TARANTO, Circuit Judge.
On remand from earlier holdings of this court, the Court of Federal Claims held that ten percent of $5.4 million in revenue (which was almost pure profit) was a reasonable royalty for the United States to
pay as damages for its unauthorized use of a distinctive copyrighted work on a postage stamp. Finding an adequate basis for the trial court’s determination, we affirm.
Background
Much of the background to the present appeal is detailed in our prior opinions,
Gaylord v. United States,
595 F.3d 1364 (Fed.Cir.2010)
(Gaylord
I), and
Gaylord v. United States,
678 F.3d 1339 (Fed.Cir.2012)
(Gaylord II).
Frank Gaylord, a World War II veteran and renowned sculptor, created
The Column,
consisting of nineteen stainless steel statues depicting a squad of soldiers on patrol, to form a central part of the Korean War Veterans Memorial located on the National Mall in Washington, D.C. The Memorial also includes a reflecting pool and a mural wall, created by others. Mr. Gaylord was paid $775,000 for his contribution, time and materials included.
In January 1996, roughly six months after the Memorial was completed and dedicated, an amateur photographer named John Alii visited the Memorial during a heavy snowstorm and photographed
The Column,
later titling the resulting photograph
Real Life.
In 2002, the United States Postal Service decided to issue a stamp to commemorate the upcoming fiftieth anniversary of the Korean War armistice. From the very outset of the process of selecting an image for the commemoration, the relevant Postal Service advisory committee fastened onto the idea of using
The Column
on the stamp. It selected Mr. Alli’s photograph of
The Column
for the stamp face, and it paid Mr. Alii a onetime fee of $1,500 for the right to use his photo. The Postal Service did not seek Mr. Gaylord’s consent to use
The Column
— the photograph was a “derivative work” of
The Column
under 17 U.S.C. § 106(2) — before issuing the stamp in 2003, and Mr. Gaylord never gave his consent.
See Gaylord I,
595 F.3d at 1370-71.
Mr. Gaylord sued the United States for copyright infringement in 2006. In
Gay-lord I,
we held that the government was liable to Mr. Gaylord for copyright infringement, because it used his work on the stamp without his permission,
The Column
was not a “joint work” (whose joint authors individually might grant permission), and its use was not protected as fair use. 595 F.3d at 1371, 1376, 1381. In
Gaylord II,
we vacated the Court of Federal Claims’ decision awarding Mr. Gay-lord $5,000 as the “reasonable and entire compensation” he was due under 28 U.S.C. § 1498(b). 678 F.3d at 1345. We remanded to determine the fair market value of a license for Mr. Gaylord’s work based on a hypothetical negotiation with the government.
Id.
at 1344-45.
On remand after
Gaylord II,
the Court of Federal Claims reopened the record to allow additional discovery and expert reports, and it then held a two-day trial on damages. As we had suggested, and the parties accept as sound, the trial court broke down its consideration of damages into three categories of infringing goods: (1) stamps used to send mail; (2) commercial merchandise featuring an image of the stamp; and (3) unused stamps purchased by collectors.
Gaylord v. United States,
112 Fed.Cl. 539, 542-43 (2013);
see Gaylord II,
678 F.3d at 1344. The first two categories are not now in dispute: the parties agreed that no damages would be awarded for stamps used to send mail and that a per-unit royalty was appropriate for the commercial merchandise,' the trial court setting the rate at 10% of revenue to produce a merchandise award of $33,092 (plus prejudgment interest), which neither side now contests.
Gaylord,
112 Fed.Cl. at 542-43.
The only question disputed in this court concerns the award regarding the third category — unused stamps. “After a full
review of the evidence presented by both sides,” the trial court determined that a 10% per-unit royalty was appropriate to calculate damages for stamps purchased by collectors.
Id.
at 542. Based on evidence from regularly conducted surveys that the Postal Service commissions and relies on in its ordinary course of business, the court determined that the Postal Service received $5.4 million in revenue— which was “almost pure profit” — from unused stamps of
The Column
sold to collectors during the (now-ended) life of the issue.
Id.
at 541. The court therefore awarded Mr. Gaylord $540,000 for the unused stamps, plus prejudgment interest.
Id.
at 542-43.
The government appeals the unused-stamp award. We have jurisdiction under 28 U.S.C. § 1295(a)(3).
Discussion
We review legal conclusions by the Court of Federal Claims de novo and its factual findings for clear error.
