Section 2 a separating equilirium is de ned as follows: 1) type I plays j I and type S plays j S 6= j I ; j T 2 fe; dg; T 2 fi; Sg. 2) V ji ( I ; I ) V js ( I ; S ) and V js ( S ; S ) V ji ( S ; I ). A pooling two-type equilirium is de ned as follows: 1) oth type play j. 2) oserving the strategy j off 6= j (o -equilirium path) the market elieves that the type is I with proaility I and the type is S with proaility S = 1 I such that V j ( I ; y) V joff ( I ; p ) and V j ( S ; y) V joff ( S ; p ), where p = I I + S S. 3) If for type T max V joff ( T ; ) < V j ( T ; y) then T = 0; T 2 fi; Sg. The rst condition means that di erent types play di erent strategies under separating equilirium and the same strategy under pooling. The second condition represents the nondeviation condition for each type (individual rationality). Finally, the third condition in the case of pooling equilirium assures that the equilirium survives the intuitive criterion of Cho and Kreps (1987). This condition means that the market o -equilirium eliefs 9

11 are reasonale in the sense that if for any type T its maximal payo from deviation is not greater than its equilirium payo then the market should place the proaility 0 on possile deviations of this type. The de nitions aove are consistent with standard PBE de nition (see, for instance, Fudenerg and Tirole, 1991) with an addition of intuitive criterion which is quite common in such kind of games (see, for instance, Nachman and Noe, 1994). Finally note that Lemma 1 insures that in descried aove equiliria the market makes zero-pro t (competitive rationality). 11 Proposition 1. 1) The situation where I plays e and S plays d is not an equilirium; 2) if and only if I 1=2, there exists a separating equilirium where I plays d and S plays e. Proof. (i) Part 1. Suppose, in opposite, that such equilirium exists. Of course, each type would have 1 2 in a separating equilirium. From Lemma 2 V e ( S ; I ) > 1 2 ecause I < S. Thus S would deviate from its equilirium strategy to e and such equilirium is impossile. (ii) Part 2. Let I 1=2. I does not mimic S. From Lemma 2 we have V e ( I ; S ) 1 2 ecause I < S. S does not mimic I. From Lemma 3, V d ( S ; I ) 1 2 ecause I < 1=2 and I < S. Now, if I > 1=2 then from Lemma 3 V d ( S ; I ) > 1 2. Thus S would mimic I. End proof. Intuitively, in the equilirium descried in Part 2, I does not deviate ecause y playing e it is not ale to capitalize on its rst-period informational advantage. The share s price in the rst period does not depend on the rm s type (Lemma 1), while the interest rate in the second period will e unfavorale. S does not deviate ecause the interest rates pro le is downward sloping or at when I 1=2, making d unpro tale for S (which performs etter with upward sloping interest rates pro le). Proposition 2. If and only if y 1=2, pooling with d is an equilirium. Proof. (i) Part 1. 1) Existence. Let y 1=2. Consider pooling equilirium where oth types play d, which is supported y o -equilirium market eliefs that the rm is S. 12 First of all, let us verify non-deviation for each type. Since 1=2 y < S ; we gets from Lemma 3 V d ( S ; y) 1 2. Thus the type S does not deviate. From Lemma 2, we have V e ( I ; y) V e ( I ; S ) (15) The condition of non-deviation for the type I is oviously V d ( I ; y) V e ( I ; S ): The latter follows from the condition (15) and Lemma 4: V d ( I ; y) V e ( I ; y) V e ( I ; S ) (16) Let us now verify that o -equilirium eliefs survive the intuitive criterion of Cho and Kreps (1987). To show this, let us calculate the maximal payo of type S in the case that it 11 Also note that y de nition of pooling, the o -equilirium eliefs are consistent (Kreps and Willson, 1982). If out of equilirium path the market elieves that the type is then it keeps the same eliefs in the second period (it follows from the de nition of the payo functions V ). Otherwise the market o -equilirium eliefs would e inconsistent. 12 Note that in terms of the de nition of pooling given aove, we have here j off = e, S = 1 and p = S. 10

