black equilibrium exchanges

CFA Institute

Equilibrium ExchangesAuthor(s): Fischer BlackSource: Financial Analysts Journal, Vol. 51, No. 3 (May - Jun., 1995), pp. 23-29Published by: CFA InstituteStable URL: http://www.jstor.org/stable/4479843 .Accessed: 08/02/2015 12:57

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Equilibrium Exchanges Fischer Black

A frictionless market for exchanges will feature an equilibrium in which traders use indexed limit orders at different levels of urgency but do not use market orders or conventional limit orders. Buy limit orders will match sell limit orders at the current price. Limit order entry will move price by an amount that increases indefinitely as urgency increases, so faster execution means higher effective cost. Dealers will lose money, so exchanges will have no specialists or market makers. Traders can signal that they have no special information by using less-urgent indexed limit orders; they cannot do better by using specialized exchanges, sunshine trading, or basket trading.

How do people trade in equilibrium? What exchanges do they use, and what kinds of

orders do the exchanges offer? How do people with and without private information trade? How do more and less patient people trade?

In a competitive equilibrium, anyone can start exchanges, anyone is free to try to take over some or all the business of other exchanges, and govern- ments do not restrict the rules that exchanges may adopt. There are no restrictions on individual traders or on exchanges. Traders can deal with one another without going through exchanges. They can trade anonymously through agents or through exchanges.

An exchange that forces all traders to use personal identification will not survive, because many people value privacy, especially in financial matters. Letting each trader choose whether to identify himself seems pointless, because a person intent on deceiving other traders can then identify his innocent trades while entering his devious trades anonymously. Therefore, all trades are anonymous.

This condition means that an informed trader can imitate an uninformed trader if he can gain by doing so, and an uninformed trader can imitate an informed trader. A trader can break an order into pieces if that reduces its impact on price. He can put in a large order on one side and a small order on the other side if that improves his expected price for the net of the two orders.

Similarly, the exchange is free to use specialists or market makers or not to use them. An

exchange can offer a wide array of order types or just a few. In particular, an exchange may choose to allow anyone to imitate a one-sided dealer when he wants to open or close a position.

In this market, every trader maximizes expected utility. Trading profits increase utility, and trading delays reduce it. Nothing else affects utility. The price impact of an order can have a big effect on trading profit.

Many traders care only about expected profit. They do not care about trading delays or risk. As a result, they are quick to act on any profit opportunities created by predictable patterns in price movements or by the arrival of new public information. In equilibrium, however, these profit opportunities do not arise.

New information, no matter how perishable, is referred to as news. Those who use it are news traders. Those who trade for other reasons are nice traders because they are willing to lose what news traders make if they cannot find a way to trade that avoids those losses.

What does the unrestricted equilibrium look like? The existing papers in the literature do not ask this question. Some limit trading to one or two rounds. All restrict traders and exchanges in ways that significantly affect the resulting equilibria. These restrictions make formal analysis easier, but the authors' conclusions cannot be generalized to more realistic models.

For example, Grossman and Stiglitz used a model with just one round of trading, so they could not explore dynamic strategies and traders' objectives could not include the disutility of waiting to trade.1 Their basic idea, however, does carry Fischer Black is a partner at Goldman, Sachs & Co. in New York.

Financial Analysts Journal / May-June 1995 23


over to a more general model. Trades by news traders and nice traders look the same, so the noise created by the nice traders acts as a cover for the news traders.

Kyle extended the Grossman and Stiglitz thinking to a world with multiple rounds of trading, but he assumed a severely restricted structure of trading.2 In his world, nice traders are unable to time their trades. They throw their trades at the market in a seemingly random pattern and without warning. News traders can time their trades, but they are not free to act as one-sided market makers. Market makers are not free to invest in information so they can trade on it.

