Full opinion text
JOSÉ A. CABRANES, Circuit Judge. Alberto Vilar and Gary Alan Tanaka appeal their judgments of conviction, entered on February 8, 2010, and February 10, 2010, respectively, in the United States District Court for the Southern District of New York (Richard J. Sullivan, Judge.) In simple terms, a jury found that Vilar and Tanaka lied to clients about the nature and quality of certain investments. This appeal raises a number of substantial issues, including a question left open after the Supreme Court’s decision in Morrison v. National Australia Bank Ltd., 561 U.S. 247, 130 S.Ct. 2869, 177 L.Ed.2d 535 (2010): whether criminal liability under Section 10(b) of the Securities Exchange Act of 1934 (“Section 10(b)”) extends to conduct in connection with an extraterritorial purchase or sale of securities. We conclude as follows. First, Section 10(b) and its implementing regulation, Rule 10b-5, do not apply to extraterritorial conduct, regardless of whether liability is sought criminally or civilly. Accordingly, a defendant may be convicted of securities fraud under Section 10(b) and Rule 10b-5 only if he has engaged in fraud in connection with (1) a security listed on a U.S. exchange, or (2) a security purchased or sold in the United States. Although the District Court did not require proof of domestic securities transactions in this case, this error was not “plain” since it did not affect the outcome of the proceedings, and therefore did not affect Vilar and Ta-naka’s substantial rights. Second, when proceeding criminally or civilly under Section 10(b) or Rule 10b-5, the government need not prove that the victims of a fraudulent scheme actually relied on the alleged material misrepresentations or omissions. Because reliance is not an element of a Section 10(b) offense, the District Court did not err by not instructing the jury on the issue of reliance. Third, although the mail fraud charge in the indictment specified that the mailing itself was false or fraudulent, the District Court’s instruction permitting the jury to convict the defendants based on a mailing that itself contained no false or fraudulent statement did not “constructively amend” the indictment. Fourth, on remand, the District Court must decide what acts constitute the offense conduct for the purposes of calculating the appropriate loss amount at sentencing, and re-determine the amount of money subject to forfeiture. Further, the District Court must, in accordance with the Mandatory Victim Restitution Act, limit its restitution order to victims of the scheme who purchased securities domestically. Fifth, with the exception of Vilar’s ineffective assistance of counsel claim, which we do not reach, Vilar and Tanaka’s remaining claims are without merit. ' BACKGROUND It is fair to say that, from the mid-1980s until their arrest in 2005, Vilar and Tanaka were prominent investment managers and advisers. Prior to the technology market crash of 2000-2001, they were responsible for managing approximately $9 billion in investments for their clients. Vilar and Tanaka managed their clients’ assets through a number of different funds and entities. In 1986, they founded and became the sole shareholders of Amerindo Investment Advisors Inc. (“Amerindo U.S.”), an investment adviser registered with the Securities and Exchange Commission (“SEC”). Amerindo U.S. was a California corporation with principal offices in San Francisco and New York City. Vilar and Tanaka also founded and were the sole shareholders of (1) Amerindo Investment Advisors, Inc. (“Amerindo Panama”), a corporation organized under the laws of the Republic of Panama that managed an off-shore investment fund offered to U.S. investors, and (2) Amerindo Investment Advisors (UK) Ltd. (“Amerindo U.K.”), a United Kingdom corporation, which managed portfolios of U.S. emerging growth stocks for U.K.-based clients. From at least July 1986 until May 2005, Vilar and Tanaka offered clients the opportunity to invest in “Guaranteed Fixed Rate Deposit Accounts” (“GFRDAs”). The GFRDA program was made available only to a select group of individual clients, who were generally close friends or family members of Vilar or Tanaka. Vilar and Tanaka promised investors in the GFRDA program that they would receive a high, fixed rate of interest over a set term, and that the overwhelming majority of the GFRDA funds would be invested in high-quality, short-term deposits, including U.S. Treasury bills. The balance of the capital in the GFRDAs—generally no more than twenty-five percent—was to be invested in publicly traded emerging growth stocks. Despite Vilar and Tanaka’s description of the GFRDA program, they invested all of the funds in technology and biotechnology stocks, presumably in the hopes of meeting or even exceeding the high “guaranteed” rates of return. The downside of this scheme, of course, was that the GFRDAs were volatile and not safe investments at all. And so, when the so-called dot-com bubble “burst” in the fall of 2000, the value of the investments held by the GFRDAs dropped precipitously. Accordingly, Vilar and Tanaka could not pay the promised rates of return and, as a consequence, several GFRDA investors lost millions of dollars. In June 2002, as the GFRDA scheme was falling apart, Vilar and Tanaka approached a long-standing client, Lily Cates, with the opportunity to invest in a type of venture known as a Small Business Investment Company (“SBIC”). Vilar told Cates that he and Tanaka had been approved for an SBIC license, which would allow the SBIC to obtain matching funds from the federal government’s Small Business Administration (“SBA”) for the SBIC’s investments. In fact, Vilar and Tanaka had never received an SBIC license and, indeed, had been denied such a license multiple times. On June 20, 2002, Cates invested $5 million in the SBIC formed by Amerindo, and her funds were deposited into an Am-erindo bank account at Bear, Stearns & Co., in the name of a Panamanian corporation called “Amerindo Management Inc.” (“AMI”). Vilar and Tanaka quickly drew on these funds in order to meet various personal and corporate obligations. Notably, Vilar and Tanaka made the following transactions: (1) on June 25, 2002, Tanaka transferred $1 million to a personal bank account held by Vilar, and Vilar immediately used that money for a variety of personal expenses, including a substantial donation to his alma mater; and (2) on July 9, 2002, Vilar and Tanaka wired approximately $2.85 million of Cates’s money from the AMI account to an account in Luxembourg, as part of a settlement agreement with a former GFRDA investor. Over the next two years—during which Vilar repeatedly assured Cates that her funds were safely in escrow—Vilar and Tanaka continued to use Cates’s SBIC investment account for their own needs. For example, in 2003, Tanaka forged Cates’s signature to authorize a $250,000 transfer from her SBIC account to one of Vilar’s personal accounts. More than $50,000 of that transfer was used by Vilar to make a personal mortgage payment. In early 2005, Cates requested that Vi-lar return her money and close her account. Vilar responded that she would have to make her request of Amerindo Panama—an organization with which she had never previously interacted. With her suspicions raised, Cates reported Vilar and Tanaka to the SEC. Vilar made several false statements in response to the SEC’s inquiries, hoping to obscure the SBIC scheme. On August 15, 2006, the