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Many firms have stockholders who face severe restrictions on their ability to sell their sharesand diversify the risk of their personal wealth. We study the costs of these liquidity restrictionson stockholders using a continuous-time portfolio choice framework. These restrictions havemajor effects on the optimal investment and consumption strategies because of the need tohedge the illiquid stock position and smooth consumption in anticipation of the eventual lapseof the restrictions. These results provide a number of important insights about the effects ofilliquidity in financial markets.

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Journal of Financial Economics 67 (2003) 385–410

Paper millionaires: howvaluable is stock to a

stockholder who is restricted from selling it?$

Matthias Kahl, Jun Liu, Francis A. Longstaff*

The Anderson School at UCLA, Los Angeles, CA 90095, USA

Received 24 October 2001; accepted 9 January 2002

Abstract

Many ?rms have stockholders who face severe restrictions on their ability to sell their shares

and diversify the risk of their personal wealth. We study the costs of these liquidity restrictions

on stockholders using a continuous-time portfolio choice framework. These restrictions have

major effects on the optimal investment and consumption strategies because of the need to

hedge the illiquid stock position and smooth consumption in anticipation of the eventual lapse

of the restrictions. These results provide a number of important insights about the effects of

illiquidity in ?nancial markets.

r 2002 Elsevier Science B.V. All rights reserved.

JEL classi?cation: G11

Keywords: Restricted stock; Valuation; Illiquidity; Lockup restrictions; Portfolio choice

1. Introduction

In recent years, the number of stockholders suffering huge losses during market

downturns while liquidity restrictions prohibited them from selling their shares has

$We are grateful for helpful discussions with David Aboody, Bradford Cornell, Darrell Daf?e, Mark

Garmaise, Robert Geske, Mark Grinblatt, Robert Merton, Lisa Meulbroek, and Richard Roll, and for the

comments of seminar participants at the University of California at Berkeley and California State

University at Fullerton. We are also grateful to Brent Longstaff and Eric Neis for research assistance. We

are particularly grateful for the comments and suggestions of the editor Bill Schwert and the referee John

Long. All errors are our responsibility.

*Corresponding author. Tel.: +1-310-825-2218; fax: +1-310-206-5455.

E-mail address: francis.longstaff@anderson.ucla.edu (F.A. Longstaff).

0304-405X/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 3 0 4 - 4 0 5 X ( 0 2 ) 0 0 2 5 8 - 1

386

M. Kahl et al. / Journal of Financial Economics 67 (2003) 385–410

skyrocketed.1 These types of restrictions are widespread, affecting entrepreneurs,

venture capitalists, private equity holders, corporate of?cers, managers, and many

others. For example, lockup restrictions are often imposed as part of the initial

public offering (IPO) process. More broadly, however, selling restrictions are usually

included in executive stock or stock-option based compensation contracts. In

addition, Rule 144 of the U.S. Securities and Exchange Commission (SEC) places

severe restrictions on the ability of most corporate insiders and af?liates to sell shares

in their ?rm. Because of these restrictions, some stockholders bear the costs of

holding an illiquid undiversi?ed portfolio for many years.

Although the bene?ts of liquidity restrictions in retaining key employees and

managers and in reducing agency con?icts are well understood, the costs imposed by

these restrictions have been largely unexplored. Accordingly, the goal of this paper is

to examine howselling or liquidity restrictions affect the welfare of stockholders.

Since these stockholders often have a substantial stake in their venture, we will refer

to them simply as entrepreneurs throughout the paper to make the intuition more

clear. To study the effects of liquidity restrictions, we model the optimal

consumption and portfolio choice problem of an entrepreneur who owns stock in

a ?rm, but is unable to sell this stock for a given period of time. In addition to this

restricted stock, the entrepreneur has liquid wealth which he can allocate between the

stock and bond markets. This feature is important since it allows the entrepreneur to

take a stock market position that offsets some of the risk of his illiquid stockholdings

and reduces the cost of the restrictions. Note that allowing the entrepreneur to invest

in other markets differentiates this paper from earlier work (primarily focusing on

executive stock options rather than restricted stock) in which agents are not allowed

to hedge their undiversi?ed positions in other markets (for example, see Hall and

Murphy, 2000b). This framework also allows us to study how the consumption level

(or lifestyle) of an entrepreneur is affected by liquidity restrictions. The welfare loss

due to the liquidity restrictions is calculated by comparing the maximal utility

achieved by the entrepreneur with that achievable if the stockholdings were fully

liquid.

The results indicate that the cost of liquidity restrictions can be surprisingly large.

For example, when stock is restricted for ?ve years and represents 50% of his wealth,

an entrepreneur would actually be better off if he could sell his restricted stock for

30% to 80% of its unrestricted market value. Furthermore, these costs can be

signi?cantly higher when nearly all of the entrepreneur’s wealth is tied up in

restricted shares and when the entrepreneur is not able to hedge his restricted shares

with offsetting stock market positions. These results contradict the widely held

practitioner viewthat restricted stock has only a minor cost to the recipient (see Wall

Street Journal, April 12, 2001, p. R1) and is a much more ef?cient form of

1 There are many examples of entrepreneurs, managers, and others with signi?cant stockholdings,

initially worth millions on paper, who lost most of their wealth without ever being allowed to sell any of

their stockholdings. See the recent articles on the effects of selling restrictions on inside stockholders in The

Wall Street Journal on March 23, 2001, April 12, 2001, April 25, 2001, and May 17, 2001, and in

Businessweek on April 17, 2000, and April 16, 2001.

M. Kahl et al. / Journal of Financial Economics 67 (2003) 385–410

387

compensation than executive stock options. These costs are roughly on the same order

of magnitude as those reported in studies of the cost of awarding executive stock

options such as Hall and Murphy (2000, 2002) and Meulbroek (2001).2 The results

also indicate that the cost of liquidity restrictions tends to be higher for agents who are

more risk averse. If the ability to innovate is not the same as the ability to bear risk,

however, this implies that liquidity restrictions may discourage risk averse (but

potentially highly productive) agents from entrepreneurial ventures. Furthermore,

these results suggest a possible basis for explaining the large valuation discounts

associated with private equity placements (see Wruck, 1989; Silber, 1992) and

contribute to the growing literature on the effects of illiquidity on security values.3

We ?nd that owning restricted shares can have a dramatic effect on the optimal

portfolio strategy for the liquid portion of the entrepreneur’s portfolio. Depending

on the ?rm’s correlation with the stock market, the entrepreneur can signi?cantly

increase or decrease his stock market holdings. This effect is largest when the

restricted shares represent an intermediate fraction of the entrepreneur’s wealth.

Interestingly, even when the correlation between the ?rm and the stock market is

zero, the entrepreneur can hold more of the stock market than he would in the

absence of liquidity restrictions. Intuitively, this is because taking additional stock

market risk helps smooth consumption variability caused by the temporary liquidity

restrictions. Finally, we show that even though the entrepreneur can borrow against

his illiquid position, he chooses to consume at a much lower rate than he would

without liquidity restrictions.

