KEARSE, Circuit Judge:

The present appeal, brought by plaintiffs Kenneth M. Krys and Margot MacInnis as, inter alia, Joint Official Liquidators of the SPhinX Ltd. family of hedge funds ("SPhinX" or "SPhinX Funds"), et al., challenges a partial final judgment of the United States District Court for the Southern District of New York, Jed S. Rakoff, Judge, in favor of defendants William T. Pigott, Liberty Corner Capital Strategies LLC, Ingram Micro Inc., and CIM Ventures Inc., dismissing the claims in plaintiffs' Amended Complaint. The Amended Complaint alleges that those defendants (collectively "appellees"), in violation of New York law, aided and abetted fraud and breach of fiduciary duty by Refco Inc. and its consolidated entities (collectively "Refco"), the brokerage and financial services firm that entered bankruptcy in 2005, and whose demise led to the bankruptcies of SPhinX and its investment manager, PlusFunds Group, Inc. ("PlusFunds"). The district court granted appellees' motions pursuant to Fed.R.Civ.P. 12(b)(6) to dismiss the claims against them for failure to state a claim on which relief can be granted, holding that the allegations in the Amended Complaint were insufficient to plead, inter alia, the requisite knowledge by appellees of Refco's wrongdoing. On appeal, plaintiffs challenge this ruling; alternatively, they argue that they should have been allowed to file a further amended complaint to cure any pleading deficiencies. For the reasons that follow, we conclude that plaintiffs' contentions lack merit, and we therefore affirm the judgment in favor of appellees.

I. BACKGROUND

The present action was commenced in New York State Supreme Court in 2008 against appellees and numerous other defendants, seeking more than $263 million in compensatory and punitive damages in connection with losses suffered by plaintiffs following the bankruptcy of Refco. The action was removed to federal court by several defendants pursuant to 28 U.S.C. §§ 1334(b) and 1452(a) on the ground that it was related to the pending bankruptcy cases of SPhinX, PlusFunds, and Refco. In the district court, plaintiffs filed their Amended Complaint.

A. The Amended Complaint

The Amended Complaint's factual allegations with respect to the claims asserted against appellees, which we take as true on appeal from the Rule 12(b)(6) dismissal, see, e.g., DiFolco v. MSNBC Cable L.L.C., 622 F.3d 104, 110-11 (2d Cir.2010), may be summarized as follows.

1. SPhinX and Refco

The SPhinX Funds, created by PlusFunds beginning in the spring of 2002, were a family of Cayman Islands-based hedge funds, offering ring investment portfolios that corresponded to the investment strategies represented in the S & P Hedge Fund Index. (See Amended Complaint ¶ 102.) The name "SPhinX" was "deriv[ed] ... from the S & P Hedge Fund Index." (Id. ¶ 100 (emphases in original).) To manage the operations of SPhinX (which had no employees or physical facilities), PlusFunds, in the summer of 2002, retained a professional service provider, Derivative Portfolio Management ("DPM"), whose co-owner and chief executive officer ("CEO") became a SPhinX board member. DPM was to perform all services necessary for the administration of the SPhinX Funds.

SPhinX used "segregated portfolio companies" organized under Cayman Islands law, allowing it to protect the assets of each of its various portfolios — which represented different trading strategies — from creditors of the other SPhinX portfolios or of SPhinX's custodian and prime broker. (Amended Complaint ¶ 3.) One of SPhinX's segregated portfolio companies was SPhinX Managed Futures Fund SPC ("SMFF"), which traded in futures and commodities.

In 2002, Refco was a leader in the financial services industry, providing execution and clearing services for exchange-traded derivatives and brokerage services in fixed income and foreign exchange markets. Prior to an initial public offering ("IPO") of stock by Refco Inc. in 2005, Refco Group Ltd. LLC ("RGL") was the parent holding company for the various Refco entities; after the IPO, Refco Inc. was the holding company and was the corporate parent of RGL. Defendant Phillip R. Bennett was the chairman, president, and CEO of RGL; he was also an owner (and after August 2004 the sole owner) of defendant Refco Group Holdings Inc. ("RGHI"), which, until August 2004, owned 90 percent of RGL. RGHI's primary asset was its shares of RGL or, after the IPO, shares of Refco Inc. After the IPO, Bennett was chairman, president, and CEO of Refco Inc. Although RGHI was related to RGL and Refco Inc., RGHI was not considered a part of Refco, was not consolidated with it for financial reporting purposes, and is not among the entities referred to in the Amended Complaint (or in this opinion) as within the term "Refco."

Refco's three primary operating subsidiaries included Refco LLC — a Delaware company regulated by the Commodity Futures Trading Commission and the Securities Exchange Commission — and Refco Capital Markets, Ltd. ("RCM"), a Bermuda company that was unregulated. In the fall of 2002, SMFF opened customer-segregated accounts for each of its portfolios with Refco LLC. Refco LLC became SMFF's exclusive broker worldwide, providing execution, clearing, and margin services in connection with futures and commodities trading activities.

