Defendants-appellants Champlain Enterprises, Inc., d/b/a Commutair, Antony von Elbe, John Arthur Sullivan, Jr., Ernest James Drollette and U.S. Trust Company of California, N.A. (collectively, “U.S.Trust”) appeal from a judgment of the United States District Court for the Northern District of New York (Hurd, J.) finding that U.S. Trust failed to satisfy its fiduciary duty under Employee Retirement Income Security Act (“ERISA”) §§ 404, 406, and 408, to ensure that plaintiffs-appellees’ Employee Stock Ownership Plan (“ESOP”) paid no more than fair market value for the convertible preferred stock it purchased from CommutAir on March 15, 1994, and awarding plaintiffsappellees $15,713,745.11 in damages, prejudgment interest, and attorney’s fees. For the reasons to be discussed, we vacate the judgment and damages award and remand the case to the district court for further proceedings consistent with this opinion.
BACKGROUND
The following facts are taken from the district court’s opinion, which was issued after an eleven-day bench trial. See Henry v. Champlain Enters., Inc. (“Henry II"), 334 F.Supp.2d 252 (N.D.N.Y.2004). CommutAir is a corporation based in Plattsburgh, New York that provides regional commuter airline services to USAir passengers pursuant to a code-sharing agreement with the larger airline. It was founded in 1989 by John Arthur Sullivan, Jr., Antony von Elbe, and Ernest James Drollette (collectively, “the sellers”), each of whom owned a one-third share of the *614company. In 1993, the investment banks Prudential Securities and Alex Brown & Sons approached CommutAir, which was thriving financially, to discuss the possibility of a strategic alliance with another airline or an initial public offering (“IPO”). Reports generated at this time indicated that CommutAir had a total equity value ranging from $140 million to $225 million; these figures were not derived from a formal valuation of the company, but from an informal ranking of CommutAir in the industry. The sellers decided not to pursue an alliance or an IPO, but in late 1993 they began exploring the possibility of establishing an ESOP — an employee benefit plan designed to encourage employee ownership through investment in securities issued by a sponsoring company. To this end, Sullivan contacted Jack Curtis, an ESOP expert and attorney with the law firm Keck, Mahin & Cate (“KMC”). Curtis informed Sullivan about the general requirements of an ESOP transaction, including the need to hire legal counsel, a financial appraiser, and a trustee to represent the ESOP. ■
I. Assembling a Team
In December 1993, the sellers decided to assemble a team to develop an ESOP for CommutAir. Sullivan contacted U.S. Trust, a company with substantial experience as an ESOP trustee, and spoke with Norman Goldberg. At this time, it was the sellers’ belief, based on the investment bankers’ estimates, that CommutAir had a total equity value of $200 million. The owners of CommutAir wanted to sell the proposed ESOP approximately 30% of their company for $60 million. Goldberg informed Sullivan that U.S. Trust would require an independent appraisal of CommutAir if it were hired to represent the ESOP and that this process would entail significant investigation of the company. Goldberg gave Sullivan the name of several companies, including Houlihan, Lokey, Howard & Zukin (“HLHZ”), that provided financial appraisal services.
On January 14, 1994, the sellers convened a meeting with the team slated to represent the proposed ESOP; this meeting was attended by representatives of CommutAir, U.S. Trust, KMC, and HLHZ, as well as investment bankers. At this meeting, a banker from Alex Brown & Sons gave a presentation regarding CommutAir and outlined the general terms of the proposed transaction. On January 17, 1994, the sellers circulated a draft offer stating that they would sell 30% of the company to the ESOP in the form of convertible preferred stock for $60 million. The offer anticipated that the purchase would be financed by a cash loan from CommutAir to the ESOP that would be paid to the sellers and by promissory notes issued to the sellers for the balance of the purchase price. On January 19th, the sellers submitted their draft offer to Goldberg and, on January 20th, a special fiduciary committee officially engaged U.S. Trust as trustee for the proposed ESOP.
U.S. Trust then retained KMC as legal counsel for the transaction. KMC’s primary responsibilities were to conduct a due diligence review of CommutAir and to assist in negotiating revisions in the terms of the securities to be purchased by the ESOP. U.S. Trust also retained HLHZ as an independent financial advisor. HLHZ’s primary responsibility was to evaluate the financial terms of the transaction, including the value of the CommutAir securities sold to the ESOP, and to render a “Fairness Opinion” that provided a detailed analysis of whether the transaction was fair to the ESOP.
