
648 F.3d 569 (2011)
UNITED STATES of America, Plaintiff-Appellee,
v.
James GREEN, Joseph Miller, and Alonzo Braziel, Defendants-Appellants.
Nos. 09-3098, 09-3482, 09-3681.
United States Court of Appeals, Seventh Circuit.
Argued March 29, 2011.
Decided August 9, 2011.
*572 Christopher J. Stetler (argued), Attorney, Office of the United States Attorney, Chicago, IL, for Plaintiff-Appellee.
Ralph J. Schindler, Jr. (argued), Attorney, Chicago, IL, for Defendant-Appellant in No. 09-3098.
Anthony R. Burch (argued), Attorney, Burch & Associates, Chicago, IL, for Defendant-Appellant in No. 09-3482.
Michael J. Petro (argued), Attorney, Chicago, IL, for Defendant-Appellant in No. 09-3681.
Before ROVNER, WILLIAMS, and HAMILTON, Circuit Judges.
HAMILTON, Circuit Judge.
Appellants Alonzo Braziel, Joseph Miller, and James Green were found guilty of participating in a fraudulent scheme to obtain mortgage loans by providing false information to lenders. The scheme involved a complex web of players: Recruiters enlisted buyers to buy properties with fraudulently obtained mortgage funds. Financiers provided funds to the buyers to facilitate the transactions. Administrators bought fake documents to enable the buyers to obtain mortgages. Loan officers prepared fraudulent mortgage applications and sent them to the lenders. Between 2003 and 2005, the group acquired over seventy properties in the Chicago area for which lenders provided $7.2 million in loans. Most of the properties went into foreclosure when the buyers could not make the mortgage payments, resulting in losses to the lenders of $2.2 million.
On February 5, 2008, a grand jury indicted Braziel, Miller, Green, and fifteen others for mail fraud and wire fraud in violation of 18 U.S.C. §§ 1341 and 1343. Braziel and Miller were tried with two of their co-defendants. A jury convicted Braziel of three counts of mail fraud and Miller of three counts of wire fraud and three counts of mail fraud. Defendant Green was tried separately and was found guilty of three counts of wire fraud. The defendants now challenge various aspects of their convictions and sentences in these consolidated appeals. We consider the issues for each defendant, and we affirm all of their convictions and sentences.

I. Alonzo Braziel

Alonzo Braziel first became involved in the scheme in 2004 as a buyer applying for mortgage loans at the direction of others. The indictment charged that Braziel participated in fraud surrounding the purchases *573 of three residential properties in the Chicago area located at 1430 Portland Avenue, 14820-22 South Hoyne Street, and 7321 South Evans Avenue.
Braziel raises two issues on appeal. He argues that the district court erred by admitting a statement made by one of his co-defendants in which Braziel was implicated, in violation of Bruton v. United States, 391 U.S. 123, 88 S.Ct. 1620, 20 L.Ed.2d 476 (1968). Second, Braziel asserts that the district court should not have applied a two-level sophisticated means enhancement in calculating his guideline sentence.

A. The Bruton Issue

In Bruton v. United States, the Supreme Court held that a defendant's Sixth Amendment right to confront witnesses against him is violated when the confession of a nontestifying co-defendant implicating the defendant as a participant in the crime is admitted in a joint trial of both defendants. 391 U.S. at 137, 88 S.Ct. 1620. A Bruton violation may be avoided, however, by redacting the reference to the defendant and substituting a generic reference such as "another person" or "another member of the group." The issue here is whether the redaction used by the government, substituting for Braziel's name the words "straw buyer," was sufficient to solve the Confrontation Clause problem. Braziel argues that, despite the redaction, the "straw buyer" reference still pointed to him as a participant in the crime. Although this is a close case, we conclude that the use of the edited statement with the "straw buyer" reference did not violate Bruton here.

