Defendants Trevor Johnson, Roberts Carnes and Daniel Hunter were convicted in the United States District Court for the Southern District of New York (Richard Conway Casey, Judge) in 1998 for conspiring to commit extortion by use of actual or threatened force in violation of the Hobbs Act, 18 U.S.C. § 1951. See United States v. Mulder, 273 F.3d 91, 98 (2d Cir.2001). The District Court applied an enhancement based upon what it found to be relevant conduct — murder committed by a co-conspirator — and sentenced defendants to “harsh” sentences, id., ranging from seventeen to twenty years’ incarceration. Defendants appealed their convictions and sentences to this Court. We affirmed their convictions, but we vacated their sentences on the basis that the District Court’s findings did not support the enhancement for the murder committed by the co-conspirator, and we remanded for resentencing. Id. On remand, the District Court resentenced the defendants and, for the first time, ordered restitution. On appeal, the defendants now challenge the District Court’s resentencing determinations and restitution orders.
The principal questions presented on appeal are (1) whether the District Court erred by enhancing Johnson’s sentence on the basis of a murder committed by a co-conspirator; (2) whether the Court exceeded the scope of our mandate on remand by applying two new enhancements to Hunter’s sentence; (3) if the Court did not exceed the scope of our mandate with respect to Hunter, whether the Court erred by applying a five-level firearm enhancement and a two-level loss-to-the-victim enhancement to Hunter’s sentence; (4) whether the Court exceeded the scope of our mandate on remand by ordering defendants to pay restitution pursuant to the Mandatory Victims Restitution Act of 1996 (“MVRA”), 18 U.S.C. § 3663A; and (5) if the Court did not exceed the scope of our mandate in ordering restitution, whether the Court erred by ordering a defendant to pay restitution to his victim, pursuant to the MVRA, when that victim declined restitution and did not assign his interest to the Crime Victims Fund as statutorily authorized.
We hold that the District Court (1) erred in enhancing Johnson’s sentence on the basis of a co-conspirator’s act of murder; (2) did not exceed the scope of our mandate either in applying two new enhancements to Hunter’s sentence or in ordering restitution for the first time on remand for all defendants; (3) did not err in applying a firearms enhancement to Hunter’s sentence; (4) made insufficient factual findings as to loss amount in determining the appropriate enhancement level for Hunter’s loss-to-the-victim enhancement; (5) did not err in ordering restitution even when a victim declined it; and (6) failed to make sufficiently specific find*234ings as to loss causation on the part of Hunter and Johnson, hence erroneously imposing unjustified and imprecise restitution orders based thereon.
Accordingly, we vacate (1) the sentencing order with respect to Johnson and direct resentencing of Johnson without consideration of the co-conspirator’s act of murder and we direct also reconsideration of the amount and apportionment of restitution; and (2) the sentencing order with respect to Hunter insofar as the loss caused by Hunter was the basis for the two-level loss-to-the-victim enhancement and the restitution order.
We remand the cause for (1) resentenc-ing of Johnson without consideration of the co-conspirator’s act of murder; (2) reconsideration of Hunter’s loss-to-the-victim sentencing enhancement following a more particularized determination of the amount of loss caused by Hunter; and (3) entry of new orders of restitution for Hunter and Johnson based on new determinations, consistent with this opinion, of the amounts of loss caused by each defendant to each of his respective victims.
In all other respects, the judgment of the District Court is affirmed.
BACKGROUND1
I. Trial
In December of 1998, following a six-week trial, a jury convicted each of the defendants of conspiring to commit extortion in violation of the Hobbs Act, 18 U.S.C. § 1951. See Mulder, 273 F.3d at 99.
The defendants belonged to Brooklyn Fight Back (“BFB”), a “labor coalition” that claims it “used time-honored tactics of the labor movement to obtain jobs for members of minority groups on construction projects.” Id. at 98. Contrary to BFB’s claims of honorable means and ends, the jury found that, from 1990 to 1998, defendants’ participation in BFB included extorting money and jobs, including the proverbial “no-show”2 jobs, from contractors at construction sites in New York City. Id.
Among the overt acts in furtherance of the conspiracy to commit extortion were Johnson’s extortion of money from Tully Construction (“Tully”) and Piccone Construction (“Piccone”), Hunter’s extortion of money from Felix Construction (“Felix”), and Carnes’s extortion of money from the DeFoe Corporation (“DeFoe”).
In September 1997, during the course of the conspiracy, fellow BFB member and co-conspirator Eric Mulder killed Erick Riddick, a member of a rival “labor coalition,” during a confrontation at a Queens construction site. None of the defendants was convicted by the jury of any charge relating specifically to the Riddick murder.
II. Sentencing Proceedings
Following the defendants’ trial, the United States Probation Office submitted presentence investigation reports (“PSRs”) to the District Court. For each of the defendants, the PSRs determined an appli*235cable base offense level of 18, see U.S.S.G. § 2B3.2(a),3 and further recommended that the District Court consider the Rid-dick murder as relevant co-conspirator conduct in sentencing the defendants, see id. § 2B3.2(c)(l). Factoring in the Rid-dick murder, each defendant was assigned a new base offense level of 43. See id. § 2A1.1.4
Johnson’s PSR suggested an additional, four-level, enhancement pursuant to U.S.S.G. § 3Bl.l(a) because Johnson was an organizer and leader within the conspiracy.5 Thus, Johnson’s PSR set his post-enhancement offense level at 47. Hunter and Carnes were left at a base offense level 43.