If we try to generalize Kyle's model by relax- ing its restrictions, its equilibrium collapses. For example, a person without special information can make a profit by first pretending to be a news trader and then acting as a one-sided market maker. He can open his position by buying aggressively and can close his position instantly by acting as the asked side of a market maker, or he can open by selling aggressively and close as the bid side of a market maker. Either way, he makes a profit. A world with profit opportunities is not in equilibrium.

Some models assume that everyone must trade through dealers. For example, Glosten and Milgrom and Grossman and Miller assumed that outside customers, both informed and uninformed, must trade through a specialist or market maker and cannot trade by quoting one side of the market.3 When we relax this assumption, their conclusions change completely.

Admati and Pfleiderer tried to describe an equilibrium that includes sunshine trading.4 They restricted their model by assuming that only a trader without special information can prean- nounce a trade and that he cannot do any other trading before executing his preannounced trade. These restrictions make no sense when trading is anonymous; without them, however, the equilibrium falls apart.

Glosten tried to describe equilibrium when traders can use both limit orders and market orders, but he restricted his model by saying that those who use market orders cannot switch to limit orders.5 In his model, a series of small market orders on the same side can move the price a lot. Thus, any trader who can switch from market orders to limit orders, even if uninformed, could move the price with market orders and then close his position, usually at a profit, with limit orders. He might even expect to gain by opening a posi-

tion with small market orders and closing with a large market order. Thus, if Glosten's model de- scribes equilibrium at all, it does so only in a very restricted world. I see no way to enforce restrictions like this when trades are anonymous.

In contrast, I am trying to describe equilibrium in an unrestricted world. I am only trying to describe equilibrium; I am not trying to judge one sort of exchange or method of trading as better or worse than another. In particular, I do not assume, as Harris seemed to, that an exchange should favor suppliers of liquidity over demanders of liquidity.6

If an equilibrium exists, it need not be Pareto- optimal. When people trade on information, they disclose it, at least in part, by moving prices. Early disclosure has external benefits because it improves the allocation of resources; so we cannot assume that people invest the right amount in gathering information on which they plan to trade. With no external benefits, any amount spent on information would be wasted (from society's point of view). Given the external benefits, the right amount is positive, but it may be either higher or lower than the equilibrium amount.

By restricting traders and exchanges heavily, other authors create models that lend themselves to formal analysis, but their conclusions do not generalize to more realistic models. I am unwilling to impose these restrictions, so I have been unable to provide a full mathematical description of equilibrium. Indeed, I cannot be sure that any equilibrium exists in my world, but I believe the resulting generality is worth the sacrifice of formal analysis.

In the end, then, this entire article amounts to a series of conjectures about the nature of equilibrium, if one exists. I have been unable to provide an exhaustive and precise analysis of the implications of my assumptions, but I would rather guess about what follows from more-relevant assumptions than derive precise conclusions from less- relevant assumptions.

MARKET ORDERS A conventional market order can execute at a single price (a single-price market order) or at a series of different prices set by the limit orders on the other side (an average-price market order). A news trader who uses a market order can trade quickly to reduce the chance that others will trade ahead of him on the same news, but the price impact of his trade is a cost to him.

If a news trader uses a single-price market order large enough to move the price by the full value of his news, he expects no profit on the

24 Financial Analysts Journal / May-June 1995


trade. Instead, he may try to use a series of single-price market orders. He continues trading until his expected profit from another small trade is zero-that is, until he has moved the price by the full value of his news.

Using a series of single-price market orders is tricky, however. Others may see what the trader is doing and trade ahead of him, reducing the profit he can make on his investment in news. To hide his trades, he may spread them out, even though this increases the risk that others may find out what he already knows. He may even switch to limit orders.

When a news trader can use average-price market orders, he prefers them to single-price market orders. An average-price market order is executed as if it were a series of tiny single-price market orders. The series of orders is executed all at once, so no other traders can interrupt it. A news trader pays or receives the average price of all the tiny market orders rather than the final price.

An average-price market order is just what a news trader wants, assuming that the exchange offers him access to all who might take the other side of his trade. He can trade quickly to minimize the chance that others trade ahead of him on the same news. He effectively breaks his trade into a series of small trades without allowing others to interrupt it, and he can choose a trade size that moves the price by the full value of his news without giving up any potential profit.

