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: finnerty@fordham.edu

© 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

References

Abbott, Ashok, 2009, “Discount for Lack of Liquidity: Understanding and Interpreting Option

Models,” Business Valuation Review 28 (Fall), pp. 144-148.

Acharya, Viral V., and Lasse H. Pedersen, 2005, “Asset Pricing with Liquidity Risk,” Journal of

Financial Economics 77 (August), pp. 375–410.

Allen, Jeffrey W., and Gordon M. Phillips, 2000, “Corporate Equity Ownership, Strategic

Alliances, and Product Market Relationships.” Journal of Finance 55 (December), pp.

2791-2815.

Alli, Kasim L., and Donald J. Thompson II, 1991, “The Value of the Resale Limitation on

Restricted Stock: An Option Theory Approach.” Valuation 36 (March), pp. 22-34.

Alois, JD, 2017, “Report: There Are 214 Unicorns in the World. A Good Number of Them Are

FinTech,” Crowdfundinsider.com, September 23, available at

https://www.crowdfundinsider.com/2017/09/122268-report-there-are-214-unicorns-in-

the-world-a-good-number-of-them-are-fintech/.

Amihud, Yakov, 2002, “Illiquidity and Stock Returns: Cross-Section and Time-Series Effects,”

Journal of Financial Markets 5 (January), pp. 31–56.

Amihud, Yakov, and Haim Mendelson, 1986, “Asset Pricing and the Bid-Ask Spread.” Journal

of Financial Economics 17 (December), pp. 223-249.

Balasooriya, Uditha, and Tilak Abeysinghe, 1994, “Selecting between Gamma and Weibull

Distributions – An Approach Based on Predictions of Order Statistics,” Journal of

Applied Statistics 21 (Number 3), pp. 17-27.

Bali, Turan G., 2007, “A Generalized Extreme Value Approach to Financial Risk Measurement,”

Journal of Money, Credit and Banking 39 (October), pp. 1613-1649.

Bali, Turan G., and Panayiotis Theodossiou, 2007, “A Conditional-SGT-VaR Approach with

Alternative GARCH Models,” Annals of Operations Research 151 (April), pp. 241-267.

Barber, Brad M., and Terrance Odean, 2000, “Trading Is Hazardous to Your Wealth: The

Common Stock Investment Performance of Individual Investors,” Journal of Finance 55

(April), pp. 773-806.

Beaulieu, Norman C., Adnan A. Abu-Dayya, and Peter J. McLane, 1995, “Estimating the

Distribution of a Sum of Independent Lognormal Random Variables,” IEEE Transactions

on Communications 43 (December), pp. 2869-2873.

40

Bekaert, Geert, Campbell R. Harvey, and Christian Lundblad, 2007, “Liquidity and Expected

Returns: Lessons from Emerging Markets,” Review of Financial Studies 20 (November),

pp. 1783–1831.

Black, Fischer, and Myron Scholes, 1973, “The Pricing of Options and Corporate Liabilities,”

Journal of Political Economy 81 (May-June), pp. 637-659.

BlackRock Private Opportunities Fund III (Delaware), L.P., 2014, Form D, Notice of Exempt

Offering of Securities, available at

https://www.sec.gov/Archives/edgar/data/1629179/000162917915000001/xslFormDX01/

primary_doc.xml.

Brorsen, Les, David Brown, Jeff Grabow, Chris Holmes, and Jackie Kelley, 2017, “Looking

Behind the Declining Number of Public Companies,” Harvard Law School Forum on

Corporate Governance and Financial Regulation, May 18, available at

https://corpgov.law.harvard.edu/2017/05/18/looking-behind-the-declining-number-of-

public-companies/.

Chaffee, David B.H., III, 1993, “Option Pricing as a Proxy for Discount for Lack of

Marketability in Private Company Valuations – A Working Paper,” Business Valuation

Review 12 (December), pp. 182-188.

Chalmers, John M.R., and Gregory B. Kadlec, 1998, “An Empirical Examination of the

Amortized Spread,” Journal of Financial Economics 48 (May), pp. 159-188.

Chen, Long, David A. Lesmond, and Jason Wei, 2007, “Corporate Yield Spreads and Bond

Liquidity,” Journal of Finance 62 (February), pp. 119-149.

Chordia, Tarun, Richard Roll, and Avanidhar Subrahmanyam, 2000, “Commonality in

Liquidity,” Journal of Financial Economics 56 (April), pp. 3–28.

