OPINION OF THE COURT
ROSENN, Circuit Judge.
After a history of fourteen years of litigation in the district court and several appeals to this court, we are confronted again with an appeal in this patent infringement proceeding. The infringement has been established and is no longer at issue. We are, however, revisited with the troublesome issue of damages. When this case was last before us on appeal from the original determination of damages, we vacated the judgment and remanded to the district court with instructions to recalculate the damages. We are now asked to decide whether the district court’s action is consistent with our holding in that earlier appeal. We hold that it is not and, therefore, again vacate the district court’s judgment.
I.
At the heart of this controversy is a patented process for removing insulation from copper wire in order to allow the copper to be salvaged. This process is covered by United States Patent No. 3,076,421, owned by Trio Process Corporation (“Trio”). In 1972 we upheld the validity of the patent and determined that it had been willfully infringed by L. Goldstein’s Sons, Incorporated (“Goldstein”). Trio Process Corp. v. L. Goldstein’s Sons, Inc., 461 F.2d 66 (3d Cir.), cert. denied, 409 U.S. 997, 93 S.Ct. 319, 34 L.Ed.2d 262 (1972). The case was remanded to the district court for a determination of damages.
On remand the district court appointed a master to assist in the determination of damages. When we last reviewed the proceedings, we observed:
The master approached the damage issue by comparing Goldstein’s costs of operating the patented process with the costs of a similar, unpatented process. He found that use of the Trio process saved Gold-stein $52,791 per furnace year in labor costs alone, and that other, smaller savings accrued to Goldstein from use of the patented method as well.
In order to reach a “reasonable royalty” for use of the patent by the infringer, the master halved Goldstein’s savings in labor costs, and concluded that $26,390 was a reasonable royalty for each furnace year. Multiplying this figure by the number of furnace years of infringement and making slight modifications, the master found damages of $1,564,804. The master recommended trebling this amount, as allowed by 35 U.S.C. § 284 (1970). After trebling and the addition of interest, the total damage figure proposed by the master was $5,062,954.
The district court viewed the damage computation not with regard to the money saved by the defendant as a result of the infringement, as the master had, but in terms of what Trio had lost. It looked first to the initial sum of $2,600 per furnace year — the amount actually charged by Trio for licenses in the 1960-1970 era. The district court then increased the $2,600 figure on the assumption that the open infringement had reduced the market price of the license, and proceeded to set damages at $7,800 per furnace year for the years prior to the decision by this Court on validity, a figure three times the rate charged by Trio during the 1960’s. Damages were set at $15,000 per furnace year for the period following the 1972 adjudication. The employment of these two figures resulted in total primary damages of $653,839. The trial judge then proceeded to use a double multiplier — in contrast to the master’s trebling figure — and denied attorneys’ fees. With interest, the total damages computed by the district court were $1,726,525.
Trio Process Corp. v. L. Goldstein’s Sons, Inc., 533 F.2d 126, 128 (3d Cir., 1976) (footnotes and citations omitted) (“Trio Process III”).
On appeal we affirmed in part and reversed in part. Trio Process III, supra. We held that there was “no error in the first step of the district court’s damage calculation, namely, focusing upon the losses suffered by the patent holder rather than upon the profits illegally made by the patent infringer.” Id. at 129. See Aro Manufacturing Co. v. Convertible Top Replacement Co., 377 U.S. 476, 84 S.Ct. 1526, 12 L.Ed.2d 457 (1964). We also affirmed the district court in its finding “that the license rate established by Trio in the 1960’s may have been artificially depressed by Gold-stein’s ongoing infringement, and that the reasonable royalty should therefore be set at a level above the actual license rate.” Trio Process III, supra, 533 F.2d at 129 (emphasis added).
