JOSÉ A. CABRANES, Circuit Judge:
On March 26, 1997, final judgment was entered in the United States District Court for the Southern District of New York (Deborah A. Batts, Judge) following a trial to the bench in this civil action involving claims for, inter alia, racketeering, fraud, and breach of fiduciary duty, counterclaims for breach of contract, and an intervening dispute over fees owed to retained counsel discharged mid-trial. The action arose from the disintegration of a contractual relationship between two corporate entities in search of means to shelter or defer tax liability and the consultant who assisted them in that effort. This [139]*139appeal, cross-appeal, and consolidated appeal raise questions regarding the calculation of damages on certain of the consultant’s breach of contract claims, the liability of the consultant for alleged breaches of fiduciary duty in connection with certain uses of falsely worded documents, and the propriety of certain hens granted to the consultant’s attorneys following their mid-trial discharge.
Plaintiffs Sequa Corporation (“Sequa”) and Sequa Capital Corporation (“SCC”)1 (collectively, the “Sequa parties” or “Sequa”) filed an appeal (No. 97-7424) from (1) that portion of the judgment awarding damages to defendants GBJ Corporation (“GBJ”), Topaz Capital Corporation (“Topaz”), and Jeffrey J. Gelmin (“Gelmin”) (collectively, the “Gelmin parties”), on their breach of contract counterclaims with respect to eight of the transactions at issue in this action, and (2) that portion of the judgment finding Gelmin not hable for breach of fiduciary duty with respect to five occasions on which he was involved in the submission of false documents to Sequa and/or SCC.
The Sequa parties’ first challenge is to the method by which the district court calculated the breach of contract damages with respect to eight transactions; they argue principally that the district court improperly rehed on estimates of certain tax effects where the parties had not had occasion to introduce evidence concerning actual tax effects. Because it is preferable to base damage awards on proof of actual data where possible, resorting to estimates if and when that proof fails, we vacate the portion of the judgment relating to the challenged damage calculations and remand with directions to afford the parties an opportunity to introduce evidence regarding actual tax effects.
The Sequa parties next challenge the district court’s conclusions of law with respect to five occasions on which Gelmin caused GBJ or Topaz to employ falsely-worded documents in dealings with Sequa and SCC. The district court held that because Gelmin had acted on those occasions with the authorization of an officer of SCC, he had not breached his fiduciary duty to the Sequa parties. Because the district court apparently did not determine whether Gelmin’s reliance on the apparent authority of this corporate agent was reasonable, we vacate the portion of the judgment relating to Gelmin’s liability for breach of fiduciary duty on the five occasions in question and remand for reconsideration.
Gelmin, GBJ, and Topaz filed a cross-appeal (No. 97-7478) from that portion of the district court’s judgment awarding damages to the Sequa parties with respect to a single occasion on which the district court held that Gelmin had breached his fiduciary duty through the use of a falsely worded document. Because we see no error in the district court’s application of the “faithless servant” doctrine in calculating this damage award, we affirm the portion of the judgment that is the subject of the cross-appeal.
The Gelmin parties also filed a separate, but subsequently consolidated, appeal (No. 97-7692) from that portion of the judgment incorporating an interlocutory order (Michael H. Dolinger, Magistrate Judge) granting charging and retaining liens securing some $2.9 million in attorney’s fees and expenses owed to the law firm of Butler, Fitzgerald & Potter (“Butler”) for work performed prior to counsel’s mid-trial dismissal on April 17, 1995.2 Because we find no merit in the Gelmin parties’ objections to these liens, we affirm the portion of the judgment challenged in the consolidated appeal.
I.
A. Factual Background
The events giving rise to this protracted litigation are set out at great length in the district court’s findings of fact and conclusions of law in Sequa Corp. v. Gelmin, No. 91 Civ. 8675 (S.D.N.Y. Dec. 31, 1996), available online at 1996 WL 745448 or 1996 U.S. Dist. LEXIS 19802, with which we assume familiarity. The facts and procedural history relevant to this appeal are as follows.
[140]*140In the mid-1980s, Sequa anticipated significant tax liabilities relating to revenue from defense contracts, and sought ways to shelter income or defer its prospective tax liability. Sequa’s vice president and treasurer, David O’Brien (“O’Brien”), suggested that Sequa engage in “tax-leveraged leasing” for this purpose. After in-house discussion and preliminary consultations with Gelmin, an individual knowledgeable in such structured leases who had previously had dealings with Sequa, Sequa formed a wholly-owned subsidiary, SCC, to serve as a vehicle for transactions that would shelter or defer Sequa’s tax liability. O’Brien was appointed as SCC’s president.