1. The Co-Defendant's Statement

The government sought a pretrial ruling on the admissibility of statements made by Braziel's co-defendants, as well as an order to limit defense counsel from eliciting portions of these statements that would give rise to a violation of Bruton. The motion stated that the government would call as a witness FBI Special Agent Donald Kaiser who would relate co-defendant Donald Thomas' confession. Thomas, also on trial with Braziel and Miller, would not testify and thus would not be subject to cross-examination on the statement. The motion summarized the statement that the government would elicit from Special Agent Kaiser. It included a description of the property transaction, but it did not mention that Braziel was incriminated in that section, nor did it provide any description of the redaction that the government intended to use.
The day before Special Agent Kaiser's testimony at trial, the government discussed with defense counsel and the court how it had redacted Thomas' statement to conceal Braziel's identity. The prosecutors noted that the original statement referred to Braziel as the purchaser of 14820-22 South Hoyne Avenue, but they had replaced his name with the term "straw buyer." Braziel's counsel made no objection at that time. At trial the next day, Special Agent Kaiser offered his testimony:
Q: Did Donald Thomas tell you about getting money for any particular properties?
A: Yes, he told me that he received $20,000 for the sale of 14822 South Hoyne in Harvey.
Q: Did he tell you anything else about that transaction?
A: Yes. Mr. Thomas explained to me that he had found a straw buyer, his term, for that property, and he explained the arrangement he had with that straw buyer. Specifically, he stated that the agreement was the straw buyer would purchase the *574 property and then deed the property back to Mr. Thomas. Mr. Thomas would, had agreed to then purchase the property from that straw buyer several months later, and in that time frame Mr. Thomas was supposed to make the mortgage payments until he was able to repurchase that property.
Q: Did Donald Thomas say whether he had paid the straw buyer?
A: Yes, he indicated that he had agreed to and that he actually did pay that straw buyer $5,000 for the purchase of that property.
Several minutes later, Braziel's counsel objected and requested a mistrial, claiming that the jury could identify Braziel as the straw buyer, so that admitting Thomas' incriminating statement without the opportunity to cross-examine him violated Braziel's Confrontation Clause rights under Bruton.[1]
Although the Thomas statement was redacted, the jury heard other witnesses read mortgage and bank records naming Braziel as the purchaser of 14820-22 South Hoyne Avenue. Heather McCartney, an employee of lender Fremont Investment & Loan, read the following from a real estate contract in Fremont's loan application for the South Hoyne property as part of her testimony:
Q: Who is the buyer?
A: Alonzo Braziel.
Q: What is the street address of the property at issue?
A: 14820 and 22 South Hoyne, Harvey, Illinois 60426.
A few minutes before Special Agent Kaiser read from Thomas' statement, he had testified about bank records he had reviewed as part of his investigation:
A. The first real estate transaction is 14822 Hoyne Avenue in Harvey, and that transaction closed or funded on 12/20/2004. The seller was the Marquette Bank Trust 16575, and the [buyer] was Alonzo Braziel.[2]
Putting these pieces together, the jury could have inferred that Braziel was the straw buyer to whom Thomas referred.
The court deferred ruling on Braziel's mistrial motion until later that day when it reviewed the transcript from the conference the day before. Acknowledging some potential confusion in their prior discussion, the court then denied the mistrial motion. Braziel renewed his motion at the close of the trial, and the court again denied the motion.

