
    U.S. ENERGY CORP., a Wyoming corporation, and Crested Corp., a Colorado corporation, d/b/a USE/CC, a joint venture, Plaintiffs-Appellees-Cross-Appellants, v. NUKEM, INC., a New York corporation, and Cycle Resource Investment Corporation, a Delaware corporation, Defendants-Appellants-Cross-Appellees.
    Nos. 03-1444, 03-1451.
    United States Court of Appeals, Tenth Circuit.
    Feb. 24, 2005.
    
      Carolyn B. Lamm (Francis A. Vasquez, Jr., and Jonathan C. Hamilton, White & Case LLP, Washington, DC, and Frances A. Koncilja and David V. Millard, Koncilja & Associates, PC, Denver, CO, with her on the briefs), White & Case LLP, for Defendants-Appellants-Cross-Appellees.
    .William R. Fishman (Kenneth A.B. Roberts, Jr., Highlands Ranch, CO, with him on the briefs), Denver, CO, for Plaintiffs-Appellees-Cross-Appellants.
    Before BRISCOE and MURPHY, Circuit Judges, and STEWART, District Judge.
    
    
      
       The Honorable Ted Stewart, District Judge, District of Utah, sitting by designation.
    
   BRISCOE, Circuit Judge.

This case involves a partnership dispute. Plaintiffs U.S. Energy Corporation and Crested Corporation brought this action in 1991 alleging that defendant Nukem, Inc., misappropriated partnership assets for its own benefit and to the detriment of the partnership. In 1994, the parties stipulated to binding arbitration. An arbitration panel issued an award in favor of plaintiffs in April 1996, which the panel amended in July 1996. Since then, the parties have been litigating issues related to the confirmation and enforcement of the amended arbitration award. The district court previously entered judgment in favor of plaintiffs in the amount of $15,677,535 plus interest, which was affirmed by this court. See U.S. Energy Corp. v. Nukem, Inc., 1998 WL 738336 (10th Cir. Oct.22, 1998) (Nukem I). The district court subsequently entered another judgment in favor of plaintiffs in the amount of $20,044,183. In this appeal and cross-appeal, the parties argue the district court erred in denying their respective post-judgment motions to alter or amend the $20,044,183 judgment. We have jurisdiction pursuant to 28 U.S.C. § 1291 and vacate and remand to the district court for further remand to the arbitration panel for clarification of the arbitration award.

I.

This case comes before this court for the third time. In an order and judgment disposing of Nukem’s first appeal, this court set out the background facts, as follows:

Plaintiff USECC is a joint venture comprised of two uranium mining companies, Plaintiffs U.S. Energy Corp. and Crested Corp., which held several mining claims in Wyoming and a number of long-term contracts to supply uranium to utility companies. [Cycle Resource Investment Corp. (“CRIC”) ] is a wholly-owned subsidiary of [Defendant] Nuk-em[, Inc.,] formed specifically to enter into the agreements at issue in this case. On December 21, 1988, USECC and CRIC entered into a transaction for the purpose of mining uranium in Wyoming and selling it to domestic utilities. In an asset purchase agreement, USECC agreed to sell a 50% interest in its mining claims and long-term supply contracts to CRIC. In a separate partnership agreement, USECC and CRIC each agreed to make capital contributions of their one-half interest in the mining claims and supply contracts to the new partnership, Sheep Mountain Partners (hereafter “SMP”). The partnership agreement contained an arbitration clause and required SMP to enter into independent contract agreements with USECC to operate the mines and with CRIC to market the uranium produced. CRIC subsequently assigned its rights and duties under the marketing agreement to Nukem.
After USECC and CRIC entered into their partnership agreement, Nukem negotiated uranium importation contracts with the Commonwealth of Independent States (hereafter “CIS”), whose members include Uzbekistan, Kazakhstan and Kirgizstan. In order to prevent members of the CIS from dumping uranium in the United States at prices below fair market value, the United States had imposed a tariff in excess of 100% of the value of the imported CIS uranium. The United States, however, agreed to permit some shipment of CIS uranium into the United States without tariff if the purchaser had a long-term utility supply contract executed prior to March 5, 1992. If a supply contract met this requirement, the contract was considered “grandfathered.” Acting in its own right, Nukem subsequently submitted utility supply contracts to the U.S. Department of Commerce for-grandfathering. These contracts included five supply contracts owned by SMP. The grandfathering of the contracts allowed Nukem to purchase CIS uranium in amounts necessary to meet the delivery requirements of those utility supply contracts. Nukem did not, however, supply the CIS uranium to SMP to satisfy the five utility supply contracts’ requirements, but instead sold the uranium to other buyers. As a result, SMP purchased uranium from other suppliers at higher prices to meet its contractual obligations.
From these and .other transactions, numerous disputes arose among the parties. In 1991, Plaintiffs filed this action in the district court. In 1994, the parties stipulated to binding arbitration. The parties raised more than 33 claims before the arbitration panel. One of those claims involved the CIS uranium contracts. Plaintiffs argued that by entering into CIS contracts on its own account, Nukem violated the marketing and partnership agreements. Plaintiffs also argued that Nukem improperly used the five SMP utility supply contracts to avoid the tariff on the CIS uranium.
The arbitration panel conducted a 73-day hearing and issued a written “Arbitration Order and Award” on April 18, 1996. The arbitration panel rejected the claim that Nukem violated the marketing and partnership agreements by entering into the CIS contracts. In regard to the five SMP utility supply contracts, the arbitration panel concluded that “Nukem without authority and without SMP’s permission or consent used the SMP uranium supply contracts, which were partnership assets, to obtain purchase rights for CIS” uranium. The panel further found that “the uranium should have been made available to SMP to meet deliveries required by SMP’s grandfathered supply contracts.” As a result of Nukem’s conduct, the panel impressed a constructive trust in favor of SMP over “those purchase rights, the uranium acquired pursuant to those rights and the profits therefrom....” The panel also determined that as a result, of Nukem’s conduct regarding the five supply contracts, SMP suffered damages in the amount of $31,355,070 and awarded Plaintiffs, as holders of a 50% interest in SMP, half that amount or $15,677,535 plus interest.
On July 3, 1996, in response to motions by Defendants, the panel amended the arbitration award to clarify the order and correct errors. On November 4, 1996, the district court entered an order and judgment confirming the arbitration award. The district court subsequently amended the order and judgment on March 11, 1997 and June 27, 1997. .