In its simplest form, a tax-leveraged lease involves the purchase of a capital asset which is then leased (sometimes to the very entity from which the asset is purchased) on a long-term basis. The asset’s purchase commonly is “leveraged” — that is, the purchaser makes only a small equity investment and finances the balance of the purchase price. The new owner3 receives tax benefits in the form of depreciation deductions (which are calculated based on the full value of the asset, not just the purchaser’s equity stake), in addition to deductions for interest paid on the financing loans.4 Some of the depreciation-based tax benefits may be paid back in the form of a tax “recapture” when the asset is later resold, but the purchaser will have deferred tax liability until that later date and will have enjoyed the concomitant benefit of the “time value” of the funds that it otherwise would have had to use earlier to pay its taxes. See generally Stephan L. Hodge, Note, Sale-Leasebacks: A Search for Economic Substance, 61 Ind. L.J. 721, 721, 730-31 (1985) (describing generic structure and tax effects).
SCC entered into a consulting agreement with GBJ (of which Gelmin was president and the sole owner) whereby the latter would find, analyze, recommend, and assist in securing financing for leveraged leasing transactions to be undertaken by SCC as a tax shelter for Sequa. In return, GBJ was to receive an up-front fee of one-half percent of the purchase price of the capital asset, as well as a payment of ten percent of the “residual value” of the asset when it was eventually resold. (As explained below, the definition of “residual value” was a point of contention at trial.) Between 1985 and 1991, SCC entered into over forty leveraged lease transactions under this agreement, for which GBJ was paid some $6.6 million.
In April 1989, Topaz was formed as a vehicle through which to engage the services of two leveraged leasing specialists. Although Topaz was originally intended to be a joint venture between SCC and GBJ, and engaged in two leasing transactions apparently on that basis in mid-1989, the joint venture was subsequently dissolved and in December 1989 Gelmin became Topaz’s sole shareholder. SCC paid some $4 million in “consulting fees” to Topaz through 1991, although some of these payments were not in fact for services rendered, but instead represented profit-sharing between SCC and Gel-min, or were used to “park” funds for SCC.
The district court found, and the parties do not contest, that on five occasions in 1989 and 1990 Gelmin — for reasons not fully explained in this record — caused GBJ or Topaz to submit falsely worded invoices to or enter into falsely worded agreements with SCC. On all five occasions, O’Brien — SCC’s president— authorized or knew of Gelmin’s use of falsely worded documentation. On a sixth occasion, the “United” transaction in 1990, Gelmin caused an advisor to a leasing transaction to submit a falsely worded invoice to SCC; there was no claim that the arrangement was [141]*141authorized by or disclosed to anyone at SCC or Sequa.5
In the fourth quarter of 1990, SCC posted losses of $59 million, O’Brien was removed from his position as president, and Sequa proceeded to reorganize SCC. By letter dated March 19,1991, SCC exercised its right to terminate the consulting agreement with GBJ, effective May 1991. After May 1991, SCC sold various capital assets whose original purchase and lease had been arranged pursuant to the 1985 GBJ consulting agreement, but did not pay GBJ the ten percent of “residual value” provided for by that agreement.
B. Procedural History
In December 1991, GBJ — through its retained counsel, the law firm of Butler, Fitzgerald & Potter — filed suit in the district court under § 10(b) of the Securities Exchange Act of 1934 (“SEA”), 15 U.S.C. § 78j(b), seeking to enjoin Sequa and SCC from marketing securities (ie., selling leveraged-leased capital assets) without notice of GBJ’s claimed ten-percent residual interest in the property. On January 23, 1992, the complaint was amended to allege claims for breach of contract under New York law.
In February 1992, Sequa and SCC filed a motion to dismiss for want of federal subject matter jurisdiction pursuant to Fed.R.Civ.P. 12(b)(1), on the ground that SCO’s capital assets were not “securities” under the SEA. On August 24, 1992, while their motion to dismiss was still pending before the district court, the Sequa parties filed counterclaims against GBJ and a third-party complaint against Gelmin and Topaz, alleging violations of the Racketeer Influenced and Corrupt Organizations Act (“RICO”), 18 U.S.C. § 1961 et seq., as well as claims under New York law for, inter alia, breach of fiduciary duty.
On October 22, 1992, the district court (Charles S. Haight, Judge) granted the Se-qua parties’ motion to dismiss GBJ’s securities claim, declined to exercise supplemental jurisdiction over GBJ’s state law breach of contract claims, and — apparently overlooking the Sequa parties’ counterclaims and third-party complaint alleging federal as well as state law causes of action — ordered the entire action dismissed for lack of subject matter jurisdiction. The parties, in a joint letter to the court dated November 2, 1992, requested that the action be retained and the parties be realigned to name Sequa and SCC as plaintiffs and the Gelmin parties as defendants; by memo endorsement on November 9, 1992, the district court entered an order granting that request.
