Lorraine GRACE, Individually and as Executrix of the Estate of Oliver R. Grace, Gerald I. White, As Executor of the Estate of Morgan H. Grace, and Gerald I. White, Trustee of the John E. Grace Trust, individually, and as stockholders of Briggs Leasing Corporation, suing on behalf of themselves and for the benefit of said corporation and for the class of all other stockholders of said corporation similarly situated, Plaintiffs-Appellants, v. Robert ROSENSTOCK, Edward Rosenstock, Briggs Leasing Corporation and Briggs Acquisition Corporation, Defendants, Robert Genser, Defendant-Appellee, Bank Leumi Trust Company Of New York, David Mack, Leo V. Berger, Apex Marine Corporation and Gary Holman, Proposed Defendants-Appellees, 228 F.3d 40

KEARSE, Circuit Judge: Plaintiffs Lorraine Grace et al., individually and representing former stockholders of defendant Briggs Leasing Corporation (“Briggs”), appeal from a judgment of the United States District Court for the Eastern District of New York, Robert M. Levy, Magistrate Judge, dismissing their amended complaint against defendant Robert Genser for alleged violations of § 10(

b) of the Securities Exchange Act of 1934 (“1934 Act”), 15 U.S.C. § 78j(b) (1994), Rule 10b-5 promulgated thereunder by the Securities and Exchange Commission (“SEC” or “Commission”), 17 C.F.R. § 240.10b-5 (1999), and New York law, in connection with the 1985 freeze-out merger of Briggs and defendant Briggs Acquisition Corp. (“BAC”). The district court, following a bench trial, dismissed the § 10(

b) and Rule 10b-5 claims on the ground that plaintiffs had failed as a matter of law to prove causation; the court dismissed plaintiffs’ state-law claims on the ground that appraisal under N.Y. Bus. Corp. L. § 623 (McKinney 1986) was their exclusive remedy. See 23 F.Supp.2d 326 (1998). On appeal, plaintiffs contend principally that they were not required to prove causation, and that if required to do so they met their burden.

Plaintiffs also challenge an interlocutory order denying their motion for leave to file a second-amended and supplemental complaint asserting new state-law claims against Genser and several proposed new defendants. For the reasons that follow, we affirm. I. BACKGROUND Prior to 1985, Briggs was a publicly-held auto-leasing company incorporated in New York. Defendants Robert Rosenstock (“Rosenstock”), his father Edward Rosen-stock, and Genser were officers and directors of Briggs and owned, respectively, approximately 64%, 3%, and 5% of its outstanding stock. Plaintiffs Lorraine Grace and Gerald I.

White, individually or as trustees or executors, are or represent former stockholders of Briggs who owned a total of 7,476 shares of Briggs, or approximately 1.25% of its outstanding stock. In early 1985, Rosenstock and Genser decided to take Briggs private in a freeze-out merger. Accordingly, they incorporated BAC, planning to merge BAC and Briggs, buy out Briggs’s minority shareholders, and make Briggs the surviving corporation. Rosenstock and Genser contributed all of their Briggs shares to BAC; Rosenstock also purchased his father’s shares and contributed those to BAC.

As a result, BAC owned 72% of the stock of Briggs, with Rosenstock and Genser owning all of the stock of BAC. Under New York’s Business Corporation Law (“BCL”), a merger may be authorized by the affirmative vote of two-thirds of the corporation’s shares. See N.Y. Bus. Corp. L. § 903(a)(2) (McKinney Supp.1999). In January 1985, Briggs sent its shareholders notice of, and a proxy statement for, a special meeting to vote on the proposed merger. The notice informed shareholders, inter alia, that (

a) because BAC owned 72% of Briggs’s shares and would vote those shares in favor of the merger, approval was assured irrespective of the votes of Briggs’s other shareholders; (

b) all minority shareholders would be forced to sell their shares to the surviving corporation for $1.50 per share; (

c) after the merger, the BAC shares would be converted to shares of Briggs; and (

d) Rosenstock and Genser, who owned all of the shares of BAC, would thus become the sole shareholders of Briggs. The proxy statement (“Briggs proxy statement” or “proxy statement”) also stated that dissenting shareholders would have the right under New York law to obtain an appraisal of, and payment for, the fair value of their shares. At the meeting, plaintiffs voted their shares against the merger, which was approved based on BAC’s vote. The merger was consummated on February 26, 1985, and Rosenstock and Genser became the sole holders of Briggs shares, owning approximately 93% and 7%, respectively. A.

The Present Action: The First Twelve Years Plaintiffs commenced the present action in June 1985 as an individual, derivative, and class action, requesting equitable and monetary relief against Briggs, BAC, Ro-senstock, Edward Rosenstock, and Genser; the amended complaint was filed a month later. Contemporaneously, plaintiffs sued in New York state court, asserting appraisal rights under BCL § 623; proceedings in that suit were stayed pending disposition of this action.

The amended complaint in the present action alleged principally that the Briggs proxy statement was false and misleading by reason of numerous material omissions and untrue statements, including vast understatements of the compensation paid by Briggs to Rosenstock, Edward Rosen-stock, and Genser; false representations that the reason for the merger was to relieve Rosenstock and Genser of their roles as personal guarantors of Briggs’s debt, when in fact the primary guarantor of such debt until January 1985 was Charlotte Rosenstock (Rosenstock’s mother) and the entrance of Rosenstock and Gen-ser as guarantors was spuriously devised to provide a facially valid reason for the merger; and the false valuation of Briggs’s real estate at $883,503, when in fact its value exceeded $2 million.

The amended complaint also alleged that the Briggs proxy statement failed to disclose, inter alia, that Rosenstock and Gen-ser pursued the merger as part of a scheme to divert the assets of Briggs to themselves; that they had plans to develop certain of Briggs’s real estate into an office-building-supermarket complex for their own benefit; that a third party had offered in excess of $1.5 million for Briggs’s real estate; that in January 1985 a third party had offered $3.00 per share for Briggs stock; and that Briggs assets had been diverted to defendants’ personal use to fund the individual defendants’ opulent lifestyles.

The amended complaint asserted that defendants’ conduct violated § 10(

b) and Rule 10b-5, as well as various provisions of New York common law and statutory law dealing principally with securities fraud and shareholders meetings. Invoking BCL § 623(k), which provides that generally an appraisal proceeding constitutes a dissenting shareholder’s exclusive remedy but makes an exception where the corporate action at issue was unlawful or fraudulent as to that shareholder, the amended complaint requested rescission or reformation of the merger and an accounting for the injuries inflicted on Briggs. The action was certified as a class action in August 1986 by Mark A.

