
714 F.Supp. 899 (1989)
WATER TECHNOLOGIES CORPORATION, Water Pollution Control Systems, Inc. and Kansas State University Research Foundation, Plaintiffs,
v.
CALCO LTD. and William J. Gartner, Defendants.
No. 82 C 4330.
United States District Court, N.D. Illinois, E.D.
January 24, 1989.
*900 W.A. Van Santen, Wegner Stellman McCord Wiles & Wood, Chicago, Ill., Harness Dickey and Pierce, Birmingham, Mich., for plaintiffs.
Robert E. Wagner, Sidney Wallenstein, Ralph R. Rath, Wallenstein Wagner Hattis Strampel & Aubel, Chicago, Ill., for defendants.

MEMORANDUM OPINION AND ORDER
ASPEN, District Judge:
This patent case comes before the District Court for a second time. Former Chief Frank J. McGarr previously found the defendants, Calco, Ltd. ("Calco") and William J. Gartner, liable to the plaintiffs, Water Technologies Corp. ("WTC"), Water Pollution Control Systems, Inc. ("WPCS") and Kansas State University Research *901 Fund ("KSURF"), for patent infringement and unfair competition. 658 F.Supp. 961 (N.D.Ill.1986); 658 F.Supp. 980 (N.D.Ill. 1987). Judge McGarr awarded plaintiffs a total of $1,416,157.16 in damages, interest, attorneys' fees, and costs. On appeal, the United States Court of Appeals for the Federal Circuit upheld Judge McGarr's determination that there had been patent infringement, but held that there was no basis for an unfair competition claim under federal law. 850 F.2d 660, 671 (Fed.Cir. 1988), cert. denied, ___ U.S. ___, 109 S.Ct. 498, 102 L.Ed.2d 534 (1988). In addition, the court held that Judge McGarr had used an improper method to calculate damages for patent infringement. Accordingly, the Federal Circuit remanded the case to us for redetermination of damages, prejudgment interest, and attorneys' fees. (The court upheld the award of costs  $5,055.16  without elaboration.) For the reasons set forth below, we award $417,976 in damages and $115,141 in prejudgment interest, for a total award of $533,117. In addition, we direct the plaintiffs to provide a summary of their request for attorneys' fees.