II. The Due Diligence Process
The ESOP team began its due diligence review of CommutAir in early January *6151994. In anticipation of the meeting on January 14th, CommutAir Vice President Andrew Price submitted management projections of the company’s financial performance to HLHZ. Price’s projections, which were prepared in November 1993, used CommutAir’s financial data from 1992 through October 1993 and forecast the company’s performance through 1995. HLHZ asked Price to submit a revised forecast that projected performance through 1998. For the revised projections, Price relied primarily on data from the final two months of 1993.
On February 2, 1994, members of the ESOP team, including Andrew Stull (a representative of HLHZ) and Michael Shea (a financial analyst from U.S. Trust), visited CommutAir headquarters to evaluate Price’s projections and assess the airline’s financial status. In preparation for this meeting, both HLHZ and KMC forwarded due diligence requests to CommutAir, and Stull and Shea began gathering company documents and receiving background information on CommutAir.1 At the meeting, representatives from U.S. Trust and HLHZ toured CommutAir’s facilities and met with the company’s senior management. Sullivan described the meetings as “rigorous” and claimed that he and other members of CommutAir’s senior management were “grilled” for hours.
Some time prior to February 28th, HLHZ provided Goldberg with a preliminary valuation report concluding that CommutAir had a total equity value of $180 million. Goldberg made handwritten notes on this document identifying areas of concern in preparation for a meeting with HLHZ to discuss the basis of its valuation. In reference to the management projections submitted by Andrew Price, Goldberg wrote, “1994 — projections look robust.” He also noted: “Equity risk high — judgment about forecasts uncertain.” Next to the schedule containing HLHZ’s comparative publicly-traded company analysis, Goldberg noted that HLHZ’s selection of above-the-median multiples would have to be justified, as would HLHZ’s use of a 9.5% long-term growth rate. Goldberg asserts that these notes represented only some of his concerns; others he communicated to HLHZ orally. U.S. Trust in-house financial analyst Michael Shea testified that he too expressed concerns during multiple phone calls with HLHZ but, like Goldberg, kept no written records indicating the content of these calls.
On February 28th, Goldberg and Shea met with Stull to discuss HLHZ’s preliminary valuation. Although Goldberg and Shea took no notes, Stull did. Stull’s notes concern, inter alia, the justification for the projected growth rate, the use of above-the-median multiples for comparable companies, the fact that the dividends were set to terminate upon payment of the debt, the threat of competition from other airlines, and the use of a weighted average cost of capital that exceeded the industry average. Stull’s notes also contain questions, descriptions of topics that would require additional research, and directives to himself to “take out” certain numbers, “explain” various calculations, and “call [somebody] about regional airlines growth.”
*616Following the February 28th meeting, Goldberg asked KMC to draft a security purchase agreement that specified the terms of the convertible preferred stock so that the parties could begin negotiations. In early March, Goldberg received a draft of HLHZ’s opinion on the fairness of the proposed transaction. This draft opinion addressed the concerns expressed in Goldberg’s notes on the preliminary valuation statement and in Stull’s notes from the February 28th meeting: it further reduced the total equity value of CommutAir from $180 to $174 million; removed the highest selected comparable company multiples; and changed the long-term growth rate from 9.5% to 9%.
III. The Terms of the ESOP Transaction
On March 8, 1994, U.S. Trust and KMC met with senior CommutAir management to negotiate improvements to the convertible preferred stock offered by the sellers. Goldberg testified that the following issues were discussed at this meeting: (1) a limitation on management compensation; (2) the insertion of an anti-dilution provision; (3) a liquidation preference and redemption rights for the preferred stock; (4) full voting rights for the preferred stock; (5) the dividend rate for the preferred stock; (6) the participating feature of the dividend rights; and (7) the perpetual nature of the dividend. There is no written record of this meeting, but the negotiations resulted in improvements favorable to the ESOP, including an increase in the stock’s cumulative dividend rate from 7% to 9%. These improvements pertained to the value of the security and not its purchase price. At trial, Goldberg explained: “It [was] what do you get for sixty million dollars or what do you give up for sixty million dollars, that was the negotiation.”
On March 10th, the day U.S. Trust’s special fiduciary committee was to determine whether to approve the proposed transaction, Goldberg received two faxes. The first was a due diligence report from KMC. The second was a memo from HLHZ stating that “[t]he heavy fare reductions instituted by USAir in February 1994 as a preemptive move against low-cost entrants, Southwest and CALite, are contributing to the company’s widening losses.” The memo asserted that cost-cutting might be necessary at USAir, but that, “as it is currently structured, USAir Express is satisfied from an economic standpoint with the arrangement it has with its commuter franchisees.” The memo also contained an amended page of the HLHZ opinion that revised the terms of the convertible preferred stock by altering the terms of the performance dividend and changing the cumulative dividend rate from 7% to 9%. Shortly after receiving these faxes, the special fiduciary committee unanimously approved the proposed transaction.
