Opinion for the Court filed by Circuit Judge SILBERMAN.
Concurring statement filed by Circuit Judge GINSBURG, in which Circuit Judge EDWARDS joins.
SILBERMAN, Circuit Judge:
Section 13(d) of the Securities Exchange Act of 1934, 15 U.S.C. § 78m(d), requires any person who has directly or indirectly obtained the beneficial ownership of more than 5 percent of any registered equity security to disclose within 10 days certain information to the issuer, the exchanges on which the security trades, and to the Securities and Exchange Commission (“SEC”).1 The SEC charged appellants, First City Financial Corporation, Ltd. (“First City”) and Marc Belzberg, with deliberately evading section 13(d) and its accompanying regulations in their attempted hostile takeover of Ashland Oil Company (“Ashland”) by filing the required disclosure statement after the 10 day period. The district court concluded that appellants had violated the statute; it then enjoined them from further violations of section 13(d) and ordered them to disgorge all profits derived from the violation. See SEC v. First City Financial Corp., et al. 688 F.Supp. 705 (D.D.C.1988). We think that the district court’s findings were not clearly erroneous and that the injunction and disgorgement orders were lawful and appropriate remedies for appellants’ violations. We therefore affirm.
I.
The SEC’s case is based on its contention that on March 4, 1986 Marc Belzberg, a vice-president of First City, telephoned Alan (“Ace”) Greenberg, the Chief Executive Officer of Bear Stearns, a large Wall Street brokerage firm, and asked Green-berg to buy substantial shares of Ashland for First City’s account. Appellants claim that Greenberg “misunderstood” Belzberg: the latter intended only to recommend that Bear Stearns buy Ashland for its own account.
[1218]*1218First City is a diversified Canadian corporation founded and controlled by the Belzberg family and engaged in, among other things, investing in the publicly-traded securities of United States corporations. Samuel Belzberg, Marc’s father, is the Chairman and Chief Executive Officer of the company. Samuel Belzberg and his two brothers own at least 70 percent of the stock of First City. Marc Belzberg managed the company’s fifteen-person New York City subsidiary. The New York office apparently evaluated potential investments for the parent First City.
On February 3, 1986, a New York stockbroker, Alan D. Alan, wrote a letter to Samuel Belzberg describing Ashland as a “sensational business opportunity” and stating that “the circumstances and timing could hardly be better for the ‘Sam Belzberg Effect.’ ” (emphasis in original). In a second letter sent several days later, Alan presented Samuel Belzberg with additional financial information on Ashland, including its break-up valuation. After Marc Belzberg received copies of this information, he instructed two financial analysts in the New York office to begin to study Ashland and its component divisions. Armed with their favorable preliminary analysis, on February 11 Marc and Samuel Belzberg purchased 61,000 shares of Ash-land stock for First City using Goldman Sachs, another large Wall Street investment banking house. Throughout the month, First City also steadily acquired large blocks of Ashland stock through Katz-Goldring, Alan’s smaller brokerage firm. By February 26, First City had accumulated more than 1.3 million shares, approximately 4.8 percent of Ashland’s total outstanding stock. Around the same time, Mare Belzberg discontinued purchasing Ashland stock through Katz-Goldring and began acquiring shares through Greenberg at Bear Stearns. Greenberg had enjoyed a longstanding business relationship with First City and Samuel Belzberg. Additional purchases of approximately 53,000 shares of Ashland stock through Green-berg between February 26 and 28 pushed First City’s holdings to 1.4 million shares, or just over 4.9 percent of all Ashland stock.
During this period, First City employed several commonly accepted measures to maintain the secrecy of its purchases, since public knowledge that First City was acquiring a large stake in Ashland would likely drive up the price of the stock. For instance, First City bought the Ashland stock through nominee accounts2 established at Katz-Goldring specifically for the Ashland purchases. Confirmations of the stock purchases were mailed under the nominee name to the home of Robi Blumen-stein, an officer in First City’s New York subsidiary.
On Friday, February 28, First City received a favorable report from Pace Consultants, a Texas consulting firm hired to assess Ashland’s petroleum-related businesses. Pace valued these segments of Ashland at $726 million. The following Monday, on March 3, Pace revised its estimate upward to $844 million. The next day, on March 4, Marc Belzberg telephoned Greenberg and engaged him in a short conversation that would be the centerpiece of this litigation. At his deposition,3 Green-berg described the conversation in the following manner:
[Marc Belzberg] called me and said something to the effect that — something like, “It wouldn’t be a bad idea if you bought Ashland Oil here,” or something like that. And I took that to mean that we were going to do another put and call arrangement that we had done in the [1219]*1219past_ I was absolutely under the impression I was buying at their risk and I was going to do a put and call.4
While Greenberg interpreted Marc Belzberg’s call as an order to purchase Ashland stock on behalf of First City, Marc Belzberg later claimed that he intended only to recommend that Greenberg buy stock for himself, that is, for Bear Stearns, and that Greenberg apparently misunderstood Belzberg. Immediately after the phone call, Greenberg purchased 20,500 Ashland shares. If purchased for First City, those shares would have pushed First City’s Ashland holdings above 5 percent and triggered the beginning of the 10 day filing period of section 13(d). In that event, First City would have been obliged to file a Schedule 13D disclosure statement on March 14 with the SEC.
Between March 4 and 14, Greenberg purchased an additional 330,700 shares of Ash-land stock for First City costing more than $14 million. Greenberg called Marc Belzberg periodically during those ten days to discuss various securities, including Ash-land. In these conversations, Greenberg reported to Marc Belzberg the increasing number of Ashland shares Greenberg had accumulated. According to Greenberg, Belzberg replied to these reports by saying, “ ‘Fine, keep going,’ or something to that effect.” Greenberg also characterized Belzberg’s response as “grunt[ing]” approvingly. Belzberg did not squarely deny that testimony; he testified that he, Belzberg, said “uh-huh, I think it’s cheap.” Over the March 15-16 weekend, Marc Belzberg met with his father and uncles in Los Angeles to discuss Ashland. On Sunday, March 16, Samuel Belzberg decided that First City should continue to buy Ash-land stock. Marc Belzberg then advised his father that Greenberg had accumulated a block of Ashland shares that “First City could acquire quickly.” Samuel Belzberg later testified that he had no prior knowledge of the Greenberg purchases.5
Returning to New York the next morning, March 17, Marc Belzberg called Green-berg and arranged a written put and call agreement for the 330,700 shares Bear Stearns had accumulated. During that conversation, Marc Belzberg did not mention a price to Greenberg. Several days later, Mare Belzberg received the written agreement with a “strike price,” or the price Bear Stearns was charging First City, of $43.96 per share. This price was well below the then market price of $45.37; thus, the total March 17 put and call price was almost $500,000 below market. Marc Belzberg apparently expressed no surprise that Bear Stearns was charging almost half a million dollars less than market value. He later testified that he believed that Bear Stearns was acting as a “Santa Claus” and that Greenberg was giving him “a bit of a break” to gain more business from First City in the future.