This analysis also has implications for several areas in corporate ?nance. The

model suggests that restricted stock can be worth substantially less to managers who

have a large fraction of their wealth invested in their company and face signi?cant

trading restrictions. This makes it a more costly corporate governance tool and less

effective at reducing agency costs. Although we focus on restricted stock, this

implication is consistent with recent results in the executive stock option literature.

For examples of this literature, see Lambert et al. (1991), Rubinstein (1995), Aboody

(1996), Carpenter (1998, 2000), Hall and Murphy (2000, 2002) Jin (2002), and

Meulbroek (2001). Moreover, the high cost of the lack of diversi?cation associated

with concentrated managerial equity ownership gives managers a strong incentive to

make diversifying acquisitions even if not in the interests of their shareholders (see

Amihud and Lev, 1981; Morck et al., 1990). Minimizing these costs can also provide

an important motivation for taking a ?rm public. Furthermore, the cost of these

restrictions helps explain the growing use of diversifying strategies such as zero-cost

collars and equity swaps documented by Bettis et al. (2001). Finally, La Porta et al.

(1999) show that in most countries, family ownership is the dominant ownership

2 Meulbroek (2001) uses an approach based on equalizing the Sharpe ratio between the market and the

illiquid stock to compute the discount. This preference-free approach is easily applied and can be viewed as

providing a lower bound on the discount for the illiquid stock. We are grateful to Robert Merton for this

insight.

3 For example, see Mayers (1972, 1973), Grossman and Laroque (1990), Amihud and Mendelson (1991),

Boudoukh and Whitelaw(1991, 1993), Kamara (1994), Longstaff (1995, 2001a, b), Vayanos (1998),

Huang (1998), and Brenner et al. (2001).

388

M. Kahl et al. / Journal of Financial Economics 67 (2003) 385–410

structure even for the largest publicly traded ?rms. Our model suggests that the costs

imposed on the family owners due to a lack of diversi?cation can be signi?cant. In an

insightful recent paper, Hong and Huang (2001) argue that investor relations efforts

by ?rms may be motivated by the goal of increasing trading volume and thereby

relaxing liquidity restrictions on corporate insiders.

The remainder of this paper is organized as follows. Section 2 describes a number

of ways in which different types of liquidity restrictions arise. Section 3 presents the

dynamic portfolio choice model. Section 4 examines the effects of liquidity

restrictions on welfare and optimal consumption and portfolio decisions. Section 5

discusses the implications of the results. Section 6 makes concluding remarks.

2. Liquidity restrictions

There are many reasons why a shareholder might not be able to sell his shares for

an extended period of time. In this section, we describe a number of common

situations in which shareholders are subject to these types of selling or liquidity

restrictions.

First, there are many situations in which selling restrictions are imposed by

contract, often to resolve moral hazard and adverse selection problems. One example

that has attracted substantial interest in the recent academic literature is that of stock

lockups in IPOs (see Brav and Gompers, 2000; Ofek and Richardson, 2000; Field

and Hanka, 2001). These lockups are not required by the SEC, but are part of the

contract between the issuer and the underwriter in the vast majority of IPOs. Most

lockups do not allowcompany insiders (of?cers, directors, employees, their friends

and family, and venture capitalists) to sell their shares for a period of 180 days. This

restriction could be lifted for individual trades by the underwriter in an early release,

but this typically affects only a small fraction of the stock held by insiders (Brav and

Gompers, 2000). The lockup period, however, can be longer than 180 days. For

example, Ibbotson and Ritter (1995) report that Morgan Stanley agreed to a two-

year lockup period in its IPO.

The literature offers several economic reasons for IPO lockup provisions. First,

they provide a signal of the value of the company, as suggested by Welch (1989), and

modeled by Brau et al. (2001). Lockups make it less likely that the shares are sold to

the public shortly before the release of negative information about the ?rm. Brav and

Gompers (2000) argue that the variation in the length of the lockup period and the

number of shares retained are systematically related to the uncertainty about the

?rm’s value. Similarly, Longstaff (1995) argues that IPO underpricing could be

partially due to the effects of lockup provisions. The lockup period gives key

employees and management an incentive to ensure good corporate performance, at

least until the insiders can sell their stock. Adding to the importance of trading

restrictions associated with insider share ownership in IPOs, it is often the case that

management and active investors (such as venture capitalists) are subject to

additional vesting agreements that go beyond the lockup period (Ofek and

Richardson, 2000).

M. Kahl et al. / Journal of Financial Economics 67 (2003) 385–410

389

Lockup or vesting periods play a similar role in managerial compensation

contracts. Many ?rms use restricted stock plans as part of the compensation

package. In these plans, managers receive a speci?ed number of shares in the ?rm,

but cannot sell these shares for a given period of time. Moreover, the shares are

forfeited if the executive leaves the ?rm before the restriction period is over. Kole

(1997) ?nds that 79 of 371 Fortune 500 ?rms in her sample have such restricted stock

plans. The average minimum holding period before any shares can be sold ranges

from 31 months for ?rms with a medium level of research and development to 74

months for ?rms with a high level of research and development. For more than a

quarter of the plans, the stock cannot be sold before retirement. The rationale for

these minimal holding periods is that it provides managers an incentive to take

actions that increase the long-term value of the ?rm, not just the short-term value.

Furthermore, this tool is used to increase managerial retention by creating

substantial switching costs since the restricted stock plan typically becomes void

upon the departure of the manager.

Minimal vesting periods also typically apply to executive stock option plans,

which require the executive to hold the options for a prespeci?ed time before

exercising them. In Kole (1997), the average minimum waiting period before any of

the options can be exercised is 13.5 months. The average waiting period before the

options can be exercised (taking into account that some fraction of the options can

be exercised after the minimum waiting period, but the remainder only after an

additional waiting period) is 23.6 months.

Another example where individuals obtain stock that cannot be sold for a certain

period is in a merger agreement where the target’s key employees and managers

obtain restricted stock in the combined company. Typically, such restricted stock

also comes with a lockup period during which it cannot be sold. The motivation is

similar to that for trading restrictions in executive compensation contracts. The

liquidity restrictions are intended to align the interests of the target’s key employees

and managers with the combined company and also give them an incentive to stay

with the combined company. This is of particular importance when the value of the

target company lies primarily in the human capital of its key employees, which is

likely the case in many start-ups. Finally, Bettis et al. (2000) ?nd that over 92% of

the ?rms in their sample impose limitations on trading by corporate insiders such as

blackout periods during which trading is not allowed.

In addition to contractual restrictions, however, corporate insiders often have

signi?cant liquidity restrictions imposed on them for legal reasons. These legal

restrictions can be even more stringent than the contractual restrictions. In some

cases, the legal restriction begins at the time the contractual restriction lapses and

signi?cantly extends the period of illiquidity. In general, a shareholder must satisfy

both the legal and contractual restrictions before selling his stock.