During sentencing proceedings, the defendants argued that they should not be held accountable for the Riddick murder. The District Court reviewed extensive submissions from all parties and heard oral arguments, disagreed with the defendants, and sentenced each of them according to the PSR-suggested base offense levels that incorporated the Riddick murder. Johnson was sentenced to the maximum term of 240 months’ incarceration,6 and Hunter and Carnes were both sentenced to 204 months’ incarceration.
III. First Appeal
Defendants appealed to this Court, challenging their convictions and sentences on a variety of grounds — including that the District Court erred when it found the Riddick murder “relevant conduct” subject to consideration at sentencing.7 We affirmed their convictions, but vacated their sentences and remanded to the District Court for resentencing consistent with our opinion. See Mulder, 273 F.3d at 120.
We vacated all three defendants’ sentences on the grounds that the District Court, when it took into account the Rid-dick murder, had not satisfied our Circuit’s standard for holding a defendant liable at the sentencing phase for the relevant conduct of a co-conspirator. We explained *236that, in order to consider the Riddick murder as relevant conduct, the District Court was required to make two particularized findings under the so-called Studley prongs: (1) that “the scope of the activity to which the defendant agreed was sufficiently broad to include the [relevant, co-conspirator] conduct in question,” Mulder, 273 F.3d at 118 (citing United States v. Studley, 47 F.3d 569, 574 (2d Cir.1995)); and (2) that the relevant conduct on the part of the co-conspirator was foreseeable to the defendant, id. (citing Studley, 47 F.3d at 574).
We could find “no evidence, direct or circumstantial,” that Hunter was a party to an agreement that included the Riddick murder. Id. at 119. We also held that, “[a]s a matter of law,” there was insufficient evidence to indicate that Carnes was a party to any agreement that included the Riddick murder. Id. Because the evidence presented to the District Court was insufficient as a matter of law to establish that either Hunter or Carnes made any agreement encompassing the Riddick murder, we vacated their sentences and remanded to the District Court to resentence both without considering Riddick’s murder.
We also vacated Johnson’s sentence, but because the evidence in the record indicated that Johnson maintained a closer, and supervisory, relationship with Mulder, we permitted the District Court the opportunity to make particularized findings to satisfy both Studley prongs. Id. We held that, although the District Court had not satisfied either Studley prong during Johnson’s initial sentencing proceeding, it might be possible to do so on remand on the basis of further, and more particularized, findings.8 We thus remanded the cause to the District Court with instructions to determine whether particularized findings could satisfy both Studley prongs, so that the Riddick murder properly could be taken into account when resentencing Johnson.
IV. Resentencing
A. Imprisonment
On remand, the District Court resen-tenced each of the defendants and ordered restitution.
As instructed, the District Court resen-tenced Carnes without considering the Riddick murder, and imposed a sentence of 57 months’ incarceration. [JA 109]
The District Court similarly resentenced Hunter without considering the Riddick murder, pursuant to our instructions. However, the District Court applied two new enhancements that are now challenged. The District Court applied a five-level enhancement pursuant to U.S.S.G. § 2B3.2(b)(3)(A)(iii)9 for brandishing or *237possessing a firearm and a two-level enhancement pursuant to U.S.S.G. §§ 2333.2(b)(2)10 and 2B3.1(b)(7)(C)11 for causing financial loss to victims of more than $50,000 but not more than $250,000. See August 2, 2002 Transcript, at 28-31. Based upon these enhancements (and a third, not relevant to this appeal) the District Court resentenced Hunter to 87 months’ incarceration. See id. at 32.
The District Court resentenced Johnson to the same sentence as it had initially imposed, 240 months’ incarceration, on the basis of its finding, by a preponderance of the evidence, that (1) Johnson’s agreement with Mulder was sufficiently broad in scope to include the Riddick murder, and (2) Mulder’s murder of Riddick was foreseeable to Johnson. See August 8, 2002 Transcript, at 28-30.
B. Restitution
In addition to determining refined periods of incarceration during resentencing proceedings, the District Court ordered, for the first time, that defendants pay restitution to the victims of their extortion conspiracy.
Following the decision of this Court on the first appeal, the Probation Office revised each defendant’s PSR. Johnson’s and Hunter’s revised PSRs advised that the MVRA, 18 U.S.C. §§ 3663A, 3664, and U.S.S.G. § 5E1.1, directs the District Court to order Johnson and Hunter to pay restitution to their victims. Carnes’s revised PSR similarly noted that the MVRA requires that Carnes pay restitution to his victim, but added that because DeFoe (Carnes’s victim) declined restitution, none was due.
Accepting the PSR recommendations with respect to Johnson and Hunter, and rejecting the PSR recommendation with respect to Carnes, the District Court ordered each defendant to pay restitution. The District Court rejected the Probation Office’s recommendation, and defense counsel’s argument, that restitution was not due in Carnes’s case because the victim had declined restitution. The District Court cited 18 U.S.C. § 3663A and held that, because Carnes was convicted of a crime of violence, the restitution order was mandatory. See July 10, 2002 Transcript, at 14.
