Sequential Financial Market Trading: The Role of Endogenous Timing

Transcription

1 Sequential Financial Market Trading: The Role of Endogenous Timing Andreas Park University of Toronto July 2004 Abstract The paper analyses a simplified version of a Glosten-Milgrom style specialist security trading model with trade-timing. In a setting where traders are differentially informed, if the best-informed investors have a sufficiently strong or weak impact on prices then the investors with the strongest impact on prices delay their investment strategically, pretending to be the low-impact types. JEL Classification: C70, D80, D83, D84, G12, G14. Keywords: Microstructure, Sequential Trade, Speculation, Trade timing. Financial support from the DAAD (D/99/13779), the ESRC (R ), and the EU Commission (TMR grant number HPMT - GH ) is gratefully acknowledged. andreas.park@utoronto.ca; web: apark/

2 1 Introduction In most models of sequential financial market trading investors are assumed to enter the market in an exogenous sequence. 1 In this paper I develop a framework so that investors can also decide when to trade and I describe the resulting trading equilibria and prices. I employ a stylised version of Glosten and Milgrom (1985) s sequential trade model with two periods, two investors, two types of informed investors and two liquidation values, and one neutrally informed market maker. An informed investor s type signifies the quality of this investors private information. A natural equilibrium then is a pure strategy equilibrium in which better informed investors move first, immediately using their informational advantage, and less informed move second. 2 If a high type s action has a sufficient impact on expectations relative to a low type s, however, this separating situation is not an equilibrium. Instead, better informed traders will play a mixed strategy equilibrium and thus strategically delay investment. This result contrasts Smith (2000), who outlines conditions so that an informed investor will be impatient. In his model, however, the informed insider was small and did not affect prices; critically, in my setting, the better informed agent fears his own informational impact on the price and thus shades his information by playing a mixed strategy. Since the high type plays a mixed strategy, there is more trading in the second period (hence delay) than if there was a separating equilibrium. The outcome is related to Kyle (1985) who shows that an informed insider tries to shade his private information. In Kyle, the insider shades information through the size of his order; here information is shaded through a mixed strategy over time. If, on the other hand, the low type has a relatively stronger impact on prices, as is the case when high types are sufficiently rare, the situation is reversed: The high type will choose a pure strategy, playing first, and the low type will mix. A pure strategy equilibrium is also possible if the model parameters are somewhere in the middle ; numerically, however, this region of parameters is very small. Thus in most cases there is strategic delay and shading of information. In Section 2 I outline the underlying setup, in Section 3 I describe the equilibrium prices and properties of trading equilibrium behavior. The first Appendix includes a numerical example that illustrates the arguments and the relevant parameter restrictions. The proof of the main result is in the second appendix. 1 See for instance Glosten and Milgrom (1985), Easley and O Hara (1987), or Avery and Zemsky (1998). 2 As, for instance, in Chamley and Gale (1994). 1 The Timing of Trades

3 2 The Basic Setup and Trading Equilibrium 2.1 Institutional Details The Security and States of the World. There are two states of the world, Θ = {G,B}, the good state θ = G, and the bad state θ = B. Asset V is an Arrow Security that pays one unit in the good state and nothing in the bad state, V (G) = 1 and V (B) = 0. Both states are equally likely, and this is common knowledge. Investors. There are two investors; investors can either be informed insiders or they are uninformed and may experience a liquidity shock. Informed Insiders. They are risk neutral, fully rational and each informed insider receives a costless, private, conditionally i.i.d. signal S Θ about the true liquidation value. There are two possible types of informed traders, τ {h,l} signifying high and low type. Types differ by their signal-precision, that is the probability of receiving a correct signal, Pr(S = θ θ,h) = q h > q l = Pr(S = θ θ,l) > 1/2. An insider s type is private information. With probability µ h a trader is a high type, with µ l he is a low type, and define µ := µ h + µ l, µ (0, 1). There are also noise traders who I will describe below. An investor s profit is the absolute difference between private expectation and transaction price. Time. There are two periods in which investors can trade, t {1, 2}. The asset pays out after trading stops. Market Maker. Trade in the market is organised by a market maker. He has no private information and is subject to competition, thus makes no profits. In both periods, he posts a bid-price Bid t at which he is willing to buy one unit of the security and an ask-price Ask t at which is willing to sell one unit of the security. Investor Actions. In Period 1, investors can buy or sell one unit of the security at prices proposed by the market maker, or they can delay until Period 2. If an investor short-sells, he is obliged to serve the asset s payoff. If the investor delays, in the second period he can again buy, sell, or not trade at all. Thus the set of actions for each agent is A = {buy, no trade, sell} = {b, nt, s}. I allow that investors attempt to play at the same moment. Thus let a t denote the attempted actions in period t, a t {b, s, bb, ss, nt}. I want to focus on the timing of a single trade. Although one can easily imagine many other trading strategies, I restrict investors to make at most one 2 The Timing of Trades

