Defendant Paul M. Daugerdas appeals from a judgment entered in the United States District Court for the Southern District of New York (Pauley, J.) following a jury trial convicting him of (1) one count of conspiracy to defraud the Internal Revenue Service (“IRS”) in violation of 18 U.S.C. § 371; see 26 U.S.C. § 7201 and 18 U.S.C. § 1343; (2) four counts of client tax evasion in violation of 26 U.S.C. § 7201 and 18 U.S.C. § 2; (3) one count of IRS obstruction in violation of 26 U.S.C. § 7212(a); and (4) one count of mail fraud in violation of 18 U.S.C. §§ 1341 and 1342. He was sentenced principally to 180 months’ imprisonment, three years’ supervised release, $164,737,500 in forfeiture, and $371,006,397 in restitution. He argues on appeal that (I) the evidence was insufficient to support his convictions; (II) the indictment was constructively amended; (III) the indictment was duplicitous; (IV) the accumulation of errors at trial violated his due process right to a fair trial; (V) the district court’s supplemental instruction on the Annual Accounting Rule misled the jury; (VI) his sentence was procedurally and substantively unreasonable; and (VII) the government failed to establish the requisite nexus between his crimes and the property sought in forfeiture. Finding no merit in his arguments, we AFFIRM.
BACKGROUND
The evidence taken in the light most favorable to the government showed the following.
Paul M. Daugerdas was a Certified Public Accountant (“CPA”) and tax attorney at Arthur Andersen through August of 1994; the law firm Altheimer & Gray from the end of 1994 through 1998; and the Chicago office of the law firm Jenkens & Gilchrist (“J & G”) from 1999 through April 2004. Throughout his career, Daugerdas developed, sold, and implemented a variety of tax-reduction strategies for wealthy clients: the so-called Short Sale Shelter, Short Option Shelter, Swaps Shelter, and HOMER Shelter. Besides Daugerdas’s employers, two other entities had significant involvement in this undertaking. The accounting firm BDO Seidman (“BDO”) referred its clients to J & G and helped to sell the shelters, and the investment bank Deutsche Bank Alex. Brown (“DB”) assisted J & G in the design of the shelters, held informational meetings with clients, and implemented the transactions that composed the shelters.
I. The Development and Sale of the Tax Shelters
Daugerdas designed and sold the shelters beginning in the early 1990s when he was a partner at Arthur Andersen. Although the transactions underlying the shelters changed over time, the facts surrounding them marketing and implementation varied little. Daugerdas designed the Short Sale Shelter, which created losses through the short sale of U.S. Treasury securities followed by transfers between a partnership, a limited liability company, and an S-corporation. In 1999, because Daugerdas and other J & G attorneys were concerned that pending legislation would render the Short Sale Shelter ineffective, they developed the Short Option Shelter as a substitute. This shelter generated losses through the sale of a digital currency option instead of through the short sale of Treasury securities but was otherwise similar to the Short Sale Shelter.
In August of 2000, the IRS announced that transactions like the Short Sale and Short Option Shelter would no longer pro*219vide the favorable tax treatment that J & G sought for its clients. To replace these two shelters, Daugerdas and his colleagues developed the Swaps Shelter, which simply replaced the digital currency option with a swap transaction. At approximately the same time as he was developing the Swaps Shelter, Daugerdas also worked on developing and implementing the HOMER Shelter. The structure of the HOMER Shelter’s underlying transactions was more complex than the structures of the other shelters.
As an essential part of the marketing of all the tax shelters,' Daugerdas and his colleagues issued “more-likely-than-not” opinion letters to clients who purchased the shelters. Such letters state that “under current U.S. federal income tax law it is more likely than not that” the transactions comprising the shelters are legal and will have the effect sought by the clients. They protect clients from the IRS’s imposition of a financial penalty in the event that the IRS does not permit the losses generated by the shelter to reduce the Ghent’s tax liability. Paralegals or attorneys who worked for Daugerdas generated these letters and Daugerdas often reviewed ■ and signed them himself. The letters stated that the clients had knowledge of the particular transactions underlying the shelter and that the clients were entering into the shelter for non-tax business reasons. Multiple clients testified that they never made representations of knowledge to Dauger-das or his associates and that, in any event, these representations were false because the clients knew little or nothing about the underlying transactions and entered into the shelters only to reduce their tax liability.
Because Daugerdas and his colleagues designed the transactions with a focus on their tax consequences rather than their profitability, they generally did not generate meaningful returns. For example, the clients who engaged in the Short Sale shelter in 1998 lost approximately $685,000 as part of the shelter. Clients who used the Short Option Shelter had a 4.6% to 37.6% chance of doubling their 1% investment. If they did not double their investment, they lost it in its entirety. Swaps Shelter clients were even less likely to profit from their transaction; of the approximately 60 participants in the trade, only two made a profit on the trade, one of whom made only one dollar. The chance of any client profiting from the HOMER shelter as it was originally designed was negligible — J & G associate attorney John Beery informed Daugerdas that an implementation issue would prevent any. of the HOMER clients from realizing a profit from the transaction. Nevertheless, Daugerdas chose to proceed and even issued “more-likely-than-not” opinion letters falsely stating that some of the transactions had a reasonable possibility of producing a profit. Moreover, the already-low profit potential of all the shelters disappeared entirely when the fees charged by J & G, BDO, and DB for the shelters were taken into account.
