Stock Options, Diversification, and Optimal Contracts Markus C. Arnold Faculty of Economics and Business Administration Georg-August University of G¨ ottingen [email protected]∗ Second Draft, April 2005

Abstract Large parts of the empirical literature on managerial diversification incentives as well as recent contributions examining the efficiency of stock versus options as compensation instruments neglect the different risk taking incentives provided by these two instruments. In our model we endogenize the manager’s diversification incentives and the options’ risk taking incentives and show that even if options are inefficient from a purely motivational perspective they can be preferred to stock in compensation contracts due to their advantages in mitigating overdiversification problems between manager and shareholders. JEL classification: G34, J33, M12, M52 Keywords: Executive stock options, diversification, risk taking incentives, incentive efficiency.

∗

The paper is based on chapter 5 of my Ph.D. thesis entitled “Anreizwirkungen von Stock Options – Eine agencytheoretische Analyse von Motivations-, Investitions- und Diversifikationsproblemen”. I thank my Ph.D. advisors Heike Y. Schenk-Mathes and Robert M. Gillenkirch for many helpful discussions and suggestions. I am also grateful to participants of the EURO DDM VIII workshop for helpful comments on earlier drafts of this paper.

Stock Options, Diversification, and Optimal Contracts Abstract Large parts of the empirical literature on managerial diversification incentives as well as recent contributions examining the efficiency of stock versus options as compensation instruments neglect the different risk taking incentives provided by these two instruments. In our model we endogenize the manager’s diversification incentives and the options’ risk taking incentives and show that even if options are inefficient from a purely motivational perspective they can be preferred to stock in compensation contracts due to their advantages in mitigating overdiversification problems between manager and shareholders.

JEL classification: G34, J33, M12, M52 Keywords: Executive stock options, diversification, risk taking incentives, incentive efficiency.

1

Introduction

Although a sizeable theoretical and empirical literature has analyzed the motives of corporate diversification and its effects on firm value the results are far from being conclusive. If firms diversify when the benefits of diversification weigh out its costs the question still remains whose benefits and costs finally determine the decision. A large part of the literature argues that the diversification decision is not made by the shareholders but by the manager of the firm.1 As Morck, Shleifer, and Vishny (1990), p. 32, put it: Boards of directors give managers considerable leeway in choosing investment projects, and do not use negative stock market reactions to investment or acquisition announcements as the definitive indicator of long-run value consequences. This view receives further support from the broad empirical evidence for the value destroying character of diversification2 which is often seen as evidence for an existing agency conflict between manager and shareholders3 since the manager may have (strong) incentives to diversify the firm despite the potential reduction in firm value. The literature usually suggests two motives for diversification: The manager may diversify the firm in order to reduce the firm specific risk inherent to his stock and option holdings or other firm risk-related wealth4 and he may derive a private benefit from the diversification. In some cases this benefit is interpreted as the prestige or improved career prospects arising from managing a more diversified firm.5 Alternatively, it is suggested that diversification may benefit the manager by entrenching him, making it more costly for the shareholders to replace him and therefore making it possible for him to realize higher compensation in the future.6 However, both explanations differ with respect to their predictions for empirical analyses. If the manager diversifies to reduce the firm specific risk of his private portfolio, increased managerial ownership will induce stronger diversification incentives. In contrast, if the manager diversifies because of his private benefit, increasing the part of his wealth tied up to the firm value will reduce his diversification incentives. The empirical evidence is mixed and supports both managerial motives.7 1

See e.g. Hermalin and Katz (2000), Aron (1988), Fluck and Lynch (1999). See e.g. Morck, Shleifer, and Vishny (1990), Lang and Stulz (1994), Berger and Ofek (1995), Comment and Jarrell (1995), John and Ofek (1995), Servaes (1996), Berger and Ofek (1999), Lamont and Polk (2002). The empirical results for Germany are inconclusive however. While Lins and Servaes (1999) do not find any significant negative effects of diversification in German firms, Schwetzler and Reimund (2003) find a diversification discount in their sample. 3 See Campa and Kedia (2002) for an alternative explanation of the diversification discount observed in empirical studies. See also Graham, Lemmon, and Wolf (2002) for a critical analysis of the excess value method usually used to determine the diversification discount. 4 See Amihud and Lev (1981). 5 See Aggarwal and Samwick (2003). 6 See Shleifer and Vishny (1989). 7 See Amihud and Lev (1981) and May (1995) for empirical support of the risk diversification hypothesis and Lewellen, Loderer, and Rosenfeld (1985), Servaes (1996) as well as Denis, Denis, and Sarin (1997) for the private benefit motive. 2