12 plays e. Its payo is evidently maximized if the market s elief places the proaility 1 on type I oserving equity, i. e. V e ( S ; I ). If o -equilirium eliefs survive intuitive criterion, this expression must e greater than V d ( S ; y). 13 It follows immediately from Lemmas 2 and 4: V e ( S ; I ) V e ( S ; y) V d ( S ; y) This completes the proof of su ciency. 2) As for the necessary condition, if y < 1=2 then pooling with d is impossile ecause type S would deviate in e (from Lemma 3 its equilirium payo would e less than 1 2). End proof. The idea ehind the Proposition 2 is simple. Only if growing rms dominate the credit market (y < 1=2) will the interest rates pro le e downward sloping, creating incentives for stagnating rms to play e. Proposition 3. 1) if y < 1=2, pooling with e is not an equilirium; 2) if y 1=2 and if pooling with e exists, then mispricing is greater under that than under pooling with d. Proof: see Appendix Intuitively, if y is low, then in the case of pooling with e second-period interest rate is low, making high pro t for the type S. In some cases this pro t is even greater than maximal possile pro t under the strategy d. This situation is not an equilirium ecause the market should set the proaility 0 on the possiility for S to play d, making o equilirium interest rates suitale for the type I; that would deviate to d. 15 Secondly, if pooling with e exists, then mispricing is greater than it is under pooling with d. Intuitively y analogy with Lemma 4, type I (undervalued under pooling equilirium, ecause S can always achieve at least rst-est using e as a last resort) prefers pooling with det over pooling with equity. Propositions 1, 2 and 3 are at the root of two major insights of this paper; they provide clues aout the link etween initial distriution of types in the economy and individual rm capital structure policy, and they show why det is a signal of a rm s increasing performance while equity is a signal of decreasing performance. The main conclusion of the aove analysis is that performance-improving rms de nitely prefer det while stagnating rms ase their strategy on the macroeconomic situation if the economy is growing, they will issue equity, and if the economy is stagnating, oth strategies can lead to equilirium. Also note that in a two-type economy, separating equilirium always dominates pooling y minimal mispricing. However, the intuition aout the existence of pooling equiliria is useful and it will e further applied in Sections 4 and 5. 4 Multiple type economy. To provide more ideas aout the role of macroeconomic situation in this game, consider a multiple type economy, and suppose that f() > 0; 8 on the support [; 1 ]. Here 13 Otherwise S should e equal to We use the standard concept of mispricing that can e found, for example, in Nachman and Noe (1994). The magnitude of mispricing in a given equilirium equals to that of undervalued types. For instance, if the strategy j is played y undervalued type T (the undervaluation is only possile in pooling) then the mispricng equals 1 2 V j (T; y). The overvaluation of overvalued type does not matter. Note that the total pro t in the economy is always the same across di erent equiliriums. 15 Remark that, since y < 1=2 implies I < 1=2, a separating equilirium exists in this case. 11

13 again we consider two types of equiliria: pooling and semi-separating (or two-strategy equilirium). A two-strategy equilirium is de ned as follows: 1) There are two sets e and d such that any 2 e plays e and any 2 d plays d, e \ d = O, e [ d = [; 1 ] and the measure of proaility of O equals 0. 2) V e (; e ) V d (; d ), 8 2 e and V d (; d ) V d (; e ), where e R = 1 M( e) 2 e f()d and d = 1 M( d ) R 2 d f()d. Note that M( j ) is the measure of proaility of j and j denotes the expected type of rms playing j 2 fe; dg. A pooling equilirium is de ned as follows: 1) all types play j. 2) oserving the strategy j off 6= j (o -equilirium path) the market eliefs are j off such that V j (; y) V joff (; j off ),8, where j off = R j off ()d. 3) if for type max V joff (; ) < V j (; y) then j off () = 0, where = R ()d. In each case the rst condition means that oth strategies are played in two-strategy equilirium and that all types play the same strategy under pooling. The second condition represents the non-deviation condition for each type (individual rationality). Finally, the third condition in the case of pooling equilirium assures that the equilirium survives the intuitive criterion of Cho and Kreps (1987). Proposition 4. 