Short Selling, Death Spiral Convertibles and The Profitability of Stock Manipulation

Posted on December 27, 2007. Filed under: Stock Manipulation? |

SHORT SELLING, DEATH SPIRAL CONVERTIBLES, AND

THE PROFITABILITY OF STOCK MANIPULATION

John D. Finnerty

Professor of Finance, Fordham University

March 2005

John D. Finnerty

Fordham University Graduate School of Business

113 West 60th Street

New York, NY 10023

Tel: 212-599-1640

Fax: 212-599-1242

e-mail: finnerty@finnecon.com

SHORT SELLING, DEATH SPIRAL CONVERTIBLES, AND

THE PROFITABILITY OF STOCK MANIPULATION

Abstract

The SEC recently adopted Regulation SHO to tighten restrictions on short selling and curb

abusive short sales, including naked shorting masquerading as routine fails to deliver. This paper

models market equilibrium when short selling is permitted and contrasts the equilibrium with

and without manipulators among the short sellers. I explain how naked short selling can

routinely occur within the securities clearing system in the United States and characterize its

potentially severe market impact. I show how a recent securities innovation called floating-price

convertible securities can resolve the unraveling problem and enable manipulative short selling

to intensify.

SHORT SELLING, DEATH SPIRAL CONVERTIBLES, AND

THE PROFITABILITY OF STOCK MANIPULATION

1. Introduction

Manipulative short selling has a long and colorful history that dates back to the origins of

organized stock markets (Allen and Gale, 1992). Bernheim and Schneider (1935) describe how

bear pools operated on the Amsterdam Stock Exchange during the late seventeenth century.

Stock manipulators carefully timed their aggressive ‘bear raids’ to exert maximum selling

pressure. The price declines attracted free riders, and the combined pressure on the prices of the

targeted stocks produced virtually assured profits. The manipulators found that they could defeat

any opposition by employing “tricks that only sly and astute speculators invent and introduce,”

such as planting false rumors about the target firm’s precarious condition in the press (Bernstein

and Schneider, 1935). When similar manipulation occurred on the London Stock Exchange in

the early eighteenth century, the British parliament passed a law prohibiting short selling in

1734. The law was not repealed until 1860, and short selling was not specifically authorized

under English law until 1893 (Bernstein and Schneider, 1935). Numerous histories document

how these and other manipulative short selling techniques have been woven into the fabric of the

stock market.1

1 Bernstein and Schneider (1935), Sobel (1965), and Wycoff (196 chronicle the history of stock market

manipulation over several decades culminating in the 1920s and 1930s when manipulative short sellers organized

into large investment pools to concentrate their short selling for maximum impact. Their descriptions of the

manipulative techniques and the destabilizing impact of bear pools on the New York Stock Exchange in the 1920s

and early 1930s are reminiscent of the Amsterdam Stock Exchange manipulations of the seventeenth century and the

London Stock Exchange manipulations of the eighteenth century. Manipulative short selling was blamed for causing

the Great Crash, although a subsequent Senate investigation found that other factors played a bigger role in causing

the crash. These histories also describe how manipulative short selling techniques have evolved. House Report

(1991) found that short sellers, sometimes including “short-selling partnerships [with] very substantial financial

resources,” were instigating SEC investigations to depress the prices of their targeted stocks. SEC (2003b) cites

short selling abuses in proposing restrictions to curb naked short selling. Later in the paper I explain how floatingprice

convertibles are one of the most recent enablers of short sale manipulation.

The SEC defines a short sale as the “sale of a security that the seller does not own or that

the seller owns but does not deliver. In order to deliver the security to the purchaser, the short

seller will borrow the security, typically from a broker-dealer or an institutional investor.”2 The

potential for abuse in short selling is a concern to market participants, regulators, and academics

alike.3 The SEC adopted Regulation SHO on July 28, 2004 to tighten the restrictions on short

selling and curb abusive short sales, such as naked short selling (SEC, 2003b, 2004).4 The SEC

proposed new Regulation SHO in October 2003 because of growing concern that naked short

selling masquerading as routine fails to deliver had impaired market efficiency:

Many issuers and investors have complained about alleged “naked short selling,”

especially in thinly-capitalized securities trading over-the-counter. Naked short selling is

selling short without borrowing the necessary securities to make delivery, thus potentially

resulting in a “fail to deliver” securities to the buyer. Naked short selling can have a

number of negative effects on the market, particularly when the fails to deliver persist for

an extended period of time and result in a significantly large unfulfilled delivery

obligation at the clearing agency where trades are settled. At times, the amount of fails to

deliver may be greater than the total public float. In effect the naked short seller

unilaterally converts a securities contract (which should settle in three days after the trade

date) into an undated futures-type contract, which the buyer might not have agreed to or

that would have been priced differently. The seller’s failure to deliver securities may also

adversely affect certain rights of the buyer, such as the right to vote. More significantly,

naked short sellers enjoy greater leverage than if they were required to borrow securities

and deliver within a reasonable time period, and they may use this additional leverage to

engage in trading activities that deliberately depress the price of a security. (SEC, 2003b,

pages 6-7.)

Used appropriately, short selling promotes market efficiency by eliminating overpricing

(Diamond and Verrecchia, 1987, D’Avolio, 2002, Duffie, Garleanu, and Pedersen, 2002, and

2 The short seller later repurchases the security in the market, presumably after its price has fallen, and returns it to

the lender to close out the short position.

3 House Report (1991) expresses Congress’s concern that abusive short selling is impairing market efficiency and

criticizes the SEC for its lax enforcement of the rules designed to prevent manipulative short selling.

4 A ‘naked’ short sale occurs when the seller has neither borrowed the shares nor made an affirmative determination

that they can be borrowed, which the securities laws require, before selling them. This failure to borrow the shares

results in a ‘fail to deliver’ until the shares can be borrowed and delivered to the purchaser. Naked shorting also has

a long history. Stedman (1905) provides colorful accounts of Jacob Little and other short sellers who amassed great

fortunes in the nineteenth century through manipulative short selling. Little, nicknamed the ‘Great Bear of Wall

Street,’ would naked short shares, spread rumors about the issuer’s pending insolvency, and then cover his short

position at the resulting depressed prices.

Jones and Lamont, 2002).5 However, when left unchecked, short selling can artificially depress

share prices and impair market efficiency (SEC, 2003b).6 Whether short selling has this

unintended effect depends on first, whether there are rules and regulations that prohibit

potentially abusive behavior and second, whether regulatory enforcement is adequate to ensure

that market participants obey these rules (SEC, 2003b).