The District Court ordered Johnson to pay Tully $216,000, Hunter to pay Felix $144,000, and Carnes to pay DeFoe $117,100.
Defendants now appeal to this Court a second time, on a variety of grounds, challenging the new sentences and restitution orders of the District Court.
DISCUSSION
I. Challenge to Johnson’s Enhancement
Johnson argues that the District Court erred by considering the Riddick murder as relevant co-conspirator conduct when resentencing him, because the Court failed to satisfy both prongs of the Studley test. We agree, and upon further consideration, including an examination of the record and a review of the District Court’s findings on remand, we hold that Johnson may not be sentenced taking into account the Riddick murder.
*238It is well established that a district court may consider the relevant conduct of co-conspirators when sentencing a defendant. See, e.g., Mulder, 273 F.3d at 118 (citing United States v. Molina, 106 F.3d 1118, 1121 (2d Cir.1997) (“[R]easonably foreseeable acts ... of others in furtherance of [a] conspiracy may be taken into account to determine a defendant’s sentence.” (internal quotation marks omitted))). It is equally well established that this Court requires a district court to make two particularized findings before a defendant may be held accountable for his co-conspirators’ acts. “[I]in order to hold a defendant accountable for the acts of others, a district court must make two findings: 1) that the acts were within the scope of the defendant’s agreement and 2) that they were foreseeable to the defendant.” Studley, 47 F.3d at 574; accord Mulder, 273 F.3d at 118.
A. First Studley Prong: Scope of Agreement
To satisfy the first Studley prong-under our original mandate, the District Court could not find that Johnson’s agreement with Mulder included the Riddick murder just because Johnson had “knowledge of [Mulder’s] criminal acts,” Studley, 47 F.3d at 575—here, Mulder’s act of murdering Riddick. Nor could the District Court find that Johnson’s agreement included the Riddick murder on the basis that Johnson was “aware of the scope of the overall operation” and therefore should be held “accountable for the activities of the whole operation,” id. Rather, a finding by the District Court that Johnson’s agreement with Mulder included the Rid-dick murder would have to be based upon a particularized finding that the murder was within “the scope of the specific conduct and objectives embraced by the ... agreement,” id. (emphasis added).
The District Court attempted to make a particularized finding that Johnson’s agreement with Mulder included murdering Riddick. The Court found that (1) Mulder worked for Johnson; (2) Johnson often instructed Mulder to page him with updates; (3) Johnson had instructed Mulder to ensure protection of BFB interests “in any way necessary”; (4) Mulder informed Johnson that there was trouble at the Queens work site (at which Mulder subsequently killed Riddick) over a period of two years; (5) Mulder told Johnson, on an occasion unrelated to the Riddick murder, that he was at a work site with his “machine gun” and that if he killed someone, it would be to solve someone else’s problem; and, most notably, (6) there was talk of an offer of payment, from Johnson to Mulder, for the Riddick murder. See August 8, 2002 Transcript, at 27-30.
Among this list of particulars, the most salient is the last: that Johnson offered a fee to Mulder in exchange for Mulder murdering Riddick. Judge Casey referred to this “offer of payment” and “fee for the murder” during the resentencing proceeding, noted that he had presided over the trial in which Mulder testified to the fee, and observed that he “found Mulder’s testimony on this issue convincing, compelling and credible. Thus, [Judge Casey found] that the scope of the agreement ... is broad enough to include” the Riddick murder. See August 8, 2002 Transcript, at 29-30.
We review the District Court’s factual findings relevant to a sentencing determination for clear error. See 18 U.S.C. § 3742(e); see also Mulder, 273 F.3d at 116. In order to vacate such findings, we must view the evidence in the light most favorable to the Government and nevertheless find to be impermissible the factual determinations based upon that favorably-viewed evidence. See United *239States v. Ruggiero, 100 F.3d 284, 291-92 (2d Cir.1996)
We are mindful that we owe the District Court strong deference when reviewing findings based on credibility determinations. See, e.g., United States v. Monzon, 359 F.3d 110, 119 (2d Cir.2004). However, Mulder’s testimony, even if credited, cannot establish that Johnson ever offered to pay a fee for the Riddick murder. The testimony on which the District Court based its findings, as quoted and recounted by the District Court, reveals only that, well after Mulder had murdered Riddick, a third party — not Johnson — had recommended to him that he approach Johnson to ask for a fee. See August 8, 2002 Transcript, at 29-30. Crediting this testimony cannot support a finding that Johnson ever offered anyone a fee directly or that he indirectly orchestrated the third party’s approach to Mulder. Indeed, there is not even a suggestion in the record that Johnson was aware of the third-party recommendation, let alone designed, condoned, or failed to condemn it. The District Court, in relying on Mulder’s testimony, clearly erred in finding that Johnson was part of an agreement to murder Riddick.
As noted, the District Court made other particularized findings in support of its ultimate determination that the scope of Johnson’s agreement with Mulder included the Riddick murder. We hold that, in the absence of support in the record for a finding that Johnson offered to pay Mulder for the murder, those other findings are not sufficient to satisfy the first Studley prong. For example, that Mulder informed Johnson of two years of trouble at the Queens work site (where Riddick was eventually murdered), or that Mulder made hypothetical comments explaining what his reasons would be if he had to kill someone, does not permit a determination that the murder of Riddick was within “the scope of the specific conduct and objectives embraced by the ... agreement” between Johnson and Mulder, Studley, 47 F.3d at 575.
