Defendants Michael Binday, James Kevin Kergil, and Mark Resnick appeal from judgments of conviction in the United States District Court for the Southern District of New York (Colleen McMahon, Judge) for conspiracy to commit mail and wire fraud, 18 U.S.C. § 1349, mail fraud, 18 U.S.C. § 1341, and wire fraud, 18 U.S.C. § 1343. Kergil and Resnick were also convicted of conspiracy to obstruct justice through destruction of records, 18 U.S.C. § 1512(k). The convictions arise from .an insurance fraud scheme whereby defendants, who were insurance brokers, induced insurers to issue life insurance policies that defendants sold to third-party investors, by submitting fraudulent applications indicating that the policies were for the applicants’ personal estate plan*565ning. Defendants argue primarily that the government did not prove that they contemplated harm to the insurers that is cognizable under the mail and wire fraud statutes. That basic argument takes several forms, including a sufficiency of the evidence challenge, a constructive amendment claim, and a jury instruction challenge. Defendants also contend that their sentences are procedurally unreasonable because the district court used an erroneous loss amount in calculating their Guidelines sentence ranges. Additionally, Res-nick and Kergil challenge their obstruction of justice convictions on various grounds.
We conclude that there was sufficient evidence that defendants contemplated a cognizable harm under the mail and wire fraud statutes; that the indictment was not constructively amended because the allegations in the indictment and the government’s proof at trial substantially correspond; and that some aspects of the defendants’ challenge to the jury instruction are waived, while the remainder fail on the merits. We reject defendants’ challenges to their sentences and to the obstruction of justice convictions.
Accordingly, for the reasons given herein, we affirm the judgments of conviction and remand the case for the limited purpose of revising the restitution amount as agreed by the parties.
BACKGROUND
I. Defendants ’ Scheme1
Defendants-appellants are insurance brokers who participated in an insurance fraud scheme involving “stranger-oriented life insurance” (“STOLI”) policies.2 A STOLI policy is one obtained by the insured for the purpose of resale to an investor with no insurable interest in the life of the insured — essentially, it is a bet on a stranger’s life. Notably, every relevant state’s law provides that, after a life insurance policy has been issued, an insured may resell that policy to an investor, who would become the policy’s beneficiary and assume payment of the premiums.3 Thus, with respect to transferability, the difference between non-STOLI and STOLI policies is simply one of timing and certainty; whereas a non-STOLI policy might someday be resold to an investor, a STOLI policy is intended for resale from before its issuance. While life insurers are required by law to permit resale of policies originally obtained for estate planning purposes, they are not obligated to issue policies intended for resale from the outset.
STOLI policies became a popular investment in the mid 2000s for hedge funds and others eager to bet that the value of a policy’s death benefits would exceed the value of the required premium payments. In response, many insurance companies— including those that issued the policies relevant here — adopted rules against issuing STOLI policies and took steps to detect them. But insurance brokers such as the defendants — who received commissions from insurers for new policies that they brokered — had a financial incentive to place STOLI policies by disguising them to *566the insurer as non-STOLI policies. By matching a potential insured with a STOLI investor, a broker could generate a commission on a policy that would not have been issued had the insurer known the policy’s true purpose.
In 2006, defendant Michael Binday assembled a network of independent brokers to assist his company, Advocate Brokerage, Inc. (“Advocate Brokerage”), in placing STOLI policies through such deceit. The team included defendant Mark Res-nick, who worked as a field agent, and defendant James Kergil, who supervised a group of field agents. Under Binday’s direction, field agents recruited older persons of modest means to act as “straw buyers” of the STOLI policies. The straw buyers were enticed to participate by promises of six-figure payments once the policies were sold to third-party investors — promises which defendants in some eases honored and in others did not. Bin-day explained to the field agents that he sought straw buyers should who were “between 69 and 85 years’ old,” and “in good enough health to get preferred health or standard health [premium] rates,” but who would not live “too long, to the point where the investors ... would be paying the premium too long.” J.A. at 699, 736.
After securing a straw buyer, defendants arranged for the necessary medical tests and submitted the results to multiple insurers for a preliminary assessment of the “risk class” in which the straw buyer would fall. (It is not alleged that the medical records were falsified.) Defendants also submitted those medical records to companies that used them to prepare reports predicting the straw buyer’s life expectancy. Based on those reports and the insurance companies’ preliminary assessments, Binday generated “illustrations” for prospective STOLI investors that projected the expected premium payments necessary to fund a given value of policy until the straw buyer’s death. The investors could then select from among the different straw buyers and policies, and the defendants would proceed to apply for the policy.4
Defendants typically sought policies worth between $3 million and $4 million: large enough to yield a lucrative commission, .but, as Kergil explained to one witness, small enough to “stay under the radar” because “anything over three to four million would require excessive documentation such as tax returns, stock reports, bank statements, that type of thing.” J.A. at 734. “[Ejxcessive documentation” would be fatal to defendants’ scheme, which depended on vastly inflating the straw buyer’s wealth without detection. Such inflation would cause the insurer to believe that the straw buyer was capable of paying the substantial premiums (typically more than $100,000 annually) herself — of course, if she was not, that would suggest that payment actually would be made by a third-party investor.5 After having the straw buyer sign a blank application, defendants supplied false financial *567information, supported by fraudulent documents prepared by an accountant relative of Binday’s and supposedly verified by an independent third-party inspector, who in reality simply “assumed [that the information] was correct.” J.A. at 721.
Along with falsifying the straw insured’s financial information, defendants lied in response to the insurers’ questions aimed at detecting STOLI policies, including the purpose of the policy, how the premiums would be paid, and whether the applicant had discussed selling the policy. Defendants also lied to the insurers by providing required certifications that, to their knowledge, the policies were not STOLI. For example, each defendant certified to Lincoln Life Insurance Company that the premiums would not be paid by financing from third parties, that there was no agreement to transfer ownership of the policy, and that the policy “does not violate the stated intent and spirit of the Lincoln Policy Regarding Investor Owned Life Insurance.” J.A. 1077-78.
Over the course of the scheme, defendants submitted at least 92 fraudulent applications, resulting in the issuance of 74 policies with a total face value of over $100 million. These policies generated for defendants a total of roughly $11.7 million in commissions, which ranged from 50-100% of the first year’s premium payments and typically surpassed $100,000 on any given policy.6
II. Indictment
On February 15, 2012, defendants were charged in a five-count indictment in the Southern District of New York. The indictment charged each defendant with one count of conspiracy to commit mail and wire fraud in violation of 18 U.S.C. § 1349; one count of mail fraud in violation of 18 U.S.C. § 1341; and one count of wire fraud in violation of 18 U.S.C. § 1343. It also charged Kergil and Resniek with conspiracy to obstruct justice through the destruction of records in violation of 18 U.S.C. § 1512(k) and Binday with obstruction of justice in violation of 18 U.S.C. § 1512(c). The obstruction of justice charge against Binday was dismissed before trial.
The indictment alleged that defendants defrauded insurers by causing them to issue STOLI policies through misrepresentations regarding: the applicants’ financial information; the purpose of procuring the policy and the intent to resell the policy; the fact that the premiums would be financed by third parties; and the existence of other policies or applications for the same applicant. According to the indictment, these misrepresentations “concerned essential elements of the agreements”— both the agreements between the insurers and the straw buyers with respect to the policies, and those between the insurers and Binday “with respect to commissions” received by the defendants — because the representations “significantly informed the [insurers’] financial expectations with respect to universal life policies.” J.A. 168, 177. Consequently, deceiving the insurers into issuing STOLI policies, when they believed they were issuing non-STOLI policies, “harmed [the insurers] in several ways” by “causing] a discrepancy between the benefits reasonably anticipated by the [companies] and the actual benefits received.” Id. at 167-68.
Four specific discrepancies or harms to the insurers were alleged in the indictment. First, by inflating the straw insured’s financial resources, the defendants *568caused the insurers to expect greater premium payments than they were likely to actually receive before the applicant’s death because it was “a standard assumption” among the insurers that “an individual with a net worth of millions of dollars [will] ... live longer than an individual with minimal net worth.” Id. at 168. Second, the insurers would receive less income from premium payments than expected, because non-STOLI policyholders for tax reasons often pay in excess of the minimum required premium, whereas STOLI policies “typically would be funded at or near the minimum amount necessary to sustain the policy.” Id. at 169. Third, insurers “built into their pricing” an assumption that a certain percentage of policies would lapse from nonpayment, but they “could not accurately assess the voluntary termination rate” for STOLI policies, whose holders “typically did not allow policies to lapse,” thereby “undermin[ing] [the insurers’] actuarial assumptions.” Id. at 170. Fourth, STOLI policyholders were more likely to avail themselves of “grace periods and other features that permitted late payment of premiums,” reducing the cash flow from premium payments avaiable to the' insurers. Id. The indictment also alleged that, to prevent these harms, the insurers “incurred significant additional underwriting, investigation and litigation expenses in attempting to detect and prevent the issuance and maintenance of STO-LI policies.” Id. at 171.
III. Trial and Sentencing
After extensive pretrial motion practice, the case proceeded to an eleven-day trial in September 2013. At trial, the government established the scheme described above through documentary evidence and testimony from cooperating witnesses and other employees of Advocate Brokerage. The government’s evidence on the effect of STOLI policies on insurers consisted primarily of the testimony of two insurance executives: James Avery, the chief executive officer of Prudential Insurance Company of America’s individual life insurance business, and Michael Burns, a senior vice president of Lincoln Financial.
Defendants did not dispute that they had submitted applications with misrepresentations in order to generate commissions by inducing the insurers to issue STOLI policies. Instead, they argued that that conduct was not fraudulent because the insurers in fact happily issued STOLI policies, while paying lip service to weeding out STOLI policies for public relations reasons. Defendants called only one witness — Jasmine Juteau, an attorney at the law firm representing Binday. Juteau identified notations by the insurers on the applications that, according to the defendants, showed that the insurers had flagged the applications as STOLI yet proceeded to issue the policies nevertheless.7
Additionally, defendants argued that they did not intend to inflict, and that the insurers had not in fact suffered, any harm that is cognizable under the mail and wire fraud statutes. Under those statutes, not every deceit is actionable. Rather, the deceit “must affect the very nature of the bargain itself,” such as by creating a “ ‘discrepancy between benefits reasonably anticipated because of the misleading representations and the actual benefits which the defendant delivered, or intended to deliver.’ ” United States v. Starr, 816 F.2d 94, 98 (2d Cir.1987), quoting United States v.- Regent Office Supply Co., 421 F.2d 1174, 1182 (2d Cir.1970). Defendants contended that them deceit had caused no “discrepancy between the benefits reason*569ably anticipated by the insurers and what they actually received,” because there was no meaningful economic difference between STOLI and non-STOLI policies. Trial Tr. 1437. Specifically, they argued that because non-STOLI policies are freely transferable once they have been issued, insurers have no reasonable expectation that a policy will not be sold to a third-party investor at the time it is issued.8 Any difference in lapse rates, defendants maintained, was .“a windfall” and “not [a] right bargained for in the contract.” Id. at 1440.
The jury was charged on October 7, 2013 and that same day returned a guilty verdict on all charges. In advance of sentencing, the government submitted memo-randa calculating the intended ioss caused by the defendants’ scheme at approximately $142 million and the actual loss at approximately $38 million. The district court elected to calculate the Guidelines loss amount based on actual loss, and adopted the government’s calculation of that figure, resulting in a 22-level increase to the base offense levels.9 That yielded a Guidelines range of 168 to 210 months’ imprisonment for Binday, 155 to 188 months for Kergil, and 87 to 108 months for Resnick. On July 30, 2014, the district court sentenced Binday principally to 144 months’ imprisonment, Kergil to 108 months, and Resnick to 72 months.
DISCUSSION
I. Mail and Wire Fraud—Cognizable Harm and Right to Control Property
The crux of defendants’ argument on appeal is that the government faded to prove that they contemplated harm to the insurers that is cognizable under the mail and wire fraud statutes. That argument takes several forms. Defendants challenge the sufficiency of the evidence. They also contend that the indictment was constructively amended because the government’s proof of harm at trial did not align with its theory of harm in the indictment. Additionally, defendants argue that the district court’s jury charge misstated the law regarding cognizable harm. Lastly, they contend that their convictions must be reversed because of improper remarks in the government’s summation.
A. Applicable Law
“Because the mail fraud and the wire fraud statutes use the same relevant language, we analyze them the same way.” United States v. Schwartz, 924 F.2d 410, 416 (2d Cir.1991). The “.essential elements of’ both offenses are “(1) a scheme to defraud, (2) money or property as the object of the scheme, and (3) use of the mails or wires to further the scheme.” Fountain v. United States, 357 F.3d 250, 255 (2d Cir.2004) (internal quotation marks and alterations omitted). It is not re-, quired that the victims of the scheme in fact suffered harm, but “the government must, at a minimum, prove that defendants contemplated some actual harm or injury to their victims.” United States v. Novak, 443 F.3d 150, 156 (2d Cir.2006) (emphasis and internal quotation marks omitted).
The parties dispute whether the requirement of contemplated harm is satis*570fied here based on the insurers’ issuance of STOLI policies when the insurers believed, because of defendants’ fraudulent representations, that they were issuing .non-STOLI policies. “Since a defining feature of most property is the right to control the asset in question, we have recognized that the property interests protected by the [mail and wire fraud] statutes include the interest of a victim in controlling his or her own assets.” United States v. Carlo, 507 F.3d 799, 802 (2d Cir.2007). Accordingly, we have held that a cognizable harm occurs where the defendant’s scheme “den[ies] the victim the right to control its assets by depriving it of information necessary to make discretionary economic decisions.” United States v. Rossomando, 144 F.3d 197, 201 n. 5 (2d Cir.1998).
