
    UNITED STATES of America, Plaintiff-Appellant (10-6136), Plaintiff-Appellee (11-5828, 11-5852, 12-5695 & 13-5440), v. Christopher Cello SMITH, Defendant-Appellee (10-6136), Defendant-Appellant (11-5828, 12-5695 & 13-5440), Michael D. Smith, Defendant-Appellee (10-6136), Defendant-Appellant (11-5852, 12-5695 & 13-5440).
    Nos. 10-6136, 11-5852, 11-5828, 12-5695, 13-5440.
    United States Court of Appeals, Sixth Circuit.
    Argued: Oct. 8, 2013.
    Decided and Filed: April 15, 2014.
    Rehearing En Banc Denied May 20, 2014.
    
      ARGUED: Paul McCaffrey, United States Attorney’s Office, Lexington, Kentucky, for Appellant United States in 10-6136 and Appellee United States in 11-5828, 11-5852, and 12-5695. Jeffrey M. Blum, Louisville, Kentucky for Appellee C. Smith in 10-6136 and Appellant C. Smith in 11-5828. Timothy J. McKenna, Timothy J. McKenna, LLC, Cincinnati, Ohio, for Appellant M. Smith in 11-5852 and 12-5695. ON BRIEF: Charles P. Wisdom, Jr., Andrew Sparks, United States Attorney’s Office, Lexington, Kentucky, for Appellant United States in 10-6136 and Ap-pellee United States in 11-5828, 11-5852, and 12-5695. Jeffrey M. Blum, Louisville, Kentucky for Appellee C. Smith in 10-6136 and Appellant C. Smith in 11-5828. Timothy J. McKenna, Timothy J. McKenna, LLC, Cincinnati, Ohio, for Appellant M. Smith in 11-5852 and 12-5695. Jeffrey M. Blum, Louisville, Kentucky, for Appellants C. Smith and M. Smith in 13-5440. Charles P. Wisdom, Jr., United States Attorney’s Office, Lexington Kentucky, for Appellee United States in 13-5440.
    Before: BOGGS, CLAY, and GILMAN, Circuit Judges.
   OPINION

RONALD LEE GILMAN, Circuit Judge.

Michael D. Smith and Christopher Cello Smith are brothers who operated Target Oil and Gas Corporation (Target Oil), a company that engaged in speculative resource drilling in Kentucky, Tennessee, Texas, and West Virginia. While serving as President of Target Oil and a related company named Kentucky-Indiana Oil and Gas Corporation (Kentucky-Indiana), Michael Smith ran Target Oil’s day-to-day operations, controlled correspondence with potential investors, and directed drilling programs. Christopher Smith was the Vice President and lead salesman of Target Oil. The Smith brothers were arrested in 2008 and accused of conspiring with others to defraud investors of millions of dollars.

After a four-week jury trial, Michael Smith was convicted on one count of conspiracy to commit mail fraud, in violation of 18 U.S.C. § 1349, and on eleven substantive counts of mail fraud, in violation of 18 U.S.C. § 1341. He was sentenced to 120 months in prison, to be followed by three years of supervised release. The district court also ordered him to pay $5,506,917 in restitution.

Christopher Smith was convicted by the same jury on seven counts of mail fraud, in violation of 18 U.S.C. § 1341. He was sentenced to 60 months in prison, to be followed by three years of supervised release. The district court also ordered him to pay $1,652,075 in restitution.

On appeal, Michael and Christopher Smith argue that (1) the government’s evidence was insufficient to support their convictions; (2) the government offered evidence that constructively amended or varied Count 1 of the indictment; (3) their sentences are procedurally and substantively unreasonable; (4) one of the forfeiture judgments is excessive; (5) the district court erred in denying their motions for a new trial; (6) the district court erred by excluding an expert witness for the defense; and (7) various items of evidence relating to the alleged fraud were erroneously admitted.

For the reasons set forth below, we AFFIRM the judgment of the district court with respect to all issues.

I. BACKGROUND

A. Factual background

Employees of Target Oil interacted with potential investors in three principal ways: (1) the potential investors were first contacted from lead sheets, (2) they were subsequently mailed information packets, and (3) they were then pressured by company “closers” to invest in oil and gas drilling programs. The fraud that the government accused the Smith brothers of perpetrating occurred in the final two stages. Information packets that were mailed to the potential investors often misrepresented Target Oil’s past successes and exaggerated the likelihood of finding oil and gas. When Target Oil salesmen followed up on the telephone, they frequently represented some wells as “the greatest thing since sliced bread” and others as virtually certain to produce consistent streams of royalties. Investors relied on these various representations in deciding to place their money with Target Oil. When some dissatisfied investors later sought a return of their investment, however, Christopher Smith engaged in stalling tactics, continued to stress the potential for investment returns, or simply would not allow the investors to “pull out.”

The image of Target Oil that its salesmen communicated was inconsistent with reality. Of the millions of dollars invested in the company, only thousands of dollars were returned as royalties. Wells that Target Oil had represented as sure-fire investments often produced virtually no oil, and numerous wells that had been drilled were never completed. Many investors lost the entirety of their investments, and investor losses totaled in the millions of dollars. In fact, from 2003 to 2008, Target Oil received approximately $15,800,000 in investor funds but, according to the postal inspector, distributed only $460,000 in royalties.

Michael and Christopher Smith were at the heart of Target Oil’s operations, but they did not act in isolation. Several individuals were employed to assist in the business, with varying degrees of experience and success. Shannon Williams was a young geologist who was hired directly from college and given the responsibility of designing and writing the information packets for Target Oil. Working at the direction of the Smith brothers, Williams often manipulated the images contained in the information packets in a misleading way. The manipulated images frequently exaggerated the likelihood of finding oil or downplayed the presence of dry wells. Eventually, despite limited experience, Williams became a salesman and field geologist for the company.

Geology reports and field recommendations were crucial to Target Oil’s operations. The company employed Ray Gar-ton, a licensed geologist, to prepare these documents. Information packets mailed to potential investors included a letter from Garton, which in the beginning recommended particular drilling sites based on field evaluations conducted by him personally. As time went by, however, Garton stopped conducting his own field evaluations and instead relied heavily on representations from Michael Smith, who would state that a particular well was “good” and would supply documents to be used in preparing the geological reports.

Information packets were the bridge between Target Oil’s initial contact with potential investors and their being called on the telephone by the company’s closers. The closers were experienced salesmen who were tasked with convincing hesitant investors to do business with Target Oil. Mark Irwin started at Target Oil as a regular salesman but eventually became a closer. He testified that all the closers followed a strict regimen. Closers and other salesmen, for example, were not to discuss investment opportunities with women. Reluctant investors were frequently asked if they were risk takers capable of managing and handling their own affairs. Target Oil salesmen often placated dissatisfied investors by offering additional shares in underperforming wells.

The closers were provided with little formal training, and many of them learned by watching and listening to more experienced closers like Christopher Smith. Learning by example, however, was not the only method of instructing new closers. At least one employee was shown the movie The Boiler Room as training material. The Boiler Room depicts a fictional New York brokerage firm that used high-pressure sales techniques to commit investment fraud. Another salesman said that Michael Smith made copies of The Boiler Room and distributed the movie to various employees.

After Target Oil solicited investors, drilling began. Under general industry practice, the initial drilling for natural gas is typically followed by a process known as completion, which occurs when the drilling area is isolated and hydraulically fractured using service companies to extract the gas. This process, which is commonly known as “fracking,” requires an additional investment of time and expense by the drilling company. Target Oil rarely engaged in fracking. Michael Smith preferred to produce gas from wells that did not require this additional expense. On at least one occasion, Michael Smith pledged to frack an underperforming well to increase its production. The fracking was never completed.

As time passed, Michael Smith faced greater difficulties in securing drilling permits and leases. Violations of environmental regulations made obtaining new permits in Target Oil’s name problematic. This caused him to obtain permits in Kentucky-Indiana’s name, but Target Oil employees continued to handle the investment solicitations for those wells.

B. Procedural background

A grand jury indicted Michael Smith, Christopher Smith, Garton, Irwin, and two other Target Oil employees — Josh Harris and Shaun Smith — on one count of conspiring to commit mail and wire fraud, twenty counts of mail fraud, two counts of wire fraud, and two counts of selling securities without registering with the United States Securities and Exchange Commission. The cases against Michael Smith, Christopher Smith, Harris, and Shaun Smith proceeded to trial. Garton and Irwin pleaded guilty to the conspiracy charge and testified against Michael and Christopher Smith. After seven days of trial, Harris and Shaun Smith likewise pleaded guilty to the charges against them.

The trial continued against Michael and Christopher Smith for a total of four weeks during the summer of 2010. In the course of the trial, the district court dismissed the securities-related counts against them. At the close of the government’s case, Michael and Christopher Smith moved for a judgment of acquittal on all the remaining counts. The court dismissed one of the mail-fraud counts, but declined to dismiss the other counts. Michael Smith was found guilty on the conspiracy count and on eleven of the substantive mail-fraud counts. Christopher Smith was found guilty on seven of the mail-fraud counts only.

After the jury returned its verdict, the district court dismissed Michael and Christopher Smith’s convictions on Count 15 (one of the mail-fraud counts). The district court also granted Christopher Smith a judgment of acquittal on Count 17 (another of the mail-fraud counts).