U.S. Energy Corp. v. Nukem, Inc., Nos. 96-1532, 97-1332, 1998 WL 738336 (10th Cir. Oct.22, 1998) (unpublished) (“Nukem I”).

Paragraph 163 of the Arbitration Award has been at the center of much of the controversy in this case. In paragraph 163, the arbitration panel stated:

It is clear that Nukem without authority and without SMP’s permission or consent used the SMP uranium supply contracts to obtain purchase rights for CIS [uranium]. Since the rights to purchase the CIS uranium were obtained through the use of SMP contracts (partnership assets), those purchase rights, the uranium, acquired pursuant to those rights and the profits therefrom are impressed until a constructive trust in favor of SMP, and we conclude that SMP is entitled to damages in the amount of $31,355,070 [half payable to Plaintiffs] to compensate it for its past and future lost profits. The uranium should have been made available to SMP to meet deliveries required by SMP’s grandfathered supply contracts. We enter awards for those past and future profits denied to the partnership, together with statutory interest at 8% per annum. The relevant amounts are as follows:
Contract Gross Profits Present Value to June 30, 1995
Duke Power $4,860,000 $ 5,543,433
IPC 12,048,414 11,843,232
TUCO 3,079,000 2,779,727
BECO 2,383,950 2,298,050
PSE & G 9,487,050 8,899,628
Total: $31,858,414 $31,355,070

Aplt.App. Vol. I, Doc. 7 at 209-10 (emphasis added).

As suggested by paragraph 163, there are two sets of contracts at issue: (1) SMP’s five “grandfathered” contracts to supply uranium to certain domestic utility companies, and (2) four contracts entered into, by Nukem, with CIS countries to buy uranium. It appears from the district court’s judgment that the district court concluded the latter set of contracts were somehow obtained by Nukem through the misappropriation of the former set of contracts.

The connection between the SMP supply contracts and the CIS uranium purchase contracts is somewhat tenuous. On appeal, Nukem denies that there is any evidence establishing a connection between the SMP supply contracts and the CIS purchase contracts. In fact, however, the arbitration panel heard the testimony of Mr. Ronald Witzel, an expert introduced by U.S. Energy. Among other things, Witzel testified that Nukem could not have obtained the CIS contracts without the existing SMP supply contracts. Aplee. Supp.App. Vol. I, Doc. 6 at 114. Plaintiffs contend that the arbitration panel adopted Witzel’s testimony on this issue, Aplee. Br. at 13, but they have not directed this court’s attention to any indication in the record that the arbitration panel officially “adopted” Witzel’s testimony regarding the connection between the SMP supply contracts and the CIS purchase contracts. The arbitration panel did, however, state that the “numbers shown in Paragraph 163” were based upon Plaintiffs’ “expert witness.” Moreover, it is undisputed that pursuant to the CIS purchase contracts, Nukem purchased much more uranium (63,312,164 pounds) than was required to fill SMP’s grandfathered supply contracts (3,076,943 pounds). App. Vol. II, Doc. 16 at 280, 285.

In Nukem I, Nukem argued that the district court erred by entering a monetary judgment in favor of the plaintiffs and imposing a constructive trust of undetermined value separate and apart from the damages award. Nukem I, 1998 WL 738336 at *3. Nukem asserted that by imposing a separate constructive trust the district court “exceeded the relief granted by the arbitration panel.” Id. This court rejected that argument, stating:

Amended Paragraph 163 clearly retains both a constructive trust and a damage award. Assuming that the arbitration panel erred by impressing , a constructive trust and awarding damages, we cannot reverse an arbitration panel’s erroneous interpretations or applications of the law. We may only reverse the panel’s award if it evidences a manifest disregard for the law. The arbitration panel’s decision to impress a constructive trust and award damages is not willful inattentiveness to the governing law.