B. Second Studley Prong: Foreseeability
We instructed the District Court on remand that, in order to find that the Riddick murder was foreseeable to Johnson, there had to be some “closer link” than that Johnson was aware that, in the past, various BFB members had used violence as means to further BFB ends. Mulder, 273 F.3d at 119. The District Court responded by relying on (1) conversations between Mulder and Johnson in which Mulder reported having a “machine gun” and contemplated the possibility of killing someone; (2) Johnson’s awareness of Mulder’s propensity for violence; and, of course, (3) the “offer of payment” testimony discussed above.
As with the first Studley prong, a review of the transcript of the resentencing proceeding reveals that while the District Court listed various facts to support its finding of foreseeability, Johnson’s purported offer of payment was the most salient among them. Because we have determined that the finding of an offer of payment is not supported by the record, and because we conclude from our review of the record that, absent sufficient evidence of an offer of payment, the District Court had no sound basis for its determination that the Riddick murder was foreseeable to Johnson, we hold that the District Court erred when it found that the second Studley prong had been satisfied.
In light of our conclusion that the record cannot support the satisfaction of either Studley prong, we hold that the District Court erred in resentencing Johnson. Accordingly, we vacate Johnson’s sentence *240and remand the cause to the District Court with directions to resentence Johnson, a third time, without considering the Riddick murder. On remand, the District Court may consider whether to apply other appropriate enhancements to Johnson’s sentence.
II. Challenges to Hunter’s Sentence
Hunter challenges his 87-month sentence on the grounds that the District Court (1) was precluded under the “mandate rule” from imposing enhancements to his sentence that were not imposed at the initial sentencing or on the first appeal; but, if not so precluded, the Court, (2) improperly imposed (a) a five-level enhancement for brandishment or possession of a firearm, and (b) a two-level enhancement for the loss to the victims resulting from his offense.
A. Mandate Rule
Hunter principally argues that the District Court exceeded the scope of our mandate when it applied, on remand, enhancements not imposed at the initial sentencing proceeding. Hunter’s argument relies on our general “mandate rule” under the doctrine of the law of the case: that “resentencing should be limited when the Court of Appeals upholds the underlying-convictions but determines that a sentence has been erroneously imposed and remands to correct that error.” United States v. Quintieri, 306 F.3d 1217, 1228 (2d Cir.2002). Hunter argues that the District Court overstepped the bounds of this general rule when it added the firearm and loss-to-the-victim enhancements at the re-sentencing hearing.
The general mandate rule can be avoided by specific instructions of this Court — for example, where we explicitly instruct the district court to resentence de novo. See, e.g., id. at 1227 (noting that the court should look to the specific dictates of the remand order). Here, there were no such instructions. See Mulder, 273 F.3d at 119 (noting only that “we vacate Hunter’s sentence and remand for resentencing at an offense level that does not take into account Riddick’s murder”).
Even without a specific instruction to resentence de novo, however, “there may be circumstances when we reverse a sentence in which[, contrary to the mandate rule,] the ‘spirit of the mandate’ requires de novo sentencing, for example when the reversal effectively undoes the entire ‘knot of calculation.’ ” Quintieri, 306 F.3d at 1228.
The Government argues that our remand on the first appeal effectively undid the Quintieri “knot of calculation” that determined Hunter’s sentence. See Government’s Br. at 74. Because the District Court chose to apply the Riddick murder enhancement at Hunter’s initial sentencing' — -thereby guaranteeing that Hunter’s Guidelines sentence was at the statutory maximum — the Government now claims that, as of that first proceeding, it had “no incentive to raise or address” other enhancements. Id. at 70; see also Quintieri, 306 F.3d at 1229 (“An issue is not considered waived, however, if a party did not, at the time of the purported waiver, have both an opportunity and an incentive to raise it ....”); id. (“[I]f a sentencing determination had no practical effect on a defendant’s sentence at the original sentencing ..., a defendant is free to challenge that sentencing determination on remand .... ”). By arguing that it had no incentive to argue for other enhancements at Hunter’s initial sentencing, the Government effectively invokes an exception to the mandate rule explained in United States v. Whren, 111 F.3d 956 (D.C.Cir.1997) — that “the district court may consider only such new arguments or new facts *241as are made newly relevant by the court of appeals’ decision,” id. at 960—which we cited with approval in Quintieri. See Quintieri, 306 F.3d at 1228 n. 6.
In support of its claim that it had no incentive or opportunity to address other enhancements at Hunter’s initial sentencing, or, alternatively, that the other enhancements were entirely irrelevant at the initial sentencing, the Government argues that the PSR’s recommendation that the District Court apply the Riddick murder enhancement rendered irrelevant any discussion of other enhancements. See Government’s Br. at 74-75. We disagree. A district court is under no obligation to accept a recommendation made in a PSR. See, e.g., United States v. Young, 140 F.3d 453, 457 (2d Cir.1998) (“[A] sentencing judge is not obliged to accept all unchallenged findings and recommendations made in the PSR.”). The Government cannot enjoy immunity from waiver solely on the basis of a recommendation in a PSR.