2. Analysis

On appeal, Braziel maintains that the district court erred by denying his motion for a mistrial. We review the district court's denial for an abuse of discretion, United States v. Tanner, 628 F.3d 890, 898 (7th Cir.2010), but we begin by reviewing the court's application of Bruton de novo. See United States v. McGowan, 590 F.3d 446, 453 (7th Cir.2009); United States v. Nash, 482 F.3d 1209, 1218 (10th Cir.2007).
*575 After the Supreme Court's further refinement of Bruton in Richardson v. Marsh, 481 U.S. 200, 107 S.Ct. 1702, 95 L.Ed.2d 176 (1987), and Gray v. Maryland, 523 U.S. 185, 118 S.Ct. 1151, 140 L.Ed.2d 294 (1998), it is clear that a redacted confession of a nontestifying co-defendant may be admitted as long as the redaction does not "obviously" refer to the defendant. This determination, focusing on the minutiae of the substituted word or phrase and surrounding context, is not always easy to make. See Gray, 523 U.S. at 195-96, 118 S.Ct. 1151. A district court's evaluation becomes especially difficult when the defendant's identity can be established through other evidence offered at trial, as here. Statements that "despite redaction, obviously refer directly to someone, often obviously the defendant, and which involve inferences that a jury ordinarily could make immediately" are prohibited under Bruton. Id. at 196, 118 S.Ct. 1151; see also United States v. Brooks, 125 F.3d 484, 501 (7th Cir.1997) (describing the Richardson Court's distinction between "specific testimony" identifying the defendant and an "inferential incrimination"). This case falls close to that subtle line.
We have navigated these murky waters in several of our prior cases. In United States v. Stockheimer, 157 F.3d 1082, 1086-87 (7th Cir.1998), we found no Bruton violation where the altered statement did not incriminate the nontestifying defendants by itself. In that case, the government used an open-ended reference ("inner circle") that avoided a one-to-one correspondence between the statement and the defendant, even if other evidence at trial incriminated the defendants as those members of the inner circle. See also United States v. Souffront, 338 F.3d 809, 829 (7th Cir.2003) (finding no Bruton violation where there was no one-to-one correspondence between the redacted statement and the defendant). In contrast, a more obvious one-to-one correspondence such as an alias or pseudonym is too transparent to pass muster. For example, in United States v. Hoover, 246 F.3d 1054, 1059 (7th Cir.2001), we concluded that substituting "incarcerated leader" and "unincarcerated leader" for the names of the two defendants did not solve the Bruton problems because those were "obvious stand-ins" for the names of the defendants. The jury in that case heard that one of the two leaders of the gang operated the gang's activities from state prison, while another served as acting leader on the outside. The Hoover court found that "incarcerated leader" and "unincarcerated leader" functioned the same way "deleted" or another similarly obvious indication of alteration would. 246 F.3d at 1059. Those terms "so closely resemble Bruton's unredacted statements that . . . the law must require the same result." Id., quoting Gray, 523 U.S. at 192, 118 S.Ct. 1151. As a general matter, we have recognized that such a delicate determination requires case-by-case consideration rather than a brightline rule. See id. (noting that "little evidence is incriminating when viewed in isolation" and that to adopt a "four-corners rule" would defeat the point of Bruton).
Here, we do not find the use of "straw buyer" in the Thomas confession to be so obvious a reference to Braziel as to violate Bruton. First, unlike an alias or a pseudonym used to disguise a single individual, "straw buyer" is more similar to an anonymous reference such as "another person" or "an individual." We agree with Braziel that "straw buyer" is not neutral insofar as it connotes some illicit activity, but the substituted word or phrase need not be neutral. In context, the Thomas statement was describing a transaction with a straw buyer, so using the phrase was not much different from using "the *576 buyer" or "the person." The statement was highly incriminating to Thomas, but his statement was not used to show that Braziel was the buyer. Most important for our analysis, the use of "straw buyer" did not facially incriminate Braziel as clearly as the terms "incarcerated leader" and "unincarcerated leader" did in Hoover. The "straw buyer" term could refer to anyone. Taken alone, nothing in Thomas' statement as told by Special Agent Kaiser suggests that Braziel was the straw buyer.
Second, although a reasonable jury member could have concluded that Braziel was the straw buyer to which Thomas referred by comparing other evidence presented at trial, the evidence required to make that connection was farther removed from the redacted statement than the clear correspondences present in Gray and Hoover. The Supreme Court has distinguished this type of acceptable indirect inference from an unacceptable immediate inference. See Gray, 523 U.S. at 195-96, 118 S.Ct. 1151; Richardson, 481 U.S. at 208, 107 S.Ct. 1702 (reiterating that only those statements that "expressly implicate" the defendant or are "powerfully incriminating" trigger Bruton). Though the case came very close to the Bruton line, the district court did not run afoul of Bruton by admitting the statement and did not abuse its discretion by denying a mistrial.

B. Sophisticated Means

At sentencing, Braziel objected to the two-level upward adjustment for sophisticated means recommended by the presentence investigation report. See U.S.S.G. § 2B1.1(b)(9)(C). The district court rejected Braziel's arguments, applied the adjustment, and imposed a within-guidelines sentence of 40 months in prison on each of his three counts of mail fraud. The court stated that it was applying the sophisticated means enhancement to Braziel, as it had to every other defendant involved in the scheme, because "the whole scheme was sophisticated." We review the district court's finding for clear error. United States v. Wayland, 549 F.3d 526, 528 (7th Cir.2008).
Braziel argues on appeal that the district court erred by applying the enhancement to him on the basis of the sophistication of the general scheme rather than his activities in particular. This argument is not consistent with the guidelines definition of relevant conduct. See U.S.S.G. § 1B1.3(a)(1)(B). Under the guideline rule for "relevant conduct," Braziel is responsible for "all reasonably foreseeable acts and omissions of others in furtherance of the jointly undertaken criminal activity," even if not charged as a conspiracy. Id. Therefore, a sophisticated means enhancement could be applied to Braziel so long as the use of sophisticated means by other criminal associates was reasonably foreseeable to him. See United States v. Crosgrove, 637 F.3d 646, 666 (6th Cir.2011) (affirming use of enhancement based on activities of others); United States v. Jenkins-Watts, 574 F.3d 950, 965 (8th Cir.2009) (same).
The district court did not err by finding this whole scheme to be sophisticated. The application note defines "sophisticated means" as "especially complex or especially intricate offense conduct pertaining to the execution or concealment of an offense." U.S.S.G. § 2B1.1, note 8(B). We have upheld the use of the enhancement "when the conduct shows a greater level of planning or concealment than a typical fraud of its kind." United States v. Landwer, 640 F.3d 769, 771 (7th Cir.2011), quoting United States v. Knox, 624 F.3d 865, 871 (7th Cir.2010). The commentary provides several examples of such conduct, illustrating the wide variety of criminal behavior covered by the theft and fraud *577 guideline. See also Wayland, 549 F.3d at 528 (elaborating on the meaning of the commentary). Here, the defendants' overall scheme lasted three years and involved numerous complex fraudulent transactions, the creation of fake documents, and the participation of nearly twenty individuals. The district court did not err in deeming it sophisticated.
Nor did the district court err by applying the enhancement on the basis of its finding that the whole scheme was sophisticated. The evidence showed that Braziel spoke with his co-defendants about several aspects of the general schemefrom the falsifications of rent and employment records to the concealment of financial transfers among the participants. On the basis of this evidence, the conspiracy's sophisticated criminal conduct was reasonably foreseeable to him. We affirm Braziel's sentence.

