Is corporate control effective when managers face investment timing decisions in incomplete markets?

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Journal of Economic Dynamics & Control

Journal of Economic Dynamics & Control 34 (2010) 1062–1076

0165-18

doi:10.1

E-m1 W

journal homepage: www.elsevier.com/locate/jedc

Is corporate control effective when managers face investment timingdecisions in incomplete markets?

Vicky Henderson

Oxford-Man Institute for Quantitative Finance, University of Oxford, Eagle House, Walton Well Rd, Oxford OX2 6ED, UK

a r t i c l e i n f o

Article history:

Received 21 November 2008

Accepted 15 January 2010Available online 3 March 2010

JEL classification:

D21

D81

G31

Keywords:

Real options

Investment timing

Incomplete markets

Corporate control

89/$ - see front matter & 2010 Elsevier B.V. A

016/j.jedc.2010.02.009

ail address: [email protected].

e abstract from capital structure issues and

a b s t r a c t

This paper presents a model of investment timing by risk averse managers facing

incomplete markets and corporate control. Managers are exposed to idiosyncratic risks

due to the dependence of their compensation on investment payoffs which are not

spanned by other assets. We show that risk averse managers invest earlier than well-

diversified shareholders would prefer, leading to significant agency costs. This effect can

be mitigated if the manager is subject to corporate control. Our main finding is that the

interaction of idiosyncratic risk and control results in two regimes. When the market is

sufficiently close to being complete, control has a strong disciplinary effect and agency

costs can be virtually eliminated. However, when idiosyncratic risk is too large,

shareholders suffer agency costs and control is ineffective. An implication is that we

would expect to see different investment behavior across industries or specific

investments as the degree of idiosyncratic risk varied. It would also suggest that both

the standard complete-markets real options model and the npv framework can proxy in

describing investment timing.

& 2010 Elsevier B.V. All rights reserved.

Agency conflicts arise when managers make decisions on behalf of shareholders. In this paper we consider the impact ofidiosyncratic risk inherent in an investment decision faced by a risk averse manager and show it leads to investmenttiming decisions which are inefficient for well-diversified shareholders. We question whether the market for corporatecontrol provides a remedy. We find in some situations, strong shareholder rights will indeed lead to a significant reductionin agency costs, however, when idiosyncratic risk is too large, corporate control is not a sufficient deterrent to inefficientinvestment, and agency costs remain high.

We consider a risk averse manager who faces a single decision of when to make an irreversible investment which has afixed cost, and which pays off in a lump-sum. The manager’s reward depends on the firm’s activities, and as such, he iscompensated with a fraction of the option to invest. In our model, investment payoffs cannot be perfectly spanned byexisting assets and so idiosyncratic or unhedgeable risks remain. The risk averse manager makes his investment timingdecision based on his own preferences, taking into account his exposure to idiosyncratic risk. Shareholders, on the otherhand, are well-diversified and do not require compensating for idiosyncratic risk. Agency conflicts arise because the riskaverse manager maximizes the value of the investment timing option based on his own preferences, whilst shareholderswant to maximize firm value.1 As is usual in real options, exercise or investment occurs when the project value reaches acertain threshold. Here, the thresholds chosen by the risk averse manager and well-diversified shareholders will bedifferent.

ll rights reserved.

ac.uk

consider a firm with no debt.

www.elsevier.com/locate/jedc

dx.doi.org/10.1016/j.jedc.2010.02.009

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V. Henderson / Journal of Economic Dynamics & Control 34 (2010) 1062–1076 1063

The manager can offset some of the risk of the investment payoff by trading in the market, however, there is someidiosyncratic or unhedgeable risk remaining. This idiosyncratic risk causes the manager to invest earlier than theshareholder’s prefer, resulting in large agency costs. He does this in order to reduce exposure to idiosyncratic risk, since theact of exercising (since he has a real option) or investing gives a one-off payment. The manager’s investment thresholddecreases away from the shareholder’s threshold as risk aversion increases, and in the limit, approaches the zero npvthreshold. Correspondingly, agency costs rise with managerial risk aversion.

We note that if the manager’s reward does not depend on the investment option itself, then (in the absence of control)the manager should not have any preference for the timing of investment, and certainly does not time investment suchthat firm value is maximized. In this sense, the compensation is partially aligning the interests of the manager with theshareholders, since the option causes the manager to have a preference for investment timing, albeit different from that ofthe shareholders. Options also act as a corporate governance mechanism (see Becht et al., 2003). Once markets areincomplete, however, options are not sufficient to discipline managers, and corporate control is needed as a disciplinarydevice in our model. In this paper we ask whether corporate control is effective in reducing agency costs to shareholderswhich arise from the manager’s timing of investment.

When the investment timing choice of the manager reduces firm or shareholder value, the manager faces the threat of acontrol challenge. Corporate control is the right to determine the management of corporate resources; to hire, fire and setcompensation (Jensen and Ruback, 1983; Fama and Jensen, 1983a, b). In our framework, control challenges result in thepossibility that the manager is dismissed if he deviates too much from the shareholder’s firm value maximization policy.This dismissal could arise from either internal or external mechanisms. Internal mechanisms involve control by the boardof directors and external control is exerted by the takeover market. Takeovers are thought to occur for disciplinary reasonsto correct for bad management practice (Jensen, 1986; Scharfstein, 1988; Morck et al., 1989). Implicit in our model is theassumption that takeovers result in the manager’s dismissal, consistent with evidence of Martin and McConnell (1991).

Although the model of corporate control described above is prevalent in the US, in most countries large firms are notwidely held, but have controlling shareholders, who are often active in management (La Porta et al., 1999; Shleifer andVishny, 1997).2 La Porta et al. (1999) find ‘‘family control of firms appears to be common, significant, and typicallyunchallenged by other equity holders’’. In firms with a large shareholder (or family member) who is also a manager,monitoring and disciplining the manager becomes difficult. We can additionally proxy for this situation in our model bytaking the strength of control challenges to be (very small or) zero.

When the manager facing idiosyncratic risks is also subject to corporate control, the risk of a control challenge alwaysreduces agency costs to shareholders, in that the manager always chooses to invest at a threshold closer to theshareholder’s preferred threshold. In fact, we show the manager subject to control invests at a threshold somewherebetween the threshold of the shareholders and the threshold of an equivalent manager who is not subject to control.

Finally, we investigate the interaction between incompleteness and corporate control and ask ‘‘is corporate controleffective when risk averse managers make investment timing decisions in incomplete markets?’’. Our main contribution isthe finding that the effectiveness of control depends on the degree of idiosyncratic risk present. We find there are tworegimes based on the correlation between the investment and the market or hedging opportunities, provided risk aversionis not too low. When the correlation is sufficiently high, there is little idiosyncratic risk and so the risk of dismissaldominates. Shareholders counteract the impact of managerial risk aversion in order to capture much of the value of theoption, as the manager invests very close to the shareholders’ threshold. In this case, control is effective as agency costs toshareholders can be virtually eliminated. The other regime is for sufficiently low correlation. In this case, idiosyncratic riskis significant and the risk of a control challenge never dominates. Shareholders lose a larger part of the option value,suffering agency costs, and control is ineffective.