However, the Government’s broader claim—that our remand to correct a sentencing error undid a “knot of calculation,” and hence merited de novo resen-tencing—is well supported by the record of the initial sentencing proceeding. The Government submitted a letter to the District Court two days before sentencing. See August 26, 1999 Transcript, at 69. In this letter, the Government addressed the possibility of several other enhancements that might apply should the Riddick murder enhancement not apply. See id. at 86. At the very least, the Government’s letter, and the subsequent discussion of the contents of that letter by the Court and Hunter’s counsel,12 support the view that the Riddick murder enhancement was part of the knot of calculation of the Court’s initial sentence. While this part of the record confirms that the Government did have an incentive to address other enhancements— and did so, to a limited extent—the issue of other enhancements was put aside, owing to the murder enhancement, and became newly relevant only upon remand. Accordingly, we find that the District Court was permitted to resentence Hunter de novo on remand, and did not exceed the scope of our mandate in doing so.
B. The Enhancements
■ Hunter argues in the alternative that, even if we hold—as we do—that the District Court did not exceed its mandate from this Court on remand, the District Court nevertheless clearly erred13 by imposing two new enhancements on remand.
1. The Firearm Enhancement
Hunter challenges the District Court’s application of a five-level enhancement pursuant to U.S.S.G. § 2B3.2(b)(3)(A)(iii) for brandishment or possession of a firearm. He asserts that the evidence in the record established, at most, that he brandished or possessed only a dangerous weapon (as opposed to a firearm), thereby meriting only a three-level *242enhancement pursuant to U.S.S.G. § 2B3.2(b)(3)(A)(v).
The District Court chose to credit the testimony of Frank Fallon, who testified that he saw the tip, or approximately two to three inches, of the handle of a gun protruding from the pocket of Hunter’s trousers. Hunter responds to this testimony by drawing to our attention a firearm expert’s testimony that the handle seen by Fallon could have been the handle to some dangerous weapon other than a firearm.
As previously noted, we owe a district court particularly strong deference when reviewing fact-findings based upon credibility determinations. See Monzon, 359 F.3d at 119. Furthermore, we have held that, upon reviewing findings subject to the even more stringent beyond-a-reasonable-doubt standard, “eyewitness testimony may be sufficient for the government to meet its burden of proof [that a defendant had a firearm] so long as it provides a rational basis for the [fact-finder] to find that the object observed by eyewitnesses was, in fact, a firearm.” United States v. Jones, 16 F.3d 487, 490 (2d Cir.1994).
Because eyewitness testimony may provide a basis for a finding that a firearm was brandished or possessed and because we defer to the District Court’s judgment that Fallon’s testimony was “convincing and credible,” we find no clear error in the District Court’s decision to enhance Hunter’s sentence five levels pursuant to U.S.S.G. § 2B3.2(b)(3)(A)(iii).
2. The Loss-to-the-Victim Enhancement
Hunter also contests the District Court’s imposition of a two-level enhancement pursuant to U.S.S.G. §§ 2B3.2(b)(2) and 2B3.1(b)(7)(C) for having caused a loss to his victim between $50,000 and $250,000.
The District Court agreed with the Probation Office’s estimate that the loss that Hunter caused Felix was approximately $144,000. The Probation Office calculated this amount by multiplying $1000, an estimated amount that Hunter received on a weekly basis from Felix, by the estimated duration of Hunter’s extortion, 144 weeks. Hunter argues that there is an insufficient evidentiary basis for this calculation.
The Government concedes that the evidence at sentencing was insufficient to justify this $144,000 calculation, but argues nonetheless that any error was harmless. The Government notes that documents produced by Hunter establish that Felix paid Hunter’s company between $1,600 and $1,700 per month from approximately 1994 to January of 1998 — enough to meet the $50,000 threshold required for a two-level enhancement. Thus, in the Government’s view, the miscalculation adopted by the District Court was harmless error. See Government’s Br. at 85; see also United States v. Saunders, 129 F.3d 925, 932 (7th Cir.1997) (holding that a loss-to-the-victim miscalculation was harmless error because it did not affect the Guideline determination).
The record does indicate some monthly $1,600 to $1,700 payments, but in an imprecise fashion. There is nothing in the record to suggest the regularity or consistency of the monthly payments, or then-precise duration.14 Absent some evidence on that point, we are unable to determine *243whether the District Court’s error in accepting the $144,000 calculation was harmless. Accordingly, we direct the District Court, upon remand, to reconsider the question of the total loss caused by Hunter to Felix before imposing an enhancement on that basis.
III. Challenges to the District Court’s Restitution Orders 15
A. Mandate Rule
All three defendants argue that the District Court exceeded the scope of the mandate from this Court by ordering, for the first time at the resentencing following remand, the payment of restitution. They note that, when we remanded, we did not direct the District Court to order restitution, and they draw our attention to the general mandate rule that “absent explicit language in the mandate to the contrary, resentencing should be limited when the Court of Appeals upholds the underlying convictions but determines that a sentence has been erroneously imposed and remands to correct that error.” Quintieri, 306 F.3d at 1228.