It is not sufficient, however, to show merely that the victim would not have entered into a discretionary economic transaction but for the defendant’s misrepresentations. The “right to control one’s assets” does not render every transaction induced by deceit actionable under the mail and wire fraud statutes. Rather, the deceit must deprive the victim “of potentially valuable economic information.” United States v. Wallach, 935 F.2d 445, 463 (2d Cir.1991). “Our cases have drawn a fíne line between schemes that do no more than cause their victims to enter into transactions they would otherwise avoid— which do not violate the mail or wire fraud statutes—and schemes that depend for their completion on a misrepresentation of an essential element of the bargain—which do violate the mail and wire fraud statutes.” United States v. Shellef, 507 F.3d 82, 108 (2d Cir.2007).
Thus, we have repeatedly rejected application of the mail and wire fraud statutes where the purported victim received the full economic benefit of its bargain.10 But we have upheld convictions for mail and wire fraud where the deceit affected the victim’s economic calculus or the benefits and burdens of the agreement.11 The *571requisite harm is also shown where defendants’ misrepresentations pertained to the quality of services bargained for, such as where defendant attorneys “consistently misrepresented to their clients the nature and quality of the legal services they were providing ... for a hefty fee.” United States v. Walker, 191 F.3d 326, 335-36 (2d Cir.1999); accord United States v. Paccione, 949 F.2d 1183, 1196 (2d Cir.1991) (“Use of the mails in furtherance of a scheme to offer services in exchange for a fee, with the intent not to perform those services, is within the reach of [18 U.S.C.] § 1341.”). Lastly, we have repeatedly upheld convictions where defendants’ misrepresentations in a loan or insurance application or claim exposed the lender or insurer to unexpected economic risk. See, e.g., United States v. Chandler, 98 F.3d 711, 716 (2d Cir.1996); United States v. Dinome, 86 F.3d 277, 284-85 (2d Cir.1996); United States v. Rodolitz, 786 F.2d 77, 80-81 (2d Cir.1986).
Significantly, defendants do not question the legal structure discussed above. Nor (except for one argument made by Kergil, discussed and rejected below) do they challenge on appeal the legal sufficiency of the indictment in light of these principles. Instead, they challenge only the sufficiency of the evidence to establish the allegations made in the indictment (and raise related alleged trial errors). They thus implicitly or explicitly concede that they are raising what is at its heart a factual question, which the jury resolved against them, on the ground that the evidence was insufficient to permit a rational jury to reach the verdict that the jury here reached.
B. Sufficiency of the Evidence
Defendants contend that the evidence of a cognizable harm was insufficient in several respects. First, they argue that there was insufficient evidence of any economic difference between STOLI and non-STOLI policies, and therefore insufficient evidence that the misrepresentations did anything more than induce transactions that the insurers would have avoided, for essentially non-economic reasons, had they known the truth. Next, assuming that there was sufficient evidence of an economic difference between STOLI and non-STOLI policies, defendants argue that those differences were mere “windfalls,” rather than essential elements of the bargain, and are therefore not a cognizable harm. Kergil then maintains that the evidence was insufficient that the harms the insurers feared from STOLI would actually result from these policies. And Binday argues that the government cannot establish a cognizable harm, having failed to show it in any other way, based on the defendants’ collection of commissions. Lastly, defendants maintain that even if the evidence showed economic differences between STOLI and non-STOLI policies that went to the heart of the bargain, there was insufficient evidence that they understood *572those differences, and thus that they intended the harm.12
In thus challenging the factual sufficiency of the government’s ease, defendants face a “heavy burden, as the standard of review is exceedingly deferential.” United States v. Brock, 789 F.3d 60, 63 (2d Cir.2015) (internal quotation marks omitted). We analyze the sufficiency of the evidence “in the light most favorable to the government, crediting every inference that could have been drawn in the government’s favor, and deferring to the jury’s assessment of witness credibility and its assessment of the weight of the evidence,” and will uphold the conviction “if any rational trier of fact could have found the essential elements of the crime beyond a reasonable doubt.” United States v. Chavez, 549 F.3d 119, 124 (2d Cir.2008) (citations, alteration, and internal quotation marks omitted).
1. Economic Difference and the Specified Harms
Defendants argue that the evidence was insufficient to show that they exposed the insurers to an unexpected risk of economic harm, because the evidence did not establish that STOLI policies were in fact any different economically than non-STOLI policies. Specifically, they argue that the testimony of the two insurance executives, Avery and Burns — essentially the only evidence the government offered on this point — failed to prove any of the four specific risks enumerated in the indictment: shorter life expectancy of the insured, lower premium payments, lower lapse rates, and greater use of grace periods. Rather, defendants contend, the testimony of Avery and Burns shows that insurers refused to issue STOLI policies for non-economic reasons — including concerns that STOLI policies were illegal, or unseemly, and therefore jeopardized the favorable tax treatment afforded to life insurance policies.13
Avery and Burns indeed testified that insurers refused to issue STOLI policies partly for reasons that had nothing to do with the profitability of individual policies, such as reputational concerns.14 But con*573trary to defendants’ assertions, Avery and Burns also testified unequivocally and at length that their companies refused to issue STOLI policies for economic reasons as well. Both testified generally that their companies expected that STOLI policies would have different economic characteristics that could reduce their profitability. Avery explained that insurers did not “price” their policies for a “group of policyholders [who] would not behave the same,” i.e., “an investor who hopes that the insured dies quicker' [rather] than later.” J.A. 475, 483. Burns reiterated the point, stating that STOLI policies “would impair profitability” because his company’s “products weren’t priced for STOLI.” Id. at 576.
According to Avery and Burns, among the reasons for this expectation of reduced profitability were the four specific harms identified in the indictment. Regarding lapse rates, Burns expressed his belief that STOLI policies “would never lapse, so always the death benefit would be paid,” id. at 577, and Avery likewise expected that the lapse rates would be lower because the policies “would be owned by investors who benefit[t]ed from death and didn’t benefit from anything else,” id. at 484. With respect to the correlation between life expectancy and wealth, Burns testified that the company based its pricing assumptions for these policies on “expectations of higher net worth mortality,” because experience showed “better overall mortality” for wealthy persons. Id. at 586. Avery, as defendants highlight, denied that his company took “the position that people with a higher net worth have a lower-mortality.” Id. at 546. But he also testified that “indirectly [the two] can be” related, because the company’s “mortality studies would indicate what mortality we get based on [the policy’s] face amount,” which is in turn “related to net worth.” Id.15 Finally, Burns testified that STOLI policies “would be funded on a minimum basis,” which would “reduce investment” available to the insurers while the policy was in effect. Id. at 577-78.16
*574Defendants describe that testimony as “pure ipse dixit” because “no statistics were offered to support [the witnesses’] belief[s],” and they contend that there was “no showing that lapse rates, minimum premium payments or use of grace periods differed between STOLI and non-STOLI policyholders.” Binday Br. 20, 28 n. 20. But defendants fail to explain why such statistics are a precondition for the jury to credit the executives’ testimony. Avery and Burns were executives in the field with decades of experience in issuing and pricing life insurance policies. Both provided specific explanations for their expectation that STOLI policies would perform differently than non-STOLI policies. These purported differences accord with what one might reasonably expect when comparing the behavior of a professional investor to an individual purchasing life insurance for personal estate planning. Defendants were free to elicit on cross-examination, or to note in their closing arguments, that the government had not provided statistical evidence supporting the witnesses’ assertions. But that was an argument for the jury.
For us, it suffices to say that the executives’ testimony provided a legally sufficient basis for a jury to find that the defendants’ misrepresentations exposed the insurers to an unbargained-for risk of economic loss, because the insurers expected STOLI policies to differ economically, to the insurers’ detriment, from non-STO-LI policies. The indictment alleged that the defendants’ misrepresentations went to an “essential element[ ] of the agreement ]” because the insurers’ belief that they were issuing non-STOLI policies “significantly informed the [insurers’] financial expectations,” J.A. 168, because the insurers expected that STOLI policies would behave differently in the four ways listed in the indictment. Avery and Burns testified specifically that the insurers held that expectation, and the jury was entitled to credit that testimony.
Because the mail and wire fraud statutes do not require a showing that the contemplated harm actually materialized, Novak, 443 F.3d at 156, the government did not need to prove that the STOLI policies defendants procured, or other such policies that slipped through the safeguards erected by the insurers to detect and reject them, in fact have lower lapse rates or insureds with shorter life-spans. Rather, it suffices that the misrepresentations were relevant to the insurers’ economic decision-making because they believed that the STOLI policies differed economically from non-STOLI policies, and thus that the defendants’ misrepresentations deprived the insurers of “potentially valuable economic information,” Wallach, 935 F.2d at 463.17
*5752. Essential Element of the Bargain
Defendants argue that, even assuming that STOLI policies differ economically from non-STOLI policies (or at least that the insurers so believed), those differences cannot support a finding of cognizable harm because they did not concern “an essential element of the bargain,” Shellef, 507 F.3d at 108. Defendants maintain that the insurers could not have “reasonably anticipated” any of the economic advantages of a non-STOLI policy as opposed to a STOLI policy, and thus “there was no discrepancy between benefits reasonably anticipated and actual benefits received.” Starr, 816 F.2d at 98-99 (internal quotation marks omitted). In this regard, defendants principally contend that because non-STOLI policies are freely transferable after issuance, the insurers could have “no reasonable expectation that the[] policies would not ultimately be purchased by hedge-fund investors.” Resnick Br. 39. Thus, defendants argue, the insurers got what they bargained for: a policy that might be sold to an investor.
That argument fails because it mistakenly equates the possibility of a future transfer with the certainty of transfer. There is a meaningful difference between a policy taken out for personal estate planning that might be transferred upon a change in the holder’s circumstances, and a policy that is from the beginning intended as a speculative investment by a third-party. As the government convincingly argues, defendants’ contention is akin to maintaining that an applicant’s income is not an “essential element” in a loan application because the bank could not revoke the loan in the event the applicant subsequently lost her job.18 Moreover, in at least one respect a STOLI policy and a non-STOLI policy subsequently sold to an investor are not economically identical: in the non-STOLI case, the insured had the means to obtain the policy and make the premium payments until resale, and in the STOLI case, the straw insured did not. Thus, if the insurer assumed a “wealth equals health” correlation, it would not be economically indifferent between a non-STOLI policy that was subsequently sold to an investor and a STOLI policy taken out on the life of a straw insured.
On this point, defendants also emphasize that they did not lie about the straw insureds’ health or age. The relevance of this point rests on the premise that only age and health information were “essential elements of the bargain.” Kergil, for instance, argues that “none of the alleged financial misrepresentations were ‘material’ or ‘essential to the bargain,’ because the insurance companies received exactly what they bargained for: legally transferable contracts on the lives of individuals of a specific age and overall health, in exchange for large premium payments.” Kergil Br. 25.
But we are not persuaded (and more importantly, we see no reason why a reasonable jury would be required to find) that the “essential elements” pertaining to a life insurance application are limited to *576age and health — even if those are the two factors with the strongest connection to life expectancy. For example, suppose that a male straw insured claimed in his application to be female. It is common knowledge that on average women live longer than men, and Avery testified that gender is a factor insurers consider in determining life expectancy. Or suppose that the applicant falsely claims not to own or ride a motorcycle, or to engage in some other similarly dangerous activity. Surely such misrepresentations would deprive the insurer of “potentially valuable economic information.” Wallach, 935 F.2d at 463. As these examples demonstrate, many factors beyond age and overall health are potentially relevant in determining life expectancy. A reasonable jury could infer that questions asked by an insurer about the insured’s characteristics, including his economic status and motivations for taking out the policy, are asked — just like questions about age and health — not out of idle curiosity, but because they are material to the insurer’s underwriting decision concerning whether, and at what price, to issue the policy. And indeed, as discussed above, the government presented specific evidence that the insurers did believe that the applicant’s wealth, like his age or health, was correlated to life expectancy. We have recognized that the “value of ... insurance transactions inherently depends on the ability of ... insurance companies to make refined, discretionary judgments on the basis of full information....” Rossomando, 144 F.3d at 201 n. 5. An insurer’s right to enter transactions based on all relevant economic information cannot be confined in the way defendants propose.
Lastly, defendants maintain that the possibility of a lapse is merely a “windfall” for the insurer and not a “reasonably anticipated” benefit or “essential element” under the policy.19 Insurance is based on managing probabilities, however, and we see no reason why the expected probability of default is not a legitimate financial consideration that the insurer is entitled to predict based on accurate information from the applicant.
3. Actual and Specific Harm
In the sole challenge raised on appeal by any defendant to the sufficiency of the indictment, Kergil maintains that the indictment’s allegations of economic harm were inadequate because they were “general and theoretical” in nature, and thus did not allege “that the misrepresentations ‘actually’ caused the harm, or would have caused the harm which the insurance companies ‘assumed’ would occur.” Kergil Br. 23. For example, the indictment alleged that the wealth to health correlation was a “standard assumption” among the insurers, J.A. 168 (emphasis added), and that third-party investors “typically took advantage of grace periods,” id. at 170 (emphasis added). According to Kergil, the indictment was insufficient because it did not allege “that the life insurance policies at issue in this case resulted in earlier payouts, minimum premiums, lower lapse rates, and later premium payments, or that such outcomes would have definitely occurred in the future.” Kergil Br. 24.
