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Measuring the Discount for Lack of Marketability:
An Empirical Investigation
John D. Finnerty
Professor of Finance, Fordham University
Earlier versions of the paper were presented at the Financial Management Association 2016 annual meeting, a 2016 Business Valuation Resources webinar, the 2018 FMA European meeting, and the 2017 World Finance Conference. I would like to thank Steven Feinstein, Adrian Pop, Loic Belze, and BVR webinar participants for helpful suggestions and Hanyu Sun, Michelle Tu, and Hang Wang for research assistance. JEL Classification: G1 Keywords: Marketability Discount, Average-Strike Put, Transfer Restrictions, Term Structure, Term Premium
June 2018
Fordham University 45 Columbus Avenue New York, NY 10023 Tel: (347) 882-8756
e-mail: [email protected]
© 2018 John D. Finnerty. All rights reserved.
Measuring the Discount for Lack of Marketability:
An Empirical Investigation
Abstract
Stock transfer restrictions limit a share’s marketability and reduce its value. The discount for lack of marketability (DLOM) has been modeled as the value of an average-strike put option based on a lognormal approximation whose accuracy worsens as the restriction period lengthens. I generalize the average-strike put option DLOM model to restriction periods of any length L by modeling the L-year DLOM as the value of the one-year DLOM compounded over L years and then adjusting for a DLOM term premium. I empirically fit the model to the implied discounts from a sample of 5,391 private equity transactions between 1985 and 2017 and derive an ex post DLOM term structure. I derive separate DLOM term structures for the last pre-IPO and earlier pre-IPO transactions. I suggest how to further generalize the model to restriction periods of uncertain length based on the empirically supported assumption that L is approximately exponentially distributed.
Measuring the Discount for Lack of Marketability:
An Empirical Investigation
The growth of private equity (PE) investing has stimulated secondary market trading in
unregistered common stocks.1 PE funds selling shares of portfolio companies, PE fund investors
selling limited partnership units, and employees and other shareholders of privately owned firms
selling shares to investors seeking such investments are just three examples.2 Privately owned
firms with greater access to private capital are also waiting longer to go public, so more
shareholders seeking liquidity for their shareholdings must sell their shares in privately negotiated
transactions.3 Secondary market transactions in private equity are infrequent, and the markets for
these shares are consequently relatively illiquid, due to the legal resale restrictions the Securities
Act of 1933 (1933 Act) imposes on unregistered common stock.4 The desire to transact in
unregistered common stock coupled with the lack of regular market prices to facilitate price
discovery or at least permit investors to gauge an appropriate discount for lack of marketability
(DLOM) have increased interest in DLOM models that can be used reliably in pricing restricted
shares.
This paper develops general DLOM models that can be used to value unregistered shares
when the length of the restriction period L can be reasonably estimated or when L is highly
uncertain but can be assumed to follow an exponential distribution. I find that the probability
distribution for the length of the restriction period is approximately exponentially distributed but
with different parameters depending on whether the private equity transaction is the last one
preceding the firm’s IPO or is earlier in the firm’s life cycle. I fit a separate DLOM term structure
for each class of pre-IPO private transactions.
Fitting a DLOM term structure is of interest because DLOM models are widely used in
valuation practice but restriction periods vary considerably in length (Pratt and Grabowski, 2014).
Financial economists and business appraisers typically value restricted stock by calculating the
freely traded value the stock would have if it traded in a liquid market free of restrictions and
subtracting an estimated DLOM to reflect the loss of timing flexibility due to any resale or transfer
restrictions during the estimated restriction period (Cornell, 1993; Pratt, 2001; Damodaran, 2012;
Pratt and Grabowski, 2014).
Resale and transfer restrictions entail a loss of timing flexibility because the shareholder
has to postpone sale or transfer until the restrictions lapse unless the shareholder can find an
2
accredited investor who is willing to buy the shares. This loss of flexibility imposes a cost, which
can be modeled as the value of a foregone put option that will expire when the marketability
restriction lapses (Longstaff, 1995, 2001; Kahl, Liu, and Longstaff, 2003; and Finnerty, 2012,
2013a). The DLOM has been modeled as the value of an average-strike put option based on a
lognormal approximation but the approximation’s accuracy worsens as the restriction period
lengthens. I generalize the average-strike put option DLOM model to restriction periods of any
length L by modeling the L-year DLOM as the value of the one-year DLOM compounded over L
years and then adjusting for a DLOM term premium. I empirically fit the model to the implied
discounts from a sample of 5,391 private equity transactions between 1985 and 2017 conducted
by firms that subsequently went public. I measure the discounts implicit in the difference between
the private equity transaction price and the firm’s IPO price after an adjustment for the expected
appreciation in the firm’s share price, based on the performance of comparable firms, between the
two dates and then use these adjusted discounts to derive an ex post DLOM term structure. The
resulting term structure would enable a business valuer to gauge a DLOM for a prospective private
placement that is consistent with past DLOMs. My data set does not include any ex ante estimates
of the (expected) restriction period, but if such a data set were to become available, my modelling
approach could be utilized to fit an ex ante DLOM term structure.
An important recent legal development has potentially increased the need for DLOM
models to price unregistered shares by expanding the availability of private equity capital and
reducing the urgency of going public to raise funds. The Regulation A small issue exemption
under the Securities Act of 1933 had become ineffectual by the early 2000s because of its low $5
million annual offering limit (Knyazeva, 2016). The Jumpstart Our Business Startups (JOBS)
Act, enacted in 2012, directed the SEC to adopt rules exempting from the registration
requirements of the 1933 Act offerings of up to $50 million of securities within a 12-month
period.5 Final SEC rules became effective June 19, 2015, which expanded Regulation A
(referred to as Regulation A+) into two tiers: Tier 1, for securities offerings of up to $20 million
in a 12-month period, and Tier 2, for securities offerings of up to $50 million in a 12-month
period.6 Selling securityholders are limited in the aggregate to selling (through Regulation A
secondary offerings) no more than 30% of the dollar amount of the offering in the following 12-
month period, and they are still subject to the overall 30% selling securityholder limitation
thereafter. Even with these marketability restrictions, a Regulation A+ issue could become a
viable alternative to a small registered initial public offering (IPO) or could complement other
3
stock offerings that are exempt from registration under the 1933 Act because the substantially
greater annual offering limit should increase the volume of transactions in unregistered shares.
The early DLOM papers reported the results of empirical studies of the discounts reflected
in the prices of letter stock,7 unregistered shares of common stock sold privately by public firms,
which purchasers could later resell subject to the resale restrictions imposed by Rule 144 under
the 1933 Act (SEC, 1971; Wruck, 1989; and Silber, 1991).8 More recent research has focused on
developing and empirically testing DLOM put option models. At least three such DLOM models
have been proposed in the finance literature (Finnerty, 2013b). The seminal BSM put option
pricing model (Black and Scholes, 1973; and Merton, 1973) was the basis for the earliest put option
DLOM model proposed by Alli and Thompson (1991). Appraisers still use it (Chaffee, 1993; and
Abbott, 2009). It measures the DLOM with respect to the loss of the flexibility to sell shares at
today’s freely traded market price when the shareholder cannot sell the stock until the end of some
restriction period. It cannot measure the DLOM accurately because its assumption of a fixed strike
price for the duration of the restriction period gives the restricted stockholder absolute downside
protection (barring default by the option writer) whereas the potential sale prices are not actually
fixed with restricted stock. Since the price of the otherwise identical unrestricted asset will change
during the restriction period due to normal market forces, the asset holder’s loss of timing
flexibility should be measured relative to the opportunity she would have to sell at any of these
market prices were there no transfer restrictions.
The lookback put and average-strike put option models avoid this shortcoming by adjusting
the strike price to reflect the changing price of the unrestricted asset during the period of non-
marketability. Longstaff (1995) obtains an upper bound on the DLOM by modeling the value of
marketability as the price of a lookback put option, which assumes that investors have perfect
market-timing ability. This assumption is generally consistent with empirical evidence that private
information enables insiders to time the market and realize excess returns (Gompers and Lerner,
1998). However, it is inconsistent with evidence that outside investors, at least on average, do not
have any special ability to outperform the market (Graham and Harvey, 1996; and Barber and
Odean, 2000). Thus, while the lookback put option model may be appropriate in the presence of
asymmetric information for equity placed with insiders who possess valuable private information,
it will overstate the discount when investors do not have valuable private information about the
stock (Finnerty, 2013a).
Finnerty (2012, 2013a) models the DLOM as the value of an average-strike put option.
4
The investor is not assumed to have any special market-timing ability; instead, he assumes that the
investor would, in the absence of any transfer restrictions, be equally likely to sell the shares
anytime during the restriction period. To be useful, a DLOM model should produce discounts that
are consistent with the DLOMs observed in the marketplace. Numerous studies have documented
average discounts between 13.5 percent and 33.75 percent in private placements of letter stock,
which is not freely transferable because of the Rule 144 resale restrictions (Hertzel and Smith,
1993; and Hertzel et al., 2002). Business appraisers typically apply DLOMs between 25 and 35
percent for a two-year restriction period and between 15 and 25 percent for a one-year restriction
period (Finnerty, 2012, 2013a). Finnerty (2012) compares the model-predicted discounts to the
discounts observed in a sample of 208 discounted common stock private placements and finds that
the average-strike put option model discounts are generally consistent with the observed private
placement discounts after adjusting for the information, ownership concentration, and
overvaluation effects that accompany a stock private placement. However, the model appears to
be increasingly less accurate for longer restriction periods beyond two years, especially for high-
volatility stocks.9
Stock transfer restrictions, such as those imposed by Rule 144, are one of the more often-
cited factors responsible for a (restricted) share’s relative lack of liquidity, in this case, due to its
lack of marketability. Liquidity refers to an asset holder’s flexibility to transfer asset ownership
through a market transaction. It is the relative ease with which an asset holder can convert the
asset into cash without losing some of the asset’s intrinsic value. A liquid market gives an asset
holder the flexibility to sell the asset at any time she chooses without sacrificing any intrinsic value.
Lack of liquidity imposes a loss of timing flexibility because an asset holder cannot dispose of the
asset quickly unless she is willing to accept a reduction in value. It takes more time to find a buyer
in an illiquid market. The consequent loss of flexibility to sell an asset freely, or equivalently, the
ability to sell it quickly only if there is some sacrifice of intrinsic value, can be modeled as the loss
of value of a foregone put option.
An asset that lacks marketability also lacks liquidity because the marketability restrictions
inhibit the holder from selling the asset quickly for its full intrinsic value. But lack of marketability
can be distinguished from lack of liquidity. An asset’s marketability refers to an asset holder’s
legal and contractual ability to sell or otherwise transfer ownership of the asset. Lack of
marketability arises when legal or contractual restrictions on transfer prevent, or at least severely
impair, an asset holder’s ability to sell the asset or transfer it until the restriction period lapses
5
(Finnerty, 2013b). The 1933 Act’s restrictions on offering unregistered securities to investors who
are not accredited (i.e., do not meet certain income and net worth tests) exemplify such legal
restrictions, and the almost absolute prohibition on transferring employee stock options
exemplifies such contractual restrictions on marketability.
It is important to appreciate that even when securities are unregistered, there may not be
an absolute prohibition on transfer. For example, a security holder can still transfer unregistered
shares of common stock by relying on one of the exemptions from registration under the 1933 Act.
Also, specialized secondary markets have developed to enable holders of unregistered common
stock to find potential buyers.10 Nevertheless, a buyer of unregistered shares faces the same
restrictions as the seller. Thus, it may still be reasonable to assume that the original holder cannot
transfer the security for the duration of the initial restriction period even though that is not strictly
correct, unless a secondary market platform offers a viable means of selling the security.
Secondary markets for restricted shares transform a security that lacks marketability into one that
is marketable but lacks liquidity. Lack of marketability and lack of liquidity thus entail similar
losses of resale or transfer flexibility but with different root causes.
The DLOM plays an equivalent economic role to the one Amihud and Mendelson’s (1986)
liquidity risk premium plays in illiquid asset pricing. Investors require a liquidity risk premium in
the required return, and by implication a discount in price, to purchase illiquid securities. Their
work has stimulated a large and growing literature documenting the importance of liquidity in asset
pricing (Chordia, Roll, and Subrahmanyam, 2000; Amihud, 2002; Pastor and Stambaugh, 2003;
Acharya and Pedersen, 2005; Liu, 2006; Sadka, 2006; and Bekaert, Harvey, and Lundblad, 2007).
It has also spawned research devoted to developing methodologies for measuring this risk premium
and investigating the factors that affect its size and variability (Chalmers and Kadlec, 1998; Chen,
Lesmond, and Wei, 2007; Koziol and Sauerbier, 2007; Goyenko, Subrahmanyam, and Ukhov,
2011; and Kempf, Korn, and Uhrig-Homburg, 2012). The model developed in this paper could
also be used to estimate illiquidity discounts that are due to factors other than marketability
restrictions provided one could reasonably estimate the effect of the share’s illiquidity on the
length of time it would take a shareholder to realize the intrinsic value of the shares in cash.
The DLOM option formulation is more challenging when there is no legal or contractual
restriction on the holder’s ability to sell or transfer the asset because the length of the restriction
period is less predictable.11 For example, the market for an asset may be poorly developed, making
it difficult, time-consuming, and therefore expensive to find a buyer for it, but the asset is still
6
marketable. The restrictions are financial, rather than legal or contractual, and there is no
expiration date. Applying a put-option-based DLOM model requires more judgment in that case,
in particular, to estimate how long the pseudo-restriction period should be expected to last.
Likewise, the expected length of the restriction period when a privately held firm’s
unregistered common shares are being valued must be estimated because the lack of registration
imposes a marketability restriction of uncertain length. If the private firm’s business is sufficiently
well developed, then it may be reasonable to estimate the amount of time that is likely to elapse
before an IPO or a change-of-control transaction might occur, or a probability distribution for the
length of that period might be assumed. I generalize my DLOM model to restriction periods of
uncertain length by assuming that the length of the restriction period is exponentially distributed.
I provide empirical support for this assumption in the empirical section of the paper.
The rest of the paper is organized as follows. Section I generalizes the average-strike put
option DLOM model to restriction periods of any length L by modeling the L-year DLOM as the
value of the one-year DLOM compounded over L years and adjusts for term premium (or
discount). Section II empirically estimates the parameters of the DLOM term structure models
and investigates their properties. Section III suggests how to extend the multi-period DLOM
model to accommodate restriction periods of uncertain length again adjusting for term premium
(or discount). Section IV concludes.
I. Generalization to Longer Restriction Periods of Specified Length
Finnerty (2012) derives the average-strike put option DLOM model by employing a
lognormal approximation, which results in an effective cap on the volatility and an effective cap
on the calculated DLOM. Calculations provided in Finnerty (2012) reveal that the lognormal
approximation can become problematic as the length of the restriction period increases beyond
about four years especially for high-volatility stocks.12 For example, the discount decreases when
the dividend yield is q = 0.02 and the restriction period is between T = 4 and T = 5. The model is
best suited to calculating discounts for restriction periods up to two years in length, which
corresponds to the longest Rule 144 restriction period in the sample of discounts Finnerty (2012)
used to test the model. Table 1 indicates that the marketability discounts calculated from the
average-strike put option model are more consistent with empirical private placement discounts
for 1-year than for 2-year (or presumably longer) restriction periods after adjusting the observed
private placement discounts for the equity ownership concentration, information, and
overvaluation effects that typically accompany common stock private placements. I will build on
7
this feature to extend the average-strike put option DLOM model to accommodate longer
restriction periods. Such an extension is important because there are numerous forms of selling or
liquidity restrictions that can prevent a shareholder from freely selling her shares for a multi-year
period, which are often designed to resolve moral hazard and adverse selection problems (Kahl,
Liu, and Longstaff, 2003).
A. Motivation for a More General Model
Selling and liquidity restrictions can inhibit share sales by corporate officers, directors,
entrepreneurs, private equity investors, venture capitalists, and certain other classes of
shareholders for periods that sometimes extend over several years. Such resale restrictions have
caused stockholders to suffer large losses during severe market downturns (Kahl, Liu, and
Longstaff, 2003). Executive restricted stock plans typically prohibit share sales for between three-
and five-year vesting periods (Simon, 2016). Rule 144 places resale restrictions on shareholders
of public firms who hold unregistered shares and on affiliates who hold control shares (SEC, 2016).
The SEC has progressively relaxed the restrictions and shortened the restriction periods on Rule
144 sales over the past 20 years making them less problematic but they are still economically
significant because of the size of the DLOM involved. Private equity investors and venture
capitalists hold unregistered shares, which they cannot sell until they either register the stock for
an IPO or sell them to another private investor or in a change-of-control transaction. Private equity
investors and venture capitalists typically hold an investment for between five and seven years,
and investors in a private equity fund typically hold their fund investment for 10 years (Metrick
and Yasuda, 2010). Entrepreneurs may hold their equity investments for even longer periods.