Roughly, market depth is the number of shares it takes to move the price by one small unit. A deep market is more attractive to news traders than a shallow market because they can execute larger trades, which means more profits. Because news trader profits are nice trader losses, a deeper market is less attractive to nice traders.

An exchange that allows average-price market orders dominates one that allows only single-price market orders. Suppose the exchange sets market depth so that news trader profits are the same as if it allowed only single-price market orders, which means nice trader losses are also the same. With average-price market orders, a news trader can execute a single trade that moves the price by the full value of his news. He need not camouflage his trades by spreading them out, which is costly to him.

In equilibrium, then, anyone who uses a market order chooses an average-price market order when he can. Exchanges might as well drop single- price market orders from their lists of available

orders. They allow average-price market orders, and their markets are shallower than if they of- fered only single-price market orders.

I can imagine an equilibrium in which all news traders use average-price market orders and all nice traders use limit orders, but only under very restrictive conditions. More generally, some news traders in any trial equilibrium want to use limit orders, and some nice traders want to use market orders.

If both news traders and nice traders use market orders, the meaning of a market order is unclear. When a news trader uses an average-price market order, he chooses a trade size that moves prices by the full value of his news. When a nice trader uses a market order, he distorts the price by an equivalent amount.

After a news trader's market order, we expect no further change in price; although after a nice trader's market order, we expect the price to revert to its original level eventually. Taking both possi- bilities into account, we expect some reversion. This reversion cannot be consistent with equilibrium, however, because it implies profit opportunities for traders who simply watch the sequence of trades. On average, the price change caused by a market order is partly reversed.

If all nice traders are sufficiently patient, we can have an equilibrium in which news traders use market orders and nice traders use limit orders. In general, though, some nice traders are impatient and some news traders are patient, so this equilibrium collapses. We can have equilibrium only if exchanges do not offer market orders at all!

LIMIT ORDERS Conventional limit orders are awkward to use. If a trader puts in a large limit order at a single price, anyone who knows about his order can take ad- vantage of him. For example, a large limit order to buy at 40 invites someone to put in a small limit order to buy just above 40. If this small order executes and the price rebounds, that trader can realize a substantial profit. If the price continues to fall, it will pause at 40, and he can close his position with a small loss. In effect, a person who puts in a large limit order at a single price gives away valuable options to traders who know about it.

A limit order at a single price rapidly becomes outdated. In fact, because market conditions change so fast, the limit price on an order is often outdated before it reaches the market. To avoid these problems, people sometimes scale their limit orders, putting in a series of sell orders at increas-

Financial Analysts Joumal / May-June 1995 25


ing prices or buy orders at decreasing prices. Scaling is costly, however. A trader starts with many orders and must add more orders when the market moves, leaving a gap between the current price and his best price.

Moreover, exchanges generally restrict limit prices to even fractions and use first come, first served as a way of allocating orders at the same price. Thus, a trader who is scaling his limit orders enters extra orders far in advance so he has a good position in the queue at each price. When he finishes trading, he cancels his extra orders.

In a frictionless market with free cancellation of conventional limit orders, there can be no equilibrium. Traders try to put in all possible orders they might ever want to use, so they will be at the head of the queue if they do want to use an order. Only one person, however, can actually be at the head of each queue.

To avoid the costs of single-price limit orders and the complications of scaling and to allow equilibrium in a frictionless world, exchanges can offer indexed limit orders. An indexed limit order has a limit price that is continually adjusted to market conditions. In effect, an indexed limit order to buy becomes an order to buy at the bid side of the market, and an indexed limit order to sell becomes an order to sell at the asked side of the market.7

Restricting trade prices to even fractions wors- ens the problem of allocating by queue, so I imagine that indexed limit orders trade at decimal prices. To reduce the incentive to put orders in early, I imagine that exchanges treat all similar orders equally. They pay no attention to an order's entry time.