Extensive discovery ensued until early 1994. On June 1, 1994, the parties filed voluminous papers in conjunction with cross-motions for summary judgment. Before responsive papers were filed, however, the action was reassigned to Judge Deborah A. Batts, who directed the parties instead to prepare for trial. A bench trial before Judge Batts commenced on October 31, 1994; following a number of adjournments, including two court-ordered rounds of settlement discussions, the seventy-nine day trial concluded on July 15,1996.
One of the trial adjournments was occasioned by the Gelmin parties’ dismissal of their retained counsel on April 17,1995. Trial proceedings were temporarily halted so that Magistrate Judge Dolinger could conduct a hearing to determine the reasonable value of Butler’s services rendered to, and costs owed by, the Gelmin parties. Under New York law, such amounts could be secured by liens in Butler’s favor — namely, a retaining lien on the Gelmin parties’ files, and a charging lien against any recovery that the Gelmin parties might ultimately obtain in the action. Magistrate Judge Dolinger conducted a six-day hearing (which spanned thirteen calendar days), following which he determined that Butler was owed some $2.9 million in fees and expenses, that it was entitled to assert a charging lien in that amount against any Gelmin recovery, and that it was entitled to either approximately $159,000 or a bond securing payment of the [142]*142$2.9 million in return for releasing its retaining lien on the Gelmin parties’ files. That interlocutory order6 was subsequently incorporated into the district court’s final judgment of March 26,1997.
As to the assorted claims and counterclaims in the main action, the district court rejected all of the Sequa parties’ claims of racketeering, conspiracy, fraud, and breach of fiduciary duty against Gelmin, GBJ, and Topaz — except with respect to the 1990 United transaction, where the district court held Gelmin liable for fraud and breach of fiduciary duty arising from the use of falsely worded documentation, and ordered principally that GBJ’s fee on the transaction be forfeited. In addition, the district court held that the Gelmin parties had been unjustly enriched with respect to four transactions.
With respect to the Gelmin parties’ counterclaims, the district court found that SCC had breached contracts with the Gelmin parties, principally by failing to pay the ten-percent residual share provided for in the consulting agreement when SCC sold capital assets after relations among the parties broke down in 1991. The district court calculated and awarded as damages to the Gel-min parties the ten-percent residual shares on some eighty-three sales; it also awarded declaratory relief providing that the Gelmin parties were entitled to residual shares upon the future sale of assets connected with eleven other leases, and specifying how those residual shares should be calculated.
II.
In the words of the district court, “the true essence of this action is a disagreement over contract interpretation.” That disagreement centered principally on the interpretation of the “back end” element of GBJ’s compensation under its consulting agreement with SCC: the “amount equal to 10% of the residual value of all such equipment” that was to be paid to GBJ upon the sale of the leased assets in addition to the up-front fee of one-half percent on each lease it arranged. The consulting agreement provided, in pertinent part, that “[rjesidual value for this purpose shall mean all proceeds from the sale of equipment after payment for and deduction of the balance of all costs and debts related to the acquisition of the equipment and the sale thereof.”
The parties offered divergent interpretations of this provision on many points, but the element now at issue in the Sequa parties’ appeal is “the balance of all costs and debts” to be subtracted from the proceeds of the sale before calculating GBJ’s ten-percent fee. The district court found that language to be unambiguous. However, certain elements of the definition settled upon by the district court were not advocated by either party — in particular, the district court determined that the costs and debts associated with the assets’ purchase should be offset by the tax benefits that SCC obtained from the leasing transaction.
Although the Sequa parties did not advocate an interpretation of “the balance of all costs and debts” that took tax benefits into consideration — presumably because any such offset would shrink “the balance of all costs and debts” that is subtracted from the sale proceeds, resulting in a larger residual value on which GBJ’s ten-percent fee would be based — on appeal they do not contest that reading of the contract language. Rather, they object to the means by which the district court proceeded to calculate those tax effects.
Because neither party advocated or anticipated the district court’s decision to consider SCO’s tax benefits in this manner, neither party had occasion or cause to introduce evidence as to SCO’s and Sequa’s actual tax benefits from the lease transactions. When, after the close of evidence, the district court arrived at a contract interpretation (and hence a damages calculation) that in part turned on those tax benefits, it looked else[143]*143where in the evidentiary record to estimate the tax benefits. More specifically, the district court relied on computerized financial projections of the lease transactions (known as “LAS runs”) that had been prepared prior to SCO’s asset purchases, -and that included projections of the tax deductions that SCC and Sequa would be able to claim if SCC undertook the purchase and lease arrangements. With respect to future asset sales governed by the consulting agreement, the district court specified in the declaratory relief portion of the judgment that “[i]f ... a calculation of benefits [including tax benefits] cannot be made or proven by SCC,” the corresponding LAS runs may be relied upon instead.