Costantino, District Judge. According to the district court docket entries, the action then lay inactive for some seven years. In 1993, pursuant to Fed.R.Civ.P. 55, defaults were entered against Briggs and BAC, neither corporation being represented by counsel. In 1994, the court, Sterling Johnson, Jr., District Judge, entered a default judgment against those defendants on the issue of liability and ordered an inquest on damages. Also in 1994, the death of Edward Rosenstock was noted on the record. No executor or administrator was substituted for him, and the claims against him were abandoned.

In August 1997, pursuant to a stipulation among plaintiffs, Rosenstock, and Briggs (into which BAC had been merged), the claims against Briggs and Rosenstock were resolved in a partial final judgment (“1997 Judgment”), David G. Trager, District Judge. The 1997 Judgment awarded the plaintiff class approximately $4 million against Briggs and approximately $6.9 million against Rosenstock. Pursuant to the stipulation, the 1997 Judgment also declared moot plaintiffs’ state-court appraisal action. In the meantime, in 1996, as discussed in

Part II. C. below, plaintiffs had sought leave to file a second-amended and supplemental complaint, alleging new state-law claims against Rosenstock, Genser, and several new proposed defendants. That motion was denied. B. The October 1998 Dismissal of the Remaining Claims The above proceedings left pending only plaintiffs’ claims against Genser. After entry of the 1997 Judgment, plaintiffs and Genser consented to have all further proceedings, including entry of judgment, conducted by a magistrate judge, and the matter was referred to Magistrate Judge Levy.

Following a bench trial, the court, in a Memorandum and Order dated October 29, 1998 (“October 1998 Order”), and reported at 23 F.Supp.2d 326, dismissed the claims against Genser in their entirety. With respect to the 1934 Act claims, the court noted that in order to succeed on a claim under § 10(

b) and Rule 10b-5, a plaintiff must prove a misrepresentation or omission as to a material fact in connection with the purchase or sale of a security, and an injury to the plaintiff caused by his reliance on the misrepresentation or omission. The district court assumed, arguen-do, that plaintiffs could prove one or more material misrepresentations or omissions in the Briggs proxy statement regarding the merger that led to the forced sale of their Briggs shares. However, it concluded that, under the principle established by Virginia Bankshares, Inc. v.

Sandberg, 501 U.S. 1083, 111 S.Ct. 2749, 115 L.Ed.2d 929 (1991), which involved claims that misrepresentations in proxy materials violated § 14(

a) of the 1934 Act, 15 U.S.C. § 78n(a) (1994), plaintiffs had not established the causation element because their votes were not needed for approval of the Briggs-BAC merger. The district court acknowledged that minority shareholders whose votes were unnecessary for approval of a freeze-out merger might nonetheless establish causation by showing that omissions or misrepresentations in the proxy statement had induced them to vote in favor of the merger and thereby caused them to forfeit an otherwise-available state-law remedy.

See Virginia Bankshares, 501 U.S. at 1107, 111 S.Ct. 2749 (leaving open the possibility of that theory of causation); Wilson v. Great American Industries, Inc., 979 F.2d 924, 931 (2d Cir.1992) (upholding liability rulings in favor of shareholders on that theory of causation). In the present case, however, the district court concluded that causation on that theory had not been established because plaintiffs had the right to an appraisal of the fair value of their shares, and they did not show that they had forfeited that or any other state-law remedy as a result of the alleged fraud in the Briggs proxy statement.

The court noted that, except in circumstances inapplicable here, New York law makes appraisal a dissenting shareholder’s exclusive remedy: [t]he enforcement by a shareholder of his right to receive payment for his shares in the manner provided herein shall exclude the enforcement by such shareholder of any other right to which he might otherwise be entitled by virtue of share ownership. N.Y. Bus. Corp. L. § 623(k). Thus, the court rejected an argument by plaintiffs that but for the misleading nature of the proxy statement they could have obtained an injunction against the merger.

Given the exclusivity of plaintiffs’ appraisal remedy, the district court also concluded that plaintiffs had no right to recover damages from Genser on their claims under state law. Although § 623(

k) goes on to state that “this

section shall not exclude the right of such shareholder to bring or maintain an appropriate action to obtain relief on the ground that such corporate action will be or is unlawful or fraudulent as to him,” id. (emphasis added), the court concluded that the exception was inapplicable because plaintiffs’ remaining claims against Genser sought only monetary relief: The “appropriate” action referred to in subdivision (

k) has been held to be “one in which a dissenting shareholder seeks some form of equitable relief.” Breed [v. Barton, 54 N.Y.2d 82, 87, 444 N.Y.S.2d 609, 429 N.E.2d 128, 130 (1981) ]; see also Burke v. Jacoby, 981 F.2d 1372, 1380 (2d Cir.1992) (“An ‘appropriate action’ within the meaning of § 623(

k) is an action seeking equitable relief, and that type of action and the exercise of the right of appraisal are the dissatisfied stockholder’s only avenues of relief.”). As the New York Court of Appeals explained in Breed, limiting the exception in subdivision (

k) to equitable relief “serves the valid function of denying dissenting shareholders the ability to reopen prior appraisal proceedings and again seek the identical relief merely by alleging fraudulent or unlawful corporate conduct in relation to the merger.” Breed, [54 N.Y.2d at 87,] 444 N.Y.S.2d 609, 429 N.E.2d at 130. Although plaintiffs alleging fraud in an equitable proceeding under subdivision (

k) may also seek money damages, the equitable relief must be the primary request, and such damages must be incidental to a grant of equitable relief. Id. [at 87, 444 N.Y.S.2d 609, 429 N.E.2d] at 131; see also Burke, 981 F.2d at 1380 (“Money damages are not available except as ancillary to a grant of traditionally equitable relief.”) ...; Direct Media/DMI, Inc. v. Rubin, 171 Misc.2d 505, 654 N.Y.S.2d 986, 989 (1997) (“[A] shareholder may not seek an appraisal and also commence a separate action to recover money damages for improper corporate action....”). Grace v. Rosenstock, 23 F.Supp.2d at 335-36.

