HOT MARKETS, INVESTOR SENTIMENT, and IPO PRICING Alexander Ljungqvist Stern School of Business, NYU and CEPR

Forthcoming, Journal of Business Preprint typeset using LATEX style emulateapj v. 6/21/04

HOT MARKETS, INVESTOR SENTIMENT, AND IPO PRICING Alexander Ljungqvist Stern School of Business, NYU and CEPR

Vikram Nanda University of Michigan Business School and Rajdeep Singh Carlson School of Management, University of Minnesota Forthcoming, Journal of Business

ABSTRACT We model an IPO company’s optimal response to the presence of sentiment investors and short-sale constraints. Given regulatory constraints on price discrimination, the optimal mechanism involves the issuer allocating stock to ‘regular’ institutional investors for subsequent resale to sentiment investors, at prices the regulars maintain by restricting supply. Because the hot market can end prematurely, carrying IPO stock in inventory is risky, so to break even in expectation regulars require the stock to be underpriced – even in the absence of asymmetric information. However, the offer price still exceeds fundamental value, as it capitalizes the regulars’ expected gain from trading with the sentiment investors. This resolves the apparent paradox that issuers, while shrewdly timing their IPOs to take advantage of optimistic valuations, appear not to price their stock very aggressively. The model generates a number of new and refutable empirical predictions regarding the extent of long-run underperformance, offer size, flipping, and lock-ups. Subject headings: Behavioral corporate finance; initial public offerings; investor sentiment; long-run underperformance.

1. INTRODUCTION little light on the long-run performance of IPOs.2 Work There are several anomalous aspects to the process by on long-run performance is primarily empirical and em- which firms go public [Ritter and Welch (2002)]. Initial phasizes the role of investor sentiment and bounded ra- public offerings (IPOs) exhibit positive first-day returns tionality in explaining the price behavior of IPO stocks. on average and so seem to be ‘underpriced’. The initial The impact of investor sentiment is regarded as partic- price run-up appears to be undone in subsequent months ularly acute in hot markets. Over time, investor exu- as IPO stocks underperform the market.1 Hence, from berance fades, resulting in long-run underperformance. the vantage of a longer horizon, IPOs can arguably be re- Loughran, Ritter, and Rydqvist (1994) go further in garded as ‘overpriced’ in the after-market. The strength claiming that issuers ‘time’ their IPOs to coincide with of these patterns varies over time, with both the ini- periods of excessive optimism, consistent with the finding tial price run-up and subsequent underperformance more in Lee, Shleifer, and Thaler (1991) that more companies dramatic in ‘hot’ periods of high IPO volume [Ritter go public when investor sentiment is high. Such patterns (1984, 1991)]. Firms may even be able to ‘time’ their IPO can persist if rational investors are dissuaded by the cost to coincide with periods of excessive valuations [Baker of implementing arbitrage strategies [Shleifer and Vishny and Wurgler (2002)]. (1997), Lamont and Thaler (2003)]. What is one to make of these patterns? The litera- The behavioral story sketched above has some obvi- ture offers no consensus. There are numerous models of ous appeal, but it raises an apparent paradox: if issuers IPO underpricing, typically based on investor rationality are regarded as rational and shrewd enough to choose in incomplete information settings, but these have shed a hot market in which to go public, why are they less than aggressive in setting the offer price? After all, it Electronic address: [email protected] (Ljungqvist), seems plausible that the presence of sentiment investors [email protected] (Nanda) and [email protected] (Singh). could lead to higher offer prices and a lower level of un- 1 There is a debate in the literature as to whether IPOs really derpricing as rational issuers take advantage of them. do underperform. Ritter (1991) shows that they underperform the index, while Brav and Gompers (1997) and Brav, Geczy, and Reconciling the simultaneous existence of underpricing Gompers (2000) show that they are not alone in doing so: small and long-run underperformance thus requires additional and high-growth companies also underperform the index. Once structure on the behavioral assumptions and the nature that is taken into account, IPO firms appear to perform no worse of the economic environment. than similar firms (i.e. small and high-growth companies). The evidence in these later papers is consistent with the possibility that The task we set ourselves in the paper is, therefore, investors get optimistic about small and high-growth firm stocks, not just IPO stocks, and firms choose to go public at that time. 2 Among these are explanations based on the ‘winner’s curse’ Our model takes this as a given and derives pricing implications [Rock (1986)], signaling [Allen and Faulhaber (1989), Welch from issuers’ optimizing behavior. (1989)], cascades [Welch (1992)], and investor incentives to reveal information truthfully [Benveniste and Spindt (1989)]. 2 Ljungqvist, Nanda and Singh to develop a model of IPO pricing in hot issue mar- is an attempt to capture the equilibrium response of is- kets that elucidates the connection between underpricing suers and underwriters in the face of divergence of opin- and long-run underperformance. We ask, what should a ion among investors. It is thus related to an empirical profit-maximizing issuer do in the presence of exuberant literature in which firms act strategically to take advan- investor demand and short-sale constraints? We argue tage of the market’s mispricing or mis-perceptions.4 that the issuer should seek to capture as much as pos- We do not attempt to rationalize the existence or be- sible of the surplus under the exuberant investors’ de- havior of exuberant investors. Biases that might lead mand curve, in a setting where demand may build over to such behavior have been studied by psychologists for time. We derive an optimal mechanism (which we argue some time and financial economists have recently intro- is consistent with institutional reality) that achieves the duced them into formal models. See Daniel, Hirshleifer, issuer’s first-best outcome. and Subrahmanyam (1998) for a discussion of the litera- The model starts with the premise that some investors ture. may, on occasion, be ‘irrationally exuberant’ about the Testing a model that relies on investor sentiment re- prospects of IPOs from, say, a particular industry. As- quires unique and refutable empirical predictions. Our suming short-sale constraints, this is consistent with model generates a number of novel predictions: long-run underperformance.3 More interestingly, the model suggests possible connections between IPO under- • Companies going public in a hot market underper- performance and the initial price run-up. We show that form, both relative to the first-day price and the value to an issuer is maximized if underwriters allocate offer price. Underperformance relative to the first- IPO shares to their regular (institutional) investors for day price is not a surprising prediction; it follows gradual sale to sentiment investors who arrive in the mar- from the twin assumptions of sentiment investors ket over time. Regulars maintain stock prices – thereby and limits to arbitrage. Underperformance relative extracting surplus from sentiment investors – by holding to the offer price is a stronger (and novel) predic- IPO stock in inventory and restricting the availability of tion. It follows because the offer price will exceed shares. Underpricing emerges as fair compensation to fundamental value by an amount equal to the is- the regulars for expected inventory losses arising from suer’s share in the surplus extracted from the sen- the possibility that sentiment demand may cease. In re- timent investors. Cross-sectionally, we predict that turn, the expropriation of value from sentiment investors the extent of underperformance relative to the of- is capitalized into a higher offer price than would other- fer price increases in the issuing firm’s bargaining wise be the case. power vis-à-visthe underwriter. For the inventory holding strategy to be implemented, • As investor sentiment grows, IPO offer size in- there must either be a dominant investor or, when there creases and lower-quality companies are taken pub- are many investors, it must be incentive compatible for lic, resulting in a decrease in average issuer quality. regular investors not to deviate by selling their IPO allocations prematurely. To deter cheating, it may be nec- • Underwriters penalize investors who engage in ex- essary for the underwriter to punish deviations from the cessive flipping. Importantly, they do so even in equilibrium strategy. We show that the degree of the IPOs that do not receive price support. Such penal- underwriter’s ability to impose penalties determines the ties are targeted primarily at retail and infrequent optimal size of an offer, the extent of underpricing, and investors. subsequent long-run performance. It is worth emphasizing that when there is a dominant • Corporate insiders are released early from their investor or when the underwriter can impose sufficient lock-up provisions, if after-market demand from costs to ensure cooperation among regular investors, the sentiment investors is unexpectedly strong, once full benefits are passed on to the issuer in the form of regular investors have unloaded their excess inven- a higher offer price. In the economic environment we tory, or if the hot market has come to an end. model, issuers cannot do better by the use of alternative Our model also addresses two hitherto puzzling empir- ways to sell equity. For instance, if the issuer were to ical findings: engage in a quick succession of equity offerings (an IPO followed by seasoned offerings), the value obtained would • Ritter (1991) documents that underpricing and not exceed the value from the inventory holding process long-run performance are negatively related, while we model. In any case, issuing stock repeatedly over a Krigman, Shaw, and Womack (1999) find a posi- short period is implausible, given significant economies tive relation. Our model shows that the relation is of scale in issuing costs and the necessity to satisfy reg- not necessarily monotonic. In particular, we show istration and disclosure requirements repeatedly. that the relation is negative only if the probability Our paper has a focus quite different from much of of the hot market ending is small. the existing work in behavioral finance. The behavioral • The empirical evidence on the relation between finance literature has tended to focus on asset pricing underwriter prestige and underpricing is mixed. anomalies, such as the predictability of returns, the eq- Consistent with evidence from the 1990s [Beatty uity premium puzzle, and under- and over-reactions [for and Welch (1996)], we predict that underpricing an exhaustive survey, see Hirshleifer (2001)]. Our model increases in underwriter prestige, but that this re- 3 In a different setting Miller (1977) shows that a divergence lation depends on the state of the IPO market. of beliefs – similar to the notion that some investors are more optimistic than others – can lead to long-run underperformance. 4 For example, see D’Mello and Shroff (2000) and Dittmar (2000) on firms’ strategic use of share repurchases. Hot Markets, Investor Sentiment, and IPOs 3

