SUMMARY ORDER
UPON DUE CONSIDERATION, IT IS HEREBY ORDERED, ADJUDGED AND DECREED that the judgment of the district court be, and it hereby is, affirmed.
Our previous decision with respect to this litigation generated three divergent opinions, each therefore without binding precedential effect. AUSA Life Ins. Co. v. Ernst & Young, 206 F.3d 202 (2d Cir.2000) (“AUSA Appeal”). However, our “mandate” had the concurrence of two judges, see id. at 225, and is therefore the law of the case. Cf. Marks v. United States, 430 U.S. 188, 193, 97 S.Ct. 990, 51 L.Ed.2d 260 (1977) (“When a fragmented [Supreme] Court decides a case and no single rationale explaining the result enjoys the assent of [a majority], ‘the holding of the Court may be viewed as that position taken by those Members who concurred in the judgment on the narrowest grounds.’ ” (quoting Gregg v. Georgia, 428 U.S. 153, 169 n. 15, 96 S.Ct. 2909, 49 L.Ed.2d 859 (1976))); see also Tyler v. Bethlehem, Steel Corp., 958 F.2d 1176, 1182-83 (2d Cir.1992).
A “mandate” from this Court consists of a “certified copy of [our] judgment, a copy of the opinion, and any direction as to *670costs,” United States v. Reyes, 49 F.3d 63, 66 (2d Cir.1995). Our judgment from the previous appeal read as follows: “[T]he judgment of said district court be and it hereby is reversed in part, vacated in part, and the case is remanded to said district court for further proceedings in accordance with the opinion of this court.” The opinion of Judge Oakes was the opinion of this Court, AUSA Appeal, 206 F.3d at 224, and so the district court was required to conduct further proceedings and arrive at a judgment in accordance with Judge Oakes’s opinion.
The following passage from Judge Oakes’s opinion contained the district court’s instructions on remand:
The ... query is whether [the defendant] could have reasonably foreseen that [its] certification of false financial information could lead to the demise of JWP, by enabling JWP to make an acquisition that otherwise would have been subjected to higher scrutiny, which led to harm to the investors. Given that the district court did not make factual findings as to foreseeability specifically, we remand for more factual findings. In accordance with the factual findings, the court is then instructed to reconsider proximate cause in the context of its factual determinations on foreseeability.
AUSA Appeal, 206 F.3d at 217. Two footnotes in this passage provided further guidance. The first defined “harm” as “losses incurred for risks which the investors did not intend to take nor of which they should have (or, in this case, even could have) been aware.” Id. at 217 n. 6. The second specified which facts were particularly relevant on remand: (1) JWP was in covenant default on its notes before the fateful Businessland acquisition; (2) the defendant knew this and concealed it; (3) notice of default would have given the plaintiffs the option to accelerate payment on their notes; and (4) the acquisition of Businessland could not have taken place without JWP’s cure, or the plaintiffs’ waiver, of the covenant default. Id. at 210, 217 n. 7.
The plaintiffs raise the same issue they raised on their previous appeal: the district court’s application of the law of loss causation. Although the plaintiffs advance multiple theories of how the defendant caused their losses, Judge Oakes’s opinion required the district court to consider only two. The first the parties call the “enabling” theory, i.e., that the defendant deprived the plaintiffs of the benefit of contractual provisions that, if enforced, would have prevented the Businessland acquisition. The second the parties call the “forbearance” theory, i.e., that the defendant’s misrepresentations caused the plaintiffs to forbear from exercising their options under the note agreements to accelerate payment of principal and thereby potentially increase their recovery.
Before turning to the specific causation theories, however, we note several important features of the posture of this appeal. First, neither party accepted the district court’s invitation on remand to submit new evidence. AUSA Life Ins. v. Ernst & Young, 119 F.Supp.2d 394, 395 (S.D.N.Y.2000) (“AUSA Remand”). Second, the plaintiffs now ask us only for judgment as a matter of law. They have not requested in the alternative a remand for further fact-finding. Thus, the plaintiffs argue that they have proven loss causation on the current record, with the facts found by the district court. Third, neither party has argued that any of the district court’s factual findings were clearly erroneous. Instead, the plaintiffs’ argument on appeal, as we understand it, is that the district court misapplied the law of loss causation to the evidence presented and facts found.
*671For these reasons, a remand for further fact-finding is not a potential disposition of this appeal. Our task reduces to answering this question: Do the facts found by the district court demonstrate loss causation as a matter of law?