Even some contractual restrictions that are usually designed for relatively short periods can
in some cases restrict share sales for much longer periods. For example, IPO lock-up periods
typically extend for 180 days from the date of the IPO but the issuer and its underwriters sometimes
agree to a much longer lock-up period (Kahl, Liu, and Longstaff, 2003). Merger agreements
typically impose multi-year lock-up periods on the selling firm’s corporate officers and key
employees to discourage them from leaving the target firm and taking with them their valuable
human capital, which may have been the main motivation behind the merger (Kahl, Liu, and
Longstaff, 2003). Lastly, Rule 144 restrictions are generally much more onerous for ‘affiliates,’
who include corporate executives, directors, and 5% or greater shareholders who exercise some
degree of control by having the power to direct the management and policies of the firm, than other
shareholders (SEC, 2016). An affiliate cannot sell her shares for at least six months after acquiring
8
her stock.13 Thereafter, the number of shares she can sell during any three-month period is limited
to one percent of the firm’s outstanding common stock or, if the shares are listed on a stock
exchange, the greater of one percent of the firm’s outstanding common stock and the stock’s
reported average weekly trading volume during the four weeks preceding the sale. Consequently,
for smaller firms with less actively traded shares, it may take several years before an affiliate can
sell her entire shareholding. In general, for privately held firms, resale limitations restrict share
sales until an IPO, a change-of-control transaction, or some other liquidity event occurs, and the
timing of such an event, and even whether it will ever occur, are uncertain.
B. Average-Strike Put Option DLOM Model Basic Assumptions
This section generalizes the average-strike put option DLOM model to restriction periods
of any specified length L by modeling the L-year DLOM as the value of the one-year DLOM
compounded over L years. I begin with a discussion of the economic rationale for using the value
of an average-strike put option to measure the one-year DLOM.14
Following Finnerty (2012), I assume the firm’s unrestricted shares trade continuously in a
frictionless market. The firm also has restricted shares outstanding, and all the firm's shares are
identical except for the transfer restrictions. Transfer restrictions prevent the investor from selling
the restricted shares for a period of length T. Any cash dividends are paid continuously during the
time interval [0, T] at a continuously compounded rate 0q ≥ that is proportional to V. V(t) is the
value of a share of common stock without transfer restrictions. I assume V(t) follows a geometric
diffusion process of the form
( ) VdZ Vdtq dV σμ +−= (1)
where µ and σ are constants that measure the continuously compounded mean return and
volatility, respectively, and Z is a standard Wiener process. The continuously compounded riskless
interest rate r is constant and the same for all maturities during [0, T]. No shareholder has any
special market-timing ability. In the absence of resale restrictions, investors would be equally
likely to sell the shares anytime during the restriction period. I assume the shareholder can sell the
registered shares at t, 0 < t < T, and reinvest the proceeds in the riskless asset until T and that the
investor would want to sell the unregistered shares prior to T were it not for the resale restrictions.
Suppose the shareholder can sell the registered shares at t, 0 < t < T, and reinvest the
proceeds in the riskless asset until T. In a risk-neutral world, the investor would be indifferent
between selling the share immediately for V(t) and selling it forward for delivery at T with forward
9
price ( )( ) ( ) . tVe t-Tq-r Suppose further that the investor would want to sell the unregistered shares
prior to T were it not for the resale restrictions. Since the investor lacks any special timing ability,
assume that the investor would be equally likely to sell unrestricted shares at N + 1 discrete points
in time and that these points are equally spaced, so that the investor considers selling at t = 0, t =
T/N, t = 2T/N,…, t = NT/N = T. In a risk-neutral world, such an investor would be indifferent
between holding a registered share and holding an unregistered share plus a series of forward
contracts all expiring at T. If the investor’s transfer restriction risk exposure could be perfectly
hedged, or if this risk is idiosyncratic with respect to the investor’s securities portfolio, then the
unregistered shares would be priced on a risk-neutral basis.
While price risk is hedgeable, liquidity risk is not. Because of the restrictions on hedging,
the investor bears an opportunity cost due to the transfer restrictions if
( ) ( ) ( )[ ] ( )
=
>+
N
0j
N/j-NTq-r TVjT/NVe1N
1 (2)
but realizes an opportunity gain if the inequality is reversed. Inequality (2) suggests that the
DLOM should be zero if the value of the investor’s potential for economic gain exactly offsets the
cost of the investor’s potential for economic loss during the restriction period T. In particular, if
(a) the stock’s return distribution is symmetrical, (b) the investor is risk neutral or can costlessly
fully hedge her risk exposure from holding the stock, and (c) she has adequate liquidity from other
sources such that the transfer restrictions do not cause her to miss any positive-NPV investment
opportunities or to bear any other illiquidity costs, then the value of the potential upside and the
cost of the potential downside offset, and the investor would not require a DLOM to purchase the
restricted share. In effect, the restricted share would be equivalent to a long forward contract for
the delivery at T of an otherwise identical unrestricted share when the transfer restrictions expire.
But actual common stock return distributions exhibit long-term negative skewness and fat
tails, investors are generally risk averse, and liquidity risk cannot be hedged, and so inequality (2)
should generally hold. Engle (2004) shows that long-term negative skewness is a consequence of
asymmetric volatility; negative returns lead to higher volatility than comparable positive returns
with the result that potential market declines are more extreme than possible market increases.15
Skewness is negative for every horizon and increasingly negative for longer horizons. Harvey and
Siddique (2000) furnish empirical evidence that U.S. equity returns for stocks in CRSP (NYSE,
AMEX, and NASDAQ) are systematically negatively skewed, this skewness is economically
10
important and commands an average risk premium of 3.60%, and the returns of the smallest stocks
by market capitalization exhibit the most pronounced negative skewness based on stock return
data for the period 1963 to 1993. Ze-To (2008) furnishes empirical evidence that the S&P 500
Index returns were negatively skewed and heavily tailed during the 1991 to 2005 period.16 Bali
(2007) and Bali and Theodossiou (2007) document the negatively skewed fat-tailed distributions
of the returns on U.S. stocks as proxied by the Dow Jones Industrial Average for the period 1896-
2000 and by the S&P 500 Index for the period 1950 to 2000, respectively. Cotter (2004)
documents the negatively skewed fat-tailed distributions of European equity index returns. Given
the economically significant negative skewness and fat tails that characterize common stock return
distributions, it is reasonable to expect a risk averse investor to require a DLOM to compensate
for the incremental cost of having to bear the excess of the cost of potential extreme downside loss
over the value of potential extreme upside gain during the restriction period, except in the special
case where he is committed to holding the stock for the entire restriction period and doing so does
not impose any potential liquidity cost.
The average-strike put option formulation models the DLOM as the value of a put option
for which the expected strike price is the arithmetic average of the risk-neutral forward prices.
This option represents the option to exchange a package of forward contracts on a share for the
underlying share. It can be evaluated as the value of an option to exchange one asset for another.
The option payoff function contains the sum of a set of correlated lognormal random variables.
Finnerty (2012) approximates its probability distribution as the probability distribution for a
lognormal random variable using Wilkinson’s method, derives the moment-generating function
for the bivariate normal distribution for the average of the risk-neutral forward prices and the price
of the underlying unrestricted share, and then applies Hull’s (2009) generalization of Margrabe’s
(1978) expression for the value of the option to exchange one asset for another when the stock is
dividend-paying to obtain the following average-strike put option model for the value, D(T), of the
DLOM:
( )
−
= −
2
Tv N-
2
TvNVTD 0
qTe (3)
[ { }[ ] [ ] ] 2/1T2T2 1 - e 2 - 1 - T -e 2 T Tv22 σσ σσ nlnl+= (4)
where ( )⋅N is the cumulative standard normal distribution function.
D(T) given by equations (3) - (4) is:
11
• Proportional to the current share price V0.
• Directly related to the stock’s price volatility σ.
• Directly related to the length of the transfer restriction period T.
• Inversely related to the stock’s dividend yield q.
• Independent of the riskless interest rate r.
The volatility v in equation (4) depends directly on the volatility σ of the underlying
unrestricted share of stock. Margrabe’s (1978) expression for the value of the option to exchange
one asset for another corresponds in the case of equations (3) and (4) to the exchange of a set of
forward contracts to deliver a restricted share for the unrestricted share where the forward contracts
have the same underlying unrestricted share. Accordingly, the volatility v in equation (4) is the
volatility of the ratio of the average forward contract price to the price of the underlying
unrestricted share. The volatility of this ratio is less than σ because the two components to the
exchange have the same underlying share, and changes in the value of the unrestricted share being
given up and changes in the value of restricted share being received in the assumed exchange pull
the value of the ratio in opposite directions, making the value of the ratio less volatile than the
price of the underlying share. Table 2 illustrates the relationship between v and σ for a range of
assumed parameter values.
The logarithmic approximation utilized in deriving equations (3) - (4) results in an
important limitation on the model’s usefulness in practice. As shown in the Appendix, the
volatility parameter behaves like 2/ forlarge , which approaches zero as T becomes very
large. The average-strike put option DLOM model has a 32.28% effective upper bound on the
percentage discount when T is large. The model may not generate reliable marketability discounts
for the stock of a private firm when the liquidity event may not occur for several years or for
valuing restricted shares when the specified restriction period extends for several years. Thus,
there is a need to modify the model before it can be applied to DLOMs for restriction periods
longer than a few years.
C. Generalization to Longer Restriction Periods
Since the marketability discounts calculated from the average-strike put option model are
more consistent with empirical private placement discounts for 1-year than for 2-year (or
presumably longer) restriction periods, I start with equations (3) - (4) to extend the average-strike
put option model to accommodate longer restriction periods. Set T = 1 in equations (3) - (4) to
12
obtain a one-period marketability discount formula:
( )
−
= −
2
vN -
2
vNV1D 0
qe (5)
[ { }[ ] [ ] ] 2/122 1 - e 2 - 1 - -e 2 v22 σσ σσ nlnl+= (6)
I will use equations (5) - (6) to generalize the average-strike put option DLOM model to obtain a
multi-period model that avoids the DLOM cap inherent in equations (3) – (4). I treat the L-
restriction-period marketability discount as the compounded value of L single-restriction-period
marketability discounts for arbitrary fixed L. I further extend the model to situations where L is
uncertain and exponentially distributed in section III.
I assume that the appropriate marketability discount for a restriction period of arbitrary
length L can be expressed as the value of the one-period marketability discount Δ compounded
forward L periods when the market for restricted common stock is in equilibrium. Since the
investor’s ability to resell the stock is assumed to be restricted, investors are not able to arbitrage
any mispricing of the discount by trading their restricted stock in the market.17 Instead, this
characterization of market equilibrium rests on the following assumptions. I assume that there is
sufficient competition among prospective issuers and investors in the market for restricted
common stock to ensure that when this market is in equilibrium, the DLOMs on sales of retricted
common stock have the property that the DLOM for an L-year restriction period is equal to the
cumulative compound DLOM from making a sequence of L one-year investments in the same
restricted stock each at a one-year DLOM plus a market-determined marketability term premium
(or discount) τ. I assume that there is no market segmentation or preferred habitat in the restricted
stock market; the market is complete in the sense that there are sufficient buyers and sellers,
including new issuers, in the market for restricted stock for each restriction period of length L such
that if this equilibrium condition is violated, there will always be sufficient sellers available to sell
and sufficient buyers available to purchase restricted stock with any length restriction period to
restore market equilibrium.18 I also assume that at the margin, neither prospective issuers nor
sellers face liquidity constraints that would make the marginal issuer or marginal seller a forced
seller. Likewise, I also assume that potential buyers do not face liquidity constraints that might
force them to withdraw from the restricted stock market. Even though investors in restricted shares
have ample liquidity, shares with marketability restrictions would still be less valuable than
otherwise identical shares without such restrictions because the restricted shares do not afford
13
investors the same degree of flexiblity as unrestricted shares to sell shares to rebalance their
investment portfolios.
To motivate the notion of a DLOM term structure, I initially assume there is no
marketability term premium (or discount). I relax this assumption later in this section when I
introduce a term premium (or discount) into the model. The one-year percentage marketability
discount is defined as P = D(1)/V0 where D(1) is given by equation (5). It is easier to work in
continuous time. The discrete marketability discount P can be expressed equivalently as a
continuously compounded percentage marketability discount for a one-year restriction period,
denoted Δ, which I will refer to hereafter as base one-year restriction period DLOM, according to
the equation
= 1 − . (7)
It follows from equations (5) and (7) that Δ can be expressed as
=− ln[1 − ] = − ln{1 −
−
−
2
vN -
2
vNqe } (8)
where v is given by equation (6).
One might think of the percentage discount per year for restriction periods of different
lengths as forming a DLOM term structure. A DLOM term structure, or term structure of
marketability restriction risk, is a natural application of Guidolin and Timmermann’s (2006) term
structure of risk. As Engle (2011) notes, risk measures are computed for horizons of varying length
extending from one day to many years. Risk measures take into account that the losses that might
result from extreme moves in the price of the underlying asset take time to unfold, which generally
leads to long-term risks exceeding short-term risks. This phenomenon suggests that it is reasonable
to expect marketability risk to increase with the length of the marketability restriction period; by
implication, the DLOM should also increase, at least eventually if not initially, as the length of the
restriction period increases.
As Engle (2011) also notes, the term structure of risk is analogous to the term structure of
interest rates and the term structure of volatility (Cox, Ingersoll, and Ross, 1981; Fama, 1984,
1990; and Fabozzi, 2016). Given its prominence in finance, I will draw on the analogy to the term
structure of interest rates in describing the DLOM term structure. If the percentage discount per
year is the same for every length restriction period, then the DLOM term structure implied by
equation (8) is flat and equal to Δ. Just as the interest rate term structure exhibits a variety of
shapes (Fabozzi, 2016) and the illiquidity risk premium term structure can be upward-sloping,
14
downward-sloping, or hump-shaped (Chen, Lesmond, and Wei, 2007), the marketability term
premium model should also permit the same variety of shapes because all three term structures are
related to the term structure of risk. The generalized DLOM model I develop in this paper is
capable of exhibiting these shapes but the empirical term structure fitted in section II is
monotonically increasing and concave.
Assume the percentage marketability discount compounds continuously at the constant
rate Δ over a restriction period of length L years. The DLOM for an L-year restriction period,
denoted D*(Δ, L), can be expressed as
∗( , ) = 1 − = 1 − 1 − − (− ) (9)
where v is given by equation (6). Note that D*(Δ, 0) = 0 and lim→ ∗( , ) = 1 for all positive
values for Δ. As one would expect, the DLOM is zero when there are no marketabilty restrictions,
and it approaches 100% in the limit as L increases without bound. For the special case of a non-
dividend-paying stock, q = 0, equation (9) simplifies to
∗( , ) = 1 − 2 (− ) (10)
Equation (9) provides the percentage discount, the loss of fair market value per dollar of
freely traded asset value, attributable to the L-period transfer restrictions. It offers another way of
characterizing the term structure of marketability risk premia by expressing these premia as
percentage price discounts that vary as a function of L. Figure 1 illustrates the behavior of D*(Δ,
L) for different values of the stock price volatility σ. D*(Δ, L) is a concave monotonically
increasing function of L. For a restriction period of any particular length, the DLOM is larger for
greater v, the volatility of the ratio of the stock price to the average futures price of the stock. One
might interpret each D*(Δ, L) curve in Figure 1 as the DLOM term structure for an issuer’s
restricted stock given the volatility σ of the issuer’s unrestricted stock. When v = 0, D*(Δ, L) = 0;
the DLOM equals zero regardless of the length of the restriction period when the volatility of the
price ratio is zero. As v increases, the DLOM increases and reaches the following maximum for
given L, which is illustrated in Figure 1:
lim→ ∗( , ) = 1 −[1 − ] (11)
D*(Δ, L) would approach, but not exceed, 100% in the limit as v becomes infinite because the
dividend yield q is always a small nonnegative quantity.19
Figure 2 compares the DLOM obtained from the Finnerty [2012] average-strike put option
15
model (3) - (4) with the DLOM obtained from the generalized model (9) with v given by equation
(6). The two models give similar DLOM values when the restriction period is two years or less
and the stock volatility is 30% or less. The two models agree on the DLOM at L =1 by design.
For restriction periods less (greater) than one year, the Finnerty (2012) model DLOM is greater
(less) than the compound model DLOM. The Finnerty (2012) model exhibits greater concavity,
which causes the DLOMs from the two models to diverge more widely for any given restriction
period L as σ increases. The DLOMs implied by equation (9) exhibit a nearly linear relationship
to L, which is a direct consequence of the flat term structure assumed for Δ. It seems more
consistent with how one would normally expect actual DLOMs to behave than the more extreme
concavity of D(T). The true behavior is, or course, an empirical question.
D. DLOM Term Premium
This section further extends the average-strike put option DLOM model to allow for a term
premium to reflect a risk averse investor’s more prolonged exposure to the risk of an increasingly
negatively skewed fat-tailed return distribution. I note that it would be possible in certain
situations, which are discussed below, for the term premium to be negative if there any special any
special benefits attributable to being able to access a particularly attractive class of superior-α
investments that impose a long period of restricted marketability. The sample I use to fit the
generalized model in this paper does not include private transactions that convey those special
benefits, so I expect the fitted term structure to exhibit a term premium for the reasons discussed
below.
If the percentage discount Δ is independent of L, as in equation (8), then the term structure
of percentage marketability discounts is flat. Building on the analogy between the interest rate
term structure and the DLOM term structure, various theories have been proposed to explain the
different observed shapes of the interest rate term structure (Fabozzi, 2016). Under a pure
expectations theory, equation (9) implies that market participants expect the one-year DLOM
given by equation (8) to remain the same each year for L years. It is most consistent with either a
broad formulation of the pure expectations theory (Lutz, 1940-1941) or a local expectations theory
formulation of a pure expectations theory of the term structure (Cox, Ingersoll, and Ross, 1981,
pp. 774-775).20 However, the DLOM term structure might be more steeply upward-sloping if risk
averse restricted stock investors require a marketability term premium to compensate for the
greater risk of an increasingly negatively skewed fat-tailed stock return distribution as the
restriction period lengthens (Engle, 2009, 2011) and the greater uncertainty associated with being
16
resale restricted for a longer period (Hicks, 1946). This term premium might not increase
uniformly and might exhibit irregular shapes if restricted stock investors self-select into restriction-
period preferred habitats (Modigliani and Sutch, 1966) or if legal investment restrictions or
investment policy constraints effectively segment the market for restricted stock into specific
maturity sectors and prevent investors from moving from one sector to another to eliminate local
supply-demand imbalances (Culbertson, 1957).
