JENNIFER WALKER ELROD, Circuit Judge

Jason Scott Varnado, Assistant U.S. Attorney (argued), Lauretta Drake Bahry, Assistant U.S. Attorney Renata Ann Gowie, Assistant U.S. Attorney U.S. Attorney's Office, Houston, TX, for Plaintiff–Appellee.

Seth Kretzer (argued), Law Offices of Seth Kretzer, Jack Benjamin Zimmermann, Esq. (argued), Jim E. Lavine, Esq., Terri Raye Zimmermann, Esq., Zimmermann, Lavine, Zimmermann & Sampson, P.C., Megan Elizabeth Smith, Houston, TX, for Defendants–Appellants.

Appeals from the United States District Court for the Southern District of Texas.

Before PRADO, ELROD, and HAYNES, Circuit Judges.

Opinion

JENNIFER WALKER ELROD, Circuit Judge:

Appellants Mark Kuhrt and Gilbert Lopez challenge their convictions and sentences on multiple counts of wire fraud and conspiracy. Appellants argue insufficiency of the evidence and errors at trial involving the district court's jury instructions and exclusion of certain expert testimony. Appellants also assert numerous procedural and substantive errors regarding their sentences. We AFFIRM.

I.

Kuhrt and Lopez were both employees of Allen Stanford's investment companies for over a decade. During that time, Stanford ran a multi-billion dollar Ponzi scheme and stole billions of dollars from his investors. See United States v. Stanford, 4:09–cr–00342–1, [Dkt. No. 808] (S.D.Tex. Mar. 6, 2012). At trial, the government alleged that Appellants actively helped Stanford hide his fraud for over a decade. The government's case consisted of documents and e-mails spanning almost a decade and witness testimony from other Stanford employees. The government's key witness was James Davis, who orchestrated much of the Ponzi scheme before he reached a plea deal with the government. After a five-week trial, Appellants were each convicted of nine counts of wire fraud and conspiracy to commit wire fraud.

A brief overview of Stanford's scheme and the institutional structure of his companies is provided here. The facts and events prior to Appellants' employment, and unrelated to Appellants' respective roles in the Ponzi scheme, are undisputed.

Stanford established Guardian International Bank (GIB), on the island of Montserrat, a British Overseas Territory in the Caribbean, in 1985. GIB sold certificates of deposit (CDs) to customers outside the United States, representing to customers that their money would be invested in liquid financial products and that the bank would not use their money to make loans. Stanford also owned a real estate company, Guardian Development Corporation (GDC), and a service company in Houston, Texas, Stanford Financial Group (SFG), which provided accounting, legal, and other services to both GDC and GIB.

Stanford soon hired James Davis, who became SFG's controller and eventually played a major role in Stanford's illegal activities. Stanford also enlisted the services of a Certified Public Accountant (CPA), Harry Failing, and an Antigua-based auditor, C.A.S. Hewlett. At the beginning of Davis's employment, Davis tracked GIB's liabilities and expenses, but he was not privy to investment information. In the early 1990s, GIB moved to Antigua and Davis became the Chief Financial Officer of SFG and GIB. Jean Gilstrap was hired as the new controller of SFG. Davis soon gained access to GIB's financial statements and learned that the reported investment returns were grossly inflated and that the missing money was being funneled to Stanford's personally owned companies. After his discovery, Davis began working to cover up the fraud.

Beginning in 1992, Stanford, Davis, and Gilstrap worked together to make false revenue entries in GIB's books. Around this time, GIB became Stanford International Bank (SIB). As SIB, the company continued to market and sell CDs with the promise that these were safe investments. Marketing materials claimed the CDs were “strong, safe and fiscally sound,” using a “conservative” investment strategy, and investing to “minimize risk and achieve liquidity.”

Several years later, Stanford founded Stanford Group Company (SGC) in Houston, Texas and Baton Rouge, Louisiana, which hired “financial advisors” to sell SIB CDs to customers in the United States. Through SGC, Stanford collected money from American investors.

In 1997, Lopez, a Certified Public Accountant, was hired as an SFG accounting manager, reporting to Gilstrap, and Kuhrt was hired as a fixed assets manager, reporting to Lopez. Kuhrt was not a Certified Public Accountant, but had a Masters of Business Administration and accounting experience. The following year, Lopez was promoted to controller. Davis testified that he was comfortable promoting Lopez into that position because “[Lopez] had demonstrated his loyalty to Ms. Gilstrap, [and] had been aware of what was happening in the financial record keeping.”

In 2000, Failing, Stanford's accountant, reported that while the IRS was looking at Stanford's personal finances, the IRS had discovered that Stanford received money from SIB. Failing indicated that the IRS could view this money as taxable income, which Stanford wanted to avoid. Failing proposed creating fake promissory notes to make it appear that Stanford had received a loan from SIB and would pay the money back. Failing explained this plan in an e-mail that Lopez received. Lopez was also mentioned by name in this e-mail (“Gil seemed to like this as well.”).

Over the next several years, there were many material inaccuracies in SIB's annual reports. The reports inflated SIB's investment numbers and did not disclose that significant sums of money were being diverted to Stanford's other, personally held companies. The information in these reports was communicated to prospective customers through the financial advisors, who were aggressively selling CDs. At trial, Davis testified that “[Lopez and Kuhrt] were responsible for the content and accuracy of the annual report.” Davis also testified that “Mr. Lopez was responsible for the content and accuracy of the [2002 annual] report,” and that “[Mr. Kuhrt] was the senior most accountant reporting to Mr. Lopez and had responsibility for [the 2002 annual report's] content and accuracy.” Lopez and Kuhrt testified that they had minimal authority over the reports and were not responsible for the reports' content and accuracy. This was one of the key factual disputes at trial.

At trial, the government introduced evidence showing that Appellants were aware of the money that Stanford had taken from the company. Specifically, in 2003, Lopez sent a spreadsheet, prepared by Kuhrt, which tracked all of the outstanding “loans” to Stanford. In addition, Davis testified that in 2004 he met with Lopez and Kuhrt and they had a lengthy discussion regarding whether the amounts diverted to Stanford should be disclosed in the annual report. Davis testified that all three men believed the amounts should be disclosed, but that they nevertheless agreed not to disclose any of the money. Appellants dispute that this meeting occurred, but assert that any discussion of “footnoting” the money diverted to Stanford ended in their objection to non-disclosure, and Davis's overruling of them.