When Blumenstein, the officer responsible for ensuring First City’s compliance with the federal securities laws, noticed the strike price, he immediately met with Marc Belzberg. Blumenstein recognized that the computation of the price reflected only the cost to Bear Stearns of acquiring the stock over the two week period before the written agreement (plus interest and commission), thus creating an inference that First City was the beneficial owner of the securities before March 17. After Blumenstein outlined the problem to Belzberg, the two men called Greenberg on a speakerphone. Belzberg later testified, “I informed Mr. Greenberg [during that conversation] that [1220]*1220the letter [the written agreement] was incorrect, that I didn’t care what the price of the stock was that he bought for himself, I didn’t care what day he made the trades for himself, that I was buying stock from him as of today.” Belzberg then testified that Greenberg said, “[Y]ou’re right, the letter's wrong, I didn't read it before it went out, throw it out and I will send you a corrected copy.” Greenberg, however, testified that Belzberg referred only to an error in the calculation of interest and not to the date on which First City acquired the stock. At the end of the conversation, Belzberg suggested he pay $44.00 per share, 4 cents per share higher than the original strike price but still $1.36, or a total of nearly $450,000, below the market price. At trial, Belzberg admitted to picking the $44 figure “out of the air” and that he “did not want the price [he] was paying to relate to [Greenberg’s] cost.” Between March 17 and 25, on Marc Belzberg's instructions, Greenberg bought another 890,-100 Ashland shares on behalf of First City using several put and call agreements.
After these purchases, Samuel Belzberg sent a letter to Ashland’s management, informing them of First City’s holdings in their stock and proposing a friendly takeover of the company. Ashland rejected the offer, and on the morning of March 25 the company issued a press release disclosing that First City held between 8 and 9 percent of Ashland’s stock. Almost immediately, the price of Ashland stock rose 10 percent to $52.25. The next day, on March 26, First City filed the Schedule 13D disclosure statement required by section 13(d). The statement indicated that First City had accumulated 9 percent of Ashland stock and intended to launch a tender offer for the remaining shares at $60 per share. The market price of Ashland stock then rose to $55, peaking at $55.75 per share the next day.
At Ashland’s urging, the Kentucky legislature passed legislation hampering First City’s ability to obtain financing for the tender offer. Soon thereafter, First City abandoned the attempted tender offer. On March 31, Ashland agreed to buy back First City’s shares for $51 per share, or $134.1 million, resulting in a $15.4 million profit for First City. In return,. First City agreed not to purchase any Ashland shares for the next 10 years.
Around the time of the buy-back agreement, the SEC began an informal inquiry into the timeliness of First City’s Schedule 13D filing and requested that both First City and Bear Stearns provide a detailed chronology of events describing their contacts with each other relating to the Ash-land purchases. After deposing Greenberg and Marc Belzberg, the SEC filed a civil complaint against Marc Belzberg and First City, alleging that they crossed the 5 percent threshold on March 4 but filed the required disclosure statement on March 26, twelve days past the section 13(d) deadline.
The district court found that Marc Belzberg and First City entered into an informal put and call agreement on March 4 and then deliberately violated the 10 day filing requirement of section 13(d). The district court, in an extensive opinion, relied primarily on First City’s acknowledged ultimate purpose to take over Ashland, Green-berg’s understanding of his March 4 telephone conversation with Marc Belzberg, the subsequent conversations between Belzberg and Greenberg, and the suspicious price of the March 17 written agreement. The court discounted Marc Belzberg’s “misunderstanding” explanation as “self-serving, inconsistent with his later actions and [not] squarpng] with the objective evidence.” 688 F.Supp. at 712. Belzberg, to put it bluntly, was not credited. The court also refused to consider Greenberg’s later testimony that there might have been an “honest misunderstanding” since Greenberg reached that conclusion based only on Belzberg’s suggestions and statements. See id. at 720.
The district court permanently enjoined appellants from future violations of section 13(d) because they violated the statute deliberately, showed no “remorse,” and were engaged in a business which presented opportunities to violate the statute in the future. See id. at 725-26. The court also ordered appellants to disgorge approximately $2.7 million, representing their prof[1221]*1221its on the 890,000 shares of Ashland stock acquired between March 14 and 25. The court reasoned that appellants were able to purchase these shares at an artificially low price due to their failure to make the section 13(d) disclosure on March 14. See id. at 726-28. Appellants appeal the district court’s finding of violation as unsupported by the evidence and a product of judicial bias. They further contend that the district court abused its discretion in ordering the injunction and disgorgement remedies.
H.
A shareholder must comply with the section 13(d) disclosure law if he beneficially owns 5 percent of a public company’s equity securities. Under Commission Rule 13d-3(a), whenever a person possesses investment or voting power through any agreement or understanding, he enjoys beneficial ownership.6 Rule 13d-3 is crafted broadly enough to sweep within its purview informal, oral arrangements that confer upon a person voting or investment power. See SEC v. Savoy Indus., Inc., 587 F.2d 1149, 1163 (D.C.Cir.1978), cert. denied, 440 U.S. 913, 99 S.Ct. 1227, 59 L.Ed.2d 462 (1979); see also Wellman v. Dickinson, 682 F.2d 355, 363-67 (2d Cir.1982), cert. denied, 460 U.S. 1069, 103 S.Ct. 1522, 75 L.Ed.2d 946 (1983). Appellants concede that a put and call agreement, even if informal, constitutes beneficial ownership to the investor of the stock subject to the agreement.