II. Joseph Miller

The government charged Joseph Miller with three counts of wire fraud and three counts of mail fraud, each related to a separate property transaction for which he acted as the loan officer. From 2001 to 2006, Miller served as a loan officer for the mortgage broker Integrity Home Mortgage, a name that seems a little ironic under the circumstances. In that position, he was responsible for compiling loan documentation on behalf of borrowers and submitting those materials to lenders. According to testimony at trial, Miller served as the loan officer for multiple home purchases arranged by co-defendants Jonathon Marchetti, Alfredo Hilado, and Larry Skrobot. In each of these transactions, Marchetti set up the sale by recruiting Hilado to serve as the buyer and sending him to Miller to obtain a loan. Skrobot provided the necessary financing for Hilado to make the purchase while Miller prepared the documentation and secured the mortgage for Hilado.
Miller makes two arguments on appeal. First, he contends that the evidence against him was not sufficient to prove his guilt beyond a reasonable doubt because testimony offered by two of his co-defendants was unreliable. Second, Miller claims that the district court erred by finding that he was involved in a single conspiracy. We disagree on both issues and affirm Miller's conviction.

A. Sufficient Evidence

A defendant challenging the sufficiency of the evidence against him must show "that no reasonable jury could have found his guilt beyond a reasonable doubt." United States v. Tavarez, 626 F.3d 902, 906 (7th Cir.2010). Under this already high standard, we consider the evidence in the light most favorable to the government. United States v. Huddleston, 593 F.3d 596, 601 (7th Cir.2010). If any rational trier of fact could have found the essential elements of the crime beyond a reasonable doubt, we will uphold the conviction. United States v. Durham, 645 F.3d 883, 892 (7th Cir.2011).
As a preliminary matter, we note that while Miller makes general claims about the insufficiency of the evidence against him, he discusses only the evidence regarding his participation in wire fraud. By not developing any argument regarding the sufficiency of evidence to support his mail fraud conviction, he waives that argument on appeal. See United States v. Collins, 604 F.3d 481, 487 n. 2 (7th Cir. 2010). We address only Miller's wire fraud convictions and conclude that the jury could reasonably have reached its guilty verdict on the evidence presented. To establish wire fraud, the government must prove (1) that the defendant participated in a scheme to defraud; (2) with the *578 intent to defraud; (3) and used interstate wire communications in furtherance of the fraud. See United States v. Howard, 619 F.3d 723, 727 (7th Cir.2010). Miller concedes that the two primary witnesses against himMarchetti and Hiladotestified that he directed them to submit false documents to him that he then forwarded to the lenders. Marchetti testified that he began working with Miller in 2004. He regularly used Miller as a loan officer in his Chicago-area property transactions, three of which were the subject of the wire fraud charges against Miller: 1920 Circle Court; 24 East 23rd Street; and 594 Andover Drive. Marchetti stated that Miller filled out false "seller second" mortgage forms for the properties at 24 East 23rd Street and 594 Andover Drive and that the two discussed creating false construction receipts for the Andover Drive property, as well.[3]
Hilado also testified that Miller was the loan officer for eleven properties he managed. The jury heard that Miller told Hilado to come up with names of people who would lease from Hilado at both the East 23rd Street property and the Circle Court property, even though Miller knew that he did not plan to have any tenants, to make the loans look more secure. These false lease forms as well as the false seller second forms were admitted into evidence. The government also provided evidence that Miller's income from commissions on loans he administered rose from $38,500 in 2002 to $49,000 in 2004, to $82,000 in 2005.
The gist of Miller's contention is that Marchetti's and Hilado's testimony cannot support proof of his guilt beyond a reasonable doubt because they were unreliable as witnesses. This argument invites us to disregard the standard of review and to substitute our judgment for that of the jury. We do not weigh the evidence on review or second-guess the jury's credibility determinations. See id. at 726; Tavarez, 626 F.3d at 906; United States v. Anderson, 580 F.3d 639, 646 (7th Cir.2009) (acknowledging that this court has repeatedly refused to question the credibility of witnesses when reviewing challenges to sufficiency of the evidence). We overturn a conviction based on a credibility determination only if the witnesses' testimony was incredible as a matter of law, see United States v. Carraway, 612 F.3d 642, 645 (7th Cir.2010), a high standard not met by the testimony here. The jury, aware of Marchetti's and Hilado's status as cooperating co-defendants and properly instructed to consider their testimony with great care, nevertheless credited the testimony they offered. We see no reason to disturb that determination.
Though the evidence against Miller was not overwhelming, his conviction is supported by witness testimony, the documentary evidence, and the reasonable inferences drawn from all of that evidence. His insufficiency of the evidence argument fails.