After the district court entered its findings of fact and conclusions of law, the Sequa parties moved for amendment of the district court’s findings, pursuant to Fed.R.Civ.P. 52(b), or, alternatively, for a new trial limited to evidence of Sequa and SCO’s actual tax benefits, pursuant to Fed.R.Civ.P. 59(a).7 The Sequa parties contended principally, as they do on appeal, that the computerized projections supplied an unreliable and inaccurate estimate of the tax effects involved in eight leasing transactions8 because they were defective in two respects. First, the LAS runs projected only tax deductions, and did not offset those tax benefits with the tax “recapture” liability that SCC incurred by selling the assets prior to the expiration of their lease terms; in effect, the LAS runs measured Sequa’s gross, not net, tax benefits on the eight transactions. Second, the LAS runs assumed a tax rate that turned out to be 14% higher than Sequa’s actual tax rate for the years in question. Both defects, Se-qua alleged, resulted in an overstatement of the tax benefits that Sequa received from the eight leveraged leases, which in turn shrank the “balance of all costs and debts” to be subtracted from the sale proceeds, and resulted in too large a residual value figure on which to base GBJ’s ten-percent fee. On March 5, 1997, the district court denied the Sequa parties’ motion because “[t]he record ... is devoid of any evidence that the Plaintiffs in fact paid the tax recapture to the I.R.S. that they seek credit for in their proposed damages recalculation”; the district court declined to reopen the record to allow the parties to present such evidence.
Rulings on motions under Rules 52(b) and 59(a) are committed to the sound discretion of the district court, and are reviewed on appeal only for abuse of that discretion. See Zenith Radio Corp. v. Hazeltine Research, Inc., 401 U.S. 321, 331, 91 S.Ct. 795, 28 L.Ed.2d 77 (1971); Dow Chem. Pac. Ltd. v. Rascator Maritime S.A., 782 F.2d 329, 342 (2d Cir.1986); 9 James Wm. Moore et al., Moore’s Federal Practice § 52.60[2] (3d ed.1998). The district court’s denial of the Rule 52(b) motion to amend its findings, on the ground that the record lacked evidentiary support for recalculations taking tax recapture into account,9 was well-founded and entirely within its discretion. Although testimony at trial indicated that Sequa had incurred recapture tax liability, that testimony furnished no specifics as to the recapture liability incurred on each of the eight transactions; Sequa’s post-trial pro[144]*144posed recalculations were based on “back-of-the-envelope” estimates of its recapture liability, not on any record evidence of Sequa’s actual tax history. The district court could reasonably, and apparently did, conclude that, in light of their weak foundation in the record, Sequa’s proposed recalculations were insufficiently reliable to support amended findings. Cf. United States Gypsum, Co. v. Schiavo Bros., Inc., 668 F.2d 172, 180 (3d Cir.1981) (“[T]he trial record must support whatever additional findings of fact or conclusions of law a party seeks under Rule 52(b) or it is certainly not entitled to them.”), cert. denied, 456 U.S. 961, 102 S.Ct. 2038, 72 L.Ed.2d 485 (1982).
The propriety of the district court’s Rule 59(a) decision, however, presents a much closer question. On balance, we conclude that in the particular circumstances of this case the district court erred in denying the Sequa parties’ motion to reopen the trial record. It is well-settled that Rule 59 is not a vehicle for relitigating old issues, presenting the case under new theories, securing a rehearing on the merits, or otherwise taking a “second bite at the apple” — and we in no way depart from that basic principle. See, e.g., United States v. Local 1804-1, 831 F.Supp. 167, 169 (S.D.N.Y.1993), aff'd sub nom. United States v. Carson, 52 F.3d 1173 (2d Cir.1995); Wallace v. Brown, 485 F.Supp. 77, 78 (S.D.N.Y.1979).
The very reason given for the denial of the Rule 52(b) motion — that the record lacked evidence of Sequa’s actual tax history, including specific evidence that it did in fact incur tax recapture liability on the eight transactions — illustrates why a limited reopening of the trial record was needed. Because neither party anticipated the district court’s reliance on Sequa’s tax benefits in calculating the amount owing to the Gelmin parties on their breach of contract claims, neither party had introduced evidence as to the details of any such tax benefits. The computerized projections to which the district court turned had been introduced at trial for other, unrelated purposes. Accordingly, this is not a case in which the district court, presented by the parties with some evidence of actual damages that is in some respect inadequate, then turns to a reasonably reliable approximation of damages, cf. Wolff & Munier, Inc. v. Whiting-Turner Contracting Co., 946 F.2d 1003, 1010 (2d Cir.1991) (noting New York standards regarding proof of damages); here, the parties had no occasion, as an initial matter, to present and address the requisite evidence of (one aspect of) actual damages.
