Ms. Vanessa A. Countryman

Agency decision

Ask Donna

What actually matters in this document.

Text

Via email

June 21, 2024

Ms. Vanessa A. Countryman

Secretary

Securities and Exchange Commission

100 F Street, N.E.

Washington, DC 20459

Re: Petition for rulemaking or guidance regarding calculation of short-swing

profits under Section 16(b) of the Securities and Exchange Act of 1934

Dear Ms. Countryman:

In Chechele v. Standard General L.P.,1 the U.S. District Court for the Southern District of

New York noted that the Commission’s “lowest-in, highest-out” method for calculating

Section 16(b) short-swing profits is “prone to error” and therefore “cannot be used”

under some circumstances.2 Despite this, the court concluded that “‘there is virtually no

chance a court will deviate from it in the absence of a statutory or rule change to the

contrary.’”3

I am writing to petition for rulemaking or guidance to the effect that a more accurate

method based on linear programming should be used where the “lowest-in, highest-out”

1 No. 20 Civ. 3177, 2022 WL 766244 (S.D.N.Y. 2022).

2 See id. at *6 & n.8 (citing Andrew Chin, The Learned Hand Unformula for Short-Swing Liability, 91

WASH. L. REV. 1523, 1552-63 (2016); Arnold S. Jacobs, An Analysis of Section 16 of the Securities Exchange

Act of 1934, 32 N.Y. L. SCH. L. REV. 209, 532-33 (1987)).

3 See id. at *6 (citing PETER J. ROMEO & ALAN L. DYE, SECTION 16 TREATISE AND REPORTING GUIDE § 12.02, at

1236 (5th ed. 2019)) (emphasis added); see also Rubenstein v. Knight-Swift Transportation Holdings Inc.,

664 F.Supp.3d 523 (S.D.N.Y. 2023) (same).

method may fail to calculate the maximum recoverable profit. 4 I am a Paul B. Eaton

Distinguished Professor of Law at the University of North Carolina, a former computer

science professor, and the author of the two attached law review articles on the accurate

calculation of short-swing profits under Section 16(b).5 I have no financial interest

concerning this matter.

The Commission’s most recent guidance regarding the matching of transactions for

calculating short-swing liability was in 2002.6 The guidance provides an example

illustrating that a sale of stock is matchable with the lowest-priced purchase that occurred

within six months of the sale.7 This is consistent with the Second Circuit’s holding in

Smolowe v. Delendo8 that Section 16(b) imposes strict liability on the insider, allowing

“an arbitrary matching to achieve the showing of a maximum profit.”9

The guidance cites Smolowe for a different proposition, however. It advises that the

calculation of recoverable profit should follow the “lowest-in, highest-out” formula,10

which the courts adopted from the Commission’s amicus briefs in Smolowe.11 This

approach is a “greedy algorithm,” which makes the locally optimal choice at each stage

but often fails to yield the globally optimal solution.12 This flaw becomes evident when

calculating short-swing profit recoveries from transactions spanning more than six

months. For instance, in the trading sequence illustrated below, the Smolowe approach

4 See Chin, supra note 2, at 1549-50 & n.126 (illustrating the more accurate method); Andrew Chin,

Accurate Calculation of Short-Swing Profits Under Section 16(b) of the Securities Exchange Act of 1934, 22

DEL. J. CORP. L. 587, 593-99 (1997) (same).

5 See supra notes 2 and 4.

6 See Commission Guidance on the Application of Certain Provisions of the Securities Act of 1933, the

Securities Exchange Act of 1934, and Rules Thereunder to Trading in Security Futures Products, S.E.C.

Release No. 8107 (June 21, 2002), 2002 WL 1677437, at *4-*10.

7 See id. at *7 (Example 4).

8 136 F.2d 231 (2d Cir. 1943), cert. denied, 320 U.S. 751 (1943).

9 Id. at 237.

10 Commission Guidance, supra note 6, at *7 n.40 (citing Smolowe) (“Under this method, recoverable

profit is computed by matching the highest sale price with the lowest purchase price within six months, the

next highest sale price with the next lowest purchase price within six months, and so on, until all shares

have been included in the computation.”); see also Interpretive Release on Rules Applicable to Insider

Reporting and Trading, S.E.C. Release No. 18114, 1981 WL 31301, at *28 n.102 (Sept. 24, 1981) (describing

the Smolowe formula as “the only rule that would … require the insider to disgorge all possible profit”).

11 See Smolowe v. Delendo Corp., 46 F.Supp. 758, 766 (S.D.N.Y. 1942), aff’d, 136 F.2d 231 (2d Cir. 1943)

(adopting “[t]he computation suggested by the Securities & Exchange Commission”); see also Brief of

Securities and Exchange Commission as Amicus Curiae at 3-5, Smolowe v. Delendo Corp., 136 F.2d 231 (2d

Cir. 1943) (No. 191) (describing the algorithm in full).

12 See, e.g., Wikipedia, Greedy Algorithm, https://en.wikipedia.org/wiki/Greedy_algorithm (visited June

21, 2024).

leaves $1,000 in recoverable profit unclaimed. In the worst case, the formula yields only

half of the recoverable profit.13

Conversely, the “lowest-in, highest-out” method is provably correct if all of the insider

trades occurred within the same six-month period, as in Smolowe.14 A close reading of

Smolowe, however, demonstrates that the court never endorsed this formula for trades

spanning a longer duration.15 Yet, courts and parties have applied it to longer trading

patterns,16 occasionally resulting in diminished recoveries.17 The Commission’s 2002

13 See Chin, supra note 2, at 1558-61.

14 See id. at 1551-58 (proving the Smolowe formula’s correctness within a single statutory six-month period).

15 See id. at 1542 (“the Commission’s formula … was designed and proposed for use only in cases involving

a single statutory six-month trading period”); id. at 1542 n.83 (“[T]he Smolowe court’s statement of the rule

must be read as limited to cases involving a single statutory six-month trading period because otherwise it

would be empirically false.”); see also ARNOLD S. JACOBS, SECTION 16 OF THE SECURITIES EXCHANGE ACT 531

(2011) (citations omitted) (“[Although it] has been widely cited and followed . . . the lowest price in-highest

price out rule is not the real holding of Smolowe.”); Gratz v. Claughton, 187 F.2d 46, 51 (2d Cir. 1951)

(Learned Hand, J.) (referring to the matching of transactions to “increase [profits] to the greatest possible

amount” as “the doctrine of Smolowe”).

As I have argued in the first attached article, Judge Learned Hand wisely did not endorse or use the

Smolowe algorithm in his famous Gratz v. Claughton decision, but simply held that the plaintiff was entitled

to choose an arbitrary matching of short-swing trades. He offered no view as to whether the “lowest-in,

highest-out” method would maximize her recovery, affirming the judgment below solely on the grounds

that the plaintiff had stipulated to it. See Chin, supra note 2, at 1545.

16 See, e.g., Adler v. Klawans, 267 F.2d 840, 847–48 (2d Cir. 1959) (spanning more than seven months);

Donoghue v. Casual Male Retail Group, Inc., 375 F. Supp. 2d 226, 237 (S.D.N.Y. 2005) (spanning more than

ten months); Segen v. Westcliff Capital Mgmt., LLC, 299 F. Supp. 2d 262, 265–66, 272 (S.D.N.Y. 2004)

(spanning more than ten months); Donoghue v. MIRACOR Diagnostics, Inc., No. 00 CIV 6696 JGK RLE,

2002 WL 233188, at *1–3 (S.D.N.Y. Feb. 11, 2002) (spanning more than thirteen months); Morales v. New

Valley Corp., 999 F. Supp. 470, 476 (S.D.N.Y. 1998) (spanning more than six months); Morales v. Mylan

Labs., Inc., 443 F. Supp. 778, 780 (W.D. Pa. 1978) (three purchases made more than two years prior to suit);

Heli-Coil Corp. v. Webster, 222 F. Supp. 831, 837 (D.N.J. 1963), modified, 352 F.2d 156 (3d Cir. 1965)

(spanning more than nine months); Ark. La. Gas Co. v. W.R. Stephens Inv. Co., 141 F. Supp. 841, 847–48

(W.D. Ark. 1956) (spanning more than thirteen months); Kogan v. Schulte, 61 F. Supp. 604, 605 (S.D.N.Y.

1945) (spanning fifteen months).

guidance also erroneously advises using the “lowest-in, highest-out” method regardless of

the trading duration.

During the time of Smolowe, the Commission did not have access to computers and linear

programming software, and could only provide a partial solution to the maximum

recovery problem.18 However, the mathematical constraints of the pre-computer age

should not be the Commission’s final word on this matter. With the necessary tools now

available, it is imperative that courts and parties be informed that the only universally valid

way “to achieve the showing of a maximum profit” is through linear programming

techniques.19

Thank you for considering this petition. Should you need any additional information or

assistance, please feel free to contact me.

Respectfully submitted,

Andrew Chin

Paul B. Eaton Distinguished Professor of Law

17 See, e.g., Chechele v. Vicis Capital, LLC, No. 11 Civ. 2191, 2012 WL 310943 (S.D.N.Y. 2012); Chin, supra

note 2, at 1561-62 (finding shortfall of $394 in Chechele).

18 See George B. Dantzig, Maximization of a Linear Function of Variables Subject to Linear Inequalities, in

ACTIVITY ANALYSIS OF PRODUCTION AND ALLOCATION 19-32 (Tjalling C. Koopmans ed. 1951) (reprinting

Dantzig’s 1947 paper introducing the simplex method for solving linear programming problems).

19 See Chin, supra note 2, at 1549-50 & n.126 (translating the profit calculation into a linear programming

problem); Chin, supra note 5, at 593-99 (translating the profit calculation into a transportation problem,

which is a particular form of linear programming problem).

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THE LEARNED HAND UNFORMULA

FOR SHORT-SWING LIABILITY

Andrew Chin

Abstract: Section 16(b) of the Securities Exchange Act of 1934 allows for the recovery of

short-swing profits realized by certain insiders from trading in a corporation’s stock within a

period of less than six months. Three generations of corporate law students have been taught

the “lowest-in, highest-out” formula that is intended to maximize the disgorgement of shortswing profits under section 16(b). Arnold Jacobs’s 1987 treatise presented two hypothetical

examples where the formula fell short of the intended maximum, but courts, commentators,

and practitioners have largely ignored these theoretical challenges to the formula’s validity.

This Article identifies Gratz v. Claughton as the first reported real-world example of the

formula’s failure. Ironically, Gratz has been taught and cited for more than sixty years as a

leading authority for the formula’s use, not least because of its distinguished author, Judge

Learned Hand. This Article argues that Gratz has been misunderstood and that Hand wisely

adjudicated this complex case without prescribing or endorsing the formula in any way. It

also shows that the formula has no need of Gratz’s endorsement, as long as the formula is

correctly interpreted as limited to simpler cases where it is mathematically valid. It

formalizes and extends Jacobs’s results by showing that the formula may fall short of the

maximum by up to fifty percent when misused in more complex cases, and has actually fallen

short in another more recent case. Finally, it provides online tools to enable practitioners and

judges to calculate short-swing liability correctly in all cases.

INTRODUCTION .............................................................................. 1524

PRELIMINARIES ................................................................... 1527

A. Short-Swing Liability Under Section 16(b)...................... 1527

B. The Smolowe Formula and Its Potential Shortcomings .... 1529

C. The Ubiquity of the Smolowe Formula and the

Misreading of Gratz ......................................................... 1533

Associate Professor, University of North Carolina School of Law; J.D., Yale; D.Phil., Mathematics

and Computer Science, Oxford. The author wishes to thank Kate Dickson, Jenica Hughes, Luke

Pettyjohn and especially Stephen Dew for their diligent and insightful research assistance; David

Adler, Kaja Coraor, Patrick Hahn, Allie Harrison, Tim Kang, Madi Pfaff and especially Kevin

Valakuzhy for their significant contributions to the development of the Web-based section 16(b)

liability calculator described in section IV.A infra; Rachel Rogers for her painstaking work in

checking the author’s transcriptions of the accounting exhibits in Gratz; and Al Brophy, Bernie

Burk, Michael Corrado, John Coyle, Deborah DeMott, William Fisher, Victor Flatt, Michael

Guttentag, Tom Hazen, Joan Heminway, James Hunter, Arnold Jacobs, Keenan Kmiec, Holning

Lau, Margaret Lemos, Marin Levy, Tom Lin, Bill Marshall, Darrell Miller, Eric Muller, Richard

Myers, Elizabeth Pollman, Arti Rai, Rob Smith, Larry Zelenak and Taisu Zhang for their helpful

comments and suggestions. The assistance of Patrick Connelly and Dave Hansen in retrieving case

materials from the National Archives is also gratefully acknowledged.

1523

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WASHINGTON LAW REVIEW

[Vol. 91:1523

THE MEANING OF HAND’S MATHEMATICAL

SILENCE ................................................................................. 1538

A. Smolowe and Hand’s Silence in Gratz ............................. 1538

B. The Judgment Below and Hand’s Silence in Gratz .......... 1546

C. Gratz’s Unsuitability for Endorsing the Smolowe

Formula ............................................................................ 1548

III. THE WISDOM OF HAND’S MATHEMATICAL

SILENCE ................................................................................. 1551

A. The Smolowe Formula Needs No Corroboration in

Simple Cases .................................................................... 1551

1. Lowest-In, Highest-Out .............................................. 1553

B. The Smolowe Formula’s Worst-Case Errors in Complex

Cases................................................................................. 1558

C. The Smolowe Formula’s Continuing Fallibility ............... 1561

IV. LEARNING FROM HAND’S MATHEMATICAL

SILENCE ................................................................................. 1564

A. An Online Solution ........................................................... 1565

B. Prospects for Change at the SEC ...................................... 1571

CONCLUSION .................................................................................. 1574

APPENDICES .................................................................................... 1575

A. Computation of Short-Swing Profits in Gratz .................. 1575

B. Computation of Short-Swing Profits in Chechele ............ 1585

II.

INTRODUCTION

Under section 16(b) of the Securities Exchange Act of 1934, 1 certain

insiders may be held liable to a corporation for any “short-swing” profits

realized from trading in the corporation’s stock within a period of less

than six months. The corporation is entitled to disgorgement of the

maximum possible profit that can be calculated by any matching of the

insider’s purchases and sales within less than six months, according to

Second Circuit case law, which has long been authoritative on the

subject. 2

In Smolowe v. Delendo Corp., 3 the Second Circuit adopted the

“lowest-in, highest-out” formula as a simple calculation intended to

maximize the disgorgement of short-swing profits under section 16(b). 4

The liability calculation in Smolowe involved a relatively simple

15 U.S.C. §§ 78a–78pp (2012).

DETLEV F. VAGTS, BASIC CORPORATION LAW 552 (3d ed. 1989) (“Opinions by the Second

Circuit in the Section 16 field are generally regarded as authoritative.”).

136 F.2d 231 (2d Cir. 1943).

See id. at 239.

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1525

sequence of insider transactions, all of which took place within a single

six-month period and within the two-year statute of limitations.5 In a

1987 article, however, Arnold Jacobs presented hypothetical examples

showing that the Smolowe formula 6 may fall short of maximizing the

short-swing profit calculation in situations in which the insider’s trades

span a period of more than six months or when some trades are not

within the statute of limitations.7 In these situations, the calendar can

preclude the recovery of profits from matching some low-priced

purchases with higher-priced sales, a complication the Smolowe formula

was not designed to take into account. 8 Courts, commentators, and

practitioners, however, have largely ignored these theoretical challenges

to the formula’s validity in adopting the Smolowe formula for use in all

section 16(b) liability calculations. 9

This Article identifies another early Second Circuit case, Gratz v.

Claughton, 10 as the first reported real-world example of the Smolowe

formula’s failure to calculate the maximum possible profit. The liability

calculation in Gratz was too complicated for the formula because it

involved a sequence of hundreds of insider transactions spanning more

than twenty-one months. 11 Ironically, Gratz has been a staple of

corporate case law and casebooks for more than sixty years as a leading

authority for the formula’s use, not least because of its distinguished

author, Judge Learned Hand. 12 However, neither the Second Circuit nor

the district court performed any calculations in Gratz. In district court

proceedings before a special master, the defendant proffered a liability

Smolowe v. Delendo Corp., 46 F. Supp. 758, 761, 766 (S.D.N.Y. 1942) [hereinafter

Smolowe I].

The formula will be referred to hereinafter as “the Smolowe formula” (or simply “the formula”

when clear from context). The more common designation “the Smolowe rule” will not be used in

order to avoid unintended connotations of legal authority in light of the formula’s questionable

applicability in complex cases.

Arnold S. Jacobs, An Analysis of Section 16 of the Securities Exchange Act of 1934, 32 N.Y. L.

SCH. L. REV. 209, 532–33 (1987).

See id.

See, e.g., Adler v. Klawans, 267 F.2d 840, 847–48 (2d Cir. 1959); LOUIS LOSS & JOEL

SELIGMAN, FUNDAMENTALS OF SECURITIES REGULATION 693 (5th ed. 2004); WILLIAM K.S. WANG

& MARC I. STEINBERG, INSIDER TRADING 924 n.12 (3d ed. 2010).

187 F.2d 46 (2d Cir. 1951).

See Def.’s Exhibits 5 & P, Gratz v. Claughton, No. 35-410 (S.D.N.Y. 1949) (hereinafter “Gratz

Master’s Report”) (listing, inter alia, 276 purchases and 101 sales of common stock and 11

purchases and 20 sales of preferred stock between December 18, 1944 and September 24, 1946).

See, e.g., Gerald Gunther, Judge Learned Hand: The Man, the Myth, the Biography, 20 J. SUP.

CT. HIST. 47, 47 (1995) (describing Hand’s opinions as “familiar to every lawyer and law student”).

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WASHINGTON LAW REVIEW

[Vol. 91:1523

calculation 13 that fell more than $50,000 short of the short-swing profits

that would have been found by the Smolowe formula. Perhaps

overwhelmed by the prospect of checking the sums, the plaintiff

stipulated to the defendant’s calculation in the district court and did not

challenge it on appeal. 14 Accordingly, Hand adjudicated Gratz without

performing a liability calculation or even mentioning the formula. 15

With the benefit of hindsight and subsequent developments in

computing, the remainder of this Article elucidates the meaning,

wisdom, and continuing significance of Hand’s mathematical silence in

Gratz. Part I of this Article sets the stage for this exposition by

introducing the short-swing liability provisions of section 16(b), the

Smolowe formula and its shortcomings, and the role Gratz has played in

sustaining the Smolowe formula.

Part II of this Article dispels the notion that Gratz in any way supports

use of the Smolowe formula. Section II.A harmonizes the Second

Circuit’s adjudication of liability in Smolowe and Gratz and shows that

Hand rightly did not read Smolowe to require use of the formula in

Gratz. Section II.B explains that Hand wisely based his affirmance on

Gratz’s acquiescence in the judgment below and not on the master’s

putative adoption of the Smolowe formula, thereby devising a form of

adjudication that might be dubbed “the Learned Hand unformula.”

