
    UNITED STATES of America, Appellee, v. Lorenzo MUÑOZ-FRANCO, Francisco Sánchez-Arán, Ariel Gutiérrez-Rodríguez, and Wilfredo Umpierre-Hernández, Defendants, Appellants.
    Nos. 04-1532 to 04-1535.
    United States Court of Appeals, First Circuit.
    Heard Dec. 6, 2006.
    Decided May 22, 2007.
    
      Mark J. Rochon, with whom Duncan N. Stevens and Miller & Chevalier were on brief, for appellant Sánchez-Arán.
    Francisco M. Dolz-Sánchez for appellant Umpierre-Hernández.
    R. Jack Cinquegrana, with whom Jennifer T. Connor, Jennifer E. Tracy, and Choate, Hall & Stewart were on brief, for appellant Muñoz-Franco. ,
    Michael S. Pasano, with whom Zucker-man Spaeder LLP was on brief, for appellant Gutiérrez-Rodríguez.
    Elizabeth A. Olson, Criminal Division, Appellate Section, U.S. Department of Justice, with whom Rosa E. Rodriguez-Velez, U.S. Attorney, and Maria A. Dominguez, Assistant U.S. Attorney, were on brief, for appellees.
    Before BOUDIN, Chief Judge, CYR, Senior Circuit Judge, and LIPEZ, Circuit Judge.
   LIPEZ, Circuit Judge.

Appellants Lorenzo Muñoz-Franco, Francisco Sánchez-Arán, Ariel Gutiérrez-Rodríguez, and Wilfredo Umpierre-Her-nández appeal their convictions for bank fraud, conspiracy, and misapplication of bank funds, stemming from conduct that persisted for nearly a decade. Muñoz-Franco and Sánchez-Arán appeal their convictions for bank fraud and conspiracy for a similar but separate series of transactions also involving the improper use of bank funds. Appellants challenge the sufficiency of the evidence and raise many other substantive and procedural claims relating to their fifteen-month trial. After careful review of this immense record, we affirm.

I.

A. Factual Overview

We provide here an overview of the relevant conduct drawn from the record, saving additional details for our sufficiency analysis. Throughout the opinion, we consider the evidence “in the light most favorable to the prosecution.” See Jackson v. Virginia, 443 U.S. 307, 319, 99 S.Ct. 2781, 61 L.Ed.2d 560 (1979).

1. The Gutiérrez Loans

During the relevant time period charged in the Third Superseding Indictment, Mu-ñoz-Franco was President and Chief Executive Officer of Caguas Central Federal Savings Bank of Puerto Rico (“Caguas”), a federally chartered savings and loan association. Sánchez-Arán was Executive Vice President and Chief Lending Officer of Caguas. Gutiérrez was a land developer who owned several companies that received loans from Caguas. Umpierre-Hernández was an officer of several companies belonging to Gutiérrez.

For nearly a decade, Muñoz-Franco and Sánchez-Arán supervised a scheme to use proceeds from various loans for purposes not authorized by Caguas’ Board of Directors (“Board”). The loans were granted for land development projects involving companies that Gutiérrez owned and Um-pierre-Hernández helped to operate, including Transglobe, Modules, and Tran-shore. In many instances the appellants used proceeds from loans to Gutiérrez-owned companies to make payments on prior loans to Gutiérrez-owned companies without Board approval. In other instances Muñoz-Franco and Sánchez-Arán submitted loans to Gutiérrez-owned companies to the Board for approval without disclosing the Gutiérrez-owned companies’ failure to complete work on previous projects. On many occasions Gutiérrez and Um-pierre-Hernández submitted certifications for construction work that had not yet been completed, and Muñoz-Franco and Sánchez-Arán accepted the certifications and ordered disbursement of funds for the projects. The Board also was not informed of this practice. In January 1988, Muñoz-Franco drafted and obtained Board approval to send a letter to Richard Denby, the Federal Home Loan Bank Board auditor supervising Caguas, which responded to Denby’s concerns regarding Caguas’ lending practices (“Denby letter”). The letter contained many misrepresentations regarding the status of Gutiérrez-related projects and loans.

As the government explained in its opening statement, this scheme contributed to the appearance that Caguas remained a financially viable institution under the leadership of Muñoz-Franco and Sánchez-Arán. If the problems with the bank became known and Muñoz-Franco and Sánchez-Arán were removed from their positions, “not only would that have deprived them of a very lucrative job, but it also would have made it very difficult for them to obtain new employment in the banking industry.” The scheme also maintained the appearance of solvency for the Gutiérrez companies, thus protecting the livelihood and professional reputation of Gutiérrez and Umpierre-Hernández.

The projects for which Gutiérrez-owned companies received loans included La Marina, Los Mameyes, Cerrovista, and Jar-dines de Villa Alba. We provide a brief overview of these projects here.

a. La Marina

In June 1980, Muñoz-Franco and Sán-chez-Arán caused Caguas to grant a $1,450,000 loan to Transglobe to finance the construction of seventy-five units of residential housing, with construction to begin within one month and to be completed within one year. As of September 1981, the loan limit had been increased four times, adding a total of $1.8 million, yet not a single house had been built. Funds were disbursed from the loan for the project based on certifications submitted by Gutiérrez and Umpierre-Hernán-dez and approved by Sánchez-Arán, and approximately $2 million was used to pay unrelated Gutiérrez debts with a different bank. In October 1984, Muñoz-Franco and Sánchez-Arán caused Caguas to finance the sale of the La Marina project to DO.W Group. The agreement for this sale provided that Transglobe would remain the contractor for the project despite its failure to build a single house in the preceding four years. In approving the loan, the Board was not informed of Trans-globe’s prior poor performance. After this sale, appellants continued to apply funds from the La Marina loan to other projects and to disburse funds for work not completed. The houses were never completed on the section of the project for which the loan was issued, and, in 1989, Caguas entered into a settlement agreement with DO.W releasing DO.W’s debt on the project, which totaled over $2 million.

b. Los Mameyes

In late 1985, Muñoz-Franco and Sán-chez-Arán caused the Board to grant Modules a commercial line of credit to build two hundred housing units. Between December 1985 and March 1986, Gutiérrez and Umpierre-Hernández submitted a series of certifications stating that a total of two hundred housing units had been built. Even though Caguas’ inspector reported that only fifty-five units had been completed, Sánchez-Arán approved disbursements totaling about $800,000. In March 1986 Gutiérrez and Umpierre-Hernández submitted a certification requesting $69,000 for “payment of subcontractors,” but attached a list of interest payments due on the Jardines de Villa Alba, Levittown, Country Club, La Marina, and Los Ca-ciques projects totalling exactly $69,000. Sánchez-Arán approved the certification and Caguas disbursed the funds.

c. Cerrovista

In the spring of 1986, John Burns, a developer, applied to Caguas for a loan to build residential housing on land he owned but his application was denied. After meeting with Umpierre-Hernández and agreeing to use Modules as a contractor, Caguas approved Burns’ application for a loan to build twenty-three units of housing in what became known as the Cerrovista project. Burns’ loan was approved on the condition that he sign a $2 million note as a down payment to Modules. Burns signed the note. Shortly thereafter, however, Umpierre-Hernández told Burns that the project needed a new “sponsor” because Burns had been “gossiping.” Um-pierre-Hernández then began to negotiate the sale of Cerrovista to Iantho, a company owned by Walter Frambes. In August or September 1986, Burns read newspaper articles indicating that the Cerrovista project might be affected by the Puerto Rico Department of Transportation’s plans to build a hospital and an expressway. He inquired with the Department and received confirmation that the development of four lots would be “frozen.” When Burns brought this information to Um-pierre-Hernández, Umpierre-Hernández told him to “hide it, boy.”

On September 10, 1986, Sánchez-Arán received Board approval to offer Iantho an $8.9 million loan, specifying that $1,412,177 would be used to purchase land for the Cerrovista project. The sale was then finalized on September 28. However, the actual land cost was only $480,000, and the remaining $932,000 was used to make principal and interest payments on other loans to Gutiérrez-owned companies. Over the next few months Gutiérrez and Umpierre-Hernández submitted certifications showing approximately $908,000 for “premanu-facture” of housing units, and Sánchez-Arán authorized disbursement of these funds. As of May 1988, however, not a single unit had been built on the project.

d. Jardines de Villa Alba

In 1985, a developer named Emilio Mon-tilla sought financing to build housing units on land he owned, and Caguas repeatedly denied his requests. Montilla then met with Umpierre-Hernández, who told him that if he used Modules as a contractor his request would be approved. In July 1985, the Board considered a proposed loan to Montilla with Modules as contractor. Although by this time Modules had failed to perform on the La Marina project for several years, the Board was not informed of this information when it considered the Jardines de Villa Alba project. Moreover, over $281,000 was disbursed before the Jardines de Villa Alba loan agreement was signed. Although the Board eventually approved the loan, this prior disbursement took place without Board knowledge or approval. Only one unit was ever completed on the Jardines de Villa Alba project. However, the Board was not informed of this fact. The Denby letter, which was drafted by Muñoz-Franco in January 1988 and signed by the Board members, refers to the “units” completed on this project.

2. The Mirandes Loans

As will be described in more detail in our sufficiency discussion, Muñoz-Franco and Sánehez-Arán supervised a similar scheme to use loan proceeds for purposes not authorized by the Board with respect to companies owned by Francisco Mi-randes. Mirandes received construction loans from Caguas for at least fourteen different projects. In December 1989, the Mirandes corporations collapsed due to insolvency, owing Caguas a total of $23 million. In 1997, Mirandes pled guilty to charges of participating in a conspiracy to defraud Caguas, the Board, and the regulatory institution, and to misapplication of bank funds. As part of his plea agreement, he testified for the government at appellants’ trial.

B. Procedural History

On November 22, 1995, the government filed an indictment charging appellants with conspiracy, bank fraud, misapplication of bank funds, and making false entries. Three superseding indictments subsequently were filed on March 5, 1997; May 18, 1997; and March 6, 1998. The third superseding indictment charged Mu-ñoz-Franco, Sánchez-Arán, Gutiérrez, Umpierre-Hernández, Enrique Gutiérrez (Ariel Gutiérrez’s brother, who was also involved in running the Gutiérrez corporations), and Rafael Dominguez Wolff (who purchased several projects from the Gu-tiérrez brothers) with bank fraud, in violation of 18 U.S.C. § 1344; conspiracy to commit bank fraud, misapply bank funds, and make false entries, in violation of 18 U.S.C. § 371; and misapplication of bank funds, in violation of 18 U.S.C. § 657. The indictment also charged Muñoz-Franco and Sánchez-Arán with bank fraud under 18 U.S.C. § 1344 and conspiracy under 18 U.S.C. § 371 with respect to the Mirandes loans.

The trial began in federal district court in Puerto Rico in January 2001 and the presentation of evidence concluded in April 2002. On May 16, 2002, the jury convicted Muñoz-Franco, Sánchez-Arán, Ariel Gu-tiérrez, and Umpierre-Hernández on all charged counts, but acquitted Enrique Gu-tiérrez. Following the verdict, appellants filed motions for a judgment of acquittal pursuant to Federal Rule of Criminal Procedure 29, raising numerous grounds, including the sufficiency of the evidence. In a thoughtful seventy-seven page opinion, the district court denied all of the motions.

Sentencing proceedings began on December 15, 2003. On February 12, 2004, the district court sentenced Muñoz-Franco and Sánchez-Arán to a term of forty-six months on the conspiracy and bank fraud counts and a concurrent term of sixty months on the misapplication count. It also imposed a fine of $50,000 on each defendant. The court sentenced Gutiérrez to a term of thirty-seven months on the conspiracy and bank fraud counts, and imposed a concurrent term of sixty months on the misapplication count. It also imposed a $60,000 fine. Finally, it sentenced Umpierre-Hernández to a term of twenty-four months on all counts.

C. Issues on Appeal

Appellants raise thirteen major issues for our consideration on appeal. Three of these claims — concerning the statute of limitations, the Ex Post Facto Clause, and the pre-indictment and pre-trial delay— implicate what might be termed the validity of the entire proceedings. Although courts often begin by addressing such issues, we find, in light of the exceptionally large record, that we must review the sufficiency of the evidence in order to assess the viability of any of appellants’ other claims. Thus, we will first resolve two evidentiary issues to determine the content of the record properly before us. We will then employ that record in conducting our analysis of the sufficiency of the evidence. Informed by a full understanding of the record, we will then turn to the host of other issues raised by appellants.

II.

We begin with two evidentiary issues: whether the district court allowed witnesses to testify beyond their personal knowledge, and whether the minutes from the Board meetings were improperly admitted.

A. Personal Knowledge

Appellants argue that the district court improperly permitted prosecution witnesses to testify beyond their personal knowledge, in violation of Federal Rule of Evidence 602. Rule 602 states, in pertinent part: “A witness may not testify to a matter unless evidence is introduced sufficient to support a finding that the witness has personal knowledge of the matter.” Fed.R.Evid. 602. Interpretation of the Federal Rules of Evidence is a question of law subject to de novo review, but the application of a rule is reviewed for abuse of discretion. United States v. Sposito, 106 F.3d 1042, 1046 (1st Cir.1997).

1. Arturo Somohano

Appellants argue that Arturo So-mohano (the vice president of Caguas’ commercial loan department until 1988, at which time he became Caguas’ chief lending officer) testified beyond his personal knowledge on three occasions: (1) in claiming that the bank’s outside auditors were replaced for improper reasons; (2) in stating that Caguas’ loan classifications for the loans to Modules were improper; and (8) in questioning the propriety of certain construction loans. The district court excluded the first of these statements after appellants objected, and we therefore do not consider it further.

With respect to Somohano’s testimony that Caguas’ loan classifications for Modules were improper, appellants objected on the ground that Somohano had previously stated that he was unaware of the bank’s classifications for those loans. So-mohano acknowledged that he did not remember seeing the loan classifications for Modules while at Caguas, but his testimony did not rely on such knowledge of Modules’ status. Instead, Somohano explained that loans at Caguas were classified from one (best) to ten (worst) based on the amount of risk involved in the loan, and that the bank would attach greater reserves to riskier loans. While still on the witness stand, Somohano then reviewed financial statements indicating that Modules had a negativo net worth of $7.6 million at the end of 1986, and that Caguas’ reserves would not cover the loans to Modules because the reserves equalled only ten or fifteen percent of the loans Modules was sustaining. Based strictly on the amount of reserves, Somo-hano would expect the Modules loan to have received a favorable classification of either two or three; however, he testified that, based on Modules’ “insolvent” financial status, such classification would have been improper.

