Under the Massachusetts Business Corporation Act (Act),3 G. L. c. 156D, as appearing in St. 2003, c. 127, shareholders have the right to dissent from a proposed merger and demand that the corporation pay “fair value” for their shares if the merger is accomplished. G. L. c. 156D, § 13.02(a). The principal issue presented by this appeal is whether the trial judge correctly applied the appraisal provision of the Act, G. L. c. 156D, § 13.30, in determining the “fair value” of stock owned by a minority shareholder of a close corporation following a cash-out merger.4 For the reasons that follow, we hold that the case law governing the determination of “fair value” under the predecessor law to the Act applies to the determination of “fair value” under § 13.30. We further conclude that the trial judge’s decision to rely on evidence she heard during a jury trial involving other claims between the parties,5 after providing the parties an opportunity to offer new evidence, was in compliance *58with the Act and the constitutional rights of the minority shareholder. Finally, we hold that in the circumstances of this case, the judge’s decision to appraise the value of the minority’s shares as the pro rata percentage of the net selling price of the business was the “fair value” for those shares under the appraisal statute, G. L. c. 156D, § 13.30(d).

Factual background. Robert Spenlinhauer (Robert), acting as the executor of the estate of Georgia Spenlinhauer (estate), brought the present case against the defendants, Spencer Press, Inc. (Spencer Press), Spencer Acquisition, Inc., and their former majority owners, Stephen Spenlinhauer and John Spenlinhauer (collectively, defendants), based on a dispute regarding the valuation of the estate’s one percent interest in Spencer Press, which Spenlinhauer was forced to sell as a result of the cash-out merger.

Spencer Press was founded in the 1940’s by John E. Spenlinhauer, Jr. (John Jr.), and his wife, Georgia M. Spenlinhauer (Georgia), parents of the three brothers who are parties in this case. John Jr. owned ninety-nine percent of the stock and Georgia owned one percent. After the death of John Jr. in 1972, his three sons each owned thirty-three percent of the shares in Spencer Press, while Georgia retained her one percent share. In 1987, after disputes arose among the brothers, John and Stephen bought out Robert’s stake in the company. Shortly thereafter, Georgia ceased active involvement in the business but remained a shareholder.

Georgia’s health deteriorated in the late 1980’s and early 1990’s. Robert began taking care of her and in 1994, Georgia executed a new will, which left substantially all of her assets to Robert, including her share in Spencer Press. Robert was also named as the executor of her will.

In December of 2004, John and Stephen received from R.R. Donnelly & Sons, Inc. (Donnelly), a competitor, a preliminary offer to purchase Spencer Press. The price was $60 million, minus outstanding debts of around $30 million in addition to other company obligations and escrows. Stephen discussed the offer with Robert.

*59On February 4, 2005, Georgia died. John initially wanted to contest Georgia’s will, which he had not previously seen, but Stephen convinced him not to based on Robert’s assurances that he would not obstruct the sale of Spencer Press and that he was satisfied with the estimated $350,000 he would receive as a pro rata share of the purchase price through the estate. However, in July of 2005, after the deadline for contesting Georgia’s will had passed, Robert informed Stephen and John that he would not accept the pro rata share and instead demanded $3 million for the estate’s one percent interest. Stephen was upset by Robert’s actions, but nonetheless continued to negotiate the purchase of the estate’s share. During the course of these negotiations, one of Stephen’s attorneys informed Robert that Spencer Press could perform a cash-out merger under Massachusetts law to obtain the estate’s interest without Robert’s consent. Nevertheless, Stephen first offered Robert $500,000 and then $750,000 in exchange for the one percent share, but Robert rejected both offers.6

On October 5, 2005, Stephen and John completed a cash-out merger, whereby Spencer Press was merged into its recently formed parent company, Spencer Acquisition, Inc., which was subsequently renamed Spencer Press, Inc. On October 15, 2005, Robert was notified for the first time of this transaction; that the estate’s share would be converted into $375,000; and of his rights under the Act. In November, 2005, the recently formed Spencer Press signed an agreement to sell its business to Donnelly.

Procedural background. On November 30, 2005, Robert brought the instant action, challenging several actions in the cash-out merger and sale. The parties agree that the defendants complied with the statutory requirements contained in G. L. c. 156D, § 11.05, in executing the cash-out merger. As required by G. L. c. 156D, § 13.30(a)7, upon demand from Robert, the *60defendants filed a counterclaim for an equitable appraisal of the estate’s stock and, inter alla, for intentional misrepresentation.

The parties conducted two years of discovery, which ended on December 31, 2007. The trial judge granted summary judgment in favor of the defendants on all of Robert’s claims except his breach of contract claim. The appraisal claim brought by the defendants was stayed at Robert’s request as executor until after a trial on Robert’s breach of contract claim and the defendants’ misrepresentation claim. A jury returned a verdict for the defendants on Robert’s claim and for Robert on the misrepresentation claim.