Manipulation is the “intentional interference with the free forces of supply and

demand.”7 A manipulative trading strategy corrupts the market’s price formation process to

generate a riskless profit (Jarrow, 1992). Market manipulation can be profitable when there is a

difference between the price elasticities of purchases and sales that the manipulator can exploit.

Stock market manipulators use a variety of devices, such as releasing false information about a

company into the market,8 and employing trading strategies that impede the price formation

process, such as naked shorting, wash sales, matched trades, and painting the tape, all of which

inject misleading trading information into the market, to move market prices in the direction that

benefits the manipulator. Illegal short selling, such as naked shorting, can distort market prices

by creating artificial supply-demand imbalances (Thel, 1994). Consequently, the securities laws

in the United States proscribe various restrictions on short selling that are designed to constrain it

so that it can not be misused to manipulate stock prices below the true asset value (Thel, 1994,

SEC, 2003b, 2004).

5 Lamont and Thaler (2003) and Ofek and Richardson (2003) furnish empirical evidence that the restricted supply of

shares available for borrowing inhibited short selling and contributed significantly to the recent dotcom bubble.

6 “New Rules to Put Squeeze on Shorts,” Wall Street Journal (January 27, 2005): C5, quotes an assistant director in

the SEC’s Division of Market Regulation, who expresses concern that massive naked shorting could create an

‘endless’ supply of shares that “could drive down the price in an abusive or manipulative way.” The article goes on

to note that Regulation SHO stemmed from instances where the short position in a stock approached or even

exceeded the firm’s entire supply of outstanding shares.

7 Pagel, Inc. v. SEC, 803 F 2d, 942, 946 (8th Circuit, 1986).

8 Placing false notices on electronic bulletin boards in Internet chat rooms is an example of the type of manipulative

behavior that is difficult for regulators to monitor.

Manipulation can occur when informed traders can take advantage of uninformed traders

who must trade to meet their liquidity needs (Glosten and Milgrom, 1985, Kyle, 1985, 1989,

Easley and O’Hara, 1987, Allen and Gale, 1992, Allen and Gorton, 1992). Allen and Gale (1992)

examine trade-based manipulation, in which a trader can manipulate a stock’s price upward

simply by buying shares and then sell them at a profit even when the purchases do not cause any

price momentum. Manipulation in their model does not require traders who take overt action to

alter the value of the firm, inject false information into the market to move the price higher, or

create a corner. Asymmetric information and the difference in the price elasticities of purchases

and sales are the key factors. Uninformed traders are uncertain whether the buyer knows that the

stock is undervalued or instead intends to manipulate the price upward. Purchases have a greater

price elasticity than sales due to the greater information content of purchases when the sellers

include liquidity traders. Uninformed liquidity traders have less freedom to time their sales, and

so informed traders, such as corporate insiders, are able to profit by exploiting both their

information advantage and the liquidity traders’ timing disadvantage. When liquidity sales are

more likely than liquidity purchases, a purchase conveys more information because it is more

likely that the trader is informed. The share price elasticity with respect to purchases exceeds the

price elasticity with respect to sales, and a pooling equilibrium can occur in which price

manipulation is profitable.

My model is in the spirit of Allen and Gale (1992) but focuses on short sales. I include

active traders (arbitrageurs), who turn out to be the critical enabling factor that facilitates

manipulative short sales in market equilibrium. I assume that active traders are uncertain whether

the seller knows that the stock is overvalued or instead intends to manipulate the stock price

downward. They are less knowledgeable than informed investors or manipulators but more

knowledgeable than uninformed traders. Active traders seek out information regarding the firm’s

prospects and look for signals in the trading behavior of informed investors, such as corporate

insiders. They sell in response to short sales by informed investors and manipulators, whom they

mistake for informed investors, which allows manipulative short selling to be profitable.

Active trader selling can resolve the unraveling problem and allow profitable

opportunities for manipulative short selling. The unraveling problem would rule out trade-based

short sale manipulation if the market consisted only of informed traders and liquidity traders. It is

more difficult to justify forced purchases than forced sales by liquidity traders, who presumably

do not have the same pressing need to buy as to sell (Allen and Gorton, 1992). The asymmetry in

price elasticities that creates an opportunity for manipulative purchases to be profitable rules out

profiting from manipulative short sales. A manipulator can repeatedly buy stocks and then sell

them to earn a profit because purchases having the greater price impact. But selling and then

buying would have the opposite effects and result in a loss.

Active traders interact with the informed investor to create downward price momentum.

Jarrow (1992) investigates how manipulation can occur when large trades create price

momentum that leads to a difference between the price elasticities of purchases and sales. Price

momentum occurs when trades are large enough to move the price and an increase in price at one

date causes an increase in price at a later date. A large trader’s purchases create upward price

momentum, and then she trades against the price trend to lock in her profit by selling to noise

traders who buy at the inflated price. Presumably this sort of manipulation could work in reverse

with the large trader selling short to stimulate downward price momentum and then covering his

short position by buying at depressed prices from noise traders. In my model active traders sell in

the next period when they observe that the informed investor has sold shares, which moves the

price downward. The informed investor can cover his short by buying from the active traders, or

he can wait until after the further drop in price to cover, depending on how costly it is to carry

the short position another period. However, I do not make any special assumptions regarding the

relative price elasticities of buys and sells. I also do not assume forced buying or selling by any

class of traders. I assume that uninformed traders are willing to buy more shares at lower prices

than those currently prevailing. Trade-based short sale manipulation is sustainable in a market

setting in which due to information asymmetries, it is unclear whether the seller has negative

information about the firm’s prospects or is simply trying to manipulate the firm’s stock price.