To keep the appearance that the diverted money was part of a loan, Failing advised that Stanford should “service” the “loan.” Therefore, according to Davis's testimony, Appellants designed a “private equity flip,” which was a sham transaction that would make it appear that Stanford had paid back a significant amount of money to SIB. To execute this “flip,” Stanford used $132 million of SIB depositors' money to purchase private equity assets from Stanford Venture Capital Holdings (SVCH), another of his companies. Stanford then “sold” these assets to SIB for $385 million, a sham value. On paper, it appeared that Stanford had injected money into SIB to service his loan interest and pay down some of the principal. Davis testified that Appellants facilitated this transaction, despite knowing that it was a sham. According to Kuhrt, “Davis and Stanford concocted [this] transaction” and “[s]everal unwitting SFGC employees, including Lopez and Kuhrt, were engaged to manage the portfolio transfer.” This was another key factual dispute at trial. By the end of 2005, Stanford had taken over $850 million out of the bank. According to Davis, Appellants were aware of these amounts.

The government also presented testimony from Rolando Roca, a budget analyst at SFG in Houston, who worked under Kuhrt. Roca testified that after Lopez and Kuhrt were promoted in 2006, Lopez had the responsibilities of a chief accounting officer and Kuhrt had the responsibilities of a controller. Roca testified that this included authority over financial decisions and disclosures. According to Roca's testimony, he found Kuhrt's and Lopez's demeanors to be odd on numerous occasions, and they reprimanded him for asking too many questions. Roca also testified that Kuhrt specifically told him that the Antigua-based auditor, Hewlett, was “the closest thing we could get [to a rubber stamp].”

The government also presented the testimony of Fran Casey, an internal auditor. Casey testified that he completed a draft audit report in May 2006, which stated that the audit was only able to verify 8% of SIB's assets and only 2% of the income statement accounts. Casey testified that when Lopez saw the report, Lopez became extremely angry and stated that the report could not be issued in its present form: stating that 92% of the assets were unverified. Casey testified that he felt “intimidated” by Lopez's demeanor. The report was eventually issued without the statements regarding the 8% and the 2% verification rates.

The following year, SIB was required to prepare and file reports with Antigua's Financial Services Regulatory Commission (FSRC). These reports included schedule “IB5s” detailing the bank's investments. Davis testified that he and Kuhrt were responsible for the content and accuracy of the schedule IB5s. According to Davis, he would send the previous quarter's IB5 to Kuhrt, who would update it and submit it. Davis also testified that Kuhrt once caught and corrected an error on an IB5, indicating that Kuhrt actually read and reviewed the document. The IB5s that SIB filed did not contain any reference to the money being diverted to Stanford.

At this point, Stanford was diverting almost $1 million per day from his companies for his personal use. Davis testified that Appellants were concerned about the amount of money being diverted. According to Davis's testimony, Kuhrt also raised a concern that the equity-to-asset ratio at SIB was too low to satisfy Antiguan regulators. Subsequently, Appellants participated in a second “private equity flip.” According to Davis's testimony, Lopez worked with the legal team—which was unaware of the purpose of the flip—to execute the private equity flip, and Davis and Kuhrt exchanged e-mails regarding the purpose of the transaction and its similarities to the prior private equity flip. The flip had two purposes: (1) it raised the Bank's equity-to-asset ratio in order to satisfy the Antiguan regulators; and (2) it showed a payment by Stanford against what he owed the bank.

During this time, the economy was in a recession and Davis testified that he, Lopez, and Kuhrt were all concerned about investor confidence. Davis also testified that both Lopez and Kuhrt knew there were financial problems at all of Stanford's companies, particularly SIB.

According to Roca's testimony, in October 2008, Kuhrt directed Roca to make an accounting entry showing $200 million of revenue. Roca testified that he asked Kuhrt for supporting documentation, but Kuhrt replied that Roca should “[j]ust forward” the entry on, indicating that Roca should execute the entry without the documentation. Davis testified that this $200 million contribution was never actually funded, meaning there was never any revenue to support the entry. Roca also testified that by this point he had begun to question several of the revenue entries, because he believed that the entries mirrored comments made by Davis and Kuhrt about what the entries should be, rather than actual returns.

To quell investor fears, Stanford announced a $1 billion capital infusion. Davis testified that when Lopez inquired as to where this money would come from, Davis responded that “the emperor has no clothes,” indicating that Stanford did not actually have the money to inject into the bank. According to Davis, he told Lopez that they would make an accounting entry showing the investments despite the fact that they were illusory and Lopez knew that there was no money to make the cash infusion.

According to Davis, in order to fund this capital infusion—which Stanford had already publicly announced as funded—he, Kuhrt, and Lopez revalued a piece of Antiguan land that Stanford had purchased for $63.8 million, at $3.2 billion—more than fifty times the purchase price. The $3.2 billion was to be used: (1) to fund the $200 million entry that was represented as already funded; (2) to fund an additional $541 million contribution; (3) to offset $1.7 billion in “loans” to Stanford; and (4) to set aside $733 million in a personal account for Stanford. In reality, Stanford had actually paid back “a miniscule amount” of money. Davis testified that both Kuhrt and Lopez actively worked to prepare the structure of this hypothetical real estate transaction.

According to Roca, when he realized that Appellants planned to backdate the transaction, he questioned Kuhrt about the financial condition of SIB. Roca testified that in response, in the days leading up to SIB's total collapse, Kuhrt made several statements indicating that Kuhrt knew SIB's true financial condition. Kuhrt's statements included: “Rolando, you're a little pig-headed. You don't understand. It's over.”; “They [the SEC] can't prove anything.”; and “we don't want to make a liar out of Mr. Stanford.”

In early 2009, the government raided Stanford's offices in Houston, Memphis, and Miami. Three days later, Roca met with the FBI and placed a monitored call to Kuhrt in which Kuhrt denied knowledge of any wrongdoing. Shortly after, Kuhrt, Lopez, Stanford, Davis, and others were charged in connection with the scheme. Stanford was convicted and sentenced to 110 years of imprisonment. Appellants were charged with wire fraud and conspiracy to commit wire fraud, stemming from the cover-up. Appellants were not charged for participation in the actual Ponzi scheme or with bribery. Appellants' trial lasted five weeks, during which the government presented evidence in the form of e-mails, spreadsheets, and testimony by Davis, Roca, Casey, victims of the scheme, and others.