We held, however, that the district court had erred in two respects. First, it had calculated not one royalty rate but two: one for the period before our decision upholding the patent’s validity and the second for the period after. We held that a single reasonable royalty rate should be calculated for the entire period of infringement. The district court has done that and the point is no longer at issue. Second, and most importantly for purposes of deciding this appeal, there was a failure to articulate the reasons underlying the determination of the royalty rate. Thus, the cause was remanded to the trial court for reconsideration of the damages issue. We noted specifically that
[O]n remand, the district court should give proper regard to the rule that the extent of the deviation of existing license fees from a reasonable royalty must be determined solely on the basis of the submitted evidence and upon an evaluation of the factors that could affect the reasonable royalty rate, not upon mere conjecture.
Trio Process III, supra, 533 F.2d at 130 (footnote omitted). This has not been done, however, and we therefore vacate the determination of damages.
II.
In calculating damages for patent infringement, a patent holder is entitled to receive compensation for the infringement but in no event less than a reasonable royalty. 35 U.S.C. § 284 (1976). An exhaustive list of factors relevant to the determination of a reasonable royalty can be found in Georgia-Pacific Corp. v. United States Plywood Corp., 318 F.Supp. 1116, 1120 (S.D.N.Y.1970), modified, 446 F.2d 295 (2d Cir.), cert. denied, 404 U.S. 870, 92 S.Ct. 105, 30 L.Ed.2d 114 (1971). The district court in this case found a number of those factors to be relevant in its own calculation of damages:
[1] [TJhe existing value of the [patented] process to the licensor as a generator of sales of his non-patented items; and the extent of such derivative or convoyed sales.
[2] The duration of the patent and the term of the license.
[3] The established profitability of the product made under the patent; its commercial success; and its current popularity.
[4] The utility and advantages of the patent property over old modes or devices, if any, that had been used for working out similar results.
[5] The nature of the patent [process] . and the benefits to those who have used the [process].
[6] The extent to which the infringer has made use of the [patented process]; and any evidence probative of the value of that use.
[7] The portion of the realizable profit that should be credited to the invention as distinguished from nonpatented elements, the manufacturing process, business risks, or significant features or improvements added by the infringer.
[8] The opinion testimony of qualified experts.
Applying the “willing buyer and willing seller” rule, the district court considered these factors in the context of hypothetical negotiations between the parties conducted in the absence of the infringing activity. See Horvath v. McCord Radiator & Manufacturing Co., 100 F.2d 326, 335 (6th Cir.), cert. denied sub nom., Carrier Engineering Corp. v. Horvath, 308 U.S. 581, 60 S.Ct. 101, 84 L.Ed. 486 (1938). The court found that the first two of the above factors would have had only a “minimal effect” in the determination of a reasonable royalty. As to the remaining factors, the court noted that “[they] all touch upon the benefits obtained by defendant through its infringing use of plaintiff’s patented process.” Thus, the court found that “the license fee the parties would have agreed upon absent defendant’s infringement would to a large extent have been determined by the economic benefits that were obtained through the use of plaintiff’s patented process.”
The court found that Goldstein had obtained four distinct benefits from its use of the Trio process: (1) a reduction in labor costs; (2) an increased recovery of copper from the scrap wire; (3) lower fuel consumption per ton of processed material; and (4) the ability to attract more electrical scrap for processing by advertising the advantages of the Trio process. The court, however, was unable to assign a dollar value to each of these benefits but indicated that “the only dollar figure available is the value of the direct and indirect labor savings achieved by defendant.”
The court began its calculation of the labor savings with expert testimony, credited by the master, which indicated that Goldstein had realized labor savings of $52,-791 per furnace year by virtue of its infringing use of the Trio process. Because Goldstein operated a number of infringing furnaces over an eight and one-half year period, the court reduced this figure to $41,-652 per furnace year, reflecting the wages prevailing in 1969, the mid-point in the infringing period. The court then found that “[i]n voluntary royalty negotiations untainted by defendant’s infringing practices, defendant might well have been willing to split this saving with plaintiff and paid plaintiff a royalty of approximately $20,000 for each furnace year.” For two reasons, however, the court further reduced this to $15,000 per furnace year. First, the court held that, as a seller of furnaces, “[Trio] would have been willing to accept somewhat less than the maximum royalty negotiable in order to promote its sales.” Second, prior to the lawsuit, “plaintiff was unaware ... of the exact extent of the labor savings that were obtainable through the use of its process.” After multiplying $15,000 by the number of infringing furnaces, the court then doubled the primary damages and added interest of 6% per annum. The total damage award was $2,901,336 plus costs.
