By Andrew Pike, CFA, ASA, RV
Discount for lack of marketability, also known as DLOM, is a discount that is often applied to the valuation of a minority stake in privately held companies. Most references to DLOM estimation studies (excluding option models) that I have seen in professional practice refer to the midpoint results of these studies. In reality, the ranges of results for these studies are rather broad.
There are four general approaches to estimating DLOM: restricted stock studies, pre-IPO transaction studies, private company discount studies, and option models. These can be layered with qualitative assessments such as Mandelbaum or van den Cruijce. Each method is discussed below.
Restricted stock studies compare the prices that investors are willing to pay for two otherwise identical securities, where one is fully liquid and the other has liquidity-related restrictions.[1] These studies show that restricted stock (stock that cannot be immediately resold due to restrictions on re-sale) indeed sells at a discount to its otherwise identical publicly traded sister stock.
Restricted stock generally consists of minority stakes in a company’s shares. Restricted stock and freely-trading stock are issued by the same company and there is no time lag in stock price observations.
The positive aspects of this method are countered by some negative aspects. Restricted stock generally has voting rights (the Depository Receipts do not) and trading restriction periods of restricted stock are generally limited to 2 years for restricted stock issued through 1997, 1 year for restricted stock issued through 2007, and 6 months or 1 year[2] for restricted stock issued thereafter. Volume limitations exist in some cases. In such cases, the holder must ‘dribble out’ the shares, generally at the greater of 1% of the outstanding securities of the class being sold and the average weekly reported volume of trading in the four weeks preceding the filing of the Form 144.[3]
Many restricted stock studies consist of companies listed on United States stock exchanges, though some of those companies are headquartered outside of the United States. The discount on restricted stock might be impacted by monitoring and information acquisition costs, passivity of investors, or investors’ willingness to provide capital when public funding would be difficult to secure on short notice.[4]
There are many restricted stock studies. John E. Elmore, JD, CPA published the mean and median restricted stock DLOMs for a total of 51 studies.[5] Most books and academic literature that discuss restricted stock DLOMs report the mean, median or both results. This understates the possibilities of DLOM levels.
One of the most widely cited restricted stock DLOM sources is the March 10, 1971 Institutional Investor Study Report of the Securities and Exchange Commission.[6] This study analyzes DLOMs in United States markets observed between the years 1966 and 1969. It was based on a survey of institutional investors about prices paid for restricted shares.
There were some differences amongst respondents about purchase date. Some respondents reported the price as of the date that they committed themselves to the purchase of a stock. Others did not consider themselves bound until settlement date and yet others reported the date when they entered the trade in their books. In some cases, this resulted in a premium paid on restricted stock due to a combination of a binding commitment and falling stock prices. Premiums also resulted from superior bargaining position of the seller. The researchers used average ask prices when calculating the DLOM, which could also have resulted in premiums in cases where the range of prices was large. This was particularly the case for low-priced stocks.
The data was divided into DLOM buckets[7] and various factors that might impact DLOM such as trading market[8], class of institutional purchaser[9], sales of issuer, and earnings of issuer. There were 398 observations for the trading market and class of institutional purchaser analyses, and 338 observations for the sales and earnings analyses. The study also showed a cross-section of average DLOMs by year and sales, and by year and earnings. For those cross-section analyses, there were 278 observations.
The midpoint DLOM is often cited from this study where the mean was 24%. The study showed average DLOMs by a cross-section of institutional purchaser class and trading market. We consider the Over the Counter Unregistered market to be the most appropriate comparison for private companies. For this market, venture capital investors reported an average DLOM of 45.7%. As this is an average, DLOM can of course be higher.
Valuation professionals and investors often overlook the range of discounts from DLOM studies and instead focus solely on the midpoint. The SEC study showed that 12% of the observations had a discount of 50.1% to 80%. This suggests two important points: 1) DLOM can be considerably higher than 24%, in fact it can be between 50.1% to 80%; and 2) DLOM can be as high as 80%.
Since the writing of this study (1971), SEC rules governing restricted stock have changed. The embargo period on selling restricted stocks has been reduced from 2 years, to up to 1 year (depending the type of investor). Shorter embargo periods and less restrictive sales volume limitations could logically be expected to have a downward impact on DLOM.
