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Errors and Omissions in DCF Calculations: A Critique of Delaware's Dr Pepper Appraisal

  • Sutter Securities Financial Services, San Francisco


This articles points out errors in the Delaware Court of Chancery's decision in the appraisal of Dr Pepper Bottling Co. of Texas. Based on this article, the Court of Chancery corrected its opinion, but the Delaware Supreme Court ruled that the correction was loo late because the parties had entered into a settlement agreement.
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Timely news, analysis, and resources for defensible valuations Excerpt from Vol. 13, No. 10, October 2007
What It’s Worth
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Errors and Omissions in
DCF Calculations:
A Critique of Delaware’s
Dr Pepper Appraisal
By Gilbert E. Matthews, CFA*
The recent statutory appraisal of Dr Pepper Bot-
tling Holdings, Inc. (“Holdings”) by the Delaware
Chancery Court in Crescent/Mach I Partnership
v. Turner
1 raises several conceptual and compu-
tational issues concerning discounted cash ow
(DCF) valuations in an appraisal context. An
analysis of the Court’s calculations reveals the
caution with which valuation practitioners should
apply the DCF method, and the enduring bene t
of using comparable market analyses as a rea-
sonableness check.
Unusual case arising from constraints on sale
The cash-out merger that led to this appraisal
was unusual because it was arms’ length; most
statutory appraisal cases stem from freeze-outs
of minority shareholders by controlling interests.
In the Crescent/Mach I case, Jim L. Turner,
Holding’s controlling shareholder and chief ex-
ecutive, decided to sell the company because he
feared “competitive challenges posed primarily by
Coke and Pepsi,” including “a price war in which
Holdings would not have the same support and
resources as its nationally-backed competition.
In addition, “troubling” economic conditions and
external constraints limited the prospects for
sale. Cadbury Schweppes PLC (“Cadbury”), the
* Gil Matthews is Chairman of Sutter Securities Incorporated
in San Francisco: He has been an invest-
ment banker for more than 40 years, and his practice centers
on litigation support, fairness opinions and valuations. The
author thanks Mark Lee for his helpful insights.
Reprinted with permission from Business Valuation Resources, LLC
franchisor for Holdings’ primary soft drink brands,
“made it clear to Turner that it would not likely
consent to a private equity acquisition.” Coca
Cola and PepsiCo could not acquire Holdings for
antitrust reasons. The only bidder was an entity
owned by Cadbury and the Carlyle Group, which
acquired Holdings in a cash merger at $25 per
share on October 1999. Turner sold his shares at
the same $25 price, subject to severe restrictions.
“Turner obtained the best price that he could from
Cadbury/Carlyle,” the Vice Chancellor noted, “but
how often will the only buyer pay full price?”
Dissenting shareholders sought appraisal2 un-
der the Delaware statute (8 Del. C. §262), and the
Court undertook what it called “an independent
valuation exercise.”
In determining fair value, the Court may look to
the opinions advanced by the parties’ experts,
select one party’s expert opinion as a framework,
fashion its own framework or adopt, piecemeal,
some portion of an expert’s model methodology
or mathematical calculations. The Court, how-
ever, may not adopt an “either-or” approach and
must use its judgment in an independent valua-
tion exercise to reach its conclusion.
At trial, experts for both sides used a compa-
rable company approach as well as a DCF; one
expert also utilized comparable transactions.
Similarly, the investment bank retained by Hold-
ings to render a fairness opinion relied on all three
The Court rejected application of the market
approach because of Holdings’ unique market
position and the lack of appropriate guideline
comparables. Citing prior Delaware decisions,
the Court reiterated its preference for the DCF
method and its general opinion that “other meth-
odologies—all based on comparables of one form
Business Valuation Update october 2007
Grant Thornton, LLP—Seattle, Wash.
Lewis & Clark Law School—Portland, Ore.
Fannon Valuation Group—Portland, Me.
Financial Research Associates—Bala Cynwyd, Pa.
Wolf, Block, Schorr & Solis-Cohen, LLP—Norristown, Pa.
FMV Opinions, Inc.—Irvine, Calif.
The Financial Valuation Group—Atlanta, Ga.
McDermott, Will & Emery—Chicago, Ill.
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Sutter Securities Incorporated—San Francisco, Calif.
Mercer Capital—Memphis, Tenn.