According to Federal Rule of Evidence 701, a lay witness may offer opinions that are “(a) rationally based on the perception of the witness, (b) helpful to a clear understanding of the witness’ testimony or the determination of a fact in issue, and (c) not based on scientific, technical, or other specialized knowledge within the scope of Rule 702.” Fed.R.Evid. 701. Under Rule 701, courts have allowed lay witnesses to express opinions about a business “based on the witness’s own perceptions and ‘knowledge and participation in the day-today affairs of [the] business.’ ” United States v. Polishan, 336 F.3d 234, 242 (3d Cir.2003)(alternation in original)(quoting Lightning Lube, Inc. v. Witco Corp., 4 F.3d 1153, 1175 (3d Cir.1993)); see also Medforms, Inc. v. Healthcare Mgmt. Solutions, Inc., 290 F.3d 98, 110-11 (2d Cir.2002)(allowing a computer programmer to testify about the meaning of terms used in the copyright registrations for programs he had helped design); Williams Enters., Inc. v. Sherman R. Smoot Co., 938 F.2d 230, 233-34 (D.C.Cir.1991)(allowing an insurance broker who had personal knowledge of an insured’s business to offer lay opinion testimony on the cause of an increase in the insured’s premiums). Here, Somohano’s testimony was based on knowledge of Caguas’ banking practices that he acquired during his employment there, and thus the opinions he expressed were properly within the scope of Federal Rule of Evidence 701.

Finally, appellants protest, without providing specific examples, that Somohano “was permitted to testify about the propriety of certain construction loans even though he was not part of the construction loan department and has no firsthand knowledge of those loans.” We find this generalized objection unfounded. Somo-hano’s opinions about the loans were based on his firsthand observations of Caguas’ practices with respect to these loans. He testified that, when the Modules loan was restructured, he did not think Caguas ever would be repaid, and he provided other examples of Gutiérrez loans receiving spe- - cial treatment. Given Somohano’s banking experience and his particular knowledge about the Gutiérrez loans, his testimony about these loans also was properly admissible as a lay opinion within the scope of Rule 701. Consequently, we find no abuse of discretion in the district court’s handling of Somohano’s testimony.

2. Anabel Enriquez

Appellants also argue that Anabel Enri-quez (the senior vice president of Caguas’ mortgage department from 1980-87, who reported directly to Sánchez-Arán) testified beyond her personal knowledge when she stated, based on her review of the minutes of Board and Executive Committee meetings, that certain information was not discussed at the meetings that should have been discussed. This objection is unfounded. At trial, Enriquez read the Board meeting minutes into evidence, including minutes for meetings at which she was not present. Appellants objected, and, after considerable debate, the district court ruled “that [Enriquez] may literally read what each paragraph [of the minutes] says. If the United States then wants an explanation as to the paragraph she must then have an independent source of knowledge.” Thus, Enriquez did not testify whether certain events actually happened at a Board meeting unless she was present; for meetings she did not attend, she testified only “whether the minutes reflect” that certain information was discussed. The district court enforced this limitation, and required the prosecutor to rephrase questions on more than one occasion. Similarly, Enriquez’s testimony that the Board “should have been told” certain information was a lay opinion properly admitted under Rule 701. Her position as a senior vice president of Caguas’ mortgage department and her regular attendance at Board meetings established her familiarity with Caguas’ business operations and made it appropriate for her testify about information the Board needed to make its decisions. Again, the district court did not abuse its discretion in allowing such testimony.

3. Victor Lugo

As part of its case, the government sought to prove that Muñoz-Franco and Sánchez-Arán concealed material information relating to the Gutiérrez loans from the Board. Victor Lugo, the Board president, offered testimony that he was not told or was not aware of significant information. Appellants claim that Lugo’s testimony showed that he had no independent recollection of the events in question, some of which occurred nearly twenty years before trial. However, appellants misconstrue Lugo’s testimony about his own memory. For example, when asked whether he was informed of Modules’ performance history when the Board was making decisions on the company’s loans in July 1985, he stated: “I don’t recall that it was discussed. I believe that if it was discussed I would remember and it is not reflected in the minutes.” Similarly, when asked whether the number of houses built on Modules projects was discussed when the Board was again making decisions on the loans in September 1986, he stated definitively: “No ... that was not discussed.” These statements show that Lugo had sufficient recall to testify from his personal knowledge about the information given to the Board.

Appellants also complain that the government improperly suggested the truth of hypothetical scenarios presented in certain questions to Lugo. The purpose of these questions was to probe whether the Board would have considered Gutiérrez companies’ past performance in deciding to approve later loans. For example, the prosecutor asked Lugo: “[I]f the Gutiér-rez company, out of these 96 [housing] units only built 22, would that have been information ... which you would have been interested in knowing for purposes of voting on the approval of this loan?” The court explained at sidebar that it would only allow such hypothetieals when the government had already introduced evidence that was the basis for the hypothetical — specifically, when “there is a document from the bank that clearly establishes a certain fact.” It also instructed the jurors that they should not take the facts that were the basis for the hypo-thetieals as true:

Okay. The fact that the Court has authorized that question, ladies and gentlemen of the jury, does not mean ... that the Court is concluding it is a foregoing fact that you should take those facts as true. Those facts are subject to your analysis and your credibility.
In other words, you have to decide in your deliberations whether or not this project called for 96 houses and you have to decide whether or not actually 22 were built or whatever number was built, that is subject to your credibility.
The fact that the Court is authorizing that this question be expressed in these terms does not at all mean that the Court has already determined that it is a fact at all. You have to decide. That’s one of the issues that you’re going to have to decide in this case.

The court reiterated this instruction several times. This detailed instruction was sufficient to advise the jury that it should not accept the facts set forth in the hypo-thetieals as true. Thus, Lugo’s response to these questions did not have the effect of allowing him to testify beyond his personal knowledge to the factual correctness of the hypothetical. The district court did not abuse its discretion in allowing such testimony.

4. Victor Kareh

Finally, appellants assert that Victor Kareh (the assistant vice president of the construction loan department from 1980 to 1990) “was permitted ... to opine about what the project documents and ledger cards indicated, without any clear memory of the events and mixing summary testimony with what was improper expert testimony.” To support this proposition, appellants cite to three thirty-page sections of the trial transcript, without specifying the testimony to which they object or the grounds for their objections. Although Kareh did not author all of the documents about which he testified, this in itself is not a basis for excluding his testimony about the documents. Without developed argumentation on this issue, we cannot conclude that the district court abused its discretion in allowing Kareh’s testimony. United States v. Zannino, 895 F.2d 1, 17 (1st Cir.1990)(“[I]ssues adverted to in a perfunctory manner, unaccompanied by some effort at developed argumentation, are deemed waived.”).

B. Admission of Minutes from Board Meetings

At trial, the prosecution introduced the Board and Executive Committee meeting minutes under Federal Rule of Evidence 803(6), which states that business records are not excluded by the rule against hearsay “if kept in the course of a regularly conducted business activity, and if it was the regular practice of that business activity to make the ... record ... unless the source of information or the method or circumstances of preparation indicate lack of trustworthiness.” Appellants do not claim that the minutes themselves were improperly admitted, but rather protest the prosecution’s reliance on the absence of certain information from the minutes to prove that the Board was not informed about such matters. Appellants argue that this use violated their rights under the Confrontation Clause and the rule against hearsay. We examine these claims in turn. We review alleged violations of the Sixth Amendment’s Confrontation Clause de novo. United States v. Rondeau, 430 F.3d 44, 47 (1st Cir.2005). As discussed above, interpretation of the Federal Rules of Evidence is subject to de novo review, but application of a rule is reviewed for abuse of discretion. Sposito, 106 F.3d at 1046.

1. Confrontation Clause

The Confrontation Clause of the Sixth Amendment guarantees defendants the right to confront adverse witnesses. The Supreme Court has explained that only “testimonial” statements “cause the declarant to be a ‘witness’ within the meaning of the Confrontation Clause.” Davis v. Washington, — U.S. -, 126 S.Ct. 2266, 2273, 165 L.Ed.2d 224 (2006)(citing Crawford v. Washington, 541 U.S. 36, 51, 124 S.Ct. 1354, 158 L.Ed.2d 177 (2004)). Thus, “[i]t is the testimonial character of the statement that separates it from other hearsay that, while subject to traditional limitations upon hearsay evidence, is not subject to the Confrontation Clause.” Id. Although the Court has yet to articulate a precise definition of “testimonial,” it is beyond debate that the Board minutes are nontestimonial in character and, consequently, outside the class of statements prohibited by the Confrontation Clause. The Court in Crawford plainly characterized business records as “statements that by their nature [are] not testimonial.” 541 U.S. at 56, 124 S.Ct. 1354. If business records are nontestimo-nial, it follows that the absence of information from those records also must be non-testimonial. Thus, the Confrontation Clause presents no bar to reliance on the absence of certain information from the Board meeting minutes to prove that the Board was not given that information.

2. Hearsay

Appellants also complain that, once the minutes were admitted, the prosecution used the absence of certain information from the minutes to demonstrate that Muñoz-Franco and Sánchez-Arán failed to inform the Board of these matters in violation of the rule against hearsay. Appellants argue that the minutes were intended to be a summary, rather than an exhaustive record, of the events that took place at Board meetings, and that, consequently, the minutes do not indicate whether such information was disclosed to the Board.

Federal Rule of Evidence 803(7) states that the rule against hearsay does not exclude

[ejvidence that a matter is not included in the ... records ... kept in accordance with the provisions of [Fed. R.Evid. 803(6) ], to prove the nonoccurrence or nonexistence of the matter, if the matter was of a kind of which a ... record ... was regularly made and preserved, unless the sources of information or other circumstances indicate lack of trustworthiness.

Fed.R.Evid. 803(7). The Advisory Committee Note to Rule 803(7) also states that the “[fjailure of a record to mention a matter which would ordinarily be mentioned is satisfactory evidence of its nonexistence.” Thus, the absence of certain information in minutes admissible under Rule 803(6) would be admissible to show that the Board was not given that information unless the circumstances indicated some reason that these omissions from the record were untrustworthy.

We note that Rule 803(6) excludes business records if “the source of information or the method or circumstances of preparation indicate lack of trustworthiness,” Fed.R.Evid. 803(6), while Rule 803(7) excludes records otherwise admissible under Rule 803(6) if “the sources of information or other circumstances indicate lack of trustworthiness,” Id. 803(7). We think this repetition indicates that even if a business record is deemed sufficiently trustworthy to be admissible for its contents under Rule 803(6), other circumstances might render omissions in that record untrustworthy to show that the events omitted did not occur. Thus, appellants’ concession that the records are admissible under Rule 803(6) does not necessarily preclude them from arguing that omissions from the records are not admissible under Rule 803(7).

The government introduced testimony from Enriquez, the vice-president who reported directly to Sánchez-Arán, explaining that the minutes were prepared by one of the executive vice presidents before 1982, by Luis Pastor (an administrative assistant to MuñozAFranco) from 1982 to 1986, and by Enriquez after that time. Enriquez stated that the records prior to 1982 “were prepared and reviewed by the board, they were set forth in a record book and the secretary of the board and the president would sign them.” The government submitted an affidavit from Pastor, dated March 21, 2001, verifying that the documents it introduced were, in fact, the minutes from 1982 to 1986 and explaining that the minutes “consist of transcriptions made from notes taken by someone who was present at each meeting” and that the minutes “were kept in the regular course of business of Caguas.... [I]t was the regular business practice of Caguas to compile and prepare [the minutes] and to maintain them in its files.”

Finally, Enriquez explained that, after 1986, “[w]hen I was present I would take notes, subject to the agenda and what was approved.” When Enriquez was not present, Eliza Balina, the executive secretary for Muñoz-Franco, would take notes at the meetings, and Enriquez would then “review the files that were prepared for use by the directors. And upon that review, if there was something I did not understand then I could ask the president.” Enriquez noted that the minutes “were always ratified at the following board meeting,” and that, after ratification, the minutes would be stored either in the president’s office or in the bank vault.

Appellants emphasize that Lugo (the president of the Board) acknowledged on cross-examination that the minutes were only a summary of the meetings and did not “word for word report discussions.” However, a review of the Board and Executive Committee minutes reveals that, while not word for word renditions of the meetings, the minutes capture more than a skeletal outline of each meeting and include descriptions of the considerations relevant to each loan rather than simply the fact that the loan was discussed and approved.

After considering this evidence, we conclude that the district court did not abuse its discretion in allowing the use of the minutes to demonstrate that the Board did not receive material information about many of the transactions it considered. Enriquez’s testimony and Pastor’s affidavit demonstrate that the business records were prepared regularly after every meeting, reviewed and ratified by the Board, and stored securely following ratification. These circumstances do not indicate lack of trustworthiness. Indeed, they permit a finding of trustworthiness. Moreover, given the minutes’ thorough description of information discussed at the meetings, we conclude that the missing information relating to the loan transactions was a “matter ... of a kind of which a memorandum, report, record, or data compilation was regularly made and preserved.” Fed. R.Evid. 803(7).

III.

We proceed to the sufficiency of the evidence, including in our analysis the previously-discussed witness testimony and the absence of certain information from the Board minutes.

In considering the sufficiency of the evidence to support a guilty verdict, “ ‘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.’ ” United States v. Woodward, 149 F.3d 46, 56 (1st Cir.1998)(quoting Jackson v. Virginia, 443 U.S. 307, 319, 99 S.Ct. 2781, 61 L.Ed.2d 560 (1979)). In performing this inquiry, we “neither weight ] the credibility of the witnesses nor attempt! 1 to assess whether the prosecution succeeded in eliminating every possible theory consistent with the defendant’s innocence.” United States v. Noah, 130 F.3d 490, 494 (1st Cir.1997). Similarly, we “review[] a district court’s denial of a defendant’s motion for a judgment of acquittal ‘using the identical standard employed to measure the sufficiency of evidence supporting a guilty verdict.’ ” United States v. Loder, 23 F.3d 586, 590 (1st Cir.1994)(quoting United States v. Sanchez, 943 F.2d 110, 114 (1st Cir.1991)).

A. Bank Fraud with Respect to the Gutiérrez Loans

The jury found all four appellants guilty of bank fraud with respect to the Gutiérrez loans. To prove bank fraud under 18 U.S.C. § 1344, the government must show that the appellants knowingly engaged in a scheme or artifice to defraud or obtain money from a federally insured financial institution by means of materially false statements or misrepresentations. See United States v. Kenrick, 221 F.3d 19, 30 (1st Cir.2000). We first consider the evidence against bank officers Muñoz-Franco and Sánchezr-Arán, and then the evidence against Gutiérrez and Umpierre-Hernández.

1. Muñoz-Franco and Sánchez-Arán

In their positions as President and Executive Vice President, respectively, Muñoz-Franco and Sánchez-Arán were responsible for keeping the Board informed of information relevant to current and prospective loans. Moreover, Sán-chez-Arán was the primary supervisor of the construction and commercial loans to Modules, a Gutiérrez-owned company involved in many of the fraudulent transactions. Muñoz-Franco supported Sánchez-Arán’s recommendations on these projects. At trial, Lugo, the president of the Board of Directors, explained that he and the other Board members relied on the reports prepared by Muñoz-Franco and Sán-chez-Arán in making decisions regarding loans. Lugo explained that it was not his job to “verify the accuracy of information that was being given” to him by the management of the bank, and added that he “trusted fully that the information that was being provided me was whole and true.”