The Stout Restricted Stock Study[10] is a more recent study including approximately 770 observations between the years 1980 and the first quarter of 2022. The average DLOM was 20.5% and some observations were greater than 75%. Stout also divided their observations into 5 quintiles based on stock price volatility. At the highest quintile, the median DLOM was 42.9% and the maximum was 91.3%.
Pre-IPO transaction studies compare initial public offering (IPO) prices to transactions in those companies preceding the IPO. When a company issues an IPO, the securities and exchange commission (SEC) requires the company to disclose all transactions in its shares for a certain period of time preceding the IPO.
Some of the discount in Pre-IPO trades may be related to contractual lock-up periods. The underwriters obtain lock-ups to reduce downward pressure on the share price following the IPO. A lock-up is a contractual restriction for the benefit of the IPO underwriters on transferring and hedging shares, usually for 180 days post-IPO.
These studies compare pricing of shares in the same company. Trades prior to the IPO are actually private company transactions, which is not the case with restricted stock studies. Pre-IPO transactions often consist of minority stakes in a company’s shares.
There are some drawbacks to these studies. Pre-IPO studies are biased upward because they represent the results of successful IPOs.[11] Pre-IPO trading might involve sophisticated investors who are more equipped to negotiate better pricing than retail investors in an IPO. Pre-IPO trading may involve related parties involved in the IPO, such as investment bankers, investors, outside directors, or key management. Such investors sometimes purchase shares pre-IPO as a form of payment for their services. These purchases are typically priced at a discount to the expected IPO price. Pre-IPO transactions may not properly consider the time value of money – transactions that took place several years before the IPO may need to be revalued to correctly calculate the discount.[12]
One of the most widely cited Pre-IPO studies is from Emory & Co., LLC. Their 2002 study[13] includes transactions between 15 December 1979 and 18 September 2000. The mean DLOM was 46% and median was 47%. For transactions involving share sales (not options), the mean and median were 50% and 52%, respectively. For grants and sales to employees, directors, officers, and management and for services, the mean discount was 54.4% and the maximum discount was 98.9%.
Another prominent Pre-IPO study is from Willamette Management Associates.[14] This paper shows results from an inhouse study by year for the period 1975 to 2000. The range of means is 18.0% to 55.6% and the range of medians is 31.8% to 73.1%. We do not have individual records and cannot drill-down within these results. However we can conclude that a midpoint median (for the year 1984) suggests a range of discounts above and below that discount.
Based on pre-IPO studies, we can conclude that DLOM can exceed 73%.
Private company discount studies (PCD) measure the difference between valuation multiples of private and public companies. Studies can be based on a comparison of: 1) valuation multiples between (not acquired) publicly traded stocks and acquired non-listed companies; or 2) valuation multiples of acquired publicly traded stocks and acquired non-listed companies. There are also some studies that combine these approaches.
Transactions in PCD studies are generally controlling stake transactions. Some studies compare valuation multiples of publicly traded company to valuation multiples of subsidiaries that they acquired.[15] Some suggested rationale for the PCD on these transactions include differences in size, growth, and/or need for liquidity by the selling shareholders.[16]
Some criticisms of the PCD approach include the following:[17]
The PCD studies base DLOM on differences between valuation multiples. There are various valuation multiples to choose from, each of which with its own strengths and weaknesses. The range of midpoint DLOMs from PCD studies based on EBITDA and revenue multiples is from 9.8% to 75%. The table below summarizes the midpoint findings:
A put option gives an option holder the right to sell a given financial asset in the future at a pre-specified price. The price of the option is based on the asset’s price at measurement date, holding period, expected dividends, and the probability of earning a profit on the sale of the asset (expressed typically as stock price volatility).