Id. at *4 (internal citations and quotations omitted; emphasis in original). Alternatively, in Nukem I, Nukem argued that the district court’s judgment imposing the constructive trust-was vague and ambiguous. This court rejected that argument, ‘stating:

The second amended judgment provides “pursuant to paragraph 163 of Arbitrators’ Order No. 1, as modified by Arbitrators’ Order No. 2, the rights to purchase those rights, and the profits therefrom are IMPRESSED WITH A CONSTRUCTIVE TRUST in favor of SMP.... ” Defendants argue that this language creates a trust of “indeterminable corpus” and must be set aside. We disagree. Amended paragraph 163 explains that the purchase rights, uranium, and profits -acquired through Nuk-em’s use of SMP’s five utility. supply contracts were impressed with a constructive trust. Paragraphs 160 and 163 of the arbitration award list the five grandfathered supply contracts: Duke, IPC, TUCO, BECO, and PSF & G. When read in conjunction with the arbitration order and award, the district court’s judgment is clear and unambiguous. The judgment impresses the trust “pursuant to paragraph 163” of the arbitration order, incorporating language identifying the proceeds from the five grandfathered -SMP supply contracts as the corpus of the constructive trust.

Id.

In its second appeal, Nukem challenged the district court’s order denying its motion, made pursuant to Fed.R.Civ.P. 60(b)(5), requesting a finding that it had fully satisfied the judgment against it. U.S. Energy Corp. v. Nukem, Inc., No. 99-1341, 2000 WL 1528682 (10th Cir. Oct.16, 2000) (“Nukem II ”). Nukem had submitted documents showing that the $15,677,535 monetary judgment had been satisfied, but the district court held that the payment would not satisfy the constructive trust obligation. Id. at *2. The district court concluded that Nukem was required, but failed, to provide an accounting of the value of the CIS purchase rights, the uranium purchased, and the profits therefrom. Id. On appeal, this court affirmed the district court, concluding that the defendants had failed to provide an accounting and, therefore, failed to demonstrate that the judgment was satisfied. Id.

Following Nukem II, plaintiffs filed a motion in the district court to obtain a valuation of the constructive trust and to enforce the judgment. The district court entered orders referring the matter to a special master, attorney Raymond Freid-lob, for an accounting. Aplt.App., Docs. 11, 12 at 235-44. The district court stated:

The Special Master shall conduct an accounting pertaining to:
a. all of Defendant’s transactions in CIS uranium based on CIS purchase rights obtained through the use of SMP uranium supply contracts;
b. the profits realized therefrom;
c. the location and poundage of all such CIS'uranium; and
d. the status of such CIS purchase rights
with reference to and consistent with: 1) Arbitration Order No. 1; 2) Arbitration Order No. 2; 3) this Court’s June 27, 1997 Second Amended Judgment; 3) Nukem I; and 4) Nukem II.

Aplt.App., Doc. 12 at 243.

On May 1, 2003, with the assistance of a Denver accounting firm and after more than 3,000 hours of work, the special master issued his report. The special master framed the issue before him as follows:

What is at issue here is what is required to do equity as a result of the Defendants’ misappropriation of partnership assets, opportunities and unjust enrichment as a result of Defendants’ gaming the ability to obtain, nurture and develop an important business relationship with the CIS for their own benefit.... The method used to calculate that value would be to impress in the constructive trust a payment to the SMP Partnership which includes such amounts to adequately compensate the SMP Partnership for the loss of the fruits of the business relationship which was developed, utilized, maintained and expanded by Defendants.

Aplt.App. Vol. II, Doc. 16 at 275-76. The special master came to several conclusions about the arbitration award. First, he concluded that the “purchase rights” were worth $12,932,727, including interest at 8% (the rate used by the arbitration panel in computing the damage award), or $6,466,364 payable to the plaintiffs. The master’s valuation of “purchase rights” was based upon an assumption that their value was equal to 1.5% of Nukem’s gross CIS uranium sales through December 31, 2001, the total of which was $616,977,221. ApltApp. Vol. II, Doc. 16 at 275-76. The special master settled on the 1.5% figure based on a 1.5% fee that Nukem had paid to a third party that facilitated Nukem’s relationship with another uranium source. Id. at 275. The special master concluded that, by December 31, 2001, the purchase rights were “extinguished” because all deliveries due under the five grandfathered SMP contracts had been made by that time. Id. at 286.

The special master concluded that “it would be equitable to find that the SMP business relationship contributed to the Defendants’ success in developing the CIS business relationship which should have inured to the benefit of the SMP Partnership.” Id. at 274. . Relatedly, the special master concluded that Nukem’s net profit from the sale of CIS uranium pursuant to four “CIS Purchase Contracts” totaled $27,155,640. Id. at 282-83. The net profit figure was arrived at by subtracting $39,243,560 in costs from Nukem’s gross profit of $66,399,200 on the sale of CIS uranium. Id. at 283. In one chart included within his report, the special master subtracted the arbitration panel’s previous damage award of $31,355,070 from Nuk-em’s net profit $27,155,640, arriving at negative $4,199,430. ApltApp. Vol. II, Doc. 16 at 283. The special master failed to explain the significance of that figure.

