May 01, 2017

Discounts for lack of marketability (DLOMs) have
frequently been the subject of controversy in
valuations. The reason: applying a DLOM – an
amount or percentage deducted from the value of
an ownership interest to reflect the relative absence
of marketability – can result in significant value
reduction compared with the pro rata value of a
business interest. Today’s valuation practitioners
use numerous methods^{1} that can be classified into
four main categories, each with its advantages
and disadvantages:

**Benchmark study**: utilizes restricted stocks and initial public offering pricing data**Security-based approach**: utilizes theoretical option pricing models, illiquidity estimates demonstrated by traded stock prices, and option prices**Analytics**: utilizes historical studies on private placement of equity**Other approaches**: Quantitative Marketability Discount Model (QMDM), Nonmarketable Investment Company Evaluation (NICE), etc.

In analyzing data sets, several key weaknesses become evident: the demonstration of a selection bias (i.e., picking data based on availability); the introduction of subjective inputs; and references to outdated data, disparate standards, and periods of significant change. Finally, results derived from data set analysis can lead to wide ranges of DLOMs that require a qualitative assessment in order to conclude specific value.

Based on this understanding, and on
empirical evidence after implementing
various techniques, the implementation
of option pricing models has been
identified as the most appropriate
method for estimating DLOMs.^{2} This
article describes the four principal
models of this type:

**Chaffe**: Black-Scholes-Merton put option**Longstaff**: lookback put option**Finnerty**: average-strike put option**Ghaidarov**: average-strike put option

The article then presents a simple case study and outlines a useful framework for valuing interests in private companies using an efficient, effective quantitative model.

David Chaffe, in his 1993 option pricing
study, highlighted a link between
a DLOM and the cost to purchase a
European put option.^{3} Chaffe’s theory
was as follows:

*If one holds restricted or
nonmarketable stock and purchases
an option to sell those shares at the
free market price, the holder has, in
effect, purchased marketability for
those shares. The price of this put is
the discount for lack of marketability.*

Because Chaffe relied on the Black-Scholes-Merton put option pricing model, the inputs to his model are the stock price, the strike price, the time to expiration, the interest rate, and volatility. In the Chaffe model, the stock price and the strike price equal the marketable value of the private company stock as of the valuation date. Due to its reliance on European options, the Chaffe model is downward-biased. Consequently, the results derived by his model should be considered a lower bound for estimating DLOMs.

Francis Longstaff, meanwhile, used a lookback put option – an exotic option with path dependency – to estimate DLOMs.4 In this case, the payoff depends on the optimal price of the underlying asset occurring over the life of the option. The options allow the holder to “look back” over time to determine the payoff. Yet the Longstaff model assumes that an investor has perfect market timing and, as a result, reflects an upper bound for DLOMs. Finally, there is disagreement over whether the Longstaff model concludes to a DLOM or to a liquidity premium that needs to be converted to a discount.

John Finnerty extended Longstaff’s
work by utilizing an average-strike put
option that is also exotic but does not
assume perfect market timing.^{5} The
Finnerty model appears to work very
well at lower volatilities, but yields
low DLOMs at higher volatilities when
compared with the restricted stock
transactions. Furthermore, because it
yielded DLOMs that exceeded 100%,
the original Finnerty model had to be
revised. The revised model produces no
discount in excess of 32.3%, regardless
of higher volatilities and longer holding
times. This limitation may significantly
understate the DLOM for volatilities
exceeding 125% and six-month holding
horizons. These inputs are relatively rare
in valuations, however.

Like Finnerty, Stillian Ghaidarov also
uses average-strike put options.^{6}
The results of the Ghaidarov model –
developed as a criticism to the original
Finnerty model – closely match the
modified Finnerty model for the sixmonth
holding period and volatilities
through 125%. Past this point though,
the Ghaidarov model can generate
DLOMs that are significantly higher
than the results indicated by restricted
share studies. These should be used with
caution.

Overall, based on our experience in the respective inputs to these models, their levels of difficulty, and the generated estimates for different asset classes, we believe that the Finnerty model is the most appropriate method to estimate DLOMs for financial reporting purposes. It is worth mentioning, though, that for tax-related valuation purposes, the Stout Restricted Stock Study™ (formerly the FMV Restricted Stock Study) is a more relevant valuation tool.

The discount associated with the illiquidity of each type of instrument included in a capital structure of a private company is directly related to its inherent rights and privileges. In each case, the level of discount applicable depends on a number of parameters, including the instrument’s overall seniority in a capital structure; whether it is considered to be in the money, at the money, or out of the money as of the measurement date (with regard to the corresponding participation thresholds); the level of equity volatility used for valuation purposes; the effective time frame for the analysis; and the company’s dividend policy. From a valuation perspective, securities with higher seniority or more protective provisions are associated with a lower illiquidity discount due to the lower risk of receiving distributions below certain thresholds or beyond certain time frames expected by potential investors. On the other hand, securities that are very junior in capital structures – and thus lack any substantial control over the company’s operating and managerial decisions – translate to a higher illiquidity discount due to the higher risks associated with achieving the expected returns over the holding period.