John Finnerty wrote that a lack of liquidity is a form of DLOM that exists when an interest holder cannot dispose of the interest quickly without a significant reduction in value. He concluded that this lack of liquidity, and by extension the DLOM, can be estimated based on a put option model:[25]
“One can also model the cost of the lack of liquidity as the value of a forgone put option. However, the option formulation is more complex than in the case of the lack of marketability because there is no legal or contractual restriction on the holder’s ability to sell or transfer the asset, and, consequently, the length of the restriction period is less clear. For example, the market for an asset may be poorly developed, making it difficult, time-consuming, and therefore expensive to find a buyer for the securities, but the assets are nevertheless marketable. The restrictions are financial, rather than legal or contractual, and there is no fixed date on which they are scheduled to lapse. It takes more time to find a buyer in an illiquid market than in a liquid market. This loss of flexibility to sell an asset freely or, equivalently, the ability to sell it quickly but only if there is some concession of intrinsic value, can be modeled as the loss of value of a put option.”
In case number 09/00340[26], the court of Amsterdam ruled that the model developed by Finnerty is the only currently available model that has been achieved [for developing DLOM] that has been tested against empirical data regarding restricted shares.
We summarize five put option studies below with their strengths and weaknesses.[27]
As a DLOM results from an inability to exercise a right to sell, the cost of the put reflects the DLOM for the shares. Like the standard Black-Scholes option pricing model, the Chaffe model has a number of assumptions:
As his model is exercisable only at expiration, thereby decreasing option value, Chaffe concluded that his findings may be viewed as a minimum applicable discount.
According to Chaffe, volatility for a small privately owned company is likely to exceed 50 percent. Chaffe reached this conclusion based on the volatility for small public companies that are traded in the over-the-counter market.
This model, developed by Marc Katsanis[28], posits that the value of a marketable share is equal to the value of a non-marketable share plus a ‘Shout out value’. It is a modified Chaffe model where the put value concluded by the Chaffe model is multiplied by an exponential adjustment factor based on the expected dividend yield of the subject company security when the risk-free interest rate exceeds the dividend yield and the dividend yield is not zero.
Katsanis explains, “Another form of put option, a shout put or shout floor option, more closely mimics marketability than do the previously mentioned forms of put option because both marketability and a shout put option give a stockholder the right to lock in a selling price (the prevailing marketable stock price) for the stock at any point in time over the term of the option. By comparison, over the term of the option the European fixed strike[29] put gives the stockholder the right to lock in a selling price equal to the current stock price; the lookback[30] put gives the stockholder the right to lock in a selling price equal to the highest stock price achieved; and the Asian put[31] gives the stockholder the right to lock in a selling price equal to the average of all stock prices achieved.”
The Shout Put model may be helpful for holding periods longer than one year, however improved accuracy over the Chaffe model for shorter periods is insignificant.[32]
While the Chaffe model is based on avoided losses, the Longstaff model is focused on unrealized gains. This model is based on a lookback option, which is an American style option that allows the holder, at expiration, to retroactively exercise an option at the most optimal price during the option holding period. This results in maximum returns. The trader is assumed to have perfect market timing and there are trading restrictions that prevent the security from being sold at the optimal time.
Longstaff[33] explains: “[Consider] a hypothetical investor with perfect market timing ability who is restricted from selling a security for T periods. If the marketability restriction were to be relaxed, the investor could then sell when the price of the security reached its maximum.”
There is some disagreement amongst practitioners and academics whether the Longstaff model produces a liquidity premium or a DLOM. The IRS’s DLOM job aid states that the Longstaff model is not often applied by valuation analysts in estimating the DLOM for privately held companies.
The Longstaff model assumes that an investor can have perfect market timing, which can be difficult to defend. Longstaff’s research is based on volatilities between 10% and 30%, while small stocks traded over the counter can have volatility exceeding 50%.[34]
The Finnerty study, which I see more frequently used in valuation practice than the other models, is an extension of the Longstaff study. Instead of assuming perfect market timing, Finnerty modelled the value of an average-strike Asian put option. The strike price equals the arithmetic average of market prices over the holding period.[35]
Many commentators find the Finnerty model to most closely reflect the results of certain restricted stock studies at volatility ranges of 50% to 100%. The Finnerty model produces DLOMs up to 32.3% regardless of volatility and holding period, which may significantly understate DLOM.[36]
Ghaidarov[37] (my preferred technique) also developed an average-strike Asian put option model that applies geometric average strike prices instead of arithmetic average strike prices. This produces results similar to Finnerty’s model for a 6-month holding period at volatilities through 125%. The Ghaidarov model is not limited to a 32.3% DLOM as in the Finnerty model, approaching 100% as volatility approaches infinity.