Despite their responsibility to keep the Board fully apprised, Muñoz-Franco and Sánchez-Arán concealed important information on many occasions. For example, in July 1985 Caguas considered the Jar-dines de Villa Alba project, for which Modules would be the contractor. As of October 29, 1984, Modules had completed none of the 212 units planned for the La Marina project, even though the original loan agreement from June 25, 1980 called for construction to begin in thirty days and for seventy-five units to be completed within twelve months. Despite the lack of progress, the loan amount had been increased four times between August 7, 1980 and September 17, 1981. Lugo testified that this poor performance history was not discussed during the loan presentation for the Jardines de Villa Alba project. Lugo stated unequivocally that he would have wanted to know Modules’ performance history in evaluating the Jardines de Villa Alba loan, demonstrating the materiality of such information.

Lugo further testified that, after the original loan presentation, Muñoz-Franco and Sánchez-Arán continued to withhold information from the Board regarding the Jardines de Villa Alba loan. Although the record shows $231,000 was paid to Modules before the Jardines de Villa Alba loan agreement was signed, even at trial Lugo expressed surprise that this had happened and stated that “it would have been difficult for me to believe that” this occurred. More than a year after the Jar-dines de Villa Alba loan was approved, in September 1986, Lugo did not know that only one house had been built on the project. Lugo also testified that as of September 1986 the performance history of Modules “as to construction of homes that were to be constructed versus the ones that were constructed ... was not discussed.” Finally, on January 10, 1988, Lugo signed the Denby letter, which was prepared by Muñoz-Franco and which referred to the “units” built on Jardines de Villa Alba, without realizing that only one house had been built on the project.

Lugo also testified that Muñoz-Franco and Sánchez-Arán did not inform him and the Board of other material information relating to Modules. For example, he did not know that Modules remained the contractor after the La Marina, Country Club, and Levittown projects were sold from Transglobe to DO.W in October 1984. He also did not know that, between 1984 and 1986, Modules did not make any principal or interest payment on any loan to Caguas with funds generated from its own business as opposed to proceeds from other loans. Likewise, he was not informed, when the Board approved the sale of Modules to Camero on September 10, 1986, in conjunction with a restructuring of its loans, that approximately seventy percent of Modules’ debt was with Caguas and that over seventy-one percent of Modules’ assets were intangibles.

Finally, Lugo testified that when the board approved a loan of $1,412,077 for land acquisition for the Cerrovista project, for which Modules also was the contractor, Muñoz-Franco and Sánchez-Arán did not advise him or the Board that the cost of the land was actually only $480,000' — -in other words, that $932,177 would be used to make principal and interest payments on other Modules loans.

Lugo’s testimony is corroborated by the absence of this and other material information from the Board minutes. Although the government did not rely heavily on the absence of such information, we have found these omissions properly admissible under Federal Rule of Evidence 803(7) and note that they lend support to the prosecution’s case.

There was substantial evidence that Mu-ñoz-Franco and Sánchez-Arán withheld all of this information from the Board with full knowledge of the problems with Modules and other Gutiérrez companies. Am-abel Enriquez, who reported directly to Sánchez-Arán, explained that the handling of Gutiérrez certifications at Caguas concerned her “[b]ecause practically monthly, each time they invoiced!,] the certifications would be ahead of the construction.” She discussed her concerns with Muñoz-Fran-eo at least by the time La Marina was going to be sold to DO.W in 1984, and with Sánchez-Arán around the same time. She also discussed her concern with Muñoz-Franco that, following the sale, Modules would remain a contractor for La Marina, Levittown, and Country Club. However, Muñoz-Franco and Sánchez-Arán took no action in response to Enriquez’s concerns.

Finally, Muñoz-Franco and Sánchez-Arán took steps to conceal their misrepresentations from both internal and external auditors. Juan Hernández, Caguas’ internal auditor, testified that, in 1981, Kareh (the assistant vice president of the construction loan department) initially provided information in connection with an audit of the construction loan department, but, after a short meeting with Sánchez-Arán, Kareh stopped providing the information Hernández requested. Hernández further testified that, although he requested the auditing committee “several times” between 1980 and 1990 to allow him to audit the construction loan department, Muñoz-Franco and Sánchez-Arán did not allow the audit to take place. In a letter dated January 22, 1985, an external auditing firm, Stephen P. Bradics and Company, recommended “that the scope of internal auditing be expanded to include ... construction loans.” After Caguas received this letter, Hernández again recommended to the audit committee on several occasions that he should be allowed to audit the construction department, but he still was not allowed to do so. The audit committee minutes reflect his recommendation at several meetings at which Muñoz-Fran-co and Sánchez-Arán were present. At one meeting, on January 28, 1988, Muñoz-Franco responded by telling Hernández that “construction loans was the area most audited by senior management.” Sán-chez-Arán heard and agreed with Muñoz-Franco’s response, and Sánchez-Arán had also made similar statements to Hernán-dez in the past. However, Hernández was never provided with any internal audit conducted by either Muñoz-Franco or Sán-chez-Arán.

Muñoz-Franco and Sánchez-Arán also concealed information regarding Caguas’ lending practices from external authorities. Critically, Muñoz-Franco drafted and obtained Board approval to send the Denby letter, which states: “This Board of Directors wishes to state in no unclear and uncertain terms that it has never considered and much less approved any policy or practice of permitting borrowers to use construction loan proceeds to satisfy or make interest payments] on other unrelated loans.” At the time of the letter, however, Muñoz-Franco and Sánchez-Arán had, as discussed above, approved loans for such purposes on many occasions.

Taken in the light most favorable to the government, the evidence demonstrates that Muñoz-Franco and Sánchez-Arán repeatedly concealed material information regarding the status of the Gutiérrez loans from the Board, approved disbursements for work that was not completed, and prevented audits. Such conduct caused Caguas to continue lending money to Gutiérrez-owned companies despite the companies’ financial instability, which demonstrates appellants’ knowing scheme to defraud Caguas by means of material misrepresentations within the meaning of 18 U.S.C. § 1344. Consequently, we affirm the district court’s finding that a rational jury could have found Muñoz-Fran-co and Sánchez-Arán guilty of bank fraud.

2. Gutiérrez and Umpierre-Hernández

The record also contained considerable evidence from which a reasonable jury could have found beyond a reasonable doubt that Gutiérrez and Umpierre-Her-nández knowingly engaged in a scheme to defraud Caguas by misrepresenting material information. The certifications they submitted and signed contained many misrepresentations. For example, on the Los Mameyes project, between December 9, 1985 and January 10, 1986, Gutiérrez and Umpierre-Hernández submitted nine certifications attesting to the manufacture of 74 housing units and requesting payments totaling $380,000, even though the inspector’s report stated that only 19 houses had been built. Between January 16 and February 21, 1986, Gutiérrez and Umpierre-Hernández submitted 10 more certifications attesting to the manufacture of 90 additional housing units and requesting payment of approximately $500,000, at which point payment had been disbursed for at least 164 housing units even though only 40 units had been manufactured. As of February 25, 1986, Gutiérrez and Um-pierre-Hernández had certified the completion of 200 units, but an inspection report dated March 23, 1986, indicated that only 55 units had been built. On March 31, 1986, Gutiérrez and Umpierre-Hernán-dez submitted a certification requesting $69,000 for “payment of subcontractors” along with a list of interest payments they had made on other projects, including Jar-dines de Villa Alba, Levittown, Country Club, La Marina, and Los Caciques, which equalled exactly $69,000. Finally, on June 26, 1986, Gutiérrez and Umpierre-Hernán-dez submitted a “special certification” of $85,000 with no justification for work allegedly completed; Kareh testified that such a certification was not a usual practice. Sánchez-Arán authorized the disbursements based on these certifications.

Gutiérrez and Umpierre-Hernández submitted similar certifications for work not completed on other projects. For example, on the Jardines de Villa Alba project, Modules had installed only one unit as of the time of the Denby letter on January 10, 1988; however, on October 22, 1985 Gutiérrez and Umpierre-Hernández certified that twenty-two units had been completed, causing $626,000 to be disbursed to Modules.

Gutiérrez and Umpierre-Hernández perpetuated their scheme by recruiting other developers to obtain loans from Cag-uas on the condition that they use Modules as the contractor. This conduct allowed Gutiérrez and Umpierre-Hernández to obtain new funding for Modules, which they would then use to pay down prior loans and keep Modules from collapsing. Developer Burns, the original owner of the Cer-rovista project, was a typical example. After unsuccessfully applying to Caguas for a loan to build residential housing on land he owned, Burns met with Umpierre-Hernán-dez, who told him that if he used Modules as his contractor and signed a $2 million note he would be approved for a loan. A week later Burns received financing for the Cerrovista project. He testified that the approval was “[d]efinitely[ ] because of the use of Modules.” Burns met with Umpierre-Hernández on “many other occasions,” and, after problems arose with the Cerrovista project, with Umpierre-Hernández and Sánehez-Arán in Sánchez-Arán’s office. Subsequently, Umpierre-Hernández told Burns that the project would be sold to Iantho, and explained that “this project was going to contribute to paying off some interest for some of the projects that were in default,” which caused Burns to understand “that was the reason ... I was rejected as sponsor.”

On September 10, 1986, Sánchez-Arán presented, and the Board approved, a loan of $8.9 million to Iantho Corporation to take over the Cerrovista project; the loan agreement specified that Modules would be the contractor for the project’s two hundred units. Burns testified that, around the time the project was being sold, he received confirmation that the Department of Transportation’s plans to build a hospital and expressway would cause the development of four lots of the project to be “frozen.” When told the news, Um-pierre-Hernández responded, “hide it, boy.” On September 28, 1986, Cerrovista was sold to Iantho with no mention of the impending development.

Taken in the light most favorable to the government, the evidence shows that Gu-tiérrez and Umpierre-Hernández knowingly concealed information relevant to the viability of the Cerrovista project and submitted many certifications for work that was not completed. To sustain their schemes, they recruited other developers and helped them receive funding from Caguas on the condition that these developers use Modules as a contractor. These activities caused Caguas to continue lending money to Gutiérrez-owned companies despite the companies’ financial instability, providing an ample basis for a jury to find that appellants knowingly schemed to defraud Caguas by means of material misrepresentations within the meaning of 18 U.S.C. § 1344. Consequently, we conclude that a reasonable jury could have found beyond a reasonable doubt that both Gu-tiérrez and Umpierre-Hernández were guilty of bank fraud.

B. Conspiracy with Respect to the Gu-tiérrez Loans

The four appellants were charged with conspiring, in violation of 18 U.S.C. § 371, to commit bank fraud (18 U.S.C. § 1344), misapply bank funds (18 U.S.C. § 657), and make false entries in the books and records of the bank (18 U.S.C. § 1006). To establish a conspiracy, the government must prove beyond a reasonable doubt that (1) a conspiracy existed; (2) the defendant knew of and voluntarily participated in the conspiracy; and (3) there was an overt act in furtherance of the conspiracy. United States v. Blasini-Lluberas, 169 F.3d 57, 67 (1st Cir.1999). The government must prove both intent to agree and intent to commit the substantive offense. United States v. Rivera-Santiago, 872 F.2d 1073, 1079 (1st Cir.1989). A formal agreement is not required, Am. Tobacco Co. v. United States, 328 U.S. 781, 809, 66 S.Ct. 1125, 90 L.Ed. 1575 (1946); rather, “[t]he agreement may be shown by a concert of action, all the parties working together understandingly, with a single design for the accomplishment of a common purpose.” Am. Tobacco Co. v. United States, 147 F.2d 93, 107 (6th Cir.1944); see also Adver. Special Nat’l Ass’n v. FTC, 238 F.2d 108, 115 (1st Cir.1956). Where, as here, the indictment alleges a conspiracy to commit multiple offenses, the charge may be sustained by sufficient evidence of conspiracy to commit any one of the offenses. See Griffin v. United States, 502 U.S. 46, 56-57, 112 S.Ct. 466, 116 L.Ed.2d 371 (1991); see also United States v. Wedelstedt, 589 F.2d 339, 341-42 (8th Cir.1978)(“[P]roof that [defendant] agreed to commit one of the multiple illegal objectives of the conspiracy sufficed to sustain the conviction on that count.”).

On the verdict form, the jury found Mu-ñoz-Franco, Sánchezr-Arán, Gutiérrez, and Umpierre-Hernández guilty of conspiracy “as charged in the indictment.” Docket Nos. 1291, 1292, 1294, 1295. In considering appellants’ motion for judgment of acquittal, the district court found that there was more than enough evidence with respect to bank fraud to sustain the conspiracy conviction, and thus did not review the evidence concerning the misapplication and false entry crimes.

We agree with the district court’s assessment. First, there was substantial evidence of an agreement among the four men to defraud the bank. Sánchez-Arán directly supervised the Gutiérrez loans and worked closely with Muñoz-Franco on the loans. Enriquez testified that Muñoz-Franco and Sánchez-Arán always reached an agreement regarding the treatment of these loans. Gutiérrez and Umpierre-Hernández submitted many certifications for work not yet completed, and Sánchez-Arán then repeatedly approved disbursements for this work. Sánchez-Arán also facilitated Gutiérrez’s efforts to secure new borrowers by ensuring that a loan would be approved if Modules was used as a contractor. Sánchez-Arán and Umpierre-Hernández met frequently regarding the status of the Gutiérrez companies. Burns testified, for example, that he met with Umpierre-Hernández many times in Sán-chez-Arán’s office regarding the Cerrovis-ta project. Finally, Muñoz-Franco and Sánchez-Arán failed to disclose material information to the Board on many occasions, and this joint failure further demonstrates their collaboration in defrauding the bank.

Given the many instances of concealing the Gutiérrez companies’ performance from the Board of Directors, submitting certifications for work not completed, authorizing disbursements for such work and preventing audits, a reasonable jury could easily find the other two elements of conspiracy: that appellants knew of and participated voluntarily in the conspiracy, and that each committed overt acts in furtherance of the conspiracy. Viewed as a whole, this synchronized pattern of conduct demonstrates appellants’ agreement to maintain the appearance of financial viability of the Gutiérrez corporations by securing new loans to make payments on outstanding loans. Thus, we find that a reasonablé jury could have found each element of conspiracy beyond a reasonable doubt with respect to each of the four appellants.

C. Misapplication of Bank Funds

The jury found all four appellants guilty of misapplication of bank funds under 18 U.S.C. § 657 for a single transaction related to the Cerrovista project. The crime of misapplication eludes easy definition. As we have previously noted, “[t]he problem that has confronted and perplexed the courts is that there is no statutory definition or common law heritage that gives content to the phrase ‘willfully misapplies.’ ” United States v. Wester, 90 F.3d 592, 595 (1st Cir.1996). However, we have held that misapplication has two key elements: (1) wrongful use of bank funds; and (2) intent to injure or defraud a bank. Blasini-Lluberas, 169 F.3d at 63. We have also explained that “the same facts can easily be the basis for deeming the conduct to be wrongful and the intent fraudulent; both misapplication and scien-ter are required.” Wester, 90 F.3d at 595. Finally, the misapplication statute applies only to officers, agents, and employees of a bank. Thus, appellants who do not hold one of these positions — here Gutiérrez and Umpierre-Hernández — may not be convicted as principals for misapplication, but may be convicted as aiders and abettors. See Giragosian v. United States, 349 F.2d 166, 167 (1st Cir.1965)(“Since [defendant] was not an officer, director, agent or employee of the bank, he could not be guilty as a principal ... for misapplication of the bank’s funds, but only as an aider and abettor.”).