Georgia-Pacific lists first among the factors relevant to the determination of a reasonable royalty “[t]he royalties received by the patentee for the licensing of the patent in suit, proving or tending to prove an established royalty.” Georgia-Pacific Corp. v. United States, supra, 318 F.Supp. at 1120. In this case, the district court chose to disregard the license fees received by Trio because “they did not show that there was an established royalty and since the fees received were artificially depressed by defendant’s ongoing infringement.” The court noted its belief that “a royalty negotiated in the absence of defendant’s infringe-j ment would have been several times higher than the license fees actually received by the plaintiff.” We have not, however, been able to discover any evidence in this record to support this conclusion.
It is true that the actual license rate does not necessarily constitute a reasonable royalty. See General Motors Corp. v. Blackmore, 53 F.2d 725 (6th Cir. 1939). Thus, when the actual license rate is artificially low, a reasonable royalty may be set above that rate. Trio Process III, supra. Nevertheless, the actual license rate is an important factor in the determination of a reasonable royalty, at least when those Royalties prove or tend to prove an established royalty. See Georgia-Pacific Corp. v. United States Plywood Corp., supra, 318 F.Supp. at 1120.
We are mindful that the district court concluded that the royalties trio received under the license agreements did not constitute an established royalty. Nevertheless, the existing license rate does tend to show an established license rate. The evidence indicates little, if any, variation in the rate charged before or after the infringement. Further, the district court, in its first consideration of the damage issue, apparently found the actual license rate to be probative, although not conclusive, evidence of a reasonable royalty. Thus, the reasonable royalty rate determined by the district court in its first consideration of the damage issue was related to the actual license rate charged. That approach was correct. As we indicated in our earlier opinion, however, the district court erred in failing to demonstrate, on the basis of the evidence, the extent of the deviation of existing license fees from a reasonable royalty. That same void continues to exist in the district court’s most recent damage calculation.
We are again unable to discover any support for the district court’s conclusion that the existing license rate was depressed by Goldstein’s infringement. Thus, its reliance on the rationale of Tights, Inc. v. Kayser-Roth Corp., 442 F.Supp. 159 (M.D.N.C.1977), is misplaced. In Tights the court disregarded the standard royalty rate, finding it had been artificially depressed “because it was established in an atmosphere of industry-wide infringement of and disrespect for the . . . patent.” Id. at 165. The court thereupon calculated a reasonable royalty based on hypothetical negotiations between a “willing licensee” and a “willing licensor.” Unlike the instant case, the depressing effect in Tights was evident. There, the low license rate had been negotiated against a background of open industry-wide infringement. Further, there was evidence that the existing license rate had dramatically declined because of that infringement. Thus, there was a substantial factual basis which justified the court’s decision to disregard the existing license rate. In the case before us, however, such factors are not present. The license rate agreed upon between Trio and Goldstein was arrived at in free and open negotiations conducted prior to any infringing activity by Goldstein. Furthermore, there are no allegations in this case of industry-wide infringement. Unlike Tights, there is no indication that the license rates here declined after Goldstein’s infringement. Indeed, even* after Trio learned of the infringement, it offered Goldstein a license at the same rate as had been earlier agreed upon. Further, in the years following our decision upholding the validity of the patent, there were apparently no new licenses granted. See Trio Process III, supra, 533 F.2d at 127. Thus, the thrust of the evidence in this case indicates the absence of a depressing effect caused by Gold-stein’s infringement. Nor have we been referred to any permissible evidentiary basis to the contrary. Thus, we are compelled to vacate the court’s assessment of damages.