Our model also allows us to answer the question of whether the fact that managers face idiosyncratic risks should be animportant issue to shareholders. This depends greatly on whether there is a well-functioning market for corporate control.If there is not, the manager will make investment timing decisions which markedly differ from firm value maximizingones, and shareholders suffer large agency costs. This is consistent with the broad evidence cited in Shleifer and Vishny(1997) on agency costs resulting from poor corporate control. It is also consistent with evidence of Gompers et al. (2003)that firms with weaker shareholder rights have lower firm value. In this case, idiosyncratic risk is a further source of agencycosts to shareholders in countries with weak corporate control. Conversely, if there is a well-functioning market forcontrol, then control can have a significant impact on agency costs when idiosyncratic risk is not too large. However, whenidiosyncratic risks are large, control has less of an impact, and such risks do have a detrimental effect on shareholderwealth.

Whilst we study agency conflicts arising from managerial risk aversion and incompleteness, there are many forms ofagency conflicts between managers and shareholders, including empire building, short-termism and overconfidence, seeStein (2003) for a review. Such studies usually concentrate on the impact of these agency issues on capital budgeting(under or overinvestment) as distinct from investment timing. Exceptions to this include the recent paper of Grenadier and

2 La Porta et al. (1999) examine data on ownership structure of large companies in 27 countries and find that almost half of these firms are controlled

by large shareholders. In total about 30% of companies are controlled by families and in 69% of family controlled firms, the family also participates in

management. A US example is Microsoft, where Bill Gates owned 23.7% of the company in the mid 1990’s.

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V. Henderson / Journal of Economic Dynamics & Control 34 (2010) 1062–10761064

Wang (2005) who extended the real options framework to account for issues of informational asymmetries and agencyissues (unobserved effort and empire building). However, their manager faces a complete market.

Dow et al. (2005) integrate free cash flow theory into an equilibrium model and study the impact of imperfect corporatecontrol on equilibrium investment and asset prices. Other papers incorporating shareholder-manager conflicts incontingent-claim models include Zweibel (1996), Morellec (2004) and Morellec and Smith (2007). However, these papersare concerned with the impact of such conflicts on the firm’s debt levels. An important distinction between agency costsarising from incompleteness and others mentioned above is that in our model the manager has a concave utility functionand maximizes expected utility. In contrast, the sources of agency costs described above either occur because the managerderives utility or benefit from private objectives or because he takes decisions based on a biased assessment of theprobabilities of the outcomes (overconfidence).

There is little work on the impact of idiosyncratic risk or control on investment timing decisions. Henderson (2007) andMiao and Wang (2007) (see also van den Goorbergh et al., 2003) have studied the effect of incomplete markets oninvestment. However, the paper most closely related to the present work is Hugonnier and Morellec (2007). They alsoconsider a manager making an investment decision whilst facing idiosyncratic risk and corporate control, however, theirconclusions differ from ours. We give a more detailed discussion of the modeling differences in Appendix C, rather here wewill highlight that they have important consequences for the economic predictions of the model. Hugonnier and Morellec(2007) only consider the joint impact of incompleteness and control and find that for reasonable levels of risk aversion, themanager invests close to the zero npv threshold, eroding almost all of the value of waiting. In this sense, they find control isineffective, since shareholder’s are not successful at encouraging the manager to invest close to their preferred threshold.In fact, if we consider a special case of their model where the manager is not subject to control, then the conclusion wouldbe that the manager invests immediately (providing the investment has positive npv). This implies all option value iseliminated by idiosyncratic risk.

Our paper differs from that of Hugonnier and Morellec (2007) in a number of ways. First, we consider the impact ofidiosyncratic risk and control separately on the manager’s investment timing. This allows us to make comparisons betweenthe potential investment behavior of managers in different corporate governance regimes, namely those with a wellfunctioning market for corporate control versus those without. Second, we reward the manager based on the investmentpayoff as well as the value of his trading in the market and riskless bonds. In contrast, Hugonnier and Morellec (2007)model the reward as purely based on the value of the manager’s financial holdings. Our formulation should be interpretedas one where the manager receives compensation based on the firm value, which in turn depends on the option to invest.This is desirable since stock and stock options are extremely popular forms of compensation (see for example, Hall andMurphy, 2002).

There are significant advantages in our modeling approach. First, an advantage is that we obtain semi-closed forminvestment thresholds which enable us to provide some comparative statics and give ordering results on the thresholdsanalytically.3 Second, and more importantly, we obtain a richer set of conclusions. We find that without corporate control,the manager invests somewhere between the zero npv threshold and the shareholder’s preferred threshold, depending onrisk aversion and the level of incompleteness. When our manager also faces the threat of corporate control, we find that thedegree of idiosyncratic risk influences the effectiveness of control. In our model, control is effective when the market issufficiently close to being complete, but less effective when the market is too incomplete.

The model set-up and solution is presented in Section 1. Section 2 analyzes the implications of the model for investmenttiming and agency costs. We offer our conclusions in Section 3. Technical developments are contained in threeappendices—Appendix A contains proofs of the results in the paper, Appendix B gives some analytical comparison resultsconcerning the thresholds, and Appendix C contains a more detailed comparison of our model with that of Hugonnier andMorellec (2007).

1. The model

In our model, shareholders are well-diversified (and are unconcerned with idiosyncratic risks) but delegate investmentdecisions to a manager who is risk averse. We assume the manager faces a single irreversible investment decision whichcan be made over an infinite horizon.4 The project pays a one-off amount Vt at time t for a cost K. (We assume all amountsare expressed in discounted units, or equivalently, we take the risk-free bond as numeraire. Note this means theinvestment cost is a constant K when expressed in discounted units.) That is, exercise or investing at time t (of themanager’s choosing) yields the difference ðVt�KÞþ .

3 In our model, we assume the manager has exponential utility, which enables us to obtain semi-closed form results for investment thresholds.

Although Hugonnier and Morellec (2007) use CRRA utility, this is not the reason for the differences in our conclusions and there is no reason to expect our

main results would change under CRRA preferences. However, employing CRRA utility in our set-up would greatly complicate the analysis since it would

require solving a two-dimensional problems, which has to be done numerically.4 For simplicity we assume there is a single investment decision so the value of the firm is the value of the investment option. There is no debt in our

model.

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Let the investment payoff V follow a geometric Brownian motion

dV ¼ nV dtþZV dW ð1Þ

where W is standard Brownian motion. Denote by x¼ n=Z the Sharpe ratio of the investment payoff. Although theinvestment payoff V is not traded, we assume there are other tradeable assets available to the manager, summarized by themarket asset, denoted P. In contrast to standard real options models (Brennan and Schwartz, 1985; McDonald and Siegel,1986, see also the textbook treatment of Dixit and Pindyck, 1994), here the payoff V is not perfectly spanned by tradedassets, so the manager makes his investment timing decision in an incomplete market (equivalently, he faces someunhedgeable or idiosyncratic risk).

The market P also follows a geometric Brownian motion, with

dP¼ mP dtþsP dB

where B is a standard Brownian motion correlated to W with correlation r 2 ð�1;1Þ. We can express the risk of V in termsof the risk of P plus some additional idiosyncratic risk. Write dW ¼ rdBþr? dZ for a Brownian motion Z independent of B

and r? ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1�r2Þ

p. Since P can be traded, B represents the hedgeable risk. The Brownian motion Z represents the

idiosyncratic risk which cannot be hedged via the market asset. Denote by l¼ m=s the Sharpe ratio of the market.We introduce some quantities which will play a major role in the solution and interpretation of the model. Define the

constant parameter d via n¼ mrZ=s�dZ. If do0, it represents the additional expected return of the project compared to itsrisk-adjusted expected return relative to the market asset. If d40, the project expected return suffers a discount relative toits risk-adjusted market return. In terms of the Sharpe ratio of the assets, we re-express d via d¼ lr�x. This represents thecorrelation-adjusted difference in the reward-to-risk (Sharpe) ratios of the market and the project itself. We also introducethe constant parameter b¼ 1�ð2ðx�lrÞÞ=Z where we can rewrite b in terms of d as b¼ 1þ2d=Z. Note if d40, then b41whilst do0 corresponds to bo1. We will shortly have more to say about these quantities. We now consider theshareholder’s and manager’s investment timing problems in turn.