Kuhrt and Lopez each were convicted on nine counts of wire fraud and one count of conspiracy to commit wire fraud. They each were acquitted of one count of wire fraud. The district court calculated each Appellant's offense level as level 43 and each Appellant's criminal history as category I (no criminal history). The Sentencing Guidelines provided for a term of life imprisonment. The district court departed downward and sentenced each defendant to 240 months of imprisonment. This appeal followed.

II.

Appellants raise numerous challenges to their convictions and to the procedural and substantive reasonableness of their sentences. They argue that: (1) the district court improperly permitted the government to make a race-based peremptory strike against a Hispanic juror; (2) the evidence is insufficient to support their convictions; (3) the district court improperly gave a deliberate ignorance instruction to the jury; (4) the district court improperly excluded Appellants' experts' testimony; (5) the district court made multiple errors in calculating Appellants' sentences; and (6) the sentences are substantively unreasonable. Each of these challenges is considered below.

A.

Kuhrt argues that the district court erred when it concluded that the government's use of a peremptory strike against a Hispanic juror was not race-based. See Batson v. Kentucky, 476 U.S. 79, 106 S.Ct. 1712, 90 L.Ed.2d 69 (1986). Under Batson, the district court conducts a three-step analysis to determine if a peremptory strike is impermissibly race-based: (1) the defendant must make a prima facie showing that the peremptory challenge was racially motivated; (2) the government may then give a race-neutral reason for exercising the challenge; and (3) the burden shifts back to the defendant to show purposeful discrimination. United States v. Pratt, 728 F.3d 463, 474 (5th Cir.2013), cert. denied, ––– U.S. ––––, 134 S.Ct. 1328, 188 L.Ed.2d 338 (2014).

We review the government's race-neutral explanation de novo. United States v. Williams, 264 F.3d 561, 571 (5th Cir.2001). Whether there was purposeful discrimination is a question of fact, which we review for clear error. See Hernandez v. New York, 500 U.S. 352, 369, 111 S.Ct. 1859, 114 L.Ed.2d 395 (1991) (plurality opinion). We are mindful that this finding “ ‘largely turn[s] on an evaluation of the credibility or demeanor of the attorney who exercise[d] the [peremptory] challenge.’ ” United States v. Davis, 393 F.3d 540, 544 (5th Cir.2004) (third alteration in original) (quoting United States v. Bentley–Smith, 2 F.3d 1368, 1373 (5th Cir.1993) ).

During voir dire, Lopez objected to the government's exercise of a peremptory challenge against juror number 38, a Hispanic female. Defense counsel stated “[Juror No. 38] is Hispanic and the defendant is Hispanic.” This is not a prima facie showing that the peremptory strike was racially motivated. Even if this were a sufficient prima facie showing of discrimination, the government here offered several race-neutral explanations for the strike. The government explained that the juror was gregarious with the district court during voir dire, that she was employed as a social worker, that she volunteered with youth services, and that she had a perceived liberal bias. The district court found the government's race-neutral explanation credible and specifically noted that the jury was racially diverse.

“[U]ltimately the inquiry boils down to whether the Government should be believed.” Williams, 264 F.3d at 572. “This is quintessentially a question of fact which turns heavily on demeanor and other issues not discernible from a cold record, such that deference to the trial court is highly warranted.” Id. On this record, we conclude that the district court did not err in overruling the Batson challenge.

B.

Appellants both challenge the sufficiency of the evidence underlying their convictions for wire fraud and conspiracy to commit wire fraud. On appeal, we view the facts in the light most favorable to the verdict. United States v. Moreno–Gonzalez, 662 F.3d 369, 372 (5th Cir.2011). “Our review of the sufficiency of the evidence is highly deferential to the verdict.”Id. (internal quotation marks omitted). “[T]he 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.” Jackson v. Virginia, 443 U.S. 307, 319, 99 S.Ct. 2781, 61 L.Ed.2d 560 (1979).

Our review is “limited to whether the jury's verdict was reasonable, not whether we believe it to be correct.” Moreno–Gonzalez, 662 F.3d at 372 (internal quotation marks omitted). We do not review the weight or credibility of the evidence, including witness testimony. United States v. Hayes, 342 F.3d 385, 389 (5th Cir.2003). We accept all credibility choices and reasonable inferences made by the jury that tend to support the verdict. Moreno–Gonzalez, 662 F.3d at 372. “[A]ny conflict in the evidence must be resolved in favor of the jury's verdict.” Id.

To prove wire fraud, the government must prove: (1) a scheme to defraud; (2) the use of, or causing the use of, wire communications in furtherance of the scheme; and (3) a specific intent to defraud. See 18 U.S.C. § 1343 ; United States v. Simpson, 741 F.3d 539, 547–48 (5th Cir.2014). To prove conspiracy to commit wire fraud, the government must prove that: (1) two or more persons made an agreement to commit wire fraud; (2) the defendant knew the unlawful purpose of the agreement; and (3) the defendant joined in the agreement willfully, i.e., with specific intent. See 18 U.S.C. § 1349 ; United States v. Grant, 683 F.3d 639, 643 (5th Cir.2012).

Appellants argue that there is insufficient evidence to support the specific intent element of the wire fraud convictions. Specifically, Kuhrt argues that the government's entire case is based upon a single footnote to the 2007 annual report (footnote 2.9) which stated that SIB “does not expose its customers to the risks associated with commercial loans. [Its] only form of lending is done on a cash-secured basis solely to existing customers.” According to Kuhrt, the footnote only promises that commercial loans are “cash-secured” and does not specifically state that SIB does not make personal loans to non-cash-secured shareholders. Kuhrt further argues that the entire basis of his conviction is the fact that SIB made shareholder loans that were not on a “cash-secured basis,” in violation of the representations in the footnote. In Kuhrt's view, his reading of the footnote—that the ‘cash-secured basis' statement applied only to commercial loans—was reasonable, and therefore that an alleged violation of the footnote's promises cannot support a conviction for fraud.