On October 8, 2013, we heard oral argument in Case Numbers 10-6136, 11-5828, and 12-5695. With the issues raised at oral argument still under consideration by this court, the government moved on February 5, 2014 to consolidate those appeals with Case Number 13-5440, a separately filed appeal raising the Smiths’ arguments under Brady v. Maryland, 373 U.S. 83, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963). We granted the government’s motion on February 18, 2014 to promote judicial economy and to provide a comprehensive record from which to render a decision.

II. ANALYSIS

A. Sufficiency of the evidence under Rule 29

Rule 29 of the Federal Rules of Criminal Procedure allows defendants to move for a judgment of acquittal on various legal grounds. One of those grounds is the alleged insufficiency of the evidence to support the jury’s verdict. In Jackson v. Virginia, 443 U.S. 307, 99 S.Ct. 2781, 61 L.Ed.2d 560 (1979), the Supreme Court established the standard for challenges based on the alleged insufficiency of the evidence, holding that “the relevant question is whether, after viewing the evidence in the light most favorable to the prosecution, any rational trier of fact could have found the essential elements of the crime beyond a reasonable doubt.” Id. at 319, 99 S.Ct. 2781 (emphasis in original); see also United States v. Warshak, 631 F.3d 266, 308 (6th Cir.2010) (en banc) (same). “This is a very heavy burden” for the defendant to meet. United States v. Jones, 641 F.3d 706, 710 (6th Cir.2011) (internal quotation marks omitted).

Reversal of a conviction is warranted “only if, viewing the record as a whole, the judgment is not supported by substantial and competent evidence.” United States v. Blakeney, 942 F.2d 1001, 1010 (6th Cir.1991). In deciding whether a conviction is supported by substantial and competent evidence, we do not “weigh the evidence, assess the credibility of witnesses, or substitute our judgment for that of the jury,” United States v. Paige, 470 F.3d 603, 608 (6th Cir.2006) (internal quotation marks omitted), and we draw “all available inferences and resolve all issues of credibility in favor of the jury’s verdict,” United States v. Salgado, 250 F.3d 438, 446 (6th Cir.2001). “We review de novo the district court’s denial of a motion for judgment of acquittal.” United States v. Algee, 599 F.3d 506, 512 (6th Cir.2010).

1. Sufficiency of the evidence on Count 1 (conspiracy to commit mail fraud, in violation of 18 U.S.C. § 1349)

The jury convicted Michael Smith on Count 1 for conspiring to commit mail fraud, but acquitted Christopher Smith of this conspiracy charge. Michael Smith argues that the government introduced no evidence of “any genuinely false statements of fact or omissions in communications with investors” and “no proof of any agreement to deceive investors or withhold material information from them.” According to Michael Smith, he did not do “anything different from what most oil and gas exploration businesses would do.”

“A conviction for conspiracy to commit mail fraud requires proof beyond a reasonable doubt that the defendant knowingly and willfully joined in an agreement with at least one other person to commit an act of mail fraud and that there was at least one overt act in furtherance of the agreement.” United States v. Cantrell, 278 F.3d 543, 546 (6th Cir.2001). Establishing a conspiracy requires only that the defendant “knew the object of the conspiracy and voluntarily associated himself with it to further its objectives.” United States v. Jamieson, 427 F.3d 394, 402 (6th Cir.2005) (internal quotation marks omitted). Mail fraud “consists of (1) a scheme or artifice to defraud; (2) use of mails in furtherance of the scheme; and (3) intent to deprive a victim of money or property.” United States v. Turner, 465 F.3d 667, 680 (6th Cir.2006) (paraphrasing 18 U.S.C. § 1341).

A scheme to defraud is “any plan or course of action by which someone uses false, deceptive, or fraudulent pretenses, representations, or promises to deprive someone else of money.” Jamieson, 427 F.3d at 402 (internal citation omitted). As is evident from this definition, a scheme to defraud is a relatively broad concept, encompassing pretenses, representations, or promises that are either “deceptive” or “false.” See id. Michael Smith thus relies on an incomplete definition of fraudulent conduct when he argues that the government presented “no evidence ... of any genuinely false statements of fact or omissions.” (emphasis added).

Fraudulent behavior clearly encompasses more than the communication of false statements. See Jamieson, 427 F.3d at 402. The government’s theory of the case — and the theory presented to the jury — is not simply that Michael Smith orchestrated a scheme to defraud by communicating false information, but that he committed mail fraud by making deceptive representations to potential and current investors. These misrepresentations include, among other things, exaggerated oil production, misleading royalty projections, and incorrect logistical information about particular wells — misrepresentations that lulled investors into entrusting Target Oil with their investment dollars.

The evidence at trial established that Michael Smith oversaw the production of Target Oil’s information packets. Maps and photographs contained in these packets regularly depicted oil pockets that did not exist or wells that were not owned by Target Oil. The evidence also established that Michael Smith supervised and condoned the use of high-pressure sales tactics — which often included gross misrepresentations of a particular well’s potential to produce resources and royalties — to encourage investors to part with their money. This evidence, which is but a sample of Michael Smith’s relevant conduct under Count 1, was more than sufficient for a rational juror to find a scheme to defraud beyond a reasonable doubt.

The testimony at trial established that the mailings were central to the conspiracy. After Target Oil’s salesmen contacted potential investors, they followed up by sending information packets, which set the stage for Target Oil’s closers to finalize the transaction. Company closers elaborated on the information contained in the packets, communicating misleading information about well production, well location, and the potential for revenue. These tactics lulled investors into believing that Target Oil’s wells would produce consistent revenue streams, but many investors ultimately found themselves with nothing. Given Michael Smith’s role in overseeing the creation of Target Oil’s investment materials, a rational juror could find that he “acted with knowledge that use of the mails would follow in the ordinary course of business, or that a reasonable person would have foreseen use of the mails.” See Cantrell, 278 F.3d at 546 (internal quotation marks omitted).

The evidence was also sufficient to establish fraudulent intent. “[A] jury may consider circumstantial evidence of fraudulent intent and draw reasonable inferences therefrom. Intent can be inferred from efforts to conceal the unlawful activity, from misrepresentations, from proof of knowledge, and from profits.” United States v. Davis, 490 F.3d 541, 549 (6th Cir.2007) (internal quotation marks omitted). Evidence that Target Oil personnel exaggerated well production and misrepresented royalty streams are the types of misrepresentations that easily fit within the Davis criteria. And a rational jury could also conclude that Target Oil’s actions in placating dissatisfied investors by giving them extra shares in new but un-derperforming wells were calculated maneuvers designed to conceal the extent of the company’s fraudulent scheme. From this evidence and more, a rational jury could conclude beyond a reasonable doubt that Michael Smith conspired to commit mail fraud.

2. Sufficiency of the evidence on the substantive counts of mail fraud, in violation of 18 U.S.C. § 1341

Both Michael and Christopher Smith challenge their convictions on the substantive counts of mail fraud, arguing that the portions of the indictment that charge mail fraud do not constitute a violation of federal law. Particularly crucial to their argument is testimony from Target Oil’s accountant, Craig Butler, who provided loss figures materially different than those provided by the government’s postal inspector. Butler testified that approximately $1,538,700 was paid to investors from 2003 to 2008, far more than the $460,000 calculated by the government’s witness. Michael and Christopher Smith also challenge several of the allegations in the indictment, including claims that Target Oil employees contacted unaccredited investors and that the company used unlicensed salesmen.

Based on the evidence presented at trial, Michael Smith was convicted on eleven counts of mail fraud arising from: (1) the receipt of a mailing from the Kentucky Department of Natural Resources about the authority to drill, deepen, or reopen the Clinton # 1, Raymond, and Reneau wells; (2) sending a Target Oil investor packet to James Smith; (3) sending a Target Oil investor packet to Louis Jeffrey; (4) Daniel Grethen’s letter and payment in response to Target Oil’s solicitation with regard to a well called Floyd Well # 6; (5) the receipt of a mailing from the Kentucky Department of Natural Resources to Target Oil about the authority to drill, deepen, or reopen the Floyd # 6 well; (6) sending sales material about the Clinton # 101, # 102, and # 103 wells to investor Ronald Knight; and (7) the receipt of four letters sent from the Kentucky Department of Resources about the authority to drill, deepen, or reopen several wells, including Bell County # 1, Bell County # 2, Bell County # 3, Knox Zephyr # 2, and the Dallas Brown well.

Christopher Smith was convicted on seven of the mail-fraud counts arising from: (1) mailing a Target Oil investor packet to James Smith; (2) mailing a Target Oil investor packet to Louis Jeffrey; (3) Daniel Grethen’s letter and payment in response to Target Oil’s solicitation with regard to a well called Floyd Well # 6; (4) the receipt of a letter from the Kentucky Department of Natural Resources to Target Oil about the authority to drill, deepen, or reopen Floyd # 6; and (5) mailing sales material about the Clinton # 101, # 102 and # 103 wells to investor Ronald Knight.

Michael and Christopher Smith do not contest that the mailings relevant to the mail-fraud counts were sent or received. The relevant question is whether the mailings were “part of the execution of the scheme as conceived by the perpetrator at the time.” See Schmuck v. United States, 489 U.S. 705, 715, 109 S.Ct. 1443, 103 L.Ed.2d 734 (1989). Being “part of’ the execution of the scheme does not mean that using the mail must be a direct part of the fraud. See id. at 710-11, 109 S.Ct. 1443. Instead, “[i]t is sufficient for the mailing to be incident to an essential part of the scheme, or a step in [the] plot.” Id. (second alteration in original) (internal quotation marks omitted).