1) In any two-strategy equilirium, there exists 2 (; 1 ) such that rms with > play e and rms with < play d; 2) If y 1=2 (non-declining economy), this equilirium exists. (i) Proof of part 1. Since oth strategies are played, the V e (; e ) and V d (; d ) must intersect (otherwise we would have a pooling equilirium like in Figure 3) and the intersection is unique since the payo s are linear in, as shown in the foregoing discussion. 16 6V V d V e H HHHH 1 2 H HHH - 0 a 6V V d 1 2 V e - 0 Figure 3. Separating and pooling equilirium in multiple type economy. Thus, the only candidates for a two-strategies equilirium are either the one descried in the Proposition or one where rms with > issue det and rms with < issue equity. If we suppose that the latter is the case, contrary to the Proposition, it is only possile if V d is increasing and cuts V e from elow. But the interest rates comination supporting this 16 Recall that j is the expected type of rms playing j = e; d. 12

14 situation is impossile. First of all, it must e that 1 2 > V e ( ; e ) = V d ( ; d ). If it does not, all rms playing det ( > ) have more than rst-est, which contradicts risk-neutral valuation. But in this case equity market makes positive pro t (all issuing equity rms have less than 1 2) which contradicts equilirium s concept. The only possile equilirium is in Figure 3a where rms with > issue equity and rms with < issue det. (ii) Proof of part 2. Let d ( ) = 1 Z F ( f()d ) and e ( ) = If y 1=2; then for any 2 (; 1 Z 1 1 (1 F ( )) f()d ); we have < d ( ) < 1=2 and: V d (; d ( )) > 1 2 > V e (; e ( )) Similarly, since d ( ) < 1=2 < 1 and e ( ) < 1 V d (1 ; d ( )) < 1 2 < V e (1 ; e ( )) Now de ne ( ) = V d ( ; d ( )) V e ( ; e ( )). From the previous results () > 0 > (1 ), hence from the intermediate value theorem there exists 2 (; 1 ) such that ( ) = 0. This proves existence ecause V e (; e ( )) is increasing in, while V d (; d ( )) is decreasing in when d ( ) < 1=2. End proof. Because the rst-est share price in the rst period is the same for all types, the types with high ene t from their informational advantages in the second period (when in actuality they are really lemons ). Hence, intuitively the separation equilirium descried in Proposition 4 should exist if the payo in the given det strategy is decreasing in. A su cient condition for existence of this equilirium is that y 1=2, which means that the economy is not declining on average. Also, in this case, pooling with det is not an equilirium. The intuition for this condition is the following: pooling with det is unprofitale for stagnating rms ecause, since there are more growing rms, interest rate pro le corresponds more with them and, as we know, stagnating rms would lose y playing det. Thus, stagnating rms tend to signal their type y playing equity. This leads to the following result. Proposition 5. 1) If and only if y 1=2, pooling with d is an equilirium; 2) if y < 1=2, pooling with e is not an equilirium; 3) if y 1=2 and if pooling with e exists, then mispricing is greater under that than under pooling with d. Proof : see Appendix 2. As we can see, the only qualitative di erence with the asic model is the fact that here the existence of separation equilirium, outlined in Proposition 4, is suject to the condition that the economy is non-declining. This is not surprising ecause in a two-type model, like in the one descried in Section 3, the grower rm is also growing asolutely, which implies that the interest rate pro le is necessarily downward sloping, making it unpro tale for the stagnating rm to mimic this type y playing det. In the multiple type case, the condition y 1=2 insures that in the case of separation, the equilirium interest rate pro le would necessarily e downward sloping. 13