In support, he cites United States v. Grossman, 117 F.3d 255 (5th Cir.1997). In Grossman, the defendant had borrowed money pursuant to a loan agreement that contained the following clause: “Borrower represents and warrants lender that the loan will be used by borrower for its business and commercial purposes and not for personal, family, household or agricultural use.” Id. at 259. The borrower used the loan money for a business purpose that was unrelated to the specific business entity that borrowed the money and was convicted of submitting false and fraudulent documents. Id. We reversed, holding that because the defendant's reading of the clause—that it permitted use of the funds by an unrelated business—was a reasonable interpretation of the clause, there was insufficient evidence to support a finding of fraudulent intent. Id. at 260.

Here, unlike in Grossman, Kuhrt does not offer a reasonable interpretation of footnote 2.9. By its terms, the footnote's second sentence represents that the only form of lending is on a cash-secured basis. The first sentence does not restrict this to the commercial-loan context. Rather, the second sentence containing the “cash-secured” representation is a fully independent representation that should be read separately from the first sentence regarding commercial loans. Each sentence is a different and independent representation about the types of loans SIB makes.

More importantly, even assuming arguendo that Kuhrt is correct that his interpretation of the footnote is reasonable, his argument fails because the footnote was just one piece of the extensive evidence offered to prove fraudulent intent. Much of the evidence of fraudulent intent comes from Davis's trial testimony. Davis testified that Kuhrt and Lopez were responsible for the annual reports, which were materially false for many years. Davis also testified that Kuhrt and Lopez were directly involved in planning and executing fraudulent transactions, and that they oversaw fraudulent accounting specifically designed to cover up Stanford's theft. In addition, Roca testified that Kuhrt knew the Antiguan auditor was not doing his job. The government also introduced e-mails and other documents showing that Kuhrt knew about Stanford's theft and that Kuhrt took specific actions to conceal it, including e-mails regarding the IB5s, documents tracking the amount of money being diverted to Stanford, and e-mails regarding the need to satisfy Antiguan regulators. Viewing this evidence in the light most favorable to the verdict, a reasonable jury could have found specific intent to commit wire fraud.

As the scheme was collapsing, Kuhrt sent several message to Davis, asking what was happening. Kuhrt contends that these text messages indicate that Kuhrt did not know about Stanford's companies' financial problems or the fraud. Kuhrt argues that “[t]he text messages Kuhrt sent to Davis (and to which Davis did not respond for days) serve to pull the evidence into equipoise such that no rational juror could find the reasonable doubt standard satisfied.” Kuhrt argues that “[w]hen the evidence is in equipoise, as a matter of law it cannot serve as the basis of a finding of knowledge.” United States v. Reveles, 190 F.3d 678, 686 (5th Cir.1999), abrogated by United States v. Vargas–Ocampo, 747 F.3d 299, 301, n. 2 (5th Cir.2014) (en banc). Simply put, the evidence is not in equipoise. There is ample evidence of Kuhrt's guilt. Even assuming it were in equipoise, Kuhrt would not be entitled to relief because this circuit has excised the use of the equipoise doctrine. See Vargas–Ocampo, 747 F.3d at 301, n. 2 (holding that this circuit does not apply the doctrine of equipoise).

Kuhrt next argues that the conspiracy charge must fail because the only evidence supporting the charge is Davis's testimony that he met with Kuhrt and Lopez in 2004 and they all agreed to not disclose the loans made to Stanford. Even if it were true that Davis's testimony was the sole evidence of conspiracy, it would be sufficient to uphold the jury verdict. United States v. Mendoza, 522 F.3d 482, 489 (5th Cir.2008) (“Evidence consisting entirely of testimony from accomplices or conspirators is sufficient.”) (citing United States v. Turner, 319 F.3d 716, 721 (5th Cir.2003) ). Moreover, there is sufficient additional evidence of conspiracy, including Roca's testimony, Casey's testimony, e-mail exchanges, and the exchange of spreadsheets and other documents needed to execute the private equity flips.

Kuhrt also argues that his conviction cannot stand on the circumstantial evidence alone. In support, he cites United States v. Maltos, 985 F.2d 743, 747–48 (5th Cir.1992), in which we held that there was insufficient evidence of conspiracy where the government only presented evidence of the defendant's association with the conspirators, but no evidence establishing his knowledge of, or participation in, the conspiracy. Maltos is inapposite. Unlike in Maltos, Davis specifically testified that Kuhrt and Lopez agreed that the loan amounts to Stanford would not be disclosed. The government presented sufficient evidence of Kuhrt's knowledge and participation in the conspiracy.

Turning to Lopez's insufficiency-of-the-evidence arguments, Lopez also attacks the “specific intent” element. He argues that there is insufficient evidence to support the wire fraud convictions because “there is no evidence that Lopez had any idea there was fraud or that Lopez intended to defraud anyone.” This argument simply ignores Davis's testimony that Lopez knew about the shareholder loans and agreed not to disclose them, the testimony and e-mails indicating Lopez's involvement in the sham transactions, and Casey's testimony about Lopez's reaction to the 2006 draft audit report.

Moreover, Lopez's reliance upon United States v. Beckner, 134 F.3d 714 (5th Cir.1998), is misplaced. In Beckner, the defendant, Beckner, was an attorney convicted of aiding and abetting fraud based upon his client's fraudulent conduct. We reversed, holding that there was insufficient evidence that the defendant was aware of his client's fraud. Id. at 715. In Beckner, the defendant learned that his client might be committing fraud because his client did not have sufficient equity in a piece of property to cover the security obligations against it. Id. at 720. Beckner, who was not his client's primary attorney, secured an appraisal that valued the property at $2.5 million. The client's primary attorney informed Beckner that there was only $300,000 of outstanding debt on the property. Id. at 719. In fact, the debts on the property exceeded the value. On these facts, we held that “[t]he government produced no evidence demonstrating that Beckner should have disbelieved this data; accordingly, a jury could not reasonably infer Beckner's criminal knowledge from this evidence alone.” Id. (emphasis added).

Here, the government produced evidence that Appellants were not only aware of the fraud, but actually helped perpetrate the fraud. The government's evidence included testimony from Davis, Roca, and Casey, and e-mails showing that Appellants helped create and execute the private equity flips. Viewing the evidence in the light most favorable to the verdict, there was sufficient evidence of fraudulent intent for a reasonable jury to convict.