An important example of a legal restriction is SEC Rule 144 which limits the

amount of stock a corporate insider or af?liate can sell without registering the

transaction. Under the Securities Act of 1933, any person who sells a security to

another person must register that security with the SEC unless a statutory exemption

can be found for the transaction. Since the registration process can be prohibitively

390

M. Kahl et al. / Journal of Financial Economics 67 (2003) 385–410

expensive and time consuming for many security holders, SEC Rule 144 was

designed to enable the public sale of limited amounts of unregistered securities under

certain conditions. These conditions are intended to avoid situations where securities

are acquired by an underwriter with a view to distributing them to the public without

going through the formal registration process. Since individual investors who are not

professionals in the securities business can be ‘‘underwriters’’ under the meaning of

the Securities Act, Rule 144 provides a safe harbor by which sales of unregistered

securities will not be construed as sales by an underwriter. Osborne (1982) discusses

the economic rationale provided by the SEC for Rule 144. The cost of achieving this

safe harbor, however, is that the security holder must hold the securities for a

number of years, presumably to signal that the securities were not acquired primarily

to distribute them to the public without making the disclosures required by the

registration process.

The holding period required under Rule 144 depends on whether the security

holder is de?ned as an af?liate of the corporation. Af?liates include of?cers of the

corporation such as the CEO, president, senior of?cers, directors, spouses of of?cers,

relatives living in the same home as the of?cer, any persons in a position to exert

in?uence such as members of an of?cer’s family or close associates, and owners of

10% or more of the voting shares. Note that the de?nition of an af?liate is somewhat

broader than that of a corporate insider. Stock held by an af?liate is termed control

stock, and af?liates are often referred to as control persons.

Control stock can be acquired in a number of ways. For example, stock can be

acquired through compensation arrangements, exercise of stock options, payment

for professional services, venture capital arrangements, partnership distributions,

private placements, or even open market purchases. Rule 144 prohibits an af?liate

from selling restricted control stock for one year after the stock is acquired. After the

one-year period, however, there are a number of limitations placed on an af?liate

who wishes to sell control shares. Speci?cally, the af?liate is only allowed to sell an

amount of stock during any three-month period equal to the greater of one percent

of the total amount of shares outstanding or, if the ?rm is listed on a stock exchange

or quoted on Nasdaq, the average weekly reported trading volume in those shares

over the four weeks preceding the potential sale. Thus, for many smaller and less-

actively traded ?rms, it can take many years before a control shareholder is able to

completely liquidate a substantial equity stake in the ?rm. In addition to these

volume restrictions, current ?nancial information must be available regarding the

company whose securities are being sold. An af?liate must also ?le Form 144 with

the SEC for larger proposed sales. For a nonaf?liate, similar liquidity restrictions

apply to their sales of restricted or unregistered stock, but only during the ?rst two

years after the stock is acquired. There are many other examples of liquidity

restrictions imposed by lawsuch as the rules prohibiting insiders from trading during

periods surrounding earnings announcements.

Finally, since the effect of liquidity restrictions on insiders is to increase the

concentration of their holdings in the ?rm, this analysis is relevant for the issue of

concentrated ownership in general. Speci?cally, ownership of many ?rms is

concentrated in the hands of a small number of investors, who often have a large

M. Kahl et al. / Journal of Financial Economics 67 (2003) 385–410

391

fraction of their wealth invested in these stocks. This is true for private equity as

documented by Moskowitz and Vissing-Jorgensen (2001). Moreover, La Porta et al.

(1999) ?nd that in most countries, the most common form of ownership is family

ownership, even for the largest publicly traded ?rms.

3. The model

In this section, we model the portfolio choice of an agent where some portion of

his wealth is in shares that he cannot sell for a given period of time. An example of

this would be a corporate manager or entrepreneur who receives compensation in the

form of shares, but is prohibited from immediately selling those shares and

rebalancing his portfolio. To make the intuition as clear as possible, we use a simple

but realistic portfolio choice framework in which there are three types of assets: a

riskless bond, a stock index fund, and the restricted stock that the entrepreneur

holds. This partial equilibrium framework is a simple generalization of the standard

Merton (1969, 1971) continuous-time framework.

Let Bt denote the value at time t of a riskless bond or money market fund with

dynamics given by

dB ¼ rB dt;

ð1Þ

where r is the constant riskless interest rate. Let Mt denote the value of a risky asset

which can be viewed either as the stock market or a share in a stock index fund. The

dynamics of Mt are given by

dM ¼ ðr þ mÞMdt þ sMdZ1;

ð2Þ

where m is the market risk premium and s is the volatility of returns. Both m and s

are positive constants.

Although the entrepreneur is not allowed to trade his shares in the ?rm, we assume

that shares in the ?rm can be traded by others who are not subject to the restriction.

Let St denote the market value of a share of the ?rm’s stock. We assume that the

dynamics of St are given by

dS ¼ ðr þ lÞS dt þ nS dZ2;

ð3Þ

where l is the excess expected return for the ?rm and n is its volatility. Again, both l

and n are positive constants. The correlation between dZ1 and dZ2 is r; where

À1oro1: This allows for the important possibility that returns on the market and

on the ?rm’s stock are (potentially highly) correlated. To focus more directly on the

effects of liquidity restrictions, we make the simplifying assumption that the risk

premium l is given by the Capital Asset Pricing Model, implying that l ¼ mrn=s: An

immediate implication of this assumption is that an unconstrained entrepreneur

would want to hold the ?rm’s stock only to the extent that it appears in the stock

index. This implies that the unconstrained optimal portfolio weight for the ?rm’s

stock is zero.

392

M. Kahl et al. / Journal of Financial Economics 67 (2003) 385–410

The entrepreneur has an investment horizon of ToN; and at time zero, is given N

shares of stock in his ?rm. To capture the essence of the liquidity restriction, we

assume that the entrepreneur cannot change the number of shares of stock he holds

until time tpT: This is consistent with actual practice where shareholders are

typically not allowed to change their position either directly by selling stock, or

indirectly by trading options or entering into equity swaps or similar types of

derivative contracts. After time t; however, the entrepreneur can trade his shares in

the ?rm without restriction. While the number of shares N the entrepreneur holds

does not change until time t; the proportion of his wealth held in the form of illiquid

stock is stochastic. Let Xt ¼ NSt=Wt denote the portfolio weight for his illiquid

stockholdings, where Wt denotes his total wealth at time t: Since N is assumed to be

positive, Xt > 0 for all tot: Longstaff (2001a) studies the optimal portfolio choice

problem in a model where an agent can only trade limited amounts of a risky

security per unit time. In an independent paper, Henderson and Hobson (2002)

develop a model similar to ours in which an agent is unable to trade shares and offer

a series-based approximation for the optimal solution that is valid only for small

values of X : Our model differs from theirs, however, in that we allow for

intermediate consumption. In addition, we study the effects on consumption and

portfolio choice for general values of X :

The entrepreneur has preferences given by

Z

T

E

eÀksU ðCsÞ ds þ eÀkT UðWT Þ ;

ð4Þ

0

where

8 x1Àg

>

<

;

xX0;

U ðxÞ ¼

1 À g

>

: ÀN; xo0;

and where C denotes consumption, k is the rate of time preference, and g is the risk-

aversion parameter. The entrepreneur’s liquid wealth is given by ð1 À XtÞWt; which

he allocates between the riskless asset and the stock market. Let ft denote

the portfolio weight (as a percentage of his total wealth) for the stock market.