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This paper is related to two large groups of the existing literature: First, it refers to the empirical studies of managerial diversification motives just described as these studies usually either completely neglect the manager’s option holdings8 or solely analyze the aggregated pay performance sensitivity of his stock and option holdings.9 Second, it is related to the literature examining the efficiency of different compensation contracts and especially the use of stock options. As recent contributions find that using options as compensation instruments cannot be efficient to motivate managerial effort compared to stock10 , this calls into question the compensation practice of virtually all large US-firms and many firms in Europe since all of them would provide managerial incentives inefficiently. However, both groups of the literature completely neglect the risk taking incentives of options intensely discussed in the stock option literature11 , and their consequences for optimal compensation contracts. Thus, the question arises whether options solve simultaneously existing risk taking problems between manager and shareholders – like diversification problems – in a way superior to stock and can therefore enter optimal compensation contracts. This is also very important for the questions how to treat stock and options in empirical studies of managerial diversification incentives and whether the manager’s diversification decision is only affected by the aggregate incentive strength of his compensation instruments or additionally by their payoff structure. The model presented in this paper is based on the basic model of Feltham and Wu (2001) but extends their setting to account for diversification decisions. The manager in this model derives both types of benefits mentioned above from the diversification, risk diversification and a private benefit. However, the compensation contract results from the shareholders’ maximization of the firm’s net value after compensation. Thus, we endogenize both the manager’s diversification incentives and the options’ risk taking incentives. Our principal findings are: If shareholders can make the diversification decision themselves in the benchmark case (i.e. the manager’s decision is verifiable) it cannot be efficient to compensate the manager with options to motivate working effort. This can be directly derived from Feltham and Wu’s findings and thus replicates the “standard” result of recent contributions to stock option compensation. However, if the manager’s decision can no longer be verified, a conflict of interest between manager and shareholders emerges as the manager overdiversifies from the shareholders’ perspective when he is compensated as before. If shareholders are restricted to issuing stock to the manager they have two alternatives, both involving additional costs for them relative to the benchmark case: They can either accept the diversification and give out the corresponding optimal number of shares or they can further adjust the number of shares 8

See all references in footnote 7. See Aggarwal and Samwick (2003). Anderson, Bates, Bizjak, and Lemmon (2000) differentiate between the manager’s stock holdings and his option compensation but do not analyze whether and how stock price based compensation differs in diversified and undiversified firms. In contrast, the study of Coles, Daniel, and Naveen (2003) explicitly accounts for potentially different risk taking incentives of stock and options. 10 See e.g. Meulbroek (2001), Jenter (2001), Hall and Murphy (2002), Dittmann and Maug (2003). 11 For theoretical and numerical analyses see e.g. Carpenter (2000), Henderson (2002), for empirical studies see e.g. Guay (1999), Cohen, Hall, and Viceira (2000), Rajgopal and Shevlin (2002), Hanlon, Rajgopal, and Shevlin (2003). 9

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such that the manager no longer diversifies. However, if shareholders can also give out options to the manager they can use the options’ risk taking incentives to induce the manager not to diversify. The analysis shows that issuing options helps to overcome the manager’s risk aversion more effectively than stock. Moreover, in some cases these advantages even more than compensate the options’ disadvantages in motivating managerial effort and make options enter optimal compensation contracts. This shows on the one hand that options can be part of optimal contracts due to their advantages in solving risk taking problems between manager and shareholders even if they are inefficient from a purely motivational perspective. On the other hand it shows that it is not feasible to neglect options in empirical studies or to combine them with the manager’s stock holdings as not only the incentive strength affects the manager’s diversification decision but also the shape of his incentive contract. A compensation contract using options can induce opposite diversification incentives for a manager compared to one using stock even if both exhibit the same incentive strength. The remainder of the paper is structured as follows: Section 2 presents the model, section 3 analyzes the benchmark solution where shareholders decide themselves about diversification. Section 4 explores the optimal contract when the manager makes the diversification decision, and section 5 concludes.

2

The model

The structure of the model we analyze is comparable to the basic model of Feltham and Wu (2001). Yet, in order to examine the effect of risk taking problems on optimal compensation contracts we add a diversification decision of the manager to the model. The firm value is given by S
= ∆ + e − I · y +
ε (1) where e represents the manager’s working effort, y ∈ [0, 1] is an exogenously given diversification parameter that cannot be influenced by the manager and
ε is a firm specific noise term which is normally distributed with mean 0 and a variance of σ 2ε = (1 − I · y)2 · σ 2 . The parameter ∆ will be explained below. I is a binary decision variable:  1 for diversification I= (2) 0 else Consequently, the variance of the firm value is equal to the variance of
ε and amounts to:
= (1 − I · y)2 · σ 2 (3) V ar(S) When deciding about diversification, the manager only has the discrete choice between the alternatives to diversify his firm (I = 1) or not (I = 0). If he diversifies he reduces both the firm value and its variance. This decrease in firm risk and firm value reproduces the empirical observations mentioned in the introduction. Moreover, the value destroying character of the diversification alternative is a necessary condition for a potential conflict of interest between manager and shareholders with respect to the manager’s decision. If the diversification decreased the firm risk but not the

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firm value, it would be always advantageous from the shareholders’ and the manager’s perspective.12 The manager’s compensation contract consists of a fixed wage WF and either s shares of restricted stock which cannot be sold before the end of the period or n European options with a deterministic exercise price X that can be exercised at the end of the period. We allow the fixed wage to become negative and to differ in the stock and the option case. Thus, the manager’s compensation in the stock case13 is given by S = WF + s · S
(4) W S and in the option case amounts to N = WF + n · max(S
− X, 0). W N

(5)

Without loss of generality the number of shares is normalized to 1. Additionally,
is large relative to σ 2 and thus we have F (S = 0) ≈ 0. This we assume that E(S) ε can be guaranteed by the choice of the parameter ∆. Though, as ∆ does not have any further effects in this model, we set ∆ = 0 for the further analysis. Similar to Feltham and Wu (2001) we assume that the manager’s preferences can be represented by the certainty equivalent of his compensation less his personal cost of effort plus a private benefit derived from the diversification.14 If the manager is compensated with restricted stock, his certainty equivalent is given by c α CES = WFS + s(e − I · y) − e2 − · s2 · (1 − I · y)2 · σ 2 + I · b 2 2