The court noted that although plaintiffs originally had requested principally equitable relief against all defendants, their claims against Briggs had been settled and their only remaining requests against Genser were for money damages: Nothing in plaintiffs’ pre-trial or post-trial submissions indicates a desire on plaintiffs’ part for the court to award equitable relief, other than “rescissory damages,” i.e., money damages, against Genser.... Plaintiffs’ counsel made no mention of any other remedies at trial, and in fact conceded that plaintiffs’ claim against Genser is for monetary damages.

See Tr. at 687 (“What we are concerned here with is a claim of breach of fiduciary duty, ... which is an action not for fair price alone, it’s an action for damages.... ”). Id. at 336 n. 8. Accordingly,, all of plaintiffs’ claims against Genser were dismissed. This appeal followed. II. DISCUSSION On appeal, plaintiffs challenge the October 1998 Order dismissing their claims against Genser, contending principally that because the proxy statement was issued in support of a freeze-out merger, resulting in a forced sale, they should not'have been required to show causation in order to recover under § 10(

b) or Rule 10b — 5; they also contend that if required to prove causation, they met their burden. In addition, plaintiffs challenge the district court’s rejection of their 1996 motion for leave to file a new complaint adding new state-law claims and new parties to the action. For the reasons that follow, we find no basis for reversal. A. The Dismissal of the Federal 'Claims Against Genser In challenging the dismissal of their federal claims against Genser, plaintiffs contend principally that Virginia Bankshares does not apply to actions brought under § 10(

b) and Rule 10b-5, and that even if Virginia Bankshares applies, they established causation because they forfeited a state-law remedy in reliance on misrepresentations in and omissions from the proxy statement. We find no merit in these contentions. 1. The Causation Requirement

Section 10(

b) of the 1934 Act, which is generally regarded as a catch-all provision with respect to securities-related fraud, see, e.g., Chiarella v. United States, 445 U.S. 222, 226, 234-35, 100 S.Ct. 1108, 63 L.Ed.2d 348 (1980), prohibits the use, in connection with the purchase or sale of any security, of any manipulative or deceptive device or contrivance in contravention of such rules and regulations as the Commission may prescribe as necessary or appropriate in the public interest or for the protection of investors, 15 U.S.C. § 78j(b).

Rule 10b-5, which impliedly provides a private right of action, see, e.g., Blue Chip Stamps v. Manor Drug Stores, 421 U.S. 723, 730, 95 S.Ct. 1917, 44 L.Ed.2d 539 (1975), makes it unlawful, in connection with the purchase or sale of any security, to make any untrue statement of a material fact or to omit to state a material fact necessary in order to make the statements made, in the light of the circumstances under which they were made, not misleading, 17 C.F.R. § 240.10b-5(b). Because the private right of action under § 10(

b) and Rule 10b-5 is implied rather than express, its boundaries are left to judicial inference. See, e.g., Blue Chip Stamps, 421 U.S. at 749, 95 S.Ct. 1917 (“No language in either of those provisions speaks at all to the contours of a private cause of action for their violation.”). In framing the contours of private rights of action implied under the 1934 Act, the courts have sought to exclude, as a matter of law, claims based on hypothetical circumstances or speculation. Early in the development of such actions under § 10(

b) and Rule 10b-5, which addresses fraud “in connection with” the purchase or sale of a security, this Court ruled that no private action under those provisions is available to persons who were neither buyers nor sellers of the relevant securities. See Bimbaum v. Newport Steel Corp., 193 F.2d 461, 462-64 (2d Cir.), cert. denied, 343 U.S. 956, 72 S.Ct. 1051, 96 L.Ed. 1356 (1952).

That principle was adopted by the Supreme Court in Blue Chip Stamps, in which the plaintiffs were persons who had been offered — but had declined to purchase — shares in Blue Chip Stamps Co. pursuant to an antitrust-related consent decree entered into between the United States and the company. See 421 U.S. at 725-26, 95 S.Ct. 1917. The plaintiffs contended that the company had “intentionally made the prospectus overly pessimistic” in order to dissuade the plaintiffs from accepting the offer, and thereby allow the rejected shares to be sold to the public at a higher price. Id. at 726-27, 95 S.Ct. 1917.

The Supreme Court ruled that the district court had properly dismissed the suit. The Court reasoned that expanding the class of persons entitled to bring suit under Rule 10b-5 to include those who had neither bought nor sold shares would vastly increase the amount of frivolous securities litigation before the -federal courts, in part because “the abolition of the Bim-baum rule would throw open to the trier of fact many rather hazy issues of historical fact the proof of which depended almost entirely on oral testimony.” 421 U.S. at 743, 95 S.Ct. 1917.

Insofar as the element of causation is concerned, we have held in a variety of contexts that in order to establish a prima facie case under Rule 10b-5, a plaintiff must show, inter alia, both “loss causation — that the misrepresentations or omissions caused the economic harm,” and “transaction causation — that the violations in question caused the [plaintiff] to engage in the transaction in question.” Bennett v.

United States Trust Co. of New York, 770 F.2d 308, 313 (2d Cir.1985) (internal quotation marks omitted) (emphases in original) (alleged misrepresentation as to margin rules), cert. denied, 474 U.S. 1058, 106 S.Ct. 800, 88 L.Ed.2d 776 (1986); see also Burke v. Jacoby, 981 F.2d 1372, 1378 (2d Cir.1992) (alleged misrepresentation in connection with corporate acquisition), cert. denied, 508 U.S. 909, 113 S.Ct. 2338, 124 L.Ed.2d 249 (1993); Citibank, N.A. v. K-H Corp., 968 F.2d 1489, 1494 (2d Cir.1992) (alleged misrepresentation in application for financing).

We have also noted that both loss causation and transaction causation must be proven in the context of a private action under § 14(

a) of the 1934 Act and SEC Rule 14a-9 promulgated thereunder. See Wilson v. Great American Industries, Inc., 979 F.2d 924, 931 (2d Cir.1992) ("Wilson"); Schlick v. Penn-Dixie Cement Corp., 507 F.2d 374, 381-82 (2d Cir.1974), cert. denied, 421 U.S. 976, 95 S.Ct. 1976, 44 L.Ed.2d 467 (1975), overruled on other grounds by Virginia Bankshares, 501 U.S. at 1100 n. 9, 1100-06, 111 S.Ct. 2749. As to the question presented here, ie., whether transaction causation must be proven in connection with a securities-fraud claim based solely on misleading proxy statements and brought under § 10(

b) and Rule 10b-5, we find guidance in the decision of the Supreme Court in Virginia Bankshares with respect to claims of misleading proxy statements brought under § 14(

a) and Rule 14a-9, given that the interest in closing the floodgates to claims that are speculative or hypothetical is the same under both sets of provisions.