II. The Backdating of Shelter Transactions
As part of the implementation of the Swaps and Short Option Shelters, Dauger-das either directly or through his team at J & G participated in the correction and backdating of certain transactions that had originally been incorrectly implemented on ■behalf of (1) Matthew Coleman and Greg Blair; (2) Michael Toporek; and (3) the Aronoff family. A lawyer who worked with Daugerdas testified that he discussed with Daugerdas the backdating of transactions and how they could justify it. Another witness testified that Daugerdas was the head of the team of lawyers with whom *220she communicated regarding backdated transactions.
In 2001, to obtain ordinary losses, business partners Coleman and Blair consulted with Daugerdas and decided to enter into the Swaps Shelter. In December 2001, DB broker David Parse’s assistant, Carrie Yackee, received authorization from J & G to complete the purchase and sale of Cisco shares as part of the shelter. In February 2002, J & G attorneys realized that this transaction had generated capital losses rather than ordinary losses. To correct this error, J & G faxed to Parse an undated letter asking him to reverse the Cisco sale, along with letters dated December 24 and 28, 2001, directing him to implement transactions that would -have the effect of generating the ordinary losses requested by Coleman and Blair. Yackee received these faxes, and ensured that DB carried out the instructions they requested. She eventually returned to J & G the account statements containing the backdated transactions. These statements were used in the preparation of Coleman and Blair’s 2001 tax returns.
For 2001 and 2002, Toporek sought from the Swaps Shelter $1.3 million in ordinary losses and $700,000 in capital losses. J & G made a mistake in the instructions given to Parse, requesting the reverse: $700,000 in ordinary losses and $1.3 million in capital losses.. In March 2002, J & G became aware of this error. After internal, discussions regarding how to proceed, Beery asked Parse to implement several transactions in April 2002 but date them to December 28, 2001. After DB implemented the corrective transactions, Daugerdas sent Toporek’s tax preparer, Judith Quedenfeld, a letter falsely indicating that the transactions .had been completed in 2001 and seeking the return of a prior opinion, which had detailed the original, erroneous transaction. Quedenfeld used documentation of the corrected transaction in the preparation of Toporek’s tax returns and, as a result, Toporek was able to claim the losses he had originally sought.
J & G attorneys and DB employees engaged' in a similar course of conduct to correct mistakes that were made in the implementation of the Short Option Shelter for the Aronoff family. This incident is not at issue in this appeal.
III. Daugerdas’s Personal Use of Tax Shelters
Daugerdas also used tax shelters to reduce his personal tax liability. From 1993 to 1998, Daugerdas received more than $26 million in income. He personally used the Short- Sale Shelter yearly during that time period, and, as a result, he paid only $7,315 in federal personal income tax. From 1999 to 2001, Daugerdas had over $79 million in income. During those years, he used the Short Option Shelter to offset this income and paid no income tax.
IV. Procedural History
In 2009, Daugerdas, Parse, and others involved in the design, marketing, and implementation. of the shelters were indicted for. conspiracy, tax evasion, obstructing the IRS, and mail fraud. Daugerdas was convicted on all of the counts with which he was charged. The results for the other defendants were mixed. On June 4, 2012, the district court granted a new trial based on juror misconduct as to all defendants except Parse, who, the district court found, had waived his right to raise the objection. After this court reversed the district court’s determination as to Parse, United States v. Parse, 789 F.3d 83 (2d Cir. 2015), Parse negotiated a deferred prosecution agreement with the government.
On July 1, 2013, the government filed a new indictment, containing charges similar to those in the prior indictment, against *221Daugerdas and a co-defendant who was later acquitted on all counts. The indictment charged Daugerdas with: (1) one count of conspiracy to defraud the IRS in violation of 18 U.S.C. § 371; see 26 U.S.C. § 7201 and 18 U.S.C. § 1343; (2) ten counts of client tax evasion in violation of 26 U.S.C. § 7201 and 18 U.S.C. § 2; (3) three counts of personal tax evasion in violation of 26 U.S.C. § 7201; (4) one count of IRS obstruction in violation of 26 U.S.C. § 7212(a); and (5) one count of mail fraud in violation of 18 U.S.C. § 1341 and 1342. Daugerdas was ultimately convicted of (1) one count of conspiracy to defraud the IRS; (2) four counts of client tax evasion (arising from the shelters implemented for Coleman and Blair and Toporek); (3) one count of IRS obstruction; and (4) one count of mail fraud. He was acquitted of six counts of client tax evasion and the three counts of personal tax evasion.
In May 2014, the district court sentenced Daugerdas to 180 months’ imprisonment, three years’ supervised release, $164,737,600 in forfeiture, and $371,006,397 in restitution.-
DISCUSSION
Daugerdas presents the following issues for our review: (I) whether the evidence was sufficient to support his convictions; (II) whether the district court’s- supplemental instruction on the Annual Accounting Rule misled the jury; (III) whether the indictment was constructively amended; (IV) whether the accumulation of errors at trial violated, his due process right to a fair trial; (V) whether the indictment was duplicitous; (VI) whether his sentence was proeedurally and substantively reasonable; and (VII) whether the government proved the requisite nexus between his crimes and the property sought in forfeiture.