The record before the district court was indeed voluminous, and the judge’s reluctance to reopen that record is understandable. Were Sequa merely ruing an oversight of its own in failing to introduce foreseeably relevant evidence, or were it merely protesting the district court’s decision to turn to an estimate after examining the evidence of actual damages, we would not be inclined to disturb the district court’s decision. In the particular circumstances presented, however — where the district court adopted, after trial, a contractual interpretation advocated and anticipated by neither party that, in turn, directed that the calculation of actual damages incorporate a financial element as to which no evidence had been introduced at trial — the introduction of limited additional evidence was warranted. Cf. Oak Hall Cap Gown Co. v. Old Dominion Freight Line, Inc., 899 F.2d 291, 295 (4th Cir.1990) (reopening record was appropriate “to correct any errors or gaps” occasioned by district court’s later adoption of different damage theory).
Should such evidence of actual tax benefits prove wanting, falling back on the LAS runs’ projections to estimate those benefits could very well be appropriate — but hypothetical calculations should not be the evidence of first resort. In essence, then, we remand for the district court to play out the scenario it envisaged in awarding declaratory relief: “[i]f ... a calculation of benefits [including tax benefits] cannot be made or proven by SCO,”10 the corresponding LAS runs may be [145]*145relied upon instead. The cause is remanded merely to give SCC the initial opportunity (which, for lack of notice, it missed at trial) to make a record for that tax benefit calculation. We note again that this opportunity is a limited one: the parties may only introduce evidence relating to SCC’s actual tax benefits11 associated with the eight challenged leasing transactions.
III.
The district court rejected the overwhelming majority of the Sequa parties’ claims against the Gelmin parties, including claims of fraud, conspiracy, and racketeering. On appeal, the Sequa parties challenge only the district court’s holding with respect to Gel-min’s alleged breach of fiduciary duties in connection with the use of falsely worded documents. As noted above, the district court found — and Gelmin does not contest— that on six occasions Gelmin caused others (principally GBJ and Topaz) to submit falsely worded invoices or enter into falsely worded agreements with SCC. The district court held, however, that Gelmin had not breached his fiduciary duties to SCC on five of those occasions12 because O’Brien, then SCC’s president, had authorized or known of Gel-min’s use of false documents. The Sequa parties claim legal error in that holding.
It is uncontested that Gelmin was a fiduciary of SCC by virtue of his role as a broker and advisor, and that as a result, he owed SCC fiduciary duties of good faith and loyalty. See BAII Banking Corp. v. UPG, Inc., 985 F.2d 685, 700 (2d Cir.1993); Mandelblatt v. Devon Stores, Inc., 132 A.D.2d 162, 168, 521 N.Y.S.2d 672 (1st Dep’t 1987). Among Gelmin’s obligations were the obvious duty of honest dealing with his principal (SCC), as well as the duty to disclose to his principal all material information concerning transactions for which he served as a broker, see BAII, 985 F.2d at 700-01. A fiduciary’s use of false documents in his dealings with his principal generally runs afoul of these obligations.
The district court held, however, that Gel-min had not breached his fiduciary duties to SCC by virtue of his use of false documents because those activities were authorized by, or at least disclosed to, O’Brien, an officer of SCC. In the district court’s view, so long as O’Brien was not himself scheming with Gel-min to act adversely to SCC’s interests, see Danish Fur Breeders Ass’n v. Furrico Furs, Inc., No. 87 Civ. 3043, 1989 WL 37664, at *2 (S.D.N.Y. Apr.12, 1989), O’Brien’s explicit or tacit authorization of Gelmin’s use of false documents was sufficient to absolve Gelmin of his alleged breaches of fiduciary duty.