Further, the special master concluded that had SMP filled its grandfathered supply contracts with CIS uranium, its “hypothetical profit” would have been $21,970,588. Id. at 364. In connection with that figure, the special master stated: “It should be noted that the Arbitration Panel awarded SMP $31,355,070 to compensate it for its past and future profits related to the SMP grandfathered supply contracts.” Id. The special master did not provide any clear-cut valuation of the constructive trust as a whole, and he did not expressly state what he considered to be the corpus of the constructive trust.

The district court adopted some of the special master’s conclusions. Specifically, the district court adopted: (1) the special master’s valuation of purchase rights; (2) the special master’s calculation of Nukem’s net profits on the CIS contracts; and (3) the special master’s conclusion that the purchase rights were extinguished on December 31, 2001. Unlike the special master, the district court declined to engage in any review of the previous damages award. As a result, the district court rejected Nukem’s argument that the constructive trust award should be reduced to reflect such a recalculation. According to the district court, the arbitration panel intended for the damages award and the constructive trust to be separaté and distinct remedies. ApltApp. Vol. II, Doc. U at 11. The district court stated that it was “way too late” for a recalculation of the damages award “through the guise of the constructive trust accounting.” Id. at 16-17. On August 1, 2003, the district court entered judgment in favor of Plaintiffs and against Nukem (1) in the amount of .$6,466,363.50, which represented 50% of the value of the CIS uranium purchase rights, and (2) in the amount of $13,577,820, which represented 50% of Nukem’s net profits on the sale of CIS uranium.

On August 15, 2003, Nukem filed a motion asking the district court to remand the matter to the arbitration panel, or, in the alternative, to alter or amend the judgment. ApltApp. Vol. V, Doc. 27 at 1377. Nukem argued that the arbitration panel’s constructive trust award was ambiguous and that the matter should be remanded to the panel for clarification, Alternatively, Nukem argued that the judgment entered by the district court was inconsistent with the findings of the special master. On the same day, plaintiffs also filed a motion to alter or amend the judgment. ApltApp. Vol. V, Doe. 26 at 1308. Plaintiffs argued that the district court had erred in concluding that the “purchase rights” were extinguished on December 31, 2001. Further, plaintiffs argued that they were entitled to statutory prejudgment interest on the award of $13,577,820, which plaintiffs calculated to be $7,901,672. The district court denied these post-judgment motions. ApltApp. Vol. VI, Docs. 32 and 33.

II.

Federal Arbitration Act (FAA)

The FAA provides that a written provision in “a contract evidencing a transaction involving commerce to settle by arbitration a' controversy thereafter arising óut of such contract ... shall be valid, irrevocable, and enforceable, save upon such grounds as exist at law or in equity for the revocation of any contract.” 9 U.S.C. § 2. Although the FAA does not creáte independent federal jurisdiction, the Supreme Court has held that the FAA creates a body of substantive federal law governing arbitration agreements within its coverage. See Bowen v. Amoco Pipeline Co., 254 F.3d 925, 931 (10th Cir.2001) (citing Allied-Brace Terminix Cos., v. Dobson, 513 U.S. 265, 270-72, 115 S.Ct. 834, 130 L.Ed.2d 753 (1995)).

Here, the parties’ partnership agreement contained an arbitration clause and the parties stipulated to binding arbitration pursuant to the FAA and the Commercial Rules of the American Arbitration Association. In consenting to arbitration, “ ‘a party trades the procedures and opportunity for review of the courtroom for the simplicity, informality,-and expedition of arbitration.’ ” Brown v. Coleman Co., 220 F.3d 1180, 1182 (10th Cir.2000) (quoting Gilmer v. Interstate/Johnson Lane Corp., 500 U.S. 20, 31, 111 S.Ct. 1647, 114 L.Ed.2d 26 (1991)). “Because the primary purpose behind arbitration agreements is to avoid the expense and delay of court proceedings, ... it is well settled that judicial review of an arbitration award is very narrowly limited.” Foster v. Turley, 808 F.2d 38, 41 (10th Cir.1986) (internal citations omitted).

Standard of review

We review de novo a district court’s order vacating or enforcing an arbitration award. NCR Corp., E & M-Wichita v. Int’l Ass’n of Machinists & Aerospace Workers, 906 F.2d 1499, 1500 (10th Cir.1990). However, great deference is owed to the arbitrator’s decision. Indeed, the standard of review of arbitral awards “is among the narrowest known to the law.” Litvak Packing Co. v. United Food & Commercial Workers, 886 F.2d 275, 276 (10th Cir.1989). An arbitral award is subject to reversal only if it evinces a “manifest disregard” of the law. Id. This court has characterized the “manifest disregard” standard as “willful inattentiveness to the governing law.” Jenkins v. Prudential-Bache Sec., Inc., 847, F.2d 631, 634 (10th Cir.1988).