The mailings forming the basis of the mail-fraud counts consisted of information sent to and from potential investors and of certain letters that Target Oil received from the Kentucky Department of Natural Resources imposing additional requirements that were not disclosed to those investors. Regarding the materials sent to and from potential investors, the proof showed that they received information packets via the mail after being initially contacted by Target Oil salesmen. These packets contained false or misleading information that increased the likelihood of an investor sending money to Target Oil. The information packets preceded interactions with the “closers,” during which salesmen for Target Oil capitalized on the false and misleading information and applied high-pressure sales techniques to finalize investment transactions. From this and other evidence, a rational juror could conclude that Michael and Christopher Smith were guilty of mail fraud beyond a reasonable doubt.

Part of this “other evidence” was proof that, as time passed, Michael Smith was unable to obtain the necessary drilling permits under Target Oil’s name, which forced him to obtain permits in the name of Kentucky-Indiana. Target Oil’s problems with the Kentucky regulators were the main reason why he had to use Kentucky-Indiana’s name. But investors were never told that drilling permits had been obtained under a different name than Target Oil, nor were they told of the problems that Target Oil was having with the Kentucky regulators. A rational juror could conclude that obtaining permits under these circumstances was incidental to or part of the fraud.

Michael and Christopher Smith nevertheless urge us to reject the jury’s verdict, claiming that the government failed to prove specific allegations of fraud in the indictment and that Smith’s accountant provided an amount of loss materially lower than that presented by the government. The specific allegations that were allegedly unproven or directly contradicted by the defense — that Target Oil hired unlicensed salesmen and preyed upon unaccredited investors — are isolated allegations that, even if the government failed to prove them, would not undermine the jury’s verdict. Overwhelming evidence in the record establishes mail fraud, and the government’s purported failure to prove that Target Oil used unlicensed salesmen or targeted unaccredited investors is inconsequential. We find nothing in the record to warrant disturbing the jury’s findings on these counts.

Michael and Christopher Smith also argue that their convictions should be overturned because the government witness who testified to the amount paid in royalties allegedly admitted that her calculations were incomplete and inaccurate. They contend that Target Oil’s accountant produced more accurate and substantiated figures than Postal Inspector Roberta Bottoms, the government’s witness. But resolving whatever differences exist between the parties’ presentation of loss calculations is an issue of witness credibility. Each party offered a witness who testified in support of the respective party’s theory-of-loss calculations, the jury weighed the testimony of these witnesses, and ultimately credited the evidence of fraud. Issues of witness credibility are not properly before us in a sufficiency-of-the-evidenee appeal. See United States v. Paige, 470 F.3d 603, 608 (6th Cir.2006) (concluding that the court does not “weigh the evidence, assess the credibility of witnesses, or substitute [its] judgment for that of the jury” when evaluating the sufficiency of the evidence) (internal quotation marks omitted).

Moreover, even if the investor loss amounts are materially inconsistent, the discrepancy is not germane. “Using the mail to execute or attempt to execute a scheme to defraud is indictable as mail fraud.” Bridge v. Ph. Bond. & Indem. Co., 553 U.S. 639, 648, 128 S.Ct. 2131, 170 L.Ed.2d 1012 (2008) (emphasis added). The amount of investor loss is thus irrelevant for suffieiency-of-the-evidence purposes. United States v. Warshak, 631 F.3d 266, 309 (6th Cir.2010) (concluding that a scheme to defraud “would constitute fraud by increasing a risk of loss, even if no actual monetary loss were shown”).

B. Constructive amendment and variance regarding Count 1

We now turn to Michael Smith’s claim of a constructive amendment to and variance from the indictment as to Count 1. The issue was first raised during a conference in the midst of the trial. Michael Smith contended that the government had offered evidence not referenced in the indictment. The evidence he challenged pertained to the unavailability of commercial gas lines in Clinton County, Kentucky and to Target Oil’s drilling in locations that were different from what had been advertised to investors. According to Michael Smith, the introduction of this evidence constitutes a constructive amendment or a variance that warrants reversal.

We generally review “de novo whether there was a constructive amendment to the indictment.” United States v. Ferguson, 681 F.3d 826, 830 (6th Cir.2012). However, “where no specific objection is raised regarding a constructive amendment or a variance before the district court, we are limited to ‘plain error’ review on appeal.” United States v. Benson, 591 F.3d 491, 497 (6th Cir.2010) (internal quotation marks omitted).

Michael Smith challenged two pieces of evidence at the Rule 29 hearing on his motion to acquit, but he challenges three on appeal. He is entitled to de novo review of his argument that the government constructively amended or varied from the indictment by introducing evidence of the absence of commercial gas lines in Clinton County, Kentucky and that Target Oil drilled in different locations than advertised. See Ferguson, 681 F.3d at 830. But because he failed to challenge below evidence that Target Oil obtained permits under Kentucky-Indiana’s name, we will review that argument under the stricter plain-error standard. See Benson, 591 F.3d at 497. When reviewing a claim under that standard, reversal is warranted only “if it is found that (1) there is an error; (2) that is plain; (3) which affected the defendant’s substantial rights; and (4) that seriously affected the fairness, integrity or public reputation of the judicial proceedings.” United States v. Barnes, 278 F.3d 644, 646 (6th Cir.2002).

An indictment by a grand jury is “itself a right guaranteed by the Constitution in federal prosecutions.” United States v. Hynes, 467 F.3d 951, 961 (6th Cir.2006). This right “also protects two other constitutional rights — the Sixth Amendment right to fair notice of the criminal charges against a defendant and the Fifth Amendment’s protection[ ] against twice placing a defendant in jeopardy for the same offense.” United States v. Hynes, 467 F.3d 951, 961 (6th Cir.2006) (third alteration in original) (internal quotation marks omitted). When allegations in an indictment differ from the evidence offered at trial, this court considers three separate but related theories of constitutional relief: an amendment, a variance, “and a third category lying between the other two — a constructive amendment.” Id.

The difference between a variance and a constructive amendment is a matter of degree. Variances occur “when the charging terms of an indictment are left unaltered, but the evidence offered at trial proves facts materially different from those alleged in the indictment.” United States v. Combs, 369 F.3d 925, 935-36 (6th Cir.2004) (internal quotation marks omitted). Constructive amendments, on the other hand, are “variances occurring when an indictment’s terms are effectively altered by the presentation of evidence and jury instructions that so modify essential elements of the offense charged that there is a substantial likelihood the defendant [was] convicted of an offense other than that charged in the indictment.” Id. at 936 (alteration in original) (internal quotation marks omitted). Michael Smith bears the burden of proving that a variance “has occurred and that the variance rises to the level of a constructive amendment.” See United States v. Hynes, 467 F.3d 951, 962 (6th Cir.2006) (internal citation omitted).

Whether the government’s proof constructively amended or varied from the indictment can be determined using this circuit’s analytical framework for variances, given that constructive amendments and variances are different degrees of the same legal theory. See id at 961-62 (stating that constructive amendments “are variances occurring when an indictment’s terms are effectively altered by the evidence and the jury instructions”) (internal quotation marks omitted). “For purposes of determining the existence of a variance, the principles applicable to conspiracies may be applied to mail fraud prosecutions.” United States v. Horton, 847 F.2d 313, 317 (6th Cir.1988). The “principles applicable to conspiracies” is a phrase that encapsulates several factors, including the existence of a common goal, the nature of the scheme, and the overlapping participants. Id. at 317-18 (internal quotation marks omitted).

Of critical importance is whether the evidence offered departs from the conspiracy alleged in such a manner that “can reasonably be construed only as supporting a finding of multiple [schemes].” Id. at 317 (alteration in original) (internal quotation marks omitted). The indictment in this case characterizes the Smith brothers’ conduct as a sophisticated conspiracy predicated on investors being “induced to enter into investment contracts with Target Oil ... and Kentueky[-]Indiana ... and to purchase shares in oil and gas well[-]drilling programs by false and fraudulent pretenses and representations.” The indictment lists numerous overt acts in furtherance of the alleged conspiracy, all of which arose from mailings sent to or from potential investors by Target Oil or received by the company from the Kentucky Department of Natural Resources. The indictment specifies that each letter from the Kentucky Department of Natural Resources related to a specific drilling permit number for Kentucky-Indiana wells— wells that were in actuality drilled and operated by Target Oil.

Given the commonality between Target Oil’s and Kentucky-Indiana’s activities, evidence that Target Oil obtained permits in Kentucky-Indiana’s name can reasonably be construed as part of the conspiracy alleged in the indictment. Michael Smith was the president of both companies. Although the permits for the wells in question were in Kentucky-Indiana’s name, the wells were marketed in Target Oil’s name by Target Oil salesmen. The drilling permit for Texas Zephyr #2, for example, was obtained in Kentucky-Indiana’s name, but in communications with investors Target Oil is the entity that is represented as marketing the well. Evidence that Target Oil obtained permits under a different name suggests a level of dishonesty that reasonably fits the overall character of the conspiracy alleged in the indictment. The overlap of personnel and common goals between Target Oil and Kentucky-Indiana thus shows that the government’s evidence did not vary from the indictment, meaning that the district court did not plainly err in admitting that evidence. See Horton, 847 F.2d at 317-18 (holding that a court will find no variance if the proof supports a common goal and overlapping participants, among other factors).