B. Single Conspiracy Determination

Miller also argues that the district court mistook several distinct conspiracies for a single conspiracy. Although Miller was not charged with a conspiracy, the court allowed the government to introduce statements made by co-conspirators based on a single conspiracy theory. We review the district court's single conspiracy determination for clear error, viewing the evidence in the light most favorable to the government. *579 See United States v. Ceballos, 302 F.3d 679, 688 (7th Cir.2002).
A single conspiracy exists if "the co-conspirators joined to effectuate a common design or purpose," id., with the focus of the court's inquiry on that common purpose. The government must demonstrate that the defendant joined the agreement alleged, not just the group. See United States v. Longstreet, 567 F.3d 911, 919 (7th Cir.2009). So long as the co-conspirators embraced a common criminal objective, a single conspiracy existed even if a participant did not know all of his co-conspirators and did not participate in every aspect of the scheme. United States v. Mojica, 185 F.3d 780, 786-87 (7th Cir.1999).
The evidence here was consistent with a single conspiracy theory. It showed an agreement to carry out a plan designed to generate income to the individuals involved by means of fraud. Over three years and the dozens of sales they completed, the schemers embraced this common goal, carrying out their different roles and responsibilities. Evidence of frequent and repeated transactions can support a single conspiracy theory. Cf. United States v. Blanding, 53 F.3d 773, 780 (7th Cir.1995). The cohorts' cooperation and coordination also indicate the same. See United States v. Handlin, 366 F.3d 584, 590 (7th Cir.2004) (trust, cooperation, and delineation of duties among participants in a common scheme "overcomes any doubt that this was anything other than a single, broad conspiracy"). Miller was a key player in at least six transactions, each of which furthered the broader scheme. Because the district court did not err in its single-conspiracy determination, this challenge to Miller's conviction also fails.

III. James Green

The government charged defendant James Green with four counts of wire fraud. Like Braziel, Green was a buyer. His cousin, co-defendant Joseph Green, recruited Green to purchase properties in exchange for cash.[4] Throughout the trial, Green argued that he was a victim of the fraudulent scheme, left holding the bag as the purchaser of properties that were supposed to be rehabilitated but were not. Green now raises five issues regarding his conviction. He also argues that the district court erred by sentencing him based on an improper loss calculation. We address each issue in turn.