Section II.C shows that Gratz could not have corroborated the Smolowe

formula because the formula was probably not used to calculate

Claughton’s liability and would have fallen short even if it had been so

used.

Part III of this Article explains why courts, attorneys, professors, and

regulators should stop relying on Gratz to justify the Smolowe formula’s

use beyond its valid and intended range. Section III.A proves the

Smolowe court’s assertion that the formula maximizes profit recovery

from trades within a single statutory six-month period, obviating six

decades of unjustified reliance on Gratz for empirical corroboration of

the formula. Section III.B extends and formalizes Jacobs’s results by

showing that the formula may fall short of calculating the maximum

short-swing profit by up to fifty percent in the worst case. Section III.C

brings Jacobs’s hypotheticals into the real world by describing a more

recent case where the Smolowe formula’s fallibility led to a diminished

recovery.

See Def.’s Exhibit C to Gratz Master’s Report, supra note 11.

See Gratz, 187 F.2d at 52.

See generally id.

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1527

It will not be easy to disrupt the six decades of case law and legal

teaching that have perpetuated overreliance on the Smolowe formula. A

leading treatise calls the formula “so firmly ingrained in the fabric of

Section 16(b) that there is virtually no chance a court will deviate from it

in the absence of a statutory or rule change to the contrary.”16 Part IV of

this Article describes two potentially disruptive efforts. Section IV.A

introduces a free online calculator on the author’s website that should

facilitate and encourage a more limited reading of Smolowe in future

short-swing liability proceedings and in law school classrooms. Section

IV.B discusses the prospect of legal change through the Securities and

Exchange Commission’s petition for rulemaking and request for amicus

participation processes. The Article concludes by summarizing its

central insight about Gratz.

PRELIMINARIES

Short-Swing Liability Under Section 16(b)

The Securities Exchange Act of 193417 aims to “insure the

maintenance of fair and honest markets” by, inter alia, regulating

transactions by officers, directors, and principal owners. 18 As a deterrent

to unfair insider trading, 19 section 16(b) of the Act allows a corporation,

or a shareholder suing on the corporation’s behalf, to recover any “shortswing” profit realized by an officer, director, or ten percent beneficial

owner from any purchase or sale, or sale and purchase, of its stock

within any period of less than six months. 20

See PETER J. ROMEO & ALAN L. DYE, SECTION 16 TREATISE AND REPORTING GUIDE § 11.02,

at 11-16 (1994).

15 U.S.C. §§ 78a–78b. (2012).

15 U.S.C. § 78b (2012).

See H.R. REP. NO. 1383, at 13 (1934) (“Men charged with the administration of other people’s

money must not use inside information for their own advantage.”).

Section 16(b) provides in relevant part:

For the purpose of preventing the unfair use of information which may have been obtained by

such [more than ten percent] beneficial owner, director, or officer by reason of his relationship

to the issuer, any profit realized by him from any purchase and sale, or any sale and purchase,

of any equity security of such issuer (other than an exempted security) or a security-based swap

agreement involving any such equity security within any period of less than six months, unless

such security or security-based swap agreement was acquired in good faith in connection with a

debt previously contracted, shall inure to and be recoverable by the issuer, irrespective of any

intention on the part of such beneficial owner, director, or officer in entering into such

transaction of holding the security or security-based swap agreement purchased or of not

repurchasing the security or security-based swap agreement sold for a period exceeding six

months. Suit to recover such profit may be instituted at law or in equity in any court of

competent jurisdiction by the issuer, or by the owner of any security of the issuer in the name

and in behalf of the issuer if the issuer shall fail or refuse to bring such suit within sixty days

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Section 16(b) is a strict liability provision in two respects. First, it

“requires insiders to disgorge these ‘short-swing’ profits ‘even if they

did not trade on inside information or intend to profit on the basis of

such information.’” 21 Second, it allows the corporation to recover the

maximum profit calculated from the matching of “any purchase and sale,

or any sale and purchase . . . within any period of less than six

months,” 22 even if the insider incurred a net loss from other trading

during the pertinent period. 23 In effect, section 16(b) demands that the

insider “pay the maximum after-the-fact value that inside information

concerning [short-term changes in the price of] the stock could have had,

given his stock transactions[,]” 24 regardless of whether or how he

actually used that information. 25 It thereby encourages insiders to

manage their companies “in ways that will cause steady appreciation of

stock prices,” while “depriv[ing] them of trading opportunities that

might lead them to manage corporate affairs in ways that will cause

prices to fluctuate or decline.” 26

after request or shall fail diligently to prosecute the same thereafter; but no such suit shall be

brought more than two years after the date such profit was realized.

15 U.S.C. § 78p(b) (2010).

Credit Suisse Securities (USA) LLC v. Simmonds, __ U.S. __, 132 S. Ct. 1414, 1417 (2012)

(quoting Gollust v. Mendell, 501 U.S. 115, 122 (1991)).

15 U.S.C. § 78p(b) (2012); see Smolowe v. Delendo Corp, 136 F.2d 231, 237 (2d Cir. 1943)

(“The fact that purchases and sales may be thus coupled, regardless of the intent of the

insider . . . points to an arbitrary matching to achieve the showing of a maximum profit.”).

See Adler v. Klawans, 267 F.2d 840, 847 (2d Cir. 1959) (“The argument that losses and profits

made by defendant . . . should be matched against each other to determine liability must be

answered in the negative . . . .”); see, e.g., Donna Darm, Short-Swing Profits in Failed Takeover

Bids—The Role of Section 16(b), 59 WASH. L. REV. 895, 912 (1984) (arguing that section 16(b)

punishes unsuccessful takeover bids too harshly); Park McGinty, Replacing Hostile Takeovers, 144

U. PA. L. REV. 983, 1061 n.205 (1996) (citation omitted) (referring to Gratz as “the most famous

example of the draconian character of [section 16(b)]’s ‘mechanical’ provisions”).

See Robert L. Davis, Note, Tax Treatment of Section 16(b) Payments, 27 STAN. L. REV. 143,

150 (1974).

See Kern Cty. Land Co. v. Occidental Petrol. Corp., 411 U.S. 582, 609 (1973) (“You hold the

director, irrespective of any intention or expectation to sell the security within 6 months after,

because it will be absolutely impossible to prove the existence of such intention or expectation, and

you have to have this crude rule of thumb, because you cannot undertake the burden of having to

prove that the director intended, at the time he bought, to get out on a short swing.”) (quoting

Hearings on Stock Exchange Practices before the Senate Committee on Banking and Currency, 73d

Cong., 2d Sess., pt. 15 at 6557 (1934) (statement of principal drafter Thomas G. Corcoran)).

Section 16(b)’s harshness has long been controversial. Ellen Taylor, Teaching an Old Law New

Tricks: Rethinking Section 16, 39 ARIZ. L. REV. 1315, 1318 (1997) (arguing that section 16(b)

should be repealed because it is ineffective, unfair, and expensive).

Steve Thel, The Genius of Section 16: Regulating the Management of Publicly Held

Companies, 42 HASTINGS L.J. 391, 411 (1991).

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1529

Short-swing profit recoveries can be considerable. For example,

during the internet bubble of the late 1990s, InfoSpace, Inc.’s CEO

Naveen Jain inflated the company’s value to more than $31 billion 27 and

cashed out millions of his own shares before the stock price plunged. 28 A

shareholder successfully sued Jain on behalf of the company 29 under

section 16(b), and Jain was ordered to disgorge more than $247 million

in trading profits and prejudgment interest. 30

The Smolowe Formula and Its Potential Shortcomings

Given a lengthy sequence of stock transactions, there can be many

ways of matching purchases and sales to calculate profits recoverable

under section 16(b). Since the Second Circuit’s decision in Smolowe v.

Delendo Corp., 31 courts have generally used the “lowest price in,

highest price out” formula 32 to calculate short-swing profits. 33 This

formula consists of iteratively “matching off against each other the

shares purchased at the lowest price during the period [of less than six

See David Heath & Sharon Pian Chan, Dot-Con Job: How InfoSpace Took Its Investors for a

Ride, SEATTLE TIMES (Mar. 6, 2005), http://www.seattletimes.com/business/dot-con-job-howinfospace-took-its-investors-for-a-ride/ [https://perma.cc/FT3T-J6NX].

See David Heath & Sharon Pian Chan, When Times Got Tough, Execs Hid Troubles, Dumped

Stock, SEATTLE TIMES (Mar. 7, 2005), http://www.seattletimes.com/business/when-times-got-toughexecs-hid-troubles-dumped-stock/ (last visited Dec. 7, 2016).

See Dreiling ex rel. Infospace v. Kellett, 281 F. Supp. 2d 1215, 1217 (W.D. Wash. 2003)

(discussing Dreiling’s suit against Jain and co-defendants on behalf of InfoSpace).

See id. at 1242 (ordering disgorgement of $202,551,696.05 in profits and $44,571,016.92 in

prejudgment interest for a total judgment of $247,122,712.97). The company eventually settled with

the Jains for approximately $83 million. Press Release, InfoSpace, Inc., Settlement Agreement

Reached in InfoSpace Derivative Case, Section 16(b) Case, and Certain Related Cases Brought by

the Jains (Dec. 22, 2004), http://www.sec.gov/Archives/edgar/data/1068875/000119312504219392

/dex991.htm [https://perma.cc/628X-66ZD].

See 136 F.2d 231 (2d Cir. 1941).

See id. at 239 (describing the formula succinctly as “lowest price in, highest price out—within

six months—as applied by the district court”).

See, e.g., Dreiling, 281 F. Supp. 2d at 1239 (“Consistent with the definition of profit and the

‘lowest in, highest out’ rule, therefore, the Jains’ profit is calculated at $202,551,696.05.”). The

court’s calculation was simplified—and arguably inflated—by the fact that it attributed a purchase

price of zero to shares of company stock Jain had transferred into his family’s brokerage accounts.

See id. at 1239; Brief of Sec. and Exch. Comm’n as Amicus Curiae at 12–13, Dreiling ex rel.

Infospace v. Kellett, 281 F. Supp. 2d 1234 (W.D. Wash. 2003) (No. 03-35710) (criticizing the

court’s characterization of the transfer). Jain was held liable for the entire proceeds of

$85,600,000.00, $17,955,000.00, and $98,966,696.05 from three corresponding sales of company

stock made within two months of the transfer, for a total liability of $202,551,696.05. See Dreiling,

281 F. Supp. 2d at 1237–39.

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months] and an equal number of shares sold at the highest price or prices

during the [same] period.” 34

Brief of Sec. and Exch. Comm’n as Amicus Curiae at 3, Smolowe v. Delendo Corp, 136 F.2d

231 (2d Cir. 1943) (No. 191) [hereinafter SEC Smolowe Brief]; see id. at 4–5 (containing the full

original statement of the formula).

In Smolowe, the defendant Kaplan purchased 15,800 shares from co-defendant I.J. Seskis on

April 4, 1940 for $2.25 per share, or $35,550. See Smolowe I, 46 F. Supp. 758, 62 (S.D.N.Y. 1942).

Of these, 15,583 were acquired in connection with a prior debt and were therefore exempt from

section 16(b) liability. See id. at 766. In addition to his purchase from Seskis, Kaplan conducted the

following transactions during the period in question:

Date

Transaction

Amount ($)

Price ($)/Share

12/1/1939

Purchase

5000

7,750.00

1.5500

2/5/1940

Purchase

200

285.00

1.4250

2/15/1940

Sale

200

308.91

1.5446

2/20/1940

Purchase

200

335.00

1.6750

3/25/1940

Purchase

400

924.00

2.3100

3/27/1940

Purchase

1,000

2,560.00

2.5600

4/11/1940

Purchase

300

768.00

2.5600

4/16/1940

Sale

15,800

35,550.00

2.2500

4/19/1940

Sale

500

750.00

1.5000

4/22/1940

Sale

500

1,312.50

2.6250

5/7/1940

Sale

200

525.00

2.6250

5/7/1940

Sale

800

2,000.00

2.5000

5/10/1940

Sale

500

1,040.20

2.0804

5/11/1940

Sale

200

250.00

1.2500

5/13/1940

Sale

2,000

7,779.03

3.8895

5/14/1940

Sale

1,000

3,889.52

3.8895

See id. at 762.

Using the Commission’s “lowest-in, highest-out” formula, the district court matched Kaplan’s

transactions as follows. First, the court identified the 200 shares purchased on February 5, 1940 as

the shares purchased at the lowest price per share ($1.4250) during the period. The court matched

these shares with 200 of the 1,000 shares sold on May 14, 1940 at the highest price per share

($3.8895) during the period. The matching process continued as shown below, yielding a total profit

of $9,161.05:

Purchase Date

Cost ($)

Sale Date

Proceeds ($)

Profit ($)

200

2/5/1940

$ 285.00

5/14/1940

777.90

492.90

800

12/1/1939

1,240.00

5/14/1940

3,111.62

1,871.62

2,000

12/1/1939

3,100.00

5/13/1940

7,779.03

4,679.03

500

12/1/1939

775.00

4/22/1940

1,312.50

537.50

200

12/1/1939

310.00

5/7/1940

525.00

215.00

800

12/1/1939

1,240.00

5/7/1940

2,000.00

760.00

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The Smolowe formula is capable of producing results that fall short of the

maximum possible profit. In a 1987 article35 that would become his section

16 treatise,36 Jacobs provided hypothetical examples to illustrate that the

formula may fail to recover the maximum possible short-swing profit when

some trades are not within the statute of limitations37 and when trades span a

period of more than six months.38 Figure 1 depicts Jacobs’s example of the

Purchase Date

Cost ($)

Sale Date

Proceeds ($)

Profit ($)

500

12/1/1939

775.00

4/16/1940

1,125.00

350.00

200

12/1/1939

310.00

4/16/1940

450.00

140.00

200

2/20/1940

335.00

4/16/1940

450.00

115.00

See id. at 766 (noting in supplemental opinion that only paired transactions resulting in profit should

be included in calculation).

Jacobs, supra note 7. Neither Jacobs nor the author is aware of any earlier acknowledgment of

the Smolowe formula’s limitations in the literature, and Jacobs claims credit for discovering them.

Personal communication with Arnold S. Jacobs.

ARNOLD S. JACOBS, SECTION 16 OF THE SECURITIES EXCHANGE ACT (2011).

Jacobs’s example considers a suit filed in month 28 attacking the following trading sequence:

Purchased

Purchase Price ($)

Per Share

Shares Sold

Sale Price ($)

Per Share

1,000

Month

The Smolowe formula would pair the purchases in months 1 and 2 with the sales in months 3 and

5, respectively; however, the statute of limitations would bar recovery of profits from the former

pair of transactions, leaving only the $3,000 proceeds from the latter pair. A higher profit of $5,000

can be calculated by instead pairing the purchases in months 1 and 2 with the sales in month 5 and

3, respectively. See Jacobs, supra note 7, at 533–34.

Jacobs’s example uses the following trading sequence:

Month

Purchased

Purchase Price ($)

Per Share

1,000

Shares Sold

Sale Price ($)

Per Share

1,000

The Smolowe formula produces a total profit of $8,000 by pairing the purchases in months 8 and

9 with the sales in months 8 and 5, respectively (leaving the transactions in months 1 and 9, which

are too far apart to be paired). A higher profit of $9,000 can be calculated by instead pairing the

purchases in 1, 8 and 9 with the sales in 5, 8 and 9, respectively. See id. at 532–33; Andrew Chin,

Accurate Calculation of Short-Swing Profits Under Section 16(b) of the Securities Exchange Act of

1934, 22 DEL. J. CORP. L. 587, 596–99 (1997) (providing another example); supra Figure 1.

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Smolowe formula’s failure to maximize recovery from a sequence of trades

spanning an eight-month period.

Figure 1:

Hypothetical example of a trading sequence spanning more than

six months for which the Smolowe formula falls short of calculating

the maximum possible short-swing profit to be disgorged to the

company. After the Smolowe formula (left) respectively matches the

two lowest-priced purchases with the two highest-priced sales within

less than six months, the remaining transactions are more than six

months apart and cannot be matched for a recoverable profit. To

achieve the maximum recovery (right), it is necessary to depart from

the matching prescribed by the “lowest-in, highest-out” formula.

Even though the Smolowe formula cannot be reliably applied to

trading sequences spanning more than six months, not every long trading

sequence results in a shortfall, as Figure 2 illustrates.

Figure 2:

Trading sequence spanning more than six months for which the

Smolowe formula correctly calculates the maximum recovery.

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Regardless of what formula is used, trades spanning more than one

statutory six-month period pose complications for section 16(b) liability

calculations that were not before the Smolowe court, as Figure 3 illustrates.39

Figure 3:

Six-month short-swing trading periods in Smolowe’s trading

sequence and in a hypothetical trading sequence. All of the trades

challenged in Smolowe (left) occurred within a single statutory sixmonth period. Even with fewer trades, the hypothetical sequence (right)

presents a more complex section 16(b) liability calculation problem

because the transaction dates span a period of more than six months.

The Ubiquity of the Smolowe Formula and the Misreading of Gratz

Despite the Smolowe formula’s computational complications and

discrepancies in trading sequences extending beyond the statute of

limitations 40 and spanning more than six months, 41 courts have not

hesitated to apply the formula in these potentially problematic

situations, 42 and courts 43 and commentators 44 have described the

See also ROMEO & DYE, supra note 16, § 10.01[2], at 10-5 (describing possible matchings of

transactions in overlapping six-month periods).

See supra note 37 and accompanying text.

See supra notes 38–39 and accompanying text.

See, e.g., Adler v. Klawans, 267 F.2d 840, 847–48 (2d Cir. 1959) (spanning more than seven

months); Donoghue v. Casual Male Retail Group, Inc., 375 F. Supp. 2d 226, 237 (S.D.N.Y. 2005)

(spanning more than ten months); Segen v. Westcliff Capital Mgmt., LLC, 299 F. Supp. 2d 262,

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Smolowe formula in unqualified terms as a correct method for

maximizing recovery in all section 16(b) cases. As one treatise puts it,

265–66, 272 (S.D.N.Y. 2004) (spanning more than ten months); Donoghue v. MIRACOR

Diagnostics, Inc., No. 00 Civ. 6696, 2002 WL 233188, at *1–3 (S.D.N.Y. Feb. 11, 2002) (spanning

more than thirteen months); Morales v. New Valley Corp., 999 F. Supp. 470, 476 (S.D.N.Y. 1998)

(spanning more than six months); Morales v. Mylan Labs., Inc., 443 F. Supp. 778, 780 (W.D. Pa.

1978) (three purchases made more than two years prior to suit); Heli-Coil Corp. v. Webster, 222 F.

Supp. 831, 837 (D.N.J. 1963) (spanning more than nine months), modified, 352 F.2d 156 (3d Cir.