The case before the district court turned on the question whether the put and call agreement between First City and Bear Stearns was entered into on March 4, as the SEC claims, or not until March 17 as First City argues. That issue, of course, is a question of fact (or of mixed fact and law), the district court’s answer to which normally may not be overturned on appeal unless clearly erroneous. See Fed.R.Civ.P. 52(a). Appellants would strip the district court of the deference that the clearly erroneous standard of review requires. The district judge allegedly exhibited bias against appellants, and therefore we should examine the district court’s finding and reverse if we think it simply erroneous — that is, if we conclude that the SEC failed to carry its burden of proving by a preponderance of the evidence that an agreement, arrangement or understanding existed on March 4. We have said, however, that even if bias were to be shown we would remand for new factfinding rather than engage in de novo review of the record. See Berger v. Iron Workers Reinforced Rodmen Local 201, 843 F.2d 1395, 1407 & n. 3 (D.C.Cir.) (per curiam), reh’ggranted on other grounds, 852 F.2d 619 (D.C.Cir. 1988), cert. denied, — U.S.-, 109 S.Ct. 3155, 104 L.Ed.2d 1018 (1989); Southern Pacific Communications Co. v. AT & T, 740 F.2d 980, 984 (D.C.Cir.1984), cert. denied, 470 U.S. 1005, 105 S.Ct. 1359, 84 L.Ed.2d 380 (1985). Alternatively, appellants claim that even if bias did not taint the factfindings, the district court’s findings must still be overturned, as clearly erroneous, because of the weakness of the SEC’s case.
A.
The charge of bias is drawn primarily from footnotes in the district court’s opinion which, taken together, appellants argue, indicate that the judge unjustifiably thought the Belzbergs were in a disreputable business and that his “populist” view unfairly colored his findings. In one footnote, the court cites 76 newspaper articles (not placed in evidence) which describe the Belzbergs as “active corporate raiders.” See 688 F.Supp. at 708 n. 1. In another footnote, the court refers to a New York magazine article which lists the Belzbergs [1222]*1222as “greenmailers.” See id. at 717 n. 12.7 In a third footnote, the district judge, appellants contend, unjustifiably criticized the Belzbergs’ extra commission payment to stockbroker Alan and Katz-Goldring, suggesting that they were paid to maintain confidentiality and therefore Belzberg committed an “outrageous abuse in management of a public corporation and a blatant violation of fiduciary duties.” See id. at 710-11 n. 4. The SEC did not challenge these payments, appellants did not seek to explain them, and the district court was thus apparently unaware that they could have been normal “finder’s fees.” 8 Appellants’ briefs also suggest that the district court’s opinion referred gratuitously and derisively to the Belzbergs’ wealth when the reference was actually to the fact that the Belzbergs are “wealthy, knowledgeable, and astute businessmen.” 688 F.Supp. at 708. Finally, appellants believe the judge expressed unwarranted disapproval of normal business techniques such as the use of nominee accounts to preserve secrecy, even though the court explicitly said it “recognizes defendants’ claim for the need to keep their business dealings discreet and from public view.” 688 F.Supp. at 710 n. 3.
We think, at the very most, these footnotes suggest that the district judge did not admire those who specialize in investing in the market for corporate control, or even that he expressed some distaste for aspects of their normal operations. That, however, is not judicial bias. Even where a judge expresses his views on law or policy, that “preconception” may not provide the basis for a reversal if the judge still “is capable of refining his views ... and maintaining a completely open mind to decide the facts and apply the applicable law to the facts.” Southern Pacific Communications v. AT & T, 740 F.2d 980, 991 (D.C. Cir.1984). We presume that a judge will set aside personal views — which given human nature are always present — and find the relevant facts solely on the evidence presented. An appellant therefore must show that a judge’s mind was “irrevocably closed” on the issue before the court. See id.; see also FTC v. Cement Inst., 333 U.S. 683, 701, 68 S.Ct. 793, 803, 92 L.Ed. 1010 (1948); United States v. Haldeman, 559 F.2d 31, 136 (D.C.Cir.1976) (en banc) (per curiam).
Appellants suggest that on argument for summary judgment the judge indicated just that degree of indifference to the relevant facts, and therefore a closed mind on the key issues, when he commented from the bench concerning First City’s $15 million profit on the Ashland buy-back agreement. The judge remarked, “Isn’t there something about that [that] shocks?” As with any good judge who recognizes the limitations of his or her own observation as matters of policy, however, the judge quickly reminded counsel and himself that “what shocks one, and when one applies the law to the situation are two different things.” He also said at that hearing, as appellants emphasize, that he had “some idea as to how [he] feel[s] the case should be decided.” But that statement was in the context of further remarking, “[Y]ou don’t necessarily infer that from my views about the case, and any questions that I have asked.”9 In other words, he prom[1223]*1223ised the parties that he would decide the case on the merits apart from his personal views. We see absolutely no indication that he betrayed that promise. Moreover, after examining his carefully crafted opinion (with the exception of only the few footnotes) as well as the record, we do not see how the district judge could have decided the factual issue any differently had he fervently believed that a vigorous market for corporate control was welfare enhancing and that if there is any fault to “greenmail” it lies in the payment, not in the receipt. Indeed, appellants’ allegations of bias in this case are barely colorable and have been constructed only by distorting and quoting out of context snippets of the judge’s comments and opinion.
B.
We now turn to the district court’s fact-findings to measure them against the clearly erroneous standard. Under that test, “[i]f the district court’s account of the evidence is plausible in light of the record viewed in its entirety,” we must uphold the factfinding even if we would have weighed the evidence differently. Anderson v. Bessemer City, 470 U.S. 564, 573-74, 105 S.Ct. 1504, 1511-12, 84 L.Ed.2d 518 (1985). Thus, the SEC’s version of the disputed events, essentially adopted by the district court, need only be “plausible” in order not to be clearly erroneous. See id. And insofar as the lower court’s finding is predicated on a credibility determination — here the court’s disbelief of Belzberg — it is even further insulated from our scrutiny. See id. at 575, 105 S.Ct. at 1512.