15 5 Conclusion. Let us summarize the analysis of the model. Two equiliriums may exist: separating and pooling. A separating equilirium exists if and only if the economy is growing on average. In separating equilirium rms issuing equity have higher performance in the rst period and lower performance in the second period. Also equity may e issued only in the case of separating equilirium ecause pooling with det dominates pooling with equity y minimal mispricing when economy is stagnating. One can see that these results contrast the standard GB approach ("good" and "ad" rms) where the solution is typically pooling with det. We elieve that the idea of signalling the performance pro le over time (in contrast or in addition) to the signalling aout total performance provides an interesting insight which can e applied in other elds of research as well. Also, the results of the paper are consistent with several nancial phenomena: (i) Firms issuing equity underperform in the long-run as compared to non-issuing rms (measured as a decline of pro t, pro t to assets ratio or pro t per share). This is implied y Propositions 1 and 4: in any equilirium, where oth det and equity are issued, only the types with low second period pro t issue equity in the rst period. 17 At the same time the performance of rms issuing equity exceeds the performance of the non-issuing rms at the time of issue (or in the near future after issue). Clearly, this also follows from Propositions 1 and Similarly, the model predicts that leverage is negatively correlated with pro taility. In separating equilirium, rst-period low-pro tale rms issue det. 19 (ii) This paper suggests a new motive for issuing equity (Propositions 1 and 4) that has not een explored in existing literature. When the rm knows that it will e high-pro tale in the near future and low-pro tale in the long-term, the entrepreneur may want to issue equity. (iii) This model provides a rationale for the link etween det-equity choice and usiness cycle. The analysis of the asic model reveals the following ideas. Growing rms prefer det. The incentives for stagnating rms depend on the macroeconomic situation. If the economy is in contraction, stagnating rms may prefer det, ut will necessarily issue equity when the economy is growing (Propositions 1, 2, 3 and 4). Thus, equity issues seem to e procyclical. Acknowledgments. Claude Fluet, Thomas Noe, Leon Petrosjan and P. Viswanath have given us especially useful comments on earlier versions of this paper. We are also grateful to seminar participants at the University of Guelph, ESADE, University of Bonn, UQAM, the Bank of Canada, the 2003 MWF, the 2002 CIRPEE and the 2003 SCSE meetings for their suggestions and comments. The nancial support awarded y the Social Sciences and Humanities Research Council of Canada and the Institut de nance mathématique de Montréal was instrumental in enaling our continued research. We also appreciate the editing assistance of Kaarla Sundström. 17 This conclusion is con rmed y empirical ndings (see for example [5], [11], [15] or [16]). 18 While this point was not the main focus of the empirical research cited aove, some authors did stress the point that issuing rms outperform non-issuing rms just efore issue, and others documented that issuing rms outperform non-issuing rms in the year of issue and in the rst year after issue (see [11] and [16]). 19 This is consistent, for example, with [9], [20] and [21]. 14

16 Appendix 1 (i) Proof of part 1. We will show that pooling with e does not survive intuitive criterion. Consider pooling with e and assume that if a type deviates to d, it is perceived as the type d. It is enough to show that min V e ( S ; y) max V d ( S ; d ) (17) y where y < 1=2 and I d S. 1) S < 1=2. From Lemma 3, V d ( S ; d ) < 1 2. Since y Lemma 2 V e ( S ; y) > 1 2, this proves (17). 