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Financial Valuation Group—Los Angeles, Calif.
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Errors and Omissions in DCF Calculations
...continued from front page
or another—are of limited value.” Adopting certain
of the experts’ inputs as to projections, discount
rates, terminal value, taxes, and debt, the Court
arrived at intermediate numbers for other inputs,
and accepted the company’s projections used
by Holdings’ expert. It determined a fair value of
$32.31 per share, or 29% more than the transac-
tion price (and far less than the dissenting share-
holders’ expert valuation at $48.69 per-share).
The Court’s conclusion was “within the range of
reason, it said, in part because it fell within the
ranges of the original fairnes s opinion: $19.32 and
$31.05 for the DCF analysis, $22.29 and $34.84
for comparable transactions, and $19.75 and
$28.53 for comparable companies. The Court fur-
ther supported its $32.31 fair value determination
by stating that Holdings “implicitly concedes that
the merger consideration was less than fair value
by sponsoring an expert who concluded that the
fair value was in excess of the merger price.”
However, in fact, the Court’s $32.31 valuation is
above the high end of two of the three very wide
fairness opinion ranges. The Court’s DCF-based
conclusion was 4% above the high of the DCF
range and 32% above its $25.19 midpoint. As to
Holdings’ implicit concession, its expert testi ed
that the fair value of Holdings based on a DCF
was $25.10 per share, only 0.4% above the $25.00
transaction price.
Errors in the Court’s valuation
A review of the inputs selected by the Court
and Holdings’ expert reveals that the differences
between them were not sizeable. In particular,
the Court used a discount rate of 9.75% (vs. 10%
by Holdings’ expert), a tax rate of 39% (vs. 40%),
and a faster use of operating loss carryforwards
($6.215 million per year vs. $4.5 million). Given
these fairly close numbers, one would not expect
the Court’s valuation to be 29% higher than that
of Holdings’ expert.
The Court helpfully appended a summary of
its calculations to the published decision. A close
reading discovered two calculation errors, the rst
one minor. In calculating the after-tax debt-free
cash ow for the 1999 stub period following the
transaction date, the Court omitted the portion of
pretax income sheltered by the loss carryforward.
(It accurately totaled the projected data for years
2000-2004.) Correcting this small miscalculation
lowers the Court’s per-share value by $0.13
The second error was material. The Court in-
cluded the post-2004 cash ow bene t of the tax
loss carryforward as a line item in the calculation
of equity value, re ecting the present value of the
Reprinted with permission from
Business Valuation Resources, LLC
october 2007 Business Valuation Update 3
carryforwards after the end of the projection pe-
riod. However, the numerator of the growth model
(the 20 04 free c ash ow number) included the full
amount of the pretax income being sheltered by
the loss carryforward. Therefore, free cash ow
for the perpetuity calculation was overstated by
$2,424,850, i.e., 39% (the assumed tax rate) of the
annual sheltered income ($6,215,000). The pres-
ent value of the second error caused the Court’s
valuation to be $2.40 per share too high.
After adjusting for these two errors, the fair
value of the transaction is $30.04 per share, 7.4%
below the Court’s award of $32.31 per-share.
Conceptual questions left opened
Based on a reading of the case, the experts
failed to analyze and address certain conceptual
issues. As a result, it appears that the Court may
have adopted conceptually awed inputs that may
have increased its valuation. Analysts should
review these factors for future valuations, where
relevant, to aid the trier-of-fact in its independent
determination of value under the DCF approach.
1. Effect on Discount rate of a weak competitive
position. The opinion acknowledged that Hold-
ings was in a weak competitive position. Faced
with the supermarket industry’s consolidation
and preference for national bottlers, the company
“would encounter substantial market dif culties.
Cadbury’s support for Holdings was about to
decline or end, further impairing Holdings’ ability
to compete.
In determining its discount rate, the Court
looked in part to testimony regarding the volatil-
ity of national bottlers (Coca Cola and PepsiCo),
which occupied a stronger market position than
Holdings’. Given the stronger positions of the
national bottlers, it would have been reasonable
to adjust Holdings’ discount rate upward.
Application of a 10% discount rate rather than
9.75% (and correcting for the two errors, dis-
cussed above) would have reduced the Court’s
calculated value of H oldings fro m $30.04 to $27.56
per share; a 10.25% discount rate would have
further reduced it to $25.27 per share.