(6)

where 2c e2 represents the manager’s disutility from the effort chosen and b ∈ [0, 1] measures the manager’s private benefit from diversification. In order to agree to the compensation contract the manager must be made at least as well off as with his external opportunities which yield CEmin . Without loss of generality we set CEmin = 0 for the further analysis. Likewise, if the manager receives n options with an exercise price X > 0 instead of restricted stock his certainty equivalent amounts to c α CEN = WFN + n · µX − e2 − · n2 · σ 2X + I · b 2 2

(7)

where the mean µX and variance σ 2X of his options under the assumptions made above are given by15 µX = (1 − I · y) · σ · [f − (1 − F )ζ] (8) 12

See the results in Diamond and Verrecchia (1982) and Aron (1988). As a caveat, note that the manager’s compensation is actually tied to the gross firm value S
and not to the value of the shareholders’ equity (which is the firm value net of compensation). However, as in this paper no result depends on the question whether the manager’s compensation is tied to the gross or the net firm value we continue to refer to s and n as number of shares and options, respectively – as commonly accepted in the literature. 14 For the linear case of the manager’s compensation this assumption is consistent with expected utility theory when the manager has exponential utility. In contrast, if the manager receives options the certainty equivalent can only approximate the expected utility. 15 For the mean and the variance of a normally distributed truncated random variable see e.g. Greene (2003), Theorem 22.3, for its application to options see Feltham and Wu (2001). 13

4

  σ 2X = (1 − I · y)2 · σ 2 · 1 − F − f 2 − f (2F − 1)ζ + F (1 − F )ζ 2 Here, ζ =

X−E(S) σε

=

(9)

X−(e−I·y) (1−I·y)σ

represents the number of standard deviations between
F = F (ζ) is the value of the the exercise price X and the expected stock price E(S), CDF at ζ and f = f (ζ) is the value of the density function at the same point. Given the compensation contract the manager maximizes his certainty equivalent according to equation (6) or (7) by choosing the optimal effort and diversification level. It is assumed that the manager’s diversification decision is not verifiable for the shareholders and thus not contractible.16 Moreover, it is not possible for the shareholders to infer the manager’s diversification decision ex post from the realization of S
due to the normally distributed noise term. Thus, shareholders not only have to consider an incentive compatibility constraint for the manager’s effort choice but also an additional diversification constraint when designing the optimal compensation contract, and their optimization problem is given by:
∗, I ∗) − W  (e∗ , I ∗ )] max Π(e∗ , I ∗ ) = E[S(e s.t. CEi (e∗ , I ∗ ) ≥ CEmin , i = S, N

(10a) (10b)

e∗ ∈ arg max CEi (e, I ∗ ) , i = S, N

(10c)

I ∗ ∈ arg max CEi (e∗ , I) , i = S, N

(10d)

e

I

To summarize, the diversification has the following effects on the manager: First, it decreases firm value and therefore his expected compensation. Second, it simultaneously decreases firm risk and thus the manager’s compensation risk and finally, the manager derives a private benefit from the diversification. Therefore, from his perspective the diversification alternative can be advantageous both for risk diversification and for private benefit reasons. Note however, that this does neither imply that the manager always diversifies nor that the diversification is always disadvantageous from the shareholders’ point of view. We will come back to this point in the next section. As the diversification has no effect on firm size in our model, it does not stand for the decision to (additionally) buy another firm in order to diversify. The manager’s private benefit can be rather interpreted as a benefit from a manager specific investment or the prestige derived from managing a (more) diversified firm. Before we analyze the entire model and in order to further illustrate the different tradeoffs inherent to this model structure, the following section will examine the case that the manager’s diversification decision is verifiable for the shareholders. 16

The assumption of the diversification’s nonverifiability seems to be unrealistic for some situations (e.g. mergers). However, it is more likely that shareholders lack the information that are necessary to control other diversification possibilities managers have in reality (when e.g. exploring new markets or researching for new products). See e.g. Hermalin and Katz (2000), pp. 19-20, Aron (1988), p. 82.

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3

The benchmark solution

3.1

Issuing restricted stock

If the manager’s diversification decision is verifiable (or equivalently if the shareholders can make the diversification decision themselves) the diversification constraint (10d) is eliminated from the shareholders’ optimization problem. If additionally the manager’s effort is verifiable, the incentive compatibility constraint (10c) is eliminated and maximizing the value of the shareholders’ equity Π yields the first best effort: ef b =

1 c

(11)

As can be seen, this effort level does not depend on the shareholders’ diversification decision. From their perspective diversification is advantageous if: Π(I = 1) > Π(I = 0) ⇔ b > y

(12)

They diversify the firm if the private benefit of the manager is larger than the decrease of the firm value due to the diversification. This is intuitive as in this model the shareholders can reap the manager’s private benefit b from the diversification via a corresponding decrease of his fixed wage.17 Thus, as the firm value decreases by y and the fixed wage is simultaneously reduced by b in the diversification case it is advantageous to diversify if b > y. If the manager’s effort is no longer verifiable the shareholders also have to consider the incentive compatibility constraint (10c) when designing the optimal contract. If the manager receives restricted stock the maximization of (6) yields his optimal effort level for a given compensation contract: e∗ =

s c

(13)

After substituting (13) and the participation constraint into (10a), the firm value net of compensation is given by ΠS =

s s2 α 2 −I ·y− − · s · (1 − I · y)2 · σ 2 + I · b, c 2c 2

(14)

and the shareholders’ maximization yields the optimal number of shares: ssb =

1 1 + c · α · (1 − I · y)2 · σ 2

(15)

Thus, for the diversification (I = 1) and the nondiversification (I = 0) we obtain: ssb d =

1 1 + c · α · (1 − y)2 · σ 2

17

(15a)

Note that this assumption might be critical when the manager has limited liability. However, this problem is already inherent to the assumption that the fixed wage is always set such that the participation constraint is satisfied as an equation and thus that the fixed wage can become negative.