I. Background[1]
The four patents at issue in this case involve "demand bactericide resins" used as disinfectants for purifying water. For twenty to thirty years before the inventions in this case, scientists had used materials such as halogens (iodine, bromine, or chlorine) to disinfect water. Of course, as swimmers know, the problem with this method was that it left relatively large amounts of the disinfectant in the water, detracting from the water's taste and risking harm to the eyes and mucous membranes.
In the late 1960's, two scientists at Kansas State University, Jack L. Lambert and Louis R. Fina, worked to eliminate this problem. Lambert and Fina were the first to develop a demand bactericide resin, which released iodine only "on demand," that is, only upon contact with bacteria or germs to be killed. We will not discuss the technical details, which we only dimly understand, but Lambert's and Fina's insight was to use a resin containing triiodide. The advantage of this process was that it disinfected the water without leaving a detectable concentration of iodine; the treated water is therefore ready for immediate use.
Lambert and Fina received two patents for their triiodide resin in 1974 and 1975, and they assigned the patents to KSURF. In 1973, while the patent applications were still pending, KSURF granted an exclusive license on the two Lambert-Fina patents to Aqua-Chem, Inc. ("Aqua-Chem"). This license will be referred to as the "KSURF-Aqua-Chem license." In 1980, an Aqua-Chem scientist, Gary L. Hatch, received two patents for improvements on the Lambert-Fina patents, and he assigned both patents to Aqua-Chem.
In September 1977, Aqua-Chem granted an exclusive license under the then-pending Hatch applications, and an exclusive sublicense under the Lambert-Fina patents to WPCS (the "Aqua-Chem-WPCS licenses") for purifier products using less than 100 cubic centimeters (cc) of the patented resins. At the time the Aqua-Chem-WPCS licenses were granted, WPCS was a wholly owned subsidiary of the Walbro Corporation, but later became a wholly-owned subsidiary of the plaintiff WTC. Using the Aqua-Chem licenses, WPCS developed, among other things, a water purifying cup.
In 1979, the defendant Gartner entered the picture. Gartner, a chemist, agreed to become a consultant for the Brunswick Corporation and to help Brunswick develop new products for sale in the water purification field. In connection with the consulting work, Gartner contacted Aqua-Chem, representing that Brunswick was interested in taking a license for products using more than 100 cc of the patented resins. Through the auspices of Aqua-Chem, Gartner met with personnel of WTC in August 1979. At this time, WTC gave Gartner technical literature on their water purifier *902 products, but did not give him any samples of the resin or any formulas or recipes for preparing the resins. But Gartner was tenacious. In the summer of 1980, he convinced officials at Aqua-Chem to tell him its preferred formula for the resin. At about the same time, Gartner applied for a patent for a straw-type purifier product using a triiodide resin. On September 24, 1980, Gartner gave the other defendant, Calco, a license to manufacture, use and sell the straw purifier product (the "Gartner-Calco license").
By September 1980, if not before, both Gartner and Calco were aware of the four patents involved in this case. Nonetheless, on November 4, 1980, Calco, with Gartner's assistance, manufactured a batch of the improved triiodide resin patented by Hatch. Calco placed the resin into the straw purifiers, which were marketed under the trademark POCKET PURIFIER. (This trademark was owned by Gartner and licensed to Calco.) Calco sold at least 400 to 500 POCKET PURIFIERS using resin from the November 4 batch. Each of the units contained less than 100 cc of the resin, therefore subverting the exclusive license that Aqua-Chem granted to WPCS.
In early 1981, Gartner developed a slightly modified resin by merely adding a small amount of potassium bromide to the Aqua-Chem formula. Gartner hoped that by adding the potassium bromide, it might be "possible to skirt both the KSU and the Aqua-Chem patents." See 658 F.Supp. at 968. Judge McGarr concluded that Gartner had no way of knowing whether the additional ingredient would have any effect on the amount of iodine left after purification or on the bactericidal abilities of the finished resin. See id. Nonetheless, Gartner applied for and subsequently received a patent for his resin formula. Sometime after April 28, 1981, Calco started using Gartner's resin in manufacturing POCKET PURIFIERS.