Lopez next argues that there is insufficient evidence of conspiracy because Davis never testified that Lopez explicitly agreed to violate the law and “the fact that Lopez handled documents that contained false information becomes irrelevant when considering that Lopez did not know, and had no way of knowing, of the documents' falsity.” However, “[a]n agreement may be inferred from concert of action, voluntary participation may be inferred from a collection of circumstances, and knowledge may be inferred from surrounding circumstances.” United States v. Stephens, 571 F.3d 401, 404 (5th Cir.2009) (internal quotation marks omitted). A reasonable jury could have found that Appellants agreed to conceal Stanford's theft and deceive investors based upon several pieces of direct and circumstantial evidence. The fact that Appellants never expressly agreed to do so is irrelevant because the jury could have viewed the joint decisions to not disclose the shareholder loans and to execute the sham transactions as an agreement to engage in fraud, regardless of whether Appellants explicitly discussed the fact that their conduct was unlawful. See Grant, 683 F.3d at 643 (“The agreement between conspirators may be silent and need not be formal or spoken.”). As discussed above, there is sufficient evidence to support the convictions.

C.

Appellants argue that the district court erred when it included a deliberate ignorance instruction and that the included instruction was incorrect as a matter of law. We review “preserved error in jury instructions under an abuse of discretion standard.” United States v. Brooks, 681 F.3d 678, 697 (5th Cir.2012) (internal quotation marks omitted). “In assessing whether evidence sufficiently supports the instruction, we view the evidence and all reasonable inferences that may be drawn from the evidence in the light most favorable to the [g]overnment.” Id. at 701 (alteration in original) (internal quotation marks omitted).

“We have often cautioned against the use of the deliberate ignorance instruction.” United States v. Mendoza–Medina, 346 F.3d 121, 132 (5th Cir.2003) (holding deliberate ignorance instruction not warranted in an actual knowledge case). We have also stated that “a deliberate ignorance instruction ‘should rarely be given.’ ” United States v. Faulkner, 17 F.3d 745, 766 (5th Cir.1994) (quoting United States v. Ojebode, 957 F.2d 1218, 1229 (5th Cir.1992) ); see also United States v. Cartwright, 6 F.3d 294, 301 (5th Cir.1993) (“Because the deliberate ignorance instruction may confuse the jury, the instruction should rarely be given.” (internal quotation marks omitted)). The proper role of the deliberate ignorance instruction is not as a backup or supplement in a case that hinges on a defendant's actual knowledge. The instruction is appropriate only in the circumstances where a defendant “claims a lack of guilty knowledge and the proof at trial supports an inference of deliberate indifference.” Brooks, 681 F.3d at 701 (internal quotation marks omitted). “The evidence at trial must raise two inferences: (1) the defendant was subjectively aware of a high probability of the existence of the illegal conduct; and (2) the defendant purposely contrived to avoid learning of the illegal conduct.” Id. (internal quotation marks omitted). “ ‘[T]he district court should not instruct the jury on deliberate ignorance when the evidence raises only the inferences that the defendant[s] had actual knowledge or no knowledge at all of the facts in question.’ ” Mendoza–Medina, 346 F.3d at 133–34 (quoting United States v. Lara–Velasquez, 919 F.2d 946, 951 (5th Cir.1990) ). Undoubtedly, the deliberate ignorance instruction is “inappropriate for an offense which requires a specific purpose by the defendant.” United States v. Chen, 913 F.2d 183, 190 (5th Cir.1990).

The government constructed its case on the premise that Appellants were criminally liable based upon their actual knowledge of the fraud and their efforts to further the fraud. Thus, it arguably was error for the district court to give the deliberate ignorance instruction.

Even assuming arguendo that this was error, we have held, nevertheless, that the giving of the instruction is harmless where there is substantial evidence of actual knowledge. See, e.g., United States v. St. Junius, 739 F.3d 193, 204–05 (5th Cir.2013) (“Even if the district court errs in its decision to give the deliberate ignorance instruction, any such error is harmless where substantial evidence of actual knowledge is presented at trial.” (internal quotation marks omitted)). That is the situation here. In Mendoza–Medina, we addressed a similar scenario where we held that there was insufficient evidence of deliberate ignorance, and therefore the district court's giving the deliberate ignorance instruction was in error. 346 F.3d at 134. However, because there was ample evidence of actual knowledge of the illegal conduct, we held that giving the instruction was harmless error. Id. at 135. Similarly, here, there was testimony that Appellants were actual participants in the illegal activity. Therefore, the error was harmless.

D.

Appellants argue that the district court erred when it excluded their accounting experts' testimony and thereby kept them from putting on their theory of the case. A district court's decision to exclude expert testimony is reviewed for an abuse of discretion and will not be disturbed unless it is “manifestly erroneous.” United States v. Wen Chyu Liu, 716 F.3d 159, 167 (5th Cir.2013) (internal quotation marks omitted). “Even after finding an abuse of discretion, [we] will not reverse harmless error.” Id.

Kuhrt and Lopez proffered accounting experts Richard Jones and Dr. Bala Dharan, respectively, to testify at trial. The government moved in limine to exclude four areas of expert testimony: (1) testimony regarding the roles and responsibilities of Appellants or the roles and responsibilities of people in comparable positions within the industry; (2) testimony regarding what Appellants knew, believed, or relied upon, or what someone in Appellants' positions would have known, believed, or relied upon; (3) testimony on whether such knowledge, belief, or reliance was reasonable; and (4) testimony that provided an opinion as to what certain documents show or do not show.

The district court's ruling on the motion in limine excluded testimony on Appellants' specific roles, responsibilities, knowledge, beliefs, or reasonableness of Appellants' subjective beliefs. The district court permitted testimony regarding common practice in the industry and reserved ruling on testimony regarding a hypothetical person in Appellants' positions to be evaluated on a question-by-question basis. However preliminary, Appellants understood this to be the scope of allowable testimony.

Prior to Jones's testimony at trial, the government objected to Jones's testimony on several additional topics. The district court held a hearing outside of the presence of the jury on three of these topics: (1) whether the shareholder loans needed to be disclosed to investors and the extent of Kuhrt's duty—under accounting ethics—to disclose them; (2) the timing of the $200 million and $541 million capital contributions; and (3) the appropriateness and frequency of footnotes in annual reports.