1. Term Structure of Illiquidity Yield Spreads
To allow for a more general DLOM term structure, I relax the restrictive assumption
underlying equations (8) and (9) and permit the DLOM to change as L increases. I assume a term
premium (discount, if negative) τ(Δ, L), which is a function of the base one-year restriction period
DLOM Δ given by equation (8) and the length of the restriction period L. The DLOM per period
is the sum of Δ and τ(Δ, L). Marketability restrictions could be expected to have an effect on the
price of a restricted share that is similar to the effect that diminished liquidity has on the price of a
fully marketable unrestricted share even though the source of the impairment in value is different.
Illiquidity yield spreads are thus suggestive of what DLOM term premia might look like.
The finance literature describes the factors responsible for and quantifies illiquidity yield
spreads for bonds, which can vary with the length of the restriction period (Amihud and
Mendelson, 1986; Chalmers and Kadlec, 1998; Chen, Lesmond, and Wei, 2007; Koziol and
Sauerbier, 2007; Goyenko, Subrahmanyam, and Ukhov, 2011; and Kempf, Korn, and Uhrig-
Homburg, 2012). The illiquidity yield spread literature suggests that there is a term structure of
illiquidity yield spreads, which can be upward-sloping, downward-sloping, or a hump-shaped
function of maturity much like the interest rate term structure (Chen, Lesmond, and Wei, 2007),
that the illiquidity yield spread tends to increase with risk as measured by the volatility of security
returns (Koziol and Sauerbier, 2007), and that the illiquidity spread, like the credit spread, is wider
for lower-rated bonds due to the greater risk (Chen, Lesmond, and Wei, 2007).
Koziol and Sauerbier (2007) apply Longstaff’s (1995) lookback put option DLOM model
and estimate a term structure for illiquidity spreads, which is hump-shaped. Chen, Lesmond, and
Wei (2007) empirically estimate illiquidity spreads for investment-grade bonds within three
maturity ranges: between 8 and 35 basis points for short maturity (1-7 years), between 24 and 70
basis points for medium maturity (7-15 years), and between 59 and 67 basis points for long
maturity (15-40 years) and for non-investment-grade bonds within the same maturity ranges:
between 2.01% and 9.33% per year for short maturity (1-7 years), between 2.59% and 9.42% per
17
year for medium maturity (7-15 years), and between 2.52% and 10.23% for long maturity (15-40
years). Non-investment-grade bond illiquidity yield spreads are more indicative of equity
investment illiquidity term premia than investment-grade bond illiquidity yield spreads because
the yields of non-investment-grade bonds tend to be more highly correlated with equity returns
than Treasury bond yields. Goyenko, Subrahmanyam, and Ukhov (2011), who focus on the
Treasury bond market and the illiquidity premia reflected in the yields of the off-the-run Treasury
issues, find that the term structure of illiquidity premia steepens during recessions and flattens
during expansions. Kempf, Korn, and Uhrig-Homburg (2012) find substantial variation in the
level and shape of the illiquidity premium term structure over the economic cycle.
The ranges of illiquidity term premia implied by the illiquidity yield spreads in Chen,
Lesmond, and Wei (2007) are from -1.16% to +0.58% per year for the term premium between
short maturity (1-7 years) and medium maturity (7-15 years) restriction periods and from -0.58%
to +0.81% per year for the term premium between medium maturity and long maturity (15-40
years) restriction periods. The ranges of illiquidity term premia for equity securities are likely to
be wider than these ranges because equity securities are riskier than non-investment-grade bonds.
2. Economic Factors Mitigating the Term Premium
There are certain economic factors that could mitigate or even offset the marketability term
premium in special circumstances. Certain classes of restricted securities that have long periods
of restricted marketability, such as private equity (PE) or hedge fund investments, might be
expected to provide superior risk-adjusted returns (positive α) that cannot be duplicated in the
public securities market or elsewhere in the private market. Investors should be willing to pay a
premium price (by accepting a smaller DLOM) for such securities to reflect their scarcity value
notwithstanding the long period of restricted marketability. There is empirical evidence that at
least some non-marketable investments, such as hedge fund or PE fund ownership interests, can
produce positive portfolio α when fund managers are highly skilled at selecting and managing
investments (Hasanhodzic and Lo, 2007; Kosowski, Naik, and Teo, 2007; Metrick and Yasuda,
2010; Fan, Fleming, and Warren, 2013; and Harris, Jenkinson, and Kaplan, 2014). However,
Phalippou and Gottschalg (2009) report that the PE fund fee structure typically absorbs about 6%
per year of the value invested, which suggests that α could be negative for some PE funds due to
high fees as well as poor investment returns. This possibility would suggest that investment risk
and the potential for negative returns net of fees could instead lead to a term premium even for
these asset classes.
18
Any asset class that regularly produces positive α could lead to term discounts
notwithstanding the long period of restricted marketability because investors should be more
willing to accept a longer restriction period when the investment is expected to earn positive α, in
effect, by being willing to pay higher prices that reflect smaller DLOMs. I leave this test for future
research.
3. Modelling the Term Premium
Denote a possible duration-specific marketability term premium (or discount, if negative)
τ(Δ, L), which is a function of base one-year restriction period DLOM Δ and the length of the
restriction period L. I interpret τ > 0 as indicative of private equity investor aversion to the
incremental riskiness of longer-duration restriction periods (including more prolonged exposure
to the risk of an increasingly negatively skewed fat-tailed return distribution as L increases).21 The
longer the restriction period, the longer the forced holding period, during which a greater number
of value-decreasing events are possible; τ > 0 implies that investors are increasingly adverse to this
negative event risk the longer the holding period, which increases their required DLOM. A longer
restriction period L for a stock investment means that it will have a longer duration, which
increases the restricted stock’s sensitivity to changes in stock prices generally, to the expected
increase in downside investment risk (due to the longer restriction period’s greater exposure to the
risk of increasingly negative skewness), to shifts in the equity risk premium, or to changes in the
riskless rate (for example, as reflected in CAPM). The length of the restriction period L could
interact with marketability risk to magnify the impact on the DLOM and increase the marketability
term premium τ as L increases.
4. Term Premium-Adjusted DLOM Model
The marketability discount D*(Δ, L) in equation (9) can be modified to incorporate a term
premium (or discount, if negative) by adding a duration-specific marketability term premium τ(Δ,
L) to the base one-year restriction period DLOM Δ. The formula for the modified marketability
discount, ( , ), is:
( , ) = 1 − ( ( , )) = 1 − ( , ) 1 − − (− ) (12)
The expressions for D*(Δ, L) for the special cases in equations (10) and (11) are similarly modifed
and the resulting formulas are interpreted similarly to equation (12).
Equation (12) allows for the possibility that investor risk aversion could lead to greater
marketability term premia for longer restriction periods (τ > 0), in which case ( , ) exceeds
19
D*(Δ, L). Figure 3 illustrates how varies as a function of Δ and L. is a concave increasing
function of both parameters. Likewise, is a concave increasing function of τ.
II. Empirical Estimation of a DLOM Term Structure
This section empirically estimates a DLOM term structure based on a 27-year sample of
private sales of unregistered common stock and quantifies the marketability term premium τ(Δ, L).
My sample consists of 5,391 private equity sales by private firms that eventually went public. I
fit term structures of ex post implied DLOMs to the full sample and to two sub-samples. I do not
have information concerning the length of the restriction period the issuers and investors assumed
at the time they negotiated the private transactions. Thus, my DLOM term structures provide the
ex post relationship between the DLOM and realized L. The models could be used to estimate a
DLOM for a pending sale of unregistered stock that is consistent with ex post DLOMs.
A. DLOM Sample
My sample consists of 5,391 U.S. private equity transactions in which privately held firms
sold unregistered common stock prior to going public. I measure the DLOM implied by the
difference between the IPO price stated in the firm’s IPO prospectus and the most recent prices at
which those same firms sold unregistered, and thus restricted, common shares or granted options
exercisable for those common shares in pre-IPO private transactions. I adjust the IPO price for
economic factors that would be likely to cause the price of the firm’s shares to change between the
private transaction date and the IPO date to control for the prices’ non-contemporaneity. I measure
the length of the restriction period L as the amount of time elapsed between the private equity
transaction date and the IPO date and empirically fit a probability distribution for L. I use the
estimated DLOMs and associated values for L to empirically fit a DLOM term structure and to
estimate the DLOM term premium as a function τ(Δ, L). I also control for industry effects and an
IPO market effect, and I test for any date of issue timing effects. The estimated DLOM term
structure may tend to understate the typical DLOMs for private common stock sales because the
DLOM sample includes DLOMs for financings by only those private firms that succeeded in going
public. Firms that go public are typically financially stronger and more profitable and tend to have
superior growth prospects -- and lower financial risk -- than private firms that do not go public
because the latter group includes firms that were not deemed sufficiently attractive by IPO
underwriters and investors to enable them to complete an IPO.
I obtained the sample of private equity transactions from the Valuation Advisors Lack of
20
Marketability Discount Study, which is available through Business Valuation Resources.22 The
Valuation Advisors database reports company name, SIC code, the private transaction date and
share price, the length of the time interval between the private transaction date and the IPO date,
the type of private equity transaction, the IPO date and IPO share price, the marketability discount
expressed as the difference between the IPO share price and the private transaction share price,
and a very limited amount of firm financial data. It provides details concerning pre-IPO primary
sales of restricted common stock, options, and convertible preferred stock, which Valuation
Advisors compiles from the historical private equity transaction data that an IPO issuer furnishes
in the IPO registration statement that it files with the U.S. Securities and Exchange Commission.
The database excludes IPOs by real estate investment trusts, master limited partnerships, limited
partnerships, closed-end funds, or mutual bank or insurance company conversions. When an issuer
makes multiple transactions within any 3-month period or multiple transactions more than one
year prior to its IPO date, Valuation Advisors includes only the highest-priced transaction (and by
implication, the one with the smallest discount) for the relevant interval. All stock and option
transactions are adjusted for stock splits between the private transaction date and the IPO date.
The Valuation Advisors database listed 13,090 private equity transactions for the period
between June 1985 and April 2017, all of which occurred prior to the issuer going public. I
excluded the 1,400 non-U.S transactions because they are less likely to be affected by
marketability restrictions imposed by U.S. securities laws than U.S. transactions. I also excluded
3,266 transactions involving convertible preferred stock because even when the conversion feature
is based on the estimated fair market value of the underlying common stock, the transaction price
of the convertible preferred reflects the optionality of the convertible security, and the fair market
value of the underlying common stock is not provided in the database. I also excluded 2,339
private equity transactions for which I could not identify the private equity isuer’s ticker symbol
because I would not be able to obtain its post-IPO share prices to calculate its share price volatility
during the one-year period post-IPO without the ticker symbol. Some private equity placements
take place at a premium to the stock’s freely traded value. Allen and Phillips (2000) find that 59
percent of what they term ‘strategic’ private placements take place at a premium stock price, and
strategic private buyers on average pay a 6 percent premium. I excluded 694 private placements
for which the calculated DLOM is negative because these premiums in price are unrelated to the
marketability of the stock.23 After these exclusions, the final sample contains 5,391 private sales
of restricted common stock and stock options of private firms for which the adjusted DLOM is
21
nonnegative.
Table 3 provides a summary description of my sample. The private equity transactions
took place between June 1985 and April 2017 and involved 1,632 firms, which went public
between March 1986 and June 2017. Panel A furnishes the annual revenue, annual net income,
and year-end total assets, book value of equity, and market capitalization of equity for the fiscal
year in which the firm went public. I have distinguished between sample firms that only sold
common stock at a calculated discount and those that had at least one private placement at a
calculated premium. The mix of firms exhibits a wide size range, for example, with revenue
varying between zero and $136 billion during the IPO year. Nearly two-thirds were unprofitable
in their IPO year, although this percentage is slightly smaller for the roughly eight percent of the
sample that had at least one private sale at a calculated premium. Roughly a tenth of the sample
firms had negative book equity at the end of their IPO year.
Panel B provides a breakdown of the sample based on the number of firms that went public
each year broken down by industry, and Panel C provides a breakdown of the private equity
transactions by year and by industry. The industry breakdown is based on the French 12-industry
classification system. Two industries predominate: business equipment (BusEq), which includes
computers and related devices (33% of the firms and 36% of the private equity transactions), and
health care (Hlth) (22% of the firms and 27% of the private equity transactions. No other industry
accounts for more than 9.5% of the firms or 8% of the transactions.
Common stock price volatility is the primary determinant of the value of the base one-year
restriction period DLOM Δ, which is critical in the DLOM and term premium models. Due to the
small number of private transactions, stock price volatility cannot be calculated reliably until a
firm goes public, so I cannot calculate the stock price volatility as of the private equity transaction
date. Table 4 provides sample firm and sample private transactions breakdowns based on the
common stock price volatility during the 12 months following the firm’s IPO. More than half the
private equity transactions and just under half the firms had 12-month common stock price
volatility between 50% and 100%, and more than a quarter of the private transactions and the firms
had a 12-month volatility greater than 100%. Given this volatility profile, after fitting each DLOM
term structure, I plot illustrative DLOM term structures for volatilities of 50%, 75%, and 100%.
B. DLOM Estimation
The Valuation Advisors database reports the “true fair market value” of the common stock
for each private equity transaction in accordance with SEC guidelines (Valuation Advisors, 2017).
22
Valuation Advisors accounts for the sale of ‘cheap stock’ or grants of ‘cheap options’ by adding
back the transaction’s deferred compensation component, which must be disclosed in the firm’s
final IPO registration statement. The deferred compensation component equals the difference
between the fair market value of the stock and either the stock issue price (for stock sales) or the
option strike price (for option grants) when the stock is underpriced.
I recalculated the DLOMs because the private transaction share price and the IPO price for
each firm are non-contemporaneous. The Valuation Advisors database does not make any
adjustment for the change in the value of the shares that would be expected to occur between the
private equity transaction date and the IPO date, for example, as a private firm grows its business
and improves its profitability as it approaches a potential IPO. I adjust for the change in equity
value by deflating the IPO price back to the private equity transaction date based on an index of
the returns on common stocks of firms in the same industry as proxied by the private firm’s SIC
code to take into account the small-firm effect and the fact that pre-IPO firms are more closely
comparable to these smaller firms than they are to the entire set of firms in an industry index.
I calculated an adjusted DLOM for each private equity transaction in my sample by
discounting the IPO price from the IPO date back to the private equity transaction date and using
the adjusted IPO price in the (adjusted) DLOM calculation. Cornell (1993) describes an essentially
equivalent approach. I employed the Fama-French 48-industry classification to assign each firm
in my sample to an industry. I ranked the firms classified in the same industry as the private firm
involved in the relevant private equity transaction by size based on latest fiscal year total sales
revenue at the beginning of each month between the month of the private equity transaction and
the month of the IPO. I calculated the weighted average monthly common stock return for the
smallest quartile of industry firms for each month and calculated the adjusted DLOM:
= ( )( )…( )( )( )…( ) (13)
The term in brackets is the IPO price adjusted for the weighted average monthly
compounded return on comparable common stocks, which proxies for the effect of market,
industry, and company-specific factors that would affect the price of the stock between the private
transaction date and the IPO date. The monthly returns and are calculated for the relevant
fraction of the month. S is the private transaction price, which incorporates any DLOM the
purchasers in the private transactions required. A very buoyant IPO market at the time of the
firm’s IPO could bias upwards in equation (13) if it tends to boost IPO prices and also accelerate
23
IPOs, both of which would affect the fitted DLOM term structure by increasing the apparent
discount when the IPO price is compared to the private equity transaction price. I control for this
IPO effect by including a control variable IPOt in the DLOM term structure estimation model,
which is equal to the number of IPOs in the year the issuer went public divided by the average
annual number of IPOs during the sample period. I have flagged the years in Panels B and C of
Table 3 that had an above-average number of IPOs. If there is an IPO effect, I expect the
coefficient of this control variable to be positive and statistically significant, which is exactly what
I find in each version of the fitted DLOM term structure model.
in equation (13) is an estimate of in equation (12). It is a function of the base one-
year restriction period DLOM Δ and the length of the restriction period L. The functional
relationship ( , ) defines the DLOM term structure, which I estimate later in this section.
C. Evidence of Sale of Cheap Stock
could be unusually large if the issuer sold the stock or options at a heavily discounted
price and the issuer failed to adjust fully for this heavy discount when it reported its pre-IPO private
equity transactions in its IPO prospectus. The implied DLOM might even exceed 100%. I control
for unreasonably large calculated DLOMs by capping the DLOMs for each L by imposing the
Longstaff (1995) model DLOM as an upper bound.
The Longstaff (1995) DLOM model provides an upper bound on the DLOM because it
assumes perfect market-timing ability. Table 5 reports the number of private equity transactions
for which the calculated DLOM was initially above the Longstaff (1995) model upper bound.
can exceed the upper bound for either of at least two reasons. The firm could sell ‘cheap’ stock or
grant ‘cheap’ stock options, which would cause the actual discount to be less than the appropriate
discount. Second, the parties to the private equity transaction could negotiate a share price
assuming that it will take much longer for a liquidity event to occur than what actually transpired.