It is not at all clear to us that the SEC’s burden in this case, as appellants claim, is to prove Belzberg’s “subjective intent” on March 4 to enter into an agreement with Bear Stearns whereby the latter purchased shares for First City. Whether a contract is formed is not normally thought to depend on the subjective intention of one of the parties. See Brown Bros. Elec. Contractors, Inc. v. Beam Constr. Corp., 41 N.Y.2d 397, 393 N.Y.S.2d 350, 361 N.E.2d 999 (1977); see generally A. FarnswoRth, Contracts § 3.6 at 114 (1982).10 If there were an actual contract a fortiori there was an “understanding.” Assuming ar-guendo, however, that appellants state the issue correctly, we think the government presented a powerful case that Belzberg did intend to enter into a put-call agreement with Greenberg on March 4 and therefore purposely sought to circumvent section 13(d)’s disclosure requirements. That some of the evidence of Belzberg’s intent is circumstantial makes it no less probative or forceful. Appellants seem to suggest incorrectly that the only competent evidence of Belzberg’s intent would be direct statements revealing that intent.
First, it will be remembered, less than two weeks after the March 4 call, First City embarked on a full scale takeover attempt of Ashland, and all of their actions beforehand seem to foreshadow that step. From the very beginning of First City’s interest in Ashland, the company was apparently eyed as a potential takeover target rather than just an investment. First City’s analysts studied the breakup value of Ashland and its component businesses, and Pace consultants were under the impression that First City was evaluating the oil company as a target. On February 26, First City switched its Ashland business from Katz-Goldring to Bear Stearns, whose larger capital accommodated put and call arrangements that enable large pre-merger accumulations and which was presumably unaware of the size of First City’s position in Ashland stock. Then Belzberg asked Greenberg to buy directly approximately 53,000 shares of Ashland for First City. Those purchases between February 26 and 28 brought First City to 4.9 percent, even closer to the 5 percent line [1224]*1224and set the stage for a large put-call agreement.11 It certainly seems more than a little strange, then, that after having directed Bear Stearns to buy for First City’s account 53,000 shares of Ashland, Belzberg, only a week later, would call Greenberg only to suggest that Bear Stearns buy Ashland stock for itself.
In light of the two men’s discussion relating to the purchase of Hartmarx stock three months earlier, First City’s decision to turn to Bear Stearns only after the Belzbergs had acquired almost 5 percent of Ashland strengthens the inference that on March 4 Marc Belzberg intended Bear Stearns to purchase further stock for First City’s benefit. Belzberg had, at that earlier time, called Greenberg and asked Bear Stearns to buy Hartmarx corporation stock on First City’s behalf through a put and call agreement. Greenberg was aware at the time that First City held just under 5 percent of Hartmarx’s total stock, and he warned Belzberg that their proposed put and call agreement might cross the section 13(d) line. After his lawyers confirmed the section 13(d) problem, Belzberg called Greenberg and explicitly and carefully recommended that Greenberg buy stock for Bear Stearns’ own account. The Hartmarx experience taught Belzberg not only that a put and call arrangement qualified as beneficial ownership under section 13(d) but also that if he only intended to “recommend” stock to Greenberg, he needed to clarify that instruction to avoid ambiguity. Belzberg also understood that Greenberg might question the legality of his trades if the latter knew (and might be charged with knowing?) that First City was close to the 5 percent line. This could explain why Bear Stearns was not brought in until First City was already on the 5 percent threshold.
Perhaps the most important piece of circumstantial evidence tending to show that Belzberg had asked Greenberg on March 4 to buy stock for First City under a put-call agreement is the price First City paid Bear Stearns for the stock. That aggregate price was $450,000 below market on March 17 (but reflects the market price on the dates that Bear Stearns purchased the stock from March 4 to March 17) and surely suggests by itself that both parties understood that Bear Stearns had previously purchased the stock as First City’s agent and therefore was only entitled to its normal costs and commission when transferring the stock to First City.
Even had there been no testimony at all from either participant on the March 4 telephone conversation, we think the SEC’s other evidence would have made out a substantial prima facie case. But the participants testified as to what was said during the call, and Greenberg’s version supports the SEC’s case. To be sure, Greenberg’s testimony — that Belzberg said something like, “It wouldn’t be a bad idea if you bought Ashland Oil here” — sounds somewhat imprecise, but Greenberg also said “I was absolutely under the impression I was buying at their risk and I was going to do a put and call.” It is more than a little difficult to imagine how Greenberg could possibly have received such a clear impression unless Belzberg conveyed the message Greenberg thought he received. After all, Greenberg was hardly a novice in the business and would not be expected immediately to buy 20,500 shares of stock (worth approximately $800,000) and to continue to purchase enormous amounts of stock for a client without any direction.
A little over a month after the March 4 phone call, Bear Stearns submitted a chronology of events in response to the SEC’s request. The March 4 entry corroborates Greenberg's testimony but puts it a bit more sharply:
Alan Greenberg ... received a phone call from Marc Belzberg ... in which Mr. Belzberg asked that Bear, Steams begin accumulating Ashland stock. Mr. Greenberg understood this to mean, as in [1225]*1225the case of other securities purchased by Bear, Stearns in which First City had an interest, that as soon as Bear, Stearns had accumulated a sizable position, we would enter into a written put and call agreement with First City, (emphasis added).