However, in our view, even if O’Brien was not engaged in a wrongful scheme with Gelmin,13 a question remains to be answered: [146]*146was Gelmin entitled to rely on O’Brien’s authorization? Put another way, could the fiduciary (Gelmin) reasonably rely on the apparent authority of an officer of his principal (O’Brien) to sanction conduct that would otherwise contravene his obligation to be truthful in dealings with that principal (SCC)? In United States v. D'Amato, 39 F.3d 1249, 1258 (2d Cir.1994), we noted that a contractor accused of fraud in connection with the use of billing paperwork that concealed the true nature of his services was entitled to “rely upon instructions [to use false paperwork] from a corporate agent with apparent authority and no ostensible conflict of interest to determine [the contractor’s] obligations and appropriate billing practices.” In Danish Fur Breeders, however, the district court determined that a broker who refrained from disclosing material information to the principal on the instructions of the principal’s vice president was not entitled to rely on the vice president’s apparent authority. See 1989 WL 37664 at *2. Critical to the propriety of the claimed reliance on an officer or agent’s apparent authority was whether the party claiming reliance could reasonably have understood the instructions or authorization to be in the principal’s best interests. See id.; cf. D’Amato, 39 F.3d at 1258 (otherwise lawful concealment using false documents may legitimately be in company’s best interests).
The district court did not consider whether Gelmin had established that his reliance on O’Brien’s apparent authority was reasonable. See Phoenix Elec. Contracting, Inc. v. Lehr Constr. Corp., 219 A.D.2d 467, 468, 631 N.Y.S.2d 146 (1st Dep’t 1995) (mem.) (burden is on party claiming reliance to establish (i) apparent authority and (ii) reasonableness of reliance), leave to appeal denied, 87 N.Y.2d 805, 640 N.Y.S.2d 878, 663 N.E.2d 920 (1995). We note that although Gelmin’s brief on appeal offers explanations of why the $2.4 million in payments triggered by the false invoices and agreements could reasonably have been understood to be in SCO’s best interests, he has offered no explanation of why it was (or could reasonably have been understood to be) in SCO’s best interests to use false documents to secure those payments. We will not, however, yield to Sequa’s urgings to resolve the question of Gelmin’s liability on appeal. Because the question of the reasonableness of Gelmin’s reliance on O’Brien’s authority is inherently fact-specific, and because we cannot begin to approach the district court’s familiarity with the circumstances of the case, we vacate that portion of the judgment holding Gelmin not liable with respect to each of the five identified uses of false documentation and remand to afford the district court the first opportunity to address that question.14
Gelmin’s cross-appeal takes issue with the sixth use of false documentation,15 which was not disclosed to O’Brien or SCC, and with respect to which the district court found Gelmin liable for fraud and breach of his fiduciary duty to SCC. He objects both to the district court’s liability conclusion, and to its holding that, as a consequence of the misdeed, he should forfeit his fee for that transaction. Gelmin’s claims of error in connection with the district court’s liability determination, based as they are principally on objections to the district court’s decision to credit one witness over another, are without merit. See Anderson v. City of Bessemer City, 470 U.S. 564, 575, 105 S.Ct. 1504, 84 L.Ed.2d 518 (1985).
Similarly, we find no error in the district court’s application of New York’s “faithless servant” doctrine to require the forfeiture of the fees Gelmin earned in connection with that transaction. Gelmin principally protests the disproportionality of the forfeiture (which he estimates at $900,000) relative to the putative harm borne by SCC [147]*147(some $26,000 in misdescribed consultant’s fees). What Gelmin fails to acknowledge, however, is that the district court’s forfeiture holding — under which he forfeited only his fees in connection with the single transaction, rather than all his fees from his entire consulting relationship with SCC — in fact represented a relatively generous interpretation of New York law. The district court relied on our decision in Musico v. Champion Credit Corp., 764 F.2d 102 (2d Cir.1985), in which we discerned a possible relaxation of New York’s strict rule demanding forfeiture of all of a faithless servant’s compensation during the entire agency relationship, in favor of a rule of apportionment under which only the fees related to the “specified items of work” as to which the agent acted faithlessly would be forfeited. Id. at 112-13. Because the New York Court of Appeals has not repudiated (though neither, for that matter, has it fulfilled) Músico’s predictions regarding apportionment of faithless servant forfeitures,16 the district court here could properly rely upon that case’s holding. However, particularly in light of the somewhat tenuous posture of the Músico rule, the current state of the law offers no support for the further relaxation of strict forfeiture — namely, the introduction of a rule of proportionality — that Gelmin seeks. Accordingly, we find no error in the district court’s forfeiture analysis with respect to Gelmin’s breach of fiduciary duty in this sixth “false document” transaction.
IV.
Interjected into the adjudication of the Sequa and Gelmin parties’ myriad claims and counterclaims was an ancillary proceeding occasioned by the Gelmin parties’ mid-trial decision to dismiss their attorneys, the law firm of Butler, Fitzgerald and Potter. Under New York law, an attorney dismissed without cause is entitled to hens in his favor to secure payment of reasonable fees and costs incurred prior to the date of substitution of counsel. See N.Y. Judiciary Law § 475 (charging hen against client’s recovery);17 In re Cooper, 291 N.Y. 255, 260, 52 N.E.2d 421 (1943) (common law retaining hen against client’s papers in attorney’s possession).