The Government does not dispute that the District Court did not order, nor did the Government request, restitution during the defendants’ initial sentence proceedings. Nevertheless, the Government asserts that the District Court was free to consider restitution orders for the first time at resentencing because of an applicable exception to the general mandate rule.
The Government correctly observes that a district court may resentence beyond the scope of the mandate, or may resentence a defendant on a limited remand using fresh considerations, if it has “cogent” or “compelling” reasons, including “the need to correct a clear error.” Id. at 1230 (internal quotation marks omitted).16 The Government argues that the District Court was permitted to order restitution during resentencing because the failure to do so previously was clear error. We agree.
The MVRA provides for mandatory restitution in criminal cases where an offense is “a crime of violence.” 18 U.S.C. § 3663A(a)(l),17 (c)(l)(A)(i).18 The extor*244tion offense of which each of appellants was convicted is a “crime of violence.” See id. § 16(b);19 see also United States v. Friedman, 300 F.3d 111, 128 (2d Cir.2002). It is therefore indisputable that the MVRA required the District Court at the initial sentencing proceedings to enter orders of restitution with respect to each defendant. Defendants do not contest this proposition; they argue only that, having failed to impose such orders at the initial sentencing proceedings, the narrow scope of the District Court’s mandate on remand for re-sentencing prohibited it from entering orders of restitution at resentencing. See Johnson’s Br. at 38-39; Hunter’s Br. at 14-18; Carnes’s Br. at 11-12.
Because the mandate rule does not forbid a district court to resentence beyond the scope of a limited remand for a cogent, compelling reason, such as correcting a clear error, and because it was error for the District Court to have failed during the initial sentencing proceedings to order mandatory restitution as required by the MVRA, we hold that the District Court did not err by ordering restitution during re-sentencing proceedings.
B. Effect of a Victim Declining Restitution
Carnes argues that the District Court erred by ordering him to pay restitution to Defoe, because Defoe declined restitution. In a related argument, Hunter argues that the District Court erred when it directed that, should Felix refuse to accept restitution payments, the funds should be paid to “the government’s Crime Victims Fund.” See August 2, 2002 Transcript, at 33. These questions of statutory interpretation are issues of first impression in our Circuit.
Contrary to Carnes’s argument, a district court may — indeed, must — impose orders of restitution on defendants convicted of crimes identified in the MVRA even if their victims decline restitution. To hold otherwise would be inconsistent with the MVRA’s statutory scheme of mandatory restitution, and it would undermine the power of the criminal justice system to punish defendants, where appropriate, through orders of restitution. Cf. United States v. Brown, 744 F.2d 905, 909 (2d Cir.1984) (“Restitution undoubtedly serves traditional purposes of punishment.”).
The central provision of the MVRA requires that, “[notwithstanding any other provision of law, ... the court shall order ... that the defendant make restitution to the victim of the offense.” 18 U.S.C. § 3663A(a)(l)20 (emphasis added). For defendants convicted of a “crime of violence,” such as Carnes and Hunter, the MVRA provides for no exception to this mandatory provision, see id. § 3663A, though it provides some exceptions for other categories of offenses, see id. § 3663A(c)(3), (c)(l)(A)(ii). The language of § 3663A expressly prohibits interpretation of “any other provision of law” to create an exception to the obligations imposed on sentencing courts. See id. § 3663A(a)(l) (“Notwithstanding any other provision of law ....”).
Hunter and Carnes draw our attention to two provisions from the following sec*245tion, 18 U.S.C. § 3664(g)(1) and (g)(2), which, they argue, indicate that when an identified victim declines restitution, the District Court is not only relieved of its obligation to order restitution under § 3663A, but, indeed, is powerless to impose an order of restitution, unless the victim himself chooses to assign his interest in restitution to the Crime Victims Fund.
It is true that the MVRA provides that orders of restitution entered under § 3663A should be administered according to procedures laid out in § 3664. See id. § 3663A(d) (“An order of restitution under [§ 3663A] shall be issued and enforced in accordance with section 3664.”). However, neither of the provisions cited in § 3664 by defendants can reasonably be interpreted as carving out an exception to the mandatory nature of § 3663A. See id. § 3664(g)(1) (“No victim shall be required to participate in any phase of a restitution order”) & (g)(2) (“A victim may at any time assign the victim’s interest in restitution payments to the Crime Victims Fund in the Treasury without in any way impairing the obligation of the defendant to make such payments.”). Indeed, these subsections state only that a victim need not accept restitution and that a victim may assign his interest in restitution to the Crime Victims Fund. Without some evidence in the text or history of the legislation to supersede the plain language of § 3663A that restitution is mandatory, the defendants’ arguments must be rejected.
Our conclusion that § 3663A requires restitution regardless of the consent of victims is consistent with our long-held understanding of the purposes of restitution, which include not only the compensation of victims, but also the punishment of offenders. See Brown, 744 F.2d at 909. In view of these two purposes of restitution and the explicit language of the MVRA rejecting any exceptions to the requirement of restitution drawn from “any other provision of law,” 18 U.S.C. § 3663A(a)(l), we conclude that defendants’ victims may not veto the obligation of the District Court to impose orders of restitution.