“The district court reviews the master’s legal conclusions de novo.” Gottlieb v. Barry, 43 F.3d 474, 486 (10th Cir.1994). “Where the district court rejects a factual finding by the master, [this court], like a majority of circuit courts, directly review[s] the findings of the special master, thereby effectively ignoring the district court’s review of the master’s findings.” Id. Although this court has not previously stated a standard of review for a district court’s review of a special master’s legal conclusions, since the district court’s review is de novo, it follows that we in turn would review the district court’s legal conclusions de novo. See NLRB v. Monfort, Inc., 29 F.3d 525, 528 (10th Cir.1994) (stating court reviews master’s conclusions of law de novo where appellate court directly reviewed special master’s report).

III.

There is no dispute that Nukem has satisfied the district court’s first judgment against it in the amount of $15,677,535. Further, this court has decisively ruled that the arbitration panel intended to impose a constructive trust as an award above and beyond that damages award. The lingering controversy involves the value of the constructive trust. And, at the core of that controversy is the question: what does “those purchase rights, the uranium acquired pursuant to those rights and the profits therefrom” mean?

On appeal, Nukem argues (1) the arbitration panel’s description of “purchase rights” was too vague to permit any valuation of “purchase rights” and that the matter should be remanded for clarification; (2) the district court’s valuation of the constructive trust erroneously included Nukem’s net profits from the sale of CIS uranium; and (3) the special master’s report was not clearly erroneous and should have been adopted in full by the district court.

Vague description of “purchase rights”

Other circuits have held there are circumstances where a district court can remand to the arbitrators for clarification. See Green v. Ameritech Corp., 200 F.3d 967, 977 (6th Cir.2000) (stating remand proper “to clarify an ambiguous award”); Colonial Penn Ins. Co. v. Omaha Indem. Co., 943 F.2d 327, 334 (3d Cir.1991) (concluding trial court had authority to remand award to arbitrators for clarification of ambiguous award). “[C]ourts have uniformly stated that a remand to the arbitration panel is appropriate in cases where the award is ambiguous.” Id. “Such a remand avoids the court’s misinterpretation of the award and is therefore more likely to give the parties the award for which they bargained.” Id. “In short, for a court to. engage in guesswork as to the meaning and application of an ambiguous arbitration award is inconsistent not only with federal policy, but also with the parties’ own agreement to submit their dispute to arbitration.” M & C Corp. v. Erwin Behr GmbH & Co., 326 F.3d 772, 782 (6th Cir.2003). When there is more than one reasonable interpretation of an arbitration award, a remand for clarification is appropriate. Mutual Fire, Marine & Inland Ins. Co. v. Norad Reinsurance Co., 868 F.2d 52, 58 (3d Cir.1989) (stating “[a] district court itself should not clarify an ambiguous arbitration award but should remand it to the arbitration panel for clarification”); Bell Aerospace Co. Div. of Textron, Inc. v. Local 516, 500 F.2d 921, 923 (2d Cir.1974) (stating “[cjonstruing ambiguous provisions of an arbitration award is the proper province of the arbitrator, not the courts”); American Postal Workers v. U.S. Postal Serv., 254 F.Supp.2d 12, 15 (D.D.C.2003) (remanding for clarification where the arbitrator’s award was “susceptible to more than one interpretation”). Such remands, however, are to be used sparingly in order not to thwart the interest of achieving finality. See Fischer v. CGA Computer Assoc., Inc., 612 F.Supp. 1038, 1041 (S.D.N.Y.1985) (recognizing remand frustrates the basic purposes of arbitration because it delays execution of the final judgment).

In Nukem I, Nukem argued “the district court’s judgment imposing the constructive trust [was] vague and ambiguous.” 1998 WL 738336, at *4. This court indicated that the district court’s judgment was clear when “read in conjunction with the arbitration order and award.” Id. However, this court was not asked to, nor did it, directly address the valuation of “purchase rights,” and the award offers no guidance on fundamental issues related to the valuation of “purchase, rights.” For example, there is no indication what the purchase rights are, how they are to be valued, or when ;or how they are to be extinguished. The special master equated the value of “purchase rights” to the value of a third party’s facilitation of a relationship between Nukem and another uranium supplier — a service for which Nukem paid the third party 1.5 percent of the resulting uranium sales. The special master did not explain how rendering such a service was comparable to “purchase rights.” Further, there is absolutely no support in the arbitration panel’s award for the special master’s formula for valuing “purchase rights” (i.e., 1.5% of $616,977,221-^-the proceeds of Nukem’s CIS uranium sales prior to December 31, 2001).

Although,the special master’s interpretation of, .the arbitration award was arguably a reasonable one, pur role is not one of interpretation; rather, it is to determine whether the arbitration panel acted within its authority and whether the arbitration panel’s decision evinces a “manifest disregard for the. law.” Because there is more than one reasonable interpretation of the arbitration panel’s award, we conclude a remand for clarification is necessary.