On March 15, 1994, HLHZ presented U.S. Trust with a 104-page final valuation report on the CommutAir stock to be purchased by the ESOP.2 HLHZ’s report discusses the commuter airline industry and CommutAir’s history and the nature of its business (including its competitors, facilities, management, and code-sharing arrangement with USAir), and provides a financial review of the company’s balance sheet and projections. The report describes HLHZ’s valuation methodology, its valuation analysis (including a discussion of CommutAir’s performance and risk factors in relation to comparable companies), *617the attributes of the securities offered to the ESOP and how they compare to similar publicly-traded securities, and the financing terms of the transaction. It concludes that the consideration the ESOP received was fair, that the price it paid for CommutAir stock was no greater than fair market value, and that the overall terms of the financing of the purchase were reasonable.
Upon receipt of this report, CommutAir, the sellers, and U.S. Trust signed a stock purchase agreement. Per the agreement, the sellers agreed to sell 540,000 shares of convertible preferred CommutAir stock to the ESOP at a price of $111.11 per share for a total of $60 million. The sale was financed through a $9 million loan from CommutAir and three promissory notes totaling $51 million issued by the ESOP to the individual sellers. The convertible preferred stock had a fixed annual 9% dividend and a non-cumulative performance dividend. Dividend payments to the ESOP were to be used to pay down its indebtedness to the sellers and shares were to be allocated to individual employees’ accounts as the debt was retired.
In 1998, the IRS issued CommutAir notices of deficiency asserting that the ESOP had overpaid for company stock in the 1994 transaction. CommutAir and the sellers challenged the notices in the United States Tax Court, and in September 1999, the parties reached a settlement that substantially reduced the alleged tax deficiencies. The parties assumed, solely for the purpose of settling the tax dispute, that the 1994 transaction shares were worth $51 million. Neither the IRS nor the sellers sought an independent valuation of the shares. Thus, the IRS settlement did not “represent a finding that the fair market value was $51 million at the time of the purchase,” but only that “the IRS believes it to be” — a belief “in no way binding” in the subsequent ERISA action. See Henry v. Champlain Enters., Inc. (“Henry I”), 288 F.Supp.2d 202, 228 n. 15 (N.D.N.Y.2003).
Plaintiffs-appellees filed their initial complaint on November 1, 2001, and an amended complaint on February 21, 2003. The only claims that survived summary judgment were brought under §§ 404 and 406 of ERISA. These claims alleged that U.S. Trust’s investigation into the terms of the CommutAir ESOP transaction failed to qualify for ERISA’s good-faith exception to the prohibition against transactions by ESOPs with interested parties, thereby rendering the transaction prohibited. On September 3, 2004, after an eleven-day bench trial, the district court issued its findings of fact and conclusions of law, finding that U.S. Trust’s investigation into the terms of the 1994 transaction failed to satisfy the good-faith requirement for the establishment of ESOPs and that the transaction was thus prohibited under ERISA § 406. U.S. Trust appeals from the district court’s judgment and award of $15,713,745.11 in damages, prejudgment interest, and attorney’s fees to the plaintiffs-appellees.
DISCUSSION
I. Standard of Review
A district court’s findings of fact following a bench trial will be set aside on appeal only if those findings are clearly erroneous. Fed.R.Civ.P. 52(a); Phansalkar v. Andersen Weinroth & Co., L.P., 344 F.3d 184, 199 (2d Cir.2003). “Under this standard, factual findings by the district court will not be upset unless we are left with the definite and firm conviction that a mistake has been committed.” F.D.I.C. v. Providence Coll., 115 F.3d 136, 140 (2d Cir.1997) (citation and internal quotation marks omitted). The district court’s appli*618cation of law to those facts, however, is subject to de novo review. Id.