Section 14(

a) makes it unlawful, by the use of the mails or any other instrumentality of interstate commerce, to solicit proxies “in contravention of such rules and regulations as the Commission may prescribe as necessary or appropriate in the public interest or for the protection of investors_” 15 U.S.C. § 78n(a).

Rule 14a-9, promulgated thereunder, provides that [n]o solicitation subject to this regulation shall be made by means of any proxy statement ... containing any statement which, at the time and in the light of the circumstances under which it is made, is false or misleading with respect to any material fact, or which omits to state any material fact necessary in order to make the statements therein not false or misleading.... 17 C.F.R. § 240.14a-9(a) (1999).

Section 14(

a) and Rule 14a-9, like § 10(

b) and Rule 10b-5, do not expressly provide a private right of action. The Supreme Court recognized an implied private right of action for injury caused by a violation of § 14(a), as implemented by Rule 14a-9, in J.I. Case Co. v. Borak, 377 U.S. 426, 84 S.Ct. 1555, 12 L.Ed.2d 423 (1964). In Mills v. Electric Auto-Lite Co., 396 U.S. 375, 90 S.Ct. 616, 24 L.Ed.2d 593 (1970), the Court ruled that the causation requirement of Rule 14a-9 would be satisfied where the challenged proxy statement “was an essential link in the accomplishment of the transaction.” Id. at 385, 90 S.Ct. 616.

In that case, the majority shareholder held just over one-half of the corporation’s outstanding shares, but the proposed merger could not be approved without the affirmative vote of two-thirds. Thus, the minority-shareholder plaintiffs were held to have satisfied the causation element because the merger could not have been authorized without the affirmative vote of at least some of the minority shares.

The Mills Court left open the question of “whether causation could be shown where the management controls a sufficient number of shares to approve the transaction without any votes from the minority.” Id. at 385 n. 7, 90 S.Ct. 616. The Supreme Court answered that question in Virginia Bankshares, ruling that minority shareholders whose votes were not required for approval of the proposed freeze-out merger had failed to show that the materially misleading representations in the proxy statement violating Rule 14a-9 caused their injury.

Although the plaintiff shareholders argued that the challenged proxy statement was an “essential link” within the meaning of Mills on the theory that their negative votes, though insufficient to defeat the merger, would have created sufficient bad publicity that the directors would not have proceeded with the merger, see 501 U.S. at 1100-01, 111 S.Ct. 2749, the Virginia Bankshares Court found that theory too speculative: [I]n Blue Chip Stamps we raised concerns about the practical consequences of allowing recovery, under § 10(

b) of the Act and Rule 10b-5, on evidence of what a merely hypothetical buyer or seller might have done on a set of facts that never occurred, and foresaw that any such expanded liability would turn on “hazy” issues inviting self-serving testimony, strike suits, and protracted discovery, with little chance of reasonable resolution by pretrial process.... These were good reasons to deny recognition to such claims in the absence of any apparent contrary congressional intent.

The same threats of speculative claims and procedural intractability are inherent in respondents’ theory of causation linked through the directors’ desire for a cosmetic vote. Causation would turn on inferences about what the corporate directors would have thought and done without the minority shareholder approval unneeded to authorize action.

A subsequently dissatisfied minority shareholder would have virtual license to allege that managerial timidity would have doomed corporate action but for the ostensible approval induced by a misleading statement, and opposing claims of hypothetical diffidence and hypothetical boldness on the part of directors would probably provide enough depositions in the usual case to preclude any judicial resolution short of the credibility judgments that can only come after trial. Reliable evidence would seldom exist.

Directors would understand the prudence of making a few statements about plans to proceed even without minority endorsement, and discovery would be a quest for recollections of oral conversations at odds with the official pronouncements, in hopes of finding support for ex post facto guesses about how much heat the directors would have stood in the absence of minority approval. The issues would be hazy, their litigation protracted, and their resolution unreliable. Given a choice, we would reject any theory of causation that raised such prospects, and we reject this one.

Virginia Bankshares, 501 U.S. at 1105-06, 111 S.Ct. 2749. Although Virginia Bankshares addressed proxy-statement-fraud claims brought only under § 14(

a) and Rule 14a-9, its express rebanee on the rationale of Blue Chip Stamps, a case involving § 10(

b) and Rule 10b-5, demonstrates that the same concerns compel the conclusion that the scope of the implied private right of action under either set of provisions does not encompass claims based on “a merely hypothetical ... set of facts that never occurred,” Virginia Bankshares, 501 U.S. at 1105, 111 S.Ct. 2749. Plaintiffs have advanced no logical rationale — and we see none — for allowing a claimant to prevail on a misleading-proxy-statement claim under § 10(

b) and Rule 10b-5 without proving causation, when causation is required with respect to such a claim under § 14(

a) and Rule 14a-9, which deal expressly with misleading proxy statements, and is required under § 10(

b) and Rule 10b-5, which deal with claims of misleading statements in any medium. We conclude that the principle announced in Virginia Bankshares is applicable to misleading-proxy-statement claims that are brought under § 10(

b) and Rule 10b-5, and that in order to prevail on such a claim, a plaintiff must present nonspeculative evidence of loss causation and transaction causation, see Wilson, 979 F.2d at 931. We reject plaintiffs’ contention that “[w]here the misrepresentations and non-disclosures under Rule 10b-5 are part and parcel of the forced sale by the plaintiffs shareholders ..., there is no separate requirement to show causation” (Plaintiffs’ brief on appeal at 43). For that proposition, plaintiffs rely on Vine v.

Beneficial Finance Co., 374 F.2d 627, 635 (2d Cir.), cert. denied, 389 U.S. 970, 88 S.Ct. 463, 19 L.Ed.2d 460 (1967). Quaere whether Vine would survive the rationale of Virginia Bankshares if Vine’s holding were as plaintiffs have characterized it; but in any event, in light of its actual holding, plaintiffs’ reliance is misplaced. In Vine, we considered the complaint of a Class A shareholder who, after being subjected to a freeze-out merger, brought an action under § 10(

b) and Rule 10b-5 alleging that the defendant tender offeror had attained the share percentage needed to approve the freeze-out merger by fraudulently inducing Class A shareholders other than the plaintiff to tender their shares to it. We did not hold that the plaintiff was not required to plead reliance or causation; rather, we held that the complaint adequately pleaded both elements.