4 trade: once an investor has traded he remains inactive. Furthermore, I allow only symmetric strategies, so that the trader s identity is irrelevant. Then an insider s strategy set is {buy in 1, (hold in 1, buy in 2), (hold in 1, hold in 2), sell in 1, (hold in 1, sell in 2)}. I will, however, not use strategies explicitly, but state verbally what strategies are assumed and restrict attention to observed (and anticipated) actions. Noise Traders. With probability 1 µ a trader is uninformed and receives no signal. In each period with probability 1/3 this trader receives a liquidity shock which forces him to buy and likewise with probability 1/3 he receives a shock that forces him to sell. Absent a shock this trader does not trade. He is thus a noise trader. In what follows, whenever I use the term trader, investor, or insider I am referring to an informed agent; the presence of noise traders is always implicitly assumed. Noise traders are necessary in order to prevent a market breakdown á la Milgrom and Stokey (1982). In any period, a noise-trader buy, hold or sale occurs with probability γ = (1 µ)/3 each. Beliefs. Prior to observing the order flow in each period, market maker and traders have to form beliefs about which type accounts for an action. Let ωt τ [0, 1] denote the belief that type τ will trade in period t, and let ω t = (ωt h,ωt). l The market maker uses public belief ω t to form bid- and ask-prices. Trader i uses the public belief to form his belief about the other trader, i; he needs this belief for his timing decision. To simplify notation and to emphasize which type is currently believed to act, I use the following shortcut-notation: If ω t = (1, 0), I use H instead of ω t to indicate that a high-type plays. Likewise if ω t = (0, 1), write L to indicate that a high type plays. Finally, for σ 1,σ 2 > 0 in ω t = (σ 1,σ 2 ), write σ 1 H,σ 2 L. Public Information. The structure of the model is common knowledge among all market participants. The identity, type and signal of a trader are private information, but everyone can observe the trade and transaction price in Period 1. Let H 2 summarize Period 1 actions a 1 ; and H 1 =. Organisation of Trading. The bid-ask spread is quoted before traders submit their orders. Imagine the following setup: The market maker stands on a stage, investors are assembled in front of him, separated by a Chinese Wall, so that cannot observe each other. In the first period the market maker posts two prices: the bid at which he is willing to buy one unit of the asset, and the ask at which he is willing to sell one unit of the security. In this period, the market maker is obliged to serve at least one unit at each posted price; he may decline to serve more than one trade. Thus an investor willing to trade may not 3 The Timing of Trades

5 be served if someone else submits an order at the same moment. The end of the period is announced publicly. In addition transactions are publicized. If an investor has traded he will leave the market. At the beginning of period two, the market maker posts two new prices. Traders may again state interest in a transaction, and the market maker is again obliged to serve at least one transaction at each posted prices. If the market maker faces more demand than he is willing to satisfy, a third round of trading with new prices will be opened. Else all trading stops after Period 2. After the last trade, the true state is revealed and the security pays out. Short-selling investors, including the market maker, have to transfer the security payoff. 2.2 The Trading Equilibrium The Market Maker s Price-Setting. Insiders are better informed than the market maker. Consequently, if the market maker sets a price below an informed trader s private expectation, he makes an expected loss on this trade. However, if he sets a price above the public expectation he gains on noise traders, as their trades bear no informative value. As he is subject to competition, he must make zero profits so that these two effects cancel. In equilibrium the market maker anticipates the information content of the time t {1, 2} trade at ask-price Ask t (H t,ω t ) or bid-price Bid t (H t,ω t ) Ask t (H t,ω t ) = E[V a t = one buy,ω t,h t ], Bid t (H t,ω t ) = E[V a t = one sale,ω t,h t ](1) For instance, if in Period 1 only noise traders and high-type informed agents with the good signal buy, S = G, ω 1 = (1, 0), then Pr(buy θ = G,ω 1 ) = γ + µ h q h and Pr(buy θ = B,ω 1 ) = γ + µ h (1 q h ). If the high type with the favourable signal buys only with probability σ in Period 1 and the low type not at all, then ω 1 (σ, 0) and Pr(buy θ = G,ω 1 ) = γ + σ µ h q h. The following likelihood ratios measure type τ s informational impact on prices and expectations l h := γ + µ h (1 q h ) γ + µ h q h, l l := γ + µ l (1 q l ) γ + µ l q l. Throughout this paper I will focus only on the buying side of the market; the selling side follows analogously. The Trader s Optimal Choice. A type τ trader buys in Period 1, if (a) his expected payoff from buying in Period 1 is positive and (b) if given the behavior of the other trader, trading in Period 2 yields smaller payoff than buying in Period 1; analogously for selling. He holds in Period 1 and delays to Period 2, if (a) trading in Period 2 yields positive 4 The Timing of Trades