Two recent papers that test some of the main predic- by the presence of optimistic investors.5 Pessimistic in- tions of our model, and that provide empirical support vestors, if present, are prevented from expressing their for it in the context of the recent ‘dot-com mania’, are demands by short-sale constraints, which are pervasive Ofek and Richardson (2003) and Dorn (2002). Ofek and in IPOs.6 As will become clear later, short-sale con- Richardson show that high initial returns occur when in- straints arise naturally in our model. Though they hold stitutions sell IPO shares to retail investors on the first excessively optimistic beliefs about the prospects of firms day, and that such high initial returns are followed by going public, s-type investors still act rationally given sizeable reversals to the end of 2000, when the bubble their beliefs, in ways we will make more precise shortly. had burst. This is precisely the pattern we predict, and The second investor type holds beliefs that correspond it highlights the importance of heterogeneous beliefs and to an unbiased estimate of the issuing firm’s future short-sale constraints in explaining both the initial IPO prospects. It may be reasonable, for instance, to regard price run-up and longer-term performance. Using Ger- institutional investors as belonging to this category. For man data on IPO trading by 5,000 retail customers of an expositional ease, we will label these investors ‘rational’. online broker, Dorn documents that retail investors over- All other market participants (issuers, underwriters) are pay for IPOs following periods of high underpricing in re- taken to be rational and value-maximizing as well. There cent IPOs, and for IPOs that are in the news. Consistent is no private or asymmetric information in the model, with our model, he also shows that hot IPOs pass from and the nature and characteristics of the market partic- institutional into retail hands. Over time, high initial ipants and their beliefs are common knowledge. Hence, returns are reversed as net purchases by retail investors sentiment and rational investors know each others’ be- subside, eventually resulting in underperformance over liefs, but still ‘agree to disagree’ on the valuation of the the first six to 12 months after the IPO. IPO shares.7 For simplicity, everyone is taken to be risk- The paper proceeds as follows. The basic model is neutral. developed in Section 2. In Section 3, we analyze the is- The model has four relevant dates: t = 0, 1, 2, and T . suer’s optimal unconstrained strategy for extracting sur- At t = 0, the period prior to the offering, the pricing plus from the exuberant investors. Since this strategy and size of the IPO are determined. Date t = 1 is the would violate regulatory rules, we derive in Section 4 initial day of trading in the IPO shares. Once trading an alternative mechanism that implements the optimal has begun, the market may continue to be hot – that is, strategy, which involves inventory-holding by a regular it may continue to be characterized by the presence of investor. We solve for the optimal issue size and offer optimistic investors – but sooner or later the hot market price, and derive the patterns of prices in the short- and will come to an end. This captures the notion that there long-run. We also analyze the comparative statics of the will eventually be incontrovertible evidence of the IPO price patterns with respect to the strength of sentiment shares being overpriced, or that the cost of shorting IPO demand and the probability of the hot market coming stock will fall to the point where prices are no longer set to an end. Section 5 considers three extensions to the by optimistic investors.8 For now, we model this by in- model: multi-period sentiment demand, multiple regular troducing a subsequent trading date t = 2 at which the investors, and multiple pre-IPO owners. In Section 6, IPO market may or may not still be hot. In Section 5.1, we discuss empirical support for various aspects of the we will explicitly extend the model to a setting with mul- model and offer new testable implications. Concluding tiple periods of trading in the after-market. Finally, T is remarks are in Section 7. All detailed proofs are collected the terminal date by which the hot market is definitely in the Appendix. over and there is no more disagreement about firm value. We denote by γ the (exogenous) probability of the hot market ending at t = 2. In addition to disagreeing about value, investors disagree about γ. Rational investors un- 2. THE MODEL derstand that the hot market may end before T with We model a firm that is going public in a ‘hot’ IPO probability γ > 0, in which case the marginal investor market, to be defined shortly. The firm’s equity is sold will be someone holding unbiased beliefs. Sentiment in- via a standard firm-commitment IPO in which an under- vestors, on the other hand, dismiss this possibility: in writer assumes responsibility for distributing the issuer’s their mind, the hot market will continue for sure. shares to investors. The offer price is finalized at the end Let VT denote the terminal payoff of the security at of bookbuilding, just prior to the start of trading, and T . There are no distributions (e.g. dividends) and the is subject to a uniform-pricing rule such that neither the discount rate is zero. At t = 1, the ‘fundamental’ or issuer nor the underwriter can price-discriminate among long-term expected value of an IPO share – the value investors [see also Benveniste and Wilhelm (1990)]. The from the perspective of rational investors – is denoted by offer size Q and price P0 will be chosen so as to maximize the owner-manager’s wealth. 5 This mirrors Miller’s (1977) divergence-of-opinion model. Our sentiment investors hold beliefs that are in the right tail of the The demand side of the IPO market consists of two distribution of beliefs. types of investors. The first type are small, unsophisti- 6 Geczy, Musto, and Reed (2002) show that borrowing IPO stock cated investors who are prone to episodes of optimistic in the early after-market is extremely expensive in general, the or pessimistic ‘sentiment’ about the stock market, es- more so, the higher was the initial day return. 7 pecially IPOs, where sentiment denotes incorrect beliefs The notion of investors ‘agreeing to disagree’ is commonly em- ployed in models with a diversity of opinions among market par- about the fundamental value of an asset arising from ticipants, a good example being Harris and Raviv (1993). treating noise as relevant information [Black (1986)]. We 8 Ofek and Richardson (2003) show that the bursting of the will label these investors sentiment or ‘s-type’ investors. dot-com bubble in March/April 2002 coincided with a substantial In our set-up, a ‘hot’ IPO market is one characterized increase in the availability of stock to borrow. 4 Ljungqvist, Nanda and Singh

VR = E(VT ). Absent sentiment investors and additional will be given by the demand curve in equation (1). We information, VR would be the market price of the IPO assume here that the quantity of shares sold is such that shares at t = 1. As we will see, the presence of senti- Q < Q. This, as we will show later, is consistent with an ment investors can affect pricing and trading patterns, optimal choice for Q. For a given quantity of shares Q R and thus the institutional arrangements that result. issued at t = 0, rational investors’ valuation is E (P2) = s The value sentiment investors place on the IPO shares γVR + (1 − γ)E (P2). Note that this exceeds their long- is not uniform. Specifically, we assume that sentiment run valuation VR, since they expect to be able to sell investors are budget-constrained and that their aggregate the security to s-types at t = 2 with probability (1 − γ). s demand curve for IPO shares can be represented as Sentiment investors value the shares at E (P2) = VR + a − λQ. V = V + a − λQ (1) s R To summarize, we model a ‘hot’ market that is char- where Q is the total number of IPO shares held by sen- acterized by the presence of optimistic investors. Not all timent investors.9 Define Q = a . For all Q < Q, s-types optimistic investors are present at t = 1 and, if the hot λ market persists, more are expected to show up at t = 2. (if they are present) place a value higher than VR on the IPO shares. Sentiment investors know the demand curve Rational investors expect the terminal value of the IPO and the value put on the shares by rational investors.10 shares to be VR. Unlike the optimists, they recognize To model the dynamics of a hot market that may come that the hot market may come to an early end at t = 2, to a premature end, we allow for the possibility that not with probability γ > 0. In the longer run, by the termi- all sentiment investors are present in the market at t = 1. nal date T , the hot market will end with certainty. All If the hot market continues at t = 2, additional sentiment investors, rational or otherwise, act in a manner consis- investors may arrive in the market.11 Thus, sentiment tent with their beliefs. demand can evolve over the two periods (or, in Section 5.1, over multiple periods). The fact that sentiment de- 3. SELLING IPO SHARES mand can build over time affects the interpretation of We consider the optimal procedure for selling IPO demand in equation (1). Acting rationally, the sentiment shares so as to maximize issuer wealth in the presence investors present in the market at t = 1 would never be of optimistic valuations. For now we maintain the as- willing to pay a price at t = 1 that is greater than the ex- sumption that the offer quantity Q is given exogenously. pected price conditional on their beliefs at t = 2. Thus, The unconstrained optimum involves selling IPO shares they properly anticipate the prices of the security in the at both t = 1 and t = 2. This is, of course, contrary to short run, by forecasting what demand will be at t = 2. the market practice of selling the shares in a single shot Conditional on the (mistaken) belief that the hot market and the requirement that investors be sold IPO shares at will continue at t = 2 for certain, s-type investors expect a uniform price. As we will see, such discretion will have demand at t = 2 to be the aggregate demands of s-types no impact if sentiment demand at t = 1 is large enough arriving at t = 1 and t = 2. It is this ‘longer-term’ de- to absorb the full offering. mand (and not just the t = 1 sentiment demand alone) Let q1 be the number of IPO shares sold at t = 1, that affects the value Vs they put on the IPO shares in while q2 is sold at t = 2. The expected proceeds, Ψ, to equation (1). the issuer are We can now determine the price of the IPO shares at s t = 2 and, thereby, the offer and trading prices at t = 0 Ψ = q1P1 + q2(E (P2) (1 − γ) + VRγ). and 1. If the hot market has ended, the price at t = 2 Given their beliefs, sentiment investors expect the price s will be set by the expectations of the rational investors at t = 2 to be E (P2) = VR + a − λ(q1 + q2). Hence, so such that P2 = VR. If the hot market persists, the price long as the sentiment investors hold all the IPO shares