We disagree. The indictment need not allege, and the government need not prove, that the specified harms had materialized for the particular policies at issue or were certain to materialize in the future. Rather, it suffices to prove that the defendants’ misrepresentations deprived the insurers of economically valuable infor*577mation that bears on their decision-making. See Wallach, 935 F.2d at 463(holding that deceit must deprive the victim of “potentially valuable economic information”). For this reason, we sustained the conviction in Chandler, where the defendant falsified information in a loan application bearing on her creditworthiness, even though she had partially repaid the loan and intended to continue repayment. 98 F.3d at 716. Regardless of any repayment, the lender had been harmed by the deceit. “[T]he immediate harm in such a scenario is the denial of [the lender’s] right to control her assets by depriving her of the information necessary to make discretionary economic decisions:” United States v. Ferguson, 676 F.3d 260, 280 (2d Cir.2011) (alterations and internal quotation marks omitted).
That approach makes particular sense in the life insurance context, where insurers enter a multitude of similar transactions based on anticipated aggregate results. Suppose an applicant obtained an insurance policy after falsely representing that he did not smoke. The deceit would fall short of Kergil’s conception of economic harm, even if it were undisputed that smokers on average die sooner than nonsmokers, because we could not know that the risk to which the insurer was exposed — namely, the applicant’s earlier-than-expected death due to smoking— would certainly materialize. Indeed, if materialization of the risk had to be shown, many types of life insurance fraud could not be punished until after the deceiver had died, since the applicant might be among that group of smokers who defied the odds and lived beyond expectations even for a non-smoker. Kergil’s formulation therefore entails a requirement of actual economic loss that we have consistently rejected. See, e.g., Novak, 443 F.3d at 156 (mail fraud statute “does not require the government to. prove that the victims of the fraud were actually injured” (emphasis omitted)).
4. Broker Fees as Economic Harm
Binday argues that the government, having failed to show economic harm in any other way, may not establish that harm based on the insurers’ payment to defendants of commissions. He argues that “[a]n insurance company pays commissions to a broker whenever [that broker] delivers a policy, and if the policy has no different economic characteristics than any other for a similarly] situated insured, then the payment of commissions is not an economic loss.” Binday Reply Br. 6. To permit conviction in such a case, he argues, would endorse the “no sale” theory of harm that this Court has repeatedly rejected.
That argument fails because, as discussed above, its premise fails; the jury was entitled to find that the STOLI policies did have different economic characteristics than non-STOLI policies. Because sufficient evidence supports a finding that the policies were not economically equivalent, this is not a case like the hypothetical offered by Binday of a real estate agent who receives commissions on the sale of an apartment after misleading its client as to the nationality of the buyer, but obtains for the client the precise economic terms of sale for which the client bargained. Rather, it is more analogous to a real estate agent who receives a broker’s fee from a buyer after arranging for the purchase of an apartment that is known by the agent, but not by the buyer, to be infested with termites.
We have repeatedly upheld convictions for mail or wire fraud where the defendant received fees for services that were not performed in the manner agreed upon, for instance where attorneys “consistently *578misrepresented to their clients the nature and quality of the legal services they were providing ... for a hefty fee.” Walker, 191 F.3d at 335; see also Frank, 156 F.3d at 335; Paccione, 949 F.2d at 1196. Thus, whether payment of commissions would constitute a standalone harm absent a showing of economic difference between STOLI and non-STOLI policies is of no consequence for the instant case. Because the jury reasonably found that the defendants deprived the insurers of economically valuable information, the payment of commissions that were not legitimately earned merely represents an additional economic harm.
5. Intent to Infíict Cognizable Harm
Defendants contend that even if their fraudulent conduct exposed the insurers to a risk of economic harm, and even if that risk concerned a reasonably expected benefit of the bargain, there was nevertheless insufficient evidence that they intended such harm. They observe that, while Avery and Burns testified that STOLI policies exposed insurers to a risk of economic harm, no witness testified that defendants intended to impose that risk, or that they understood the insurers’ pricing assumptions, expectations about lapse rates, or other beliefs that led them to find STOLI policies economically undesirable. Thus, defendants argue, they might have believed that insurers sought to avoid STOLI on general principle or for other reasons unrelated to the economics of the policies. Binday maintains that he believed insurance companies saw STOLI policies as “unseemly ... or perhaps illegal ... but not unprofitable” because “[t]o him, the economics of a STOLI policy were no different from those of a non-STOLI policy that an owner decided to sell soon after acquiring it.” Binday Br. 28 1 (emphasis omitted).20
“Misrepresentations amounting only to a deceit are insufficient” to support conviction for mail or wire fraud because “the deceit must be coupled with a contemplated harm to the victim.” Starr, 816 F.2d at 98. “Where the false representations are directed to the quality, adequacy or price of the goods themselves, the fraudulent intent is apparent because the victim is made to bargain without facts obviously essential in deciding whether to enter the bargain.” Regent Office Supply, 421 F.2d at 1182. Fraudulent “[i]ntent may be proven through circumstantial evidence, including by showing that defendant made misrepresentations to the victim(s) with knowledge that the statements were false.” United States v. Guadagna, 183 F.3d 122, 129 (2d Cir.1999).
We have affirmed such an inference where the defendant’s misrepresentations foreseeably concealed economic risk or deprived the victim of the ability to make an informed economic decision. For example, in Chandler, 98 F.3d at 711, the defendant was charged with bank fraud after she applied for a line of credit using a pseudonym. She argued that she had no intent to cause harm to the bank because she made her first two payments and would have continued to do so but for her arrest. Id. at 716. We rejected that argument because “[ijntent to harm ... can be inferred from exposure to potential loss” and the defendant’s “intentionally deceptive conduct [was] inexplicable other than as a *579means of intentionally exposing [the bank] to an unwanted risk.” Id.
Similarly, we have explained that to sustain a mail fraud conviction based on a fraudulent insurance claim, it is not necessary to show that the defendant intended to recover “more from the insurance company than that to which he was entitled,” but only that he “employed a deceptive scheme intending to prevent the insurer from determining for itself a fair value of recovery.” Rodolitz, 786 F.2d at 80-81. And in United States v. Carlo, we upheld a conviction for wire fraud where the defendant, in hopes of earning a financing fee, misrepresented to real estate developers the likelihood of obtaining financing, inducing them to continue their projects at additional expense. 507 F.3d at 801. We held that the fact that the defendant hoped that the financing would indeed be obtained “does not negate his intent to inflict a genuine harm on the victims by depriving them of material information necessary to determine for themselves whether to continue their development projects.” Id. at 802.
As these cases demonstrate, it is not necessary that a defendant intend that his misrepresentation actually inflict a financial loss—it suffices that a defendant intend that his misrepresentations induce a counterparty to enter a transaction without the relevant facts necessary to make an informed economic decision. Defendants attempt to distinguish the instant case from our precedent. They contend that, while the materiality of misrepresentations of health or age in an insurance application, or credit history or income in a loan application, is sufficiently obvious that an intent to defraud may be inferred, the effect of lapse rights and minimum payments were not so obvious, and therefore intent to defraud cannot be inferred.
Sufficient evidence supports an inference of fraudulent intent in this case. “[T]he value of credit or insurance transactions inherently depends on the ability of banks and insurance companies to make refined, discretionary judgments on the basis of full information.” Rossomando, 144 F.3d at 201 n. 5. Whether or not defendants understood the precise nature of the economic differences between STO-LI and non-STOLI policies, they were aware that the hedge funds investing in the STOLI policies were betting that the value of the policies would exceed the premiums paid on those policies, contrary to the interests of the insurers. As Binday puts it, his business model involved selling STOLI policies “to investors who believed that there was an opportunity for an arbitrage profit” based on their “betting that the insureds would die sooner than the insurance companies were estimating.” Binday Br. 31. And indeed, the evidence made clear that defendants marketed the policies to investors on the theory that the policies would prove profitable to them, precisely because the straw insureds would not live long enough for the premiums paid to exceed the death benefit. In other words, the defendants knew that their misrepresentations induced the insurers to enter into economic transactions—ones that entailed considerable financial risk—without the benefit of accurate information about the applicant and the purpose of the policy.
The defendants were also aware that the insurers refused to issue and attempted to detect STOLI policies, including by requiring brokers to represent that the policies were not intended for resale. Defendants then took elaborate steps to evade those detection efforts by insurers—entities that exist for the purpose of generating profit. On these facts, the jury reasonably could infer that the defendants intended to withhold information relevant to the insurers’ *580economic decision-making, and not simply to the insurers’ “general principles.”
Lastly, Kergil contends that there was insufficient evidence of his intent to defraud the insurers because the evidence showed his belief that the insurers, despite their claims to the contrary, wanted to issue STOLI policies, while only pretending to attempt to avoid them.21 To support this proposition, Kergil points to the testimony of cooperating witness Paul Krupit that Kergil told him “that insurance companies wanted to issue these policies,” Trial Tr. 1037, and to the insurers’ financial statements indicating that universal life insurance sales increased dramatically in the years that STOLI policies became popular.22
There is no evidence in the record indicating that Kergil had reviewed the insurers’ financial statement and inferred from them that STOLI business was welcome. What is in the record is that Kergil signed certifications required by the insurers that were specifically designed to avoid issuing STOLI policies. Despite Kergil’s unsupported and self-serving statement to Kru-pit, the jury was certainly entitled to infer, based on those certifications and the other facts of the case, that Kergil was aware that the insurers did not want to issue STOLI policies, and that he intended that the numerous misrepresentations in the applications would cause the insurers to do so against their wishes.
C. Jui'y Instruction—Cognizable Harm
Defendants argue that the district court’s jury charge failed to convey the requirement of a cognizable harm, and thus erroneously permitted conviction on a “no sale” theory, or at minimum failed to convey that requirement clearly enough for the jury to understand it. The government counters that defendants have waived any challenge to the instructions, and that defendants are mistaken in any event.
“To secure reversal based on a flawed jury instruction, a defendant must demonstrate both error and ensuing prejudice.” United States v. McIntosh, 753 F.3d 388, 392 (2d Cir.2014) (internal quotation marks omitted). We review de novo a properly preserved challenge to a jury in*581struction, reversing “where the charge, viewed as a whole, either failed to inform the jury adequately of the law or misled the jury about the correct legal rule.” United States v. White, 552 F.3d 240, 246 (2d Cir.2009) (internal quotation marks omitted). Where a challenge to a jury instruction has not been preserved, we review for plain error. United States v. Ghailani 733 F.3d 29, 52 (2d Cir.2013).23 Lastly, where a defendant has “invited” the instruction he seeks to challenge, he “has waived any right to appellate review of the charge.” United States v. Giovanelli, 464 F.3d 346, 351 (2d Cir.2006).
The challenged instruction went as follows:
Now, as I told you a few minutes ago, a scheme to defraud is a course or a plan of action to deprive someone of money or property. What does that mean, deprive someone of money or property? Well, obviously a person is deprived of money or property when someone else takes his money or property away from him. But a person can also be deprived of money or property when he is deprived of the ability to make an informed economic decision about what to do with his money or property. We referred to that as being deprived of the right to control money or property.
Because the government need only show that a scheme to defraud existed, not that it succeeded, it is not necessary for the government to prove that any insurance company actually lost money or property as a result of the scheme. Such a loss must, however, have been contemplated by the defendant.
In considering whether loss was contemplated, keep in mind that the loss of the right to control money or property constitutes deprivation of money or property only when the scheme, if it were to succeed, would result in economic harm to the victim. Economic harm is not limited to a loss on the company’s bottom line.
In order for the government to prove a scheme to defraud, it must prove that the scheme, if successful, would have created a discrepancy between what the insurance companies reasonably anticipated and what they actually received. If all the government proves is that under the scheme the insurance companies would enter into transactions that they otherwise would not have entered into, without proving that the ostensible victims would thereby have suffered some economic harm, then the government will not have met its burden of proof.
J.A. 889-90.
Defendants contend that the jury charge permitted conviction on a showing of nothing more than that the insurers avoided STOLI policies as “unseemly” — that is, on a “no-sale” theory.24 They maintain that *582their challenge is preserved, even though they jointly submitted the charge language with the government, because they did so subject to their previous objections to the ■ government’s theory of guilt — made in pre-trial motion practice and again in their Rule 29 motion — that the government needed to prove the four specific harms alleged in the indictment, and that the loss of a right to control property was insufficient.
We assume without deciding that defendants’ prior arguments are sufficient to preserve their challenge that the jury instructions permitted conviction absent a showing of cognizable harm, for that challenge fails in any event. Indeed, the charge states explicitly that “the loss of the right to control money or property constitutes deprivation of money or prop- • erty only when the scheme, if it were to succeed, would result in economic harm to the victim.” J.A. 889 (emphasis added). The instruction then reiterates that the government would not meet its burden if it showed only that the insurers “enter[ed] into transactions that they otherwise would not have entered into, without proving that the ostensible victims would thereby have suffered some economic harm.” Id. at 890. Thus, far from permitting conviction on a “no sale” theory, the charge , directly explained that proving such a theory would be insufficient to support conviction.
Defendants counter that even if the instruction required a showing of economic harm, that requirement was confusingly conveyed and undermined by other portions of the instruction. For example, Kergil contends that, “[wjhile the jury did hear that ‘economic harm’ was required, the court failed either to define this term or provide examples of what might constitute ‘economic harm.’ The instruction told the jury what was not required, but left it guessing as to what would constitute economic harm.” Kergil Br. 3. Defendants protest that the requirement that the harm be “economic” was undermined by the statement that such harm is not limited to a loss on the company’s bottom line.” They also contend that the charge’s statement that the government must prove that the scheme “would have created a discrepancy between what the insurance companies reasonably anticipated and what they actually received” might be interpreted to require only that the insurers received economically identical STOLI policies when they had bargained for non-STOLI policies.