Overestimating L would result in an ex post discount that exceeds the DLOM that would have been
agreed if the parties had estimated L accurately.
Firms sometimes sell cheap stock or grant cheap options to officers, directors, or affiliates
prior to their IPO. However, SEC Regulation S-K requires the firm to disclose in its IPO
prospectus the terms of the sale or grant and the sale/grant date aggregate fair market value for
each such transaction within the past five years for which there is a ‘substantial disparity’ between
the IPO public offering price and the effective sale price per share (for a stock sale) or strike price
24
(for an option grant). The SEC staff often issues comment letters on the IPO registration statement
asking firms to provide a justification for transaction prices that seem unusually low relative to the
IPO price (Melbinger, 2014). They typically focus on the most recent pre-IPO transaction for
which any intentional underpricing is likely to be most evident The SEC normally requires the
firm to report any discount the SEC finds excessive as share-based compensation to the extent the
firm has not already reported it voluntarily. Valuation Advisors accounts for cheap stock by adding
back to the reported share price or option strike price the amount of the deferred compensation per
share the firm reports its final IPO registration statement for any underpriced pre-IPO stock or
option transactions (Valuation Advisors, 2017). The threat of such disclosure should at least partly
restrain the practice of unreported underpricing, but it seems unlikely that this disclosure
requirement will completely eliminate undisclosed cheap stock sales or option grants due to the
lack of transparency in pricing illiquid shares. I find evidence of such unreported underpricing,
especially in the DLOMs for the last private equity transactions that precede sample firms’ IPOs.
Table 5 reports the number of private equity transactions with above the upper bound
(before applying the upper bound). The tendency to overestimate L should be less problematic for
the last pre-IPO transactions because presumably the firm’s IPO should be within sight. As
illustrated in Figure 4, L for those transactions is bunched within zero to 1 year and very few
exceed 2 years. If firms do sell cheap stock and grant cheap options, I would expect the incidence
of underpricing to be greater because it is the last opportumity pre-IPO to grant chaep stock but I
also expect the degree of underpricing to be smaller in the last pre-IPO transaction than in earlier
transactions because the last pre-IPO transaction is closer in time to the IPO and thus any
underpricing should be easier for the SEC to detect. As indicated in Table 5, 37.07% of the last
pre-IPO transactions had above the upper bound but only 22.05% of the earlier pre-IPO
transactions did, and the difference of 15.02% is statistically significant at the .01 level.
Interestingly, the median excess is 15.09% for the last pre-IPO transactions but 16.92% for the
earlier pre-IPO transactions, and this difference of 1.83% is statistically significant at the .01 level.
The significantly greater underpricing in the earlier pre-IPO transactions is consistent with the
SEC’s focusing its monitoring on the most recent pre-IPO private equity transactions.
D. Probability Distributions for L
The restriction period for unregistered common stock will last until the stock is registered
for public resale or until it is exchanged for cash or registered securities in a change-of-control
transaction. Regarding going public, the length of the restriction period is uncertain. Its expected
25
value and range should both decrease as a firm’s IPO approaches. A business valuer might use
the observed elapsed time between past private equity sales by similarly situated firms (including
with respect to going public) and their IPOs to estimate a reasonable value for L, although this
approach is less likely to be successful the younger the firm and consequently the more distant its
IPO.
To frame the issue, suppose an investor wants to purchase the common stock of a privately
held firm that intends to go public after the firm has developed its product line sufficiently that the
stock market would be receptive to its IPO. An IPO would be highly uncertain as it is contingent
on successfully developing the new product line. The investor might estimate an IPO date after
identifying similar past IPOs and assume that L will end on the date following the firm’s IPO when
the lockup on selling insider shares could be expected to expire.24 Similarly, the venture capital
backers of a young firm can plan for a liquidity event perhaps several years in the future, but the
date is dependent on how the firm develops, and of course, it will never occur if the firm fails
before the liquidity event can take place. In either case, it might be reasonable to assume a
particular probability distribution for L based on historical experience for similar firms under
similar economic conditions. Nevertheless, the IPO market is notoriously fickle, and a firm’s
business fortunes could change for the worse or investor interest in the firm’s industry could cool
off before an IPO or a change-of-control transaction could take place. So even specifying a
probability distribution for L might be difficult.
The distribution of elapsed times for L might also be quite different for firms that are doing
one last round of financing before going public. The last pre-IPO private equity transactions in
my sample have restriction periods that are concentrated in the interval L ≤ 1. Inspection of the
DLOM data revealed that the distributions of DLOMs appear fundamentally different depending
on whether L ≤ 1 or L > 1. I performed a Chow test of the null hypothesis that there is no structural
break at L = 1 against the alternative hypothesis that there is such a structural break and obtained
F = 18.58, which is statistically significant at the 1% level. I conclude that the distribution of the
sample DLOMs has a break point at L = 1. The more interesting economic issue is whether the
last pre-IPO transactions and the earlier pre-IPO transactions have different DLOM distributions.
One would expect that if a firm is going to sell cheap stock or grant cheap stock options to key
employees or other insiders, it is more likely to do so in the last private placement that precedes
its IPO because that transaction affords the last opportunity to do so privately. The results reported
in section C support this hypothesis. Figure 4 compares the distributions of L for the last pre-IPO
26
private equity transaction and the earlier pre-IPO private equity transactions.
Metrick (2007) models the same liquidity events in calculating the value of a venture
capitalist’s (VC) random-expiration call option. A VC holds a call option on the equity value of
each VC portfolio company. The option exercise date corresponds to a liquidity event, which is
random because of the uncertainty concerning when an IPO or the sale of the portfolio company,
which will trigger the exercise of the VC’s call option, will occur (or whether the option will expire
worthless before it ever goes into the money). Metrick assumes an exponential distribution for the
exercise date. For the same reason, the date L the restricted stock becomes unrestricted (exits the
marketability restriction period) due to a liquidity event might be expected to follow an exponential
distribution.
The exponential distribution with constant parameter λ is a memoryless process. Finance
theory identifies several firm characteristics that could cause the value of λ to change over time
and thus affect the timing of a particular firm going public, such as improvement or deterioration
in profitability or a change in growth prospects. In addition, IPOs are affected by herding; the
likelihood that a firm goes public tends to increase when the firm belongs to an industry that
becomes ’hot’ and to decrease when its ‘hot’ industry suddenly cools off.
The insights from Metrick (2007), confirmed by a data plot, suggested that the exponential
distribution is a good candidate for the distribution of L. The gamma and Weibull distributions
were also considered because they are generalizations of the exponential distribution (Balasooriya
and Abeysinghe, 1994). Further insights from Siswadi and Quesenberry (1982), again confirmed
by a data plot, suggest that the log-normal distribution is also a good candidate for the distribution
of L. Thus, I compared four candidates for modeling the probability distribution for L, the log-
normal, exponential, gamma, and Weibull distributions.
I fitted the four alternative probability distributions separately to the sample data for the
last pre-IPO private equity transactions (1,632 transactions) and the sample data for the earlier pre-
IPO private equity transactions (3,759 transactions). I applied the following procedure based on
the predictions of order statistics methodology described in Balasooriya and Abeysinghe (1994).
First, I sorted the sample from smallest to largest and divided the ordered sample X1 < X2 < X3 <
…< Xn into two sub-samples, S1 consisting of the smallest three-quarters of the ordered
observations, denoted from 1 to r (1,224 transactions for the last pre-IPO sub-sample and 2,819
transactions for the earlier pre-IPO sub-sample), and S2 consisting of the remaining largest one
quarter, denoted from r + 1 to n (408 transactions in the last pre-IPO sub-sample and 940
27
transactions in the earlier pre-IPO sub-sample). Next, for each candidate distribution Fi, i = 1, …,
4, I applied maximum likelihood estimation to sub-sample S1 to estimate the distribution
parameters for Fi. Next, I used the fitted distribution for Fi and a randomly generated set of
parameter values to generate a sample of n predicted values, which I ordered from smallest to
largest < < < … < . Next, I divided the ordered sample of predicted values into two
sub-samples, consisting of the smallest three-quarters and consisting of the remaining largest
one quarter of the predicted values, again denoted from r + 1 to n. Then I calculated the sum of
the absolute differences and the sum of the squared differences between the sub-sample of ordered
predicted values and the sub-sample of actual values S2 for each candidate distribution:
=∑ , − , (14)
=∑ , − , (15)
The candidate distribution that provides the best fit to the data is the one for which Di is a minimum.
Table 6 provides the parameter estimates for each candidate distribution and the calculated
sum of the absolute differences and sum of the squared differences between each fitted distribution
and the actual distribution of L. The log-normal distribution provides a slightly better fit than the
exponential distribution and both fit the data much better than the gamma or Weibull distributions
for the last pre-IPO private equity transactions. The exponential distribution provides a
substantially better fit than the other three distributions for the earlier pre-IPO private equity
transactions. Moreover, the two exponential distributions have very different shape parameters
(0.68 versus 2.34). Due to this difference, I estimate the DLOM term structure separately for the
last pre-IPO private equity transactions and for the earlier pre-IPO private equity transactions.
However, in view of the similar fit of the log-normal and exponential distributions, one could
model the length of the restriction period as having an exponential distribution for both classes of
pre-IPO transactions if there is an advantage to picking a common form of distribution.
One further calculation is necessary to obtain the rate of the exponential distribution for L
for each sub-sample of pre-IPO transactions. I use the full sub-sample for each parameter
estimation. In Table 6, the sample mean of the truncated distribution is μ = 0.68 for the last pre-
IPO transactions and μ = 2.34 for the earlier pre-IPO transactions. Denote the rate of the
exponential process λ. The rate of the exponential distribution is = 1/μ. Thus, μ = 0.68 implies = . = 1.47 and μ = 2.34 implies = 1/ = 0.43. The probability distribution for L for
the last pre-IPO transactions is approximately exponentially distributed with parameter = 1.47,
28
and the probability distribution for L for the earlier pre-IPO transactions is also exponentially
distributed but with parameter = 0.43.
E. Marketability Term Premium
The marketability term premium τ plays a critical role in determining the term structure
(Δ, L) in equation (12). I assume Δ has a constant value through time given by equation (8),
which will vary from one private equity transaction to another depending on the volatility v and
the dividend yield q of the stock. I assume the following general functional form for the
continuously compounded DLOM with the terms to the right of ΔL reflecting the lack of
marketability term premium:
− 1 − = + ( , ) ∙ (16)
In equation (17), τ(Δ, L) is general enough to permit the DLOM term structure to be upward-
sloping, downward-sloping, concave, or convex. Allowing τ to be a function of Δ permits the lack
of marketability term premium to increase with the stock’s volatility because Δ is directly related
to σ. This behavior would be consistent with the documented tendency of the illiquidity yield
spread to increase with the volatility of security returns (Koziol and Sauerbier, 2007). Solving
equation (16) for τ gives
( , ) = − − ln[1 − ]/ (17)
Τhe term premium τ is expressed on a continuously compounded basis in equation (17).
Table 7 provides the median values for τ within successive restriction period ranges and
separately for 12 industry classifications based on the full sample of adjusted DLOMs. The median
values for τ are positive, as expected, for each industry within each restriction period range. The
values of τ are very similar for the two largest industries, business equipment and health care, and
they decline more or less steadily as L increases. They start between about 0.6 and 0.8 for a 1.0-
to 1.5-year restriction period and end under 0.10 for a 10- to 12-year restriction period. There are
industries with smaller median values for τ but it is impossible to draw any firm conclusions
because of the small sample sizes.
F. Ex Post DLOM Term Structure Empirical Estimation
A plot of the DLOMs calculated from equation (13) versus L illustrates the general shape
of the DLOM term structure. Figure 5 plots the median value of for each restriction period for
the last pre-IPO transactions sample, the earlier pre-IPO transactions sample, and the full sample,
in each case without controlling for industry effects. Each DLOM term structure is generally
29
gradually upward-sloping.
I control for industry effects in the term structure regression models by including industry
dummies because there appear to be systematic industry effects in the τ(Δ, L) calculations reported
in Table 7. I allow for possible interactions between each industry dummy and the length of the
restriction period. I also allow for the possibility that a buoyant IPO market in the year the firm
that issues the private placement goes public might bias the calculated value for in equation (13)
upward. I define the variable as the ratio of the number of U.S. IPOs in the year t in which
the private equity issuer went public to the average annual number of IPOs for the years in the
sample period, 1995 to 2017, when any of the issuing firms in the sample went public. With these
modifications to equation (16), I fit the following equation to the adjusted DLOM data for the full
sample and each sub-sample:
− 1 − = + + + + + +
+ +⋯+ + (18)
When c4 differs from 1, the empirical term structure conforms to equation (16). are the industry
dummies. The omitted industry is the miscellaneous category, Industry 12 (Other). Comparing
equations (16) and (18), I expect to be statistically insignificant.
The sample exhibits clustering of error terms because most of the sample firms have more
than one private equity transaction. Table 8 reports the number of private equity placements by
each firm in the sample. All private equity transactions conducted by a particular firm have the
same IPO date in the calculation of in equation (13). Any bias that remains after controlling for
the IPO effect in the regression model will affect all the error terms for those private equity
transactions. I adjust for the clustering of error terms by calculating asymptotically consistent
estimators applying the Liang-Zeger procedure (Liang and Zeger, 1986).
Table 9 provides the regression results of fitting equation (18) to the full sample of 5,391
transactions, the sub-sample of 1,632 last pre-IPO private equity transactions, and the sub-sample
of 3,759 earlier pre-IPO private equity transactions. The adjusted R2 are within the range from
0.3793 to 0.4733, and F statistic for each regression is significant at the .01 level. The coefficient
of the IPOt control variable is highly statistically significant at the .01 level in all three regressions,
which indicates the important effect the quality of the IPO market has on the pricing of IPOs. The
regression results for the full sample differ from those for the sub-samples. The intercept is not
statistically significant at the .10 level for the full sample but is statistically significant at the .05
30
level for the last pre-IPO sub-sample and at the .01 level for the earlier pre-IPO sub-sample. All
three restriction period coefficients are statistically significant at the .01 level in the full sample
regression but none are statistically significant at the .10 level for the last pre-IPO transaction sub-
sample regression. The coefficients of L and L2 are statistically significant at the .01 level in the
earlier pre-IPO transaction regression. The coefficient of L3 is not statistically significant at the
.10 level.
The coefficient of the ΔL interaction term is highly statistically significant at the .01 level
in all three regressions and is also significantly greater than 1 at the .01 level in all three
regressions,25 which indicates that the term premium is positively related to the volatility of the
firm’s stock. In other words, the term premium increases with the stock’s risk as proxied by its
volatility, which is consistent with Koziol and Sauerbier (2007). The coefficients of the restriction
period terms are generally consistent with the values and the more or less steadily declining pattern
of τ that is evident in Table 7.
Figure 6 plots the DLOM term structure and the term premia τ(Δ, L) for the full transaction
sample, the last pre-IPO transaction sample, and the earlier pre-IPO transaction sample in all three
cases for σ = 0.50, 0.75, and 1.00. The three DLOM term structures are all monotonically
increasing concave functions of L at the higher volatilities σ = 0.75 and σ = 1.00). All three term
structures shift upward as the risk proxy Δ increases. The DLOM term structure for the full sample
in Panel A for σ = 0.50 is relatively flat for L between 4 and 12 years before increasing, and the
term structure converges to = 1 as L increases beyond about 18 years. The term structure
behavior illustrated in Panel A appears reasonable. The term structure for the last pre-IPO
transactions in Panel C for σ = 0.50 reaches a maximum about year 6 and then declines. However,
the sample is concentrated in the range L ≤ 1 and has very few observations greater than 2.0. Thus,
its behavior over the relevant range is monotonically increasing and concave even for σ = 0.5. The
term structure for the earlier pre-IPO transactions in Panel E for σ = 0.50 reaches a maximum about
year 8 and then declines slightly out to year 12, which is the longest restriction period in the sample.
The term premia in Figure 6 exhibit different patterns. The term premia for the full sample
in Panel B are generally increasing except for a middle range where they are flat or declining,
especially at the lower volatilities. The generally increasing pattern of term premia is consistent
with the liquidity preference theory of the behavior of the term structure. It is reasonable that the
liquidity preference is more evident at higher volatilities due to the higher risk. Likewise, the term
premia for the last pre-IPO transactions in Panel D are monotonically increasing over the relevant
31
range, again consistent with the liquidity preference theory of the term structure. The term premia
for the earlier pre-IPO transactions in Panel F are concave and positive over the range of the
sample, in which the longest restriction period is 12 years. Beyond the range of the sample, the
term premia would eventually turn negative.
There is evidence of significant industry effects in the regression models for the full sample
and for the earlier pre-IPO transactions in Table 6. The industry effects are strongest for non-
durables (industry 1), durables (industry 2), manufacturing (industry 3), energy (industry 4),
utilities (industry 8), retail shops (industry 9), and financial firms (industry 11), all of which have
a negative coefficient that is statistically significant at the .05 level in at least one of the models in
Table 9. The two largest components of the sample, business equipment (industry 6) and health
care (industry 10), do not exhibit statistically significant industry effects. The DLOMs for firms
in industries 1, 2, 3, 4, 8, 9, and 11 are generally smaller than the DLOMs for firms in industries 6
and 10, which is not surprising in view of the comparatively higher technology and other business
risks in business equipment (computers and other electronic devices) and health care. None of the
dummy variable coefficients are significant at the .05 level in the last pre-IPO transactions
regression, which may be due to the relatively short restriction periods limiting the industry effect
that is evident for longer restriction periods.