Ghaidarov, as with the other studies mentioned in this report, developed a closed-form equation for option pricing, meaning a fixed formula for pricing the option. With a Monte Carlo simulation one can more explicitly model inputs and conditions – it provides more flexibility. When testing his closed form formula against a Monte Carlo simulation, Ghaidarov found results similar to those resulting from a Monte Carlo simulation under normal conditions (i.e. results deviated by more than 1% at a 5-year restriction period above 40% volatility and at a 4-year restriction period at about 50% volatility).[38] At a 5 year holding period and 80% volatility, the Ghaidarov model produces a DLOM of 44.29% vs a 38.36% DLOM derived from a Monte Carlo simulation.
In case number 09/00340, the court of Amsterdam ruled that empirical market evidence from similar situations should be applied as much as possible and that the Finnerty model has successfully tested against empirical evidence from SEC 144 restricted stock studies.[39]
“Wel is het Hof van oordeel dat, voor zover mogelijk, gebruik dient te worden gemaakt van relevante empirische (markt)gegevens in vergelijkbare situaties, en dat uitkomsten daarvan, voor zover betrouwbaar en representatief voor belanghebbende” […] “Het artikel van Finnerty introduceert een theoretisch model dat empirisch wordt getoetst aan feitelijk opgetreden discounts bij private placements van aandelen van Amerikaanse ondernemingen, welke aandelen op grond van SEC-Rule 144 beperkt verhandelbaar zijn. Het Hof is van oordeel dat het artikel van Finnerty empirische grondslag biedt voor het bestaan van een discount uit hoofde van beperkte verhandelbaarheid en empirisch gefundeerde aanknopingspunten geeft voor een modelmatige benadering van de omvang van die discount.”
The court accepted the Finnerty model because of its correlation with actual restricted stock studies, however they cautioned that [differences between] the shares involved in Finnerty’s research and [the shares of the subject company] (…) must be taken into account. The Chaffe and Shout Put models are not mentioned in the transcript of this court case. Chaffe and Shout Put both overestimate DLOM at lower volatilities relative to restricted stock studies and have theoretical weaknesses (ignore risk of price increases).
This case further mentions that results from the Finnerty model are similar to results from a restricted stock study from FMV Opinions. The following was part of a reaction of the defendant:
“[De gemachtigde] heeft met zijn reactie een publicatie meegestuurd getiteld Determining Discounts for Lack of Marketability, van FMV Opinions Inc. () FMV probeert evenals Finnerty langs empirische weg inzicht te geven in de mate van Discount for Lack of Marketability (). Dit is de gemeten () afwaardering als gevolg van handelsrestricties op grond van SEC Rule 144. () De uitkomsten van de FMV Study en Finnerty vertonen gelijkenis, met name waar het betreft de kleinere ondernemingen met hoge volatiliteiten.”
The Ghaidarov model was not mentioned in the transcript of this case. As demonstrated below, Finnerty and Ghaidarov result in similar DLOMs at the one-year holding period, in particular at lower volatility levels. The gap between Finnerty and Ghaidarov increases with volatility. Given the holding periods prescribed by SEC rule 144 (6 months to 2 years in most cases and depending on the grant date), approximately 1 year would be a relevant holding period for comparison restricted stock studies.
The Finnerty model tops off at a DLOM of 32.3%. A restricted stock study by FMV Opinions for the period 1980 to 2015 shows a median DLOMs for five quintiles, arranged by stock price volatility. At the 5th quintile, where volatility is 104%, the median DLOM is 43.2% (and can achieve a high of 91.3%).[40] A similar restricted stock study by Stout Risius Ross LLC[41] shows a 5th quintile median DLOM of 42.9% (high DLOM of 91.3%). The 5th quintile volatility in that study was 99.4%. If the restricted stock studies can be viewed as empirical evidence of DLOM, the Finnerty model understates DLOM at elevated volatility levels.