A reasonable jury could have concluded that appellants’ handling of the Cerrovista loan constituted misapplication. The original loan offering, which was prepared according to Sánchez-Arán’s instructions, designated $1,412,177 for land costs and $855,323 for “partial assumption of other loans.” This document did not disclose the purpose for which the funds were actually used. Of the $1,412,177 designated for land, only $480,000 was used to purchase land for the project. The remaining $932,177 was disbursed to the Gutiérrez-owned Quintas de Humacao company and immediately applied to other Gutiérrez projects: principal and interest on the Quintas de Fajardo loan, interest on the Las Gaviotas loan, and interest on a Modules commercial loan.

The documents associated with the Cer-rovista loan demonstrate appellants’ intent to defraud Caguas by redirecting these funds. Although the settlement statement for the Cerrovista loan, dated September 29, 1986, lists a disbursement of $932,177 for “repayment of other loan” and lists the payee as “Quintas de Humacao Inc. & Caguas Federal Savings,” these disclosures appear in a markedly different type, as does the total loan amount. Ka-reh testified that this different type indicates that “the entry was made later or at another place,” allowing an inference that the document was amended to make it appear that the Board had approved the use of loan proceeds to make payments on other loans. This inference is bolstered by the fact that the loan settlement statement was not prepared until after the Board had approved a loan. Kareh also testified that, although the $855,323 amount designated for “partial assumption of other loans” was listed on the loan offering and disbursed on the date of closing, the $855,323 amount was omitted from the settlement statement at Sánchez-Arán’s instruction. This omission concealed the fact that two separate disbursements, in the amounts of $932,177 and $855,323, were used to pay down other loans.

Each appellant was sufficiently involved in concealing the use of funds to allow an inference of intent to defraud. Sánchez-Arán drafted the original loan offering, which did not disclose that the $932,177 difference between the amount allocated for land purchase and the amount spent on land would be used to pay principal and interest on other loans. Moreover, according to Kareh, Sánchez-Arán explicitly instructed him to omit the $855,323 amount from the loan settlement document. A jury could infer that Muñoz-Franco helped to conceal this wrongful use of funds, thereby intentionally defrauding Caguas, based on his supervisory role on the Gu-tiérrez loans and his regular responsibility of conveying information to the Board. A jury also could infer that Gutiérrez and Umpierre-Hernández aided and abetted Muñoz-Franco and Sánchez-Arán in the transaction: Umpierre negotiated the transaction involving the sale of Cerrovis-ta, and the funds were used to pay down loans held by companies that Gutiérrez owned and supervised. Although appellants assert that the loan settlement documents demonstrate that the transaction was disclosed to the Board, the record indicates that the loan settlement statement was prepared after the loan had already been approved.

Appellants attempt to counter the charge by segregating the activities relating to the Cerrovista loan into two separate transactions: the disbursement of the $932,177 land purchase differential to Quintas de Humacao, and the use of those funds to pay down other Gutiérrez loans. They argue that, at the time the funds were disbursed to the Gutiérrez-owned Quintas de Humacao, they ceased to be “bank funds” under 18 U.S.C. § 657 and thus cannot support a misapplication violation.

We do not find this technicality pertinent. The disbursement of funds from the Cerrovista loan to Gutiérrez-owned companies, and the subsequent use of these loans to pay down other Gutiérrez loans, was a single unified transaction. The district court explained:

The funds did not cease to be controlled by the express purpose for which they were disbursed. The fact the $932,177 check was endorsed does not change the express purpose for which said funds were to be used. The second transfer, the Gutiérrez’s endorsement back to Caguas Central for the repayment of four separate loans, is still the use of bank funds and subject to misapplication.

By expressly designating these funds for “repayment of other loan[s],” Muñoz-Franco and Sánchez-Arán retained control over the funds even if the funds nominally changed hands. Moreover, Muñoz-Franco and Sánchez-Arán supervised the entire transaction, as demonstrated by four entries in Caguas’ ledgers showing the transfer of funds to the Gutiérrez accounts.

Critically, appellants also ignore the separate concealment of the $855,323, which, although it was listed on the original loan application, was left off of the loan settlement statement of September 29, 1986 at Sánchez-Ai'án’s explicit instruction. Although the disbursement of the $855,323 was disclosed to the Board, the fact that this disbursement was left off the loan settlement statement indicates the bank officials’ efforts to conceal the fact that two separate disbursements, in the amounts of $855,323 and $932,177, were used to pay down other loans. In short, there was more than sufficient evidence for a reasonable jury to conclude that appellants engaged in wrongful use of bank funds with the intent to defraud the bank.

D. Bank Fraud with Respect to Mi-randes Loans

The jury also found appellants Muñoz-Franco and Sánchez-Arán guilty of bank fraud in violation of 18 U.S.C. § 1344 with respect to the Mirandes loans. The Mi-randes scheme involved many of the same elements as the Gutiérrez loans. In September 1981, Muñoz-Franco asked Mi-randes to take over ownership of a project called Repai'to Valenciano, and Mirandes, then the contractor for the project, agreed. At the time the project was in debt to Caguas for $2.2 million, and delays and other problems continued after Mirandes assumed ownership. On an ongoing basis, Muñoz-Franco and Sánchez-Arán supervised the transfer of funds from other projects to pay down the debt on the Re-parto Valenciano project. For example, on September 30, 1986, approximately $2.2 million was applied to the Reparto Valenci-ano project from other Mirandes projects. Muñoz-Franco and Sánchez-Arán did not disclose these transfers of money among various Mirandes projects to the Board. After the Reparto Valenciano debt increased to $3.2 million, Mirandes received five contracts from the Puerto Rico Housing Department to build basic housing. Although Caguas initially denied Mi-randes’ application for funding for these projects, it subsequently agreed to finance the Villas de Gurabo project if Mirandes would agree to apply the profits of that project to the debt and interest of the Reparto Valenciano project. These transfers temporarily sustained the Mirandes projects, but the decreased funds available to successive projects ultimately made it impossible for those projects to be completed or to yield profits.

Both Muñoz-Franco and Sánchez-Arán were intimately involved with the administration of the Mirandes loans. Sánchez-Arán met with Mirandes on many occasions, including, during one seven or eight month period, meetings every fifteen days. Sánchez-Arán was involved in most of the disbursements for the Mirandes projects, whereas for other loans he was typically involved only if a problem arose. Muñoz-Franco was also involved in the Mirandes loans: he originally met with Mirandes to get him to take over the loans, met with Mirandes several times thereafter, was regularly informed of the loans’ progress, and reported on the loans’ status to the Board.

Beginning thirty days after the Reparto Valenciano loan agreement was signed, Sánchez-Arán authorized many certifications for completed work. With two exceptions, no work on the project had been completed at the time of the authorizations. On several occasions, a transfer of funds from one loan to another or an increase in the amount of a loan was authorized without the approval of the loan committee or the Board. Mirandes also testified that the disbursement schedule for loans “normally ... wasn’t complied with because money was taken out to pay interest.” Thus, the disbursement schedule that the Board approved was not the schedule that Muñoz-Franco and Sán-chez-Arán subsequently followed.

Enriquez testified such practices were not standard at Caguas; along with the Gutiérrez loans, the Mirandes loans were the only ones for which disbursements were authorized for work that was not completed. According to Enriquez, the Mirandes projects commonly received “special” certifications, which meant that “[w]hen one project didn’t have a line [of credit] it would be taken from another project.” When regular certifications were submitted, they would have an itemization of expenses attached; special certifications would not include such an itemization and thus would not disclose the destination of the funds.

Sánchez-Arán also caused Caguas to finance the sale of land from one Mirandes company to another in order to pay interest on the Reparto Valenciano loan. On September 11, 1984, Mirandes’ company Deproco purchased property for $60,000, and, three months later, sold the property to Bubao, another Mirandes company, for $94,000. The proceeds from the sale were used to pay interest on the Reparto Valen-ciano loan. The sale occurred despite the fact that, according to Mirandes, the land was not worth the purchase price and no improvements to the land were made in the interim. The minutes did not reflect that the Board was informed of the transaction.

After Mirandes was unable to improve the finances of the Reparto Valenciano project, he was awarded five projects from the Housing Department, including Villas de Gurabo, but was denied funding from Caguas. At that point, Mirandes met with Muñoz-Franco and Sánchez-Arán and asked to be released from the debt on the Reparto Valenciano project because he “felt that the debt was not my problem” and “needed to seek financing for the project from another bank.” Muñoz-Franco and Sánchez-Arán refused to release Mi-randes from the debt. Subsequently, however, Kareh informed Mirandes that Cag-uas would give him financing on Villas de Gurabo, but that the proceeds from that loan would have to be used to pay off the debt on the Reparto Valenciano project. Mirandes reluctantly agreed, and this plan was implemented. Again, the minutes did not reflect that the Board was informed of this transaction.

The Denby letter provides further support for the jury’s finding of fraud. Drafted and signed by Muñoz-Franco, it explicitly denies that Caguas’ Board approved the practice of borrowers using construction loan proceeds to pay interest on unrelated loans. At the time of the letter, however, Muñoz-Franco and Sánchez-Arán had on many occasions authorized the payment of debts on the Reparto Va-lenciano project with funds from other Mi-randes loans. The fact that the Denby letter was signed by all the members of the Board would allow a reasonable jury to conclude beyond a reasonable doubt that Muñoz-Franco misrepresented the practices associated with the Mirandes loans to the Board.

This circumstantial evidence of the requisite intent was bolstered by more explicit evidence of appellants’ knowledge. Mi-randes was concerned that, on the Reparto Valenciano project, “some projects were being emptied out in order to deal with others” and worried that “it was going to become paralyzed.” Mirandes testified that he and Sánchez-Arán “spoke constantly” about his concerns, and that he also expressed his concerns to Muñoz-Franco. Despite Mirandes’ repeated statements of concern, Sánchez-Arán explicitly told Mirandes that the practices had to continue. In many cases, the bank authorized the disbursements directly rather than seeking authorization from Mi-randes, and Mirandes testified that, although he knew of these disbursements, he was not in agreement with them. The evidence that appellants continued with their scheme even over Mirandes’ objections demonstrated that they acted deliberately to defraud the bank.

Taken in the light most favorable to the government, the evidence shows that Mu-ñoz-Franco and Sánchez-Arán repeatedly authorized disbursements for work that was not completed and concealed relevant information from the Board. As a result of these activities, Caguas continued lending money to Mirandes-owned companies despite them financial instability. This course of conduct constitutes a knowing scheme to defraud Caguas by means of material misrepresentations within the meaning of 18 U.S.C. § 1344. Consequently, we conclude that a reasonable jury could have found beyond a reasonable doubt that Muñoz-Franco and Sánchez-Arán were guilty of bank fraud with respect to the Mirandes loans.

E. Conspiracy with Respect to Mi-randes Loans

The jury also found Muñoz-Franco and Sánchez-Arán guilty of conspiracy with respect to the Mirandes loans. Although there was no formal agreement among Muñoz-Franco, Sánchezr-Arán and Mi-randes, the three acted in concert to defraud the bank. As described above, Muñoz-Franco originally approached Mi-randes to assume the Reparto Valenciano loans, and Mirandes agreed. After this initial agreement, Mirandes met frequently with Sánchez-Arán and on several occasions with Muñoz-Franco.

The appellants participated voluntarily in the conspiracy and took overt actions in its furtherance. When the Reparto Valen-ciano project was unable to pay its debts, Muñoz-Franco and Sánchez-Arán always attempted to find solutions to sustain the project, frequently by transferring funds from one project to another. Such transfers, as well as their approval of certifications for work not completed, constituted overt acts in furtherance of the conspiracy. Thus, we find sufficient evidence to sustain the jury’s verdict on charges of conspiracy to commit bank fraud in violation of 18 U.S.C. § 1344 with respect to the Mi-randes loans.

The district court held that the evidence of conspiracy to commit bank fraud was sufficient in itself to sustain the conspiracy charges, but also noted that there was sufficient evidence that appellants conspired to misapply bank funds in violation of 18 U.S.C. § 657 and make false entries in violation of 18 U.S.C. § 1006. However, because we agree with that court’s assessment of the bank fraud conspiracy, we need not discuss the other two bases for the conspiracy conviction.

IV.

Having found sufficient evidence to support the convictions, we must now address three of appellants’ claims that challenge the validity of the proceedings. First, appellants claim that the charges filed against them were barred by the statute of limitations. Second, they argue that their convictions violated the Ex Post Facto Clause. Finally, they contend that their convictions were invalid because of pre-indictment and pre-trial delay.

A. Statute of Limitations

Under 18 U.S.C. § 3293, an individual may not be charged with bank fraud or conspiracy to commit bank fraud unless the indictment is returned within ten years after the commission of the offense. Appellants raise three arguments regarding this statute of limitations. First, they argue that the original indictment was untimely. Next, they argue that the First, Second, and Third Superseding Indictments substantially amended the original indictment, do not relate back, and consequently were untimely. Finally, they argue that the district court erred in failing to instruct the jury on the statute of limitations.

1. Timeliness of Original Indictment

We review de novo a district court’s decision not to dismiss on statute of limitations grounds. López-González v. Mun. of Comerío, 404 F.3d 548, 551 (1st Cir.2005). Here, the original indictment was returned on November 22, 1995. Appellants contend that the Board was fully informed of the practices at issue in two of the alleged violations — bank fraud and conspiracy on the Gutiérrez loans — by a regulator report dated August 31, 1985. Consequently, they argue, any deception in the scheme ended more than ten years prior to the original indictment, making the indictment untimely for those charges.

In light of the activities discussed in Section III, supra, most of which took place after November 22, 1985, appellants’ argument overstates the significance of the regulator report in question. The report describes several Modules loans and expresses general concern about Modules’ financial state. The report notes that the loans “are considered to be substandard loans subject to special comment since the future viability of the main borrower, Modules, could be impaired by uncertainties involving the collectibility of an accounts receivable due from an affiliated party.” These general statements do nothing to inform the Board of such practices as the use of proceeds from one loan to make payments on another loan or the ongoing certification of and payment for work that was not completed. Moreover, the report itself contains certain misleading statements by Sánchez-Arán. For example, despite his knowledge of Modules’ uncertain financial status and poor performance on several projects, he defended the loans to Modules by explaining that “the borrower should have sufficient funds to satisfy the loans in a timely manner.” Thus, the district court did not err in ruling that the original indictment was timely.

2. Relation Back of Superseding Indictments

Appellants next argue that each of the three superseding indictments materially broadened and substantially amended the charges against them and, consequently, do not relate back to the date of the original indictment. As a result, they argue that these indictments are untimely because they allege conduct that concluded more than ten years previously.