1965); Ark. La. Gas Co. v. W.R. Stephens Inv. Co., 141 F. Supp. 841, 847–48 (W.D. Ark. 1956)

(spanning more than thirteen months); Kogan v. Schulte, 61 F. Supp. 604, 605 (S.D.N.Y. 1945)

(spanning fifteen months).

See, e.g., Credit Suisse Secs. LLC v. Simmonds, __ U.S. __, 132 S. Ct. 1414, 1418–21 (2011);

Whittaker v. Whittaker Corp., 639 F.2d 516, 532–33 (9th Cir. 1981) (“We believe the Smolowe rule

is in accord with the absolute and thoroughgoing nature of liability under § 16(b). This statute is

intended to be a deterrent to a type of activity which Congress realized was subject to much abuse.

In some cases the Smolowe rule can be criticized for harshness and artificiality. But other methods

would be equally artificial. The Smolowe rule assures full recovery of profits for the corporation.”);

Morales v. Lukens, Inc., 593 F. Supp. 1209, 1213 (S.D.N.Y. 1984) (quoting Blau v. Lehman, 286

F.2d 786, 791 (2d Cir. 1960), aff’d 368 U.S. 403 (1962)) (“The purpose of the [lowest-in, highestout] rule is to ‘squeeze every penny of profit’ from the defendant.”); Roth v. Jennings, No. 03 Civ.

7760(DAB), 2009 WL 1440670, at *5 (S.D.N.Y. May 21, 2009) (citing Nat. Microsystems Corp.,

198 F. Supp. 2d at 492 (“[T]he lowest-in, highest-out rule maximizes damages to be assessed

against a short-swing trader, rendering potential losses that might otherwise be recognized

irrelevant.”)); Segen ex rel. KFX Inc. v. Westcliff Capital Mgmt., LLC, 299 F. Supp. 2d 262, 272

(S.D.N.Y. 2004) (citing Smolowe v. Delendo Corp., 136 F.2d 231, 239 (2d Cir. 1943)) (“[T]he

trades must be matched in a manner that maximizes the disgorgeable amount to [the issuer]. This is

accomplished by matching the highest sale prices with the lowest purchase prices within the six

month period.”); Mayer v. Chesapeake Ins. Co. Ltd., 877 F.2d 1154, 1164 (2nd Cir. 1989) (citing

Smolowe, 136 F.2d at 239); Synalloy Corp. v. Gray, 816 F. Supp. 963, 971 (D. Del. 1993) (citing

Mayer, 877 F.2d at 1164); Dreiling ex rel. Infospace v. Kellett, 281 F. Supp. 2d 1234, 1238–39

(W.D. Wash. 2003) (citing Whittaker, 639 F.2d at 533); Casual Male Retail Group,,375 F. Supp. 2d

at 237 (citing Donoghue v. Nat. Microsystems Corp., 198 F. Supp. 2d at 492); Huppe v. Special

Situations Fund III QP, L.P., 565 F. Supp. 2d 495, 502 (S.D.N.Y. 2008) (citing Nat. Microsystems

Corp., 198 F. Supp. 2d 487).

See, e.g., WANG & STEINBERG, supra note 9, at 924 n.12 (“The formula established [in

Smolowe] matches the lowest price in with the highest price out, thus ensuring recovery of all

possible profits.”); Robert L. Davis, Note, Tax Treatment of Section 16(b) Payments, 27 STAN. L.

REV. 143, 150 (1974) (citing Smolowe, 136 F.2d 231) (“Only by computing the ‘profit’ in this

manner is all potential for trading on inside information within a six-month period removed.”);

Michael Rosenzweig, Note, Section 16(b) Liability for Profits Realized from a Cash Purchase and

Sale Within Six Months of the Securities of Two Issuers Involved in an Intervening Reorganization,

75 COLUM. L. REV. 1323, 1326 n.23 (1975) (citing Smolowe, 136 F.2d at 239) (“Had another

method of calculation been chosen, liability for violation of the section would not be as great.”);

Steve Thel, The Genius of Section 16: Regulating the Management of Publicly Held Companies, 42

HASTINGS L.J. 391, 404 & n.36 (1991) (stating that through Smolowe’s “lowest-in, highest-out”

formula, “[t]he courts have given section 16(b) teeth by computing profit so as to maximize the

forfeiture”); Timothy Tomlinson, The Application of Section 16(b) to Tax-Qualified Employee

Benefit Plans, 33 STAN. L. REV. 231, 232 n.7 (1981) (citing Smolowe, 136 F.2d 231) (“‘Profits’ are

normally calculated so as to recover the maximum amount from trading insiders. Thus, the highest

sale price is matched with the lowest purchase price within the 6-month period.”).

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the Smolowe formula “has reigned supreme” among methods for

calculating short-swing profits. 45

One of famed jurist Learned Hand’s final decisions as an active judge,46

Gratz v. Claughton, has been instrumental in the Smolowe formula’s

ubiquity. Casebooks have used Gratz to introduce three generations of law

students to short-swing profit calculation,47 often in connection with

problems or worked examples to illustrate the Smolowe formula’s

operation.48 Of the many section 16(b) cases that could be used for this

purpose, Gratz stands out both for its author’s illuminating analysis 49 and

See LOSS & SELIGMAN, supra note 9, at 693.

Learned Hand retired from active status on May 15, 1951, but continued to serve as a senior

judge until his death in 1961. See GERALD GUNTHER, LEARNED HAND: THE MAN AND THE JUDGE

504–05, 548–49 (1994). Hand’s significance in American jurisprudence is unquestioned. See

HENRY J. ABRAHAM, JUSTICES, PRESIDENTS, AND SENATORS: A HISTORY OF U.S. SUPREME COURT

APPOINTMENTS FROM WASHINGTON TO BUSH II 45 (5th ed. 2008) (“To date, Learned Hand served

longer, a total of 52 years, and arguably, perhaps with more distinction, than any other federal jurist

in our history.”); James A. Thomson, Learned Hand: Evaluating a Federal Judge, 22 N. KY. L.

REV. 763, 794 (1995) (“Unanimity prevails on one proposition: Hand’s influence on American law

was wide and deep.”).

See, e.g., WILLIAM T. ALLEN ET AL., COMMENTARIES AND CASES ON THE LAW OF BUSINESS

ORGANIZATION 627 (3d ed. 2009) (note case); WILLIAM L. CARY & MELVIN ARON EISENBERG,

CASES AND MATERIALS ON CORPORATIONS 593–97 (concise 6th ed. 1988) (principal case); JAMES

D. COX ET AL., SECURITIES REGULATION: CASES & MATERIALS 894 (4th ed. 2004) (note case);

MELVIN ARON EISENBERG & JAMES D. COX, CORPORATIONS & OTHER BUSINESS ORGANIZATIONS:

CASES & MATERIALS 1009–12 (10th ed. 2011) (principal case); ALEXANDER H. FREY ET AL., CASES

AND MATERIALS ON CORPORATIONS 762–64 (1966) (principal case); ROBERT W. HAMILTON,

CASES AND MATERIALS ON CORPORATIONS INCLUDING PARTNERSHIPS AND LIMITED LIABILITY

COMPANIES 1001–02 (7th ed. 2001) (note case); THOMAS LEE HAZEN & JERRY W. MARKHAM,

CORPORATIONS & OTHER BUSINESS ENTERPRISES 887–90 (standard 3d ed. 2009) (note case);

NORMAN D. LATTIN ET AL., CORPORATIONS CASES AND MATERIALS 695–700 (4th ed. 1968)

(principal case); VAGTS, supra note 2, at 551–53 n.7 (note case); see generally Gunther, supra note

12, at 47 (describing Hand’s opinions as “familiar to every lawyer and law student”).

See, e.g., ALLEN, supra note 47, at 629 (exercise); CARY & EISENBERG, supra note 47, at 598

(example); COX, supra note 47, at 894–95 (exercise); EISENBERG & COX, supra note 47, at 1013

(examples); HAMILTON, supra note 47, at 1000–03; HAZEN & MARKHAM, supra note 47, at 889

(exercises); VAGTS, supra note 47, at 562 (exercise).

See JOHN R. VILE, 1 GREAT AMERICAN JUDGES: AN ENCYCLOPEDIA 319 (2003) (Judge Hand

“has been quoted in Supreme Court opinions and scholarly publications more often than any lower

court judge in the United States. . . . He could take a mass of cases, unorganized splinters and shards

of ideas, and painstakingly fit them into a glittering stained glass window that illuminated an entire

field for the rest of the legal world”); THE ART AND CRAFT OF JUDGING: THE DECISIONS OF JUDGE

LEARNED HAND 1 (Hershel Shanks ed. 1968) (“[F]requently, a case attained significance because

the opinion was written by Learned Hand—because of his ability to fathom the principle on which

decision depended and illuminate its meaning. In this way, he created his legacy: a light for the

future, to guide lawyers and judges in applying the law to cases yet unborn.”).

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for its draconian judgment of $300,000 against an insider who had

already suffered a net overall loss of $400,000. 50

Courts have also widely cited Gratz in connection with the formula.

Along with Smolowe, Gratz has been cited as one of the two leading

authorities for the formula’s use in section 16(b) decisions spanning

from the 1950s to the present day. 51 Commentators have followed suit. 52

See Adler v. Klawans, 267 F.2d 840, 847–48, (2d Cir. 1959) (citing Gratz v. Claughton, 187

F.2d 46 (2d Cir. 1951)); COX & HILLMAN, supra note 47, at 894 (citing Gratz, 187 F.2d at 52–53)

(“Under this [lowest-in, highest-out] approach, an insider can be liable for large amounts of profits,

even where he lost money on his purchase and sale activity in the aggregate.”); EISENBERG & COX,

supra note 47, at 1014 (noting that the “Smolowe/Gratz formula” may impose liability based on the

mere “possibility” that a defendant may have profited by limiting his loss through the use of inside

information); Park McGinty, Replacing Hostile Takeovers, 144 U. PA. L. REV. 983, 1061 n.205

(1996) (citation omitted) (referring to Gratz as “the most famous example of the draconian character

of [section 16(b)]’s ‘mechanical’ provisions”).

See, e.g., Whittaker v. Whittaker Corp., 639 F.2d 516, 531 (9th Cir. 1981) (citing Smolowe v.

Delendo Corp., 136 F.2d 231 (2d Cir. 1943)); Anderson v. Comm’r, 480 F.2d 1304, 1307 (7th Cir.

1973) (same); Adler v. Klawans, 267 F.2d 840, 847–48 (2d Cir. 1959) (same); Falco v. Donner

Found., 208 F.2d 600, 602 (2d Cir. 1953) (same); Gratz, 187 F.2d 46, abrogated on other grounds,

Credit Suisse Secs. LLC v. Simmonds, 132 S. Ct. 1414, 1418–21 (2012); Huppe v. Special

Situations Fund III, 565 F. Supp. 2d 495, 502–03 (S.D.N.Y. 2008) (citing Gratz, 187 F.2d 46);

Donoghue v. Casual Male Retail Grp., Inc., 375 F. Supp. 2d 226, 237 (S.D.N.Y. 2005) (same);

Donoghue v. MIRACOR Diagnostics, Inc., No. 00 Civ. 6696, 2002 WL 233188, at *1–3 (S.D.N.Y.

Feb. 11, 2002) (same); Donoghue v. Nat. Microsystems Corp., 198 F. Supp. 2d 487, 492 (S.D.N.Y.

2002) (same); Tyco Labs., Inc. v. Cutler-Hammer, Inc., 490 F. Supp. 1, 9 n.7 (S.D.N.Y. 1980)

(same); Lewis v. Levinson, 77 Civ. 1481, 1978 WL 1087, at *3 (S.D.N.Y. May 8, 1978) (same);

Lewis v. Riklis, 446 F. Supp. 582, 584 (S.D.N.Y. 1978) (same); Makofsky v. Ultra Dynamics Corp.,

383 F. Supp. 631, 638–39 (S.D.N.Y. 1974) (same); W. Auto Supply Co. v. Gamble-Skogmo, Inc.,

231 F. Supp. 456, 460–61 (D. Minn. 1964) (same), rev’d on other grounds, 348 F.2d 736 (8th Cir.

1965); Heli-Coil Corp. v. Webster, 222 F. Supp. 831, 837 (D.N.J. 1963) (same); Kornfeld v. Eaton,

217 F. Supp. 671, 673–74 (S.D.N.Y. 1963) (same); Blau v. Lehman, 173 F. Supp. 590, 595 n.3

(S.D.N.Y. 1959) (same); Ark. La. Gas Co. v. W.R. Stephens Inv. Co., 141 F. Supp. 841, 847 (W.D.

Ark. 1956) (same). See generally EISENBERG & COX, supra note 47, at 1013 (“The formula adopted

in Smolowe and Gratz has been generally approved by the courts.”); VAGTS, supra note 47, at 552

(“Opinions by the Second Circuit in the Section 16 field are generally regarded as authoritative.”).

Unlike many of the liability calculations discussed in this Article, the calculation of Jain’s

liability was trivially simple, see supra note 33, and relied on Gratz only indirectly. See Dreiling ex

rel. Infospace v. Kellett, 281 F. Supp. 2d 1215, 1238 (W.D. Wash. 2003) (citing Whittaker, 639 F.2d

at 522, 533); Whittaker, 639 F.2d at 531 (citing Gratz, 187 F.2d at 50–52).

See, e.g., LOSS & SELIGMAN, supra note 9, at 694 (“Eight years later [in Gratz] the Second

Circuit reasserted the lowest-in, highest-out formula after independent analysis.”); ROMEO & DYE,

supra note 16, at 11-8 (1994) (“The ‘lowest-in, highest-out” method was reasserted, with

independent analysis, by the Second Circuit in [Gratz].”); Donald C. Cook & Myer Feldman,

Insider Trading Under the Securities Exchange Act, 66 HARV. L. REV. 612, 614 n.151 (1953)

(stating that Gratz reaffirmed the Smolowe formula); Michael H. Dessent, Weapons to Fight Insider

Trading in the 21st Century: A Call for the Repeal of Section 16(b), 33 AKRON L. REV. 481, 481 n.3

(2000) (“The [Gratz] court followed Smolowe . . . , which stated that to give section 16(b) its full

effect, the calculation would be the shares with the lowest purchase price, matched against those

with the highest sale prices.”); Maureen S. Duggan, Annotation, Proper Measure and Elements of

Recovery for Insider Short-Swing Transaction, 86 A.L.R. FED. 16, § 4 (1988) (“In Gratz . . . the

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This historically dominant reading of Gratz has always been strained

at best. Hand did cite Smolowe 53 and agreed with its strict approach to

fiduciary liability, 54 but he said nothing about the Smolowe formula, did

not use it, and did not even comment on the calculations in the record on

appeal: “the plaintiff has not appealed, so that she is not entitled to any

more than she has recovered. On this account we have not examined the

[special] master’s computations in detail and are not to be understood to

have passed upon them.” 55 Hand may have been famously fond of

algebra, 56 but in Gratz, he skipped the math. 57

As the remainder of this Article will show, recent developments in

computer science and technology have brought to light the meaning and

wisdom of Hand’s mathematical silence in Gratz. By “examin[ing] the

[special] master’s computations in detail,” today’s computers can

determine that the Smolowe formula was probably not used to calculate

defendant Edward N. Claughton’s short-swing profits and would have

fallen short of maximizing those profits even if it had been used (section

II.C). Modern computer science has also made it possible to characterize

the Smolowe formula’s worst-case errors (section III.B) and to identify a

court affirmed the adoption of the lowest in-highest out rule for computing short-swing profits when

there are multiple purchases and sales . . . .”); Roger J. George, Jr., Comment, The Application of

Section 16(b) to Mergers: A Hidden Hazard, 47 TEX. L. REV. 1417, 1421 n.34 (1969) (same);

Robert W. Hamilton, Convertible Securities and Section 16(b): The End of an Era, 44 TEX. L. REV.

1447, 1448 n.7 (1966) (citing Gratz, 187 F.2d 46, as authority for the formula); Timothy Tomlinson,

Section 16(b): A Single Analysis of Purchases and Sales—Merging the Objective and Pragmatic

Analyses, 1981 DUKE L.J. 941, 941 n.5 (1981) (same); Rosenzweig, supra note 44, at 1326 n.23

(same); Recent Development, Second Circuit Limits Insider-Partner’s 16(b) Liability, 14 STAN. L.

REV. 192, 194 n.10 (1961) (same); but cf. ARNOLD S. JACOBS, SECTION 16 OF THE SECURITIES

EXCHANGE ACT 531 (2011) (citations omitted) (“[Although it] has been widely cited and

followed . . . the lowest price in-highest price out rule is not the real holding of Smolowe [or

Gratz].”).

Gratz, 187 F.2d at 49 n.4, 50, 52 (citing Smolowe, 136 F.2d 231).

See id. at 51–52.

See id. at 52 (emphasis added).

See United States v. Carroll Towing Co., 159 F.2d 169, 173 (2d Cir. 1947) (“[I]f the probability

be called P; the injury, L; and the burden, B; liability depends upon whether B is less than L

multiplied by P: i.e., whether B [less than] PL.”). This algebraic rule is taught to every first-year

torts student as the “famous Learned Hand formula.” Neal Kumar Katyal, Criminal Law in

Cyberspace, 149 U. PA. L. REV. 1003, 1080 (2001); see also Patrick J. Kelley, The Carroll Towing

Company Case and the Teaching of Tort Law, 45 ST. LOUIS U. L.J. 731, 732 n.4 (2001) (citing

casebooks that prominently feature Carroll Towing). It is “arguably the most prominent approach

used to determine negligence.” Arden Rowell & Jessica Bregant, Numeracy and Legal Decision

Making, 46 ARIZ. ST. L.J. 191, 215 (2014).

Accordingly, there is no basis for referring to the “lowest-in, highest-out” formula as “[t]he

formula adopted in Smolowe and Gratz.” See EISENBERG & COX, supra note 47, at 1013; supra

notes 51–52 (citing cases and commentaries that attribute the formula to Gratz, 187 F.2d 46).

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costly error from the formula’s use in a recent case (section III.C). Even

by 1987, Jacobs had shown that the Smolowe formula could not reliably

be applied to Claughton’s twenty-one month trading sequence. 58 In

1951, however, Hand could not have feasibly calculated the maximum

value of Claughton’s short-swing profits or assessed the Smolowe

formula’s accuracy (section II.B). Prudently, Hand adhered to Smolowe’s

strict fiduciary liability doctrine (section II.A) and resolved the issue of

Claughton’s liability (section II.B) without prescribing the Smolowe

formula or any other method of liability calculation. Now that the

requisite technology is available to calculate and verify an insider’s

maximum short-swing profits in all cases (section IV.A), there is no

longer any reason to rely on Gratz (or any other case law) as an authority

for the Smolowe formula’s use (section III.A). It is time for Hand’s

mathematical silence to be heard (section IV.B).

II.

