OPINION OF THE COURT
ROSENN, Circuit Judge.
The issues raised on this appeal reflect some of the subtle tax hazards lurking in corporate liquidations. The issues turn on the application of section 334(b)(2) of the Internal Revenue Code of 1954, 26 U.S.C. § 334(b)(2) (1976), dealing with the basis of property received in liquidation of a subsidiary. Section 334(b)(2) provides that “[i]f property is received by a corporation in a distribution in complete liquidation of another corporation,” and if certain requirements are met, “then the basis of the property in the hands of the distributee shall be the adjusted basis of the stock with respect to which the distribution was made.” Basis in the assets is to be allocated in proportion to the fair market values of the assets. 26 C.F.R. 1.334 — l(c)(4)(viii) (1978). In this case, both parties agree that the prerequisites to the applicability of section 334(b)(2) have been met. The dispute hinges on the proper allocation of basis to the property transferred.
I. THE PREDICATE FOR ASSET BASIS
In early 1970, R. A. Gilmour and Robert M. Smith entered into an agreement for the sale of all assets of the Gilmour Company, a manufacturer of lawn and garden equipment, to R. M. Smith, Inc. (“Smith”). At the request of Gilmour, the transaction was changed to a sale of all Gilmour Company stock for a total price of $3,780,550. The agreement did not include identification of the individual values of the assets being transferred. Payment for the stock was as follows: cash in the amount of $780,550; a promissory note for $300,000 at 71/2 percent; and an installment note for $2,700,000 at 4 percent. Smith also assumed existing liabilities of Gilmour Company which, by the time of liquidation, totalled $272,180.93. Although the transfer was effective as of February 1, 1970, the transaction was not consummated until March 24,1970. Within a week, Gilmour Co. was liquidated with Smith retaining the assets. After the liquidation, Smith continued to use the trade names, trademarks, and customer lists of Gilmour Company.
Pursuant to section 334(b)(2) and the underlying Treasury regulations, Smith allocated the refined adjusted basis in the stock to the assets received upon liquidation. Included in this allocation was $1,833,392.53 assigned to depreciable intangible assets, namely six patents and an unpatented invention. No allocation was made to nondepreciable intangible assets (e. g., goodwill) which Smith regarded as having no fair market value. In subsequent tax returns, Smith claimed deductions for the amortization of the patents and invention based on the amount assigned.
In reviewing Smith’s tax return the Internal Revenue Service (“IRS”) concluded that only one patent had any basis and that amounted to $10,000. Thus, the Commissioner viewed the amortization deductions as excessive and issued a notice of deficiency. Smith petitioned the tax court for review of this determination.
The primary issue at trial was the fair market value of the patents and the invention. Each side presented an expert witness who employed the identical method of computing the values but used different variables resulting in substantially differing figures of values. The tax court determined that although the identical method employed by the experts was appropriate, each had either overestimated or underestimated certain variables. The tax court fixed the correct valuation at $860,000, a sum between the figures proposed by the witnesses.
The tax court proceeded to find that Smith had failed to assign basis to nondepreciable intangible assets — goodwill, trademark, trade name, and customer lists. The court determined that the value of these intangible assets must be equal to the total purchase price of the stock less the sum of the known values of all other assets. This is based on the assumption that the best evidence of the fair market value of all of the assets is the total price paid for the stock. The court thus determined that the goodwill was worth $1,225,697.41. No attempt was made to fix specific values to the individual components of this total as all are nondepreciable.
The net effect of the court’s holdings is that approximately $1.23 million in basis previously assigned to depreciable intangible assets are now assigned to nondepreciable assets, resulting in a reduction in allowable depreciation and amortization deductions.
On appeal, Smith challenges the tax court determinations of the values of the patents and of the nondepreciable intangibles. We affirm.
II. PATENTS AND INVENTION
In support of its assessment of the fair market value of the patents and invention, Smith presented two witnesses qualified to address this issue. Smith’s president adopted a method of calculating this figure which the court found “wholly unacceptable” and refused to give weight to the testimony. However, the other witness, a financial appraiser, utilized an approach which the court found appropriate. His analysis yielded an aggregate fair market value for the six patents of $1,867,000.
The Commissioner presented one expert witness, a patent lawyer. The witness employed the same method as Smith’s expert, but his gross value for the patents was only $424,647.
The court found that correct valuation of the patents lay in between the two figures presented by the experts. Specifically, it found the royalty rates and projections of future sales used by Smith’s expert to be overly optimistic and those variables applied by the IRS expert to be unduly pessimistic. The judge concluded:
After giving careful consideration to testimony received, the valuation reports submitted, and the parties’ arguments on brief, we have done the best we could to make a reasonable determination of the fair market value of the patents . . ., and, using the same approach used by [both experts], have computed an aggregate amount of $745,000 ....
