This is an appeal from criminal convictions for perjury, conspiracy to commit perjury, and aiding and abetting perjury. The defendant’s allegedly false statements were made in an affidavit submitted in response to a summary judgment motion in a civil proceeding.
The defendant makes the following contentions on appeal: (1) that the evidence presented at trial was insufficient to sustain a conviction for perjury because the statements at issue were literally true; (2) that because the statements were literally true, the evidence was also insufficient to support a conviction for aiding and abetting or for conspiracy to commit perjury; (3) that the evidence was insufficient to prove that the defendant’s affidavit was signed under oath; and (4) that the district court erred in admitting testimony by the federal judge who handled the civil action. Finding none of these assignments of error persuasive, we shall affirm the challenged judgment.
I
The defendant, Howard Nicholas Johnson, was the president of an entity called the Foundation for Advancement, Education and Employment of American Indians. William Preston Campbell was engaged in soliciting donations to the foundation in return for a percentage of the amounts donated.
In 1992, an international businessman by the name of John Mousourakis was introduced to Messrs. Johnson and Campbell by his attorney, Norm Rollins, an acquaintance of Campbell. Mr. Mousourakis was having federal income tax problems, and Mr. Johnson suggested that if Mr. Mousourakis gave the foundation several parcels of real estate he owned in Chattanooga, Tennessee, Mr. Mousourakis *233would be eligible for a “carry-back” tax deduction that could be used to reduce his 1990 tax liability. It was agreed that Mousourakis would convey the property to the foundation, which was allegedly to hold the property in escrow pending receipt of a favorable ruling from the IRS concerning the carry-back deduction.
After the delivery of the deed, part of the property was conveyed to a third party. Having consulted with several accountants and attorneys, Mr. Mousourakis became increasingly dubious about the availability of a carry-back deduction. He then began tape-recording his telephone conversations with the defendant. In 1994, no favorable ruling having been received from the IRS, Mr. Mousourakis brought a federal action against the foundation for breach of contract and misrepresentation.
Mr. Mousourakis promptly moved for summary judgment. In support of the motion, he submitted an affidavit that included the following statements:
“3. That I was induced to donate to the Defendants in this matter by their assurances that I would be able to carry back my tax deduction for the properties described in this action at 100% of the donated volume [sic] or that they would be returned to me. I was further promised that all properties were to be held in escrow pending the production of a ‘Private Letter Ruling’ from the IRS to this effect.”
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“5. That the property was valued by the Defendant at the sum of $210,000.00 by Malcolm Crossland, the in-house counsel of Defendant. ... That I have respectfully made demand of Defendant for the return of our property or its value of $210,000.00, which Defendant has consistently promised but not carried out.”
In response to the motion for summary judgment, Messrs. Johnson and Campbell submitted their own affidavits. Mr. Johnson’s submission was titled “AFFIDAVIT OF H. NICHOLAS JOHNSON.” The first paragraph stated, “My name is H. Nicholas Johnson. I reside at 1259 Windsor Drive, Gallatin, Tennessee 37066.” The affidavit then continued, in relevant part:
“6. The statements contained in Paragraph 3 of John Mousourakis’ Affidavit are categorically denied. I never told him he would be would be [sic] able to carry back his deduction for the properties described in this action at 100% of the donated amount or that the properties would be returned to him. Furthermore, I never promised that these properties were to be held in escrow pending the production of a ‘private letter ruling’ from the IRS to that effect.”
“7. I have never promised that I would see that their property was returned or its value of $210,000.00.... ”
The relevant portions of Mr. Campbell’s affidavit were nearly identical:
“5. The statements contained in Paragraph 3 of John Mousourakis’ Affidavit are categorically denied. I never told Mr. Mousourakis nor did I ever hear H. Nicholas Johnson tell him in my presence that he would be would be [sic] able to carry back his deduction for the properties described in this action at 100% of the donated amount or that the properties would be returned to him. Furthermore, I never heard Mr. Johnson, nor did I promise that these properties were to be held in escrow pending the production of a ‘private letter ruling’ from the IRS to that effect.”
“6. I have never heard Mr. Johnson promise the plaintiffs that he would see *234that their property would be returned or its value of $210,000.00.... ”
Beneath the affiant’s signature on each affidavit was a jurat bearing the signature and seal of a notary public. The jurat on the Johnson affidavit read as follows:
“STATE OF TENNESSEE
COUNTY OF DAVIDSON
Sworn to and subscribed before me this 25th day of January 1995.