In approximately May 1981, Gartner told Aqua-Chem that the proposed licensing deal with Brunswick had fallen through. In the fall of 1981, when WTC was on the verge of entering a marketing agreement with an unidentified "large U.S. national corporation[]," see 658 F.Supp. at 967, it learned for the first time that the POCKET PURIFIER was on the market. As a result, WTC's marketing agreement collapsed. Shortly thereafter, WTC asked its licensor Aqua-Chem to challenge the accused infringing product, but Aqua-Chem refused. Accordingly, WTC withheld further royalties from Aqua-Chem, and in July 1982, WTC and its subsidiary WPCS brought this suit against Gartner, Calco, and Calco's president Robert Kalnitz. The suit also named Aqua-Chem as an involuntary plaintiff and cross-defendant.
In response to WTC's suit, Aqua-Chem terminated the Aqua-Chem-WPCS licenses in September 1982, and filed cross claims against WTC and WPCS for unpaid royalties under those licenses. In addition, Aqua-Chem filed a cross-claim against Gartner for misappropriation of alleged trade secrets.
WTC, WPCS and Aqua-Chem settled their claims in an agreement signed in May 1985. Pursuant to the agreement, Aqua-Chem assigned the two Hatch patents to KSURF, who was also a signatory to the agreement, and relinquished the KSURF-Aqua-Chem license on the two Lambert-Fina patents. Thus, at the time of trial, KSURF owned all four of the patents and all rights for past infringement, and therefore joined WTC and WPCS as a plaintiff.
In accordance with the agreement, Aqua-Chem was dismissed "with prejudice" from the lawsuit on the first day of trial. Judge McGarr also dismissed defendant Kalnitz. After the case was realigned in this way, the Court found that Calco and Gartner had infringed the four patents by making and using both the first batch of resin and the later batches with the negligible amount of potassium bromide thrown in; that Gartner had induced infringement by Calco and the public; that Calco and Gartner were jointly liable; that their infringement was willful; and that Calco and Gartner were liable for unfair competition. Judge McGarr awarded $420,000 for damages based on lost profits and $100,000 based on lost marketing opportunities, and *903 then doubled both of these figures because of the defendants' willful infringement, as permitted by 35 U.S.C. § 284 (1982). The Court also concluded that the plaintiffs merited attorneys' fees under 35 U.S.C. § 285 (1982) due to the exceptional nature of the case. However, the court reduced the plaintiffs' request for fees from approximately $245,000 to $150,000, in part because the plaintiffs' law firm had lost certain billing records and in part because of its "equitable instinct" about the case. See 658 F.Supp. at 985. Finally, the court awarded $221,102 in prejudgment interest and $5,055.16 in costs.
On appeal, the Federal Circuit affirmed Judge McGarr's finding that Calco had directly infringed the plaintiffs' patents, and held that Gartner had at least induced infringement by Calco and the public. The court did not consider whether Gartner had infringed the patents himself, but held that Gartner's inducement of infringement was enough to justify the holding of joint liability. However, the Court rejected Judge McGarr's holding that Calco and Gartner were liable for unfair competition; according to the Federal Circuit, federal law provided only a limited basis for relief from unfair competition, and any state law claim could only have been invoked by Aqua-Chem. Accordingly, the Court directed us "to vacate the judgment to the extent it upholds WTC/WPCS's unfair competition claim and includes an award of damages thereon." 850 F.2d at 671. In addition, the Federal Circuit held that damages for patent infringement should not have been based on lost profits, since the plaintiffs had failed to prove that they would have made sales "but for" Calco's sales. Rather, the correct measure of damages was a reasonable royalty, and the Federal Circuit therefore directed us to redetermine damages on that basis. The Court also held that attorneys' fees were proper in this case, but rejected Judge McGarr's "equitable instinct" and concluded that there were insufficient findings to determine if the $150,000 awarded was reasonable. The Court therefore vacated the award of attorneys' fees and remanded for redetermination. Finally, the Court vacated the award of prejudgment interest, since it was based on an erroneous award of damages, and affirmed the award of costs without comment.
After the Federal Circuit remanded the case, Gartner moved for the release of the supersedeas bond he had posted pending appeal. We granted this motion in an opinion dated September 20, 1988. 694 F.Supp. 1328 (N.D.Ill.1988). In addition, the plaintiffs petitioned the United States Supreme Court for a writ of certiorari to the Federal Circuit, but the Court denied the petition on November 28, 1988. ___ U.S. ___, 109 S.Ct. 498, 102 L.Ed.2d 534 (1988).