I. Sufficiency of the Evidence
We review de novo a challenge to the sufficiency of the evidence supporting a criminal conviction by “view[ing] the evidence in the light most favorable to the government, drawing all inferences in the government’s favor and deferring to the jury’s assessments of the witnesses’ credibility.” United States v. Pierce, 785 F.3d 832, 837-38 (2d Cir. 2015) (internal quotation marks omitted). “We will sustain the jury’s verdict if any rational trier of fact could have found the essential elements of the crime beyond a reasonable doubt.” Id. at 838 (emphasis and internal quotation marks omitted).
A. The Evidence Supporting Dauger-das’s Mens Rea
Daugerdas first argues that the government failed to prove that he had the necessary mens rea to be guilty of tax evasion, mail fraud, and obstruction of the IRS. The convictions on these counts turned upon the misrepresentations in the tax returns of (1) Coleman and Blair; and (2) Toporek.
Daugerdas’s conviction on these,counts will stand only if the evidence was sufficient to enable a rational jury to conclude beyond a reasonable doubt that Daugerdas knew that the tax returns of Coleman and Blair, and Toporek misrepresented then-tax liability in some way. See United States v. Regan, 937 F.2d 823, 827 (2d Cir.) amended, 946 F.2d 188 (2d Cir. 1991) (stating that the crime of tax evasion requires the “voluntary, intentional-violation of a known legal. duty” (internal quotation marks omitted)); United States v. Alkins, 925 F.2d 541, 550 (2d Cir. 1991) (“If an individual believes that the information set forth in a mailing is true, it follows that he cannot have the requisite intent to [commit mail fraud].”); Parse, 789 F.3d at 121 (“To act or endeavor ‘corruptly,’ within the *222meaning of [the statute criminalizing obstruction of the IRS], means to act or endeavor with the intent to secure an unlawful advantage or benefit either for one’s self or for another.” (some internal quotation marks omitted)).
Daugerdas attacks the sufficiency of the government’s proof that he knowingly developed and sold shelters that violated the so-called economic substance rule. Under this rule, “sham transactions that [cannot] with reason be said to have purpose, substance, or utility apart from them anticipated tax consequences” cannot form the basis of legitimate tax losses under the Internal Revenue Code. Bank of N.Y. Mellon Corp. v. Comm’r of Internal Revenue, 801 F.3d 104, 113 (2d Cir. 2015) (internal quotation marks omitted).
For a jury to conclude that Dau-gerdas had the necessary mens rea to commit tax evasion, mail fraud, and obstruction of the IRS on the basis of violations of the economic substance rule, they would have to find both that he knew the rule and knew that the transactions lacked economic substance. See Parse, 789 F.3d at 121. Our inquiry into whether a transaction lacks economic substance requires us to examine “(1) whether the [clients] had an objectively reasonable expectation of profit, apart from tax benefits, from the transaction; and (2) whether the [clients] had a subjective non-tax business purpose in entering the transaction.” Bank of N.Y. Mellon, 801 F.3d at 115. This inquiry does not involve “a rigid two-step process,” but instead calls for a “flexible analysis where both prongs are factors to consider in the overall inquiry.” Id. (internal quotation marks omitted).
1. Knowledge of the Economic Substance Rule
In Parse, we found the following facts sufficient to support a conclusion that Parse knew the economic substance rule: (1) he was an experienced CPA with a master’s degree in business administration; and (2) in connection with his personal use of a similar tax shelter, he received an opinion letter describing the economic substance rule in detail. 789 F.3d at 122. As in Parse, on the facts of this case, a rational jury was entitled to conclude that Daugerdas was well-acquainted with the economic substance rule based on evidence that he was an experienced CPA and tax lawyer who explained the economic substance requirement to his clients.
2. The Economic Substance of the Swaps Shelter
a. Coleman’s, Blair’s, and Toporek’s Objectively Reasonable Expectation of Profit
In arguing that the government failed to prove that he knew that Coleman and Blair and Toporek did not have an objectively reasonable expectation of making a profit from the Swaps Shelter, Daugerdas relies on the legal ambiguity surrounding the question of whether the tax advisory fees associated with each shelter should be factored into the analysis. Daugerdas argues that, because it is not clear whether or not the fees should be included in the calculation, he had a good-faith belief that the fees should not be considered. The lack of economic substance is only apparent, he argues, if the costs associated with the tax advisory fees are included in the calculation.
This argument ignores the fact that, even if the tax advisory fees were not included in the analysis, the chance of any Swaps Shelter “investor” earning a profit was still extremely low. The government’s expert testified that Coleman and Blair had a 1% chance and Toporek a 3% chance of earning a profit even before the inclu*223sion of the fees. In the- end, only two of the approximately 60 clients who invested in the Swaps Shelter made a profit, and one of these earned a profit of just one dollar.