2. The Refco Fraud and SMFF's Loss

The Amended Complaint alleges that in 1997 Refco, despite its apparent success, had begun to suffer hundreds of millions of dollars in trading losses. These, notwithstanding Refco's public representations that its business "did not involve proprietary trading" (Amended Complaint ¶ 159(q) (emphases added)), included losses from its own trading (see, e.g., id. ¶¶ 220, 223). They also included losses from trading by its customers to whom Refco had extended credit and from whom it did not receive reimbursement. In order to conceal the losses and fund its operations, Refco diverted customer assets, including cash from SMFF accounts, and "had no ability and no intention of returning [those] customer funds" (id. ¶ 205).

The account agreement between SPhinX and Refco required that Refco LLC maintain the assets sent by SMFF in regulated, customer-segregated accounts. Refco was well aware of the significance of that requirement, as Refco itself promoted the SPhinX Funds and created several funds that invested in SPhinX. (See id. ¶ 200; see also id. ¶ 41 ("Bennett was an investor in SPhinX").) Notwithstanding the account agreement, and notwithstanding the account-segregation promises in the marketing materials used by SPhinX to attract investors, DPM, at Refco's request, authorized Refco "to transfer SMFF's cash and other assets" that had been deposited with Refco LLC "interchangeably among Refco entities without regard to segregation or protection of assets" (id. ¶ 174). As a result of that authorization, "hundreds of millions of dollars of SMFF's excess cash was [sic] transferred on a regular basis from Refco LLC, [the] regulated entity where SPhinX's assets were maintained in regulated, customer-segregated accounts, to non-customer-segregated, commingled accounts at [RCM], Refco LLC's unregulated ... affiliate." (Id. ¶ 6.) Thus, "[m]ore than 70 percent of SMFF's cash was held at RCM," despite there being "no bona fide business reason for allowing cash to be maintained at RCM. The movement of SMFF's excess cash to RCM subjected the cash to risk of loss in the event of insolvency and resulted in the commingling of SMFF's cash in RCM's account, but offered no offsetting advantage to SPhinX." (Id. ¶ 197; see also id. ¶¶ 240-245.)

The Amended Complaint alleges that "[o]nce SMFF's cash was moved to RCM, Refco14 related parties and conspirators" — including some SPhinX and PlusFunds officials — "used RCM's customer assets for their own benefit in fraudulent activities designed to conceal Refco losses, bolster Refco's financial statements and enrich individuals." (Amended Complaint ¶ 7; see id. ¶¶ 38-40, 178-179; see also id. ¶ 242 ("Refco simply took the money and property entrusted to RCM by its customers, including SPhinX and PlusFunds, and sent the funds to other Refco entities."); id. ¶ 243 ("The diverted RCM customer assets were used by various Refco affiliates that would not have been able to sustain their operations without RCM customer funds.").)

In October 2005, "the Refco fraud collapsed when Refco announced that it had `discovered' a $430 million receivable owed to it by an entity controlled by Bennett ([defined as] the RGHI Receivable) and that Refco's financial statements could no longer be relied upon." (Amended Complaint ¶ 11; see id. ¶ 7.) Refco filed for bankruptcy. (See id. ¶ 11.) As a result of the wrongful transfers of SPhinX cash from Refco LLC where it had been held in segregated-portfolio and customer-segregated accounts that were protected against insolvency of other customers or of Refco, to RCM where it was commingled and unprotected, SPhinX lost approximately $263 million and suffered other damages as well.

3. Plaintiffs' Claims Against Appellees

In addition to diverting client money to its own use in order to fund its operations, Refco took steps to conceal its losses from clients and the authorities. First, instead of writing off Refco's trading losses and its uncollectible debts from customers, Bennett caused those losses and bad debts to be transferred from Refco to his own company, RGHI, in order to remove them from Refco's financial statements (see Amended Complaint ¶¶ 222-223), and "[a] corresponding receivable of several hundred million dollars from RGHI (the `RGHI Receivable') was recorded on Refco's books" (id. ¶ 7). Thus, "Refco `converted' hundreds of millions of dollars in customer and proprietary trading losses into what appeared to be a legitimate and collectible receivable from RGHI." (Id. ¶ 223.)

Second, to avoid having its audited reports and financial statements reveal that one of its supposed assets was a massive receivable from RGHI — a related entity whose principal asset was shares of Refco Inc. or its predecessor, RGL (see Amended Complaint ¶ 223) — Refco entered into allegedly "sham" loan arrangements (id. ¶ 230) with a number of its customers who were unaffiliated with Refco (see id. ¶¶ 231, 8), including defendants Liberty Corner Capital Strategies LLC and its owner William T. Pigott (collectively "Liberty Corner") and defendants Ingram Micro Inc. and its subsidiary CIM Ventures Inc. (collectively "Ingram Micro"), to replace the RGHI Receivable. Refco "timed" these transactions "to straddle Refco's reporting and audit periods." (Id. ¶ 230.) Characterizing the allegedly sham loan transactions by Refco with appellees (and with several other defendants (see id. ¶ 79)) as "`round-trip loan' transactions," or "RTLs" (id. ¶ 8) and calling appellees and those defendants "RTL Participants" (id. ¶ 79), the Amended Complaint alleges that numerous multi-step RTLs occurred as follows. First,

(Id. ¶ 233.) Second,

(Id. ¶ 234.) Plaintiffs allege that the RTL Participants "agreed to serve, for a fee, as conduits in the RTLs." (Id. ¶ 231; see, e.g., id. ¶ 235 ("For agreeing to participate in the RTLs and conceal the RGHI Receivable, the RTL Participants received payment of the `spread' between the interest rates of the two `loans.'").) Plaintiffs also allege that the RTL Participants were motivated to enter into the RTLs in order to further their business relationships with Refco and/or personal relationships with Bennett. (See id. ¶ 1015.)