When measuring the appropriate
illiquidity discount in complex capital
structures, one crucial complexity
involves the search for a quantitative
and qualitative way to differentiate the
applicable discount among the various
securities analyzed. This article relies
on the framework introduced by Dwight
Grant in “Thoughts on Calculating
DLOMS,” where the discount applicable
in a random security is positively
correlated to the volatility of the
specified security within the framework
of the given capital structure under
examination.^{7}
The first step in calculating the
appropriate volatility of a security
included in a given capital structure
is to develop a Black-Scholes option
pricing model (BSOPM): an arbitrage
pricing model developed using the
premise that two assets with identical
payoffs must have identical prices to
prevent arbitrage. The BSOPM, which
relies on such variables as asset price,
strike price, expected term, risk-free
rate, volatility, and dividend yield, is
basically a contingent claim analysis
that treats equity as a combination of
call options associated with the claims
of each security included in the capital
structure. The strike prices of these
options correspond to the participation
thresholds of these securities, and the
respective call options represent the
value attributable to these securities
above the specified strike prices
(or breakpoints) based on the set of
required assumptions.

After calculating the values of the call options, the next step is to calculate the delta that corresponds to each security. The delta measures the changes in the value of the call options relative to the change in the value of the asset price – in this case, the equity value of the company. In other words, the delta measures the relationship between the volatility of the equity value and the value of the specific security, which is considered to be a derivative instrument on the equity value of the company. The delta spreads of the different call options are used to estimate the aggregate delta associated with each security class, which is equal to the sum of each security class’ ownership claim on the calculated delta spreads. The volatility for each security class can then be expressed as the product of the aggregate delta, the leverage ratio (based on the underlying asset price relative to the total claims of each security class), and the assumed equity volatility. Based on the concluded asset volatility for each security, the appropriate DLOM is finally calculated based on the revised Finnerty model.

Below, we present a simple case with a hypothetical capital structure that includes only one type each of preferred security, common units, and options on common stock. Figure 1 shows a summary of the assumed capital structure.

Figure 2 presents the BSOPM framework vis-à-vis the rights and privileges of the securities analyzed, based on the assumed capital structure.

Figure 3 presents both the individual security volatility analysis and the calculation of the appropriate DLOM applicable to each security.

As presented in line 8 (in Figure 3), the concluded asset volatility is lower for the most senior securities and higher for the more junior securities – a reasonable conclusion based on the expectations described earlier. The same relationship is observed between the seniority of the securities and the concluded DLOM based on the Finnerty model, which relies on the concluded asset volatility for each security along with the assumptions for the expected holding term and the dividend yield. The preferred units, which enjoy greater seniority in the capital structure, are subject to a lower illiquidity discount compared with lower seniority attributed to the “options @$1.00.” The difference between the concluded illiquidity discounts of these securities might increase or decrease based on the set of assumptions used for the purpose of the BSOPM analysis.

Figure 4 presents a sensitivity analysis based on different volatility, exit timing, and asset price assumptions, and on the impact on the concluded DLOM among the different securities. Figure 4 reveals a positive relationship between the volatility or holding period assumption and the concluded DLOM. Higher volatility or holding period assumptions lead to higher applicable illiquidity discounts due to the greater uncertainty and risk concerning future realized distribution levels. The opposite relationship can be observed concerning the applicable asset price allocated: higher asset price inputs produce lower illiquidity discounts because of the lower risk associated with the probability of the analyzed securities being out of the money at the end of the holding period.

Finally, Figure 5 presents the results of the different quantitative methods described earlier, based on different holding periods and an assumed volatility of 40%.

The framework outlined here is an expedient tool for valuing interests in private companies with a quantitative model that differentiates securities and assigns different illiquidity discounts based on their relative rights and privileges (and on the remaining assumptions linked to the BSOPM framework and management inputs). However, qualitative factors associated with ownership control premiums, voting rights, or other protective provisions should always be considered in order to avoid determining an illiquidity discount that over- or understates the value of the subject interest.

- IRS. Discount for Lack of Marketability: Job Aid for IRS Valuation Professionals. September 2009.
- Aaron Rotkowski and Michael Harter, “Current Controversies Regarding Option Pricing Models,”
*Taxation Planning and Compliance Insights*, Autumn 2013. - David Chaffe, “Option Pricing as a Proxy for Discount for Lack of Marketability in Private Company Valuations.”
*Business Valuation Review*, 12, 4:182-188, 1993. - John Elmore, “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. - John D. Finnerty, “An Average-Strike Put Option Model of Marketability Discount.”
*The Journal of Derivatives*, 19, 4:53-69, 2012. - Stillian Ghaidarov, ‘‘The Use of Protective Put Options in Quantifying Marketability Discounts Applicable to Common and Preferred Interests.’’
*Business Valuation Review*, 28, 2:88-99, 2009. - Dwight Grant, “Thoughts on Calculating DLOMs,”
*Business Valuation Review*, 33, 4:102-112, 2014.