2) S 1=2. From (9) minimal value of V e ( S ; y) on feasile support is otained when y = 1=2 (and equals (1 From (10) S)). V d ( S ; d ) = 1 ( S + 1 S ) d 1 d (18) Local minimum of expression in rackets under positive d is dmin = S the maximal payo of S, if it were playing d, would e: p S (1 S ) 2 S 1. Thus, S 1 (2 S 1)( p S S (1 S ) + 1 p S ) S (1 S ) 1 + S Hence (17) is equivalent to the following: ( S) > 1 (2 S 1)( p S S (1 S ) + 1 p S ), S (1 S ) 1 + S (1 )(2 S 1) 2p S (1 S ) (2 2 S 2 S + p S (1 S ))( S 2 S ) > 0 Since S 1=2, the left side is decreasing in. Thus it is enough to show that this inequality holds under = 1 S (ecause 1 S ). It can e veri ed that S (2 S 1) 2p S (1 S ) (2 2 S 2 S + p S (1 S ))(6 S 4 2 S 1) > 0 on feasile support of S. Thus, the market would attriute the proaility 0 to the possiility of playing d y S. The market would elieve that rm is type I after oserving det. In this case, type I would certainly deviate from pooling with e to d. (ii) Part 2 follows from Lemma 4. End proof. Appendix 2. (i) Proof of part 1. 1) Su ciency. Consider pooling equilirium where all types play d, which is supported y o -equilirium market eliefs that the rm is the type 1. First of all, let us verify non-deviation for each type. Consider the case y. Since 1=2 y ; we gets from Lemma 3 V d (; y) 1 2. From Lemma 2 we have V e (; 1 ) < V e (; ) = 1 2. Thus the type does not deviate. Now, consider the case < y. From Lemma 2, we have S V e (; y) V e (; 1 ) (19) The condition of non-deviation for the type is oviously V d (; y) V e (; 1 ): Given the condition (19), is is enough to show that V d (; y) V e (; y). This oviously follows from Lemma 4. Let us now verify that o -equilirium eliefs survive the intuitive criterion of Cho and Kreps (1987). To show this, let us calculate the maximal payo of type 1 in the case 15

17 that it plays e. Its payo is evidently maximized if the market s elief places the proaility 1 on type oserving equity, i. e. V e (1 ; ). If o -equilirium eliefs survive intuitive criterion, this expression must e greater than V d (1 ; y). From Lemmas 2 and 4 we get immediately: V e (1 ; ) V e (1 ; y) > V d (1 ; y) This completes the proof of su ciency. 2) As for the necessary condition, if y < 1=2 then pooling with d is impossile ecause type 1 would deviate in e (its equilirium payo would e less than 1 2). (ii) Proof of part 2. We will show that pooling with e does not survive intuitive criterion when y < 1=2. First, for such that > y it can e shown (analogously to Proposition 4) that min V e (; y) max V d (; ) (20) y if y < 1=2. Thus, the market would attriute the proaility 0 to the possiility of playing d y if > y. Evidently in this case d < y. Thus V d ( d ; d ) = 1 2. Also, V e ( d ; y) < 1 2. Thus d would deviate and pooling with e is impossile. (iii) Proof of part 3. Let y 1=2. According to part 1, pooling with d exists. Assume that pooling with e exists. From Lemma 2 V e (; y) < 1 2 for and only for < y. As implied y Lemma 4 for such types V d (; y) V e (; y). Thus mispricing under pooling with e is larger that under pooling with d. End proof. References [1] Akerlo G. (1970). The Market for Lemons: Quality Incertainty and the Market Mechanism. Quarterly Review of Economics. Vol. 74 (n. 3), [2] Brennan, M., & Kraus, A. (1987). E cient Financing under Asymmetric Information. The Journal of Finance, Vol. XLII, n. 5, decemer, [3] Brick, I., Frierman M., & Kim, Y. K. (1998). Asymmetric information Concerning the Variance of Cash Flows: The Capital Structure Choice. International Economic Review, Vol. 39, n. 3 (Aug.), [4] Cadsy, C.B., Frank, M. & Maksimovic, V. (1998). Equilirium Dominance in Experimental Financial Markets. The Review of Financial Studies, Vol. 11, No. 1. (Spring), [5] Cai, J., & Wei, K. (1997). The Investment and Operating Performance of Japanese Initial Pulic O erings. Paci c-basin Finance Journal, 5, [6] Cho, I. K., & Kreps, D. (1987). Signalling Games and Stale Equiliria. Quarterly Journal of Economics, 102 (2), [7] Dittmar, A., Mahrt-Smith, J. & Servaes, H. (2003). International Corporate Governance and Corporate Cash Holdings. Journal of Financial and Quantitative Analysis, Vol. 38, n. 1 (march), [8] Easterrook, F. (1984). Two Agency-cost Explanation of Dividends. American Economic Review, Vol. 74 (4), [9] Fama, E.,& French, K. (2002). Testing Trade-o and Pecking Order Predictions aout Dividends and Det. The Review of Financial Studies, Vol. 15, n.1,

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