Moreover, Daugerdas was aware that the possibility of success was low. He acknowledged to Coleman that the transaction was “highly unlikely” to produce a profit and this would happen “only if all the moons were lined up.” Taken together, this evidence permitted a rational jury to infer that, regardless of Daugerdas’s purported belief that advisory fees were properly excluded, Daugerdas knew that Coleman and Blair and Toporek could not have had an objectively reasonable expectation of profiting from their investment.
b. Coleman’s, Blair’s, and Toporek’s Subjective Non-Tax Business Purpose in Entering the Transaction
The evidence of the low likelihood of Coleman, Blair, and Toporek profiting from the Swaps- Shelter provided circumstantial support for inferences that they did not' have a subjective non-tax business purpose in entering the transaction, and that Daugerdas knew it.
The latter inference is also supported by Coleman’s testimony about his conversations with Daugerdas. Coleman testified at trial that Daugerdas advised him to “focus” on “making the investment for profit purposes” if he were to be audited by the IRS. Tr. 5847. Coleman viewed this advice as an effort to have him “stretch the truth” about why he was entering into the shelter. Id. at 5904. This advice was also relevant to the question of whether Daugerdas knew that Toporek did not have a subjective non-tax business purpose in entering the transaction. If Daugerdas could infer, based on the low profit potential, that Coleman and Blair were not intending to make a profit, then he could have inferred the same about Toporek, given that his profit potential'was similarly low.
A rational jury could thus have concluded that Daugerdas - (1) was familiar with the economic substance rule and (2) knew that the transaction's lacked economic substance because he knew that (a) Coleman and Blair and Toporek had no objectively reasonable expectation of making a profit from the Swaps Shelter and (b) they did not have a subjective non-tax business purpose in entering the' transaction. Therefore, the evidence was sufficient for a jury to find that Daugerdas was guilty of tax evasion, mail fraud, and obstruction of the IRS.
B. The Evidence Supporting Dauger-das’s Convictions of Conspiracy and Mail Fraud
Daugerdas next argues that the evidence was insufficient to sustain his convictions for conspiracy to defraud the IRS and the same mail fraud count previously discussed because (1) there was no evidence that he participated in the scheme involving illegal backdating; (2) the government failed to prove that the mail fraud affected a financial institution; and (3) the indictment did not allege that the mail fraud affected a financial institution. Each of these contentions is meritless.
1. Evidence of Daugerdas’s Participation in Backdating
The record belies Daugerdas’s argument that there was no evidence that he participated in the scheme involving illegal backdating. Erwin Mayer, a lawyer who worked with Daugerdas for several years, testified that he had discussed backdating with Daugerdas on multiple occasions and that the discussions concerned “fixes that [he does] for their client situations” and' ways to “potentially justify^ ]” backdating the transactions. Tr. 2339-40.
*224Further, Yackee testified extensively about her communications with J &. G employees regarding backdated transactions effected on behalf of Coleman and Blair, Toporek, and. the Aronoff family. Although Yackee did not specifically name Daugerdas as a participant in these communications, she stated that he was “the head of that team of people” at J & G with whom she communicated regarding the backdating. Id. at 4Q31-32,
Finally, Quedenfeld, who prepared tax returns for Toporek that reflected the backdated transaction, also provided testimony describing Daugerdas’s direct involvement in the backdating. She told the jury that she had first prepared returns for Toporek based on the opinion letter, drafted and sent by J & C, that contained the original, incorrect transaction. She sent these returns to Daugerdas, and then received a revised opinion letter signed by Daugerdas that included the backdated transaction and a request to return the original, incorrect opinion letter. She prepared returns based on the new, opinion letter and then provided the new returns to Beery, who worked for Daugerdas, .for Beery’s review.
A rational jury could have concluded from the testimony of Mayer, Yackee, and Quedenfeld that Daugerdas participated in the scheme involving illegal backdating by directing his subordinates to effect backdated trades. His argument on this point is therefore without merit.
2. Stipulation that the Mail Fraud Affected a Financial Institution
We easily reject Daugerdas’s argument that the government failed to prove that the mail fraud affected a .financial institution. Without this element, the mail fraud charged would be barred by the five-year statute of limitations. Section 3282 of Title 18 of the United States Code sets a five-year statute of limitations for non-capital federal crimes, whereas § 3293 sets a ten-year statute of limitations for mail fraud “affect[ing] a financial institution.” 18 U.S.C. § 3293. We agree with the district court that the government must prevail because Daugerdas entered into a pretrial stipulation that Deutsche Bank “was a financial institution that was ‘affected’ ,.. by the [shelters].” J.A. 972.
3. Indictment’s failure to allege that the mail fraud affected a financial institution
Daugerdas claims error because the indictment did not allege that the mail fraud affected a financial institution. Dau-gerdas did not raise this claim until after trial, and therefore this challenge is subject to plain error review. United States v. Nkansah, 699 F.3d 743, 752 (2d Cir. 2012), abrogated on other grounds by United States v. Bouchard, 828 F.3d 116 (2d Cir. 2016). To obtain a reversal on this basis, Daugerdas “must show (1) there is an error; (2) the error is clear or obvious ...; (3) the error affected [his] substantial rights ...; and (4) the error seriously affects the fairness, integrity or public reputation of judicial proceedings.” Id. at 751 (alterations and internal quotation marks omitted).
