
353 N.W.2d 374 (1984)
Julie H. RICHARDSON and William B. Richardson, Executors of the Estate of Burdick N. Richardson, Deceased, Parker H. Ficke, Successor Trustee of the Trust Under the Will of G.H. Ficke, Deceased, Charles Blair, Executor of the Estate of Lucy R. Walsh, Deceased, Appellees,
v.
PALMER BROADCASTING COMPANY, an Iowa Corporation, and Palmer Broadcasting Company, A Delaware Corporation, Appellants.
No. 83-168.
Supreme Court of Iowa.
July 18, 1984.
Rehearing Denied September 17, 1984.
Charles Blair and Edward B. Harris of Nagle, Blair & Harris, Davenport, for appellees.
R. Richard Bittner and Robert D. Lambert of Betty, Neuman, McMahon, Hellstrom & Bittner, Davenport, and Albert H. Turkus of Dow, Lohnes & Albertson, Washington, D.C., for appellants.
Considered by REYNOLDSON, C.J., and McCORMICK, SCHULTZ, CARTER and WOLLE, JJ.
CARTER, Justice.
The issue on this appeal concerns the determination of the "fair value" of 128 shares of stock in Palmer Broadcasting Company, an Iowa corporation, which were owned by shareholders who dissented from a plan of merger of that corporation with Palmer Broadcasting Company, a Delaware corporation. The dissenting shareholders are entitled to such determination of fair value under Iowa Code section 496A.78 (1977). The district court found the fair value to be $1115 per share and both the dissenting shareholders and the merged corporations have appealed from this determination.
The merger in question was effective on September 9, 1977. Prior to that time, plaintiffs were notified and offered $350 per share for the surrender of their certificates. *375 Plaintiffs rejected that offer and demanded that the corporation pay them the fair value of said shares to be determined under Iowa Code section 496A.78 (1977) which provides in part:
[S]uch corporation shall pay to such shareholder, upon surrender of the certificate or certificates representing such shares, the fair value thereof as of the day prior to the date on which the vote was taken approving the proposed corporate action....
If ... the dissenting shareholder and the corporation do not agree [upon the fair value of such shares], then the dissenting shareholder may ... file a petition... asking for a finding and determination of the fair value of such shares, and shall be entitled to judgment against the corporation for the amount of such fair value.
At the time of the merger there were 25,450 outstanding shares of stock in Palmer Broadcasting Company, an Iowa corporation. The corporation owned the WOC Broadcasting Company of Davenport, consisting of stations WOC-TV, WOC-AM, and KIIK-FM; WHO Broadcasting Company of Des Moines, consisting of WHO-TV, WHO-AM, and KLYF-FM; Gulf Coast Television and Radio Naples of Naples, Florida, comprised of cable television franchises and WNOG-AM and WCVU-FM; and Coachella Valley Television of Palm Desert, California, a cable television company.
Palmer Broadcasting Company stock has never been listed on any stock exchange. In 1976 and 1977 it paid a dividend of $3.00 per share. During the period 1967 and 1977, the corporation redeemed stock at prices between $200 and $268 per share, total shares redeemed being 1593, with sales of 4 shares to 848 shares per transaction. There have been no other sales except these redemptions. These shares were redeemed at "book value" as determined by an audit and were not necessarily "fair value." In May of 1977, 4 shares of stock were redeemed by the corporation at $250 per share.
The "book value" for Palmer Broadcasting Company for years ending 1975, 1976 and 1977 was $419.32 per share, $488.11 per share, and $562.03 per share, respectively. Net income for Palmer Broadcasting Company was $869,386 in 1974, $1,219,559 in 1975, and $1,863,660 in 1976, for a three-year average of $1,317,535. The corporation had retained earnings of $8,024,281 in 1975, $9,774,774 in 1976, and $11,733,883 in 1977.
Arthur Holt, Jr., testified as an expert witness on behalf of the plaintiffs on the issue of fair value. After reviewing the corporate balance sheets, Federal Communications Commission reports, and comparable sales, he concluded that the "fair value" of the assets of the merging corporation was $66,950,000, resulting in a per share valuation of $2630.64.
Mr. Holt was in the business of appraising in the broadcasting and electronic media industry. After earning a B.A. degree in radio and television broadcasting in 1956, he worked in and managed radio and television stations throughout the United States. He has been licensed by the Federal Communications Commission as an engineer, program director, and general manager and has owned radio stations.