6 expected payoffs and (b) a Period 2 trade yield higher expected profits than Period 1 trades. Information Updating. Market maker and informed traders observe the Period 1 trade, and know the informational content through the equilibrium bid-ask-spread. An Example for Equilibrium Prices. Suppose, that in equilibrium both types use pure strategies so that type h trades in Period 1 and type l in Period 2. Note, however, that if there are two buys in Period 1, the market maker will serve only one trade and turn down the other. In the second period, he will then post a price that anticipates an arriving trader to be a high type. All potential prices can then actually be computed; they are Ask 1 (H) = l h, Ask 2 (a 1 = bb; H) = Ask 2 (a 1 = s; L) = l, Ask 1 2(a 1 = b; L) =, h 1 + l h l l l h, Ask 2 (a 1 = b; L) = 1, l h + l l 1 + l l and Ask 3 (a 1 = nt,a 2 = bb; L) = l l 2. 3 Equilibrium Analysis There are four candidates for pure strategy equilibria: (i) Both types trade in Period 1, (ii) both trade in Period 2, (iii) low types trade in Period 1, high types in Period 2, and (iv) high types trade in Period 1 and low types in Period 2. The first two constellations, (i) and (ii), clearly cannot be equilibria: The lower the informativeness of a trade, the lower the spread. If the market maker anticipates only a noise trader action, the spread is zero and any informed trade at a zero spread is more profitable than at a positive spread. Thus informed investors would deviate and trade in the noise-trader -period. As it turns out, if a high type s action has a sufficiently strong or weak impact on expectations relative to a low type s action, none of these can be an equilibrium. A High Type Trader s Payoff from Playing in Period 1 Suppose for now, that in equilibrium both types use pure strategies and consider the candidate pure strategy equilibrium (iv) where type h trades in Period 1 and type l in Period The argument concerning the other candidate pure strategy equilibrium, (iii), is conceptually identical. 5 The Timing of Trades

7 Ask 2 Ask 1 E[V a 1 ] Bid 2 Ask 1 Ask 1 Ask 2 E[V ] E[V ] E[V ] E[V a 1 ] Bid 1 Bid 1 Ask 2 E[V a 1 ] Bid 2 Bid 1 Bid 2 t = 1 t = 2 a 1 = single buy t = 1 t = 2 a 1 = single sale t = 1 t = 2 a 1 = both hold Ask 2,E[V a 1 ] Ask 1 Ask 1 Ask 1 E[V ] E[V ] E[V ] E[V a 1 ] Bid 1 Bid 1 Bid 1 t = 1 t = 2 Bid2,E[V a1] t = 1 t = 2 t = 1 t = 2 a 1 = both buy a 1 = both sell a 1 = one buy, one sale Figure 1: Development of Prices. First-period bid and ask prices are fixed (there are no pricedemand schedules). If there are two trade-attempts on the same side of the market, there will be no trade on the other side of the market in Period 2. If trades occur on different sides of the market, there is no more trade in Period 2 (and thus no prices). Consider a high type h with signal S = G (notation S will henceforth be omitted), and name this investor i; the other is i. Trader i considers three potential outcomes: i can also buy, i can sell, and i can delay for one more period. If (iv) were an equilibrium types get screened over time, and in each trading period signals get screened through the bid-ask-spread. In this context, a t obviously denotes i s action. Then for the high type buying in Period 1 yields 1. E[V a 1 = b,g,τ = h, H] Ask 1 (H) if i also buys and i is successful. This occurs with probability Pr(a 1 = b G,h, H)/2. If i is successful, i s payoff is E[V a 1 = b,g,h, H] Ask 2 (H). 2. E[V a 1 = s,g,h, H] Ask 1 (H) if i sells. This occurs with probability Pr(a 1 = s G,h, H). 3. E[V G,h, H] Ask 1 (H) if i delays. This occurs with probability Pr(a 1 = nt G,h, H). A delay here carries no informational value. 6 The Timing of Trades