A. Admissibility of Business Records

Green appeals the district court's admission of certain loan documents under Federal Rule of Evidence 902(11). Rule 902(11) streamlines the admission of certain inherently reliable documents by allowing a party to introduce a record of regularly conducted activity without live testimony from a records custodian so long as the record is accompanied by a proper written certification from a custodian or otherwise qualified person. See Fed. R.Evid. 902(11). The Rule requires that the record be admissible under Rule 803(6), which creates a hearsay exception for business records and reports "unless the source of information or the method or circumstances of preparation indicate lack of trustworthiness." Fed.R.Evid. 803(6).
In advance of trial, the government filed notice of its intent to offer evidence pursuant to Rule 902(11). Green and his co-defendants objected, claiming that the records did not meet the Rule's requirements. Relevant to this appeal, they objected to *580 the introduction of records certified by Charlene Batalla, a former employee of Equity Express mortgage brokerage firm, who was also a defendant in the scheme. They argued that Batalla, as a co-defendant, was not trustworthy as a custodian. In 2008, Batalla pled guilty to falsifying documents in the defendants' conspiracy. In 2009, Batalla certified as records made in the regular course of business the Equity Express loan files for five Chicago-area properties: 8544 South Givins Court, 1418 Portland Avenue, 155 East 153rd Street, 6851 South Prairie Avenue, and 1436 Parnell Avenue. Defendant Green was charged for his involvement in the purchases of the first four of those properties.
At a pretrial conference, the district court reviewed the certificates and overruled the objection, finding the certificates to be sufficient under Rule 902(11). After severance, Green renewed his trustworthiness objection to the Batalla certification. At a second pretrial conference, Green's counsel further argued that he had a right to cross-examine Batalla about "the fact that James Green didn't put any of this information on these papers." The government replied that if Batalla were to testify, she would answer only three questions"Were these documents created in the ordinary course, were they maintained, and were they created at or near the time of the documents?"and that anything else regarding the documents would be outside the scope of her testimony. The district court again overruled the objection and admitted the documents and certification. Despite all this attention to the certification and attached loan files, they were hardly used at trial. The government moved the files into evidence at the start of trial but never referred to them.
Rule 902(11) is a powerful and efficient short-cut, but it includes important built-in safeguards that cannot be taken lightly. Those safeguards include providing opposing counsel with advance notice of any Rule 902(11) certifications to give that party "a fair opportunity to challenge" the certifications, which could involve calling the certificate's signer to testify as Green demanded here. See Fed.R.Evid. 902(11); see also United States v. Adefehinti, 510 F.3d 319, 327-28 (D.C.Cir.2008) (amended opinion). We have noted that in some circumstances a Rule 902(11) certification will not implicate a defendant's Confrontation Clause rights because the certificate itself is not testimonial. See United States v. Ellis, 460 F.3d 920, 927 (7th Cir.2006). But the Rule does not give a party license to dump business records into evidence without giving an adverse party an opportunity to question the certificate's signer where such questioning may be warranted. See Adefehinti, 510 F.3d at 328 (commenting that "[i]n an appropriate case," the Rule's "opportunity to challenge" may include cross-examination, while also noting that the Rule 902(11) certificate does not fall within the guarantee provided by the Confrontation Clause). The government was treading on dangerous ground by using Rule 902(11) here to introduce not just these but hundreds of other records to prove the truth of the matters asserted without regard to the many layers of hearsay and the Confrontation Clause rights that those records may have implicated.[5]
In any event, we need not decide whether the district court erred by admitting evidence based on the Batalla certification because any such error would have *581 been harmless. At oral argument, we asked the government to file a supplemental memorandum identifying the portions of the record that addressed counsel's objections to the Batalla certification, as well as the portions of the record that corroborate or support the contents of the Equity Express documents admitted based on her certification. The parties' supplemental materials show that many of the documents within each file were duplicates of business records maintained by other lenders that were also admitted without objection. The most relevant loan application materials (the applications purportedly signed by Green containing false information) were also included in other exhibits that came in through trustworthy certifications to which Green did not object. Given this overlap and the limited use of the files, Green has not shown that he was prejudiced by the Equity Express records certified by Batalla. In the absence of prejudice, we need not reach Green's arguments that Batalla's certificate of authenticity failed to meet the requirements of Rules 803(6) and 902(11) or that its use violated his Sixth Amendment right.

B. Sworn Statements in Loan Applications

Green also contends that certain testimony offered by witnesses for the government prejudiced him by misleading the jury about the charges against him. The jury heard three government witnesses read slight variations of the following passage from loan application materials submitted in connection with Green's loans: "Borrower understands that it is a federal crime punishable by fine or imprisonment or both to knowingly make any false statement concerning any of the above facts, as applicable under the provisions of Title 18 USC, Section 1014. I declare that the foregoing agreement is true and correct. . . ." At its first mention, defense counsel requested a limiting instruction directing the jury that Green had not been charged with violating Section 1014. After the government responded that it was offering the evidence to show Green's knowledge, the court asked both counsel to review the issue overnight and raised the possibility of striking the reference or giving a limiting instruction the following day.
The next morning, prosecutors and defense counsel agreed on an instruction that the court gave the jury. The court told the jury that it had "heard some evidence regarding statements in loan documents about that it's unlawful to violate particular statutes. That evidence, the particular statute actually isn't at issue here other than it'sthe government is offering it with respect to knowledge and intent." Similar passages referencing 18 U.S.C. § 1014 were read and referenced again by two witnesses and by the government during closing argument.
On appeal, Green argues that the government's repeated references to section 1014 confused the jury. He contends that, as a result of this testimony, the jury convicted Green based on statutory violations not charged. But Green forfeited this argument by not objecting to the admissibility of the statements when they were offered. Rather, upon their introduction, Green's counsel requested, and the court gave, a limiting instruction to avoid the very confusion Green claims still permeated his trial.
In light of Green's forfeiture, we review the introduction of the testimony for plain error only. United States v. Jaimes-Jaimes, 406 F.3d 845, 847 (7th Cir.2005). We find none. Although the references to another provision of law perhaps posed some risk of confusing the jury, the court's limiting instruction alleviated that risk. The references were not so *582 prejudicial so as to render the instruction ineffectual. See United States v. Curry, 538 F.3d 718, 728 (7th Cir.2008) (noting that we assume a jury follows an instruction unless the evidence is so "powerfully incriminating that they cannot reasonably be expected to put it out of their minds"). Even if Green had not forfeited his objection, we would have little trouble agreeing with the district court that the passages were admissible to help prove that Green knew he was required to provide true information and knew it was wrong to provide false information, and finding that the limiting instruction was sufficient to protect against any potential prejudice.