Any error could not have affected Dau-gerdas’s- substantial rights, because his stipulation that Deutsche Bank “was a financial institution that was ... affected by the' [shelters]” put him on notice of this element of mail fraud. See United States v. Doe, 297 F.3d 76, 88 n.12 (2d Cir. 2002) (stating that a defendant’s substantial rights are not affected where information is missing from an indictment if the defendant otherwise had notice of the missing information). Therefore, he cannot establish plain error,
*225II. Constructive Amendment of the Indictment
Daugerdas argues that the prosecution’s rebuttal summation and the court’s supplemental instructions constructively amended the indictment to include a theory of criminal liability based on his violation of the Annual Accounting Rule by including backdated transactions in tax returns that were submitted to the IRS. This rule requires that the tax consequences of a transaction generally be assessed in the returns of the year in which the transaction took place. See United States v. Shelly Oil Co., 394 U.S. 678, 684, 89 S.Ct. 1379, 22 L.Ed.2d 642 (1969).
We review dé novo the question of whether an indictment was constructively amended. Pierce, 785 F.3d at 844. A constructive amendment is a per se violation of the Fifth Amendment, but “significant flexibility in proof’ is constitutionally permissible, as long as the indictment provides notice to the defendant “of the core of criminality to be proven at trial.” United States v. D ’Amelio, 683 F.3d 412, 417 (2d Cir. 2012) (emphasis , and internal quotation marks omitted). Because the “core of criminality involves the essence of a crime, in general terms,” and excludes “the particulars of how a defendant effected the crime,” id. at 418 (internal quotation marks omitt.ed), there is no constructive amendment when the proof at trial does no more than supply the particulars.
The indictment in this case was not constructively amended. Daugerdas’s claim that the government ignored backdating as a separate theory of liability until its rebuttal summation is contradicted by the text of the indictment, which describes backdating repeatedly. The backdating of tax returns, described in Paragraph 47 of the indictment, is in the section of the indictment that describes the various fraudulent schemes that were involved in the development, sale, and implementation of the tax shelters. Backdating' is later listed in Paragraph 63-i as one of the methods by which the conspiracy was carried out. Paragraph 63-i is incorporated into the obstruction and mail fraud counts.
Daugerdas correctly notes that, in its opening argument, the prosecution described backdating as evidence that the shelters lacked economic substance. Much of the trial testimony also focused on the economic substance theory of liability. However, presenting backdating as an alternate theory of liability on some of the counts did not change “the essence of the crime, in general terms,” which remained throughout a massive tax fraud orchestrated by Daugerdas. The incidents of backdating and corresponding violations of the Annual Accounting Rule were simply “particulars of how [Daugerdas] effected the crime,” and the government’s reliance on this theory of liability did not amount to a constructive amendment of the indictment. Id. at 417-18.
III. Duplicitous Indictment
Daugerdas argues that Count Thirteen, which alleges obstruction of the IRS, was duplicitous because it involves two separate schemes: one based on the use of tax shelters by clients and the other based on Daugerdas’s personal use of the shelters.
We review properly preserved challenges to an indictment de novo. See United States v. Vilar, 729 F.3d 62, 79 (2d Cir. 2013). As we explained in United States v. Aracri, 968 F.2d 1512, 1518 (2d Cir. 1992), an indictment is duplicitous if it includes multiple, crimes in one count, but not if it includes in that count multiple ways of committing a single offense.
*226Under the Amcri standard, Count Thirteen is not duplicitous. It alleges a single crime: obstruction of the IRS. It then alleges that this crime was committed in many different ways. The fact that the various methods of committing a single offense can be divided into two separate categories — those that relate to Dauger-das’s work on behalf of his clients and those that relate to Daugerdas’s personal use of the shelters — does not create duplicity. Cf. id. (“[I]t is well established that the allegation in a single count of a conspiracy to commit several crimes is not duplicitous, for the conspiracy is the crime and that is one, however diverse its objects.” (alterations and internal quotation marks omitted)).
IV. Violation of Due Process
Daugerdas argues that (A) the admission of evidence of a non-party’s guilty plea to a tax evasion charge; (B) the government’s improper summation arguments; and (C) the government’s inconsistent positions at summation and sentencing created a trial so riddled with errors that his due process right to a fair trial was violated.
A. The Admission of John Ivsan’s Guilty Plea
Daugerdas first objects to the admission of the fact of John Ivsan’s guilty plea. Ivsan was a tax attorney and part of a group of advisors who were unaffiliated with Daugerdas but who consulted with Larry Morgan, a Daugerdas client and cooperating witness, about his decision to participate in the Short Sale Shelter. On cross-examination, Daugerdas elicited from Morgan that Ivsan was one of the advisors who concluded that the shelter was legal. Over Daugerdas’s objection, the district court allowed the government to introduce, pursuant to Federal Rules of Evidence 609 and 806, the fact, that Ivsan had in 2013 pleaded guilty to unrelated tax crimes. When this evidence was introduced, the district court instructed the jury that the jury’s consideration of Iv-san’s guilty plea was limited to what it revealed about Ivsan’s credibility. Ivsan’s credibility was relevant to the jury’s assessment of the truthfulness of his statement that he believed that the Short Sale Shelter was legal, which is itself relevant to the question of whether Daugerdas, also a tax attorney, had a similar belief. In its summation, the prosecution referred to the guilty plea.