Since portfolio weights sum to one, the portfolio weight for the riskless asset is

1 À f À

t

Xt:

In this framework, we allow the entrepreneur to take unlimited short positions in

both the riskless asset and the stock market. A reviewof industry practice indicates

that some investment ?rms allowinvestors to borrowa limited amount of funds on

the security of their restricted stockholdings. In fact, a number of ?nancial ?rms

specialize in what is termed Rule 144 lending. In actuality, however, it is easily shown

that if the entrepreneur allows his liquid wealth to become negative, then there is a

possibility of reaching negative total wealth at T. Intuitively, this happens because

once the value of the liquid part of the entrepreneur’s portfolio becomes negative,

there is a non-zero probability that it will remain negative. Furthermore, there is

always a possibility that the illiquid stock will decline in value towards zero before

M. Kahl et al. / Journal of Financial Economics 67 (2003) 385–410

393

the liquidity restriction lapses. Thus, the entrepreneur’s total wealth at T could

become negative if X becomes greater than one. Since this implies an expected utility

of negative in?nity in this model, the entrepreneur never chooses an investment

strategy that allows liquid wealth to become negative. Thus, the entrepreneur never

borrows against illiquid stock, which implies that 0oX p1 for all tot:

Following Merton (1969, 1971), the entrepreneur’s wealth follows the dynamic

process

dW ¼ ððr þ mf þ lX ÞW À CÞ dt þ sfW dZ1 þ nXW dZ2:

ð5Þ

The entrepreneur’s dynamic decision problem is to choose his consumption Ct and

the portfolio weight for the stock market ft in a way that maximizes his expected

utility subject to the dynamic budget constraint in Eq. (5). Allowing the entrepreneur

to make optimal portfolio choices is essential in estimating the cost of liquidity

restrictions. In particular, simple certainty-equivalence approaches which do not

allowagents to select portfolios optimally can actually produce negative estimated

costs, implying the counterfactual result that restricted securities are worth more

than unrestricted securities. For example, Hall and Murphy (2002) provide a further

discussion. As in Merton, we de?ne the entrepreneur’s indirect utility of wealth

function to be

Z T

JðW ; X ; tÞ ¼ max E

eÀksU ðCsÞ ds þ eÀkT UðWT Þ :

ð6Þ

f;C

t

The Appendix shows that when tot; JðW ; X ; tÞ can be expressed in the form

W 1Àg

JðW ; X ; tÞ ¼

F ðX ; tÞ;

ð7Þ

1 À g

and that the ?rst-order conditions for the optimal consumption level Cn and the

optimal investment in the stock market fn are, respectively,

X

À1=g

Cn ¼ W ekt F À

FX

;

ð8Þ

1 À g

Àðm=s2Þ ð1 À gÞF þ ðgrn=s þ m=s2ÞXFX þ ðrn=sX Þ2 FXX

rn

fn ¼

À

X :

ð9Þ

Àgð1 À gÞF þ 2gXFX þ X 2FXX

s

The function F ðX ; tÞ satis?es a Hamilton-Jacobi-Bellman equation which is given in

the Appendix. Because expected utility equals ÀN if X exceeds one, FX ð1; tÞ ¼ N is

required to hold at the boundary X ¼ 1: Although F ðX ; tÞ cannot be solved in closed

form, standard ?nite difference or simulation techniques can be applied to provide

numerical solutions for JðW ; X ; tÞ and the values of Cn and fn: Because of the

nonlinearity of the Hamilton-Jacobi-Bellman equation, standard existence and

uniqueness results for the solution cannot be applied. Thus, it is important to

acknowledge that in providing numerical estimates of the solution, we are implicitly

assuming that a solution exists and abstracting from uniqueness concerns. When

tXt; the liquidity restriction is no longer binding, X equals its unconstrained optimal

394

M. Kahl et al. / Journal of Financial Economics 67 (2003) 385–410

value of zero, and JðW ; X ; tÞ takes on the functional form JðW ; tÞ since it no longer

depends on X (the solution for JðW ; tÞ is given at the end of this section).

From the ?rst-order conditions, the entrepreneur’s optimal consumption level Cn

and portfolio weight fn are nonlinear functions of X : Despite this, some intuition

about the optimal strategies can be obtained by considering the structure of the

problem. In particular, when the entrepreneur faces no liquidity restrictions, utility is

maximized at every instant by solving a local mean-variance optimization problem.

In contrast, when the entrepreneur faces liquidity restrictions, the decision problem

can be viewed as a blend of a buy-and-hold problem with a standard problem of

continuous rebalancing, which means that the entrepreneur must now also consider

global portfolio changes.

Another way of seeing this is by noting from Eq. (5) that the dynamics of the

entrepreneur’s wealth are completely determined by the values of f; X ; and C: When

the entrepreneur faces no liquidity restrictions, the entrepreneur is free to choose any

values of f; X ; and C; which gives full control over the distribution of wealth.

Because one can optimize choices of f; X ; and C individually, the optimal values of

these controls have the simple functional forms obtained by Merton (1969). When

there are trading restrictions, however, the initial value of X is given exogenously

and the entrepreneur can only choose the values of f and C: In this case, f and C

nowplay dual roles in maximizing the entrepreneur’s utility. Speci?cally, f and C

affect the dynamics of wealth directly as before. However, the values of f and C

affect the behavior of X over time, which has an indirect effect on the dynamics of

wealth. For example, choosing a lower level of current consumption tends to reduce

future values of X : When there are liquidity restrictions, both the direct and indirect

effects of f and C on the distribution of wealth must be considered in maximizing

the entrepreneur’s utility. Not surprisingly, this makes the optimal values of f and C

depend on X in very subtle and complex ways.

Despite this complexity, however, several comparative statics results can be given.

For example, as X -0; the optimal portfolio weight fn converges to the constant

portfolio weight m=gs2 given in Merton (1969). As X -1; both fn and Cn converge

to zero. The intuition for this is that if the entrepreneur’s liquid wealth were to reach

zero, the entrepreneur would need to prevent liquid wealth from becoming negative.

Thus, the entrepreneur would avoid any further market risk in the liquid portfolio by

placing zero weight in the stock market. Furthermore, the entrepreneur would

forego consumption rather than borrowing against illiquid wealth and creating a

negative liquid wealth position. In actuality, the entrepreneur’s optimal consumption

and investment strategies serve to insure that liquid wealth remains positive. By

guaranteeing that liquid wealth is always nonnegative, the entrepreneur’s optimal

consumption and portfolio strategies also insure that total wealth is always

nonnegative. Dybvig and Huang (1988) showthat requiring w

ealth to be

nonnegative eliminates unrealistic strategies such as the doubling strategy discussed

by Harrison and Kreps (1979).

