Felix Nkansah appeals from his conviction after a jury trial before Judge Rakoff for: (i) conspiracy to file false claims with the Internal Revenue Service (“IRS”) in violation of 18 U.S.C. § 286 (“Count One”); (ii) filing false claims with the IRS in violation of 18 U.S.C. § 287 (“Count Two”); (iii) bank fraud in violation of 18 U.S.C. § 1344 (“Count Three”); (iv) aggravated identity-theft related to the bank fraud charged in Count Three in violation of 18 U.S.C. § 1028A(a)(l), (c)(5) (“Count Four”); and (v) identity-theft in violation of 18 U.S.C. § 1028(a)(7), (b)(1) (“Count Five”).
We hold that there was insufficient evidence for appellant’s conviction on Count Three and that his conviction on Counts Three and Four must, therefore, be vacated. We find appellant’s other arguments to be without merit but do not address his claim regarding the substantive reasonableness of his sentence.
BACKGROUND
Viewing the evidence in the light most favorable to the government, see United States v. Chavez, 549 F.3d 119, 124 (2d Cir.2008), appellant was part of a group that, beginning in early 2005 through August 2008, stole names, dates of birth, and social security numbers from foster care, hospital, and childcare databases. This information was used to file thousands of fraudulent tax returns in the victims’ names with fictitious income figures, resulting in tax refunds either sent as a check to the address of a group member or electronically deposited into a bank account controlled by a group member. The group, expecting about half the refunds to be approved by the IRS, filed for $2.2 million in fraudulent refunds and ultimately obtained $536,167. When a refund was received in the form of a check made out to an identity-theft victim, a group member would forge the payee’s signature along with an endorsement over to the group member. The particular group member would deposit the check into a controlled bank account and would soon thereafter withdraw the money.
Appellant was linked to deposits at Commerce Bank, HSBC, and Bank of America. A search of his home and car revealed stolen identity information, tax refund correspondence to identity-theft victims sent to appellant’s address, and a computer with a partially-completed tax refund made out in an identity-theft victim’s name. Other evidence linked nearly 70 fraudulent tax returns to an IP address registered to appellant. On the same day, one of appellant’s bank accounts received two federal tax refunds of several thousand dollars each. Evidence linked appellant to checks made out to identity-theft victims and endorsed over to a “William K. Arthur.” These checks were deposited into a Bank of America account that appellant controlled under that name.
Appellant was arrested on September 9, 2008 and ultimately charged with the five *747counts described above. Count Five of the indictment for identity-theft did not include a reference to interstate commerce, an element of the crime for which he was convicted. While on bail prior to trial, appellant fled to Canada where he was later apprehended and returned to the United States. After an aborted plea deal, he was found guilty by a jury on January 29, 2010, on all five counts. He was sentenced principally to 51 months’ imprisonment on each of Counts One through Three and Five to run concurrently and 24 months’ imprisonment on Count Four to run consecutively to the other counts.
At trial, the government sought to show that the banks in which deposits were made by the group were at a risk of loss. Special Agent Peck of the Secret Service testified:
When the bank transmits the funds to be collected and it comes back as not accurate or a counterfeit check or a fraudulent check, they no longer will get those funds back and they, most of the time, have already given out the funds to a payee or someone else. It is withdrawn.
He also testified that Commerce Bank suffered financial losses “for not just Mr. Nkansah himself but the combined total in the case” as a result of the scheme. However, when pressed about specific losses suffered by banks as a result of appellant’s specific use of accounts, Agent Peck could not confirm that such losses occurred. He also testified that while he thought that such banks bore the loss from accepting for deposit fraudulently obtained Treasury checks, he was “unsure” if that theory was correct.
During jury deliberations, the government inadvertently provided the jury with documents that had not been introduced into evidence. In particular, a standard form proffer agreement was included. Upon being notified of this by the government, the district court sent the courtroom deputy into the jury room, which had already been vacated for the evening, to retrieve the exhibit. It was still in the manila folder in which it had been originally housed. The folder was still inside the Redweld in which it had been given to the jury. Because of the circumstances surrounding the exhibit’s location, the court concluded that it was likely that the jury had not seen the proffer agreement. The court further found that, even if the jury had seen the proffer, it contained nothing in evidence and, in any event, nothing material to any issue not already established in the case — usually from Nkansah’s own testimony. Other documents mistakenly given to the jury were found to be similarly duplicative of evidence already before the jury.