II. The Obligations of an ERISA Fiduciary
The central question presented in this case is whether the district court erred when it determined that U.S. Trust failed to satisfy its fiduciary duty. ERISA provides a comprehensive scheme to govern the conduct of ERISA fiduciaries. The general obligations of an ERISA fiduciary are set forth in § 404, which requires that a fiduciary investigate proposed transactions with “the care, skill, prudence, and diligence under the circumstances then prevailing that a prudent man acting in a like capacity and familiar with such matters would use in the conduct of an enterprise of a like character and with like aims.” 29 U.S.C. § 1104(a)(1)(B). In determining whether a fiduciary has satisfied this requirement, “[t]he court’s task is to inquire whether the individual trustees, at the time they engaged in the challenged transactions, employed the appropriate methods to investigate the merits of the investment and to structure the investment.” Katsaros v. Cody, 744 F.2d 270, 279 (2d Cir.1984) (citation and internal quotation marks omitted).
Section 406 of ERISA supplements the general fiduciary obligations set forth in § 404 by prohibiting certain categories of transactions believed to pose a high risk of fiduciary self-dealing. To this end, § 406 prohibits transactions involving the “sale or exchange ... of any property between the plan and a party in interest,” including the “acquisition, on behalf of the plan, of any employer security.” 29 U.S.C. § 1106(a)(1)(A), (E); see also Commissioner of Internal Revenue v. Keystone Consol. Indus., Inc., 508 U.S. 152, 160, 113 S.Ct. 2006, 124 L.Ed.2d 71 (1993) (stating that § 406(a) was enacted “to bar categorically a transaction that was likely to injure the pension plan”). It is undisputed that each of the sellers in this case was a “party in interest,” see 29 U.S.C. § 1002(14)(H), (C) (defining a “party in interest” as “an employee, officer, director, ... or a 10 percent or more shareholder directly or indirectly” of “an employer any of whose employees are covered by [the] plan”), and that the transaction at issue involved the sale of CommutAir stock to the company’s ESOP. Thus, the parties agree that the March 15, 1994 sale of convertible preferred stock was a prohibited transaction under § 406.
“Doubtlessly recognizing that [the] absolute prohibitions [in Section 406] would significantly hamper the implementation of ESOPs, particularly by small companies, Congress enacted in Section 408 a conditional exemption from the prohibited transaction rules for acquisition of employer securities by ESOPs and certain other plans.” Donovan v. Cunningham, 716 F.2d 1455, 1465 (5th Cir.1983). To encourage employees’ ownership of their employer company, § 408(e) permits the sale of employer stock by a party in interest to an ESOP if the purchase is made for “adequate consideration.” 29 U.S.C. § 1108(e). In transactions involving securities with no known market value, as is the case here, ERISA defines “adequate consideration” as “the fair market value of the asset as determined in good faith by the trustee or named fiduciary pursuant to the terms of the plan and in accordance with regulations promulgated by the Secretary [of Labor].” Id. at § 1002(18)(B).
In 1988, the Department of Labor (“DOL”) proposed a regulation that elaborated on the definition of “adequate consideration.” The proposed regulation states:
First, the value assigned to an asset must reflect its fair market value.... Second, the value assigned to an asset *619must be the product of a determination made by the fiduciary in good faith.... The Department will consider that a fiduciary has determined adequate consideration in accordance with section 3(18)(B) of the Act ... only if both of these requirements are satisfied.
See Proposed Regulation Relating to the Definition of Adequate Consideration, 53 Fed.Reg. 17,632, 17,633 (May 17, 1988) (to be codified at 29 C.F.R. § 2510-3(18)(b)) (“Proposed Regulation”). Although proposed regulations have no legal effect, Sweet v. Sheahan, 235 F.3d 80, 87 (2d Cir.2000), numerous circuit courts have adopted the DOL’s proposed definition of adequate consideration. See, e.g., Keach v. U.S. Trust Co., 419 F.3d 626, 636 (7th Cir.2005) (“In order to rely on the adequate consideration exemption, a trustee or fiduciary has the burden to establish that the ESOP paid no more than fair market value for the asset, and that the fair market value was determined in good faith by the fiduciary.”); Chao v. Hall Holding Co., Inc., 285 F.3d 415, 436 (6th Cir.2002) (“[T]he definition of ‘adequate consideration’ has two distinct parts. First, there is the ‘fair market value’ part, then there is the ‘as determined in good faith by the trustee’ part.”); Howard v. Shay, 100 F.3d 1484, 1488 (9th Cir.1996) (“To enforce [ERISA fiduciary rules], the court focuses not only on the merits of the transaction, but also on the thoroughness of the investigation into the merits of the transaction.”).
Although fair market value and good faith are often stated as distinct requirements, they are closely intertwined. In keeping with the established meaning of “fair market value” in the area of asset valuation, the DOL’s Proposed Regulation defines the term
as the price at which an asset would change hands between a willing buyer and a willing seller when the former is not under any compulsion to buy and the latter is not under any compulsion to sell, and both parties are able, as well as willing, to trade and are well-informed about the asset and the market for that asset.