After trial, the judge gave the parties an opportunity to comment on the procedure she should follow in adjudicating the appraisal claim,8 and directed them to submit an offer of proof as to new evidence they would like to offer beyond what was presented at trial. In particular, the judge stated to counsel for Robert:

“[I]f it’s Mr. Marshall [plaintiff’s expert] that you would be offering, ... or something else, if there’s some documents or some testimony, or whatever it is you might want to offer, I want to know what it is, and I’d like it in affidavit form or documentary form or some form beyond just a synopsis and a memorandum. The point to focus on I think is what should I do . . . as a procedural matter, other than simply mak[e] findings based on the evidence presented at trial, and why should I ... do something other than just male[e] findings.”

The trial judge confirmed this request in an order dated August 7, 2009. The defendants responded by stating that they would rely on the evidence presented at trial. Robert, on the other hand, requested additional discovery without making an offer of proof that there was any new evidence.9 The judge indicated that she *61had just completed a trial in which there was substantial evidence relating to the value of the stock of Spencer Press and the fairness of the purchase price, and in the absence of an offer of proof of any further evidence there was no reason to believe that the purchase price for the stock was “anything other than fair and favorable.”10 The judge concluded that the fair value of the estate’s interest was one percent of the net sale price of Spencer Press to Donnelly.

Discussion. 1. Appraisal procedure. Robert claims that the trial judge violated the Act’s appraisal provision, G. L. c. 156D, § 13.30, and deprived him of due process under the State and Federal Constitutions by not allowing an evidentiary hearing on his appraisal claim following the jury trial.

a. Obtaining an appraisal. When a subsidiary corporation is merged with its parent under G. L. c. 156D, § 11.05, a shareholder who is dissatisfied with the amount of the payment under G. L. c. 156D, § 13.26(a), is entitled to an equitable proceeding pursuant to § 13.30 to determine the “fair value” of his or her shares as defined by § 13.01. G. L. c. 156D, § 13.02(a). This was the procedure by which Spencer Press was merged with Spencer Acquisition, Inc., and the estate’s appraisal rights were correspondingly triggered.

b. Appraisal notice and form. When corporate action is taken *62that triggers appraisal rights under G. L. c. 156D, § 13.02(a), the corporation must deliver a written appraisal notice and form to all shareholders who meet certain criteria within ten days after the corporate action. G. L. c. 156D, § 13.22(a), (b). In mergers between subsidiaries and parents pursuant to G. L. c. 156D, § 11.05, the parent company must deliver a written appraisal notice and form “to all record shareholders who may be entitled to assert appraisal rights.” G. L. c. 156D, § 13.22(a). Such a notice must contain “the corporation’s estimate of the fair value of the shares.” G. L. c. 156D, § 13.22(¿>)(2)(iii). Depending on whether a shareholder certifies beneficial ownership of a share, the corporation must either pay shareholders or make an offer of payment. G. L. c. 156D, §§ 13.23, 13.24, 13.25. A shareholder “who is dissatisfied with the amount of the payment” may instead demand payment of the shareholder’s own estimate of the fair value of the shares, plus interest. G. L. c. 156D, § 13.26(a).11 If such a demand for payment remains unsettled, “the corporation shall commence an equitable proceeding” and “petition the court to determine the fair value of the shares and accrued interest.” G. L. c. 156D, § 13.30(a). The parties followed these steps, which culminated in the appraisal proceeding that is the subject of the present appeal.

c. Judicial appraisal proceedings. Robert alleges that the judge violated the appraisal provision of the Act, which provides, in part, that:

“The court may appoint [one] or more persons as appraisers to receive evidence and recommend a decision on the question of fair value. The appraisers shall have the powers described in the order appointing them, or in any amendment to it. The shareholders demanding appraisal rights are entitled to the same discovery rights as parties in other civil proceedings.”

G. L. c. 156D, § 13.30(d)12 The statute does not specify how to *63conduct an appraisal, except that litigants must be afforded “the same discovery rights as parties in other civil proceedings.” G. L. c. 156D, § 13.30(d). The statute permits, but does not require, the judge to appoint one or more appraisers.13 The procedures followed by the trial judge did not violate the Act.

d. Determining fair value. The “fair value” of the shares being appraised is defined as “the value of the shares immediately before the effective date of the corporate action to which the shareholder demanding appraisal objects, excluding any element *64of value arising from the expectation or accomplishment of the proposed corporate action unless exclusion would be inequitable.”14 G. L. c. 156D, § 13.01. In keeping with the equitable nature of the proceeding, G. L. c. 156D does not require the judge to follow any specific method for the determination of “fair value.” See G. L. c. 156D, § 13.30(a). This is in accord with the discretion judges have enjoyed under prior Massachusetts law.15