Gaylord II,
678 F.3d at 1342;
Gargoyles, Inc. v. United States,
113 F.3d 1572, 1576-77 & n. 4 (Fed.Cir.1997). “A finding is ‘clearly erroneous’ when[,] although there is evidence to support it, the reviewing court on the entire evidence is left with the definite and firm conviction that a mistake has been committed.”
United States v. U.S. Gypsum Co.,
333 U.S. 364, 395, 68 S.Ct. 525, 92 L.Ed. 746 (1948). In the patent context, we have treated the royalty determination as a factual finding but certain methodological questions for determining a fair market value as subject to abuse-of-discretion review.
See SmithKline Diagnostics, Inc. v. Helena Labs. Corp.,
926 F.2d 1161, 1164 & n. 2 (Fed.Cir.1991). Any difference in the standard of review does not matter here, however, as we find no clear error or abuse of discretion in the Court of Federal Claims’ determinations supporting its royalty award.
A
In 28 U.S.C. § 1498(b), Congress waived the sovereign immunity of the United States to allow “the recovery of [the copyright owner’s] reasonable and entire compensation as damages” for copyright infringement. In
Gaylord II,
we held that “the methods used to determine ‘actual damages’ under the copyright damages statute, 17 U.S.C. § 504, are appropriate for measuring the copyright own-er’s loss” under § 1498(b). 678 F.3d at 1343. Consistent with the conclusions of other circuits that have considered the issue, we held that actual damages for copyright infringement may be based on a reasonable royalty representing “ ‘the fair market value of a license covering, the defendant’s use.’ ”
Id.
(quoting
On Davis v. The Gap, Inc.,
246 F.3d 152, 172 (2d Cir.2001));
see also Dowagiac Mfg. Co. v. Minn. Moline Plow Co.,
235 U.S. 641, 648-50, 35 S.Ct. 221, 59 L.Ed. 398 (1915) (affirming that, in the patent context, a reasonable royalty “afford[s] a basis for measuring the damages”);
On Davis,
246 F.3d at 168-69 (summarizing copyright cases).
To calculate the fair market value, a court deciding a copyright case may use a tool familiar from patent law, without necessarily following every aspect of patent law’s use of that tool. It may hypothesize a negotiation between the parties before the infringement occurred and determine “‘the reasonable license fee on which a willing buyer and a willing seller would have agreed for the use taken by the infringer.’”
Gaylord II,
678 F.3d at 1343 (quoting
On Davis,
246 F.3d at 167);
see Sid & Marty Krofft Television Prods., Inc. v. McDonald’s Corp.,
562 F.2d 1157, 1174 (9th Cir.1977) (approving instruction to jury to determine “what a willing buyer would have been reasonably required to pay to a willing seller for plaintiffs’ work.”). Several cases have stressed the
use of “objective considerations” in the determination of a copyrighted work’s market value using that method.
Jarvis v. K2 Inc.,
486 F.3d 526, 534 (9th Cir.2007);
see Oracle Corp. v. SAP AG,
765 F.3d 1081, 1088 (9th Cir.2014);
On Davis,
246 F.3d at 166-67. The court is not constrained to accept particular practices of the parties on either side — either to allow owners to charge what they “would like to have charged if unconstrained by reality,”
Oracle,
765 F.3d at 1088, or to “shield [infringers] from paying fair market value for what they took,”
Gaylord II,
678 F.3d at 1343 (citing to patent law).
Determining a reasonable royalty does not require “mathematical exactness,” but a “reasonable approximation” under the circumstances of a given case.
See Dowagiac,
235 U.S. at 647, 35 S.Ct. 221 (discussing the degree of precision required in the sometimes-similar process of apportioning profits). Other circuits have recognized the often-unavoidable uncertainties in establishing the fair market value of a reasonable license fee as well as the importance of avoiding undue speculation.
Oracle,
765 F.3d at 1088-89;
On Davis,
246 F.3d at 166. “[S]ome difficulty in quantifying the damages attributable to the infringement should not bar recovery.”
On Davis,
246 F.3d at 167.
The hypothetical-negotiation determination must be tied to the particular work at issue and its marketplace value— much as, in patent law, the determination must be tied to the particular patented technology and its footprint in the market.