Mr. Holt's definition of "fair value" is what a willing, well-informed buyer would pay a willing, well-informed seller for the property in question. Using this criterion, he separately valued each of the broadcasting components owned by the merging corporation based upon the gross sale price which he believed could be obtained if that component was sold separately. This valuation produced the following results:


   WOC-TV                     $15,500,000
   WOC-AM                       1,500,000
   KIIK-FM                      1,750,000
   WHO-TV                      18,500,000
   WHO-AM                       5,750,000
   KLYF-FM                      3,250,000
   WNOG-AM                        750,000
   WCVU-FM                      1,850,000
   Florida Cable System         9,500,000
   California Cable System      8,600,000

Mr. Holt's opinion of fair value was derived by dividing the $66,950,000 total of the *376 above sums by the number of shares outstanding in the corporation. On their cross-appeal, the plaintiffs urge that the trial court should have completely accepted the Holt testimony as the measure of fair value.
Defendant presented Charles H. Kadlec, executive vice president of Frazier, Gross, & Kadlec, Inc., a financial appraisal firm in Washington, D.C. as an expert valuation witness. Mr. Kadlec's interpretation of "fair value" under section 496A.78 is the extent to which a shareholder has been deprived of an opportunity to share in the future earnings of a going corporation. Using a price-earnings ratio method, he calculated the stock price per share based upon a multiple of earnings. That multiple was developed by analyzing the price-earnings ratio of eight publicly traded corporations in the broadcast industry. These eight corporations were chosen from an original list of thirteen after five had been eliminated as being "atypical."
The stock prices for these eight corporations were matched with year-end earnings for the years 1974, 1975 and 1976 to produce the price-earnings ratio which the witness found to be relevant. Based on these comparisons, the witness fixed the price-earning ratio for the merging Iowa corporation at .072 which, when applied to what the witness believed to be a representative annual return based on adjusted averaging of several years earnings produced a price per share of $527. After making this calculation, Mr. Kadlec then attempted to discount the value which it produced because the stock was not readily convertible to cash, and represented a minority interest in a closely held corporation. He believed these circumstances justified a discount in a thirty to forty percent range.
In weighing the competing testimony, the trial court determined that plaintiffs' witness, Arthur Holt, made a number of mathematical errors in the underlying data upon which his conclusions were based. Although the witness subsequently noted these errors, it was his view that they made no difference in his ultimate conclusions. The trial court believed that this witness's valuations were overstated and, in addition, that the method employed by the witness in arriving at his conclusions did not produce an accurate measure of "real value" for purposes of the determination required by section 496A.78.
The trial court concluded that "fair value" under section 496A.78 was intended to compensate the dissenting shareholder for the extent to which he was deprived of an ownership interest in a going concern. As such, the trial court concluded that this was not simply a right to future earnings but a proportionate interest in a going concern to be determined by balancing the net asset value of the company and its investment value.
In applying these conclusions, the trial court found a corrected net asset value to allow for Holt's overstatements was $1335.95 per share. With respect to investment value, the trial court determined that the Kadlec ratio of .072 was not an accurate indicator and that a price-earnings ratio of .10 was more acceptable. The trial court applied that ratio to a three-year average net income for the period 1974-76 and produced a valuation of $570.70 per share. This figure was adjusted upward by adding retained earnings which the trial court believed to be an appropriate adjustment for reaching a final value by use of the price-earnings ratio method. The value thus produced was $967.78 per share. Using a weighted balancing process which attributed sixty percent to the adjusted price-earnings ratio determination and forty percent for the adjusted net asset valuation, the trial court ultimately arrived at the price of $1115 per share as the fair value under section 496A.78.
Nowhere in the Iowa Business Corporation Act is the term "fair value" defined. The section is derived from Model Business Corporations Act section 74 and the official comment to that act simply states: "The cases indicate that there is no definite rule for determining `fair value' but that the proper result in each case will depend upon the particular circumstances of the corporation *377 involved." In 13 W. Fletcher, Cyclopedia of the Law of Private Corporations, section 5906.12, at 286 (rev. perm. ed. 1980), the author states:
Courts have generally held that no one factor governs the valuation of the shares but that all factors, such as market value, asset value, future earning prospects, etc., should be considered. Generally under the statutes, the value of the stock of a dissenting minority shareholder is to be determined as of the day prior to the date on which the vote was taken authorizing the sale of the corporate assets. The wording of the statutes is not uniform as to the valuation of the shares of the dissenters, some referring merely to "value," while others refer to "fair value," "fair cash value," "market value," or "full market value." The term "fair value" as used in the valuation of shares of a dissenting stockholder is not a rigid criterion but establishes a flexible general standard for fixing the value between parties who are unwilling or unable to agree. The judicial determination of value must be an informed judgment, but "fair value" is not susceptible of determination by any precise mathematical computation and no one formula or figure is binding or conclusive. Precise rules as to the method of determining the value cannot be laid down. In fact most of the statutes are silent as to the meaning of "value" or the tests to be applied by the appraisers or the courts in determining value. The effectiveness of the remedy, in fact its very existence, thus depends on the meaning which the courts ascribe to the word value.