In the special case where m ¼ grsn; fn reduces to rnð1 À X Þ=s; which implies that

the optimal portfolio strategy is a simple linear function of X : Finally, note that the

second term on the right hand side of Eq. (9) represents minus the amount of the

Document Outline

Paper millionaires: howvaluable is stock to a

stockholder who is restricted from selling it?$

Matthias Kahl, Jun Liu, Francis A. Longstaff*

The Anderson School at UCLA, Los Angeles, CA 90095, USA

Received 24 October 2001; accepted 9 January 2002

Abstract

Many ?rms have stockholders who face severe restrictions on their ability to sell their shares

and diversify the risk of their personal wealth. We study the costs of these liquidity restrictions

on stockholders using a continuous-time portfolio choice framework. These restrictions have

major effects on the optimal investment and consumption strategies because of the need to

hedge the illiquid stock position and smooth consumption in anticipation of the eventual lapse

of the restrictions. These results provide a number of important insights about the effects of

illiquidity in ?nancial markets.

r 2002 Elsevier Science B.V. All rights reserved.

JEL classi?cation: G11

Keywords: Restricted stock; Valuation; Illiquidity; Lockup restrictions; Portfolio choice

1. Introduction

In recent years, the number of stockholders suffering huge losses during market

downturns while liquidity restrictions prohibited them from selling their shares has

$We are grateful for helpful discussions with David Aboody, Bradford Cornell, Darrell Daf?e, Mark

Garmaise, Robert Geske, Mark Grinblatt, Robert Merton, Lisa Meulbroek, and Richard Roll, and for the

comments of seminar participants at the University of California at Berkeley and California State

University at Fullerton. We are also grateful to Brent Longstaff and Eric Neis for research assistance. We

are particularly grateful for the comments and suggestions of the editor Bill Schwert and the referee John

Long. All errors are our responsibility.

*Corresponding author. Tel.: +1-310-825-2218; fax: +1-310-206-5455.

E-mail address: francis.longstaff@anderson.ucla.edu (F.A. Longstaff).

0304-405X/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 3 0 4 - 4 0 5 X ( 0 2 ) 0 0 2 5 8 - 1

386

M. Kahl et al. / Journal of Financial Economics 67 (2003) 385–410

skyrocketed.1 These types of restrictions are widespread, affecting entrepreneurs,

venture capitalists, private equity holders, corporate of?cers, managers, and many

others. For example, lockup restrictions are often imposed as part of the initial

public offering (IPO) process. More broadly, however, selling restrictions are usually

included in executive stock or stock-option based compensation contracts. In

addition, Rule 144 of the U.S. Securities and Exchange Commission (SEC) places

severe restrictions on the ability of most corporate insiders and af?liates to sell shares

in their ?rm. Because of these restrictions, some stockholders bear the costs of

holding an illiquid undiversi?ed portfolio for many years.

Although the bene?ts of liquidity restrictions in retaining key employees and

managers and in reducing agency con?icts are well understood, the costs imposed by

these restrictions have been largely unexplored. Accordingly, the goal of this paper is

to examine howselling or liquidity restrictions affect the welfare of stockholders.

Since these stockholders often have a substantial stake in their venture, we will refer

to them simply as entrepreneurs throughout the paper to make the intuition more

clear. To study the effects of liquidity restrictions, we model the optimal

consumption and portfolio choice problem of an entrepreneur who owns stock in

a ?rm, but is unable to sell this stock for a given period of time. In addition to this

restricted stock, the entrepreneur has liquid wealth which he can allocate between the

stock and bond markets. This feature is important since it allows the entrepreneur to

take a stock market position that offsets some of the risk of his illiquid stockholdings

and reduces the cost of the restrictions. Note that allowing the entrepreneur to invest

in other markets differentiates this paper from earlier work (primarily focusing on

executive stock options rather than restricted stock) in which agents are not allowed

to hedge their undiversi?ed positions in other markets (for example, see Hall and

Murphy, 2000b). This framework also allows us to study how the consumption level

(or lifestyle) of an entrepreneur is affected by liquidity restrictions. The welfare loss

due to the liquidity restrictions is calculated by comparing the maximal utility

achieved by the entrepreneur with that achievable if the stockholdings were fully

liquid.

The results indicate that the cost of liquidity restrictions can be surprisingly large.

For example, when stock is restricted for ?ve years and represents 50% of his wealth,

an entrepreneur would actually be better off if he could sell his restricted stock for

30% to 80% of its unrestricted market value. Furthermore, these costs can be

signi?cantly higher when nearly all of the entrepreneur’s wealth is tied up in

restricted shares and when the entrepreneur is not able to hedge his restricted shares

with offsetting stock market positions. These results contradict the widely held

practitioner viewthat restricted stock has only a minor cost to the recipient (see Wall

Street Journal, April 12, 2001, p. R1) and is a much more ef?cient form of

1 There are many examples of entrepreneurs, managers, and others with signi?cant stockholdings,

initially worth millions on paper, who lost most of their wealth without ever being allowed to sell any of

their stockholdings. See the recent articles on the effects of selling restrictions on inside stockholders in The

Wall Street Journal on March 23, 2001, April 12, 2001, April 25, 2001, and May 17, 2001, and in

Businessweek on April 17, 2000, and April 16, 2001.

M. Kahl et al. / Journal of Financial Economics 67 (2003) 385–410

387

compensation than executive stock options. These costs are roughly on the same order

of magnitude as those reported in studies of the cost of awarding executive stock

options such as Hall and Murphy (2000, 2002) and Meulbroek (2001).2 The results

also indicate that the cost of liquidity restrictions tends to be higher for agents who are

more risk averse. If the ability to innovate is not the same as the ability to bear risk,

however, this implies that liquidity restrictions may discourage risk averse (but

potentially highly productive) agents from entrepreneurial ventures. Furthermore,

these results suggest a possible basis for explaining the large valuation discounts

associated with private equity placements (see Wruck, 1989; Silber, 1992) and

contribute to the growing literature on the effects of illiquidity on security values.3

We ?nd that owning restricted shares can have a dramatic effect on the optimal

portfolio strategy for the liquid portion of the entrepreneur’s portfolio. Depending

on the ?rm’s correlation with the stock market, the entrepreneur can signi?cantly

increase or decrease his stock market holdings. This effect is largest when the

restricted shares represent an intermediate fraction of the entrepreneur’s wealth.

Interestingly, even when the correlation between the ?rm and the stock market is

zero, the entrepreneur can hold more of the stock market than he would in the

absence of liquidity restrictions. Intuitively, this is because taking additional stock

market risk helps smooth consumption variability caused by the temporary liquidity

restrictions. Finally, we show that even though the entrepreneur can borrow against

his illiquid position, he chooses to consume at a much lower rate than he would

without liquidity restrictions.

This analysis also has implications for several areas in corporate ?nance. The

model suggests that restricted stock can be worth substantially less to managers who

have a large fraction of their wealth invested in their company and face signi?cant

trading restrictions. This makes it a more costly corporate governance tool and less

effective at reducing agency costs. Although we focus on restricted stock, this

implication is consistent with recent results in the executive stock option literature.