Trial proceedings were temporarily suspended so that Magistrate Judge Dolinger could conduct a hearing to fix the amount of the hens, based principally on an assessment of the reasonable value of Butler’s services to date. Magistrate Judge Dohnger determined that Butler was owed some $2.9 million in fees and expenses. On appeal, the Gelmin parties allege error in the magistrate judge’s (i) case management and discovery decisions; (ii) refusal to allocate Butler’s services rendered to allegedly distinct “actions”; (iii) use of “lodestar” analysis in calculating a reasonable fee; and (iv) imposition of joint and several liability upon Topaz. All four claims are unavailing.
(i) A district court’s rulings regarding discovery and the conduct of a trial (including the conduct of an ancillary evidentiary hearing) will not be disturbed on appeal absent an abuse of discretion. See United States v. Carson, 52 F.3d 1173, 1188 (2d Cir.1995) (conduct of trial), cert. denied 516 U.S. 1122, 116 S.Ct. 934, 133 L.Ed.2d 861 (1996); Goetz v. Crosson, 41 F.3d 800, 805 (2d Cir.1994) (discovery), cert. denied 516 U.S. 821, 116 S.Ct. 80, 133 L.Ed.2d 39 (1995). We are particularly sohcitous of a district court’s ruling on a motion to adjourn the [148]*148scheduled start of a trial proceeding. Such a ruling will not be disturbed unless clear abuse is shown; to make that showing, the complaining party must establish both that the denial of the adjournment was arbitrary, and that it substantially impaired the presentation of his case. See Dow Chem. Pac. Ltd., 782 F.2d at 341-42.
In our view, the Gelmin parties have not made this showing with respect to the magistrate judge’s refusal, in response to an oral request on the first day of the hearing, to adjourn the proceeding so that they could review all of Butler’s work product files. The magistrate judge promised, and provided (in the form of an adjournment spanning six full days), ample time for Gelmin’s new lawyers to review Butler’s contemporaneous billing records prior to their cross-examination of Butler’s witnesses. Moreover, the Gelmin attorneys also had access at that time to a substantial portion of the Butler work product files (by virtue of a separate private arrangement). In these circumstances, we are not persuaded that the refusal to adjourn substantially impaired the presentation of the Gelmin parties’ ease. Similarly, the magistrate judge was well within his discretion to limit discovery and to conduct the hearing as relatively expeditiously as he did; his ruling has not been shown to be arbitrary.
(ii) The Gelmin parties also claim legal error in the magistrate judge’s grant of a charging lien based on legai services rendered for the duration of Butler’s representation (from December 1991 through the April 1994 dismissal). They contend that the Gel-min-Sequa proceedings in fact comprised two separate “actions” — as separated by the district court’s (subsequently retracted) October 22, 1992 dismissal of all claims — only the latter of which generated a recovery for Gel-min against which Butler could assert a charging lien for fees.
We will not, however, reach this objection (whose factual and legal merits are tenuous in any event) because it was not properly preserved for appeal. See Amalgamated Clothing & Textile Workers Union v. Wal-Mart Stores, Inc., 54 F.3d 69, 73 (2d Cir.1995). We are not persuaded by the Gelmin parties’ claims that this argument can be divined by implication from certain comments at the hearing and certain statements of law in their submissions to the magistrate judge; they have not identified a cogent articulation before the magistrate judge of the argument they now press on appeal. Nor has there been shown any “extraordinary need” for this Court to disregard that omission. Id. Accordingly, we will not do so; the claim of legal error is deemed waived.
(iii) It is undisputed that it was proper to determine the amount of Butler’s § 475 charging lien on a quantum meruit basis, ascertaining the reasonable value of the legal services rendered up to the date of the Gelmin parties’ substitution of new counsel. See Lai Ling Cheng v. Modansky Leasing Co., 73 N.Y.2d 454, 457, 541 N.Y.S.2d 742, 539 N.E.2d 570 (1989). Among the factors to be considered in such an analysis are the difficulty of the matter, the nature and extent of the services rendered, the time reasonably expended on those services, the quality of performance by counsel, the qualifications of counsel, the amount at issue, and the results obtained (to the extent known). See, e.g., Spano v. Scott, 166 A.D.2d 917, 917, 561 N.Y.S.2d 678 (4th Dep’t 1990) (mem.), leave to appeal denied, 77 N.Y.2d 801, 566 N.Y.S.2d 586, 567 N.E.2d 980 (1991); N.Y.Code of Professional Responsibility DR 2-106, 22 N.Y.C.R.R. § 1200.11(b); see also 520 East 72nd Commercial Corp. v. 520 East 72nd Owners Corp., 691 F.Supp. 728, 739 (S.D.N.Y.1988), aff'd 872 F.2d 1021 (2d Cir.1989).