Naked short selling can increase the manipulator’s profit. A short seller, who profits by

buying the shares to cover her short position at lower prices than the selling prices, can drive the

price of a stock lower by selling short a larger number of shares. Without enforceable restrictions

requiring short sellers to borrow the shares before they can commit to sell, a short seller might

destabilize the market for a particular stock through naked shorting.9 While some naked shorting

may take place for benign reasons, for example because it lowers the cost of short selling (Evans,

Geczy, Musto, and Reed, 2003), Regulation SHO reflects the SEC’s concern that previous

restrictions on short selling had not been effective in preventing its use as a manipulative device

(SEC, 2003b, 2004).10 There is mounting evidence that manipulative short selling has seriously

disrupted the market for some over-the-counter stocks.11

9 Naked shorting creates so-called phantom shares, which give rise to a potential corporate governance problem. The

buyer of the phantom shares usually does not realize they are not real shares and believes she has the same voting

rights as the holders of real shares. Her broker will record the shares as a long position in her account and as a fail to

receive on its books. If brokers send the proxy materials to owners of phantom shares, who then vote them, there

could be more votes cast for directors than actually exist. See Curry. The SEC’s proposed Regulation SHO (SEC,

2003) is designed to address the problem of naked short selling. In June 2004, the SEC announced a pilot program

that would allow unrestricted short sales of 1,000 actively traded stocks for one year. At the same time, it announced

a proposal to require broker-dealers to locate shares available for borrowing before engaging in any short sale. This

rule was designed to curb naked short selling. “SEC Is Set to Approve Plan to Ease Short-Selling Curbs for One

Year,” Wall Street Journal (June 23, 2004).

10 House Report (1991) expresses the same concern. The SEC recently adopted Regulation SHO to curb abusive

short selling (SEC, 2003, 2004).

11 Securities and Exchange Commission v. Rhino Advisors, Inc. and Thomas Badian, United States District Court,

Southern District of New York, February 26, 2003, describes the naked short sale manipulation of the common

stock of Sedona Corporation.

The unraveling problem should impose a constraint on naked shorting. There are two

mechanisms for avoiding this constraint. Since a firm’s common stock claims are extinguished if

it liquidates, a manipulative short seller can effectively cover its short position at zero cost by

forcing the firm into liquidation (House Report, 1991). Second, a popular private equity

financing instrument, floating-price convertible securities (Hillion and Vermaelen, 2004), can

resolve the unraveling problem because the manipulator does not have to buy back shares in the

open market. He can obtain as many conversion shares as he needs by short selling the price

downward just prior to the conversion notice date. The flawed structure of the floating-price

convertible’s contract may actually give security holders an incentive to manipulate the issuer’s

share price downward.

The rest of the paper is organized as follows. Section 2 describes the model and

characterizes the market equilibrium when there are no manipulators. Section 3 describes the

market equilibrium when manipulators can enter the market. I assess the impact of short sale

manipulation by comparing the two equilibriums. Section 4 explains how naked short selling can

destabilize the market for a stock. Section 5 shows how floating-price convertibles resolve the

unraveling problem, so that even trade-based short sale manipulation is profitable. Section 6

concludes.

2. The Market Model

This section characterizes the market equilibrium when there are no manipulators.

2.1 Institutional Details on Short Selling

A short sale is the sale of stock that the seller does not own.12 The seller borrows the

shares from a broker-dealer or an institutional investor. She establishes the short position by

selling the borrowed shares and closes it out by buying the stock at a later date and returning the

shares to the stock lender to extinguish the loan. Short sales increase the number of shares that

are beneficially owned by investors and hence the stock’s float.13 As a result, the total number of

shares beneficially owned and eligible to vote exceeds the number of shares the firm has

issued.14

Short sales are heavily regulated in the United States both because of the riskiness of the

strategy and also because of its potential for abuse as a manipulative device.15 In the United

States, many institutional investors are either prohibited by policy or regulation from short

selling or tightly restricted as to the size of the short positions they can maintain. Many brokerdealers

severely restrict short selling by their retail customers. However, the SEC has expressed

concern that enforcement of the restrictions on short selling, and especially naked short selling,

appears lax due to broker-dealers’ tolerance of extended fails to deliver (SEC, 2003b, Boni,

2004).

The regulation of short selling in the United States has evolved from the recognition that

unrestricted short selling could impair market efficiency by causing the price of a stock to spiral

downward (Dechow et al., 2001). Regulation constrains short selling in several ways. SEC Rule

12 Asquith and Meulbroek (1996), and Dechow, Hutton, Meulbroek, and Sloan (2001) describe the institutional

arrangements of short selling in great detail. D’Avolio (2002) and Geczy, Musto, and Reed (2002) describe the

market for stock loans. I provide just a brief summary.

13 A common stock’s float is equal to the number of outstanding shares minus the number of insider shares plus the

short position in the stock.

14 This has potentially significant corporate governance implications, which are beyond the scope of this paper

(House Report, 1991, and SEC, 2003). The process of nominal share expansion through short selling and stock

lending is very similar to the process of money supply expansion through bank lending, except that there is no

‘reserve requirement,’ only the clearing firm’s willingness to arrange stock loans to cover the fails to deliver so that

it can clear the trades, to control it.

15 Because of these concerns, short selling is severely restricted in many foreign stock markets. Japanese securities

regulators introduced a rule in February 2002 forbidding short sales at or below the current market price (Lilico,

2002). Taiwan regulations prohibit short selling by foreigners. All short selling in Hong Kong must be declared, and

failure to do so is punishable by imprisonment.

10a-1 permits investors to sell short stocks listed on a national securities exchange only on either

a “plus tick” or a “zero plus tick” (SEC, 2003)16 The NASD has a similar bid test under NASD

Rule 3350 but it only applies to Nasdaq National Market (NNM) securities when the trades are

executed on either SuperMontage or over the NASD’s Alternative Display Facility (ADF). The

bid test does not apply to Nasdaq SmallCap, OTC Bulletin Board, or other over-the-counter

stocks or to NNM securities traded away from SuperMontage or ADF unless the market in which

they are traded has adopted its own price test. The short seller must place the proceeds from the

short sale in an escrow account, which collateralizes the stock loan. The short seller can not use

the short sales proceeds to hedge the short position. The short seller receives interest from the

stock lender at a below-market interest rate, called the rebate rate, with the difference between

the market rate and the rebate rate, the rebate spread, compensating the lender.17 Federal

Reserve Regulation T requires short sellers to post additional collateral in a margin account when

the stock is shorted. The initial margin requirement is 50% of the market value of the shorted

shares. The maintenance margin requirement is 25%.18 Broker-dealers often set higher margin

requirements, and large broker-dealers typically require at least 30% equity. The short seller has

to top up the escrow account if the price of the stock rises but can reduce it if the price of the

stock falls.