The court also found the unpatented invention to have a value of $115,000. Smith’s expert set the figure at $130,000, as opposed to $10,000 by the IRS expert. (The value of this asset is not an issue on appeal.) Thus, the total value of the depreciable intangible assets was determined by the tax court to be $860,000.
Smith’s contention on appeal attacks the appropriateness of the tax court assigning this value to the patents. Smith states that it had the burden of proving that the Commissioner’s determination of the gross values, which initially amounted to $10,000, was arbitrary. Having established that the calculation was in error, Smith was not required to prove the correct figure. The Commissioner had the burden of establishing the proper valuation and thus the actual tax owed. This much is an accurate statement of the law. Helvering v. Taylor, 293 U.S. 507, 55 S.Ct. 106, 79 L.Ed. 623 (1935); Federal National Bank v. Commissioner of Internal Revenue, 180 F.2d 494, 497 (10th Cir. 1950).
Where Smith’s argument fails is in suggesting that the refusal of the tax court to accept the Commissioner’s evidence requires this court to reverse the trial judge with instructions to expunge the deficiencies. The teaching of Helvering v. Taylor, supra, and Federal National Bank v. Commissioner of Internal Revenue, supra, is that the appropriate remedy in the absence of evidence of proper valuation is a remand to allow for additional evidence to be presented. In this case, however, sufficient evidence was introduced to allow the tax court to reach a reasonable conclusion. The court is not limited to simply choosing one of the two values proffered. It is appropriate for it to evaluate all of the evidence and to make an independent determination that does not necessarily accept the valuation of either party. This is not the first time the tax court has used this approach. See Philadelphia Steel & Iron Corp. v. Commissioner, 23 T.C.M. 558, 564-65 (1964), aff’d per curiam, 344 F.2d 964 (3d Cir. 1965). The court’s finding of fact with respect to valuation is to be reviewed under the clearly erroneous standard, Commissioner of Internal Revenue v. Duberstein, 363 U.S. 278, 291, 80 S.Ct. 1190, 4 L.Ed.2d 1218 (1960), and in this case must be sustained.
III. NONDEPRECIABLE INTANGIBLE ASSETS
A. Is the Residual Value Method Appropriate for Determining the. Value of Goodwill in this Case?
According to Treasury Regulation 1.334-l(c)(4)(viii) (1978), the allocation of taxable basis to the assets received in a section 334(b)(2) transaction shall be made in proportion to the fair market value of the assets. Thus, before basis is allocated, a determination of fair market value of all of the assets must be made. Having found that Smith received assets not listed on its balance sheet (goodwill, trademark, trade name, and customer lists), the tax court proceeded to assess their worth.
Rather than attempting an independent appraisal of these intangible assets, the court held that their value could be ascertained by the residual valuation method. Simply put, this method assumes that the sum of the fair market values of all assets acquired equals the price paid for the acquisition of these assets. The formula is the price paid minus known values of assets equals unknown values of the remaining assets. Because the only assets without known values are the nondepreciable intangible assets, this formula can be applied readily to this case and yield the following results:
total consideration paid
(price plus liabilities assumed) $4,052,730.93
less known fair market values of assets
(as of date of liquidation) - 2,827,033.52
fair market value of intangible assets $1,225,697.41
The tax court also found that the evidence justifies allocating this sum to nondepreciable intangibles. Upon liquidation, Smith received an ongoing business that had been profitable for a number of years. It inherited a sound marketing program established in North and South America. It continued to use Gilmour Co.’s trade names which the tax court found were “well and favorably known to the users of lawn and garden equipment and promoted sales of these products.”
Smith challenges the appropriateness of the utilization of the residual value method to the facts of this case. Smith presents basically two arguments in support of its contention.
First, Smith points out that two months elapsed between acquisition of the stock and liquidation of the corporation. During this time the value of the assets had to change as a result, for example, of normal appreciation and depreciation. If the value of the tangible assets decreased, then, by applying the residual value method (using total price paid on February 1 and the fair market value of assets on March 31), the assigned value to goodwill would be greater (and vice versa), even though the value of goodwill should remain constant. Although the argument is logically sound, the determination of fair market value, as a practical matter, is not an act of precision but only a matter of approximation. Minor fluctuations in asset values during the two months would be insignificant. Although proof of substantial variations would require adjustments in the resulting goodwill figure, this is not a sufficient reason for altogether abandoning the residual value method. In any event, no such proof was presented in this case.