Cornell, Bradford, 1993, Corporate Valuation: Tools for Effective Appraisal and Decision

Making, New York: McGraw-Hill, chapter 8.

Cotter, John, 2004, “Downside Risk for European Equity Markets,” Applied Financial

Economics 14 (February), pp. 707-716.

Cox, John C., Jonathan Ingersoll, Jr., and Stephen A. Ross, 1981, “A Reexamination of

Traditional Hypotheses About the Term Structure of Interest Rates,” Journal of Finance

36 (September), pp. 769-799.

Culbertson, J.M., 1957, “The Term Structure of Interest Rates,” Quarterly Journal of Economics

71 (November), pp. 489-504.

41

Damodaran, Aswath, 2012, Investment Valuation: Tools and Techniques for Determining the

Value of Any Asset, 3rd ed., Hoboken, NJ: Wiley, chapter 24.

Desjardins, Jeff, 2017, “The 57 Startups that Became Unicorns in 2017,” Visualcapitalist.com,

December 29, available at http://www.visualcapitalist.com/57-startups-unicorns-in-2017/.

Engle, Robert F., 2009, “The Risk that Risk Will Change,” Journal of Investment Management 7

(Fourth Quarter), pp. 24-28.

Engle, Robert F., 2011, “Long-Term Skewness and Systemic Risk,” Journal of Financial

Econometrics 9 (Summer), pp. 437-468.

Erdogan, Begum, Rishi Kant, Allen Miller, and Kara Sprague, 2016, “Grow Fast or Die Slow:

Why Unicorns Are Staying Private,” McKinsey & Company, May, available at

https://www.mckinsey.com/industries/high-tech/our-insights/grow-fast-or-die-slow-why-

unicorns-are-staying-private.

Fabozzi, Frank J., 2016, Bond Markets, Analysis, and Strategies, 9th ed., Boston: Pearson, ch. 5.

Fama, Eugene F., 1984, “Term Premiums in Bond Returns,” Journal of Financial Economics 13

(December), pp. 529-546.

Fama, Eugene F., 1990, “Term-Structure Forecasts of Interest Rates, Inflation, and Real

Returns,” Journal of Monetary Economics 25 (January), pp. 59-76.

Fan, Frank Jian, Grant Fleming, and Geoffrey J. Warren, 2013, “The Alpha, Beta, and

Consistency of Private Equity Reported Returns,” Journal of Private Equity 16 (Fall), pp.

21-30.

Finnerty, John D., 2012, “An Average-Strike Put Option Model of the Marketability Discount,”

Journal of Derivatives 19 (Summer), pp. 53-69.

Finnerty, John D., 2013a, “The Impact of Stock Transfer Restrictions on the Private Placement

Discount,” Financial Management 42 (Fall), pp. 575-609.

Finnerty, John D., 2013b, “Using Put Option-based DLOM Models to Estimate Discounts for

Lack of Marketability,” Business Valuation Review 31 (Fall), pp. 165-170.

Gompers, Paul, and Josh Lerner, 1998, “Venture Capital Distributions: Short-run and Long-run

Reactions,” Journal of Finance 53 (December), pp. 2161-2183.

Goyenko, Ruslan, Avanidhar Subrahmanyam, and Andrey Ukhov, 2011, “The Term Structure of

Bond Market Liquidity and Its Implications for Expected Bond Returns,” Journal of

Financial and Quantitative Analysis 46 (February), pp. 111-139.

42

Graham, John R., and Campbell R. Harvey, 1996, “Market Timing Ability and Volatility Implied

in Investment Newsletters’ Asset Allocation Recommendations,” Journal of Financial

Economics 42 (November), pp. 397-421.

Guidolin, Massimo, and Allan Timmermann, 2006, “Term Structure of Risk under Alternative

Econometric Specifications,” Journal of Econometrics 131 (March-April), pp. 285-308.

Harris, Robert S., Tim Jenkinson, and Steven N. Kaplan, 2014, “Private Equity Performance:

What Do We Know?” Journal of Finance 69 (October), pp. 1851-1882.

Harvey, Campbell R., and Akhtar Siddique, 2000, “Conditional Skewness in Asset Pricing

Tests,” Journal of Finance 55 (June), pp. 1263-1295.

Hasanhodzic, Jasmina, and Andrew W. Lo, 2007, “Can Hedge-Fund Returns Be Replicated?