The Amended Complaint contains several paragraphs — discussed in greater detail in Part II.C.1. below — alleging that the RTL Participants "knew and/or consciously avoided knowing" (e.g., Amended Complaint ¶¶ 236, 1020(d), 1021) that the RTL loan transactions were "designed to" (id. ¶¶ 7, 1017, 1021), and did, assist Refco to issue fraudulent financial statements and conceal its insolvency. Plaintiffs asserted claims against appellees and the other RTL Participants under New York law for aiding and abetting fraud (see Amended Complaint ¶¶ 1320-1327 (Count XXIII)), and aiding and abetting breach of fiduciary duty (see id. ¶¶ 1328-1333 (Count XXIV)).

B. The District Court Decision

Liberty Corner and Ingram Micro moved pursuant to Fed.R.Civ.P. 12(b)(6) to dismiss the claims against them, arguing principally that the Amended Complaint failed to state a claim on which relief can be granted and failed to plead fraud with the particularity required by Fed.R.Civ.P. 9(b). The district court referred the motions to a special master, who recommended that they be granted.

Noting that in order "[t]o be liable on an aiding and abetting claim, [a] defendant must have had knowledge of the underlying wrongful conduct," Report and Recommendation of the Special Master on Motions To Dismiss Brought by the [] Ingram Defendants and the Liberty Corner Defendants, dated February 10, 2012, at 8, the special master stated that the Amended Complaint sufficiently alleged underlying wrongs; but he concluded that it did not sufficiently allege the requisite knowledge on the part of Liberty Corner or Ingram Micro. The special master reasoned in part, citing several cases, that "the kind of back-to-back loans at issue in this case are not per se fraudulent." Id. at 10.

With respect to the claims for aiding and abetting breach of fiduciary duty, the special master concluded that the Amended Complaint did not allege that Liberty Corner or Ingram Micro knew of Refco's fiduciary relationship with SPhinX and PlusFunds, much less knew of any breach of Refco's fiduciary duties to them. The special master also concluded that the Amended Complaint did not sufficiently allege that Liberty Corner or Ingram Micro provided substantial assistance to the perpetrators of the Refco fraud or the fiduciaries who violated their duties. (See id. at 13-15, 17-19.)

In a Memorandum Order dated July 28, 2012, see In re Refco Inc. Securities Litigation, 2012 WL 3126834 (S.D.N.Y. July 30, 2012) ("July 2012 Order"), the district court reviewed the motions and the Amended Complaint de novo and accepted the special master's recommendation that plaintiffs' claims against Liberty Corner and Ingram Micro be dismissed, although the court's analysis differed somewhat from that of the special master with respect to the claims of aiding and abetting fraud. The court noted that

July 2012 Order, 2012 WL 3126834 at *1 (internal quotation marks omitted).

While distinguishing the authorities relied on by the special master in his conclusion that back-to-back loans were not per se fraudulent, the district court agree[d] with the Special Master's ultimate recommendation to dismiss the claims premised on the round-trip loans because the plaintiffs here failed to allege facts that plausibly suggest that the defendants here had knowledge of the fraud they are accused of aiding and abetting. ... [T]he most the [p]laintiffs have alleged is that the [d]efendants were aware of some red flags that might have raised some vague suspicion that Refco might have been up to something in these RTL transactions.... Vague suspicions are far removed from reckless disregard, let alone actual knowledge.

Id. at *2-*4 (emphases added). Accordingly, the July 2012 Order dismissed, with prejudice, plaintiffs' claims against Liberty Corner and Ingram Micro. Plaintiffs filed notices of appeal in August 2012.

C. The Partial Final Judgment as to Appellees

The district court subsequently amended the July 2012 Order to instruct that a partial final judgment dismissing the claims against Liberty Corner and Ingram Micro be entered "in accordance with Rule 54(b), the Court having determined there is no just reason for delay." Order dated October 27, 2012 ("October 2012 Order"). At oral argument of the appeal in May 2013, however, we questioned, inter alia, the basis for the district court's entry of a partial final judgment in favor of appellees, noting that many of the claims asserted against other defendants remained pending — including, apparently, claims against the other defendants named in paragraph 79 of the Amended Complaint who, like Liberty Corner and Ingram Micro, were sued as RTL Participants in Counts XXIII and XXIV of the Amended Complaint.