II. Redetermination of Damages

A. Damages Based on the Unfair Competition Claim

As noted previously, the Federal Circuit in remanding the case instructed us to "vacate the judgment to the extent that it uphelds WTC/WPCS's unfair competition claim and includes an award of damages thereon." 850 F.2d at 671. In their briefs on the release of the supersedeas bond, both sides agreed that this passage required us to vacate the $200,000 award for lost marketing opportunities. However in our opinion ordering the release of the bond, we concluded that the parties were incorrect. 694 F.Supp. at 1329 n. 1. We noted that Judge McGarr had found both patent infringement and unfair competition by the defendants, but did not apportion between those two claims; rather, he apportioned only between lost profits and lost marketing opportunity. Put another way, we concluded that the $840,000 award for lost profits and the $200,000 award for lost marketing opportunity did not correspond to the patent infringement and unfair competition claims. We note that the Federal Circuit apparently reached the same conclusion. If the Federal Circuit thought that the award for lost marketing opportunities compensated for the unfair compensation claim, it would have ordered us to vacate the award for lost marketing opportunities when it reversed the unfair competition claim. Instead, the Court directed us to vacate the judgment only "to the extent *904 that it upholds [the] unfair competition claim and includes an award of damages thereon."
Nonetheless, in their brief on the recalculation of damages, the plaintiffs again assert that the Federal Circuit "reversed outright the $200,000.00 lost marketing opportunities (cf. unfair competition) portion of Judge McGarr's award." Plaintiffs Brief at 2. The conclusion we reached in our earlier opinion has not changed and we believe, therefore, that the plaintiffs' assertion is still wrong. Had the plaintiffs paid heed to our earlier opinion, they might have attempted to show that some of the award for lost marketing opportunities actually compensated for Gartner's and Calco's patent infringement. But the plaintiffs chose not to do so and we therefore vacate the $200,000 award for lost marketing opportunities in its entirety.