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1 (15b) 1 + c · α · σ2 If the manager is risk neutral (α = 0), we have s = 1 in both cases. This of course represents the standard solution of such agency problems and risk neutrality of the manager and it directly follows that both the first best effort level and the first best firm value net of compensation costs are realized. Obviously, the optimal number of shares does not depend on the shareholders’ diversification decision since the manager is risk neutral and therefore the different levels of firm risk do not influence the optimal incentive level. However, if the manager is risk averse the optimal number of shares and consequently the optimal effort level in the diversified firm is larger than in the undiversified sb 18 This is due to the fact that the firm risk is lower when the firm is firm (ssb d > su ). diversified and thus more shares can be issued to the manager in order to elicit higher effort from him. Finally, comparing the value of the shareholders’ equity with and without diversification yields the following condition for the advantage of diversification from the shareholders’ perspective: ssb u =

ΠS (I = 1) > ΠS (I = 0) ssb − ssb u b > y− d (16) 2c sb For the case of the manager’s risk neutrality, i.e. ssb d = su =1, shareholders again prefer diversification if b > y. Thus, if the manager is risk neutral not only the first best effort level is achieved but also the diversification decision remains the same. Though, if the manager is risk averse the shareholders’ diversification decision sb changes. As ssb d > su the right hand side of (16) is smaller than y and the condition b > y is still sufficient for the diversification to be advantageous from the shareholders’ perspective but no longer necessary. It is now possible that b < y but that the diversification is still advantageous for the shareholders. The reason for this is straightforward: As the optimal number of shares with diversification is larger than without, the manager provides a higher effort in the first case thereby increasing firm value. Thus, it is more often advantageous to diversify from the shareholders’ perspective than in the case of the manager’s risk neutrality. So far, the analysis has shown that the diversification is not always disadvantageous from the shareholders’ perspective although it reduces the (gross) firm value. Even if the manager did not derive a private benefit from the diversification (i.e. b = 0) the effect of risk reduction and the corresponding increase in the manager’s effort can make the diversification advantageous from the shareholders’ perspective. However, the shareholders are not restricted to issuing shares to the manager but can also give out options. The following section will analyze how this alternative affects the benchmark solution.

3.2

Issuing options

If the manager’s diversification decision is verifiable the only problem shareholders have to resolve is to motivate the manager to provide working effort. In this case, the model 18

See Diamond and Verrecchia (1982) and Aron (1988) for similar results.

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structure analyzed in this paper is equivalent to the basic model of Feltham and Wu (2001) and we can directly state the following proposition: Proposition 1 If the manager is risk averse (α > 0) restricted stock is always preferred to options in the benchmark case. The costs of inducing any given effort level e are strictly increasing in X. Proof. See Feltham and Wu (2001), Proof of Proposition 1. Thus, if the shareholders can decide themselves about the diversification it can never be optimal for them to give out options. This shall be briefly explained in the following in order to better illustrate the further tradeoffs of the model. Feltham and Wu (2001) proceed by showing that for every given effort level, the costs of inducing this effort level increase in X. This is equivalent to showing that for a given effort level the manager’s risk premium increases in X since for a given e the firm value net of compensation only differs with respect to the risk premium that has to be paid to the manager for bearing the compensation risk. If the manager receives n options the necessary condition for a maximum of his expected utility is given by19 : n · (1 − F ) − c · e − α · n2 · F · µX = 0

(17)

Rearranging equations (13) and (17) yields the number of shares and options necessary to induce a given effort level: s=c·e and n=

2·c·e  (1 − F ) + (1 − F )2 − 4 · c · e · α · F · µX .

(18)

(19)

If the manager is risk neutral (α = 0) equation (19) simplifies to: n=

c·e 1 = ·s 1−F 1−F

(20)

1 As can be easily seen, the shareholders have to give out 1−F options if they want to induce the same effort as with one share of restricted stock. This increase in the number of options relative to the restricted stock is due to the fact that the manager bears the additional cost of his increased effort with certainty but obtains the additional compensation only with probability 1−F (i.e. if S
> X). Thus, the marginal premium 1 in order to provide the same incentives as restricted stock. has to be multiplied with 1−F However, the fact that options do not end up in the money with certainty not only increases the expected compensation but also the compensation risk when the ∂σ 2 manager’s effort increases ( ∂eX > 0). The risk neutral manager is indifferent with regard to this increase in risk and it can be easily shown that therefore stock and options are equivalent compensation instruments. However, the increase in risk implies 19

Note that this condition is not necessarily sufficient for a global maximum of the manager’s expected utility.

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a lower valuation of the risky cash flows by the risk averse manager. Equation (19) shows that in order to offset this decrease in value the manager has to receive more than 1 options per share of restricted stock to provide the same effort level. Moreover, 1−F ∂n the more risk averse the manager, the more options have to be issued to him ( ∂α > 0). Note that this sort of “risk premium” in terms of additional options for risk averse managers does not correspond to the usual risk premium shareholders have to pay to the manager in such models for his exposure to firm risk. Here, the risk premium does not emerge from the participation constraint but from the incentive compatibility constraint and represents an additional premium for the risk averse manager to make him provide the same effort as his risk neutral counterpart.20 These additional options eventually cause the manager’s risk premium α2 · n2 · σ 2X to increase in the exercise price for a given effort level21 and therefore make stock options an inefficient compensation instrument in the benchmark case. To summarize: The basic structure of the model analyzed in this paper is extremely unfavorable for the use of options as compensation instruments since it can never be efficient from the shareholders’ perspective to issue options to the manager if they can make the diversification decision themselves. Thus, whenever the optimal compensation contract includes options in case the manager’s decision is no longer verifiable, this can be directly traced back to the additional diversification problem. This question will be analyzed in the following section.