Accordingly, pursuant to the procedure set out in United States v. Jacobson, 15 F.3d 19, 22 (2d Cir.1994), we entered an order, see Krys v. Pigott, 531 Fed.Appx. 44, 47 (2d Cir.2013), remanding for the district court to supplement the record with a reasoned, even if brief, explanation for why an immediate appeal of the dismissal of the claims against Liberty Corner and Ingram Micro is appropriate. See generally Curtiss-Wright Corp. v. General Electric Co., 446 U.S. 1, 8, 100 S.Ct. 1460, 64 L.Ed.2d 1 (1980) (in light of "the historic federal policy against piecemeal appeals," "[n]ot all final judgments on individual claims should be immediately appealable, even if they are in some sense separable from the remaining unresolved claims" (internal quotation marks omitted)); id. at 10, 100 S.Ct. 1460 (although "the decision to certify" pursuant to Fed. R.Civ.P. 54(b) is "left to the sound judicial discretion of the district court," that decision is "reviewable by the Court of Appeals" for abuse of discretion (internal quotation marks omitted)); Harriscom Svenska AB v. Harris Corp., 947 F.2d 627, 629-30 (2d Cir.1991) (a statement by the district court only of its conclusion that "there is no just reason for delay," unaccompanied by any explanation of the assessments that led to that conclusion, is insufficient to permit this Court to conduct abuse-of-discretion review).

In an order dated August 22, 2013 ("August 2013 Order"), the district court supplemented the record with an explanation for its October 2012 Order directing the entry of a partial final judgment on the claims against Liberty Corner and Ingram Micro. The court stated, inter alia, that although similar RTL claims "w[ere] also brought against certain other defendants," they are no longer being litigated against the other defendants, August 2013 Order at 2 n. 1; and that there is virtually no overlap between the issues in this appeal and those likely to be presented on any potential appeal relating to the issues remaining to be resolved in the district court, see id. at 2-3.

This appeal was reinstated on September 19, 2013. Given the findings that no other RTL claims in this matter are likely to be the subject of future appeals, and that there is no likely overlap between the present appeal involving these four appellees and any potential appeals by the several dozen other defendants named in the Amended Complaint, we conclude that the district court's reasoned explanation supports the exercise of its discretion to enter the Rule 54(b) certification.

II. DISCUSSION

On appeal, plaintiffs contend, inter alia, that their Amended Complaint is sufficient to plead appellees' knowledge of Refco's fraud and breach of fiduciary duty. For substantially the reasons stated by the district court, we disagree. We also reject plaintiffs' alternative request for permission to file a further amended complaint.

A. Aiding and Abetting Fraud or Breach of Fiduciary Duty

To establish liability for aiding and abetting fraud under New York law, "the plaintiffs must show (1) the existence of a fraud; (2) [the] defendant's knowledge of the fraud; and (3) that the defendant provided substantial assistance to advance the fraud's commission." Lerner v. Fleet Bank, N.A., 459 F.3d 273, 292 (2d Cir. 2006) ("Lerner") (internal quotation marks omitted). "[A]ctual knowledge is required to impose liability on an aider and abettor under New York law." Id. (internal quotation marks omitted). A company's "reject[ion of certain] transactions on the basis that they were potential vehicles for fraud" does not constitute a "factual basis for the assertion that [the company's] officials actually knew that the fraud was, in fact, occurring." Id. at 293 (internal quotation marks omitted); see, e.g., Oster v. Kirschner, 77 A.D.3d 51, 55, 905 N.Y.S.2d 69, 72 (1st Dep't 2010) ("A plaintiff alleging an aiding-and-abetting fraud claim must allege the existence of the underlying fraud, actual knowledge, and substantial assistance." (emphasis added)).

To be "distinguished" from actual knowledge is "constructive knowledge," which is "[k]nowledge that one using reasonable care or diligence should have, and therefore that is attributed by law to a given person." BLACK'S LAW DICTIONARY 950 (9th ed.2009) (emphasis added). However, under New York law, a complaint adequately alleges the knowledge element of an aiding and abetting claim when it pleads "not ... constructive knowledge, but actual knowledge of the fraud as discerned from the surrounding circumstances." Oster v. Kirschner, 77 A.D.3d at 56, 905 N.Y.S.2d at 72. A failure to allege sufficient facts to support the inference that the alleged aider and abettor had actual knowledge of the fraudulent scheme warrants dismissal of the aiding and abetting claim at the pleading stage. See, e.g., Lerner, 459 F.3d at 292-93; National Westminster Bank USA v. Weksel, 124 A.D.2d 144, 149, 511 N.Y.S.2d 626, 630 (1st Dep't) ("National Westminster"), lv. denied, 70 N.Y.2d 604, 519 N.Y.S.2d 1027, 513 N.E.2d 1307 (1987).

There must also be a "nexus between the primary fraud, [the alleged aider and abettor's] knowledge of the fraud[,] and what [the alleged aider and abettor] did with the intention of advancing the fraud's commission...." Franco v. English, 210 A.D.2d 630, 633, 620 N.Y.S.2d 156, 159 (3d Dep't 1994), abrogated on other grounds, Eurycleia Partners, LP v. Seward & Kissel, LLP, 12 N.Y.3d 553, 561, 883 N.Y.S.2d 147, 151-52, 910 N.E.2d 976 (2009).