Appellants vigorously object to the admission of the chronology as hearsay since it relies on Greenberg’s out-of-court declarations. The district court admitted the chronology under the “residual” exception to the hearsay rule. When a statement is not specifically exempted from the general hearsay prohibition, Rule 803(24) allows the introduction of the statement if it is invested with “equivalent circumstantial guarantees of trustworthiness,” is more probative than other evidence that the proponent can reasonably procure, and serves the interests of justice. We recognize that the legislative history of this exception indicates that it should be applied sparingly. See United States v. Kim, 595 F.2d 755, 765 (D.C.Cir.1979). But we also acknowledge the broad discretion a trial court enjoys in assessing the probity and trustworthiness of documents. See United States v. Reese, 561 F.2d 894, 903 n. 18 (D.C.Cir.1977). Since the residual hearsay exception depends so heavily upon a judgment of reliability, typically we would be particularly deferential to the trial court’s determinations under Rule 803(24). See, e.g., Balogh’s of Coral Gables, Inc. v. Getz, 798 F.2d 1356, 1358 (11th Cir.1986); Huff v. White Motor Corp., 609 F.2d 286, 291 (7th Cir.1979). Appellants had ample opportunity to cross-examine Greenberg about his out-of-court statements during his two depositions to probe for weaknesses. They also could challenge David Hyman’s preparation of the chronology during Hyman’s deposition. Thus, the primary rationale for the hearsay rule — the inability to cross-examine the out-of-court declarant on the veracity of his statement — was at least partially offset here. See United States v. Iaconetti, 406 F.Supp. 554, 558-60 (E.D.N.Y.1976), aff'd, 540 F.2d 574 (2d Cir.1976); see also United States v. Scrima, 819 F.2d 996, 1001 (11th Cir.1987); J. Weinstein & M. Berger, Weinstein’s Evidence § 803(24)[01] at 803-375 (1988). Furthermore, any false statements in the chronology would be subject to criminal prosecution under 18 U.S.C. § 1001. See United States v. White, 611 F.2d 531, 537-38 (5th Cir.) (concluding that out-of-court statements can be trustworthy if made under threat of prosecution for false statements), cert, denied, 446 U.S. 992, 100 S.Ct. 2978, 64 L.Ed.2d 849 (1980).12 Finally, before the chronology was sent to the SEC, Green-berg apparently reviewed the document for accuracy. The chronology thus represented the most contemporaneous account of the March 4 conversation.13
We conclude then that the district court did not commit error in admitting into evidence the Bear Stearns chronology under the residual hearsay exception. But even had the chronology been improperly admitted, the statements were merely cumulative and corroborating evidence of Green-berg’s understanding of Belzberg’s intent during the March 4 phone call and therefore would be harmless error. See Fed.R. Civ.P. 61 (“No error in either the admission or exclusion of evidence ... is ground for [1226]*1226granting a new trial ... unless refusal to take such action appears to the court inconsistent with substantial justice.”); Coughlin v. Capitol Cement Co., 571 F.2d 290, 306-07 (5th Cir.1978).
Nevertheless, appellants argue that Belzberg could not have intended to direct Greenberg to purchase Ashland stock for First City’s account on March 4, because there was no discussion of price and quantity. In fact, Greenberg testified that Belzberg “must have” mentioned a price and quantity. In any event, if the parties wished a put-call agreement, Belzberg’s intended price was not a necessary term, for it was understood that under that arrangement Bear Stearns would buy at market and charge First City only its costs and normal commission. Greenberg, moreover, kept in close telephone contact with Belzberg, advising him as to the quantity of shares Greenberg was purchasing. Greenberg testified, for example, that he would tell Belzberg “I now own 150,000 and [Belzberg would] say ‘Fine.’ ” Or, at other times, Belzberg responded “Fine, keep going” or something to that effect. Belzberg had continuous and at least approximate information as to Bear Stearns’ purchases of Ashland, both as to price and amount.
Appellants’ last attack on the SEC’s evidence is a variant of the “dog that doesn’t bark.” According to appellants, the parties could not have reached a put-call agreement on March 4 because the customary “paper trail” that accompanies those transactions was absent. Bear Stearns typically would send confirmation slips, as well as a formal letter agreement, soon after a trade and would require its customers to send a “margin” or deposit within seven days of the agreement. Greenberg explained that in this case, however, he was accumulating Ashland shares until he had a sizable block at which point he intended, as he did on March 17, to formalize the put-call agreement. And as the district court found, “there was no consistent pattern in the time interval between an oral put/call agreement, the preparation and execution of the agreement, and its dispatch to First City.” 688 F.Supp. at 719 (emphasis added). In other words, Greenberg’s put-call dealings with First City were “informal,” as the SEC put it, apparently because Greenberg felt no need for formal legal protection.
The government’s affirmative case against First City, in sum, was quite strong, and not, as appellants claim, predicated on unwarranted inferences and weak circumstantial evidence. Marc Belzberg testified in his own behalf and gave a competing account of the events and their appropriate interpretation. The district judge, as we have indicated, did not credit the testimony, and we find unassailable the district court’s assessment of Belzberg’s account as implausible.
At the threshold of Belzberg’s version of the events is his claim that on March 4 he had “recommended” to Greenberg that Bear Stearns buy Ashland stock for Bear Stearns’ own account, because if it subsequently turned out that First City wished to acquire Ashland, the takeover would be more easily accomplished if Ashland stock were held “loosely” by arbitrageurs (short-term speculators more inclined to cash in on a quick profit) rather than remain in what appellants refer to as “forgotten safe-deposit box[es].” This explanation for Belzberg’s call to Greenberg is unconvincing. First, Bear Stearns is in the business of recommending stocks to customers (for which it is compensated through trading commissions), not vice versa.14 As we noted, only a week before the March 4 conversation, Greenberg, performing as a trader, had purchased about 53,000 Ashland shares directly for First City. Second, although it may well be true that after a tender offer is launched the acquiring company would wish as much stock as possible to be gath[1227]*1227ered by arbitrageurs, who typically buy for the very purpose of tendering, we had not previously heard the theory that it would be helpful for arbitrageurs to start their buying before the acquiring company publicly signals its intention. Appellants’ brief cites as support for this novel theory 1 M. Lipton & E. Steinberger, Takeovers & Freezeouts § 1.07[2] at 1-51 & 1-51 (1987) and 1 A. Fleischer, Tender Offers: Defenses, Response, and Planning 3-4 (Supp. 1985). But the former citation discusses only the activities of arbitrageurs generally, while the latter suggests that the conventional wisdom, that tender offers are more successful if prior to the tender institutional investors hold large blocks of stock in the target company, is outmoded. Neither author directly addresses appellants’ dubious proposition that it would be useful to give arbitrageurs a head start. If arbitrageurs were to be induced to buy before an acquirer is obliged to signal or announce a tender offer, the resulting trading would move the price up prematurely, thereby putting pressure on the acquirer to raise the anticipated tender offer price. Nor do we think that in today’s market (if it was ever the case) much stock in publicly traded companies is held in forgotten safe deposit boxes and is therefore price inelastic. See 1 M. Lipton & E. Steinberger, Takeovers & Freezeouts § 1.09[2] at 1-95 (1989) (noting that pension funds alone, not including mutual funds, insurance companies, and other institutional investors, own 50 percent or more of the shares of large U.S. public corporations). We do not, however, rely on our own perception and understanding of market behavior to reject Belzberg’s explanation as incredible, for Belzberg’s account is inconsistent with First City’s own behavior.