/s/ Gregory J. Cullen
Notary Public My Commission Expires July 26,1997.”
Based on these affidavits, Judge James Jarvis, to whom the civil suit had been assigned, concluded that there were genuine issues of material fact. Accordingly, the motion for summary judgment was denied.
On March 3,1995, Mr. Mousourakis filed a renewed motion for summary judgment that was supported by transcripts of tape-recorded conversations between himself and Messrs. Johnson and Campbell. In one of these conversations, Mr. Johnson was recorded telling Mr. Mousourakis the following:
“When we did the deal with you we went ahead and filed it in our books that we did the deal with you, and we also filed an agreement that we had a commitment that if for some reason that didn’t work, then we would restore the property or pay the damages or pay for the property or whatever.”
Later in the same conversation, Mr Johnson added this:
“And you came to us and we told you we would either get it done for you, or we’ll give your property back or give you the money back.”
Mr Johnson was also recorded assuring Mr. Mousourakis of the following:
“And if your ruling turns wrong, we’ll make it right, and if your ruling turns right you got a home run. So you’re not going to lose. It’s nothing to lose sleep over. You’re in no position to lose. You either get what you got or get your property back____”
The defendant failed to respond to the renewed motion. Concluding that no genuine issue of material fact remained, Judge Jarvis granted summary judgment in favor of Mousourakis for $210,000.
On April 13, 1999, a grand jury handed up an indictment against the foundation, Mr. Johnson, and Mr. Campbell. A second superseding indictment was filed on October 27, 1999. Count I charged Mr. Johnson and the Foundation with conspiracy to commit perjury, in violation of 18 U.S.C. § 1621 and 18 U.S.C. § 371. Count II charged Mr. Johnson with perjury, in violation of 18 U.S.C. §§ 1621, 1622, citing 11116 and 7 of his affidavit as the false statements at issue. Count III charged Mr. Johnson with aiding and abetting Mr. Campbell in the commission of perjury and Mr. Campbell with perjury, in violation of 18 U.S.C. §§ 1621,1622.
On November 12, 1999, a motion in limine was filed seeking to exclude the proposed testimony of Judge Jarvis. The court denied the motion, holding that “Judge Jarvis can testify for the government, and may testify, if he can, that the statements made by the defendant in the ease before him either influenced his decision in that case, or were capable of doing so. * * * However, Judge Jarvis will not be permitted to testify as an expert witness as to his opinion about whether the alleged false statements were material.” The court’s order further stated that the jury would be given a cautionary instruction in this regard.
Following the oral argument on the motion, and after the court had announced its ruling, the court raised the possibility of a stipulation as to materiality. Although such a stipulation would have obviated any *235need to call Judge Jarvis as a witness, defense counsel rejected the suggestion.
The same day, Mr. Campbell, without benefit of a plea agreement, pleaded guilty to Count III of the indictment.
The defendant’s trial commenced on November 16 before a jury. The government presented the testimony of three witnesses: Judge Jarvis, Mr. Mousourakis, and Mr. Campbell. We turn to Judge Jarvis first.
Judge Jarvis was asked to define for the jury the terms “motion for summary judgment” and “affidavit.” He did so. He also stated that “by definition,” an affiant “has sworn that the contents of [the affidavit] are true.” He then testified, in pertinent part, that the affidavits submitted by Messrs. Johnson and Campbell were “determinative” of the summary judgment motion, which he explained meant, “They decided the issue of whether or not there was a question of fact. They raised a question of fact. And so, therefore, I denied the motion for summary judgment and set the case for trial____” Judge Jarvis also testified that after receiving transcripts of the tape-recorded conversations made by Mr. Mousourakis, and after Mr. Johnson and the foundation failed to respond, “... I granted the plaintiffs motion for summary judgment because these statements that I had, the transcripts of these statements were in direct contradiction to the statements that [Mr. Johnson and Mr. Campbell] made under oath.” There was no objection at this stage to any of the judge’s testimony.
Following the testimony, the court gave the following cautionary instruction:
“Ladies and Gentlemen of the jury, you’ve heard the testimony here of Judge Jarvis. You should consider the testimony just as you would consider the testimony of any other witness in the case. He has testified about certain facts. However, he has not testified about the law that is to be applied by you in this case. Remember that the law that you apply in the case comes only from me as the presiding judge in the case.”