Proposed Regulation, 53 Fed.Reg. at 17,-634. As this definition indicates, “fair market value” is an imprecise term. See Rhodes v. Amoco Oil Co., 143 F.3d 1369, 1372 (10th Cir.1998) (“There is no universally infallible index of fair market value. There may be a range of prices with reasonable claims to being fair market value.” (internal quotation marks, citation, and alteration omitted)); Alvary v. United States, 302 F.2d 790, 795 (2d Cir.1962) (noting the “inherent inexactness of the concept of fair market value”). Whether a fiduciary has made a proper determination of “fair market value” depends on whether the parties “are well-informed about the asset and the market for that asset.” Thus, in practice, the “fair market value” inquiry overlaps considerably with the “good faith” inquiry; both are “expressly focused upon the conduct of the fiduciaries.” Cunningham, 716 F.2d at 1467 (emphasis in original).
Under ERISA, the fiduciary bears the burden of proving by a preponderance of the evidence that the ESOP received “adequate consideration” for its purchase of company stock. 29 U.S.C. §§ 1002(18), 1108(e). The role of courts in reviewing the adequacy of consideration in an ERISA case is to determine whether the fiduciary can show that the price paid represented a good faith determination of the fair market value of the asset, “not to redetermine the appropriate amount for itself de novo.” Chao, 285 F.3d at 437 (internal citation and quotation marks omitted); see also Eyler v. Comm’r, 88 F.3d 445, 455 (7th Cir.1996) (“ESOP fidu*620ciaries will carry the burden of proving that adequate consideration was paid by showing that they arrived at their determination of fair market value by way of a prudent investigation in the circumstances then prevailing. Thus, the adequate consideration test focuses on the conduct of the fiduciaries in determining the price, not the price itself.” (internal citation and quotation marks omitted)).
III. U.S. Trust’s Conduct as an ERISA Fiduciary
The district court found that U.S. Trust failed to demonstrate that it had satisfied its obligations as an ERISA fiduciary because it produced insufficient documentation of the investigation it undertook in the months leading up to the ESOP transaction on March 15, 1994. See Henry II, 334 F.Supp.2d at 272-73. The court’s analysis focuses entirely on the February 28, 1994 meeting between U.S. Trust and HLHZ. Detailed notes taken at this meeting by Andrew Stull indicate that the parties discussed numerous aspects of HLHZ’s preliminary valuation, including, inter alia, the concerns Goldberg noted on his copy of the preliminary valuation. The district court did not challenge the authenticity of Stull’s notes. Rather, the court found that Stull’s notes did not prove that representatives of U.S. Trust had raised the issues contained therein. See id. at 272 (“[T]he notes themselves do not state that they were made in response to any question from Goldberg or Shea.”). The court further found that because Goldberg and Shea did not keep any notes, they had “no evidence, created by themselves, that they actually posed the question to HLHZ about these issues.” Id. In the final analysis, the court determined that Goldberg’s notes on the preliminary valuation and the notes Shea made on the revised projections and in connection with the due diligence meeting, in addition to “the word of U.S. Trust and its witnesses,” were insufficient to demonstrate a good-faith investigation and disqualified the CommutAir ESOP transaction from the ERISA § 408 exemption. Id. at 272-73.
The district court erred as a matter of law when it found that U.S. Trust failed to qualify for the § 408 exception as a result of its failure to produce notes showing that it “properly evaluated HLHZ’s report, asked the difficult questions, and assessed the responses to those questions.” Id. at 272. The focal point of our inquiry under ERISA is not whether a fiduciary took adequate notes of its investigation, but whether it acted with the prudence required of a fiduciary under the prevailing circumstances at the time of the transaction. See 29 U.S.C. § 1104(a)(1)(B); Katsaros, 744 F.2d at 279. It is impossible to determine, solely on the basis of Goldberg and Shea’s lack of notes, whether U.S. Trust “at the time [it] engaged in the challenged transactions, employed the appropriate methods to investigate the merits of the investment and to structure the investment.” Katsaros, 744 F.2d at 279 (citation and internal quotation marks omitted). First, the fact that the issues Goldberg identified as areas of concern on his copy of the preliminary valuation are echoed in Stull’s notes from the February 28th meeting gives rise to a strong inference that U.S. Trust raised these issues at the meeting. Even if Stull generated these questions himself, however, that fact alone would not suggest that U.S. Trust was an inadequate fiduciary. Regardless of who raised the questions at the February 28th meeting, the district court recognized that U.S. Trust: (1) reviewed HLHZ’s preliminary valuation report; (2) met with HLHZ to discuss its concerns; and (3) saw the report changed in ways that reflected those concerns. The court recognized also that U.S. Trust witnessed significant improvements in the ESOP’s *621position over the course of its trusteeship and that these improvements were incorporated into the final valuation of CommutAir. Under these circumstances, U.S. Trust’s failure to produce notes documenting the steps it took during its investigation of CommutAir does not in and of itself permit a determination that U.S. Trust failed to engage in a good-faith determination of the fair market value of the security at issue.