Noting that the complaint alleged a single two-phase scheme to eliminate Class A shareholders, we stated that “[w]hat must be shown is that there was deception which misled Class A stockholders and that this was in fact the cause of plaintiffs claimed injury. The allegations of this complaint meet that test.” Id. at 635. In the present case, plaintiffs did not meet that test. Rosenstock and Genser at all pertinent times owned sufficient shares, either directly or through their ownership of BAC, to approve the Briggs-BAC merger, and there is no suggestion that they acquired those shares by means of fraud.

No vote, sale of shares, or other action was required of minority shareholders in order to accomplish the merger. In sum, Virginia Bankshares applies to claims of proxy fraud brought under § 10(

b) and Rule 10b-5, and a plaintiff may not recover on such a claim without proving that his injury was caused by the fraud. Plaintiffs here, because minority-shareholder votes were not required for approval of the merger that was the subject of the proxy statements, did not show causation based on the approval of the merger. 2.

Causation Based on Forfeiture of a State Remedy The Virginia Bankshares Court left open the possibility that even minority shareholders whose votes were not necessary for approval of the proposed merger might nonetheless satisfy the causation requirement if they could show that the violations induced them to forfeit a state remedy. See 501 U.S. at 1107, 111 S.Ct. 2749.

This Court addressed that issue in Wilson, in which the management had controlled sufficient shares to approve the proposed merger without the minority’s votes, the plaintiff minority shareholders had been induced to vote in favor of the merger, and “their deceptively procured vote in favor of the merger deprived them of their state appraisal rights under” New York law, 979 F.2d at 930.

We upheld the liability ruling in favor of the Wilson plaintiffs, stating as follows: We recognize that loss causation or economic harm to plaintiffs must be shown, as well as proof that the misrepresentations induced plaintiffs to engage in the subject transaction, that is, transaction causation. Here loss causation may be established when a proxy statement prompts a shareholder to accept an unfair exchange ratio for his shares rather than recoup a greater value through a state appraisal.

And transaction causation may be shown when a proxy statement, because of material misrepresentations, causes a shareholder to forfeit his appraisal rights by voting in favor of the proposed corporate merger. Wilson, 979 F.2d at 931. We noted that allowing that type of action does not pose a threat of “speculative claims and procedural intractability,” ... because the forfeiture of state [remedies] is a question separate from the effectuation of the merger and does not require courts to guess how or whether the majority shareholders would have proceeded in the face of minority dissent.

Id. at 932 (quoting Virginia Bankshares, 501 U.S. at 1105, 111 S.Ct. 2749). In the present case, however, plaintiffs did not vote in favor of the merger and did not show that they relied on the proxy materials in any way that caused them to forfeit their rights to appraisal. Rather, plaintiffs contend, relying on Lynch v. Vickers Energy Corp., 429 A.2d 497, 501 (Del.1981), overruled on other grounds, Weinberger v. UOP, Inc., 457 A.2d 701, 703-04 (Del.1983), that causation was established because they forfeited their state-law rights to obtain “rescissory damages” (Plaintiffs’ brief on appeal at 51).

We are unpersuaded for several reasons. First, as the district court noted, see

Part I.B. above, under New York law governing freeze-out mergers, except for an “appropriate” equitable action as envisioned by BCL § 623(k), appraisal is the dissenting shareholders’ exclusive remedy. Although the Lynch court held that a res-cissory-damages remedy was available under Delaware law to the plaintiffs before it, the transaction at issue was not a freeze-out merger but a tender offer by the corporation’s majority shareholder, and no appraisal remedy was available.

We are not persuaded that Lynch, in which there was no forced sale and the tendering shareholders had no appraisal remedy, is authority for the proposition that rescissory damages were available under New York law in the present case where there was a forced sale, there was an appraisal remedy, and the pertinent statute provided that appraisal was the exclusive remedy. Plaintiffs cite no case applying New York law, and we know of none, that authorizes an award of rescissory damages for a freeze-out merger as to which a dissenting shareholder had a right of appraisal.

Second, though Lynch held that rescis-sory damages may be available as an equitable substitute for rescission where, because of mergers, the latter remedy is as a practical matter no longer feasible, plaintiffs have cited no authority suggesting that a monetary award in favor of a shareholder against a director would be viewed as anything other than an award of damages at law in circumstances where, as here, the plaintiffs have already settled their claims against the party that purchased their shares- — i.e., the only entity against which rescission would be even a theoretically permissible remedy.

Third, we note that under Lynch, rescis-sory damages may reflect increases in the corporation’s value subsequent to the challenged transaction and that plaintiffs in the present action apparently seek to benefit from a “surge in land values” through a date at least “30 months after the [Briggs] merger” (Plaintiffs brief on appeal at 51). Here, however, the pertinent defendants lawfully owned sufficient shares to approve the merger and freeze out other shareholders without the approval of any minority shares.

Lynch did not involve such circumstances, and plaintiffs have not cited any New York case to support their suggestion that, in the circumstances here, an action for rescissory damages encompassing post-merger surges in land values would be considered an “appropriate” action within the meaning of § 623(k). Finally, we note that although plaintiffs no longer have appraisal rights, that is not the result of any reliance on the Briggs proxy statements.

Rather, their appraisal suit was declared moot pursuant to their stipulation leading to the 1997 Judgment, which awarded the plaintiff class some $11 million from Briggs and Rosenstock. We see no forfeiture there. In sum, plaintiffs have cited no New York law, or any other relevant authority, for the proposition that there was any available state-law remedy that they forfeited in reliance on the Briggs proxy statement. Accordingly, since their votes were not needed for approval of the merger, the district court correctly concluded that they failed to prove causation as required by § 10(

b) and Rule 10b-5. B. The Dismissal of the State-Law Claims Plaintiffs contend that the district court erred in finding their state-law claims against Genser barred by BCL § 623(

k) because it mistakenly believed that those claims sought primarily money damages rather than equitable relief. Plaintiffs argue that those claims must be viewed as requests for equitable relief because a February 1986 order of Judge Costantino effectively so characterized them. We disagree. First, even if Judge Costantino’s characterization of the claims constituted an order, it plainly was interlocutory. All interlocutory orders remain subject to modification or adjustment prior to the entry of a final judgment adjudicating the claims to which they pertain. See Fed. R.Civ.P. 54(b); Cullen v.