Mr. Mousourakis was called as the second government witness. He testified that Mr. Johnson and Mr. Campbell had “told me that under the current, the then laws if you were to donate to the Foundation, you could carry the amount to either two years back or take it now or carry it three years forward. In other words, you can utilize the value or a portion of the value of those properties to reduce your overall tax liability.” Concerning the private letter ruling from the IRS, Mr. Mousourakis testified that if such a ruling could not be obtained. “They had promised me they would either return the properties back or give us value for the properties.” Mr. Mousourakis explained that he understood that pending an IRS ruling, the land would be held in escrow: “That was my understanding, that they would be holding them until, as I said, I received my credit or deduction, I’m not sure which is the correct term.” Once he discovered the land had been sold, Mr. Mousourakis testified, “They said don’t worry about this, we will make good. In other words, we will pay for the value, for the value of that property to you.” Mr. Mousourakis also testified that the defendant’s attorney had appraised the property at issue at $210,000 and that Mousourakis had received a paper from him reflecting this figure.
The government’s final witness was Mr. Campbell, who had already pleaded guilty to committing perjury in executing an affidavit nearly identical to the defendant’s. Mr. Campbell testified that despite telling the defendant that he believed his affidavit to be untrue, the defendant “persuaded me that this was the only avenue that we had in order to — when I say ‘we,’ the Founda*236tion, to buy adequate time to make a proper appeal.” Mr. Campbell then testified that he had understood that he was making the statements in his affidavit under oath. The questioning by Assistant United States Attorney Gary Humble continued as follows:
“MR. HUMBLE: And your affidavit, and we’ll go back to it now, you say, the affidavit, the statements contained in Paragraph 3 of Mousourakis’s affidavit are denied. When you denied that, was that a true statement?
MR. CAMPBELL: When I denied that it was not a true statement.
MR. HUMBLE: So Mr. Mousourakis’s allegations were true?
MR. CAMPBELL: They are true.
MR. HUMBLE: Now, you go on to say here that, T never told Mr. Mousourakis, nor did I ever hear H. Nicholas Johnson tell him in my presence that he would be able to carry back his tax deduction for the properties described in this action at 100 percent of the donated amount or that the properties would be returned to him.’ Now, is that a true or a false statement?
MR. CAMPBELL: That’s a false statement.
MR. HUMBLE: And you knew that was false at the time you signed the affidavit?
MR. CAMPBELL: I did, sir.
MR. HUMBLE: And you discussed with Mr. Johnson that that was a false statement?
MR. CAMPBELL: Yes.
MR. HUMBLE: Were you there, in fact, when Mr. Johnson told Mr. Mousourakis words to that effect?
MR. CAMPBELL: That was all discussed that first meeting that we had with Mr. Mousourakis at Norm Rollins’ office.
MR. HUMBLE: Okay. Where you say at the bottom, ‘Furthermore, I have never heard Mr. Johnson, nor did I promise that these properties were to be held in escrow pending the production of a “private letter ruling” from the IRS to that effect’ Was that a true or false statement?
MR. CAMPBELL: That was a false statement, sir.
MR. HUMBLE: Paragraph 6, you say, T have never heard Mr. Johnson promise the plaintiffs that he would see that their property was returned or its value of $210,000.’ Is that true or false?
MR. CAMPBELL: That’s a false statement, sir.”
Following the government’s case-in-chief, the defendant moved for a judgment of acquittal. The motion was denied. The defense rested without calling any witnesses.
The jury found Mr. Johnson and the foundation guilty on all three counts of the indictment. Johnson was sentenced to five months imprisonment, an assessment of $300, and three years of supervised release. The foundation received no punishment as it was unable to pay any fine. This appeal followed.
II
A. The Literal Truth Defense
The defendant first argues that the allegedly perjurious statements in 11116 and 7 of his affidavit were literally true and therefore could not serve as the basis of a perjury prosecution. The defendant also maintains that if the statements were true, they could not serve as the basis of a prosecution for aiding and abetting or conspiracy to commit perjury.