In United States v. O’Bryant, 998 F.2d 21, 23 (1st Cir.1993), we held that “a superseding indictment which supplants a timely-filed indictment ... is itself to be regarded as timely ... so long as it neither materially broadens nor substantially amends the charges against the defendant.” Consequently, “the superseding indictment relates back to the filing date of the original indictment so long as a strong chain of continuity links the earlier and later charges.” Id. at 24. We emphasized that “notice-related concerns ... comprise the touchstone for determining when a superseding indictment materially broadens or substantially amends earlier charges,” explaining that a timely indictment serves notice by apprising defendants “ ‘that they will be called to account for their activities and should prepare a defense.’ ” Id. (quoting United States v. Grady, 544 F.2d 598, 601 (2d Cir.1976)).

In this case, the original indictment charged the four appellants with bank fraud and conspiracy, alleging fifty-eight overt acts spanning nearly a decade and involving unlawful disbursement of bank funds by Muñoz-Franco and Sánchez-Arán, unlawful receipt of the funds by Gutiérrez, Umpierre-Hernández, and Mi-randes, and concealment of these activities from Caguas’ Board of directors and regulatory agencies. The First Superseding Indictment, returned on March 5, 1997, added allegations concerning five additional loan projects and forty-four overt acts and added two additional defendants. It also changed the alleged starting date of the conspiracy from December 1981 to June 1980, lengthening the duration of the conspiracy from eight years and five months to nine years and eleven months. The Second Superseding Indictment, returned on May 13, 1997, added allegations about two more loan projects and ten overt acts. Finally, the Third Superseding Indictment, returned on March 6, 1998, separated the allegations relating to the Gutiérrez and Mirandes loans, which the previous indictments had presented as a single unitary conspiracy, into two separate conspiracies set forth in separate counts.

We conclude that these revisions do not materially broaden or substantially amend the original indictment. The government “is not limited in its proof at trial to those overt acts alleged in the indictment.” United States v. Adamo, 534 F.2d 31, 38 (3d Cir.1976). Therefore, allegations of additional loan projects and overt acts in the superseding indictments do not broaden the original indictment; they simply provide more specific examples to substantiate the original fraud and conspiracy charges.

Similarly, the addition of eighteen months to the beginning of a conspiracy spanning nearly a decade does not materially broaden the scope of the indictment. Virtually none of the conduct supporting appellants’ convictions occurred between June 1980 and December 1981, and the dates were expanded primarily to allow factual completeness by including the dates of the original loans Caguas extended to various companies. Thus, appellants had adequate notice of the charges against them despite the expanded time frame.

Finally, we find that the separation of the original unitary bank fraud and conspiracy scheme into two separate schemes, one alleging bank fraud and conspiracy with respect to the Gutiérrez loans and one alleging bank fraud and conspiracy with respect to the Mirandes loans, did not materially broaden or substantially amend the indictment. Muñoz-Franco and Sán-ehez-Arán still had to explain the same set of actions. With respect to Gutiérrez and Umpierre-Hernández, the division also provided adequate notice. The only difference was that they no longer had to contend with the allegations that related only to the Mirandes loans. Thus, we conclude that the Third Superseding Indictment relates back to November 22, 1995, the date of the original indictment, and find no error in the district court’s refusal to dismiss the indictment for untimeliness.

3. Jury Instruction on Statute of Limitations

Appellants also argue that the district court erred in failing to instruct the jury on the statute of limitations on the ground that the jury could have relied on conduct outside the limitations period in finding defendants guilty. At trial, however, Gu-tiérrez’s attorney explicitly declined to seek a statute of limitations instruction, and the other appellants made no request for such an instruction.

We addressed a similar situation in United States v. Walsh, 928 F.2d 7, 11-12 (1st Cir.1991). There, we rejected the defendant’s claim that the district court should have issued an instruction on the statute of limitations, explaining that the defendant “failed to identify any point in the record where such an instruction was requested. Absent such a request, the point was waived.” Id. at 12. This holding is logical because “[a] statute of limitations claim presents ‘an affirmative defense’ that is ‘not cognizable on appeal unless properly raised below.’ ” United States v. Spero, 331 F.3d 57, 60 n. 2 (2d Cir.2003)(quoting United States v. Walsh, 700 F.2d 846, 855-56 (2d Cir.1983)). The situation here is identical. We hold that because appellants failed to request an instruction, the district court had no obligation to instruct sua sponte on the limitations issue.

B. Ex Post Facto

Appellants claim that their convictions for bank fraud and conspiracy violated the Ex Post Facto Clause, U.S. Const, art. I, § 9, cl. 3, because the bank fraud statute, 18 U.S.C. § 1344, was not enacted until October 12, 1984, and the jury could have convicted them entirely on the basis of conduct that occurred prior to that date.

The Ex Post Facto Clause “ ‘forbids the application of any law or rule that increases punishment for pre-existing conduct.’ ” United States v. Regan, 989 F.2d 44, 48 (1st Cir.1993)(quoting United States v. Havener, 905 F.2d 3, 5 (1st Cir.1990)). For “continuing offenses” such as the bank fraud and conspiracy charges at issue here, however, the critical question is when the conduct ended. As we have explained, “[w]here a ‘continuing offense’ straddles the old and new law ... applying the new is recognized as constitutionally sound.” Id. In other words, “[a] conviction for a continuing offense straddling enactment of a statute will not run afoul of the Ex Post Facto clause unless it was possible for the jury, following the court’s instructions, to convict ‘exclusively’ on pre-enactment conduct.” United States v. Monaco, 194 F.3d 381, 386 (2d Cir.1999) (citation omitted). As suggested, “the question of whether the violation extended beyond the effective date of [the statute is] one that ha[s] to be resolved by the jury.” United States v. Tykarsky, 446 F.3d 458, 480 (3d Cir.2006).

Appellants contend that the district court should have issued instructions informing the jury that it must find beyond a reasonable doubt that the bank fraud and the conspiracy to commit such fraud continued past October 12, 1984. In the absence of such an instruction, they argue, the jury’s verdict was ambiguous because it could have been premised exclusively on conduct that occurred prior to the date of enactment. Under such circumstances, a conviction either for bank fraud or conspiracy to commit bank fraud would violate the Ex Post Facto Clause.

Appellants failed to raise the possibility of an ex post facto violation at any time prior to or during trial, including in their proposed jury instructions. They also failed to raise the issue in their post-trial Rule 29 motions. In their pre-sentencing briefs, they argued for the first time that their convictions should be vacated on ex post facto grounds. The district court rejected the claim.

An unpreserved ex post facto claim is subject to plain error review. See, e.g., Tykarsky, 446 F.3d at 479; United States v. Julian, 427 F.3d 471, 481 (7th Cir.2005). Under such review, an appellant must show: “(1) that an error occurred (2) which was clear or obvious and which not only (3) affected the defendant’s substantial rights, but also (4) seriously impaired the fairness, integrity, or public reputation of judicial proceedings.” Unit ed States v. Duarte, 246 F.3d 56, 60 (1st Cir.2001).

In its brief, the government agrees that the district court should have issued an instruction that the jury must find that the conduct continued past the enactment date of the bank fraud statute. Its position is consistent with recent circuit court decisions holding that the failure to issue such an instruction satisfies the first two prongs of the plain error analysis by establishing that an error occurred that was obvious. See Tykarsky, 446 F.3d at 479-80 (“[B]ecause the communications spanned two different versions of the statute with different minimum penalties, the question of whether the violation extended beyond the effective date of the amended version was one that had to be resolved by the jury.... We also conclude that the error in not requiring a special jury finding was ‘plain’ in that it was an obvious mistake in retrospect.”); Julian, 427 F.3d at 482 (“This was a plain error in the sense of being an obvious mistake in retrospect.”). Thus, our task is to determine whether this error affected appellants’ substantial rights, and, if so, whether the fairness, integrity or public reputation of the judicial proceedings was impaired.

The other circuits have taken varying approaches to applying these prongs of the plain error test in assessing a claimed ex post facto violation. In United States v. Calabrese, 825 F.2d 1342, 1346 (9th Cir.1987), the Ninth Circuit held that the “jury’s verdict represents a finding that a crime was committed as alleged in the indictment.” Because the indictment alleged an offense that continued past the date of enactment, the court concluded that there was no plain error in the court’s failure to issue an instruction. Id.

The Fifth Circuit has examined the record to see whether the bulk of the evidence occurred after the enactment of the statute. In United States v. Todd, 735 F.2d 146 (5th Cir.1984), that court held that where “[m]ost of the evidence focused on events that occurred within the effective date of the amendments,” and “the record ... clearly establishes violations of the amended act by the appellants during the relevant time period,” the failure to give the jury instruction did not “result in the likelihood of a grave miscarriage of justice or circumstances that seriously affect the fairness, integrity, or public reputation of judicial proceedings.” Id. at 150.

The Second and Third Circuits have adopted an approach in which prejudice is established if there was a reasonable possibility that the jury convicted appellant on the basis of pre-enactment conduct. In Tykarsky, for example, the Third Circuit explained that a district court’s failure to issue an instruction means that a defendant “has been prejudiced if there is a reasonable possibility that a jury, properly instructed on this point, might have found [the defendant] guilty based exclusively on acts that occurred before the increased penalty took effect.” 446 F.3d at 480. The court noted that “[t]he most that can be said here is that it is improbable, rather than impossible, as a factual matter, that the jury convicted Tykarsky exclusively on the basis of’ pre-enactment conduct, which was “insufficient to persuade us that [the defendant’s] substantial rights were not affected.” Id. at 482. The court concluded that this prejudice “tainted the integrity and reputation of the judicial process” and consequently vacated the defendant’s sentence and remanded to the district court. Id. at 483.

Finally, in Julian, the Seventh Circuit considered an alleged conspiracy and found, on the third prong of the plain error analysis, that a court’s failure to issue an ex post facto instruction “affected [the defendant’s] substantial rights in the sense that it exposed him to a longer sentence.” 427 F.3d at 482. It then focused on the fourth prong, explaining that “[i]f a jury, properly instructed on this point, might have found that the conspiracy had come to an end before the increased penalty took effect or that [defendant] had withdrawn from the conspiracy before that date, then the error is one that implicates the fairness, integrity, or public reputation of the judicial process.” Id. It concluded that a reasonable jury could not have found that the conspiracy at issue ended prior to the enactment date of the statute. Id. at 483.

Ultimately, we need not settle on a rule here. The plain error analysis used by the Second and Third Circuits is most favorable to defendants because it requires a finding of prejudice if there was a “reasonable possibility” that the jury convicted appellants solely on the basis of pre-enactment conduct. Here, even under that test, appellants cannot demonstrate an ex post facto violation because no reasonable jury would have convicted appellants based exclusively on conduct that occurred prior to the enactment date. Moreover, even if we were to apply the “miscarriage of justice” approach of the Seventh Circuit, we would conclude there was no miscarriage of justice here because the government presented so much evidence of the involvement of the defendants in the ongoing conspiracy and bank fraud following the enactment of the bank fraud statute in 1984.

As the district court correctly noted, the government presented overwhelming evidence of appellants’ conduct, the majority of which occurred after October 12, 1984. Relatively few of the overt acts for any of the charges alleged in the indictment occurred prior to October 12, 1984: only twelve of fifty-eight overt acts on the charges of conspiracy and bank fraud with respect to the Gutiérrez loans, and only ten of forty-seven acts on the charges of conspiracy and bank fraud with respect to the Mirandes loans. The government presented considerable evidence of activities occurring after October 12, 1984, including all of the transactions involving the Los Mameyes project, all of the transactions involving the Jardines de Villa Alba project, and many of the transactions involving the Reparto Valenciano project. The Denby letter, which contained several material misrepresentations, was not signed until 1987. Indeed, the bulk of our sufficiency analysis details conduct occurring after the enactment date. See Section III, supra.

Moreover, we find nothing to differentiate appellants’ pre-enactment conduct from subsequent conduct. The government’s key witnesses — Lugo, Mirandes, Enriquez, Somohano and Kareh — testified to conduct that occurred both before and after the enactment date. Given that the testimony of these witnesses spanned the enactment date, we find it implausible that the jury would find such testimony compelling only for events that occurred prior to October 12, 1984. Similarly, no transfor-mative event occurred prior to October 12, 1984 that would alter a reasonable jury’s perception of the nature of any appellant’s involvement. Although appellant Gutiér-rez protests that he sold his construction loans with Caguas on October 29, 1984, and sold Modules on December 31, 1985, it is uncontested that he and Umpierre-Her-nández remained employed by the company, and the record contains evidence that their culpable conduct (such as submitting certifications for incomplete work) continued uninterrupted after the sale of the company.

Because we see no reason to differentiate the conduct occurring prior to the enactment date, we find that a reasonable jury would not have convicted the appellants based solely on pre-enactment conduct. Thus, appellants were not prejudiced, nor did their convictions implicate the fairness or integrity of the judicial process. We conclude that there is no plain error from the absence of a limiting instruction on the ex post facto issue.

C. Pre-Indictment and Pre-Trial Delay

Appellants argue that the district court erred in refusing to dismiss on the grounds that (1) the five-year pre-indictment delay violated their Fifth Amendment right to due process, and (2) the five-year pre-trial delay violated their Sixth Amendment right to a speedy trial. We review both claims for abuse of discretion. United States v. Picciandra, 788 F.2d 39, 43 (1st Cir.1986)(“[T]he district court did not abuse its discretion in holding that the pre-indictment delays did not violate the defendants’ fifth amendment rights of due process.”); United States v. Salimonu, 182 F.3d 63, 69 (1st Cir.1999)(“This circuit reviews a district court’s ruling on a Sixth Amendment speedy trial claim for abuse of discretion.”).

1. Pre-Indictment Delay

In United States v. Lovasco, 431 U.S. 783, 97 S.Ct. 2044, 52 L.Ed.2d 752 (1977), the Supreme Court noted that “statutes of limitations, which provide predictable, legislatively enacted limits on prosecutorial delay, provide the primary guarantee! ] against bringing overly stale criminal charges.” Id. at 789, 97 S.Ct. 2044 (citations and internal quotation marks omitted). However, the Court also acknowledged that “the Due Process Clause has a limited role to play in protecting against oppressive delay.” Id. A prosecutor is not obliged to file charges as soon as probable cause exists, id. at 791, 97 S.Ct. 2044, and has “wide discretion” in delaying indictment “limited only by the requirement that it not violate those ‘fundamental conceptions of justice which lie at the base of our civil and political institutions,’ ” United States v. Ciampaglia, 628 F.2d 632, 639 (1st Cir.1980)(quoting Lovasco, 431 U.S. at 790, 97 S.Ct. 2044). To show a violation of “those ‘fundamental conceptions of justice’ a defendant must prove that (1) pre-indictment delay caused substantial prejudice to his right to a fair trial and, (2) the Government intentionally delayed indictment in order to gain a tactical advantage over the accused.” Picciandra, 788 F.2d at 42 (citation omitted). With respect to prejudice, a defendant must do more than allege that witnesses’ memories had faded or that evidence had been lost that might have been helpful to him. United States v. Lieberman, 608 F.2d 889, 902 (1st Cir.1979).

Appellants have failed to meet that burden here. They assert that they suffered prejudice from the unavailability of nineteen potential defense witnesses. However, we find that the unavailability of these nineteen individuals did not impair appellants’ defense, either because appellants could have, but did not, obtain testimony from similarly situated individuals who were available to testify; or because appellants failed to demonstrate that the testimony of these individuals would have included exculpatory evidence.