See VirnetX, Inc. v. Cisco Sys., Inc.,
767 F.3d 1308, 1327 (Fed.Cir.2014);
LaserDynamics, Inc. v. Quanta Computer, Inc.,
694 F.3d 51, 67, 79 (Fed.Cir.2012);
Uniloc USA Inc. v. Microsoft Corp.,
632 F.3d 1292, 1315-18 (Fed.Cir.2011);
ResQNet.com, Inc. v. Lansa, Inc.,
594 F.3d 860, 869 (Fed.Cir.2010). Different kinds of evidence may be relevant, as long as it is viewed for what it says about the work-specific value. For example, past arms-length licensing practices by the copyright owner or the infringer for similar uses and “benchmark” licenses by others in the industry may be useful.
See Oracle,
765 F.3d at 1093;
Jarvis,
486 F.3d at 534-35. But the use of past licenses as evidence must always take account of economically relevant differences between the circumstances of those licenses and the circumstances of the matter in litigation-as we have said in the related patent — law context.
Ericsson, Inc. v. D-Link Sys., Inc.,
773 F.3d 1201, 1227-28 (Fed.Cir.2014);
VirnetX,
767 F.3d at 1330-31;
Finjan, Inc. v. Secure Computing Corp.,
626 F.3d 1197, 1211-12 (Fed.Cir.2010). Thus, in the copyright context, the unique features of a particular work (including its recognized stature and symbolic value) may be important in assessing the ultimate significance of past practices in licensing other works.
B
The basic premise of the hypothetical negotiation in this case would have been the opportunity for making substantial profits if the two sides were willing to join forces, which we must assume they were.
See Gaylord II,
678 F.3d at 1343 (“‘a willing buyer and a willing seller’ ”). The Court of Federal Claims in this case determined that the negotiators, presented such an opportunity and acting under assumptions designed to identify market value, would have agreed to a 90/10 split of the revenue from retained stamps, which, here, is in substance a 90/10 split of profits, because the revenue for the unused stamps is almost pure profit to the Postal Service. The question for us is whether that result — giving the Postal Service 90% of the profits and Mr. Gaylord 10% — is within the range of reasonable findings from the evidence.
See Polar Bear Prods., Inc. v. Timex Corp.,
384 F.3d 700, 709 (9th
Cir.2004) (affirming jury award that was “within the range of the fair market value” justified by the evidence).
In answering that question, “[w]e presume that a fact finder reviews all the evidence presented unless he explicitly expresses otherwise.”
Medtronic, Inc. v. Daig Corp.,
789 F.2d 903, 906 (Fed.Cir.1986). Here, the trial court, far from stating otherwise, expressly stated that it conducted “a full review of the evidence presented by both sides.”
Gaylord,
112 Fed. Cl. at 542. We therefore consider the entirety of the evidence, not just the evidence expressly recited in the trial court’s opinion.
Plant Genetic Sys., N.V. v. DeKalb Genetics Corp.,
315 F.3d 1335, 1343 (Fed.Cir.2003) (“The fact that the district court did not in its opinion recite every piece of evidence does not mean that the evidence was not considered.”). We conclude that the trial court’s determination must be upheld.
As a threshold matter, the trial court could reasonably find that a per-unit royalty, and not a one-time lump-sum payment, would have been the outcome of the negotiation. The familiar advantages of a per-unit royalty can readily be found present here. A per-unit royalty is a logical way to tie the amount paid for the asset to the marketplace success it helps produce, which fits the objective of measuring market value. Moreover, by using a per-unit royalty, the licensee avoids the risk of making a fixed payment that overvalues the asset before its market performance occurs, and the licensor avoids the risk that a sure up-front payment might undervalue the asset. In an important respect, it is the licensor that takes more of the risk in such an arrangement, because it typically retains little or no control over the efforts required to produce the revenue that would generate per-unit royalties.
Lucent Techs., Inc. v. Gateway, Inc.,
580 F.3d 1301, 1325-26 (Fed.Cir.2009).
No evidence required the trial court to reject a per-unit royalty here. For example, there is no overriding evidence of the kinds of difficulties of monitoring the number of sales or amount of revenue that would necessarily persuade rational negotiators that a per-unit royalty was inefficient. To the contrary, the evidence shows that the Postal Service has long conducted quarterly surveys to track the retention rate of commemorative stamps, surveys the Postal Service considered statistically accurate and relied on its business. J.A. 1844, 6605 n. 4.
The government argues that industry practice, by which it means its own past licensing practices, shows that it would not have agreed to a per-unit royalty. In the first trial on damages, the Court of Federal Claims found that Mr. Gaylord would have received a flat fee of $5,000, based primarily on the government’s evidence that it had never agreed to a per-unit royalty for stamps.
