Full opinion text
GERARD E. LYNCH, Circuit Judge: 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 planning. 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 ’ Scheme Defendants-appellants are insurance brokers who participated in an insurance fraud scheme involving “stranger-oriented life insurance” (“STOLI”) policies. 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. 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 the 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. 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. After having the straw buyer sign a blank application, defendants supplied false financial information, 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. 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 caused 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. 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 reasonably 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. 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. 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 satisfied 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. 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. The requisite 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 those differences, and thus that they intended the harm. 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. 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. But contrary 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. 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. Defendants 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. 2. 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. 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 age 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. 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 information 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 misrepresented 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). “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 means 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’ economic 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. 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. 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 instruction, 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). 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. They maintain that their 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. The 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. 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. The 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). 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 correspond.” 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 prop