THE MEANING OF HAND’S MATHEMATICAL SILENCE

Smolowe and Hand’s Silence in Gratz

It may seem difficult at first to reconcile Hand’s silence regarding the

“lowest-in, highest-out” formula in Gratz with the district court’s and Second

Circuit’s explicit adoption of the “lowest-in, highest-out” formula in

Smolowe. The two cases, however, presented very different facts. Because

Gratz involved hundreds of transactions spanning more than twenty-one

months,59 not all pairs of the defendant’s low-priced purchases and highpriced sales would yield a recoverable short-swing profit, but only such pairs

occurring within six months of each other.60 Smolowe involved a far simpler

sequence of six purchases and nine sales between December 1, 1939 and

May 14, 194061 (i.e., all within a single statutory six-month period).62

Smolowe was therefore more amenable to use of the formula than was Gratz,

See supra note 38 and accompanying text.

See Gratz Master’s Report, supra note 11 (listing, inter alia, 276 purchases and 101 sales of

common stock and 11 purchases and 20 sales of preferred stock between December 18, 1944 and

September 24, 1946).

To be more precise, section 16(b) requires disgorgement of profit “from any purchase and sale,

or any sale and purchase . . . within any period of less than six months.” 15 U.S.C. § 78p(b) (2012)

(emphasis added). Neither Smolowe nor Gratz discussed the fine points of measuring the statutory

six-month period or the implications of section 16(b)’s “less than” provision. See ROMEO & DYE,

supra note 16, § 10.01, at 10-2 to 10-4 (surveying case law on measuring the short-swing period).

Smolowe I, 46 F. Supp. at 762.

The complaint was filed October 28, 1940, i.e., within the statute of limitations. See Smolowe

v. Delendo Corp., 36 F. Supp. 790, 791 (S.D.N.Y. 1940).

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as Figure 3 suggests, and the adjudication of liability in the two cases

confirms this theory.

In Smolowe, the Securities and Exchange Commission filed an amicus

brief to the district court expressly “confine[d] . . . to a single question—the

measure of damages to be applied in cases where, as here, numerous

purchases and sales have been made in differing sized lots and at different

prices during the period in respect of which relief is sought.”63 The

Commission proposed the following formula:

[T]he plaintiff in any case under Section 16(b) is entitled to list in

one column all purchases made during the period in respect of which

he seeks relief, and in another column all sales made within the same

period. As a measure of the recovery to which he is entitled, he may

start by matching off against each other the shares purchased at the

lowest price during the period and an equal number of shares sold at

the highest price or prices during the period, the measure of

recovery in respect of this “purchase and sale” being the difference

between the two prices. Then, the purchase price of the shares

purchased at the next lowest price may be similarly matched off

against the highest share price of any remaining equal number of

shares sold during the period. The same process may be continued

until all shares purchased have been matched off, so far as possible,

against an equal number of shares sold at higher prices. The gross

recovery is the sum of the several differentials thus determined.64

In this definitive statement of the formula,65 each of the italicized

occurrences of the term “the period” refers to the antecedent term “the period

in respect of which he seeks relief,” so they are all synonymous. Because it is

permissible to match shares purchased and shares sold during “the period”

for a recoverable profit only if the transactions occur within six months of

each other,66 “the period” logically must refer to a single statutory six-month

SEC Smolowe Brief, supra note 34, at 4.

Id. at 4–5 (emphases added).

It bears noting that the “lowest-in, highest-out” formula was first suggested much earlier in two

preliminary drafts of the 1934 Act. See Smolowe v. Delendo Corp., 136 F.2d 231, 237 n.11 (2d Cir.

1943) (“H.R. 7852 and S. 2693 contained the provision that ‘profit shall be calculated on the sale or

sales by such person of such security made at the highest price or prices and on the purchase or

purchases made by such person of such security at the lowest price or prices during the six months’

period . . . .’”). The Smolowe court, however, found these drafts to be minimally relevant to the

interpretation of the enacted statute, see id., and explicitly affirmed the district court’s adoption of

the Commission’s version of the formula. See id. at 239; Smolowe I, 46 F. Supp. at 766. Even if the

draft language were to be taken as definitive, it still refers to a single “six months’ period,” so the

limited scope of the Commission’s formula would apply to the legislative version of the formula

with equal force. See Smolowe, 136 F.2d at 237 n. 11.

See 15 U.S.C. § 78p(b) (2012).

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period. The Commission’s formula therefore addresses only cases in which

“the period in respect of which relief is sought” is a single statutory sixmonth period, and says nothing regarding the additional complexities of

matching transactions that span a longer time frame.67 In particular, the

Commission’s formula is facially inapplicable to the twenty-one-month

trading sequence challenged in Gratz.

In addition to the Commission’s formula, the district court also considered

defendant Henry C. Kaplan’s alternative proposals to allow only the

matching of purchases and sales occurring “first in and first out” within the

trading sequence or involving identical stock certificates.68 After finding

Kaplan’s proposals inconsistent with section 16(b)’s text and purpose, the

court immediately proceeded to adopt the Commission’s formula without

further comment or analysis:

The subsection [16(b)] carefully states that profits are to be

computed from “any” purchase and sale or “any” sale and purchase

within the six months. It does not say that any purchase is to be set

off against the next sale nor that any rule of “first in and first out”

shall be adopted. The purpose of the statute was to make

unprofitable short swings by persons in a position to have inside

information. If they saw fit to disobey the law, there is no reason

why the recovery should be minimized. The rule to be adopted must

disregard the identity of the certificates, as I have previously stated.

The computation suggested by the Securities & Exchange

Commission is, therefore, adopted as fixing the amount of profits

recoverable from the defendant Kaplan.69

The Second Circuit, in an affirmance authored by Judge Charles Edward

Clark, similarly found Kaplan’s proposals inconsistent with section 16(b)’s

See supra Figure 3 (illustrating that longer transaction sequences give rise to multiple partially

overlapping statutory six-month periods within which pairs of transactions can be matched for a

recoverable profit). The following example illustrates the complexity introduced by overlapping

short-swing periods:

A plaintiff may match transactions in overlapping six-month periods. Suppose, for example,

that an insider makes a purchase of 100 shares of stock on January 1, followed by a sale of 300

shares on May 1 and a purchase of 200 shares on September 30. Both the purchase on January

1 and the purchase on September 30 may be matched with the sale on May 1. The period from

January 1 through June 29 may be considered one short-swing period, permitting the January 1

purchase to be matched with the sale of 100 of the shares sold on May 1. Similarly, the period

from May 1 through October 30 may be considered a separate short-swing period, permitting

the May 1 sale of the remaining 200 shares to be matched with the September 30 purchase.

However, if the two purchases exceeded 300 shares (the number of shares sold), only 300

shares would be matched.

ROMEO & DYE, supra note 16, § 10.01[2], at 10-5 (citation omitted).

Smolowe I, 46 F. Supp. at 766.

Id.

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“any purchase and sale, or any sale and purchase” provision.70 Clark drew an

even stronger conclusion from the statute’s expansive language, finding that

“its generality permits and points to . . . . an arbitrary matching to achieve the

showing of a maximum profit.”71 Clark then proceeded to set forth the “only

rule” that would recover the maximum profit attributable to an insider’s stock

transactions:

We must suppose that the statute was intended to be thoroughgoing, to squeeze all possible profits out of stock transactions, and

thus to establish a standard so high as to prevent any conflict

between the selfish interest of a fiduciary officer, director, or

stockholder and the faithful performance of his duty. The only rule

whereby all possible profits can be surely recovered is that of lowest

price in, highest price out—within six months—as applied by the

district court. We affirm it here, defendants having failed to suggest

another more reasonable rule.72

Clark explicitly identified “[t]he only rule whereby all possible profits can

be surely recovered” as the formula “applied by the district court” in

Smolowe: namely, the formula “suggested by the Securities & Exchange

Commission.”73 Accordingly, the instruction “lowest price in, highest price

out”74 was simply an elegantly succinct paraphrase of the Commission’s

formula for matching off “the shares purchased at the lowest price during the

period and an equal number of shares sold at the highest price or prices

during the period.”75 The accompanying qualifier “within six months”76

referred to the statutory six-month period during which all of the challenged

transactions occurred, as set forth in the Commission’s formula.77

Smolowe, 136 F.2d at 237–38.

Id. at 237.

Id. at 239 (citations omitted).

Smolowe I, 46 F. Supp. at 766.

Smolowe, 136 F.2d at 239.

See SEC Smolowe Brief, supra note 34, at 4–5.

Smolowe, 136 F.2d at 239.

SEC Smolowe Brief, supra note 34, at 3 (emphasis added). By 1981, the Smolowe formula had

been employed in enough cases involving longer trading sequences that the Commission

reinterpreted the “within six months” provision as referring to each pair of matched transactions,

rather than the length of the entire trading sequence. See Interpretive Release on Rules Applicable to

Insider Reporting and Trading, 46 Fed. Reg. 48147, 48161 n.102 (1981) (citing Smolowe, 136 F.2d

231) (stating that “profit is computed by matching the highest sale price with the lowest purchase

price within six months, the next highest sale price with the next lowest purchase price within six

months, and so on, until all shares have been included in the computation”). As the foregoing

discussion has shown, this reinterpretation has no basis in Smolowe.

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The Smolowe formula “lowest price in, highest price out—within six

months”78 therefore amounted to nothing more or less than the

Commission’s formula,79 which in turn was designed and proposed for use

only in cases involving a single statutory six-month trading period.80 Thus,

despite Clark’s sua sponte reference to the formula as “[t]he only rule,”81 and

contrary to the dominant reading of Smolowe,82 the Smolowe court did not

endorse the formula for application to the twenty-one-month trading

sequence challenged in Gratz.83

Gratz involved a sequence of more than 400 transactions in MissouriKansas-Texas Railroad Company stock spanning from December 18, 1944

to September 24, 1946.84 In district court proceedings before a special

master, the plaintiff Stella Gratz had sought liability under the Smolowe

formula.85 The defendant Edward N. Claughton had argued for more lenient

methods of calculation, including a modification of the Smolowe formula

involving “matching the highest prices out against the lowest prices in for

three months before or three months after each sale.”86 The master rejected

Claughton’s alternative proposals because they did not “conform to or satisfy

the statute as I view it, or the rule of damages in the Smolowe case which I

find plaintiffs have correctly adopted.”87

Claughton had also submitted various calculations, including an

accounting purporting to show:

[T]he damages, though not conceding the correctness of the theory

of such calculation, which might be awarded to the plaintiffs, in the

sum of [$308,417], upon the basis of highest price out and lowest

price in during the period of his trading, as to purchases and sales

Smolowe, 136 F.2d at 239.

See SEC Smolowe Brief, supra note 34, at 4–5.

Id.

Smolowe, 136 F.2d at 239.

See, e.g., Whittaker v. Whittaker Corp., 639 F.2d 516, 533 (9th Cir. 1981), abrogated on other

grounds by Credit Suisse Secs. LLC v. Simmonds, __ U.S. __, 132 S. Ct. 1414, 1418–21 (2011)

(stating without qualification that “[t]he Smolowe rule assures full recovery of profits for the

corporation”); EISENBERG & COX, supra note 47, at 1013 (discussing the predominance of the

formula in Smolowe and Gratz in case law).

For further discussion, see section III.A (arguing that the Smolowe court’s statement of the rule

must be read as limited to cases involving a single statutory six-month trading period because

otherwise it would be empirically false).

See Gratz Master’s Report, supra note 11.

Id. ¶ 16.

See id.; Brief of Defendant-Appellant at 15, Gratz v. Claughton, 187 F.2d 46 (2d Cir. 1951)

(No. 147 Docket 21660).

See Gratz Master’s Report, supra note 11, ¶ 16.

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and sales and purchases . . . within any period less than six

months . . . .88

A detail from Claughton’s accounting is reproduced in Figure 4.

Figure 4:

Detail of Claughton’s profit calculation. 89

Gratz and the master were both content to let Claughton handle the math.

Gratz stipulated that Claughton’s accounting was correct according to the

Smolowe formula,90 and the master entered a finding that “the profits made

by Claughton during the less than six months periods have been shown to

amount altogether to the sum of [$308,417], under the [Smolowe] rule of

damages.”91 The district court adopted the master’s report in all respects.92

Hand affirmed the district court’s judgment in a unanimous decision for

the Second Circuit.93 Hand began his analysis of Claughton’s liability by

See id. Various reports of Claughton’s calculation exhibited small typographical and/or

rounding discrepancies. Cf. Gratz Master’s Report, supra note 11, ¶ 16 (stating the result of

Claughton’s calculation as $308,417.50 and as $308,417.09); Def.’s Exhibits C & N to Gratz

Master’s Report, supra note 11 (showing Claughton’s calculation of profits totaling $308,417.05).

These errors are negligible, and fractional dollars have been omitted hereinafter where warranted for

clarity of exposition.

Def.’s Exhibit N to Gratz Master’s Report, supra note 11, at 1.

See Gratz Master’s Report, supra note 11, ¶ 16.

See id.

Order ¶ 2, Gratz v. Claughton (S.D.N.Y. 1949) (No. 35-410).

Gratz v. Claughton, 187 F.2d 46, 52 (2d Cir. 1951).

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observing that section 16(b)’s expansive language provided “no principle by

which to select any two transactions which are to be matched,” thereby

forcing a choice between matching trades

in such a way as to reduce profits to their lowest possible amount, or

in such a way as to increase them to the greatest possible amount.

The master adopted the second course, following what he supposed

to be the doctrine of Smolowe. . . . We think that he was right for the

following reasons.94

Hand reasoned that any uncertainty in the liability calculation must be

resolved against the fiduciary, Claughton, in accordance with the traditional

common law doctrine of spoliation:

As we have said, the statute makes all such dealings unlawful, and

makes the fiduciary accountable to the corporation. Although it is

impossible in the case at bar to compute the defendant’s profits,

except that they must fall between two limits—the minimum and the

maximum—the cause of this uncertainty is the number of

transactions within six months: that is, the number of defendant’s

derelictions. The situation falls within the doctrine which has been

law since the days of the “Chimney Sweeper’s Jewel Case,” that

when damages are at some unascertainable amount below an upper

limit and when the uncertainty arises from the defendant’s wrong,

the upper limit will be taken as the proper amount.95

After rejecting Claughton’s alternative calculation method as falling short

of this “upper limit,”96 Hand observed that the plaintiff was free to recover

this maximum amount by matching purchases and sales of equal numbers of

shares in any way that would produce a short-swing profit:

If one is seeking an equation of purchase and sale, one may take any

sale as the minuend and look back for six months for a purchase at

less price to match against it. On the other hand, if one is looking for

an equation of sale and purchase, one may take the same sale and

look forward for six months for any purchase at a lower price.

Although obviously no transaction can figure in more than one

Id. at 51.

Id. at 51–52 (emphasis added); but see John E. Munter, Section 16(b) of the Securities

Exchange Act of 1934: An Alternative to “Burning Down the Barn in Order to Kill the Rats,” 52

CORNELL L. REV. 69, 83 n.64 (1966) (“The validity of the analogy is dubious in cases where the

defendant would be able to prove the exact amount of his actual profit if the court gave him a

chance, for then the damages would no longer be ‘unascertainable.’”).

See Gratz, 187 F.2d at 52 (“This results in looking for six months both before and after any

sale, and not for three months only, as the defendant insists.”).

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equation, with that exception we can see no escape from what we

have just said.97

Hand’s analysis thus led to precisely two legal conclusions: first, that “the

proper amount” of section 16(b) liability is given by “the upper limit” of

short-swing profits attributable to the defendant’s trading, and second, that a

section 16(b) plaintiff is entitled to recover this maximum amount by

arbitrarily matching pairs of purchases and sales within six months of each

other.98

Hand’s opinion offered no view as to whether specifically matching pairs

of trades according to the “lowest-in, highest-out” formula would yield the

maximum amount of profits recoverable from Gratz’s twenty-one-month

trading sequence.99 Hand also expressly declined to review Claughton’s

calculation and affirmed the sufficiency of the judgment below solely on the

grounds that the plaintiff had stipulated to it:

[T]he plaintiff has not appealed, so that she is not entitled to any

more than she has recovered. On this account we have not examined

the master’s computations in detail and are not to be understood to

have passed upon them.100

Hand’s analysis concluded: “[t]herefore, not only will we follow Smolowe

v. Delendo Corporation, supra, as a precedent; but as res integra and after

independent analysis we reassert its doctrine.”101

It is notable that in undertaking his “independent analysis” of what he took

to be Smolowe’s doctrine, Hand saw no need to discuss or even mention the

“lowest-in, highest-out” formula.102 Hand instead focused on and reasserted

two other doctrinal aspects of the Smolowe decision: the strict character of

fiduciary liability103 and the determination that section 16(b)’s expansive

Id. (emphasis added).

See supra text accompanying notes 96–97.

As it turns out, the Smolowe formula would not have maximized Claughton’s liability. See infra

app. A, tbls. 1 & 2.

100

Gratz, 187 F.2d at 52.

101

Id.

102

For more discussion of Hand’s view of his Second Circuit colleague Clark’s jurisprudence,

see, for example, MARVIN SCHICK, LEARNED HAND’S COURT 304 (1970) (quoting Letter from

Learned Hand to Charles Edward Clark (Feb. 23, 1950)) (“Of course, we have positive differences;

we should not be worth our salt if we did not . . . . Between ourselves we may say, what I think we

all believe in secret, that we have a fine court and that each of us contributes to it a part which would

make the sum much poorer if it were absent.”).

103

See Smolowe v. Delendo Corp., 136 F.2d 231, 239 (2d Cir. 1943) (“We must suppose that the

statute was intended to be thorough-going, to squeeze all possible profits out of stock transactions,

and thus to establish a standard so high as to prevent any conflict between the selfish interest of a

fiduciary officer, director, or stockholder and the faithful performance of his duty.”).

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language warranted “an arbitrary matching to achieve the showing of a

maximum profit.”104

In the final analysis, the Second Circuit’s contrasting adjudications of

short-swing liability in Smolowe and Gratz are easily harmonized. It suffices

to observe that the historically dominant interpretation of Gratz as an

authority in support of the Smolowe formula is incorrect. Contrary to popular

belief, the formula was not among the doctrines from Smolowe that Hand

“independently examined and adhered to in Gratz.”105 Gratz may have

followed Smolowe as a precedent with respect to its other doctrines, but

Hand’s analysis and conclusions provided no support for the Smolowe

formula.

The Judgment Below and Hand’s Silence in Gratz

It should be clear at this point that Hand’s decision to “follow

Smolowe . . . as a precedent” and to “reassert its doctrine” 106 did not

involve an endorsement of the Smolowe formula. Even so, it might be

possible to interpret Hand’s affirmance of the judgment below as

encompassing the master’s characterization of the Smolowe formula as

“[t]he only rule whereby all profits can be ‘squeezed out’ of the 10%

stock trader [Claughton].” 107

Such a reading, however, would belie Gratz’s historical context. In

reviewing an accounting of more than 400 transactions over a twentyone month period 108 in an era when spreadsheets were calculated by

hand 109 and transcribed on a typewriter, 110 Hand could not have

confidently based his affirmance of the district court’s $308,417

judgment on the proposition that this sum actually represented the

104

Id. at 237.