For examples of this literature, see Lambert et al. (1991), Rubinstein (1995), Aboody

(1996), Carpenter (1998, 2000), Hall and Murphy (2000, 2002) Jin (2002), and

Meulbroek (2001). Moreover, the high cost of the lack of diversi?cation associated

with concentrated managerial equity ownership gives managers a strong incentive to

make diversifying acquisitions even if not in the interests of their shareholders (see

Amihud and Lev, 1981; Morck et al., 1990). Minimizing these costs can also provide

an important motivation for taking a ?rm public. Furthermore, the cost of these

restrictions helps explain the growing use of diversifying strategies such as zero-cost

collars and equity swaps documented by Bettis et al. (2001). Finally, La Porta et al.

(1999) show that in most countries, family ownership is the dominant ownership

2 Meulbroek (2001) uses an approach based on equalizing the Sharpe ratio between the market and the

illiquid stock to compute the discount. This preference-free approach is easily applied and can be viewed as

providing a lower bound on the discount for the illiquid stock. We are grateful to Robert Merton for this

insight.

3 For example, see Mayers (1972, 1973), Grossman and Laroque (1990), Amihud and Mendelson (1991),

Boudoukh and Whitelaw(1991, 1993), Kamara (1994), Longstaff (1995, 2001a, b), Vayanos (1998),

Huang (1998), and Brenner et al. (2001).

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M. Kahl et al. / Journal of Financial Economics 67 (2003) 385–410

structure even for the largest publicly traded ?rms. Our model suggests that the costs

imposed on the family owners due to a lack of diversi?cation can be signi?cant. In an

insightful recent paper, Hong and Huang (2001) argue that investor relations efforts

by ?rms may be motivated by the goal of increasing trading volume and thereby

relaxing liquidity restrictions on corporate insiders.

The remainder of this paper is organized as follows. Section 2 describes a number

of ways in which different types of liquidity restrictions arise. Section 3 presents the

dynamic portfolio choice model. Section 4 examines the effects of liquidity

restrictions on welfare and optimal consumption and portfolio decisions. Section 5

discusses the implications of the results. Section 6 makes concluding remarks.

2. Liquidity restrictions

There are many reasons why a shareholder might not be able to sell his shares for

an extended period of time. In this section, we describe a number of common

situations in which shareholders are subject to these types of selling or liquidity

restrictions.

First, there are many situations in which selling restrictions are imposed by

contract, often to resolve moral hazard and adverse selection problems. One example

that has attracted substantial interest in the recent academic literature is that of stock

lockups in IPOs (see Brav and Gompers, 2000; Ofek and Richardson, 2000; Field

and Hanka, 2001). These lockups are not required by the SEC, but are part of the

contract between the issuer and the underwriter in the vast majority of IPOs. Most

lockups do not allowcompany insiders (of?cers, directors, employees, their friends

and family, and venture capitalists) to sell their shares for a period of 180 days. This

restriction could be lifted for individual trades by the underwriter in an early release,

but this typically affects only a small fraction of the stock held by insiders (Brav and

Gompers, 2000). The lockup period, however, can be longer than 180 days. For

example, Ibbotson and Ritter (1995) report that Morgan Stanley agreed to a two-

year lockup period in its IPO.

The literature offers several economic reasons for IPO lockup provisions. First,

they provide a signal of the value of the company, as suggested by Welch (1989), and

modeled by Brau et al. (2001). Lockups make it less likely that the shares are sold to

the public shortly before the release of negative information about the ?rm. Brav and

Gompers (2000) argue that the variation in the length of the lockup period and the

number of shares retained are systematically related to the uncertainty about the

?rm’s value. Similarly, Longstaff (1995) argues that IPO underpricing could be

partially due to the effects of lockup provisions. The lockup period gives key

employees and management an incentive to ensure good corporate performance, at

least until the insiders can sell their stock. Adding to the importance of trading

restrictions associated with insider share ownership in IPOs, it is often the case that

management and active investors (such as venture capitalists) are subject to

additional vesting agreements that go beyond the lockup period (Ofek and

Richardson, 2000).

M. Kahl et al. / Journal of Financial Economics 67 (2003) 385–410

389

Lockup or vesting periods play a similar role in managerial compensation

contracts. Many ?rms use restricted stock plans as part of the compensation

package. In these plans, managers receive a speci?ed number of shares in the ?rm,

but cannot sell these shares for a given period of time. Moreover, the shares are

forfeited if the executive leaves the ?rm before the restriction period is over. Kole

(1997) ?nds that 79 of 371 Fortune 500 ?rms in her sample have such restricted stock

plans. The average minimum holding period before any shares can be sold ranges

from 31 months for ?rms with a medium level of research and development to 74

months for ?rms with a high level of research and development. For more than a

quarter of the plans, the stock cannot be sold before retirement. The rationale for

these minimal holding periods is that it provides managers an incentive to take

actions that increase the long-term value of the ?rm, not just the short-term value.

Furthermore, this tool is used to increase managerial retention by creating

substantial switching costs since the restricted stock plan typically becomes void

upon the departure of the manager.

Minimal vesting periods also typically apply to executive stock option plans,

which require the executive to hold the options for a prespeci?ed time before

exercising them. In Kole (1997), the average minimum waiting period before any of

the options can be exercised is 13.5 months. The average waiting period before the

options can be exercised (taking into account that some fraction of the options can

be exercised after the minimum waiting period, but the remainder only after an

additional waiting period) is 23.6 months.

Another example where individuals obtain stock that cannot be sold for a certain

period is in a merger agreement where the target’s key employees and managers

obtain restricted stock in the combined company. Typically, such restricted stock

also comes with a lockup period during which it cannot be sold. The motivation is

similar to that for trading restrictions in executive compensation contracts. The

liquidity restrictions are intended to align the interests of the target’s key employees

and managers with the combined company and also give them an incentive to stay

with the combined company. This is of particular importance when the value of the

target company lies primarily in the human capital of its key employees, which is

likely the case in many start-ups. Finally, Bettis et al. (2000) ?nd that over 92% of

the ?rms in their sample impose limitations on trading by corporate insiders such as

blackout periods during which trading is not allowed.

In addition to contractual restrictions, however, corporate insiders often have

signi?cant liquidity restrictions imposed on them for legal reasons. These legal

restrictions can be even more stringent than the contractual restrictions. In some

cases, the legal restriction begins at the time the contractual restriction lapses and

signi?cantly extends the period of illiquidity. In general, a shareholder must satisfy

both the legal and contractual restrictions before selling his stock.

An important example of a legal restriction is SEC Rule 144 which limits the

amount of stock a corporate insider or af?liate can sell without registering the

transaction. Under the Securities Act of 1933, any person who sells a security to

another person must register that security with the SEC unless a statutory exemption

can be found for the transaction. Since the registration process can be prohibitively

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M. Kahl et al. / Journal of Financial Economics 67 (2003) 385–410

expensive and time consuming for many security holders, SEC Rule 144 was

designed to enable the public sale of limited amounts of unregistered securities under

certain conditions. These conditions are intended to avoid situations where securities

are acquired by an underwriter with a view to distributing them to the public without

going through the formal registration process. Since individual investors who are not

professionals in the securities business can be ‘‘underwriters’’ under the meaning of

the Securities Act, Rule 144 provides a safe harbor by which sales of unregistered

securities will not be construed as sales by an underwriter. Osborne (1982) discusses

the economic rationale provided by the SEC for Rule 144. The cost of achieving this

safe harbor, however, is that the security holder must hold the securities for a

number of years, presumably to signal that the securities were not acquired primarily

to distribute them to the public without making the disclosures required by the

registration process.