III.
Having done so we must now decide whether to once again remand to the district court for a determination of damages or whether, as Goldstein urges, to set the level of damages ourselves. The time and expense already consumed by this litigation strongly argue against remand. Furthermore, the record appears complete and sufficient for such a determination. In briefs and oral argument before this court on their last appeal, both parties agreed that no additional evidence was necessary for the calculation of damages. Trio Process III, supra, 533 F.2d at 131 n.24. Nor did the district court find it necessary on remand to receive additional evidence.
This court was presented with a similar situation in Randolph Laboratories, Inc. v. Specialties Development Corp., 213 F.2d 873 (3d Cir.), cert. denied, 348 U.S. 861, 75 S.Ct. 91, 99 L.Ed. 678 (1954). There this court set aside the district court’s determination of damages on the ground that it was “grossly excessive.” 213 F.2d at 875. The court, however, declined to send it back to the district court but set the damages itself. There, as here, the controversy had been pending for a number of years. Writing for the court, Judge Maris noted that “[t]he record . . . contains all the evidence and other materials necessary to a proper judgment.” Id. We believe it is appropriate that, in view of the protracted litigation and having vacated for the second time the district court’s determination of damages, we should now determine the appropriate level of damages ourselves. However, we will not perform the mathematical calculations as was done in Randolph but will leave that task to .the district court.
IV.
We begin with the rule that we noted in our last opinion, that the extent of the deviation of the actual licensing rate from a reasonable royalty must be explained solely on the basis of the submitted evidence. Trio Process III, supra, 533 F.2d at 130. In the absence of such an explanation, we must examine the record ourselves to determine whether it contains such evidence.
Trio itself did not utilize the patented process. Instead, its only use of the patent was to license it for use by others. The licenses sold were for five year periods. The first license was sold for $20,000. This amount covered the license and the furnace necessary to utilize the process; $7,000 represented the cost of the furnace and $13,000 the cost of the license, i. e., $2,600 per furnace year. In 1960, Goldstein purchased two sets of licenses and furnaces, one for $20,000 and the other for $15,000. Between 1962 and 1969 four more buyers purchased licenses and furnaces at the $20,000 rate. In 1967, another company bought the package with a modified furnace for $25,000. Later that year the package was purchased by another buyer for $19,500. After a decision by Trio to raise the price, two more were sold in 1972 to purchasers other than Goldstein, for a price of $25,000. Thus, throughout this period, the license rate of $2,600 per furnace year appears to have remained relatively constant.
Goldstein’s infringing activities began in 1965. However, Trio and Goldstein had in free and open negotiations previously agreed to a license rate of $2,600 per furnace year. The license rate Trio charged other licensees did not decline after Gold-stein’s infringement began. Consequently, if the infringing activity did have a depressing effect on the license rate it could only have been in deterring Trio from charging the rate it otherwise would have negotiated in the open market. The district court, however, disregarded the license fees received by Trio, because it believed they did not reveal an established royalty, and they were artificially depressed by the ongoing infringement. The court found that Trio was a seller of furnaces and thus in negotiating a royalty rate (prior to the infringement) “would have been willing to accept somewhat less than the maximum royalty negotiable in order to promote its sales.” It further observed that prior to this lawsuit, Trio was unaware of the exact extent of the labor savings effected by the patented process. But the record indicates that even after learning of the infringement, Trio offered Goldstein a license for the infringing furnace “under the same terms and conditions as the previous two incinerators.” (Emphasis added.)