National Westminster, 124 A.D.2d at 149, 511 N.Y.S.2d at 630.

As we have noted in an appeal in a related case, "New York law [also] recognizes a cause of action for aiding and abetting another's breach of fiduciary duty," Krys v. Butt, 486 Fed.Appx. 153, 157 (2d Cir.2012) (summary order). For that cause of action too, the plaintiff must establish the aider and abettor's actual knowledge:

Krys v. Butt, 486 Fed.Appx. at 157 (other internal quotation marks omitted).

B. Rule 12(b)(6) Standards

A district court's ruling under Rule 12(b)(6) that the complaint fails to state a claim on which relief can be granted against a given defendant is reviewed de novo. See, e.g., Taylor v. Vermont Department of Education, 313 F.3d 768, 776 (2d Cir.2002). In reviewing the sufficiency of a complaint, we accept only its factual allegations, and the reasonable inferences that can be drawn therefrom, as true. See, e.g., Rothstein v. UBS AG, 708 F.3d 82, 94 (2d Cir.2013). "[W]e `are not bound to accept as true a legal conclusion couched as a factual allegation,'" Ashcroft v. Iqbal, 556 U.S. 662, 678, 129 S.Ct. 1937, 173 L.Ed.2d 868 (2009) ("Iqbal") (quoting Bell Atlantic Corp. v. Twombly, 550 U.S. 544, 555, 127 S.Ct. 1955, 167 L.Ed.2d 929 (2007) ("Twombly")); nor are we required to accept as true allegations that are wholly conclusory, see, e.g., Iqbal, 556 U.S. at 678-79, 681, 686, 129 S.Ct. 1937.

"To survive a motion to dismiss, a complaint must contain sufficient factual matter, accepted as true, to `state a claim to relief that is plausible on its face.'" Iqbal, 556 U.S. at 678, 129 S.Ct. 1937 (quoting Twombly, 550 U.S. at 570, 127 S.Ct. 1955).

Iqbal, 556 U.S. at 678, 129 S.Ct. 1937 (other internal quotation marks omitted); see, e.g., Twombly, 550 U.S. at 555, 127 S.Ct. 1955 ("[f]actual allegations must be enough to raise a right to relief above the speculative level," i.e., the complaint "must contain something ... more ... than a statement of facts that merely creates a suspicion of a legally cognizable right of action" (internal quotation marks omitted)).

In asserting claims of fraud — including claims for aiding and abetting fraud or a breach of fiduciary duty that involves fraud — a complaint is required to plead the circumstances that allegedly constitute fraud "with particularity," Fed. R.Civ.P. 9(b); see, e.g., Lerner, 459 F.3d at 293 (aiding and abetting fraud); Kennedy v. Venrock Associates, 348 F.3d 584, 593 (7th Cir.2003) (breach of fiduciary duty involving fraud), cert. denied, 541 U.S. 975, 124 S.Ct. 1889, 158 L.Ed.2d 470 (2004). Although the Rule permits a plaintiff to plead knowledge "generally," Fed.R.Civ.P. 9(b), "`generally'" is merely "a relative term" that allows knowledge to be pleaded with less particularity than is required for the pleading of fraud, Iqbal, 556 U.S. at 686, 129 S.Ct. 1937; "`generally'" is not the equivalent of conclusorily, id. "[T]he Federal Rules do not require courts to credit a complaint's conclusory statements without reference to its factual context." Id. Thus, "[a]lthough Rule 9(b) permits knowledge to be averred generally, plaintiffs must 3 still plead the events which they claim give rise to an inference of knowledge." Devaney v. Chester, 813 F.2d 566, 568 (2d Cir.1987).

C. The Amended Complaint's Allegations as to Appellees' Knowledge

The Amended Complaint contains many paragraphs that purport to allege that appellees "knew and/or consciously avoided knowing" (e.g., Amended Complaint ¶¶ 236, 1020(d), 1021) that they were aiding and abetting fraud and breach of fiduciary duty by Refco by reason of their participation in "`round-trip loan' transactions" (id. ¶ 8) that were "designed to conceal Refco losses, bolster Refco's financial statements and enrich individuals" (id. ¶ 7), "designed to conceal a significant related-party receivable at the end of Refco's reporting and audit periods" (id. ¶ 1021), and "designed to ... conceal the nature and extent of the RGHI Receivable" (id. ¶ 1017). The Amended Complaint asserts that the RTL Participants "directly and knowingly participated in and facilitated the RTLs, knowingly helping Refco manipulate its financial statements and conceal its losses." (Id. ¶ 1014.)

1. What Appellees Are Alleged To Have Known

As to appellees' knowledge of particular facts, the Amended Complaint alleges as follows:

(Id. ¶ 1020; see also id. ¶ 1018 ("each of the RTL Participants knew[] the RTLs did not serve any legitimate business purpose for RCM, RGL, or any other Refco affiliate" (emphasis added)).) The Amended Complaint also alleges that "[t]he RTL Participants had existing business with Refco and/or a personal relationship with Bennett and each had an incentive to further their relationship with Bennett and Refco by agreeing to serve as conduits in the RTLs" (id. ¶ 1015).