In reaching this conclusion, we note that neither the passage of time nor the fact that one of the three arbitration panel members has died precludes remand. The parties . in this case stipulated that the arbitration proceeding would.be-governed by the Commercial Rules of the American Arbitration Association (“AAA”) and the FAA. AAA Rule 19 sets forth a procedure to address any vacancies that may occur in the panel:

(a) If for any reason an arbitrator is unable to perform the duties of the office, the AAA may, on proof satisfactory to it, declare the office vacant. Vacancies shall be filled in accordance with the applicable provisions of these rules.
(b) In the event of a vacancy in a panel of neutral arbitrators after the hearings have commenced, the remaining arbitrator or arbitrators may continue with the hearing and determination of the controversy, unless the parties agree otherwise.
(c) In the event of the appointment of a substitute arbitrator, the panel of arbitrators shall determine in its sole discretion whether it is necessary to repeat all or part of any prior hearings.

AAA, Commercial Arbitration Rules, Rule 19 (2003).

Further, in Trade & Transport, Inc. v. Natural Petroleum Charterers Inc., 931 F.2d 191 (2d Cir.1991), a contract dispute, one member of a three-member arbitration panel died after the panel issued a decision regarding liability but before the panel issued a damages award. On appeal, the party that lost in arbitration argued that the FAA required that the arbitration begin anew before a whole new panel. Id. at 194. The parties’ arbitration agreement did not set forth a procedure for continuing after the death or disability of an arbitrator. The Second Circuit observed that the general rule is that where one member of a three-person arbitration panel dies before the rendering of an award and the arbitration agreement does not anticipate that circumstance, the arbitration must commence anew with a full panel. Id. The court concluded, however, that where one of three arbitrators died after an award of liability, it was not necessary to appoint a new panel or to begin arbitration anew. Instead, the court concluded the district court had authority to appoint a replacement arbitrator to consider the issue of damages. Id. at 195-96. The court relied on Section 5 of the FAA, which provides:

If in the agreement provision be made for a method of naming or appointing an arbitrator or arbitrators or an umpire, such method shall be followed; but if no method be provided therein, or if a method be provided and any party thereto shall fail to avail himself of such method, or if for any other reason there shall be a lapse in the naming of an arbitrator or arbitrators or umpire, or in filling a vacancy, then upon the application of either party to the controversy the court shall designate and appoint an arbitrator or arbitrators or umpire, as the case may require, who shall act under the said agreement with the same force and effect as if he or they had been specifically named therein; and unless otherwise provided in the agreement the arbitration shall be by a single arbitrator.

9 U.S.C. § 5.

Here, as in Trade & Transport, the three-member panel has already issued an award, and the only questions remaining concern the amount of damages. Aplt. App. Vol. I, Doc. 2. Unlike Trade & Transport, the parties in this case expressly agreed that the arbitration procedure would be conducted pursuant to AAA Rules, in addition to the FAA. As outlined above, AAA Rule 19 sets forth the procedure to be followed in the event of a vacancy on the arbitration panel. Assuming the arbitration panel in this case was a “neutral panel,” the two surviving members of the panel could continue pursuant to AAA Rule 19(b) without a third member. Alternatively, if the district court determines that the vacancy must be filled, the vacancy should be filled pursuant to the AAA Rules. However, if there were a .“lapse” in filling the vacancy, then the district court would have the authority to appoint a new member of the panel pursuant to 9 U.S.C. § 5. If a replacement arbitrator were appointed, then, under AAA Rule 19(c), the panel would determine in its discretion whether it was necessary to repeat all or part of any hearings.

Valuation of the constructive trust

Nukem argues that the constructive trust does not include the profits that Nukem realized from the sale of CIS uranium. According to Nukem, the constructive trust consists of, at most, “purchase rights,” which the special master valued at $12,932,727 (50% of which would be payable to plaintiffs). Nukem contends, therefore, that the judgment “must be reversed and reduced by $13,577,820.” Nuk-em’s position fails to account for the explicit language of paragraph 163, which impressed a constructive trust on “purchase rights, the uranium acquired pursuant to those rights and the profits therefrom ...Aplt-App. Vol. I, Doc. 7 at 209-10 (emphasis added); see also 'Nukem I, 1998 WL 738336 at *3 (stating that Nukem was “[discounting the explicit language of [Paragraph 163]” in arguing that “the panel did not mean to impress a constructive trust”).

Relatedly, Nukem, argues that there is language in this court’s Nukem I decision that indicates that the corpus of the trust is limited -to the profit lost by SMP on the grandfathered supply contracts. As is set forth above, in Nukem I, Nukem argued that the district court’s judgment was vague to the extent that it impressed a constructive trust.-- In rejecting that argument, this court described the corpus of the trust as “the proceeds from the five grandfathered SMP supply contracts.” Nukem argues that this court’s description of the trust in Nukem I should be read as limiting the corpus of the trust to “the proceeds from the uranium that could have been imported into the United States in connection' with the SMP Contracts.” Aplt. Br. at 32.