A witness testifying under oath or affirmation violates 18 U.S.C. § 1621 if, with willful intent, he gives false testimony concerning a material matter. See United *237States v. Nash, 175 F.3d 429, 438 (6th Cir.), cert. denied, 528 U.S. 888, 120 S.Ct. 210, 145 L.Ed.2d 176 (1999). It is important to recognize that “the perjury statute is not to be loosely construed, nor the statute invoked simply because a wily witness succeeds in derailing the questioner— so long as that witness speaks the literal truth. The burden is on the questioner to pin the witness down to the specific object to the questioner’s inquiry.” Bronston v. United States, 409 U.S. 352, 360, 93 S.Ct. 595, 34 L.Ed.2d 568 (1973).1
The defendant contends that 11116 and 7 of his affidavit directly responded to UK 3 and 5 of the Mousourakis affidavit and, on those terms, were literally true. That is, as the defendant would have it, H 3 of the Mousourakis affidavit contains two statements of fact: (1) that Mr. Mousourakis was assured by the defendant that he “would be able to carry back [his] tax deduction for the properties described in this action at 100% of the donated [value] or that they would be returned;” and (2) that he was promised that “all properties were to be held in escrow pending the production of a ‘Private Letter Ruling’ from the IRS to this effect.”2 (Emphasis added.) The defendant claims that the government failed to prove that a tax deduction at “100% of the donated [value]”— rather than simply a tax deduction of some amount — was ever promised by the defendant. Accordingly, the defendant asserts that H 6 of his affidavit, denying that Mr. Mousourakis was promised that (1) “he would be able to carry back his tax deduction for the properties described in this action at 100% of the donated amount or that the properties would be returned to him;” and (2) “that these properties were to be held in escrow pending the production of a ‘private letter ruling’ from the IRS to that effect,” was literally true.
Similarly, the defendant claims that 1Í 7 of his affidavit was also literally true. Paragraph 5 of the Mousourakis affidavit averred, as we have seen, that counsel associated with Johnson had valued the real estate at $210,000 and that “I [Mousourakis] have respectfully made demand of Defendant for the return of our property or its value of $210,000.00 which Defendant has consistently promised but not carried out.” The defendant -maintains that the government failed to prove by competent evidence that a “value of $210,000” was ever promised and, therefore, failed to prove that his specific denial of such a promise in 117 of his affidavit was not literally true.
The defendant’s indictment referenced both 11116 and 7 of his affidavit as containing allegedly perjurious statements. *238Mr. Johnson’s conviction must therefore be upheld if a reasonable jury could find that either of the two denials in 116 or the denial in U 7 was not literally true. As to 117, at least, it seems clear to us that a reasonable jury could so find. We deem it unnecessary to decide the question raised as to 11 6.
It was in IT 7 that the defendant denied having promised that he “would see that their property was returned or its value of $210,000.00.” The evidence that this denial was untrue including the testimony of Mr. Mousourakis, who testified that Messrs. Johnson and Campbell “had promised me they would either return the properties back or give us value for the properties.” The government also offered into evidence the tape recordings made by Mr. Mousourakis of his conversations with the defendant. Among the admissions made by the defendant there were the following: “we would restore the property or pay the damages or pay for the property or whatever;” “we’ll give your property back or give you the money back;” “you either get what you got or get your property back.” Finally, Mr. Campbell testified that when in his own affidavit he denied having heard the defendant promise Mr. Mousourakis the return of the property or its value of $210,000, his denial was “a false statement.”
The defendant argues that this evidence failed to establish that the precise figure of $210,000, rather than an unspecified “value,” was ever promised. In context, however, the $210,000 valuation figure allegedly having come from the defendant’s own lawyer, H 7 can fairly be read as recognizing that the property had a value of $210,000 but denying any promise to return the property or its value. In any event, a reasonable jury could well interpret Mr. Campbell’s testimony to mean that he personally heard the defendant promise that the specific value of $210,000 would be reimbursed.
Taking the evidence in the light most favorable to the government, a reasonable jury could conclude that 117 of the Johnson affidavit was not literally true. Having concluded that the literal truth defense fails in this case, we will also uphold the defendant’s convictions for aiding and abetting and conspiracy to commit perjury.
B. A Sworn Affidavit.
The defendant next argues that the government failed to introduce evidence from which a jury could conclude beyond a reasonable doubt that the affidavit was made under oath. The defendant points out that there was no language in the body of the document stating that the affiant had been duly sworn. He also contends that the presence of a notary’s jurat is insufficient to show that an oath was administered.