We review the district court’s ruling for abuse of discretion, United States v. Taubman, 297 F.3d 161, 164 (2d Cir. 2002) (per curiam), which requires us to ask if the ruling was “arbitrary and irrational,” United States v. Mercado, 573 F.3d 138, 141 (2d Cir. 2009) (internal quotation marks omitted).
Assuming that the statements that Dau-gerdas elicited from Morgan about Ivsan’s opinion on the Short Sale Shelter constituted hearsay within the meaning of Federal Rule of Evidence 801, it was permissible to use Ivsan’s guilty plea to impeach that opinion under Rules 609 and 806, subject to Rule 403’s balancing test. The district court did not abuse its discretion in determining that the probative value of Ivsan’s guilty plea in impeaching Ivsan’s opinion was not “substantially outweighed” by its “danger of ... unfair prejudice.” Fed. R. Evid. 403. The probative value of the evidence lay in its relevance to the jury’s assessment of Ivsan’s credibility, which was itself relevant to its assessment of whether Daugerdas had a good-faith belief in the legality of his actions. To be sure, the fact that Ivsan was convicted of a tax crime ran a risk of unfairly prejudicing Daugerdas, but here this prejudice was mitigated by the district court’s limiting *227instruction and by the fact that Ivsan himself was unaffiliated with Daugerdas or his firm. The admission of Ivsan’s guilty plea did not implicate Daugerdas’s right to a fair trial.
B. Errors in the Prosecution’s Summation
Daugerdas argues that the prosecutor’s summation was so prejudicial that he must receive a new trial.
We review de novo a claim that a prosecutor’s summation unfairly prejudiced a defendant. See United States v. Bular, 567 F.2d 192, 199-200 (2d Cir. 1977). But such a claim presents a significant hurdle for Daugerdas. “The prosecution and the defense are generally entitled to wide latitude during closing arguments, so long as they do not misstate the evidence.” United States v. Tocco, 135 F.3d 116, 130 (2d Cir. 1998). “An improper summation will only warrant a new trial when the challenged statements are shown to have caused substantial prejudice to the defendant; rarely will an improper summation meet the requisite level of prejudice.” United States v. Mapp, 170 F.3d 328, 337 (2d Cir. 1999) (citation omitted).
1. Commentary on Daugerdas’s Failure to Produce Evidence
Daugerdas identifies two examples of what he believes to be impermissible statements in the prosecution’s summation concerning his failure to present evidence: (1) a statement that the defense had not introduced any evidence that anyone had convinced the IRS that the shelter transactions were legitimate; and (2) a statement that Daugerdas did not record his views about the validity of the shelters, seek other legal opinions on the validity of the shelters he was selling, or litigate them in court.
The district court sustained the defense objection to the first line of argument and not to the second. Because we are required to look at the entire argument in context, see United States v. Caracappa, 614 F.3d 30, 41 (2d Cir. 2010), we consider both arguments and conclude that both were permissible. Although the government cannot comment on a defendant’s failure to testify, it is permissible to draw the jury’s attention to the fact that-a defendant did not call witnesses to contradict the government’s case or support his own theory of what happened. United States v. McDermott, 918 F.2d 319, 327 (2d Cir. 1990). A prosecutor’s commentary about a defendant’s lack of evidence becomes prejudicial only if the jury would “naturally and necessarily interpret the Government’s summation as a comment on the defendant’s failure to testify” or if the evidence that the defendant has not produced was exclusively in his control. Id.
Here, although the missing evidence is relevant to Daugerdas’s subjective belief in the legality of the shelters, it is not evidence that is only in his control, nor would the jury otherwise interpret the prosecutor’s statements as comments on Dauger-das’s failure to testify. A witness who had successfully defended one of the shelters before the IRS would provide evidence of a legal ambiguity that could provide circumstantial support for Daugerdas’s contention that he had a good faith belief in the legality of the shelters. The same is true of evidence that Daugerdas had either recorded or publicly sought confirmation of this belief. Many witnesses other than Daugerdas could have testified about these actions. Because Daugerdas was not the exclusive source of this evidence, there was no suggestion that he had a duty to testify or produce evidence solely within his control.
*2282. Commentary on the Law
Daugerdas next points to the prosecutor’s comments twice during summation that the jurors would' learn from the district court that Daugerdas’s legal views about the calculation of profit under the economic substance doctrine were incorrect. These statements were not an improper comment on the law because Dau-gerdas’s subjective view of the law was a fact that was at issue in the case. The government was' required to prove that Daugerdas did not have a good faith belief that the shelters at issue were legal. See supra Section I.A in Discussion. Therefore the state of the law at- the time of the transactions as it bore on Daugerdas’s state of mind was relevant to the government’s case'and was an appropriate subject for the prosecution’s comment.