Some traders who use indexed limit orders want to trade faster than others, so exchanges offer indexed limit orders that vary in urgency. An urgent order executes faster and costs more than the typical order. Urgency takes the place of time priority in allocating orders at a single price, but even the most urgent orders share all executions with other orders. When allocating scarce resources, using cost is almost always more efficient than using waiting time in a queue.

Indexed limit orders seem to dominate conventional limit orders because they take account of current market conditions and because they eliminate the incentive to put orders in early. In equilibrium, I expect all traders to use indexed limit orders at varying levels of urgency.

A news trader is likely to be less patient than a nice trader because he wants to trade before his news becomes public. Thus, urgent orders come

more from news traders than from nice traders, and orders that execute slowly but at low cost come more from nice traders.

Because any limit order may come from a news trader, arrival of an order moves the price: A buy limit order moves it up, and a sell limit order moves it down. Because news traders make up a larger share of urgent orders, arrival of an urgent order moves the price more than arrival of a typical order.

An exchange can distinguish among more and less urgent orders only when it penalizes limit order cancellation. With free cancellation, a trader can enter a very large limit order with low urgency and then cancel when he has the shares he wants. He has fast execution at low cost, thus avoiding the penalty for urgency.

Similarly, an order with constant urgency must have a fractional expected execution rate that does not depend on how much has executed so far. If the expected execution rate in shares is constant, the order's effective urgency rises as it executes, so its expected execution cost must rise too.

Because a trader is free to break up his order into small pieces or to join his order with other traders' orders before entering it, the price moves caused by pieces of an order must add up to the price move caused by the full order. Roughly, this fact means that entry of a limit order at a given level of urgency causes a price move proportional to order size.

When all traders use indexed limit orders, we can define market depth for each level of urgency as the number of shares it takes to move the price one small unit. (Earlier, when considering the possibility that traders use market orders, we de- fined depth in a related but different way.) Depth is a local property that can depend on market conditions, including the amounts of buy and sell limit orders at various levels of urgency. Buy and sell depth must be the same. Depth can appear to depend on order size only through its dependence on market conditions.

Similarly, a buy limit order and a sell limit order with the same urgency should have the same effect on price as the net of the two orders. If the effects are different, a trader who cares only about profits will see opportunities.

If we could hold market conditions fixed, the price impact of a limit order at a given level of urgency would be proportional to order size. Local depth would also be global depth. A 10,000-share order would move the price ten times as much as a 1,000-share order.



EXCHANGES An exchange has indexed limit orders at various levels of urgency on both sides of the market. Market depth depends on urgency and market conditions, including conditions at other exchanges. In fact, entering an order on one exchange changes the price at all exchanges trading the same security.

The market is shallower for more-urgent orders; market depth approaches zero as urgency continues to rise. Thus, when new public information arrives in the market, a few small, urgent orders move the price by almost the full value of the information.

In setting the depth schedule, an exchange is acting as a kind of Walrasian auctioneer. It balances supply and demand for urgency, and more-urgent orders cost more because they attract more news traders. At each level of urgency, the auctioneer follows the size of the book of limit orders and the price changes following entry of new orders. He chooses the speed at which he matches orders so that the book neither grows nor shrinks, on average. He chooses market depth so that the price neither rises nor falls, on average, following entry of a new order.

The exchange matches buy and sell limit orders continuously at the current price. On each side of the market, the fractional execution rate is higher for more-urgent orders; also, the execution rate rises as orders build up on both sides of the market. Because the exchange may have more orders on one side of the market than on the other, the actual fractional execution rates for orders of given urgency may differ between the two sides of the market.

When a news trader puts in an order, he moves the price by less than the full value of his news. His profit depends on the amount he can execute before his news becomes public and on how fast other news traders arrive in the market with the same information.

When a nice trader puts in an order, his expected trading cost rises with urgency because more-urgent orders move the price more and because they execute faster, which means the price has less chance to revert toward its original level.