2. Perpetual growth rate. In calculating terminal
value, the Court applied a perpetual growth rate
of 3.5% to free cash ow, although it used a 3%
EBITDA growth rate for the projection period.
Based on the accepted management projections,
the growth rate in projected free cash ow from
2000 to 2004 was 3.98%, but this resulted from
using at capital expenditures and increasing de-
preciation. If depreciation, capital expenditures,
and changes in working capital in the projection
had been increased by 3.0% annually, free cash
ow would have increased at 2.7% per annum.3
Applying a 3.0% (rather than 3.5%) perpetual
growth rate and correcting for the two calculation
errors, the Court’s valuation of Holdings would
have declined from $30.04 to $26.21 per share.
It is rare (except in turnaround situations) to nd
a long-term growth rate exceeding the anticipated
growth rate for the medium-term projection pe-
riod. Competitive factors, changes in consumer
preferences, and obsolescence should normally
cause the long-term growth rate to be lower than
the medium-term rate.
3. Capital expenditures and depreciation. The
Court adopted management’s projection that
capital expenditures would be $25 million each
year of the projection period and that depreciation
would increase each year. Although increasing
revenues normally require increasing capital
expenditures, the Court’s (and management’s)
assumption that capital expenditures would not
grow was not supported by a schedule of future
capital expenditures. The assumption of flat
capital expenditures is clearly inconsistent with
management’s projection that depreciation would
increase each year, since it is almost impossible
for depreciation to increase unless capital expen-
ditures increase.4
In a perpetuity model (such as the Gordon
Growth Model that the Court used), capital expen-
ditures must be materially higher than depreciation
because, in an in ationary economy, new capital
expenditures must re ect increasing capital asset
costs, while depreciation must re ect the amor-
Continued to next page...
Errors and Omissions in DCF Calculations
Reprinted with permission from Business Valuation Resources, LLC
Business Valuation Update october 2007
tization of lower historical capital asset costs.5
In a perpetuity model with a 3% growth rate and
assuming a 10-year average life for xed assets,
capital expenditures would exceed depreciation
by 15.5% using straight line depreciation and
11.6% using the double-declining method.6 The
Court’s opinion gave no indication that the experts
considered the appropriate relationship between
capital expenditures and depreciation, which is a
common error of omission in DCF analyses.
If capital expenditures and depreciation were
both projected to increase at 3% per year after
the rst full year of the projection period (using the
Court’s 9.75% discount rate and correcting for the
two errors), the calculated value of Holdings would
drop from $30.04 to $27.20 per share. Combining
this adjustment with a 3% perpetual growth, the
calculated value of Holdings would have fallen to
$23.56, less than the $25.00 transaction price.
In evaluating any projections, the practitioner
should consider both the reasoning behind the
projected capital expenditures and the relationship
between capital expenditures and depreciation.
When possible, management should explain its
assumptions. It should always be able to calcu-
late depreciation accurately based on historical
and projected capital expenditures. When trial
experts later examine management projections,
they should consider making appropriate adjust-
ments if the assumptions are questionable, and
be prepared to support any revisions in court.
4. Amortization has a limited life. The Court’s
valuation model included $5.4 million of annual
tax-deductible amortization as a non-cash charge.
Amortization necessarily has a limited life, but
the Court, by including it in the free cash ow
in the growth model, effectively assumed that it
was perpetual, thereby understating taxes and
overstating value. (Without knowing how long
the amortization was scheduled to continue, the
overstatement cannot be quanti ed.) Moreover,
applying a growth rate to free cash ow errone-
ously assumes that amortization will grow at the
same growth rate, thereby further overstating
terminal value.
In fact, amortization in a projection should
normally be a constant or declining number.7
The appropriate manner to value amortization
subsequent to the projection period is to exclude
it from the growth mo del calculation, and, instead,
to determine the present value of the scheduled
amortization over its life.
Since the Court did not explain the amortiza-
tion’s speci c application or its scheduled life, it is
not possible to quantify its impact on the valuation.
However, assuming that the amortization contin-
ued for ve years beyond the end of the projection
periods, the calculated value of Holdings (cor-
rected for the two errors discussed above) would
be reduced from $30.04 to $28.44 per share.