C. Inconsistent Theories at Trial and at Sentencing
Daugerdas relies on Bradshaw v. Stumpf, 545 U.S. 175, 125 S.Ct. 2398, 162 L.Ed.2d 143 (2005), to argue that the government’s separate theories at summation and sentencing violated his right 'to due process. Bradshaw suggested that the prosecutor’s inconsistent theories in two separate prosecutions that each of two defendants had pulled the trigger in a murder had not “affect[ed] the knowing, voluntary, and intelligent nature” of one defendant’s guilty plea, but that the inconsistency could have affected the defendant’s sentence. Id. at 187, 125 S.Ct. 2398. This argument fails here because the government did not' rely on inconsistent theories. Instead, the government at trial advanced multiple theories that could have supported conviction, but then focused on one of these theories at sentencing. In any event, given the factual differences between these two cases, Bradshaw is inapposite. It therefore bears no relationship to the purported inconsistency in this case.
In sum, Daugerdas points to no error or combination of errors, in the prosecution’s summation or elsewhere, that were so prejudicial as to warrant a new trial.
V. Supplemental Instruction
Daugerdas argues that the district court committed prejudicial error in giving a supplemental instruction about the Annual Accounting Rule.
We review jury instructions de novo, looking at the entirety of the jury charge. Hudson v. New York City, 271 F.3d 62, 67-68 (2d Cir. 2001). Error occurs if the instructions “mislead the jury as to the correct legal standard or do not adequately inform the jury of the law.” Id. at 67 (alterations and internal quotation marks omitted). We have cautioned that a trial judge must be especially careful with supplemental instructions in response to jury questions because they are often provided to the jury at crucial moments of deliberation. United States v. Kopstein, 759 F.3d 168, 172-73 (2d Cir. 2014). A flawed supplemental instruction can undermine and even invalidate a charge that is otherwise correct if the supplemental instruction is “sufficiently incomplete and misleading.” Id. at 172 (internal quotation marks omitted).
During his full charge to the jury, Judge Pauley informed the jurors that “the income tax laws ... are administered on the basis of an annual accounting system, which prohibits the reopening of a prior year’s tax return to take account of events occurring in later years.” J.A. 250. Neither party disputes that this is an accurate statement of the law.
During deliberations, the jury sent a note to the judge that referred back to that portion of the instructions and then *229asked: “What is defined as ‘a prior year’s tax return’? Does this specifically mean a tax return that has already been filed?” Tr. 7758. Judge Pauley responded: “[I]n singling out [this part of-the instructions], you’re focusing on the [A]nnual [Accounting [R]ule. The [A]nnual [A]ccounting [R]ule prohibits, in connection with the preparation of a tax return for a particular year, consideration of transactions that occur in a subsequent year. Therefore, the answer to that question is no.” Id. It is this explanation to which Daugerdas objects.
The next day, the jury asked Judge Pauley: “Is there a law or rule within the internal revenue laws that can be considered contradictory to the annual accounting rules cited in the supplemental charge ... ?” Id. at 7791. Judge Pauley responded:
[T]he answer to that question is that the law I gave you concerning the annual accounting rule [in the original charge] and in my supplemental instruction is the law that should govern your deliberations. The application of this law depends on the particular facts of each case. In some instances you have heard about as-of reporting. And it .may be proper under the internal revenue laws, if the transaction reported on the as-of date actually occurred on that date. On the other hand, it would violate the internal revenue laws to report a transaction that occurred in a subsequent year as occurring during the prior year.
Id. at 7791-92.
We can resolve Daugerdas’s first two arguments—that the district court erroneously implied that the Rule was absolute and that the district court refused to instruct the jury that the good faith defense applies to this rule—by an examination of the supplemental instruction in the context of the charge as a whole. Hudson, 271 F.3d at 67-68. The judge’s second supplemental instruction, in which he told the jury that “[t]he application of [the Annual Accounting Rule] depends on the particular facts of each case,” rectified any misunderstanding as to whether the Rule was absolute that could have resulted from the first supplemental instruction. See United States v. Velez, 652 F.2d 258, 262 (2d Cir. 1981) (“[E]ven if some error can be found in the supplemental charge, it will not ordinarily require reversal ... where a subsequent supplemental instruction cures the defect.”).
The context in which Judge Pau-ley gave the supplemental instruction also made it' unnecessary for him to reiterate his instruction on the good faith defense. In his charge before the jury retired to deliberate, he stated that “good faith is a complete defense to each of the charges in the indictment” and elaborated on what good faith could mean in this context. J.A. 544-544.1. In light of this explication of the good faith defense, there was no need for him to reiterate it. This is particularly so because he was not tying this guidance to any particular charge, but rather was providing information about a background principle of law- that applied broadly to all of the charges.
Daugerdas’s third and fourth objections to the supplemental instructions are equally meritless. His argument that the supplemental instruction misstated the required mens rea and thus improperly instructed the jury on an element of the offense that the government must prove only makes-sense if the Rule itself is a criminal prohibition that can be violated simply by-the inclusion of later transactions in the tax return. But the Rule, on its own, is not a criminal prohibition. Instead, the Rule is an accounting principle the violation of which can provide a basis for determining that a defendant violated particular criminal statutes, such as those at *230issue in -this appeal, assuming all of the elements- of -these crimes, including mens rea, are -proven. It is the statutes prohibiting these acts that provide the heightened scienter requirement and it is their violation, not simply a violation of the Annual Accounting Rule, that creates criminal liability.