B. Reasonable Royalty

The Federal Circuit has instructed us to determine damages based on a reasonable royalty as applied to Calco's sales of the infringing product. The parties do not contest what Calco's sales were, but they do contest what a reasonable royalty would be. A reasonable royalty has been defined as the royalty a willing licensor and a willing licensee would have agreed to at the time the infringement began. See TWM Manufacturing Co. v. Dura Corp., 789 F.2d 895, 899 (Fed.Cir.), cert. denied, 479 U.S. 852, 107 S.Ct. 183, 93 L.Ed.2d 117 (1986); Radio Steel & Mfg. Co. v. MTD Products, Inc., 788 F.2d 1554, 1557 (Fed. Cir.1986); Panduit Corp. v. Stahlin Bros. Fibre Works, Inc., 575 F.2d 1152, 1158 (6th Cir.1978). In practice, this definition provides little help; it essentially requires a court to reconstruct the past and determine what might have happened had there been no infringement. This is, of course, an impossible task, and even constructing a reasonable facsimile of what might have been is quite difficult. This difficulty is exacerbated if the patent owner has granted few or no licenses before the infringement began.[2] Moreover, what the willing licensor would have accepted probably will not match what the willing licensee would have paid. Rather, there will be a range, with the least the licensor would have been willing to accept on the low end, and the most the licensee would have paid on the high end. The parties would have ended up somewhere within this range, depending on the relative negotiating skills of each side and how many cards each side has laid on the table.[3]
Apparently recognizing the difficulties inherent in the task, the Federal Circuit has held that the determination of a reasonable royalty lies in the sound discretion of the trial court. Nickson Industries, Inc. v. Rol Manufacturing Co., 847 F.2d 795, 798 (Fed.Cir.1988). When the amount of damages is not ascertainable with precision, reasonable doubt is resolved against the infringer. Del Mar Avionics, Inc. v. Quinton Instrument Co., 836 F.2d 1320, 1327 (Fed.Cir.1987); Lam, Inc. v. Johns-Manville Corp., 718 F.2d 1056, 1065 (Fed.Cir.1983). This only makes sense, since it is the infringer who makes the reconstruction of history necessary.
The amount of a reasonable royalty turns on the facts of each case, Panduit Corp., 575 F.2d at 1159, and not surprisingly, *905 each side in this case emphasize different facts. The plaintiffs emphasize the 1977 Aqua-Chem-WPCS license, which required WPCS to pay Aqua-Chem a minimum yearly advance royalty of $40,000 for the year beginning May 11, 1980, and $50,000 for each thereafter. Plaintiffs' Exhibit (PTX) 32 at 9 (attached as Appendix to Plaintiffs' Brief). Plaintiffs also note that the 1973 KSURF-Aqua-Chem license required Aqua-Chem to pay KSURF an advance yearly royalty of $20,000. PTX 49 at 12. Aqua-Chem therefore increased the royalty by $20,000 to $30,000 in sublicensing to WPCS for the years in question, and the plaintiffs argue that WPCS would have demanded at least $10,000 a year more to sublicense Calco and Gartner. Accordingly, the plaintiffs ask that the royalty be calculated at $60,000 a year.
As an alternative, the plaintiffs ask for a reasonable royalty of thirty percent of Calco's sales. The plaintiffs contend that the case law permits a district court to award a reasonable royalty based on the licensor's or infringer's profit margin. See TWM Manufacturing Co. v. Dura Corp., 789 F.2d 895, 899 (Fed.Cir.1986). Since Judge McGarr found that Calco's profit margin was seventy-three percent, see 658 F.Supp. at 978,[4] the plaintiffs argue that thirty percent is a reasonable royalty.
By contrast, Calco and Gartner emphasize the 1980 licensing agreement they entered into, in which Calco agreed to pay Gartner a five percent royalty. PTX 63 ¶ 5 (attached as Appendix to Defendants' Response). Calco and Gartner rely in part on Tights, Inc. v. Kayser-Roth Corp., 442 F.Supp. 159 (M.D.N.C.1977) and argue that the Gartner-Calco license is compelling evidence of the value they placed on the patents at issue. In addition, Calco and Gartner look to the per product royalties in the KSURF-Aqua-Chem and Aqua-Chem-WPCS licenses. In the Aqua-Chem-WPCS license, WPCS agreed to pay ten cents for each water purifier sold, and five mills ($0.005) for each cc of resin in such a purifier. PTX 32 at 8. Similarly, in the KSURF-Aqua-Chem license, Aqua-Chem agreed to pay KSURF two percent of net sales of the patented resin. PTX 49 at 9. These per product warranties were charged against the minimum royalties described above; therefore, the licensee had to pay the minimum every year and had to pay an additional amount only if the per product warranties that were earned exceeded the minimum.[5]
Both sides' figures present problems. The plaintiffs' figures seem too high, largely because the plaintiffs had limited commercial success with the patents. See 658 F.Supp. at 977. In addition, setting a royalty based on Calco's profit margin is inappropriate, since it essentially brings the lost profits standard, which the Federal Circuit has already rejected for this case, in through the back door.
But Calco's and Gartner's figures are equally inappropriate. First, we believe the five percent figure in the Gartner-Calco *906 license should be given little credence. Both parties entered into the agreement even though they were already aware of the plaintiffs' patents. See 658 F.Supp. at 967. As the plaintiffs suggest, see Plaintiffs' Reply at 3, adopting Calco's and Gartner's approach would allow two infringers to enter into a sham licensing agreement with a low royalty figure, and then claim that this manufactured figure represents a reasonable royalty. Even if the Gartner-Calco license is not a sham  and we have at least some doubt, since Gartner and Calco proceeded in the face of the plaintiffs' known property rights  we are unwilling to establish such a rule for the future.
We also note that the case which the defendants rely on, Tights, Inc. v. Kayser-Roth Corp., 442 F.Supp. 159 (M.D.N.C. 1977), does not support the use of the five percent royalty in the Gartner-Calco agreement. In Tights, the Court stated:
In determining a reasonable royalty, it is improper to give substantial evidentiary weight to existing license agreements having a royalty rate which was arrived at under conditions of open, industry-wide infringement and lack of respect for the patent.
Id. at 165. Accordingly, the Court increased the royalty by a factor of six. To be sure, there was not widespread infringement in the present case, but there was a "lack of respect" for the patents. The five percent figure is therefore too low.
In addition, the five percent figure does not reflect the royalties WPCS had to pay Aqua-Chem.[6] It is reasonable to infer that WTC and WPCS would have attempted to recover some of this money in negotiations with Calco and Gartner. By contrast, Gartner had invested little in acquiring the patent information and would not have felt the same necessity to recoup losses.
Finally, although the plaintiffs had relatively little commercial success with the triiodide resins, they more than likely thought their fortunes were looking up at the time infringement began. Shortly before infringement began, the Brunswick Corp., through the good offices of defendant Gartner, had expressed an interest in triiodide products. Shortly after infringement began, but before plaintiffs had learned of it, WTC nearly reached an agreement with an unidentified "large U.S. national corporation[]" to market and sell WTC's patented product. See 658 F.Supp. at 967. Given this more optimistic business outlook, it is unlikely that the plaintiffs would have settled for the five percent that Calco and Gartner suggest.
Thus, the thirty percent or greater royalty proposed by the plaintiffs is too high, and the five percent proposed by the defendants is too low. We must determine a figure somewhere in between, but it is difficult to know where. The triiodide resin involved in this case was really a new product, not just an improvement on an old one. See 658 F.Supp. at 970. As a result, plaintiffs were not able to present evidence on the royalties or profit margin on comparable products. Nonetheless, we are required to establish a figure, and we conclude that twenty percent is a reasonable royalty. Obviously, this is closer to the plaintiffs' figure than to the defendants, but, as noted previously, reasonable doubts are to be resolved against the infringing party.
The following table applies the twenty percent royalty to Calco's undisputed sales. (All figures are rounded to the nearest dollar.)