G. Tests for a Date of Issue Effect
The size of the DLOM could depend on the state of the private equity market at the date of
issue. I have already controlled for an IPO effect, which controls for market conditions at the time
of the IPO on in equation (13) and which I found to be highly statistically significant. The date
of issue effect tests for the impact of market conditions at the issue date on . I tested all three
DLOM term structures in Table 9 for a date of issue effect. I augmented the term structure
regression models to include annual time dummy variables to test for a date of issue effect.
Table 10 tests the stability of the DLOM term structure by testing for a date of issue effect
in the three regression models. I code the time dummy variables based on the date of the private
equity transaction, which requires 28 annual dummy variables for the years in my sample when
the transactions occurred, 1985 and 1989 through 2017. The years 2016 and 2017 (with a very
small number of transactions) are reserved as the holdout period. As reported in Table 10, only
four of the date of issue dummies are statistically significant at the .05 level in the regression for
the full sample, only five are statistically significant at the .05 level in the regression for the earlier
pre-IPO private equity transactions sample, and only one is statistically significant at the .05 level
32
in the regression for the last pre-IPO private equity transactions sample. There is evidence of only
a very weak date of issue effect in any of the sample regressions. All three DLOM term structures
appears relatively stable over time.
H. Properties of the Term Premium
Setting aside the industry effects and the weak date of issue effect and suppressing the IPO
control variable, the regression model in Table 9 for the full sample simplifies to
− 1 − = − 0.0072 + 0.4311 − 0.0801 + 0.0034 + 0.6049 (19)
The DLOM term premium τ(Δ, L) is
( , ) = − . + 0.4311 − 0.0801 + 0.0034 + 0.6049 (20)
Next, I investigate the properties of τ(Δ, L). The term premium has the following
sensitivities to variation in L and Δ:
= 0.0072/ − 0.0801 + 0.0068 = − . + 0.0068 (21A)
= 0.0649 = 0 (21B)
The term premium is a linear function of Δ. Ignoring the statistically insignificant regression
constant, τ(Δ, 0) = 0.4311 + 0.6049Δ > 0 since Δ > 0. The term premium is initially positive (except
for extremely small L) and increasing in Δ. It is greater for riskier stocks across the entire range
of restriction periods, as one would expect. δτ/δL = 0 when L = 0.17 or L = 11.74. The term
premium reaches a minimum at L = 11.74 years. Since τ(Δ, 11.74) = -0.0413 + 0.6049Δ, the
minimum value of τ(Δ, L) depends on the value of the base one-year restriction period DLOM Δ.
τ(Δ, 11.74) ≥ 0 provided Δ ≥ 0.0682. The term premium is positive provided the stock’s base one-
year restriction period DLOM is greater than 6.82%. Δ is specific to each stock and depends on
the stock’s price volatility. However, Δ is restricted to the range [0, 0.3228], as indicated in the
Appendix. Δ = 0.0682 in equation (8) when ν = 0.1654, which corresponds to a stock price
volatility of 28.85% in Table 2.
Next, I consider the term structure for the last pre-IPO transactions in Table 9. Again
ignoring the industry effects and the weak date of issue effect and suppressing the IPO control
variable, the regression model in Table 9 for the last pre-IPO transactions sub-sample simplifies
to
− ln 1 − = + 0.0546 + 0.1198 − 0.0433 − 0.0002 + 1.3648 (22)
The regression constant is statistically significant at the .05 level but none of the coefficients of
33
the three restriction period terms is statistically significant at the .10 level. The DLOM term
premium τ(Δ, L) is
( , ) = . + 0.1198 − 0.0866 − 0.0006 + 1.3648 (23)
The term premium for the last pre-IPO transactions is a convex function of L and a linear
function of Δ over the relevant range for L since = − . − 0.0866 − 0.0012 = . − 0.0012 (24A)
= 1.3648 = 0 (24B)
τ(Δ, 0) is positive due to the first term in equation (23). Δτ/δL < 0 for all L, and > 0 for L <
4.5 years, which captures the relevant range for the last pre-IPO transactions sub-sample (L ≤ 2).
The term premium is initially positive and increasing in Δ. It is greater for riskier stocks across
the entire range of restriction periods, as one would expect. The term premium is positive at least
within the range [0,2] provided τ(Δ, 2) = -0.0285 + 1.3684 Δ > 0, which holds so long as Δ >
0.0208. The term premium is positive provided Δ exceeds 2.08. Δ = 0.0208 in equation (8) when
ν = 0.0516, which corresponds to a stock price volatility of 8.95% in Table 2.
Last, I consider the term structure for the earlier pre-IPO transactions in Table 9. Ignoring
the industry effects and the weak date of issue effect and suppressing the IPO control variable, the
regression model in Table 9 for the earlier pre-IPO transactions sub-sample simplifies to
− ln 1 − = + 0.1426 + 0.2682 − 0.0442 + 0.0012 + 0.5383 (25)
The regression constant and the first two restriction period coefficients are all statistically
significant at the .01 level but the coefficient of the cubed restriction period term is not statistically
significant at the .10 level. The implied DLOM term premium τ(Δ, L) is
( , ) = . + 0.2682 − 0.0442 + 0.0012 + 0.5383 (26)
The term premium for the last pre-IPO transactions term structure is a convex function of
L and a linear function of Δ over the entire range for L. Taking partial derivatives, = − . − 0.0442 + 0.0024 = . + 0.0024 > 0 (27A)
= 0.5383 = 0 (27B)
Again, τ(Δ, 0) is positive due to the first term in equation (26). The term premium is initially
positive and increasing in Δ. It is greater for riskier stocks across the entire range of restriction
34
periods, as one would expect. δτ/δL < 0 for all L < 18.59, and > 0 for all L. The term premium
is initially positive but decreasing for restriction periods shorter than 18.59 years. It could become
negative depending on the magnitude of Δ. The sum of the middle three terms in equation (26)
reaches a minimum at about 18.42 years, at which point the term premium is τ(Δ, 18.42) = -0.1311
+ 0.5383Δ, which is positive provided Δ > 0.2435. The term premium is positive provided Δ
exceeds 24.35. Δ = 0.2435 in equation (8) when ν = 0.5485, which corresponds to a stock price
volatility of 104.64% in Table 2.
III. Generalization to Restriction Periods of Uncertain Length
Equation (12) calculates the DLOM when the length of the restriction period is known, for
example, due to specific contractual or legal restrictions on transfer of the security that will lapse
on a date certain, or when L can be reasonably predicted and therefore treated as having a particular
value L with certainty. A more challenging problem arises when L is so uncertain that a point
estimate for L cannot be reasonably estimated. This section suggests how to further extend the
average-strike put option DLOM model to accommodate restriction periods of such highly
uncertain length. To motivate the discussion, I begin with the simple case in which τ is a constant.
A. Constant τ
The results of fitting a probability distribution to the sample L in Table 6 indicate that it is
reasonable to assume that L is approximately exponentially distributed. According to this
distribution, the instantaneous probability that the restriction period will expire L years after the
valuation date given that the stock is still restricted at that time is a constant λ, and the probability
that the stock will still be restricted after L years is . The probability the restriction period
will end L years after the valuation date is the product of these two probabilities. The probability
density function for L is f( ) = for ≥ 0, which is the exponential pdf. The exponential
distribution covers the entire domain [0,∞). An infinite value for L could be interpreted as
signifying that the firm never goes public, for example, because it has failed.26
The parameter λ can also be interpreted as the rate at which a firm progresses toward
achieving a liquidity event for the investors in its shares, such as an IPO or a change-of-control
transaction. The mean restriction period is = 1/λ. Inverting this equation leads to a method for
estimating the parameter λ. If can be estimated from historical data for comparable restricted
securities, then = 1/ . The marketability discount is the exponentially weighted average
of the marketability discounts ( ) in equation (12) for restriction periods of length L.
35
If τ is a constant and L is assumed to be exponentially distributed with parameter λ
(depending on whether the private equity transaction is expected to be the firm’s last before its
IPO or instead that it is an earlier pre-IPO transaction), the restriction-period-weighted
marketability discount can be obtained by calculating the expected value of ( ) given by
equation (12):
= 1 − ( ) = 1 − (28)
where Δ is given by equation (8). is an increasing function of Δ and τ and a decreasing function
of λ. In particular, increasing the rate λ at which the firm progresses toward achieving a liquidity
event for its investors effectively shortens the expected length of the restriction period and reduces
the DLOM.
Table 11 illustrates the range of values for the DLOM for a representative range of
parameter values. Table 11 also shows the relationship between the volatility parameters υ and σ
according to equation (6). increases with the volatility of the security’s returns σ and with the
expected length of the restriction period = 1/ , both of which are consistent with the option
character of the DLOM. For example, a 5% per year (τ = 0.05) term premium increases the DLOM
by about one-third from 19.25% to 25.29% for a two-year restriction period for a non-dividend-
paying stock (q = 0) with 50% volatility and by about one-sixth from 44.67% to 51.39% for a five-
year restriction period for a stock yielding 3% (q = .03) with 70% volatility.
Figure 7 illustrates how varies as a function of λ and σ, and Figure 8 illustrates how
varies as a function of Δ and . is a convex decreasing function of λ and an increasing concave
function of Δ, τ, and σ. Doubling roughly doubles when is less than about 1.5 years and σ
is less than about 40%. Beyond those ranges is a little less sensitive to increases in and σ.
B. Variable τ(Δ, L)
When L is so uncertain that a point estimate cannot be reasonably determined, estimating
a DLOM is more challenging. Employing an estimate for is a possibility but substituting
equation (17) or even equation (19) for τ(Δ, L) on the left-hand side of equation (28) leads to an
expression that cannot be integrated. The integral can be evaluated using numerical methods,
which is complex.
There is one special case worth noting. An approximate closed-form expression for for
the last pre-IPO private equity transactions can be obtained by suppressing the variables in
equation (22) that have statistically insignificant coefficients, re-estimating the regession,
36
obtaining a new expression for τ(Δ, L), substituting into the expression for , and evaluating the
resulting integral. In the DLOM regression model (22) for the last pre-IPO private equity
transactions, none of the coefficients of the three restriction period variables is statistically
significant at the .10 level. Suppressing these variables and re-estimating the DLOM model gives
− 1 − = 0.1320 + 2.0903 (29)
and thus,
= 1 − ( . . ) = 1 − .. (30)
Applying this approach to the earlier pre-IPO equity transactions leads to a simpler
integral but not to a closed-form expression. In the DLOM regression model (25) for the earlier
pre-IPO private equity transactions, the regression constant and the first two restriction period
coefficients are all statistically significant at the .01 level but the coefficient of the cubed restriction
period variable is not statistically significant at the .10 level. Suppressing this variable and re-
estimating the DLOM model gives
− 1 − = 0.1812 + 0.2287 − 0.0276 + 1.5421 (31)
= 1 − ( . . . . ) (32)
Unfortunately, the integrand cannot be integrated, so the integral would have to be evaluated by
applying numerical methods.
C. Applying the Models to Estimate a DLOM
To estimate a DLOM when L is either fixed or can be estimated, apply equation (12).
Apply the appropriate DLOM model depending on whether the private placement is expected to
be the last private equity transaction prior to the firm’s anticipated IPO. Usually, if the transaction
is expected to be the last pre-IPO, then L can be reasonably estimated. For earlier pre-IPO
transactions, L may be highly uncertain.
When a value for L can be estimated, use equation (20), equation (23), or equation (26), as
appropriate, to estimate a value for τ(Δ, L). Substitute the coefficient values from the appropriate
model in Table 9, depending on whether the firm’s IPO is imminent. If the private equity
transaction is expected to be the firm’s last before it goes public, use equation (23) and apply the
coefficients for the last pre-IPO transactions in Table 9. If the IPO is a long way off but the
approximate timing can be reasonably estimated, then use equation (26) and apply the coefficients
for the earlier pre-IPO transactions in Table 9. If L can be reasonably estimated but it is unclear
whether the private equity transaction will be the firm’s last before its IPO, then use equation (20)
37
and apply the coefficients for the full sample in Table 9. If the firm’s IPO is not imminent and the
length of the restriction period is so uncertain that a point estimate for L cannot be reasonably
estimated, then one could apply in equation (28) with τ(Δ, L) in place of τ and evaluate the
resulting integral using numerical methods.
Based on the regression results in Table 9, there are seven industries with statistically
significant industry effects. The common stocks of firms in these industries are affected by non-
diversifiable economic risks, which necessitate adjustments to the DLOM. Equations (20), (23),
and (26) all adjust for industry effects. If more up-to-date data are available, an appraiser can re-
estimate equation (18) before applying any of the DLOM models.
IV. Conclusions
Previous research has shown that the average-strike put option DLOM model calculates
marketability discounts that are generally consistent with the discounts observed in letter stock
private placements, which have relatively short marketability restriction periods. Building on this
prior work, I developed a general DLOM model and empirically fitted the model to a sample of
private equity transaction implied marketability discounts with a wide range of restriction periods.
I specified an ex post DLOM term structure and investigated its properties, including quantifying
the relationship between the term premium and the length of the restriction period. Finally, I
incorporated the DLOM term structure in the general DLOM model and explained how to use the
general DLOM model.
I generalized the average-strike put option DLOM model to restriction periods of any
specified length L by modeling the L-year DLOM as the value of the one-year average-strike put
option DLOM compounded over L years. I extended the model to restriction periods of uncertain
length by assuming the length of the restriction period is exponentially distributed and furnished
empirical support for that assumption. I also adjusted for a DLOM term premium to reflect a risk
averse investor’s more prolonged exposure to the risk of an increasingly negatively skewed fat-
tailed return distribution.
My data set does not include any ex ante estimates of the (expected) restriction period, but
if such a data set were to become available, my modelling approach could be utilized to fit an ex
ante DLOM term structure. The DLOM model developed in this paper could also be extended to
allow for a possible DLOM term discount to reflect the value of access to a particular class of
longer-duration assets generally offering positive-α investments but imposing a long period of
restricted marketability. It could also be extended to allow for the increasing effect on the DLOM
38
of any idiosyncratic exposure to investment-specific agency costs or the decreasing effect on the
DLOM of any idiosyncratic benefit from investment-specific α due to superior fund manager
investment skill or to strategic partner equity investment value. I leave these extensions for future
research.
Due to the impossibility of arbitrage when trading restricted shares, there may be additional
share-specific factors that could affect share valuation but are not captured in equations (12) and
(13). For example, restricted stock investors may have preferred habitats due to their particular
liquidity preferences or to the liquidity characteristics of the restricted securities, which could
cause the DLOM term structure to deviate from the term structure implied by equation (12) or
equation (13). I also leave that question for future research.
Appendix. Upper Bound on the DLOM in the Average-Strike Put Option Model
Finnerty’s (2012, 2013a) average-strike put option model has an effective upper bound on
the percentage discount equal to 32.28% when T is large. Rewrite equation (4) as = + 2 − − 1 − 2 − 1
= + 2 + − − 1 − 2 − 1
< + 2 + − 1 − 2 − 1 for T > 0
= 2 + − − 1
Note that since lim→ − 1 = it follows that lim→ = 2. Thus, for large T,
behaves like 2/ , which approaches zero as T becomes large.
Next rewrite equation (3) as
( )
= − 1-
2
Tv2NVTD 0
qTe . (3)
For large T, D(T) approaches V0 2 √ − 1 = 0.3228V0 < 0.3228V0 . The
percentage discount cannot exceed 32.28% for large T, which is unrealistic.
39
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Endnotes
1 The growing importance of private equity is reflected in the fact that 921 private equity funds raised $453.3 billion
globally in 2017 (Thomson Reuters, 2018) and invested $1,270 billion in 8,100 private deals globally, up from $190
billion in 2,200 private deals in 1990 (McKinsey, 2018). By comparison, public equity offerings raised $184.4 billion
through 856 new issues in the U.S. and $598.7 billion through 4,715 new issues globally in 2017 (Bloomberg, 2018). 2 The growth of the PE industry has led to the development of secondary markets for PE fund investments and for PE
fund LP units. PE fund managers have formed funds specifically to purchase the investments that other PE funds
wish to cast off and the PE fund LP units that institutional investors wish to dispose of for liquidity or other reasons.
BlackRock Private Opportunities Fund III, L.P. (2014) is one such example. Likewise, a secondary market has
developed for shares of privately held firms that are expected to go public. Two leading secondary markets for such
shares in the U.S. are Nasdaq Private Market and SharesPost. 3 Private equity market participants refer to privately owned companies whose total equity value exceeds $1 billion as
‘unicorns.’ As of year-end 2017, there were approximately 214 unicorns globally with 57 added in 2017 alone (Alois,
2017; Desjardins, 2017). Unicorns are staying private longer due to the increased availability of private equity,
especially for technology firms; the fourfold increase in the maximum number of shareholders before a firm must
publish its financial statements publicly under the U.S. Jumpstart Our Business Startups (JOBS) Act of 2012; and the
generally higher IPO multiples the public equity market accords larger tech firms (Erdogan, Kant, Miller, and Sprague,
2016). More than half the unicorns are in the U.S., and about a quarter are in China. Likewise, the number of public
companies in the U.S. has declined. Between the dot.com peak in 1996 and 2017, the number of U.S. corporations
listed on a U.S stock exchange roughly halved from more than 8,000 to a little over 4,000 (Brorsen et al., 2017). 4 For example, Nasdaq Private Market reported transaction volume of $3.2 billion in 2017 (up from $1.6 billion in
2015 and $1.1 billion in 2016) whereas Nasdaq reported that the total dollar value of common stock traded through
its electronic order book in 2017 amounted to $11,336 billion. 5 Section 401 of the JOBS Act added new Section 3(b)(2) to the Securities Act of 1933 to effect this change. The
1933 Act governs the requirements for registering securities that are to be offered to public market investors. 6 Regulation A is the so-called ‘small issue’ exemption from the 1933 Act’s securities registration requirements. The
Regulation A offering limitation had not been raised (to $5 million) since 1992. Tier 2 issuers face tighter restrictions
than Tier 1 issuers. Tier 2 issuers must include audited financial statements in their offering documents and must file
regular reports with the SEC. Tier 2 offerings also have investor restrictions. Unless the securities will be listed on a
national securities exchange, purchasers in a Tier 2 offering must be accredited investors or else they are limited to
investing no more than the greater of 10% of their annual income or 10% of their net worth in the offering. 7 Letter stock is not registered for resale under the 1933 Act and therefore cannot be freely traded in the public market.