Shareholder’s problem: If well-diversified shareholders were to make the investment timing decision they wouldmaximize firm value. Let SðvÞ denote the shareholder’s value of the investment option, where v denotes the current(discounted) value of the project. Using standard arguments (Dixit and Pindyck, 1994), SðvÞ solves

0¼1

2Z2v2 @

2S@v2þZðx�lrÞv @S

@v

subject to boundary, value-matching and smooth pasting conditions:

Sð0Þ ¼ 0; SðV sÞ ¼ V

s�K;

@S@v

��V

s¼ IfV

s4Kg

This gives the usual first passage time criteria where the shareholders want investment to occur the first time the(discounted) investment payoff Vt is greater than or equal to a constant threshold level V

s. We solve for this threshold and

associated value of the option to invest in the standard way to give the following result.

Proposition 1. Denote by b¼ 1�ð2ðx�lrÞÞ=Z¼ 1þ2d=Z the non-zero root of the quadratic fðf�1ÞZ2=2þZfðx�lrÞ ¼ 0.

(i)
Suppose b41 (equivalently d40). Investment/exercise takes place at the first passage time t¼ infft : Vt Zvg for some
constant v, and the shareholder’s value of the investment option (under any v) is given by

Sv ðvÞ ¼ðv�KÞ

v

v

� �b; vov

v�K; vZv

8<:

Maximizing the option value over possible thresholds v gives

v� ¼ Vs¼

bb�1

K ð2Þ

The shareholder’s value of the investment option under their optimal threshold Vs

(or equivalently firm value maximization)is

SðvÞ � SV

s ðvÞ ¼ðV

s�KÞ

v

Vs

� �b

; voVs

v�K; vZVs

8><>: ð3Þ

Suppose br1 (equivalently dr0). Then the threshold Vs¼1 and investment is postponed indefinitely. If bo1, the value of
(ii)
the investment option is infinite. If b¼ 1, the option value is v.

The expression for the threshold in (2) gives the well known conclusion of real options that waiting to invest has value,since V

s4K. Likewise, the formula in (3) takes the usual form—the payoff upon investment V

s�K , multiplied by a

stochastic discount factor which reflects the probability that investment will occur.

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The condition b41 or d40 is required on parameters to ensure that shareholder’s will invest at a finite threshold. Wecan interpret d as an implicit dividend and note that when d40, the expected return on the option on the project V isdecreasing.5 In this case there is a reason to exercise the option as on average, waiting results in a lower amount. Theimplicit dividend yield represents the opportunity cost of waiting. If dr0 (or br1), there is no reason to exercise theoption early and it is optimal to keep waiting. Another interpretation of d is as a measure of the life of an investmentopportunity. In a competitive market or with patent protection that is short-lived, competitive advantages and positiveNPV projects are likely to be short-lived. This corresponds to a large positive value for d, and equivalently, a value of b inexcess of one. This interpretation suggests that positive d is typical in competitive markets. A positive d can result eitherwhen: (i) lr40 and xolr where x may be positive or negative to satisfy the inequality or (ii) lro0 and xo�jlrj, so x isalso negative. The first of these possibilities seems the more likely.

We also comment here on the standard assumption that (the appropriate) d is regarded as a fixed parameter,independent of project volatility Z. In particular, Dixit and Pindyck (1994) fix d when varying Z to give the well-knowncomparative static result that the value of an option to invest is increasing in volatility.6 McDonald and Siegel (1986) notethat not doing so ‘‘leads to ambiguity in the comparative statics results’’ (p. 722).7 In a similar vein, we will regard d as afixed parameter. The value of the parameters d and Z will set a value for b.

The shareholder’s preferred investment timing and firm value will represent a benchmark against which the manager’sbehavior can be compared. It will also allow us to compute agency costs arising from the manager choosing a differenttrigger level at which to invest.

The Manager’s timing problem under corporate control: We now consider the manager’s investment timing decision. Therisk averse manager chooses when to pay the investment cost to receive the project value V. Since V is not tradeable, and itsvalue is uncertain, the manager faces risk whilst waiting. However, the manager can also invest in the market asset P. Thisenables him to partially hedge the risk from the investment timing option. Let X denote the manager’s wealth from hisholdings in the market asset P and the risk-free bond, discounted by the bond. The dynamics of X are dX ¼ ytdP=P where yt

is the cash amount invested in the market asset at time t.The risk averse manager chooses an investment time t and a position y in the market asset to maximize his expected

utility. At the investment time, his position consists of two components—the payoff ðVt�KÞþ in addition to his accruedwealth Xt from his holdings in the market. Implicit in this set-up is the assumption that the manager’s private wealth iscontingent on the value of the investment option or equivalently the value of the firm.8 This assumption is usually satisfiedbecause most managers will be compensated in the form of stock and stock options and thus a significant proportion oftheir own private wealth will be dependent on the success of the company.

The manager will be penalized with a greater chance of facing a control challenge if his investment timing choicedeviates further from the shareholders’ firm value maximizing choice, given in Proposition 1. Let the probability themanager faces a control challenge at time t be given by

pðVtÞ ¼ 1�e�FðVtÞ ð4Þ

where FðvÞ ¼ c½SðvÞ�ðv�KÞ�=K. The form of F reflects the shareholder value lost when the wrong threshold is chosen.9 Theparameter c represents the strength of the control challenge. A proxy for 1�c might be the governance index constructedby Gompers et al. (2003). High levels of their governance index are representative of firms with high management power(and thus weak shareholder rights) which corresponds to low values of our parameter c. As the strength of the challengeincreases, at a fixed choice of exercise threshold, the probability of a challenge is higher. As the manager’s thresholddeviates from V

s, the probability of a challenge increases.

If the manager is subject to a control challenge, he is dismissed and no longer receives any compensation based on theoutcome of the real investment. This is modeled via the indicator function It. The manager solves the following problem:

suptZ t

supðyuÞt r u r t

E½e�ztUðXtþðVt�KÞþ ItÞjXt ¼ x;Vt ¼ v� ð5Þ

where It ¼ 1 if the manager is not dismissed, It ¼ 0 if he is dismissed, and z is a discount factor. We assume the risk aversemanager has exponential utility, so UðxÞ ¼�ð1=gÞe�gx.

5 It should be noted that d¼ lr�x reduces to the corresponding parameter called d of Dixit and Pindyck (1994) when, as they assume, r¼ 1 and

volatilities are equal.6 Note this depends heavily on the assumption of complete markets, and as Henderson (2007) shows, volatility may increase or decrease option value

in incomplete markets.7 Wong (2007) points out that when the risk-adjusted rate of return is not invariant to volatility, there is a non-monotonic relationship between

volatility and the investment trigger.8 Note we could have rewarded the manager with a fraction of the option to invest, however, doing this would simply result in a rescaling of the

problem.9 Note that F depends on the optimal threshold of the shareholders V

s, and that FðvÞ ¼ 0 for vZV

s. This implies there is no chance of a control

challenge if the manager invests at (or above) the shareholder’s preferred threshold, Vs. In fact, as we will see later, the manager never chooses a

threshold higher than Vs

so this condition is for convenience and does not affect the results. Note also that although F is well defined for vZ0, the

manager will never choose a threshold below K.