Current regulations prohibit naked short sales except under limited circumstances. New

York Stock Exchange (NYSE) Rule 440c and NYSE Information Memorandum 91-41 (1991)

require a short seller to make an affirmative determination that it will be able to borrow shares

16 A “plus tick” occurs when the last trade occurred at a price higher than the last previous trade. A “zero plus tick”

occurs when the last trade occurred at a price equal to the price of the last previous trade and the last prior trade that

took place at a different price occurred at a higher price.

17 The stock lending market is not a well-functioning competitive market (Ofek, Richardson, and Whitelaw, 2003).

It is more appropriate to treat the rebate spread as an indicator of how difficult it is to borrow a stock, rather than as

a competitively determined borrowing rate. Even though it is not a market price, it can still serve in the model as a

useful proxy for the cost of borrowing stock.

18 The stock exchanges and the NASD set the minimum maintenance margin requirements for their members. NYSE

Rule 431 sets a 25% minimum for NYSE members.

before it can make a short sale unless the short seller is a market maker, specialist, or odd-lot

broker who is selling short in connection with its normal market-making responsibilities.

National Association of Securities Dealers (NASD) Rule 3370, NASD Rules of Fair Practice,

Article III, Section 1, and SEC Release No. 34-35207 (1995) impose a similar affirmative

determination requirement for NASDAQ stocks, and SEC Release No. 34-37773 (1996) imposes

a similar requirement for American Stock Exchange-listed stocks.

The SEC recently adopted Regulation SHO to curb abusive short selling (SEC, 2003b,

2004). Rule 203 under Regulation SHO, which became effective January 3, 2005, prohibits a

broker-dealer from accepting a short sale order unless it has arranged to borrow the security or

has reasonable grounds to believe that it will be able to borrow it before the settlement date. It

also requires the broker-dealer to enter into a bona-fide borrowing arrangement before executing

an order to short sell any equity security that has been identified as a ‘threshold security’. The

threshold list includes any equity security that is either exchange-traded or is issued by a public

reporting company for which aggregate fails to deliver at a registered clearing house amount to

(a) at least 10,000 shares which represent (b) at least one-half of one percent of the issuer’s

outstanding shares.19 It also requires the clearing house member or the clearing house to take

action to cure all fails to deliver threshold stocks that persist for 10 days after the normal

settlement date. The SEC proposed Regulation SHO out of concern that the existing rules

restricting naked shorting had not been effective in preventing abuses (SEC, 2003b). However,

the existing affirmative determination rules and the new rules under Regulation SHO except

short sales executed by specialists and market-makers engaged in bona-fide market-making

19 There are firms whose shares are quoted in the Pink Sheets but which are not subject to the public reporting

requirements of the Securities Exchange Act of 1934. Such stocks are not covered by Regulation SHO.

activities (SEC, 2003b, 2004), which provides a potential loophole.20 Boni (2004) finds that

naked short sales are pervasive in the U.S. stock market, which supports the SEC’s concern that

broker-dealers have not been diligent in enforcing the existing short sale restrictions.21

Borrowing shares is costly. In addition to the cost implicit in receiving a below-market

rebate rate, stock loan agreements typically require the borrower to reimburse the lender in full

for any dividends or other distributions the issuer makes to its stockholders, which imposes a real

cost (Frank and Jagannathan, 1998). Third, the Internal Revenue Code taxes all profits from

short sales at the short-term capital gains rate, regardless of the length of time the position is

open. Fourth, stock borrowers are exposed to the risk of a squeeze.22

Stock loan agreements usually provide that the loan must be repaid on demand. A short

squeeze can occur when the lender demands the return of the shares but the borrower can not

find a substitute lender and must therefore repurchase the shares in the open market. If the stock

is thinly traded, or if there are a relatively large number of short sellers trying to cover their short

positions, the resulting demand for shares can force the price higher and impose an added cost on

short sellers. A short seller can mitigate this risk by borrowing on a term basis. However, term

stock loans are unusual, and they are expensive (Geczy et al., 2002). Instead, market participants

may use strategic fails to deliver (i.e., naked shorting) when stock borrowing is costly or

impossible (Evans et al., 2003, Boni, 2004).

2.2 The Model

20 The affirmative determination rules do apply even to market-makers when a stock has settlement failures that

exceed the greater of (a) 0.5% of the stock’s float and (b) 10,000 shares.

21 Boni (2004) finds that 42% of listed stocks and 47% of unlisted stocks had fails of five days or more, and about

4% of the stocks had fails that would have classified them as ‘threshold securities’ under Regulation SHO. However,

the median fails as a percentage of the outstanding shares was only 0.01% for NYSE, AMEX, and Nasdaq stocks

and only 0.03% for OTCBB and Pink Sheet stocks. Both distributions are skewed because the mean fails as a

percentage of outstanding shares was 0.19% for NYSE, AMEX, and Nasdaq stocks and 1.56% for OTCBB and Pink

Sheet stocks.

22 Dechow et al. (2001) cite as an example a short squeeze in the shares of Amazon.com in June 1998. Ofek and

Richardson (2003) provide empirical evidence that rebate rates for Internet stocks were very high during the

DotCom bubble, which implies a limited supply of shares available for loan and a relatively high risk of a squeeze.

The model is a simplified depiction of an actual stock market that still is able to capture

the essence of manipulative short selling in actual stock markets. The model also gains

considerable clarity without losing generality by assuming a non-dividend-paying stock and a

zero interest rate. I assume that the intrinsic value of the stock to be revealed in the future can

have either of two possible values, high (H) or low (L). I also assume that aside from the initial

shareholders, stock market investors are of four types.23

First, there is an informed investor (subscripted I) who possesses information about the

firm that enables him to know what the value of the stock will be when it is revealed to the

market in the future. The informed investor could be a hedge fund or some other sophisticated

investor. Insiders are also informed but are prohibited from short selling by corporate

restrictions and the Securities Exchange Act of 1934.24 One could also think of the informed

investor as a professional short seller who has reliable information about the firm’s future

business prospects, which he gained through research (Diamond and Verrecchia, 1987). To

simplify the model, I assume a single informed investor.

Because of the risks and the cost involved, short sellers are likely to be better informed

than holders of long positions about the prospects for a stock (Diamond and Verrecchia, 1987).

A short sale is the most direct way for an investor to bet that a stock’s price will fall.25 Short

sellers expect the share price to fall sufficiently to compensate them for their costs and risks.