To qualify as a prudent fiduciary under § 408, U.S. Trust must show that the CommutAir ESOP paid no more than “the fair market value of the asset as determined in good faith by the trustee or named fiduciary....” 29 U.S.C. § 1002(18)(B). As we noted above, there is considerable overlap between the “fair market value” inquiry and the “good faith” inquiry: Whether a fiduciary has made a proper determination of the “fair market value” of an asset purchased by an ESOP depends on whether the parties are “well-informed about the asset and the market for that asset.” Proposed Regulation, 53 Fed.Reg. at 17,634. The district court in this case did not identify any area where U.S. Trust was ill-informed about the CommutAir security or the market for that security. Nor did it identify any flaws that U.S. Trust should have detected in the assumptions or methodologies HLHZ used to determine the fair market value of the CommutAir stock. In the absence of such findings, we cannot conclude that U.S. Trust failed to satisfy its burden of demonstrating a good faith investigation. In fact, the findings actually made by the district court tend to show that U.S. Trust conducted a good faith investigation.
The court’s failure to pinpoint any errors in HLHZ’s valuation of CommutAir makes it impossible for us to determine whether U.S. Trust should have detected those errors in the course of a “good faith” investigation. Thus, until the district court identifies which, if any, errors in HLHZ’s valuation U.S. Trust failed to ascertain that a prudent fiduciary would have ascertained in the months leading up to the CommutAir ESOP transaction, it will not be possible to determine whether U.S. Trust qualifies for the § 408 exception. We remand this case to the district court to identify the specific errors, if any, in HLHZ’s valuation of CommutAir and to determine whether a prudent fiduciary would have detected these errors under the circumstances prevailing at the time of the ESOP transaction.3
IY. Damages and Prejudgment Interest
The district court found, “based on an evaluation of the credibility and back*622ground of all the witnesses; the difference between the opinions of the expert witnesses; and a review of all the testimony and the exhibits received into evidence ... that the total equity value of CommutAir as of March 15, 1994, was $145 million.” Henry II, 334 F.Supp.2d at 268. Based on this calculation, the court found that the value of the convertible preferred stock purchased by the ESOP was $52.25 million and not $60 million, the amount the ESOP had agreed to pay for the stock. The district court determined, based on this discrepancy, that the damages caused by the prohibited transaction amounted to $7.75 million. Id. at 274.
Courts are not required to provide lengthy analyses in support of their damages findings but they are required under Fed.R.Civ.P. 52(a) to “adequately explain the subsidiary facts and methodology underlying the ultimate finding.” Davis v. N.Y. City Hous. Auth., 166 F.3d 432, 436 (2d Cir.1999). The district court’s determination that CommutAir was valued at $145 million at the time of the ESOP transaction was not accompanied by any explanation of the facts or methodology used to reach this figure. In the absence of a specific explanation for the district court’s findings, we are unable to determine whether this valuation, and the resulting damages award, are justified.4
As we noted above, without knowing which errors led to the purported overvaluation of CommutAir, we cannot determine whether U.S. Trust fulfilled its obligation to “‘question the methods and assumptions that do not make sense.’ ” Bussian v. RJR Nabisco, Inc., 223 F.3d 286, 301 (5th Cir.2000) (quoting Shay, 100 F.3d at 1490). Only after the court has identified incorrect assumptions or flaws in HLHZ’s valuation methodology will it become possible to determine whether U.S. Trust can “establish that the ESOP paid no more than fair market value for the asset, and that the fair market value was determined in good faith by the fiduciary.” Keach, 419 F.3d at 636. We vacate both the district court’s finding that U.S. Trust failed to qualify for the § 408 exception and its damages award so that the court may specify how HLHZ erred in its valuation of CommutAir and why a prudent fiduciary would have detected the errors. Assuming that the district court finds defects in the valuation report that a prudent fiduciary would have detected, it should then assess damages and set forth its specific findings.