Of the nineteen unavailable witnesses, appellants contend that the deaths of two individuals — Phillip Saffold and Luis Dor-ta — prevented cross-examination relating to reports these -witnesses prepared. Saf-fold was an investigator who authored reports that appellants claim would have allowed them to impeach Roman Cruz, a plant manager who testified that buildings were not built at the Modules plant; however, Cruz’s testimony was a minor part of the government’s case against appellants and appellants’ inability to impeach Cruz did not significantly prejudice them. Similarly, Dorta inspected the Transglobe and Modules sites and prepared reports that were admitted at trial; appellants emphasize that his death prevented them from cross-examining him, but do not specify how this opportunity would have helped their case.

Appellants claim that nine other unavailable witnesses (Gilberto Pol Segarra, Luis Acevedo Gonzalez, Jaime Sitiriche, Juan Hernández, Celestino Seneriz, Pedro Bull Nater, Francisco Susoni Lens, Jose Felici-ano, and Rafael Toro Nazario) were members of the Board who “could have testified to the extent of the disclosures made to the Board, in response to Lugo[ ]’s imprecise testimony.” However, appellants offer no reason to believe that these Board members would have offered testimony contradictory to Lugo’s. Moreover, appellants do not explain why they did not call as witnesses other Board members who were available to testify throughout the trial.

Finally, appellants contend that the remaining eight unavailable witnesses (Gilberto Mayo Aguayo, Nelson Soto Velasquez, Stephen P. Radies, Emil de Pentima, Troy Chapman, Stanley Orenstein, Hugo Lopez, and Fernando Rivera) “could have testified” or “could have confirmed” information relating to financial practices at Caguas, without offering any evidence that these individuals would, in fact, have been able to provide such testimony.

Similarly, appellants protest the unavailability of three sets of documents: inspection reports, commercial loan files, and miscellaneous Modules documents. With respect to the first two sets of documents, however, appellants make no attempt to establish that the reports disappeared after indictment but prior to trial. With respect to the third, appellants claim only that “the unavailability of the Modules documents made it impossible to fairly cross examine [Transhore and Modules accountant Jorge] Fabrigas” and that, generally, the evidence might have allowed them to refresh witness recollection and corroborate their defense. However, Fabrigas’ testimony was a minor part of the trial, and these generalized objections do not establish prejudice resulting from an impaired defense.

Even if appellants had demonstrated some degree of prejudice from the delay, they have made no concomitant showing that the government intentionally delayed indictment to gain tactical advantage. Thus, we conclude that the district court did not abuse its discretion in denying their motion to dismiss for pre-indictment delay.

2. Pre-Trial Delay

The Sixth Amendment right to a speedy and public trial attaches upon arrest or indictment, whichever occurs first. United States v. MacDonald, 456 U.S. 1, 6-7, 102 S.Ct. 1497, 71 L.Ed.2d 696 (1982). Thus, the time frame relevant to appellants’ claim extends from the date of indictment, November 22, 1995, until the date of trial, January 29, 2001 — a span of over five years. We note that appellants do not raise a claim under the Speedy Trial Act, 18 U.S.C. § 3161, and thus their claim is based exclusively on constitutional rather than statutory grounds.

In Barker v. Wingo, 407 U.S. 514, 92 S.Ct. 2182, 33 L.Ed.2d 101 (1972), the Supreme Court established a four-part balancing test to determine whether a defendant’s Sixth Amendment right to a speedy trial has been violated. Under Barker, a court should consider: (1) the length of the delay; (2) the reasons for the delay; (3) the defendant’s assertion of his speedy trial right; and (4) the prejudice to the defendant caused by the delay. Id. at 530, 92 S.Ct. 2182. However, the Court identified the first factor, the length of the delay, as “to some extent a triggering mechanism. Until there is some delay which is presumptively prejudicial, there is no necessity for inquiry into the other factors that go into the balance.” Id. The Court also has indicated that post-accusation delay approaching one year is presumptively prejudicial. Doggett v. United States, 505 U.S. 647, 652 n. 1, 112 S.Ct. 2686, 120 L.Ed.2d 520 (1992). Consequently, the five-and-a-half year delay in this case creates a presumption of prejudice.

Under Barker, we must then consider the remaining three factors. We have described the second factor, the reasons for the delay, as “ ‘the focal inquiry.’ ” See, e.g., United States v. Santiago-Becerril, 130 F.3d 11, 22 (1st Cir.1997) (citation omitted). Here, there are two main reasons for the pre-trial delay. First, the appellants filed numerous requests for continuances, some of which the district court granted, as well as many motions requiring time and consideration by the district court. Second, two days before the trial was originally scheduled to begin on May 26, 1998, the government filed a motion to disqualify the trial judge assigned to the case on the ground that she had a banking relationship with Caguas that might prejudice her in the trial of the case. The district court originally denied the motion to disqualify and the government sought mandamus from this court. We denied mandamus, but noted that “the judge would have been well-advised either to bow out of the case or to ask that the recusal motion be assigned to a different judge for hearing” and that “the government hardly can be faulted for bringing” the motion. In re United States, 158 F.3d 26, 33-34 (1st Cir.1998). The trial judge assigned to the case then recused herself voluntarily, resulting in a delay in the proceedings of approximately one year. In sum, the pre-trial delay arose largely from the complexity of the case and the events related to the recusal.

The third Barker factor weighs in favor of appellants. They raised their speedy trial objection three times during the proceedings below, and now renew that objection.

The Supreme Court has held that the final Barker factor — prejudice to appellants — “should be assessed in the light of the interests of defendants which the speedy trial right was designed to protect.” Barker, 407 U.S. at 532, 92 S.Ct. 2182. The Court identified three such interests: “(i) to protect oppressive pretrial incarceration; (ii) to minimize anxiety and concern of the accused; and (iii) to limit the possibility that the defense will be impaired.” Id. The first of these interests has no bearing here because appellants remained free prior to trial. With respect to the second interest, “considerable anxiety normally attends the initiation and pendency of criminal charges; hence only ‘undue pressures’ are considered.” Santiago-Becerril, 130 F.3d at 23 (citations omitted). Appellants allege no such “undue pressure,” and therefore we assign negligible weight to this interest. Finally, in our discussion of pre-indictment delay, we have already analyzed appellants’ arguments that their defense was impaired. These arguments are equally applicable to the pre-trial time frame, and we find no material prejudice resulting from the delay.

The five years that elapsed between indictment and trial is a troublesome length of time. Nonetheless, our inquiry has revealed no constitutional violation. Thus, after considering the factors listed in Barker, we conclude that the district court did not abuse its discretion in denying appellants’ motion to dismiss for pre-trial delay.

V.

Appellants also raise several challenges to evidence they contend was improperly admitted. We address these claims in turn.

A. Evidence Regarding Loss

Appellants challenge three aspects of the admission and treatment of evidence regarding financial loss to Caguas. First, they protest that such evidence was irrelevant and prejudicial in violation of Federal Rule of Evidence 403. Second, they argue more specifically that the government’s Exhibit 40, aspects of which were later shown to be inaccurate, was erroneously admitted. Finally, they argue that, even if evidence of loss and Exhibit 40 were properly admitted, the district court erred in refusing to guide the jury with an instruction about the enactment of the Financial Institutions Reform, Recovery, and Enforcement Act of 1989 (“FIRREA”), Pub.L. No. 101-73, 103 Stat. 183 (codified in scattered sections of 12 U.S.C.).

1. General Evidence of Loss

We review evidentiary rulings on relevance and unfair prejudice for abuse of discretion. United States v. Richardson, 421 F.3d 17, 37-38 (1st Cir.2005). Loss is not an element of bank fraud. United States v. Blasini-Lluberas, 169 F.3d 57, 65 (1st Cir.1999). However, courts have held repeatedly that loss is relevant in fraud cases to demonstrate a defendant’s knowledge or intent to commit fraud. See, e.g., United States v. Heimann, 705 F.2d 662, 669 (2d Cir.1983)(“While technically the success or failure of a scheme to defraud is irrelevant in a mail fraud case, realistically, when the contested issue is intent, whether or not victims lost money can be a substantial factor in a jury’s determination of guilt or innocence.” (citation omitted)). Thus, while “an ultimate purpose of either causing some financial loss to another or bringing about some financial gain to oneself is not the essence of fraudulent intent,” United States v. Kenrick, 221 F.3d 19, 29 (1st Cir.2000) (citation and internal quotation marks omitted), the knowledge that one’s actions are, in fact, bringing about such losses may demonstrate one’s intent to commit fraud.

In this case, the government referred to loss throughout the trial to demonstrate appellants’ knowledge of the consequences of their ongoing practices of using loan proceeds to make principal and interest payments on unrelated loans, authorizing disbursements for work not completed, and using funds for purposes not authorized by the Board. For example, during its opening statement and closing argument, the government noted that federal regulators closed Caguas in 1990 due to the bank’s lack of funds. On a few occasions, the government also questioned witnesses about the amount of loss that certain projects sustained and whether those losses would have caused concern. However, these references did not dominate the evidence because the government also presented considerable other evidence of defendants’ conduct, as we have discussed at length in Section III, supra. Moreover, the district court carefully managed the effect of evidence relating to loss on the jury by preventing both parties from addressing loss in their closing arguments and instructing the jury that loss was not an element of the offenses charged. Finally, the court permitted appellants to cross-examine vigorously the government witnesses who discussed loss.

In sum, the general references to loss were relevant as a means of demonstrating appellants’ intent to defraud Caguas, and, given the carefully limited presentation of this evidence to the jury, were not unduly prejudicial. The district court did not abuse its discretion in admitting this evidence.

2. Exhibit 40

Following Caguas’ closure in 1990, Banco Santander acquired many of its assets, and prepared a document — government Exhibit 40 — listing hundreds of Cag-uas’ loans, the outstanding balances on those loans, Banco Santander’s valuation of the loans, and the resulting discount (the difference between the outstanding balance of the loan and the loan’s value). At trial, Banco Santander’s comptroller attested that the bank had acquired all of the loans summarized in Exhibit 40, and the exhibit served as part of the basis for expert testimony regarding accounting practices by Kathy McKinless, a partner at an accounting firm. By the time of sentencing, however, the government discovered that the Resolution Trust Company (“RTC”), a government-owned asset management company, had retained many of the loans. Thus, the exhibit could not actually have reflected valuations of the loans made by Banco Santander upon purchase, and was not a reliable means of establishing the magnitude of the loss experienced by Caguas.

Appellants now contend that Exhibit 40 was both testimonial, in violation of the Confrontation Clause, and that the exhibit was improperly admitted under the Federal Rules of Evidence. The government’s acknowledgment that Exhibit 40 contained inaccurate information indicates that the exhibit should have been excluded on that basis alone. However, we agree with the district court that any error in its admission was harmless. Under Fahy v. Connecticut, 375 U.S. 85, 86-87, 84 S.Ct. 229, 11 L.Ed.2d 171 (1963), the critical question in assessing harmless error is “whether there is a reasonable possibility that the evidence complained of might have contributed to the conviction.”

Several circumstances indicate that there is no reasonable possibility that Exhibit 40 contributed to the convictions here. First, the exhibit was relevant only to prove appellants’ fraudulent intent, and the government presented considerable evidence of this intent from other sources. Second, the court explicitly instructed the jury that loss was not an element of bank fraud. Third, the exhibit was the subject of direct examination for less than one-half of one day of a trial spanning fifteen months. Fourth, defense counsel subjected McKinless to searching cross-examination regarding the exhibit, revealing several typographical errors and inaccuracies and eliciting McKinless’ acknowledgment that she could not confirm the document’s completeness. Finally, even if the loss calculations in Exhibit 40 were not entirely accurate, other evidence at trial demonstrated that Caguas suffered large losses on many of the loans at issue. To some extent, the Exhibit 40 calculations replicated evidence already in the record. Taking into account all of these circumstances, there is no reasonable possibility that the isolated use of Exhibit 40 could have contributed to appellants’ convictions, and thus any error in its admission was harmless.

3. Jury Instruction on Loss

Appellants also challenge the district court’s refusal to give an instruction explaining that, when FIRREA was enacted in 1989, “many institutions immediately fell out of compliance with regulatory capital requirements, making them subject to seizure by thrift regulators.” United States v. Winstar Corp., 518 U.S. 839, 857-58, 116 S.Ct. 2432, 135 L.Ed.2d 964 (1996). Appellants contend that, given the admission of evidence relating to financial loss, the jury should have been instructed that bank failures were common in the aftermath of FIRREA’s enactment. In other words, the jury should have been told that banks commonly lost money in the absence of fraud.

A district court’s refusal to issue a jury instruction “ ‘constitutes reversible error only if the requested instruction was (1) correct as a matter of substantive law, (2) not substantially incorporated into the charge as rendered, and (3) integral to an important point in the case.’ ” White v. N.H. Dep’t of Corrs., 221 F.3d 254, 263-64 (1st Cir.2000) (citation omitted). No such error occurred here. The FIRREA instruction was not integral to an important point in the case because loss is not an element of bank fraud. Moreover, the jury was instructed that the government need not prove loss. Thus, we find no reversible error in the district court’s decision.

B. Forgery and Check-Kiting Evidence

Appellants assert that the admission of evidence regarding forged signatures on checks and evidence of a check-kiting scheme constructively amended the indictment. A constructive amendment occurs “ ‘when the charging terms of the indictment are altered’ at trial so that they are different from those handed up by the grand jury.” United States v. Rodriguez, 215 F.3d 110, 118 (1st Cir.2000)(quoting United States v. Portela, 167 F.3d 687, 701 (1st Cir.1999)). Appellants also argue that this evidence constituted an improper variance from the indictment. A variance occurs when the proof differs from the indictment’s allegations, and “is material and reversible only if it has affected the defendant’s ‘substantial rights’: to be informed of the charges; and to prevent a second prosecution for the same offense.” United States v. Vavlitis, 9 F.3d 206, 210 (1st Cir.1993) (citation omitted).

At trial, the government introduced evidence that numerous checks issued by Caguas and made payable jointly to Gu-tiérrez companies and third party contractors were deposited in the accounts of the Gutiérrez companies after the contractor’s endorsement was fraudulently added to the checks. The government elicited this testimony as evidence of appellants’ intent to defraud the bank.

The district court addressed the forgery evidence in an order denying appellants’ motion to exclude it. The court found that such evidence caused neither a constructive amendment nor a variance, explaining that, “[s]ince some of the checks issued by Caguas [that were ultimately used to make payments on failing loans] were jointly issued to innocent third party payees, these payees’ endorsement would have been required before the checks could actually be cashed or deposited in the bank.” Since “disbursement of the funds could not be completed until the checks were cashed ... in some cases forged endorsements would necessarily be required.” The court concluded that, rather than constructively amending or varying the indictment, the forged endorsements were direct evidence of one component of appellants’ scheme. Finally, to foreclose any possibility that the jury would convict appellants based on the forgery evidence, the court issued thorough instructions regarding the elements of bank fraud and conspiracy as well as an instruction specifically tailored to the evidence of forgery: “You have heard evidence about checks with allegedly forged endorsements. The crime of forgery is not charged.... The fact that endorsements may be forged may only be used to assist you, as you may find useful, in your evaluation of all the evidence in the case....”