It must be placed privately with accredited (sophisticated) investors. Under Rule 144, the holder cannot sell the shares
during a specified minimum holding period, which is measured from the issue date, except through another exempted
transaction. Prior to February 1997, the minimum holding period for a stockholder who was not affiliated with the
47
issuer was two years. The SEC shortened the Rule 144 minimum holding period in 1997 to one year (SEC, 1997) and
again in December 2007 to six months (SEC, 2007). Longer restriction periods apply if the stockholder is an affiliate
of the issuer or if the firm is not current with its SEC filings. After the minimum holding period, the shares can be
sold in the public market without first registering them but only after the firm agrees to remove the restrictive stock
legend on the share certificate, which usually requires an opinion from counsel that the sale complies with Rule 144. 8 More recent noteworthy empirical studies include Hertzel and Smith (1993) and Hertzel et al. (2002). Collectively,
these studies have documented significant discounts in private placements of unregistered common stock averaging
between 13 percent and 34 percent. The more recent studies generally find smaller private placement discounts, which
may reflect the fact that the early studies predated the development of organized stock option markets and the hedging
opportunities they afford. 9 There is also a tendency for the model to understate (overstate) the discount when the stock’s volatility is less than
45 percent or greater than 75 percent (between 45 percent and 75 percent). 10 Nasdaq Private Market (previously SecondMarket) and SharesPost are two of the more prominent secondary
markets for unregistered shares of private firms. Nasdaq Private Market facilitates transactions in unregistered shares
of private firms among accredited investors and operates a web-based auction platform that supports liquidity
programs through which private firms can tender for or buy back unregistered stock from shareholders and employees.
It has completed more than $6.1 billion of transactions since 2013, including $3.2 billion in 2017. SharesPost, which
developed the first online secondary market for private firm shares, provides company research and data, facilitates
secondary trading in unregistered shares among accredited investors, and operates a private stock loan program all in
the shares of leading private technology firms. It has completed over $4 billion of transactions in more than 200
technology stocks since 2009. Sponsors of both markets state that they comply with U.S. securities laws, which
provide exemptions from the registration requirements of the 1933 Act provided potential individual purchasers are
limited to accredited investors. Information concerning these markets is available at
https://www.nasdaqprivatemarket.com/ and https://www.sharespost.com/, respectively. 11 Even with a marketability restriction, the length of the restriction period should be adjusted upward to reflect how
long it is expected to take to sell all the shares after the resale restriction lapses. For example, the larger the block of
shares, the longer it is likely to take to sell them, and the greater should be the length of the restriction period assumed
in the DLOM calculation. 12 See in particular Exhibit 1 in Finnerty (2012). 13 The minimum holding period is six months so long as the firm is subject to the reporting requirements under the
Securities Exchange Act of 1934 (the 1934 Act) and is current with its SEC filings. If the firm is not subject to the
reporting requirements under the 1934 Act or is not current with its filings, the minimum holding period is one year. 14 I also briefly summarize the derivation of the average-strike put option DLOM model for the reader’s convenience.
The details are in Finnerty (2012).
48
15 Guidolin and Timmermann (2006) define a term structure of risk. It has an important implication for stock return
distributions: the risk of unexpected extreme stock price movements increases over longer time horizons and negative
skewness consequently increases with longer time horizons (Engle, 2009, 2011). 16 Ze-To (2008) finds that the returns on the Hang Seng Index and the Hang Seng China Enterprise Index were more
heavily fat-tailed than the S&P 500 Index returns but that the two Hang Seng index return distributions exhibited
positive skewness in contrast to the negative skewness of the S&P 500 return distribution. 17 Investors can sell restricted shares subject to an exemption from registration under the 1933 Act but such sales must
be arranged privately, and the issuer will normally require the purchaser to sign an investment representation letter
(Hicks, 1998). The issuer may also require an opinion of counsel that the sale is exempt from the 1933 Act’s
registration requirements. Both steps are time-consuming and involve legal fees and other transaction costs. 18 Equivalently, I could assume that there are always sufficient numbers of issuers and investors in the restricted stock
new issue market who are flexible enough to adjust their choice(s) of restriction period to restore market equilibrium. 19 According to equation (11), when q = 0, D*(L) = 1 in the limit for all values of L as v → ∞. 20 The local expectations formulation is the only one of the interpretations of the pure expectations theory that is
consistent with a sustainable market equilibrium (Cox, Ingersoll, and Ross, 1981, pp. 774-775). 21 τ < 0 would be indicative of any special benefits attributable to being able to access a particularly attractive class of
longer-duration positive-α investments. A willingness to accept a longer investment holding period might create
opportunities to invest in special situations that require longer holding periods but provide superior risk-adjusted
returns (independent of idiosyncratic manager-specific skill or investment-specific agency risks, which would have to
be captured separately in the DLOM calculation), which can decrease the DLOM. For example, investment
opportunities that require time to accumulate assets, such as real estate, natural resource deposits, or agricultural land,
and then additional time to manage the asset portfolio to optimize its value, could be associated with longer duration
and superior returns that go hand in hand. 22 Information regarding this database is available at https://www.bvresources.com/products/valuation-advisors-lack-
of-marketability-study. Last accessed November 18, 2017. 23 This premium might be the result of an actual private placement premium, as for example when the stock is
purchased by a strategic buyer (Allen and Phillips, 2000), but it might instead be due to the value of the stock declining,
or at least not increasing as fast as the value of stocks in the smallest industry quartile of comparable common stocks. 24 A six-month post-IPO lockup period is typical. 25 The t-statistics for the test of the null hypothesis that the coefficient of ΔL equals 1.0 against the alternative
hypothesis that it is greater than 1.0 are 3.97 for the full sample, 2.91 for the lase pre-IPO transactions sample, and
3.46 for the earlier pre-IPO transactions sample, all of which are significant at the .01 level. 26 Alternatively, since firms are not infinite-lived, a truncated exponential distribution could be assumed for L instead.
0.0 - 29.9 4 19.47 % 7.84 % 9.54 % 10.95 %
30.0 - 44.9 0 - - - -
45.0 - 59.9 4 10.51 16.97 20.15 22.55
60.0 - 74.9 9 13.82 19.18 22.03 23.96
75.0 - 89.9 10 24.15 23.16 26.52 28.64
90.0 - 104.9 2 34.97 26.10 29.07 30.64
105.0 - 120.0 1 61.51 28.40 30.73 31.70
> 120.0 6 44.10 30.88 31.95 32.20
Average: 36 24.50 % 21.37 % 23.97 % 25.62 %
0.0 - 29.9 7 12.66 % 5.22 % 6.99 % 8.12 %
30.0 - 44.9 19 18.56 7.95 10.75 12.64
45.0 - 59.9 17 15.92 11.74 16.15 19.22
60.0 - 74.9 28 19.21 14.83 19.84 23.00
75.0 - 89.9 25 21.37 17.82 23.50 26.83
90.0 - 104.9 16 21.61 20.28 26.07 29.05
105.0 - 120.0 6 24.89 22.74 28.29 30.65
> 120.0 28 29.71 27.72 31.20 32.02
Average: 146 21.31 % 17.02 % 21.45 % 23.86 %
Source: Finnerty [2012], Exhibit 6.
Panel B: Offerings Announced After February 1997(One-Year Minimum Restriction Period)
σ Range (%)
Number of
Offerings
Mean Implied Marketability
Discount(day prior)
Mean Model-Predicted Discount(T = 1)
Mean Model-Predicted Discount(T = 2)
Mean Model-Predicted Discount(T = 3)
Table 1
Panel A: Offerings Announced Prior to February 1997 (Two-Year Minimum Restriction Period)
σ Range (%)
Number of
Offerings
Mean Implied Marketability
Discount(day prior)
Mean Model-Predicted Discount(T = 2)
Mean Model-Predicted Discount(T = 3)
Mean Model-Predicted Discount(T = 4)
Market Test of the Accuracy of the Average-Strike Put Option Model of the Discount for Lack of Marketability
The Model-Predicted Discount is calculated from the average-strike put option model (3) - (4), and the Implied Marketability Discount is from Exhibit 6 in Finnerty [2013]. Panel A applies to private placements of common stock that were announced prior to February 1997, and Panel B applies to private placements of common stock that were announced after February 1997. The sample includes 182 U.S. private placements between April 1, 1991 and March 8, 2007 that were priced at a discount to the price at which the unrestricted shares were trading in the market. T is the total effective restriction period after allowing for the additional time required to dispose of the shares when complying with Rule 144.
vσ T = 0.5 T = 1.0 T = 2.0 T = 5.0
10% 5.77% 5.77% 5.76% 5.75%
20% 11.53% 11.51% 11.47% 11.35%
30% 17.26% 17.19% 17.06% 16.67%
40% 22.94% 22.79% 22.47% 21.54%
50% 28.57% 28.26% 27.66% 25.83%
60% 34.12% 33.60% 32.54% 29.43%
70% 39.59% 38.75% 37.08% 32.26%
80% 44.95% 43.70% 41.22% 34.31%
90% 50.20% 48.42% 44.91% 35.67%
100% 55.31% 52.88% 48.12% 36.48%
Table 2
This figure indicates the relationship between the volatility parameters ν and
σ . The volatility ν is calculated from equation (6).
Relationship between the Volatility Parameters v and σ in the Average-Strike Put Option DLOM Model
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
D*(Δ,
L)
L
Figure 1D*(Δ, L) Given by Equation (9)
D*(Δ, L; σ = 10%)D*(Δ, L; σ = 20%)D*(Δ, L; σ = 30%)D*(Δ, L; σ = 40%)D*(Δ, L; σ = 50%)D*(Δ, L; σ = 60%)D*(Δ, L; σ = 70%)D*(Δ, L; σ = 80%)D*(Δ, L; σ = 90%)D*(Δ, L; σ = 100%)
Note: Assumes q=0.03 for illustrative purposes.
∗ , ; → ∞ 1 1
D*(Δ, L; σ = 10%)
D*(Δ, L; σ = 30%)
D*(Δ, L; σ = 50%)
D*(Δ, L; σ = 70%)
D*(Δ, L; σ = 90%)
D(T; σ = 10%)
D(T; σ = 30%)
D(T; σ = 50%)
D(T; σ = 70%)
D(T; σ = 90%)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.00 0.50 1.00 1.50 2.00
Mar
keta
bil
ity
Dis
cou
nt
Length of Restriction Period
Figure 2Comparison of the Two Average-Strike Put Option DLOM Models
D*(Δ, L; σ) [1]
D(T; σ) [2]
Note: Assumes q=0.03 for illustrative purposes.[1] D*(Δ, L) is given by equation (9) with v given by equation (6), and σ is the stock price volatility.[2] D(T) is given by equation (3) with √ given by equation (4), and σ is the stock price volatility.
10%
30%
50%
70%
90%
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
10.00 9.00 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00
Δ
L
Figure 3Sensitivity of D̂ Given by Equation (12) to L and Δ When τ = 0
90%-100%80%-90%70%-80%60%-70%50%-60%40%-50%30%-40%20%-30%10%-20%0%-10%
Range of
Sensitivity of Given by Equation (12) to L and Δ When τ = 0
Table 3
Sample Description
Panel A provides a financial description of the firms in the marketability discount sample. The financial data pertain to the fiscal year the firm went
public. IPO dates range from March 1986 to June 2017. In the first column under each header, the sample contains 1506 firms for which all the pre-
IPO private transactions were at an implied marketability discount. IPO dates range from March 1986 to June 2017. In the second column under each
header, the sample contains 126 firms with IPO dates ranging from May 1996 to April 2017. Panel B provides a breakdown of the sample based on the
number of firms that went public each year broken down by industry. Panel C provides a breakdown of the sample based on the number of private
transactions each year broken down by industry.
Panel A. Firm Descriptive Financial Data (Millions of Dollars)a
Annual Revenue Annual Net Income Total Assets at Year-End Book Value of Equity at
Year-EndMarket Capitalization at
Year-End
Adjusted Discount Discounts
only
Discounts and
Premiums
Discounts only
Discounts and
Premiums
Discounts only
Discounts and
Premiums
Discounts only
Discounts and
Premiums
Discounts only
Discounts and
Premiums
Number of Firms 1486 125 1486 125 1486 125 1486 125 1486 125
Minimum 0.00 0.00 -3445.07 -468.83 1.93 17.33 -8258.01 -2111.01 3.37 29.84
First Quartile 19.24 37.23 -27.09 -28.44 71.11 99.61 43.67 43.87 179.00 220.94
Median 69.42 103.10 -4.60 -1.29 139.61 145.19 84.83 80.41 437.94 356.31
Mean 511.83 284.69 -5.65 -5.20 831.30 593.51 191.38 118.19 1181.49 671.93
Third Quartile 227.68 273.50 8.28 11.24 386.08 300.26 167.11 153.04 1075.51 679.80
Maximum 135592.00 4738.00 6172.00 217.00 151828.00 18242.88 24277.02 1347.76 63141.93 5452.04
Percentage Negative --
64.52% 57.50% --
10.11% 10.92% --
Panel B. Sample Breakdown by Number of Firms Going Public per Year by Industry
Year 1 NoDur 2 Durbl 3 Manuf 4 Enrgy 5 Chems 6 BusEq 7 Telcm 8 Utils 9 Shops 10 Hlth 11 Money 12 Other Total
1986b 0 0 0 0 0 1 0 0 0 0 0 0 1
1994b 0 0 0 0 0 1 0 0 1 0 0 0 2
1995b 0 1 1 0 1 6 1 0 2 3 0 0 15
1996b 2 0 4 2 1 28 0 0 9 10 1 8 65
1997b 4 2 3 0 0 26 0 0 5 5 3 11 59
1998b 3 1 1 0 0 45 6 0 13 6 8 20 103
1999b 5 1 6 0 0 91 14 1 21 10 3 39 191
2000b 0 1 5 1 2 61 7 0 5 18 0 19 119
2001 1 1 1 0 0 12 0 0 0 3 2 4 24
2002 1 0 0 1 0 7 0 0 8 6 4 7 34
2003 1 0 1 0 0 15 1 0 7 10 8 8 51
2004 1 2 3 4 1 17 3 1 10 24 11 7 84
2005 3 2 3 2 1 6 3 0 8 15 10 8 61
2006 3 1 6 4 5 17 3 1 5 19 10 16 90
2007 1 1 3 2 1 31 2 1 4 24 7 16 93
2008 0 0 2 0 0 2 0 1 0 1 1 4 11
2009 0 0 2 0 1 10 1 0 3 3 0 7 27
2010 1 3 4 1 2 17 2 0 5 10 8 10 63
2011 0 1 2 1 3 18 2 0 7 9 2 6 51
2012 3 1 5 2 0 31 1 0 9 13 5 9 79
2013 0 1 1 2 2 29 1 0 9 41 10 16 112
2014 1 1 3 1 1 27 0 0 11 56 15 9 125
2015 1 1 4 0 1 19 0 0 8 41 9 8 92
2016 4 0 1 1 1 14 0 1 3 24 4 3 56
2017 0 1 1 3 0 8 0 0 2 5 1 3 24
Total 35 22 62 27 23 539 47 6 155 356 122 238 1632
Industry Percentage 2.14% 1.35% 3.80% 1.65% 1.41% 33.03% 2.88% 0.37% 9.50% 21.81% 7.48% 14.58% 100.00%
Panel C. Sample Breakdown by Number of Private Transactions per Year by Industry
Year 1 NoDur 2 Durbl 3 Manuf 4 Enrgy 5 Chems 6 BusEq 7 Telcm 8 Utils 9 Shops 10 Hlth 11 Money 12 Other Total
1986b 0 0 0 0 0 2 0 0 0 0 0 0 2
1994b 0 0 0 0 0 1 0 0 1 0 0 0 2
1995b 0 3 2 0 2 9 1 0 5 4 0 0 26
1996b 3 0 7 4 3 62 2 0 16 14 1 14 126
1997b 7 2 4 0 0 55 1 0 16 15 8 16 124
1998b 6 2 1 0 0 104 9 0 22 13 12 35 204
1999b 9 2 13 0 0 211 29 1 48 18 8 88 427
2000b 0 5 14 1 8 158 19 0 6 47 1 50 309
2001 3 3 3 1 0 31 0 0 1 11 6 8 67
2002 2 0 0 1 0 23 0 0 12 13 9 15 75
2003 4 0 1 0 0 23 4 0 11 16 14 17 90
2004 1 8 4 9 1 51 7 3 42 67 28 19 240
2005 5 7 5 4 3 31 14 0 22 47 29 14 181
2006 4 2 14 7 11 54 9 2 7 44 22 30 206
2007 1 5 9 4 1 115 5 2 7 70 11 43 273
2008 0 0 3 0 0 8 0 1 1 4 2 5 24
2009 0 0 4 0 1 43 1 0 12 8 0 13 82
2010 1 10 14 1 11 71 8 0 9 40 32 32 229
2011 0 5 3 2 17 133 18 0 31 57 9 26 301
2012 15 1 23 4 0 226 11 0 46 100 17 36 479
2013 0 5 1 2 8 154 2 0 40 241 46 68 567
2014 3 2 13 2 2 151 0 0 45 300 60 47 625
2015 1 2 18 0 4 104 0 0 24 182 38 26 399
2016 12 0 5 1 4 75 0 2 9 96 19 17 240
2017 0 2 1 6 0 36 0 0 8 33 1 6 93
Total 77 66 162 49 76 1931 140 11 441 1440 373 625 5391
Industry Percentage 1.43% 1.22% 3.01% 0.91% 1.41% 35.82% 2.60% 0.20% 8.18% 26.71% 6.92% 11.59% 100.00%
aNot all firms have financial data available on Compustat or Bloomberg for their IPO year. In the first column, data were available for 1486 of 1506 firms. In the second column, data were available for 125 of 126 firms. bYear with an above annual-average number of IPOs for the sample period.