While not as high as the 5th quintile restricted stock studies, the Ghaidarov model shows higher DLOMs at higher volatility levels. As such, we can conclude that the Ghaidarov DLOM is superior to Finnerty at elevated volatility levels.
The gap between Ghaidarov DLOM and Finnerty DLOM also increases with holding period. See the graphs below.
DLOM studies can be applied to roughly estimate DLOM for a valuation object. This can be further finetuned by applying qualitative assessments. Two such approaches that I would like to highlight are Mandelbaum and van den Cruijce.
The United States case of Mandelbaum v. Commissioner[42] concerned shares gifted to family members in a women’s apparel chain called Big M. The shares were closely held within the family and the shareholders agreement aimed to keep the shares within the family by limiting the ability to seek an outside buyer. During the period within which shares were gifted, the company restructured so that a limited number of class-A shares with voting rights remained in the hands of the board and the remaining shares (class-B, no voting rights) were held by other family members. Only class-B shares were gifted.
Each side of the case introduced their own expert witnesses. Mandelbaum introduced Paul Mallarkey – and American Society of Appraisers (ASA) senior appraiser in business valuation and CFA charterholder. The commissioner of internal revenue introduced Roger Grabowski, who is currently a fellow with the American Society of Appraisers (ASA).
Judge David Laro developed a nine-factor adjustment process for evaluating DLOMs and employed this is in a qualitative fashion (i.e. above average DLOM, average DLOM, below average DLOM) to adjust DLOM within its range:
Van den Cruijce and Endres (2022) tested the impact of 9 non-volatility factors on DLOM.[43] They developed a multiple regression of these 9 factors against DLOMs determined by United States courts in estate and gift tax cases. In this case, DLOMs were drawn directly from valuations of private companies. Unlike restricted stock studies where there is an expectation of the ability to sell shares after a lock-up period, there is no liquid market for the shares in the Van den Cruijce and Endres study.
The range of 137 DLOMs in this study are from 0% to 50% with a mean of 23.7%. As discussed previously in this report, we apply a 5-year restriction period for the Client. The cases used in this study probably have restriction periods less than 5 years. As such, it would not be unusual for our concluded DLOM, after non-volatility adjustments, to exceed 50%.
Van den Cruijce and Endres applied the following independent variables in their analysis:
The meaning of “value” can change depending on the purpose of the valuation. The type of value and its definition is referred to as “standard of value”. The range of acceptable DLOMs can change depending on the standard of value. In Mandelbaum, one of the experts argued for a 70% to 75% DLOM. Judge Laro criticized this conclusion, stating that in the expert’s determination of fair market value, they focused only on a hypothetical willing buyer and did not reflect the view of a hypothetical willing seller.
The judge went further to state, “Although the record indicates that the petitioners adamantly desire to keep the ownership of Big M within their family, the test of fair market value rests in the concept of a hypothetical willing buyer and a hypothetical willing seller. Ignoring the views of a willing seller is contrary to this well-established test … He did not consider whether such a seller would sell his or her Big M stock for at least 70% less than its freely traded value. We find it incredible the proposition that any shareholder of Big M would be willing to sell his or her stock at such a large discount.”
Most standards of value that I work with in professional appraisal practice prohibit the presence of a compelled or forced sale. One should apply a measure of logic, and if possible empirical evidence, when considering the level of DLOM to be applied.
While restricted stock studies show a DLOM midpoint of 20% to 24%, DLOM can reach up to 91%. Pre-IPO studies show a range of DLOM midpoints, one of which shows 46% to 47%, and DLOM can exceed 73%. Private company discount studies show a variety of midpoints, one of which is approximately 24% and DLOM can reach 75%. Notwithstanding the foregoing ranges, one must logically evaluate whether a DLOM near the upper end of the ranges is applicable to the standard of value being applied.
The court of Amsterdam ruled in a case involving a minority stake in shares and the Dutch tax authorities that DLOM should be based, as far as possible, on empirical data from comparable situations and that the Finnerty model has been tested with a reasonable degree of success against empirical data. I have demonstrated that the Finnerty model understates DLOM at higher volatilities and longer holding periods relative to the Ghaidarov model. The court also ruled that differences between the shares involved in Finnerty’s study and the shares of the subject company must be taken into account. The Van Den Cruijce Transformation, as an additive to option models, can achieve these differences.