The district court’s award of prejudgment interest to the ESOP is similarly devoid of explanation. See Henry v. Champlain Enters., Inc. (“Henry III”), 342 F.Supp.2d 122, 123 (N.D.N.Y.2004) (“Plaintiff is awarded prejudgment simple interest at the rate of 9% from March 15, 1994 to October 29, 2004, in the sum of $7,410,937.50.”). The issue of “whether or not to award prejudgment interest [in ERISA cases] is ordinarily left to the discretion of the district court.” Jones v. *623UNUM Life Ins. Co. of Am., 223 F.3d 130, 139 (2d Cir.2000). The court must, however, explain and articulate its reasons for any decision regarding prejudgment interest. Connors v. Conn. Gen. Life Ins. Co., 272 F.3d 127, 137 (2d Cir.2001); Jones, 223 F.3d at 140 (“The district court here ... made no findings as to why the [chosen interest] rate would adequately compensate [the plaintiff]. In the absence of such findings, meaningful review is forestalled .... ”). We vacate the award of prejudgment interest to provide the district court with an opportunity to articulate its reasons for awarding prejudgment interest in the amount and for the period it did in this case.
V. Dismissal of the Sellers
The district court dismissed most of the claims against the sellers at the summary judgment stage but ordered them to remain in the case in order to defend against the ESOP’s claims for disgorgement of profits under ERISA § 502(a)(3), 29 U.S.C. § 1132(a)(3). Following the bench trial, the district court dismissed the sellers from the case entirely. See Henry II, 334 F.Supp.2d at 255 n. 1. On appeal, U.S. Trust argues that the plaintiffs’ disgorgement claim against the sellers should be reinstated; the sellers argue that the plaintiffs’ disgorgement claim has been abandoned. The district court’s factual findings regarding the actions taken by the plaintiffs in regard to their disgorgement claim against the sellers are reviewed for clear error. Benjamin v. Fraser, 343 F.3d 35, 43 (2d Cir.2003). The question of whether the plaintiffs’ actions constituted an abandonment of their disgorgement claim against the sellers is a question of law, which is subject to de novo review. Contship Containerlines, Ltd. v. PPG Indus., Inc., 442 F.3d 74, 77 (2d Cir.2006).
The record shows that after the court dismissed the plaintiffs’ disgorgement claim against the sellers, the plaintiffs neither objected to nor appealed from that dismissal. Rather than appeal the dismissal, the plaintiffs opted to pursue other claims against the sellers (chiefly, the claim that the sellers violated § 5.7 of the Stock Purchase Agreement) in a separate action. See Henry v. von Elbe (“Henry IV”), No. 1:04-CV-0201(DNH), slip op. at 5-6 (N.D.N.Y. Aug. 25, 2005).
U.S. Trust seeks to overturn the plaintiffs’ decision not to pursue their disgorgement claim, arguing on appeal that this Court should reinstate the claim. Only the plaintiffs, however, may appeal the district court’s dismissal of their disgorgement claim because the plaintiffs are the ones who asserted this claim in the first instance. As the plaintiffs opted not to appeal, we have no cause to revive the claim at this stage of the litigation. See, e.g., Castellano v. City of New York, 142 F.3d 58, 72 (2d Cir.1998) (issues not raised on appeal are deemed abandoned). U.S. Trust cannot pursue a claim abandoned by the plaintiffs, and because it did not assert a cross-claim against the sellers below, it has no right to ask for the reinstatement of the disgorgement claim against the sellers. See, e.g., AA. ex rel. J.A. v. Philips, 386 F.3d 455, 459 n. 3 (2d Cir.2004) (declining to consider claims raised for the first time on appeal). Thus, we find that the plaintiffs’ disgorgement claim against the sellers has been abandoned and may not be revived by U.S. Trust.