Margiotta, 811 F.2d 698, 708 (2d Cir.), cert. denied, 483 U.S. 1021, 107 S.Ct. 3266, 97 L.Ed.2d 764 (1987), overruled on other grounds, Agency Holding Corp. v. Malley-Duff & Associates, 483 U.S. 143, 107 S.Ct. 2759, 97 L.Ed.2d 121 (1987); United States v.

LoRusso, 695 F.2d 45, 53 (2d Cir.1982) (“whether the case sub judice be civil or criminal[,] so long as the district court has jurisdiction over the case, it possesses inherent power over interlocutory orders, and can reconsider them when it is consonant with justice to do so” (internal quotation marks omitted)), cert. denied, 460 U.S. 1070, 103 S.Ct. 1525, 75 L.Ed.2d 948 (1983). Second, the posture of the case changed significantly in the 12 years between Judge Costantino’s characterization of it in 1986 and the magistrate judge’s analysis in 1998.

Judge Costantino dealt only with the pleadings, and, aside from a typical catchall paragraph that mentioned damages, the only relief requested by the amended complaint was equitable. On the claims against Briggs, it requested rescission of the merger or reformation of its terms; on the derivative claim on behalf of Briggs, it sought from the individual defendants an accounting and disgorgement of profits. By the time of trial in 1998, however, plaintiffs had settled their claims against Briggs, the purchaser of their shares; plaintiffs thus could no longer seek rescission.

Nor were there any remaining derivative claims, as Briggs was no longer a party. Thus, as the district court observed in its October 1998 Order, plaintiffs’ counsel stated at trial that “ ‘[w]hat we are concerned here with is a claim of breach of fiduciary duty, ... which is an action not for fair price alone, it’s an action for damages....’” 23 F.Supp.2d at 336 n. 8 (quoting Tr. 687). We note that, in the wake of that ruling, plaintiffs made no effort to suggest, by way of a motion for reconsideration by that court, that the court had misapprehended the nature of their remaining claims against Genser.

We see no error in the district court’s characterization or dismissal of plaintiffs’ state-law claims against Genser. C. The Proposed Second Amended and Supplemental Complaint In January 1996, more than a decade after commencing this action, plaintiffs sought leave to file a “Supplemental and Second Amended Complaint” (“Supplemental Complaint”), which would have added state-law claims against (

a) Rosen-stock and Genser, (

b) Bank Leumi Trust Company of New York (“Bank Leumi”), David Mack, Leo V. Berger, and Apex Marine Corp. (collectively the “proposed creditor defendants”), and (

c) Gary Holman, an attorney for Briggs. Plaintiffs asserted, inter alia, that after consummation of the freeze-out merger, Rosenstock and Genser had engaged in a scheme to “subvert the jurisdiction of this federal Court ... [and] to make any judgment rendered by this Court in this action unenforceable and of no practical value” (Supplemental Complaint ¶ 47) by causing Briggs to transfer assets to themselves and to the proposed creditor defendants.

The Supplemental Complaint alleged that Rosenstock and Genser, in furtherance of that scheme, looted Briggs between November 1986 and August 1989 by mortgaging its property to secure loans from the proposed creditor defendants and using the proceeds of the loans primarily for their own personal benefit rather than for the benefit of Briggs. Those acts were alleged to have violated common law and various provisions of New York’s Debtor and Creditor Law (“DCL”), see N.Y. Debt. & Cred.

L. (McKinney 1990), to wit, id. § 273 (conveyance by a person who is insolvent); id. § 273-a (conveyance by a person who is a defendant in a pending action or who has a judgment docketed against him); id. § 274 (conveyance by a person in business); and id. § 276 (conveyance made with intent to defraud). The proposed pleading also alleged that in January 1990, Briggs and Rosenstock sold Briggs real estate for approximately 20% below its fair market value in order to make payments to the proposed creditor defendants.

The Supplemental Complaint further alleged that each of the proposed creditor defendants accepted notes and mortgages from Briggs, knowing that the proceeds of the loans were being diverted to Rosen-stock and Genser for their personal use. Bank Leumi was alleged to have known that Briggs would not be able to meet its obligations, would be forced to borrow additional funds, and would be forced to sell its real estate under duress.

The Supplemental Complaint alleged that Holman had committed professional malpractice and breached fiduciary and contractual duties to Briggs and to plaintiffs by representing Briggs while also serving as attorney to Rosenstock and Genser in their personal capacities, and by supervising the allegedly fraudulent conveyances while knowing that they were fraudulent as to Briggs and plaintiffs. The Supplemental Complaint principally sought awards of damages to plaintiffs and Briggs, including punitive damages from Rosenstock and Genser.

It also sought nullification of a 1992 state-court foreclosure judgment that Bank Leumi had obtained against Briggs. The proposed new defendants opposed plaintiffs’ motion to file the Supplemental Complaint, arguing principally that there was no basis for federal jurisdiction over the claims against them and that the claims were barred by the applicable statutes of limitations. Magistrate Judge Levy (to whom the matter had not yet been referred for all purposes) denied the motion, and plaintiffs’ objections to that denial were rejected by Judge Trager.

Without deciding the jurisdictional questions, the court denied the motion on the grounds of futility and delay, as explained in the magistrate judge’s Memorandum and Order dated September 30, 1996 (“September 1996 Order”), reported at 169 F.R.D. 473 (1996). In that decision, the court noted that the new claims that plaintiffs moved to add were subject to a six-year statute of limitations.

Because plaintiffs’ motion was made in 1996 and the new claims focused on conduct alleged to have occurred principally in 1986-1989, the court concluded that filing of the Supplemental Complaint would likely be futile because most of the new claims appeared to be time-barred. Plaintiffs argued that the statute of limitations was no bar because the filing of their new claims should be deemed to “relate[ ] back” to the time of filing of their amended complaint, Fed.R.Civ.P. 15(c).

The court rejected that contention, observing that the proposed pleading was not based on the events challenged in the amended complaint but rather “related to the notes and mortgages that Briggs executed years later”; hence the amended complaint could not have provided notice of the new claims. September 1996 Order, 169 F.R.D. at 481-82.