The holding period required under Rule 144 depends on whether the security

holder is de?ned as an af?liate of the corporation. Af?liates include of?cers of the

corporation such as the CEO, president, senior of?cers, directors, spouses of of?cers,

relatives living in the same home as the of?cer, any persons in a position to exert

in?uence such as members of an of?cer’s family or close associates, and owners of

10% or more of the voting shares. Note that the de?nition of an af?liate is somewhat

broader than that of a corporate insider. Stock held by an af?liate is termed control

stock, and af?liates are often referred to as control persons.

Control stock can be acquired in a number of ways. For example, stock can be

acquired through compensation arrangements, exercise of stock options, payment

for professional services, venture capital arrangements, partnership distributions,

private placements, or even open market purchases. Rule 144 prohibits an af?liate

from selling restricted control stock for one year after the stock is acquired. After the

one-year period, however, there are a number of limitations placed on an af?liate

who wishes to sell control shares. Speci?cally, the af?liate is only allowed to sell an

amount of stock during any three-month period equal to the greater of one percent

of the total amount of shares outstanding or, if the ?rm is listed on a stock exchange

or quoted on Nasdaq, the average weekly reported trading volume in those shares

over the four weeks preceding the potential sale. Thus, for many smaller and less-

actively traded ?rms, it can take many years before a control shareholder is able to

completely liquidate a substantial equity stake in the ?rm. In addition to these

volume restrictions, current ?nancial information must be available regarding the

company whose securities are being sold. An af?liate must also ?le Form 144 with

the SEC for larger proposed sales. For a nonaf?liate, similar liquidity restrictions

apply to their sales of restricted or unregistered stock, but only during the ?rst two

years after the stock is acquired. There are many other examples of liquidity

restrictions imposed by lawsuch as the rules prohibiting insiders from trading during

periods surrounding earnings announcements.

Finally, since the effect of liquidity restrictions on insiders is to increase the

concentration of their holdings in the ?rm, this analysis is relevant for the issue of

concentrated ownership in general. Speci?cally, ownership of many ?rms is

concentrated in the hands of a small number of investors, who often have a large

M. Kahl et al. / Journal of Financial Economics 67 (2003) 385–410

391

fraction of their wealth invested in these stocks. This is true for private equity as

documented by Moskowitz and Vissing-Jorgensen (2001). Moreover, La Porta et al.

(1999) ?nd that in most countries, the most common form of ownership is family

ownership, even for the largest publicly traded ?rms.

3. The model

In this section, we model the portfolio choice of an agent where some portion of

his wealth is in shares that he cannot sell for a given period of time. An example of

this would be a corporate manager or entrepreneur who receives compensation in the

form of shares, but is prohibited from immediately selling those shares and

rebalancing his portfolio. To make the intuition as clear as possible, we use a simple

but realistic portfolio choice framework in which there are three types of assets: a

riskless bond, a stock index fund, and the restricted stock that the entrepreneur

holds. This partial equilibrium framework is a simple generalization of the standard

Merton (1969, 1971) continuous-time framework.

Let Bt denote the value at time t of a riskless bond or money market fund with

dynamics given by

dB ¼ rB dt;

ð1Þ

where r is the constant riskless interest rate. Let Mt denote the value of a risky asset

which can be viewed either as the stock market or a share in a stock index fund. The

dynamics of Mt are given by

dM ¼ ðr þ mÞMdt þ sMdZ1;

ð2Þ

where m is the market risk premium and s is the volatility of returns. Both m and s

are positive constants.

Although the entrepreneur is not allowed to trade his shares in the ?rm, we assume

that shares in the ?rm can be traded by others who are not subject to the restriction.

Let St denote the market value of a share of the ?rm’s stock. We assume that the

dynamics of St are given by

dS ¼ ðr þ lÞS dt þ nS dZ2;

ð3Þ

where l is the excess expected return for the ?rm and n is its volatility. Again, both l

and n are positive constants. The correlation between dZ1 and dZ2 is r; where

À1oro1: This allows for the important possibility that returns on the market and

on the ?rm’s stock are (potentially highly) correlated. To focus more directly on the

effects of liquidity restrictions, we make the simplifying assumption that the risk

premium l is given by the Capital Asset Pricing Model, implying that l ¼ mrn=s: An

immediate implication of this assumption is that an unconstrained entrepreneur

would want to hold the ?rm’s stock only to the extent that it appears in the stock

index. This implies that the unconstrained optimal portfolio weight for the ?rm’s

stock is zero.

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M. Kahl et al. / Journal of Financial Economics 67 (2003) 385–410

The entrepreneur has an investment horizon of ToN; and at time zero, is given N

shares of stock in his ?rm. To capture the essence of the liquidity restriction, we

assume that the entrepreneur cannot change the number of shares of stock he holds

until time tpT: This is consistent with actual practice where shareholders are

typically not allowed to change their position either directly by selling stock, or

indirectly by trading options or entering into equity swaps or similar types of

derivative contracts. After time t; however, the entrepreneur can trade his shares in

the ?rm without restriction. While the number of shares N the entrepreneur holds

does not change until time t; the proportion of his wealth held in the form of illiquid

stock is stochastic. Let Xt ¼ NSt=Wt denote the portfolio weight for his illiquid

stockholdings, where Wt denotes his total wealth at time t: Since N is assumed to be

positive, Xt > 0 for all tot: Longstaff (2001a) studies the optimal portfolio choice

problem in a model where an agent can only trade limited amounts of a risky

security per unit time. In an independent paper, Henderson and Hobson (2002)

develop a model similar to ours in which an agent is unable to trade shares and offer

a series-based approximation for the optimal solution that is valid only for small

values of X : Our model differs from theirs, however, in that we allow for

intermediate consumption. In addition, we study the effects on consumption and

portfolio choice for general values of X :

The entrepreneur has preferences given by

Z

T

E

eÀksU ðCsÞ ds þ eÀkT UðWT Þ ;

ð4Þ

0

where

8 x1Àg

>

<

;

xX0;

U ðxÞ ¼

1 À g

>

: ÀN; xo0;

and where C denotes consumption, k is the rate of time preference, and g is the risk-

aversion parameter. The entrepreneur’s liquid wealth is given by ð1 À XtÞWt; which

he allocates between the riskless asset and the stock market. Let ft denote

the portfolio weight (as a percentage of his total wealth) for the stock market.