Department of Justice indicted Vilar and Tanaka, charging them in twelve counts with: (1) conspiracy to commit securities fraud, investment adviser fraud, mail fraud, wire fraud, and money laundering, in violation of 18 U.S.C. § 371 (Count One); (2) securities fraud, in violation of 15 U.S.C. §§ 78j(b), 78ff and 17 C.F.R. § 240.10b-5 (Counts Two and Three); (3) investment adviser fraud, in violation of 15 U.S.C. §§ 80b-6 and 80b-7 (Count Four); (4) mail fraud, in violation of 18 U.S.C. § 1341 (Count Five); (5) wire fraud, in violation of 18 U.S.C. § 1343 (Counts Six and Seven); (6) money laundering, in violation of 18 U.S.C. § 1957 (Counts Eight through Eleven); and (7) the making of false statements to the SEC, in violation of 18 U.S.C. § 1001(a) (Count Twelve). Trial began before Judge Sullivan and a jury on September 22, 2008. On November 19, 2008, after a nine-week trial, the jury convicted Vilar on all twelve counts and convicted Tanaka on Counts One (conspiracy), Three (securities fraud relating to the GFRDA scheme), and Four (investment adviser fraud). Tanaka was acquitted on the other nine counts. On February 8, 2010, the District Court sentenced Vilar principally to a term of 108 months’ imprisonment. Two days later, Tanaka was sentenced to a term of sixty months’ imprisonment. On April 5, 2010, the District Court ordered both defendants to pay almost $35 million in restitution, including a 9% compounding interest rate, and to forfeit more than $54 million. Vilar and Tanaka now appeal. DISCUSSION Vilar and Tanaka raise what can only be described as a host of challenges to their convictions and sentences. For ease of comprehension, we address, first, Vilar and Tanaka’s claim that the conduct underlying their convictions for securities fraud was “extraterritorial,” and therefore not criminal under Section 10(b) or Rule 10b-5; second, their challenges to the indictment; third, their various challenges to the admission of evidence introduced by the government at trial; fourth, their challenges to the District Court’s jury instructions; fifth, their challenges to the sufficiency of the evidence supporting their convictions; sixth, their challenges to their sentences; and, finally, seventh, their claims of ineffective assistance of counsel. I. “Extraterritoriality” Vilar and Tanaka contend that their respective convictions for securities fraud must be reversed because their conduct was extraterritorial, meaning that it “occurred] in the territory of [a] sovereign [other than the United States],” Kiobel v. Royal Dutch Petroleum Co., — U.S. -, 133 S.Ct. 1659, 1669, 185 L.Ed.2d 671 (2013), and therefore was not proscribed by Section 10(b) or Rule 10b-5. They rely on the Supreme Court’s holding in Morrison, which was decided after Vilar and Tanaka were convicted, and which limited Section 10(b) and Rule 10b-5 to prohibiting fraud committed in connection with “transactions in securities listed on domestic exchanges, and domestic transactions in other securities.” Morrison, 130 S.Ct. at 2884. Observing that Momson was a civil lawsuit, the government responds that Morrison’s geographic limit on the reach of Section 10(b) and Rule 10b-5 applies only in the civil context and therefore is no bar to Vilar and Tanaka’s criminal convictions. In the alternative, the government argues that Vilar and Tanaka’s illegal conduct was “territorial” within the meaning of Morrison, inasmuch as at least some of the transactions were “domestic transactions in other securities.” Id. Although we conclude that Morrison does apply to criminal cases brought pursuant to Section 10(b) and Rule 10b-5, we agree that the record in this case confirms that Vilar and Tanaka did perpetrate fraud in connection with domestic securities transactions, and we therefore affirm their convictions. A. Standard of Review It is an axiom of appellate review that “[a] federal court of appeals normally will not correct a legal error made in criminal trial court proceedings unless the defendant first brought the error to the trial court’s attention.” Henderson v. United States, - U.S. -, 133 S.Ct. 1121, 1124, 185 L.Ed.2d 85 (2013). Accordingly, where, as here, defendants present us with a claim that they did not give the district court an opportunity to consider, we limit our review to curing “plain error.” United States v. James, 712 F.3d 79, 96 (2d Cir.2013). This standard is met when “(1) there is an error; (2) the error is clear or obvious, rather than subject to reasonable dispute; (3) the error affected the appellant’s substantial rights, which in the ordinary case means it affected the outcome of the district court proceedings; and (4) the error seriously affects the fairness, integrity or public reputation of judicial proceedings.” United States v. Marcus, 560 U.S. 258, 130 S.Ct. 2159, 2164, 176 L.Ed.2d 1012 (2010) (internal quotation marks and brackets omitted). Plain error review applies equally where the defendant did not object before the trial court because he failed to recognize an error, and where the defendant did not object because the trial court’s decision was correct at the time but assertedly became erroneous due to a supervening legal decision. See Johnson v. United States, 520 U.S. 461, 466-67, 117 S.Ct. 1544, 137 L.Ed.2d 718 (1997). In all cases, we look not to the law at the time of the trial court’s decision to assess whether the error was plain, but rather, to the law as it exists at the time of review. See Henderson, 133 S.Ct. at 1129-30. In other words, a decision of the trial court that was perfectly proper when issued may nonetheless be considered “plainly erroneous” on appeal due to a supervening change in the law. B. Analysis 1. “Clear or Obvious Error” The question presented here requires us to consider yet another issue created by the Supreme Court’s far-reaching holding in Morrison. Section 10(b) and Rule 10b-5 together proscribe the use of fraudulent schemes “in connection with the purchase or. sale of any security registered on a national securities exchange or any security not so registered....” 15 U.S.C. § 78j (b). Prior to the Supreme Court’s decision in Morrison, “[i]n determining whether § 10(b) and Rule 10b-5 could apply extraterritorially, this Court had ... applied the so-called conduct and effects test, which focused on: (1) whether the wrongful conduct occurred in the United States, and (2) whether the wrongful conduct had a substantial effect in the United States or upon United States citizens.” Absolute Activist Value Master Fund Ltd. v. Ficeto, 677 F.3d 60, 65 (2d Cir.2012) (internal quotation marks omitted). Morrison did away with this test. Relying on the “longstanding principle of American law that legislation of Congress, unless a contrary intent appears, is meant to apply only within the territorial jurisdiction of the United States,” Morrison, 130 S.Ct at 2877 (internal quotation marks omitted), the Supreme Court rejected any extraterritorial application of Section 10(b) and 10b-5, and instructed that “Section 10(b) reaches the use of a manipulative or deceptive device or contrivance only in connection with the purchase or sale of a security listed on an American stock exchange, and the purchase or sale of any other security in the United States.” Id. at 2888. Despite the supposed bright-line nature of Morrison’s holding, there has been no shortage of questions raised in its wake. This appeal requires us to decide whether Morrison’s limit on the scope of Section 10(b) liability extends to criminal prosecutions brought under that provision. Although we have not yet addressed this specific issue, we have no problem concluding that Morrison’s holding applies equally to criminal actions brought under Section 10(b), and that this result is “clear or obvious” for the purposes of plain error review. We reach this result for two reasons: (1) the presumption against extraterritoriality applies to criminal statutes, and (2) the presumption against extraterritoriality applies to Section 10(b). i. Criminal Statutes First, the government is incorrect when it asserts that “the presumption against extraterritoriality for civil statutes ... simply does not apply in the criminal context.” Gov’t Br. 96. As we have observed, “[i]t is beyond doubt that, as a general proposition, Congress has the authority to ‘enforce its laws beyond the territorial boundaries of the United States.’ ” United States v. Yousef, 327 F.3d 56, 86 (2d Cir.2003) (quoting EEOC v. Arabian Am. Oil Co., 499 U.S. 244, 248, 111 S.Ct. 1227, 113 L.Ed.2d 274 (1991)). But the courts have also recognized the “eom-monsense notion that Congress generally legislates with domestic concerns in mind,” Smith v. United States, 507 U.S. 197, 204 n. 5, 113 S.Ct. 1178, 122 L.Ed.2d 548 (1993), and “the presumption that United States law governs domestically but does not rule the world,” Kiobel, 133 S.Ct. at 1664 (internal quotation marks omitted). For these reasons, “[w]hen a statute gives no clear indication of an extraterritorial application, it has none.” Morrison, 130 S.Ct. at 2878. The government contends, relying on United States v. Bowman, 260 U.S. 94, 43 S.Ct. 39, 67 L.Ed. 149 (1922), that the presumption against extraterritoriality has no place in our reading of criminal statutes. To the contrary, no plausible interpretation of Bowman supports this broad proposition; fairly read, Bowman stands for quite the opposite. In Bowman, the Supreme Court was asked to consider criminal charges brought against sailors who, while at sea, conspired to defraud a company owned by the United States. See id. at 95-96, 43 S.Ct. 39. The Court explained: Crimes against private individuals or their property, like assaults, murder, burglary, larceny, robbery, arson, embezzlement and frauds of all kinds, which affect the peace and good order of the community, must of course be committed within the territorial jurisdiction of the government where it may properly exercise it. If punishment of them is to be extended to include those committed outside of the strict territorial jurisdiction, it is natural for Congress to say so in the statute, and failure to do so will negative the purpose of Congress in this regard. Id. at 98, 43 S.Ct. 39. The Court nonetheless concluded that charges could be brought in district court in that case, because “the same rule of interpretation should not be applied to criminal statutes which are, as a class, not logically dependent on their locality for the Government’s jurisdiction, but are enacted because of the right of the Government to defend itself against obstruction, or fraud ivherever perpetrated, especially if committed by its own citizens, officers, or agents.” Id. (emphasis supplied). In other words, the presumption against extraterritoriality does apply to criminal statutes, except in situations where the law at issue is aimed at protecting “the right of the government to defend itself.” Id. Indeed, we have repeatedly observed that Bowman makes precisely this distinction. For example, in United States v. Gatlin, 216 F.3d 207 (2d Cir.2000), we observed that “[sjtatutes prohibiting crimes against the United States government may be applied extraterritorially even in the absence of ‘clear evidence’ that Congress so intended,” but that “the Bowman Court explicitly stated that the presumption against extraterritoriality does apply to ‘[cjrimes against private individuals or their property----’ ” Id. at 211 n. 5 (quoting Bowman, 260 U.S. at 98, 43 S.Ct. 39) (emphases in Gatlin); see also Yousef, 327 F.3d at 87-88. And, we have explicitly recognized that “although there is no general bar against the extraterritorial application of our criminal laws to American citizens, the Supreme Court has long recognized a presumption against such applications.” United States v. Kim, 246 F.3d 186, 188-89 (2d Cir.2001); see also Small v. United States, 544 U.S. 385, 388-89, 125 S.Ct. 1752, 161 L.Ed.2d 651 (2005) (using the presumption against extraterritoriality as guidance in the criminal context); United States v. Ayesh, 702 F.3d 162, 166 (4th Cir.2012) (“Whether Congress has exercised that authority is a matter of statutory construction and, generally, statutes enacted by Congress, including criminal statutes, apply only within the territorial jurisdiction of the United States.”). The government nonetheless argues that we have previously interpreted Bowman as limiting the presumption against extraterritoriality to civil statutes. The government draws our attention to United States v. Siddiqui, 699 F.3d 690 (2d Cir.2012), where we stated that “[tjhe ordinary presumption that laws do not apply extraterri-torially has no application to criminal statutes,” id. at 700, and United States v. Al Kassar, 660 F.3d 108 (2d Cir.2011), where we similarly wrote that “[tjhe presumption that ordinary acts of Congress do not apply extraterritorially ... does not apply to criminal statutes,” id. at 118. However broadly worded, these statements must be understood in their context. In Al Kassar, we considered the extraterritorial application of four counts of conviction that relied on statutes with “explicit provisions applying them extraterritorially,” and one count for “conspiracy to kill U.S. officers or employees,” which we held applies extraterritorially in light of “the nature of the offense—protecting U.S. personnel from harm when acting in their official capacity.” Al Kassar, 660 F.3d at 118. Siddiqui also addressed statutes designed to protect U.S. personnel engaged in the performance of their duties from assault or interference. Siddiqui, 699 F.3d at 701. In other words, Al Kassar and Siddiqui both followed Bowman’s rule precisely—they required that the relevant statutes either contain a clear indication of Congress’s intent to provide for extraterritorial application or relate to crimes against the United States government. Inasmuch as these cases relied upon Bowman to reach this result, they surely do not require that the presumption against extraterritoriality be laid aside in all criminal cases. They are simply applications of Bowman’s holding. Further, the government provides little reason, beyond its misplaced reliance on Bowman, for why the presumption against extraterritoriality should not apply to criminal statutes. The government argues that criminal statutes “are concerned with prohibiting individuals ... from defrauding American investors and from using the infrastructure of American commerce to defraud investors” and that applying the presumption against extraterritoriality to criminal statutes “would create a broad immunity for criminal conduct simply because the fraudulent scheme culminates in a purchase or sale abroad.” Gov’t Br. 98-99. But much the same could be said of civil fraud statutes: Applying the presumption against extraterritoriality immunizes thieves and swindlers from civil liability for defrauding Americans abroad. More to the point, the distinction the