Appellant’s second argument is that the tax court permitted “the Commissioner to introduce as new matter residuary value goodwill after the record was closed” and, as a result, Smith was denied an opportunity to present testimony concerning goodwill. If true, this claim is significant.
As pointed out earlier, the theoretical underpinning of the residual value method is that the total price paid for the stock equals the sum of the fair market values of all of the underlying assets. Although this is a sound principle in economic theory, in reality it has its shortcomings. Specifically, it fails to take into account the common situation when one party to the transaction achieves a bargain. If the purchaser of the stock obtains a “good deal,” then the residual value method would undervalue the goodwill. If, on the other hand, the price paid is too high, then the computation will result in a correspondingly inflated goodwill figure. This problem does not require rejection of the residual value method — price paid is strongly probative, albeit not conclusive, of fair market value. However, it does suggest that the court consider evidence which would require alterations in the figure or abandonment of the formula altogether.
The cases applying the residual value method recognize this limitation. In Jack Daniel Distillery v. United States, 379 F.2d 569, 180 Ct.Cl. 308 (1967), the parties to the sale of stock had agreed to a value for goodwill after actively bargaining in good faith and settling on the value of the other assets. The court rejected an alternative computation for goodwill advocated by the IRS and used the residual method to confirm the figure agreed to by the parties. “The residuary method, though lacking in precision for use in all cases, may in a proper case be accepted as the reasonable way to value goodwill.” Id. at 579.
In Moss American, Inc. v. Commissioner, 33 T.C.M. 1121 (1974), the question was whether the price of the stock was equal to the fair market value of the underlying assets. The IRS argued that the aggregated fair market values exceeded the price by $12.5 million. However, the court concluded that the Commissioner’s evidence was insufficient to treat the purchase of stock as a bargain purchase. “If such a purchase had, in fact, occurred, we would, of course, be faced with a different question and respondent’s position would have merit.” Id. at 1127-28.
In Plantation Patterns, Inc. v. Commissioner, 29 T.C.M. 817 (1970), aff’d 462 F.2d 712 (5th Cir. 1972), and Philadelphia Steel & Iron Corp. v. Commissioner, supra, as in the cases above, the court allowed evidence demonstrating the inappropriateness of the residual method to be presented. Only after deciding that the evidence was unpersuasive did the court determine the value of goodwill by this formula. See also 10 Mertens, Law of Federal Income Taxation § 59.37 (residual method of valuing goodwill “should be followed with caution and in most cases the value of the goodwill should be supported by other collateral evidence”).
The rule to be drawn from these cases is that the residual value method is an appropriate means for deriving the value of intangible assets as long as the total price paid and the values of all other assets are known. The resultant figure, however, is not to be deemed conclusive proof of the unknown value. Evidence suggesting that a fair deal was not reached should be permitted. When appropriate, adjustments should be made in the value assigned to the intangible assets, or a different basis for deriving the figure should be substituted. Collateral evidence supporting a goodwill valuation by the residual method also should be received.
It is evident that in this case the tax court found support for its conclusion that the residual method was appropriate. It considered carefully the evidence which proved that intangible assets of substantial value had been received by Smith upon liquidation. If, however, the court precluded Smith from presenting contrary evidence to show that the price paid was not equal to fair market value, then it committed error. Despite its protestations, our review of the record leads us to the conclusion that Smith did have the opportunity to present such evidence. At trial, Smith argued that any intangible assets received upon liquidation were of nominal value only. The tax court stated in its opinion:
And, although, petitioner steadfastly maintains that we should sustain its asserted valuation of the patents and invention, petitioner belatedly argued that if we should decide that the fair market values of the patents and invention were of lesser amounts than petitioner has claimed, nevertheless, the value of the trademark and goodwill should not be determined under the residual method. Rather, petitioner urges, as best we can understand it, that a specific determination of the fair market value of the trademark and goodwill must be made . (Emphasis supplied.)
Thus, it appears that Smith anticipated the possibility that the court might find value for goodwill and it should have presented evidence on this point at trial.
B. Did the Tax Court Calculate the Residua] Value Goodwill Correctly?
Smith argues that the tax court erred in the starting point for the residual method— the price paid for the stock. The form of payment for the stock was as follows:
1. $780,550 cash;
2. $300,000 promissory note at l}h percent for 3 years;
3. $2,700,000 installment note at 4 percent with no principal payments during first three years;
4. $272,180.92 liabilities assumed.