C. Ostrich Instruction

Green next contests the "ostrich" instruction given by the court at the government's request. The instruction explained to the jury members that in their consideration of Green's knowledge about the fraud, the legal definition of knowledge includes the deliberate avoidance of knowledge"a combination of suspicion and indifference to the truth." We review a decision to give an ostrich instruction for abuse of discretion, viewing all evidence in the light most favorable to the government. United States v. Severson, 569 F.3d 683, 689 (7th Cir.2009).
A district court may give an ostrich instruction "where (1) a defendant claims to lack guilty knowledge, i.e., knowledge of her conduct's illegality, and (2) the government presents evidence from which a jury could conclude that the defendant deliberately avoided the truth." United States v. Garcia, 580 F.3d 528, 537 (7th Cir.2009). The purpose of an ostrich instruction is to inform the jury "that a person may not escape criminal liability by pleading ignorance if he knows or strongly suspects he is involved in criminal dealings but deliberately avoids learning more exact information about the nature or extent of those dealings." United States v. Carani, 492 F.3d 867, 873 (7th Cir.2007), quoting United States v. Carrillo, 435 F.3d 767, 780 (7th Cir.2006). In Garcia, we explained that a case in which a defendant admits his association with a group but denies knowledge of its illegal activity despite circumstantial evidence to the contrary is a "paradigm case" for use of the instruction. 580 F.3d at 537.
The evidence presented at Green's trial could be interpreted to show exactly that: Green deliberately avoided determining conclusively that he was engaged in criminal activity. Green was aware that his co-defendants had offered to obtain false documents for him and that they had done so for others in the past. In fact, Green testified that he questioned his co-defendants about the legality of what they were doing on several occasions. This evidence alone, demonstrating knowledge of his cohorts' involvement in suspicious activities, warranted an ostrich instruction. See United States v. Ramirez, 574 F.3d 869, 877 (7th Cir.2009). Green did more. He also signed blank loan applications and accompanied his codefendants to banks where they would withdraw money and obtain cashier's checks in an effort to make it appear as though Green was presenting his own money for the properties. He did not inspect the properties before he bought them, and he never met some of the individuals who he represented on his loan materials would be renting the properties. On the basis of this ample evidence, the district court acted well within its discretion to provide an ostrich instruction.

D. Cumulative Error

Green asserts that the cumulative effect of the errors made by the district court deprived him of a fair trial. Since we find *583 no error on the part of the district court, the cumulative error doctrine does not apply.

E. Sufficiency of Evidence

The jury convicted Green of fraud in connection with the property transactions at 8544 South Givins Court, 1418 Portland Avenue, and 155 East 153rd Street, and acquitted him of the charge in connection with 6851 South Prairie Avenue. He argues that the government failed to prove that he had the intent to defraud any lender and that therefore he should have been acquitted on all counts. He concedes that he participated in the scheme and that a wire fraud occurred, but he contends he was actually a victim of the schemeunaware of the illegality of the transactions and not intending to defraud the lenders. Green's challenge fails because the testimony at trial, along with the documentary evidence, was sufficient for a jury to find beyond a reasonable doubt that Green engaged in wire fraud for each of the three properties underlying his convictions.
Early on, Green worked through his cousin and recruiter Joseph Green. Joseph Green testified that he told defendant Green in their earliest conversations about real estate that Green would be paid for his transactions and that he could get assistance obtaining false pay stubs and W-2s when necessary. The jury heard that one of Green's co-schemers, James Robert Thomas, told Green that he had to purchase false documents to qualify for a loan for the South Givins property in early 2005. James Robert Thomas also told Green that Green did not have to repay the "seller second" mortgage that had been arranged for him and that was listed on his loan application for the South Givins house.
Beginning with the transaction at 1418 Portland Avenue, Green worked more closely with the financiers of the scheme rather than through his cousin. James Robert Thomas testified that he paid Green after the closings at the Portland property and East 153rd Street in April and May 2005. At trial, Green himself admitted that he accepted money for his role as the purchaser of all three properties without notifying lenders about these transactions. Further, the jury heard that, at his cousin's instruction, Green falsely represented to lenders that he worked for a company called The Art of Construction. Viewing this testimony in the light most favorable to the prosecution, a reasonable fact finder could have found that Green intended to defraud lenders through his participation in the scheme.