government attempts to draw between civil and criminal laws is no response to the fundamental purposes of the presumption. The Supreme Court has emphasized that we apply this rule of statutory interpretation because we understand that “Congress generally legislates with domestic concerns in mind,” Smith, 507 U.S. at 204 n. 5, 113 S.Ct. 1178, and because the presumption “serves to protect against unintended clashes between our laws and those of other nations which could result in international discord,” Kiobel, 133 S.Ct. at 1664 (internal quotation marks omitted). We discern no reason that these concerns are less pertinent in the criminal context. Finally, the government argues, that Section 10(b) belongs to the “class [of statutes that are] not logically dependent on their locality for the Government’s jurisdiction, but are enacted because of the right of the Government to defend itself against obstruction, or fraud wherever perpetrated, especially if committed by its own citizens, officers or agents.” Bowman, 260 U.S. at 98, 43 S.Ct. 39. We are not persuaded. Although Section 10(b) clearly forbids a variety of fraud, its purpose is to prohibit “[c]rimes against private individuals or their property,” which Bowman teaches is exactly the sort of statutory provision for which the presumption against extraterritoriality does apply. Id. In sum, the general rule is that the presumption against extraterritoriality applies to criminal statutes, and Section 10(b) is no exception. ii. Section 10(b) Second, even if it were the case that we do not generally apply the presumption against extraterritoriality to criminal statutes, Section 10(b) would still not apply extraterritorially in criminal cases. The reason is simple: The presumption against extraterritoriality is a method of interpreting a statute, which has the same meaning in every case. The presumption against extraterritoriality is not a rule to be applied to the specific facts of each case. See Morrison, 130 S.Ct. at 2877. A statute either applies extraterritorially or it does not, and once it is determined that a statute does not apply extra-territorially, the only question we must answer in the individual case is whether the relevant conduct occurred in the territory of a foreign sovereign. The Supreme Court has already interpreted Section 10(b), and it has done so in unmistakable terms: “Section 10(b) reaches the use of a manipulative or deceptive device or contrivance only in connection with the purchase or sale of a security listed on an American stock exchange, and the purchase or sale of any other security in the United States.” Id. at 2888. To permit the government to punish extraterritorial conduct when bringing criminal charges under Section 10(b) “would establish ... the dangerous principle that judges can give the same statutory text different meanings in different cases.” Clark v. Martinez, 543 U.S. 371, 386, 125 S.Ct. 716, 160 L.Ed.2d 734 (2005). The government nonetheless insists that Section 10(b) is interpreted differently in the criminal and civil contexts because different elements are required to prevail in each. More specifically, the government observes that only private plaintiffs must prove reliance, economic loss, and loss causation, whereas only the government (in criminal cases) must prove that the fraud was committed willfully. Critically, however, none of these differences relate to the conduct proscribed by Section 10(b). Reliance, economic loss, and loss causation relate to who (other than the government) may bring suit and not to the conduct prohibited by Section 10(b). The Supreme Court has made this distinction clear, explaining that [i]n our cases addressing § 10(b) and Rule 10b-5, we have confronted two main issues. First, we have determined the scope of conduct prohibited by § 10(b). Second, in cases where the defendant has committed a violation of § 10(b), we have decided questions about the elements of the 10b-5 private liability scheme: for example, whether there is a right to contribution, what the statute of limitations is, whether there is a reliance requirement, and whether there is an in pan delicto defense. The latter issue, determining the elements of the 10b-5 private liability scheme, has posed difficulty because Congress did not create a private § 10(b) cause of action and had no occasion to provide guidance about the elements of a private liability scheme. Cent. Bank of Denver, N.A. v. First Interstate Bank of Denver, N.A., 511 U.S. 164, 172-73, 114 S.Ct. 1439, 128 L.Ed.2d 119 (1994) (internal brackets and citations omitted). Accordingly, “when it comes to ‘the scope of the conduct prohibited by Rule 10b-5 and § 10(b), the text of the statute controls our decision’ ” and “[i]t is only with respect to the additional ‘elements of the 10b-5 private liability scheme’ that we ‘have had to infer how the 1934 Congress would have addressed the issues had the [civil] 10b-5 action been included as an express provision in the 1934 Act.’ ” Morrison, 130 S.Ct. at 2881 n. 5 (quoting Cent. Bank of Denver, 511 U.S. at 173, 114 S.Ct. 1439) (brackets omitted). As for the element of willfulness in criminal cases, it comes directly from Section 32 of the Securities Exchange Act of 1934, which permits criminal liability to attach to a violation of Section 10(b), only when the violation is willful. 15 U.S.C. § 78ff(a). But like the elements relevant only to private plaintiffs, the requirement of proving willfulness has nothing to do with the text or interpretation of Section 10(b). In other words, Section 32 provides no basis for expanding the conduct for which a defendant may be held criminally liable under Section 10(b). Nor, for that matter, can Rule 10b-5 provide for the extraterritorial reach that its underlying statute lacks. Although the Supreme Court has approved the delegation of authority to the SEC to create rules with criminal penalties, see United States v. O’Hagan, 521 U.S. 642, 650-51, 117 S.Ct. 2199, 138 L.Ed.2d 724 (1997), “criminal liability under SEC regulations for insider trading may not extend beyond the conduct that Congress intended to encompass in § 10(b) of the 1934 Act,” United States v. Gansman, 657 F.3d 85, 90 n. 5 (2d Cir.2011). 2. “Substantial Rights” Having concluded that it would be clear or obvious error to apply Section 10(b) and Rule 10b-5 to extraterritorial criminal conduct in light of Morrison, we now turn, in our analysis of the asserted “plain error,” to whether “the error affected the appellant’s substantial rights, which in the ordinary case means it affected the outcome of the district court proceedings.” Marcus, 130 S.Ct. at 2164 (internal quotation marks omitted). Under Morrison, a defendant may be convicted of securities fraud under Section 10(b) and Rule 10b-5 if he has engaged in fraud with respect to either (1) a security listed on an American exchange, or (2) a security purchased or sold in the United States. Morrison, 130 S.Ct. at 2888. There is no claim that any of the securities sold in this ease were listed on an American exchange. We ask, therefore, whether the jury would have found, beyond a reasonable doubt, that Vilar and Tanaka engaged in fraud in connection with a domestic purchase or sale of securities, in violation of Section 10(b) and Rule 10b-5. Cf. Needham, 604 F.3d at 679-80 (explaining that a defendant has not demonstrated plain error where, absent the asserted error, the government still would have proved its case