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We now remark on the formulation in (5). There are two decisions to be made by the manager—when to exercise theoption to invest, and the portfolio choice problem of how much cash to hold in the market asset. It is appropriate toformulate the model such that there are no biases arising from the underlying portfolio choice problem influencing themanager’s choice of exercise/investment time, t. For a moment, consider only the portfolio choice component of theproblem in (5), which can be written as

supðyuÞt r u r t

E½e�ztUðXtÞjXt ¼ x� ð6Þ

where now, t is understood to be the horizon of the portfolio choice problem. We do not want the solution of thisunderlying problem to depend on the horizon time, t. That is, we do not want the manager to have a preference forparticular times t in absence of the option to invest. He should be indifferent over horizons t if he just faces the underlyingportfolio choice problem in (6). We show in Appendix A, for the solution of (6) to be independent of the horizon t, we needto have the particular specification of discount factor, z¼� 1

2 l2. This choice means that when the manager also has the real

investment option and is choosing both the exercise time and the portfolio holdings according to (5), his underlyingportfolio optimization problem is not biasing his choice of exercise time. A different choice of z would create artificialincentives to exercise early, or may even lead to a degenerate situation where the investment option should never beexercised.10 In fact, in Appendix C we show that it is their discounting choice that is driving the conclusions of Hugonnierand Morellec (2007). Define

Hðx;vÞ ¼ suptZ t

supyu ;trurt

E �1

geð1=2Þl2

ðt�tÞe�gðXt þðVt�KÞ þ ItÞ

��Xt ¼ x;Vt ¼ v

�

Finding H (x, v) at t=0 is equivalent to solving (5) (with z¼� 12 l

2). By time-homogeneity, we deduce the managerinvests at the first passage time of V to a constant threshold V

c, t¼ infft : Vt ZV

cg.

Proposition 2. In the continuation region, H solves the following non-linear HJB equation

0¼1

2l2HþxZvHvþ

1

2Z2v2Hvv�

1

2

ðlHxþrZvHxvÞ2

Hxx

subject to boundary, value-matching and smooth-pasting conditions:

Hðx;0Þ ¼ �1

ge�gx ð7Þ

Hðx;VcÞ ¼�

1

g e�gx½1þe�FðVcÞðe�gðV

c�KÞ þ �1Þ� ð8Þ

Hvðx;VcÞ ¼

1

ge�gxe�FðV

cÞIðV

c4KÞfge�gðV

c�KÞ þ þF0ðV

cÞðe�gðV

c�KÞ þ �1Þg ð9Þ

We see from the above proposition that the investment timing problem of a manager facing incomplete markets andcontrol can be expressed as the solution to a HJB equation with appropriate conditions, just as in the standard real optionsframework, see Proposition 1. In fact, there is a semi-closed-form solution to the above problem which we express in thefollowing proposition.

Proposition 3. If b40, define kðvÞ ¼ 1�e�gðv�KÞ and DðvÞ ¼ 1�e�FðvÞkðvÞ. The manager’s constant investment threshold Vc

is

the solution Vc4K to

bV

c DðVcÞr2

½1�DðVcÞ1�r2

� ¼ ð1�r2Þe�FðVcÞ½ge�gðV

c�KÞ�kðV c

ÞF0ðVcÞ� ð10Þ

and the value function H (x, v) is given by

Hðx;vÞ ¼ �1

ge�gx 1þfDðV

cÞ1�r2

�1gv

Vc

� �b" #1=ð1�r2Þ

If br0, then Vc¼1 and the investment is postponed indefinitely.

The Manager’s timing problem without corporate control: We can recover the investment timing behavior of the managerwho is not subject to control challenges, but does face incomplete markets. This proxies for situations where shareholderrights are weak (Gompers et al., 2003), for example, a firm with a large shareholder-manager who is not disciplined byminority shareholders (La Porta et al., 1999). By setting c=0 (or equivalently the function FðvÞ ¼ 0), there is no chance ofthe manager being subject to a control challenge. We obtain the following result (see Henderson, 2007).

10 Note that the specification z¼� 12l

2 is not essential to solve the model. We could solve the model in (5) for a general discount factor z, although

the resulting investment times would be biased towards early or late exercise.

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Proposition 4. Assume b40. The manager facing idiosyncratic risks (but no control challenges) invests at the first passage time

of V to the constant threshold V0, t¼ infft : Vt ZV

0g where V

0is the unique solution V

04K to

V0�K ¼

1

gð1�r2Þln 1þ

gð1�r2ÞV0

b

" #ð11Þ

Notice in both Propositions 3 and 4, the condition b40 is required for the exercise of the investment option. In themodel with no control challenges (c=0), Henderson (2007) explores the implications of this requirement versus thecorresponding (stronger) requirement in the shareholder model that b41. In this paper, we will simply proceed by alwaystaking b41 so that we are ruling out situations where the shareholders have an infinite threshold. These are clearly lessinteresting and realistic in our context.

2. Model implications

2.1. Investment timing

We focus first on the investment timing by the manager who is not subject to control.

Proposition 5.

(i)
limg-0 V0¼ V
sand limg-1 V

0¼ K.

0 0 s
(ii) V is decreasing in g, hence KoV oV .
Proposition 5 says a risk averse manager always chooses an investment threshold which is below the threshold theshareholders prefer. That is, the risk averse manager facing idiosyncratic risk invests earlier than the shareholders want.This occurs because waiting to invest in our model involves being exposed to fluctuations in the payout received uponinvestment and hence waiting involves exposure to idiosyncratic risk. Investing locks in a value for the project and reducesrisk. Since the manager is risk averse, he prefers to act earlier to lock in a certain value for the investment. He therefore actstoo early compared with what the shareholders prefer. Similar observations were made by Henderson (2007), see alsoMiao and Wang (2007). Higher levels of risk aversion lead to earlier investment times, and if the manager is extremely riskaverse, he carries this behavior to its extreme and invests at the zero npv threshold. In this case, all option value of waitingis eliminated. On the other hand, if his risk aversion is close to zero, he selects the shareholder’s threshold as he isunconcerned with idiosyncratic risk.

Now consider the investment threshold for the manager subject to corporate control. Fig. 1 plots the three thresholdsagainst the manager’s risk aversion level. In each panel, the bold horizontal dotted line represents V

s¼ 1:5, the

shareholder’s investment threshold level. As expected, this does not vary with risk aversion. The zero npv threshold in allpanels is K=1. The dashed line represents the threshold V

0at which a manager facing idiosyncratic risks (but not corporate

control) would invest. This threshold, as predicted by Proposition 5, is below the shareholder’s threshold, is decreasingwith risk aversion, and approaches the zero npv threshold as risk aversion gets large.

The solid lines on each panel represent the threshold Vc

for the manager facing both incomplete markets and corporatecontrol, for varying strengths of control challenges, c. Notice first that (in each panel) V

clies between the other two

threshold levels—the corporate control causes the manager to moderate his investment timing to be closer to the time theshareholders prefer. In Appendix B, we give analytical results in support of this ordering of the thresholds. Observe alsothat as the strength of the challenge rises, the threshold V

cmoves closer to the shareholder’s threshold. The risk of control

by the shareholders causes the manager to accept more exposure to idiosyncratic risk than he would otherwise want. Thisis as we would expect, given the chance of dismissal was higher the larger the deviation from firm value maximization.