Asquith and Meulbroek (1996) furnish empirical evidence that supports Diamond and

Verrecchia (1987). They find a strong negative relation between the amount of short interest and

subsequent stock returns, during both the period the stocks are shorted and the following two

23 Aggarwal and Wu (2002) assume a similar market structure to model manipulative purchases.

24 I assume that corporate blackout periods and the insider trading laws prohibit them from buying if they

believe the firm’s stock is undervalued. Thus, they do not buy shares to counter the short seller’s

manipulation. A more general model could allow for this behavior.

25 According to Asquith and Meulbroek (1996), hedge fund managers and other professional investors have found

that the option market is more expensive than short selling, especially for stocks that are hard to borrow.

years. They also find that those stocks that are heavily shorted for more than one month have the

most negative returns.

Second, at times there is a manipulator (subscripted M), who I assume can also determine

the stock’s intrinsic value either through research or by observing the trading behavior of the

informed investor. To simplify the model, I assume a single manipulator. The manipulator takes

actions that are designed to drive down a stock’s price, hoping to profit from the lower future

price. The manipulator is capable of mimicking the informed investor, for example, by

duplicating his volume of short sales, so as to conceal his manipulative intent from active traders

and uninformed traders.

Manipulative strategies are of two general types. My model focuses on trade-based

manipulation (Allen and Gale, 1992). The manipulator sells shares to drive down the price and

hopes to profit by buying them back at lower prices in the future. Second, the manipulator could

also engage in information-based manipulation by spreading rumors (Allen and Gale, 1992),

engage in wash sales, or employ other manipulative devices without actually selling any shares

to drive the price down. Such behavior violates Rule 10b-5 under the Securities Exchange Act of

1934 but it probably accounts for a significant portion of stock manipulation. The two strategies

are complementary. By spreading false negative information after establishing the short position,

a manipulator can further depress a stock’s price and increase her profit. Reducing the price

further gives the manipulator greater opportunity to cover her short position without driving the

price up so much that it eliminates her profit. These non-trading devices could also be used to

resolve the unraveling problem. In my model, the existence of active traders and the variable

price feature of floating-price convertibles can both resolve the unraveling problem.

The manipulator can behave like an informed investor and as a manipulator at different

times. He could act like an informed investor by selling short in anticipation of the stock’s price

falling to L. He can also act like a manipulator by selling short to drive down the price and

covering his short position before the share price is revealed to be H. In addition, in Section 4, I

allow for the possibility that the manipulator can switch modes of behavior, at times borrowing

shares to make routine short sales and at other times intentionally effecting naked short sales by

failing to make delivery. Alternating between these two modes of behavior to exploit his

information asymmetry disguises the manipulator’s behavior and makes it more difficult for the

regulators to detect his misbehavior and for the other market participants to interpret the signals

in his trading decisions.

Third, there are N active traders (subscripted An , n = 1,….,N). Active traders, who may

include market makers, search for information about whether the firm’s stock price will be high

or low in the future.26 As part of their information gathering, they monitor the behavior of other

traders to look for value signals. They observe market price and trading volume but they do not

know the identities of buyers and sellers, which makes them incapable of distinguishing perfectly

between sales by a manipulator and an informed investor.27 They do not have complete

information about the firm. Instead, they infer information from prices, trading volumes, and the

trading behavior they observe in the market to decide whether they should buy the stock or sell

it. They interpret sales by an informed investor (or by a manipulator they mistake for an

informed investor) as a negative signal and sell shares the following period in response to the

negative signal.

Fourth, there is a continuum of uninformed (or noise) traders (subscripted U). They

initially have negligible holdings of the stock and behave like price takers. They do not

26 Market makers may also be informed investors, depending on their access to information about the firm, or

manipulators, depending on their trading motivation. I explain later in the paper that the manipulator has an

incentive to register as a market-maker because of the exceptions to the short sale restrictions that apply to marketmakers

(but only to the extent of bona-fide market-making activities).

27 It is certainly possible, of course, that the manipulator is also an insider. However, this is less likely when the

insiders have large stock ownership because the manipulative short selling would also decrease the value of their

shares.

condition their purchases on any specific information but instead, stand ready to buy more shares

at lower prices, which provides liquidity to sellers. The uninformed traders’ willingness to hold

Q shares at time t is summarized in the following demand curve:

P = D(Q) = A − BQ, H>A>L ≥ 0, B>0 (1)

P is the market price of the stock at time t, and A and B are constants. The demand curve for the

stock is downward-sloping (Shleifer, 1986, Kaul, Mehrotra, and Morck, 2000, and Liu, 2000).

At time 0, the firm’s shares are held by insiders and passive investors who view their

shareholdings as long-term investments. A portion of the firm’s shares are held in margin

accounts with broker-dealers where they are available for lending to short sellers.28 If no one

wishes to sell the stock, then its price is A. The total number of shares outstanding is (A – L)/B.

If the time zero shareholders wish to sell all the outstanding shares to uninformed traders, then

the price would fall to L.

Share transactions occur in the market in the following sequence. At time 1, either the

informed investor or the manipulator can initiate a short sale. Since neither the informed investor

nor the manipulator owns any shares, each must borrow them. I relax this assumption later when

I consider the possibility of naked short sales. The informed investor sells shares if and only if

the future stock price will be L. The probability that the future stock price will be L is p (and the

probability that it will be H is 1 – p). One can think of A, the current market price, as the

expected present value of the share price at time 3:

A = pL + (1− p)H (2)

28 Shares held in cash accounts are not available for lending without the account holder’s permission. Shares held in

margin accounts are freely lendable. I assume that the margin account holders are uninformed investors.

Alternatively, it could be assumed that a portion of the shares are held by a fifth class of shareholders, passive

investors, such as stock index funds or mutual funds, who intend to hold them for the long term and are willing to

lend them to short sellers in order to earn extra income in the form of stock loan rebates.

The manipulator observes the informed investor’s trading. She will not sell the stock

short if the informed investor does, and she may decide not to enter the market even if the

informed investor is not selling.29 The manipulator sells shares with probability q < 1 - p.30 There

is a probability 1 – p – q that neither the informed investor nor the manipulator will engage in

short selling.

Active traders observe the stock price and trading volume at time 1. They sell shares at

time 2 based on what they learn at time 1 conditioning their decision to sell on whether they

observe an informed investor (or the manipulator whom she mistakes for the informed investor)

selling.31 The manipulator or the informed investor can buy or sell shares at time 2. The

uninformed traders stand ready to buy shares at time 1 and also at time 2. The stock’s value is

revealed to be H or L per share at time 3.