With fractional execution of limit orders, we can end up with lots of very small orders. To avoid this, an exchange can use a system by which small orders do not participate in every trade. For a small order, the fractional expected execution rate repre- sents the chance that the order executes in full

rather than the fraction of the order that executes in each trade.

Similar exchanges can compete by sharing orders and executions. If the execution speed or expected cost of trading on an exchange deterio- rates, nice traders migrate to a competing exchange, and news traders soon follow. As long as different exchanges are equally efficient, this pro- cess will make all exchanges equally attractive. An exchange that is more efficient than its competitors will attract more order flow.

An exchange may choose to specialize in certain kinds of orders. As long as it allows both buy and sell orders of those kinds, such specialization is consistent with equilibrium. For example, an exchange can specialize in orders with little urgency. It executes its orders slowly, and its market is deep. It attracts mostly nice traders, with a few news traders whose news has a very long half-life.

A nice trader who uses such an exchange can expect only modest losses to news traders, but he will do just as well by putting a low-urgency order into a full-service exchange. Specialized exchanges have no particular advantages in equilibrium.

AN EXAMPLE OF EQUILIBRIUM Imagine that limit orders from news traders and nice traders arrive at the same rate. All orders from nice traders have x shares, and news traders use orders of x shares so they will look like nice traders. Market depth is such that an order of x shares moves the price by p. All news has value v.

News (or lack of news in an order from a nice trader) becomes public at the continuous probabil- ity rate A. Orders execute at the continuous fractional rate ,u. All orders are equally urgent because news traders want to look like nice traders in this respect.

A nice trader expects the price to revert to its original level at rate A. Thus, his expected cost per share traded, c, is

c= pelAtie-td +dt ) Jo ~~~(A + u)

A news trader expects the price to continue to move until it reflects the full value of his news. He moves the price by p by entering his order. He expects it to move by an additional amount, v - p. Thus, his expected profit per share, a, is

f = (v - p)e- At,e- JLtdt =(v -



These equations are approximate because actual random arrivals of orders on each side of the market will change the fractional execution rates on the two sides.

A condition of equilibrium is that news trader profits must equal nice trader losses. Because all orders are the same size, a = c. Therefore, based on the equations above, p = v/2. In other words, market depth must be such that any order causes a price move of half the value of a piece of news. If the order is from a news trader, we expect the price to continue an equal amount in the same direction. If it is from a nice trader, we expect the price to revert to its original level.

The condition that news trader profits equal nice trader losses is equivalent, in this case, to the condition that price moves randomly enough to eliminate trading profits for those who simply watch the sequence of trades.

Dears Dealers have no role in this equilibrium. Trad-

ers do not use market orders. If they did use market orders, limit orders would provide the market's depth. Exchanges would not need dealers to provide depth.

Exchanges simply match indexed buy and sell limit orders. All a dealer can do is put in his own limit orders. If he does this without any special information, he expects to lose money on his trades. Market making is unprofitable, so in equilibrium, dealers vanish.

Single-Pce Auctions Some researchers such as Amihud and Men-

delsen have suggested a series of single-price auctions, or batch markets, instead of or in addition to continuous trading.8 In the equilibrium outlined above, the matches between indexed buy and sell limit orders seem similar to single-price auctions.

In fact, an exchange that matches its buy and sell orders at regular intervals differs little from one that matches its orders continuously. The people who care are those who want to use very urgent orders at odd times, but the market is very shallow for orders like that, so omitting them does not affect the equilibrium much. Similarly, whether an exchange remains open all the time or closes overnight does not matter much.

Those people who propose single-price auctions seem to imagine that traders put in schedules of amounts they are willing to buy or sell at different prices. Like conventional limit orders, such schedules are outdated by the time they reach

the market. They are also hard to formulate in the first place. Thus, matching of indexed limit orders dominates a market of conventional single-price auctions.

When the market for exchanges is frictionless, periodic single-price auctions have no advantages over periodic or continuous matching of indexed limit orders at different levels of urgency.