The behavior of the manager’s threshold Vc

is due to two influences—the impact of idiosyncratic risk and the threat ofcontrol by shareholders. The relative importance of these effects depends on the level of correlation, the manager’s riskaversion, and the strength of the control threat. We now investigate how these effects interact. Compare the behavior ofthe threshold V

cin the upper panel (where r¼ 0:95) to that of the lower panel (where r¼ 0:5). We see in the top panel, V

c

is non-monotonic and tends to the shareholder’s threshold as risk aversion becomes large. In contrast, the lower panelshows the threshold (for various values of c) is monotonically decreasing with risk aversion. We can explain thesedifferences as follows.

When risk aversion is low, the impact of each risk on the threshold is relatively small, and neither risk is dominant. Forlow risk aversion, the threshold V

clies somewhere between the two limiting cases V

0and V

s. When risk aversion is high,

we have two possibilities, depending on the level of correlation. If correlation is high, there is little idiosyncratic risk andcontrol risk is the dominant effect, and the manager selects a threshold V

cclose to V

swhich maximizes firm value. This

results in the threshold being non-monotonic in risk aversion, as in the upper panel of Fig. 1. If correlation is low, thenidiosyncratic risk is large, and it outweighs the impact of control risk if the strength of the control threat is low. In this case,the threshold approaches V

0(which approaches K, the zero npv threshold) as risk aversion increases. We can see this in the

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0 2 4 6 8 10 12 14 16 18 201.3

1.32

1.34

1.36

1.38

1.4

1.42

1.44

1.46

1.48

1.5

γ

Inve

stm

ent t

hres

hold

s

0 2 4 6 8 10 12 14 16 18 201.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

γ

Inve

stm

ent t

hres

hold

s

Fig. 1. Investment thresholds for shareholders and manager, as a function of managerial risk aversion. In each panel—the bold dotted horizontal line is

Vs¼ 1:5, the dashed curve is V

0and the solid lines are V

cfor values of c=1,3,7,10 (from lowest to highest). Parameters are K=1, d¼ 0:3, Z¼ 0:3 giving

b¼ 3. The top panel takes r¼ 0:95 whilst the lower panel takes r¼ 0:5. Also shown in the lower panel are the limiting values for c=7,10.


lower panel for the lower values of c. However, if the strength of the control threat is large, then this can balance with thelarge idiosyncratic risk, and result in a threshold which asymptotes to some level between V

0and V

sas risk aversion

increases. In this case, neither risk has dominated the other.11 In the lower panel, we see this for the higher values of c

where we have marked the limiting values on the graph. We observe that when correlation is low, the threshold ismonotonically decreasing in risk aversion, regardless of the strength of the control (see the lower panel).

We find the critical value of correlation where the behavior of the thresholds switches is jrj ¼ffiffiffiffiffiffiffi0:5p

.12 We call values ofcorrelation such that jrjo

ffiffiffiffiffiffiffi0:5p

the ‘‘low’’ correlation regime, and values such that jrj4ffiffiffiffiffiffiffi0:5p

the ‘‘high’’ correlationregime.

We now comment on the implications of our findings for the effectiveness of corporate control. As we saw in Fig. 1 (andin the analytical results of Appendix B), control risk causes the risk averse manager to moderate his investment timing tobe closer to what the shareholders prefer. We say corporate control is more effective if this occurs to a greater degree.Clearly one factor influencing the effectiveness of control is the strength of the challenge. More importantly, we have seenthat the effectiveness of control also depends greatly on the level of correlation between the investment and the market.We investigate this further in Fig. 2 where we plot thresholds against the value of correlation. We first observe that thethresholds for the manager without control are increasing in correlation. As the correlation increases, there is less exposureto idiosyncratic risk whilst waiting to invest, and the manager can wait longer to invest at a higher threshold. When risk

11 In Appendix A, we give results on the critical value of c for which this occurs, and the limiting value for the threshold.12 The precise value of this critical value of correlation depends on the behavior of the function F near the shareholder’s threshold. A different

specification for F would give a different critical value. However, the important observation is that there is a critical value, rather than its precise value.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

ρ

Inve

stm

ent t

hres

hold

s

Fig. 2. Investment thresholds as a function of correlation. The dashed lines represent the threshold for the manager who is not subject to control, V0.

The solid lines are thresholds for the manager subject to control, Vc, with the strength of control being c=4. The pair of bold lines are for risk aversion

level g¼ 10. The non-bold lines take g¼ 2. Other parameters are K=1, d¼ 0:3, Z¼ 0:3, giving b¼ 3. The shareholder’s threshold is Vs¼ 1:5.


aversion is small, control causes the manager to choose a threshold between his preferred threshold V0

and that of theshareholders. Consistent with Fig. 1, for small levels of risk aversion, the threshold V

0is not that far away from the

shareholder’s threshold. However, for higher levels of risk aversion, we can see the two regimes based on the level ofcorrelation. The switch between the regimes can be seen in the figure when the threshold changes from being convex toconcave in correlation. When correlation is in the low regime, control increases the manager’s choice of threshold, but it isstill far from the shareholder’s threshold. For the low correlation regime, the corporate control is relatively ineffective. Inthe high correlation regime, control causes the manager’s threshold to move much closer to the shareholder’s threshold.Control in the high correlation regime is very effective since it results in the manager investing close to the firm valuemaximizing threshold.

We can compare our findings with those of Hugonnier and Morellec (2007). They give a single threshold at which themanager subject to both idiosyncratic risks and control invests (comparable to our threshold V

c). As risk aversion increases

in their model, the manager’s threshold approaches the npv threshold, K. Thus in their model it appears that control isineffective in moderating the investment behavior of the manager. Our conclusion is in stark contrast to their finding. Wefind that effectiveness of corporate control depends on the degree of incompleteness, and that for sufficiently high valuesof correlation and risk aversion, control significantly alters the manager’s investment timing to be much closer to theshareholder’s optimum.

2.2. Agency costs

We now consider the agency costs that are experienced by shareholders as a result of the risk averse manager choosingthe timing of investment. This enables us to evaluate whether the variations in investment thresholds translate intosignificant dollar-values and are therefore of economic importance. We can evaluate agency costs as being the difference inthe shareholder’s value of the option, and the shareholder value given the manager makes the investment timing decision:AcðvÞ ¼ SðvÞ�SV

c ðvÞ. Similarly, we can also calculate the agency costs in the case where there is no corporate control:A0ðvÞ ¼ SðvÞ�S

V0 ðvÞ. A comparison of these two costs will allow us to draw conclusions on the impact of control on agency

costs to shareholders.In the absence of control risk, agency costs only depend on risk aversion and correlation through the product gð1�r2Þ

(see Eq. (11)). For a fixed correlation, these costs increase monotonically with risk aversion, since the manager’s thresholdV

0is decreasing with risk aversion (see Proposition 5). If there is no well functioning market for corporate control

(shareholder rights are weak) then agency costs due to idiosyncratic risks can be significant. For parameter values of g¼ 10and r¼ 0:5, the agency costs without control are around 25% of the shareholder value.