DISCUSSION
We address each of appellant’s challenges in turn.
a) Bank Fraud Conviction
We turn first to the sufficiency of the evidence regarding appellant’s bank fraud conviction.
We note the familiar standard that sufficiency challenges are reviewed de novo, United States v. Leslie, 103 F.3d 1093, 1100 (2d Cir.1997), but a defendant challenging the sufficiency of the evidence bears a “heavy burden,” United States v. Gaskin, 364 F.3d 438, 459 (2d Cir.2004) (internal quotation marks omitted). Appellant’s claim, however, turns largely upon the legal definition of the defendant’s state of mind that must be proven for purposes of a bank fraud conviction.
The federal bank fraud statute, 18 U.S.C. § 1344, provides:
*748Whoever 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.
Appellant knowingly used deception with regard to the bank accounts he controlled: (i) he opened them in the names of other people as well as the fictional William K. Arthur; and (ii) he deposited in the accounts checks fraudulently obtained from the United States Treasury causing the bank to seek reimbursement from the Treasury. Appellant argues, however, that the government was required to prove that he intended to victimize the banks as opposed to the Treasury. He claims that there was no evidence of such an intent or even that the banks had actually lost money. In essence, he argues that the banks were no more victims of his deceptions than a bank in which someone opens an account under a false identity to conceal funds from a spouse or business partner.
Appellant is correct that the bank fraud statute is not an open-ended, catch-all statute encompassing every fraud involving a transaction with a financial institution. Rather, it is a specific intent crime requiring proof of an intent to victimize a bank by fraud. See United States v. Rubin, 37 F.3d 49, 54 (2d Cir.1994). “[A] federally insured or chartered bank must be the actual or intended victim of the scheme.” United States v. Stavroulakis, 952 F.2d 686, 694 (1992); see also United States v. Blackmon, 839 F.2d 900, 906 (2d Cir.1988) (“Where the victim is not a bank and the fraud does not threaten the financial integrity of a federally controlled or insured bank, there seems no basis in the legislative history for finding coverage under section 1344(a)(2).”); S.Rep. No. 98-225, at 377 (1983), reprinted in 1984 U.S.C.C.A.N. 3182, 3517(bank fraud Statute designed to “assure a basis for federal prosecution of those who victimize these banks through fraudulent schemes.”) Therefore, convictions for bank fraud are limited to situations where “the defendant (1) engaged in a course of conduct designed to deceive a federally chartered or insured financial institution into releasing property; and (2) possessed an intent to victimize the institution by exposing it to actual or potential loss.” United States v. Barrett, 178 F.3d 643, 647-48 (2d Cir.1999).
Our concurring colleague takes serious issue with the need to prove intent to harm a financial institution, albeit he concedes that this element is well-established in our caselaw. We note only that the government has argued none of the points he makes and begins its discussion of this issue with the following statement: “The bank fraud statute was enacted to ‘protect ] the financial integrity of [federally guaranteed financial] institutions, and ... assure a basis for Federal prosecution of those who victimize these banks through fraudulent schemes.’ S.Rep. No. [98-225], [at] 377 (1983), reprinted in 1984 U.S.C.C.A.N. 3182, 3517.” Brief of Appellee at 16, United States v. Gyanbaah (Nkansah), 10-2441 (2d Cir. Apr. 13, 2011). The ensuing discussion then goes on to underline the need to prove the intent to harm a financial institution.1
*749The government had to prove beyond a reasonable doubt that appellant intended to expose the banks to losses.2 Were that intent proven, the actuality, or even possibility, of losses would be irrelevant. However, there is no direct evidence of appellant’s intent to victimize the banks at which he opened accounts under the name of others or the fictitious William K. Arthur and deposited fraudulently obtained Treasury checks. The government therefore relies on inferences to be drawn from two pieces of circumstantial evidence: (i) conversations between appellant and other participants in the scheme; and (ii) the actual exposure of the banks to loss as a result of appellant’s deceptions, based largely on the testimony of Agent Peck.
Appellant and other participants in the scheme discussed which banks would be the least likely to detect the scheme and/or the quickest to make the proceeds from the deposits available for withdrawal. While these concerns surely support an inference of an intent to avoid detection, on this record they have no probative value as to an intent to injure the banks. A bank’s detection of appellant’s depositing a fraudulently obtained tax refund check would lead to his arrest whether or not the bank was exposed to a loss. If the Treasury detected the fraud first, it would notify the bank, and the bank’s efficiency at detection would be irrelevant. The conversations are, therefore, not probative of appellant’s intent to injure the banks.