The Linear Case,” Journal of Investment Management 5 (Second Quarter), pp. 5-45.

Hertzel, Michael, Michael Lemmon, James S. Linck, and Lynn Rees, 2002, “Long-run

Performance Following Private Placements of Equity,” Journal of Finance 57

(December), pp. 2595-2617.

Hertzel, Michael, and Richard L. Smith, 1993, “Market Discounts and Shareholder Gains for

Placing Equity Privately,” Journal of Finance 48 (June), pp. 459-485.

Hicks, J. William, 1998, Resales of Restricted Securities, New York: West Group.

Hicks, John R., 1946, Value and Capital, 2nd ed., London: Oxford University Press, pp. 141-145.

Hribar, P., and N.T. Jenkins, 2004, “The Effect of Accounting Restatements on Earnings

Revisions and the Estimated Cost of Capital,” Review of Accounting Studies 9 (June),pp.

337-356.

Hull, John C., 2009, Options, Futures, and Other Derivatives, 7th ed., Upper Saddle River, NJ:

Pearson Education, pp. 566-567.

Kahl, Matthias, Jun Liu, and Francis A. Longstaff, 2003, “Paper Millionaires: How Valuable Is

Stock to a Stockholder Who Is Restricted from Selling It?” Journal of Financial

Economics 67 (March), pp. 385-410.

Karpoff, Jonathan M., D. Scott Lee, Gerald S. Martin, 2008, “The Cost to Firms of Cooking the

Books,” Journal of Financial and Quantitative Analysis 43 (September), pp. 581-611.

Kempf, Alexander, Olaf Korn, and Marliese Uhrig-Homburg, 2012, “The Term Structure of

Illiquidity Premia,” Journal of Banking & Finance 36 (May), pp. 1381-1391.

Knyazeva, Anzhela, 2016, “Regulation A+: What Do We Know So Far?” Securities and

Exchange Commission, white paper (November), available at

43

https://www.sec.gov/dera/staff-papers/white-papers/18nov16_knyazeva_regulation-a-

plus-what-do-we-know-so-far.html.

Kosowski, Robert, Narayan Y. Naik, and Melvyn Teo, 2007, “Do Hedge Funds Deliver Alpha?

A Bayesian and Bootstrap Analysis,” Journal of Financial Economics 84 (April), pp.

229-264.

Koziol, Christian, and Peter Sauerbier, 2007, “Valuation of Bond Illiquidity: An Option-

Theoretical Approach,” Journal of Fixed Income 16 (Spring), pp. 81-107.

Kravet, T., and T. Shevlin, 2010, “Accounting Restatements and Information Risk,” Review of

Accounting Studies 15 (June), pp. 264-294.

Liang, Kung-Yee, and Scott L. Zeger, 1986, “Longitudinal Data Analysis Using Generalized

Linear Models,” Biometrika 73 (April), pp. 13-22.

Liu, Weimin, 2006, “A Liquidity-Augmented Capital Asset Pricing Model,” Journal of

Financial Economics 82 (December), pp. 631–671.

Longstaff, Francis A., 1995, “How Much Can Marketability Affect Security Values?” Journal of

Finance 50 (December), pp. 1767-1774.

Longstaff, Francis A., 2001, “Optimal Portfolio Choice and the Valuation of Illiquid Assets,”

Review of Financial Studies 14 (Summer), pp. 407-431.

Lutz, Frederich A., 1940-41, “The Structure of Interest Rates,” Quarterly Journal of Economics

55 (November), pp. 36-63.

Margrabe, William, 1978, “The Value of an Option to Exchange One Asset for Another,”

Journal of Finance 33 (March), pp. 177-186.

McKinsey & Company, 2018, The Rise and Rise of Private Markets: McKinsey Global Private

Markets Review 2018, Exhibit 11, p. 17, available at

https://www.mckinsey.com/~/media/McKinsey/Industries/Private%20Equity%20and%20

Principal%20Investors/Our%20Insights/The%20rise%20and%20rise%20of%20private%

20equity/The-rise-and-rise-of-private-markets-McKinsey-Global-Private-Markets-

Review-2018.ashx.

Melbinger, Michael S., 2014, SEC Is Tightening the Screws on Pre-IPO Stock Awards, Winston

& Strawn (February 20), available at https://www.winston.com/en/executive-

compensation-blog/sec-is-tightening-the-screws-on-pre-ipo-stock-awards.html.