In general, in determining “fair value” under G. L. c. 156D, § 13.30(d), the judge “is not to reconstruct an ‘intrinsic value’ of each share of the enterprise but, rather, to determine what a willing buyer realistically would pay for the enterprise as a whole on the statutory valuation date.” BNE Mass. Corp. v. Sims, 32 Mass. App. Ct. 190, 197 (1992) (citation omitted). “[Tjypically, in an appraisal ... the price the corporation must pay for minority shares is fair value determined, as the judge did here, by finding the value represented by all the shares and *65apportioning pro rata.” Shear v. Gabovitch, 43 Mass. App. Ct. 650, 677 (1997).16 When expert witness testimony is offered by one or more parties, as in this case, the judge may credit the valuation method utilized by an expert witness or reject it and rely on another reasonable method. See Fechtor v. Fechtor, 26 Mass. App. Ct. 859, 863 (1989).

2. Constitutional challenge. The appraisal proceedings also did not violate Robert’s right to due process under art. 12 of the Massachusetts Declaration of Rights and the Fourteenth Amendment to the Constitution of the United States. Due process fundamentally requires “notice and the opportunity to be heard ‘at a meaningful time and in a meaningful manner.’ ” Matter of Angela, 445 Mass. 55, 62 (2005), quoting from Armstrong v. Manzo, 380 U.S. 545, 552 (1965).

*66Robert alleges that he was given constitutionally insufficient notice of how the appraisal would proceed. In particular, he argues that the trial judge did not indicate what procedure she would follow or that she would rely on evidence presented at the jury trial. However, the trial judge plainly stated at the motion hearing on August 7, 2009, that she was going to proceed on the basis of evidence presented at trial, unless Robert showed good reason to do otherwise.17 Robert’s claim that he was given constitutionally insufficient notice is thus belied by the record.

Robert also asserts'that the appraisal proceeding itself violated his right to due process. “Due process does not require any particular type of hearing, and ‘many matters can lawfully — and satisfactorily — be heard on the papers.’ ” Demoulas v. Demoulas, 428 Mass. 555, 589 (1998), quoting from EEOC v. Steamship Clerks Union, Local 1066, 48 F.3d 594, 608 (1st Cir.), cert, denied, 516 U.S. 814 (1995).18 Much like the plaintiff contended in Demoulas, Robert claims that his due process rights were violated because he did not have the opportunity to conduct additional discovery and present additional evidence on the appraisal claim at an evidentiary hearing. However, both Robert and the defendants presented evidence which pertained to the valuation of the stock at the summary judgment proceedings and at trial. When the trial judge gave Robert the opportunity to submit an offer of proof for the appraisal claim, he failed to identify any evidence that he would present beyond that which was offered at trial.19 In addition, Robert did not submit any affidavits or documentary evidence in support of his offer of proof, as the trial judge specifically requested. As in Demoulas, the trial judge acted properly and constitutionally in *67ruling on the appraisal claim without an additional evidentiary hearing where Robert failed to comply meaningfully with the judge’s requests to indicate what new evidence he would present at such a hearing.

3. Judge’s decision. Robert argues that the trial judge failed to make required findings of fact and conclusions of law in her summary adjudication of the appraisal claim. Under Mass. R.Civ.R 52(a), as amended by 423 Mass. 1402 (1996), “[i]n all actions tried upon the facts without a jury, the court shall find the facts specially and state separately its conclusions of law thereon.” This rule “does not require extensive detail, but does impose on the judge an independent duty to articulate the essential grounds of [her] decision.” Leader v. Hycor, Inc., 395 Mass. 215, 224 (1985), quoting from Schrottman v. Barnicle, 386 Mass. 627, 638 (1982). Here, the trial judge stated, on the record, that her decision was based on the evidence presented at the trial of the contract claim; specifically addressed arguments made by Robert; and concluded that the fair value of the estate’s stock was one percent of the net sale price. Accordingly, the trial judge satisfied the requirements of rule 52(a).

4. Determination of “fair value” of stock. Robert also claims that the trial judge erred in determining that the fair value of the estate’s stock was a one percent pro rata share of the net sale price of Spencer Press. See Shear v. Gabovitch, 43 Mass. App. Ct. at 677. Valuation is a question of fact that is reviewed only for clear error. See Sarrouf v. New England Patriots Football Club, Inc., 397 Mass. 542, 550 (1986); Chokel v. First Natl. Supermkts., Inc., 421 Mass. 631, 641 (1996); Demoulas v. Demoulas Super Mkts., Inc., 424 Mass. 501, 541 n.47 (1997). The judge’s “decision imports every fact essential to sustain it if there is evidence to support it.” Mailer v. Mailer, 390 Mass. 371, 373 (1983).