The government also relies on the banks’ claimed exposure to losses resulting from appellant’s scheme as circumstantial evidence of appellant’s intent to victimize the banks. It relies in this regard on cases in which we have affirmed bank fraud convictions based on such exposure absent direct evidence of the defendant’s *750state of mind. However, these cases all involved a defendant who fraudulently sought to cause a bank to pay out to the defendant some of a depositor’s account in that bank, e.g., cashing a forged check, see, e.g., United States v. Crisci, 273 F.3d 235 (2d Cir.2001); Barrett, 178 F.3d 643, or to release funds for which the releasing institution was liable, e.g. presenting a falsely certified draft, see United States v. Jacobs, 117 F.3d 82 (2d Cir.1997). Although the rationale of these decisions is generally not elaborated, we read them to hold where the direct legal exposure to losses is sufficiently well-known, a jury may infer that the defendant intended to expose the bank to the loss.
However, the widely understood exposure of a bank in such a case is only a fact sufficient to support an inference of the requisite state of mind. Someone may well forge a check believing that only the account holder will suffer a loss. The inference is, therefore, not mandatory, but permissible. Such a permissible inference cannot be extended to cases in which evidence of the state of mind is absent and the actual exposure of a bank to losses is unclear, remote, or non-existent. See United States v. Rodriguez, 140 F.3d 163, 169 (2d Cir.1998) (holding that when bank had no risk of loss because it was a holder in due course and where no other evidence showed intent, a bank fraud conviction must be overturned).
In the instant matter, there is no clear, much less well-known, exposure of the banks to loss. Indeed, until alerted by the Treasury to the scheme, the banks may well have been holders in due course with the risk of loss borne entirely by the Treasury. See id.
For example, appellant opened the William K. Arthur account by providing a false passport as identification. He used the account from October 2006 through at least May 2008. From October 2007 through April 2008 he deposited several fraudulently-obtained but genuine tax refund checks each month, often withdrawing the balance soon thereafter. The tax refund checks were made out to an identity-theft victim, and appellant would forge the victim’s signature on the check with an endorsement over to William K. Arthur. Appellant would then deposit the check in the Arthur account. Some deposits were made by ATM; others by teller.
The checks were genuine Treasury checks. The signature of the final endorsee, William K. Arthur, was the authorized signature for the account and was the only signature the bank needed to verify to take the checks as a holder in due course. There is no evidence of the Treasury dishonoring the checks or seeking reimbursement from any of the banks. Our analysis does not depend on dispositively finding that Bank of America was not exposed to risk of financial loss.3 It is sufficient to say that there is not the well-known exposure to loss that might support a finding beyond a reasonable doubt of appellant’s intent to victimize Bank of America.4
*751Agent Peck’s testimony does not alter our conclusion. He testified generically that banks lose money if a check is returned and that Commerce Bank lost funds as a result of the scheme. However, he also stated that he did not know whether any banks lost money as a result of appellant’s specific depositing of Treasury checks, did not know of a specific case where the Treasury dishonored a check or sought reimbursement from a bank used by appellant, and was “unsure” of whether a bank was exposed to reimbursing the Treasury for accepting such checks. Such murky testimony cannot establish a sufficiently well-known exposure to loss by a bank to prove appellant’s intent to victimize banks beyond a reasonable doubt.
Therefore, appellant’s conviction on Count Three must be overturned.5 As a consequence, his conviction on Count Four for aggravated identity-theft, which required the use of identity-theft in connection with bank fraud, must also be overturned.
b) Extrinsic Evidence Provided to Jury
Appellant next challenges his convictions on the ground that “highly incriminating extrinsic material” — his proffer agreement — was sent to the jury room during deliberations. He argues that the district court should have held an evidentiary hearing to determine the impact on the jury. However, appellant not only failed to ask for such a hearing but also agreed with the district court’s handling of the issue. We, therefore, review only for plain error. To establish plain error, appellant must show
(1) there is an error; (2) the error is clear or obvious, rather than subject to reasonable dispute; (3) the error affected the appellant’s substantial rights, which in the ordinary case means it affected the outcome of the district court proceedings; and (4) the error seriously affect[s] the fairness, integrity or public reputation of judicial proceedings.