To the extent that defendants argue not that the instruction did not require a showing of economic harm, but that the instruction failed to clearly explain what would constitute economic harm, they have “waived any right to appellate review,” Giovanelli, 464 F.3d at 351, by agreeing to the language of the instruction. After a dispute arose at the charge conference regarding the proposed instruction on economic harm, the parties conferred and Binday’s counsel stated “I think we can agree on language here.” J.A. 840. That evening, the government wrote the district court that, “[t]o resolve the outstanding Starr language issue, the parties have agreed that the attached should replace the first three paragraphs of the current [economic harm charge].” Gov. Add. I.25 *583The attached language agreed to by the parties is substantially identical to the economic harm charge ultimately provided by the district court. Compare Add. 2, with J.A. 889-90. Thus, the parties jointly submitted the language which defendants now contend was insufficiently clear.
Even assuming that the defendants’ earlier motions preserved the general challenge that economic harm must be required, those earlier objections do not preserve a claim that the specific language of the jury instruction did not convey that requirement with sufficient clarity. “[W]hen a defendant, as here, objects only generally to the issuance of a jury instruction, and not to the specific language used by the District Court, the objection to the formulation of the charge is not preserved.” Ghailani 733 F.3d at 52. That applies with even greater force where, as here, the defendant jointly submitted the specific language. While defendants maintain that they were confined by the district court’s erroneous conception of cognizable harm, none of the district court’s prior rulings foreclosed defendants from seeking the clarification they now claim was necessary.26
D. Constructive Amendment Claim
Defendants argue that the indictment was constructively amended because the government’s theory of economic harm broadened from the indictment through the trial. This issue first arose when, before trial, the government moved in li-mine to preclude defendants from offering evidence relating to how the insurers “actually fared, economically, in the wake of defendants’ false representations.” D. Ct. Doc. 230 at 18. It argued that such evidence was irrelevant because it need prove merely “that defendants contemplated harm—if only to the [Insurers’] right to control their assets through discretionary economic decisions.” Id. at 19. Defendants opposed that motion and also moved to dismiss the indictment for constructive amendment, arguing that the government sought to change course from the “economic harm” theory of harm alleged in indictment to a “right to control” theory. D. Ct. Doc. 233 at 23-26. The district court granted the government’s motion in limine and denied defendants’ motion to dismiss. It explained that the government could not “prevail simply by establishing loss of the ‘right to control’ the Insurers’ assets” because “[t]hat would be tantamount to proving only that the Insurers would not have issued the policies if they had known the truth.” J.A. 292. Rather, the court explained, the government must “introduce evidence that the Insurers suffered, for example, the harms outlined at Paragraph [10] of the Indictment—which qualify as ‘financial harm’ as pleaded in Paragraph 4.” Id. at 293.
*584The issue resurfaced at the close of the government’s case-in-chief, when defendants moved unsuccessfully for a judgment of acquittal pursuant to Rule 29 of the Federal Rules of Criminal Procedure. Defendants argued that the government had failed to prove “the harms in the indictment” or that the insurers had suffered “economic harm.” Id. at 815. In response, the government argued that it had shown a scheme to deprive the insurers of commissions they would not have paid, policies they would not have issued, and “costs they [would not] have incurred had they known the truth.” Id. at 819-20. The government referenced the testimony of Avery and Burns to argue that the “specific harms alleged in the indictment were proved at trial” — including lapse rates, reduced premiums, and a “link between mortality and net worth.” Id. at 821. The government also noted that Burns and Avery had testified about other “economic harms [insurers] were facing as a result of STOLI,” including “reduced profitability, ... tax consequences, ... [and] higher prices resulting from having to get reinsurers’ approval.” Id. at 820. Lastly, the government argued that the insurers had “incurred massive economic costs, not quantifiable necessarily, [that] they described as soft costs, to try to limit STOLI.” Id.
Defendants maintain that the indictment alleged that STOLI policies inflicted only four specific harms on insurers: (i) the “wealth equals health” effect; (ii) minimum premium payments; (iii) lower lapse rates; and (iv) greater use of grace periods. But at trial, defendants argue, the government broadened its theory of economic harm by eliciting that the insurers suffered harm in ways not alleged in the Indictment. Defendants contend that, as underscored in the government’s argument opposing the Rule 29 motion, the economic harm alleged at trial also stemmed from payment of commissions on STOLI policies, jeopardizing the insurers’ favorable tax treatment, and forcing the insurers to incur “soft costs” to detect STOLI. Defendants contend that the broadening of proof was all the more significant because the jury charge “did not mention [the four specific] harms and instead told the jury that the concept of ‘economic harm’ was not ‘limited to a loss to the company’s bottom line.’ ” Binday Br. 34. Thus, defendants argue, the evidence at trial and the jury charge in combination permitted conviction on a ground not charged in the indictment.
A constructive amendment occurs “when the trial evidence or the jury charge operates to broaden the possible bases for conviction from that which appeared in the indictment.” United States v. McCourty, 562 F.3d 458, 470 (2d Cir.2009) (internal quotation marks omitted).27 To prevail on a constructive amendment claim, defendants must show “that the terms of the indictment are in effect altered by the presentation of evidence and jury instructions which so modify essential elements of the offense charged that there is a substantial likelihood that the defendant may have been convicted of an offense other than that charged in the indictment.” United States v. Vilar, 729 F.3d 62, 81 (2nd Cir.2013) (internal quotation marks and emphasis omitted). “The critical determination is whether the allegations and the proof substantially corre*585spond.” United States v. Danielson, 199 F.3d 666, 670 (2d Cir.1999) (internal quotation marks omitted). We have “consistently permitt[ed] significant flexibility in proof’ of the charges, “provided that the defendant was given notice of the core of criminality to be proven at trial.” United States v. Agrawal, 726 F.3d 235, 259-60 (2d Cir.2013). (alteration, emphasis, and internal quotation marks omitted).
As an initial matter, some of the harms that defendants contend broadened the indictment were in fact alleged in the indictment. With respect to commission payments, the indictment alleged that the “purpose of procuring the policies was to generate millions of dollars in commissions and other profits.” J.A. 171. Commission payments were not identified as a type of economic harm, but that is because, as the government explains, the commissions were not a stand-alone economic harm, but the object of the scheme: commissions “were the ‘money or property’ implicated by the scheme,” whereas the “economic harms to which the defendants’ scheme exposed the Insurers ... were what made the defendants’ misrepresentations fraudulent as opposed to merely deceptive.” Gov. Br. 100 n. 28 (emphasis omitted). With respect to the “soft costs” imposed by the scheme, the indictment alleged — in the same paragraph identifying the four specific harms — that the insurers “incurred significant additional underwriting, investigation and litigation expenses in attempting to detect and prevent the issuance and maintenance of STOLI policies.” J.A. 171.
That leaves only the jeopardizing of the insurers’ preferential tax treatment and the increased cost of reinsuring the policies, neither of which was specifically referenced in the indictment. But on these facts, the proof regarding those two harms does not rise to the level of a constructive amendment. This is not a case where “the allegations and the proof [did not] substantially correspond,” Danielson, 199 F.3d at 670 (internal quotation marks omitted)— such as where the government failed to offer support of the specific harms alleged in the indictment, or where those harms were not the core of the government’s proof at trial. As recounted above, Burns and Avery testified at length regarding the four specified harms. They explained that their companies feared the precise economic harms that were specifically alleged in the indictment, and also identified tax consequences and increased cost of reinsurance as additional harms. Tangential evidence of two additional specific harms did not deprive the defendants of “notice of the core of criminality to be proven,” Agrawal, 726 F.3d at 260 (emphasis omitted) — namely, that defendants submitted fraudulent applications to deceive the insurers into issuing policies that they considered less economically attractive than the policies that they believed they were issuing.
From the indictment through the trial, the government consistently maintained that defendants sought to obtain money (in the form of commissions) from the victim insurers, by an elaborate scheme of deliberate falsehoods that were designed to deceive the insurers into issuing policies that reasonable insurers would have and did believe were economically disadvantageous, and that defendants knew that the insurers were attempting to detect and avoid, and' that defendants deliberately marketed to investors as policies on which the investors would profit at the insurers’ expense. There was no constructive amendment of the indictment.
E. Challenge to Government’s Rebuttal Summation
Binday argues that his conviction must be vacated because the district court erred *586in denying his request for a mistrial or, in the alternative, a curative instruction based on what he contends were improper remarks made by the government in its rebuttal summation.
A defendant seeking to overturn a conviction based on an improper comment in summation bears the “heavy burden” of showing that “the comment, when viewed against the entire argument to the jury, and in the context of the entire trial, was so severe and significant as to have substantially prejudiced him, depriving him of a fair trial.” United States v. Farhane, 634 F.3d 127, 167 (2d Cir.2011) (citations and internal quotation marks omitted). In assessing whether a defendant has been substantially prejudiced, we consider “the severity of the misconduct, the curative measures taken, and the certainty of conviction absent the misconduct.” United States v. Rosa, 17 F.3d 1531, 1549 (2d Cir.1994).
In his summation, Binday argued that he contemplated no economic harm because he “intended [that] everyone involved in the investment make money, everyone, and not lose money, including the insurance companies.” Trial Tr. 1436. He noted that “the insurance companies, who say they didn’t want to issue these policies, nevertheless, got extremely high premiums from them.” Id. at 1446. The government responded in its rebuttal summation that the STOLI policies were not a “win-win for everyone.” Id. at 1511. It argued, referencing the insurance executives’ testimony, that although the insurers received premium payments, these payments were expected to be more than offset by the ultimate death benefits payable, due to the disruption of the insurers’ actuarial assumptions created by STOLI policies. To refute Binday’s “win-win” argument, the government noted a chart prepared by Binday contained in Government Exhibit (“GX”) 1408, which projected that a $4 million policy would require roughly $2 million in premium payments by the investor. Pointing to the chart, the government argued, “Yes, the insurance company gets premiums, they get about $2 million in premiums. But for what, ladies and gentlemen? For the opportunity to pay $4 million when that insured dies.” Id. at 1516. After closing arguments, Binday submitted a letter to the district court objecting to the government’s argument regarding GX 1408 as “misleading or known not be true” because “the government failed to mention evidence in the record that insurance companies invest in life insurance premiums they receive and make returns on those investments.” Id. at 1477, 1481-82. Binday moved unsuccessfully for a mistrial or, alternatively, for a curative instruction.
On appeal, Binday again argues that the government’s reference to GX 1408 and its accompanying argument were misleading because “[a]dding up yearly premiums and comparing the total to the death benefit ... is no way to calculate an insurance company’s gain or loss,” since it fails to account for the return generated by the yearly premiums prior to the death benefit. Binday Reply Br. 21. Binday maintains that an “accurate calculation requires one to know the company’s rate of return on investment, and that number is nowhere in the record.” Id. For its part, the government acknowledges a “failure to caveat its argument from GX 1408 with a reference to interest earned on premiums.” Gov. Br. 118. But, it contends, that failure was insignificant because its point stands that this was not a “win-win” game, particularly because GX 1408 “was merely offered as support for a point already established by ample other evidence.” Id. at 120.
*587Binday is correct that an insurance company’s gain or loss on a policy cannot be determined simply by comparing the amount of the death benefit to the sum of the premium payments, because that method fails to account for the time value of money for the premium payments received prior to payment of the death benefit. But to show substantial prejudice that would warrant vacating his conviction, Bin-day must show, at minimum, that the $4 million to $2 million comparison was so inaccurate that it misled the jury about the point for which the government offered it — namely, that the policies were not a “win-win” for the investor and the insurer. Binday offers no reason to believe that accounting for a return on the premium payments would undermine that central point, and indeed there is reason to suspect that it would not. The government’s intended loss calculation for sentencing purposes incorporated a 20 percent discount to account for the insurers’ investment of premiums, which the defendants do not challenge on appeal.28 Applying a 20 percent discount to the $2 million loss implied by the government’s rebuttal statement suggests a loss of $1.6 million rather than the original $2.0 million— which hardly affects the government’s argument that if all policies operated like STOLI policies, the insurers would be out of business.
Binday also argues that the government’s comparison was misleading because the $2 million of expected premium payments shown in GX 1408 was based on the life expectancy used by the investors, not the insurer. Using the insurer’s estimate for the policy illustrated in GX 1408— which is not in the record, but which Bin-day asserts to have been higher and more accurate than the investors’ — might therefore shrink or eliminate the gap between the expected premium payments and the $4 million death benefit, particularly when accounting for returns on the premiums. Even if insurers made some money from Binday’s STOLI policies, -however, those policies were still less advantageous to the insurers than the non-STOLI ones they thought they were issuing. Binday’s argument rests on the false premise that the applicant’s age and health are the only information on which an insurer is entitled to rely in issuing policies. Whether or not Binday intended, for the insurers to lose money, he sought to induce insurers to issue policies based on fraudulent information, see Binday Reply Br. 16, which is itself a harm to insurers.29
In short, while Binday has shown that the government’s comparison of death benefit to premium payments' mistakenly failed to account for investment returns, he has not shown that the reference to GX 1408 misleadingly supported the government’s point that the policies were not “win-win,” and that Binday’s own presentations to investors could be taken by rational jurors as evidence that he himself understood that they were not. Accordingly, he has failed to demonstrate the substantial prejudice required to reverse his conviction on this ground.
II. Resnick’s Obstruction of Justice Challenges
Resnick, joined in relevant part by Ker-*588gil,30 challenges his conviction for conspiracy to obstruct justice through destruction of records, in violation of 18 U.S.C. § 1512(c) and (k).31 The indictment alleges that Kergil and Resnick violated' that statute when, “upon learning of a criminal investigation, [they] agreed with each other to destroy documents and electronic files relevant to a Federal grand jury investigation regarding the fraudulent procurement of life insurance policies.” J.A. 191-92.