Hunter also urges us to vacate the District Court’s restitution award on the basis that it was error for the District Court to “assign” Felix’s interest in restitution, should Felix renounce it, to the Crime Victims Fund. Hunter argues that § 3664(g)(2) provides the statutory authority to assign interest in restitution only to victims, and that the Court therefore had no authority to do so. We disagree. Although § 3664(g)(2) authorizes victims to make such an assignment, it does not preclude the Court from doing so. See id. § 3664(g)(2) (“A victim may at any time assign the victim’s interest in restitution payments to the Crime Victims Fund in the Treasury without in any way impairing the obligation of the defendant to make such payments.”).21
The District Court did not err in directing that, if the victim renounced its inter*246est in Hunter’s restitution payments, those funds would be assigned to the Crime Victims Fund.
CONCLUSION
To summarize our holdings:
(1) The District Court erred in considering the Riddick murder when sentencing Johnson pursuant to U.S.S.G. §§ 2B3.2(c)(l) and 2A1.1.
(2) The District Court did not exceed the scope of its mandate on remand either by imposing new sentencing enhancements on Hunter or by ordering restitution, for the first time, when resentencing the defendants.
(3) The District Court did not err in finding that Hunter had brandished or possessed a firearm for the purposes of imposing a five-level enhancement under U.S.S.G. § 2B3.2(b)(3)(A)(iii).
(4) The District Court erred by accepting the imprecise and insufficiently supported conclusion that Hunter’s loss caused to his victim was in excess of $50,000.
(5) The MVRA does not preclude a court from imposing an order of restitution on a defendant whose victim has declined restitution. Accordingly, the District Court did not err in ordering that Carnes pay restitution.
(6) The MVRA does not preclude a court from directing that restitution funds be paid to the Crime Victims Fund if declined by a defendant’s victim. Accordingly, the District Court did not err by providing that the proceeds from Hunter’s order of restitution under the MVRA be directed to the Crime Victims Fund if Hunter’s victim declined restitution.
(7) Upon a review of the record, and by stipulation of the Government, see note 12 ante, we conclude that the record supports neither the amount of restitution ordered from Hunter nor the amount and apportionment among victims ordered from Johnson.
Accordingly, we vacate the sentences (including restitution orders) with respect to Johnson and Hunter.
We remand the cause for (1) resentenc-ing of Johnson without consideration of the Riddick murder; (2) reconsideration of Hunter’s loss-to-the-victim sentencing enhancement following a more particularized determination of the amount of loss caused by Hunter; and (3) entry of new orders of restitution for Hunter and Johnson based on new determinations, consistent with this opinion, of the amounts of loss caused by each defendant to his victims.
Having considered defendants’ remaining arguments on appeal and concluded that they are without merit, we affirm the judgment of the District Court in all respects other than those specifically noted above.
. The factual background of this case is thoroughly recounted in our earlier opinion in United States v. Mulder, 273 F.3d 91 (2d Cir.2001). Accordingly, we restrict our recitation of the facts to those relevant to the issues now on appeal, and consider those facts in the light most favorable to the Government, see Jackson v. Virginia, 443 U.S. 307, 318-19, 99 S.Ct. 2781, 61 L.Ed.2d 560 (1979); United States v. Dhinsa, 243 F.3d 635, 648-49 (2d Cir.2001).
. A no-show is “a person who draws a salary for a job that requires no actual work or that does not truly exist.” J.E. Lighter, Random House Historical Dictionary of American Slang, Vol. 2, at 682 (1997).
. U.S.S.G. § 2B3.2 covers "Extortion by Force or Threat of Injury or Serious Damage,” and subsection (a) lists 18 as the appropriate base offense level.
. U.S.S.G. § 2B3.2(c)(l) states that § 2A1.1 is to be applied to a defendant's sentence determination "[i]f a victim was killed under circumstances that would constitute murder under 18 U.S.C. § 1111.” The PSRs stated that the Riddick murder fell under 18 U.S.C. § 1111, and therefore applied U.S.S.G. § 2A 1.1. That section dictates that the base offense level for first degree murder is 43.
. U.S.S.G. § 3Bl.l(a) provides that "[i]f the defendant was an organizer or leader of a criminal activity that involved five or more participants or was otherwise extensive, increase by 4 levels.”
. Although the Sentencing Guidelines call for a life sentence for any person with an offense level of 43 or above, see U.S.S.G. Ch. 5, Pt. A, Sentencing Table, the statutory maximum sentence for a Hobbs Act violation is twenty years, see 18 U.S.C. § 1951(a); see also Mulder, 273 F.3d at 115. Because the Sentencing Guidelines stipulate that "[w]here the statutorily authorized maximum sentence is less than the minimum of the applicable guideline range, the statutorily authorized maximum sentence shall be the guideline sentence,” U.S.S.G. § 5Gl.l(a), the defendants' maximum sentences could only have been incarceration for twenty years (or 240 months).
.U.S.S.G. § 1B1.3(a)(1)(B) describes relevant conduct:
[I]n the case of a jointly undertaken criminal activity (a criminal plan, scheme, endeavor, or enterprise undertaken by the defendant in concert with others, whether or not charged as a conspiracy), all reasonably foreseeable acts and omissions of others in furtherance of the jointly undertaken criminal activity, that occurred during the commission of the offense of conviction, in preparation for that offense, or in the course of attempting to avoid detection or responsibility for that offense.