Sunshine Trading A sunshine trade is an order to buy or sell a

certain number of shares at a set future time at whatever price suffices to clear the market.9 Pre- announcing the trade signals that you are a nice trader and thereby reduces your expected trading cost. We can think of a sunshine trade as a preannounced indexed limit order that becomes urgent once it starts to execute.

Why do we need sunshine trades when we have indexed limit orders? The delay in executing a limit order with low urgency plays the same role as the delay in executing a sunshine trade. Indeed, indexed limit orders dominate sunshine trades because they execute at a constant expected fractional rate. That means an indexed limit order has no specific time horizon. Exchanges do not have to say what other trades people who use limit orders can or cannot do. In an anonymous market, restrictions like these are difficult or impossible to enforce.

If exchanges allow sunshine trades but do not restrict other trading before the specified time, such trades lose their role in signaling lack of information. A person can put in a large sunshine trade to sell and then move the price using a small but urgent order to buy just before the time of the sunshine trade. This action makes the sunshine trade into a news trade.

In other words, sunshine trades add nothing to a market in which exchanges offer indexed limit orders at low levels of urgency. In an unrestricted equilibrium, sunshine trades vanish.

Basket Trading Subrahmanyam and Gorton and Pennacchi

have suggested that traders without information may try to signal that they have no news by trading index futures contracts or a whole basket of securities at once.10 Does this strategy mean that, in equilibrium, a nice trader faces better terms if he is trading a basket of securities than if he is trading individual securities? I think not. If a basket trade moves prices less than a collection of individual trades, then any trader can move the price without taking a net position. He can simul-



taneously enter opening orders on many individual securities and a closing basket order, choosing levels of urgency that cause the opening and closing orders to execute at the same speed. The likelihood that the opening orders move prices more than the closing order should cause a change in the prices of all the securities and the basket. Knowing this, the trader can add in another trade that makes the whole program profitable.

In equilibrium, the only way to signal that you have no news is to use an indexed limit order with a very low level of urgency.

CONCLUSIONS What will exchanges look like when we reach a competitive equilibrium? I conjecture that they will

offer noncancelable indexed limit orders at different levels of urgency but will not offer market orders or conventional limit orders. Buy limit orders will match sell limit orders at the current price and at an expected fractional rate that increases with urgency. Limit order entry will move the price by an amount that is roughly proportional to order size and that increases indefinitely as urgency increases. Thus, larger orders and more-urgent orders will usually cost more. Dealers will have no special role and will lose money if they use limit orders; so they will vanish. Traders without special information will use less-urgent limit orders; they cannot do better by using specialized exchanges, sunshine trading, or basket trading. "

FOOTNOTES

1. Sanford J. Grossman and Joseph E. Stiglitz, "Information and Competitive Price Systems," The American Economic Review, vol. 66, no. 2 (May 1976):246-53; and "On the Impossibility of Informationally Efficient Markets," The American Economic Review, vol. 70, no. 3 (June 1980):393- 408.

2. Albert Kyle, "Continuous Auctions and Insider Trading," Econometrica, vol. 53, no. 6 (November 1985):1315-35; and "On Incentives to Produce Private Information with Con- tinuous Trading," working paper, Princeton University, 1985.

3. Lawrence R. Glosten and Paul R. Milgrom, "Bid, Ask and Transaction Prices in a Specialist Market with Heteroge- neously Informed Traders," Journal of Financial Economics, vol. 14, no. 1 (March 1985):71-100; and Sanford J. Gross- man and Merton H. Miller, "Liquidity and Market Struc- ture," The Journal of Finance, vol. 43, no. 3 Uuly 1988):617- 33.

4. Anat R. Admati and Paul Pfleiderer, "Sunshine Trading and Financial Market Equilibrium," The Review of Financial Studies, vol. 4, no. 3 (Fall 1991):443-81.

5. Lawrence R. Glosten, "Is the Electronic Open Limit Order Book Inevitable?" The Journal of Finance, vol. 49, no. 4 (September 1994):1127-61.