During the period before it unveiled its takeover plan, First City, acting quite typically for a putative acquirer, see id. § 1.04[2] at 1-24 (emphasizing the importance of pre-tender offer confidentiality), took pains to keep its interest in Ashland secret. The very last thing Belzberg would do in light of those efforts would be to tout Ashland to a prospective speculator as a good investment.15 Indeed, if his purpose was to induce arbitrageurs to buy Ashland, he presumably would have contacted more than just Bear Stearns; he would have openly disclosed his interest in a possible takeover of Ashland to all of Wall Street.
Moving on to other elements in Belzberg’s story, we are struck by his incongruous explanation for his reaction to the $450,000 discount off the market price that Greenberg offered him on March 17 if, in actuality, there had been no put-call agreement on March 4. Belzberg testified that he was not surprised by the price since he thought Bear Stearns was acting like “Santa Claus” by offering “a bit of a break” to gain more First City business. Bit of a break indeed! And unsolicited at that. With apologies to Virginia, we thought that Wall Street Santa Clauses were confined to the sidewalk during Christmas time. Greenberg testified that if he gave clients a half million dollar break on stock for which Bear Stearns bore the market risk, he would “go broke within a week.” Greenberg added, quite convincingly we might add, that “[he does not] run [that] risk for anybody.” More probable, although not particularly helpful to appellants, is Belzberg’s explanation of the price modification of March 17. It will be recalled he moved the price up a mere 4<t a share to the round number $44 in order to rebut the inference that the strike price of $43.96 was the product of a prior put-call agreement. That was a rather thin (if at least inexpensive) disguise which seems only to make all the more obvious that an agreement was, in fact, reached on March 4.
Finally, the “misunderstanding” explanation came suspiciously late in the SEC in[1228]*1228vestigation. Neither Bear Stearns’ nor the First City’s chronology, submitted only a month after the March 4 call, even suggests that Greenberg may have misunderstood Belzberg on that earlier date. And it is quite unlikely (indeed incredible) that after the Hartmarx incident three months before, when Belzberg was carefully instructed as to the risks of imprecise communications concerning put-call agreements crowding section 13(d) deadlines, Belzberg would be cavalier about the subject. Instead, the misunderstanding explanation seems rather clearly to be a post-hoc offering designed to avoid the impact of Greenberg’s admissions. The district court, recognizing this point, refused to admit into evidence Greenberg’s July 3, 1986 affidavit — in which Greenberg first mentioned the possibility of a “misunderstanding” — because it was the product of Belzberg’s suggestion to Greenberg just prior to Greenberg’s deposition. See 688 F.Supp. at 720. The district judge also presumably ignored Greenberg’s subsequent deposition testimony to the same effect.
After carefully examining appellants’ lengthy briefs and the record in this case, then, we conclude the district court’s finding, that appellants deliberately violated section 13(d), should be affirmed, and we turn to the remedies the trial court fashioned.
III.
Appellants challenge the injunction the district court granted as unwarranted on the facts of this case. Since a district judge has wide latitude in fashioning a remedy, we will not disturb the trial court’s remedial choice unless there is no reasonable basis for the decision. See United States v. W.T. Grant Co., 345 U.S. 629, 634, 73 S.Ct. 894, 898, 97 L.Ed. 1303 (1953); SEC v. Bausch & Lomb, Inc., 565 F.2d 8, 18 (2d Cir.1977). Appellants claim such an abuse of discretion here because the SEC did not show “that there [was] a reasonable likelihood of further violations] in the future.” SEC v. Savoy Indus., 587 F.2d 1149, 1168 (D.C.Cir.1978) (emphasis in original) (quoting SEC v. Commonwealth Chem. Sec., Inc., 574 F.2d 90, 99-100 (2d Cir.1978)); see also SEC v. Bausch & Lomb, Inc., 565 F.2d at 18.
Under our Savoy Industries test, the district judge should consider whether a defendant’s violation was isolated or part of a pattern, whether the violation was flagrant and deliberate or merely technical in nature, and whether the defendant’s business will present opportunities to violate the law in the future. See Savoy Indus., 587 F.2d at 1168; Commonwealth Chem., 574 F.2d at 100; SEC v. National Student Mktg., 457 F.Supp. 682, 715-16 (D.D.C.1978).16 No single factor iS determinative; instead, the district court should determine the propensity for future violations based on the totality of circumstances. See SEC v. Youmans, 729 F.2d 413, 415 (6th Cir.), cert. denied, 469 U.S. 1034, 105 S.Ct. 507, 83 L.Ed.2d 398 (1984); SEC v. Bausch & Lomb, Inc., 565 F.2d at 18.17
Appellants raise two troubling points about the injunction order. First, in explaining the injunction the district court said, “[T]here is also a public perception [1229]*1229that defendants have been recent and active participants in heated takeover battles and unfriendly mergers.” 688 F.Supp. at 725.18 Justifying an injunction, even in part, in terms of propitiating public sentiment, is objectionable as a matter of law. Second, the district court misconstrued our precedents in suggesting that another basis for justifying the injunction was appellants’ “lack of remorse.” See SEC v. Savoy Indus., 587 F.2d 1149, 1168 (D.C.Cir.1978). The district court compared Belzberg’s “unrepentant demeanor” with defendants in other cases who “express[] immediate and vociferous remorse for [their] errors.” 688 F.Supp. at 726.19 The government’s appellate brief compounds this mistake by urging us to consider, as further evidence of lack of remorse, appellants’ counsel’s arguments and appellants’ willingness to pursue this case on appeal. The securities laws do not require defendants to behave like Uriah Heep in order to avoid injunctions. They are not to be punished because they vigorously contest the government’s accusations. We think “lack of remorse” is relevant only where defendants have previously violated court orders, see SEC v. Koenig, 469 F.2d 198, 202 (2d Cir.1972), or otherwise indicate that they did not feel bound by the law, see Savoy Indus., 587 F.2d at 1168. As such, it is really only another indication as to whether it is “reasonably likely” that future violations will occur in the absence of an injunction.