Gaylord II,
678 F.3d at 1341. The trial court has now found otherwise. The government challenges the new finding, pointing to witness testimony that, in the past, the Postal Service had found alternatives rather than agree to what it deemed excessive demands from a rights holder. But the issue is “first and foremost a question of fact,” with the trial court’s finding reviewed only for clear error based on what the record says about the particular circumstances defining the hypothetical negotiation.
Bruce v. Weekly World News,
310 F.3d 25, 29-30 (1st Cir.2002) (finding no clear error in trial court’s adoption of lump-sum license based on im dustry practice). In this case, the record does not allow us to conclude that the Court of Federal Claims committed a clear error.
We do not question the government’s suggestion that alternatives available to a potential licensee provide an important constraint in a hypothetical negotiation. “[T]he buyer will not ordinarily pay more for a license than its anticipated benefit,” a benefit measured relative to available alternatives.
Oracle,
765 F.3d at 1089;
see Aqua Shield v. Inter Pool Cover Team,
774 F.3d 766, 771 & n. 1 (Fed.Cir.2014) (applying the same concept in the patent context). But given the uniqueness of the work at issue here, including its status as a distinctively recognized symbol of the Korean War for most Americans, the trial court could properly discount the significance of potential alternatives. In fact, the-government’s witnesses testified that the Memorial was the immediate and fixed focus of the stamp-selection committee as the subject for commemorating the Korean War armistice and that all of the images considered by the committee were photographs of
The Column,
not of less recognizable portions of the Memorial or other subjects.
Accordingly, the trial court could plausibly discount as insufficiently analogous the government’s examples from other stamp-selection processes — such as replacing a stamp honoring one celebrity with a stamp honoring another. As already noted,
see supra
pp. 1368-69, taking account of significant differences in circumstances is essential to a sound evaluation of evidence of payments for other works in a hypothetical negotiation. There is only one nationally recognized Korean War memorial, and the evidence readily allows the finding that, by 2003, that memorial — and particularly
The Column
within it — was a distinctively valuable subject for a commemoration of the veterans who sacrificed through service in that war. Under these circumstances, the trial court did not err in concluding that, faced with limited alternatives, the Postal Service would have agreed to a per-unit license,
As to the particular amount of the royalty, the trial court awarded a 10%-per-unit royalty amounting to a 90/10 split of profits between the Postal Service and Mr. Gaylord. We see no clear error in the trial court’s determination of the royalty amount, considering the perspectives of the two parties to the hypothetical negotiation.
We begin with Mr. Gaylord’s perspective. In his other licenses for derivative works incorporating
The Column,
one of which was entered as a litigation settlement, Mr. Gaylord had obtained a per-unit royalty of
revenue
on t-shirts, miniature replicas, and other collectibles. J.A. 1180-81 (10% of “Product Revenue”); J.A. 1201 (10% of “wholesale sales”); J.A. 1721 (10% of “gross proceeds”). And the government now accepts that rate for the merchandise portion of the award in this case. Moreover, in a 1998 agreement, the contractor that oversaw the design and construction of the Memorial demanded and obtained a royalty “of 10% of the retail sales price” from Mr. Alii for the latter’s sales of
Real Life,
Mr. Alii believing the contractor to own the copyright to
The Column.
J.A. 1311, 1784. ' Consistent with that pre-liti-gation result, Mr. Alii agreed in a 2007 litigation settlement — of less significance for that reason — to pay Mr. Gaylord 10% of “net retail sales” for his direct sales of
Real Life.
J.A. 1599.
The per-unit royalty here gives Mr. Gay-lord, if anything, a lower share of the joint gains from the stamp — the profits — than
The Column
fetched in the foregoing licenses. In those other circumstances, where costs of production are substantial, revenues do not effectively equal profits, so that 10% of revenues represented a
higher percentage of profits, leaving the licensee with less than 90% of the jointly earned profits. For example, if a manufacturer’s costs of creating a miniature of
The Column
accounted for 50% of its revenue from that product, Mr. Gaylord’s 10% share of revenue would amount to a 20% share of the profits, with the licensee keeping 80%. For the unused stamps at issue, in contrast, the revenues are almost pure profit, and the trial court’s royalty left the Postal Service with 90% of profits.
Under the trial court’s royalty award, therefore, Mr. Gaylord would actually come out of the negotiation with a smaller share of the net gains from the joint project than was paid by the licensee of the same work in other licenses.