The district court’s order accurately assesses the value of the forgery evidence, and the admission of such evidence did not constructively amend the indictment or result in a variance. We further conclude that its instruction was more than sufficient to inform the jury of the proper role of that evidence.

We view the check-kiting evidence similarly. The disputed evidence consisted of testimony from Somohano (the vice president of Caguas’ commercial loan department) that in 1983 he detected a check-kiting scheme involving two Gutiérrez corporations’ accounts at Caguas and Banco de Ponce. When he informed Muñoz-Franco and Sánchez-Arán, Muñoz-Franeo scolded Somohano and denied that check-kiting was occurring. Somohano further testified that Sánchez-Arán continued to honor the checks sent to Caguas even after Banco de Ponce stopped honoring checks deposited with them. The check-kiting continued, resulting in overdrafts of more than $900,000 on the Gutiérrez-owned companies’ accounts at Caguas. This overdraft ultimately was converted into a loan.

The district court did not err in admitting this evidence. The reactions of Mu-ñoz-Franco and Sánchez-Arán when So-mohano told them about the check-kiting scheme demonstrate both their goal of preserving the Gutiérrez-owned companies’ financial status and the preferential treatment that these companies received. This evidence was directly relevant to the charges of bank fraud and conspiracy, and was not outside the scope of the original indictment. Thus, we conclude that no constructive amendment or improper variance occurred.

C. Evidence of Civil and Regulatory Violations

Appellants also protest that the district court improperly admitted evidence of civil and regulatory banking violations. Again, we review this evidentiary ruling for abuse of discretion. Richardson, 421 F.3d at 37-38.

Evidence of civil or regulatory violations is admissible so long as “the evidence is not presented in such a way that the jury’s attention is focused on the civil violations rather than the criminal ones.” United States v. Stefan, 784 F.2d 1093, 1098 (11th Cir.1986). The concern is that such evidence could create an “alternative standard of guilt.” United States v. Christo, 614 F.2d 486, 491 (5th Cir.1980)(prohib-iting the introduction of civil banking statutory violations solely for the purpose of proving criminal misapplication).

Courts have allowed the introduction of evidence of civil and regulatory violations in a variety of situations. In United States v. Devin, 918 F.2d 280, 287 (1st Cir.1990), we found no abuse of discretion in a district court’s decision to allow evidence of violations of police department regulations to demonstrate the knowledge and intent of a police officer charged with RICO violations and attempted extortion. We explained that “[s]uch evidence was unquestionably helpful to the jury in determining whether ... [the officer] was willing to disregard his sworn obligations and accept things of value which influenced his performance of official duties.” Id. Similarly, in Stefan, the district court permitted the government to introduce testimony regarding violations of a civil regulatory banking statute and charts illustrating how certain loans at issue contravened the statute. It also allowed the government to refer to the statutory violations in its opening statement and closing argument, and even “focused the jury’s attention” on the statute in the instructions. 784 F.2d at 1097. On appeal, the Eleventh Circuit found that the defendant’s scheme included efforts to avoid this statute, and that the district court therefore did not abuse its discretion in admitting such evidence to help the jury understand the scheme. In both of these cases, the courts placed emphasis on a jury instruction which made clear that civil or regulatory violations did not equate to criminal conduct. See Devin, 918 F.2d at 288 n. 9; Stefan, 784 F.2d at 1099.

Here, the court allowed the government to introduce evidence that defendants violated certain regulations of the Federal Home Loan Bank to provide context for their actions and to establish their knowledge of the impropriety of their activities. Although such evidence was discussed from time to time, it was not central to the government’s case, as our lengthy discussion of the sufficiency of the evidence demonstrates, see Section III, supra.

Moreover, the district court issued a limiting instruction on the use of evidence of civil and regulatory violations:

You have heard testimony and evidence that certain civil rules and/or regulations which govern the conduct of banks may not have been followed at times at Caguas.
You are hereby instructed that a violation of any of these rules and regulations is not a crime in and of itself, and is not determinative of a defendant’s guilt or innocence.
The fact that certain civil rules and regulations may have been violated at Caguas may only be used by you to assist you, as you may find useful, in determining a particular defendant’s motive or intent as to the crimes charged in the third superseding indictment.

This instruction clearly conveys that civil and regulatory violations do not establish criminal culpability. Appellants protest that the instruction exacerbated the problem because the reference to “the fact that certain rules and regulations may have been violated” improperly suggested that such rules and regulations had been violated. However, we see no such suggestion because the phrase “may have been violated” makes clear to the jury that, if it found that such violations had occurred, it could use those violations as one piece of evidence in determining the ultimate question of criminal guilt or innocence. Given this appropriate limiting instruction, we find that the district court did not abuse its discretion by admitting evidence of regulatory violations to provide context and to demonstrate appellants’ knowledge of their criminal conduct.

YI.

Appellants challenge other aspects of the proceedings against them, including the length of the trial, allegedly disparaging and misleading statements by the prosecutor, the jury instruction on misapplication, and alleged error in sentencing. We now turn to these remaining issues.

A. Length of Trial

Appellants argue that the sheer length of the trial and the repetitive nature of the government’s evidence denied them due process. In denying appellants’ motions for judgment of acquittal on these grounds, the district court devoted more than ten pages of its seventy-seven page opinion to a thorough examination of this issue. We largely agree with its analysis. Thus, we merely summarize its conclusions here and note only a few additional points.

The trial lasted for fifteen months and the prosecution’s case lasted far longer than appellants’ case. However, appellants bore much of the responsibility for the length of the trial. They either requested or agreed to several continuances after the trial began. In response to the jury’s request that the trial occur five days per week, they requested that trial occur only four days per week. Moreover, much of the time devoted to the government’s case consisted of the vigorous and lengthy cross-examination to which the defense subjected government witnesses. Finally, nearly all of the witnesses testified in Spanish and required the use of interpreters, which contributed significantly to the length of the trial.

The government’s presentation was not repetitive. For example, although Enri-quez (the senior vice president of the mortgage department) and Kareh (the assistant vice president of the construction loan department) both testified at length about the same projects, Kareh offered more detailed information about the loans themselves, while Enriquez provided insight into appellants’ knowledge about the loans and whether appellants provided certain information to the Board. Although appellants can point to instances in which the same story was told more than once, such repetition often encompassed new and relevant details, and was not unduly frequent given the complexity of the violations alleged and the length of the trial.

Finally, there is no indication that the length of the trial and the arguably cumulative nature of the evidence affected the outcome of the trial in any way. In fact, the jury convicted the four appellants while acquitting Enrique Gutiérrez on all counts, which suggests that the jury was able to carefully weigh the evidence against each defendant. See United States v. LiCausi, 167 F.3d 36, 49 (1st Cir.1999)(explaining that a jury’s acquittal of one defendant on two counts indicates that it “was not prevented from making reliable judgments about guilt or innocence”). Given appellants’ failure to demonstrate any prejudice resulting from the length of the trial, we find no due process violation under these circumstances.

B. Prosecutorial Misconduct

Appellants claim that, throughout the trial, the prosecutor made disparaging and misleading comments that prejudiced the jury against them and require us to vacate their convictions. The district court denied motions for a mistrial on such grounds on at least six occasions.

We must first resolve the threshold question of whether these comments were improper. Darden v. Wainwright, 477 U.S. 168, 180-81, 106 S.Ct. 2464, 91 L.Ed.2d 144 (1986). If we answer that question in the affirmative, we must determine whether the impropriety warrants vacating the convictions. The issue is not whether “ ‘the prosecutors’ remarks were undesirable or even universally condemned’ rather, “[t]he relevant question is whether the prosecutors’ comments ‘so infected the trial with unfairness as to make the resulting conviction a denial of due process.’ ” Id. at 181, 106 S.Ct. 2464 (citations omitted). We have held that the relevant factors include: “(1) whether the prosecutor’s misconduct was isolated and/or deliberate; (2) whether the trial court gave a strong and explicit cautionary instruction; and (3) whether any prejudice surviving the court’s instruction likely could have affected the outcome of the case.” United States v. Auch, 187 F.3d 125, 129 (1st Cir.1999). We may overturn the convictions only if “the prosecutor’s misguided tactic ... could have affected the outcome of the trial.” Id. at 130.

1. Disparagement

Appellants first object to comments, made in the presence of the jury, that they term “disparagement.” For example, the prosecutor, on one occasion, stated that defense counsel “just lied to the court” and referred to his statement as a “misrepresentation.” On another occasion, the prosecutor asked the court to “please direct [defense counsel] to refrain from his offensive odious comment that [has] characterized [] this trial,” concluding, “I don’t think that I need to endure this sort of treatment from him.” On several occasions, the prosecutor interrupted defense counsel’s objections with such comments as “say something new,” and “Improper question. First year law school. Improper question.”

Such comments are unfortunate and unprofessional. However, the trial was highly contentious, and the record reveals that appellants’ trial counsel made a number of similar comments. Although such exchanges between the prosecution and defense occurred intermittently, their impact was diluted by the length of the trial. Immediately following such exchanges, the court usually commented on their inappropriateness. For example, the court on one occasion ordered “all counsel to measure their words, measure their conduct, and measure their civility towards each other,” noting, “I have observed both sides step over the line.” On another occasion the court admonished counsel for improper comments and stated: “And by counsel I included both sides. This is not an admonition to one side or the other, both sides are included.” Such statements from the court adequately informed the jury that such conduct was inappropriate and assigned blame equally to both sides. Finally, any prejudice that resulted from the prosecutor’s role in these exchanges and survived the court’s instruction could not have influenced the outcome in this case because, as we explained in Section III, swpra, the evidence of appellants’ guilt was overwhelming. Thus, the prosecutor’s occasional remarks in this long and contentious trial, although unfortunate, did not infect the trial with such unfairness as to make the resulting convictions a denial of due process. Darden, 477 U.S. at 181, 106 S.Ct. 2464.

2. Misleading Statements

Appellants also assert that the prosecutors made misleading statements and referred to information outside the record. Many of appellants’ characterizations of the prosecution’s presentation are simply inaccurate. For example, appellants claim that the prosecutor falsely stated that Lugo (the Board president) had testified that the Board was not told of Modules’ performance history, when Lugo had at most said he could not remember those statements. This claim is unsupported by the record. Lugo in fact testified: “I don’t recall that it was discussed. I believe that if it was discussed I would remember and it is not reflected in the minutes.... The best of my recollection is that it was not discussed.” In another instance, appellants protest that the prosecutor improperly suggested to Joseph Gonzalez (an examiner for the Federal Home Loan Bank) that bank management, rather than the Board, was responsible for Caguas’ lending practices, when in fact the prosecutor only elicited testimony that the Board relied on Muñoz-Franco and Sánchez-Arán to provide information relevant to its decisions, and often accepted Muñoz-Franco’s recommendations regarding these decisions. Such comments were not improper, let alone prejudicial.

In other instances, even if we found the challenged statements improper, any resulting prejudice was negated either by appellants’ own cross-examination or by a curative instruction from the district court. For example, appellants assert that the prosecution elicited testimony from Montil-la (the developer on the Jardines de Villa Alba project) that Caguas sued him but then dismissed its claim, when in fact the bank settled with Montilla and obtained title to all but one of the Jardines de Villa Alba lots. However, appellants then elicited the terms of the settlement on cross-examination, remedying any prejudice. On another occasion, the court addressed defense counsel’s allegations that the prosecution’s closing argument mentioned loans not charged in the indictment and mischaracterized Victor Kareh’s testimony with a detailed curative instruction.

In sum, any impact resulting from these alleged misleading statements must be evaluated in light of the overwhelming evidence presented at this fifteen-month trial. Under such circumstances, we again conclude that the prosecutor’s comments could not have caused unfairness that resulted in a denial of due process.

C. Jury Instruction on Misapplication

Appellants argue that the jury instructions improperly permitted the jury to convict them on the misapplication charge if it found them merely reckless. We review de novo a claim that the district court erred in instructing the jury on the intent necessary to support a conviction. United States v. Woodward, 149 F.3d 46, 68-69 (1st Cir.1998).

The contested instruction stated that the government must prove beyond a reasonable doubt

that the defendants acted with the intent to injure, defraud or deceive the bank. You may also consider whether defendants acted recklessly, that is, in reckless disregard of the interests of the bank. If you find that defendants acted recklessly, with respect to the alleged misapplications, you may find that the defendants acted with intent to injure, defraud or deceive the bank.

Relying on a Fifth Circuit case, United States v. Adamson, 700 F.2d 953, 965 (5th Cir.1983), appellants contend that, contrary to this instruction, reckless behavior is insufficient to establish misapplication under 18 U.S.C. § 657. Contrary to the Fifth Circuit, however, we have specifically held that “ ‘the sine qua non of charges of willful misapplication of bank funds is action taken with the knowledge of harm to, intent to harm, or reckless disregard for, the financial health of the bank.’ ” United States v. Brennan, 994 F.2d 918, 923 (1st Cir.1993)(emphasis added)(quoting United States v. Fusaro, 708 F.2d 17, 21 (1st Cir.1983)); see also United States v. London, 66 F.3d 1227, 1241 (1st Cir.1995)(“[T]he cases applying § 656 [the misapplication statute that deals with banks rather than savings and loans institutions] ... have generally held reckless disregard to establish the requisite intent to defraud.”); United States v. Rodriguez-Alvarado, 952 F.2d 586, 590 (1st Cir.1991)(“A reckless disregard by a bank officer of his bank’s interest ... is sufficient to establish the requisite intent to defraud .... ”). In light of these binding decisions, appellants’ arguments are futile. We find no error in the district court’s instruction on the misapplication charge.

D. Booker Error

Appellants were sentenced pursuant to a mandatory sentencing scheme that has since been deemed unconstitutional, United States v. Booker, 543 U.S. 220, 245, 125 S.Ct. 738, 160 L.Ed.2d 621 (2005), but they did not preserve their claims that they were improperly sentenced. We have held that such unpreserved claims are reviewed only for plain error. United States v. Antonakopoulos, 399 F.3d 68, 76 (1st Cir.2005). Appellants challenge their sentences on the ground that “the Antonako-poulos plain error analysis does not comport with the Supreme Court’s decisions in [United States v. Olano, 507 U.S. 725, 734-36, 113 S.Ct. 1770, 123 L.Ed.2d 508 (1993) ] and Booker because it imposes on defendants the burden of proving that error materially affected the proceedings below.” Absent en banc or Supreme Court review, however, we are bound to follow the Anto-nakopoulos framework, and thus we acknowledge only that appellants have preserved this issue for later review.

VII.

After concluding our review of the immense record in this complicated case, we agree with the district court that overwhelming evidence supported the jury verdicts. Also, appellants have raised no other grounds that would justify vacating their convictions.

We have been greatly aided in our review by the district court’s sensible management of this lengthy trial, as well as its careful and thorough analysis of the issues that arose throughout the proceedings. Its handling of this burdensome case was exemplary.

We affirm, the convictions and sentences of the appellants on all counts.