beyond a reasonable doubt.). On this issue, we are guided by our test, recently enunciated in the aftermath of Morrison, for determining whether a security not listed on an American exchange was purchased or sold in the United States: “[A] securities transaction is domestic when the parties incur irrevocable liability to carry out the transaction within the United States or when title is passed within the United States.” Absolute Activist, 677 F.3d at 69. More specifically, a domestic transaction has occurred when “the purchaser [has] incurred irrevocable liability within the United States to take and pay for a security, or ... the seller [has] incurred irrevocable liability within the United States to deliver a security.” Id. at 68 (emphasis supplied). As to the GFRDA fraud (Counts One and Three), the government contends that one set of victims, the Mayer family, entered into and renewed their agreement in Puerto Rico, and another victim, Graciela Lecube-Chavez, did so in New York. See Gov’t Br. 101. As to the SBIC scheme (Count Two), the government points to evidence that Lily Cates “executed the documents necessary to invest in the SBIC in her own New York apartment and handed those documents to a New York messenger.” Gov’t Sur-Reply 12. In light of these domestic transactions, we are persuaded that, based on the record evidence, a jury would have found that Vilar and Tanaka engaged in fraud in connection with a domestic purchase or sale of securities pursuant to Section 10(b) and Rule 10b-5. In particular, the record contains correspondence between Vilar and the Mayer family, indicating that they met in Puerto Rico, where the Mayers lived, to discuss the GFRDA program and that the Mayers committed to the investment while in Puerto Rico. See Gov’t Supp. App’x 876-79; see also Trial Tr. 849 (explaining that all of the Mayers’ meetings with Vilar to discuss the GFRDA investment occurred in Puerto Rico). Indeed, the Mayers’ GFRDA application lists their address as Santurce, Puerto Rico, and Vilar and Ta-naka sent a letter confirming the Mayers’ investment to that address. See Tanaka App’x 534-37. The evidence similarly shows that Lisa Mayer was in Puerto Rico when she re-invested her family’s money in the GFRDA program, once the original investment had expired. See Gov’t Supp. App’x at 753-54. Much the same can be said of the evidence demonstrating that Lecube-Chavez irrevocably committed herself to her GFRDA investment while in New York. A series of letters sent by Vilar to Lecube-Chavez demonstrate that Lecube-Chavez was in New York when she received and signed the commitment forms for her GFRDA and sent the money required for opening her account. See Tanaka App’x 465-69, 520-24; Gov’t App’x 772; see also Trial Tr. 306-09 (testimony confirming that Lecube-Chavez sent a check to Amer-indo after receiving these letters). In sum, with respect to the purchases of GFRDAs by the Mayers and Lecube-Cha-vez, the record contains facts “concerning the formation of the contracts” and “the exchange of money,” which are precisely the sort that we indicated may suffice to prove that irrevocable liability was incurred in the United States. See Absolute Activist, 677 F.3d at 70. Despite this evidence of domestic securities transactions with the Mayers and Le-cube-Chavez, our analysis of whether the change in law created by Morrison would have affected the outcome of this case as to Counts One and Three is complicated by the fact that the jury rendered a general verdict covering all victims of the GFRDA scheme. In other words, it is possible that, in responding to a carefully drawn special verdict form, the jury would have found Vilar and Tanaka guilty only of defrauding victims outside of the United States, and not of defrauding the Mayers or Lecube-Chavez. Nonetheless, upon a review of the record, we have no doubt that the jury would have found Vilar and Tanaka guilty of violating Section 10(b) and Rule 10b-5 with respect to the Mayers and Lecube-Chavez specifically. The record contains evidence in the form of offering materials and prospectuses, as well as testimony by Lisa Mayer and Lecube-Chavez confirming beyond a reasonable doubt that Vilar and Tanaka, willfully, and with the intent to defraud, made material misrepresentations as to the nature of the GFRDA program in connection with their purchases of GFRDAs. See, e.g., Tanaka App’x 447-48 (offering circular); Gov’t Supp. App’x 878 (letter from Vilar to Dr. Herbert Mayer); Trial Tr. 301-04 (testimony of Graciela Lecube-Chavez); id. at 831-38 (testimony of Lisa Mayer). We therefore conclude that, notwithstanding the fact that the jury returned general verdicts as to Counts One and Three, the error made manifest by Morrison’s change in law would not have altered the outcome of the jury’s determinations as to Counts One or Three, and therefore did not affect Vilar and Tanaka’s substantial rights. Cf. Hedgpeth v. Pulido, 555 U.S. 57, 61, 129 S.Ct. 530, 172 L.Ed.2d 388 (2008) (harmless error analysis applied where a jury returned a general verdict after being instructed on alternative theories of guilt, one of which was erroneous). Count Two presents no such difficulties inasmuch as the jury convicted Vilar of defrauding Cates, and the record makes clear that Cates made her investment in the SBIC scheme while in the United States. Cates testified that she met with Vilar at his apartment in New York in June 2002 to discuss the investment opportunity, and shortly thereafter signed the commitment papers. See Trial Tr. 2098-104. We do not doubt that, on this record, the jury would have found that Cates incurred irrevocable liability to purchase her SBIC investment while in the United States. In sum, there was no plain error in Vilar and Tanaka’s convictions on Counts One, Two, or Three with respect to the territoriality of their conduct. II. Sufficiency of the Indictment Vilar and Tanaka challenge the indictment on two grounds. First, they claim that Count One (Conspiracy) was “duplicitous,” by which they mean that the indictment alleged one conspiracy where, in fact, there were two smaller conspiracies. Second, they claim that Count Four (Investment Adviser Fraud) was not sufficiently definite, and was later constructively amended at trial. We review de novo the denial of a motion to dismiss the indictment. United States v. Daley, 702 F.3d 96, 99-100 (2d Cir.2012). Neither claim has merit. As for the “duplicity” of Count One, Vilar and Tanaka argue that the GFRDA frauds and SBIC frauds were separate conspiracies, and should have been charged as such. The District Court rejected this argument, see United States v. Vilar, No. 05 Cr. 621(RJS), 2008 WL 4298545, at *1 (S.D.N.Y. Sept. 5, 2008), and we do so as well. We have held that “[a]n indictment is impermissibly duplicitous where: 1) it combines two or more distinct crimes into one count in contravention of Fed.R.Crim.P. 8(a)’s requirement that there be a separate count for each offense, and 2) the defendant is prejudiced thereby.” United States v. Sturdivant, 244 F.3d 71, 75 (2d Cir.2001) (internal quotation marks omitted). However, “this [CJourt has long held that acts that could be charged as separate counts of an indictment may instead be charged in a single count if those acts could be characterized as part of a single continuing scheme.” United States v. Olmeda, 461 F.3d 271, 281 (2d Cir.2006) (internal quotation marks omitted). It has been established for at least seventy years that “ ‘[t]he allegation in a single count of a conspiracy to commit several crimes is not duplicitous, for [t]he conspiracy is the crime, and that is one, however diverse its objects.’ ” United States v. Williams, 705 F.2d 603, 624 (2d Cir.1983) (quoting Braverman v. United States, 317 U.S. 49, 54, 63 S.Ct. 99, 87 L.Ed. 23 (1942)). The record leaves no doubt that the GFRDA and SBIC schemes can be characterized as part of one conspiracy to defraud investors by (1) lying about the nature of their investments and (2) continuing to mislead them into believing that their money was safe and invested in accordance with the representations they had received from Vilar and Tanaka. See, e.g., United States v. Tutino, 883 F.2d 1125, 1141 (2d Cir.1989) (“As long as the essence of the alleged crime is carrying out a single scheme to defraud, then aggregation is permissible.”). As for the definiteness of Count Four (Investment Adviser Fraud), Vilar and Tanaka argue that the indictment was not sufficient because it did not specify that the SBIC fraud, as opposed to the GFRDA fraud, formed the basis of Count Four. Pursuant to Federal Rule of Criminal Procedure 7, “[t]he indictment or information must be a plain, concise, and definite written statement of the essential facts constituting the offense charged.... ” Fed.R.Crim.P. 7(c)(1). As we have explained, “[a]n indictment is sufficient when it charges a crime with sufficient precision to inform the defendant of the charges he must meet and with enough detail that he may plead double jeopardy in a future prosecution based on the same set of events.” United States v. Yannotti, 541 F.3d 112, 127 (2d Cir.2008) (internal quotation marks omitted). “Moreover, an indictment need do little more than to track the language of the statute charged and state the time and place (in approximate terms) of the alleged crime.” Id. (internal quotation marks omitted). The indictment in this case sufficiently sets out the time and circumstances of the conspiracy and tracks the language of the statute charged. See Tanaka App’x 116. While it is true that the indictment does not explicitly refer to the SBIC scheme, we have little trouble concluding that it contained “sufficient precision” to inform Vilar and Tanaka of the charges to be met and to enable them to plead double jeopardy in the future. Yannotti, 541 F.3d at 127. Vilar and Tanaka nonetheless argue that the indictment was constructively amended by the government when it narrowed Count Four to the SBIC scheme because the indictment was worded more generally and could have encompassed broader conduct. We have only recently had occasion to recall that, “[t]o prevail on a constructive amendment claim, a defendant must demonstrate that the terms of the indictment are in effect altered by the presentation of evidence and jury instructions which so modify essential elements of the offense charged that there is a substantial likelihood that the defendant may have been convicted of an offense other than that charged in the indictment.” United States v. D’Amelio, 683 F.3d 412, 416 (2d Cir.2012) (internal quotation marks omitted; emphasis in original). Although “a constructive amendment is a per se violation of the Grand Jury Clause,” we have “consistently permitted significant flexibility in proof, provided that the defendant was given notice of the core of criminality to be proven at trial.” United States v. Banki 685 F.3d 99, 118 (2d Cir.2012) (internal quotation marks and citation omitted; emphasis in Banki). We have little doubt that Vilar and Tanaka were on notice of the “core of criminality” alleged in Count Four—namely that, between 2002 and 2005, Vilar and Tanaka defrauded Lily Cates, to whom they were investment advisers, as part of the SBIC scheme. Even if it were true that Count Four of the indictment originally contemplated both the GFRDA and SBIC schemes, rather than the latter scheme alone, “where a generally framed indictment encompasses the specific legal theory or evidence used at trial, there is no constructive amendment.” Id. (internal quotation marks and brackets omitted); see also United States v. Miller, 471 U.S. 130, 136, 105 S.Ct. 1811, 85 L.Ed.2d 99 (1985) (“As long as the crime and the elements of the offense that sustain the conviction are fully and clearly set out in the indictment, the right to a grand jury is not normally violated by the fact that the indictment alleges more crimes or other means of committing the same crime.”). Indeed, in contrast to the alleged narrowing of Count 4 here, we have specifically noted that a constructive amendment occurs “when the trial evidence or the jury charge operates to broaden the possible bases for conviction from that which appeared in the indictment.” Banki 685 F.3d at 118 (internal quotation marks omitted; emphasis supplied). In sum, Vilar and Tanaka have established no error based on the sufficiency of the indictment. III. Evidentiary Challenges Vilar and Tanaka claim that at trial the District Court erred by not suppressing (1) evidence obtained pursuant to an overbroad warrant to search Amerindo’s New York office; (2) evidence seized in a search of Amerindo’s London storage facility; and (3) testimony concerning statements made by Renata Tanaka, Tanaka’s wife, who worked in Amerindo’s London office. When considering a district court’s order denying a motion to suppress evidence, “we review the district court’s factual findings for clear error, viewing the evidence in the light most favorable to the government,” and review its legal conclusions de novo. United States v. Moreno, 701 F.3d 64, 72 (2d Cir.2012) (internal quotation marks and brackets omitted). We review a district court’s rulings about the admissibility of trial evidence for “abuse of discretion,” United States v. Coplan, 703 F.3d 46, 80 (2d Cir.2012), which means that we set aside its decision if “it based its ruling on an erroneous view of the law or on a clearly erroneous assessment of the evidence, or rendered a decision that cannot be located within the range of permissible decisions,” In re Sims, 534 F.3d 117, 132 (2d Cir.2008) (internal quotation marks and citation omitted) (explaining term of art “abuse of discretion”). A. The U.S. Search 1. Background On May 25, 2005, Magistrate Judge Frank Maas signed a search warrant for Amerindo’s office in New York City. The warrant permitted authorities to seize broad categories of documents, including all corporate records and client files, without any temporal limitation. The next day, investigators executed the warrant and seized approximately 170 boxes of documents, as well as 30 computers. Several hours into the search, counsel to Amerindo arrived at the office and indicated that he would cooperate with the government’s investigation. He offered to put into place a preservation policy for Amerindo’s documents and suggested that, because it appeared that the search would not be completed in a day, Amerindo would agree to accept service of a grand jury subpoena for the documents; Amerindo could then, in his view, disclose the requested documents in an orderly fashion. By early afternoon, the government had faxed the grand jury subpoena and Amerindo’s attorney had accepted service. In return for this acceptance, the investigators stopped their search. But Amerindo did not comply with the subpoena. Instead, Vilar and Tanaka filed a motion to quash the subpoena and suppress any evidence obtained during the