The informed investor or the manipulator can sustain a short position until time 3 but it is

less expensive to sustain it to time 2 (unless the manipulator naked shorts). One might think of

this assumption in any of three ways. First, the rebate spread represents a direct cost of carrying

the short position. This cost can exceed the market rate of interest when the stock is on special

and extremely hard to borrow (D’Avolio, 2002, Duffie, Garleanu, and Pedersen, 2002, Geczy,

Musto, and Reed, 2002). Second, time 3 represents the long run, and it may be very costly for the

informed investor or the manipulator to borrow the shares to maintain the short position, for

example, because she is unable to borrow the stock continuously over an extended period.32

29 In a market equilibrium in which the informed investor sells the profit-maximizing number of shares, I show later

in the paper that incremental short sales by the manipulator will not be profitable.

30 Later in the paper I determine the optimal probability of manipulation and show that if the probability of

manipulation is too high, then the active traders refuse to sell shares and the manipulative scheme fails.

31 If there are no stock sales at time 1, then it is reasonable to assume that active traders will purchase shares at time

2 until they raise the price to H. I do not address this possibility in my model because my focus is on what happens

when there are short sales at time 1.

32 A stock lender can get the shares back on demand. In that case, the short seller’s broker must try to borrow

replacement shares from some other shareholder to keep the short position open. If the broker can not borrow the

shares, then it must close out the short position.

Third, while there is no uncertainty in my model, I could motivate a cost to maintaining the short

position that risk-averse investors face by making the distribution of time 3 prices uncertain.

Instead, I model the cost of holding the short position until time 2 as a scalar C per share and to

time 3 as 2C per share. D’Avolio (2002) finds that the overall value-weighted cost to borrow

stocks is 25 bp per year; 91% of the stocks (“general collateral” stocks) cost less than 1% per

year to borrow with a mean-weighted fee of only 17 bp; but the other 9% (“special” stocks) have

a mean fee of 4.3% per year; and less than 1% (“extremely special” stocks) have negative rebate

rates as high as 50%. If the stock price at time 3 is L, then the informed investor’s cost of

shorting a share until time 3 is L + 2C. Unless A – L – 2C > 0, the informed investor would

never sell shares at a price less than or equal to the time 0 price and maintain the short position

until time 3. To simplify the model, I also assume that active traders incur at most a negligible

cost to holding a short position.33

2.3 Market Equilibrium

I investigate the impact of short sale manipulation on stock market equilibrium by

comparing two market settings. In the first, there is an informed investor and active traders but

no manipulator. Both can sell shares short. They sell short when they expect the equilibrium

price of the shares to drop to L at time 3. In the next section, I permit a manipulative short seller

to enter the market and examine how her trading alters the market equilibrium. Unlike legitimate

short sellers, the manipulators sell short in the hope that their selling drives the share price below

the shares’ intrinsic value and attracts other sellers from whom they can buy shares after the

price drop to cover their short positions.

33 Market makers have lower shorting costs than other market participants because they can sell on a downtick and

also because they do not have to make an affirmative determination that they will be able to borrow shares before

they sell short. Market makers are granted these exceptions to facilitate their market-making activities. A strategy a

manipulator can employ to reduce its cost of shorting is to register as a market maker for the target stock (SEC,

2003). With the assumption of zero cost for active trader shorting, C can be thought of as the incremental shorting

cost the informed investor and the manipulator must pay as compared to active traders. Later in the paper I consider

naked shorting, which I assume to have zero cost.

The informed investor might sell (1) I Q shares short at time 1 that he plans to repurchase

at time 3, an additional (1)

I Q shares short at time 1 that he plans to repurchase at time 2, and a

further (2) I Q shares short at time 2, which he would repurchase at time 3.34 I show that when

shorting is expensive, the optimal strategy for the informed investor is to sell shares at time 1 but

neither to buy nor to sell shares at time 2. When shorting is inexpensive, the informed investor

will sell shares short in both periods. Initially, I assume that there are N symmetric active traders

but no manipulator. Each active trader sells ) 2 ( i

A Q shares short to the uninformed investors at

time 2 if she observes what she believes to be the informed investor selling at time 1.35 There are

no limits on the number of shares short sellers can borrow.36

The active traders believe that the informed investor has negative information about the

firm’s prospects when they have observed him selling at time 1. Each active trader realizes that

she is competing against N – 1 other active traders to sell her shares. The aggregate number of

shares the active traders offer for sale is:

Q (2) Q (2)

i N

A A Σ∈

= (3)

where Qi (2)

A is active trader i’s offer to sell at time 2. All the outstanding shares at time 2 are

available for sale because the uninformed investors can sell the QI(1) = QU(1) shares they

purchased from the informed investor at time 1.37

Each active trader solves the following problem:

34 I assume that the informed investor does not buy shares when he realizes that the time 3 price will be H

either because his charter limits him to short selling or because he believes he can find other short selling

opportunities that are more profitable.

35 I develop this case further when I introduce a manipulator into the market. To distinguish her behavior from that

of the manipulator, suppose he has no shares. I assume that the informed investor in that case would want to release

any credible negative information he has concerning the true value of the shares into the market before he buys any

shares. Since he is not a manipulator, I assume that any information he releases is credible.

36 Later in the paper I consider the impact of a limitation on the number of shares that are available for short sellers

to borrow.

37 As long as there is at least one active trader, the aggregate number of shares offered by active traders at time 2 will

exceed the number of shares demanded by the informed investor, QA(2) > QI(2).

max ( [ (1) (2) (2)]) (2) (2) (2)

iA

iA

i N

iA

Q I I A B Q Q Q Q LQ i

A

− + + − Σ∈

(4)

subject to . 0 ) 2 ( ≥ i

A Q Solving the N first order conditions gives

( 1)

* (2) (1) (2)

− − −

B N

Q A L BQI BQI

A (5)

Given the assumed symmetry of the active traders, equation (5) holds for each. The equilibrium

market price is

1

* (2) (1) (2)

+ − −

N

P A NL BQI BQI (6)

The informed investor sells shares at time 1 and repurchases them either at time 2 or at

time 3. He decides how many shares to sell at time 1 by solving the following problem:

max ( [ (1)) (1)])( (1) (1)) ( 2 ) (1) ( (2) ) (1)

^ ^

(1), (1)

^ I I I I I I

Q Q

A B Q Q Q Q L C Q P C Q

I I

− + + − + − + (7)

subject to (1) ≥ 0 I Q and ˆ (1) ≥ 0. I Q Applying the Kuhn-Tucker conditions, equation (7) has the

following solutions. Either (1) = 0 I Q or

ˆ (1)

2 2

2 1

(1) 2 I I Q

N

N

B

Q A L C

−

− −

= (

Either ˆ (1) = 0 I Q or

(1)

(1) (2)

^ *

I I Q

B

Q A P C −

− −

= (9)

(1) ≥ 0 I Q provided C ≤ (3N + 3)(A − L) /(2N 2 + 5N + 5) , and (1) 0

≥ I Q provided

C ≥ (A − L) /(2N).