However, these allegations that appellees knew of and agreed to participate in Refco's fraud and breach of fiduciary duty are conclusory because the Amended Complaint lacks any allegation that appellees had actual knowledge of certain key facts, itemized in Part II.C.2. below, that could give rise to a reasonable inference of such knowledge and agreement.

2. Facts That Appellees Are Not Alleged To Have Known

The facts as to which the Amended Complaint fails to allege knowledge on the part of the RTL Participants include principally the fact that Refco was insolvent — and the facts relating to its insolvency — and the fact that RGHI would use the proceeds of the loans from appellees to pay down the RGHI Receivable on Refco's books. For example, the Amended Complaint contains the following allegations of fact, which we must accept here as true; but it contains no allegation that Ingram Micro or Liberty Corner had any knowledge of:

3. The Effect of the Missing Allegations of Knowledge

In the absence of any allegations in the Amended Complaint that appellees had knowledge of the facts itemized in Part II.C.2., plaintiffs' assertion that appellees had sufficient knowledge to warrant the imposition of aider-and-abettor liability was properly rejected. Every allegation that an RTL Participant "agreed" (Amended Complaint ¶ 231; see id. ¶¶ 1015, 1019) to serve as a "conduit" for the RTLs (id. ¶¶ 1008-1009, 1012, 1019; see id. ¶¶ 231, 1015) is conclusory because, as the Amended Complaint itself recognizes, there was no "round trip" until RGHI paid down the RGHI Receivable: "The RTL was completed when RGHI used the `loan' to pay down the debt it owed Refco" (id. ¶ 233). If RGHI had not used the money received from an RTL Participant to pay down the receivable at Refco, there would not have been a "round trip"; but nowhere does the Amended Complaint allege that any RTL Participant knew or even had reason to suspect that RGHI would in fact use the loan proceeds in that way. The conclusory allegation that a defendant "agreed" to participate in a transaction, an essential part of which (a) must take place between two other entities and (b) is not alleged to have been known by the defendant, need not be accepted as true. The allegation that appellees "knew[] the RTLs did not serve any legitimate business purpose for RCM, RGL or any other Refco affiliate" (id. ¶ 1018) is similarly conclusory, given the absence of any allegation that the RTL Participants knew or had reason to know that RGHI would take the final step that completed the "round trip" circle.

To the extent that plaintiffs assert that appellees "should have" known, from the fact that their loan transactions straddled the ends of Refco's reporting periods (see Amended Complaint ¶¶ 236, 1020(a)), that the RTLs were being used to conceal Refco's fraud and financial condition (e.g., id. ¶ 1333 (the RTL Participants "should have reasonably foreseen that RCM's customers, whose assets were diverted to fund the Refco fraud, would be unable to recover their assets as a result of the Refco fraud")), that contention invokes the standard for constructive knowledge, rather than actual knowledge. Constructive knowledge, as discussed in Part II.A. above, is insufficient to sustain a claim for aiding and abetting fraud or breach of fiduciary duty. In any event, nothing in the Amended Complaint raises any reasonable inference that the RTL Participants' knowledge that the short-term loans they received from Refco and made to RGHI straddled the Refco audit and reporting dates caused them to know of, or gave them reason to suspect, the misuses of customer funds.

Nor is the Amended Complaint's failure to allege that appellees had any knowledge of the facts set out in Part II.C.2. rescued by its compound allegations that appellees "knew and/or consciously avoided knowing" (e.g., Amended Complaint ¶¶ 236, 1020(d), 1021 (emphasis added)) various facts. As we noted in Krys v. Butt, it is unclear whether a claim of aiding and abetting under New York law is sustainable on a basis not of actual knowledge but of conscious avoidance of knowledge. See 486 Fed.Appx. at 157 n. 5. Here, as in Butt, however, we need not reach that question because the conscious-avoidance concept is used where a "defendant was aware of a high probability of the [relevant] fact ... and consciously avoided confirming that fact," United States v. Ebbers, 458 F.3d 110, 124 (2d Cir.2006) (internal quotation marks omitted), cert. denied, 549 U.S. 1274, 127 S.Ct. 1483, 167 L.Ed.2d 244 (2007). Most of the facts that plaintiffs deem so "suspicious" that the RTL Participants should be found to have known, "and/or" to have consciously avoided knowing, that Refco was engaged in fraud and breach of fiduciary duty (e.g., Amended Complaint ¶ 1020(c)) are instead quite commonplace. They include the fact that each loan "was for a specific ... amount" (id.); that the "dollar amount" of the loan was "large" (id.); that the "dollar amounts" were "round[ ]figure[s]" (id. ¶ 236); that the loan amounts "increas[ed] over time" (id. ¶ 1020(c)); that the loans were short-term (see id. ¶¶ 236, 1020(a)); and that the loan "documentation ... identified the repayment date" (id. ¶ 236). Each of these facts is entirely consistent with normal, lawful business practices.