Kornfeld v. Eaton, 217 F. Supp. 671, 674 (S.D.N.Y. 1963) (“[The Smolowe court] reached an

empirical judgment that ‘[t]he only rule whereby all possible profits can be surely recovered is that

of lowest price in, highest price out—within six months . . . .’ This doctrine was independently

examined and adhered to in Gratz . . . .”); cf. ROMEO & DYE, supra note 16, § 11.02, at 11-8 (“The

‘lowest-in, highest-out’ method was reasserted, with independent analysis, by the Second Circuit in

Gratz v. Claughton eight years after its adoption.”).

106

Gratz, 187 F.2d at 52.

107

See Gratz Master’s Report, supra note 11, ¶ 16. For such an interpretation, see Duggan, supra

note 52, § 4 (“In Gratz . . . the court affirmed the adoption of the lowest in-highest out rule for

computing short-swing profits when there are multiple purchases and sales . . . .”).

108

See Gratz Master’s Report, supra note 11.

109

No pun intended. See supra Figure 4.

110

See Def.’s Exhibits C & N to Gratz Master’s Report, supra note 11 (providing handwritten and

typewritten versions of Claughton’s liability calculations).

105

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maximum possible profit that could be “squeezed out” of Claughton’s

transactions. 111

Finding a profit-maximizing matching of purchases and sales is an

example of what Lon Fuller called a “polycentric task,” 112 a problem

whose complexity stems from the fact that each decision point “is a

distinct center for distributing tensions.” 113 Six years after Gratz, in what

would become his classic article, The Forms and Limits of

Adjudication, 114 Fuller illustrated this concept with the example of a

probate court’s division of an art collection into two equal shares where:

[T]he disposition of any single painting has implications for the

proper disposition of every other painting. If it gets the Renoir,

the Gallery may be less eager for the Cezanne but all the more

eager for the Bellows, etc. . . . . Any judge assigned to hear such

an argument would be tempted to assume the role of mediator or

to adopt the classical solution: Let the [Metropolitan] divide the

estate into what he regards as equal shares, let the [Gallery] take

his pick. 115

Section 16(b) liability calculation is similarly polycentric, in that any

matching of a purchase P1 with a sale S1 may affect the profits

recoverable from sales that otherwise might have been matched with P1

and purchases that otherwise might have been matched with S1 .

Prefiguring Fuller’s probate judge and his “cut and choose” solution,

Hand adopted a form of adjudication—giving Gratz the entitlement to

choose an arbitrary matching of short-swing trades 116—that elegantly

elided the limits of the court’s computational powers. 117 It might aptly be

dubbed “the Learned Hand unformula.” 118

111

An accurate computational method for calculating the maximum short-swing profit attributable

to a sequence of transactions was first published in 1997. See Chin, supra note 38.

112

See Lon L. Fuller, The Forms and Limits of Adjudication, 92 HARV. L. REV. 353, 394 (1978).

113

See id. at 395.

114

See id. at 353 (explaining that the initial version of the article was circulated at Harvard Law

School in 1957). As of Nov. 22, 2016, the query “Fuller /p ‘The Forms and Limits of Adjudication’”

yielded 1,023 hits in Westlaw’s secondary sources database.

115

See id. at 394.

116

See Gratz v. Claughton, 187 F.2d 46, 52 (2d Cir. 1951).

117

This form-of-adjudication approach to the resolution of polycentric disputes has continued to

inspire a burgeoning game theory literature on mechanism design. See, e.g., STEVEN J. BRAMS &

ALAN D. TAYLOR, FAIR DIVISION: FROM CAKE CUTTING TO DISPUTE RESOLUTION (1996)

(surveying applications of mechanism design to dispute resolution); Steven J. Brams & Joshua R.

Mitts, Law and Mechanism Design: Procedures to Induce Honest Bargaining, 68 N.Y.U. ANN.

SURV. AM. L. 729, 773–89 (2013) (applying mechanism design to improve blockholder disclosure

under section 13(d) of the Securities Exchange Act of 1934); Lee Ann Fennell, Revealing Options,

118 HARV. L. REV. 1399 (2005) (surveying applications of option mechanisms to dispute resolution

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Hand understood that Gratz’s stipulation to Claughton’s calculation

made it unnecessary to verify its correctness and maximality. 119 The

district court’s $308,417 judgment could be affirmed solely on the

grounds that Gratz was entitled to an arbitrary matching of purchases

and sales within six months of each other, 120 and Gratz had exercised

this entitlement by assenting to the matching set forth in Claughton’s

accounting. 121 Hand therefore had no reason in Gratz to rely on or

endorse the master’s adoption of the Smolowe formula, even implicitly.

Given Gratz’s historical context and Hand’s famous adherence to

judicial restraint, 122 there is no basis for reading into Hand’s opinion an

endorsement of the master’s characterization and adoption of the

Smolowe formula.

Gratz’s Unsuitability for Endorsing the Smolowe Formula

Hand did explicitly endorse a different aspect of the master’s analysis:

namely, its adherence to Smolowe’s doctrine of strict fiduciary

liability. 123 Hand also specifically found that the master was right to

“adopt[] the . . . course” of matching trades “in such a way as to increase

[profits] to the greatest possible amount, . . . following what he supposed

to be the doctrine of Smolowe.” 124 Hand’s independent analysis of the

Smolowe doctrine confirmed his conclusion that “the proper amount” of

section 16(b) liability is given by “the upper limit.” 125

It can now be seen that the calculation of Claughton’s liability was

unsuitable as a vehicle for endorsing the Smolowe formula, because

and regulation); Eric L. Talley, Note, Contract Renegotiation, Mechanism Design, and the

Liquidated Damages Rule, 46 STAN. L. REV. 1195 (1994) (using mechanism design to suggest more

efficient contract renegotiation procedures).

118

Cf. supra note 56 (describing the Learned Hand formula).

119

See Gratz, 187 F.2d at 52.

120

See id.

121

See id.

122

See GUNTHER, supra note 46, at xi (foreword by Ruth Bader Ginsburg) (citation omitted)

(describing Hand’s approach to judging as “heedful of limitations stemming from the judge’s own

competence”); Zachary Baron Shemtob, Following Thayer: The Many Faces of Judicial Restraint,

21 B.U. PUB. INT. L.J. 61, 71 (2011) (“Few jurists followed judicial restraint as closely as Hand.”);

Justin Zaremby, Learned Hand’s Two Concepts of (Judicial) Liberty, 65 RUTGERS L. REV. 787, 790

(2013) (“Hand maintains a reputation as a judge whose jurisprudence epitomizes restraint.”).

123

See Gratz Master’s Report, supra note 11, ¶ 14 (citation omitted) (finding that section 16(b)

“was intended ‘to be thorough-going, to squeeze all possible profits out of stock transactions . . . and

thus to establish a standard so high as to prevent any conflict between the selfish interest of a

fiduciary officer, director or stockholder, and the faithful performance of his duty’”).

124

See Gratz, 187 F.2d at 51.

125

See id. at 51–52.

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Hand could not have endorsed the formula’s use without destabilizing

Smolowe’s strict fiduciary liability doctrine. It suffices to compare a

correct calculation of Claughton’s liability using the Smolowe formula

with a linear programming method that actually “squeeze[s] all possible

profits out of” a sequence of transactions. 126 Using modern

126

The latter method derives from a 1997 article in which I identified the section 16(b) liability

calculation problem as a special case of the transportation problem in the field of management

science. See Chin, supra note 38, at 593–99. The transportation problem is, in turn, a special case of

the linear programming problem. See Alexander Schrijver, On the History of Combinatorial

Optimization, in HANDBOOKS IN OPERATIONS RESEARCH AND MANAGEMENT SCIENCE: DISCRETE

OPTIMIZATION 13 (K. Aardal et al. eds. 2005), http://homepages.cwi.nl/~lex/files/histco.pdf

[https://perma.cc/97B7-W6FE]. Any trading sequence can therefore be translated into a linear

programming problem whose solution represents “the upper limit” of section 16(b) liability.

For example, consider the following sequence of trades:

Date

Transaction

Amount ($)

Price ($)/Share

Jan. 1

Purchase

1,000

$ 9,000

Feb. 15

Sale

400

3,200

Mar. 1

Purchase

2,000

16,000

May 1

Purchase

800

5,600

June 15

Sale

1,200

12,000

Sept. 1

Purchase

1,000

6,000

Oct. 15

Sale

2,400

21,600

For i=1,2,3,4 and j=1,2,3, let pij denote the per-share profit recoverable under section 16(b) from

pairing the i-th purchase and j-th sale in this table (counting chronologically). For example, pairing

the shares purchased on May 1 for $7/share (i.e., the third purchase) with the shares sold on

February 15 for $8/share (i.e., the first sale) yields a recoverable profit of $1/share; this fact may be

expressed as p31=1. On the other hand, the first purchase on January 1 and third sale on October 15

are more than six months apart, so p13=0. Thus we form the vector

P=(p11,p12,p13,p21,p22,p23,p31,p32,p33,p41,p42,p43)=(0,1,0,0,2,0,1,3,2,0,4,3).

To maximize the total recoverable profit, one must find the number of shares xij for each pair of

purchases and sales for which the total recoverable profit

p x is maximum,

∑

i, j

subject to the constraints:

∑ x ≤ 1,000 ∑ x ≤ 400

∑ x ≤ 2,000 ∑ x ≤ 1,200

∑ x ≤ 800 ∑ x ≤ 2,400

∑ x ≤ 1,000 ∀i, j : x ≥ 0

This linear programming problem may be solved by standard techniques, such as the simplex

method. See MOKHTAR S. BAZARAA ET AL., LINEAR PROGRAMMING AND NETWORK FLOWS 91–150

(4th ed. 2010). The solution vector is:

X=( x11,x12,x13,x21,x22,x23,x31,x32,x33,x41,x42,x43)=(0,0,0,0,1200,0,0,0,800,0,0,1000),

for a maximum recoverable profit P·X of $ 7,000.

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computational tools to apply the Smolowe formula to the sequence of

Claughton’s common stock transactions yields a liability of $337,599.127

By comparison, the linear programming method applied to the same

sequence of transactions produces a liability of $337,800, 128 or $201

more than the result from the Smolowe formula.

While this is a small difference, it does demonstrate that Hand could

not have endorsed the formula’s use while adhering to “the upper limit”

of section 16(b) liability. 129 Even if the discrepancy might have gone

unrecognized, 130 Hand’s opinion would have carried within it a latent

irreconcilable tension. 131 Allowing the Smolowe formula to trump the

plaintiff’s entitlement to “an arbitrary matching to achieve the showing

of a maximum profit” 132 in Gratz would have opened the door to much

larger discrepancies in other cases. The formula may fall short of the

maximum by up to fifty percent when trades span a period of more than

six months, as section III.B will show. 133

The $337,599 result from the Smolowe formula deviates even further

from Claughton’s accounting, which showed a total liability of only

$283,835 from common stock trades. 134 This latter discrepancy casts

doubt on the master’s finding that Claughton used the formula in his

While linear programming problems had been formulated by 1939, see L.V. Kantorovich,

Mathematical Methods of Organizing and Planning Production (1939), cited in SAUL I. GASS &

ARJANG A. ASSAD, AN ANNOTATED TIMELINE OF OPERATIONS RESEARCH: AN INFORMAL HISTORY

50 (2005), and the simplex method was known in 1947, see George B. Dantzig, Maximization of a

Linear Function of Variables Subject to Linear Inequalities, in ACTIVITY ANALYSIS OF

PRODUCTION AND ALLOCATION 19–32 (Tjalling C. Koopmans ed. 1951), the application of linear

programming to section 16(b) liability was not publicly available until fifty years later. See Chin,

supra note 38, at 596–99.

127

See infra app. A, tbl. 1.

128

See infra app. A, tbl. 2.

129

It also falsifies the master’s characterization of the formula as “[t]he only rule whereby all

profits can be ‘squeezed out’ of [Claughton] . . . .” See Gratz Master’s Report, supra note 11, ¶ 16.

130

See, e.g., Kornfeld v. Eaton, 217 F. Supp. 671, 674 (S.D.N.Y. 1963) (describing Gratz’s

liability calculation as adhering to both the Smolowe formula and Smolowe’s strict fiduciary liability

doctrine).

131

Cf. Stuart Benjamin, Stepping Into the Same River Twice: Rapidly Changing Facts and the

Appellate Process, 78 TEX. L. REV. 269, 281 (1999) (“If the facts on which the opinion relied no

longer describe the world, then the opinion purports to lay down the current status of the law but in

fact misdescribes the world, and thus creates an intolerable tension.”).

132

Smolowe v. Delendo Corp., 136 F.2d 231, 237 (2d. Cir. 1943).

133

See supra text accompanying note 38.

134

See Def.’s Exhibit C to Gratz Master’s Report, supra note 11 (showing recoverable profit of

$282,572.91 from matching of purchases and sales prior to April 4, 1946, and $1,261.43 from

matching of purchases and sales after April 4, 1946). Claughton also submitted an accounting

showing $24,582.71 from preferred stock trades for a total liability of $308,417. See Def.’s Exhibit

N to Gratz Master’s Report, supra note 11.

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liability calculations and strongly suggests that Gratz and the railroad left

at least $53,764 on the table by not challenging that finding.

When considered together, these discrepancies reveal a deep

incongruity in the notion that Hand used Gratz as a vehicle to endorse

the Smolowe formula, even beyond the demonstrated absence of

evidence that he had any reason to do so. 135 It must be remembered that

Judge Clark provided no mathematical justification for his assertion in

Smolowe that the formula was “[t]he only rule whereby all possible

profits can be surely recovered”; 136 it was, in the words of another court,

merely an “empirical judgment.” 137 As a factual predicate for Hand’s

adjudication of Gratz, Claughton’s accounting was so inaccurate that it

probably did not result from the Smolowe formula’s use, and even a

corrected accounting would have fallen $201 short of corroborating

Clark’s empirical assertion. The dominant reading of Gratz as an

authority for the Smolowe formula’s applicability thus proves to be both

doctrinally and mathematically unjustifiable.

III. THE WISDOM OF HAND’S MATHEMATICAL SILENCE

The Smolowe Formula Needs No Corroboration in Simple Cases

Until now, through case law, casebooks, and commentary, 138 the

dominant reading of Gratz has played a significant role in ensuring that

the Smolowe formula has become “firmly ingrained in the fabric of

Section 16(b).” 139 This role has largely been necessitated by the tenuous

justification for the formula provided by the Smolowe case itself. By

offering the formula as an “empirical judgment” 140 with no mathematical

rationale, Judge Clark put the formula on a path to be corroborated over

time through the common law process, rather than proved once and for

all as a mathematical proposition. In the dominant reading of section

16(b) case law, Gratz has served long and well as Smolowe’s vital

buttress, putatively carrying the gravitas of Learned Hand’s independent

135

See supra sections II.A and II.B.

Smolowe, 136 F.2d at 239.

137

Kornfeld v. Eaton, 217 F. Supp. 671, 674 (S.D.N.Y. 1963) (citing Smolowe, 136 F.2d at 239).

138

See supra section I.C.

139

See ROMEO & DYE, supra note 16.

140

Kornfeld, 217 F. Supp. at 674.

136

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analysis 141 and extending the formula’s applicability beyond six-month

trading sequences. 142

The findings in Part I have called into question Gratz’s role as an

auxiliary authority for the Smolowe formula’s use. As it turns out,

however, the Smolowe formula no longer has any need of such empirical

corroboration. The formula states a mathematical fact, not merely an

empirical judgment, as long as Smolowe’s “within six months” provision

is correctly interpreted as a limit on the formula’s range of application.143

What follows is the first known proof that the “lowest-in, highest-out”

formula correctly produces the maximum profit attributable to a

sequence of transactions falling within a single statutory six-month

period. This provides the Smolowe formula with the mathematical

justification it has lacked for more than seventy years.

The proof follows a standard technique for software verification

known as a loop invariant. Loop invariant methods for software

verification have been formally shown to be sound, 144 and various

introductory texts provide clear explanations and illustrative examples of

loop invariant proofs. 145 For present purposes, it suffices to explain that

“[a] loop invariant expresses important relationships among the variables

that must be true at the start of every iteration and when the loop

terminates.” 146 As illustrated in Figure 5, a correctness proof must

show, 147 given that the input satisfies the specified precondition, that: the

loop invariant (a) is true before executing the loop for the first time 148

and (b) remains true after each iteration.149 In addition, the proof must

show that (c) the loop’s exit condition is eventually met, 150 and that (d)

141

See id. (stating that the Smolowe formula was “independently examined and adhered to in

Gratz”).

142

See, e.g., Adler v. Klawans, 267 F.2d 840, 847–48 (2d Cir. 1959) (spanning more than seven

months); Donoghue v. Casual Male Retail Group, Inc., 375 F. Supp. 2d 226, 237 (S.D.N.Y. 2005)

(spanning more than ten months).

143

See supra text accompanying notes 78–80.

144

See KRZYSZTOF R. APT & ERNST-RÜDIGER OLDEROG, VERIFICATION OF SEQUENTIAL AND

CONCURRENT PROGRAMS 57–66 (David Gries & Fred B. Schneider eds., 2d ed. 1997).

145

See, e.g., JEFF EDMONDS, HOW TO THINK ABOUT ALGORITHMS 12–26 (2008) (explaining loop

invariant proofs and providing examples); DERRICK G. KOURIE & BRUCE W. WATSON, THE

CORRECTNESS-BY-CONSTRUCTION APPROACH TO PROGRAMMING 55–93 (2012) (providing

examples).

146

EDMONDS, supra note 145, at 8 (emphasis omitted).

147

See id. at 20.

148

See id. at 17–18.

149

See id. at 16–17.

150

See id. at 19.

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the required result, or postcondition, is achieved when this occurs and

the loop is exited. 151

Figure 5:

Structure of a correctness proof that uses a loop invariant (LI).

Given that input F satisfies the precondition @pre, it is necessary to

prove that (a) the loop invariant LI is true initially; (b) LI remains

true after each iteration of the algorithm steps S; (c) the predicate P

is eventually false; and (d) when P is false, the postcondition @post

is true. 152

To formalize the result, it is necessary first to provide the following

mathematical specification of the “lowest-in, highest-out” algorithm,

heavily commented to facilitate comparison with less formal descriptions

of the Smolowe formula in the legal literature.

Lowest-In, Highest-Out

Precondition: Purchases

( p1 , q1 ), ( p 2 , q 2 )2 , ( p m , q m ) and sales

(P1 , Q1 ), (P2 , Q2 )2, (Pn , Qn ) (listed in nondecreasing and nonincreasing

151

See id.

This diagram was taken from the course blog for CS207: Systems Development for

Computational Science at Harvard University’s School of Engineering and Applied Sciences. See

Cris Cecka & Ray Jones, CS207 Systems Development for Computational Science: Loop Invariants,

HARVARD SCH. ENG’G AND APPLIED SCIS. (Oct. 5, 2014), http://iacs-courses.seas.harvard.

edu/courses/cs207/blog/index.php [https://perma.cc/XC2D-CJR5].