VI. U.S. Trust’s Windfall Claim
U.S. Trust argues that the district court’s $15.5 million award for damages and prejudgment interest greatly exceeds the ESOP’s actual economic loss and thereby result in a windfall. U.S. Trust alleges that the district court made two *624errors that resulted in a windfall for the plaintiffs-appellees. First, it argues that the court erroneously assumed that the ESOP paid $60 million (the amount agreed upon in the stock purchase agreement) for shares worth $52.25 million (the value determined by the district court). The parties jointly stipulated that as of January 2001, the ESOP had paid only $54,519,801 million in principal for the shares it received in the transaction. U.S. Trust argues that the ESOP’s actual economic loss of principal is at most $2.27 million — the difference between the amount the ESOP actually paid and the amount the court determined the ESOP should have paid— and not the $7.75 million awarded by the court.5
Second, U.S. Trust argues that the ESOP may already have been fully compensated for its economic loss. Section 5.7 of the Stock Purchase Agreement between the sellers and the ESOP provides that if the IRS, the DOL, or a court makes a final determination that the ESOP paid more than the fair market value of the CommutAir shares, the sellers must pay the ESOP an amount equal to the difference between the purchase price and the fair market value of the shares, plus reasonable interest. Under the Agreement, the sellers are permitted to satisfy the difference in cash or in the form of shares valued in accordance with their actual fair market value as of March 15, 1994. In February 2004, after the district court found that the IRS had made a final determination that the ESOP had overpaid for its CommutAir stock, Henry I, 288 F.Supp.2d at 228, the sellers executed the § 5.7 remedy, agreeing to provide the ESOP all the shares it was entitled to receive on March 15, 1994, see Henry IV, No. 1:04-CV-0201(DNH), slip op. at 5 (N.D.N.Y. Aug. 25, 2005). U.S. Trust maintains that even if it is determined ultimately that the ESOP overpaid for the CommutAir stock it received on March 15, 1994, this stock payment restores the ESOP to the position it would have occupied absent the overpayment and thus constitutes a complete remedy. As such, U.S. Trust contends, it renders any further recovery by the ESOP a prohibited windfall.
The aim of ERISA is “to make the plaintiffs whole, but not to give them a windfall.” Jones, 223 F.3d at 139 (citation and internal quotation marks omitted). If the district court determines on remand that the plaintiffs-appellees are entitled to recover damages from U.S. Trust, the court should explain why those damages do not result in a windfall to the ESOP.
CONCLUSION
For the reasons stated above, we vacate the district court’s judgment and damages award and remand the case to the district court for further proceedings consistent with this opinion.
. In the course of conducting due diligence, HLHZ and KMC requested detailed financial and corporate data from CommutAir, including copies of CommutAir's audited financial statements and draft financials, income tax returns, budgetary reports, financial projections, aircraft leasing and maintenance agreements, information relating to its code-sharing arrangement with US Air, as well as information about CommutAir's officers and employees, its route system and chief competitors, and pending or threatened litigation involving the company.
. On this date, KMC sent U.S. Trust an opinion letter stating that the ESOP satisfied the requirements of ERISA and the IRS, and a companion, letter stating that U.S. Trust had complied with ERISA’s fiduciary rules and fulfilled its duties as a trustee to the ESOP.
. Although we remand this case to the district court to apply the proper legal standard, we write also to correct the district court's clearly erroneous finding that because "HLHZ was advised before it even reviewed the matter [ ] that the sellers wished to receive $60 million from the ESOP for a 30% interest in CommutAir ... HLHZ did not start with an open mind or an independent approach to valuation.” Henry II, 334 F.Supp.2d at 268. Although the sellers initially expressed a desire to sell 30% of their company for $60 million, and ultimately did just that, the transaction that occurred in March 1994 involved substantially different terms than the sellers had envisioned in the fall of 1993 because the 30% of CommutAir that the ESOP actually received comprised shares significantly more valuable than those originally contemplated by the sellers. Thus, although the final price approved by U.S. Trust was the same as the seller's initial price, the final offer was significantly more beneficial to the ESOP than the sellers' initial offer. ERISA § 408 requires only that the fiduciary insure, through prudent investigation, that the ESOP receive "adequate consideration.” U.S. Trust was entitled to accomplish this objective by, inter alia, negotiating either a lower share price or a higher share value. The fact that U.S. Trust chose the latter option does not compromise its efforts on behalf of the ESOP.
. Plaintiffs-appellees suggest that the court’s valuation is justified because it falls within the $106 million valuation they advanced and the $174 million valuation advanced by U.S. Trust, but the fact that the court's valuation falls between these figures is irrelevant. Rule 52(a) requires courts in all cases to explain the facts and methodology used to reach their conclusions. Plaintiffs-appellees further argue that because the court chose a figure that fell between those offered by the parties, its valuation fell within a "reasonable range." In actuality, the court found both the plaintiffs’ and the defendants' valuations incredible; its valuation of $145 million is no more justified simply because it (roughly) splits the difference between these figures. Finally, the fact that the plaintiffs-appellees can find ways of adjusting the relevant variables to reach a valuation of $145 million does not tell us which of these formulas, if any, the court employed.
. U.S. Trust argues that the prejudgment interest award should be adjusted for the same reason.