4 4.1

Delegation of the diversification decision Potential conflicts of interests

If the manager is compensated with restricted stock, equation (13) shows that the effort chosen by the manager for a given compensation contract does not depend on his diversification decision. He prefers diversification for a given compensation contract if CES (I = 1) > CES (I = 0) α ⇔ b > sy − · s2 · σ 2 · [1 − (1 − y)2 ] 2

(21)

As the inequation shows, the manager’s decision depends on the number of shares issued to him. Hence, goal congruence between manager and shareholders with regard to the diversification decision requires that the manager would always diversify if he sb received ssb d shares and that he would always forego diversification if he received su shares. Thus, for goal congruence we need CES (I = 1) ≥ CES (I = 0) if ΠS (I = 1) > ΠS (I = 0) and vice versa. sb If the manager is risk neutral, inequation (21) reduces to b > sy and with ssb d = su = 1 the diversification becomes advantageous for the manager if b > y which corresponds exactly to the shareholders’ condition in the benchmark case when the manager is risk 20

Recall that in the stock case the number of shares necessary to induce a given effort level does not depend on the manager’s risk aversion as can be seen from equation (18). 21 See Feltham and Wu (2001), Proof of Proposition 2.

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neutral. Thus, in the case of the manager’s risk neutrality there is no conflict of interest with regard to the diversification decision. sb However, if the manager is risk averse things are getting more difficult as ssb d = su . For this case we will proceed as follows: First, we will explore whether the manager always decides the way the shareholders prefer. As will be shown, this is not the case. Instead, the manager overdiversifies from the shareholders’ perspective. Subsequently, it will be analyzed how this undesired diversification can be avoided by varying the number of shares in the compensation contract and how this affects the value of the shareholders’ equity. Finally, section 4.2 compares this solution to the case that shareholders accept the manager’s diversification and issue the corresponding optimal number of shares. The following proposition summarizes the results for the manager’s diversification decision if the shareholders compensate the manager with the same number of shares as in the benchmark solution. Proposition 2 Potential conflicts of interest between manager and shareholders arise if and only if the diversification is disadvantageous from the shareholders’ perspective in the benchmark solution. The manager overdiversifies from their point of view. Proof. First, it is shown that the manager always diversifies if this is optimal from the shareholders’ perspective in the benchmark solution and they give out ssb d shares to the manager. If we rearrange equation (16), the shareholders’ condition for the diversification to be advantageous is given by ssb − ssb u b > bSH = y − d . 2c

(22)

Thus, there exists a critical bSH and shareholders prefer diversification for every b > bSH . Similarly, if the manager receives ssb shares he diversifies if d α 2 2 2 b > bMd = ssb · (ssb d ·y− d ) · σ · [1 − (1 − y) ] 2  α sb 2 2 · s · y − · σ · [1 − (1 − y) ] = ssb d 2 d

(23)

Now, if bSH ≥ bMd , the manager would always diversify if this were advantageous from the shareholders’ perspective. As ssb d ≤ 1 for y ∈ [0, 1], it is sufficient to show that ssb − ssb α u y− d · σ 2 · [1 − (1 − y)2 ]. (24) > y − · ssb 2c 2 d For y = 0, the left hand side (LHS) equals the right hand side (RHS) of (24) as sb sd = ssb u for y = 0 and therefore both sides are equal to 0. For y > 0, rearranging (24) yields sb sb 2 2 (24a) ssb d − su < α · c · sd · σ · [1 − (1 − y) ] and if we substitute   sb sb sb 2 2 ssb − s = s · s · α · c · σ · [1 − (1 − y) ] d u d u 10

(25)

into this inequation, (24a) simplifies to ssb u < 1 which is always fulfilled according to equation (15b). Thus, there is no conflict of interest between manager and shareholders in this case. Vice versa, if ssb − ssb u (26) b < bSH = y − d 2c the diversification is disadvantageous from the shareholders’ perspective and they want the manager to forego the diversification when they give out ssb u shares to him. From the manager’s point of view it would now be advantageous not to diversify if  α sb 2 2 · s · y − · σ · [1 − (1 − y) ] . (27) b < bMu = ssb u 2 u Thus, goal congruence between manager and shareholders would exist if bSH ≤ bMu . Substituting equations (26) and (27) for bMu and bSH , respectively, yields  sb α sb 2 ssb d − su 2 ssb · y − · σ · [1 − (1 − y) ] ≥ y − · s . u 2 u 2c

(28)

Again, the LHS and the RHS equal 0 for y = 0. For y > 0 we necessarily must have that sb ssb α d − su 2 2 (29) y − · ssb u · σ · [1 − (1 − y) ] > y − 2 2c sb sb as ssb u < 1 for all risk averse managers. Rearranging (29) and substituting again sd −su according to equation (25) however leads to 1 < ssb d which is not possible as ssb d ≤ 1 for y ∈ [0, 1]. It directly follows that bSH > bMu for y > 0 and that there is a parameter range b u < b < bSH for every y > 0 where the diversification is disadvantageous for the M shareholders in the benchmark solution but the manager diversifies if he receives ssb u shares. Thus, if the manager is risk averse there is no longer goal congruence between manager and shareholders with respect to the diversification decision as the manager not only diversifies if this is optimal from the shareholders’ perspective but also in some cases in which this is disadvantageous for the shareholders in the benchmark solution. The analysis in the case of the manager’s risk neutrality has shown that the manager’s private benefit alone cannot introduce distortions in the two parties’ evaluation of the diversification alternative as the shareholders can reap the manager’s benefit via a decrease in his fixed wage. Although the manager’s risk aversion also increases the attractiveness of the diversification alternative for the shareholders – as has been shown above (see equation (22)) – the manager’s evaluation of the diversification shifts in an even stronger way. 11