As noted, G. L. c. 156D, § 13.30(d), does not specify the valuation method that the judge must use and should be interpreted in a manner that is consistent with existing decisional law.20 This is in accord with the consensus of courts in other jurisdictions acting under comparable appraisal statutes that the minority shareholder receives its proportionate share of a going *68concern without artificial discounts.21 Robert argues that the judge erred because she oversimplified the appraisal determination and undervalued the “fair value” of the minority shareholder’s stock by basing it exclusively on the pro rata share of the net sale price. While it is true that, in a close corporation that is not insolvent, factors other than the net sale price may be significant in assessing its value, in the absence of any credible evidence of such factors offered by the minority shareholder, the pro rata share of the net sale price is “fair value.” See Sarrouf, supra at 544 (the value of a “going concern ... is the price a knowledgeable buyer would pay for the entire corporation”).

The evidence that was relevant to the factual question of “fair value” presented during the jury trial included a certification filed by Robert with the Probate and Family Court five months after his mother’s death (and while this litigation was pending) in which he declared that the value of his mother’s one percent stockholding was $377,000. At his pretrial deposition, Robert testified that the purchase price paid by Donnelly was “a great price.” Although Robert testified at trial that in his opinion his mother’s stock was worth “at least one million dollars,” none of the evidence presented at trial indicated that the fair value was anything other than one percent of the net sale price. The judge was warranted in concluding that the $60 million sale price paid by Donnelly for Spencer Press as a going concern was arrived at as a result of an arm’s-length transaction. Moreover, the testimony of Philip Taub of Nixon Peabody, LLC, about the adjustments and deductions made from the purchase price established the net value of all the shares as $36 million.22

Since the parties in this case were afforded the opportunity to *69engage in discovery on all of their claims prior to the stay of the appraisal claim, the judge was warranted in concluding that Robert’s discovery rights under G. L. c. 156D, § 13.30(<¿), had been satisfied, and that “there is not and will not be anything to indicate that the overall purchase price being paid for all of the stock is anything other than fair and favorable.” Accordingly, the judge was warranted in concluding that the “fair value” of the estate’s one percent stock interest in Spencer Press at the time of the merger was $361,540 and that the estate had already been paid that amount.

While Robert cites a variety of factors that may be considered in assessing the fair value of shares, with two exceptions, he does not give any reason why the judge should have used them in this case. Robert only substantively asserts that the estate’s one percent interest is worth more than its pro rata share because it would give either Stephen or John a controlling interest in the company; and the purchase of the share was necessary in order to sell one hundred percent of the stock to Donnelly. Both of these arguments pertain to the value of the estate’s stock to Stephen or John, rather than the fair value of the stock as a pro rata share of the net value of the business. Thus, Robert has not shown that the trial judge’s valuation determination was clearly erroneous.23

5. Breach of fiduciary duty and conspiracy claims. Robert further claims that the trial judge erred in granting summary judgment on his claims for breach of fiduciary duty and conspiracy. A grant of summary judgment is reviewed de nova, with all facts construed in favor of the nonmoving party. Miller v. Cotter, 448 Mass. 671, 676 (2007). Robert, in essence, argues that the fiduciary duties owed to shareholders in a close corporation entitled the estate to notice of the cash-out merger before it *70happened, which is sooner than the estate received notice pursuant to G. L. c. 156D, § 13.02.

Stockholders in a close corporation owe each other a duty of utmost good faith and loyalty. Donahue v. Rodd Electrotype Co. of New England, Inc., 367 Mass. 578, 593 (1975). Majority shareholders violate this duty when they frustrate the minority’s “reasonable expectations of benefit from their ownership of shares.” Brodie v. Jordan, 447 Mass. 866, 869 (2006) (illustrating so-called “freeze-out” techniques). Georgia had not participated actively in the management of the corporation or received financial benefit from her ownership of the stock for approximately eighteen years. Thus, she had no reasonable expectation of benefit from ownership of the stock other than its fair value, which the estate received as a result of the cash-out merger. Robert’s personal expectations of benefit are irrelevant, as he controlled the stocks only as fiduciary of the estate. See Lathrop v. Merrill, 207 Mass. 6, 10 (1910). Accordingly, the trial judge did not err in concluding that the defendants were entitled to judgment as a matter of law on Robert’s claim for breach of fiduciary duty.24

Similarly, the trial judge did not err in finding that the defendants were entitled to judgment as a matter of law on Robert’s claim for conspiracy because he had not proved an independent basis for tort liability and did not allege coercion. See Kurker v. Hill, 44 Mass. App. Ct. 184, 188-189 (1998) (civil conspiracy requires either independent tort liability or coercion). Robert’s claim that the defendants conspired to commit the tort of breach of their fiduciary duties fails because the defendants did not breach their fiduciary duties.

Amended judgment affirmed.