Smith contends that in determining the price paid, the $2,700,000 note should not have been assigned face value, but that it should have been discounted at its market value, $1,990,943. If this argument is accepted, then the resultant value of the intangible assets would be lower. This would mean that there would be a greater allocation of basis to the depreciable assets, and, thus, greater depreciation deductions. Smith, however, fails to discern the adjustment that then must be made to the $2,700,000 note when calculating the basis in the stock. If the adjustment is made in both places, then the net effect would be nil.
It appears that the difference of over $700,000 between the face value of the note and its asserted market value results from a decision by Smith and Gilmour to incorporate this economic or unstated interest into the principal part of the installment note. If the face value of the note did reflect its market value and the note carried a market rate of interest, then the $700,000 would have been included in interest payments on the note and would have been deductible by Smith. Of course, an equal amount would have to be treated as ordinary income by R. A. Gilmour. By including the sum in the principal portion of the note, the $700,000 becomes a capital gain to Gilmour. The question involving Smith’s tax return is whether this amount should be allocated to basis spread among all of the assets received in liquidation allowing Smith to depreciate much of it, or whether it should be assigned to goodwill, a nondepreciable asset. We believe including it in goodwill is the appropriate result in this case.
Our independent research has failed to uncover a case which addresses the issue: under the residual method, in calculating the total purchase price for capital stock partially paid by a promissory note, should adjustments be made to the face value of the note when it is apparent from the low interest rate on the obligation that the face value does not equal the market value? The argument in favor of adjustment has intuitive appeal in that greater precision is being used to derive the fair market value of the unknown. However, the effect desired by the taxpayer (increase in bases of depreciable assets) only comes about because less precision is demanded in calculating the basis in the stock. The Internal Revenue Code, 26 U.S.C. § 483 (1976), and the underlying regulations, 26 C.F.R. §§ 483.1-483.2 (1978), require that the treatment of notes for tax purposes be adjusted for unstated interest when the rate assigned is less than four percent. Because the interest rate of the note in question is four percent, no adjustment is required; nor do we believe one is necessary. We hold that both calculations should be made at face value. See Commissioner of Internal Revenue v. Danielson, 378 F.2d 771 (3d Cir. 1967). Thus, the tax court did not err.
The final contention made by Smith is that the tax court erroneously included in the total price paid for the stock the federal income tax liability ($59,766) of the acquired Gilmour Co. for the period between acquisition of the stock and liquidation of the corporation. By application of the residual value method, the increase in the price paid results in an increase in the value assigned to goodwill (see formula, supra at 252). As noted earlier, an increase in the amount treated as goodwill results in a lower allocation of basis to depreciable assets. Because the tax court was attempting to derive the value of goodwill as of the date of acquisition, the inclusion of the subsequent tax liability in total price paid is inappropriate. The interim income tax liability in no way would increase the value of goodwill at the time of acquisition.
Reversal is unnecessary, however, because the tax court also erred in deducting from the total price paid the interim earnings of the Gilmour Co. ($109,700). The interim earnings flowed into the company after acquisition and became part of the value of the assets known (the subtrahend in the residual value formula). Increasing the subtrahend resulted in a reduction of the value assigned to goodwill. The events transpiring during the interim period could not have altered the value of goodwill as of the date of acquisition and again should not have been included in applying the residual value formula. The amount of interim earnings exceeded the interim income tax liability and more than offset the error claimed by the taxpayer. The IRS has not appealed this error in favor of the taxpayer, and thus, the tax court’s judgment need not be disturbed.
IV. CONCLUSION
We perceive no reversible error by the tax court and its decision will be affirmed. Costs to be taxed against the appellant.
. “Refined adjusted basis” is a term adopted by the tax court referring to basis that has gone through two adjustments.
. Prior to trial the parties agreed to the fair market value of all the tangible assets received by Smith upon liquidation.
. The president of Smith, Robert M. Smith, also testified on this issue; however, his testimony was discredited by the trial judge.
. The Commissioner abandoned his original position that the total value of these assets was $10,000.
. It seems incongruous for the court to approve as large a sum as $1,225,697.41 for the goodwill of a manufacturing concern whose gross revenue only approximates $2,000,000 per annum. This allocation, however, was the direct result of the amount fixed in the installment note by the parties to the sale and purchase. Smith’s president was a practicing accountant fully familiar with the affairs of the Gilmour Company, having been its accountant. He should have recognized the possibility that this result might flow from the agreement. When the parties themselves have deliberately fixed the values to be placed on the elements of a transaction, they are bound to accept the Commissioner’s construction of these values, unless they can prove mistake, undue influence, fraud, or duress. Commissioner of Internal Revenue v. Danielson, 378 F.2d 771, 775 (3d Cir. 1967).