PRADO, Circuit Judge:

Former Enron Corporation CEO Jeffrey K. Skilling was convicted of conspiracy, securities fraud, making false representations to auditors, and insider trading. After we affirmed his convictions, the Supreme Court invalidated one of the objects of the conspiracy charge—honest-services fraud—and remanded, instructing us to determine whether the error committed by the district court in submitting the honest-services theory to the jury was harmless as to any of Skilling's convictions. Because we find that the error was harmless, we affirm the convictions. In addition, for the reasons stated in our previous opinion, we vacate the sentence and remand for resentencing.

I. BACKGROUND

In May 2006, Skilling was convicted by a jury of one count of conspiracy, twelve counts of securities fraud, five counts of making false representations to auditors, and one count of insider trading. The indictment alleged several possible objects of the conspiracy, including securities fraud and honest-services fraud, and the district court's jury instructions permitted the jury to convict on any of the alleged theories of guilt. The jury returned a general verdict of guilty on the conspiracy charge without identifying the specific object of the conspiracy. The district court sentenced Skilling to 292 months of imprisonment and three years of supervised release, and assessed $45 million in restitution.

Skilling appealed, arguing, among other things, that his conspiracy conviction was premised on an improper theory of honest-services fraud. We affirmed the convictions, holding that the Government's honest-services theory was proper under Fifth Circuit case law. See United States v. Skilling, 554 F.3d 529, 595 (5th Cir.2009), vacated in part, ___ U.S. ___, 130 S.Ct. 2896, 177 L.Ed.2d 619 (2010). We also vacated the sentence and remanded for resentencing because the district court had incorrectly applied a sentencing enhancement for substantially jeopardizing a "financial institution." See id.

On appeal, the Supreme Court reduced the scope of the honest-services fraud statute and invalidated the Government's honest-services theory in this case. See Skilling, 130 S.Ct. at 2907 ("Because Skilling's alleged misconduct entailed no bribe or kick-back, it does not fall within [the honest-services fraud statute]'s proscription."). The Court did not, however, reverse any of Skilling's convictions, but remanded the case to us to determine whether the honest-services instruction amounted to harmless error. Id. at 2934-35.

II. STANDARD OF REVIEW

In Hedgpeth v. Pulido, 555 U.S. 57, 129 S.Ct. 530, 172 L.Ed.2d 388 (2008) (per curiam), the Supreme Court recently confirmed that an alternative-theory error— i.e., where a jury rendering a general verdict was instructed on alternative theories of guilt and may have relied on an invalid theory—is subject to harmless-error analysis "so long as the error at issue does not categorically `vitiat[e] all the jury's findings.'" Id. at 532 (alteration in original) (citation omitted); see Skilling, 130 S.Ct. at 2934 n. 46 (extending the holding of Pulido, which was a case on collateral review, to this case and other cases on direct appeal). The Court did not specifically identify the harmless-error standard that is applicable to alternative-theory errors, but it cited to a string of cases that apply a common harmless-error standard to other types of instructional errors. See Pulido, 129 S.Ct. at 532 (citing Neder v. United States, 527 U.S. 1, 119 S.Ct. 1827, 144 L.Ed.2d 35 (1999) (omission of an element of an offense); California v. Roy, 519 U.S. 2, 117 S.Ct. 337, 136 L.Ed.2d 266 (1996) (per curiam) (erroneous aider-and-abettor instruction); Pope v. Illinois, 481 U.S. 497, 107 S.Ct. 1918, 95 L.Ed.2d 439 (1987) (misstatement of an element of an offense); Rose v. Clark, 478 U.S. 570, 106 S.Ct. 3101, 92 L.Ed.2d 460 (1986) (erroneous burden-shifting as to an element of an offense)). The Court declared that "[a]lthough these cases did not arise in the context of a jury instructed on multiple theories of guilt, one of which is improper, nothing in them suggests that a different harmless-error analysis should govern in that particular context." Pulido, 129 S.Ct. at 532.