8 A High Type Trader s Payoff from Playing in Period 2 If the high type trader decides to delay to Period 2, there are again three outcomes, 1. If i bought in Period 1, i s payoff now will be E[V a 1 = b,g,h, L] Ask 2 (b; L). 2. If i sold in Period 1, buying will now yield E[V a 1 = s,g,h, L] Ask 2 (s; L). 3. If i delayed in Period 1, then three cases analogue to those in Period 1 arise. (a) i delays again. Only a noise trader would do this. When buying now, under (iv) the h-type is perceived as an l-type. The payoff then is E[V G,h, L] Ask 2 (nt; L) and it occurs with probability Pr(a 1 = nt G,h, H) γ. (b) i sells. Then the payoff is E[V a 1 = nt,a 2 = s,g,h, L] Ask 2 (nt; L). This occurs with probability Pr(a 1 = nt G,h, H) Pr(a 2 = s G,h, L). (c) i buys. If i is successful, i will get the asset only in after Period 2 at a price taken him as a low type. i s payoff then is E[V a 1 = nt,a 2 = b,g,h, L] Ask 2 (nt; L). This happens with probability Pr(a 1 = nt G,h, H) Pr(a 2 = b G,h, L)/2. If, on the other hand, i is successful himself, i pays a lower price and gets E[V a 1 = nt,a 2 = b,g,h, L] Ask 2 (nt; L). Lemma (The Law of Iterated Expectations) An insider s Period 1 expectation over his expected payoff from the Arrow Security in Period 2 is the same as his expected payoff in Period 1. The lemma s statement is equivalent to E[V a i t = b,g,τ]pr(a i t = b G,τ) + E[V a i t +E[V a i t = s,g,τ]pr(a i t = h,g,τ]pr(a i t = s G,τ) = h G,τ) = E[V G,τ], a standard result in statistics. However, I will show in the next subsection that a trader s expectation about the future return is not equal to his current return: the insider may well expect prices to increase or decrease (depending on his signal and type). Strategic Excess Delay First observe, that the bid-ask spread is formed around the public expectation, thus agents do not change the side of the market, or in other words, whoever contemplates buying in Period 1 would not sell in Period 2 and vice versa. Since the law of iterated expectation applies to expected payoffs from the Arrow security, it suffices to analyse the development 7 The Timing of Trades

9 of prices. In what follows in a semi-mixed strategy equilibrium, one type mixes and the other plays a pure strategy. Two effects generate semi-mixed strategy equilibria: If i sells in Period 1, by delaying the other investor gets a lower price. If i buys, then i will have to pay a higher price if i does not get the security. When avoiding this clash by delaying to in Period 2, the insider will be perceived as a low type. If the high type has a sufficiently High Impact on prices, Condition (HI), expected Period 2 prices are preferred. If, on the other hand, the high type has a sufficiently Low Impact on prices relative to the low type, Condition (LI), then the situation is reversed in the following sense: The low type wants to deviate and play a mixed strategy between Period 1 and 2, and the high type plays with certainty in Period 1. Condition (HI): A high type s action is sufficiently informative relative to a low type s so that l l > 1(1 + l 2 h).04. Condition (LI): A high type s action is sufficiently uninformative relative to a low type s so that l l < l 2 h. Proposition (No Pure Strategy Equilibria with Endogenous Timing) (i) If Condition (HI) or Condition (LI) holds, there exists no pure strategy equilibrium. (ii) If Condition (HI) holds, in the equilibrium high types play a mixed strategy between both periods, low types play only in the second. (iii) If Condition (LI) holds, in equilibrium, low types play a mixed strategy between both periods, high types play only in the first. The proof follows counterfactually: Hypothesizing that there could be pure strategy equilibria, I show that in all these cases the high type has an incentive to deviate. Condition (HI) comes into play as follows: if two agents try to trade simultaneously on the same side of the market, only one can be served. The other will only get a price that includes this agent s signal. Then if l l is sufficiently larger than l h, that is, a high type has a sufficiently strong impact on prices, the average price in this situation, 1/2(Ask 1 (H)+Ask 2 (bb; H)), is smaller than the price achieved if the other moves first, Ask 2 (b; L). It is obvious that the ask-price decreases if the other sells. Condition (HI) also ensures, that prices decrease in case of a hold, too. The situation is reversed if the low type has a sufficiently high impact on prices: Condition (LI) ensures, that even when there is a sale in the first period, the ask price in the second period is higher than in the first. Observation (Pure Strategy Equilibria) If neither (HI) nor (LI) are satisfied, there might exist a pure strategy equilibrium. In 8 The Timing of Trades