Daugerdas’s final argument, that the district court should have confined the supplemental instruction to the conspiracy charge, relies on the incorrect premise that the indictment limited the backdating allegations to the conspiracy charge. This is belied by the text of the indictment, which, as discussed above, describes backdating repeatedly as one of the fraudulent schemes that facilitated the tax shelters and one of the methods by which the conspiracy was carried out and incorporates the backdating into the obstruction and mail fraud counts. Because the jury could have used the backdating to convict on counts other than the conspiracy, it would have been incorrect for the district court to have confined the supplemental instruction to the conspiracy charge.
VI. Reasonableness of the Sentence
We review challenges to a sentence under a “reasonableness” standard, which is “a particularly deferential form of abuse-of-discretion review.” United States v. Broxmeyer, 699 F.3d 265, 278 (2d Cir. 2012) (internal quotation marks omitted). A sentence must be both procedurally and substantively reasonable. Id. We address each requirement in tui-n.
A sentence is procedurally unreasonable if the district court “fails to calculate the Guidelines range ..., makes a mistake in its Guidelines calculation, ... treats the Guidelines as mandatory ...[,] does not consider the § 3553(a) factors, ... rests its sentence on a clearly erroneous finding of fact ...[,] fails adequately to explain its chosen sentence, [or fails to] include an explanation for any deviation from the Guidelines range.” United States v. Cavera, 550 F.3d 180, 190 (2d Cir. 2008) (en banc) (internal quotation marks omitted). When we review for substantive reasonableness, “we will set aside a district court’s ... determination only in exceptional cases where the trial court’s decision cannot be located within the range of permissible decisions.” United States v. Rigas, 583 F.3d 108, 122 (2d Cir. 2009) (alteration and internal quotation marks omitted).
A. Procedural Reasonableness
Daugerdas argues that the district court procedurally erred by ignoring the interviews his attorneys had conducted with two members of the jury after the end of the trial, in which the jurors said that they believed he was guilty only of the backdating of transactions, not of the entirety of the fraud charged in the indictment. Because this information was relevant to the district court’s determination of the length of Daugerdas’s sentence, Dau-gerdas alleges, it committed procedural error by failing to consider it.
Daugerdas fails to identify a procedural error that we have recognized, see Cavera, 550 F.3d at 190, and, furthermore, he misrepresents the record. Judge Pauley did not ignore this information; he acknowledged that the interviews existed but determined that they did not justify the imposition of a shorter sentence. Because the district court’s conclusion was reasonable, Daugerdas’s procedural challenge fails.
B. Substantive Reasonableness
Daugerdas argues that his sentence was substantively unreasonable because it was based primarily on conduct for which he was acquitted — his participation in a massive $400 million tax fraud. *231It is well-established that a district judge can take into account acquitted conduct in determining a sentence. See United States v. Gomez, 580 F.3d 94, 105 (2d Cir. 2009). Daugerdas’s citations to the concurrences in Rita v. United States, 551 U.S. 338, 127 S.Ct. 2456, 168 L.Ed.2d 203 (2007), and to a law review article do not persuade us that the law is otherwise or that we should somehow embark on a quest to change it. Therefore, his substantive reasonableness challenge also fails.
VII. Errors in Forfeiture Order
Daugerdas argues that the forfeiture order was flawed because the government did not trace the fees paid by clients into the J & G bank account from which Daugerdas was paid by J & G. Instead, he argues, the government established only that (1) J & G deposited Daugerdas’s compensation into an account controlled by Daugerdas; and (2) Daugerdas transferred funds from this account to the accounts sought to be forfeited. To obtain forfeiture of these funds, according to Daugerdas, the government was also required to establish the origin of the fees deposited into the J & G bank account and separate the fees paid as a result of the fraudulent conduct from the remainder.
We review a district judge’s legal conclusions regarding forfeiture de novo and his factual determinations for clear error. United States v. Sabhnani, 599 F.3d 215, 261 (2d Cir. 2010). For a. criminal forfeiture order to pass muster, the government must establish, by,a preponderance of the evidence, the “requisite nexus between the property and the offense.” Fed. R. Crim. P. 32.2(b)(1)(A); see United States v. Bellomo, 176 F.3d 580, 595 (2d Cir. 1999). In a criminal forfeiture order issued pursuant to 18 U.S.C. § 982 following a defendant’s conviction for mail fraud, the requisite nexus is that the “proceeds [must have been] obtained directly or indirectly”, through such fraud.
The district court correctly concluded that the money sought to be forfeited had been obtained through Daugerdas’s mail fraud. The J & G account from which Daugerdas was paid held only the funds received by the Chicago office. See Docket Entry 558 Ex. 6 § 2b. The trial evidence established that the entirety of the tax-shelter fee income received by J & G’s Chicago office — the pool of money from which Daugerdas was paid — was generated by Daugerdas’s criminal acts. See Docket Entry 839 p. 11-12. Based on this evidence, the district court did not clearly err in concluding that the funds located in Daugerdas’s various accounts were the proceeds of his frauds.
We have considered Daugerdas’s remaining arguments and find them to be without merit. We take this opportunity to express our appreciation for the commendable way that Judge Pauley handled this litigation.
CONCLUSION
For the reasons stated above, we AFFIRM the judgment of the district court.