United States v. Marcus, — U.S. -, 130 S.Ct. 2159, 2164, 176 L.Ed.2d 1012 (2010) (alteration in original) (internal quotation marks and citations omitted). Trial courts have “wide discretion in deciding how to pursue an inquiry into the effects of extra-record information,” United States v. Hillard, 701 F.2d 1052, 1064 (2d Cir.1983), and “the trial judge’s conclusions regarding the effect of the extra-record evidence on the jury are entitled to substantial weight,” United States v. Hansen, 369 Fed.Appx. 215, 216 (2d Cir.2010) (citing United States v. Weiss, 752 F.2d 777, 783 & n. 2 (2d Cir.1985)).
There is no plain error. The district court determined that the mistaken submission of the proffer agreement did not affect appellant’s substantial rights. The basis for that conclusion was that it was highly likely the jury did not view the proffer agreement before it was recovered and that, even if seen by the jury, its contents were either immaterial or already on the record and therefore harmless. We agree.
*752c) Omission of Interstate Commerce Element from Identity-Theft Count
Appellant next argues that omission of the requisite interstate commerce element from the indictment on Count Five for identity-theft, see 18 U.S.C. § 1028(a)(7) & (c), requires reversal of that conviction. We review such an omission for “constitutional infirmities, most notably whether the alleged defect offends the Sixth Amendment right of the accused to be informed of the charges against him, the Fifth Amendment right not to be prosecuted without indictment by grand jury, or the Fifth Amendment protection against double jeopardy.” United States v. Wydermyer, 51 F.3d 319, 324 (2d Cir.1995) (citations omitted). When, as here, the argument is raised only after trial, “we interpret the indictment liberally in favor of sufficiency, absent any prejudice to the defendant,” id., and require “a clear showing of substantial prejudice to the accused — such as a showing that the indictment is so obviously defective that by no reasonable construction can it be said to charge the offense for which conviction was had,” id. at 325 (quoting United States v. Thompson, 356 F.2d 216, 226 (2d Cir. 1965), cert. denied, 384 U.S. 964, 86 S.Ct. 1591, 16 L.Ed.2d 675 (1966)). We have consolidated these elements into a requirement that the omission is reversible only for plain error, the standard of which is discussed above. United States v. Cotton, 535 U.S. 625, 631, 122 S.Ct. 1781, 152 L.Ed.2d 860 (2002); United States v. Doe, 297 F.3d 76, 81-82 (2d Cir.2002).
Even if omission of the interstate commerce language constitutes error that is plain, it did not impact appellant’s substantial rights. When “notice adequate to allow [a defendant] to prepare a defense” is provided, omissions in the indictment do not affect substantial rights. Doe, 297 F.3d at 88 n. 12. At least ten days before trial, the government submitted proposed jury instructions in which the issue of interstate commerce was specifically discussed with reference to the identity-theft count. Appellant’s counsel neither objected to the proposed instruction nor expressed surprise at the interstate commerce language’s inclusion. Even if this were the first time appellant’s counsel became aware of the interstate commerce element, he had adequate time to address it before trial or to ask for additional time. There was therefore no harm to appellant’s substantial rights.
Nor can appellant prevail on his argument that “[a] defendant is deprived of his right to be tried only on the charges returned by a grand jury when an essential element of those charges has been altered.” When there is “overwhelming” evidence in support of the missing indictment element, the grand jury surely would have found the missing element, and the right to be tried on only charges returned by the grand jury is not violated. Cotton, 535 U.S. at 633, 122 S.Ct. 1781. The evidence of appellant engaging in interstate commerce via the internet in support of the identity-theft is overwhelming. For example, IP addresses registered to appellant were used to prepare fraudulent tax returns.
d) Unreasonableness of Sentence
Finally, appellant argues that his sentence is unreasonable because of: (i) procedural unreasonableness in the district court’s failure to use an “actual loss” measure — $536,167—rather than a modified intended loss measure — over $1 million; and (ii) substantive unreasonableness because his sentence of 75 months was much greater than that of his co-defendants.
Sentences are reviewed for procedural and substantive unreasonable*753ness using a deferential abuse-of-discretion standard. See Gall v. United States, 552 U.S. 38, 46, 128 S.Ct. 586, 169 L.Ed.2d 445 (2007). Issues of law are reviewed de novo, issues of fact are reviewed for clear error, and mixed issues of law and fact are either reviewed de novo or for clear error depending on whether the question is predominantly legal or factual. See United States v. Thorn, 446 F.3d 378, 387 (2d Cir.2006).