The court likewise rejected plaintiffs’ contention that relation back was appropriate on the theory that plaintiffs were challenging a “continuing course of conduct”; the court pointed out that the amended “complaint alleged, in essence, a specific act of wrongdoing, i.e., a freeze-out merger, which is a distinct transaction from the subsequent note and mortgage transactions plaintiffs now seek to add.” Id. at 482 (internal quotation marks omitted).

Plaintiffs also argued that in 1990, less than six years prior to their January 1996 motion to file the Supplemental Complaint, Briggs made mortgage payments to the proposed creditor defendants, raising the cash for those payments by selling real estate below fair market value, and that the fraudulent-conveyance claims under DCL §§ 273, 274, and 276 based on those payments were not time-barred.

The district court disagreed, reasoning that “[t]he January 1990 loan payments cannot, by themselves, constitute fraudulent conveyances, since they are not events that are independent or separate from the original note and mortgage transactions.” September 1996 Order, 169 F.R.D. at 482.

In addition, the court stated that even if that argument had merit, it would deny leave to add those claims to the suit in order to avoid delaying resolution of this hoary action: Even if the January 1990 payments could themselves be considered fraudulent conveyances, ... this court would nonetheless deny plaintiffs’ motion to amend and supplement the pleadings to include allegations regarding those payments.

At this late date in the proceedings[,] granting plaintiffs’ motion would cause substantial further delays and prejudice to the proposed new defendants, who would now be required to reopen all of the depositions that have taken place thus far in the litigation and to conduct full document discovery. Id. at 482 n. 5. The court also rejected several other timeliness arguments advanced by plaintiffs, including equitable tolling and equitable estoppel.

Noting that “the doctrine of equitable tolling applies Vhere the facts show that the defendant engaged in conduct, often itself fraudulent, that concealed from the plaintiff the existence of [the] cause of action,’ ” id. at 483 (quoting Long v. Frank, 22 F.3d 54, 58 (2d Cir.1994), cert. denied, 513 U.S. 1128, 115 S.Ct. 938, 130 L.Ed.2d 883 (1995)), the court pointed out that this case was then more than 11 years old and that plaintiffs did not dispute that they had known of the alleged fraudulent conveyances at least as early as 1993.

The court found that plaintiffs “offer[ed] no adequate explanation as to why they were unable to conduct full discovery ... prior to 1993.” September 1996 Order, 169 F.R.D. at 484. In addition, noting that the doctrine of equitable estoppel is applied “ ‘in cases where the plaintiff knew of the existence of his cause of action but the defendant’s conduct caused him to delay bringing his lawsuit,’ ” id. (quoting Dillman v.

Combustion Engineering, Inc., 784 F.2d 57, 61 (2d Cir.1986)), the court found that doctrine inapplicable because plaintiffs “d[id] not contend that any of the defendants or proposed defendants engaged in conduct that caused plaintiffs to delay bringing suit once they were fully aware of their claims,” September 1996 Order, 169 F.R.D. at 484. On this appeal, plaintiffs raise various challenges to the district court’s denial of leave to file the Supplemental Complaint.

In reviewing their challenges, we are mindful that leave to amend is to be granted “freely ... when justice so requires,” Fed.R.Civ.P. 15(a); see also Foman v. Davis, 371 U.S. 178, 182, 83 S.Ct. 227, 9 L.Ed.2d 222 (1962); John Hancock Mutual Life Insurance Co. v. Amerford International Corp., 22 F.3d 458, 462 (2d Cir.1994). However, in determining whether leave to amend should be granted, the district court has discretion to consider, inter alia, the apparent “futility of amendment,” Foman v. Davis, 371 U.S. at 182, 83 S.Ct. 227; see also Ruffolo v. Oppenheimer & Co., 987 F.2d 129, 131 (2d Cir.1993) (per curiam).

Amendment would likely be futile if, for example, the claims the plaintiff sought to add would be barred by the applicable statute of limitations. See, e.g., McGill v. Goff, 17 F.3d 729, 734 (5th Cir.1994). The court also has discretion to deny leave to amend “where the motion is made after an inordinate delay, no satisfactory explanation is offered for the delay, and the amendment would prejudice” other parties, Cresswell v. Sullivan & Cromwell, 922 F.2d 60, 72 (2d Cir.1990); see also Tokio Marine & Fire Insurance Co. v.

Employers Insurance of Wausau, 786 F.2d 101, 103 (2d Cir.1986), or where the belated motion would unduly delay the course of proceedings by, for example, introducing new issues for discovery, see, e.g., Ansam Associates v. Cola Petroleum, Ltd., 760 F.2d 442, 446 (2d Cir.1986). The district court’s denial of leave to amend is reviewed only for abuse of discretion. See, e.g., Peterson v. Insurance Co. of North America, 40 F.3d 26, 31 (2d Cir.1994); United States v. Continental Illinois National Bank and Trust Co. of Chicago, 889 F.2d 1248, 1254 (2d Cir.1989). 1.

The Refusal To-Allow Assertion of the DCL § 273-a Claims Plaintiffs contend that their claims under DCL § 273-a were not barred by the statute of limitations because at the time of plaintiffs’ motion to file the Supplemental Complaint, those claims had not yet even accrued. For the reasons that follow, we agree that those claims had not yet accrued, but we conclude that this is not a basis for reversal.

Section 273-a provides that [e]very conveyance made without fair consideration when the person making it is a defendant in an action for money damages or a judgment in such an action has been docketed against him, is fraudulent as to the plaintiff in that action without regard to the actual intent of the defendant if, after final judgment for the plaintiff, the defendant fails to satisfy the judgment. N.Y. Debt. & Cred. L. § 273-a (emphasis added).

To prevail on a claim under § 273-a, a plaintiff must establish “(1) that the conveyance was made without fair consideration; (2) that the conveyor is a defendant in an action for money damages or that a judgment in such action has been docketed against him, and (3) that [the] defendant has failed to satisfy the judgment.” Taylor-Outten v. Taylor, 248 A.D.2d 934, 935, 670 N.Y.S.2d 295, 296 (4th Dept.1998) (internal quotation marks omitted). Because “[t]he existence of an unsatisfied judgment is an essential element of [§ 273-a],” Carey v.