The resultant valuations have generally concentrated on three principal elements: market value, net asset value, and investment value.
(Footnotes omitted.) Other authorities supporting the view that market value, net asset value, and investment value may all be relied on in determining "fair value" include the following. General Securities Corp. v. Watson, 251 Ark. 1066, 477 S.W.2d 461 (1972); Gibbons v. Schenley Indus., Inc., 339 A.2d 460 (Del.Ch.1975); Francis I. duPont & Co. v. Universal City Studios, Inc., 312 A.2d 344 (Del.Ch.1973), aff'd 334 A.2d 216 (Del.1975); TriContinental Corp. v. Battye, 31 Del.Ch. 523, 74 A.2d 71 (1950); Flarsheim v. Twenty-Five Thirty-Two Broadway Corp., 432 S.W.2d 245 (Mo.1968); Tome Land & Improvement Co. v. Silva, 83 N.M. 549, 494 P.2d 962 (1972); Endicott Johnson Corp. v. Bade, 37 N.Y.2d 585, 376 N.Y.S.2d 103, 338 N.E.2d 614 (1975); Brown v. Hedahl'sQ B & R, Inc., 185 N.W.2d 249 (N.D.1971); In the Matter of Northwest Greyhound Lines, 41 Wash.2d 672, 251 P.2d 607 (1952); N. Lattin, The Law of Corporations § 166 (1971); Note, Valuation of Dissenters' Stock Under Appraisal Statutes, 79 Harv. L.Rev. 1453 (1966); Comment, 55 Mich.L. Rev. 689 (1957); Annot., 48 A.L.R.3d 430 (1973).
Some cases have indicated that a weighted formula involving application of each of the elements of market price, investment value and net asset value must be applied in order to produce a correct finding of "fair value." Root v. York Corp., 29 Del.Ch. 351, 50 A.2d 52 (1946); Lucas v. Pembroke Water Co., 205 Va. 84, 135 S.E.2d 147 (1964); see Annot., 48 A.L.R.3d 430, 439 (1973). This view has developed from the application of statutes which provide for a strictly prescribed system of appraisal as a condition to judicial review. These systems have been described as follows:
The percentages are understood to be only crude estimates and are commonly expressed in round numbers. Nevertheless, such a procedure forces the appraiser to analyze the relative importance of each element and to justify his conclusions, and it affords a more concrete basis for review by the courts. It guards to some extent against the possibility that the figures will be merely a compromise between the contentions of the parties or an intuitive judgment.
Note, Valuation of Dissenters' Stock, 79 Harv.L.Rev. 1453, 1469 (1966).
*378 In Multitex Corporation of America v. Dickinson, 683 F.2d 1325, 1328-29 (11th Cir.1983), the court noted the distinction between statutory appraisal schemes and de novo fact-finding by a court. Courts, unlike appraisers, are forced to decide cases based upon the evidence presented. Parties to "fair value" cases in the courts may be unable to find witnesses whose testimony will correspond to each of the recognized elements of fair value. Moreover, testimony as to a particular element may appear to the court to be so unreliable that it has no place in influencing the final result. Diefenthal v. United States, 343 F.Supp. 1208, 1213-14 (E.D.La.1972).
We believe that in relying on net asset value, investment value, and market value of the shares of dissenting stockholders, courts should view these three approaches to valuation as relevant factors rather than essential components. This accords with the view which we announced in Robbins v. Beatty, 246 Iowa 80, 91, 67 N.W.2d 12, 18 (1954) in determining the value of dissenting shareholders stock under Iowa Code section 491.25 (1946). That statute used the phrase "real value" rather than "fair value," but because the purpose of the valuation thereunder is basically indistinguishable from the purpose of section 496A.78, we believe our interpretation in Robbins is persuasive in the present case. We said in that case "[i]t is unwise to attempt to state every factor that may bear on value of stock in a particular case." 246 Iowa at 91, 67 N.W.2d at 18.
We applied the valuation concept announced in Robbins in Woodward v. Quigley, 257 Iowa 1077, 133 N.W.2d 38 (1965). The Woodward case was also decided with reference to the phrase "real value" employed in Iowa Code section 491.25 (1946). We considered each of the three popular approaches to valuation, i.e., market value, investment value, and net asset value and concluded:
[W]e recognize this matter cannot be determined by any exact mathematical computation and no one formula or figure is binding or conclusive.... The decision is, in the final analysis, a matter of judgment based on all material evidence and consideration of all relevant factors which may influence the valuation.