Resnick contends that the evidence was insufficient and that his conviction must be vacated because of improper remarks in the government’s rebuttal summation. He also argues that the district court erred by not suppressing audio tapes tending to prove his guilt that were recorded by a cooperating witness.
A. Sufficiency of the Evidence
1. Destruction of Hard Drive
In June 2010, Binday learned that the FBI was investigating Advocate Brokerage and had begun interviewing straw insureds. Binday initially instructed his brokers to ask their straw insureds not to speak to investigators. After he was “told that we can get in trouble for telling people not to speak to [the FBI],” Binday modified his instructions: “We cannot advise not to speak; yet nothing good can come from any conversation.” Id. at 1458. On June 21, 2010, Resnick was interviewed by FBI agents concerning policies he had issued, after which Resnick emailed Bin-day the names of the agents.
After learning of the FBI investigation in June 2010, Kergil instructed Resnick to “get rid” of his “hard drive” and to “get rid of everything with the name of Advocate Brokerage, Michael Binday’s name on it ... and get rid of it.” J.A. 766. Res-nick then flew from New York to his primary residence in Orlando, Florida. There, on June 26, 2010, he took his desktop computer to an Apple Store, where he had technicians “wipe” his hard drive and transfer its contents to a portable device. Notably, Resnick confirmed to the technicians that he wanted to have the contents of his hard drive erased, rather than simply cloned to the portable device, even though that would require an additional fee. Several days later, on July 2, 2010, Resnick’s attorney called Apple to request “information regarding specific services [Apple] had performed.” J.A. 785.
On July 23, 2010, Resnick spoke on the phone with broker Paul Krupit, who had begun cooperating with the FBI investigation and whom Kergil had also instructed to destroy evidence. Krupit secretly recorded their conversation, at the government’s direction. On that call, Krupit stated that he had “deleted stuff because [Kergil] told me to,” and that he “would never have done that[,]” to which Resnick responded, “me too.... I got back on a plane and ... went back home the next day and ... did it.. v it was stupid to do.” J.A. 1462-63.
In response to a grand jury subpoena dated December 23, 2011, Resnick’s com*589pany, MAR Group, Inc., produced more than 2,000 documents Bates-stamped “MR.” The government never executed a search warrant to obtain the portable device.
2. Resnick’s Challenge
Resnick contends that the “only reasonable inference to draw from this evidence was that [he] sought only to secure, not destroy, his documents, and did so at the direction of his attorneys, or with a view to handing it over to his attorneys’ custody for safe-keeping.” Resnick Br. 19. He argues that he “did the only thing Appl[e] offered to preserve the hard-drive in its current state: he imaged it,” because any further use of the computer “would have resulted in the over-writing (i.e. deletion) of original meta-data.” Id. at 18-19. Thus, he argues, the evidence showed at most not that he intended to destroy evidence, but rather that he intended to transfer the evidence such that it could be turned over on his own (lawful) terms. An intent to preserve the evidence is further demonstrated, Resnick maintains, by the fact that he ultimately turned over thousands of documents in response to a subpoena.
Even absent additional incriminating evidence, it can hardly be said that an intent to preserve evidence is the “only reasonable inference” the jury could draw from Resnick flying to another state to wipe his hard drive and transfer its'contents to a portable device days after Kergil instructed him to “get rid of everything” in response to an FBI investigation. A reasonable juror could easily conclude that the transfer of the flies to a more easily concealed external device, and their deletion from the most obvious place for investigators to seek them, bespeaks an intention to conceal or destroy evidence.
But there is additional incriminating evidence. Most importantly, Resnick did not simply “back up” his hard drive onto a portable device; he specifically agreed to pay an extra fee to have its contents erased. That conduct cannot be explained by a desire to “preserve the hard drive in its current state” by preventing gradual “over-writing” from its continued use. Moreover, Resnick’s recorded conversations with Krupit undermine his innocent explanations of his conduct. Resnick did not tell Krupit that he had acted to preserve, rather than destroy, evidence. Rather, he said he had done the same thing Krupit had done, that is, “deleted stuff because [Kergil] told me to.” J.A. 1462. The jury could also reasonably interpret Resnick’s statements to Krupit that his conduct was “stupid” as strongly probative that he took those steps with an intent to conceal or destroy, rather than preserve, evidence. And while Resnick argues that his attorney’s subsequent phone call to the Apple Store evinces an intent to ensure that the evidence was preserved, that phone call just as plausibly supports an inference that his attorney was attempting to assess the damage done by conduct Resnick had already taken without consulting her. “[I]t is the task of the jury, not the court, to choose among competing inferences.” United States v. Martinez, 54 F.3d 1040, 1043 (2d Cir.1995).
Lastly, the fact that Resnick ultimately may have preserved and turned over all of the hard-drive’s contents — something the government contends it had no way to definitively establish — is not dispositive, because he was convicted of conspiracy to obstruct justice through destruction of records, not the completed offense. A conspiracy conviction may be sustained where there is “some evidence from which it can reasonably be inferred that the person charged with conspiracy knew of the existence of the scheme alleged in the indict*590ment and knowingly joined and participated in it.” United States v. Morgan, 385 F.3d 196, 206 (2d Cir.2004) (internal quotation marks omitted). Here, there is no dispute that Resnick was aware of a scheme to destroy evidence promoted by Kergil, and there was ample evidence for the jury to conclude that Resnick “joined and participated” in the scheme when he flew from New York to Florida to delete the contents from his hard drive.32
3. Nexus to Grand Jury Proceeding
Resnick also challenges the sufficiency of the evidence establishing the requisite nexus between his conduct and a grand jury proceeding.33 “The touchstone for the nexus requirement ... is an act taken that would have the natural and probable effect of interfering with a judicial or grand jury proceeding that constitutes the administration of justice; that is, the act must have a relationship in time, causation, or logic with the judicial proceedings.” United States v. Quattrone, 441 F.3d 153, 171 (2d Cir.2006) (internal quotation marks omitted). To violate § 1512, “an official proceeding need not be pending or about to be instituted at the time of the offense.” 18 U.S.C. § 1512(f)(1). Rather, we have found the nexus requirement satisfied where a grand jury proceeding was “foreseeable” because the defendant was aware “that he was the target of an investigation.” United States v. Persico, 645 F.3d 85, 108 (2d Cir.2011); accord United States v. Desposito, 704 F.3d 221, 230-31 (2d Cir.2013).
Resnick argues that no grand jury proceeding was foreseeable when he erased his hard drive because the evidence showed only “that agents had questioned [him] about his Lincoln policies,” and not that there was “any mention of the word ‘grand jury,’ ” or even that he was the target of a criminal investigation “as opposed to some kind of regulatory” one. Resnick Br. 22. To find the nexus requirement satisfied in this case would, according to Resnick, treat “a mere FBI inquiry [as] the equivalent of an ‘official proceeding’ under the statute.” Id. He contends that “[a] lay person cannot be expected to leap to the conclusion that a grand jury will be convened simply because FBI agents are asking questions about one’s business activities.” Id.
That every inquiry from the FBI might not render a grand jury investigation reasonably foreseeable is of no avail to *591Resnick, as there was sufficient evidence of foreseeability in this case. Resnick knew that the subject of the FBI’s inquiries was in fact a large insurance fraud scheme in which he participated and about which he possessed incriminating documents. That a grand jury had not been commenced or specifically discussed with Resnick at the time of the destruction does not render a grand jury proceeding unforeseeable. Indeed, to conclude otherwise would undermine the statute’s provision that “an official proceeding need not be pending or about to be instituted at the time of the offense.” 18 U.S.C. § 1512(f)(1). That Resnick and Kergil conspired to destroy documents relevant to a massive fraud once they learned that the fraud-had become the subject of an FBI investigation provides a sufficient basis for the jury to find that their conduct had “the natural and probable effect of interfering with a judicial or grand jury proceeding,” Quattrone, 441 F.3d at 171.
B. Government’s Rebuttal Summation
Resnick also argues that the district court erred in denying his motion for a mistrial based on statements the government made in its rebuttal summation pertaining to the obstruction of justice charge. As noted, the documents produced by Res-nick’s company in response to the government’s subpoena were Bates-stamped “MR.” Neither the government nor Res-nick introduced into evidence the entirety of those documents. Toward the end of the government’s case-in-chief, Resnick’s counsel stated her intention to call an FBI agent to testify that Resnick’s company had produced thousands of pages of emails in response to the subpoena, and that the government never sought to obtain Res-nick’s hard drive. The government then informed Resnick’s counsel that if the FBI agent were called, the government would elicit on cross-examination that Resnick’s document production did not include several incriminating emails to or from him that had been obtained from other sources.
Resnick’s counsel did not call the FBI agent. In her summation, however, she referenced Resnick’s extensive document production — presumably inferred from a Bates number greater than 2000 — to refute that he had destroyed documents: “Where is the evidence of a conspiracy to destroy documents so they’re not available for a federal grand jury proceeding when you’re being asked to convict Mr. Resnick on documents he himself supplied, over 2,000 of them? If he is a document-destroyer, I will say to you he is a very incompetent one.” J.A. 870-71. In its rebuttal summation, the government responded that Resnick’s counsel “didn’t mention the emails that [Resnick] didn’t produce that the government obtained through other means.” Id. at 879. Over Resnick’s objection, the government then referenced five exhibits that had been admitted into evidence, noting that “[tjhere’s no MR stamp” on the exhibits and asserting “[fit’s not there because he didn’t produce it.” Id. at'880. At the close of the rebuttal summation, Resnick moved for a mistrial based on these remarks.
In a letter supporting his motion for a mistrial, Resnick argued that the government’s summation was misleading because four of the five exhibits either pre-dated the produced documents or post-dated the deletion of Resnick’s hard-drive, and because there was no evidence that the remaining exhibit, which “was indeed not included in the MAR Group production,” was “deleted as part of a document destruction conspiracy, rather than [in] the *592regular course of business.” Id. at 1494.34 The district court denied Resnick’s motion for a mistrial, but issued the following curativé instruction:
You are instructed that there is no evidence in the record showing the universe of what Resnick’s company, MAR Group, produced in response to the grand jury subpoena. You, therefore, cannot infer that every e-mail to or from • or copying Mark Resnick that was introduced into evidence in this case was produced in some form by MAR Group, nor can you infer that any such e-mail was not produced by MAR Group.
Id. at 885.
As noted above, a defendant seeking to overturn a conviction based on an improper comment in summation bears the heavy burden of demonstrating substantial prejudice, considering “the severity of the misconduct, the curative measures taken, and the certainty of conviction absent the misconduct.” Rosa, 17 F.3d at 1549.
Resnick’s argument falls short on each of these factors. As to the severity of the misconduct, the challenged remarks were a brief and limited (though impermissible) rebuttal to Resnick’s attempt to have the jury infer based on the Bates stamps that he had produced all relevant documents — an inference itself at best tenuously linked to evidence in the record, and arguably an effort quite similar to the government’s to suggest to the jury facts about which Resnick chose for strategic reasons not to present evidence. More importantly, the district court’s curative instruction guarded against precisely the prejudice that Resnick alleges, explaining that the jury could not infer from the government’s exhibits whether Resnick had produced the documents shown therein. And the challenged remarks are not so inflammatory or prejudicial that a proper curative instruction would be an insufficient remedy.
Lastly, we have no reason to believe that the challenged remarks contributed to Resnick’s conviction. As discussed above, Resnick could be convicted for conspiracy to obstruct justice even if he ultimately preserved and produced all documents, so long as he “knowingly joined and participated in,” Morgan, 385 F.3d at 206 (internal quotation marks omitted), a conspiracy to destroy documents. Indeed, the' government emphasized in its summation that it did not “have to prove that there is actual destruction of records or documents,” but only “an agreement to destroy records.” J.A. 878-79. Particularly in light of the curative instruction, we have no reason to think that the challenged remarks contributed to the jury’s finding of guilty, in light of ample evidence of the same.
C. Suppression Motion
Resnick argues that the district court erred in denying his motion to suppress the recordings in which he acknowledged that he had deleted his hard drive at Kergil’s direction and that his conduct was “stupid” and “wrong.” At *593minimum, Resnick argues, the district court erred in denying him an evidentiary hearing regarding the tapes. We review a district court’s denial of a motion to suppress de novo on questions of law and for clear error on findings of fact. United States v. Stewart, 551 F.3d 187, 190-91 (2d Cir.2009). We review the denial of an evidentiary hearing for “abuse of discretion.” United States v. Bonventre, 720 F.3d 126, 128 (2d Cir.2013).35
At the time of the recordings, Resnick had not been indicted, and the government was aware that he was represented by counsel. Under New York Code of Professional Conduct Rule 4.2(substantially similar to former Disciplinary Rule 7-104), a lawyer may not “communicate or cause another to communicate about the subject of the representation- with a party the lawyer knows to be represented by another lawyer in the matter, unless the lawyer has the prior consent of the other lawyer or is authorized to do so by law.” 22 N.Y.C.R.R. § 1200.0, 4.2(a). That rule applies to federal prosecutors in New York State, see 28 U.S.C. § 530B(a), and to “non-attorney government law enforcement officers when they act as the alter ego of government prosecutors,” United States v. Jamil, 707 F.2d 638, 645 (2d Cir.1983).