. As to the first Studley prong — the breadth of the scope of Johnson's agreement with Mulder — we explained that “the district court reasonably could find that [Johnson and Mulder] jointly planned methods to take control of the Queens site [at which Mulder killed Riddick], albeit not specifically the murder of Rid-dick. ... [A] reasonable fact finder could find, but would not be required to find, that ... Johnson entered into a joint agreement with Mulder to maintain control of the Queens work site.” Mulder, 273 F.3d at 119.
As to the second Studley prong — whether Mulder's murder of Riddick was foreseeable to Johnson — we noted that the basis for the District Court's factual determinations during the initial sentencing proceeding was insufficient to satisfy the standard in our Circuit. That Johnson was aware of "the use of violence by coalition members in other circumstances ... was inadequate. Our cases contemplate a closer link.” Id. We therefore directed the District Court to determine whether such a closer link existed. Id. at 120.
. U.S.S.G. § 2B3.2(b)(3)(A)(iii) states that “if a firearm was brandished, displayed, or possessed, increase by 5 levels .... ”
. U.S.S.G. § 2B3.2(b)(2) states that “[i]f the greater of the amount demanded or the loss to the victim exceeded $10,000, increase by the corresponding number of levels from the table in § 2B3.1(b)(7).”
. U.S.S.G. § 2B3.1(b)(7) states, in pertinent part, that "[i]f the loss exceeded $10,000, increase the offense level as follows: ... (C) [m]ore than $50,000 ... [a]dd 2 ... (D) [m]ore than $250,000 ... [a]dd 3 ....”
. At sentencing, Hunter's counsel called attention to the several other enhancements raised by the Government in its letter to the Court, arguing that, if the Court declined to apply the murder enhancement, it would have to disregard the other enhancements that addressed essentially the same conduct as the Riddick murder. August 26, 1999 Transcript, at 87. Hunter’s counsel also discussed particularly and at some length the firearm enhancement, putting forth many of the same arguments that are before us today, including casting doubt upon Frank Fallon's testimony and questioning whether possession of a ''firearm” can be discerned from just the tip of the weapon showing from Hunter’s trouser pocket. See id. at 87-90.
. We review the District Court's enhancement for clear error. See 18 U.S.C. § 3742.
. Hunter’s third, revised, PSR states that Hunter’s company performed "consulting services” for three different companies. It also notes that these services spanned 1994 to January 1998. The record does not indicate if Felix, in particular, was making payments to Hunter’s company for the entirety of this period, or how regularly. It is possible, on our reading of the PSR, that (for example) Hunter had intermittent relationships with *243each of these three companies and irregularly received payments. There is nothing to suggest that Felix, in particular, made payments to Hunter every month during the entire period.
. We do not address the restitution issues on which the Government has consented to a remand, which include the value of restitution owed by Hunter to Felix and by Johnson to Tully and Piccone. See Government’s Br. at 99-100; Conclusion, post.
. In Quintieri, we noted the "need to correct a clear error” as a sufficient reason for an appellate court to "depart from the law of the case and reconsider the issue,” 306 F.3d at 1230 (citing United States v. Tenzer, 213 F.3d 34, 39 (2d Cir.2000)), and we also held that "district courts may similarly depart from the law of the case and reconsider their own decisions for cogent and compelling reasons if those decisions have not been ruled on by the appellate court," id. (citing DiLaura v. Power Auth., 982 F.2d 73, 77 (2d Cir.1992) (emphasis added)).
. 18 U.S.C. § 3663A(a)(l), entitled "Manda-, tory restitution to victims of certain crimes,” provides in relevant part:
Notwithstanding any other provision of law, when sentencing a defendant convicted of an offense described in subsection (c), the court shall order, in addition to, or in the case of a misdemeanor, in addition or in lieu of, any other penalty authorized by law, that the defendant make restitution to the victim of the offense or, if the victim is deceased, to the victim’s estate.
. 18 U.S.C. § 3663A(c)(l) provides, in part:
This section shall apply in all sentencing proceedings for convictions of, or plea agreements relating to charges for, any offense ... (A) that is ... (i) a crime of violence, as defined in section 16 ....
. 18 U.S.C. § 16, entitled “Crime of violence defined,” provides:
The term of “crime of violence” means—
(a) an offense that has as an element the use, attempted use, or threatened use of physical force against the person or property of another, or
(b) any other offense that is a felony and that, by its nature, involves a substantial risk that physical force against the person or property of another may be used in the course of committing the offense.
. For the full text of 18 U.S.C. § 3663A(a)(l), see note 17, ante.
. The Senate considered a version of § 3664(g)(1) that read: "No victim shall be required to participate in any phase of a restitution order. If a victim declines to receive restitution made mandatory by this title, the . court shall order that the victim's share of any restitution owed be deposited in the Crime Victims Fund in the Treasury ...." S.Rep. No. 104-179, at 6 (Dec. 6, 1995), reprinted in 1996 U.S.C.C.A.N. 924, 925 (emphasis added). However, by the time of the statute’s enactment, this italicized text had been removed. See Pub.L. 104-132, § 206(a), 110 Stat. 1214, 1234 (1996). By removing this text before enacting § 3664(g)(1), Congress may have suggested that assignment to the Crime Victims Fund is not mandatory. It certainly did not suggest that assignment to that fund is prohibited.