search. In a thorough and well-reasoned Opinion and Order, Judge Kenneth M. Karas, who was then the presiding judge, determined that, although there was probable cause to search the New York office, certain portions of the warrant were not supported by probable cause and lacked sufficient particularity. See United States v. Vilar, No. S3 05-CR-621(KMK), 2007 WL 1075041, at *19-23 (S.D.N.Y. Apr. 4, 2007) (“Warrant Decision ”). Judge Karas went on to conclude that the evidence obtained could not be rescued from suppression by the good faith of the executing officers, but that certain valid portions of the warrant could be severed, and evidence seized pursuant to those valid portions would be admissible. Id. at *23-24, *31-34. Finally, Judge Karas found that the grand jury subpoena was not an improper extension of the search and, after modifying the subpoena to address certain areas that were overbroad, insufficiently particular, or overly burdensome, he denied the motion to quash. Id. at *39-50. In practical terms, the government was permitted to obtain the documents from Amerindo’s New York office through its grand jury subpoena. The saga of the evidence obtained from Amerindo’s New York office did not end there, however. Prior to trial, Vilar and Tanaka moved to exclude the documents produced pursuant to the grand jury subpoena on the ground that these documents represented so-called fruits of the unlawful search. Judge Sullivan, who by then had taken over the case, disagreed and held that both the “inevitable discovery” and “independent source” doctrines permitted the admission of the documents obtained through the grand jury subpoena. See United States v. Vilar, 530 F.Supp.2d 616, 626-34 (S.D.N.Y.2008) (“Subpoena Decision ”). 2. Analysis Vilar and Tanaka now argue that Judge Karas erred in severing the valid portions of the warrant and that Judge Sullivan erred in admitting the evidence obtained pursuant to the grand jury subpoena. We need not address Judge Karas’s invalidation of parts of the warrant, nor the independent source doctrine, because it is clear that Judge Sullivan properly admitted all of the evidence pursuant to the inevitable discovery doctrine. As a preliminary matter, Vilar and Tanaka contend, as they did before Judge Sullivan, that the government forfeited its claim that the materials obtained under the subpoena were admissible pursuant to the inevitable discovery doctrine. In particular, they argue that the documents admitted pursuant to the subpoena were the same ones suppressed pursuant to the overbroad warrant and, therefore, the government was required to assert the inevitable discovery doctrine before Judge Karas in opposition to Vilar and Tanaka’s motion to suppress evidence obtained by the warrant. This argument neglects the fact that the government explained its position on the admissibility of the documents to Judge Karas, see Subpoena Decision, 530 F.Supp.2d at 622; indeed, Judge Karas permitted the government to move forward with the subpoena, recognizing that the admissibility of the documents might be subject to challenge at a future date, see id. at 622-23. Even if the government did not incant the phrase “inevitable discovery” when defending before Judge Karas the admissibility of the documents seized pursuant to the warrant, Vi-lar and Tanaka have advanced no plausible authority for the notion that it was not perfectly appropriate to reserve those arguments for the time when Vilar and Ta-naka challenged the admissibility of the subpoenaed evidence itself. In short, the government did not waive its “inevitable discovery” argument in the proceedings before Judge Karas. Vilar and Tanaka’s substantive arguments fare no better. It is correct, of course, that the so-called exclusionary rule, “when applicable, forbids the use of improperly obtained evidence at trial.” Herring v. United States, 555 U.S. 135, 139, 129 S.Ct. 695, 172 L.Ed.2d 496 (2009). But “[u]nder the ‘inevitable discovery’ doctrine, evidence obtained during the course of an unreasonable search and seizure should not be excluded ‘if the government can prove that the evidence would have been obtained inevitably’ without the constitutional violation.” United States v. Heath, 455 F.3d 52, 55 (2d Cir.2006) (quoting Nix v. Williams, 467 U.S. 431, 447, 104 S.Ct. 2501, 81 L.Ed.2d 377 (1984)). Put another way, whether this exception to the exclusionary rule applies depends on whether “the disputed evidence inevitably [would] have been found through legal means ‘but for’ the constitutional violation[.]” Id. To prevail under the inevitable discovery doctrine, the government must prove “by a preponderance of the evidence” presented to the district judge that the evidence inevitably would have been discovered. Nix, 467 U.S. at 444, 104 S.Ct. 2501. In United States v. Cabassa, 62 F.3d 470 (2d Cir.1995), we acknowledged that using the preponderance-of-the-evidence standard to prove inevitability creates a problem of probabilities, and observed that even if each event in a series is individually more likely than not to happen, it still may be less than probable that the final event will occur. Id. at 474. For this reason, we subsequently explained “that illegally-obtained evidence will be admissible under the inevitable discovery exception to the exclusionary rule only where a court can find, with a high level of confidence, that each of the contingencies necessary to the legal discovery of the contested evidence would be resolved in the government’s favor.” Heath, 455 F.3d at 60. In other words, the government must prove that each event leading to the discovery of the evidence would have occurred with a sufficiently high degree of confidence for the district judge to conclude, by a preponderance of the evidence, that the evidence would inevitably have been discovered. . Judge Sullivan, in a careful and clear Memorandum and Order of January 17, 2008, found that (1) the subpoena was not issued on the basis of any information unlawfully seized from Amerindo’s New York office; (2) the government was actively investigating Amerindo and inevitably would have conducted a substantial search of the New York office; (3) Amerindo’s attorney inevitably would have raised the alternative of a grand jury subpoena; (4) the government inevitably would have issued the grand jury subpoena; and (5) Vilar and Tanaka inevitably would have produced the documents requested by the subpoena (which they, in fact, did). See Subpoena Decision, 530 F.Supp.2d at 627-32. We identify no error in these findings, let alone clear error. See Moreno, 701 F.3d at 72. Indeed, we can be confident that this sequence is what would have happened, because it did happen. This case presents the unusual scenario where the actual events played out exactly as they would have “but for” the over-breadth of the warrant, because the government actually obtained the evidence through alternative means that did not depend on any invalidity of the warrant. As Judge Sullivan noted, “it is beyond doubt that the Government had a lawful basis to be present in Amerindo’s office on the date of the search in order to execute the lawful portions of the Warrant,” Subpoena Decision, 530 F.Supp.2d at 627, because, as Judge Karas found—and Vilar and Tanaka do not contest—there was “[c]learly” probable cause to conduct a search of Amerindo’s New York office, Warrant Decision, 2007 WL 1075041, at *20. Further, the offer of Amerindo’s counsel to accept