F. Loss Calculation in Sentencing

Green was sentenced to 37 months in prison based in part on an aggregate loss amount of $189,500. He contests the district court's calculation of the loss amount attributable to him. We review loss calculations for clear error. See United States v. Powell, 576 F.3d 482, 497 (7th Cir.2009). We have stated on many occasions that loss calculations need only be a reasonable estimate of the loss. See United States v. Borrasi, 639 F.3d 774, 783 (7th Cir.2011); U.S.S.G. § 2B1.1, note 3(C) ("The court need only make a reasonable estimate of the loss."). For Green to succeed, he must show that the court's loss calculations "were not only inaccurate but outside the realm of permissible computations." United States v. Radziszewski, 474 F.3d 480, 486 (7th Cir.2007), quoting United States v. Lopez, 222 F.3d 428, 437 (7th Cir.2000).
At his sentencing hearing, Green introduced evidence showing that some of the properties involved in the fraud were sold at public auction and requested that proceeds *584 from the sales, at which the lenders were the highest bidders, be credited against the loss. The district court rejected Green's calculations. Green re-asserts his argument on appeal, claiming that the district court improperly calculated the loss amount by not using the prices at which the lenders obtained title to the properties at the public auctions.
Green's suggested calculation misses the mark. Where a lender forecloses and acquires the property at public auction by making a credit bid (i.e., a bid that offers to cancel the outstanding principal, interest, and related fees in return for title to the property), the credit bid is not a reliable measure of the actual market value of the property. See generally River Road Hotel Partners, LLC v. Amalgamated Bank, ___ F.3d ___, ___ (7th Cir.2011); In re Philadelphia Newspapers, LLC, 599 F.3d 298, 320-21 (3d Cir.2010) (Ambro, J., dissenting) (explaining credit bidding in Chapter 11 bankruptcy context).[6] In a typical fraudulent mortgage scheme, a credit bid is highly likely to overvalue the property. The whole point of the fraud was to fool the lender into lending far more than the market value of the property, and then to disappear, leaving the lender with a property worth far less than the loan. Using a credit bid based on the fraudulently inflated loan amount to measure loss would surely understate the actual loss. Thus, in this situation, it would have been an error for the district court to use Green's proposed method of calculating loss.
Here, the district court correctly determined the appropriate loss amount using the formula we outlined in United States v. Radziszewski. The court subtracted the sale price the lender received after it recovered possession of the property from the amount of its original loan, as in Radziszewski. See 474 F.3d at 486-87; see also United States v. Serfling, 504 F.3d 672, 679 (7th Cir.2007) (upholding district court's loss calculation which subtracted the price obtained for collateral from the amount of loan proceeds, and rejecting calculation proposed by defendant based on fraudulent appraisal); U.S.S.G. § 2B1.1, note 3(E)(ii) (loss shall be reduced by "the amount the victim has recovered at the time of sentencing from disposition of the collateral, or if the collateral has not been disposed of by that time, the fair market value of the collateral at the time of sentencing"). The district court used this method that we have previously upheld for the same situation and that properly captures the loss suffered by the lenders. We find no error and uphold Green's sentence.

IV. Conclusion

The convictions and sentences of defendants Braziel, Miller, and Green are AFFIRMED.
NOTES
[1]  The government asserts that Braziel forfeited any objection to the use of the statement when he did not object on two occasions before trial when it was discussed, nor did he immediately object when the statement came in at trial. We defer to the district court's discretionary judgment, however, that Braziel sufficiently preserved his objection by raising it shortly after the introduction of the statement during Special Agent Kaiser's testimony.
[2]  We have inserted "[buyer]" where the trial transcript says "lawyer." The summary chart from which Special Agent Kaiser was reading names Braziel as the buyer. In context it is clear that Special Agent Kaiser must have said "buyer."
[3]  "Seller second" mortgages are junior mortgage loans, subordinate to a first mortgage on a property, taken by the buyer from the seller. See Black's Law Dictionary 1103 (9th ed.2009).
[4]  We refer to defendant James Green as Green and when necessary refer to his cousin as Joseph Green.
[5]  Our caution does not apply when records, especially of fraudulent activity, are introduced not to prove the truth of the matters asserted in the records but to show the course of the transaction or scheme, such as the communication of false information. Such uses are outside the scope of the definition of hearsay. Fed.R.Evid. 801(c).
[6]  In River Road Partners, we agreed with Judge Ambro's dissent in Philadelphia Newspapers.