For a broad discussion of minority stake valuations in the Netherlands, see our article, “How do you value a minority interest in the Netherlands?” For a detailed discussion of court cases involving the Dutch tax authorities and minority interest valuations, see my article, “Minority Interest Valuations in Dutch Tax Court Cases.” For more information on the business valuation capabilities of AN Valuations, see our Company Valuation services page.
DLOM should generally be applied when valuing a non-controlling stake in a privately held company. Control can be achieved through voting rights, board seats, and/or contractual rights. A careful assessment of the company’s shareholder agreements and company statutes is an important step in assessing whether DLOM should be applied.
In cases where there is clearly a lack of control, DLOM might not be required if there is an active and liquid market for non-controlling stakes in similar companies. In certain industries and markets and for certain types of companies, venture capital and private equity regularly invest less than a controlling stake. One must, however, look carefully at the other contractual arrangements that these professional investors require, which in some cases result in substantial control.
DLOM means discount for lack of marketability. It reflects the extra time, expense and effort, relative to a controlling stake, to sell shares in a minority or non-controlling stake in a privately held company.
The DLOM method means applying a discount to the share value in a privately held company when a non-controlling stake is being valued. The formula is V x (1 – DLOM), where V = marketable value.
DLOC mean discount for lack of control. DLOM means discount for lack of marketability. DLOC reflects a valuation discount that a typical investor may require when they do not have financial decision-making powers. DLOM reflects the extra time, expense and effort, relative to a controlling stake, to sell shares in a minority or non-controlling stake in a privately held company.
DLOM can be calculated using restricted stock studies, pre-IPO transaction studies, private company discount studies, and option models. These can be adjusted with qualitative assessments such as Mandelbaum or Van den Cruijce.
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[1] Aaron M. Stumpf, Robert L. Martinez, and Christopher T. Stallman; “The Stout Risius Ross Restricted Stock Study: A Recent Examination of Private Placement Transactions from September 2005 through May 2010”; Business Valuation Review Volume 30, No. 1, Spring 2011; pp. 7–19. https://bvr.kglmeridian.com/view/journals/bvrv/30/1/article-p7.xml
[2] 6 months or 1 year depending on certain conditions relating to the issuer and seller. See SEC Rule 144.
[3] Fleisher, A; Simmons, C; Hamilton, E; Povilonis, J; “Post-IPO Liquidity Roadmap for Pre-IPO Equity Investors”; Alert Memorandum; ©2024 © Cleary Gottlieb Steen & Hamilton LLP. January 4, 2024. https://www.clearygottlieb.com/-/media/files/alert-memos-2024/liquidity-roadmap-a-post-ipo-guide-for-pre-ipo-investors.pdf
[4] Van den Cruijce, Johan; “The impact of control on the discount for lack of marketability”; taxnotes federal, volume 175, number 4; April 25, 2022. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4141152
[5] Elmore, J; “Determining the Discount for Lack of Marketability with Put Option Pricing Models in View of the Section 2704 Proposed Regulations”; Valuation Practices and Procedures Insights, Winter 2017; Willamette Management Associates. https://www.willamette.com/assets/files/2017%20Winter%20-%20Estate%20and%20Gift%20Tax.pdf
[6] Institutional Investor Study Report of the Securities and Exchange Commission, Volume 5; 92nd Congress, 1st session; House Document No. 92-64, Part 5; pgs 2445 – 2475. https://www.sechistorical.org/collection/papers/1970/1971_0310_SECInstitutionalInvestor_14.pdf
[7] -15% to 0%, 0.1% to 10%, 10.1% to 20%, 20.1% to 30%, 30.1% to 40%, 40.1% to 50%, 50.1% to 80%
[8] NYSE, American Stock Exchange, Over the Counter registered, Over the Counter unregistered, and Unknown
[9] Banks, Investment Advisors, Property & Liability Insurance Companies, Life Insurance Companies, Self-Administered Employee Benefit, Foundations, Educational Endowments, and Venture Capital
[10] “Stout Restricted Stock Study Companion Guide”; © 2023 Stout Risius Ross, LLC. https://www.bvresources.com/docs/default-source/free-downloads/rss-companion.pdf?sfvrsn=b0ebc8b2_24
[11] Pratt, Shannon P; Grabowski, Roger J; Cost of Capital: Applications and Examples, 3rd edition; ©2008 John Wiley & Sons, Inc; pg 424.