152

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order of per-share prices, respectively; 153 i.e., p1 ≤ p ≤  ≤ pm ,

P1 ≥ P ≥  ≥ Pn , qi , Q j > 0 for all i, j ), all of which occurred within

the same period of less than six months and within the statute of

limitations under section 16(b).

Comment: Recoverable profit M is accumulated by iteratively

matching blocks of previously unmatched shares ( ui , U j ) at the lowest

remaining purchase price and the highest remaining sale price until no

further shares can be profitably matched.

Postcondition: M is the maximum possible profit that can be attained

from any matching of the given purchases and sales. That is, for all

 

q ′, Q ′ with

0 ≤ qi' ≤ qi , 0 ≤ Q j ≤ Q j for all i, j , such that

j =1

i =1

j =1

∑ qi' = ∑ Q 'j ,

M ≥ ∑ Pj Q 'j − ∑ pi qi' .

B, S , M , x ← 0

i0 , j 0 ← 1



u←q



U ←Q

r , x ← min{u1 , U 1 }

while ((Pj > pi ) ∧ (r > 0 )) do

153

In the case where all purchases and sales take place within the same period of less than six

months, transaction dates are immaterial to matching, and transactions can be listed in any

convenient order.

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B ← B + rp i0

S ← S + rPj0

M ←S-B

1555

{update cost of purchases}

{update proceeds from sales}

u i0 ← u i0 - r

U j0 ← U j0 - r

i0 ← min{i : u i > 0}

{update profits}

{update unmatched shares purchased}

{update unmatched shares sold}

{find lowest - price unmatched purchase}

j 0 ← min{j : U j > 0} {find highest - price unmatched sale}

r ← min u i0 , U j0

{determine number of matchable shares}

x← x+r

{update total number of matched shares}

end while

{

}

It is now possible to prove the following.

Theorem. Algorithm Lowest-In, Highest-Out terminates with the

specified postcondition.

Proof. We use the following loop invariant:

 





For all non − negative real − valued q ′, Q′ such that ∑ qi =∑ Q j = x, 

i =1

j =1



there exist non − negative integers k , l such that



k



k +1

l +1

 q ≤x< q ,



and

≤

∑

i ∑

∑



i =1

j =1









 m

(1)

 B = ∑ pi qi + pk +1  x − ∑ qi  ≤ ∑ pi qi



i =1



 i =1





 n

 S = ∑ Pj Q j + Pl +1  x − ∑ Q j  ≥ ∑ Pj Q j (2)



j =1



 j =1



(3)

M = S − B ≥ ∑ Pj Q j − ∑ pi qi



j =1

i =1





In the above loop invariant, the expression B = B (x) represents the cost

of purchasing a total of x shares in nondecreasing order of per-share

price. Inequality (1) states that there is no lower-cost list of purchases



q ′ totaling x shares. We present a full proof only for the truth of (1).

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The proof of (2) is analogous, and (3) follows immediately from (1) and

(2).

(a) The loop invariant is initially true: Before the while loop

(x = min{u1 , U1 }) , (1) is true because B = p1 x ≤ p1q1' + p2 (x − q1' ) ≤ ∑ pi qi'

i =1



for any q ′ such that ∑ qi' = x .

i =1

(b) The truth of the loop invariant is maintained: Now suppose (1)

holds at the beginning of the while loop; thus i0 = k + 1 . There are two

cases, each of which will imply (1) also holds at the end of the while

loop.

k +1



Case 1: r = ui . Then xnew ← x + ui = ∑ qi , and for any q ′ such

i =1

that ∑ qi' = xnew , we have

i =1

k +1

B = ∑ pi qi

i =1

k +1





≤ ∑ pi qi' + pk +1  xnew − ∑ qi' 

i =1





k +1

= ∑ pi qi' + pk +1 ∑ qi'

i =1

i =k + 2

≤∑pq

i =1

Case

i i

r = U j 0 < u i0 .

∑q ≤ x

new

= x + r < ∑ qi .

∑q = x

new

= x + r , we have

i =1

Then

k +1

i =1

For

any

xnew ← x + U j0



q′

such

and

that

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1557





B = ∑ pi qi + pk +1  x + r − ∑ qi 

i =1









≤ ∑ pi qi' + pk +1  xnew − ∑ qi' 

i =1





= ∑ pi qi' + pk +1 ∑ qi'

i =1

≤ ∑ pi q

i =1

i = k +1

The proof of (2) similarly breaks into Case 1, where r = U j , so that

l +1

xnew ← ∑ Q j ; and Case 2, where r = ui0 < U j0 , so that xnew ← x + ui0 .

j =1

Each case in the proof of (2) proceeds analogously to its counterpart case

in the proof of (1).

condition ((Pj > pi ) ∧ (r > 0 )) fails, i.e., when either all remaining

unmatched purchases were at a higher per-share price than that of all

remaining unmatched sales, or when there are no remaining unmatched

purchases or sales. Progress toward termination is guaranteed by the fact

that during each iteration, Case 1 of the proof of either (1) or (2) applies,

so that the r matched shares must exhaust the remaining unmatched

shares of at least one transaction, i.e., the (k + 1) -st purchase or the

(l + 1) -st sale, respectively. Because there are only m + n transactions

to exhaust, the loop must terminate after at most m + n iterations.

(d) The postcondition is met upon exit: the postcondition follows from

(3) and the failure of the exit condition. The postcondition is trivially

true if all transactions can be matched (eventually r = 0 ) or if none can

be matched (P1 ≤ p1 ) . We show that the postcondition also holds when

(Pj > pi ) fails after it has held at least once.

Let xT denote the final total number of matched shares, i.e, the value

of x at the beginning of the last iteration of the while loop. Let (iT , jT )

and (iF , j F ) denote the respective values of (i0 , j0 ) when (Pj > pi )

last holds and fails, respectively; thus Pj > pi and Pj ≤ pi . Also,

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denote

x′ = ∑ qi' = ∑ Q 'j

i =1

[Vol. 91:1523

j =1

i =1

M ′ = ∑ Pj Q 'j − ∑ p i q i' . Then (3)

and

j =1

simply states the postcondition in the case x′ = xT . For the case x′ < xT ,

(3) implies

j =1

i =1

(

M ≥ ∑ Pj Q 'j − ∑ pi qi' + (xT − x′) PjT − piT

≥ M′

)

and for the case x′ > xT , (3) implies

j =1

i =1

M ′ = ∑ Pj Q 'j − ∑ pi qi'

(

≤ M + ( x′ − xT ) PjF − piF

≤M

)

Thus the postcondition holds in all cases. Q.E.D.

This theorem clarifies the Smolowe formula’s mathematically valid

range—“within six months”—and obviates six decades of unjustified

reliance on Gratz for empirical corroboration of the formula.

The Smolowe Formula’s Worst-Case Errors in Complex Cases

The fact that the Smolowe formula is always correct when applied to

statutory six-month trading sequences does not, of course, imply that it is

always erroneous when applied to longer sequences.154 The formula’s

$201 shortfall in Gratz 155 does, however, demonstrate its potential for

material inaccuracy in complex cases.

The legal community should discontinue the practice of citing Gratz

to support the Smolowe formula’s use, not only because it is untenable 156

154

See supra Figure 2 (illustrating with a hypothetical example). For cases where the Smolowe

formula correctly calculated the maximum liability attributable to a trading sequence spanning more

than six months, despite questionable authority for the formula’s use, see, e.g., Adler v. Klawans,

267 F.2d 840, 847–48 (2d Cir. 1959) (more than seven months); Donoghue v. Casual Male Retail

Group, Inc., 375 F. Supp. 2d 226, 237 (S.D.N.Y. 2005) (more than ten months); Segen v. Westcliff

Capital Mgmt., LLC, 299 F. Supp. 2d 262, 265–66, 272 (S.D.N.Y. 2004) (more than ten months);

Donoghue v. MIRACOR Diagnostics, Inc., No. 00 Civ. 6696, 2002 WL 233188, at *2 (S.D.N.Y.

Feb. 11, 2002) (more than thirteen months); Morales v. New Valley Corp., 999 F. Supp. 470, 476

(S.D.N.Y. 1998) (more than six months); Heli-Coil Corp. v. Webster, 222 F. Supp. 831, 837 (D.N.J.

1963) (more than nine months), modified, 352 F.2d 156 (3d Cir. 1965); Ark. La. Gas Co. v. W.R.

Stephens Inv. Co., 141 F. Supp. 841, 847 (W.D. Ark. 1956) (more than thirteen months); Kogan v.

Schulte, 61 F. Supp. 604, 605 (S.D.N.Y. 1945) (fifteen months).

155

See supra text accompanying note 128.

156

See supra section I.C.

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and superfluous, 157 but because it could lead to a significant error in the

amount of a section 16(b) judgment. While $201 pales in comparison to

the $53,764 deficiency in Claughton’s calculations, 158 it is worth

considering how much larger the Smolowe formula’s errors might

become in the worst case. 159

As Jacobs pointed out with his hypothetical examples, the Smolowe

formula may fall short of calculating the maximum possible short-swing

profit when some trades are not within the statute of limitations160 and

when trades span a period of more than six months. 161 These two kinds

of problematic trading sequences give rise to different worst-case

scenarios, which can be illustrated with the following variations on

Jacobs’s examples.

As a worst-case scenario involving trades outside the statute of

limitations, consider a suit filed in month 28 attacking the following

trading sequence:

Month Purchased

1,000

Purchase Price ($)

Per Share

1,000

Sold

Sale Price ($)

Per Share

1,000

1,002

1,001

The Smolowe formula would pair the purchases in months 1 and 2

with the sales in months 3 and 5, respectively; however, the statute of

limitations would bar recovery of profits from the former pair of

transactions, leaving only the $1,000 proceeds from the latter pair. A

higher profit of $1,002,000 can be calculated by instead pairing the

purchases in months 1 and 2 with the sales in months 5 and 3,

respectively. It should be apparent from this example that the formula’s

157

See supra section III.A.

See supra text accompanying note 134.

159

Even though the formula’s $202 million short-swing profit calculation in Dreiling v. Jain, 281

F. Supp. 2d 1234 (W.D. Wash. 2003) was accurate, the court’s citation to Whittaker v. Whittaker

Corp., 639 F.2d 516, 522, 533 (9th Cir. 1981) as primary authority for the formula’s use was

unsound. The Whittaker decision features one of the most comprehensive and unqualified

endorsements of the Smolowe formula in section 16(b) case law, in which it inaccurately states that

the Gratz court “considered the profit computation issue and, after an independent analysis,

affirmatively reasserted the Smolowe [formula].” Whittaker v. Whittaker Corp., 639 F.2d 516, 522,

531 (9th Cir. 1981).

160

See Jacobs, supra note 7, at 533.

161

See id. at 532–33.

158

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error in cases where some trades fall outside the statute of limitations

may be arbitrarily close to 100 percent.

A worst-case scenario involving a trading sequence spanning more

than six months might resemble the following:

Month

Purchased

1,000

Purchase Price ($)

Per Share

Sold

Sale Price ($)

Per Share

1,000

1,001

1,002

Here, the Smolowe formula would pair the purchase in month 1 with

the sale in month 5, yielding a recovery of $1,001,000 (leaving the

transactions in months 2 and 9 unpaired as more than six months apart).

A higher profit of $2,000,000 can be calculated by instead pairing the

purchases in months 1 and 9 with the sales in months 2 and 5,

respectively. It should be apparent from this example that the formula’s

error in cases covering more than six months may be arbitrarily close to

fifty percent.

The following theorem shows that fifty percent is also an upper limit

on the formula’s error in such cases.

Theorem 1. For any sequence of trades within the statute of limitations,

the recovery calculated by the Smolowe formula is at least half as much as

the recovery calculated by any other method.

Proof. Assume to the contrary that there exist trading sequences for which

there is a pairing of trades that results in more than twice the amount of profit

recovered by the Smolowe formula. Among these trading sequences,

consider one in which the formula’s pairing involves a minimal number of

shares (a “Smolowe-minimal” trading sequence). Let G = (( X , Y ), E ) be the

bipartite graph corresponding to this Smolowe-minimal trading sequence,

wherein each vertex x ∈ X represents one share purchased, each vertex

y ∈ Y represents one share sold, and edge ( x, y ) ∈ E is present with

weight w = w( x, y ) if a pairing of x with y would yield a recoverable

profit w > 0 . 162

162

It may be assumed, without loss of generality, that all of the challenged trades involve whole

numbers of shares; if any fractional shares are involved, all share quantities may be multiplied by

their lowest common denominator before proceeding with the construction of G without affecting

the proof.

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Let S = (( X ( S ), Y ( S )), E ( S )) be the subgraph of G corresponding

to the pairing of transactions produced by the Smolowe formula, and let

w(S ) denote the total weight of S. By the assumption, there exists a

subgraph T = (( X (T ), Y (T )), E (T )) of G corresponding to a different

pairing of transactions such that w(T ) > 2 w( S ) .

Let ( x1 , y1 ) ∈ E ( S ) be an edge of maximal weight in S. Then the

share purchased at x1 and the share sold at y1 must be part of the first

purchase and sale, respectively, paired by the Smolowe formula, and

( x1 , y1 ) must also be an edge of maximal weight in G. Let G1 denote

the subgraph of G induced by ( X \ x1 , Y \ y1 ) . Because reducing the first

purchase

and

sale

one

each

leaves

the

“lowest-in,

highest-out”

sequence

intact,

follows

that

S1 = (( X ( S ) \ x1 , Y ( S ) \ y1 ), E ( S ) \ ( x1 , y1 )) is the subgraph of G1

corresponding to the pairing of transactions produced by the Smolowe

formula, and w( S1 ) = w( S ) − w( x1 , y1 ) .

From among the edges in E(T) incident to x1 and y1 , arbitrarily

choose representatives ( x1 , y ′) and ( x ′, y1 ) . (Without loss of generality,

these

exist

and

are

and

the

following

distinct;

otherwise

inequality

holds

{( x1 , y ′), ( x ′, y1 )} < 2

fortiori.)

Then

T1 = (( X (T ) \ {x1 , x ′}, Y (T ) \ { y1 , y ′}), E (T ) \ {( x1 , y ′), ( x ′, y1 )}) is a

subgraph of G1 corresponding to a different pairing of transactions such that

w(T1 ) ≥ w(T ) − (w( x1 , y′) + w( x′, y1 ) )

≥ w(T ) − 2 w( x1 , y1 )

(since w( x1 , y1 ) is maximal in G )

> 2 w( S ) − 2 w( x1 , y1 )

= 2 w( S1 ),

but S1 < S , contradicting the assumption that G represents a Smoloweminimal trading sequence. Q.E.D.

The Smolowe Formula’s Continuing Fallibility

Modern technology may have facilitated the accurate calculation 163

and verification 164 of short-swing trading liability, but it still has not

eliminated the risk of error when the Smolowe formula is used

163

164

See Chin, supra note 38, and accompanying text.

See supra section III.A.

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improperly. 165 In Chechele v. Vicis Capital, 166 a shareholder of Bond

Laboratories, Inc. sued one of the company’s former directors, Elorian

Landers, over a sequence of 252 purchases and 81 sales of the

company’s stock between August 2009 and October 2010, a period

spanning more than thirteen months. 167 The complaint alleged that

Landers had realized short-swing profits of “not less than $30,000”

calculated using the Smolowe formula. 168 The claim settled before trial,

with the company recovering $30,000. 169

Actual calculations of Landers’s short-swing profits using the

Smolowe formula and, alternatively, using linear programming methods

would have yielded $34,967 170 and $35,361, 171 respectively.

Interestingly, the Smolowe formula’s small shortfall of $394 in Chechele

resembles the formula’s small $201 error in Gratz. It also appears that

Bond Laboratories did not attempt an actual calculation of the

defendant’s short-swing profits under the Smolowe formula and left a

significant fraction of the potential recovery on the table, just as Gratz

did sixty years earlier.172

Even though Claughton’s handwritten accounting has given way to

Excel spreadsheets, plaintiffs and their attorneys still might not consider

careful liability calculations to be worth the effort, because “[r]ecovery

runs not to the stockholder, but to the corporation.” 173 Maximizing the

165

See supra section III.B.

Chechele v. Vicis Capital, LLC, No. 11 Civ. 2191, 2012 WL 310943 (S.D.N.Y. 2012).

167

See Complaint ¶¶ 19–20, Chechele v. Vicis Capital, LLC, No. 11 Civ. 2191, 2011 WL

7566992 (S.D.N.Y. Mar. 30, 2011) (listing trades). Chechele also sued an investment fund that had

traded in the company’s stock. See id. ¶¶ 21–25 (stating claim against Vicis Capital Master Fund

and Vicis Capital, LLC). The claim against the fund was dismissed without prejudice. Chechele,

2012 WL 310943, at *1.

168

See Complaint ¶ 29, Chehele, 2011 WL 7566992 (“Under the ‘lowest-in, highest-out’ method

for computing realized profits pursuant to Section 16(b) of the Act, Defendant Landers realized

recoverable profits as a result of the transactions described in paragraphs 19–20 above in an

aggregate amount not less than $30,000.”).

169

BOND LABORATORIES, INC. ANNUAL REPORT (FORM 10-K) 23 (April 13, 2012),

http://www.sec.gov/Archives/edgar/data/1374328/000141588912000538/bnlb10k12312011.htm

[https://perma.cc/UU4D-DL3U] (noting that $30,000 of Landers’s consulting fees “was setoff

against amounts owed to the Company as a result of violations of Section 16(b)”).

170

See infra app. B, tbl. 3.

171

See infra app. B, tbl. 4.

172

See supra text accompanying note 134.

173

Smolowe v. Delendo Corp., 136 F.2d 231, 239 (2d. Cir. 1943). Out of a $18,894.85 recovery

in Smolowe, the plaintiffs received about three dollars based on their ownership share, and the

attorney was awarded $3,000 in fees and $78.98 in expenses. Id. at 241; cf. Louis Kaplow & Steven

Shavell, Accuracy in the Determination of Liability, 37 J.L. & ECON. 1 (1994) (suggesting a tradeoff between accurate liability calculation and enforcement effort); Louis Kaplow & Steven Shavell,

166

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short-swing recovery from a sequence of 333 transactions over a

thirteen-month period is still a polycentric task, 174 and the path from the

Smolowe formula to a matching that actually “squeeze[s] all possible

profits out of [those] stock transactions” 175 is not always direct or

intuitive. As Figure 6, below, illustrates by reference to Landers’s

transactions, the facial differences between a profit-maximizing

matching of trades found by the linear programming method and a

matching according to the Smolowe formula are complex and subtle. It is

not readily apparent to a casual observer that the Smolowe formula’s

matching is deficient, let alone how it can be improved. In light of these

complexities, the cost-benefit calculus in section 16(b) litigation may not

yet support the adoption by plaintiffs of a more accurate alternative to

the Smolowe formula.