Caution when using DCF
This discussion demonstrates that practitioners
must use caution in valuing a business based on
discounted cash ow. A DCF analysis assumes
the validity of the nancial projections. However,
any projection depends on the reasonableness of
the underlying inputs, such as growth rate, pro t
margins, and capital expenditures. A DCF valua-
tion is highly dependent on terminal value, which
is a direct function of the nal year of a projection;
the nal year is clearly more dif cult to forecast
than the near future. The selection of a discount
rate is also subjective, particularly for smaller, less
diversi ed companies.
Discounted cash ow is a valuable tool, but
small changes in the inputs can materially affect
the valuation conclusion. As this critique shows,
with all other inputs in the case held constant,
adjusting the perpetual growth rate assumption
from 3.5% to 3.0% reduces the calculated value by
13%, and adjusting the discount rate from 9.75%
to 10.25% reduces the calculated value by 16%.
The subjectivity of many DCF inputs frequently
makes the approach unreliable, producing a wide
range of calculated results and rendering any
mathematical precision illusory.
When analysts employ different approaches
that lead to materially different conclusions,
Continued to next page...
Errors and Omissions in DCF Calculations
...continued from previous page
Reprinted with permission from Business Valuation Resources, LLC
october 2007 Business Valuation Update 5
they should examine the inputs to determine the
causes of the discrepancy. Alternative valua-
tion methods serve as helpful “reality checks” to
con rm whether a speci c conclusion is reason-
able. An examination of the implied multiples
of data points in the nal year of the underlying
projection should help to establish whether a DCF
valuation is reasonable. In valuing Holdings, it
appears that the Vice Chancellor did not use the
EBITDA multiple inherent in his DCF valuation as
a reasonableness check. He called the dissent-
er s’ exper t’s valuation of Holdings at 8.9x EBITDA
“somewhat high” because the comparable com-
panies had “an implicit value premium stemming
from the liquidity advantage of a closer relationship
with their ‘parent’ companies.” However, based
on EBITDA in the nal year of the projection, the
EBITDA multiple was 8.6x, not far from the 8.9x
that the Court rejected. If the Court’s calculation is
corrected for the two errors and if a 3% perpetual
growth is applied, then the EBITDA multiple would
be 7.6x.
In its seminal Weinberger decision, the Dela-
ware Supreme Court held that “the methodology
to be used for measuring fair value should be gen-
erally accepted techniques used in the nancial
community.8 A review of the summaries of invest-
ment bankers’ fairness opinions in proxy state-
ments shows that an overwhelming majority of
public company transactions rely on comparable
companies and comparable transactions analy-
ses. In recent years, several Delaware Chancery
Court decisions have used DCF as the sole ana-
lytical measure of value. Sound practice, however,
calls for using more than one method. Experts
and courts should consider the widely used com-
parable company and comparable transaction
methods whenever possible. Expert witnesses
should be prepared to explain the basis for their
selection of comparables and the adjustments
that they consider appropriate and necessary in
applying the comparables’ multiples to the subject
company. A well-reasoned and appropriately
adjusted comparables analysis should be helpful
to a court both as a basic valuation approach and
as a crosscheck on DCF calculations.
1 2007 Del. Ch. LEXIS 63 (Del Ch. May 2, 2007). The
case abstract appears in the Sept. 2007 Business Valua-
tion Update.
2 Former shareholders also claimed that Turner breached
his duciary duties, but the Court dismissed these claims.
3 Free cash ow would increase at a slower rate than
EBITDA because amortization and the loss carryforward
were projected to be constant.
4 Theoretically, depreciation might increase when capital ex-
penditures are at if a substantial portion of capital expendi-
tures are for shorter-life assets than prior capital expenditures
had been, e.g., buying trucks instead of building plants.
5 See M. Mark Lee, “The Ratio of Depreciation and Capital
Expenditures in DCF Terminal Values,” Financial Valuation
and Litigation Expert, August-September 2007, pp. 7-8.
See also, Daniel L McConaughy and Lorena Bordi, “The
Long Term Relationships between Capital Expenditures and
Depreciation Across Industries: Important Data for Capital-
ized Income Based Valuations,” Business Valuation Review,
March 2004, pp. 14-24.
6 Gilbert E. Matthews, “Fairness Opinions: Common Errors
and Omissions,” in The Handbook of Business Valuation and
Intellectual Property Analysis, R. Reilly and R. Schweihs,
eds. (McGraw Hill, 2004), pp.223-4.