The threat of control always results in a reduction of agency costs to shareholders. However, the magnitude of thisreduction varies with parameters. As c increases, agency costs fall, since the manager’s threshold is closer to theshareholder’s threshold. Recall, we said control was more effective if it caused a larger change in the manager’s threshold,taking it closer to the shareholders’ threshold. We can equivalently say control is more effective if it causes a largerreduction in agency costs to shareholders. We compare the impact of control on agency costs in two scenarios wheregð1�r2Þ ¼ 1 and hence the agency costs without control are identical. We can show that when correlation is high, agencycosts are virtually eliminated by control. However, when correlation is low, control only reduces agency costs by around

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40%. As expected, the impact of control on agency costs depends on the level of correlation. This is exactly as we wouldexpect, given our earlier analysis of the thresholds. If the correlation is sufficiently high, there is little idiosyncratic risk, andcorporate control is very effective in reducing agency costs. In the high correlation regime, agency costs are non-monotonicin risk aversion. However, when correlation is in the low regime, agency costs increase monotonically in risk aversion. Wefinally remark that our conclusions on agency costs are richer than those made in Hugonnier and Morellec (2007). In theirmodel, agency costs are monotonically increasing in risk aversion and control is relatively ineffective.

3. Conclusions

We present a model of investment timing by risk averse managers facing idiosyncratic risks (and incomplete markets)and corporate control. Idiosyncratic risks and control risks have opposing influences on the manager’s investmentbehavior, and the manager trades-off between them. In fact, we show that the manager’s trade-off results in two regimesbased on the degree of idiosyncratic risk his investment poses. The risk of dismissal dominates over the effect ofidiosyncratic risk when the latter is small, causing the manager to invest close to the shareholder’s threshold. Ifidiosyncratic risk is not small, then if the control challenge is fairly weak, idiosyncratic risk dominates and the managerselects a threshold close to the zero npv threshold. However, for stronger control threats, neither effect dominates. Ourfindings are in contrast to those of Hugonnier and Morellec (2007), as idiosyncratic risk always dominated in their setting.Consequently, our model predictions are also different.

Our model predicts that when there is a well functioning market for corporate control, and when risk averse managersface a fairly ‘‘complete’’ market, they invest close to the time resulting in firm value maximization. Any evidence in favor ofthe standard real options model of investment timing (which assumes a complete market) is also evidence consistent withour prediction, provided it is gathered in a situation which is fairly complete, and where there was control risk. Althoughthere are a number of papers supporting real options models, we mention Paddock et al. (1988) and Moel and Tufano(2002). The former use offshore oil reserve data, and the latter gold mines, both of which are likely to be relativelycomplete situations, given trading can be done in commodity futures. In addition, takeovers were commonplace in the oiland mining industries in the 1980’s (see Jensen, 1986).

On the other hand, if a risk averse manager faces a relatively incomplete investment situation, then our model predictsthe manager invests closer to the zero npv threshold since control is relatively ineffective. Increasing the strength of thecontrol challenges improves effectiveness by encouraging the manager to choose a higher threshold. However, in reality, itis costly for shareholders to increase the control threat. There are costs of internal monitoring, as well as costs associatedwith terminating the manager (golden parachutes and other severance payouts) and hiring a new manager (both in termsof the search itself and the initial reduction in productivity whilst the manager settles in). It seems plausible that withinthe likely constraints on the strength of control, that control remains at levels which are ineffective when the investment’scorrelation with traded assets is low. The implication is that in situations where there are strong shareholder rights, wewould expect to see differences in investment behavior across industries and indeed over specific investments, as the levelof idiosyncratic risk varied. In this case, both the standard real options model and the npv framework can play a role indescribing investment behavior of risk averse managers.

If there is no well-functioning market for corporate control, then we would expect to see the manager investing muchearlier than shareholders prefer, resulting in large agency costs. The implication is that in many countries where largeshareholder-managers are the norm, or for investments for which there are no highly correlated traded assets in which tohedge, the npv framework may become a reasonable proxy for investment behavior.

It would be challenging to empirically test our main result, although we delay this for future research. We mentionbriefly some related empirical studies. Using a governance index to proxy for the balance of power between managers andshareholders, Gompers et al. (2003) find that firms with stronger shareholder rights had higher firm value. In addition, ourmodel says that the impact of governance on firm value will depend on the degree of idiosyncratic risk the manager faces.In the absence of governance issues, work by Durnev et al. (2004) links higher idiosyncratic volatilities with increasedinvestment efficiency. However, since their benchmark for investment efficiency is net present value (or a marginal Tobin’sq of one) rather than real options (or a marginal Tobin’s q greater than one), their findings neither confirm nor refute ourmodel’s predictions.

Of course the investment timing choice of managers will reflect other features not included here. However, it could beargued that exposure to idiosyncratic risk is a more fundamental effect than many other types of agency conflicts since itdoes not require any special assumptions on managers’ preferences, other than that they are risk averse. More importantly,we have demonstrated in this paper that the interaction of idiosyncratic risk and corporate control can result in a variety ofinvestment behavior by risk averse managers.

Acknowledgments

We would like to thank Albina Danilova, Jana Fidrmuc, Stewart Hodges, Anthony Neuberger, Elizabeth Whalley,participants at the ICMS Workshop ‘‘Further Developments in Quantitative Finance’’, the EFM Symposium on CorporateGovernance and Shareholder Activism (Bocconi University), the Bachelier Finance Society Fifth World Congress, and at

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seminars at the University of Oxford and University of Warwick. Some of this work was carried out whilst the author wasat Princeton University and was partially supported by NSF Grant DMI 0447990.

Appendix A. Remarks and proofs

Remarks on the Manager’s optimization problem in (5): In this section we provide justification for the choice of discountfactor z¼� 1

2 l2 in the manager’s optimization problem in (5). A more general framework for these problems is outlined in

Henderson and Hobson (2007). As we discussed already, if we separate out the portfolio choice problem, then we do notwant the manager to have a preference over the horizon t. The choice z¼� 1

2 l2 ensures the manager facing the portfolio

choice problem in (6) is indifferent over the choice of the horizon t. The intuition is that under this formulation, themanager with the investment option to exercise does not already have an in-built preference for early or late exercisearising from the underlying portfolio choice problem. If he were to have such a preference, this would bias the exercisetime of the option to invest. Let Jðt; xÞ ¼ eð1=2Þl2tUðxÞ so (6) can be written as

supðyuÞt r u r t

E½Jðt;XtÞjXt ¼ x�

If we can show that J(t, Xt) is a super-martingale in general, and a martingale for the optimal y (Jr0), then

Jðt; xÞ ¼ supt

supðyuÞt r u r t

E½Jðt;XtÞjXt ¼ x�

since J(t, x) does not depend on the horizon t.We now show these properties. Applying Ito’s formula to J (t, Xt) and integrating gives

Jðt;XtÞ ¼ Jðt;XtÞþ

Z t

t

Jðs;XsÞ

2½l�gyss�2 ds�

Z t

tgyssJðs;XsÞdBs

It follows that EJðt;XtÞr Jðt;XtÞ for any y, and using the optimal strategy solving the problem (6), y�s ¼ l=gs, we have

supðyuÞt r u r t

E½Jðt;XtÞ� ¼ Jðt;XtÞ

Hence J(t, Xt) is a super-martingale in general and a martingale for the optimal y.