The district court, then, appeared to rely in part on two inappropriate factors to support the injunction. Typically we would remand to the district court under these circumstances for reconsideration of the injunction. We do not do so here, however, because we have made an independent determination that the injunction is appropriate, i.e. a reasonable likelihood of future violations exist.20 Appellants do not even deny that their business will offer many opportunities in the future to violate the securities laws. Instead, appellants quarrel primarily with the district court’s determination that Belzberg deliberately violated section 13(d). As our prior discussion— particularly our appraisal of Belzberg’s testimony — should make clear, we think the district court’s determination as to Belzberg’s state of mind is not subject to serious challenge. That this is First City’s first section 13(d) violation is no bar to the issuance of an injunction. See SEC v. National Student Mktg. Corp., 360 F.Supp. 284, 299 (D.D.C.1973) (stating that a single violation may justify an injunction if committed willfully and knowingly); see also SEC v. Bausch & Lomb, Inc., 565 F.2d 8, 18 (2d Cir.1977) (approving injunctions for single acts in some circumstances).
IY.
The district court directed disgorgement of profits, and appellants’ challenge to this aspect of the order presents an issue of first impression — whether federal courts have the authority to employ that remedy with respect to section 13(d) violations and whether it is appropriate in this sort of case. Appellants also claim that the [1230]*1230amount ordered disgorged is excessive. We reject both arguments and affirm the district court’s order on these issues as well.
Appellants, by claiming that Congress did not explicitly authorize a monetary remedy for section 13(d) violations, misapprehend the source of the court’s authority. Disgorgement is an equitable remedy designed to deprive a wrongdoer of his unjust enrichment and to deter others from violating the securities laws. See SEC v. Tome, 833 F.2d 1086, 1096 (2d Cir.1987), cert. denied, - U.S. -, 108 S.Ct. 1751, 100 L.Ed.2d 213 (1988); SEC v. Blavin, 760 F.2d 706, 713 (6th Cir.1985); SEC v. Texas Gulf Sulphur Co., 446 F.2d 1301, 1307 (2d Cir.), cert. denied, 404 U.S. 1005, 92 S.Ct. 561, 30 L.Ed.2d 558 (1971). “Unless otherwise provided by statute, all the inherent equitable powers of the District Court are available for the proper and complete exercise of that jurisdiction.” Porter v. Warner Holding Co., 328 U.S. 395, 398, 66 S.Ct. 1086, 1089, 90 L.Ed. 1332 (1946); see also Mitchell v. Robert DeMario Jewelry, Inc., 361 U.S. 288, 291-92, 80 S.Ct. 332, 334-35, 4 L.Ed.2d 323 (1960). We see no indication in the language or the legislative history of the 1934 Act that even implies a restriction on the equitable remedies of the district courts. See Mills v. Electric Auto-Lite Co., 396 U.S. 375, 391, 90 S.Ct. 616, 625, 24 L.Ed.2d 593 (1970). Disgorgement, then, is available simply because the relevant provisions of the Securities Exchange Act of 1934, sections 21(d) and (e), 15 U.S.C. §§ 78u(d) and (e), vest jurisdiction in the federal courts.
Indeed, appellants concede that disgorgement is rather routinely ordered for insider trading violations despite a similar lack of specific authorizations for that remedy under the securities law. See e.g., SEC v. Tome, 833 F.2d at 1096; SEC v. Materia, 745 F.2d 197, 201 (2d Cir.1984); see generally L. Loss, Fundamentals of Securities Regulation 1004-11 (1988). But they seek to distinguish section 13(d) violations as a “technical” transgression of reporting rules that really do not cause injury. In contrast, they argue, insider trading under modern theory is tantamount to theft; an actual injury is inflicted on the individual or institution entitled to confidentiality. Section 13(d), however, is the pivot of the entire Williams Act regulation of tender offers.21 To be sure, some may doubt the usefulness of that statute generally or the section 13(d) requirement specifically, but it is hardly up to the judiciary to second-guess the wisdom of Congress’ approach to regulating takeovers. Suffice it for us to note that section 13(d) is a crucial requirement in the congressional scheme, and a violator, it is legislatively assumed, improperly benefits by purchasing stocks at an artificially low price because of a breach of the duty Congress imposed to disclose his investment position. The disclosure of that position — a holding in excess of 5 percent of another company’s stock — suggests to the rest of the market a likely takeover and therefore may increase the price of the stock. Appellants circumvented that scheme, and the theory of the statute, by which we are bound, is that the circumventions caused injury to other market participants who sold stock without knowledge of First City’s holdings. We therefore see no relevant distinction between disgorgement of inside trading profits and disgorgement of post-section 13(d) violation profits.
There remains, of course, the question of how the court measures those illegal profits. Appellants vigorously dispute the $2.7 million figure that the district court arrived at by simply calculating all of the profits First City realized (in its eventual sale back to Ashland) on the 890,000 shares First City purchased between March 14 and 25. See 688 F.Supp. at 728 n. 24. The SEC’s claim to disgorgement, which the district court accepted, is predicated on the assumption that had First City made its section 13(d) disclosure on March 14, at the [1231]*1231end of the statutory 10 day period,22 the stock it purchased during the March 14-25 period would have been purchased in a quite different and presumably more expensive market. That hypothetical market would have been affected by the disclosure that the Belzbergs had taken a greater than 5 percent stake in Ashland and would soon propose a tender offer.