                  Table I
Year               Sales          Royalties
1981            $   91,682        $ 18,336
1982            $  367,213        $ 73,443
1983            $  281,979        $ 56,396
1984            $   98,438        $ 19,688
1985            $   42,288        $  8,458
1986            $  163,336        $ 32,667
                __________        ________
  TOTALS        $1,044,936        $208,988
(1/1/81 to 10/22/86)



*907 C. Multiplication of Damages

As noted previously, Judge McGarr doubled the award of damages to reflect his finding of willful infringement. See 658 F.Supp. at 979. Both sides argue, rather halfheartedly, that the doubling of damages was incorrect. The plaintiffs cite cases in which damages were trebled and argue that trebling is justified in this case as well. The defendants, on the other hand, argue that Judge McGarr should not have increased the damages at all.
We reject both arguments. Judge McGarr was in the best position to evaluate all of the circumstances in this case and to determine whether an increase of damages was warranted. Neither side has presented a sufficient reason for us to reconsider his holding. Accordingly, the award of $208,988 calculated in the previous section is doubled, for a total damage award of $417,976.

III. Prejudgment Interest
Prejudgment interest is to be assessed only on the compensation part of the award, not on the multiplied portion. See Lam, Inc. v. Johns-Manville Corp., 718 F.2d 1056, 1066 (Fed.Cir.1983). Judge McGarr assessed damages at the prime rate plus one percent for the years 1981-1986. The parties agree that this is again appropriate. They also agree that we should use the more conservative prime rate for the years 1987 and 1988. Calco and Gartner argue that the plaintiffs would not have received their royalties all at once at the beginning of the year, but would have probably received quarterly payments. The defendants argue that we should adjust for this by assuming that the plaintiff would have had half of their royalties from the beginning of the year. This is not exact, but it does give a rough and reasonable approximation. We adopt this suggestion in the table below. We calculate interest on the "average principal balance," which is the sum of the "principal sum" for the previous years, plus one half of the royalty due in the given year. Thus, the average principal balance for 1984 is the principal sum from 1983 ($148,175) plus one-half of the royalty for 1984 (.5 × $19,688 = $9,844) for a total of $158,019. This accounts for two facts. First, the plaintiffs would have had all of the principal sum for 1983 in their hands for the entire year of 1984. Second, at any given time during 1984, the plaintiffs would have had an average of about half of the twenty percent royalty due during that year.
Applying this methodology, we come up with a prejudgment interest award of $115,141.


                                Table II
                                          Average               Interest
        Calco's      20%     Principal   Principal   Interest   For the
         Sales     Royalty      Sum       Balance      Rate       Year
1981   $ 91,682    $18,336   $ 18,336    $  9,168     16.75%    $  1,536
1982   $367,213    $73,443   $ 91,779    $ 55,058      12.5%    $  6,882
1983   $281,979    $56,396   $148,175    $119,977      12.0%    $ 14,397
1984   $ 98,438    $19,688   $167,863    $158,019      11.5%    $ 18,172
1985   $ 42,288    $ 8,458   $176,321    $172,092      10.5%    $ 18,070
1986   $163,336    $32,667   $208,988    $192,655       8.5%    $ 16,376
1987       -          -      $208,988    $208,988       9.0%    $ 18,809
1988       -          -      $208,988    $208,988      10.0%    $ 20,899
                                                                ________
                         TOTAL PREJUDGMENT INTEREST             $115,141



*908 IV. Attorneys' Fee

A. Background

As noted above, Judge McGarr found that this was an exceptional case and therefore awarded attorneys' fees under 35 U.S. C. § 285 (1982). Judge McGarr did not make a detailed analysis of the fees required, but rather relied on his "equitable instinct" and awarded $150,000 in fees. See 658 F.Supp. at 985. The Federal Circuit held that an award of fees was proper, but found that Judge McGarr's instinct was an insufficient basis for setting the level of fees. Accordingly, the Court vacated the award of fees, but with leave to award fees on remand if appropriate findings were made. See 850 F.2d at 674.

B. Motion to Strike

On remand, the plaintiffs have presented two items for consideration in the determination of the new fee award. The first consists of twenty months of billing records that the plaintiffs' attorneys had previously misplaced. The second is a "Renewed Application for Attorneys' Fees," a "lodestar" document that sets out the fees charged by plaintiffs' attorneys for each month of the litigation.
The defendants Calco and Gartner, contend that we should not consider these documents and accordingly have moved to strike them. The defendants note that at a September 16, 1988 status hearing, the following colloquy took place between this Court and an attorney for the defendants.
MR. CHRISTUS: I assume that what you are saying then is that the evidentiary record that is in right now on the issue of damages, is the evidentiary record that is to be used and no further evidence should be submitted on that point.
THE COURT: That's right.
Calco and Gartner argue that our assent to Mr. Christus' statement means that the plaintiffs cannot submit any additional evidence on the question of attorneys' fees.
We disagree. We note that the argument is rather moot with regard to the twenty months of billing documents, because we do not intend to work our way through a one-and-one-half-inch thick stack of unindexed documents.[7] More importantly, however, our discussion with Mr. Christus did not foreclose our consideration of new evidence on the question of attorneys' fees. Mr. Christus' statement only referred to the issue of damages, and does not refer to attorneys' fees. Moreover, even if new evidence were not permitted on the question of fees, the lodestar document is not new evidence, despite defendants' protestations to the contrary. Rather, it is more like a brief that merely summarizes the evidence which is already available. Accordingly, the defendants' motion to strike is denied.