10 numerical simulations, the parameter region allowing pure strategy equilibria, however, turns out to be very small. Conditions (HI) and (LI) are, in fact, relatively coarse. Even when they are not satisfied, there can be breakdown of separation. The Appendix A contains a numerical example that illustrates this point. 4 Discussion of the Result In classic models of sequential trade, agents cannot choose the time of market entry. In these models private and public information converge, thus the bid-ask spread gets smaller. Furthermore, as Smith (2000) pointed out, an investor with a favourable signal expects transaction prices to increase (for him it is a sub-martingale). Casual intuition then suggests, that profits from speculation are largest early thus investors should invest rather earlier than later. My result is somewhat surprising as it states the opposite: under Condition (HI), intertemporal separation of types would lead a high-type investor to expect ask-prices to fall in the second period, so he should delay investment. 4 A A Numerical Example The following numerical example illustrates the proposition graphically: The model depends on the values of four parameters: the qualities of signals and the proportion of low and high types. The graphs were generated for a constant level of informed trading, set to µ = µ h + µ l = 0.5, and q l = 0.6. All plots are for expected prices of traders with signal S = G. The right panels always plot the intersection of the curves in the left with the 0-surface. Although some may seem identical, they are not as the theoretical results suggest, they also do not intersect. Some curves in the left panels have been scaled upwards for illustrative purposes. The left panel in Figure 2 plots Conditions (HI) and (LI), and also the difference of a high type s Period 2 and Period 1 expected prices under the assertion that low types trade in Period 2, and high types in Period 1. Whenever this curve is positive, the high type should deviate and play in Period 2. The conditions indicate, that whenever the relative effect of correct high buys is large (thus spreads would be large) prices in Period 2 are smaller and thus it is better to delay. The left panel of Figure 3 plots the differences of Period 2 and Period 1 prices for both types for the situation where the high type plays first. The left panel in Figure 4 plots 4 Of course, in the mixed strategy equilibrium he is indifferent. 9 The Timing of Trades

11 pi mu_h q_h Figure 2: Conditions (HI) and (LI), and the difference between Period 1 and Period 2 prices for the high type in the equilibrium where the high type moves first, the low type second. the difference of Period 1 and Period 2 prices for the situation where the low type plays first. In Figure 3, the high s assessment of price-differences is the lower curve, in Figure 4 it is the upper curve. Let σ denote the probability of the high type playing in the opposite period than the low type. To find the equilibrium strategy one would have to find σ so that for given q h and µ h, the h type price-difference is zero. For instance take Figure 3: If σ traverses from 1 to 0, the two curves shift downwards, but the low s curve always stays above the high s curve. Thus when the high type is indifferent between playing in Period 1 or 2, the low type prefers to play in Period 2. The opposite holds in the left panel of Figure 4, as the low type s curve stays below the high type s: If the high type is indifferent between both periods, the low type strictly prefers the second. This contradicts the assertion that the low type plays in Period 1 and thus underlines the Proposition s claim. The right panel of the Figure 3 also indicates the parameter region which allows a pure strategy equilibrium: It is the area between both curves. For parameter in this region, Period 2 price are higher than Period 1 prices for the high type and the reverse for the low type. The situation is different in the right panel of Figure 4, whenever the high type prefers Period 1, the low type prefers Period 2; thus there is no pure strategy equilibrium in which the low plays first and the high second. 10 The Timing of Trades