The morning after the hearing, outside the presence of the jury, the district court made oral rulings on the admissibility of expert testimony on each of these additional topics. The district court: (1) excluded Jones's testimony about whether disclosure of the shareholder loan was required; (2) permitted Jones's testimony regarding the timing of the funding of the capital contributions; (3) permitted Jones's testimony regarding the frequency of footnotes in annual reports; and (4) excluded Jones's testimony regarding whether the exhibits showed that interest was paid on the $1.7 billion loan.

The district court further excluded testimony regarding “the actual knowledge or belief of either Defendant” and excluded testimony regarding whether Appellants' subjectively held beliefs were reasonable. The district court stated that it would address questions regarding the knowledge, beliefs, and reasonableness of those knowledge and beliefs, of a hypothetical person in Kuhrt's position on a question-by-question basis. As a result, at trial, Jones did not testify about whether Kuhrt was responsible for, or required to disclose, the shareholder loan. Jones did testify that posting capital contributions prior to their funding was common in the industry. Kuhrt's counsel did not question Jones regarding the use of footnotes in annual reports.

Defendants are generally free to present any expert testimony that complies with Rule 702. “The Supreme Court in Daubert v. Merrell Dow Pharmaceuticals, Inc. [509 U.S. 579, 113 S.Ct. 2786, 125 L.Ed.2d 469 (1993) ] laid down the analytical framework for determining whether expert testimony is admissible under Federal Rule of Evidence 702.” United States v. Tucker, 345 F.3d 320, 327 (5th Cir.2003) (footnote omitted). “ ‘Under Daubert, Rule 702 charges trial courts to act as gate-keepers, making a preliminary assessment of whether the reasoning or methodology underlying the testimony is scientifically valid and of whether that reasoning or methodology properly can be applied to the facts in issue.’ ” Id. (quoting Pipitone v. Biomatrix, Inc., 288 F.3d 239, 243–44 (5th Cir.2002) (internal quotation marks omitted)); see also, e.g., United States v. Valencia, 600 F.3d 389, 423–24 (5th Cir.2010). Daubert “imposes a special obligation upon a trial judge to ensure that any and all scientific testimony ... is not only relevant, but reliable.” Kumho Tire Co. v. Carmichael, 526 U.S. 137, 147, 119 S.Ct. 1167, 143 L.Ed.2d 238 (1999) (alteration in original) (internal quotation marks omitted).

The Daubert standard applies “to all species of expert testimony, whether based on scientific, technical, or other specialized knowledge.” Tucker, 345 F.3d at 327 (internal quotation marks omitted). “[The] language [of Rule 702 ] makes no relevant distinction between ‘scientific’ knowledge and ‘technical’ or ‘other specialized’ knowledge.” Kumho Tire Co., 526 U.S. at 147, 119 S.Ct. 1167. “Federal Rules 702 and 703 grant expert witnesses testimonial latitude unavailable to other witnesses on the ‘assumption that the expert's opinion will have a reliable basis in the knowledge and experience of his discipline.’ ” Id. at 148, 119 S.Ct. 1167 (quoting Daubert, 509 U.S. at 592, 113 S.Ct. 2786 ).

To be admissible, expert testimony must be relevant and reliable. Wen Chyu Liu, 716 F.3d at 167. “The basic standard of relevance ... is a liberal one.” Daubert, 509 U.S. at 587, 113 S.Ct. 2786. The relevance prong requires the proponent to demonstrate that the expert's “reasoning or methodology can be properly applied to the facts in issue.” Curtis v. M & S Petroleum, Inc., 174 F.3d 661, 668 (5th Cir.1999). The “ ‘expert testimony proffered in the case [must be] sufficiently tied to the facts of the case that it will aid the jury in resolving a factual dispute.’ ” Tucker, 345 F.3d at 327 (quoting Daubert, 509 U.S. at 591, 113 S.Ct. 2786 ).

Federal Rule of Evidence 702 permits opinion testimony from a witness qualified as an expert by knowledge, skill, experience, training, or education if such testimony will assist the trier of fact and (1) the testimony is based on sufficient facts or data, (2) the testimony is the product of reliable principles and methods, and (3) the expert has reliably applied the principles and methods to the facts of the case.

Wen Chyu Liu, 716 F.3d at 167 (internal quotation marks omitted).

“In a criminal case, an expert witness must not state an opinion about whether the defendant did or did not have a mental state or condition that constitutes an element of the crime charged or of a defense. Those matters are for the trier of fact alone.” Fed.R.Evid. 704(b). Cf. Tucker, 345 F.3d at 330 (holding that “[the defendant's] efforts to elicit factual testimony [about his mental state] through his expert were impermissible.”). However, an expert “opinion is not objectionable just because it embraces an ultimate issue.” Fed.R.Evid. 704(a). The proponent of the expert testimony has the burden of establishing its admissibility. Moore v. Ashland Chemical, Inc., 151 F.3d 269, 276 (5th Cir.1998).

The topics for expert testimony at issue in this appeal can be categorized as: (1) the roles, responsibilities, and reasonable beliefs of persons in a similar position to Appellants and Appellants themselves; and (2) whether Appellants had a duty to disclose the fraud under any accounting ethics rules or norms. Based on our review of the proffered testimony, we conclude that the district court should have permitted some of the expert testimony on these topics. Nevertheless, because any error was harmless, we affirm.

At trial, Kuhrt attempted to elicit specific testimony from Jones regarding Kuhrt's role and responsibilities. The government objected and the district court sustained the government's objections regarding Kuhrt's actual roles and responsibilities for SIB's annual report. The district court also sustained the government's objection to the question of whether a hypothetical person could be responsible for the content and accuracy of the annual report if that person did not have the information that would indicate whether the report was accurate. The second of these rulings was conceivably in contrast to the district court's earlier rulings on the government's motion in limine.

Kuhrt argues that Jones's testimony is the type of “pattern” testimony traditionally admissible in securities cases, analogizing to United States v. Russo, 74 F.3d 1383 (2d Cir.1996). In Russo, the expert was permitted to testify regarding stock patterns and impacts on the marketplace. Id. at 1389, 1397. Kuhrt argues that Jones should have been permitted to testify regarding the typical roles and responsibilities of someone in Kuhrt's position and explain how responsibilities of different officers exist within large corporations.