It is true that “[a] patentee who has attempted to avoid costly and time-consuming litigation by settling for less than a reasonable royalty should not be penalized when an infringer forces full litigation.” Tights, Inc. v. Kayser-Roth Corp., supra, 442 F.Supp. at 165. Here, however, there is no reason to believe that the license rate negotiated by the parties was anything other than a balanced consideration by both Goldstein and Trio of those competing concerns that normally enter into the determination of price in an open marketplace economy. Trio consistently offered licenses at the rate of $2,600 per furnace year. Thus, the possibility that Trio, had it chosen to do so, might have obtained a higher license rate than that actually charged, is irrelevant. We believe the rate fixed by the parties prior to any infringement is pertinent and highly persuasive. Further, our examination of the record has not disclosed any reason to distrust the existing license rate as a measure of actual damages. Thus, we hold that the $2600 per furnace year rate negotiated between Trio and Goldstein prior to the infringement, constitutes a reasonable royalty.
V.
35 U.S.C. § 284 provides that the court may increase damages up to three times the amount found or assessed. We agree with the district court that the multiplication of damages depends on the degree of bad faith exhibited by the defendant. Although the master recommended trebling the damages the district court, in its first consideration of damages, determined that “the facts of this case do not warrant trebling the full amount of the primary damages.” Thus, the court used a double multiplier. In Trio Process III we affirmed that decision as being “within the sound discretion of the district court.” 533 F.2d at 131. In its next consideration, the court again used a double multiplier and we find no reason to disturb that decision.
On remand, the district court should calculate damages based on a reasonable royalty of $2,600 per furnace year of infringement. Primary damages should then be multiplied by a factor of two with 6% interest from the date of the infringement, plus costs. Accordingly, the judgment of the district court will be vacated and the case remanded for further proceedings not inconsistent with this opinion. Each party to bear its costs of this appeal.
. Trio Process Corp. v. L. Goldstein’s Sons, Inc., 461 F.2d 66 (3d Cir.), cert. denied, 409 U.S. 997, 93 S.Ct. 319, 34 L.Ed.2d 262 (1972).
. Trio Process Corp. v. L. Goldstein’s Sons, Inc., 533 F.2d 126 (3d Cir. 1976).
. For a more detailed discussion of the patented process and the factual background of this case, see Trio Process Corp. v. L. Goldstein’s Sons, Inc., 461 F.2d 66, 68-70 (3d Cir.), cert. denied, 409 U.S. 997, 93 S.Ct. 319, 34 L.Ed.2d 262 (1972).
. Contrary to Trio’s assertion, however, we did not hold that the reasonable royalty rate was, as a matter of law, higher than the actual license rate. We held only that it might be higher if the actual license rate had, in fact, been artificially depressed by Goldstein’s infringement. Thus, we held in Trio Process III that the reasonable royalty should be set at a level above the actual license rate if it was demonstrated, on the basis of the submitted evidence, that Goldstein’s infringing activities had artificially depressed the actual license rate established by Trio.
. The district court had set damages based on a reasonable royalty of $7,800 per furnace year for the period prior to our adjudication of the patent’s validity and $15,000 per furnace year for the period thereafter.
. The dissent asserts that this court in Trio Process III trespassed upon the fact-finding authority of the district court and that the majority here has once again arrogated that function to itself. Our perception of the court’s role in this case, however, differs from that of the dissent. In Trio Process III the court remanded, not because of its desire to induce the fact-finder to award a particular amount of damages but because of its concern with a legal precept in patent law, that the extent of the deviation of license fees from a reasonable royalty be determined solely on the basis of the evidence and not conjecture. Here, we vacate the determination of damages because it does not comply with the rule of law set forth in Trio Process III and not, as the dissent asserts, because we have set ourselves up as a fact-finding authority.
. This figure was arrived at by comparing the cost of operating a similar, noninfringing furnace. Goldstein disputes the basis for this comparison but we need not address this issue in view of our disposition of the case.
. See section IV infra.
. The court recognized that “a prevailing royalty rate may be probative of what constitutes a reasonable royalty.” However, as we have previously indicated it fixed a rate of $7,800 per furnace year as a reasonable royalty.