Prior to indictment, however, the government is “authorized by law,” and thus permitted under Rule 4.2, to “employ legitimate investigative techniques in conducting or supervising criminal investigations, and the use of informants to gather evidence against a suspect will frequently fall within the ambit of such authorization.” United States v. Hammad, 858 F.2d 834, 839 (2d Cir.1988). In Hammad, we held that a prosecutor’s eliciting statements from a represented defendant prior to his indictment through the use of a sham grand jury subpoena fell outside of the “authorized by law” exception. Id. at 840. “The court in Hammad was very careful, however, to urge restraint in applying the Rule in the pre-indictment context so as not to unduly hamper legitimate law enforcement investigations.... ” Grievance Comm. for S. Dist. of N.Y. v. Simels, 48 F.3d 640, 649 (2d Cir.1995). Since Hammad, this Court, in considering an alleged violation of Rule 4.2, has not found government conduct to fall outside the “authorized by law” exception. Siméis makes clear that a pre-indictment undercover communication with a represented person does not ipso facto violate the rule. See id.
Resnick offers several arguments why thé government’s conduct was not “authorized by law,” each without merit. First, he maintains that the recordings were “in direct contravention with [his] invocation of his right to remain silent,” expressed when hie previously declined the government’s invitation to proffer and cooperate. Resnick Br. 26. A defendant must be apprised of his right to remain silent “before a custodial interrogation may begin,” United States v. Plugh, 648 F.3d 118, 125 (2d Cir.2011), but Resnick’s telephone conversations with Krupit were of course not a custodial interrogation.
Second, Resnick argues that the government conduct was impermissible because “the goal of the investigation was not on*594going criminal and/or obstructive activity, but rather a discussion of ... past conduct.” Resnick Br. 27 (emphasis omitted). But as we explained in Hammad, the “use of informants by government prosecutors in a preindictment, non-custodial situation ... will generally fall within the ‘authorized by law exception to [Disciplinary Rule 7-104].” 858 F.2d at 840. The limitation that Resnick proposes is not supported by our case law, and would swallow the rule generally permitting use of informants, as “past conduct” will often be the focus of such investigations.
Third, Resnick argues that “Krupit actively acted as a ‘spy’ in ... Resnick’s camp” by, “while feigning loyalty[,] expressly ask[ing] ... Resnick to reveal his lawyer’s advice and strategies.” Resnick Br. 27. But the record demonstrates that the discussion of lawyers’ advice and strategies was not extensive, and was primarily instigated by Resnick, not Krupit.36 Thus, Krupit’s limited questions regarding Res-nick’s discussions with his attorney were “part of the natural flow of conversation,” rather than an “attempt to elicit privileged information.” United States v. Nouri, 611 F.Supp.2d 380, 387 (S.D.N.Y.2009). Notably, Resnick and Krupit did not share an attorney and were not contemplating a joint defense, and there is no indication that sensitive defense strategies were actually disclosed to Krupit.
Finally, Resnick maintains that “Krupit repeatedly tried to give [him] legal advice, undermining [his] faith in his own defense and potentially his counsel.” Resnick Br. 28. But that supposed “legal advice” consisted only of Krupit’s suggestions to Res-nick that they had done something wrong in submitting fraudulent applications. For example, Krupit stated that his lawyer asked “how can you check ‘no’ [that the policies were not for resale] when you’ve talked to these people about the possible sale of the policies?” J.A. 205. Such statements constitute legal advice or undermine Resnick’s confidence only in the sense that they suggest that Resnick had engaged in wrongdoing. Such a suggestion is not impermissible, since that is precisely what might elicit an incriminating statement from Resnick.
Thus, the district court did not err in concluding that the government’s use of an undercover informant to question Resnick, before indictment, in the knowledge that he was represented by counsel, did not fall outside the “authorized by law” exception of Rule 4.2. While the specific questions asked by Krupit in at least one instance *595give us pause, and in several respects suggest the pitfalls and risks inherent in investigative approaches of this kind, we see no indication that the government intended to invade the attorney-client relationship. As importantly, Resnick does not argue that any of the portions of the transcript that touch on the topic of attorneys’ advice were introduced into evidence or prejudiced him in any other way. Under the circumstances, the district court did not err in denying Resnick’s suppression motion. For the foregoing reasons, it also did not “abuse its discretion” in denying an evidentiary hearing that would have had no effect on its ultimate decision. See United States v. Forbes, 790 F.3d 403, 411 (2d Cir.2015).
III. Sentencing Challenges
Defendants argue that their sentences are procedurally unreasonable because the district court erred in calculating the amount of loss their scheme imposed. Re-latedly, Resnick contends that the government should have been judicially estopped from arguing at sentencing that the scheme imposed a quantifiable loss on the insurers, after maintaining in á motion in limine that the ultimate financial performance of the STOLI policies could not be determined. Lastly, Kergil argues that his sentence is substantively unreasonable. We conclude that the loss amount calculation was not erroneous, that Resnick’s judicial estoppel argument fails, and that the sentences are not substantively unreasonable.
A. Loss Amount
In challenging their sentences, defendants primarily argue that the district court used an erroneous loss amount in calculating their Guidelines ranges. “Loss for purposes of the fraud guideline [of the United States Sentencing Guidelines] ... is defined as ‘the greater of actual loss or intended loss.’ ” United States v. Certified Envtl. Servs., Inc., 753 F.3d 72, 103 (2d Cir.2014) (quoting U.S.S.G. § 2B1.1 cmt. 3(A)). “ ‘Actual loss’ means the reasonably foreseeable pecuniary harm that resulted from the offense,” U.S.S.G. § 2B1.1 cmt. 3(A)(i), whereas “ ‘[ijntended loss’ ... means the pecuniary harm that was intended to result from the offense,” id. § 2B1.1 cmt. 3(A)(ii). A district court is not required to calculate loss with “absolute precision,” but need only by a preponderance of the evidence “make ‘a reasonable estimate of the loss’ given the available information.” See United States v. Coppola, 671 F.3d 220, 250 (2d Cir.2012) (quoting U.S.S.G. § 2B1.1 cmt. 3(C)). We review a district court’s factual findings as to loss amount for clear error and its legal conclusions de novo. United States v. Uddin, 551 F.3d 176, 180 (2d Cir.2009).
In advance of sentencing, the government submitted memoranda calculating as to each defendant the intended losses of the scheme at $141,947,88037 and the aetu*596al losses at $38,153,631. While the Guidelines permit calculating loss at “the greater of actual loss or intended loss,” U.S.S.G. § 2B1.1 cmt.. 3(A), the court elected to calculate the Guidelines ranges based on actual loss and adopted the government’s calculation, resulting in a 22-level enhancement.38 See U.S.S.G. § 2B1.1. That yield ed a Guidelines range of 168 to 210 months’ imprisonment for Binday,39 155 to 188 months for Kergil,40 and 87 to 108 months for Resnick.41 The district court sentenced each defendant to terms of imprisonment below the bottom of the Guidelines range: Binday to 144 months, Kergil to 108 months, and Resnick to 72 months.
Defendants challenge the method used to calculate actual losses, and thus their Guidelines ranges. To reach actual losses, the government examined the 74 policies that were issued pursuant to the scheme proved at trial. It added all the commissions and death benefits that had been paid under these policies, and subtracted from that amount “any premiums [the] Insurers received either before death or before termination by lapse or otherwise on a policy the outcome of which is known.” J.A. 1566-67. That is, the government did not consider in the actual loss figure any policies that remained outstanding — except to consider the commission defendants received on those policies.42
Defendants argue that that approach was fundamentally flawed because it ignores the nature of insurance by considering only those policies that have already terminated at the time of sentencing: They contend that this “counting the losers” is not how anyone in the insurance business would assess the performance of a pool of policies, and is therefore not a permissible means of calculating the actual losses caused by the pool of STOLI policies. For support, they invoke Avery’s *597testimony: ‘You have the individual who pays one premium and dies six' months later. You lose a lot of money on that policy but we don’t consider that a loss.... [T]hat’s the benefit of insurance because there’s another 900 people who paid a premium who didn’t die.” Binday Br. 44-45 (emphasis omitted) (ellipsis in ' original) (quoting J.A. 486).43 Defendants argue that we cannot calculate actual losses because the insureds whose policies are still in effect might outlive their life expectancy, reducing or perhaps eliminating the losses incurred on those who died sooner.
We agree that the government’s approach is unlikely to yield an accurate measure of the ultimate performance of the pool of policies. Presumably, in any collection of policies some insureds will die ■ earlier than expected and some later. Tallying the insurer’s gains and losses by referencing only those insureds that died in the first few years of the policies therefore provides a distorted view of the group’s ultimate overall performance.
But the fact that this is not a method that would be used by insurers to calculate the ultimate outcome of the policies is not necessarily dispositive. In determining loss amount, the district court need not find the amount beyond a reasonable doubt and with exact precision. Rather, the district court may find the loss amount by a preponderance of the evidence, making a reasonable estimate based on the information available. Uddin, 551 F.3d at 180.
Under defendants’ logic, a district court would never be able to determine the actual loss on a group of fraudulently obtained life insurance policies while some (or at least a substantial portion) of those policies remained in effect. Notably, the defendants have not offered an alternative calculation for actual loss, nor is one readily apparent. Indeed, the alternative for which defendants seem to argue is zero, because the actual losses cannot currently be determined. To be sure, it is not defendants’ obligation to establish loss amount. ' Yet unless we conclude, which we hesitate to do, that actual loss caused by frauds of this nature are categorically outside the reach of the loss Guideline even where there has clearly been some loss, the absence of a better alternative weighs in favor of concluding that the method used here is a reasonable one.
Moreover, if defendants’ misrepresentations indeed significantly upset the insurers’ actuarial assumptions, then it does not follow from the fact that the actual loss calculation only accounted for the worst-performing policies that the insurers would profit from the remaining policies, let alone in a way that would significantly recoup their earlier losses. Indeed, as the government emphasizes, some of the insurers “claimed projected losses on the in-force scheme policies.” Gov. Br. 132 (citing J.A. 1567 n. 28). Additionally, the same arguments -that defendants use to question limiting consideration of loss to the pool of policies on which death benefits have already been paid counsel consideration of the more speculative, and much larger, intended loss amount, which does attempt to project losses across the entire pool of policies.
We conclude that the loss amount calculation was not clearly erroneous. Initially, it bears noting that the approximately $11 *598million of that amount attributable to the commissions defendants received constitutes a loss to the insurers. With respect to the remaining amount, the calculation was .a “reasonable estimate of the loss given the available information.” Coppola, 671 F.3d at 250 (internal quotation marks omitted). While the insurers might recoup their losses on the expired policies from the policies that remain in effect, that is a possibility and not a certainty. The policies for which the results are already known are in that sense the only ones for which an “actual” gain or loss to the insurer can be calculated. We leave open the possibility that on different facts, the method employed here may not provide a reasonable estimate of actual losses. But where, as here, the district court reasonably found that the insurers’ actuarial assumptions had been disrupted by the defendants’ misrepresentations — and thus the insurers might not recoup their losses on the remaining policies — calculating actual losses based on those policies that have in fact terminated was not a clearly erroneous approach.44
To the extent that we find the method of calculating the loss amount to be imperfect, we take comfort in the district court’s emphatic statement that it would have imposed the same sentence regardless of the loss amount, which renders any error in the loss calculation harmless. The district court explained that it would “calculate the guidelines, consider the guidelines, and then sentence in the old-fashioned way,” J.A. 1592. Indeed, the court found the case to be “a perfect example of why the [Guidelines] should be abolished,” id. at 1595, and described the time and effort invested in arguing over the Guidelines calculations as “really extraordinary, and ... completely unnecessary.” Id
Instead of relying on the loss amount, the court explained that it would “[e]mpha-siz[e] who [the defendants] are, what they did ... and send a message to the industry- that this sort of conduct will not be tolerated.” Id. at 1625. The court found that, “whatever [the loss amount] ..., this was a scheme perpetrated over a span of years, brazen, ... and characterized by a number of truly horrible behaviors on the defendants’ part.” Id. at 1626. The district court emphasized that the scheme involved “rampant mendacity, the creation *599of false documents, [and] obstruction of efforts by the victims [and the government] to ascertain the truth.” Id.
Moreover, this was not a situation in which the district court imposed a 22-level increase based on an actual loss amount of $38 million, where arguably there should have been no increase because no loss could be identified.45 At minimum, even if no loss is attributed to the policy outcomes on the STOLI policies that defendants fraudulently imposed on the insurers, the approximately $11.7 million that the defendants received in commissions constitutes an actual loss to the insurers.46 An $11.7 million loss- — as one between $7 million and $20 million — results in a 20-level increase in the guideline sentence. See U.S.S.G. § 2B1.1. Subtracting two levels from each of the defendants’ offense levels would result in a Guidelines range of 135-168 months’ imprisonment (instead of 168-210) for Binday, 121-151 months (instead of 155-188) for Kergil, and 70-87 months (instead of 87-108) for Resniek. See U.S.S.G. § 5, Pt. A. The terms of imprisonment imposed by the district court of 144, 108, and 72 months, respectively, remain either below or on the lower end of the Guidelines range that would result were we to include as actual losses only the commissions defendants received. Under these circumstances, we have no reason to question the district court’s statement that its sentences were not affected by the Guidelines calculation.
B. Judicial Estoppel Argument
Resniek contends that the district court erred in rejecting his argument that the government was judicially estopped from arguing at the sentencing stage that the insurers suffered a determinable, quantifiable loss. As an initial matter, Resnick’s judicial estoppel argument attacks the use of a loss amount calculation that, as discussed above, did not affect the sentences imposed. Thus, even if Resniek were correct, that would not necessarily undermine his sentence. But in any event, Resnick’s judicial estoppel argument fails.