[12] Van den Cruijce, Johan; “The impact of control on the discount for lack of marketability”; Tax Notes Federal, volume 175, number 4; April 25, 2022. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4141152
[13] Emory, John D, Sr; Dengel, F. R. III; Emory, John D, Jr; Discounts for Lack of Marketability, Emory Pre-IPO Discount Studies 1980-2000 as Adjusted October 10, 2002. https://emoryco.com/wp-content/uploads/2022/09/Adjusted_Emory_Studies_1980_2000_10_10_02.pdf
[14] Reilly, Robert F, CPA; Nicholls, Samuel S; “Discount for Lack of Marketability in the Professional Practice Valuation”; Willamette Management Associates, a Citizens Company; Summer 2022. https://willamette.com/resources/perspectives-insights/insights-summer-2022.html
[15] See for example Koeplin, J; Sarin, A; Shapiro, A; “The Private Company Discount”; Bank of America Journal of Applied Corporate Finance 12, no. 4 (Winter 2000); pg. 94-101. https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1745-6622.2000.tb00022.x
[16] Goetz, S.; “Public versus Private: New Insights into the Private Company Discount”; Journal of Business Valuation and Economic Loss Analysis, 2021, vol. 16, issue 1, 15-40. https://ideas.repec.org/a/bpj/jbvela/v16y2021i1p15-40n2.html
[17] Van den Cruijce, Johan; “The impact of control on the discount for lack of marketability”; taxnotes federal, volume 175, number 4; April 25, 2022. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4141152
[18] Koeplin, J.; Sarin, A.; & Shapiro, A. C.; “The Private Company Discount”; ©2000 Journal of Applied Corporate Finance, 12(4). https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1745-6622.2000.tb00022.x
[19] Kooli, M.; Kortas, M.; & L’Her, J.-F.; “A New Examination of the Private Company Discount: The Acquisition Approach”; ©2003 The Journal of Private Equity. https://www.jstor.org/stable/43503345
[20] Block, S.; “The Liquidity Discount in Valuing Privately Owned Companies”; ©2007 Journal of Applied Finance, 17(2), 33-40. https://scispace.com/papers/the-liquidity-discount-in-valuing-privately-owned-companies-c7yisg65yx
[21] Officer, M. S.; “The price of corporate liquidity: Acquisition discounts for unlisted targets”; ©2006 Journal of financial economics, 571-598. https://www.researchgate.net/publication/222838090_The_Price_of_Corporate_Liquidity_Acquisition_Discounts_for_Unlisted_Targets
[22] Paglia, J. K.; Harjoto, M. A.; “The Discount for Lack of Marketability in Privately Owned Companies: A Multiples Approach”; ©2010 Journal of Business Valuation and Economic Loss, 5(1), 1-26. https://www.researchgate.net/publication/46555514_The_Discount_for_Lack_of_Marketability_in_Privately_Owned_Companies_A_Multiples_Approach
[23] De Franco, G.; Gavious, I.; Jin, J.; Richardson, G. D.; “Do Private Company Targets that Hire Big4 Auditors Receive Higher Proceeds?”; ©2011 Contemporary Accounting Research, 28(1), 215-262. https://utoronto.scholaris.ca/server/api/core/bitstreams/4ff855e5-9061-443d-9204-a22a316e61c7/content
[24] Van Oeveren, Kim; “The Private Company Discount: An acquisition study on private firms within the US”; August 2018. https://arno.uvt.nl/show.cgi?fid=147774
[25] John D. Finnerty, “Using Put Option-Based DLOM Models to Estimate Discounts for Lack of Marketability,” Business Valuation Review 31, no. 4 (Winter 2013): 166. https://www.researchgate.net/publication/269870448_Using_Put_Option-based_DLOM_Models_to_Estimate_Discounts_for_Lack_of_Marketability