Unfortunately, it is not clear what this court meant by “the proceeds from the five grandfathered SMP supply contracts.” Arguably, however, this court intended to indicate that the corpus of the trust consisted of any enrichment to Nukem resulting from its misappropriation of SMP’s five grandfathered contracts. Such an interpretation would be consistent with this court’s conclusion that the arbitration panel intended to make two separate awards: (1) damages for SMP’s lost profits related to the grandfathered supply contracts, and (2) a constructive trust. Arguably, such an interpretation would also be consistent with the language of paragraph 163, which impressed a constructive trust over not only “purchase rights,” but also the “profits therefrom.” In any event, in Nukem I, the question of the value of the constructive trust was not before this court.

The root difficulty with resolving this issue is that the arbitration award does not define “purchase rights.” As a result, any attempt on our part to define or value the “profits therefrom” would involve a great deal of guesswork and is better left to the arbitration panel on remand.

Partial rejection of special master’s report

Finally and alternatively, Nukem argues that the district court erred by rejecting some of the special master’s findings and conclusions. The flaw in Nukem’s argument is that the district court did not reject any of the special master’s factual findings or legal conclusions. To the contrary, the district court’s order and judgment relied on three of the special master’s conclusions, namely: (1) that the value of the “purchase rights” was $12,923,727; (2) that the purchase rights were extinguished on December 31, 2001; and (3) that Nukem’s net profits from the sale of CIS uranium totaled $27,155,640.

Although Nukem contends that the district court valued the constructive trust differently than the special master, the special master stopped short of placing a value on the constructive trust. Nukem deduces from the report that the special master valued the constructive trust at $12,932,727. It appears, however, the special master merely valued “purchase rights” at $12,932,727. The special master never expressly concluded that the corpus of the trust was limited to “purchase rights.” Further, it appears that such a limitation would be contrary to the arbitration panel’s impression of a constructive trust over “purchase rights, the uranium acquired pursuant to those rights and the profits therefrom ...Aplt.App. Vol. I, Doc. 7 at 209-10 (emphasis added). Although the special master calculated Nuk-em’s net profits from the sale of CSI uranium, he never addressed whether or not he believed those profits to be part of the constructive trust.

Further, Nukem contends that the report indicated a reduction of the constructive trust award to account for Nukem’s previous “overpayment” of $4,692,241 in damages. Indeed, in one chart, the special master subtracted the arbitration award from Nukem’s net profit on CIS sales, and the resulting figure was negative $4,199,430. . ApltApp. Vol. II, Doc. 16 at 283. The special master further concluded that had SMP filled its grandfathered supply contracts with CIS uranium, its “hypothetical profit” would have been $21,970,588, and, in connection with that conclusion, he noted that the arbitration panel had awarded $31,355,070 in damages. However, the special master never used the term “overpayment” and never expressly indicated that the district court ought to reduce the judgment in favor of plaintiffs in consideration of such an overpayment. In sum, the special master never reached the ultimate question of the value of the constructive trust. The district court, in placing a value on the trust, relied on some of the special master’s conclusions and ignored others, but did not reject any of the special master’s findings or conclusions regarding the value of the constructive trust.

IV.

In their cross-appeal, plaintiffs argue that the special master and the district court erred: (1) by concluding that the constructive trust terminated on December 31, 2001, even though Nukem’s CIS purchase contracts continue past that date; (2) in calculating Nukem’s net profits on the sale of CIS uranium — specifically by deducting Nukem’s actual sales and overhead costs even though pursuant to a marketing agreement Nukem was to receive only a $300,000 annual marketing fee; and (3) by failing to award prejudgment interest on the award of $13,577,820.

Termination of the trust

As stated, the special master concluded that “purchase rights” were extinguished on December 31, 2001, when, according to the special master, all deliveries due under the five grandfathered SMP supply contracts had been made. Aplt.App. Vol. II, Doc. 16 at 286. Further, the special master calculated Nukem’s profits on the sale of CIS uranium only through December 31, 2001. The district court adopted the special master’s conclusion that purchase rights were extinguished on December 31, 2001, as well as the special master’s calculation of Nukem’s profit on the sales of CIS uranium. Plaintiffs do not dispute that the SMP supply contracts were filled by December 31, 2001. Instead, plaintiffs contend that the duration of the constructive trust must be coextensive with Nukem’s allegedly ill-gotten CIS purchase contracts — not the grandfathered SMP contracts. In response, Nukem restates its argument that the constructive trust' consists of the- hypothetical profit from the grandfathered SMP supply contracts and not the actual profit from CIS uranium purchase contracts. Aplt. Reply Br. at 6.

Because it is unclear what the arbitration panel intended “purchase rights” to consist of, it is not clear when or how those rights should be extinguished. The arbitration panel should therefore clarify this point on remand.

Deduction of actual sales, general and administrative costs from net profit

The special master concluded that Nuk-em’s gross profit from the sale of CIS uranium was $66,399,200. From gross profits, the special master subtracted $39,243,560 in costs, including $35,134,972 in “sales, general and administrative costs” and finance charges of $4,108,588. Thus, the special master concluded that Nukem’s net profit was $27,155,640. The district court entered judgment in favor of plaintiffs and against Nukem for half of that amount. Plaintiffs contend that the special master and the district court should have deducted only $2,414,546 in sales, general and administrative costs.