Accuracy in the Assessment of Damages, 39 J.L. & ECON. 191 (1996) (arguing that plaintiffs may

inefficiently overinvest in accurately calculating liability when there are potential gains from doing

so).

174

See Fuller, supra note 112, at 394–95.

175

Smolowe, 136 F.2d at 239.

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Figure 6:

Landers’s purchases (down-arrows) and sales (up-arrows) of

Bond Laboratories stock, matched according to the linear

programming method (top graph) and the Smolowe formula (bottom

graph).

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IV. LEARNING FROM HAND’S MATHEMATICAL SILENCE

An Online Solution

This Article’s sole normative concern is for mathematical correctness.

It does not take sides in the longstanding debate over the statute’s

harshness. 176 Nor does it address the merits of Smolowe and Gratz,

except to urge a more careful reading of their statements and silences.

Proponents of section 16(b)’s repeal might dismiss the pursuit of

accurate liability calculation as akin to fine-tuning a sledgehammer, 177

perfecting a trap for the unwary, 178 or abetting the creation of a

176

See, e.g., John C. Coffee, Jr., The SEC and the Institutional Investor: A Half-Time Report, 15

CARDOZO L. REV. 837, 895 (1994) (noting that section 16(b) supports the public policy of

encouraging a “longer time horizon” on the part of corporate managers and investors); Donna Darm,

Short-Swing Profits in Failed Takeover Bids—The Role of Section 16(b), 59 WASH. L. REV. 895,

912 (1984) (arguing that section 16(b) punishes unsuccessful takeover bids too harshly); Dessent,

supra note 52 (arguing that section 16(b)’s strict liability approach is out of step with other legal

standards developed under Rule 10b-5 to address insider trading, warranting repeal); Jesse M. Fried,

Reducing the Profitability of Corporate Insider Trading Through Pretrading Disclosure, 71 S. CAL.

L. REV. 303, 361–65 (1998) (arguing that section 16(b) should be abolished in favor of pretrading

disclosure); Kanji Ishizumi, Insider Trading Regulation: An Examination of Section 16(b) and a

Proposal for Japan, 47 FORDHAM L. REV. 449, 484 (1979) (arguing that “[t]he costs of the section

exceed its benefits”); Marleen A. O’Connor, Toward a More Efficient Deterrence of Insider

Trading: The Repeal of Section 16(b), 58 FORDHAM L. REV. 309, 323 (1990) (noting that

commentators began criticizing the statute as soon as it was enacted); Karl Shumpei Okamoto,

Rereading Section 16(b) of the Securities Exchange Act, 27 GA. L. REV. 183, 186 (1993) (defending

section 16(b) under a reconception of the statute as “a device primarily concerned with price

manipulation by insiders through stock trading”); Ellen Taylor, Teaching an Old Law New Tricks:

Rethinking Section 16, 39 ARIZ. L. REV. 1315, 1318 (1997) (arguing that section 16(b) should be

repealed because it is ineffective, unfair, and expensive); Thel, supra note 44, at 397–99 (conceding

that “Section 16 is ill-tailored for the task of preventing insiders from taking advantage of inside

information,” but arguing that it is “an extraordinarily precise measure for getting those in charge of

publicly held companies to operate them in ways that will benefit the general public”).

177

See, e.g., O’Connor, supra note 176, at 372–75 (arguing that section 16(b)’s “sledge hammer”

approach is both overinclusive and underinclusive, and therefore inefficient); cf. Provident Secs. Co.

v. Foremost-McKesson, Inc., 331 F. Supp. 787, 792 (N.D. Cal. 1971) (describing section 16(b) as

“an extremely crude rule of a most deformed and misshapen thumb”), aff’d, 506 F.2d 601 (9th Cir.

1974), aff’d, 423 U.S. 232 (1976).

178

See, e.g., RICHARD W. JENNINGS & HAROLD MARSH, JR., SECURITIES REGULATION 1402

(David L. Shapiro et al. eds., 6th ed. 1987) (“Judging solely from the facts stated in the opinions in

the decided cases, the function of Section 16(b) would appear to be to impose unjust liability upon

entirely innocent persons.”); O’Connor, supra note 176, at 373 (“Section 16(b) . . . does not provide

much deterrence because its arbitrary restrictions are easy to evade.”); but see Merritt B. Fox,

Insider Trading Deterrence Versus Managerial Incentives: A Unified Theory of Section 16(b), 92

MICH. L. REV. 2088, 2093 (1994) (arguing that insider trading may be deterred by the six-month

waiting period to make a corresponding trade).

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monstrosity. 179 Defenders of the statute might concede at least some of

these characterizations, yet take a more appreciative view. 180

One need not take sides on the (probably moot) question of section

16(b)’s repeal, however, to acknowledge the importance of ensuring that

“this rule of thumb is no cruder than it needs to be.” 181 A matching of

trades produced by an erroneous application of the Smolowe formula

does not correspond to any articulable theory of insider trading

deterrence, does not advance anyone’s ideal approach to securities

regulation, and does not lend itself to coherent jurisprudence. It is

problematic for everyone, even proponents of repeal. If sound public

policy favors faster traffic, the solution is not to use defective radar guns,

but to raise the speed limit.

For any federal judges (especially those in the Second Circuit),

members of the section 16(b) plaintiffs’ bar, and corporate law

professors willing to consider using and teaching a more accurate

alternative to the Smolowe formula, a free online tool may now shift the

cost-benefit calculus in their favor. With the able assistance of

undergraduate computer science students enrolled in the Fall 2014 and

Spring 2016 software engineering laboratory courses at the University of

North Carolina, I have made a “Short-Swing Profit Liability Calculator”

179

See LOUIS LOSS, SECURITIES REGULATION 1088 n.212 (2d. ed. 1961) (quoting James D.

Calderwood, Section 16(b): Another Noble Experiment Gone Wrong 32 (address before American

Society of Corporate Secretaries, Apr. 21, 1960) (“[T]he SEC has gotten so fascinated with the

algebraic formulae which a fertile mind can conceive under Section 16(b) that it has never walked

away a hundred paces and taken a good look at the monstrosity which has been created.”).

180

See, e.g., Thel, supra note 44, at 414–15 (“Automatic forfeiture of short-swing profits eliminates

the incentive to speculate for the short swing, and thus helps to keep corporate managers from being

distracted from the business of running publicly held companies.”); Byron D. Woodside, Resumé of

the Report of the Special Study of Securities Markets and the Commission’s Legislative Proposals,

19 BUS. LAW. 463, 476 (1964) (“Section 16(b) is about as subtle as a sledge hammer . . . [t]herein,

in part, lies its virtue. The clamor for certainty is pretty well satisfied in this section of the law.”);

see also Reliance Elec. Co. v. Emerson Elec. Co., 404 U.S. 418, 422 (1972) (stating that section

16(b) is a “relatively arbitrary rule capable of easy administration.”) (quoting Bershad v.

McDonough, 428 F.2d 693, 696 (7th Cir. 1970)); Blau v. Lamb, 363 F.2d 507, 515 (2d Cir. 1966)

(“It might be said that [in enacting section 16(b)] Congress decided in order to throw out the

bathwater that the baby had to go too.”); Hearings on Stock Exchange Practices Before the Senate

Committee on Banking & Currency, 73d Cong., 2d Sess., 6557–58 (1934) (statement of principal

drafter Thomas G. Corcoran) (“You have to have a general rule. In particular transactions it might

work a hardship, but those transactions that are a hardship represent the sacrifice to the necessity of

having a general rule.”).

181

See Fox, supra note 178, at 2201–02 (reaching no conclusion as to “whether section 16(b)

should be retained” and stating that “section 16(b) is unlikely to be repealed in the foreseeable

future” because of popular opposition to insider trading, but concluding “we must be sure that this

rule of thumb is no cruder than it needs to be”).

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publicly available on the web. 182 If this tool succeeds in making it easy

to use accurate linear programming methods to calculate short-swing

profits and to detect the Smolowe formula’s errors when they occur, then

courts, attorneys, and professors will have less reason to perpetuate the

misreading of Gratz and the misapplication of the formula. Corporate

law professors in particular may find the calculator helpful as a reminder

to students that the formula is not the exclusive method for calculating

section 16(b) liability.

In addition to accepting manually inputted transaction data, the

calculator provides the ability to search the SEC’s public EDGAR

database for any insider’s Form 4 filings to compile a list of trades

during any given time period. Figure 7 illustrates how a plaintiff might

search for trades by Peter Huntsman, CEO of Huntsman Corporation,

that took place between March and September 2009.

Figure 7:

The section 16(b) liability calculator’s integrated EDGAR

database search engine interface.

The search engine retrieves a sequence of four purchases (three of

which were at a price of zero) and three sales of Huntsman Corporation

stock.

182

Andrew Chin, Short-Swing Profit Liability Calculator, UNIV. OF N.C. SCH. OF LAW,

http://16b.law.unc.edu.

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Figure 8:

The section 16(b) liability calculator’s data input interface

populated by the result of the EDGAR database search depicted in

Figure 7.

By clicking on the adjacent “Link to filing” links, the user can see

that the first zero-price purchase was a grant of restricted stock that

would not vest until March 2, 2010, 183 and the other two zero-price

183

The plaintiff in Bennigson v. Huntsman, No. 13 Civ. 452, 2013 WL 5348461 (S.D.N.Y. Sept.

24, 2013), apparently concluded that the grant satisfied the requirements for exemption under Rule

16b-3 and did not refer to it in the complaint. Benningson, 2013 WL 5348461, at *4. The

requirements for exemption of restricted stock under the rule are quite detailed and beyond the

scope of this Article. See STANTON P. EIGENBRODT, A PRACTICAL GUIDE TO SECTION 16:

REPORTING AND COMPLIANCE, § 11.05[C], at 11-16 (2013).

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purchases were withdrawals for the benefit of Huntsman Family

Holdings and not Peter Huntsman. All three of these zero-price

purchases can be eliminated (using the adjacent “Remove” buttons) as

not matchable with any of the listed sales. The resulting trading sequence

is shown in Figure 9 The bottom of the input interface shown in Figure 9

provides buttons to launch calculations based on the Smolowe formula

(“lowest-in, highest-out,” or “LIHO”) and linear programming (“LP”)

methods.

Figure 9:

The search result depicted in Figure 8 after deletion of exempt

transactions.

This happens to be a case in which the Smolowe formula produced the

same result as the linear programming method even though the trading

period spanned (slightly) more than six months. Using either of the two

methods, the calculator produces the result shown in Figure 10.

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Figure 10:

The section 16(b) liability calculator’s output interface providing

matched trades and recoverable profit from the data in Figure 9.

The result is reflected in the amended complaint in Bennigson,

which sought a recovery of $549,030.00. 184

A distinctive feature of the calculator is that it addresses the intricate

problem of measuring the statutory six-month period in light of the

complications created by months of differing lengths. According to the

calendar, the interval that begins on October 30 and ends on April 29 is a

“period of less than six months,” inasmuch as April 29 precedes the date

(i.e., April 30) that falls exactly six calendar months after October 30.185

While a section 16(b) plaintiff could argue for this “matching date”

interpretation, courts have read the statutory period more narrowly. 186

184

See Bennigson, 2013 WL 5348461, at *4. The district court dismissed the complaint, finding

that the challenged sales were merely “transfer[s] of shares by a Trust of which [the defendant] is

simply a trustee, to an independent LLC” and therefore not “sales” within the meaning of section

16(b). See id.

185

See, e.g., Stella v. Graham-Paige Motors Corp., 132 F. Supp. 100 (S.D.N.Y. 1955), remanded

on other grounds, 232 F.2d 299 (2d Cir. 1956).

186

Id. The court adopted a construction of the term “period of less than six months” to require that

the midnight preceding the start date and the midnight following the end date be less than six

months apart. See id. at 103. Trades on January 1 and June 29 could therefore be paired for shortswing profit recovery, but trades on January 1 and June 30 could not. See id. at 103–04. According

to a leading treatise, the Stella method “has been followed by all courts that have considered the

question.” ROMEO & DYE, supra note 16, § 10.01, at 10-3.

In Jammies Int’l, Inc. v. Nowinski, 700 F. Supp. 189 (S.D.N.Y. 1988), the court considered the

situation where, due to the varying lengths of months, there was no date six months following and

numerically corresponding to the first date in a period. Jammies, 700 F. Supp. at 191. The plaintiffs

argued for “May 1 as the date most closely corresponding to October 31, because it is one day after

the thirtieth day of the month.” Id. at 192. The court, however, held that in such cases, “the

corresponding date for the last day of a month is the last day of the month six months hence.” Id.

The Jammies court also regarded Stella as controlling precedent. See id. Consequently, under

Jammies, the first permissible trade date in a non-leap year following a transaction on August 29,

30, or 31 is February 27. The Jammies rule addresses the measurement of short-swing periods that

begin on March 31, May 31, August 29 (in non-leap years), 30, and 31, October 31, and December

31.

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The calculator’s attention to these calendrical details may seem

arcane, but it simply reflects the level of precision that has emerged from

six decades of case law on the calculation of section 16(b). This

illustrates a final point about the cost-benefit calculus of adopting the

calculator. If the courts have taken such pains to address the

measurement of short-swing periods that begin on seven exceptional

calendar dates, 187 it seems more than worthwhile for the legal

community to adopt a freely available alternative calculation method in

cases where a formula with a worst-case error of fifty percent cannot be

validly used as a rule for calculating maximum short-swing profit.

Prospects for Change at the SEC

Rules of law need less and less to rely on computational rules of

thumb. As Larry Zelenak has pointed out, tax rules are rarely drafted

with simplicity in mind, now that almost ninety percent of federal

income tax returns are prepared on computers. 188 Zelenak tells the story

of the “Rule of 78’s,” a simple but inaccurate method of calculating

interest on short-term installment notes. 189 The IRS had historically

permitted taxpayers to use the rule, but reversed its position in a 1983

revenue ruling, concluding that it could no longer be used “because it

fails to reflect the true cost of borrowing.” 190 Zelenak notes that the

Hewlett-Packard 12C, “the world’s first mass-market handheld financial

calculator,” was introduced in 1982, 191 and writes that “it is unlikely that

the appearance of the ruling shortly after the appearance of the calculator

was a coincidence.” 192

With the introduction of a free online tool for accurately calculating

section 16(b) liability, the time is now ripe for the Securities and

The calculator provides three options for measurement of the statutory “period of less than six

months”: (1) the calendar method, applying the Jammies plaintiff’s rule for differing lengths of

months; (2) the Stella method, applying the Jammies plaintiff’s rule; and (3) the Jammies method,

which incorporates Stella. The Jammies method is selected by default, as it is apparently the only

reported case on the question of varying lengths of months, but plaintiffs in jurisdictions where

Stella and Jammies are not controlling may want to consider the calculation of section 16(b) liability

under other rules.

187

See Jammies, 700 F. Supp. at 192 (regarding the Jammies rule, which specifies certain

calendar dates that especially affect the calculation of short-swing profits).

188

Lawrence Zelenak, Complex Tax Legislation in the TurboTax Era, 1 COLUM. J. TAX. L. 91, 95

(2010).

189

See id.

190

See id. at 96 & n.16.

191

See id. at 95–96.

192

See id. at 96.

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Exchange Commission (“Commission”) to consider updating its

guidance regarding such calculations. Section 16(b) does not give the

Commission standing to sue 193 and expressly acknowledges the

Commission’s rulemaking authority 194 only with respect to rules and

regulations exempting transactions from the subsection’s coverage. 195

Nevertheless, there is a strong argument for engaging the Commission in

the effort to encourage the legal community to adopt more accurate

short-swing liability calculation methods.

Importantly, the Commission has used these muscles before. 196 It was

the Commission’s amicus brief in Smolowe that provided the courts with

the “lowest-in, highest-out” formula that would bear the case’s name. 197

The Commission also filed an amicus brief to the Second Circuit in

Gratz 198 in which it asserted without mathematical justification that the

Smolowe formula “was the rule for the calculation of profits applied by

the court below” 199 and that “the intention ‘to squeeze all possible profits

out of stock transactions’ can only be accomplished by the adoption of

the measure of damages applied in the Smolowe case and in the court

below.” 200 While Hand wisely decided Gratz without endorsing either of

these dubious assertions, 201 the Commission remains on record as an

193

See 15 U.S.C. § 78(p)(b) (2012) (granting standing to “the issuer” and “the owner of any

security of the issuer”).

194

Section 23(a) of the Securities Exchange Act of 1934 authorizes the Commission “to make

such rules and regulations as may be necessary or appropriate to implement the provisions of this

chapter for which they are responsible.” 15 U.S.C. § 78(w)(a) (2012).

195

See 15 U.S.C. § 78(p)(b) (2012) (“This subsection shall not be construed to cover . . . any

transaction or transactions which the Commission by rules and regulations may exempt as not

comprehended within the purpose of this subsection.”).

196

In the 1991 comprehensive revision to its section 16 rules, see Ownership Reports and Trading

by Officers, Directors and Principal Security Holders, Release No. 34-28869, 56 Fed. Reg. 7242

(Feb. 21, 1991), the Commission promulgated Rule 16b-6(c) addressing the calculation of shortswing profits recoverable from transactions involving derivative securities, see id. at 7272–73

(promulgating 17 C.F.R. § 240.16b-6(c)). See generally Joan MacLeod Heminway, Rock, Paper,

Scissors: Choosing the Right Vehicle for Federal Corporate Governance Incentives, 10 FORDHAM J.

CORP. & FIN. L. 225, 288 (2005) (“Substantive competence is, however, acquired through repeated

relevant rulemaking experience over an extended period of time. The SEC has this experience in

securities regulation . . . .”); but cf. HARRY MARKOPOLOS, NO ONE WOULD LISTEN: A TRUE

FINANCIAL THRILLER 63–64 (2010) (arguing that the SEC suffers from an “unbridgeable

[quantitative] skills gap” in regulating capital markets and must rely on the intervention of

mathematically sophisticated outsiders).

197

See SEC Smolowe Brief, supra note 34, at 4–5.

198

Memorandum for the SEC as Amicus Curiae, Gratz v. Claughton, 187 F.2d 46 (2nd Cir. 1951).

199

Id. at 10.

200

Id. at 11.

201

See Gratz v. Claughton, 187 F.2d 46, 52 (2d Cir. 1951); supra note 134 and accompanying text

(showing that Claughton probably did not use the Smolowe formula to produce the calculation

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THE LEARNED HAND UNFORMULA

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1573

advocate for the use of the Smolowe formula beyond its intended and

valid scope. The Commission has subsequently issued two interpretive

releases describing the application of the Smolowe formula to trading

sequences spanning multiple six-month short-swing periods, 202 again

without mathematical justification 203 and without acknowledging the

formula’s fallibility when used in this way. 204 It does not seem

unreasonable to request that the Commission now set the record straight.