a grand jury subpoena had nothing to do with any illegality in the warrant; at the time, he thought the warrant was valid. The record, therefore, confirms that the government would have validly discovered the documents produced pursuant to the subpoena “but for” the over-breadth of the warrant. Vilar and Tanaka nonetheless insist that this material should have been excluded because the subpoena was issued after the government began its search pursuant to the (partly) invalidated warrant. They rely on our decision in United States v. Eng, 971 F.2d 854 (2d Cir.1992), in which we cautioned that, “[particular care is appropriate where ... subpoenas are issued after or at the time of the unlawful search,” and “subpoenas must not serve as an after the fact insurance policy to validate an unlawful search under the inevitable discovery doctrine.” Id. at 860-61 (internal quotation marks omitted). Put simply, Vilar and Tanaka assert that the inevitable discovery doctrine cannot apply where the subpoena was sought after the government initiated its partly unlawful search. We cannot agree. First, Eng created no such rule and, in fact, it rejected a per se rule “that the subpoena power never may be relied upon by the government to meet the inevitable discovery burden of proof.” Id. at 860. Indeed, we explained quite clearly that we could find “no reason why the government may not rely upon the subpoena power as one way it might meet the burden of proving inevitable discovery by a preponderance of the evidence.” Id. Second, based on the facts of this case, the temporal or causal relationship between the search of Amerindo’s New York office and the issuance of the subpoena has no bearing on whether “the disputed evidence inevitably [would] have been found through legal means ‘but for’ the constitutional violation[.]” Heath, 455 F.3d at 55. And, as we have explained, there is no doubt that the government would have discovered the evidence through the grand jury subpoena, irrespective of the invalidity of parts of the warrant. Accordingly, we identify no error in the admission of the documents from Amerin-do’s New York office. B. The Search in the United Kingdom 1. Background On October 13 and 14, 2005, British and American authorities searched the Cado-gan Tate warehouse in London for documents stored by Amerindo U.K. Warrant Decision, 2007 WL 1075041, at *17. In order to gain access to Amerindo’s documents in the United Kingdom, American authorities first filed a “Mutual Legal Assistance Treaty” (“MLAT”) request, in which they sought the assistance of U.K. law enforcement. Id. at *12. After receiving the request, a U.K. detective conducted his own investigation to ensure that any warrant application would comport with U.K. law and that it would be based on sufficient information to justify the issuance of a warrant. Id. at *11-13. Once these prerequisites had been satisfied, the detective sought and obtained a valid warrant from a magistrate. Id. at *16. As part of his normal procedure, the detective requested that American officials attend the search to provide advice on what materials were relevant. Id. at *13. Although American investigators did attend the search and offered their opinions on the relevance of various documents, Judge Karas found-—and we have no reason to doubt—that the authority to determine what documents could validly be seized under U.K. law remained at all times with the U.K. authorities. Id. at *16-17. Vilar and Tanaka subsequently sought to suppress the evidence obtained at the Cadogan Tate warehouse. Judge Karas, then presiding, denied their motion, holding (1) that the Fourth Amendment’s warrant requirement did not apply because the search was executed abroad, and (2) that the evidence did not have to be suppressed because the search was valid under U.K. law and was reasonable. See id. at *51-58. Vilar and Tanaka now renew their challenge to the admission of the evidence obtained in the United Kingdom. 2. Analysis After Judge Karas issued his opinion, we clarified the law applicable to searches of United States citizens conducted abroad by United States authorities, holding “that the Fourth Amendment’s warrant requirement does not govern searches conducted abroad by U.S. agents; such searches of U.S. citizens need only satisfy the Fourth Amendment’s requirement of reasonableness.” In re Terrorist Bombings of U.S. Embassies in E. Africa, 552 F.3d 157, 167 (2d Cir.2008) (“In re Terrorist Bombings ”). “To determine whether a search is reasonable under the Fourth Amendment, we examine the ‘totality of the circumstances’ to balance ‘on the one hand, the degree to which it intrudes upon an individual’s privacy and, on the other, the degree to which it is needed for the promotion of legitimate governmental interests.’ ” Id. at 172 (quoting Samson v. California, 547 U.S. 843, 848, 126 S.Ct. 2193, 165 L.Ed.2d 250 (2006)). Judge Karas’s undisputed factual findings provide ample support for the conclusion that the U.K. search was reasonable, and that the Cadogan Tate materials were therefore properly admitted. Indeed, Vilar and Tanaka’s only substantive challenge to this determination is a misguided effort to confine In re Terrorist Bombings to its facts. They contend that “[t]he interest in obtaining evidence of possible white collar crimes—particularly involving only a handful of investors—cannot compare to the national security interest in preventing murderous, terrorist attacks, the interest at stake in Terrorist Bombings.” Tanaka Br. 86-87. In re Terrorist Bombings was not limited to cases of suspected terrorism or instances of similarly horrific crime. We could not have been clearer in stating that “the Fourth Amendment’s warrant requirement does not govern searches conducted abroad by U.S. agents.” In re Terrorist Bombings, 552 F.3d at 167. The government has demonstrated that the U.K. search was reasonable; it need do no more. For this reason, the District Court properly admitted the evidence obtained in the U.K. C. Statements of Renata Tanaka The final evidentiary claim of Vilar and Tanaka may also be disposed of readily. They protest the admission of statements made by Renata Tanaka, the wife of defendant Tanaka, who worked at Amerindo U.K., to Stephen Gray, the attorney for several GFRDA clients. Judge Sullivan permitted Gray to relate the statements pursuant to Federal Rule of Evidence 801(d)(2)(D), which excludes from the definition of hearsay any statement “offered against an opposing party” that “was made by the party’s agent or employee on a matter within the scope of that relationship and while it existed.” Vilar and Tana-ka each argue that Renata Tanaka was not his agent and that the statements were not made within the scope of any agency relationship. However, after a lengthy colloquy, Judge Sullivan found, by a preponderance of the evidence, that Renata Tanaka was, in fact, an agent of both Vilar and Tanaka, and that the challenged statements were made within the scope of that relationship. See Tanaka App’x 345-61. We identify no error, let alone clear error, in Judge Sullivan’s findings, see Coplan, 703 F.3d at 80, nor in his conclusion that these findings satisfied the requirements of Rule 801(d)(2)(D), see United States v. Lauersen, 348 F.3d 329, 340 (2d Cir.2003). IV. Jury Instructions Defendants charge two main errors in the District Court’s jury instructions. First, they both argue that the District Court erroneously omitted a relia