Lopez similarly argues that Dharan's testimony was “essential to the jury's understanding of Lopez's defense,” because Lopez was not responsible for the fraud. Lopez states that Dharan reviewed “millions of documents that pertained ... [to] Lopez's role within SFGC ... [and that] “[a] component of [the failure to disclose the shareholder loan] was who had the responsibility for the content and accuracy of [SIB]'s annual report.”

Even assuming arguendo that Appellants' experts should have been permitted to testify regarding the roles and responsibilities of a typical person in Appellants' positions, and that excluding this testimony was “manifestly erroneous,” any error here was harmless. An error is only harmful if it affects “a substantial right of the complaining party.” Wen Chyu Liu, 716 F.3d at 169. “When assessing whether an error affected a substantial right of a defendant, the necessary inquiry is whether the trier of fact would have found the defendant guilty beyond a reasonable doubt with the additional evidence inserted.”Id. (internal quotation marks omitted).

Appellants argue that their experts' testimony could have definitively established that Kuhrt “did not know, nor should he have known, of Stanford's and Davis's multi-billion dollar frauds,” and that Lopez “was not responsible for [SIB]'s financials or the content and accuracy of its annual reports.” Appellants' experts, at most, could have testified about what a typical person in Appellants' positions would have likely known, based upon common practice in the industry. What Appellants actually knew, and whether they were actually responsible for SIB financials or annual reports, are questions of fact. Because the government presented ample evidence as to what Appellants actually did in this case, any error was harmless. The experts could have not provided any new information on Appellants' actual roles and responsibilities within SIB.

Appellants next argue that their experts would have testified that under accounting ethics and principles, they did not have a duty to disclose Stanford's fraud. Lopez argues that the exclusion of this testimony violated his constitutional right to present a complete defense. “[A] complete defense challenge [to excluded evidence] is meritorious when two factors are present: the excluded evidence is indispensable to the theory of defense; and the district court fails to provide a rational justification for its exclusion.” United States v. McGinnis, 201 Fed.Appx. 246, 252 (5th Cir.2006) ; accord United States v. Miliet, 804 F.2d 853, 859 (5th Cir.1986).

In the district court, both Appellants made an offer of proof on this point. Kuhrt's expert would have testified that “if [Kuhrt] advised his superior that [the shareholder loan] needed to be disclosed and [the supervisor] ma[d]e the decision not to disclose it, [Kuhrt had] no further duty at that point.” Lopez specifically stated that he wished to call his expert to “opine on the duty of an accountant who makes a recommendation about an accounting standard that is not followed by his superior, the chief financial officer.” Lopez's expert would have testified that “there is no duty to go to the police, to the FBI, to the SEC, to go above the CFO's head because that is not what is required by the standards in the accounting profession.” Lopez cites United States v. Cavin, 39 F.3d 1299 (5th Cir.1994), in support of this position. In Cavin, we held that the district court committed reversible error when it excluded expert testimony regarding a lawyer's ethical obligations to his client, including duties of loyalty, zealous representation, and confidentiality. Id. at 1308. We stated that “a lawyer accused of participating in his client's fraud is entitled to present evidence of his professional, including ethical, responsibilities, and the manner in which they influenced him. Exclusion of such evidence prevents the lawyer from effectively presenting his defense.” Id. at 1309 (footnote omitted).

This case is distinguishable from Cavin because this is not a case about Appellants' failure to disclose or blow the whistle on Stanford's fraud. If Appellants were being charged because of failures to make accounting disclosures, because they remained silent, or because they violated accounting rules or ethics, this proffered expert testimony might be critical. However, that is not this case.

As discussed above, the entire theory of the government's case was that Appellants actively participated in the fraud and took affirmative action to perpetuate and conceal Stanford's theft. The government presented evidence that Appellants knew of Stanford's theft and oversaw the creation of financial reports that hid that theft. The government also presented evidence that Appellants designed, orchestrated, and carried out multiple sham transactions designed to hide Stanford's theft from investors, regulators, and auditors. These transactions included two “private equity flips,” false accounting entries to boost the Bank's appearance of solvency, and the fraudulent revaluing of the Antiguan land at more than fifty times its purchase price.

Regardless of whether Appellants complied with accounting rules and ethics regarding reporting obligations, there was ample evidence that they participated directly in the fraud. Because there was ample evidence supporting the government's theory of the case—that Appellants actively committed criminal fraud—the testimony regarding accounting-related duties was not indispensable, nor was its exclusion harmful error. See Wen Chyu Liu, 716 F.3d at 169 (“[E]ven if the court erred in excluding the testimony, we will not vacate a conviction ... unless the error was harmful, affecting a substantial right of the complaining party.”).

E.

Kuhrt and Lopez challenge both the procedural and substantive reasonableness of their below-Guidelines 20–year sentences. “This Court reviews federal sentences under an abuse-of-discretion standard, ‘[r]egardless of whether the sentence imposed is inside or outside the Guidelines range.’ ” United States v. Fraga, 704 F.3d 432, 437 (5th Cir.2013) (alteration in original) (quoting Gall v. United States, 552 U.S. 38, 51, 128 S.Ct. 586, 169 L.Ed.2d 445 (2007) ). “Our inquiry involves two steps. First, we must ensure that the district court committed no significant procedural error.” Id. (internal quotation marks omitted). “Second, if the district court's sentencing decision is procedurally sound, we consider the substantive reasonableness of the sentence imposed under an abuse-of-discretion standard.” Id. (internal quotation marks omitted). “Our review of sentencing decisions is limited to determining whether they are reasonable.” Id. (internal quotation marks omitted).

Appellants argue that there were several procedural errors in their sentence calculation. First, they both dispute the district court's loss calculation. The district court applied a 30–level enhancement under United States Sentencing Guidelines Manual (U.S.S.G.) § 2B1.1(b)(1)(P) for a loss amount in excess of $400 million, calculating a total loss of just over $2 billion—the amount of money that Stanford had taken from SIB.

Actual loss is the “reasonably foreseeable pecuniary harm that resulted from the offense.” United States v. Umawa Oke Imo, 739 F.3d 226, 240 (5th Cir.2014) (quoting U.S.S.G. § 2B1.1(b)(1) cmt. n. 3(A)(i)). The amount of loss is a factual finding which need only be found by a preponderance of the evidence and is reviewed for clear error. Simpson, 741 F.3d at 556. The district court “need only make a reasonable estimate of the loss.” U.S.S.G. § 2B1.1, cmt. n. 3(C).