9 issued at t = 1, the marginal investor is a sentiment The linear form of the demand curve is chosen to simplify the s presentation and does not affect our results materially. investor and the price at t = 1 will be E (P2). Let Q1 ≤ 10 Our results do not depend on any particular reason why senti- Q denote the total optimistic demand present at t = 1. ment investors are willing to pay more than fundamental value VR If q1 > Q1, the marginal investor is a rational investor for the shares at t = 1. For example, rather than holding optimistic R beliefs about VR, sentiment investors may hold optimistic beliefs and the price at t = 1 will be E (P2). Thus we have: about their ability to sell out to other sentiment investors in the ½ future. Our results go through as long as i) there exist a limited V + a − λ(q + q ) if q ≤ Q P = R 1 2 1 1 number of sentiment investors with a downward sloping demand 1 γV + (1 − γ)(V + a − λ(q + q )) if q > Q curve at t = 1 who value the shares above fundamental value; ii) R R 1 2 1 1 more such investors may arrive in the market at t = 2; and iii) the (2) sentiment investors may disappear at t = 2. Assuming the firm does not need to raise a particular 11 The fact that not all sentiment investors are present in the level of financing, the owner-manager’s objective is sim- market at t = 1 may be the result of the time required for information to disseminate among the less informed investors; for ply to maximize the ‘profit’ from selling IPO shares, that enthusiasm about the IPO to build while the market stays hot; or is, the excess value Π of the proceeds over his own val- ∗ ∗ the additional time needed for some sentiment investors to raise uation VRQ. The optimal (q1 , q2 ) can, therefore, be re- resources and bid for IPO shares, especially when many ‘hot’ IPOs garded as the solution to the following constrained opti- come to market around the same time. It is also possible that sentiment investors may have noisy information about their value func- mization problem: tions and about the existence of other sentiment investors. High max Π ≡ Ψ − VRQ underpricing at t = 1, caused by investors with positive signals, q ,q may lead even those with negative signals to join the crowd [e.g. 1 2 Welch (1992)] at t = 2. We hope future work will provide insights s.t. q1 + q2 = Q into the relation between the existence of sentiment investors and their propensity to form rational cascades. Its solution is given in the following proposition. Hot Markets, Investor Sentiment, and IPOs 5

∗ ∗ Proposition 1 For a given number of shares to be is- are given by q1 and q2 , respectively. The staggered sale ∗ ∗ ∗ sued, Q, the optimal choice of q1 and q2 is such that strategy requires the regular investor to hold q2 shares ½ in inventory from t = 1 to t = 2, when the quantity to ∗ ∗ ¡(Q, 0) ¢ if Q ≤ Q1 be sold is such that Q > Q . Given our assumption of (q1 , q2 ) = . (3) 1 Q1,Q − Q1 if Q > Q1 a monopolist profit-maximizing regular investor, there is no incentive to deviate by selling the shares early. Proposition 1 shows that the issuer may do better by Increases in the supply of stock in the market under- staggering the sale of equity over two time periods in- mine the inventory-holding strategy, creating an incen- stead of one. By restricting the initial supply of shares, tive to limit short sales. In practice, short sellers must the issuer ensures that the marginal investor at t = 1 borrow stock from investors who are long, which at t = 1 is a sentiment investor. If, however, the total quantity in our model means the owner-manager or the regular in- Q to be sold is less than the demand by sentiment investor. The owner-manager almost invariably is ‘locked vestors at t = 1, then the issuer optimally chooses to set up’ and the regular has no incentive to lend stock. Thus ∗ q2 equal to zero. The intuition is straightforward. In our there will be no short sales until the regular begins to set-up there is no price advantage from delaying the sale trade out of the security, and the short-sale constraint of equity if it can be sold to sentiment investors at t = 1. relaxes over time.13 Delay exposes the issuer to the risk of the market crashing at t = 2. Hence, the issuer is strictly better off selling 4.1. Optimizing Offer Size and Price to the sentiment investors at t = 1 and thus taking ad- In equilibrium, a regular investor will invest in IPOs vantage of their mistaken belief that the hot market will only if he does not expect to lose as a consequence. If persist at t = 2. As we will discuss later, a similar result an IPO share is bought at an offer price P0, the regular holds when the model is extended to consider the arrival investor’s participation constraint is of sentiment investors over a larger number of periods. ∗ ∗ s Proposition 1 indicates that it may be optimal to sell −QP0 + q1 P1 + q2 [(1 − γ)E (P2) + γVR] ≥ 0 (4) an offering in stages. However, as mentioned, laws and ∗ ∗ where q1 and q2 are as given in (3). The first term is regulations effectively prevent issuers and their under- the cost of purchasing all the shares in the IPO. The writers from conducting firm commitment offerings in a second and third terms represent the cash flows received staggered fashion. In the U.S., for instance, NASD rule from selling at t = 1 and t = 2. The bracketed part of IM-2110-1 on “Free-riding and Withholding” prevents the third term is the price at which the regular investor an underwriter who holds IPO shares in inventory from 12 expects to be able to sell IPO shares at t = 2. selling them in the after-market above the offer price. Assuming, as before, that the issuer does not need to Thus, there is considerable downside risk without upside raise a particular level of financing, the objective remains potential. We now consider an alternative arrangement to maximize the excess value, Π, of offered shares over by which an underwriter can achieve the same ends with- their ‘true’ (long-term) value, subject to the participa- out directly selling the IPO in stages. tion constraint defined in (4). Thus, the issuer solves

4. INVENTORY HOLDING BY INSTITUTIONAL max Π ≡ Q (P0 − VR) INVESTORS P0,Q ∗ ∗ s Given constraints on the underwriter’s ability to (di- s.t. −QP0 + q1 P1 + q2 [(1 − γ)E (P2) + γVR] ≥ 0 rectly) stagger the sale, we suggest that institutional (or other ‘regular’) investors can be delegated the task of Lemma 1 The participation constraint will always be holding inventory in the after-market for resale to sen- binding. timent investors. Specifically, we assume (for now) that there exists a monopolist regular investor who purchases Proof. Suppose not. That is, the optimal P0 and Q Q shares at the offer price P0 and then sells q1 shares are such that the constraint has slack. Then the issuer at t = 1 and the remainder q2 at t = 2, when the full can increase P0 and so increase his profits, which contra- demand by s-type investors is established (so long as the dicts the optimality of P0 and Q. ∗ ∗ s hot market persists). The assumption of a single (or Using the lemma, Π = [q1 P1 + q2 E (P2) (1 − γ) + ∗ ∗ ∗ dominant) regular investor simplifies the exposition and q2 VRγ] − QVR, where q1 and q2 are given by (3). The abstracts from concerns about free-riding among regular bracketed term is the maximum amount a regular in- investors. The case with a multitude of regular investors vestor is willing to pay for the IPO shares, from the par- is discussed later, with the threat of punishment dissuad- ticipation constraint in (4). From Proposition 1, we know ∗ ing regulars from engaging in free-riding behavior. q1 ≤ Q1. Thus, P1 is determined by s-type investors, on Once the shares have been allocated, the regular in- the basis of their expectation regarding P2. Substituting s vestor’s problem is no different from that of the issuer. for P1 = E (P2) = VR + a − λQ in Π and simplifying, Thus, the regular investor will find it optimal to follow the issuer’s objective function can be written as the staggered sale strategy, where the aggregate quan- ∗ ∗ ∗ ∗ max [q1 (Q) + (1 − γ) q2 (Q)] [a − λ (q1 (Q) + q2 (Q))] , tities sold in the secondary market at t = 1 and t = 2 Q

12 Countries where staggered sales are possible include Germany. 13 In practice, not all IPO shares are allocated to the regular; Though rare and usually conﬁned to small companies, such oﬀer- some are allocated to retail investors. Retail investors could in ings proceed as follows. Rather than allocating stock to investors principle lend stock to short sellers. Interestingly, the most un- at t = 0, the issuer announces a quantity Q it intends to sell via derpriced IPOs are associated with the smallest retail allocations the stock exchange, in one or more trading sessions, at the market- (Aggarwal, Prabhala, and Puri (2002)) and the least amount of clearing price. This closely resembles our mechanism. short selling (Geczy, Musto, and Reed (2002)). 6 Ljungqvist, Nanda and Singh

∗ where we explicitly recognize the dependence of q1 and Fig. 1.— Issuer Surplus ∗ q2 on Q. We can now derive the issuer’s optimal offer size. The figure illustrates two different selling mechanisms when Proposition 2 With a single regular investor, the is- the optimal quantity chosen (Q∗) is strictly greater than the ∗ ∗ ∗ suer’s optimal choice of quantity Q = q1 + q2 , where sentiment demand at t = 1 (Q1). First, suppose the issuer ( ³ ³ ´´ can sell in stages directly to the investors (as modeled in a γ a(1−γ) Section 3). The s-type investors present at t = 1 rationally ∗ ∗ Q1, 2λ − Q1 1 + 2(1−γ) if Q1 < λ(2−γ) (q1 , q2 ) = ¡ ¢ anticipate demand at t = 2 and price the security at P1. At a , 0 otherwise t = 2, the hot market persists with probability (1 − γ), in 2λ ∗ which case the issuer sells quantity (Q − Q1) at price P2 = Proof. We obtain the above expressions from first- P1. If the hot market ends, he is forced to sell the shares at their fundamental value VR. The rectangle GHIJ represents order conditions obtained by taking the derivative of the the expected surplus obtained at t = 2, which is equal to firm’s objective function with respect to q2. It can be ∗ (1 − γ)(P1 − VR)(Q − Q1). The issuer’s total surplus is given shown that there is a unique maximum because the sec- by the area in the two rectangles ABJK and GHIJ. Second, ond order condition with respect to q2 is negative. if the issuer is prevented from directly selling in stages, but We now turn to pricing. The issuer needs the regular he can obtain the cooperation of a regular investor, we have the case modeled in Section 4. The issuer sells Q∗ shares to investor to hold inventory if Q1 is small (relative to to- the regular investor at price P0. At t = 1, the regular investor tal sentiment demand), i.e. less than a(1−γ) . So long obtains a profit equal to the rectangle ABEF and at t = 2 λ(2−γ) suffers an expected loss equal to the rectangle DEGH. The as the hot market persists, the regular investor sells his zero profit condition on the investor ensures that the gain at inventory to newly-arriving sentiment investors at t = 2. t = 1 is equal to the expected loss at t = 2, leaving the issuer If the hot market ends, he is left with shares priced at with the same profits as in the earlier case. Underpricing arises because the regular investor needs to be compensated VR. For a regular investor to accept this negative-valued for the expected inventory loss, and so P0 < P1. Long-term gamble, the initial offer price needs to be set at a discount underperformance arises because the issuer always extracts relative to the price at which the shares are expected to some surplus from the sentiment investors, and so P0 > VR. s trade initially, so that P0 < E (P2) = P1. In our model, the share price will eventually drift to VR, where VR < P0 from the binding participation constraint of the investor. Thus, with a regular investor holding inventory that he Price VR + disposes of over time, both an initial price run-up (un- a − λQ derpricing) and long-run underperformance will be ob-