C. Inadequacy of the Lodestar Document

Unfortunately, the plaintiffs' lodestar document still provides too much material and too little guidance to our determination of a reasonable attorneys' fee. When it remanded this case, the Federal Circuit indicated that the important factor in the determination of a reasonable fee was "the number of hours worked by specific attorneys." 850 F.2d at 674. Plaintiffs have now given us this information, but not in a very helpful form. Specifically, they break down the figures month-by-month but never summarize "the number of hours worked by specific attorneys." If we wanted that information, we would have to go through the figures month by month, for the approximately eighty months since the beginning of this litigation, and for the five major attorneys and a number of minor attorneys and law clerks, and add up the hours worked and the fees paid. We do not intend to do this.
Therefore, we direct the plaintiffs to provide us with a summary of their lodestar document. The summary should be broken down year-by-year and should include for *909 each attorney the total hours worked in that year on the case, the total fees billed, and the average hourly fee. Plaintiffs, if they choose, may lump together the attorneys who played only a minor role in each year, but there should be separate entries for the leading attorneys. Similar information should be provided for the law clerks, although the information need not be broken down individually. Plaintiffs should also provide a yearly breakdown of expenses. This information will make it easier to determine the reasonableness of the fee request.
We note here that many of the entries in the lodestar document for local counsel do not list the number of hours worked but only the fees billed. This is inadequate. We will deny any request for fees if it does not include at least an approximation of the hours worked.

D. Other Issues Involved in the Award of Fees

The parties raise a number of issues concerning the plaintiffs' request for fees. Even though we will not make a final determination of damages in this opinion, we will consider four of those issues here.[8]

1. Award of Appellate Attorneys' Fees
Plaintiffs seek reimbursement for legal work required on the appeal to the Federal Circuit. A district court has the power to grant appellate attorneys' fees. PPG Industries, Inc. v. Celanese Polymer Specialties, Co., 840 F.2d 1565, 1569 (Fed.Cir.1988). However, the Federal Circuit has read 35 U.S.C. § 285 (1982) ("The court in exceptional cases may award reasonable attorney fees to the prevailing party.") to mean that appellate fees are appropriate only if the appeal itself is exceptional. Rohm & Haas Co. v. Crystal Chemical Co., 736 F.2d 688, 692 (Fed.Cir. 1984). Usually, exceptional is equivalent to frivolous, see Porter v. Farmers Supply Service, Inc., 790 F.2d 882, 887 (Fed.Cir. 1986), but other exceptional appeals may also justify fees. See PPG Industries, 840 F.2d at 1569 (award of fee for appeal in parallel case proper where district court case was stayed pending the parallel appeal). Since Calco and Gartner prevailed on some of its claims on appeal, we cannot say that the appeal was exceptional, and we cannot award fees. Note that the fairness of this rule is somewhat questionable; after all, the plaintiffs had to defend the appeal or lose what they had gained at trial. However, that is a matter for the Federal Circuit to consider; in the meantime, Rohm & Haas prevents an award of attorneys' fees for the appeal.

2. Attorneys' Fees Incurred on This Remand
Although we have found no cases directly considering the issue, similar reasoning prohibits an award of fees for remand. Rohm & Haas indicates that attorneys' fees are justified for a particular phase of litigation only if that phase is exceptional. Because there is nothing exceptional about the remand, attorneys' fees are not justified.

3. Prejudgment Interest on the Attorneys' Fees
The plaintiffs also seek prejudgment interest on the award of attorneys' fees. The Federal Circuit has recently held that a district court has an "inherent equitable power" to award prejudgment interest on an attorneys' fees award under 35 U.S.C. § 285 (1982) when there is "`bad faith or other exceptional circumstances.'" Mathis v. Spears, 857 F.2d 749, 761 (Fed. Cir.1988) (quoting General Motors Corp. v. Devex Corp., 461 U.S. 648, 653, 103 S.Ct. 2058, 2061, 76 L.Ed.2d 211 (1983)). The plaintiffs argue that since Judge McGarr has already found that this was an exceptional case, and since that finding was upheld on appeal, we should exercise our power and award prejudgment interest on the fee award.
Although the issue is not as free from doubt as plaintiffs suggest, we conclude that prejudgment interest is appropriate *910 here. We note that in Mathis, the plaintiff had brought an unwarranted suit and two unwarranted appeals. See Mathis, 857 F.2d at 754. Accordingly, the case might be read to allow prejudgment interest on fees only when the losing party's prosecution of the litigation is frivolous or otherwise "exceptional." We do not believe, however, that Mathis should be read so narrowly. The court stated that the "district court's inherent equitable power and informed discretion remain available in determining the level of exceptionality rising out of the offender's particular conduct, and in then determining, in light of that conduct, the compensatory question of the award [under section 285], including the amount of attorney fees ... and the rate of prejudgment interest, if any, on the award." Id. Since the defendants' willful infringement made it necessary for the plaintiffs to bring this suit, an award of prejudgment interest is proper to fully compensate the plaintiffs for the expenses they incurred during litigation.