Since portfolio weights sum to one, the portfolio weight for the riskless asset is

1 À f À

t

Xt:

In this framework, we allow the entrepreneur to take unlimited short positions in

both the riskless asset and the stock market. A reviewof industry practice indicates

that some investment ?rms allowinvestors to borrowa limited amount of funds on

the security of their restricted stockholdings. In fact, a number of ?nancial ?rms

specialize in what is termed Rule 144 lending. In actuality, however, it is easily shown

that if the entrepreneur allows his liquid wealth to become negative, then there is a

possibility of reaching negative total wealth at T. Intuitively, this happens because

once the value of the liquid part of the entrepreneur’s portfolio becomes negative,

there is a non-zero probability that it will remain negative. Furthermore, there is

always a possibility that the illiquid stock will decline in value towards zero before

M. Kahl et al. / Journal of Financial Economics 67 (2003) 385–410

393

the liquidity restriction lapses. Thus, the entrepreneur’s total wealth at T could

become negative if X becomes greater than one. Since this implies an expected utility

of negative in?nity in this model, the entrepreneur never chooses an investment

strategy that allows liquid wealth to become negative. Thus, the entrepreneur never

borrows against illiquid stock, which implies that 0oX p1 for all tot:

Following Merton (1969, 1971), the entrepreneur’s wealth follows the dynamic

process

dW ¼ ððr þ mf þ lX ÞW À CÞ dt þ sfW dZ1 þ nXW dZ2:

ð5Þ

The entrepreneur’s dynamic decision problem is to choose his consumption Ct and

the portfolio weight for the stock market ft in a way that maximizes his expected

utility subject to the dynamic budget constraint in Eq. (5). Allowing the entrepreneur

to make optimal portfolio choices is essential in estimating the cost of liquidity

restrictions. In particular, simple certainty-equivalence approaches which do not

allowagents to select portfolios optimally can actually produce negative estimated

costs, implying the counterfactual result that restricted securities are worth more

than unrestricted securities. For example, Hall and Murphy (2002) provide a further

discussion. As in Merton, we de?ne the entrepreneur’s indirect utility of wealth

function to be

Z T

JðW ; X ; tÞ ¼ max E

eÀksU ðCsÞ ds þ eÀkT UðWT Þ :

ð6Þ

f;C

t

The Appendix shows that when tot; JðW ; X ; tÞ can be expressed in the form

W 1Àg

JðW ; X ; tÞ ¼

F ðX ; tÞ;

ð7Þ

1 À g

and that the ?rst-order conditions for the optimal consumption level Cn and the

optimal investment in the stock market fn are, respectively,

X

À1=g

Cn ¼ W ekt F À

FX

;

ð8Þ

1 À g

Àðm=s2Þ ð1 À gÞF þ ðgrn=s þ m=s2ÞXFX þ ðrn=sX Þ2 FXX

rn

fn ¼

À

X :

ð9Þ

Àgð1 À gÞF þ 2gXFX þ X 2FXX

s

The function F ðX ; tÞ satis?es a Hamilton-Jacobi-Bellman equation which is given in

the Appendix. Because expected utility equals ÀN if X exceeds one, FX ð1; tÞ ¼ N is

required to hold at the boundary X ¼ 1: Although F ðX ; tÞ cannot be solved in closed

form, standard ?nite difference or simulation techniques can be applied to provide

numerical solutions for JðW ; X ; tÞ and the values of Cn and fn: Because of the

nonlinearity of the Hamilton-Jacobi-Bellman equation, standard existence and

uniqueness results for the solution cannot be applied. Thus, it is important to

acknowledge that in providing numerical estimates of the solution, we are implicitly

assuming that a solution exists and abstracting from uniqueness concerns. When

tXt; the liquidity restriction is no longer binding, X equals its unconstrained optimal

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M. Kahl et al. / Journal of Financial Economics 67 (2003) 385–410

value of zero, and JðW ; X ; tÞ takes on the functional form JðW ; tÞ since it no longer

depends on X (the solution for JðW ; tÞ is given at the end of this section).

From the ?rst-order conditions, the entrepreneur’s optimal consumption level Cn

and portfolio weight fn are nonlinear functions of X : Despite this, some intuition

about the optimal strategies can be obtained by considering the structure of the

problem. In particular, when the entrepreneur faces no liquidity restrictions, utility is

maximized at every instant by solving a local mean-variance optimization problem.

In contrast, when the entrepreneur faces liquidity restrictions, the decision problem

can be viewed as a blend of a buy-and-hold problem with a standard problem of

continuous rebalancing, which means that the entrepreneur must now also consider

global portfolio changes.

Another way of seeing this is by noting from Eq. (5) that the dynamics of the

entrepreneur’s wealth are completely determined by the values of f; X ; and C: When

the entrepreneur faces no liquidity restrictions, the entrepreneur is free to choose any

values of f; X ; and C; which gives full control over the distribution of wealth.

Because one can optimize choices of f; X ; and C individually, the optimal values of

these controls have the simple functional forms obtained by Merton (1969). When

there are trading restrictions, however, the initial value of X is given exogenously

and the entrepreneur can only choose the values of f and C: In this case, f and C

nowplay dual roles in maximizing the entrepreneur’s utility. Speci?cally, f and C

affect the dynamics of wealth directly as before. However, the values of f and C

affect the behavior of X over time, which has an indirect effect on the dynamics of

wealth. For example, choosing a lower level of current consumption tends to reduce

future values of X : When there are liquidity restrictions, both the direct and indirect

effects of f and C on the distribution of wealth must be considered in maximizing

the entrepreneur’s utility. Not surprisingly, this makes the optimal values of f and C

depend on X in very subtle and complex ways.

Despite this complexity, however, several comparative statics results can be given.

For example, as X -0; the optimal portfolio weight fn converges to the constant

portfolio weight m=gs2 given in Merton (1969). As X -1; both fn and Cn converge

to zero. The intuition for this is that if the entrepreneur’s liquid wealth were to reach

zero, the entrepreneur would need to prevent liquid wealth from becoming negative.

Thus, the entrepreneur would avoid any further market risk in the liquid portfolio by

placing zero weight in the stock market. Furthermore, the entrepreneur would

forego consumption rather than borrowing against illiquid wealth and creating a

negative liquid wealth position. In actuality, the entrepreneur’s optimal consumption

and investment strategies serve to insure that liquid wealth remains positive. By

guaranteeing that liquid wealth is always nonnegative, the entrepreneur’s optimal

consumption and portfolio strategies also insure that total wealth is always

nonnegative. Dybvig and Huang (1988) showthat requiring w

ealth to be

nonnegative eliminates unrealistic strategies such as the doubling strategy discussed

by Harrison and Kreps (1979).

In the special case where m ¼ grsn; fn reduces to rnð1 À X Þ=s; which implies that

the optimal portfolio strategy is a simple linear function of X : Finally, note that the

second term on the right hand side of Eq. (9) represents minus the amount of the

Document Outline

- Paper millionaires: how valuable is stock to a stockholder who is restricted from selling it?
- Introduction
- Liquidity restrictions
- The model
- The effects of liquidity restrictions
- The cost of liquidity restrictions
- The optimal portfolio strategy
- Optimal consumption

- Discussion
- Conclusion
- References

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