Yet, if the diversification is disadvantageous from the shareholders’ perspective they do not have to accept the manager’s diversification decision but have the possibility to u shares in order to change the manager’s decision. deviate from ssb u and to give out s In the following, it will be analyzed how s has to be adjusted, but first we will address the question how this deviation from ssb u affects the value of the shareholders’ equity. causes additional costs for the shareholders relative Apparently, a deviation from ssb u to the benchmark solution as su induces a suboptimal exposure of the manager to firm risk. These costs are given by the difference between the value of the shareholders’ equity in the benchmark solution (i.e. the manager would forego diversification when u receiving ssb u shares) and the corresponding value of the shareholders’ equity when s shares are issued: ∆K1 = ΠS (ssb su ) u ) − ΠS ( 

sb sb 2 (su ) α su )2 α su su ( sb 2 2 2 2 − − · (su ) · σ − − − · ( su ) · σ = c 2c 2 c 2c 2 (ssb u )2 u −s = 2c · ssb u

(30)

Equation (30) shows that the cost of deviating from ssb u is a quadratic function of su independent of whether su is larger or smaller than ssb u . It directly follows that it is always optimal for the shareholders to deviate from ssb u by the smallest possible amount. Whether the number of shares has to be increased or decreased relative to ssb u follows from the analysis of the manager’s certainty equivalents in the diversification and the nondiversification case: CES (I = 1) − CES (I = 0) ≶ 0

α 2 2 · s · σ · 2y − y 2 − sy + b ≶ 0 ⇔ (21a) 2 The manager diversifies the firm if this difference is positive for a given s. This is the case for every s < sL and every s > sH , where    y − 2b · α · σ 2 · (2 − y) 1 sH,L = 1± (31) α · σ 2 · (2 − y) y Thus, whenever there is a conflict between manager and shareholders with regard to the diversification decision, this can be either due to ssb u < sL and the shareholders would have to increase the number of shares to make the manager forego diversification ( su = sL ), or it can be due to ssb u > sH and the number of shares has to be decreased ( su = sH ). This can be directly related to the two diversification motives mentioned in the introduction. From s = 0 to sL the private benefit motive obviously dominates the manager’s decision. Choosing a higher s then shifts the manager’s preferences towards nondiversification. Yet, at the minimum the risk diversification motive begins to dominate the term (21a) and a further increase in s makes the manager tend towards diversification again.22 22

See also Denis, Denis, and Sarin (1997) who find in their empirical study that the risk diversification motive apparently only dominates for high managerial ownership.

12

However, it is easy to see that for some parameter combinations the manager always chooses diversification as there is no solution for sL and sH . Particularly, if b>b=

2α ·

σ2

y · (2 − y)

(32)

the combined effects of the manager’s private benefit and the risk diversification obtained by the diversification are always too strong to make the manager forego diversification by adjusting the number of shares issued to him. Moreover, it is easy to construct examples with b < bSH and consequently, there exists a conflict of interest between manager and shareholders with respect to the diversification decision but simultaneously no possibility for the shareholders to adjust s such that the manager foregoes diversification. This directly leads to the question what alternatives the shareholders dispose of in designing compensation contracts for b ∈ [ bMu , bSH ] – especially if they do not have the possibility to avoid the diversification – and how the optimal contract for this parameter range finally looks like.

4.2

Optimal contracts using restricted stock only

Given that the shareholders do not want to or are not able to adjust s such that the manager does not diversify, their alternative is to accept diversification and to choose the optimal number of shares for this case. In general, this will be ssb d according to sb sb < s < s equation (15a). However, there are some cases in which ssb L u d , i.e. su is too small for the manager to forego diversification but simultaneously ssb d is too large for the manager to diversify the firm. In such a case shareholders have to deviate from ssb d and have to adjust the number of shares to make the manager diversify. Similarly to the strategy to give out su shares to make the manager omit the diversification, the strategy to accept diversification and to issue the corresponding optimal number of shares also leads to additional costs for the shareholders relative to the benchmark case. Here, these costs correspond to the difference between the value of the shareholders’ equity without diversification (ΠS (ssb u )) and its value with diversification (ΠS (sd )) that just made the nondiversification advantageous for the shareholders in the benchmark case. The question how the optimal contract for b ∈ [ bMu , bSH ] is characterized thus depends on the comparison between the values of the shareholders’ equity or equivalently between the additional costs of the (forced) deviation from the benchmark solution in both cases. The following proposition summarizes the optimal solution: Proposition 3 For b ∈ [ bMu , b∗ ] it is optimal to deviate from ssb u shares u and give out s ∗ u to the manager. For b ∈ (b , bSH ] it is optimal to let the manager diversify and to give out the corresponding optimal number of shares. Proof. It has been shown that the strategy to deviate from ssb u and to make the manager forego diversification by giving out su shares leads to additional costs of ∆K1 =

(ssb u )2 u −s . 2c · ssb u 13

Similarly, the strategy to let the manager diversify and to give out the optimal number of shares for this case induces additional costs for the shareholders which are given by ∆K2 = ΠS (ssb u ) − ΠS (sd ) sb s − ssb d = u +y−b+Ω 2c