The Amended Complaint's allegation that Liberty Corner or any other RTL Participant, by demanding that some of its loans to RGHI be guaranteed by RGL (see Amended Complaint ¶¶ 236, 1009, 1020(d)), demonstrated knowledge or conscious avoidance of knowledge "that the loan transactions were guaranteed by" an entity that was part of an "insolvent" enterprise (id. ¶ 1020(d)), is both conclusory and implausible. It is conclusory because there is no allegation that RTL Participants were aware of or had reason to suspect Refco's insolvency. It is entirely implausible for at least two reasons. First, Liberty Corner, Ingram Micro, and the other RTL Participants were customers of Refco; and as the Amended Complaint itself notes, if customers had known of Refco's true financial condition they would have ceased to do business with Refco (see, e.g., id. ¶ 220). Second, plaintiffs' contention is implausible because a guarantee from an insolvent guarantor would be, in whole or in part, worthless. The only reasonable inference to be drawn from demands by Liberty Corner or Ingram Micro for loan guarantees from Refco is that, although having questions about RGHI, they believed Refco was financially sound.

For the same reasons, the allegation that Liberty Corner and Ingram Micro knowingly participated in Refco's fraud and breach of fiduciary duty because they wished "to further their relationship with Bennett and Refco" (Amended Complaint ¶ 1015) does not provide a basis for any reasonable inference that Liberty Corner or Ingram Micro had actual knowledge of Refco's fraud or breach of fiduciary duty. The notion that a customer wishes to participate in "a massive fraud" (id. ¶ 1) in order to further its relationship with a company it knows to be insolvent is not plausible.

We note that plaintiffs allege that Ingram Micro itself expressed "concerns about the[] propriety" of the RTLs, leading it to decline to participate in more than two. (Amended Complaint ¶ 1012.) The Amended Complaint quotes an Ingram Micro email to a Refco executive stating that "`the Enron debacle is putting pressure on the SEC to increase the level of financial disclosure by large companies like [Ingram Micro].'" (Id.) In light of the absence of allegations as to Ingram Micro's knowledge of any of the facts itemized in Part II.C.2. above, however, such concerns are inadequate to create an inference that Ingram Micro had knowledge of actual wrongdoing. As discussed, a company's refusal to enter into proffered "transactions on the basis that they were potential vehicles for fraud" is not a "factual basis for the assertion that [company] officials actually knew that the fraud was, in fact, occurring," Lerner, 459 F.3d at 293 (internal quotation marks omitted).

We also note that the Amended Complaint's allegations as to the participation in RTLs by Ingram Micro cannot support plaintiffs' claims of aiding and abetting the alleged breach of Refco's fiduciary duties to SPhinX and PlusFunds. Ingram Micro is not alleged to have participated in any RTLs after 2001. SPhinX did not come into existence until 2002.

Finally, the Amended Complaint's allegations that the RTL Participants participated in transactions that they knew were "designed to ... conceal the nature and extent of the RGHI Receivable" (Amended Complaint ¶ 1017; see id. ¶ 1020(c) ("the RTL Participants who participated in multiple RTLs ... were aware that the annual aggregate amount of the RTLs was increasing over time")) are implausible in light of factual allegations in the pleading itself. As to the "extent" of the RGHI Receivable, for most of the years in which there were RTLs the Amended Complaint alleges that there were multiple Refco customers who were RTL Participants (see id. ¶ 1016); further, the Amended Complaint alleges participation in many additional RTLs — "to conceal the RGHI Receivable" (id. ¶ 962) — by another entity, a Refco part-owner (see id. ¶¶ 237-238, 953, 957, 962-965). But the Amended Complaint alleges that the RTL Participants had "no apparent relation to Refco, Bennett or RGHI" (id. ¶ 233); it does not allege that any RTL Participant collaborated or communicated with any other RTL Participant or with the Refco part-owner that also participated in RTLs; and it alleges that Refco entered into "a number of RTLs" precisely "[i]n order to conceal the size of the RGHI Receivable" (id. ¶ 232). Thus, the Amended Complaint provides no factual basis for its conclusory allegation that any Refco customer knew the size of the RGHI Receivable.

More importantly, there is no basis in the Amended Complaint for inferring that the RTL Participants knew the RGHI Receivable's fraudulent "nature." Although the Amended Complaint alleges that the RTLs were "designed to conceal Refco losses" and "bolster Refco's financial statements" (Amended Complaint ¶ 7; see also id. ¶ 1014), such concealment had already been achieved, according to the Amended Complaint, by the very creation of the RGHI Receivable (see, e.g., id. ¶¶ 202-203 (Refco's "trading losses," its "operating expenses" that were funded through the misappropriation of "assets belonging to customers of RCM, including SMFF," and its "true financial condition" were "conceal[ed]" by "recording them as receivables owed by RGHI")). The Amended Complaint alleges that the receivable "from RGHI — a related party," "appeared to be a legitimate and collectible receivable." (Id. ¶ 223.)