According to plaintiffs, the amount of money that Nukem was to receive from SMP for all costs was contractually established at $300,000 per year by their marketing and trading agreement (“MTA”). •Under the MTA, Nukem was to obtain buyers for SMP uranium and to acquire uranium on SMP’s behalf. For those services, Nukem was to receive a flat fee of $300,000 per year. The agreement was in effect from December 21, 1988 through June 5, 1995, when it was terminated. Aplt. Appl. Vol. Ill, Doc. 18 at 728-36 (agreement); Vol. I, Doc. 6 at 160 (arbitration award, stating that agreement was terminated on June 5, 1995). Plaintiffs argue that “it would be consistent with equity for the parties’ contractual arrangements to continue to utilize the annual fee of $300,000 throughout the term of the Constructive Trust.” Aplee. Br. at 49. Once again, the problem with plaintiffs’ argument is that the arbitration award is silent regarding “sales, general and administrative costs.”

Failure to award pre-judgment interest

Plaintiffs appeal the district court’s denial of pre-judgment interest. According to plaintiffs, they are entitled to prejudgment interest from some unspecified time through July 30, 2003, on the $13,577,820 portion of the judgment (50% of Nukem’s net profit from CSI sales) at the rate of 8% per annum for a total of $7,901,672. The arbitration panel awarded interest at 8% per annum on its “damages award.” The arbitration panel was silent, however, regarding interest related to the constructive trust. Accordingly, this question as well should be addressed by the arbitration panel on remand.

V.

While there are some factors that weigh against remand in this case, it is not the role of the courts to interpret vague arbitration awards. The arbitration award in this case is silent as to the definition of “purchase rights” and the “profits therefrom,” including the valuation of either. Also unstated in the award is the duration of the constructive trust and whether and what costs should be deducted when eom-puting the value of the constructive trust. Further, the arbitration panel failed to address whether prejudgment interest should be awarded on the value of the constructive trust. As a result, the district court’s valuation of the constructive trust was based upon extensive guesswork. Therefore, a remand to the arbitration panel for clarification is necessary, despite the long and tortured procedural history of this case.

We VACATE the judgment of the district court and REMAND with instructions that the matter be remanded to the arbitration panel for clarification of the arbitration award. 
      
      . Nukem had two contracts with Uzbekistan, “Uzbek I & II.” Nukem also had a contract with Kazakhstan and a contract with Kirgiz-stan.
     
      
      . There is no fixed deadline for a motion to remand for purposes of obtaining a clarification of the award. See Employers Ins. of Wausau v. El Banco Le Seguros Del Estado, 357 F.3d 666, 670 (7th Cir.2004); Hyle v. Doctor’s Assoc., Inc., 198 F.3d 368, 371 n. 1 (2d Cir.1999); United Steelworkers of America, Local 4839 v. New Idea Farm Equip. Corp., 917 F.2d 964, 968 (6th Cir.1990).
     
      
      . As is discussed later, Nukem contends that the arbitration panel’s reference to “purchase rights’’ is too vague to allow a valuation of those rights.
     
      
      . Plaintiffs argue that Nukem has made a "judicial admission” that the corpus of the trust includes profits from Nukem's CIS contracts because, as an alternative-position discussed in detail later in this opinion, Nukem contends that the district court erred by not fully adopting the master’s report. We reject plaintiffs’ argument. Judicial admissions are "formal, deliberate declarations which a party or his attorney makes in a judicial proceeding for the purpose of dispensing with proof of formal matters or of facts about which there is no real dispute.” Kempter v. Hurd, 713 P.2d 1274, 1279 (Colo.1986). Statements in briefs "may be considered admissions at the court's discretion.” Guidry v. Sheet Metal 
        
        Workers Int'l Assoc., 10 F.3d 700, 716 (10th Cir.1993). However, inconsistent statements made in the alternative are not formal, deliberate declarations that could reasonably be construed as judicial admissions.
     
      
      . Half, of the difference between these two figures is $4,692,241 — the amount which Nukem claims it has "overpaid.”
     
      
      . It is undisputed that two of the CIS .purchase contracts have been extended beyond December 31, 2001: the “Uzbek II” to 2006 and the Kazak contract to 2010.
     
      
      . The plaintiffs do not dispute the deduction for finance charges.
     
      
      . Plaintiffs assert in their opening brief that the marketing and trade agreement was terminated on June 1, 1998. They fail, however, to cite any evidence that supports that assertion. Aplee. Br. at 9, 49.
     
      
      . It appears from an exhibit submitted with plaintiffs' motion to alter or amend the judgment that they seek prejudgment interest from some time in 1992 through 2001. Aplt. App. Vol. V, Doc. 26 at 1364. On appeal, however, it appears from plaintiffs' brief that they seek interest through July 30, 2003. Aplee. Br. at 51.
     