(33)

(ssb −s
)2

with Ω = d2c·ssbd if the shareholders additionally have to deviate from ssb d and to give d out sd shares in order to make the manager diversify the firm. In Proposition 2 it was shown that we have bMu < bSH . At b = bMu the manager is indifferent between the alternatives of diversification and nondiversification and consequently we have ∆K1 = 0 and ∆K2 > 0. Vice versa, at b = bSH the shareholders are indifferent between both alternatives and therefore ∆K2 = 0 but ∆K1 > 0 (if bSH < b and a solution for this strategy exists). Differentiating ∆K1 and ∆K2 with respect to b yields: su (ssb − su ) ∂ ∂∆K1 = − u sb · ∂b c · su ∂b

(34)

∂Ω ∂∆K2 = −1 + ∂b ∂b

(35)

s
u s
u = ∂s∂bL > 0 for ssb u = sL and ∂∂b = ∂s∂bH < 0 for ssb u = sH , it directly As ∂∂b u < s u > s follows that (34) is positive and ∆K1 is strictly increasing in b. (ssb −s
)2 In contrast, we have ∂∆∂bK2 < 0 independent of whether Ω = 0 or Ω = d2c·ssbd . For d

ssb
d ∂ s
d d −s · ∂b < 0. c·ssb d This follows from the fact that sd = sL if the distance between sL and ssb d is smaller ∂ s
d ∂sL sb and then we have = > 0 and s > s  = sL .23 Vice than between sH and ssb d d d ∂b ∂b sb versa, if the distance between sH and ssb d is smaller than between sL and sd we have ∂ s
d ∂sH = ∂b < 0 and ssb d = sH . Thus, (35) is strictly negative. d < s ∂b Consequently, there is a unique b∗ up to which ∆K1 ≤ ∆K2 , and it is optimal to avoid diversification by giving out su shares to the manager for all b ∈ [bMu , b∗ ] and to

Ω = 0 the second term of (35) disappears. Otherwise we have

∂Ω ∂b

=−

make the manager diversify and give out the corresponding optimal number of shares u ]. for b ∈ (b∗ , bSH Note that this does not imply that ∆K1 and ∆K2 necessarily have a point of intersection. It is easy to construct examples where ∆K1 (b) < ∆K2 (b) and the shareholders u shares as long as this is possible. In this case, would deviate from ssb u and give out s we have b∗ = b. Proposition 3 shows that at the beginning of the parameter range that induces a conflict of interest between manager and shareholders (in the environment of bMu ) it is optimal to deviate from ssb u to make the manager forego diversification. However, the larger b, the larger are the additional costs ∆K1 to induce the desired diversification decision as the deviation from ssb u increases and therefore the manager’s suboptimal 23

Note that always

∂sH ∂b

L = − ∂s ∂b .

14

exposure to firm risk. Thus, at a certain point the shareholders’ alternative solution becomes advantageous and it is optimal to let the manager diversify the firm in the environment of bSH .

4.3

Effects of options on the manager’s diversification decision

Section 3.2 has shown that it cannot be optimal for the shareholders to compensate the manager with options if they can decide themselves about the diversification. Furthermore, it can be shown for the case analyzed in Proposition 3 that it can never be optimal for the shareholders to let the manager diversify the firm if s has to be further adjusted to sd , i.e. Ω > 0. Thus, whenever the strategy to let the manager diversify the firm is superior, the optimal number of shares for this case (ssb d ) is issued and giving out options instead cannot improve the shareholders’ position. Consequently, options cannot enhance the shareholders’ position unless they change the manager’s diversification decision for b ∈ [ bMu , bSH ] and thus solve the conflict of interest between manager and shareholders in a way superior to restricted stock. Studies exploring risk taking incentives of options24 suggest that options might play an important role in solving such risk taking problems. Although they find that managers not being allowed to hedge the firm specific risk do not always prefer to increase firm risk – as the risk taking incentive derived from option pricing theory would suggest –, the fundamental risk taking incentives of option compensation are nevertheless confirmed in these studies. Therefore, we will analyze in the following how options influence the diversification decision of the risk averse manager and how this in return affects the optimal contract. According to equation (17) the necessary condition for the choice of the manager’s effort level is given by n · (1 − F ) − c · e − α · n2 · F · µX = 0, whereas the manager diversifies the firm if c α c α n · µX (ζ d ) − e2d − · n2 · σ 2X (ζ d ) + b ≥ n · µX (ζ u ) − e2u − · n2 · σ 2X (ζ u ). 2 2 2 2

(36)

Here, ζ d and ζ d represent the number of standard deviations between the exercise price and the expected firm value if the manager diversifies (d) or not (u). As ed = eu and therefore the incentive compatibility and the diversification constraint are no longer independent from each other in the option case, we have to determine the global maximum of the manager’s certainty equivalents with and without diversification for every compensation contract. As an analytical solution of this problem is no longer possible, we will concentrate on numerical solutions in the following. Table 1 displays the effect of options on the manager’s diversification decision. It shows the exercise price from which the manager foregoes the diversification if the shareholders want to induce different effort levels, as well as the corresponding number of options. For the whole table we have c = 1, σ = 0.3 and y = 0.25. As we know from Proposition 1 that the costs of inducing a given effort level are strictly increasing 24

See e.g. Carpenter (2000), Henderson (2002), Ju, Leland, and Senbet (2002), Nohel and Todd (2004).