Resniek notes that the government successfully moved in limine to preclude the defendants from submitting evidence that the insurers were not actually harmed by the fraud, on the ground that actual harm was not required. In support of that motion, the government argued that it had not alleged that the insurance companies suffered a loss on the STOLI policies because “this is a long game ... [and] [t]he *600way this all shakes out in the end is not clear, but there is no question that there was an intent to defraud and a scheme to defraud.” J.A. 267. The government repeated this argument in opposing the defendants’ motion for a judgment of acquittal under Federal Rule of Criminal Procedure 29.
Resnick concedes that “there is no necessary inconsistency between arguing that ultimate loss need not be proved at trial, and arguing that losses can be proved at sentencing.” Resnick Br. 45. But, he argues, the government maintained not simply that actual loss need not be proven, but that the amount of loss, if any, was unknowable. According to Resnick, under the rule of judicial estoppel the government cannot take the position at trial that the harm is unknowable, only to argue at sentencing that the actual losses inflicted by the scheme can be calculated and amount to $38 million. See New Hampshire v. Maine, 532 U.S. 742, 749, 121 S.Ct. 1808, 149 L.Ed.2d 968 (2001) (“[Jjudicial estoppel[ ] generally prevents a party from prevailing in one phase of a case on an argument and then relying on a contradictory argument to prevail in another phase.” (internal quotation marks omitted)).
Contrary to Resnick’s argument, the government’s positions at trial and at sentencing are perfectly consistent. At both stages, the government maintained that it could not know what the ultimate loss figure on all of the policies will be. Indeed, that uncertainty is why in calculating actual losses the government included only those policies which were no longer in effect, due to death, lapse, or other termination. That method—whether it provided a reasonable estimate based on available information, as the government maintains, or was, as defendants maintain, akin to counting only those houses that caught on fire—was ' consistent with the government’s theory throughout the case.
C. Substantive Unreasonableness Argument
Kergil, alone among the defendants, challenges his sentence as substantively unreasonable. He contends that his sentence is excessive because at the time of defendants’ conduct fraudulent STOLI policies were “a matter for civil litigation rather than criminal indictment,” and because his sentence of nine years’ imprisonment far exceeds what was necessary to deter similar fraud among other brokers. Kergil Br. 47. We reject those arguments.
Substantive review of sentences “provide[s] a backstop for those few cases” where allowing the sentence imposed to stand “would ... damage the administration of justice because the sentence imposed was shockingly high, shockingly low, or otherwise ¿insupportable as a matter of law.” United States v. Rigas, 583 F.3d 108, 123 (2d Cir.2009). We will reverse a sentence for substantive unreasonableness only “where the trial court’s decision cannot be located within the range of permissible decisions.” United States v. Cavera, 550 F.3d 180, 189 (2d Cir.2008) (en banc) (internal quotation marks omitted).
Kergil cannot meet our high bar for vacating a sentence as substantively unreasonable. He took part in a sophisticated, multi-million dollar fraud scheme. And when the FBI began investigating the scheme, he directed co-conspirators to obstruct justice by destroying incriminating documents. Notably, Kergil’s sentence fell below his Guidelines range (even when adjusting the loss amount to include only commissions). See United States v. Fernandez, 443 F.3d 19, 27 (2d Cir.2006) (“[I]n the overwhelming majority of cases, a Guidelines sentence will fall comfortably within the broad range of sentences that *601would be reasonable in the particular circumstances.”)47 Given Kergil’s culpability, and the district court’s reasonable determination that the sentence should serve as a deterrent to other brokers, the 84-month sentence does not shock the conscience.
D. Restitution Amount
After the judgment in the case was entered, the parties agreed that the total restitution award should be reduced to $37,433,914.17. Because the defendants had by that point appealed their convictions, the district court was without jurisdiction to amend the amount as agreed upon by the parties. Accordingly, the government has requested, without objection from defendants, that we remand the case for the limited purpose of permitting entering a revised restitution order in the amount agreed upon by the parties.
CONCLUSION
For the foregoing reasons, we AFFIRM the defendants’ judgments of conviction and REMAND for the limited purpose of entering a revised restitution order.
. Given the jury’s verdict of guilty, "we view the evidence in the light most favorable to the government.” United States v. Mergen, 764 F.3d 199, 202 (2d.Cir.2014) (internal quotation marks omitted).
. Such policies are sometimes referred to as “IOLI” (“investor-originated life insurance”) policies.
. Alienability could be useful if, for example, the insured fell ill and her need of money for healthcare or other living expenses outweighed her estate considerations.
. Some investors agreed to purchase the STO-LI policies as soon as they were issued, while others funded the premium payments immediately but did not purchase the policies until the two-year “contestability period” had run, after which an insurer cannot deny benefits or rescind a policy on the basis of misstatements in the application.
. To disguise the source of the funds used to pay the premiums, the brokers and investors typically held the policies and paid the premiums through trust funds established in the straw buyer’s name but funded by the investor. The brokers’ friends and family members often served as trustees of the trusts, and received a fee for ensuring that the trust’s funds were used to meet the premium payments.
. In addition to the commission profits, defendants in two cases used their inside knowledge of a straw insured's deteriorating health to repurchase policies from their original investors, making a substantial profit when the straw insured died shortly thereafter.
. In its rebuttal case, the government presented evidence that those notations from, the insurers signified questions rather than conclusions, which defendants allayed through fraudulent documentation.
. As defense counsel put it: "So, the difference is in STOLI, when you sign it, [you] intend to sell it[.] [T]hat the life insurance companies say, oh, that’s terrible. Bad for business, bad social policies, bad everything. But, a minute later [you] can decide to sell it into the life settlement market and ... that’s okay. There's no social problem with that.” Trial Tr. 1443-44.
. Binday’s total offense level was calculated at 35, Kergil’s at 34, and Resnick's at 29. Each defendant had a Criminal History Category of I.
. For example, in United States v. Starr, defendants processed bulk mailing for their customers, underpaid the Post Office by concealing high-rate mail in low-rate mail packages, but charged their customers the full price and kept the difference. 816 F.2d at 96. That conduct, we held, did not constitute mail fraud against the customers, because they had "received exactly what they paid for” and "there was no discrepancy between benefits 'reasonably anticipated' and actual benefits received.” Id. at 99. Similarly, in United States v. Novak we held that where the defendant’s counterparties had “received all they bargained for,” it was not sufficient to support conviction for mail fraud that the coun-terparties might have refused the bargain "had they been aware that [defendant] would receive a portion of the money” as a personal kickback. 443 F.3d at 159.
In United States v. Mittelstaedt, where a government employee concealed his ownership interest in property that his department agreed to purchase, we held that it was not sufficient to show that the government, had it known the truth, "would have refused to deal with him on general principles.” 31 F.3d 1208, 1218 (2d Cir.1994). Rather, "[t]o convict, the government had to establish that the omission caused (or was intended to cause) actual harm to the [purchaser] of a pecuniary nature or that the [purchaser] could have negotiated a better deal for itself had it not been deceived.” 31 F.3d at 1217 (emphasis in original). And in United States v. Shellef, defendants induced a company to sell them its products by falsely representing that they would not resell the products domestically. 507 F.3d at 107-08. We vacated defendants’ conviction for wire fraud because the indictment alleged only that defendants' misrepresentation induced the seller “to enter into a transaction it would otherwise have avoided” and not that the misrepresentation "had relevance to the object of the contract.” Id. at 108-09.
. Shellef distinguished itself from the factually similar case of United States v. Schwartz, 924 F.2d 410 (2d Cir.1991), where we upheld a conviction for mail fraud, as a case concerning the sufficiency of the evidence, rather *571than the sufficiency of the indictment, at issue in Shellef. 507 F.3d at 108. In Schwartz, defendants induced a company to sell them military equipment by falsely representing they would not resell the equipment to nations that U.S. law prohibited from purchasing them. 924 F.2d at 414-16. We concluded that there was sufficient evidence that the misrepresentations "were not simply fraudulent inducements,” because the deceit cost the victim “good will because equipment [that the victim], a government contractor, sold was exported illegally.” Id. at 421. We explained that no "pecuniary harm” need be inflicted or intended, so long as the deceit goes to "an essential element of the bargain.” Id. Similarly, we have found contemplated harm proven where defendant waste disposers made misrepresentations to their customer that "could have subjected the [customer] to fines and to the loss of its environmental permit.” United States v. Frank, 156 F.3d 332, 335 (2d Cir.1998).
.Defendants do not renew on appeal their argument at trial that there was insufficient evidence that the insurers were actually deceived by the defendants' misrepresentations, because the insurers in fact wanted to issue STOLI policies, while only pretending to take steps to avoid them. (Resnick, however, argues that there was insufficient evidence of his intent to defraud because he believed that insurers secretly wanted to issue STOLI policies.) To the extent that defendants' recounting in the background section of their briefs the evidence at trial on that point could be construed as a. challenge to the sufficiency of the evidence, we reject it. The government presented evidence that the insurers’ internal notes regarding “IOLI” detection were only an indication of the insurers' suspicion, which defendants allayed through additional fraudulent documentation. There was ample evidence for the jury to conclude that the insurers' efforts to detect STOLI policies were in earnest, and that defendants in fact deceived the insurers.
. 'Because we conclude that sufficient evidence established the four specific harms identified in the indictment, and because the indictment did not include among those harms the jeopardizing of the insurers’ tax treatment, we do not address whether jeopardizing the victim's tax treatment could constitute a cognizable harm.
. For example, Burns testified that STOLI policies might compromise the insurance industry "as it- was related to the social benefits,” because "STOLI took what is a financial instrument that is intended to protect families and individuals and turned it into an investor commodity!,] and [Lincoln Financial] felt that that could potentially put some of the social and tax benefits that life insurance has at risk.” J.A. 577. Avery likewise expressed his company's concern that STOLI policies might be illegal without an insurable interest and "would have insureds or beneficiaries feeling that they somehow were duped, and it would *573be bad for our reputation to be involved in these transactions.” Id. at 485. Burns also testified that issuing STOLI policies would cause the insurers' reinsurance costs to rise. Id. at 606. As with the tax concerns referenced in note 13 above, because we reject defendants’ argument for the reasons set forth in the text, we can and do assume without deciding, for purposes of this opinion, that these reasons for avoiding STOLI policies can be classified as non-economic. We note, however, that while none of those reasons affect the short-term economic benefits of issuing a particular insurance policy, they all suggest that insurance companies chose to avoid STOLI policies for reasons related to their long-term economic interests. That is no surprise; insurance companies are economic entities, which can be expected to act in their own perceived interest, and not to reject a potentially profitable line of business for non-business reasons. We need not decide here whether a fraud prosecution can be based on deceptively inducing a company to accept transactions that cause no short-term losses — or even can be expected to turn a profit — but that the company reasonably fears will be harmful to it in the long run.
. The jury could reasonably interpret that testimony as a diplomatic way of stating that the insurers had certain life expectancy assumptions about the group of people who were in a position to take out such a policy, and that those assumptions might not hold for STOLI straw-insureds. It is a, reasonable inference that an insurance executive might be reluctant to say flatly that he prefers richer customers, because poorer ones die sooner and thus cut into profits.
. Defendants emphasize that both executives testified that they had not seen negative economic consequences arise from STOLI and that Burns identified the "social aspects” of STOLI as "the primary risk of concern.” Id. at 614-15. Defendants contend that this testimony supports their theory that STOLI policies were no different economically from non-STOLI policies, and that the insurers only *574wanted to avoid the policies on general principle. But that argument rests on a distortion of the testimony. Avery and Burns testified that STOLI policies did in fact have different economic characteristics, but that the consequences of those difference was limited because of the insurers' efforts to avoid issuing STOLI policies. Avery testified that Prudential “would incur losses” if it “sold a large number of these policies,” but that the company “didn’t see [a] different experience [with STOLI policies] because [it] tried [its] best ... not to sell any.” Id. at 484, 485. Similarly, Bums testified that because Lincoln Financial "largely screenfed] out unwanted ... STOLI business ... there shouldn't have been a significant number of policies ... [with] the adverse economics” of STOLL Id. at 614. In any event, to the extent that portions of the executives' testimony could be construed as supporting arguments made by the defendants, or as contradicting or undermining other portions relied on by the government, it was for the jury to sort out the contradictions and decide what the testimony, taken as a whole, did or did not prove.
. We do not address whether a victim’s belief that information was economically valuable would suffice if that belief were entirely *575unreasonable or idiosyncratic. But a jury is entitled to credit the informed and plausible business decision of a sophisticated entity as to what constitutes, for its purposes, potentially valuable economic information.
. The fact that transferability of these policies is required by state law bolsters our conclusion that the two are not identical. Because insurers are required to allow transfers of insurance policies, but are not required to issue STOLI policies, we cannot infer that they are indifferent between allowing and prohibiting transfers of STOLI policies. We similarly cannot infer that they are indifferent between a policy that has the possibility of being transferred someday and a STOLI policy that is certain to be transferred.
. The same argument would also seem to apply equally to the expectation that STOLI policyholders would be more likely to make only the minimum payments or to avail themselves of grace periods. '
. Binday did not testify at trial. To the extent he offers these assertions as representing his actual beliefs, they are unsupported by any evidence in the record. We thus assume that they are presented as a hypothesis of a possible innocent state of mind that Binday maintains the government's evidence did not sufficiently exclude, and that necessarily would raise a reasonable doubt in the mind of a rational juror.
. While Binday does not explicitly attack the sufficiency of the evidence regarding his intent on this basis, he too emphasizes evidence apparently meant to suggest that the insurers secretly wanted STOLI business.