. Hartford Nat’l Bank & Trust Co. v. E. F. Drew & Co., 188 F.Supp. 353, modified, 188 F.Supp. 347 (D.Del.1960), aff'd per curiam, 290 F.2d 589 (3d Cir.), cert. denied, 368 U.S. 825, 82 S.Ct. 45, 7 L.Ed.2d 29 (1961), which Trio argues supports the district court’s opinion, is also distinguishable. There the court stated: “It is a fact of economic life that an open infringement tends to reduce a patentee’s fees from its subsequent licensees who must meet the infringer’s competition, and such infringement deters potential licensees from taking a license.” 188 F.Supp. at 362 n.50. Here there is no indication that the negotiated fees were adversely affected by subsequent infringement or that potential licensees were deterred from seeking license agreements.
. In Georgia-Pacific, supra, the court also determined damages on the basis of the “willing buyer and willing seller” rule. This was used, however, only after the parties agreed there was no established royalty for the patented item. Indeed, the apparent policy of the patent holder was not to enter into licensing agreements but rather, to maintain its patent monopoly. See id. at 1123.
. The royalty rate in Tights had declined from a nominal rate of $.50 per dozen garments to $.02 per dozen. Tights, Inc. v. Kayser-Roth Corp., supra, 442 F.Supp. at 162.
. Goldstein and Trio entered into two license agreements in 1960. There is no indication in the record that Goldstein’s infringing activities began any earlier than 1964 when it contracted with a metal fabricator for the construction of a copy of a furnace Goldstein had purchased from Trio.
. The record indicates that, on March 8, 1965, Trio’s counsel wrote to Goldstein requesting that it cease its infringing activities. That letter stated, inter alia:
[T]his will formally charge you with infringement of Pat. No. 3,076,421. Further, your activities in this matter appear to directly violate the license agreement between L. Goldstein’s Sons, Inc. and Trio Process Corporation dated January 2, 1960.
* * * * * *
This letter is written in the interest of having you license the additional incinerator under the same terms and conditions as the previous two incinerators.
Exhibit DM-15 (emphasis added).
. Trio brought its original action for patent infringement in May 1965. The present appeal marks the fourth time the case has been before this court. See Trio Process Corp. v. L. Goldstein's Sons, Inc., 533 F.2d 126 (3d Cir. 1976); Trio Process Corp. v. L. Goldstein’s Sons, Inc., 493 F.2d 1401 (3d Cir. 1974); Trio Process Corp. v. L. Goldstein’s Sons, Inc., 461 F.2d 66 (3d Cir.), cert. denied, 409 U.S. 997, 93 S.Ct. 319, 34 L.Ed.2d 262 (1972).
. In its first consideration of damages the district court found that this increase in price included an increase in the royalty rate charged by Trio to about $3,600 per furnace year. At oral argument on this appeal, Goldstein disputed this, asserting that the increase in price was attributable, instead, to an increase in the cost of the furnace. Assuming that the rate was increased, however, only two licenses were granted at this figure. Further, no licenses were granted after our decision upholding the validity of the patent. Even after learning of Goldstein’s infringement. Trio indicated, in a letter of March 8, 1965, its willingness to continue licensing Goldstein at the rate of $2,600 per furnace year. See note 13 supra. Thus, it appears that the license rate charged by Trio over the span of years commencing in 1960, up to and including the last consideration by the district court in September 1978, reflects little change.
. See note 13 supra.
. “Actual damages must be calculated, not imagined, and an arithmetical calculation cannot be made without certain data on which to make it.” Mayor of New York v. Ransom, 64 U.S. (23 How.) 487, 488, 16 L.Ed. 515. See Trio Process III, supra, 533 F.2d at 130.
. Trio argues that, as a penalty for having brought a frivolous appeal, we should require Goldstein to pay substantial additional interest on the judgment and double costs on this appeal. See 28 U.S.C. § 1912 (1976). Given our disposition of this case, however, Goldstein’s appeal is clearly not frivolous. Thus, Trio’s argument is without merit.