Crescenzi, 923 F.2d 18, 21 (2d Cir.1991) (internal quotation marks omitted); see also Frybergh v. Weissman, 145 A.D.2d 531, 531, 536 N.Y.S.2d 465, 466 (2d Dept.1988), the limitations period, which is six years, see NY. C.P.L.R. § 213(1) (McKinney 1990), begins to run, at the earliest, on the date of the judgment, see Carey v. Crescenzi, 923 F.2d at 21. The unsatisfied judgment at issue here is the judgment entered against Briggs in August 1997. That judgment had not yet been entered when plaintiffs moved in 1996 to file the Supplemental Complaint.

Accordingly, the limitations period had not yet begun to run, and we agree with plaintiffs that the district court erred in viewing their DCL § 273-a claims as time-barred. Two factors, however, persuade us that that error does not warrant reversal. First, plaintiffs did not make their present nonaccrual argument with respect to the § 273-a claim before the magistrate judge. Nor did they make any timely presentation of that argument to the district judge, for it was not mentioned to Judge Trager in their objections to the magistrate judge’s September 1996 Order.

To the contrary, plaintiffs’ statement of their objections to the magistrate judge’s statute-of-limitations ruling asserted, without any exception for claims under § 273-a, that “the statute of limitations on a claim for fraudulent transfers is measured from the date of the transfer.” (Letter from Sidney Bender to Judge Trager, dated October 23, 1996, at 1.) Plaintiffs made no mention of their § 273-a nonaccrual contention until January 1997, when they unsuccessfully asked the district court to certify statute-of-limitations issues to this Court.

Accordingly, plaintiffs waived their nonaccrual objection to the magistrate judge’s ruling. Second, we are disinclined to remand for the court to exercise pendent jurisdiction over this state-law claim because all of the federal claims have now been dismissed. The decision to exercise pendent jurisdiction is vested in the sound discretion of the district court. United Mine Workers of America v. Gibbs, 383 U.S. 715, 725-28, 86 S.Ct. 1130, 16 L.Ed.2d 218 (1966) .... The discretion is limited, however, by the consideration that “[n]eedless decisions of state law should be avoided....” Id. at 726, 86 S.Ct. 1130....

Fay v. South Colonie Central School District, 802 F.2d 21, 34 (2d Cir.1986) (reversing district court’s exercise of pendent jurisdiction as abuse of discretion where dismissal of federal claim was required and state-law questions were unsettled). In United Mine Workers v.

Gibbs, 383 U.S. 715, 86 S.Ct. 1130, 16 L.Ed.2d 218 (1966), the Supreme Court stated that “if it appears that the state issues substantially predominate, whether in terms of proof, of the scope of the issues raised, or of the comprehensiveness of the remedy sought, the state claims may be dismissed without prejudice and left for resolution to state tribunals,” id. at 726-27, 86 S.Ct. 1130; see also id. at 726, 86 S.Ct. 1130 (“Certainly, if the federal claims are dismissed before trial, even though not insubstantial in a jurisdictional sense, the state claims should be dismissed as well.”); 28 U.S.C. § 1367(c) (expressly authorizing district court, in actions commenced after December 1, 1990, see Judicial Improvements Act of 1990, Pub.L.

No. 101-650, 104 Stat. 5089, 5114, to “decline to exercise supplemental jurisdiction” over a state-law claim over which it has no independent jurisdiction if “the district court has dismissed all claims over which it has original jurisdiction”).

Given that plaintiffs did not timely raise their § 273-a nonaccrual argument in the district court, that all federal issues have now been disposed of, and that the § 273-a claims are substantially different from the federal claims that have been adjudicated, we conclude that it would be inappropriate to remand to the district court for the belated introduction and litigation of these state-law claims in federal court. 2. The Other Proposed Claims Plaintiffs also challenge the district court’s rejection of their theories as to why their proposed claims under other DCL sections were timely.

We see no error in the district court’s rulings with respect to those claims. Plainly, most of the claims were based on transactions that occurred more than six years prior to plaintiffs’ motion to amend. Further, as to the 1990 payments made less than six years prior to the motion, the district court stated that even if those payments themselves could be considered fraudulent conveyances within the limitations period, the court would not allow plaintiffs to add those claims, given, inter alia, the belated timing of the motion and the fact that discovery would have to be reopened.

We see no abuse of discretion in that ruling. Moreover, even if we concluded that the district court’s views of the timeliness of the proposed claims under DCL §§ 273, 274, and 276 were flawed, we would decline, for the reasons just discussed with respect to § 273-a, to remand for the district court to adjudicate those state-law issues now that all federal issues have been resolved.

We note that plaintiffs, concerned with the possible preclusive effect of the district court’s denial of leave to file the Supplemental Complaint, contend that the district court could not properly consider whether their new claims would be barred by the statute of limitations without first determining that it had subject matter jurisdiction over those claims. For that proposition, plaintiffs rely on Arrowsmith v. United Press International, 320 F.2d 219 (2d Cir.1963) (en banc).

Their reliance is misplaced, however, as their contention misperceives both the nature of the decision on a motion for leave to amend and the nature of pendent jurisdiction. Arrowsmith did not involve a plaintiffs motion to amend the complaint; nor did it involve pendent jurisdiction. Rather, it concerned a defendant’s motion to dismiss the complaint on the grounds of, inter alia, lack of personal jurisdiction and failure to state a claim on which relief can be granted.

We held that the court should have resolved the jurisdictional issue, determining whether it had the power to adjudicate the action against that defendant, before dismissing for failure to state a claim, because a dismissal on the latter ground has preclusive effect. Neither of those grounds, however, was discretionary. If the action was flawed in either respect, the court was required to dismiss.

In contrast, as discussed above, both the decision whether to exercise pendent jurisdiction and the decision whether to allow a new pleading are committed to the sound discretion of the district court, see, e.g., United Mine Workers v. Gibbs, 383 U.S. at 725-28, 86 S.Ct. 1130 (pendent jurisdiction); Foman v. Davis, 371 U.S. at 182, 83 S.Ct. 227 (amended pleading), and we have suggested that generally the denial of leave to bring in new defendants does not have preclusive effect with respect to the claims against those defendants, see Northern Assurance Co. of America v. Square D Co., 201 F.3d 84 (2d Cir.2000).

In any event, both types of decision in question here are discretionary, and we see no requirement, in circumstances such as these, that either branch of discretion be exercised before the other. CONCLUSION We have considered all of plaintiffs’ arguments on this appeal and have found in them no basis for reversal. The judgment of the district court is affirmed.