257 Iowa at 1103, 133 N.W.2d at 53. In Woodward, we rejected valuation techniques which were unsupported by evidence and on their face unreliable. We took the view of the case "which is the easiest to understand and furnishes a reliable guide to value."
The positions of the parties on appeal may best be described as urging the adoption of the valuation testimony of the party's own expert witness and challenging the testimony of the opponent's witness as unreliable. Because the action in the district court was tried by equitable proceedings as required by section 496A.78, our review under rule 4 of the Iowa Rules of Appellate Procedure is de novo.
As was the case in Woodward, we find certain of the approaches to valuation which are contained in the record do not present a reliable measure of what the plaintiffs have lost because of their unwillingness to go along with the controlling shareholders. There was no evidence of market value presented. With respect to net asset value, the district court recognized that Mr. Holt's determination was "overstated." We agree that this witness's values were overstated and, in addition, we do not believe the methods which were used lead to net asset value of the corporation. Unlike the trial court, however, we are unable to come up with our own adjusted net asset values based upon any principled application of the record.
There is some indication in the district court's opinion that its weighted formula approach was designed to reflect its view that a dissenting shareholder is entitled to both a right to future earnings and a proportionate interest in a going concern. Based on this conclusion, it viewed the two approaches as leading to components of fair value. We believe that all the major methods of appraising dissenters' stock are designed to independently produce the full *379 measure of the fair value of the stock rather than a component of fair value. Accordingly, rather than use an arbitrary adjustment as a weighted component in arriving at our ultimate determination, we simply disregard net asset value on the basis that it may not be determined on the existing record.
The testimony of Mr. Kadlec with respect to an investment value approach to the measure of fair value employed verifiable data to arrive at a value pursuant to a two-step investment approach. This is an approved investment valuation technique. In re Valuation of Common Stock of Libby, McNeill & Libby, 406 A.2d 54, 65-66 (Me.1979). It bears some similarity to the method of valuation we applied in Woodward. The record indicates a price-earnings ratio for the merging corporation of between .053 and .083. Mr. Kadlec used an average price-earning ratio of .072 in his calculations. The district court believed the .072 ratio was not an accurate indicator. The basis for this determination does not appear in the court's findings other than the fact that the comparable data was derived from publicly held corporations as contrasted with the defendant, which is a closely held corporation.
Assuming some adjustment upward of the price-earnings ratio is required, the trial court's action in adopting a .10 priceearnings ratio finds no support in the evidence. There is no evidence concerning price-earning ratios of greater than .083.[1] Giving deference to the trial court's determination that the price-earnings ratio should be adjusted upward, we believe that it should not be permitted to exceed the highest price-earnings ratio for which there is testimony in the record. Accordingly, we approve a ratio of .083 which, when applied to the three-year average net income for the period 1974-76, produces a value of $623.73 per share.
We agree with the trial court's conclusion that Mr. Kadlec's efforts to adjust downward the value produced by a priceearnings ratio because plaintiffs' shares represent a minority interest is, as we recognized in Woodward, contrary to the spirit of "fair value" determinations for dissenting minority shareholders. We cannot, however, completely accept the court's additional conclusion that the per share value which is yielded by the price-earnings ratio should be adjusted upward based upon retained earnings of the corporation. The injection of retained earnings into a valuation based on price-earnings ratio is not typical of the two-step investment value approach. See In re Valuation of Common Stock of Libby, McNeill & Libby, 406 A.2d at 65-66.
We did approve an upward adjustment of a two-step investment value appraisal in Woodward based on excessive retained earnings. Perhaps the trial court viewed some of the retained earnings of the merging corporation in the present case as excessive based upon the pattern of dividend payment. Such conclusion, however, would not justify the court's per share adjustment which was produced by dividing the total retained earnings by the number of shares.
If .083, the maximum price-earnings ratio which we find to have support in the evidence, is applied to the average earnings for the years 1974-76, this produces a per share value of $623.73. Giving some deference to the trial court's views on excessive retained earnings warrants our adjusting this value upward to $725 per share. We find this provides a reliable measurement of the value of plaintiffs' shares. As recognized in Woodward, the decision is in the final analysis a matter of judgment, and the foregoing opinion constitutes our finding as to the per share value of the stock which is most clearly supportable by the record.
The judgment from which the parties have appealed is modified so as to reflect a fair value of $725 per share for each of the plaintiffs 128 shares. As so modified, the *380 judgment is affirmed. Costs on appeal are divided equally between plaintiffs and defendants.
AFFIRMED AS MODIFIED.
NOTES
[1]  Reference in the trial court's findings to testimony of a .125 price-earnings ratio by plaintiffs' expert witness finds no support in the record.