12 pi mu_h q_h mu[h] Figure 3: The difference between Period 1 and Period 2 prices for the high and the low type type in the equilibrium where the high type moves first, the low type second. The top curve is the difference for the low type. B Proof of the Proposition I will only consider the buy side of the market and thus all traders I mention in this proof are assumed to have a favourable signal, s = G. (i)[no Pure Strategy Equilibria] Low Type plays second. Consider first the case where the low type plays in Period 2. I need to show that the ask- price is smaller in Period 2. In Period 1, i can either (a) sell, (b) buy or (c) not trade at all. I will now show that in each case Period 2 expected prices are smaller than period prices. In Case (a), clearly prices decrease 5 Ask 1 (H) Ask 2 (s; L) = l h l h l h + l l l l l h 2 > 0, (2) where the last inequality follows from Condition (HI). Now suppose that the other investor buys in Period 1. When also buying in Period 1, the expected price agent i faces is is 1/2(Ask 1 (H) + Ask 2 (bb; L)). Period 2 s price will be Ask 2 (b; L). Thus 1 2 (Ask 1(H) + Ask 2 (bb; L)) Ask 2 (b; L) = l h l 1 (3) h 1 + l h l l 5 I use the symbol as follows: term in front of has the same sign as the term behind it. 11 The Timing of Trades

13 pi mu_h q_h mu[h] Figure 4: The difference between Period 1 and Period 2 prices for the high and the low type type in the equilibrium where the low type moves first, the high type second. The top curve is the difference for the high type. Manipulating and rearranging, one can see that (3) is positive if 1 + l h l > 1 l l l l > 2l h 2 + l h h l l l h 2 + l h + l. h Numerically it is straightforward to check that Condition (HI) ensures that the inequality also holds, thus, again, Period 2 s price is smaller. Suppose now that there is no trade in Period 1. Period 1 s price is Ask 1 (H). In Period 2, the other can buy, sell or not trade. If i does not trade or sell, investor i has to pay Ask 2 (nt; L). It is straightforward to see that Ask 1 (H) Ask 2 (nt; L) = l h l l l l l h > 0 (4) because of Condition (HI). If i buys in Period 2, when buying the trader faces price 1/2(Ask 2 (nt; L) + Ask 3 (nt, bb; L). Clearly, price Ask 3 (nt, bb; L) is the highest. However, Ask 1 (H) Ask 3 (nt, bb; L) = l h l l 2 l l 2 l h. Since l l,l h [0, 1], it is straightforward to check that Condition (HI) implies l l 2 l h > 0. Thus again the ask-price is expected to drop in Period 2. Thus overall if the low type is expected to play in Period 2, under Condition (HI) prices are lower in Period 2. Consequently it cannot be an equilibrium that the high type plays in Period 1 and the low in Period The Timing of Trades

14 Low Type plays first. Consider now the reverse situation where the low type moves first, the high type second. First, suppose that i sells. Then Ask 1 (L) Ask 2 (s; H) = l l l l l l + l h l h (1 l l ) < 0, so Period 2 s price is larger. Suppose now that i buys. Buying in Period 1, player i has to pay 1/2(Ask 1 (L)+Ask 2 (bb; L)); when delaying until in Period 2 to buy then, the player is charged Ask 2 (b; H). Clearly, Ask 1 (L) Ask 2 (b; H) = l l and Ask 2 (bb; L) Ask 2 (b; H) = l 2 l 1 l h 1 < l h l l (5) 1 l h l l < 0, 1 + l h l l (6) where the last inequality holds because of Condition (HI). Finally, suppose there is no trade in Period 2. Then the smallest possible ask price in Period 2 is Ask 2 (nt; H). But Ask 1 (L) Ask 2 (nt; H) = l l l h l h l l < 0. (7) To summarize, when the low type plays first the high type second, Period 1 prices are always smaller than Period 2 prices. Consequently, this cannot be an equilibrium. (ii) [Mixed Strategy Equilibrium] The proof for (ii) follows in three steps, (iii) is analogous. In the first step to prove (ii), I will describe properties of a candidate semi-mixed strategy equilibrium. In this equilibrium, the high type mixes, the low type plays a pure strategy. In the second step I will show that there exists such a semi-mixed equilibrium in which the high type mixes between Period 1 and Period 2, the low plays in Period 2. Finally, I argue that under Condition (HI) it cannot be an equilibrium that the low type mixes and the high plays a pure strategy. 6 Step 1: [Setup for Mixed Strategy Equilibria] Consider the two candidate semimixed strategy equilibria where the high type mixes between both periods and the low type plays in Period 1 or 2 respectively. Then I define that the high type plays buys with probability σ in the other period than the low type. For instance, if the low type plays in Period 2, the high type plays with probability σ in Period 1, and 6 As a more elaborate model one can imagine that agents receive a continuous, conditionally i.i.d. signal. Potential types can then be identified by their signal directly. Separation means that there exists two threshold signals, a high and a low bound. Every player with a signal above the high bound buys in Period 1 and below the low bound sells in Period 1, everyone between the bounds delays until Period 2. The threshold types are indifferent. Then in my setting mixed strategy play essentially can be understood as creating a virtual threshold type. 13 The Timing of Trades