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§ § § § § § § § a necessary but not sufficient condition for the first-day § VR return. The long-run return (VR − P1) /P1, on the other K J I hand, is always negative in our set-up. It results from the overly optimistic valuation of sentiment investors and represents market inefficiency – sustained by the diffi- ∗ Quantity culty and cost of establishing short positions in the stock. Q1 Q By implication, we do not expect a monotonic relation between underpricing and the long-run price drift. Proposition 3 summarizes the above discussion regarding the predicted price patterns. Figure 1 illustrates. Though optimistic about the issuer’s stock, sentiment Proposition 3 With a single regular investor, investors, in our model, are still acting rationally given their beliefs: they correctly anticipate the arrival of more ∗ 1. if Q1 is small enough such that q2 > 0, then sentiment investors at t = 2 (albeit with the wrong prob- the IPO shares will exhibit an initial price run-up: ability) and price the stock accordingly. If the sentiment P0 < P1; investors were not forward-looking in this sense, then the price at t = 1 would be determined by the marginal sen- 2. if Q1 is large, then the shares will not exhibit an timent investor present at t = 1. In that case, the price initial price run-up: P0 = P1; run-up would, in fact, be even higher than that predicted by our existing set-up. 3. ∀ Q1 the long-run return will be negative: VR < P1. We can make a more precise prediction regarding the relative magnitudes of underpricing and long-run perfor- 14 It is important to note that our model does not predict that mance: underpricing increases with offer size. Underpricing is required only if the optimal offer size derived in Proposition 2 exceeds the sentiment demand that is available in the market at that time, Proposition 4 With a single regular investor, the ini- which could be true for either ‘large’ or ‘small’ IPOs. tial price run-up [P1 − P0] and long-run price drift Hot Markets, Investor Sentiment, and IPOs 7

[P1 − VR] will be related as follows: a reduction in the quantity issued increases the price at t = 1, thus worsening long-run performance. γq2 P1 − P0 = (P1 − VR). An increase in γ has two opposing effects on the first- Q day return. First, it increases the regular investor’s required compensation due to the direct effect of an in- For expositional ease, we will refer to the ratio of the crease in the probability of a crash. Second, the indirect initial price run-up and the long-run price drift as the effect of a reduction in the quantity issued is to reduce ‘price reversal ratio’. From Proposition 4, the price re- the inventory the regular investor holds. Proposition 5 versal ratio is proportional to the inventory carried by shows that the first effect dominates for low γ as the per- the regular investor as a fraction of offer size. Thus, ∗ centage change in q2 for low γ is small. For high enough P − P q γ, q∗ goes to zero and so the first-day return disappears. Price Reversal Ratio ≡ 1 0 = γ 2 . 2 P − V Q For intermediate levels of γ, the change in the first-day 1 R return is ambiguous. Similar characteristics are inherited 4.2. Comparative Statics by the price reversal ratio. We now study the properties of the first-day return, 4.3. Discussion long-run performance, and the price reversal ratio. We In this section, we have developed an alternative to focus on two parameters of interest: the intercept of the the direct-sale mechanism described in Section 3. Our sentiment investors’ demand function (a) and the prob- alternative mechanism requires the regular investor to ability of the hot market coming to an end (γ). In the carry inventory for sale in the secondary market. It is context of the model, both parameters are exogenous and important to understand that both mechanisms give the affect the nature of the hot market. issuer exactly the same expected proceeds, even though the delegated-inventory mechanism involves underpric- Proposition 5 With a single regular investor, ing. This simply follows from the zero-profit condition in Lemma 1. In words, in the delegated-inventory mech- 1. the number of shares issued, the first-day return, anism, the issuer underprices the stock to compensate and the price reversal ratio are all increasing, while the regular investor for bearing the risk of sentiment de- long-run performance is decreasing, in investor mand evaporating too soon. Thus, underpricing is not a sentiment (a); value transfer from the issuer to the regular investor; it 2. long-run performance and the number of shares is- is a fair payment for the regular’s expected loss. In the sued is decreasing in γ; and direct-sale mechanism, the issuer bears the exact same risk himself. Noting that everyone is risk-neutral, it is 3. the first-day return and the price reversal ratio are straightforward to show that the expected proceeds from increasing in γ for low γ. the two mechanisms are equivalent. Figure 1 illustrates. In some sense, the direct-sale mechanism described in An increase in the intercept of the demand function, a, Section 3 resembles an IPO followed – if the sentiment can be interpreted as an increase in sentiment investors’ demand survives – by an SEO. Couldn’t the issuer im- optimism. As one might expect, issuers in our model re- prove on the delegated-inventory mechanism by conduct- spond by increasing the size of the offering. The predic- ing an SEO shortly after the IPO? The answer is no: the tion on the first-day return, however, is not obvious. It expected proceeds are at best the same (ignoring trans- may seem anomalous that a more bullish market does not action costs for the SEO) or, more realistically, strictly translate into a smaller first-day return: why don’t is- lower (net of transaction costs). Leaving aside trans- suers take advantage of the bullishness of the market and action costs, if sentiment demand develops over several increase the offer price, resulting in a smaller first-day re- periods (perhaps stirred by the buzz of the IPO), it is turn? The reason why the first-day return increases with clearly impractical for the issuer to take advantage of it market sentiment is that underpricing is a way of com- via a sequence of possibly small SEOs. The regular in- pensating the regular investor for taking on the risk of vestor, on the other hand, faces no constraints on the the hot market crashing at t = 2. As offer size increases, frequency or size of after-market sales, and so can opti- the fraction of the offering carried over to t = 2 also mally take advantage of sentiment investors as and when increases. Consequently, the regular investor needs to they arrive in the market. Thus, while we do not rule be compensated more (on a per share basis) for taking out an SEO soon (within a few weeks) after the IPO, we on the risk of carrying this inventory. Thus we predict argue the issuer can better take advantage of developing that companies going public in a ‘hot’ market are more sentiment demand by obtaining the regular investor’s co- underpriced.15 operation than by planning to do multiple SEOs. An increase in γ, the probability of market sentiment 5. EXTENSIONS turning sour, reduces the expected gain from holding inventory until t = 2. As a consequence, the issuer is bet- We now outline three extensions to the model. In the ter off reducing the quantity of shares issued. However, previous section we analyzed a very tractable model to understand the properties of the initial price run-up when 15 Note also that IPO volume tends to increase in hot markets. issuers optimally take advantage of the sequential arrival Thus, in hot markets, multiple issuers compete for a resource that is of sentiment investors. In Section 5.1, we generalize the in short supply in the short-run, namely sentiment investors. So as model to show that similar results obtain if sentiment more companies go public in a hot market, the need to underprice increases unless the supply of s-types grows faster than the supply investors arrive over many periods. In Section 5.2, we of IPO shares. examine the strategy of underwriters who have to pay 8 Ljungqvist, Nanda and Singh rents to induce cooperative behavior among multiple reg- From these expressions it is easy to see that our results ular investors. In Section 5.3, we relax the assumption of on the existence of an initial price run-up and long-run a single owner-manager and link the pre-IPO ownership underperformance will go through in a multiple-period structure to the magnitude of underpricing and long-run setting. The following proposition summarizes without underperformance. proof.