7 Amortization could increase during the projection period
where the forecast assumed future events, such as acquisi-
tions, that will lead to additional amortization.
8 Lawrence A. Hamermesh & Michael L. Wachter, The Fair
Value of Corn elds in Delaware Appraisal Law, 31 Iowa J.
Corp. L. 119, 123-4 (2005), citing Weinberger v. UOP, Inc.,
457 A.2d 701, 712 (Del. 1983).
Errors and Omissions in DCF Calculations
Reprinted with permission from Business Valuation Resources, LLC
Full-text available
Topics: 1. The perpetual growth rate and firm mortality 2. The relationship between capital expenditures and depreciation 3. The appropriate treatment of amortization 4. Projections, normalization, and steady state growth 5. The trend toward using lower long-term growth rates 6. The relevance of multiples for terminal value
This chapter calls attention to some of the errors frequently encountered in estimation and applications of the cost of capital. It points out these errors partly so that readers will not fall into the same traps themselves when estimating or using cost of capital. Another reason is to help readers readily identify such errors when reviewing the work of others, understand the impact such errors have on the resulting values, and understand how the errors should be corrected. Still another aim is to call attorneys' attention to common errors when reviewing valuation reports, settling cases, or cross-examining expert witnesses. Some of errors mentioned in the chapter are: confusing discount rates with capitalization rates; substituting earnings before interest, income taxes, depreciation, and amortization (EBITDA) for net cash flow; using both the discounting and capitalizing methods and weighting them; and mistaking historical rates of return for expected rates of return.
Praise for Cost of Capital in Litigation: Applications and Examples "After revising their existing text, Cost of Capital: Applications and Examples, the authors have endeavored to expand their treatment of the subject and further explore the role of cost of capital in the courts. Addressing various methods for calculating value and different standards of value, the authors explain how courts may differ in their decisions based on the topic, jurisdiction, or available evidence. This new compilation is an important contribution to the field of valuation and will serve as an imminently helpful resource for attorneys and judges. Shannon Pratt and Roger Grabowski have provided yet another thoughtful, helpful, and excellent resource for all practitioners interested in how appraisers and courts have, and should, arrive at value."? -From the Foreword by David Laro, Judge, United States Tax Court "Cost of Capital in Litigation is a very helpful resource for litigators faced with valuation disputes involving an assessment of discounted cash flows. In the area with which I am most familiar-Delaware valuation law-it is thorough and insightful. More important to me, its introductory material nicely lays out the conceptual structure of the valuation issue, and the remaining chapters gather lots of information about the treatment of the issue in areas with which I am less familiar, (such as bankruptcy law and domestic property litigation)." -Lawrence A. Hamermesh, Ruby R. Vale Professor of Corporate and Business Law, Widener University School of Law, Wilmington, Delaware "Cost of Capital in Litigation comprehensively deals with a difficult subject, pointing out common grounds as well as differences among experts in the treatment of this subject. It is a must first stop for the uninitiated in the subject and the ultimate checklist that every practitioner should consult. Particularly useful are the chapters focusing with the eyes of experts on the treatment of cost of capital in specific areas of litigation. Notably, the authors alert the reader to those aspects of the various methods for determining cost of capital that may differentiate among the suitability of the methods based on available data, the circumstances presented, and the legal issue under consideration." -Reeves Westbrook, Esq., Partner and Chair of the Tax Group, Covington & Burling LLP "Shannon Pratt and Roger Grabowski have consolidated information on important theoretical valuation frameworks, practical applications, and case law needed by finance professionals, attorneys, and judges in these challenging times. Cost of Capital in Litigation serves as a solid well-written reference on cost of capital and valuation that I recommend to both practitioners and students." -Theodore Barnhill, Professor of Finance and Director of the Global and Entrepreneurial Finance Research Institute, The George Washington University "Valuation in litigation settings requires careful analysis, solid evidence, and defensible positions. Cost of Capital in Litigation details the nuances and subtle but critical elements of crafting a defens-ible position for experts, attorneys, and others who need to understand how and why valuation cases are won and lost." -Hal Heaton, Professor of Finance, Marriott School of Management, Brigham Young University "Discount rates are often crucial components in calculating damages. Even small changes in these numbers can have large effects on the amounts in dispute-and the magnitude of damages won or lost. Cost of Capital in Litigation explains the underlying economic theory. It then offers separate chapters elaborating on rate of return approaches in their widely varied forensic contexts. This helpful compendium even includes an outline of cost of capital questions for use by attorneys and their financial experts. No business litigator should be without it." -Peter V. Baugher, Esq., Partner, Schopf & Weiss LLP.