We are not persuaded. The document at issue is styled “AFFIDAVIT OF H. NICHOLAS JOHNSON.” By definition, an affidavit is a “written or printed declaration or statement of facts, made voluntarily, and confirmed by the oath or affirmation of the party making it, taken before a person having authority to administer such oath or affirmation.” Black’s Law Dictionary, 58 (6th ed.1990). The jurat contained at the end of the document states. “Sworn to and subscribed before me this 25th day of January, 1995.” (Emphasis added.) It is signed by the notary, Gregory J. Guilin, and bears the appropriate seal. These facts are uncontested, and we see no reason why the jury should not have been entitled to take the document at face value.
C. Judge Jarvis’ Testimony.
Finally, the defendant contends that the district court erred in letting Judge Jarvis take the stand. The defendant asserts *239that the judge’s testimony invaded the jury’s province as fact-finder, invaded the trial court’s role in stating the law, and unduly prejudiced the proceeding.
The question before us is whether the district court abused its discretion in holding that the probative value of the evidence to be given by Judge Jarvis was not “substantially outweigh[ed] by the danger of unfair prejudice, confusion of the issues, or misleading the jury....” Fed R. Evid. 403. This discretion in determining the proper balance is interpreted broadly. See United States v. Quinn, 230 F.3d 862, 867 (6th Cir.2000).
It is an essential element of a perjury charge that the alleged false statement be material. See United States v. Gaudin, 515 U.S. 506, 512, 115 S.Ct. 2310, 132 L.Ed.2d 444 (1995). Materiality is a question for the jury. See id. It was thus the government’s burden to prove that the alleged false statements were material. Absent a stipulation by the defendant (which he refused), testimony by Judge Jarvis represented the most natural means of establishing the crucial facts as to materiality. Mindful of the fact that the district court gave the jury an appropriate cautionary instruction, thereby minimizing any potential prejudice, we conclude that there was no abuse of discretion here.
United States v. Zipkin, 729 F.2d 384 (6th Cir.1984), on which the defendant relies, does not compel a different conclusion. In Zipkin, a bankruptcy judge was permitted to testify, over objection, as to questions of law. In the case at bar, by contrast, the district court explicitly ruled that Judge Jarvis could not give opinion testimony, as an expert, on the materiality of Johnson’s allegedly false statements. The judge was to testify, rather, on whether— as a matter of fact — he was or could have been influenced by the Johnson affidavit in his decision to deny the summary judgment motion.
If defense counsel believed that the testimony actually given by Judge Jarvis went beyond the boundaries established by the district court in its ruling on the motion in limine, it was incumbent on counsel to object at the time. See the Advisory Committee notes to Fed.R.Evid. 103(a), stating that “if the opposing party violates the terms of the initial ruling [on an evidentiary issue], objection must be made when the evidence is offered to preserve the claim of error for appeal.” There was no such objection here, and there was no obvious error which, although not objected to, seriously undermined the fairness, integrity or public reputation of the judicial proceedings. See United States v. Koeberlein, 161 F.3d 946, 949 (6th Cir.1998). cert. denied, 526 U.S. 1030, 119 S.Ct. 1278, 143 L.Ed.2d 371 (1999).
AFFIRMED.
. In Bronston, the defendant was charged with perjury based upon the following answers he gave to questions in bankruptcy proceedings:
"Q: Do you have any bank accounts in Swiss banks, Mr. Bronston?
A: No, sir.
Q: Have you ever?
A: The company had an account there for about six months, in Zurich.” Id., 409 U.S. at 354.
Although Bronston had maintained personal accounts in a Swiss bank for five years, his answers were literally true since he did not have a Swiss bank account when he was questioned and since his business did have the account he described. See id. The jury convicted and the Second Circuit affirmed, finding that "for the purposes of 18 U.S.C. § 1621, a nonresponsive answer containing half of the truth which also constitutes a lie by negative implication, when intentionally given in place of the responsive answer called for by a proper question, is perjury.” Id. at 356. The Supreme Court reversed, holding that an unresponsive but literally true answer could not form the basis of a perjury charge. See id. at 360.
. Both the government and the defendant are in agreement that the phrase "to this effect" means to the effect that a deduction for 100% of the donated value could be obtained.