5.1. Multi-Period Sentiment Demand Proposition 6 If sentiment demand evolves over mul- We now extend the model to incorporate sentiment tiple periods and the underwriter has access to a single demand that arises over several periods, say weeks or regular investor, then months. The set-up captures the notion that as poten- ∗ tial sentiment investors hear the ‘buzz’, some are likely 1. if Q1 is sufficiently small such that S > 1, then to invest in the stock. The arrival of future sentiment the IPO shares will exhibit an initial price run-up: investors, though likely, is still uncertain. This will be P0 < P1; reflected in the setting of the offer price. 2. ∀ Q the long-run return will be negative: V < P . We assume that new sentiment investors may arrive 1 R 1 every period after the IPO. The demand, however, decays Obtaining the comparative statics in closed-form is not over time at rate α. Specifically, we assume feasible in general. Given that the multi-period model is not as tractable as the two-period model analyzed ear- Qt = αQt−1, α < 1. lier, we resort to providing numerical solutions for se- As in the two-period model analyzed earlier, if the senti- lected parameter values. We solve the problem for the ment demand has survived up to period t then with prob- following parameter values: the long-term value VR is 5; ability γ it will disappear in that period. We maintain the probability of the hot market ending in any period, all other assumptions of the two-period model developed γ, is 10%, which is roughly equivalent to a 10-period ex- earlier. Thus, the marginal sentiment investor’s reserva- pected length of the hot market; sentiment demand is tion value is given by VR + a − λQ, sentiment investors assumed to decay at rate α = 10%; the initial demand account for the arrival of future sentiment investors, and Q1 is normalized to 1 unit; and the slope of the demand they do not share the regular investor’s belief about the curve λ is 0.5. Given the above parameter values we nu- possibility of the hot market ending. We assume that merically solve for the optimal S and plot the predicted the number of shares issued is sufficient to satisfy sen- price patterns as a function of the level of optimism (a). timent demand for up to S periods. Note that we are Figure 2 shows that the first-day return and the price characterizing the optimal quantity to be sold in terms reversal ratio are both increasing in a. Long-run perfor- of the number of periods. The reason we can do this is mance is always negative and decreasing in the level of that, for a given quantity to be sold, the optimal selling optimism. The intuition is similar to the one provided strategy (as in Proposition 1) is to sell whatever can be earlier. An increase in optimism among sentiment in- absorbed by sentiment investors each period till the hot vestors makes it optimal for the issuer to increase issue market ends or else the allocation is fully sold. size, which implies that the regular investor has to carry Thus, the number of shares issued, Q, is given by more inventory and bear a greater expected loss if the hot market ends. S The multi-period model developed in this section can 2 S−1 1 − α Q = Q1 + αQ1 + α Q1 + ... + α Q1 = Q1 (5) also be used to study the relation between the first-day 1 − α return and the expected length of the hot market. In A single regular investor, who is allocated Q shares at our model, the expected length of the hot market is P∞ n−1 n 1 price P0, sells Q1 shares at t = 1, if the hot market n=1 n (1 − γ) γ = γ . In Figure 3, we plot the first- persists αQ1 shares at t = 2, and so on. The break-even day return as a function of the expected length of the hot condition implies market for the same parameter values as in Figure 2. In (P − V ) Q = Q (P − V ) + (1 − γ) αQ (P − V ) addition, we assume that the intercept of the sentiment 0 R 1 1 R 1 1 R demand curve (a) is 5. The first-day return is initially S−1 S−1 + ... (1 − γ) α Q1 (P1 − VR) increasing and then decreasing in the expected length of the hot market. This result is consistent with the pre- Substituting for (P − V ) = a − λQ and using (5), we 1 R diction in Proposition 5. A decrease in γ increases the obtain expected length of the hot market, which has two oppos- µ ¶ 1 − (1 − γ)S αS 1 − αS ing effects. First, it decreases the risk that the hot mar- (P − V ) Q = a − λ Q Q . 0 R 1 − (1 − γ) α 1 − α 1 1 ket will end with the regular investor holding inventory, (6) which implies less underpricing is required to compen- Thus, the issuer’s problem is to maximize the right-hand sate the regular investor. Second, the issuer will choose side of (6) by the choice of S. We denote the optimal S to increase the quantity issued, which increases the regu- by S∗. At the optimum, lar investor’s inventory risk. As in the two-period model, for low values of γ (i.e. high expected length of the hot ∗ 1 − αS market) the first-day return decreases as the expected Q∗ ≡ Q (S∗) = Q 1 − α 1 length of the hot market increases. Figures 2 and 3 suggest, therefore, that the qualitative S∗ ∗ µ ¶ 1 − (1 − γ) αS 1 − α nature of our results is unaffected by an extension to P = V + (a − λQ∗) 0 R 1 − (1 − γ) α 1 − αS∗ many periods. We also believe that the multiple period Hot Markets, Investor Sentiment, and IPOs 9

Fig. 2.— Plot of ﬁrst-day return, long-term performance and 5.2. Limited Ability to Obtain Cooperation from price reversal ratio Institutional Investors

In this figure we plot the first-day return [ P1−P0 ], the long-term We have so far considered the case of a monopolist reg- P0 ular investor. Being a monopolist, the investor has an performance [ VR−P1 ], and the price reversal ratio [ P1−P0 ]. We P1 P1−VR incentive to cooperate with the underwriter, by holding solve for the optimal S and calculate P1 and P0 at the optimal inventory and delaying the sale of part of his IPO allo- S for the following parameter values: the long term value (VR) is 5; the probability of the hot market ending in any period, γ, is cation. However, if there are many regular investors, say 10%, which is equivalent to a 10-period expected length of the hot N, they face a free-rider problem. Collectively, regular market; the demand is assumed to decay at a rate (α) of 10%; the investors are better off holding on to their inventory until initial demand (Q1) is normalized to 1 unit; and the slope of the t = 2. However, individually each can benefit by unload- demand curve (λ) is 0.5. ing his entire allocation at t = 1. Hence, an underwriter’s ability to induce cooperative behavior is determined by 0.4 the extent to which he can offer inducements or threaten P −P 0.3 1 0 punishment. A likely form of punishment is the threat P1−VR 0.2 of exclusion of regular investors from future IPOs (or other desirable deals). Such an exclusion will impose a 0.1 P1−P0 0 P0 cost on the regular investors only if they obtain non-zero -0.1 rents from IPO allocations. Given the clamor to obtain -0.2 IPO allocations witnessed in the late 1990s, it seems rea- -0.3 VR−P1 sonable that regular investors do obtain rents. In this -0.4 P1 section, we generalize the analysis to explicitly allow for -0.5 such rents. For analytical tractability, we use the two- -0.6 period model of Section 4 rather than the multi-period -0.7 model introduced in Section 5.1. 2 4 6 8 10 We assume the underwriter can extract some rents on Sentiment demand intercept (a) behalf of his regular investors. These rents can be viewed as the outcome of a bargaining game between the issuer and the underwriter and in general would depend on the Fig. 3.— Plot of first-day return level of competition in the IPO market. We denote the per share rent by r. Given these rents, an underwriter In this figure we plot the first-day return [ P1−P0 ] as a function of can impose penalties on regular investors by excluding P0 1 them from future allocations of IPO shares – thereby the expected length of the hot market, which is given by γ . Thus, expected length equal to 2 corresponds to a 50% probability of the deterring deviation from the inventory holding strategy. hot market ending in every period and expected length equal to 5 The extent of punishment depends on the magnitude of periods corresponds to an 20% probability. We solve for the opti- r and the anticipated frequency of future IPO alloca- mal S and calculate P1 and P0 at the optimal S for the following tions. Specifically, we assume regular investors’ valua- parameter values: the long term value (VR) is 5; the demand is tion of such future benefits is rπ, where π is the multiple assumed to decay at a rate (α) of 10%; the initial demand (Q1) is normalized to 1 unit; the slope of the demand curve (λ) is 0.5; and that accounts for the probability and timing of future the sentiment demand intercept (a) is 5. IPOs. One would expect an investment bank with a bigger market share to have a higher π. Similarly, if the market believes the hot market to continue for some time, one would expect π to be high. Conversely, if the near- 0.18 term outlook for the IPO market is bleak, or if the un- 0.17 derwriter’s market share is small, exclusion from future 0.16 IPOs will provide only limited incentives for inventory holding. 0.15 Let Pˆ0 be the offer price that incorporates the rent r. 0.14 P1−P0 Thus, Pˆ = P − r, where P , as defined in Section 3, is P 0 0 0 0.13 0 the offer price for r = 0. On the margin, regulars can 0.12 choose to sell a share at price P1 at t = 1, or sell at R s 0.11 t = 2 and expect to get E (P2) = γVR + (1 − γ)E (P2). The potential loss from future exclusion from the IPO 0.1 process, rπ, must be large enough to deter deviation from 0.09 the inventory holding strategy. Therefore, we need 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 q2 ¡ ¢ Expected Length of Hot Market ( 1 ) rπ ≥ P − ER (P ) γ N 1 2 q2 where N represents the inventory each investor carries to t = 2. Denoting R ≡ rπN we can express the above constraint as s R ≥ q2 (P1 − γVR − (1 − γ)E (P2)) . (7) extension better captures the notion that the disposal of Substituting for P1 from (2) in (7) the constraint reduces shares by the regular investor is gradual, taking place to over a number of periods. R ≥ γq2(a − λ(q1 + q2)). (8) 10 Ljungqvist, Nanda and Singh

The analysis presented in Section 4 corresponds, there- The next proposition analyzes the impact of R on the fore, to the case where the above constraint has slack. IPO price patterns when the inventory holding constraint We now consider the situation in which the constraint is is binding. ∗ binding, i.e. in which (8) is violated at the optimal q1 ∗ and q2 . The next proposition shows that the constraint is Proposition 9 If the number of shares issued Q is such more likely to be violated when market sentiment is more that regular investors’ inventory holding constraint is exuberant or when the market has a higher probability binding, then the first-day return (P1 − Pˆ0)/Pˆ0, long-run of crashing. performance (P1 − VR) /P1, and the price reversal ratio (P − Pˆ )/ (P − V ) are all increasing in R. Proposition 7 The expected gain to regular investors 1 0 1 R of deviating from the inventory holding strategy and sell- The positive relation between the first-day return and ing shares at t = 1, R predicted in Proposition 9 may seem surprising, for it ∗ ∗ ∗ γq2 (a, γ)[a − λ(q1 (a, γ) + q2 (a, γ))] implies that IPOs lead-managed by more active or more prestigious underwriters are more underpriced.16 Recall is increasing in a and γ. Thus, if constraint (8) is vio- ∗ ∗ that underpricing serves as a form of compensation to lated at (q1 , q2 ) for some a =a ˆ and γ =γ ˆ, then it will the regulars for carrying inventory. An underwriter with be violated for all a > aˆ and γ > γˆ. a lower R can induce only a relatively small amount of inventory holding qc, as shown in Proposition 8. The less The gain from deviating from the inventory holding 2 inventory is carried, the less need there is for the offering strategy depends on the product of q and (P − V ). 2 1 R to be underpriced. An increase in a increases both (Proposition 5), increas- That lower R offerings are associated with worse long- ing the incentive to deviate as indicated in Proposition run performance is immediate, since the decrease in q 7. Similarly, an increase in the probability of a crash γ 2 (and thus in Q) increases the P = V + a − λQ that increases the incentive of regular investors to deviate by 1 R sentiment investors are willing to pay. This prediction selling their entire allocation of IPO shares at t = 1. is generally consistent with the empirical evidence that The issuer’s constrained problem is to solve the follow- IPOs done by larger, more established underwriters tend ing: to exhibit better long-term performance. max (q1 + (1 − γ) q2)(a − λ (q1 + q2)) q1,q2 5.3. Ownership structure and bargaining power s.t. R ≥ γq2(a − λ(q1 + q2)). So far, we have implicitly assumed that the owner- Let the solution to the above programming problem be manager has all the bargaining power relative to the un- c c (q1, q2). The next proposition characterizes the quanti- derwriter and the regular investor, so that the surplus ties chosen by the issuer. extracted from sentiment investors is fully incorporated into the offer price. In this section, we study the impact ∗ ∗ Proposition 8 If the optimal (q1 , q2 ) defined in Propo- of bargaining power on the first-day return and long-run sition 2 are such that (8) is violated, then the optimal performance. c c choice of shares issued (q1, q2) is given by We assume that the issuing firm’s ownership structure is such that β of the extracted surplus is captured by the qc = Q 1 ³ q 1 ´ issuing firm and 1−β is captured by a combination of the qc = 1 (a − q λ) − (a − q λ)2 − 4Rλ . (9) regular investor and the investment bank. For a firm with 2 2λ 1 1 γ highly concentrated ownership, we believe β will be close to 1, reflecting the high incentive to bargain hard over The optimal quantity sold in the secondary market at the surplus, while for a firm with dispersed ownership t = 1 is the same as that in the earlier unconstrained or other agency problems β will be significantly smaller case. This is because if more than Q were sold at t = 1, 1 than 1. From the issuer’s perspective, the key choice the marginal investor would no longer be a sentiment variables are the quantity to be issued Q∗ and the offer investor but instead a rational investor. However, con- price P . Figure 1 shows that the total amount of surplus straint (8) does decrease the quantity sold at t = 2, and 0 extracted, which equals the sentiment investors’ expected consequently the total issue size. This distortion in q is 2 loss, is a function of Q∗: highest for underwriters with a small R: with a smaller ¡ ∗ ¢ amount of potential rent at stake, incentive compatibil- (P1 − VR) Q1 + (1 − γ) Q − Q1 (P1 − VR) (10) ity requires regular investors to carry fewer IPO shares in inventory. Thus, banks with small R have less IPO Even when the issuer captures only a fraction of the sur- placing capacity and so do smaller deals. plus, it is still in his best interest to maximize the total surplus. Therefore, Q∗ is independent of β and, conse- The positive relation between R and q2 in equation (9) has one further implication. If periods of high IPO quently, P1 is also independent of β. The issuing firm ∗ volume imply increases in R, the size of the IPOs will chooses P0 such that its surplus (P0 − VR) Q is β times also be larger, ceteris paribus. Similarly, underwriters the total surplus given in (10). Thus, ¡ ¢ who gain (or are expected to gain) larger market shares ∗ Q1 + (1 − γ) Q − Q1 can impose bigger penalties, i.e. they have larger R. All P0 − VR = β (P1 − VR) else equal, this allows them to increase the size of their Q∗ offerings. Thus, growth will beget more growth and a hot 16 As we will discuss later, recent empirical evidence tends to market will get hotter. This suggests that a hot market support this prediction. However, certification arguments imply can have a certain self-fulfilling logic. the opposite relation. Hot Markets, Investor Sentiment, and IPOs 11