Table 4
Common Stock Volatility During the First 12 Months Following the IPO
The table shows the common stock volatility during the 12 months following the firm’s IPO for the 5391 private
transactions in Panel A and the 1632 firms in Panel B broken down by industry.
Panel A. Sample Private Transactions
Industry σ≤25% 25%<σ≤50% 50%<σ≤75% 75%<σ≤100% 100%<σ≤125% σ>125% Total
1 NoDur 2 14 23 11 3 24 77
2 Durbl 3 13 24 7 7 12 66
3 Manuf 5 71 39 26 11 10 162
4 Enrgy 2 35 12 0 0 0 49
5 Chems 5 20 22 21 0 8 76
6 BusEq 1 229 620 447 236 398 1931
7 Telcm 4 34 21 39 12 30 140
8 Utils 0 7 0 4 0 0 11
9 Shops 3 139 128 57 41 73 441
10 Hlth 11 211 611 299 179 129 1440
11 Money 64 191 68 21 20 9 373
12 Other 23 141 189 86 62 124 625
Total 123 1105 1757 1018 571 817 5391
Percent of Sample 2.28% 20.50% 32.59% 18.88% 10.59% 15.15% 100.00%
Panel B. Sample Firms
Industry σ≤25% 25%<σ≤50% 50%<σ≤75% 75%<σ≤100% 100%<σ≤125% σ>125% Total
1 NoDur 1 9 9 6 3 7 35
2 Durbl 1 7 6 3 2 3 22
3 Manuf 1 27 20 7 4 3 62
4 Enrgy 1 18 8 0 0 0 27
5 Chems 1 9 6 5 0 2 23
6 BusEq 1 61 144 119 75 139 539
7 Telcm 1 11 7 10 4 14 47
8 Utils 0 4 0 2 0 0 6
9 Shops 1 50 47 20 13 24 155
10 Hlth 2 66 154 67 36 31 356
11 Money 21 66 24 3 4 4 122
12 Other 9 62 66 31 21 49 238
Total 40 390 491 273 162 276 1632
Percent of Sample 2.45% 23.90% 30.09% 16.73% 9.93% 16.91% 100.00%
Table 5
Implied Marketability Discounts Exceeding the Longstaff Model Upper Bound
This table reports the number of implied DLOMs that exceed the Longstaff (1995) model upper bound. Number with MD Above Upper Bound is the
number of DLOMs that exceed the Longstaff (1995) model upper bound. Percentage with MD Above Upper Bound is the percentage of DLOMs that
exceed the Longstaff (1995) model upper bound. The difference between the percentage of last pre-IPO transaction DLOMs and the percentage of
earlier pre-IPO transaction DLOMs that exceed the Longstaff (1995) model upper bound is tested based on Pearson’s chi-squared test. Excess is the
difference between the implied DLOM and the Longstaff (1995) model upper bound. Mean Excess (Median Excess) is the mean (median) of the Excess
for those private equity transactions with a DLOM that exceeds the Longstaff (1995) model upper bound. The difference of Mean Excesses between
the last pre-IPO equity transactions and earlier pre-IPO equity transactions is tested using the Pooled-t test, and the difference of Median Excesses is
tested using the Wilcoxon-Mann-Whitney U test. The p-value is given to the right of the difference within parentheses.
ALL Pre-IPO Transactions Last Pre-IPO Transactions Earlier Pre-IPO Transactions
Number Upper Case Percentage Number
Upper Case Percentage Number
Upper Case Percentage
Difference Between Last and Earlier Transactions
(p value)Number of Private Issues 5391 1632 3759Number with MD Above
Upper Bound 1434 605 829
Percentage with MD Above Upper Bound
26.60% 37.07% 22.05% 15.02% (< .0001)
Private Issues with MD Above Upper Bound:
Mean Excess 0.1978 0.1891 0.2041 -1.50% (0.0685)
Median Excess 0.1606 0.1509
0.1692 -1.83% (< .0001)
0 < Excess ≤ 5% 244 17.02% 114 18.84% 130 15.68%5% < Excess ≤ 10% 211 14.71% 96 15.87% 115 13.87%
10% < Excess ≤ 15% 217 15.13% 91 15.04% 126 15.20%15% < Excess ≤ 20% 184 12.83% 82 13.55% 102 12.30%20% < Excess ≤ 25% 129 9.00% 43 7.11% 86 10.37%
Excess > 25% 449 31.31% 179 29.59% 270 32.57%
Figure 4
Comparison of Distributions of L for the Last Pre-IPO Transactions and the Earlier Pre-IPO Transactions
The figures plot the actual distribution and the four fitted distributions estimated for the restriction periods in the private equity transaction sample. The
four fitted distributions are the exponential distribution, log-normal distribution, gamma distribution, and Weibull distribution. The full sample contains
5,391 transactions between June 1985 and April 2017. The sample of last pre-IPO private equity transactions contains 1,632 transactions between
December 1985 and April 2017. The sample of earlier pre-IPO private equity transactions contains 3,759 earlier pre-IPO private transactions between
June 1985 and January 2017.
Last Pre-IPO Equity Transactions Earlier Pre-IPO Transactions
Table 6
Comparison of Distributions of L for the Last Pre-IPO Transactions and the Earlier Pre-IPO Transactions
The table compares four candidate probability distributions for the actual distribution of the restriction periods L for the sample of private equity
transactions and provides the parameter estimates for each distribution. The four candidate distributions are the exponential distribution, log-normal
distribution, gamma distribution, and Weibull distribution. The full sample contains 5,391 transactions between June 1985 and April 2017. Panel A fits
the candidate distributions to the 1,632 last pre-IPO private equity transactions, which took place between December 1985 and April 2017. Panel B fits
the candidate distributions to the 3,759 earlier pre-IPO private equity transactions, which took place between June 1985 and January 2017. The
parameters were estimated using the smallest 75% of the sample. The tables provide the results of the sum of absolute differences and the sum of
squared differences between fitted values and empirical values of the largest 25% of the sample.
Panel A. Last Pre-IPO Private Equity Transactions
Exponential Log-Normal Gamma Weibull
Shape Parameter/Mu 0.68 -0.90 1.10 0.99
Scale Parameter/Sigma -- 1.06 0.62 0.68
Sum of Absolute Differences 362.27 361.45 419.83 431.00
Sum of Squared Differences 624.19 561.27 830.10 888.85
Panel B. Earlier Pre-IPO Private Equity Transactions
Exponential Log-Normal Gamma Weibull
Shape Parameter/Mu 2.34 0.49 1.41 1.17
Scale Parameter/Sigma -- 0.91 1.73 2.58
Sum of Absolute Differences 2364.14 2862.21 3018.22 3017.04
Sum of Squared Differences 6271.75 9404.96 10877.21 11045.42
Table 7
Distribution of Tau and Sample by Industry and Length of Restriction Period
Panel A provides the distribution of the median tau for the sample firms by industry and length of restriction period. Panel B provides a sample
breakdown of the number of private equity transactions by industry and length of restriction period. The sample contains 5391 private equity
transactions between June 1985 and April 2017.
Panel A. Median Tau by Industry and Length of Restriction Period
Median Tau
Length of Restriction Period
1 Non-
Durables
2 Durables
3 Manufact-
uring
4 Energy
5 Chemicals
6 Business Equip.
7 Telcomm.
8 Utilities
9 Retail
Wholesale
10 Healthcare
11 Finance
12 Other
0 < L <= 0.1 . 3.80 3.98 0.31 8.16 3.80 7.06 31.55 5.35 4.19 2.00 2.95
0.1 < L <= 0.25 1.80 1.64 1.74 9.80 3.30 1.71 1.61 . 1.69 2.12 1.11 1.54
0.25 < L <= 0.5 1.44 1.41 0.97 1.83 0.94 1.04 1.53 -0.04 0.99 1.40 0.50 1.45
0.5 < L <= 0.75 2.07 1.28 1.06 1.33 0.46 0.93 1.53 1.15 0.59 1.14 0.43 0.81
0.75 < L <= 1 0.42 0.63 0.78 0.87 1.11 0.87 1.56 . 0.45 0.88 0.51 0.62
1 < L <= 1.5 0.75 0.67 0.51 0.20 0.87 0.67 0.97 0.44 0.49 0.78 0.28 0.53
1.5 < L <= 2 0.31 0.46 0.54 0.23 0.96 0.41 0.33 0.45 0.33 0.63 0.15 0.49
2 < L <= 2.5 0.30 0.36 0.47 0.45 0.82 0.46 0.47 0.32 0.20 0.58 0.16 0.32
2.5 < L <= 3 0.12 0.28 0.23 0.58 0.49 0.36 0.51 0.25 0.17 0.51 0.08 0.26
3 < L <= 4 0.30 0.43 0.23 0.21 0.42 0.34 0.47 0.29 0.15 0.36 0.06 0.21
4 < L <= 5 -0.07 0.21 0.46 0.06 0.92 0.29 0.31 . 0.14 0.27 0.05 0.12
5 < L <= 6 -0.04 -0.05 0.10 . 0.40 0.25 0.57 . 0.07 0.17 0.07 0.24
6 < L <= 7 0.07 0.08 0.11 0.00 . 0.26 0.16 . 0.09 0.17 0.07 0.16
7 < L <= 8 -0.13 . -0.06 0.02 0.10 0.19 0.47 . 0.08 0.22 0.10 0.12
8 < L <= 9 -0.07 -0.09 0.01 . 0.34 0.17 0.17 . -0.06 0.17 0.10 0.27
9 < L <= 10 -0.12 . 0.00 . 0.18 0.13 0.07 . 0.00 0.13 0.04 -0.03
10 < L <= 12 . . 0.01 . . -0.01 . . 0.00 0.05 0.05 0.09
Panel B. Number of Private Equity Transactions by Industry and Length of Restriction Period
Number of Transactions
Length of Restriction Period
1 Non-
Durables
2 Durables
3 Manufact-
uring
4 Energy
5 Chemicals
6 Business Equip.
7 Telcomm.
8 Utilities
9 Retail
Wholesale
10 Healthcare
11 Finance
12 Other Total
0 < L <= 0.1 0 1 1 1 2 75 4 1 11 34 11 14 155
0.1 < L <= 0.25 5 4 10 1 5 145 16 0 23 70 15 50 344
0.25 < L <= 0.5 9 9 23 6 11 276 22 1 54 173 36 102 722
0.5 < L <= 0.75 10 8 25 6 13 246 21 2 61 176 40 82 690
0.75 < L <= 1 10 10 21 8 10 255 12 1 55 187 43 84 696
1 < L <= 1.5 10 12 24 8 12 292 19 1 72 193 51 107 801
1.5 < L <= 2 4 4 11 5 5 61 6 2 28 65 23 31 245
2 < L <= 2.5 5 5 9 6 5 125 10 2 34 96 27 32 356
2.5 < L <= 3 3 3 8 2 3 75 4 0 20 57 11 21 207
3 < L <= 4 9 4 11 3 4 129 7 1 33 104 33 37 375
4 < L <= 5 3 2 5 1 1 84 5 0 19 81 21 27 249
5 < L <= 6 2 1 4 1 2 66 4 0 16 63 20 16 195
6 < L <= 7 2 2 3 1 1 40 6 0 9 52 15 10 141
7 < L <= 8 1 1 2 0 0 28 2 0 2 37 13 3 89
8 < L <= 9 2 0 2 0 1 21 2 0 2 30 7 6 73
9 < L <= 10 2 0 2 0 1 10 0 0 1 16 4 2 38
10 < L <= 12 0 0 1 0 0 3 0 0 1 6 3 1 15
Total 77 66 162 49 76 1931 140 11 441 1440 373 625 5391 Industry
Percentage 1.43% 1.22% 3.01% 0.91% 1.41% 35.82% 2.60% 0.20% 8.18% 26.71% 6.92% 11.59% 100.00%
Figure 5
Distribution of for Restriction Periods of Different Lengths
This figure plots the median value of the adjusted marketability discounts for restriction periods of different lengths L. is calculated by applying
equation (14) but is set equal to the Longstaff (1995) upper bound if it would otherwise exceed this upper bound. The sample consists of 5,391 private
equity transactions for which ≥ 0. The median value of is calculated for each L and plotted separately for the full sample of private equity
transactions, the 1,632 last pre-IPO private equity transactions, and the 3,759 earlier pre-IPO private equity transactions.
Table 8
Clustering of Private Equity Transactions by Issuing Firm
This table reports the clustering of private equity transactions based on the number of private
equity transactions for each firm in the sample. The sample of private equity transactions consists
of 5,391 private equity transactions between June 1985 and April 2017 for which ≥ 0.
No. of Private Equity Transactions
No. of Firms
1 378 2 381 3 287 4 205 5 132 6 73 7 64 8 49 9 27 10 19 11 13 12 1 13 0 14 1 15 2
Total Firms 1632
Table 9
Parameter Estimation for the DLOM Term Structure Model
This table reports the results of fitting the regression model Equation (17) − ln 1 − = + + + + ∆ + +⋯ + ,
which includes the continuously compounded marketability discount as the dependent variable, four independent variables measuring the effect of the
length of the restriction period, industry fixed effects control variables, and the IPO control variable. The standard errors in the table have been adjusted
for clustering by using the asymptotically consistent estimator described in Huber (1967); White (1980); Liang and Zeger (1986); and Diggle, Liang,
and Zeger (1994).
The continuously compounded discount is calculated as the difference between the adjusted IPO price (the IPO price discounted by the industry returns)
and the transaction price divided by the adjusted IPO price. Holding period is the difference between the IPO date and the private equity transaction te.
The continuously compounded discounts are winsorized using the Longstaff (1995) model price as the upper bound. Δ is calculated from equation (7).
The industry dummy variables are based on the Fama-French 12-industry classification system. The IPO control variable is the ratio of the number of
IPO offerings in the year the private equity transaction occurred to the average annual number of IPO offerings during the time period in which private
equity transactions in the sample took place,1985 to 2017.
The sample in the left three columns contains 5,391 private equity transactions betweem June 1985 and April 2017. The private equity transaction
Adjusted Discount is nonnegtive for every transaction. The sample in the middle three columns includes only the last private equity transactions prior
to the firm’s IPO and consists of 1,632 transactions. The sample in the right three columns includes all the earlier pre-IPO private equity transactions
by the sample firms and consists of 3,759 transactions.
All Private Equity Transactions Last Pre-IPO Private Equity Transactions Earlier Pre-IPO Private Equity Transactions
Variable Coefficient Standard Error t-Statistic Coefficient Standard Error t-Statistic Coefficient Standard Error t-StatisticIntercept -0.0072 0.0305 -0.24 0.0546 0.0222 2.47b 0.1426 0.0483 2.95a
L 0.4311 0.0536 8.04a 0.1198 0.1053 1.14 0.2862 0.0677 4.23a
-0.0801 0.0128 -6.28a -0.0433 0.0390 -1.11 -0.0442 0.0158 -2.80a
0.0034 0.0010 3.33a -0.0002 0.0049 -0.04 0.0012 0.0012 1.02 ∆ 1.6049 0.1524 10.53a 2.3684 0.4707 5.03a 1.5383 0.1557 9.88a
-0.2570 0.0654 -3.93a 0.0090 0.1195 0.07 -0.2875 0.0671 -4.28a
-0.1578 0.0715 -2.21b 0.1400 0.0748 1.87c -0.1935 0.0747 -2.59a
-0.0929 0.0437 -2.12b 0.1383 0.0817 1.69c -0.1123 0.0460 -2.44b
-0.1280 0.0439 -2.92a -0.0244 0.0832 -0.29 -0.1167 0.0474 -2.46b
0.1690 0.1015 1.67c -0.0588 0.0911 -0.65 0.1679 0.1066 1.58
0.0445 0.0379 1.18 0.0315 0.0796 0.40 0.0319 0.0399 0.80
0.1144 0.0821 1.39 0.3343 0.2062 1.62 0.0887 0.0875 1.01
-0.1220 0.0407 -3.00a 0.0778 0.1633 0.48 -0.1287 0.0448 -2.87a
-0.0900 0.0408 -2.21b -0.0032 0.0825 -0.04 -0.1024 0.0434 -2.36b
0.0222 0.0368 0.60 -0.0747 0.0798 -0.94 0.0180 0.0389 0.46
-0.0693 0.0380 -1.83c 0.0276 0.0822 0.34 -0.0825 0.0405 -2.04b
0.2368 0.0228 10.40a 0.1104 0.0155 7.12a 0.3301 0.0332 9.94a
N 5391 1632 3759
Adj R2 0.4733 0.3793 0.413
F 303.68a 63.29a 166.25a
a Significant at the .01 level. b Significant at the .05 level. c Significant at the .10 level.