Consistent with this line of cases, there are two ways to prove the harmlessness of an alternative-theory error. First, as set forth in Neder v. United States (which is the most recent of the line of cases cited in Pulido), an error is harmless if a court, after a "thorough examination of the record," is able to "conclude beyond a reasonable doubt that the jury verdict would have been the same absent the error." 527 U.S. at 19, 119 S.Ct. 1827. If the defendant "raised evidence sufficient to support a contrary finding," then the error was not harmless. Id. Thus, under the so-called Neder standard, a reviewing court, "in typical appellate-court fashion, asks whether the record contains evidence that could rationally lead to [an acquittal] with respect to the [valid theory of guilt]."1 Id.

Second, as we held in United States v. Holley, 23 F.3d 902 (5th Cir.1994), and United States v. Saks, 964 F.2d 1514 (5th Cir.1992), an alternative-theory error is harmless if the jury, in convicting on an invalid theory of guilt, necessarily found facts establishing guilt on a valid theory. See United States v. Howard, 517 F.3d 731, 738 (5th Cir.2008) (stating that Holley and Saks stand for the proposition that "legally erroneous jury instructions [are] harmless in fraud cases when the inevitable result of the fraudulent activity proved at trial established that the defendants participated in the scheme that justified their convictions on legally correct instructions"). Our rulings in Holley and Saks pre-date the Supreme Court's decision in Pulido, but they apply a harmless-error test that is consistent with the Neder standard, and therefore we affirm their continuing vitality in our case law.2

III. ANALYSIS

A. The Conspiracy Conviction

The Government asserts that the invalid honest-services instruction was harmless with respect to the conspiracy conviction. Specifically, it argues that the evidence presented at trial proved that Skilling participated in a scheme to deceive the investing public about Enron's financial condition in order to maintain or increase Enron's stock price. If so, then we would be able to conclude beyond a reasonable doubt that absent the honest-services instruction, the jury would have convicted Skilling under a valid theory of guilt—conspiracy to commit securities fraud.3

Before examining the evidence presented at trial, however, we must address two nonevidentiary arguments raised by Skilling. First, Skilling argues that the district court's jury instructions, by permitting the jury to convict on either the honest-services theory or the securities-fraud theory, are dispositive evidence of the harmfulness of the error. We disagree. The instructions were clear and easy to understand, and they did not contain any statement that gave a preference to one theory over another. Moreover, the fact that the jury may have relied upon an invalid theory of guilt shows only that an alternative-theory error occurred, not that the error was not harmless. See Paredes v. Thaler, 617 F.3d 315, 318 (5th Cir.2010) (confirming that a jury instruction on alternative theories of guilt, one of which is invalid, is not a fatal "structural" error, but instead is "subject to harmless-error analysis").

Second, Skilling contends that the Government's opening and closing statements made it more likely that the jury would rely on the honest-services theory rather than on the securities-fraud theory. Again, we disagree. The Government's opening and first closing statements both mentioned honest-services fraud only in relation to Skilling's co-defendant, Ken Lay, who, unlike Skilling, was charged with several counts of honest-services wire fraud. With respect to Skilling, both statements focused exclusively on conduct that constitutes securities fraud. In its rebuttal closing statement, the Government made reference to the honest-services allegations against both defendants, but it mentioned the honest-services theory in relation to Skilling only once. Further, it never argued that the jury should convict Skilling solely on the honest-services theory, nor did it tell the jury that it should disregard the evidence of securities fraud in reaching a conviction. This single reference to Skilling's honest services, in light of the Government's extensive argument on securities fraud, merely permitted the jury to decide the case on the wrong theory. It did not force or urge it to do so, and therefore, it shows only that an alternative-theory error occurred, not that the error was not harmless.

Having disposed of these two preliminary arguments, we next turn to the crux of the matter: whether, under the Neder standard, the evidence presented at trial proves that Skilling conspired to commit securities fraud. Based on our own thorough examination of the considerable record in this case, we find that the jury was presented with overwhelming evidence that Skilling conspired to commit securities fraud, and thus we conclude beyond a reasonable doubt that the verdict would have been the same absent the alternative-theory error.4