Given that P0 is increasing in β, the next proposition is They document that underpricing increases in time-to- immediate and is provided without proof. first-trade, consistent with Prediction 2. Cornelli, Goldreich, and Ljungqvist (2004) use trading Proposition 10 An increase in the surplus β captured data from European ‘pre-issue markets’, in which small by the issuing firm leads to: (i) no change in the optimal investors bet on the share prices of companies that are in number of shares issued Q∗; (ii) an increase in the offer the process of going public. Their results are consistent price P0; (iii) a decrease in the initial price run-up; (iv) a with a positive correlation between divergence of opinion decrease in the price reversal ratio; (v) no change in long- and IPO offer prices. run performance with respect to the first-day closing price Prediction 3 As the difference in opinion between ra- P1; and (vi) worse long-run performance with respect to tional and sentiment investors increases, long-run per- the offer price P0. formance worsens. This follows from Propositions 5 and 9. Using Aggar- The last two items in the proposition are novel. The wal and Conroy’s (2000) proxy for differences in opinion, stronger is the issuing firm’s bargaining power relative to Houge et al. (2001) show that late-opening IPOs signifi- the underwriter, the more its stock will underperform rel- cantly underperform over the subsequent three years. ative to the offer price, while long-run returns relative to Rajan and Servaes (1997) look at analyst following the first-day close are invariant to its bargaining power. after the IPO and find that not only were analysts If bargaining power can be approximated with ownership over-optimistic about earnings and long-term growth structure, these novel predictions are easily testable. prospects, but issuers may also have taken advantage of windows of opportunity: more companies went 6. EMPIRICAL IMPLICATIONS public when analysts where particularly over-confident Our model has a number of empirical implications, about recent IPOs in the same industry. Interestingly, some of which have already been mentioned. We now IPOs with low forecast growth rates subsequently out- collect these and other empirical implications. Several performed IPOs with high forecast growth rates, by a of them are consistent with existing empirical evidence, margin of more than 100% over five years. To the extent while others are novel and untested. the forecasts reflected some of the optimism of the sen- Prediction 1 (Long-run performance) Firms taken timent investors, these findings are consistent with our public in a hot market subsequently underperform, both prediction. relative to the first day trading price P1 and the offer Prediction 4 The relation between long-run perfor- price P0. mance and the first-day return is non-monotonic. It is Underperformance relative to P1 is not a surprising negative if the probability of the hot market ending is prediction; it follows from the twin assumptions of sen- small. timent investors and limits to arbitrage. Underperfor- This follows from Propositions 5 and 9. Prediction 4 mance relative to P0 is a stronger claim. It follows be- may explain the relatively mixed extant evidence on this cause the offer price will exceed fundamental value VR point. Ritter (1991) finds weak evidence that under- by an amount equal to the issuer’s share in the surplus pricing and long-run performance are negatively corre- extracted from the sentiment investors. Purnanandam lated. In particular, he shows that long-run performance and Swaminathan (2003) lend support to our prediction is particularly poor among smaller issuers, which tend to that the offer price can exceed fundamental value. They have the highest initial returns. Focusing on the recent show that compared to its industry peers’ multiples, the boom in internet IPOs, Ofek and Richardson (2003) find median IPO firm in 1980-1997 was overpriced at the offer a strong negative relation between first-day returns and by 50%. Interestingly, it is the firms that are most over- future excess returns to the end of 2000. Krigman, Shaw, priced in this sense which subsequently underperform. and Womack (1999), on the other hand, find a positive In a cold market, there are no exuberant investors and relation between underpricing and one-year returns, ex- so prices are set by rational investors at fundamental cept for ‘extra-hot’ IPOs: offerings with initial returns value. Thus, in our model, there is neither underpricing in excess of 60% have the worst one-year performance in nor long-run underperformance in a cold market. The their sample. empirical evidence is consistent with this concurrence Prediction 5 The greater the issuing firm’s bargaining of hot markets and poor long-run performance. Ritter power relative to the underwriter, the higher is the offer (1991) shows that companies that went public in the hot price and the lower is the first-day return. Greater bar- market of the early 1980s experienced high underpricing gaining power also leads to worse long-run performance and performed poorly in the long-run, a result corrob- relative to the offer price but not relative to the first-day orated more generally by Loughran and Ritter (1995), closing price. Helwege and Liang (1996), and Cook, Jarrell, and Ki- Bargaining power is unobservable, but it seems reason- eschnick (2003). able to assume that agents will bargain harder over the Prediction 2 (Partial adjustment) As the difference outcome of a negotiation the greater their claim over the in opinion between rational and sentiment investors in- surplus. Ljungqvist and Wilhelm (2003) show that first- creases, both the offer price and underpricing increase. day returns decrease in the size of pre-IPO equity hold- A test of Prediction 2, which follows directly from ings of key shareholder groups, namely the CEO, venture Proposition 5, requires proxies for differences in opinion. capitalists, and corporations, consistent with our predic- Aggarwal and Conroy (2000) propose time-to-first-trade tion. They also show that companies with more con- as a proxy: delaying the first trade may enable the un- centrated ownership at the time of the IPO suffer lower derwriter to better gauge market demand and could thus underpricing. be an indication of greater initial divergence of opinion. The first part of Prediction 5 does not require the pres- 12 Ljungqvist, Nanda and Singh ence of sentiment investors: whatever the source of the long-run performance. surplus, greater bargaining power should lead to reduced The evidence on underpricing is mixed. Contrary to underpricing. For instance, in a principal-agent setting, our prediction, Carter and Manaster (1990) and Carter, issuers may bargain harder with their underwriters to Dark, and Singh (1998) find that more prestigious under- reduce the agency cost of delegating the pricing decision writers are associated with lower underpricing. Beatty to a better-informed agent (Baron (1982)). The second and Welch (1996), on the other hand, point out that part of Prediction 5, on the other hand, does require the this relation appears to be reversed in the 1990s. Habib presence of sentiment investors: without them, there is and Ljungqvist (2001) show that the apparent reversal is no reason why the underwriter should tolerate pricing driven, at least in part, by the failure to treat the choice the issue above fundamental value. We are aware of no of underwriter as endogenous. Prediction 7 applies in existing evidence relating long-run IPO performance to particular to hot markets whereas none of the above pa- ownership structure (or bargaining power). pers control for the state of the IPO market. Benveniste, Loughran and Ritter (2002) argue that during 1998- Ljungqvist, Wilhelm, and Yu (2003) find a positive re- 2000, the bargaining power between issuers and under- lation between underpricing and underwriter prestige in writers shifted in favor of the investment banks which the 1999/2000 hot market, consistent with our predic- then engaged in manipulative and self-interested behav- tion. Carter, Dark, and Singh (1998) show that IPOs ior. Prediction 5 suggests that this shift should have been lead-managed by more prestigious underwriters are asso- accompanied not only by an increase in first-day returns, ciated with lower underperformance over the next three but also by better long-run performance relative to the years. offer price. This remains to be tested. Supporting the equilibrium The dynamics of the IPO market cycle Prediction 8 (Allocation policy) Underwriters have Prediction 6 As the optimism of sentiment investors a preference for selling to regular (typically institutional) increases, more companies have an incentive to go public investors. (to take advantage of the optimistic investors) and offer This prediction follows because the repeated interac- sizes increase. tion with regular investors, and the ease of tracking larger Lee, Shleifer, and Thaler (1991) show that the annual positions, will lower the costs of sustaining the equi- number of IPOs is negatively related to the discount on librium. Empirical evidence suggests that IPO alloca- closed-end mutual funds, which they argue is a measure tions are heavily skewed in favor of institutional investors of the sentiment of retail investors. Similarly, Lowry and [Hanley and Wilhelm (1995)] and that regular investors Schwert (2002) show that following periods of ‘unusually’ are favored over infrequent investors [Cornelli and Gol- high underpricing, both IPO volume and IPO registra- dreich (2001)]. tions increase and that companies which are already in The extension to a setting of multiple regular investors SEC registration accelerate the completion of their IPOs. considered in Section 5.2 suggests that underwriters also In a related context, Baker and Wurgler (2000) show that have a preference for targeting a select, probably small companies in aggregate issue relatively more equity than group of institutions. All else equal, and for a given debt just before periods of low market returns, suggest- amount of rent, a smaller group makes it easier to obtain ing timing ability. This is consistent with the first part the institutions’ cooperation. Targeting a narrow subset of Prediction 6. of investors is a common feature of U.S.-style bookbuild- Ljungqvist and Wilhelm (2003) document increases in ing. mean and median issue size among U.S. IPOs in every Prediction 9 (Flipping) Underwriters penalize in- year as the IPO market became hotter between 1996 and vestors who engage in excessive flipping (relative to the 2000, which is consistent with the second part of Predic- optimal selling strategy). tion 6. The prediction is consistent with the use of penalty We also conjecture that as the IPO market heats up, bids [Aggarwal (2000)] which underwriters impose on lower-quality companies may go public for opportunis- syndicate members whose clients flip their allocations. tic reasons, resulting in a decline in the quality of the Subtler penalties include exclusion from future IPO of- average issuer. The hot market of 1998-2000 may be a ferings. Such penalties are usually viewed as part and good illustration of the evolution of issuer quality over parcel of price support. Our model predicts that the the IPO market cycle. According to Ljungqvist and Wil- penalties should occur more widely than in IPOs which helm (2003), 61.6% of firms listing in the U.S. in 1997 receive price support. This remains to be tested. had 12-month track records of earnings; by 1999 this had Prediction 10 (Flipping) Penalties for excessive flip- fallen to just 23.6%. Helwege and Liang (1996) specifi- ping are targeted more heavily at retail and infrequent cally examine the quality of IPO firms in ‘hot’ and ‘cold’ investors. markets. Interestingly, and contrary to our conjecture, In order to sustain the equilibrium, underwriters need they find no difference in operating performance (their to ensure that their regular investors do not make (ex- measure of issuer quality) between hot-market and cold- cessive) losses on their holdings between dates 1 and 2. market issuers. Cook, Jarrell, and Kieschnick (2003), Competing selling pressure from investors who are not on the other hand, report that the five-year mortality party to the equilibrium strategy would therefore under- rate among hot-market IPOs is much higher than among mine the equilibrium. Though there is anecdotal support cold-market IPOs. for Prediction 10, we know of no systematic evidence on The role of the underwriter this point. Prediction 7 More prestigious underwriters have ac- Prediction 11 (Post-IPO sales) Over time institu- cess to larger IPO deal flow and so have higher R. Higher tional investors unload their excess inventory. Hence, R, in turn, leads to larger first-day returns and better we predict a gradual divestment of IPO shares held by Hot Markets, Investor Sentiment, and IPOs 13 institutions and an increase in the shares held by retail tions about the IPO process. investors. The model raises interesting issues for future research Boehmer and Fishe (2001) find that more than 92% of as well. Consider, for instance, the extension of the all first-day flipping transactions by investors who were model to a situation with multiple issuers, competing for allocated stock in the IPO are smaller than 10,000 shares. a fixed supply of sentiment investors. In a competitive This strongly suggests that the buyers in these transac- environment issuers may be unable to expropriate rents tions are retail investors. There is more flipping in more from the sentiment investors. If investment banks have underpriced offerings, consistent with our model. Krig- market power, however, they may restrain the number man, Shaw, and Womack (1999) show that large (pre- of firms going public and the quantity of shares offered, sumably institutional) investors are more active flippers in order to maintain IPO prices. Because of the obvi- (consistent with Prediction 10), and that they flip IPOs ous temptation to cheat among the banks, we expect to that perform the worst in the future. Field (1995) shows see punishment strategies that enforce a collusive equilib- that long-run performance is better, the larger institu- rium. We speculate that the use of investment banking tional stockholdings at the end of the first quarter of list- syndicates may play a role, with punishment strategies ing. Field does not have data on allocations, but her ev- taking the form of excluding cheating banks from future idence is consistent with the prediction that institutions underwriting syndicates. quickly sell out of the more marginal IPOs, so that by While our model does not specifically address social quarter’s end they hold more stock in the higher-quality welfare, the possible expropriation of sentiment investors companies. Dorn’s (2002) German data provide direct does give rise to some policy issues. To the extent that evidence in support of Prediction 11, by showing that such expropriation subsidizes risk-taking by young firms, the kinds of IPOs that retail investors overpay for the social welfare may be enhanced. The downside, of course, most, the hot IPOs, subsequently pass from institutional is that, as the market heats up, some firms may go public investors to retail investors. for opportunistic reasons, purely to extract surplus from Prediction 12 (Lock-ups) Insiders will be released sentiment investors. This may involve firms with neg- early from their lock-up provisions, a) if after-market de- ative NPV investment opportunities. After sentiment mand from sentiment investors is unexpectedly high, b) collapses, the IPO market may effectively be shut for all once regular investors have unloaded their excess inven- but the most ‘blue-chip’ issuers, and some positive NPV tory, or c) if the hot market has come to an end. projects may go unfunded. The social consequences of Several recent papers have documented that share exuberant investors and their possible expropriation is, prices fall significantly upon the expiry of lock-up pro- therefore, an open question. Do the exuberant provide visions [e.g. Field and Hanka (2001), Brav and Gompers subsidy to the socially productive – or are they merely (2003)], but their purpose has not previously been mod- lunch for the avaricious? eled. In our setting, lock-ups may serve to reassure institutional investors of their ability to sell at high prices in subsequent periods if the hot market persists, without having insiders compete to satisfy sentiment investors’ demand. According to Brav and Gompers (2003), early release is common: 60% of the firms in their sample have insiders sell shares prior to the lock-up expiry. The determinants of early release remain to be investigated.