15 with 1 σ in Period 2. Naturally, in this case beliefs are ω h 1 = σ and ω h 2 = 1 σ; for simplicity I write σh and (1 σ)h. The informational content of a buy in Period 1, given θ = G, is thus σµ h q h + γ, in Period 2 it is (1 σ)µ h q h + µ l q l γ. Consequently, the likelihood ratios now depend on the value of σ. In what follows I will use l h (σ) = γ + σ µ h (1 q h ) γ + σ µ h q h, l l (σ) = γ + (1 σ) µ h (1 q h ) + µ l (1 q l ) γ + (1 σ) µ h q h + µ l q l. (8) Step 2: [Structure and Existence of a Semi-Mixed Equilibrium] The idea is simple though. In (i) above, I have argued that from the perspective of the high type the ask price is always smaller in the period when the low type is expected to play. If both high and low type play in Period 2 and only noise traders in Period 1, then the bid-ask-spread is 0 and thus ask-prices minimal. Consequently, the high type would prefer to play in Period 1 with some positive probability. As expected prices are continuous in σ, by the middle-value theorem, there exists a probability σ so that he is indifferent. In a (semi-) mixed strategy equilibrium, σ must be so that the high type is indifferent between playing in Period 1 and 2. Thus σ must be such that for the high type, expected Period 1 and expected Period 2 price are the same. I now have to show that if the high type is indifferent, the low type prefers to play in Period 2. Thus, again, I have to run through the prices in the different cases of a Period 1 buy, sale or no trade and then the same for Period 2. When σ traverses from 1 to 0, l l (σ) decreases and l h (σ) increases. If there is a sale in Period 1, the ask-price in Period 2 will be smaller. However, the difference between this new ask price and the Period 1 ask-price gets smaller if the high type plays with more weight in Period 2. Thus the difference decreases if σ falls. The same holds for a buy in Period 1. In contrast to (i), I now have to take the investor s probability assessment of every trade into account. Suppose now that the high type is indifferent. It now holds that Pr(buy in 1 τ = h,σh,s = G) = q h (σµ h q h + γ) + (1 q h )(σµ h (1 q h ) + γ) > Pr(buy in 1 τ = l,σh,s = G) = q l (σµ h q h + γ) + (1 q l )(σµ h (1 q h ) + γ), and for sales the situation is reversed, i.e. the low type with the good signal considers a sale more likely than the high type with a good signal. In general, if there is a buy, prices increase, for a sale the reverse. The high type considers a buy more likely, thus when the high type is indifferent, the low type strictly prefers Period 2 s 14 The Timing of Trades