The informed investor’s strategy at time 2 must be optimal given the N active traders’

demand for shares at that time. The informed investor solves the problem:

− + + Σ − +

i N

I I

iA

Q I I A B Q Q Q Q L C Q I

max ( [ (1) (2) (2)]) (2) ( ) (2) (2) (10)

subject to (2) ≥ 0. I Q The solution to equation (10) is (2) = 0 I Q or

( 2)

(2) (1) ( 1)

− − − +

B N

Q A L BQI N C

I (11)

(2) ≥ 0 I Q provided C ≤ (A − L) /(2N).

2.4 Equilibrium When Selling Short Is Expensive

When the informed investor’s cost of shorting shares is high enough and the number of

active traders is large enough that C ≥ (A − L) /(2N), then (1) = 0 I Q , (2) = 0, I Q and (1) 0

> I Q .

The informed investor only sells shares short at time 1 and repurchases at time 2 all the shares he

shorted at time 1. Each active trader sells short

B(N 1)

Q* (2) A L

A +

−

= (12)

shares. The aggregate number of shares the N active traders offer to sell at time 2 is:

A L

N 1

Q (2) N *

−

= (13)

The time 2 price is:

1 1

(2) (2) * *

−

= +

= − =

N

L A L

N

P A BQ NL A A (14)

Each active trader expects to earn profit of

( 1)

( )

−

B N

A L π Ai (15)

Table 1 shows how the market equilibrium depends on the cost of shorting and the number of

active traders.

As the number of active traders becomes large, the aggregate short position converges to

all the outstanding shares, and P*(2) converges to the shares’ intrinsic value:

B

Q A L N A

−

= →∞ lim (2) * (16)

lim P* (2) L

N = →∞ (17)

Competition among active traders promotes market efficiency. However, if there is only a small

number of active traders, competition is limited, and each will try to extract a per-share rent

equal to

P* (2) − L = (A − L) /(N + 1) ≥ C (1

The informed investor sells

2 ( 1)

( ) ( 1)

(1) (2)

^ * *

− − +

− −

B N

N A L N C

B

Q A P C I (19)

shares at time 1 at a market price of

( )

2 2

1 2 2

* (1) A L C

N

A L C L N

N

P A N − +

+

+

+ = +

−

+

= − (20)

She will not find it profitable to sell shares short at time 2 nor to maintain the short position until

time 3. With this strategy, the informed investor realizes a profit equal to

4 ( 1)

[ ( ) ( 1) ]

− − +

B N

N A L N C

I π (21)

The informed investor will not find it profitable to sell shares at time 2. If he were going

to sell additional shares, he would be better off selling them at time 1 because P* (1) > P* (2) .

The informed investor also will not buy any shares at time 2 beyond what would be required to

cover his short position because he would lose P* (2) − L on each net share he purchased at time

2 and held to time 3.38 He will not sell any shares at time 2 because he would lose L + C − P* (2)

on each share he sold short at time 2 and held to time 3. He will not hold his short position to

38 Unless the holding period exceeds six months, Rule 16b under the Securities Exchange Act of 1934 would

obligate any 10 percent shareholder, officer, or director to return to the firm the so-called short swing trading profits

earned from selling and repurchasing the stock within a six-month period.

time 3 because the cost of holding it and closing it out at time 3 is L + 2C. Since

C ≥ (A − L) /(2N), this strategy is less profitable than repurchasing the shares at time 2 because

in that case his profit would only be:

(1) (1) ( 2 ) (1)

* *

I I I π = P Q − L + C Q

2 2 ( 1)

[ ( 1) ][ ( ) ( 1) ]

4 ( 1)

[ ( ) ( 1) ]

B N I

A L N C N A L N C

B N

N A L N C ≤ π

− − + − − +

− − +

= (22)

Finally, each active trader’s strategy is optimal given all the other active traders’ strategies and

the informed investor’s strategy. Thus, no active trader can deviate profitably.

2.5 Equilibrium When Selling Short Is Inexpensive

When the informed investor’s cost of maintaining the short position is low enough and

the number of active traders is small enough that C ≤ (3N + 3)(A − L) /(2N 2 + 5N + 5), then

(1) ≥ 0 I Q , (2) ≥ 0 I Q , and (1) 0.

= I Q The informed investor sells additional shares short at time

2 and waits until time 3 to cover his entire short position.39

The informed investor sells more shares short at time 1 when the cost of shorting is low

because

2 ( 1)

( ) ( 1)

(1) (1) 2

− − +

− −

+ =

B N

N A L N C

B

Q Q A L C I I (23)

The active traders sell fewer shares short at time 2 because

B

A L

N

N

N

N

B

Q A L C A

−

− +

2 2 1

(2) 4 * (24)

provided N > 1. But the total number of shares shorted at time 2 is greater, and as a result, P*(2)

is lower than in the high-shorting-cost case:

39 When (A − L) /(2N) < C < (3N + 3)(A − L) /(2N 2 + 5N + 5) , the informed investor repurchases at time

2 a portion of the shares initially sold short and the rest at time 3 but will not sell short any additional shares at time

2( 2) 1

* (2) 4

−

< +

− +

= +

N

L A L

N

P L A L C (25)

provided N > 1.

P*(1) is also lower, and therefore, closer to the shares’ intrinsic value, because

2 1 2 2

* (1) A L C

N

P A L C A N +

−

+

+ < −

+

= (26)

provided N > 1 due to the heavier short selling by the informed investor at time 1 when the cost

of shorting is lower. Less expensive short selling facilitates arbitrage and promotes market

efficiency.