Nor does evidence of unavailable gas lines in Clinton County, Kentucky vary from the conspiracy alleged in the indictment. Drilling for natural gas in an area with no commercial access to gas lines establishes that generating royalties would have been impossible because there was essentially no market for the gas in Clinton County. No gas lines, in other words, meant no profits. Soliciting investments in wells that could not generate profits is consistent with the overall conspiracy charged in the indictment, which took the form of inducing investors to “purchase shares in oil and gas well-drilling programs by false and fraudulent pretenses and representations.”

Finally, drilling in locations other than as advertised is closely connected to the overt acts listed in the indictment, particularly the mailing of information packets that provided false or misleading information and were intended to instill false confidence in the investors. Evidence that Target Oil drilled elsewhere than represented can be characterized as showing an overt act in furtherance of the conspiracy alleged in the indictment. When allegations of an overt act are not specifically listed in an indictment, there is no variance if “the theory of the case [is] not changed, the defendant [is] not charged with a different substantive crime, and the elements of the crime charged [are] not altered.” United States v. Atisha, 804 F.2d 920, 927 (6th Cir.1986). Admitting evidence of drilling in different locations than advertised did not alter the underlying elements of conspiracy to commit mail fraud or charge a new crime. Michael Smith simply needed to rebut the government’s contention that drilling in different locations amounted to fraud.

Given these factors, we find no merit in the argument that the government’s evidence rises to the level of a variance. Nor did the evidence change the defense’s theory of the case because Michael Smith’s contention was always that he operated a legitimate family business that used sound business and geological practices. All these points suggest that, at most, the government’s evidence was an “alternative method” of proving the conspiracy alleged, not a completely different conspiracy. See United States v. Hynes, 467 F.3d 951, 964 (internal quotation marks omitted). Because we conclude that there was no variance, we have no need to address the issue of constructive amendment.

C. Procedural and substantive reasonableness of sentences and forfeiture

Michael and Christopher Smith next argue that their sentences were procedurally and substantively unreasonable. At sentencing, the district court calculated Michael Smith’s Guidelines range at 262 to 327 months of imprisonment, which was determined by adding enhancements for the amount of loss caused by the conspiracy, the number of victims, the sophisticated means of Michael Smith’s crimes, and his criminal leadership. These enhancements increased the base offense level of 7 to a total offense level of 39.

Christopher Smith’s Guidelines range was calculated at 135 to 168 months of imprisonment. The district court added enhancements for the amount of loss caused by Christopher Smith and the conspiracy, the sophisticated means of his crimes, and the number of victims. These enhancements increased the base offense level of 7 to a total offense level of 33. Although Christopher Smith was acquitted of the conspiracy charged in Count 1 of the indictment, the court justified the enhancements by concluding, based on a preponderance of the evidence, that he was liable for many of the actions taken on behalf of the conspiracy.

We review sentences imposed by the district court for reasonableness. United States v. Smith, 474 F.3d 888, 892 (6th Cir.2007), abrogated on other grounds by Gall v. United States, 552 U.S. 38, 128 S.Ct. 586, 169 L.Ed.2d 445 (2007). Sentences within the applicable Guidelines range are afforded a presumption of reasonableness, whereas a sentence “outside the Guidelines carries with it no legal presumption.” United States v. Vowell, 516 F.3d 503, 509 (6th Cir.2008). We must “review every sentence that is free from significant procedural error under a deferential abuse-of-discretion standard, regardless of whether the sentence is within the Guidelines range or significantly outside it.” Id. (citing Gall, 552 U.S. at 49, 128 S.Ct. 586).

Michael and Christopher Smith argue that the district court improperly calculated the sentencing enhancements for the amount of loss and the number of victims. Specifically, they contend that the court erred by basing the enhancement for loss on conduct for which they were acquitted. They also argue that the court erred in calculating the number of victims — and thus incorrectly calculated the resulting Guidelines range — by considering their conduct on the many counts for which they were acquitted.

The Smith brothers further claim that their sentences were substantively unreasonable because the court improperly balanced the sentencing factors set forth in 18 U.S.C. § 3553(a). Although the court varied substantially downward from the applicable Guidelines range in imposing a 120-month sentence on Michael Smith, he argues that the § 3553(a) factors warrant an even greater downward variance. Christopher Smith adopts Michael Smith’s arguments, contending that his own 60-month sentence was likewise substantively unreasonable.

1. The sentences were procedurally reasonable

Our “reasonableness analysis begins with a robust review of the factors evaluated and the procedures employed by the district court in reaching its sentencing determination.” United States v. Bolds, 511 F.3d 568, 578 (6th Cir.2007) (internal quotation marks omitted). Procedural reasonableness turns on whether the district court: “(1) properly calculated the applicable advisory Guidelines range; (2) considered the other § 3553(a) factors as well as the parties’ arguments for a sentence outside the Guidelines range; and (3) adequately articulated its reasoning for imposing the particular sentence chosen.” Id. at 581. Because Michael and Christopher Smith focus their challenge to procedural reasonableness on the district court’s calculation of their applicable Guidelines ranges, we will limit our review to that issue.

The Guidelines range for each of the Smith brothers was particularly affected by the district court’s determination of the amount of loss and the number of victims. In calculating the amount of loss, the court started from the premise that “the entire program that Target Oil and its subsidiary, Kentucky[-Indiana], was running at that time was designed to raise money, it was designed for the sole benefit of the Target family.” After reciting numerous facts that were established at trial, the court concluded that whether Target Oil “hit oil or not, hit gas or not was immaterial.” The court stated that it was “concerned with valuing the harm to the society as a whole suffered from the defendant’s fraud.” Using Target Oil’s receipts from 2003 to 2008, which were prepared by the company’s accountant, the court determined that the appropriate amount of loss equaled the total amount of investor funds taken in by Target Oil, minus any royalties paid. That net amount — $14,276,681—led to a 20-level enhancement under the Guidelines.

The district court employed similar logic in determining the number of victims:

[Consistency would say in this particular case that the number of — if I find that every oil well program was pitched to these people with an intent maybe to defraud them, probably mislead them anyway, that I have done to determine the amount of loss, then consistency would obviously require that anybody who did invest in that or was subjected to that may be a victim, I think that the number of victims ... exceeds 250, and precisely 323. So the six-level enhancement urged by the government I think is appropriate.

Michael and Christopher Smith also contend that the district court erred by considering acquitted conduct, but they concede that binding precedent in this circuit holds to the contrary. As this court has stated, “so long as the defendant receives a sentence at or below the statutory ceiling set by the jury’s verdict, the district court does not abridge the defendant’s right to a jury trial by looking to other facts, including acquitted conduct, when selecting a sentence within that statutory range.” United States v. White, 551 F.3d 381, 385 (6th Cir.2008) (en banc). We thus find no error in the district court’s imposition of the below-Guidelines sentences for Michael and Christopher Smith that were calculated in part by using acquitted conduct.

Nor did the district court err in calculating the amount of loss. A district court “need only make a reasonable estimate of the loss,” a directive based on the sentencing court’s “unique position” in assessing the evidence to arrive at that estimate. U.S.S.G. § 2B1.1 cmt. n. 3(C). Calculating the gross amount of loss attributable to Michael Smith at $15,815,371 was not unreasonable in light of the court’s finding that Target Oil’s primary purpose was to generate money for its principals rather than for its investors. See id. § 2B1.1 cmt. n. 3(A)(i). The court properly offset from that amount “the fair market value of the property returned ... by the defendant ... to the victim before the offense was detected,” which produced a net loss of $14,276,681. Id. § 2B1.1 cmt. n. 3(E)(i). Because the total loss exceeded $7,000,000, the court did not err in adding a 20-point enhancement to Michael Smith’s Guidelines offense level. See id. § 2B1.1(b)(1)(K).

Nor did the district court err in concluding that the net loss attributable to Christopher Smith was $9,767,770. Because Christopher Smith left the conspiracy earner than his brother, the court properly limited its loss calculations to the time frame during which Christopher Smith was a member of the conspiracy, minus any royalties paid. During his participation in the conspiracy, however, Christopher Smith could be held accountable for “the reasonably foreseeable harm that was intended to result from the offense.” See U.S.S.G. § 2B1.1 cmt. n. 3(A)(i). Offsetting the total funds received by Target Oil with the investments received after Christopher Smith had withdrawn from the conspiracy therefore reflects the loss caused by his behavior.

A similar logic applies in calculating the number of victims attributable to the fraudulent conduct of each defendant. Under the Guidelines, a victim is “any person who sustained any part of the actual loss determined.” Id. § 2B1.1 cmt. n. 1. The phrase “any part” casts a wide net and applies “[s]o long as a person suffers reasonably foreseeable pecuniary harm as a result of an offense.” United States v. Stokes, 392 Fed.Appx. 362, 368 (6th Cir.2010). Because Michael Smith directed Target Oil’s operations for the full duration of the conspiracy, he was properly held accountable for all investors who were adversely affected by “any part” of the company’s fraudulent activity.

The same analysis applies in calculating the number of victims attributable to Christopher Smith. At sentencing, the district court acknowledged that Christopher Smith left the conspiracy early. It therefore reasoned that if Christopher Smith “is not working for the company, then he shouldn’t be—should not be held responsible for the loss that occurred while he—subsequent to him leaving, or he should not be held for a number of victims subsequent to him leaving; so that’s exactly what I did.” Consequently, the court properly applied a four-level enhancement based on Christopher Smith defrauding 213 victims, two-thirds of the total victims in the conspiracy.