Kuhrt first argues that the loss calculation failed to give credit for $327 million in private equity “repayments” that offset the $2 billion misappropriated by Stanford. However, the trial evidence showed that the $327 million was not a real repayment and the district court did not err by failing to give credit for a fictitious repayment. Furthermore, even assuming arguendo that the district court's failure to credit this amount was error, such error was harmless because the total loss amount would still have exceeded $400 million and, therefore, would still have resulted in a 30–level enhancement.

Kuhrt further argues that “it was simply unforeseeable that Davis and Stanford would parlay the absence of a footnote about the loan into a macro-level investment fraud.” Similarly, Lopez argues that he was unable to foresee harm to the victims because factors beyond his control (such as the global economy and Davis's independent actions in furtherance of the fraudulent scheme) contributed to Stanford's downfall. These foreseeability arguments fall flat in light of the e-mails and other documents that the government produced at trial showing that Appellants knew the precise amount of SIB money that Stanford had taken and how he was using that money. Appellants were not charged with the totality of Stanford's fraudulent scheme, but rather only for their participation in the scheme. Because both of them participated in Stanford's misappropriation of the $2 billion of SIB money, Appellants could reasonably foresee the loss of that money.

Both Kuhrt and Lopez also assert that the district court should have used their gain as an alternative measure of loss because the loss could not be determined. See U.S.S.G. § 2B1.1 cmt. n. 3(B) (“The court shall use the gain that resulted from the offense as an alternative measure of loss only if there is a loss but it reasonably cannot be determined.”). In particular, Lopez argues that the government's inability to calculate restitution for individual victims demonstrates the court's inability to calculate the total loss that resulted from the offense. This contention conflates two wholly distinct questions—how much each individual victim lost for restitution purposes, and how much total loss resulted from Appellants' crimes. Difficulty in calculating individual loss amounts does not automatically signal difficulty in calculating the total loss. Here, as explained above, the total loss calculation was quite straightforward: the amount of SIB money that Appellants helped Stanford misappropriate. Kuhrt also argues that the district court erred when it applied a two-level enhancement for perjury under U.S.S.G. § 3C1.1. A perjury enhancement is appropriate if a defendant gives false testimony at trial “concerning a material matter with the willful intent to provide false testimony, rather than as a result of confusion, mistake, or faulty memory.” United States v. Dunnigan, 507 U.S. 87, 94, 113 S.Ct. 1111, 122 L.Ed.2d 445 (1993). The district court explicitly adopted the PSR, which identifies several specific incidents of perjury, including Kuhrt's statements that he sent particular spreadsheets via FedEx and that he never reviewed the IB5s. Kuhrt provides no rebuttal evidence on these points, merely asserting that he testified truthfully. The district court did not err in applying this sentencing enhancement. Furthermore, assuming arguendo that the district court erred, the error was harmless. As Kuhrt conceded in his objections to the PSR, removing the perjury enhancement would reduce Kuhrt's offense level from 51 to 49, and his sentence would not change. See U.S.S.G. ch. 5, pt. A, cmt. n. 2 (“An offense level of more than 43 is to be treated as an offense level of 43.”).

The district court also applied a two-level enhancement for abusing a “position of trust” to facilitate the crime under U.S.S.G. § 3B1.3. Kuhrt argues that the position of trust enhancement was improper because he “had no discretionary judgment.” The public trust enhancement is warranted “[i]f the defendant abused a position of public or private trust, or used a special skill in a manner that significantly facilitated the commission or concealment of the offense.”U.S.S.G. § 3B1.3 ; accord United States v. Willett, 751 F.3d 335, 344 (5th Cir.2014). A “special skill” is defined as a “skill not possessed by members of the general public and usually requiring substantial education, training or licensing.” U.S.S.G. § 3B1.3 cmt. n. 4. Accounting is specifically listed as an example. Id. To determine whether a defendant “significantly facilitate[d]” the commission of an offense, the question is “whether [the defendant's] position afforded him an opportunity not enjoyed by the general public.” United States v. Powers, 168 F.3d 741, 752 (5th Cir.1999). Kuhrt was one of the top two accountants in Stanford's companies and had been working as an accountant for over a decade. The district court did not err in applying this sentencing enhancement.

Kuhrt further argues that he should have received a reduction in his sentence based upon his mitigating role under U.S.S.G. § 3B1.2(b). To qualify for a mitigating role reduction, “[i]t is not enough that a defendant does less than other participants; in order to qualify as a minor participant, a defendant must have been peripheral to the advancement of the illicit activity.” United States v. Villanueva, 408 F.3d 193, 204 (5th Cir.2005) (internal quotation marks omitted). The district court adopted the PSR's finding that Kuhrt had an “average,” rather than a “minor,” role in the scheme. Kuhrt relies on the fact that he did not receive compensation for his role in the fraud beyond his salary and bonus, and that much of the fraud was intentionally withheld from him and Lopez. Neither of these arguments renders him “peripheral” to the scheme. Whether he was compensated directly for the fraud is not dispositive. Because much of the fraud—such as the bribery of financial officials—was withheld from Kuhrt and Lopez, they were not charged in connection with the entire fraud. Appellants were only charged and convicted based upon their participation.

Finally, Kuhrt and Lopez contend that their 240–month sentences are substantively unreasonable, particularly in contrast to the 60–month sentence that Davis received. The district court's correct application of the Guidelines yielded an advisory sentence of life imprisonment. Instead, the district court departed downward and imposed sentences of 240 months. “Appellate review is highly deferential as the sentencing judge is in a superior position to find facts and judge their import under § 3553(a) with respect to a particular defendant.” United States v. Campos–Maldonado, 531 F.3d 337, 339 (5th Cir.2008). Furthermore, our “concern about unwarranted disparities is at a minimum when a sentence is within the Guidelines range.” United States v. Willingham, 497 F.3d 541, 545 (5th Cir.2007).

Appellants make much of the fact that they did not personally profit from Stanford's scheme, while Davis, who made millions of dollars throughout the course of the scheme, received a much shorter sentence. However, Davis received a lesser sentence in exchange for pleading guilty, accepting responsibility, and testifying at multiple trials, include Stanford's trial. We will not second-guess the district court on these facts.

III.

For the foregoing reasons, the district court's judgment and sentence are AFFIRMED.