Proof of Proposition 2. In the continuation region, eð1=2Þl2tHðx;vÞ is a martingale under the optimal strategy and asupermartingale otherwise. The HJB equation is derived using Ito’s formula, giving

0¼1

2l2HþxZvHvþ

1

2Z2v2Hvvþsup

yylsHxþ

1

2y2s2HxxþysrZvHxv

�

Optimizing over y gives

0¼1

2l2HþxZvHvþ

1

2Z2v2Hvv�

1

2

ðlHxþrZvHxvÞ2

Hxx

We solve subject to the associated boundary condition, as well as at the constant exercise threshold Vc, we must have

value-matching and smooth-pasting conditions as given in the proposition. &

Proof of Proposition 3. We want to solve the non-linear pde subject to the associated boundary, value matching andsmooth pasting conditions. Proposing a solution of the form Hðx;vÞ ¼�ð1=gÞe�gxGðvÞg gives

0¼ vGvZðx�lrÞþ1

2Z2v2Gvvþ

1

2

G2v

GZ2v2ðgð1�r2Þ�1Þ

" #ð12Þ

Choosing g ¼ 1=ð1�r2Þ eliminates the non-linear term completely, leaving

0¼ ½vGvZðx�lrÞþ12Z

2v2Gvv� ð13Þ

with corresponding conditions on GðvÞ (translated from those in Proposition 2).

We propose a solution of the form GðvÞ ¼ Lvc, for some constant L which results in the fundamental quadratic in c,

cðc�1ÞZ2

2þcZðx�lrÞ ¼ 0

The two roots of the quadratic are c¼ b¼ 1�ð2ðx�lrÞ=ZÞ, c¼ 0. That is, there are one non-zero and one zero root. It can be

seen that the general form of the solution must be GðvÞ ¼ LvbþB, and (7) gives B=1. We now have to decide when we can

build a solution that value-matches and smooth-pastes. There are two possibilities. If br0 then L=0, smooth pasting fails

and there is no solution. In this case, the manager postpones indefinitely. If b40 the manager will exercise at time t.

(We will, however, concentrate in this paper on the situation b41 so that the corresponding shareholder’s problem is well

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defined and Vso1.) In the case b40, (8) gives the constant Lo0, and thus

GðvÞ ¼ 1þf½1þe�FðVcÞðe�gðV

c�KÞ þ �1Þ�1�r

2

�1gv

Vc

� �b

ð14Þ

Recalling Hðx;vÞ ¼�ð1=gÞe�gxGðvÞg , substitution gives

Hðx;vÞ ¼ �1

g e�gx 1þfDðVcÞ1�r2

�1gv

Vc

� �b" #1=ð1�r2Þ

Differentiating H (x, v) with respect to v yields

Hvðx;vÞ ¼�1

ge�gx 1

1�r21þfDðV

cÞ1�r2

�1gv

Vc

� �b" #r2=ð1�r2Þ

fDðVcÞ1�r2

�1gv

Vc

� �b bv

and upon substitution of v¼ Vc

we have

Hvðx;VcÞ ¼ �

1

ge�gx 1

1�r2DðV

cÞr2

fDðVcÞ1�r2

�1gb

Vc

From the above expression and the smooth-pasting condition in (9), we obtain the result in (10). &

Proof of Proposition 5. (i) It is straightforward to take limits as g-0 in (11) and g-1 in (11) to obtain the stated results.(ii) A tedious differentiation shows that V

0is decreasing in g. &

Under the case where correlation is zero (and all risk from fluctuations in the investment value is idiosyncratic) we canderive the limiting results and the value of c at which the threshold behavior switches between npv limit of K and higherlimiting values. We have

Proposition 6. Suppose r¼ 0. Let c� ¼ b=1�ððb�1Þ=bÞb�1. If crc� then as g-1, Vc-K. If c4c� then as g-1, V

c- ~vðcÞ 2

ðK;VsÞ where ~vðcÞ solves

b~vðcÞ�

c

K1�

~vðcÞ

Vs

� ¼ 0

Proof of Proposition 6. When r¼ 0 we have from (10) that Vc

solves

bwkðwÞ ¼ ge�gðw�KÞ þkðwÞjF0ðwÞj ð15Þ

We look for a solution near K and try w¼ KþFðgÞ. Then kðwÞ ¼ 1�e�gFðgÞ. Suppose FðgÞ-0 but gFðgÞ-1 so that kðwÞ-1.Then we have b=K ¼ ge�gFðgÞ þjF0ðKÞj giving

FðgÞ ¼ 1

g lng

b=K�jF0ðKÞj

� ¼

1

g lnðAgÞ

where A¼ 1=ðb=K�jF0ðKÞjÞ.Provided b=K4 jF0ðKÞj or equivalently A40, then gFðgÞ-1 but FðgÞ-0 as required. Otherwise FðgÞ does not exist. Since

jF0ðKÞj ¼c

K1�

b�1

b

� �b�1" #

the requirement that A40 is equivalent to coc� where c� ¼ b=1�ðb�1=bÞb�1.

Now suppose c4c�. Let y* solve M (y)=0 where

MðyÞ ¼by�

c

K1�

y

Vs

� �b�1" #

ð16Þ

Note for c4c�

MðKÞ ¼1

Kb�c 1�

K

Vs

� �b�1 !" #

¼bK

1�c

c�

h io0

and MðVsÞ ¼ b=V

s40 so there exists a solution to M (y)=0. Also

M0ðyÞ ¼�by2þ

c

K

b�1

y

y

Vs

� �b�1

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and using (16),

M0ðy�Þ ¼�bðy�Þ2

þc

K

b�1

y�y�

Vs

� �b�1

¼1

Kðy�Þ2½cðb�1Þy��b2K�

We see M0ðy�Þ40 for large enough y, and M0ðy�Þo0 for small y. Thus there is a unique root in ½K ;Vs�. Now try w¼ y�þLðgÞ.

Then since w solves (15) we have

bw�jF0ðwÞj

� ¼ kðwÞ�1ge�gðw�KÞ

where kðwÞ ¼ 1�e�gðy��KÞ�gLðgÞ. Letting g-0 gives kðwÞ�1ge�gðw�KÞ-0 and hence the left-hand-side must tend to zero.

Hence we conclude that w-y�. &

Appendix B. Threshold comparison results

Proposition 7. When the correlation r¼ 0, the thresholds are ordered as V0rV

crV

s.

This result is for the special case of the model where there is effectively no market asset in which to hedge risk and so allrisk from the fluctuating investment value is idiosyncratic. In this case, the expression for the threshold V

csimplifies

dramatically and it is straightforward to prove the result.For any r, we have result that the thresholds are ordered for small levels of risk aversion.13

Proposition 8. For small g, the thresholds are ordered as V0rV

crV

s.

We now prove each of the above results.

Proof of Proposition 7. We know from Proposition 5 that V0rV

sfor all values of r. When r¼ 0, the expression for V

cin

(10) simplifies to Vc�K ¼ ð1=gÞlnð1þgV

c=bþV

cF0ðV

cÞÞ. Similarly, the expression for V

0in (11) simplifies to

V0�K ¼ ð1=gÞlnð1þgV

0=bÞ. Now suppose V

c4V

s. Then F0ðV

cÞ ¼ 0. This implies V

c¼ V

0, and combined with V

0rV

sgives

Vc¼ V

0rV

s, which is a contradiction. So we must have V

crV

s. In this case, F0ðV

cÞr0. Observe the function

Fðy;aÞ ¼ lnð1þgy=ðbþayÞÞ where g;b40, is decreasing in a. Observe we can write the expression for Vc

in terms of F asV

c�K ¼ ð1=gÞFðV c

;F0ðVcÞÞ and the expression for V

0as V

0�K ¼ ð1=gÞFðV 0

;0Þ. Since a¼F0ðVcÞr0, so Fðy;aÞ4Fðy;0Þ. This

gives VcZV

0. Thus V

0rV

crV

sas desired. &

Proof of Proposition 8. First we expand the threshold V0

in powers of g. Rewrite (11) as

egð1�r2ÞðV

0�KÞ�1¼

gð1�r2Þ

bV

0

Expanding in g and putting y¼ V0�K gives

yþ1

gð1�r2Þ

b�1

b

� �2

¼1

g2ð1�r2Þ2

b�1

b

� �2

1þ2Kgbð1�r2Þ

ðb�1Þ2

" #

so

y¼1

gð1�r2Þ

b�1

b

� �Kbgð1�r2Þ

ðb�1Þ2�

1

2K2b2g2 ð1�r2Þ

2

ðb�1Þ4þ � � �

( )

giving

V0¼ V

s�gð1�r2ÞbK2

2ðb�1Þ3þOðg2Þ ¼ V

s�gd1þOðg2Þ

where d1 ¼ ð1�r2ÞbK2=2ðb�1Þ3 ¼ ð1�r2ÞK2=2ðb�1Þ2ðb=ðb�1ÞÞ. Since b41, d140, so V0oV

s. This is consistent with the

result in Proposition 5 earlier.