2. The sentences were substantively reasonable

For a sentence to be substantively reasonable, “it must be proportionate to the seriousness of the circumstances of the offense and offender, and sufficient but not greater than necessary, to comply with the purposes of § 3553(a).” United States v. Vowell, 516 F.3d 503, 512 (6th Cir.2008) (internal quotation marks omitted). “A properly calculated advisory [G]uidelines range represents a starting point for substantive-reasonableness review because it is one of the § 3553(a) factors and because the [Guidelines purport to take into consideration most, if not all, of the other § 3553(a) factors.” United States v. Holcomb, 625 F.3d 287, 293 (6th Cir.2010) (internal quotation marks omitted). A sentence will be found substantively unreasonable “when the district court selects a sentence arbitrarily, bases the sentence on impermissible factors, fails to consider relevant sentencing factors, or gives an unreasonable amount of weight to any pertinent factor.” Id. (internal quotation marks omitted).

When sentencing Michael Smith, the district court considered the various factors under § 3553(a) and granted a downward variance, concluding that a sentence within the Guidelines range of 262 to 327 months was unnecessary. The court instead imposed a sentence of 120 months of imprisonment. In doing so, the court concentrated on the seriousness of the offense and the need to deter future conduct. Michael Smith’s abuse of the fiduciary trust of Target Oil’s investors underscored the judgment of the court that the fraud perpetrated was serious and that similar conduct needed to be deterred. Hundreds of victims—including one whose infirmities were such that he “[c]ouldn’t even remember his own age”—had their trust violated by Michael Smith, which the district court concluded warranted substantial punishment.

The fact that the district court specified the relevant § 3553(a) factors and explained the reasons for the particular sentence imposed shows that “the district court did not arbitrarily choose a sentence, but chose a sentence it considered sufficient but not greater than necessary to comply with the purposes of § 3553(a).” See Vowell, 516 F.3d at 512. “On abuse of discretion review, we will give ‘due deference to the [district [c]ourt’s reasoned and reasonable decision that the § 3553(a) factors, on the whole, justified the sentence.’ ” Id. (citing Gall v. United States, 552 U.S. 38, 59-60, 128 S.Ct. 586, 169 L.Ed.2d 445 (2007)). The court could easily have sentenced Michael Smith within his applicable Guidelines range, so the actual 120-month sentence— which is 142 months below the low end of that range—can hardly be said to be substantively unreasonable.

Turning next to Christopher Smith’s sentence, the district court explicitly stated that it did not “plan to sentence within the guideline range. Matter of fact, ... I will ... grant the defendant a variance.” The need for a 60-month sentence was explained by detailing the seriousness of Christopher Smith’s offense, including his abuse of investor trust. According to the court, Christopher Smith was the person who talked to potential investors and was the person who convinced them to place their trust and money in Target Oil. Deterring future conduct was at the forefront of the court’s reasoning, with the court declaring that “I want people to think, if I am going to sell those things, I remember Chris Smith got in trouble and he got some time.” The court sought to make Christopher Smith “the poster boy for what happens” when one misleads investors or misrepresents the potential of speculative oil drilling. ^

In sum, the district court properly calculated Christopher Smith’s applicable Guidelines range of 135 to 168 months of imprisonment and discussed the relevant § 3553(a) factors, giving each an appropriate weight. Although the court did not explicitly state why it imposed a below-Guidelines sentence on Christopher Smith, there is nothing in the record to suggest that the court made its decision arbitrarily. The court considered the deterrent effects of the sentence, concluding that Christopher Smith’s sentence would make him an example for what happens to those who engage in fraudulent conduct. His sentence was only half of what Michael Smith received, but that lower sentence still reflects Christopher Smith’s role as an integral — though lower-level — member of the Target Oil operation. The sentence imposed by the district court is therefore substantively reasonable.

3. The sentences do not violate Ap-prendi and its progeny

Michael and Christopher Smith next argue that the district court violated Apprendi v. New Jersey, 530 U.S. 466, 120 S.Ct. 2348, 147 L.Ed.2d 435 (2000), when it increased their applicable Guidelines ranges based on findings related to the amount of loss and the number of victims. In Ap-prendi, the Supreme Court held that any fact that increases the prescribed statutory maximum sentence for an offense is an “element” that must be found by the jury. Id. at 483 n. 10, 120 S.Ct. 2348 (“[F]aets that expose a defendant to a punishment greater than that otherwise legally prescribed were by definition ‘elements’ of a separate legal offense.”).

The Supreme Court’s decision in Alleyne v. United States, — U.S. —, 133 S.Ct. 2151, 2163, 186 L.Ed.2d 314 (2013), extended Apprendi by holding that “facts that increase mandatory minimum sentences must be submitted to the jury.” But both Apprendi and Alleyne took care not to disturb the district court’s discretionary fact-finding in other circumstances. See id. (“Our ruling today does not mean that any fact that influences judicial discretion must be found by a jury.”); see also Apprendi 530 U.S. at 481, 120 S.Ct. 2348 (“[N]othing in this history suggests that it is impermissible for judges to exercise discretion — taking into consideration various factors relating both to offense and offender — in imposing a judgment within the range prescribed by statute.”) (emphasis in original).

Neither Michael nor Christopher Smith faced a statutory maximum or statutory minimum sentence, meaning that Apprendi and Alleyne are inapplicable. In related contexts, this court has concluded that “Apprendi does not purport to apply to penalties in excess of any particular range or based on any particular offense level under the Sentencing Guidelines.” United States v. Garcia, 252 F.3d 838, 843 (6th Cir.2001). If Apprendi does not apply to penalties in excess of “any particular range” under the Guidelines, simple logic dictates that it also does not apply to sentences imposed below the Guidelines range. We therefore reject the Smiths’ Apprendi-based argument.

4. The forfeiture judgment against Michael Smith was supported bg sufficient evidence

Michael Smith farther argues that the evidence was insufficient to support the $3,192,793.50 forfeiture judgment against him. We “review the district court’s interpretation of the federal forfeiture laws de novo.” United States v. O’Dell, 247 F.3d 655, 679 (6th Cir.2001) (italics omitted). “The Government must prove forfeiture by a preponderance of the evidence.” United States v. Jones, 502 F.3d 388, 391 (6th Cir.2007). Findings of fact will not be set aside unless clearly erroneous, but whether those facts are “sufficient to constitute a proper criminal forfeiture” is a determination that we review de novo. O’Dell, 247 F.3d at 679.

Although Michael Smith broadly contends that the forfeiture judgment against him should be reduced by approximately two million dollars, his argument on this issue is limited to two cashier’s checks payable to Vernon Smith, his father, that were seized from a safe located in his father’s residence. The jury ordered the forfeiture of the two cashier’s checks in question: a $60,649.64 cashier’s check drawn on Target Oil’s account at First Southern National Bank and a $100,000.00 cashier’s check drawn on Target Oil’s account at Whitaker Bank.

“A district court must order the forfeiture of a defendant’s interest in property when a nexus is drawn between the defendant’s criminal conduct and the property.” Jones, 502 F.3d at 391-92 (internal citation omitted); see also 28 U.S.C. § 2461(c) (“If the defendant is convicted of the offense giving rise to the forfeiture, the court shall order the forfeiture of the property as part of the sentence in the criminal case pursuant to the Federal Rules of Criminal Procedure and section 3554 of title 18, United States Code.”). Criminal forfeiture judgments are mandatory for mail-fraud convictions. See 18 U.S.C. § 982(a)(2) (listing mail fraud as an offense subject to mandatory property forfeiture).

To demonstrate that property is “subject to forfeiture under 18 U.S.C. § 982(a)(2), the government must show that the property constituted, or was derived from, proceeds the [defendant] obtained directly or indirectly, as the result of [certain illegal conduct or a conspiracy to commit certain illegal conduct].” United States v. Warshak, 631 F.3d 266, 332 (6th Cir.2010) (alteration in original) (internal quotation marks omitted). The property at issue must be traceable to particular offenses, including conspiracies to commit a crime. Id.

In this case, there is evidence that the checks in question are proceeds that “resulted, whether directly or indirectly, from unlawful activity.” Id. Banking records introduced at trial show that Michael Smith deposited investor funds into accounts at First Southern National Bank and at Whitaker Bank, and the proof established that the checks were drawn from the funds placed into these accounts. Because the deposits received greatly exceeded the amounts payable in the checks, the district court could reasonably infer that sufficient liquidity existed to have had the checks honored.

Ample evidence also supports the district court’s finding that Target Oil was used as a vehicle to commit fraud. See Part II.A.1. above. The court concluded that fraud touched everything — from Target Oil’s banking accounts to its day-to-day operations — meaning that the various items connected to Target Oil’s revenue stream are subject to forfeiture because the transactions “all resulted directly or indirectly from a conspiracy to commit fraud.” See Warshak, 631 F.3d at 332 (internal quotation marks omitted).