16 prices. Now consider the case of no trades in Period 1. Both types consider this equally likely. In Period 2, the high type (with s = G) again considers a buy more likely than the low type and the reverse for sales. Both also consider Period 2 holds equally likely. Thus I omit the price in case of a Period 2-no trade. Define B l := Pr(buy in 2 θ = G, (1 σ)h, L) = (1 σ)µ h q h + µ l q l + γ, S l := Pr(buy in 2 θ = B, (1 σ)h, L) = (1 σ)µ h (1 q h ) + µ l (1 q l ) + γ. Then, omitting Period 2 holds, the expected Period 2 price after no trade in Period 1 is Pr(buy in 2 τ = h,(1 σ)h,l) 1 2 (Ask 2(nt;(1 σ)h,l) + Ask 3 (nt,bb;(1 σ)h,l)) +Pr(sale in 2 τ = h,(1 σ)h,l)ask 2 (nt;(1 σ)h,l) Pr(buy in 2 τ = l,(1 σ)h,l) 1 2 (Ask 2(nt;(1 σ)h,l) + Ask 3 (nt,bb;(1 σ)h,l)) +Pr(sale in 2 τ = l,(1 σ)h,l)ask 2 (nt;(1 σ)h,l) (q h B l + (1 q h )S l ) 1 2 (Ask 2(nt;(1 σ)h,l) + Ask 3 (nt,bb;(1 σ)h,l)) +(q h S l + (1 q h )B l )Ask 2 (nt;(1 σ)h,l) (q l B l + (1 q l )S l ) 1 2 (Ask 2(nt;(1 σ)h,l) + Ask 3 (nt,bb;(1 σ)h,l)) +(q l S l + (1 q l )B l )Ask 2 (nt;(1 σ)h,l) B l ( 1 2 (Ask 2(nt;(1 σ)h,l) + Ask 3 (nt,bb;(1 σ)h,l)) Ask 2 (nt;(1 σ)h,l)) S l ( 1 2 (Ask 2(nt;(1 σ)h,l) + Ask 3 (nt,bb;(1 σ)h,l)) Ask 2 (nt;(1 σ)h,l)). But obviously B l S l, and Ask 3 (nt, bb; (1 σ)h, L) Ask 2 (nt; (1 σ)h, L), so the inequality holds indeed. Consequently, the low type has a lower expectation of Period 2 prices than the high type in the case with no trade in Period 1. To summarise: (1) The high type is indifferent. (2) Sales lead to lower Period 2 prices and the low type considers this scenario more likely than the high type. (3) To have indifference for the high type, when there is a buy in Period 1, prices must be increasing in Period 2 relative to Period 1. (Thus Condition (HI) must not hold.) The high type considers this scenario more likely, thus the low type puts less weight on this effect. (4) For holds, relative to the low type, the high type believes that prices in Period 2 will be higher. Therefore, if the high type is indifferent, the low type believes that Period 2 prices are lower and thus plays only in Period 2. Step 4: [Low Type cannot Play First] If the low type plays first, the situation of Step 3 is exactly reversed: Prices charged in Step 3 if there is a hold in Period 2, are now the prices to be charged in Period 1, and vice versa. Consequently, under the 15 The Timing of Trades

17 same conditions yielding profitable deviations from Period 1 to Period 2 in Step 2, here the high type would deviate to play in Period 1. This implies that for σ = 1, Period 1 prices are lower than Period 2 prices. Suppose now σ equilibrates the high type s Period 1 and Period 2 payoffs. Then the assessment from Step 3 is reversed the low type believes that the high-price-states will be more likely and thus to a low type, Period 2 prices are lower. Therefore it cannot be an equilibrium that the low type plays first. Step 5: [Low type cannot Mix] Suppose under Condition (HI) the high type plays a pure strategy and the low type mixes. Then the other Period is always more attractive for the high type by the arguments from Step 2 the low s mixing strengthen s Condition (HI). The proof for (iii) follows analogously: Condition (LI) straightforwardly ensures that Ask 1 (bh) < Ask 2 (sh, hb), as (2) turns negative. Thus even in the best case scenario, when a sale should push prices down, the ask-price does not fall. References Avery, C., and P. Zemsky (1998): Multi-Dimensional Uncertainty and Herd Behavior in Financial Markets, American Economic Review, 88, Chamley, C., and D. Gale (1994): Information Revelation and Strategic Delay in a Model of Investment, Econometrica, 62, Easley, D., and M. O Hara (1987): Price, Trade Size, and Information in Securities Markets, Journal of Financial Economics, 19, Glosten, L., and P. Milgrom (1985): Bid, Ask and Transaction Prices in a Specialist Market with Heterogenously Informed Traders, Journal of Financial Economics, 14, Kyle, A. (1985): Continuous Auctions and Insider Trading, Econometrica, 53, Milgrom, P., and N. Stokey (1982): Information, Trade and Common Knowledge, Journal of Economic Theory, 26, Smith, L. (2000): Private Information and Trade Timing, American Economic Review, 90, The Timing of Trades