Full-text available
The cost of capital is a central issue in judicial business valuations in statutory appraisal, stockholder oppression, and “entire fairness” cases. The Delaware courts have effectively set the standards for valuations related to corporate disputes because Delaware law is widely accepted on corporate legal issues. This article primarily discusses Delaware Court of Chancery and Delaware Supreme Court opinions involving the discounted cash flow (DCF) method and its crucial component, the cost of capital. Most of the Delaware decisions discussing cost of capital have come from statutory appraisal cases, and the Court does not differentiate in its approach to cost of capital in fairness cases. The Delaware Court of Chancery has declared its preference for the DCF method of valuation, including all elements of the expanded capital asset pricing model (CAPM) to determine the cost of capital. It has, however, rejected the company-specific adjustment in the calculation of weighted average cost of capital (WACC) unless there are unusual circumstances to validate it.
A CFC tile 5 from the JET outer divertor, and CFC tiles from neutral beam shine-through and re-ionisation regions were coated with tungsten and exposed during the 2005-7 JET campaigns in preparation for the ITER-like wall project. Approximately 1.6 microns of coating were eroded from the tile 5 during high-delta discharges when the outer strike-point is on the tile. The coatings on the other tiles were unaffected by NB-heating and divertor discharges, however a tile mounted near the centre of the Inner Wall Guard Limiter lost all its coating from the surface within 10 mm of the tile leading edge; this probably occurred during the ramp-up phase of JET discharges. © 2009 J.P. Coad.
Full-text available
Fairness opinions almost always involve an assessment of the value of a company. The question in most fairness opinions is whether a merger or acquisition offer is within an objectively determined range of values for the company or for its shares. In evaluating the fairness of an offer, the analyst is required by law to use valuation methods that are generally accepted. The analyst providing a fairness opinion must expect that the opinion will be scrutinized by the parties to whom it is addressed and will perhaps even be tested in the crucible of litigation. This chapter discusses numerous issues and problems in fairness opinions and valuations, ranging from simple mathematical errors to methodological flaws to broad conceptual issues. Errors that can be laid at the door of the analyst include (I) misinterpretation or misuse of data, (2) failure to recognize mathematical inconsistencies in the data used, and (3) misapplication of generally accepted methods. Some errors, however, derive from flaws in methodologies that have been generally accepted by investment bankers and valuation practitioners. Since analytical methods continue to evolve, it is necessary to be aware of current thinking, to recognize the strengths and weaknesses of various approaches, and to understand when traditional methods need to be reexamined. When an approach is flawed, these flaws should be recognized and corrected. When a concept or method is demonstrably incorrect, either statistically or conceptually, it should be rejected. This chapter points out several areas in which widely used approaches to valuation and to determining fairness should be subjected to critical examination. Some of the broader issues that this chapter will examine include the scope of fairness opinions, the updating of fairness opinions, and the relationship between fairness opinions and the fiduciary duties of directors and control parties. Corporate directors should examine not only the financial aspects of fairness, but also the nonfinancial aspects. In addition, both corporate directors and analysts should also give greater attention to the question of when it is appropriate to update fairness opinions.
The Delaware Supreme Court's opinions in Weinberger and Technicolor have left a troublesome uncertainty in defining the proper approach to the valuation of corporate shares. That uncertainty - increasingly important as going private mergers become more frequent - can be resolved by a blend of financial and doctrinal analysis. The primary problem--the potential opportunism by controlling shareholders in timing going private mergers--can be addressed by a more complete understanding of corporate finance. The definition of fair value must include not only the present value of the firm's existing assets, but also the future opportunities to reinvest free cash flow, including reinvestment opportunities identified, even if not yet developed, before the merger. This issue has been incompletely articulated by the courts. On the other hand, value created by the merger that can only be achieved by means of the merger itself - such as reduced costs of public company compliance - should not be included in determining fair value. We also show that except in the case of acquisitions by third parties (where actual sale value, minus synergies, is a useful measure of fair value), hypothetical third party sale value does not and should not ordinarily be taken as a measure of fair value.