Beyond the forfeiture of the two cashier’s checks, Michael Smith argues that his total forfeiture amount should be no more than $1,344,629.77, but he offers no support or argument for his assertion. “[I]t is a settled appellate rule that issues adverted to in a perfunctory manner, unaccompanied by some effort at developed argumentation, are deemed waived.” United States v. Keller, 498 F.3d 316, 326 (6th Cir.2007) (internal quotation marks omitted). We therefore find no basis to disturb the district court’s determination regarding the amount of forfeiture.

D. Motions for new trial based on alleged Brady violations

We will now turn our attention to the Smith brothers’ claims that they were improperly denied a new trial. They base their arguments on alleged Brady violations by the government.

1. Standard of review

We review the “denial of a motion for new trial based on Brady violations under an abuse of discretion standard.” United States v. Graham, 484 F.3d 413, 416 (6th Cir.2007) (internal citation omitted). “However, the district court’s determination as to the existence of a Brady violation is reviewed de novo.” Id. at 416-17.

“[T]he suppression by the prosecution of evidence favorable to an accused ... violates due process where the evidence is material either to guilt or to punishment, irrespective of the good faith or bad faith of the prosecution.” Brady v. Maryland, 373 U.S. 83, 87, 83 S.Ct. 1194, 10 L.Ed.2d 215 (1963). The principles in Brady have been extended to the disclosure of impeachment evidence in cases where the “reliability of a given witness may well be determinative of guilt or innocence.” Giglio v. United States, 405 U.S. 150, 154, 92 S.Ct. 763, 31 L.Ed.2d 104 (1972) (internal quotation marks omitted). To establish a violation of Brady, a defendant “has the burden of showing that the Government suppressed evidence, that such evidence was favorable to the defense, and that the suppressed evidence was material.” Graham, 484 F.3d at 417.

But there is no Brady violation “if the defendant knew or should have known the essential facts permitting him to take advantage of the information ... or if the information was available to him from another source.” Id. (internal quotation marks omitted). Moreover, reversal is warranted only when “there is a ‘reasonable probability’ that, had the evidence been disclosed to the defense, the result of the proceeding would have been different. A reasonable probability is a probability sufficient to undermine the confidence in the outcome.” United States v. Mullins, 22 F.3d 1365, 1371 (6th Cir.1994).

2. The preservation of the motions for new trial on appeal

Michael and Christopher Smith filed four post-trial motions based on Brady and Giglio, but only two of them are preserved for appellate review. On March 23, 2011, Christopher Smith first moved for post-trial disclosure of Brady and Giglio material premised on Irwin’s admission that he had filed false financial affidavits and that he had failed to disclose his participation in an oil and gas company called Summit Energy Corporation. Michael Smith joined Christopher Smith’s motion. The district court denied the motion, but neither Michael nor Christopher Smith filed a timely appeal. This ruling is therefore not before us for consideration.

Approximately one month later, on April 29, 2011, Michael Smith filed a motion for a new trial, which his brother joined, arguing that Irwin’s false declarations to the district court necessitated a new trial. The district court denied the motion, and Michael Smith (but not his brother) timely appealed. Although Christopher Smith later filed a Notice of Appeal in which he sought to preserve review of “any prior orders or judgments including but not limited to ... every order denying any motion for new trial,” this Notice of Appeal was filed more than one month after the Amended Judgment was entered and approximately one year after the motion had been denied. Michael Smith — but not his brother — has thus preserved review of the first motion for a new trial. See Fed. R.App. P. 4(b)(3)(A) (stating that a notice of appeal “must be filed within 14 days after the entry of the order disposing of the last such remaining motion, or within 14 days after the entry of the judgment of conviction, whichever period ends later”).

On April 18, 2012, Michael Smith filed a second motion for a new trial, accusing the government of pursuing “five quack theories” about oil and gas exploration. The memorandum in support of Michael Smith’s motion indicated that it was in support of “defendants Michael Smith and Christopher Smith’s motion for new trial,” but Christopher Smith never filed separately to indicate that he wished to join his brother’s motion. The district court denied this second motion for new trial on May 11, 2012, but neither Michael nor Christopher Smith appealed. Although the Smith brothers later filed a joint Notice of Appeal on June 16, 2012 after the court entered amended judgments on May 17, 2012, this Notice of Appeal is insufficient to preserve review of their second motion for a new trial because it was filed more than one month after the denial of that motion. See id.

On June 16, 2012, Michael and Christopher Smith jointly filed a third motion for new trial, arguing that the government improperly suppressed a report from the Kentucky Board of Registration for Professional Geologists (the Board). The Smith brothers again raised the argument that the government presented “quack” geological theories to the jury, claiming that newly discovered evidence discredited the government’s positions. The district court denied this third motion on April 1, 2013, and the Smith brothers filed a joint notice of appeal on April 4, 2013. This Notice of Appeal is timely because it was filed within 14 days after the district court denied their third motion for a new trial. See Fed. R.App. P. 4(b).

3. The fírst motion for a new trial

Michael Smith argues that the prosecutor violated Brady by failing to disclose that Irwin had engaged in fraudulent activity when he worked at Summit Energy Corporation following his employment at Target Oil. But this information was available to Michael Smith “in time for use at trial.” See United States v. Presser, 844 F.2d 1275, 1283 (6th Cir.1988). An email from the Assistant United States Attorney handling the prosecution to the lawyers for Michael Smith dated May 31, 2010 informed the defense that “Irwin was involved for a short time with another oil and gas company after leaving Target Oil and may have engaged in conduct arguably similar to the activities at Target Oil.”

Although the email did not specifically name Summit Energy, the communication put Michael and Christopher Smith on notice that Irwin had engaged in potentially fraudulent conduct at another energy company. And the prosecutor’s disclosure in the email suggests that she did not knowingly conceal or. withhold that impeachment evidence. “[S]howing that the prosecution knew of an item of favorable evidence unknown to the defense does not amount to a Brady violation, without more.” Kyles v. Whitley, 514 U.S. 419, 437, 115 S.Ct. 1555, 131 L.Ed.2d 490 (1995) (emphasis added); accord United States v. Ross, 245 F.3d 577, 584 (6th Cir.2001) (same). Michael Smith has not pointed to anything in the record that could be considered “more,” and nothing in the record suggests that the government acted improperly.

Nor is a new trial always necessary to account for newly discovered evidence. A new trial may be ordered if “new evidence was discovered after the trial; the evidence could not have been discovered earlier with due diligence; the evidence is material and not merely cumulative or impeaching; and the evidence would likely produce an acquittal.” United States v. Braggs, 23 F.3d 1047, 1050 (6th Cir.1994) (internal citation and numbering omitted). Evidence that Irwin had lied about his involvement with Summit Energy was discovered after trial, but the prosecutor’s email on May 31, 2010 clearly put the Smiths on notice that Irwin worked with- — and engaged in questionable activity at — another company after leaving Target Oil. This was more than a week before trial.

Moreover, Michael Smith’s counsel thoroughly cross-examined Irwin at trial, with the defense exploring issues as varied as Irwin’s drug addiction and his sexual promiscuity. The jury appears to have credited Irwin’s testimony despite the cross-examination, and nothing in the record suggests that conducting a new trial to include evidence of Irwin’s activity at Summit Energy would produce an acquittal.

4. The third motion for a new trial

Similarly, the district court did not err in denying Michael and Christopher Smith’s third motion for a new trial, which was premised on newly discovered emails allegedly establishing that (1) the government prosecution was based on “quack” geological theories, (2) the government suppressed a report prepared by the Board, (3) the government concealed correspondence with Marvin Combs, Assistant Director of the Kentucky Division of Oil & Gas, and (4) the government failed to advise the Smiths of exculpatory witness testimony. The Smiths also argue that the court erred in not holding an evidentiary hearing to address their motions for a new trial.

With regard to the government’s allegedly “quack” geological theories, the Smiths point to a series of emails between the prosecutor and various investigators. In one email, dated February 9, 2009, the prosecutor sought expert testimony about the Kentucky oil and gas industry. Another email, dated February 18, 2009, contained a list of nine possible board-certified geological experts. One of these potential geological experts, Marvin Combs, emailed one of the investigators and said “I won’t be able to give expert testimony in the upcoming trial against Target.” According to the Smiths, these emails establish that no expert agreed with the government’s “quack” geological theories of the case.

But the emails are insufficient to support the Smiths’ conclusion. Rather than exposing the alleged weakness of the pros-ecutorial theories at issue, the emails simply show that the government considered, but ultimately did not retain, the services of a geological expert. In an attempt to rebut this reading of the evidence, the Smiths’ attorney claims to have spoken with six of the nine potential government experts, declaring that four of the geologists disagree with the government’s theories of the case. But even the Smiths’ attorney admits in his affidavit that “[n]one of the six was even contacted by the government,” and that he did not contact the other three experts. We therefore cannot accept the Smith’s argument that the government knew of and attempted to conceal its allegedly weak geological theories of the case.

Nor did the prosecutor violate Brady with regard to the report from the Board. That report, which was written more than a year after trial, speaks to whether Garton engaged in substandard geological practices. Brady requires that the prosecution disclose all exculpatory and impeachment evidence that is in the government’s possession “in time for use at trial.” Presser, 844 F.2d at 1283. This obligation entails “a duty to learn of any favorable evidence known to others acting on the government’s behalf in the case.” United States v. Graham, 484 F.3d 413, 417 (6th Cir.2007) (emphasis added). But “Brady clearly does not impose an affirmative duty upon the government to take action to discover information which it does not possess.” Id.