The Gelmin parties object to the magistrate judge’s use, as part of his quantum meruit assessment, of “lodestar” analysis, see Hensley v. Eckerhart, 461 U.S. 424, 433-34, 103 S.Ct. 1933, 76 L.Ed.2d 40 (1983), to assist in calculating the reasonable value of Butler’s services. We find no error. The magistrate judge first carefully considered all the factors relevant to a quantum meruit fee analysis, and only then turned to lodestar analysis to reach a specific dollar figure for the value of the services rendered by Butler. We agree with Magistrate Judge Dolinger that “the lodestar approach is sufficiently congruent with the criteria applicable under [149]*149New York law to justify such a choice” of calculation methodology. See F.H. Krear & Co. v. Nineteen Named Trustees, 810 F.2d 1250, 1263 (2d Cir.1987) (listing factors and employing lodestar analysis to award reasonable value of attorney’s services); Peter Fabrics, Inc. v. S.S. Hermes, 765 F.2d 306, 314-20 (2d Cir.1985) (affirming quantum meruit attorney’s fee award based on modified lodestar analysis). Moreover, the determination of the reasonable value of the attorney’s services “is a matter within the sound discretion of the trial court.” Chemofsky & DeNoyelles v. Waldman, 212 A.D.2d 566, 566, 622 N.Y.S.2d 560 (2d Dep’t 1995) (mem.). We find no abuse of that discretion in the magistrate judge’s resort to lodestar analysis, or in his application18 of that analytical framework.
(iv) The Gelmin parties’ final objection to the magistrate judge’s determination of Butler’s charging lien concerns Topaz, which only became a formal party to the action on August 24, 1992, when it was named as a defendant in Sequa’s third-party complaint, and became a claimant in November 1992 when the action was realigned. The magistrate judge’s opinion and order awarding the charging lien directs that the Gelmin parties (including Topaz) “are jointly and severally responsible” for the full $2.9 million owed to Butler. Topaz contends that it may not properly be held jointly and severally liable for whatever portion of that sum was incurred before it was brought into the action in 1992.
Even if Topaz is correct that its liability should only extend to the period during which Butler was engaged to represent it,19 we hold the magistrate judge’s putative omission to be harmless. Topaz’s “liability” for the $2.9 million is embodied in the charging lien, which by definition attaches only to the proceeds obtained as a result of the action. See N.Y. Judiciary Law § 475; In re Petition of Rosenman & Colin, 850 F.2d 57, 61 (2d Cir.1988). That is, Topaz’s joint and several responsibility is limited to the amount of its recovery in the action— specifically, the roughly $23,000 it obtained on a contractual counterclaim against SCO. By any measure, that portion of the charging lien for which Topaz is properly held jointly and severally responsible (i.e., at the very least, Butler’s services rendered from November 1992 through April 17, 1995 — two and a half of the three and a half years for which Butler was owed $2.9 million) must exceed Topaz’s maximum liability of $23,000. To remand for apportionment of the charging lien would be a pointless exercise, because in no event will Topaz be required to convey to Butler a sum exceeding that portion of the fees for which it is properly held liable.
In sum, we have carefully considered all of the Gelmin parties’ assorted objections to the magistrate judge’s opinion and order awarding Butler charging and retaining liens to secure payment of $2.9 million for the reasonable value of services and disbursements rendered prior to counsel’s dismissal without [150]*150cause. Having done so, we find no grounds for disturbing that determination.
V.
To conclude:
(1) We vacate that portion of the judgment relating to the calculation of the Gelmin parties’ damages in connection with eight specific leasing transactions,20 and remand for a limited reopening of the record so that the district court may hear the parties’ evidence regarding SCO’s actual tax benefits, and for such recalculation of those damages as may prove appropriate.
(2) We vacate that portion of the judgment holding Gelmin not liable for alleged breaches of his fiduciary duty to SCO in connection with five specific instances21 in which he employed or caused others to employ falsely worded documents in dealings with SCO, and remand so that the district court may consider in the first instance whether Gelmin’s reliance on the authorization of SCO’s then-president to employ those falsely worded documents was reasonable, and thereafter may determine Gelmin’s liability accordingly.
(3) We affirm the judgment of the district court in all other respects not implicated by the foregoing.
(4) The parties to the appeal (No. 97-7424) and cross-appeal (No. 97-7478) are to bear their own costs. Costs to appellees in the consolidated appeal (No. 97-7692).