First, the evidence overwhelmingly proved that Skilling and his co-conspirators transferred losses and the risk-management books from Enron's struggling retail division, Enron Energy Services ("EES"), to Enron's Wholesale division, which accounted for most of Enron's revenue, so that EES would appear to be more profitable than it really was. The testimony at trial established that under the mark-to-market accounting rules that EES professed to follow, it should have booked hundreds of millions of dollars in losses in the first quarter of 2001. These losses arose from bad-debt write-offs, errors in how EES had originally booked the value of its retail contracts, and unanticipated expenses that could not be passed on to EES's retail customers. According to the testimony, Skilling knew about the extent of the losses attributable to EES and approved a plan to (1) shift the losses and EES's risk-management books (and therefore most of EES's money-losing components) to Wholesale, even though the losses arose from EES's own retail contracts, not from Wholesale's business; and (2) justify this "resegmentation" as an operational change, even though it resulted in no efficiencies, Wholesale had no experience in handling retail contracts, and Wholesale and EES had different contracting procedures, origination policies, risk assessments, and marketing efforts. Dave Delainey, the CEO of EES at the time, testified that he warned Skilling that the resegmentation was fraudulent because it had no real business purpose and was designed to hide EES's losses from investors, and that Skilling approved the resegmentation anyway.

In his brief, Skilling argues that he "established through each of the government witnesses, as well as his own, that the accounting for this transaction was `rock solid' and complied with the disclosure rules." His record citations, however, do not substantiate his claim. In fact, his argument relies almost exclusively on his own testimony at trial, which was that he has been told that the EES resegmentation adhered to generally accepted accounting principles and that he had not been told that it might be illegal. The jury, by finding him guilty, necessarily determined that his own self-serving testimony, in which he contested his liability under any theory of guilt, including the honest-services theory, was not worthy of belief. Therefore, we too decline to give Skilling's testimony any weight in our harmless-error review when unsupported by other evidence or testimony in the record.

Skilling also argues that he presented evidence at trial that EES's losses "had either not occurred, had not occurred in the way the government's witnesses described, or were only speculative losses that had to be reserved against, and that proper reserves had been taken on all accounts." The record also proves that this claim is false. The testimony at trial clearly showed that EES's losses were real, recognizable under mark-to-market accounting rules, and due to be booked. Skilling points to no evidence, other than his own say-so, that disputes this finding.

Second, the evidence overwhelmingly proved that Skilling and his co-conspirators falsely portrayed Wholesale to the investing public as a low-risk company that made sustainable profits by delivering gas and electric power to customers (i.e., a "logistics company"), even though they knew that Wholesale actually made most of its profits from its highly volatile trading operations. The evidence presented at trial proved that the majority of Wholesale's earnings in 2000 and early 2001 came from trading activities, especially from speculative trading during the California energy crisis. This trading activity created huge gains (as much as $485 million in a single day) and losses (as much as $551 million in a single day). Witnesses testified that Skilling knew about the riskiness of Wholesale's business, but falsely represented to investors and analysts that Wholesale was a "logistics company," not a "trading company."

In addition, from 1999 to the analysts conference during the first quarter of 2001, Skilling repeatedly prohibited other Enron managers from calling Wholesale a "trading company" or referring to any of Wholesale's employees as "traders," often citing the negative effect such language would have on Enron's stock price. That is, if the market perceived Wholesale to be a trading company, it would lead to a decrease in Enron's price—earnings multiplier, which, in turn, would drag its stock price down. The managers obeyed Skilling and referred to Wholesale as a "logistics company" at their own engagements with investors and analysts.

Skilling argues that he presented sufficient contradictory evidence at trial proving that Wholesale actually was a "logistics company" because it owned one of the largest pipeline and energy distribution systems in the world, which allowed Wholesale to meet supply and demand for its energy customers and cover its trading positions. Yet again, this evidence is from Skilling's own self-serving testimony. Moreover, even if we were to believe Skilling's assertions, they do not undermine the Government's proof at trial, which showed that Skilling fraudulently kept from the investing public the reality that Wholesale was driving up its profits through highly risky and volatile trading operations, not through its energy distribution system.