An additional common-sense factor supports the 10% figure from Mr. Gaylord’s perspective. At the time of the hypothetical negotiation here, Mr. Gaylord was not in a position like that of other copyright
owners
— e.g., The Walt Disney Company, an example the government cites — who might obtain appreciable indirect economic benefits from the publicity generated by a stamp and who, therefore, might have a strong economic incentive to accept less than their usual amount as a royalty. Before July 2003, when the parties agree the hypothetical negotiation would have occurred, Mr. Gaylord had retired, closed his studio, and was no longer seeking new opportunities as an artist. The record here permits a finding that Mr. Gaylord would have focused on a direct monetary. royalty as the primary compensation for licensing his work on a stamp.
We turn to the perspective from the other side of the negotiating table. The trial court could reasonably conclude that the Postal Service had sufficient incentive to agree to a 10% per-unit royalty for sales of unused stamps to collectors. The evidence shows that the Postal Service knew that retained stamps were a source of significant revenue, which was almost pure profit, and tracked estimates of that revenue on a quarterly basis. The Postal Service knew, too, that past military-themed stamps had performed well with collectors, and it expected the Korean War Veterans Memorial stamp to sell well, as evidenced by its choice to print roughly 50% more of the Memorial stamps than was typical for its commemorative stamps (86.8 million compared to 50 to 60 million). An arrangement under which it kept 90% of the profits from this opportunity was a .good economic deal.
The government argues that the availability of alternatives and its past licenses show that the royalty rate would have been lower. We have already explained why the trial court could appropriately discount the alleged effect of alternatives on the hypothetical negotiation in this case. The trial court could likewise justifiably find that the government’s past negotiations regarding other stamps are of limited probative value. As already noted, respecting economically relevant differences in evaluating evidence about other works is critical to the royalty assessment.
The government points to past licenses of sculptural and architectural works as examples of licenses containing a much lower royalty. But the works can easily be found to be quite different for licensing purposes. Some, such as the sculptures
Akari 25N
or
Mother and Child, 1944-45,
do not have the household recognition and
symbolic value of
The Column.
Others, such as the Walt Disney Concert Hall, might not have copyright protection for pictorial representations at all,
see Gaylord 1,
595 F.3d at 1380-8Í (discussing the Architectural Works Copyright Protection Act, applicable to buildings but not
The Column,
a sculptural work), and involve owners with apparent indirect interests in stamp-generated publicity. The government has not pointed to such clear evidence of past
similar
licensing situations as could warrant overturning the trial court’s finding regarding the distinctive work at issue here.
The remaining issue is whether the trial court erred in using $5.4 million as the base for the Postal Service’s revenue from sold but unused stamps. We find no clear error in the trial court’s determination.
The government argues that the trial court should not have relied on the survey data estimating $5.4 million in revenue from retained copies of
The Column
stamp, because a later study conducted by the same survey company concluded that its earlier methodology overestimated the number of unused stamps for some types of stamps. The trial court had sufficient reason not to alter the $5.4 million base. The later study focused on the retention of “forever stamps,”
J.A. 6605, and the Postal Service itself did not apply the results to non-forever stamps, like the stamp at issue here, in its internal accounting procedures, J.A. 6603. That decision squares with the Postal Service’s position that non-forever stamps “have different consumer behavior characteristics” from forever stamps. J.A. 6375. On these facts, the trial court did not err by relying on the same data the Postal Service has relied on for decades to track the retention of stamps.
The government also challenges use of the $5.4 million figure by invoking our patent-law decisions involving multicomponent products, where we have insisted that the base of a running-royalty calculation presented to a jury generally be tied realistically to the component embodying the invention.
Ericsson,
773 F.3d at 1226-27;
Virnetx, Inc.,
767 F.3d at 1327-28;
LaserDynamics,
694 F.3d at 67. But even aside from the fact that this is a copyright case, and one not tried to a jury, it is enough to say that the principles-of the cited line of patent-law authorities do not undermine the use of the $5.4 million figure based on the price of the stamp here. The stamp consists, essentially in full, of the image of Mr. Gaylord’s work and is not a multi-component product in a meaningful sense. There are two copyrights in that' stamp, but Mr. Alii has already been paid, and the government does not seek reversal to subtract that $1,500 payment. What remains is to apportion the revenues, here equaling the gains, to Mr. Gaylord’s contribution. The 90/10 split accomplishes that goal.
Conclusion
For those reasons, we affirm the judgment of the Court of Federal Claims.
No costs.
AFFIRMED.