. In the background portion of .his brief, Kergil notes two additional pieces of evidence that appear to be offered to support the proposition that the insurers secretly wanted to issue STOLI policies. First, Kergil points to Avery’s testimony that in 2004-05, Avery was approached by brokers who suggested that the company issue and market STOLI policies. See J.A. 469.(Avery went on to testify that his company decided not to issue STOLI policies, explaining that, "[b]eing an actuary, I understood pricing, I also understood risk and quickly understood this was not the type of transaction[ ] thát I would allow at [the company] in my capacity.” Id.) Second, Ker-gil notes that several of the policies were issued despite being flagged by the insurers with an internal notation stating ‘TOLL” But Kergil does not note these facts in the portion of his brief addressing the sufficiency of the evidence as to his intent, and for good reason: because there is no evidence that Kergil was privy to the insurance companies' internal deliberations as to whether to market STOLI policies, and because he did not see the insurers’ internal notations regarding "IOLI” detection, those pieces of evidence are not probative of his intent. As noted above, defendants do not challenge the sufficiency of the evidence concerning whether their misrepresentations actually deceived the insurers, and to the extent their appeals could be construed to raise such a challenge, it fails because there was ample basis for the jury to conclude that the insurers’ efforts to avoid STOLI policies were sincere.
. Under that standard, for this Court to correct an error defendants must show that "(1) there is an error; (2) the error is clear or obvious, rather than subject to reasonable dispute; (3) the error affected [their] 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, 560 U.S. 258, 262, 130 S.Ct. 2159, 176 L.Ed.2d 1012 (2010) (second alteration in original) (internal quotation marks and alteration omitted).
. In an attempt to show that the jury charge erroneously permitted conviction on a no-sale theory, Binday poses a hypothetical: if an insurer refused to issue STOLI policies because it found them "unseemly,” and a broker through the mail received commission from deceiving that insurer into issuing STOLI policies, but those policies were no different economically than non-STOLI policies, would the jury charge here permit conviction of the bro*582ker for mail fraud? Binday Br. 20. He, of course, implies that it would.
. After the government submitted that agreed-upon language, the only additional modification proposed by the defendants was to “add at the end 'and you must find the defendants not guilty.’ ” Add. 3. The next day, Binday's counsel stated "[w]e are in agreement on that submission that the government made last night.... [0]bviously, subject to the *583objéctions we made on the record yesterday, we will deal with that charge.” J.A. 862.
. For example, defendants might have suggested the following modifications, shown in italics: (1) “If all the government proves is that under the scheme the insurance companies would enter into transactions that they otherwise would not have entered into—for example, because the insurers found the transactions morally objectionable—without proving that the ostensible victims would thereby have suffered some economic harm, then the government will not have met its burden of proof.”; or (2) “If all the government proves is that under the scheme the insurance companies would enter into transactions that they otherwise would not have entered into without proving that the ostensible victims would thereby have suffered some economic harm— such as a compromising of the insurers’ pricing assumptions that made the policies less economically desirable—then the government will not have met its burden of proof.” No such modifications were proposed to the district court.
. The Grand Jury Clause of the Fifth Amendment provides in relevant part that “[n]o person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a Grand Jury.” U.S. Const. amend. V. Accordingly, “a court may not alter or amend the indictment, literally or constructively, once it has been returned by the grand jury.” McCourty, 562 F.3d at 470.
. See post notes 37 and 44.
. Binday notes that he “did not have a horse in the race” between the insurer and the STOLI investor because he "earned his commission no matter which ‘bettor’ ... got it right.” Binday Reply Br. 13. But that does not negate an intent to inflict a cognizable loss on the insurers. Indeed, we have affirmed conviction where a broker induces a victim to make an economic decision based on fraudulent information, even though the defendant hoped that the transaction would succeed for the victim. See Carlo, 507 F.3d at 801.
. Kergil does not explicitly address his obstruction of justice conviction but indicates that he joins in the arguments made by his co-defendants.
. Subsection (c) punishes '‘[w]hoever corruptly ... alters, destroys, mutilates, or conceals a record, document, or other object, or attempts' to do so, with the intent to impair the object's integrity or availability for use in an official proceeding; or ... otherwise obstructs, influences, or impedes any official proceeding, or attempts to do so.” Subsection (k) provides that "[w]hoever conspires to commit any offense under [§ 1512] shall be subject to the same penalties as those prescribed for the offense the commission of which was the object of the conspiracy.”
. Resnick also notes that "[e]ven [his] alleged co-conspirators described his commitment to any document destruction scheme as 'light.' ” Resnick Br. 16. But that is of no matter. Whether Resnick’s participation was reluctant and limited, indeed whether he eventually reversed course, the jury could conclude that he took an overt action in furtherance of the conspiracy, and that is sufficient to sustain the conviction. Resnick did not request a jury instruction that if the jury found he had joined the conspiracy, he should nevertheless be acquitted if he later "thwarted the success of the conspiracy, under circumstances manifesting a complete and voluntary renunciation of his criminal purpose.” American Law Institute, Model Penal Code § 5.03(6). We therefore have no occasion to address whether any such defense exists, but note that we have not held "that withdrawal ends a conspirator's liability for the conspiracy prior to the time of withdrawal,” United States v. Dallas, 229 F.3d 105, 110 (2d Cir.2000), and have not adopted the Model Penal Code's defense, see United States v. LoRusso, 695 F.2d 45, 55 n. 5 (2d Cir.1982)
. We have previously assumed without deciding that the requirement of a nexus between the obstructive act and the official proceeding that is required under subsections (b)(2) and (c)(2) of § 1512 likewise applies to subsection (c)(1). See United States v. Ortiz, 220 Fed.Appx. 13, 16 (2d Cir.2007). Because Resnick's claim fails in any event, we likewise assume here that the nexus requirement applies.
. Moreover, Resnick argued, because no comprehensive account of the documents produced by Resnick was admitted into evidence, there was no evidence establishing that Res-nick had not produced the documents in the government exhibits. For its part, the government argued that its remarks were “a proper response to the argument proffered by Resnick’s counsel in summation,” because it was "entitled to point out, using the logic of counsel’s own argument, that there were incriminating emails admitted into evidence as Government Exhibits that did not bear the ’MR’ Bates stamp and therefore had not been produced by MAR Group." Id. at 1488 (emphasis omitted). In the alternative, the government proposed that the district court provide a curative instruction.
. "A district court has abused its discretion if it based its ruling on an erroneous view of the law or on a clearly erroneous assessment of the evidence, or rendered a decision that cannot be located within the range of permissible decisions.” In re Sims, 534 F.3d 117, 132 (2d Cir.2008) (internal quotation marks, alteration, and citation omitted); see also In re City of New York, 607 F.3d 923, 943 n. 21 (2d Cir.2010) (explaining that "abuse” is a nonpejorative "term of art”).
. Consider the following exchange:
[Resnick]: Well Paul, it's a scary, it’s a scary thing. And um, you know I almost wish it would you know the process would get started, so you know we could start dealing with things. You know, the waiting is the hardest part. You know?
[Krupit]: Yeah.
[Resnick]: So, I mean, your attorney hasn’t been able to find anything out?
[Krupit]: No, he's just fact finding, you know, faxing stuff back and forth, that kind of thing.
[Resnick]: Right nothing about the arrests or any of that stuff?
[Krupit]: No, what does yours say?
[Resnick]: Well, it’s gonna happen pretty soon.
J.A. 201. In only one instance did Krupit make an unsolicited inquiry touching on legal advice or strategy, asking, "Is your attorney asking about the commissions on [the policies]?” Id. at 207. Resnick responded that he had "told [his attorney] 'every one of the cases I know of, I never received a commission' ” Id. Thus, nothing that Resnick’s attorney said was conveyed to Krupit, and the statement to his attorney that Resnick conveyed to Krupit was not itself incriminating. Resnick then stated, “we were paid by Michael [Binday], we were never paid by the insurance company,” and that Binday had not paid Resnick a commission "in the last almost 3 years now.” Id. at 207-08. Resnick and Krupit then discussed the "formula” by which the commissions were to be divided.
. To arrive at intended loss, the government considered the 92 applications submitted by defendants that were "squarely connected to the scheme charged and proved at trial” for which the insurers had "still-extant records.” J.A. 1563. The government calculated "the difference between the death benefit sought and the expected total premium outlay on the applied-for policy” for each of these policies. Id. at 1564. For the 41 of the 92 policies for which Binday had prepared investor presentations, the "expected total premium outlay” was taken directly from these presentations. For the remaining 51 policies, the expected total premium outlay was calculated "us[ing] the 41 charts that do exist to estimate the average losses expected to be inflicted upon Insurers in cases for which no charts could be located.” Id. at 1565. After determining the difference between the death benefit sought and the expected total premium outlay for each of the 92 policies, the government dis*596counted this amount by 20% in order to "account for interest the Insurers could earn on premiums paid into the policies until death .. .• — intended to correspond roughly to an annually compounded rate of 4%.” Id. The government then added to this amount the $11,695,523 in commissions the insurers paid to defendants on these policies. Id. at 1566.
. The court explained that it was “easy to calculate [losses] known ... to date[,] but difficult to estimate future losses.” J.A. 1622. While the court "understood] that the guidelines embody a preference for intended loss, [it] favor[ed] calculating actual loss, which betters any reasonable estimate by virtue of being tethered in fact.” Id.
. Binday’s total offense level was 35, consisting of: a base offense level of 7, a 22-level enhancement based on loss amount, a four-level adjustment for his leadership role in the scheme, see U.S.S.G. § 3Bl.l(a), and a two-level adjustment for obstruction of justice, see U.S.S.G. § 3C1.1. His criminal history category was I.
. Kergil’s total offense level was 34, consisting of: a base offense level of 7, a 22-level enhancement based on loss amount, a three-level adjustment for his supervisory role in the scheme, see U.S.S.G. § 3B 1.1(b), and a two-level adjustment for obstruction of justice, see U.S.S.G. § 3C1.1. His criminal history category was I.
. 'Resnick’s total offense level was 31, consisting of: a base offense level of 7, a 22-level enhancement based on loss amount, and a two-level adjustment for obstruction of justice, see U.S.S.G. § 3C1.1. His criminal history category was I.
. According to the government's memorandum, some of the insurers disagreed with this omission and contended "that they [were] entitled to estimated losses on in-force policies as a part of restitution.” J.A. 1567 n. 28. But the government explained that it was "unaware of precedent for awarding projected losses in these circumstances or any closely analogous circumstances,” and that it had made no "attempt to quantify expected losses” on those policies as part of actual .loss for Guidelines or restitution purposes. Id. at 1567 &n. 28.
. Binday argues that counting only the policies where the insured as died before sentencing "is akin to looking at the financial health of a fire insurance company by considering only those homes that caught fire.” Binday Br. 43. Binday’s hypothetical somewhat overstates the case, because while a relatively few buildings with fire insurance will ever catch fire, most life insurance policies that do not lapse will eventually pay a death benefit.
. The actual loss amount calculation, unlike the intended loss amount calculation, does not discount the difference between the premium payments the insureds received and the death benefits to account for the fact that the premiums are received before the death benefits are paid. Binday argues in a footnote that that is erroneous because it "continues to ignore the fact that insurance companies invest the premiums that they receive." Bin-day Br. 43 n. 28. Assuming that a discount should have been applied, its effect on the calculation would have been insignificant for sentencing purposes. Of the $38,153,631 actual loss amount, $11,695,523 consisted of commissions, while the remaining $26,458,108 consisted of the difference between premiums received and death benefits paid. Discounting the latter component by 20% — the rate, which defendants do not challenge on appeal, used in calculating the intended loss amount — would reduce the figure to $21,166,486. (There would be no reason to discount losses attributable to commissions, because they are paid early in the policy and before the later years' premiums.) Adding back the commissions, the total loss amount after discount would still be $32,862,009, far above the $20 million threshold for the 22-level increase that was applied to defendants. See U.S.S.G. § 2B1.1. Even under the Guidelines, the recommended sentence is based on a table that groups together losses in a very wide range (here $20 million to $50 million), rather than on an impossibly precise calculation of loss. As the district court well understood, to the extent the culpability of the defendants is assessed, in part, by the magnitude of their scheme, there can be no doubt that this was a fraud of significant proportions.
. We have explained that "[w]e are especially wary” of concluding that a Guidelines calculation error did not affect a sentence where, if the appellant’s procedural challenge "were correct, his Guidelines sentencing range ... would have been cut by more than half.” United States v. Feldman, 647 F.3d 450, 460 (2d Cir.2011); see also United States v. Malki, 609 F.3d 503, 511 (2d Cir.2010) (“Although the [district court] stated that a lesser sentence would be ‘inappropriate,’ we cannot be confident that [it] would have imposed the same sentence had [it] understood that the bottom of the correct guideline was 58 months less than the bottom of the guideline [it] thought was applicable.”). Here, as explained below, the effect on the Guidelines recommendation was far less significant.
. As discussed above, had the insurers received the full benefit of their bargain, the fact that defendants profited from deceitfully inducing that bargain does not establish a stand-alone harm under the mail and wire fraud statutes. See Starr, 816 F.2d at 96; Novak, 443 F.3d at 159; Mittelstaedt, 31 F.3d at 1217. But where the defendants’ deceit deprives the victim of the benefit of its bargain, or exposes it to hidden economic risk, the defendants' receipt of commissions or other payment for services from the victim undoubtedly constitutes a loss to the victim. See Walker, 191 F.3d at 335; Frank, 156 F.3d at 335. The insurers here paid more than $11 million in commissions for generating policies that they did not want because they expected them to be economically unprofitable.
. As evidence that addressing frauds of this sort was previously left to civil litigation, Ker-gil cites as examples nine civil cases in which insurers sought to rescind STOLI policies based on fraudulent applications. But that STOLI frauds continued despite repeated civil enforcement supports the district court’s conclusion that more significant deterrence was appropriate.