7. CONCLUSIONS Our model of the IPO process links some of the main empirical IPO ‘anomalies’ – underpricing, hot issue markets, and long-run underperformance – and traces them to a common source: the presence of a class of irrationally exuberant investors. The existence of such investors, coupled with short-sale restrictions, leads to long-run underperformance. More interestingly, we resolve the ap- We are grateful to Aydo¯ganAlti, Peter Bossaerts, Robert parent paradox that underpricing and long-run under- Bloomfield, Rachel Croson, Thomas Hellmann, David performance can coexist: after all, it is not obvious a Hirshleifer, Pete Kyle, Ross Levine, Katharina Lewellen, priori why issuers do not take advantage of exuberance James Montier, Lasse Pedersen, Enrico Perotti, Jay Rit- by raising offer prices, thus eliminating underpricing. ter, Ann Sherman, Neal Stoughton, Bill Wilhelm, Jeff We show that the optimal selling policy, from the is- Wurgler, an anonymous referee, and seminar partici- suer’s point of view, usually involves staggered sales. pants at the 12th Utah Winter Finance Conference, the Such staggered sales can be implemented by allocating Fourth Texas Finance Festival, the First EVI Conference the IPO to cooperative regular investors who hold in- (Yale 2002), the RFS Conference on Experimental Fi- ventory for resale in the after-market. IPO underpricing nance (Mannheim 2002), INSEAD, New York University, compensates regulars for the losses expected from hold- American University, the University of Amsterdam, and ing inventory, given the risk that the hot market will Tilburg/CentER for helpful comments. We also thank end prematurely. The model is shown to be consistent the NYU Glucksman Institute for Research in Securities with much of the – at times seemingly contradictory – Markets for a 2001-2002 Prize for Best Faculty Research evidence on IPOs. It also generates new, testable predic- in Finance. 14 Ljungqvist, Nanda and Singh

APPENDIX PROOFS ∗ Proof of Proposition 1: For a given Q > Q1, suppose q1 > Q1. Considerq ˆ1 = Q1 and adjusting q2 toq ˆ2 such that ∗ ∗ qˆ1 +q ˆ2 = Q. P2 is unchanged, as it is a function of Q. However, from (2), P1 (ˆq1) > P1 (q1 ). Thus, Π (ˆq1) > Π(q1 ). ∗ Similarly, for a given Q > Q1, suppose q1 < Q1. Considerq ˆ1 = Q1 and adjusting q2 toq ˆ2 such thatq ˆ1 +q ˆ2 = Q. s R E (P2) is unchanged, as it is a function¡ of Q¢. In this case P1 = E (P2), which is greater than E (P2). Thus, expected s ∗ ∗ Π increases by (E (P2) − E (P2)) Q1 − q1 . Hence, q1 6= Q1 cannot be optimal. If Q ≤ Q1, the non-negativity for q2 ∗ ∗ implies q1 = Q and q2 = 0.

Proof of Proposition 4: From equation (4), the highest price P0 a regular investor is willing to pay is P0 = q1 q2 P1 Q + Q (P1(1 − γ) + VRγ). Substituting this in P1 − P0, we obtain h i q1 q2 P1 − P0 = P1 − P1 Q + Q (P1(1 − γ) + VRγ) h i q1 q2 q2 = P1 1 − Q − Q (1 − γ) − VRγ Q

γq2 = Q (P1 − VR) s Now, P1 = E (P2) > VR. Thus, (P1 − VR) > 0. Therefore (P1 − P0) > 0 if and only if q2 > 0 (or if and only if Q1 is small enough).