[26] ECLI:NL:GHAMS:2012:BW1517
[27] Elmore, J; “Determining the Discount for Lack of Marketability with Put Option Pricing Models in View of the Section 2704 Proposed Regulations”; Valuation Practices and Procedures Insights, Winter 2017; Willamette Management Associates. https://www.willamette.com/assets/files/2017%20Winter%20-%20Estate%20and%20Gift%20Tax.pdf
[28] Katsanis, M; “Stand Up and Shout—It Is Another DLOM Put Model!”; Business Valuation Review 31, no. 1 (Spring 2012): 48–52. https://chaffe-associates.com/wp-content/uploads/2025/04/Stand-Up-and-Shout-It-Is-Anoher-DLOM-Put-Model.pdf
[29] Chaffe model
[30] Longstaff model
[31] Finnerty and Ghairadov models
[32] Elmore, J; “Determining the Discount for Lack of Marketability with Put Option Pricing Models in View of the Section 2704 Proposed Regulations”; Valuation Practices and Procedures Insights, Winter 2017; Willamette Management Associates. https://www.willamette.com/assets/files/2017%20Winter%20-%20Estate%20and%20Gift%20Tax.pdf
[33] Francis A. Longstaff, “How Much Can Marketability Affect Security Values?” Journal of Finance 1, no. 5 (December 1995): 767–774. https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1540-6261.1995.tb05197.x
[34] Elmore, J; “Determining the Discount for Lack of Marketability with Put Option Pricing Models in View of the Section 2704 Proposed Regulations”; Valuation Practices and Procedures Insights, Winter 2017; Willamette Management Associates. https://www.willamette.com/assets/files/2017%20Winter%20-%20Estate%20and%20Gift%20Tax.pdf
[35] John D. Finnerty, ‘‘An Average-Strike Put Option Model of the Marketability Discount,’’ The Journal of Derivatives 19 (Summer 2012): 52–69. https://www.researchgate.net/publication/275904945_An_Average-Strike_Put_Option_Model_of_the_Marketability_Discount
[36] Elmore, J; “Determining the Discount for Lack of Marketability with Put Option Pricing Models in View of the Section 2704 Proposed Regulations”; Valuation Practices and Procedures Insights, Winter 2017; Willamette Management Associates. https://www.willamette.com/assets/files/2017%20Winter%20-%20Estate%20and%20Gift%20Tax.pdf
[37] Stillian Ghaidarov, “Analysis and Critique of the Average Strike Put Option Marketability Discount,” workpaper (September 24, 2009). https://www.researchgate.net/publication/228290661_Analysis_and_Critique_of_the_Average_Strike_Put_Option_Marketability_Discount_Model
[38] Ibid.
[39] ECLI:NL:GHAMS:2012:BW1517
[40] Elmore, J; “Determining the Discount for Lack of Marketability with Put Option Pricing Models in View of the Section 2704 Proposed Regulations”; Valuation Practices and Procedures Insights, Winter 2017; Willamette Management Associates. https://www.willamette.com/assets/files/2017%20Winter%20-%20Estate%20and%20Gift%20Tax.pdf
[41] “Stout Restricted Stock Study Companion Guide”; © 2023 Stout Risius Ross, LLC. https://www.bvresources.com/docs/default-source/free-downloads/rss-companion.pdf?sfvrsn=b0ebc8b2_24
[42] Mandelbaum v. Commissioner; 69 T.C.M. (CCH) 2852; (1995).
[43] Van den Cruijce, J; Endres, S; “The Impact of contractual transfer restrictions and micro liquidity on the discount for lack of marketability”; Dissertation presented to obtain the degree of Doctor in Business Administration; 2022 KU Leuven. https://openurl.ebsco.com/EPDB%3Agcd%3A13%3A37587293/detailv2?sid=ebsco%3Aplink%3Ascholar&id=ebsco%3Agcd%3A163322611&crl=c&link_origin=www.google.com