Figure 6
Fitted Term Structures for Marketability Discounts and Term Premia
These figures show the term structures of fitted marketability discounts and term premia based on the estimated parameters of the DLOM term structure
models in Table 9 for different restriction periods (L) and volatilities of stock returns (σ). The term premia are calculated from the equation ,1 ∆ . The Adjusted Discount ≥ 0% for all private equity transactions, and Adjusted Discount is equal to the Longstaff (1995) upper bound
when Adjusted Discount would otherwise exceed this upper bound. The DLOM term structures and implied term premia are plotted for three
representative stock price volatilities, σ = 0.5, σ = 0.75, and σ = 1.0, which reflect the volatilities of these stocks in the 12 months after they go public.
Panel A plots the DLOM term structure for all transactions in the sample, and Panel B plots the associated implied term premia. Panel C plots the
DLOM term structure for the last pre-IPO private equity transactions, and Panel D plots the associated implied term premia. Panel E plots the DLOM
term structure for the earlier pre-IPO private equity transactions, and Panel F plots the associated implied term premia.
Panel A. Term Structure for All Pre-IPO Private Equity Transactionsa Panel B. Term Premia τ(Δ, L) for All Pre-IPO Private Equty Transactions
Panel C. Term Structure for Last Pre-IPO Private Equity Transactionsa Panel D. Term Premia τ(Δ, L) for Last Pre-IPO Private Equity Transactions
Panel E. Term Structure for Earlier Pre-IPO Private Equity Transactionsa Panel F. Term Premia τ(Δ, L) for Earlier Pre-IPO Private Equity Transactions
aThe Adjusted Discount ≥ 0% for all private equity transactions, and Adjusted Discount is equal to the Longstaff (1995) upper bound when Adjusted Discount would otherwise exceed this upper bound.
Table 10
Test for a Date of Issue Effect in the DLOM Term Structure Model
This table reports the results of testing for a date of issue effect in the sample of private equity transactions that
occurred between June 1985 and April 2017, which includes 33 years. None of the private equity transactions in
the sample took place in 1986, 1987, or 1988, and 2016 and 2017 (very small number of observations) are the
holdout years. The model has 28 annual time dummy variables, one for each year other than 2016 and 2017 in
which at least one private equity transaction in the sample occurred. For each private equity transaction, the
dummy variable for year j is 1 if the transaction closed in year j and the other dummies are zero for that
observation. The modified model is − 1 − = + + + + ∆ + + +⋯++ +⋯+ . ≥ 0 for all private equity transactions. The standard errors have been adjusted
for clustering by using the asymptotically consistent estimator described in Huber (1967); White (1980); Liang
and Zeger (1986); and Diggle, Liang, and Zeger (1994).
The sample in the left three columns contains 5,391 private equity transactions between June 1985 and April 2017.
The sample in the middle three columns includes 1,632 last pre-IPO private equity transaction between December
1985 and April 2017. The sample in the right three columns includes 3,759 earlier pre-IPO private equity
transactions, which took place between June 1985 and January 2017.
All Transactions All Last Transactions All Earlier Transactions Variable Coefficient Error t-Statistics Coefficient Error t-Statistics Coefficient Error t-Statistics Intercept -0.0355 0.1236 -0.29 0.0984 0.1173 0.84 -0.0720 0.2102 -0.34
L 0.4345 0.0537 8.09a 0.0694 0.0944 0.74 0.3248 0.0690 4.71a
-0.0744 0.0130 -5.73a -0.0899 0.0309 -2.90b -0.0466 0.0163 -2.85b
0.0028 0.0011 2.65b 0.0067 0.0030 2.28b 0.0011 0.0012 0.86 ∆ 1.5668 0.1516 10.34a 2.8113 0.4576 6.14a 1.4825 0.1543 9.61a
-0.2486 0.0640 -3.89a 0.0613 0.1192 0.51 -0.2789 0.0651 -4.29a
-0.1641 0.0699 -2.35b 0.2111 0.0680 3.10b -0.1949 0.0737 -2.65b
-0.0993 0.0459 -2.16b 0.1884 0.0700 2.69b -0.1228 0.0482 -2.55b
-0.1321 0.0432 -3.05b 0.0622 0.0757 0.82 -0.1254 0.0472 -2.66b
0.1849 0.1037 1.78c 0.0199 0.0879 0.23 0.1859 0.1086 1.71c
0.0465 0.0379 1.23 0.0481 0.0743 0.65 0.0352 0.0399 0.88 0.1098 0.0810 1.35 0.3698 0.2076 1.78c 0.0851 0.0857 0.99 -0.1145 0.0474 -2.42b 0.1450 0.1529 0.95 -0.1155 0.0585 -1.97b
-0.0821 0.0409 -2.01b 0.0275 0.0863 0.32 -0.0927 0.0433 -2.14b
0.0321 0.0370 0.87 0.0276 0.0685 0.40 0.0271 0.0391 0.69 -0.0665 0.0380 -1.75c 0.0797 0.0566 1.41 -0.0770 0.0404 -1.91c
-0.1738 0.1466 -1.19 -0.0091 0.1730 -0.05 -0.3482 0.1751 -1.99b
0.0000 . . 0.0000 . . 0.0000 . . 0.0000 . . 0.0000 . . 0.0000 . . 0.0000 . . 0.0000 . . 0.0000 . . 0.9795 0.4637 2.11b 0.0000 . . 1.4852 0.4608 3.22b
-3.7272 0.2633 -14.15a -5.3368 1.0370 -5.15a 0.0000 . . -0.0639 0.2826 -0.23 0.6913 0.5342 1.29 0.0000 . . 0.0678 0.6416 0.11 0.0000 . . -0.0659 0.7179 -0.09 -0.2706 0.5789 -0.47 0.1721 0.8373 0.21 -0.5363 0.7761 -0.69 -0.0567 0.4257 -0.13 0.4127 0.6038 0.68 -0.2724 0.5608 -0.49 0.0867 0.4514 0.19 0.0592 0.5018 0.12 -0.0457 0.6245 -0.07 -0.1230 0.7071 -0.17 0.1555 0.7954 0.20 -0.4643 1.0043 -0.46 0.1004 0.4667 0.22 0.1733 0.5276 0.33 -0.0378 0.6504 -0.06 0.5419 0.2611 2.08b 0.3701 0.3008 1.23 0.5226 0.3447 1.52 0.0565 0.4621 0.12 0.1889 0.5257 0.36 -0.1496 0.6441 -0.23 -0.1307 0.3439 -0.38 0.1533 0.3921 0.39 -0.4412 0.4767 -0.93 -0.1182 0.0910 -1.30 0.0498 0.0735 0.68 -0.0801 0.1297 -0.62 -0.0107 0.0872 -0.12 0.0076 0.0970 0.08 0.1560 0.1278 1.22 -0.0440 0.0739 -0.60 -0.0451 0.0543 -0.83 0.0811 0.1226 0.66 0.0306 0.1154 0.26 -0.0175 0.1302 -0.13 0.0955 0.1318 0.72 0.1059 0.1012 1.05 -0.0211 0.1113 -0.19 0.2030 0.1061 1.91c
0.1784 0.0986 1.81c 0.0003 0.1053 0.00 0.2703 0.0989 2.73b
0.0340 0.1007 0.34 0.0413 0.1065 0.39 0.0938 0.1051 0.89 -0.3196 0.1209 -2.64b -0.2350 0.1343 -1.75c -0.1361 0.1878 -0.72 -0.1346 0.0946 -1.42 -0.0394 0.1030 -0.38 -0.0041 0.1505 -0.03 0.0502 0.0595 0.84 -0.0280 0.0412 -0.68 0.1583 0.0677 2.34b
0.0636 0.0553 1.15 -0.0280 0.0518 -0.54 0.1775 0.0792 2.24b
-0.0140 0.0508 -0.28 -0.0330 0.0549 -0.60 0.0589 0.0525 1.12 0.0698 0.0929 0.75 0.0039 0.1053 0.04 0.0924 0.0970 0.95 0.0606 0.1610 0.38 0.0484 0.1870 0.26 0.0267 0.2010 0.13 0.0328 0.0613 0.53 0.0049 0.0622 0.08 0.0883 0.0738 1.20 0.1949 0.2860 0.68 -0.0140 0.3160 -0.04 0.4498 0.4201 1.07
N 5391 1632 3759 adjR2 0.4936 0.4232 0.4324
F 120.55 29.71 69.38
a Significant at the .01 level. b Significant at the .05 level. c Significant at the .10 level.
Table 11Tabulated Values for D̅ Given by Equation (28)
Panel A: τ = 0q = 0%
σ vλ = 12.00L = 0.083
λ = 6.00L = 0.167
λ = 4.00L = 0.250
λ = 2.00L = 0.500
λ = 1.00L = 1.000
λ = 0.67L = 1.500
λ = 0.50L = 2.000
λ = 0.33L = 3.000
λ = 0.25L = 4.000
λ = 0.20L = 5.000
λ = 0.10L = 10.000
10% 6% 0.19% 0.39% 0.58% 1.15% 2.27% 3.37% 4.45% 6.53% 8.52% 10.43% 18.88%20% 12% 0.39% 0.78% 1.16% 2.29% 4.49% 6.58% 8.59% 12.35% 15.82% 19.02% 31.96%30% 17% 0.59% 1.17% 1.74% 3.43% 6.63% 9.62% 12.43% 17.55% 22.11% 26.19% 41.50%40% 23% 0.79% 1.56% 2.32% 4.54% 8.68% 12.48% 15.98% 22.19% 27.55% 32.22% 48.74%50% 28% 0.98% 1.95% 2.89% 5.63% 10.65% 15.17% 19.25% 26.34% 32.29% 37.35% 54.38%60% 34% 1.18% 2.33% 3.46% 6.68% 12.52% 17.68% 22.26% 30.05% 36.42% 41.72% 58.88%70% 39% 1.37% 2.70% 4.00% 7.70% 14.30% 20.01% 25.02% 33.35% 40.02% 45.48% 62.52%80% 44% 1.56% 3.07% 4.53% 8.67% 15.96% 22.17% 27.53% 36.30% 43.17% 48.71% 65.51%90% 48% 1.74% 3.42% 5.04% 9.60% 17.51% 24.16% 29.81% 38.91% 45.93% 51.50% 67.98%
100% 53% 1.91% 3.75% 5.52% 10.47% 18.95% 25.97% 31.87% 41.23% 48.33% 53.90% 70.05%
q = 3%
σ vλ = 12.00L = 0.083
λ = 6.00L = 0.167
λ = 4.00L = 0.250
λ = 2.00L = 0.500
λ = 1.00L = 1.000
λ = 0.67L = 1.500
λ = 0.50L = 2.000
λ = 0.33L = 3.000
λ = 0.25L = 4.000
λ = 0.20L = 5.000
λ = 0.10L = 10.000
10% 6% 0.19% 0.37% 0.56% 1.12% 2.21% 3.28% 4.32% 6.35% 8.29% 10.15% 18.42%20% 12% 0.38% 0.75% 1.13% 2.23% 4.36% 6.40% 8.35% 12.02% 15.41% 18.55% 31.30%30% 17% 0.57% 1.13% 1.69% 3.32% 6.44% 9.35% 12.09% 17.11% 21.58% 25.59% 40.75%40% 23% 0.76% 1.51% 2.25% 4.40% 8.44% 12.14% 15.56% 21.66% 26.93% 31.54% 47.95%50% 28% 0.95% 1.89% 2.81% 5.46% 10.35% 14.76% 18.76% 25.73% 31.60% 36.60% 53.59%60% 34% 1.14% 2.26% 3.35% 6.48% 12.18% 17.22% 21.71% 29.37% 35.67% 40.94% 58.10%70% 39% 1.33% 2.62% 3.88% 7.47% 13.90% 19.50% 24.41% 32.63% 39.24% 44.67% 61.75%80% 44% 1.51% 2.97% 4.39% 8.41% 15.52% 21.61% 26.87% 35.54% 42.36% 47.88% 64.76%90% 48% 1.68% 3.31% 4.88% 9.31% 17.04% 23.55% 29.11% 38.12% 45.10% 50.66% 67.25%
100% 53% 1.85% 3.63% 5.35% 10.16% 18.44% 25.33% 31.14% 40.42% 47.49% 53.06% 69.33%
This table provides values for given by equation (13) for restriction periods varying between 0.083 and 10 years. Volatility is calculated from equation (6). The marketability term premium is 0 in Panel A and 0.05 in Panel B.
Table 11Tabulated Values for D̅ Given by Equation (28)
Panel B: τ = 0.05q = 0%
σ vλ = 12.00L = 0.083
λ = 6.00L = 0.167
λ = 4.00L = 0.250
λ = 2.00L = 0.500
λ = 1.00L = 1.000
λ = 0.67L = 1.500
λ = 0.50L = 2.000
λ = 0.33L = 3.000
λ = 0.25L = 4.000
λ = 0.20L = 5.000
λ = 0.10L = 10.000
10% 6% 0.61% 1.21% 1.80% 3.53% 6.83% 9.90% 12.78% 18.02% 22.67% 26.81% 42.29%20% 12% 0.80% 1.59% 2.37% 4.62% 8.84% 12.70% 16.24% 22.54% 27.95% 32.65% 49.23%30% 17% 1.00% 1.98% 2.94% 5.70% 10.79% 15.36% 19.48% 26.63% 32.61% 37.69% 54.74%40% 23% 1.19% 2.36% 3.50% 6.76% 12.67% 17.87% 22.49% 30.33% 36.72% 42.04% 59.20%50% 28% 1.39% 2.74% 4.06% 7.80% 14.47% 20.24% 25.29% 33.67% 40.36% 45.83% 62.85%60% 34% 1.58% 3.12% 4.61% 8.81% 16.19% 22.47% 27.87% 36.69% 43.59% 49.13% 65.89%70% 39% 1.77% 3.49% 5.14% 9.78% 17.82% 24.54% 30.25% 39.41% 46.44% 52.02% 68.44%80% 44% 1.96% 3.84% 5.66% 10.71% 19.35% 26.46% 32.43% 41.85% 48.97% 54.54% 70.58%90% 48% 2.14% 4.19% 6.15% 11.60% 20.78% 28.24% 34.41% 44.04% 51.20% 56.74% 72.40%
100% 53% 2.31% 4.52% 6.63% 12.43% 22.11% 29.86% 36.21% 45.99% 53.17% 58.66% 73.95%
q = 3%
σ vλ = 12.00L = 0.083
λ = 6.00L = 0.167
λ = 4.00L = 0.250
λ = 2.00L = 0.500
λ = 1.00L = 1.000
λ = 0.67L = 1.500
λ = 0.50L = 2.000
λ = 0.33L = 3.000
λ = 0.25L = 4.000
λ = 0.20L = 5.000
λ = 0.10L = 10.000
10% 6% 0.60% 1.20% 1.78% 3.50% 6.77% 9.82% 12.68% 17.88% 22.50% 26.63% 42.06%20% 12% 0.79% 1.57% 2.33% 4.56% 8.72% 12.54% 16.04% 22.28% 27.65% 32.33% 48.86%30% 17% 0.98% 1.94% 2.88% 5.61% 10.62% 15.12% 19.20% 26.27% 32.21% 37.26% 54.29%40% 23% 1.17% 2.31% 3.43% 6.64% 12.45% 17.57% 22.14% 29.89% 36.25% 41.54% 58.70%50% 28% 1.36% 2.68% 3.97% 7.64% 14.20% 19.89% 24.87% 33.17% 39.83% 45.28% 62.33%60% 34% 1.55% 3.05% 4.50% 8.62% 15.87% 22.06% 27.39% 36.14% 43.01% 48.54% 65.35%70% 39% 1.73% 3.40% 5.02% 9.56% 17.45% 24.08% 29.72% 38.81% 45.82% 51.39% 67.89%80% 44% 1.91% 3.75% 5.52% 10.46% 18.95% 25.96% 31.86% 41.22% 48.32% 53.89% 70.04%90% 48% 2.08% 4.08% 6.00% 11.32% 20.34% 27.70% 33.81% 43.38% 50.53% 56.08% 71.86%
100% 53% 2.25% 4.40% 6.46% 12.13% 21.64% 29.29% 35.57% 45.30% 52.48% 57.99% 73.41%
Note: The volatility v is calculated from equation (6).
10%
30%
50%
70%
90%
0%
10%
20%
30%
40%
50%
60%
70%
80%
0.10 0.20 0.25 0.33 0.50 0.67 1.00 2.00 4.00 6.00 12.00
σ
λ
Figure 7Sensitivity of D ̅ Given by Equation (13) to λ and σ When τ = 0.05
70%-80%
60%-70%
50%-60%
40%-50%
30%-40%
20%-30%
10%-20%
0%-10%
Range of
Note: Assumes q = 0.03 for illustrative purposes.
D ̅
Sensitivity of Given by Equation (28) to λ and σ When τ = 0.05
10%
30%
50%
70%
90%
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
10.00 9.00 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00
Δ
Figure 8Sensitivity of D ̅ Given by Equation (13) to L̅ and Δ When τ = 0
80%-90%
70%-80%
60%-70%
50%-60%
40%-50%
30%-40%
20%-30%
10%-20%
0%-10%
Range of
D ̅
1/
Sensitivity of Given by Equation (28) to and Δ When τ = 0