Third, the evidence overwhelmingly proved that Skilling and his co-conspirators used LJM and LJM2, which were two partnerships run by Andy Fastow, who also served as Enron's CFO at the time, to hide Enron's nonperforming assets and book earnings to meet its earnings targets. Specifically, the testimony showed that Skilling was intimately involved in the Enron-LJM "Cuiaba" deal involving the sale of Enron's interest in a power plant in Brazil and the Enron-LJM "Nigerian Barges" deal involving the sale of Enron's interest in power-generating barges off the coast of Nigeria. In both cases, Fastow testified that Skilling personally asked him, on behalf of LJM, to buy money-losing assets at a price that no third party would be willing to pay and, in "secret" oral side deals, guaranteed that LJM would make a certain rate of return (or at least not lose any money) while it owned the assets. Skilling also agreed that Enron would buy the assets back from LJM if a permanent buyer could not be found, which it was eventually forced to do because both assets continued to deteriorate in value. For both deals, Enron booked gains on the sales of the assets to LJM, thereby allowing Enron to meet its earnings targets, even though neither transaction was a true sale because Skilling's "secret" guarantee eliminated any risks LJM might have suffered in the transactions. This caused Enron's financial statements to be false and misleading.

Skilling makes a number of arguments about the proof at trial, none of which are especially relevant. He argues that the Government's theory at trial was (1) that the very creation of the partnerships was fraudulent; (2) that the conflict of interest inherent in the formation of the partnerships was illegal; (3) that by creating and using these partnerships Skilling was opening Enron up to excessive risk and was therefore not doing his job properly; and (4) that the deals themselves made no business sense and therefore Skilling was not doing his job properly. We disagree with Skilling's characterizations of the Government's case, although we do not need to discuss them since we do not rely on them. Rather, we find that the evidence is overwhelming that Skilling was personally involved in the Cuiaba and Nigerian Barges deals and that he used those deals to cause Enron to book fake profits and hide money-losing assets from its investors.

Nonetheless, the Government's other arguments (as characterized by Skilling) are supported by strong evidence, and this evidence, in turn, bolsters the assertion that Skilling engaged in the two deals on which we rely. Likewise, we do not rely on any of the other dozen transactions that LJM and LJM2 executed with Enron, although there is also strong evidence that Skilling and his co-conspirators committed securities fraud in relation to those transactions as well. Such evidence further bolsters the assertion that Skilling engaged in the Cuiaba and Nigerian Barges deals. For all of Skilling's many counterarguments, his personal involvement with respect to those two deals is not controverted.

Fourth, the evidence overwhelmingly proved that Skilling and his co-conspirators underreported the projected losses of Enron's broadband division, Enron Broadband Services ("EBS"), and when those losses became too large to hide within EBS, merged EBS into Wholesale. Enron had marketed EBS to the investing public as an important part of its future growth strategy, but the evidence presented at trial showed that EBS lost money in every quarter of its existence and only met its earning targets in 2000 by engaging in a series of transactions that fell outside of its core businesses: (1) the sale of part of its fiber-optic network to LJM2 at a price that no third party would pay; (2) the hedging of a gain on its investment in Avici, an Internet start-up, into the Raptor special-purpose entity, which allowed EBS to recognize the gain as earnings under mark-to-market accounting; and (3) the "monetization" of a video-on-demand contract with Blockbuster, which allowed EBS to book anticipated future earnings from the contract. Skilling knew about all of these transactions, but failed to tell investors about their impact on EBS's bottom line.

During the first quarter of 2001, EBS's managers informed Skilling that EBS was making almost no revenue from its core businesses and would likely suffer a loss of $146 million, which was well short of its first quarter target of a loss of $30 million. After remarking that Enron was getting pressure from analysts to improve its return on invested capital, Skilling refused to adjust the unrealistic earnings target. Instead, he authorized a second monetization of EBS's contracts and approved a cost-cutting plan that included the dismissal a number of EBS employees. Again, Skilling, while selling investors on the growth potential of EBS, failed to disclose that EBS had almost no revenue from its core businesses. When EBS continued to deteriorate during the second quarter of 2001 (EBS was due to report a loss of $102 million), witnesses testified that Skilling approved a plan to merge EBS with Wholesale, knowing that, as a result, the losses attributable to EBS would be untraceable.