∗ Proof of Proposition 5: We analyze the case where Q1 is small enough such that q2 > 0. Substituting for q1 and q in P = a − λ (q + q ) + V we obtain P = a + λγ Q + V , which is increasing in γ and a. 2 1 1 2 R ³ 1 ´ 2 2(1−γ) 1 R Now consider long-run performance VR−P1 . P1 · µ ¶¸ ∂ V − P h i h i R 1 ∂P1(a) VR ∂P1(a) sign = sign − 2 = −sign ∂a P ∂a P1(a) ∂a · µ 1 ¶¸ ∂ V − P h i h i R 1 ∂P1(γ) VR ∂P1(γ) sign = sign − ∂γ P (γ)2 = −sign ∂γ ∂γ P1 1 Thus, long-run performance is decreasing in γ and a. From Proposition 4, the price reversal ratio is given by P − P γq γq 1 0 = 2 = 2 . P1 − VR Q q1 + q2 Taking the derivative with respect to a, we get µ ¶ ³ ´ ∂ P1 − P0 ∂ q2 = γ ∂a q +q ∂a P1 − VR 1 2 γq1 = 2 > 0 (q1+q2)

a−2q1λ The price reversal ratio is not monotonic in γ. To see this, note that q2 = 0 at γ = a−q λ . Thus, ½ 1 P1 − P0 γq2 0 for γ = 0 = = a−2q1λ P1 − VR q1 + q2 0 γ = a−q λ ³ ´ 1 and P1−P0 > 0 for γ ∈ 0, a−2q1λ . The derivative of the ratio is P1−VR a−q1λ µ ¶ Ã ! ∂ γq2 q2 q1 ∂q2 = + γ 2 ∂γ q1 + q2 q1 + q2 (q1 + q2) ∂γ · µ ¶¸ ∂ γq h i sign 2 = sign q (q + q ) + γq ∂q2 ∂γ q + q 2 1 2 1 ∂γ 1 2 h i γq2 = sign q (q + q ) − 1 2 1 2 2(1−γ)2

a−2q1λ The above is positive at γ = 0 and negative at γ = as q2 = 0. The second derivative is given by a−q1λ 2 µ ¶ µ ³ ´ ¶ 2 2 ∂ γq2 q1 ∂q2 ∂ q2 ∂q2 2 = (q +q )3 2 (q1 + q2) ∂γ + γ (q1 + q2) ∂γ2 − 2g ∂γ ∂γ q1 + q2 1 2 2 ∂q2 ∂ q2 q1 < 0 [as ∂γ < 0 and ∂γ2 = − (1−γ)3 < 0]. Hot Markets, Investor Sentiment, and IPOs 15 ³ ´ ³ ´ Thus, there exists aγ ˆ such that for all γ < γˆ, ∂ P1−P0 > 0 and for γ > γˆ, ∂ P1−P0 < 0. ∂γ P1−VR ∂γ hP1−³VR í h ³ í To sign the derivative of the first-day return with respect to a, notice that sign ∂ P1−P0 = −sign ∂ P0 . ∂a P0 ∂a P1 γq2 Using P0 = P1 − Q (P1 − VR) and dividing both sides by P1, we can write · µ ¶¸ h ³ ´ ³ ´ ³ í ∂ P1 − P0 sign = −sign − 1 − VR ∂ q2 + VRq2 ∂ 1 ∂a P P1 ∂a q1+q2 q1+q2 ∂a P1 0 h³ ´ ³ ´ ³ í = sign 1 − VR ∂ q2 − VRq2 ∂ 1 P1 ∂a q1+q2 q1+q2 ∂a P1 = positive ³ ´ ³ ´ ³ ´ ∂ q2 ∂q2(a) q1 ∂ 1 VR The above uses = 2 > 0, < 0, and 1 − > 0. Similarly, ∂a q1+q2 ∂a (q1+q2(a)) ∂a P1 P1 · µ ¶¸ h ³ í ∂ P1 − P0 sign = −sign ∂ P0 ∂γ P ∂γ P1 0 h³ ´ ³ ´ ³ í = sign 1 − VR ∂ γq2 − VRγq2 ∂ 1 P1 ∂γ q1+q2 q1+q2 ∂γ P1

The second term is positive and so is the ﬁrst for γ < γˆ. Thus, for low γ the ﬁrst-day return is increasing in γ.

Proof of Proposition 7: Diﬀerentiating the right-hand side of the constraint in (8) with respect to a we obtain £ ¤ ∂ ∗ ∗ ∗ sign ∂a γq2 (a, γ)(a − λ(q1 (a, γ) + q2 (a, γ))) h¡ ¢ ∗ i ∗ ∂q2 = sign a − λQ1 − 2q2 λ ∂a = positive

Thus, if the constraint binds for some a, it will also bind for all higher a. The constraint can bind only if q2 > 0, i.e. a(1−g) a(1−g) Q1 < λ(2−g) . Let Q1 = β λ(2−g) where β < 1. Substituting in the constraint we get γq∗(a, γ)(a − λ(q∗(a, γ) + q∗(a, γ))) ³ 2 ³ 1 ´´ ³ 2 ´ a γ a γλ = γ 2λ − Q1 1 + 2(1−γ) 2 + Q1 2(1−γ) ³ ³ ´´ ³ ´ a a(1−γ) γ a a(1−γ) γλ = γ 2λ − β λ(2−γ) 1 + 2(1−γ) 2 + β λ(2−γ) 2(1−γ) ³ ´ ³ ´ γ 2a(1−γ) aγ aγ = 4λ a − β (2−γ) − β (2−γ) a + β (2−γ) Taking the derivative of the above with respect to γ, we obtain 4 (1 − γ) + γ2 (1 − β) + 4βγ a2 (1 − β) > 0 (2 − γ)2

∗ ∗ ∗ ∗ Proof of Proposition 8: If the constraint binds at (q1 , q2 ) then q2 ≥ 0, which implies that q1 = Q1. Similar to the c ∗ argument provided in Proposition 1, it is easy to see that q1 = q1 = Q1. The optimal q2 if the penalty constraint binds is one of the two solutions to the following quadratic: R = q2γ (a − λ (q1 + q2)) . Solving for q2 we obtain Ã s ! 1 4Rλ q = (a − q λ) ± (a − q λ)2 − . 2 2λ 1 1 γ

Calculating the objective function³ at the aboveq two solutions,´ we can show that the diﬀerence in value of the 1 2 4Rλ objective function at q2 = (a − q1λ) − (a − q1λ) − and the value of the objective function at q2 = ³ q 2λ ´ q γ 1 2 4Rλ 2 4Rλ 2λ (a − q1λ) + (a − q1λ) − γ is q1 (a − q1λ) − γ , which is positive. Hence, the constrained optimal value of q2 is the one given in the statement of the proposition. 16 Ljungqvist, Nanda and Singh

c c Proof of Proposition 9: In the proof we use q1³and q2 insteadq of q1 and q2, respectively.´ When the constrain on 1 2 4Rλ R is binding then γq2 [P1 − VR] = R and q2 = (a − q1λ) − (a − q1λ) − . Substituting q2 in P1 we obtain q 2λ γ a−q1λ 1 2 4Rλ P1 = 2 + 2 (a − q1λ) − γ + VR, which is decreasing in R. · µ ¶¸ " # · ¸ ∂ VR − P1 ∂P1 (a) VR ∂P1 (a) sign = −sign 2 = −sign ∂R P1 ∂R P1 (a) ∂R Thus, long-run performance is increasing in R. Now consider the price reversal ratio: " Ã !# · µ ¶ µ ¶¸ ∂ P − Pˆ ∂ P − P ∂ r sign 1 0 = sign 1 0 + ∂R P1 − VR ∂R P1 − VR ∂R P1 − VR ³ ´ ∂ γq2 γq1 ∂q2 The first term is equivalent to = 2 , which is positive. The second term, ∂R q1+q2 (q1+q2) ∂R · µ ¶¸ · ¸ ∂ r ∂P sign = −sign 1 = positive ∂R P1 − VR ∂R ³ ´ ˆ Thus, ∂ P1−P0 > 0. To prove the comparative statics on the first-day return, examine ∂R P1−VR " Ã !# h ³ í ∂ P1 − Pˆ0 sign = sign ∂ P1−P0+r ∂R P0−r ∂R Pˆ0 h i ∂(P1−P0) ∂P0 = sign ∂R (P0 − r) − ∂R (P1 − P0 + r) h i ∂(P1−P0) ∂P0 ∂P1 = sign ∂R P0 − ∂R (P1 − P0) − r ∂R ³ ´ ˆ Given that ∂P1 > 0, to show ∂ P1−P0 > 0, it is sufficient to show that ∂R ∂R Pˆ0

∂ (P1 − P0) ∂P0 P0 − (P1 − P0) > 0 ∂R ∂Rµ ¶ ∂ P − P ⇐ 1 0 > 0 ∂R P0

P1−P0 γq2 P1−P0 R To show the above, we use = and R = γq2 (P1 − VR) to obtain = . Thus, P1−VR q1+q2 P0 P0(q1+q2) · µ ¶¸ ∂ P − P h ³ ´i sign 1 0 = sign ∂ R ∂R P ∂R P0(q1+q2) 0 £ ¤ ∂ = sign P0 (q1 + q2) − R (P0 (q1 + q2)) h ∂R i ∂ ∂q2 = sign P0 (q1 + q2) − R ∂R ((P0 − VR)(q1 + q2)) − RVR ∂R h i ∂ ∂q2 = −sign ∂ (P0 − VR)(q1 + q2) + VR ∂a h i ∂ ∂q2 = −sign ∂a Π(q2 (γ)) + VR ∂a h i h i ∂Π ∂q2 ∂q2 ∂Π = −sign + VR = sign + VR ∂q2 ∂a ∂a ∂q2 = positive Hot Markets, Investor Sentiment, and IPOs 17 References

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