A report prepared more than one year after trial was obviously not available for a timely disclosure by the prosecution. And although the Smiths argue that the prosecutor deliberately stalled the report, the Board was not working “on the government’s behalf.” The prosecutor was therefore under no obligation to disclose any “favorable evidence” that was in the Board’s report. See id. In any event, the report found that Garton had “knowledge and complicity” regarding Target Oil’s “alteration, misuse, and fraudulent intent of the [geological] reports.” Garton’s knowledge of Target Oil’s operations was certainly known to the Smith brothers at the time of trial, thus obviating the effect of any Brady violation. See id. (finding no Brady violation “if the defendant knew or should have known the essential facts”).

The report also casts aspersions on the Smiths’ business practices. According to the report, “[i]t was not the original content of the Target reports but the alteration, misuse, and fraudulent intent of the reports that became the legal issue. Ray Garton had a long standing relationship with Target Oil and likely knew about Target’s business practices.” (emphases added) We therefore conclude that the report would have been of little help to the Smiths, so that its absence hardly “undermined the confidence in the outcome” of the trial. See United States v. Mullins, 22 F.3d 1365, 1371 (6th Cir.1994).

The Smiths next argue that emails between Marvin Combs, the Assistant Director of the Kentucky Division of Oil and Gas Conservation, and Chad Harlan, an examiner for the Kentucky Division of Securities, show that regulators knew about and approved Target Oil’s efforts to obtain permits in Kentucky-Indiana’s name. But these emails are not exculpatory in nature; they simply show that Combs knew that the Smiths “intended] to file new applications under Ky-Indiana ... [because] they couldn’t get any new permits under [the name] Target ... due to field violations.” At no point in the emails does Combs make any judgment on the propriety of filing for permits under Kentucky-Indiana’s name. And even if Combs’s emails did contain favorable information, this evidence would simply be cumulative of other evidence that the Smiths submitted to show that they had acted properly. These emails would not likely have changed the outcome. See United States v. Blackwell, 459 F.3d 739, 769 (6th Cir.2006) (rejecting a defendant’s motion for a new trial because the “[d]efendant presented no new evidence likely to produce an acquittal”).

Nor did the prosecution err in failing to disclose its interviews with surveyors Gary Coldiron and John Small-wood. The Smiths knew or should have known that Coldiron and Smallwood, who worked on several Target Oil projects and claimed to have never seen the company engage in fraudulent activities, were available to be called as witnesses in this case. Their allegedly exculpatory testimony could therefore have been “discovered earlier with due diligence.” United States v. Braggs, 23 F.3d 1047, 1050 (6th Cir.1994). And even if this evidence could not have been discovered earlier with due diligence, their testimony was cumulative of the testimony of other witnesses regarding the propriety of Target Oil’s practices.

Finally, we reject the Smiths’ contention that the district court erred by not holding an evidentiary hearing on their motions for a new trial. “The question of whether to decide a motion on the supporting evidence filed with the motion or to hold an evidentiary hearing is within the discretion of the trial court.” United States v. O’Dell, 805 F.2d 637, 643 (6th Cir.1986). In deciding the Smiths’ motions, the court benefited from the parties’ thorough presentation of the issues in their briefs. The court therefore did not abuse its discretion in declining to hold an evidentiary hearing.

E. The district court properly excluded expert testimony

Michael and Christopher Smith next argue that the district court’s decision to exclude the testimony of Jack Wheat— an independent geologist retained to rebut the theories advanced by the government—violated their due process and Confrontation Clause rights. A district court’s decision to exclude expert evidence is reviewed under the abuse-of-discretion standard. Gen. Elec. Co. v. Joiner, 522 U.S. 136, 143, 118 S.Ct. 512, 139 L.Ed.2d 508 (1997).

On appeal, Michael and Christopher Smith expend considerable effort in arguing that the district court erred by excluding Wheat’s testimony, and that Wheat should have been allowed to testify as an expert. Their argument is flawed, however, because the Smiths never sought to introduce Wheat as an expert. During the colloquy preceding Wheat’s testimony, the following exchange occurred:

Def. Counsel: Well, this one I am going to put on next is a—he is a—he is a geologist, that’s right, he is going to take a little bit, take a little time with him. The Court: He is a geologist.
Def. Counsel: Yep.
The Court: What’s he going to testify about? Is he going to testify about what?
Gov’t Counsel: We have not received any expert notification about a geologist. Def. Counsel: It’s not an expert. He is much like Mr.—the guy who made a lot of typos, what’s his name? He is much like him. He is not an expert. He is just an analyst. What was that guy’s name?
Gov’t Counsel: You mean Mr. Cannon who traced the money back.
The Court: Be careful, you are going to put him on as an expert.
Def. Counsel: We are not going to put him on as an expert.
The Court: Oh, no, you are not. You have not made reciprocal discovery. When he starts giving opinions, I am going to be on him like ugly on an ape because he can’t do it.
Def. Counsel: He can give opinions from my understanding about Target’s operations .... I think we can ask questions of a person who is trained in geology about certain aspects of the oil and gas business or geological—
The Court: You cannot ask him anything about these wells because he didn’t do anything with these wells. If you do, you are starting to get into the opinion of an expert.
Def. Counsel: Well, now, he did do some — he did a couple things with these wells.
The Court: What did he do? He can testify to that.
(Emphases added.)

Seeking to admit expert testimony provides the logical foundation for later objecting that a court excluded such evidence. But defense counsel here did not lay that foundation. The colloquy between the district court and defense counsel makes clear that Wheat was not offered as an expert, a fact that defense counsel conceded three separate times. A party “should not be rewarded by pursuing one strategy at trial ... and then allowing him to use his appeal rights to evade that strategy’s ultimate failure.” United States v. Cowart, 90 F.3d 154, 158 (6th Cir.1996). This conclusion is reinforced by the colloquy that occurred immediately after defense counsel admitted that Wheat was not being tendered as an expert witness:

Def. Counsel: Well, we may have to have a couple minutes to talk to this fellow before we put him on to make sure we are going to put him on.
The Court: I mean, you can put on anybody you want to put on. I am not keeping you. [The Assistant United States Attorney] is just telling you she is going to cross examine him if you are saying Target—
Def. Counsel: You just got done saying ugly on ape or something.
The Court: If he starts to give an opinion because what he’s not done — you haven’t given him — you have him give an opinion — or notice of opinion of expert testimony.
Def. Counsel: No, I agree with that. I mean, I am 100 percent behind that. The Court: When he starts talking about the appropriateness of what Target did, then that starts to get to be an opinion.
Def. Counsel: Yeah.
The Court: It started to rank right up there with an opinion, I think.
Def. Counsel: I was just alerting you to the fact we were putting this guy on. / didn’t want to talk to you about the expert part of that. You just brought that up.
(Emphasis added.)

Contrary to the argument made by Michael and Christopher Smith, the district court did not categorically prohibit them from calling Wheat to testify. The court explicitly stated that the defense could call Wheat, but that Wheat could not give expert testimony. Defense counsel, in response, stated “I am 100 percent behind that.” That representation is an unequivocal agreement with the court’s course of conduct, which makes review of this issue inappropriate on appeal. See United States v. Aparco-Centeno, 280 F.3d 1084, 1088 (6th Cir.2002) (“[A]n attorney cannot agree in open court with a judge’s proposed course of conduct and then charge the court with error in following that course.”) (internal quotation marks omitted).

The district court explained why Wheat would not be permitted to provide expert testimony, defense counsel unequivocally agreed with the district court’s explanation, and Wheat’s testimony was ultimately not offered. Based on this sequence of events, Michael and Christopher Smith have waived the issue. See id. (concluding that the defense counsel’s agreement with the judge’s proposed course of conduct “waived his claim on this issue”).

F. The district court’s evidentiary rulings

Michael and Christopher Smith’s final arguments focus on the district court’s decision to admit (1) a Target Oil sales brochure that had been defaced with anonymous negative comments, (2) several clips from the movie The Boiler Room, and (3) seven Cease and Desist Orders. We “typically review a district court’s evidentiary rulings for an abuse of discretion.” United States v. Clay, 667 F.3d 689, 693 (6th Cir.2012) (citing Gen. Elec. Co. v. Joiner, 522 U.S. 136, 141, 118 S.Ct. 512, 139 L.Ed.2d 508 (1997)). An abuse of discretion is deemed to exist when a reviewing court is “firmly convinced that a mistake has been made,” United States v. Whittington, 455 F.3d 736, 738 (6th Cir.2006), or when a district court makes errors of law or clear errors of factual determination. See Clay, 667 F.3d at 694. In all cases, “[a]n erroneous admission of evidence that does not affect the substantial rights of a party is considered harmless, and should be disregarded.” United States v. Cope, 312 F.3d 757, 775 (6th Cir.2002) (internal quotation marks omitted).

1. The admission of the sales brochure was harmless

At trial, the district court admitted a Target Oil sales brochure that had been defaced with various negative comments relating to the truthfulness of statements made therein. Michael and Christopher Smith objected on hearsay grounds. The brochure was a Target Oil promotional pamphlet that was found on a sales desk during the government’s investigation of the company’s call center. Across the front of the brochure was the phrase “Let’s do our due diligence,” and several other statements were inside the brochure questioning the truthfulness of the representations made by Target Oil. The district court admitted the evidence, reasoning that the comments in the brochure, although apparently written by someone other than a Target Oil employee, were “being offered [as] something that Target got, apparently somebody sent them; and it goes to the fraudulent intent, and, therefore, it can come in.”