4. Postjudgment Interest on the Attorneys' Fees
Calco and Gartner argue that we exercise our discretion to deny postjudgment interest on the fee award, but we have no discretion which to exercise. Under 28 U.S.C. § 1961(a) (1982 & Supp. IV 1986), "Interest shall be allowed on any money judgment in a civil case recovered in a district court." (emphasis added). The Federal Circuit has made clear that "any money judgment" includes a judgment awarding attorney fees. Mathis, 857 F.2d at 760. Interest on an attorneys' fee award runs from the date of the judgment establishing the right to the award, not the date of the judgment establishing its quantum. Id. Judge McGarr's first opinion, dated October 22, 1986, established the entitlement to attorneys' fees. Accordingly, once the plaintiffs submit their summary of the lodestar document, we will award postjudgment interest on the attorneys' fee award from that date.

V. Conclusion
For the reasons set forth above, we vacate Judge McGarr's previous award of $200,000 for lost marketing opportunities in its entirety. We award a total of $417,976 in damages and $115,141 in prejudgment interest. The plaintiffs are directed to provide a summary of their lodestar document, as outlined in section IVC, by February 1, 1989; the defendants may respond by February 10, 1989. Status hearing previously set for January 18, 1989, is reset for March 3, 1989, at 10:00 a.m. It is so ordered.
NOTES
[1]  Our summary here is taken from both Judge McGarr's extensive findings of facts and the Federal Circuit's opinion.
[2]  A greater number of licenses may create an established royalty rate, which is usually the best measure of what is a reasonable royalty. Nickson Industries, Inc. v. Rol Manufacturing Co., 847 F.2d 795, 795 (Fed.Cir.1988); Studiengesellschaft Kohle m.b.H. v. Dart Industries, Inc., 666 F.Supp. 674, 680 n. 6 (D.Del.1987). However, the established rate is not dispositive. Nickson Industries, 847 F.2d at 798.
[3]  At least one judge had recognized the unreality of the willing licensor  willing licensee rule. Judge Markey, now the Chief Judge of the Federal Circuit, wrote:

Determination of a "reasonable royalty" after infringement, like many devices in the law, rests on a legal fiction. Created in an effort to "compensate" when profits are not provable, the "reasonable royalty" device conjures a "willing" licensor and licensee, who like Ghosts of Christmas Past, are dimly seen as "negotiating" a "license." There is, of course, no actual willingness on either side, and no license to do anything, the infringer being normally enjoined ... from further manufacture, use, or sale of the patented product.
Panduit, 575 F.2d at 1159.
[4]  Calco and Gartner assert in their sur-reply brief that this finding was incorrect. It is true that there was conflicting testimony on this issue, but Judge McGarr resolved this conflict and found that the profit margin was seventy-three percent. Even if we thought this issue was properly raised, we would not be willing to disturb Judge McGarr's finding, since he presided at trial and was therefore in the best position to judge credibility.
[5]  Calco and Gartner wish us to consider certain documents that indicate what per product royalties were earned in given years. These documents are two letters which were exhibits to the deposition of L.L. Davis, the President of WTC. Judge McGarr admitted Davis' deposition into evidence, and Calco and Gartner assert that this means the deposition exhibits were admitted also. The plaintiffs disagree. They note that they had previously objected to the exhibits in question and that these exhibits were not included in the pre-trial order. Neither party cites any supporting case law for its position, but we conclude that we should not consider these exhibits. We note that Judge McGarr never ruled on the objections to these exhibits. Because the court must admit an exhibit before it becomes evidence, we think this is the best indication that the exhibits were not evidence, even though the depositions were admitted and despite Calco's and Gartner's protestations to the contrary. Yet even if we are wrong, we note that Calco and Gartner only argue that these documents support a royalty in the five percent range. Thus, the disputed evidence is cumulative at best  another reason to deny the admission of the documents.
[6]  Calco and Gartner assert that there is no evidence that WPCS paid these royalties, but Judge McGarr found that it paid approximately $150,000 in license fees. See 658 F.Supp. at 977. Based on the royalty schedule in the Aqua-Chem-WPCS license, it appears that $50,000 of this figure was paid after infringement began on November 4, 1980. Still, that means on the date the hypothetical negotiations would have begun, WPCS would have paid $100,000 for royalties.
[7]  We would like to meet the district judge who has the time, let alone the inclination, to wade through a morass like this. We frankly think that the submission of these billing documents in this form shows a lack of judgment on the part of plaintiffs' attorneys.
[8]  The other challenges involve the amount of damages and will be considered when the plaintiffs provide a summary of their request for fees.