A petition for rulemaking may be a productive avenue for eliciting the

Commission’s interest and support. The Commission has been singled

out for praise among federal agencies for the transparency and efficiency

of its petition for rulemaking process. 205 The findings in this Article

could provide the principal basis for a petition for rulemaking or

interpretive guidance on short-swing liability calculation. 206

It is also possible to seek the Commission’s support by requesting that

it participate as an amicus curiae in a pending case involving an

important securities law issue. Given the six decades of case law that

have incorrectly cited Gratz as an authority in support of the

adopted by the district court); supra section II.C (showing that the Smolowe formula would not have

maximized calculation of Claughton’s short-swing profits).

202

See Commission Guidance on the Application of Certain Provisions of the Securities Act of

1933, the Securities Exchange Act of 1934, and Rules Thereunder to Trading in Security Futures

Products, Securities Act Release No. 34-46101, 2002 WL 1677437, at *7 & n.40 (June 21, 2002)

(stating that under the Smolowe formula, “profit is computed by matching the highest sale price with

the lowest purchase price within six months, the next highest sale price with the next lowest

purchase price within six months, and so on, until all shares have been included in the

computation”); Interpretive Release on Rules Applicable to Insider Reporting and Trading,

Securities Act Release No. 34-18114, 46 Fed. Reg. 48147, 48161 n.102 (1981) (same).

203

See ROMEO & DYE, supra note 16, § 10.01[2], at 10-5 (explaining the complexity added by

multiple short-swing periods).

204

See supra section III.B (demonstrating the formula’s fallibility and worst-case errors when

multiple short-swing periods are involved in the Smolowe formula calculation).

205

As Jason Schwartz and Richard Revesz recently reported to the Administrative Conference of

the United States:

After receiving and initially screening petitions, SEC sends the petitioner an acknowledgment

and transmits the petition to the appropriate division of the agency, as well as to its web staff

for posting. Stakeholders report this docketing typically happens fairly promptly. The agency

then continues to update the docket with all comments it receives from the public on the

petition. SEC reports that even with a relatively high volume of petitions, public comments,

and other documents to process, its small web team has managed the volume well.

JASON A. SCHWARTZ & RICHARD L. REVESZ, PETITIONS FOR RULEMAKING: FINAL REPORT TO THE

ADMINISTRATIVE CONFERENCE OF THE UNITED STATES (Nov. 5, 2014), https://www.acus.gov/sites/

default/files/documents/Final%2520Petitions%2520for%2520Rulemaking%2520Report%2520%25

5B11-5-14%255D.pdf [https://perma.cc/27WU-FNER].

206

Cf. Joan MacLeod Heminway, Just Do It! Specific Rulemaking on Materiality Guidance in

Insider Trading, 72 LA. L. REV. 999, 1000 (2012) (urging the Commission “to adopt clarifying

guidance on materiality—one unclear area of insider trading law”).

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[Vol. 91:1523

unwarranted and erroneous use of the Smolowe formula, 207 the potential

precedential impact of a case addressing the scope of the Smolowe

formula’s applicability would likely be substantial enough to warrant the

Commission’s participation. 208 The findings in this Article may prove

helpful to future parties in making such a request.

CONCLUSION

Gratz has finally reached its teachable moment. The takeaway lesson

is that Gratz should no longer be read as endorsing the Smolowe

formula, but as wisely declining to prescribe a formula the court was not

yet technologically competent to validate.209 Given the complexity of the

modern regulatory state and the pace of recent technological change, the

Learned Hand unformula’s silent jurisprudential insights might come to

inform the path of the law in this century as pervasively as the Learned

Hand formula did in the last.

207

See, e.g., Falco v. Donner Found., 208 F.2d 500, 502 (2d Cir. 1953); Huppe v. Special

Situations Fund III, 565 F. Supp. 2d 495, 502–03 (S.D.N.Y. 2008).

208

See Request for Commission Amicus Participation in a Pending Case, U.S. SECURITIES AND

EXCHANGE COMMISSION, https://www.sec.gov/litigation/briefs/amicusrequest.htm [https://perma.

cc/AXH6-EVBT] (“In deciding whether to recommend that the Commission file an amicus brief,

the staff generally considers the following factors: (a) whether the decision in the case is likely to

have substantial precedential impact; (b) whether the case raises issues important to the

Commission’s ability to carry out its statutory objectives or other important securities law issues; (c)

whether there is a potential conflict between the securities laws and other federal or state laws

involved; and (d) whether the brief might provide an opportunity to convince the court to adopt a

narrow or moderate holding, rather than a broad and potentially damaging one.”).

209

Even without Gratz’s endorsement, the Smolowe formula can still validly be applied to trading

sequences falling “within six months,” as the Smolowe court said. See supra section III.A.

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APPENDICES

Computation of Short-Swing Profits in Gratz

Table 1:

Matching of Edward N. Claughton’s common stock trades 210 according

to the Smolowe formula, as performed by the online Short-Swing Profit

Liability Calculator. 211

600

200

400

600

100

900

1200

1000

500

150

850

100

1000

1500

600

1300

Purchase Date

12/20/1944

12/21/1944

12/18/1944

12/22/1944

12/26/1944

12/21/1944

12/18/1944

12/26/1944

12/27/1944

12/21/1944

12/19/1944

Cost ($)

4.22

4.34

4.47

4.59

Sale Date

2/28/1945

6/19/1945

2/28/1945

6/20/1945

6/19/1945

2/28/1945

6/20/1945

6/19/1945

2/28/1945

3/1/1945

2/27/1945

Proceeds ($)

8.29

8.16

13.01

12.89

12.76

8.16

16.38

16.34

12.76

8.16

16.34

16.25

12.76

8.16

7.91

7.79

Profit ($)

2,444.88

2,369.88

1,734.32

3,418.76

3,368.88

2,294.88

1,191.30

10,687.41

14,249.88

8,297.20

1,849.60

1,762.49

9,913.89

1,166.34

8,172.20

5,362.05

2,144.82

4,322.11

4,159.61

210

See Pl.’s Exhibit 5 to Gratz Master’s Report, supra note 11 (listing Claughton’s common stock

trades between December 18, 1944 and September 9, 1946 in chronological order).

211

See Andrew Chin, Short-Swing Profit Liability Calculator, UNIV. OF N.C. SCH. OF LAW,

http://unclaw.com/chin/16b [https://perma.cc/Q87G-VVK7]; supra section IV.A (describing the

calculator). All monetary values have been rounded to the nearest cent. See supra note 88.

Somewhat anachronistically, but without loss of generality, short-swing periods have been measured

according to two subsequent district court decisions that have clarified the matching of trades

separated by almost six full months. See generally Jammies Int’l Inc. v. Nowinski, 700 F. Supp. 189

(S.D.N.Y. 1988); Stella v. Graham-Paige Motors Corp., 132 F. Supp. 100 (S.D.N.Y. 1955),

remanded on other grounds, 232 F. 2d 299 (2d Cir. 1956).

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500

600

500

800

600

1500

300

1550

700

200

350

150

250

400

2300

500

100

400

500

800

150

500

600

700

400

800

1000

900

300

100

600

1000

12/20/2016 12:50 PM

WASHINGTON LAW REVIEW

Purchase Date

12/19/1944

12/22/1944

12/26/1944

12/27/1944

12/21/1944

12/19/1944

12/27/1944

12/21/1944

12/27/1944

12/29/1944

1/23/1945

1/8/1945

12/29/1944

1/5/1945

2/19/1945

Cost ($)

4.59

4.72

4.84

5.48

5.73

5.85

5.98

6.11

7.62

Sale Date

3/26/1945

4/25/1945

3/27/1945

4/26/1945

3/26/1945

3/31/1945

6/20/1945

6/19/1945

3/31/1945

1/30/1945

1/29/1945

6/20/1945

6/21/1945

6/19/1945

6/26/1945

7/3/1945

6/26/1945

7/3/1945

8/9/1945

[Vol. 91:1523

Proceeds ($)

7.79

7.68

7.67

6.58

16.25

16.13

16.00

15.88

15.75

12.76

12.64

12.51

12.01

6.58

6.56

6.18

15.75

15.51

15.38

12.01

11.76

10.64

12.50

13.64

12.50

13.64

14.00

Profit ($)

1,599.55

1,850.94

1,542.35

2,467.76

1,850.22

2,983.20

3,498.72

17,884.52

8,076.81

2,282.74

564.45

3,951.15

1,674.65

558.22

2,759.90

3,218.88

18,221.75

3,898.90

729.92

745.44

922.30

1,175.68

1,637.19

5,332.65

5,270.30

6,324.36

5,021.94

2,769.88

4,642.08

7,025.00

6,097.50

2,335.50

766.00

652.50

4,515.00

6,380.00

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300

100

400

100

700

300

100

200

300

500

400

200

400

200

100

800

400

200

100

900

100

300

400

100

900

200

800

200

600

100

300

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THE LEARNED HAND UNFORMULA

Purchase Date

4/26/1945

4/27/1945

10/4/1946

9/24/1946

9/9/1946

6/5/1945

9/9/1946

6/5/1945

9/8/1946

9/9/1946

6/18/1945

Cost ($)

7.87

8.00

8.13

8.63

8.88

9.01

9.14

9.52

9.64

Sale Date

10/15/1945

9/19/1946

3/12/1946

11/20/1945

11/19/1945

3/12/1946

11/20/1945

11/19/1945

11/29/1945

9/25/1945

10/10/1945

9/24/1945

9/25/1945

9/21/1945

9/24/1945

9/25/1945

9/21/1945

11/2/1945

3/12/1946

3/13/1946

12/11/1945

12/12/1945

12/11/1945

11/2/1945

9/21/1945

Proceeds ($)

14.76

10.39

9.77

14.38

14.26

15.01

14.88

14.26

14.88

14.76

14.63

14.51

14.38

14.26

13.26

13.01

12.89

15.01

14.88

14.84

14.76

14.38

14.26

1577

Profit ($)

2,066.16

688.72

252.19

239.69

655.64

113.91

3,852.17

1,613.52

600.15

1,175.38

1,576.02

2,870.90

2,246.88

2,246.28

1,123.14

2,197.00

1,098.50

536.78

4,294.24

2,097.24

1,048.62

524.31

4,718.16

524.24

1,535.52

1,648.36

387.16

3,484.44

3,372.21

1,098.02

4,293.20

1,064.38

1,048.14

2,920.44

474.24

1,385.52

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1578

100

1300

400

100

200

600

200

300

200

100

200

100

200

100

200

100

400

200

400

200

500

100

600

100

400

200

400

200

12/20/2016 12:50 PM

WASHINGTON LAW REVIEW

Purchase Date

6/18/1945

6/19/1945

7/25/1945

6/19/1945

7/25/1945

7/27/1945

8/22/1945

8/21/1945

8/1/1945

8/3/1945

8/9/1945

8/21/1945

8/23/1945

8/1/1945

8/21/1945

8/23/1945

8/1/1945

8/2/1945

8/23/1945

8/2/1945

8/9/1945

8/23/1945

8/9/1945

Cost ($)

9.64

9.77

9.89

10.15

11.15

11.40

11.65

11.78

11.90

12.03

12.16

12.28

12.41

12.53

12.66

12.78

12.91

13.03

Sale Date

10/22/1945

10/17/1945

9/21/1945

1/14/1946

1/15/1946

9/21/1945

1/15/1946

1/14/1946

1/15/1946

1/16/1946

1/14/1946

1/15/1946

1/14/1946

[Vol. 91:1523

Proceeds ($)

14.26

15.51

15.26

14.26

15.26

15.13

15.01

14.88

Profit ($)

461.84

6,003.27

1,797.16

449.28

873.56

2,464.68

870.56

1,231.02

621.06

385.28

770.40

360.13

347.60

335.21

645.36

310.15

595.24

570.18

260.10

1,040.40

520.20

495.14

990.28

470.06

1,175.15

222.50

1,335.00

222.50

839.88

419.94

790.00

369.94

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100

300

200

1000

200

100

200

100

200

100

200

400

200

400

100

900

100

200

300

500

200

900

200

1500

200

1000

1100

200

700

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THE LEARNED HAND UNFORMULA

Purchase Date

8/21/1945

7/19/1945

8/9/1945

8/17/1945

2/28/1946

8/17/1945

8/3/1945

8/17/1945

9/5/1945

8/3/1945

8/17/1945

9/5/1945

8/6/1945

8/14/1945

9/5/1945

8/6/1945

8/14/1945

12/12/1945

7/18/1945

8/6/1945

8/14/1945

9/5/1945

8/7/1945

7/18/1945

8/14/1945

9/5/1945

7/16/1945

7/18/1945

8/6/1945

8/10/1945

8/14/1945

9/5/1945

9/10/1945

Cost ($)

13.03

13.16

13.28

13.41

13.53

13.66

13.78

13.86

13.91

14.03

14.16

Sale Date

1/14/1946

1/15/1946

1/16/1946

1/14/1946

1/15/1946

9/21/1945

1/16/1946

Proceeds ($)

14.88

14.76

14.26

14.76

1579

Profit ($)

184.97

184.96

517.29

517.26

344.84

1,724.20

319.78

147.36

294.72

147.36

134.83

269.66

134.83

244.58

487.16

439.04

219.52

410.16

97.23

763.02

84.78

169.56

254.34

423.60

144.50

650.25

44.62

895.80

119.44

597.20

656.92

119.44

418.04

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1580

1700

900

100

2400

200

100

200

500

100

850

12/20/2016 12:50 PM

WASHINGTON LAW REVIEW

Purchase Date

9/11/1945

12/6/1945

2/25/1946

2/27/1946

7/13/1945

12/7/1945

9/17/1945

11/26/1945

9/17/1945

11/26/1945

Cost ($)

14.16

14.29

14.41

Sale Date

1/16/1946

9/21/1945

1/16/1946

1/23/1946

[Vol. 91:1523

Proceeds ($)

14.76

14.26

14.76

Profit ($)

1,015.24

537.48

59.72

1,433.28

19.56

59.59

94.38

173.30

34.66

294.61

Table 2:

Matching of Edward N. Claughton’s common stock trades 212 according

to the linear programming method, 213 as performed by the online ShortSwing Profit Liability Calculator. 214

100

1,100

150

350

500

100

500

950

700

300

250

500

200

500

1,000

212

Purchase Date

12/20/1944

12/21/1944

12/18/1944

12/26/1944

12/22/1944

12/21/1944

12/26/1944

12/21/1944

12/18/1944

12/26/1944

Cost ($)

4.22

4.34

4.47

4.59

Sale Date

3/27/1945

3/31/1945

6/19/1945

2/28/1945

6/20/1945

6/19/1945

1/29/1945

6/20/1945

See Pl.’s Exhibit 5 to Gratz Master’s Report, supra note 11.

See Schrijver, supra note 126.

214

See supra section 0.

213

Proceeds ($)

7.67

6.58

12.64

12.01

10.64

8.29

8.16

16.34

16.25

16.00

12.76

12.64

12.51

11.76

6.18

16.34

Profit ($)

345.98

2,600.29

1,244.63

2,685.97

3,151.30

394.98

1,912.40

11,281.16

8,251.81

3,461.70

2,489.16

2,043.13

4,023.90

1,459.94

859.65

11,749.90

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100

200

600

2,700

600

1,300

500

600

800

500

300

1,850

200

1,300

1,450

450

350

200

300

500

400

800

100

650

200

400

500

600

100

1,000

900

300

100

600

1,000

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THE LEARNED HAND UNFORMULA

Purchase Date

12/27/1944

12/21/1944

12/19/1944

12/22/1944

12/26/1944

12/22/1944

12/26/1944

12/22/1944

12/21/1944

12/22/1944

12/21/1944

12/27/1944

12/19/1944

12/27/1944

12/21/1944

12/29/1944

1/23/1945

1/8/1945

1/3/1945

1/23/1945

2/19/1945

Cost ($)

4.59

4.72

4.84

5.48

5.73

5.85

5.98

6.11

7.62

Sale Date

6/20/1945

6/19/1945

2/28/1945

3/1/1945

2/27/1945

3/26/1945

4/25/1945

3/26/1945

3/31/1945

1/30/1945

1/29/1945

6/20/1945

6/19/1945

2/28/1945

3/27/1945

4/26/1945

6/20/1945

6/21/1945

6/19/1945

6/26/1945

7/3/1945

6/26/1945

7/3/1945

8/9/1945

Proceeds ($)

16.00

13.01

12.89

12.76

8.16

7.91

7.79

7.68

7.67

6.58

6.56

6.18

16.25

16.34

16.25

16.13

12.89

12.76

12.64

12.01

11.76

10.64

8.29

7.67

16.38

16.25

15.88

15.75

15.51

15.38

12.76

12.01

12.50

13.64

12.50

13.64

14.00

1581

Profit ($)

1,141.40

1,684.32

1,659.38

4,903.32

9,651.69

2,144.82

4,322.11

4,159.61

1,599.55

1,850.94

1,850.22

1,591.04

984.90

478.44

3,498.72

3,487.47

21,346.04

2,282.74

1,634.38

10,461.36

11,487.63

3,565.13

2,554.72

1,409.94

1,778.28

1,787.25

1,183.80

2,367.60

1,153.80

7,418.71

2,207.86

4,365.84

5,332.65

5,270.30

6,324.36

4,753.32

717.42

7,025.00

6,097.50

2,335.50

766.00

652.50

4,515.00

6,380.00

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1582

300

100

200

100

1,000

300

400

200

800

400

900

800

100

900

1,500

400

500

100

200

500

400

200

100

1,700

400

100

200

300

200

100

500

100

300

100

200

12/20/2016 12:50 PM

WASHINGTON LAW REVIEW

Purchase Date

4/26/1945

4/27/1945

10/4/1946

9/24/1946

9/9/1946

6/5/1945

9/9/1946

6/5/1945

9/9/1946

6/5/1945

9/9/1946

6/5/1945

9/8/1946

9/9/1946

6/18/1945

6/19/1945

7/25/1945

6/19/1945

7/25/1945

7/27/1945

8/22/1945

Cost ($)

7.87

8.00

8.13

8.63

8.88

9.01

9.14

9.52

9.64

9.77

9.89

10.15

11.15

11.40

11.65

11.78

11.90

Sale Date

10/15/1945

9/21/1945

9/19/1946

3/13/1946

9/21/1945

3/12/1946

11/19/1945

11/29/1945

9/25/1945

3/12/1946

9/21/1945

11/2/1945

10/22/1945

3/12/1946

9/21/1945

3/13/1946

11/20/1945

12/11/1945

11/19/1945

11/20/1945

12/11/1945

11/20/1945

12/11/1945

12/12/1945

10/17/1945

12/11/1945

12/12/1945

10/10/1945

9/25/1945

9/24/1945

9/25/1945

1/16/1946

9/24/1945

9/21/1945

1/14/1946

1/14/19

This text is long and has been trimmed here. Open the source document for the complete record.

This is a copy of a public record, reproduced as it was published. It is not legal advice, and it may not be the version a court would rely on. Check the official source before you cite it.