The Enabling Theory
The plaintiffs first argue that the defendant’s misrepresentations deprived them of the benefit of certain contractual provisions that, if enforced, might have blocked the Businessland acquisition that led to JWP’s payment default. The relevant provisions include the general power of creditors to throw a defaulting debtor into bankruptcy proceedings, the threat of which potentially gives creditors the ability to influence significantly debtor management decisions. See AUSA Appeal, 206 F.3d at 217 (framing the issue as whether the defendant’s misrepresentations potentially “enabl[ed] JWP to make an acquisition that otherwise would have been subjected to higher scrutiny, which led to harm to the investors”). They also include specific clauses in the note agreements preventing JWP from engaging in certain investments while in default. Finally, they include clauses in the JWP-Businessland merger agreement that forbade JWP from consummating the merger while a petition for bankruptcy was filed against it. In light of these provisions, Judge Oakes observed that “the acquisition of Business-land could not have taken place without a cure of the default or the investors’ waivers of the default.” AUSA Appeal, 206 F.3d at 210.
In analyzing the enabling theory, we are mindful that the purpose of the laws prohibiting securities fraud is to restore to a defrauded individual the “benefit of the bargain,” i.e., “the excess of what he paid over the value of what he got.” McMahan & Co. v. Wherehouse Entm’t, Inc., 65 F.3d 1044, 1049 (2d Cir.1995), quoting Levine v. Seilon, Inc., 439 F.2d 328, 334 (2d Cir.1971) (Friendly, J.); accord Chemical Bank v. Arthur Andersen & Co., 726 F.2d 930, 943 (2d Cir.1984) (“The purpose of § 10(b) and Rule 10b-5 is to protect persons who are deceived in securities transactions—to make sure that buyers of securities get what they think they are getting----”) (Friendly, J.). Thus, the plaintiffs can succeed on their “enabling” theory only if recovery would restore to them “what they [thought] they [were] getting.” Chemical Bank, 726 F.2d at 943. This means that the plaintiffs were required to demonstrate that it is more likely than not that, if they had been apprised of JWP’s covenant defaults, they would have sought to block the Businessland merger. The various contractual provisions that might have served to block the merger were options, maturing upon a covenant default, that the plaintiffs might or might not have exercised. It is entirely possible that the plaintiffs would have lost confidence in JWP’s management upon learning of its accounting misrepresentations, and blocked further mergers as a result. But it is also plausible that the plaintiffs would have viewed JWP’s business strategy and accounting irregularities as independent concerns, and therefore would have decided that the better course was to allow JWP to continue with a theretofore successful acquisition strategy. If the latter is true, then recovery under the “enabling” theory would be a windfall because it would place the plaintiffs in a position superior to where they would be had the defendant never made misrepresentations in the first place. As indicated above, our mandate required the district court to determine if the defendant had caused “losses incurred for risks which the investors did not intend to take....” AUSA Appeal, 206 F.3d at 217 n. 6. Even with the default options, the plaintiffs assumed risks in buying the *672notes, including that the decisions they might make should those options mature could prove unwise. For this reason, the plaintiffs were required to demonstrate that, with the information available at the time, it is more probable than not that they would have exercised their options in a way that would have ultimately obviated their losses.
We hold that the plaintiffs have failed to demonstrate that it is more likely than not that they would have exercised their default options to block the Businessland merger. Judge Oakes noted in his opinion that, on the record before us then, the determination of what the plaintiffs and JWP would have done had the defendant revealed the defaults before the Business-land merger involved pure speculation. AUSA Appeal, 206 F.3d at 210. The plaintiffs chose not to seek to introduce new evidence on remand that might have transformed such speculation into fact-based analysis. The burden is on the plaintiffs to prove the elements of their claim, including loss causation. See Citibank, N.A. v. K-H Corp., 968 F.2d 1489, 1494 (2d Cir.1992). Because the plaintiffs failed to produce such evidence, their enabling theory fails.
The plaintiffs argue that the defendant should be estopped from alleging that their enabling theory lacks evidentiary support because the defendant successfully moved in this case’s original proceedings to disallow the introduction of evidence regarding what the plaintiffs might have done had they learned of JWP’s covenant defaults. Whatever force this argument may have had on this case’s first appearance before us, it is unavailing now. The enabling theory was contemplated by our mandate on remand, as was further fact-finding. Indeed, the district court invited the introduction of new evidence. The plaintiffs declined.