Merton, Robert C., 1973, “Theory of Rational Option Pricing,” Bell Journal of Economics and

Management Science 4 (Spring), pp. 141-183.

44

Metrick, Andrew, 2007, Venture Capital and the Finance of Innovation, Hoboken, NJ: Wiley.

Metrick, Andrew, and Ayako Yasuda, 2010, “The Economics of Private Equity Funds,” Review

of Financial Studies 23 (June), pp. 2303-2341.

Modigliani, Franco, and Richard Sutch, 1966, “Innovations in Interest Rate Policy,” American

Economic Review 56 (March), pp. 178-197.

Murphy, Deborah L., Ronald E. Shrieves, and Samuel L. Tibbs, 2009, “Understanding the

Penalties Associated with Corporate Misconduct: An Empirical Examination of Earnings

and Risk,” Journal of Financial and Quantitative Analysis 44 (February), pp. 55-83.

Pastor, Lubos, and Robert F. Stambaugh, 2003, “Liquidity Risk and Expected Stock Returns,”

Journal of Political Economy 111 (June), pp. 642–685.

Phalippou, Ludovic, and Oliver Gottschalg, 2009, “The Performance of Private Equity Funds,”

Review of Financial Studies 22 (April), pp. 1747-1776.

Pratt, Shannon P., 2001, The Market Approach to Valuing Businesses, New York: Wiley, chapter

12.

Pratt, Shannon P., and Roger J. Grabowski, 2014, Cost of Capital: Applications and Examples,

5th ed., Hoboken, NJ: Wiley, pp. 611-629.

Ritchken, Peter, L. Sankarasubramanian, and Anand M. Vijh, 1993, “The Valuation of Path

Dependent Contracts on the Average,” Management Science 39 (October), pp. 1202-

1213.

Sadka, Ronnie, 2006, “Momentum and Post-Earnings-Announcement Drift Anomalies: The Role

of Liquidity Risk,” Journal of Financial Economics 80 (May), pp. 309–349.

Securities and Exchange Commission, 1971, Institutional Investor Study Report of the Securities

and Exchange Commission (U.S. Government Printing Office, Washington, D.C.).

Securities and Exchange Commission, 1997, Revision of Holding Period Requirements in Rules

144 and 145, Release No. 33-7390 (February 20).

Securities and Exchange Commission, 2007, Revisions to Rules 144 and 145, Release No. 33-

8869 (December 6).

Securities and Exchange Commission, 2015, Amendments for Small and Additional Issues

Exemptions under the Securities Act (Regulation A) (March 25).

Securities and Exchange Commission, 2016, Rule 144: Selling Restricted and Control Securities.

Available at https://www.sec.gov/investor/pubs/rule144.htm. Last accessed August 15,

2016.

45

Silber, William, 1991, “Discounts on Restricted Stock: The Impact of Illiquidity on Stock

Prices,” Financial Analysts Journal 47 (July/August), pp. 60 - 64.

Simon, Matt, 2016, “RSUs: Essential Facts,” Charles Schwab, San Francisco, CA, available at

http://www.schwab.com/public/eac/resources/articles/rsu_facts.html, last accessed on

August 18, 2016.

Siswadi, ____, and C.P. Quesenberry, 1982, “Selecting Among Weibull, Lognormal and Gamma

Distributions Using Complete and Censored Samples,” Naval Research Logistic

Quarterly 29 (_____), pp. 557-569.

Thomson Reuters, 2018, “Fundraising on the Rise,” available at

http://fingfx.thomsonreuters.com/gfx/editorcharts/PRIVATEEQUITY-

FUNDRAISING/0H0010KE07M/index.html.

Valuation Advisors, 2017, Valuation Advisors Lack of Marketability Discount Study Frequently

Asked Questions (FAQs), Business Valuation Resources, Portland Oregon, available at

https://www.bvresources.com/products/faqs/valuation-advisors-lack-of-marketability-

study.

Wruck, Karen, 1989, “Equity Ownership Concentration and Firm Value: Evidence from Private

Equity Financings,” Journal of Financial Economics 23 (June), pp. 3 - 28.

Ze-To, Samuel Yau Man, 2008, “Value at Risk and Conditional Extreme Value Theory via

Markov Regime Switching Models,” Journal of Futures Markets 28 (December), pp.

155-181.

46

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