Now we focus on the threshold Vc

and again expand in powers of g. First rewriting (10) and expanding in g gives

eFðVcÞ 1þgðV c

�KÞþ1

2g2ðV

c�KÞ2

� �1�e�FgðV c

�KÞþe�F1

2g2ðV

c�KÞ2

� �r2

�1

( )þgðV c

�KÞþ1

2g2ðV

c�KÞ2

¼ð1�r2Þ

bV

c g�F0gðV c�KÞ�F0

1

2g2ðV

c�KÞ2

�

13 Due to the complicated nature of the formulae, it was not straightforward to prove the ordering for general g values, although our numerical

investigations support the conclusion that the thresholds maintain their ordering.

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where we abbreviate F¼FðVcÞ and F0 ¼F0ðV

cÞ. Expanding and collecting terms lower than Oðg2Þ gives

ðVc�KÞþ

ð1�r2e�FÞgðV c�KÞ2

2¼

Vc

b1�ðV

c�KÞF0�

g2ðV

c�KÞ2F0

n oð17Þ

Now propose that Vc¼ V

s�gd2þOðg2Þ for some d2. We aim to calculate d2. Substitution into (17) gives

K

b�1�gd2

� �þ

1�r2

2g K

b�1�gd2

� �2

¼K

b�1�gd2

b

� �1�

K

b�1�gd2

� �F0�

g2

K

b�1�gd2

� �2

F0( )

Matching terms on each side of this equation shows terms cancel to O(1). To OðgÞ, we need an expression for F0.Differentiating F, substituting and expanding gives

F0ðVcÞ �

c

K

ðb�1Þ

Kb

� �b�1 bb�1

K�gd2

� �b�1

�1

" #��ðb�1Þ2gd2

bK

c

K

Comparing terms of OðgÞ gives

d2 ¼ð1�r2ÞK2

2ðb�1Þ2b

ðb�ð1�cÞÞ

We now obtain comparisons between the thresholds. Since b41, d240 and so VcoV

s. Also b=ðb�ð1�cÞÞob=ðb�1Þ so

d2od1 and hence V0oV

c. We thus have V

0oV

coV

sas desired. &

Appendix C. Comparison with Hugonnier and Morellec (2007)

In this appendix, we explore in more detail the differences in our framework and that of Hugonnier and Morellec(2007). These differences in formulation distinguish the conclusions of this paper from theirs, and contribute neweconomically important insights into the effects of incompleteness and corporate control upon investment timing andagency costs.

Hugonnier and Morellec (2007) consider simultaneously the impact of incomplete markets and corporate control oninvestment timing. They find that the manager invests close to the zero npv threshold, eroding almost all of the value ofwaiting. In this sense, they find control is ineffective, since shareholder’s are not successful at encouraging the manager toinvest closer to their preferred threshold.

In this Appendix, we highlight that two features of their model drive their results—first, their choice of discount factorand second, their manager’s reward does not depend upon the investment payoff,14 rather it depends on wealth arisingfrom holding the market and riskless bond. This wealth is scaled up or down (via a factor yt) depending on whether he isterminated. By highlighting the role of these factors in their conclusions, we can explain why our model gives morerealistic outcomes.

We first draw an analogy between the treatment of discounting in Hugonnier and Morellec (2007) and this paper. InHugonnier and Morellec’s model, there is a subjective discount factor (which is analogous to our parameter z) and aparameter q which is a function of the subjective discount factor and other parameters of the model. We want to obtain acomparison to q in our setup. Since Hugonnier and Morellec use CRRA preferences, we take the limit as R-1 (where R isthe coefficient of constant relative risk aversion) of their parameter q to obtain the equivalent parameter under exponentialutility, which we denote qe.15 We obtain (converting other model parameters to our notation)

qe ¼ limR-1

q¼ ð1�r2Þ½zþ12l

2�

We now note that there is a link between the sign of qe and the sign of zþ 12 l

2. If qe40, then this corresponds to z4� 12 l

2

in our model. Recall (see Appendix A and the discussion in the main text) we chose z¼� 12 l

2 in our model, whichcorresponds to the choice qe=0. Hugonnier and Morellec (2007) assume q40, or in exponential utility terms, qe40. Wewill elaborate now on the impact of this choice on their conclusions.

Although not considered in their paper, we can take the special case of the Hugonnier and Morellec (2007) model wherethere is no risk of a control challenge. In this case, the scale factor y should be set to yt ¼ 1. In this case, the manager simplysolves a portfolio choice problem to choose holdings in a market asset and a bond. In contrast to our model, their manager’sreward does not depend on the investment payoff itself, and therefore, their manager should be indifferent over theinvestment timing (since it does not affect in any way his reward when there is no chance of termination). After somecalculations, their manager’s problem becomes (see their Theorem 1 with yt ¼ 1) UðxÞsupt Ee�ðq=FÞt, where F40 and U (x)are constants, x represents initial wealth. Under Hugonnier and Morellec’s assumption that q40, the supremum above istrivially obtained at t¼ 0. Their manager is not indifferent to investment timing (as he certainly should be in this case) asthis would require q=0. Under the special case of their model without corporate control, their manager invests according to

14 Provided we make the reasonable assumption that the manager only invests in positive npv situations.15 Alternatively, we could derive qe directly by re-solving their model with exponential utility.

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the npv rule, and all option value is eliminated. This has arisen because the assumption q40 means that the discounting istoo great, inducing earlier (in this case, immediate) investment.

The reason we draw attention to this special case is that the assumption that q40 leads to a bias towards earlierinvestment times in the model with corporate control, which explains why the conclusions of Hugonnier and Morellec(2007) were that the manager facing both incompleteness and control invests close to the zero npv threshold.

Now let us consider what would happen in their model if we took q=0. In the special case without control risks, thiswould imply the manager is indifferent over choice of investment times. However, in the model where there is control risk,the choice q=0 would lead to the manager always choosing the shareholder’s investment threshold. That is, the conclusionwould be that there are no agency costs since the manager would always do what the shareholders wanted (see Hugonnierand Morellec, 2007, Theorem 2 with q=0).

We need to explain why our formulation does not give this degenerate conclusion, despite the fact that we choose theequivalent discounting to q=0 (the choice z¼� 1

2 l2) in order to ensure there is no bias from the underlying portfolio choice

problem. The reason is that our manager’s reward depends on the investment payoff, see (5). In contrast, the manager inHugonnier and Morellec (2007) has a reward which does not depend on the investment performance (except indirectly viadismissal). Our formulation has realistic conclusions both in the case where there is no corporate control and where themanager is subject to control challenges.

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Is corporate control effective when managers face investment timing decisions in incomplete markets?