Because the plaintiffs have not established that it is more likely than not that they would have sought to block the Businessland merger had they known of JWP’s covenant default, we do not reach the potentially independent question whether they have demonstrated that it was foreseeable that they would have exercised their options in a way to reduce their losses.
The Forbearance Theory
The plaintiffs also advance on appeal a “forbearance” theory of causation, in which the defendant allegedly caused their loss by depriving them of their bargained-for right to accelerate payment on the notes upon default by JWP. By accelerating payment upon notification that JWP’s books did not comply with GAAP, the argument goes, the plaintiffs might have recovered significantly more than they ultimately fetched when they sold their notes on the market after JWP’s bankruptcy. This theory might have been encompassed by the references in Judge Oakes’s opinion to the “enabling” question because, presumably, JWP would have been unable to pay for Businessland if it also had to meet its principal obligations on its notes.
Our analysis of the forbearance theory is parallel to that of the enabling theory, and reaches the same conclusion. The forbearance theory fails for lack of evidentiary support because the plaintiffs have not proven that it is more likely than not that they would have exercised their acceleration options had they learned of the misrepresentations in JWP’s financial statements. In deciding whether to accelerate payment, the plaintiffs likely would have considered several factors, including market interest rates, the perceived ability of JWP to meet its obligations, and their expected recovery from the bankruptcy proceeding that acceleration likely would *673have precipitated. Recovery here thus requires evidence that the plaintiffs would more likely than not have exercised their options had the defendant revealed that JWP’s books were not in compliance with GAAP. But the plaintiffs did not introduce such evidence on remand. We thus see no basis upon which to disagree with the district court’s factual finding that, even if the defendant had issued revised financial statements and thereby revealed that JWP was in covenant default, “there is no reason to expect that plaintiffs would have exercised the option to declare a default and accelerate the notes.” AUSA Remand, 119 F.Supp.2d at 404. For this reason, the plaintiffs’ forbearance theory also fails for lack of proof.
Again, we need not reach the potentially separate question whether acceleration by the plaintiffs was foreseeable on the facts of this case.
The “Buy and Hold” Theory
Finally, the plaintiffs advance a “buy and hold” theory, premised largely on our decision in Marbury Mgmt., Inc. v. Kohn, 629 F.2d 705 (2d Cir.1980). In that case, investors purchased and then declined to sell shares based on the advice of a stock broker who had misrepresented his credentials. Id. at 707. We allowed the investors to recover the amount their stock value fell in value after the broker dissuaded them from selling it. Id. at 716. Along similar lines, the plaintiffs argue that the repeated fraud by the defendant induced them not only to buy the notes, but also to hold them (rather than sell them or accelerate payment), and thus the plaintiffs are entitled to recover the amount by which the notes lost value.
We do not think that this theory of recovery survived our mandate in AUSA Appeal. Judge Oakes’s opinion, while discussing Marbury, ultimately focused on the question whether the defendant’s fraud somehow enabled the Businessland acquisition. This is separate from the question whether the note holders would have sold their notes on the market had they learned of JWP’s covenant default. But even if this theory of recovery was preserved on remand, it fails as a matter of law in this case.
The dissent in Marbury argued that the Marbury majority failed to distinguish between transaction causation and loss causation. Id. at 720. The Marbury majority did not contest this objection, but instead argued that the distinction was inappropriate given the particular type of fraud involved in that case:
Differentiating transaction causation from loss causation can be a helpful analytical procedure only so long as it does not become a new rule effectively limiting recovery for fraudulently induced securities transactions to instances of fraudulent representations about the value characteristics of the securities dealt in. So concise a theory of liability for fraud would be too accommodative of many common types of fraud, such as the misrepresentation of a collateral fact that induces a transaction.
Marbury, 629 F.2d at 710 n. 3. In other words, in Marbury we emphasized that the type of fraud involved was not about the “value characteristics” of securities, but rather about a “collateral” issue—the quality of the broker giving investment advice. But the defendant’s misrepresentations in this case were about the value characteristics of the securities, as they related directly to the financial condition of their issuer. Thus, Marbury contemplates that, in claims such as the plaintiffs’, loss causation is a necessary element, and thus the plaintiffs may not recover merely by showing that fraud induced them to buy and hold securities. Our mandate required a showing of loss causation, and thus we *674decline to allow recovery without such a showing based on a decision that, by its own terms, distinguishes itself factually from a case in which such a showing is required.
For the foregoing reasons, the judgment of the district court is hereby AFFIRMED.