In sum, the allegations of the Amended Complaint are not sufficient to give rise to any reasonable inference that Liberty Corner or Ingram Micro knew or even suspected, inter alia, that Refco was insolvent, that RGHI's primary asset was stock in the insolvent Refco, that RGHI would use the money borrowed from them to pay down the receivable at Refco, or that the RGHI Receivable, which "appeared to be... legitimate and collectible" (Amended Complaint ¶ 223), was not.

We conclude that the claims against appellees were properly dismissed for failure of the Amended Complaint to contain sufficient allegations that appellees had actual knowledge of Refco's fraud and breach of fiduciary duty.

D. Plaintiffs' Request To Amend the Amended ended Complaint

Plaintiffs ask us, if we conclude that the Amended Complaint is insufficient, to rule that the district court was required to grant their request, embedded in their supplemental memorandum in opposition to appellees' motions to dismiss, to file a second amended complaint to augment their allegations based on deposition testimony given by defendant Santo Maggio, a former Refco executive. We decline to do so.

A district court's denial of a request for leave to amend a pleading is reviewed for abuse of discretion. See, e.g., Foman v. Davis, 371 U.S. 178, 182, 83 S.Ct. 227, 9 L.Ed.2d 222 (1962); Starr v. Sony BMG Music Entertainment, 592 F.3d 314, 321 (2d Cir.2010), cert. denied, ___ U.S. ___, 131 S.Ct. 901, 178 L.Ed.2d 803 (2011). "An abuse of discretion may consist of an erroneous view of the law, a clearly erroneous assessment of the facts, or a decision that cannot be located within the range of permissible decisions." Anderson News, L.L.C. v. American Media, Inc., 680 F.3d 162, 185 (2d Cir.2012), cert. denied, ___ U.S. ___, 133 S.Ct. 846, 184 L.Ed.2d 655 (2013); see, e.g., Sims v. Blot, 534 F.3d 117, 132 (2d Cir.2008). Leave to amend may properly be denied if the amendment would be futile, see, e.g., AEP Energy Services Gas Holding Co. v. Bank of America, N.A., 626 F.3d 699, 726 (2d Cir.2010), as when the proposed new pleading fails to state a claim on which relief can be granted, see, e.g., Ricciuti v. N.Y.C. Transit Authority, 941 F.2d 119, 123 (2d Cir.1991). "The adequacy of a proposed amended complaint to state a claim is to be judged by the same standards as those governing the adequacy of a filed pleading." Anderson News, L.L.C. v. American Media, Inc., 680 F.3d at 185.

We see no abuse of discretion in the district court's dismissal of the Amended Complaint's claims against appellees without leave to amend. Maggio was, at pertinent times, president of RCM and executive vice president of RGL (see Deposition of Santo C. Maggio ("Maggio Dep.") at 26); under the direction of Bennett, he supervised Refco's RTL process (see, e.g., id. at 107-08, 174). Although Maggio ventured an opinion that anyone with "half a brain" would know that the RTLs had no legitimate purpose for Refco (id. at 174), he also testified that in fact Refco made efforts to conceal, not only from the public at large but also from the customers who entered into the RTL transactions, both the purpose of the RTL transactions and the very existence of the RGHI Receivable.

For example, Maggio testified that when he was asked by customers about the purpose of the RTL transactions he simply "lied to them" (Maggio Dep. 927-28). He did so precisely because "they were Refco's customers," and Refco "didn't want a Refco customer to know about Refco's financial problems." (Id. at 928.) If the existence of and truth about the RGHI Receivable had been known, Refco "could have lost [its] customers." (Id. at 105-06; see, e.g., id. at 928-30.) "So," Maggio testified, "[i]f a customer asked," the customer was "lied to" (id. at 612):

(Id. at 611-12 (internal quotation marks omitted).) Maggio testified that "Refco wanted to mask the existence of the RGHI receivable from the customers who were engaging in the [RTL] transactions...." (Id. at 612 (emphases added).)

Helping to mask the existence of the RGHI Receivable were two key facts about the RTL transactions: (1) the "last part" of any RTL transaction was "[t]he actual application by RGHI to the consolidated entities to get rid of the [RGHI] receivable" (Maggio Dep. 643; see also id. at 642-43 (it is "only at that point" that "the receivable disappears" and "there is financial statement manipulation")), and the RTL Participants "were not involved in that last part of the transaction" (id. at 643); and (2) the RTL transaction documents did not disclose that last step (see id. at 646-47). This perhaps explains why there is no allegation in the Amended Complaint that the RTL Participants knew RGHI would use the proceeds of the loans from the RTL Participants to pay down the RGHI Receivable.

In sum, Maggio explained, "[t]he idea was to keep all the customers in the dark about" the RGHI Receivable and the fraud. (Maggio Dep. 940-41.) Thus, the Maggio testimony helps to demonstrate why further amendment of the Amended Complaint could not cure the absence of factual allegations as to actual knowledge on the part of appellees sufficient to state a claim against them for aiding and abetting Refco's fraud and breach of fiduciary duty. We see no abuse of discretion in the district court's dismissal of those claims without leave to amend.

CONCLUSION

We have considered all of plaintiffs' arguments in support of their appeal and have found them to be without merit. The judgment of the district court is affirmed; the request for leave to amend the Amended Complaint is denied.