Michael and Christopher Smith argue that the anonymous comments in the brochure are irrelevant and inadmissible hearsay. We agree that the annotations are clearly hearsay — out-of-court statements being offered for the truthfulness of their commentary on the brochure’s alleged falsehoods. See Fed.R.Evid. 801. The defendants would certainly have had an interest in cross-examining the author of those statements. We thus conclude that the district court erred in admitting the brochure, but further hold that the error was harmless because the evidence of Michael and Christopher Smith’s guilt from other sources is overwhelming. Excluding the brochure, in other words, would not have changed the outcome at trial and was therefore harmless. See Cope, 312 F.3d at 776.

2. Showing clips of The Boiler Room movie was not unfairly prejudicial

During direct examination by the prosecution, Irwin described the sales techniques taught by Michael and Christopher Smith. Irwin testified that Michael Smith initially listened to Irwin’s interactions with potential investors and “would talk over [Irwin] and kind of tell [him] what to say, as [Michael Smith] could hear what [the potential investors] were saying.” Salesmen faced with difficult or insistent questions were instructed to talk over the investors, telling them “about this well or that well.” These tactics were employed to avoid answering questions and to communicate Target Oil’s desired message.

Irwin also testified that Michael Smith made copies of The Boiler Room, a movie in which salesmen working for a fictional investment firm employ high-pressure sales tactics to defraud clients, and he “passed them out to everybody.” Based on this testimony, the district court allowed the government to play several clips from The Boiler Room to the jury. Williams, the young geologist-turned-salesman for Target Oil, stated that he too was shown The Boiler Room as training to be a salesman.

Michael and Christopher Smith argue that admitting these clips into evidence was unfairly prejudicial because there was no evidence that Michael or Christopher Smith presented the movie to encourage high-pressure sales tactics, and that the scenes shown to the jury had no relevance to Target Oil. The clips depicted salesmen lying to potential investors, generating artificial demand for potential investments, and demonstrating calculated indifference to oversight by the Securities and Exchange Commission. Michael and Christopher Smith contend that the very notion that they used The Boiler Room for instructional purposes is absurd, given that the salesmen in the movie were depicted as having been ultimately incarcerated.

A district court “is granted very broad discretion in determining whether the danger of undue prejudice outweighs the probative value of the evidence.” United States v. Vance, 871 F.2d 572, 576 (6th Cir.1989) (internal quotation marks omitted). Broad discretion, however, does not mean that evidentiary decisions will be rubber-stamped. In United States v. Gamory, 635 F.3d 480, 492-93 (11th Cir.2011), the Eleventh Circuit determined that the district court had abused its discretion by admitting a rap video and showing it to the jury. At trial on the defendant’s drug-related charges, the district court had permitted the government to play a rap video produced by the defendant’s recording studio in which a rapper referred to the defendant’s supposed alias, mentioned drug use, and displayed large quantities of money. Id. at 493. There were superficial similarities between the musicians in the rap video and the defendant, but “the substance of the rap video was heavily prejudicial” because the lyrics “contained violence, profanity, sex, promiscuity, and misogyny and could reasonably be understood as promoting a violent and unlawful lifestyle.” Id. Even so, the Eleventh Circuit ultimately concluded that admitting the video was harmless error.

We find no error in admitting the film clips in question. Unlike in Gamory, in which the connection between the defendant’s conduct and the conduct portrayed in the video clip was tenuous, here there are direct connections between The Boiler Room and the fraud committed by Michael and Christopher Smith. Irwin and Williams explicitly testified that the movie was provided to employees as a training device. In Williams’s case, he was shown the movie to motivate him to “want to be a salesman.” Evidence establishing that Target Oil’s employees were provided with copies of The Boiler Room for training and motivational purposes indicates that Michael and Christopher Smith at least somewhat “adopted or shared” the views expressed in the movie. See Gamory, 635 F.3d at 493.

True enough, there are differences between the fictional investment firm in The Boiler Room and Target Oil. The film clips portrayed successful salesmen being rewarded with prostitutes, depicted supervisors calling federal enforcement officers “chimps,” and showed salesmen being explicitly instructed to lie to investors. There is no evidence in the record to suggest that Target Oil did any of these things, although Michael Smith did admit to Williams that Target Oil “operate[d] in the grey area.” In any event, the testimony at trial established that Michael and Christopher Smith made copies of The Boiler Room, and encouraged salesmen to learn from the movie. This makes the core conduct shown in the movie, although not identical to what happened at Target Oil, at least probative of knowledge and intent to defraud. For that reason, we find no error in the admission of the clips. And even if the district court had erred, the error was harmless given the overwhelming amount of other evidence demonstrating guilt.

3. Admitting the Cease and Desist Orders was not erroneous

The final evidentiary objection raised by Michael and Christopher Smith is that the district court erred by admitting the seven Cease and Desist Orders. They claim that the orders were unfairly prejudicial because jurors could have been “confused into thinking the orders had been [issued] seeking to prevent the [investment] transactions with Target’s investors.” The orders in question prohibited Target Oil from soliciting investors in certain states other than Kentucky without first registering with the appropriate state authorities.

Particularly troubling to Michael and Christopher Smith were the implications that the government drew from the Cease and Desist Orders. During closing arguments, the prosecutor contended that the Cease and Desist Orders were evidence of Target Oil’s knowledge that it was not allowed to conduct business in certain states, and that Target Oil’s failure to inform investors about the orders was a “material concealment of information, information that was material to these investor’s [sic] decisions when they were trying to decide to invest in Target Oil.” Michael and Christopher Smith responded by arguing that Target Oil did not materially conceal the Cease and Desist Orders because they were available on the internet, and that such orders are common in the industry.

The Smiths’ arguments are without merit. In United States v. Benson, 79 Fed.Appx. 813 (6th Cir.2003), this court evaluated the prejudicial effect of cease- and-desist orders and concluded that they were admissible as evidence of a defendant’s knowledge and intent to defraud. Id. at 820 (citing United States v. Gold Unlimited, Inc., 177 F.3d 472, 486-87 (6th Cir.1999)). Here, the district court admitted the Cease and Desist Orders as tending to show Michael and Christopher Smith’s knowledge and intent to defraud, and it further provided a limiting instruction before allowing the jury to see the evidence. The court’s limiting instruction informed the jury that it may “consider the evidence only as it relates to the defendant’s intent, motive and knowledge. You must not consider it for any other purpose.” Because the court admitted the Cease and Desist Orders for the limited purpose of demonstrating knowledge and intent to defraud, there was no error.

G. The district court correctly dismissed Count 17 against Christopher Smith

We now turn to the government’s cross-appeal. The jury acquitted Christopher Smith of conspiring to commit mail fraud, but convicted him on the substantive charge of committing mail fraud, as set forth in Count 17, with regard to an oil well known as Bell County # 1. Christopher Smith subsequently moved for a judgment of acquittal on Count 17, reasoning that the evidence was insufficient to find him guilty of mail fraud on that count. The district court agreed. This has caused the government to cross-appeal, arguing that the “district court erred by considering the jury’s inconsistent verdicts and by failing to consider circumstantial evidence of Christopher Smith’s personal guilt and evidence that he could have reasonably contemplated the underlying mailing.”

To the extent the government argues that the district court granted Christopher Smith’s motion for a judgment of acquittal based on inconsistent verdicts, the argument is without merit. The district court explicitly stated that it based its decision on the insufficiency of the evidence, not on inconsistent verdicts. Nor could it have properly relied on inconsistent verdicts in making its decision. Inconsistent jury verdicts are unreviewable except for “a situation in which a defendant receives two guilty verdicts that are logically inconsistent,” which is not at issue here. See United States v. Ruiz, 386 Fed.Appx. 530, 533 (6th Cir.2010).

“On appeal from a judgment of acquittal under Rule 29, we must determine whether, viewing the evidence in the light most favorable to the government, any rational trier of fact could have found the elements of the crime beyond a reasonable doubt.” United States v. Baggett, 251 F.3d 1087, 1095 (6th Cir.2001) (internal quotation marks omitted). “Due to the close relationship between the substantive and conspiracy crimes, which was created by the Pinkerton instruction, an automatic consideration of the viability of the substantive convictions should [be] undertaken by the district court when the defendant was acquitted of the conspiracy conviction.” United States v. Henning, 286 F.3d 914, 920 (6th Cir.2002).

“Withdrawal terminates the defendant’s liability for postwithdrawal acts of his coconspirators, but he [can] remain[ ] guilty of conspiracy.” Smith v. United States, — U.S. -, 133 S.Ct. 714, 719, 184 L.Ed.2d 570 (2013). Christopher Smith left Target Oil approximately eight months before the document at issue in Count 17 — a permit connected to Bell County # 1 — was mailed. He had clearly withdrawn from the conspiracy before Target Oil received the permit that formed the basis for Count 17.

Christopher Smith was acquitted of the conspiracy charged in the indictment. And no evidence linked him personally to the interactions with the landowner who dealt with Target Oil in connection with Bell County # 1. This leaves no basis for a jury to find Christopher Smith guilty of mail fraud in connection with that well. The district court therefore did not err in granting Christopher Smith’s motion for a judgment of acquittal on Count 17.

III. CONCLUSION

For all the reasons set forth above, we AFFIRM the judgment of the district court.  