Again, Skilling's counterarguments are unavailing. It is not true, as Skilling claims, that the Government's theory at trial was that Skilling made bad business decisions; its argument was that Skilling hid those bad business decisions from investors. Moreover, we disagree with Skilling that the Government's case relied upon the selective editing of his statements. The record shows that Skilling's comments to investors were more than just hopeful views about a troubled new venture. Rather, given that EBS was sinking and Skilling knew it, his comments were deceitful. There was no reason for hope about EBS's prospects, and Skilling failed to disclose fundamentally important facts about EBS's problems.

Fifth, the evidence overwhelmingly proved that Skilling and his co-conspirators manipulated Enron's accounting reserves for contingent liabilities in order to hit specific earnings targets (known as "consensus estimates"). The evidence showed that Skilling knew that missing the consensus estimate by even a small amount would have a significant negative effect on Enron's stock price and, conversely, that exceeding the consensus estimate would have a significant positive effect on the stock price. Toward the end of the fourth quarter of 2002, Delainey told Skilling that Wholesale, which was making huge profits from its speculative trading, "had a couple of quarters [of reportable earnings] in its pocket." Skilling was pleased to hear this news and later said at an Enron management committee meeting that Enron had significant reserves that were available to meet its earnings targets for 2000. In fact, Wholesale set aside $873 million in trading income as reserves for contingent liabilities, which the testimony showed was a significant over-reservation. After the end of the accounting year, but before Enron had reported earnings, Wholesale told Richard Causey (Enron's Chief Accounting Officer at the time), who then told Skilling, that Wholesale had extra reserves available. Skilling, through Causey, communicated to Wholesale that it should shift as much reserves to earnings as was necessary to report 41 cents per share, which was 7 cents more than the fourth quarter earnings target of 34 cents per share. Wholesale shifted the money from a "gas and power valuation" contingent-liabilities reserve account, even though there had been no changes in the circumstances associated with the reserve account. Enron then produced a document for its outside auditors that falsely stated other reasons for the reserve transfer. Enron's stock price went up after Enron reported earnings for that quarter.

In response, Skilling argues that there was extensive evidence at trial that the final reserve amount accurately reflected contingent liabilities and that any differences were immaterial from an accounting perspective. We disagree that the record shows this. A Wholesale accountant testified that, on Skilling's command, he released money from the reserve account in order to make earnings go up, without consideration of the correct reserve amount. Such an act is fraudulent: when a company establishes reserves based on the earnings it wants, its investors do not know if the reserve accounts accurately reflect the contingent liabilities facing the company, and they do not know that the company is meeting or exceeding its earnings targets only by moving reserves from one bucket to another.5 Further, the amount transferred from reserves to earnings was not immaterial, since the transfer allowed Enron to exceed its earnings targets and caused Enron's stock price to increase.

Lastly, we note that we do not rely on the Government's allegation that Enron fraudulently misstated earnings for the fourth quarter of 1999 and fraudulently transferred contingent-liability reserves to beat the consensus estimate for the second quarter of 2000. Although there is strong evidence for a conviction on these allegations, there is not sufficient evidence to find harmlessness. Nonetheless, both allegations show that Enron executives had a pattern of manipulating earnings targets.

These five fraudulent schemes, which formed a large part of the basis for the Government's proof at trial, all represent efforts by Skilling and his co-conspirators to manipulate Enron's reported earnings or conceal Enron's losses from the investing public with the intent and result of affecting Enron's stock price. Because they are supported by overwhelming evidence, we find that the honest-services instruction was harmless error beyond a reasonable doubt.

B. The Other Convictions

Skilling also challenges his other convictions for securities fraud, making false statements to auditors, and insider trading. He argues that because the district court gave the jury a Pinkerton instruction— which permitted the jury to hold Skilling vicariously liable for the securities-fraud charges if they found him guilty of the conspiracy charge—the alternative-instruction error on the conspiracy charge also taints the other convictions. At oral argument, however, counsel for Skilling conceded that this challenge is predicated on the existence of a harmful error that invalidates the conspiracy conviction. Because we find that the alternative-instruction error in this case was harmless with respect to the conspiracy conviction, it follows that Skilling has no basis on which to challenge the remaining convictions.6

IV. CONCLUSION

The alternative-instruction error in this case was harmless beyond a reasonable doubt. Accordingly, we AFFIRM the convictions on all counts, and, for the reasons set forth in our previous opinion, we VACATE the sentence and REMAND for resentencing.