6. Lawrence Harris, "Liquidity, Trading Rules, and Electronic Trading Systems," working paper, University of Southern California, 1990.

7. I have called this type of order a participating order. See Fischer Black, "Toward a Fully Automated Exchange," Financial Analysts Journal, vol. 27, no. 4 Uuly/August 1971): 28-35 and 44; and "A Fully Computerized Stock Ex- change," Financial Analysts Journal, vol. 27, no. 6 (Novem- ber/December 1971):24-28 and 86-87. Brown and Holden discuss a related order called a quote adjusted limit order. See David P. Brown and Craig W. Holden, "The Design of Limit Orders," working paper, Indiana University, 1993.

8. Yakov Amihud and Haim Mendelson, "The Effects of

Computer Based Trading on Volatility and Liquidity," in Henry C. Lucas, Jr., and Robert A. Schwartz, eds., Infor- mation Technology for the Securities Markets (Homewood, Ill.: Dow Jones-Irwin, 1989):59-85.

9. See Admati and Pfleiderer, "Sunshine Trading." 10. Avanidhar Subrahmanyam, "A Theory of Trading in Stock

Index Futures," The Review of Financial Studies, vol. 4, no. 1 (Spring 1991):17-51; and Gary Gorton and George Pennac- chi, "Security Baskets and Index-Linked Securities," The Journal of Business, vol. 66, no. 1 (January 1993):1-28.

11. I am grateful for conversations on these issues and com- ments on earlier drafts to Yakov Amihud, Gilbert Bee- bower, George Benston, Donald Collat, Thomas Cooper, Ian Domowitz, Philip Dybvig, Eugene Fama, Steven Fen- ster, Kenneth French, James Gammill, Gerard Gennotte, Sanford Grossman, Lawrence Harris, Gur Huberman, Dwight Jaffee, Ravi Jagannathan, Robert Jones, Eugene Kandel, Alan Kraus, Michael Levine, Harry Markowitz, Merton Miller, Mark Mitchell, Paul Pfleiderer, David Romer, Jose Scheinkman, Myron Scholes, Alan Schwartz, Robert Schwartz, Erik Sirri, Clifford Smith, Stephen Smith, Hans Stoll, James Stone, Avanidhar Subrahmanyam, Rob Trevor, Jerold Warner, Mark White, and especially Lawrence Glosten, Joel Hasbrouck, Albert Kyle, and Rene Stulz. Thanks also to participants in workshops at the Anaheim meetings of the American Finance Association, Baruch College, Boston University, Columbia University, Emory University, Harvard University, the Institute for Quantitative Research in Finance, McGill University, New York University, Princeton University, the Stockholm School of Economics, the University of California at Berke- ley, the University of Chicago, the University of Colorado, the University of New South Wales, the University of Pennsylvania, the University of Rochester, and Yale Uni- versity. Earlier versions of this paper had several different titles.



Article Contentsp. 23p. 24p. 25p. 26p. 27p. 28p. 29

Issue Table of ContentsFinancial Analysts Journal, Vol. 51, No. 3 (May - Jun., 1995), pp. 1-80Front Matter [pp. 1-7]Guest SpeakerThe Currency Hedging Decision: A Search for Synthesis in Asset Allocation [pp. 8-17]

On the Risk of Stocks in the Long Run [pp. 18-22]Equilibrium Exchanges [pp. 23-29]Analyst Forecasting Errors and Their Implications for Security Analysis [pp. 30-41]The Three Types of Factor Models: A Comparison of Their Explanatory Power [pp. 42-46]The Effects of Rebalancing on Size and Book-to-Market Ratio Portfolio Returns [pp. 47-57]The Day-of-the-Week Anomaly: The Role of Institutional Investors [pp. 58-67]The Time-Diversification Controversy [pp. 68-76]A Note on the Measurement of Equity Duration and Convexity [pp. 77-79]Back Matter [pp. 80-80]

black equilibrium exchanges

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