With regard to the claimed procedural unreasonableness, the Sentencing Guidelines provide that district courts are to use the greater of actual or intended loss. See U.S.S.G. § 261.1, cmt. n. 3(A). It was therefore not error for the district court to use intended loss, even if this number was greater than actual loss. Furthermore, a district court “need only make a reasonable estimate of the loss” given the “available information.” U.S.S.G. § 261.1, cmt. n. 3(C). Appellant and his co-conspirators filed returns totaling $2.2 million, but a co-conspirator testified that about half the claims were expected to be rejected by the IRS. Therefore, an intended loss figure of over $1 million is not clear error, because it was reasonable to find that appellant expected half of the $2.2 million in filed returns, or $1.1 million, to succeed.
We need not address appellant’s substantive unreasonableness argument at this time because the convictions on Counts Three and Four must be vacated and resentencing is required. Nevertheless, we note for purposes of future proceedings that several factors support a sentence for appellant that is in considerable excess of those of his co-defendants. Appellant went to trial, whereas his co-defendants agreed to plea deals. Also, appellant fled the country while on bail before trial, and, during the trial, he repeatedly lied. A significant disparity between appellant’s and his co-conspirators’ sentences is, therefore, to be expected.
CONCLUSION
For the foregoing reasons, we vacate the convictions on Counts Three and Four and remand for resentencing. Otherwise, we affirm.
. Our colleague notes that our opinion does not follow the literal language of the statute. *749We agree but also note that our colleague's construction of the statute suffers from a similar failing. Read literally, the statute would encompass a fraudulent scheme inducing a victim to write a valid check to the perpetrator who then cashes it to obtain "moneys ... under the custody or control” of a bank. However, our colleague avoids this reading of the statute by adding an element not in the statutory language, namely, that the scheme include a lie to the bank, albeit, in his view, a lie not harmful to the bank. In contrast, the language of the statute requires only fraudulent "representations” without any requirement that they be made to the financial institution. Requiring a lie to the financial institution thus adds an element not in the statutory language that is presumably inferred from the statutory purpose. The further element of a lie intending harm to the institution, required by our decisions, is an inference drawn more accurately from that same purpose.
. Our concurring colleague expresses concern that this reading of the statute leaves a gap in the tools of federal law enforcement enabling many fraudsters to escape prosecution. We believe that concern to be exaggerated. The statute in question was designed to expand criminal liability beyond the wire or mail fraud statutes for persons who were also endangering the financial health of financial institutions. See S.Rep. No. 98-225, at 377-78 (1983), reprinted in 1984 U.S.C.C.A.N. 3182, 3517-18. It is hardly the case that by limiting the statute to its intended purpose — prosecution of schemes intended to harm financial institutions — prosecutors are left bereft of necessary authority. For an immediate example, our opinion leaves appellant's convictions for three federal felonies standing.
The present issue arises only because the U.S. Treasury is not within the statute's definition of a "financial institution.” See 18 U.S.C. § 20 (listing exhaustively the entities that qualify as financial institutions for the purposes of title 18). Our decision leaves untouched the caselaw allowing a finding of intent to injure where the liability of drawee banks is clear, thereby fulfilling the congressional intent to protect banks from check-kiting schemes. See S.Rep. No. 98-225, at 378 (1983), reprinted in 1984 U.S.C.C.A.N. 3182, 3518. Moreover, we leave untouched the statute's protection of banks from fraudulent practices in the securing of loans.
. We have recognized that banks can sometimes suffer various reputational or other indirect harms that will satisfy bank fraud’s intent element even if they are a holder in due course. See Barrett, 178 F.3d at 648 n. 3 (recognizing that "banks face practical adverse consequences and potential liability problems when they cash checks over forged endorsements because ... they may suffer a loss of customer good will” or "institutional embarassment.”) In the cases where these nonpecuniary harms are recognized, the holder in due course is also the drawee's bank, and enforcing tire holder in due course status against the drawee could negatively impact the bank’s relationship with its drawee customer. See id. In the present matter, there is no evidence, or argument by the government, that any bank was at risk of any such loss.
. Appellant was associated with similar schemes at other banks, but as is the case in the Bank of America scheme, there was no evidence that these banks were exposed to risk of loss.
. The government also argues that appellant could have been convicted for bank fraud as an aider and abettor. However, the court's jury instructions on aiding and abetting liability were given only on an unrelated count. Therefore, aiding and abetting could not have been a basis for appellant's bank fraud conviction. See Napier v. United States, 159 F.3d 956, 960 (6th Cir.1998).