So ordered. 
      
      . The Board was a body of ten to fifteen members responsible for establishing policy for the operation of Caguas and overseeing the general operation of the bank. The Board oversaw the Executive Committee (which reviewed and approved commercial and real estate construction loans in excess of $500,000) and the Loan Committee (which reviewed and approved commercial and real estate loans under $500,000).
     
      
      . Similar conduct also occurred on other Gu-tiérrez projects, including Levittown, Country Club, Los Caciques, Quintas de Humacao, and Quintas de Fajardo.
     
      
      . The Third Superseding Indictment did not charge Gutiérrez and Umpierre-Hernández with involvement in this scheme.
     
      
      . Appellants protest the delay both preceding and resulting from this series of superseding indictments on Fifth and Sixth Amendment grounds. We will discuss the events surrounding this series of indictments in more detail in Section IV.C, infra.
      
     
      
      .Wolff died prior to trial.
     
      
      . In addition to the sufficiency of the evidence, the district court considered and rejected claims of prosecutorial misconduct, improper jury instructions, inconsistent verdicts, and violation of the right to a speedy trial, as well as challenges to the admission of certain evidence and testimony.
     
      
      . With respect to the replacement of the bank's outside auditors, appellants object to Somohano’s testimony that he knew why Cag-uas’ outside auditors were replaced because "someone told me.” This characterization is incomplete. Upon further questioning, So-mohano stated that he also knew this information based on ”[m]y own personal observations,” specifically, the fact that "the reserves were totally inadequate to guarantee that the bank would not shake if a loss could occur.” After the government asked how Somohano’s observation related to the auditors’ removal, however, appellants objected, and, after a lengthy sidebar and research, the court excluded the testimony altogether.
     
      
      . For example, when the prosecutor asked whether the Board was told certain information about Modules, the court interrupted: "You don’t know what the board was told unless you’re there. The most you can state is whether or not the minutes reflect that information.’’ The prosecutor rephrased the question, and Enriquez then responded that "from what I remember of having reviewed the minutes, there is no mention anywhere of that fact."
     
      
      . Appellants filed their briefs before the Court's decision in Davis, and thus do not discuss its significance.
     
      
      . The Court has held that the term "testimonial” includes "at a minimum ... prior testimony at a preliminary hearing, before a grand jury, or at a former trial; and ... police interrogations.” Crawford, 541 U.S. at 68, 124 S.Ct. 1354. In the context of police interrogations, the Court indicated that a key factor is whether "the primary purpose of the interrogation is to establish or prove past events potentially relevant to later criminal prosecution.” Davis, 126 S.Ct. at 2274. In United States v. Hansen, 434 F.3d 92, 100 (1st Cir.2006), we also found that statements that "the declarant would not reasonably expect to be available for use at a later trial” were nontestimonial in character.
     
      
      . The Advisory Committee Notes leave open the possibility that the absence of evidence from a record is not hearsay at all. Fed. R.Evid. 803(7) advisory committee notes (“While probably not hearsay as defined in Rule 801, decisions may be found which class the evidence not only as hearsay but also as not within any exception. In order to set the question at rest in favor of admissibility, it is specifically treated here.’’). Moreover, at least one court recently has noted the possibility that “evidence that a record does not exist arguably is not hearsay at all.” United States v. Cervantes-Flores, 421 F.3d 825, 832 n. 4 (9th Cir.2005). For present purposes we assume that such evidence is hearsay, but admissible under Rule 803(7).
     
      
      . In a typical example involving a loan to a developer on a project that did not involve any of the appellants, the Executive Committee Minutes report: "Mr. Kareh explained that as a result of discrepancies between the developer and the builder, this project was par[y]lized for many months. The developer requested that he be allowed to substitute builders in order to finish the project. Original builder was Las Américas Construction and the new builder would be a local small contractor known as D.C. Inc. with ample experience in this field. Mr. Kareh also explained that further delay in the conclusion of the project would probably affect the final outcome of the project.”
     
      
      . Appellants wisely do not attempt to argue that it was unnecessary for them to present such information to the Board. As we will discuss more thoroughly in Section III, infra, this information was highly material to the Board's decisions regarding the loans.
     
      
      . In full, 18 U.S.C. § 1344 provides: “Whoever knowingly executes, or attempts to execute, a scheme or artifice (1) to defraud a financial institution; or (2) to obtain any of the moneys, funds, credits, assets, securities, or other property owned by, or under the custody or control of, a financial institution, by means of false or fraudulent pretenses, representations, or promises; shall be fined not more than $1,000,000 or imprisoned not more than 30 years, or both.’’
     
      
      . The high percentage of assets that were intangible might raise concerns about the ability of Modules to pay off its debts. Such information would be relevant to the Board's decision process.
     
      
      . Hernández testified that "senior management” consisted of Muñoz-Franco and Sán-chez-Arán.
     
      
      . At trial, Joseph Gonzalez (an examiner for the Federal Home Loan Bank) testified that "[a] loan that has a different borrower, different property to me would be an unrelated loan.”
     
      
      . Gutiérrez and Umpierre-Hernández implemented similar schemes involving developer Montilla on the Jardines de Villa Alba project and developer Santiago on the Caciques project. The contours of these schemes are similar to that involving Burns, and thus we need not recount them in detail here.
     
      
      . In pertinent part, 18 U.S.C. § 371 states: "If two or more persons conspire either to commit any offense against the United States, or to defraud the United States, ... and one or more of such persons do any act to effect the object of the conspiracy, each shall be fined under this title or imprisoned not more than five years, or both.”
     
      
      . In pertinent part, 18 U.S.C. § 657 provides: "Whoever, being an officer, agent or employee of or connected in any capacity with the Federal Deposit Insurance Corporation ... or savings and loan corporation or association authorized or acting under the laws of the United States ... embezzles, abstracts, purloins or willfully misapplies any moneys, funds, credits, securities or other things of value belonging to such institution ... [is guilty of misapplication of bank funds].''
     
      
      
        .Giragosian construed 18 U.S.C. § 656, which deals with banks rather than savings and loan associations. However, "[i]n the absence of legislative history to suggest that there is any substantive difference in meaning, we find the reasoning in cases construing § 656 equally applicable to our reasoning” in cases involving § 657. Blasini-Lluberas, 169 F.3d at 63 n. 8.
     
      
      . The loan offering was a document prepared by bank management (often Sánchez-Arán) that proposed the terms of a loan to be offered to a borrower and was then submitted to the Board for approval.
     
      
      . Kareh testified that the settlement statement "sums up the conditions for the loan, the amount of the money, the payouts that have been made for the loan and whatever pending balances may remain for the loan in process” and was “created and maintained in the regular course of business of the construction loan department.”
     
      
      . The district court's discussion of this issue suggests that $932,177 was disbursed instead of $855,323, when in fact both amounts were disbursed but the $855,323 was not disclosed on the settlement statement.
     
      
      . These transfers occurred on the last day of the bank's fiscal year, thus improving the appearance of Caguas' finances to anyone reviewing the bank's records.
     
      
      . Appellants do not contend that the misapplication charges against them were outside the statute of limitations.
     
      
      . Defendant Enrique Gutiérrez was acquitted by the jury, and defendant Rafael Dominguez Wolff died before trial.
     
      
      . Courts have recognized an expanded time span as a substantial amendment only when the expansion was far more significant than the one here, and then only in conjunction with numerous other factors. See, e.g., United States v. Ratcliff, 245 F.3d 1246, 1253-54 (11th Cir.2001)(finding that a superseding indictment materially broadened or substantially amended the original indictment when it increased the length of time of the conspiracy from six or seven months to thirteen years, increased the number of conspirators from five to fifteen, and increased the amount of marijuana imported from 1,500 pounds to 6,800 pounds).
     
      
      . Appellants present various arguments that the alleged conduct concluded more than ten years prior to the dates of the First, Second, and Third Superseding Indictments. However, because we find that the superseding indictments did not materially broaden or substantially amend the charges, we need not address these arguments.
     
      
      . We also note that there was more than sufficient evidence to allow a reasonable jury to conclude that the alleged offenses were completed within the limitations period, see supra Section III, and thus our holding that appellants waived their right to request an instruction on the statute of limitations in no way risks a miscarriage of justice.
     
      
      .With respect to the Gutiérrez loans, the indictment alleged a conspiracy to commit bank fraud among all four appellants from approximately June 25, 1980 through May 25, 1990. It also charged those appellants with bank fraud with respect to those loans "on or about the dates listed below” and listed a series of fifty-eight overt acts, of which the earliest took place on March .24, 1977, and the latest took place on December 19, 1989. With respect to the Mirandes loans, the indictment alleged a conspiracy to commit bank fraud between Muñoz-Franco and Sánchez-Arán from approximately December 1981 through May 25, 1990. It also charged Mu-ñoz-Franco and Sánchez-Arán with bank fraud with respect to the Mirandes loans "on or about the dates listed below” and listed a series of forty-seven overt acts, of which the earliest took place in December 1981 and the latest took place on December 7, 1989.
     
      
      . We have held that conspiracy is "often a continuing offense,” United States v. Welch, 15 F.3d 1202, 1207 (1st Cir.1993), and, in the context of determining venue, we have acknowledged that "[blank fraud is also a continuing offense,” United States v. Scott, 270 F.3d 30, 36 (1st Cir.2001). The other circuits that have considered the issue generally have agreed that both conspiracy and bank fraud are continuing offenses. See, e.g., United States v. Lennon, 372 F.3d 535, 541 n. 8 (3d Cir.2004)(conspiracy); United States v. Nash, 115 F.3d 1431, 1440-41 (9th Cir.1997)(bank fraud); United States v. Duncan, 42 F.3d 97, 104 (2d Cir.1994)(bank fraud and conspiracy). Under the circumstances present here, there is no question that the conspiracy and bank fraud charges reflect continuing offenses.
     
      
      . The court applied plain error review, finding that: (1) the overwhelming weight of the evidence established bank fraud continuing beyond the date § 1344 was enacted; (2) the conspiracy statute, § 371, was valid prior to any of the conduct alleged; and (3) the jury’s conviction on the misapplication counts occurring in 1986 substantiated a finding of culpable conduct continuing beyond the date § 1344 was enacted.
     
      
      . The Third Circuit's holding is similar to that of the Second Circuit. See United States v. Harris, 79 F.3d 223, 229 (2d Cir.1996)("Be-cause the [] statute is a continuing crime statute, we must determine whether it was possible for the jury, following the district court's instructions, to convict [the defendant] exclusively on pre-[ ]enactment conduct. After examining the jury instructions, we think that such a scenario was impossible.”). Ap-pedants also refer to United States v. Torres, 901 F.2d 205 (2d Cir.1990). However, that case is distinguishable because appellants did raise a general ex post facto challenge before the district court. Id. at 228. On appeal, the Second Circuit discussed plain error review but did not explicitly apply such review. Id. It concluded only that "it is inappropriate to impose a mandatory life sentence upon [the defendants] where there was an ex post facto violation in the instruction actually given, and the defendants brought the general ex post facto question to the attention of the district court.” Id. at 229.
     
      
      . Of these nineteen unavailable witnesses, fifteen had died, three were seriously ill, and one could not be located.
     
      
      . For example, on August 27, 1997, Muñoz-Franco filed seventeen separate motions, including one entitled "Motion Requesting Order Concerning Additional Motions.” Similarly, on March 24, 1998, Gutiérrez filed what the district court termed a "deluge” of "ten separate and distinct motions of varying complexity.”
     
      
      . Appellants imply that the government acted in bad faith by waiting to file the recusal motion until immediately prior to trial. However, the government explains that it did not learn about the banking relationship until one year prior to trial, and was delayed in learning the details of the relationship due to resistance to the subpoenas it filed. In fact, the government still had not been able to obtain some of the documents it requested by the time it filed the recusal motion. In light of this explanation, and the fact that appellants produced no evidence of bad faith on the part of the government below, we still have no basis for questioning the government's explanation.
     
      
      . We note that, of the nineteen unavailable witnesses that appellants name in their brief, five of these witnesses (Hugo Lopez, Celestino Seneriz, Pedro Bull Nater, Francisco Susoni Lens, and Jose Feliciano) died prior to indictment and, consequently, their unavailability does not implicate the Sixth Amendment analysis under Barker. See MacDonald, 456 U.S. at 7, 102 S.Ct. 1497 C'[N]o Sixth Amendment right to a speedy trial arises until charges are pending.”).
     
      
      . For example, the government elicited testimony from Luis Garate, the former vice president of Bermudez and Longo, Inc., a construction contractor in electrical and building mechanics. Bermudez and Longo worked with the Gutiérrez company Transglobe on the Modules project. Garate testified that he did not recognize the endorsements on fifteen checks, totalling over $67,000, that were issued by Caguas and made payable jointly to Transglobe and Bermudez and Longo.
     
      
      . Appellants did not object to the instruction, and consequently the objections to the instruction that they raise on appeal are reviewable only for plain error. See Fed. R.Crim.P. 30(d)("A party who objects to any portion of the instructions or to a failure to give a requested instruction must inform the court ... before the jury retires to deliberate.”). We see no such error in the clear instruction issued here.
     
      
      . For example, the instruction in Devin stated: "[RJules and regulations of the Boston Police Department were admitted into evidence in this case. And while they may be relevant in order to show a guideline or a code of conduct to a particular job, the defendant is not here because he violated a rule or regulation. He is charged here with violations of certain specific laws, not rules or regulations.” 918 F.2d at 288.
     
      
      . In July 2001, the trial judge notified the parties that he had a vacation scheduled for two weeks in September but stated that he was "willing to forego, if there is absolutely any objection from anyone,” emphasizing that “if [there] is any problem whatsoever, be it a due process claim or some sort of mistrial, I want to know now.” Defense counsel responded that there was "[n]o objection whatsoever.”
     
      
      . Although procedures vary from trial to trial, questions from attorneys are usually translated into Spanish upon completion. With respect to witness testimony, the interpreter typically instructs the witness to pause every few sentences to allow translation of the testimony into English, and will stop the witness if the witness does not remember to pause in giving a longer answer.
     
      
      . Appellants did not object to this testimony at trial, and thus we review their claim only for plain error. See United States v. Duarte, 246 F.3d 56, 60 (1st Cir.2001).
     
      
      . The complete instruction spanned several pages of trial transcript, but a representative paragraph will demonstrate the level of detail: “You heard reference to the Coamo loan dur- . ing the government’s closing argument and entries on the Coamo ledger card. Because the Coamo loan is not part of the charges in this case I instruct you to disregard the prosecutor’s comment about the Coamo ledger card as they refer specifically to the Coamo loan. You may consider, in evaluating all the evidence, whether any payments from other loans were used or not used to make any payment in the Coamo loan.”
     
      
      .In Adamson, the trial court instructed the jury: "A reckless disregard of the interest of the bank is the equivalent of intent to injure or defraud the bank.” 700 F.2d at 965. In reversing the misapplication conviction, the court held that, "[i]n order to convict a defendant for willfully misapplying funds with intent to injure or defraud a bank, the government must prove that the defendant knowingly participated in a deceptive or fraudulent transaction.” Id.
      
     