Also, as in the high-cost case, the aggregate short interest converges to all the shares

outstanding and P*(2) converges to the shares’ intrinsic value as the number of active traders

becomes large:

B

Q Q Q A L N I I A

−

+ + = →∞ lim (1) (2) (2) * (27)

P L N = →∞ lim * (2) (2

Thus, in both cases, competition among active traders promotes market efficiency.

2.6 Timing of Short Covering

The informed investor will hold the short position until time 3, rather than cover it at time

2, provided

2( 1)

* (2) 2

− +

+ < = +

N

L C P L A L C (29)

which is satisfied when C < (A − L) /(2N) . So long as C is small or N is small, the informed

investor finds it more profitable to cover the short position at time 3. He also sells short more

shares at time 2 and covers those short sales at time 3. But as the number of active traders grows,

eventually the sign in equation (29) reverses. The informed investor stops selling shares short at

time 2, and he repurchases some at time 2 and the rest at time 3. When N grows large enough

that C > (A − L) /(2N), the informed investor only sells shares short at time 1 and covers the

entire short position at time 2.

Active traders have two opposing effects on the informed investor’s profit. First, they sell

shares at time 2, which reduces P*(2) and the informed investor’s information rent from short

selling, P*(2) – L. The lower P*(2) allows the informed investors to repurchase shares more

cheaply at time 2 in the expensive-shorting case (IV). As a result, they sell more shares ˆ * (1)

I Q .

Increasing the number of active traders in the expensive-shorting case increases *

I π .40 On the

other hand, the active traders compete with the informed investor to sell shares short at time 2

when short selling is inexpensive enough (and the number of active traders is small enough) that

the informed investor wants to sell shares short at time 2 (case II). Greater competition reduces

the informed investor’s information rent. This reduces (2) I Q and the profitability of the

informed investor’s short selling in the low-shorting-cost case (II), since *

I π in the low-shortingcost

case depends on N:

2 2

4 ( 2)

( 2 )

− −

B N

A L NC

B

A L C

I π (30)

If the number of active traders becomes large enough – and the competition becomes

sufficiently intense – the informed investor’s profit-maximizing strategy shifts from selling

shares to buying them back at time 2. Short selling at time 2 becomes less profitable as N

increases and P*(2) falls. Competition from more active traders eventually makes it unprofitable

to sell short at time 2. He buys rather than sells at time 2. Further increases in the number of

active traders raise ˆ (1) I Q and increase the informed investor’s profit. This equilibrium

40 *

I π increases with N in case IV except when A – L is very small, which is not an interesting case because the

profit potential in short selling is small.

corresponds to the high-shorting-cost case. The change in strategy in response to the increase in

the number of active traders makes the informed investor worse off because even though he only

incurs one period’s shorting cost, he must pay P*(2) > L to repurchase the shares. His profit is

greater in the low-shorting-cost case because *

I π in equation (30) exceeds *

I π in equation (21).41

2.7 Effect of Short Sale Restrictions

Suppose the number of shares available for borrowing is capped at H. Then

Q Q H I I (1) + ˆ (1) ≤ (31)

Q Q Q H I I A (1) + (2) + (2) ≤ (32)

As a result, P*(2) in equation (6) includes a shadow price for short sales λ . The time 2 market

price in equations (17) and (2 converges to L + λ . The amount of short sales at time 2 in

equations (16) and (27) converges to H < (A – L)/B. The borrowing restriction reduces market

efficiency by preventing short sellers from arbitraging away the mispricing (Dechow, Hutton,

Meulbroek, and Sloan, 2001, D’Avolio, 2002, and Geczy, Musto, and Reed, 2002).

The market equilibrium in the simplified market structure exhibits the behavioral

properties one would expect in a market that is free of manipulation.42 Figure 1 illustrates the

sensitivity of market prices to the cost of shorting and to the number of active traders, and

Figure 2 illustrates the sensitivity of *

I π to N.

3. Market Equilibrium When Manipulators Are Present in the Market

Next, I consider how a manipulator entering the market affects the market equilibrium. If

the informed investor is shorting shares optimally and the manipulator has the same cost of

41 Since C ≤ (A − L) /(2N), *

I π in equation (30) is no less than ( 1) 2 ( )2 /(4 2 ). N − A − L BN Since

C ≥ (A − L) /(2N) for *

I π in equation (21), in that case * (N 1)2 (A L)2 /(4BN 2 ) I π < − − for N > 0.

42 Table 1 contains a third case, which might be termed the ‘moderate cost of shorting case.’ It is easily verified that

the market equilibrium in this case also exhibits the expected behavioral properties.

shorting, then it will be unprofitable at the margin for the manipulator to sell shares short.43

Therefore, I assume that the manipulator does not short shares if the informed investor is selling

shares short.44 If the informed investor does not sell any shares short at time 1, then the

manipulator knows that the price will be H at time 3. In that event, the manipulator sells shares

short with probability q/(1 – p). The active traders continue to condition their behavior at time 2

on what they observe at time 1. Market equilibrium can be either a pooling equilibrium or a

separating equilibrium, depending on the cost of shorting.

3.1 Pooling Equilibrium

A pooling equilibrium can occur in case IV but not in the other three cases in Table 1.

The manipulator can imitate the selling behavior of the informed investor by shorting (1)

^ *

I Q

shares at time 1. The manipulator must cover his short position at time 2 because holding the

shares until time 3 is unprofitable.

If the manipulator sells (1)

^ *

I Q shares at time 1, the same number as the informed

investor, then the active traders will assess the likelihood that the seller is a manipulator as:

β = q /( p + q) (33)

Each active trader solves the following problem at time 2 conditional on observing a sale at time

max (1 )[( (2)) (2) (2)] [( (2)) (2) (2)] (2)

iA

iA

i N

iA

iA

iA

i N

iA

Q A B Q Q LQ A B Q Q HQ

A

− − Σ − + − Σ −

∈ ∈

β β (34)

The N active traders sell fewer shares short at time 2 as a result of the risk of manipulation:

B

A L

N

N

B

A L H

N

Q N A

−

− − −

(1 )

(2) * β β

(35)

43 Later in the paper I show that it is profitable for the manipulator to sell shares short if he has lower shorting costs

than the informed investor. Naked shorting satisfies this condition.

44 The manipulator could spread rumors or engage in manipulative trading to drive the informed investor from the

market. I do not consider the implications of such behavior in this paper.

Consequently, the market price at time 2 is higher due to manipulation:

(2)

1 1

* (2) * (2) ( ) P*

N

L A L

N

P A BQ L A L N H L M A =