Since disgorgement primarily serves to prevent unjust enrichment, the court may exercise its equitable power only over property causally related to the wrongdoing. The remedy may well be a key to the SEC’s efforts to deter others from violating the securities laws, but disgorgement may not be used punitively. See SEC v. Blatt, 583 F.2d 1325, 1335 (5th Cir.1978); SEC v. Manor Nursing Centers, Inc., 458 F.2d 1082, 1104 (2d Cir.1972). Therefore, the SEC generally must distinguish between legally and illegally obtained profits. See CFTC v. British Am. Commodity Options Corp., 788 F.2d 92, 93 (2d Cir.), cert. denied, 479 U.S. 853, 107 S.Ct. 186, 93 L.Ed.2d 120 (1986). Appel lants assert that the hypothetical market between March 14-25 that the SEC urged and the district court accepted was simplistic, quite unrealistic, and so de facto punitive. It did not take into account other variables — besides the section 13(d) disclosure — which caused the post-March 25 price of the stock to rise above that which prevailed during the March 14-25 period. At trial appellants’ expert witness testified that four independent factors combined to increase the stock price to the level it reached on March 25 and that these factors were not present on March 14, when the defendants should have disclosed. He identified these factors as: (1) the Belzbergs by the 25th held between 8 and 9 percent of Ashland, (2) the Belzbergs had prior to the 25th communicated to Ashland the size of their holdings, (3) Ashland publicly disclosed the Belzbergs’ position on the 25th before the 13(d) disclosure, and (4) by the 25th, rumors swirled of an imminent takeover bid at $55 per share. In an attempt to hypothesize how the Belzbergs would have acted — had they disclosed on March 14 — and how the market would have responded to those actions, the witness presented three alternative scenarios that in his view more accurately measured the impact of the nondisclosure and which yielded disgorgement figures of zero, $496,050, and $864,588. Perhaps not surprisingly, appellants’ witness testified that the most realistic scenario required no disgorgement at all.23
If exact information were obtainable at negligible cost, we would not hesitate to impose upon the government a strict burden to produce that data to measure the precise amount of the ill-gotten gains. Unfortunately, we encounter imprecision and imperfect information. Despite sophisticated econometric modelling, predicting stock market responses to alternative variables is, as the district court found, at best speculative. Rules for calculating disgorgement must recognize that separating legal from illegal profits exactly may at times be a near-impossible task. See, e.g., Elkind v. Liggett & Myers, Inc., 635 F.2d 156, 171 (2d Cir.1980).
Accordingly, disgorgement need only be a reasonable approximation of profits causally connected to the violation. In the insider trading context, courts typically require the violator to return all profits made on the illegal trades, see, e.g., SEC v. Texas Gulf Sulphur Co., 446 F.2d 1301, 1307 (2d Cir.), cert. denied, 404 U.S. 1005, 92 S.Ct. 561, 30 L.Ed.2d 558 (1971), and [1232]*1232have rejected calls to restrict the disgorgement to the precise impact of the illegal trading on the market price. See Elkind, 635 F.2d at 171; cf CFTC v. British Am. Commodity Options Corp., 788 F.2d 92 (2d Cir.) (concluding that a nexus between the unlawful conduct and the disgorgement figure need not be shown because of the pervasiveness of the fraud), cert. denied, 479 U.S. 853, 107 S.Ct. 186, 93 L.Ed.2d 120 (1986).
Although the SEC bears the ultimate burden of persuasion that its disgorgement figure reasonably approximates the amount of unjust enrichment, we believe the government’s showing of appellants’ actual profits on the tainted transactions at least presumptively satisfied that burden. Appellants, to whom the burden of going forward shifted, were then obliged clearly to demonstrate that the disgorgement figure was not a reasonable approximation. Defendants in such cases may make such a showing, for instance, by pointing to intervening events from the time of the violation. In SEC v. MacDonald, 699 F.2d 47 (1st Cir.1983) (en banc), the First Circuit reversed a district court order requiring the defendant to disgorge all profits from an illegal insider trade when the defendant had held on to the stock for more than a year. The court restricted the amount to a figure based on the price of the stock “a reasonable time after public dissemination of the inside information.” Id. at 55. Similarly, the Second Circuit in SEC v. Manor Nursing Centers, Inc., 458 F.2d 1082 (2d Cir.1972), refused to extend the disgorgement remedy to income subsequently earned on the initial illegal profits. In those cases, the defendant demonstrated a clear break in or considerable attenuation of the causal connection between the illegality and the ultimate profits.
Here, appellants took a different approach using a sophisticated expert witness. As we noted, they maintained that the post-March 25 price was influenced by four other independent factors besides the belated section 13(d) disclosure, so even if First City had disclosed on March 14, the price would not have run up then to the extent it did after March 25. The difficulty we see with appellants’ argument is that none of the four factors are independent of the section 13(d) disclosure determination. Thus, although by March 25 First City had accumulated 8-9 percent of Ashland, whereas on March 14 it had slightly over 5 percent (and although the market might react more strongly to the higher figure), we do not see why we should not assume that First City would have acquired 8-9 percent before March 14 — if they knew they had to disclose on the earlier date. Second, it seems likely that' First City would have notified Ashland’s management on March 13 if they had planned to disclose on the next day, just as they did on March 25. Third, Ashland’s premature disclosure of First City’s holdings (prior to the section 13(d) notice) would likely have also occurred. And finally, the March 25 market takeover rumors were probably associated with all of the above activity, which is inextricably linked with the impending section 13(d) notice. We therefore agree with the district court that appellants’ efforts to hypothesize both the takeover efforts of a First City that complied with section 13(d) and the market reaction to that are impossibly speculative.
Placing the burden on the defendants of rebutting the SEC’s showing of actual profits, we recognize, may result, as it has in the insider trader context, in actual profits becoming the typical disgorgement measure. But the line between restitution and penalty is unfortunately blurred, and the risk of uncertainty should fall on the wrongdoer whose illegal conduct created that uncertainty. See SEC v. MacDonald, 699 F.2d 47, 55 (1st Cir.1983) (en banc); Elkind v. Liggett & Myers, Inc., 635 F.2d 156, 171 (2d Cir.1980); cf. Bigelow v. RKO Radio Pictures, 327 U.S. 251, 265, 66 S.Ct. 574, 580, 90 L.Ed. 652 (1946) (placing the risk of uncertainty on the wrongdoer in the antitrust context).24
[1233]*1233We conclude that the district court’s fact-findings were neither clearly erroneous nor the product of judicial bias, and we uphold the permanent injunction and disgorgement orders. Accordingly, the judgment of the district court is
Affirmed.