15

in X under the given model structure, it is optimal for every effort level to choose the lowest exercise price that just changes the manager’s decision. Table 1 further shows the number of shares necessary to avoid diversification for the given parameter set ( su ) sb and contrasts it to su , the optimal number of shares in the benchmark case. As can be seen from the table, su is larger than ssb u in any case, thus we have a case where su = sL . The table reveals the fundamental advantage of options versus restricted stock in solving the risk taking incentive problem between manager and shareholders as well as the basic tradeoff inherent to the choice between different compensation alternatives. If shareholders want the manager to forego diversification, the effort levels en that can be induced by giving out options are smaller than eu that is induced by giving out restricted stock and therefore are closer to the optimal effort level in the benchmark case (esb u ). This risk taking incentive of the options is due to the fact that the expected value of an option increases when the variance of the firm value increases and thus
the expected value of the option is larger without diversification for a given E(S) than with. Table 1 shows that these risk taking incentives enable the shareholders to induce nondiversification for smaller deviations from the second best effort level in the benchmark case. Similarly, (results not reported) it can be shown for the opposite case sb (i.e. su = sH < ssb u and the number of shares has to be decreased relative to su to make the manager forego diversification) that options can induce nondiversification for effort levels larger than eu and thus the induced effort level is again closer to the optimal effort level in the benchmark case. Hence, independent of whether the private benefit or the risk diversification motive dominates the manager’s valuation of the diversification alternative, the options’ risk taking incentives reduce his desire for diversification and thus allow the deviation from esb u to be smaller than in the stock case. However, Table 1 also shows that the smaller the induced effort level, the larger the exercise price has to be in return.25 Thus, the stronger the manager tends towards diversification (i.e. the lower the effort level that is to be induced), the stronger the risk taking incentives of the options have to be to make the manager omit diversification and thus the larger the exercise price has to be to induce this effort level. This reveals the fundamental tradeoff inherent to the decision between different options with different exercise prices as well as to the decision between restricted stock and options: The closer the effort level induced by the compensation contract is to the optimal effort level in the benchmark case, the smaller are the costs of the manager’s suboptimal exposure to firm risk. Though, the smaller the difference between these two effort levels, the larger the exercise price has to be to induce the desired effort which in return makes inducing this effort level more costly for the shareholders. The same tradeoff is also inherent to the decision between stock and options. Hence, the options’ advantage relative to restricted stock should be particularly large when the (additional) costs of the manager’s suboptimal participation at the firm value are large, thus for b close to bSH . However, the analysis of section 4.2 has shown 25

Note that the lowest effort level displayed for every (α ,b)-combination is among the smallest effort levels that can be induced for the parameter constellation considered. Although it is possible to induce a local maximum of the manager’s certainty equivalents for even smaller effort levels, the manager’s global maximum for these cases is located at marginal effort and diversification. This is due to the fact that the first order approach used in this model to represent the manager’s effort choice can no longer be unconditionally applied.

16

that for these b-values the strategy to adjust the number of shares in order to make the manager forego diversification does not represent the optimal solution from the shareholders’ perspective, but they should accept diversification in this case and give out the corresponding optimal number of shares. Consequently, the question arises whether issuing options instead of restricted stock can indeed improve the shareholders’ position or whether the option solution is always dominated by one of the two stock solutions.

4.4

Optimal contracts issuing restricted stock or options

As in the following analysis we will optimize both over the exercise price and the number of options it could be argued that options dominate restricted stock anyway as restricted stock seems to be a special case of the more general option compensation. However, the analysis will reveal that the intuition about option compensation is confirmed and we will obtain optimal solutions with strictly positive exercise prices. The subsequent numerical analysis allows us to state the following proposition: Proposition 4 Options can be part of the optimal contract for b ∈ [ bMu , bSH ] if both solutions using restricted stock simultaneously lead to comparatively large additional costs for the shareholders, i.e. in the environment of b∗ . Issuing options leads to less diversification than the issue of restricted stock. Tables 2 and 3 as well as Figure 1 display the overall optimal solution for the shareholders. If the cells in both tables do not contain any value for the option solution this implies that the option solution corresponds to the restricted stock solution that induces nondiversification. In contrast “no sol.” means that b > b and shareholders cannot make the manager forego diversification by giving out options or varying s, respectively. For both tables, we again have σ = 0.3 and y = 0.25. Additionally, in Table 2 the manager’s parameter for his cost of effort is given by c = 1 and in Table 3 by c = 0.4. The asterisk in every line designates the optimal solution for the corresponding (α, b)-combination. Table 2 represents a case where the manager’s private benefit dominates his diversification decision and s has to be increased relative to ssb u to make him omit diversification whereas Table 3 displays a case in which the risk diversification motive dominates the manager’s decision at the origin and s has to be decreased. Comparing first the stock and the option solution for the nondiversification case su ) and ΠN (nsb (ΠS ( u )) confirms the intuition about the relative advantage of options versus restricted stock: The larger b, i.e. the larger the costs of the manager’s suboptimal participation at the firm value, the larger is the comparative advantage of giving out options instead of restricted stock to make the manager forego diversification. Additionally, Table 2 shows that the smaller b, i.e. the closer b is to bMu , the lower is the exercise price and the more the option solution resembles to the stock solution. The exercise price of the optimal option solution increases in b. In Table 3 though, the exercise price of the optimal option solution sometimes increases or decreases with growing b. This is due to the fact that under these parameter constellations it is optimal for the shareholders to choose the largest effort level inducible (i.e. the effort level 17

which is closest to esb u ) from a certain b-value. However, if b becomes larger it simultaneously becomes more difficult to induce higher effort levels as the global maximum of the manager’s certainty equivalents shifts to marginal effort plus diversification if the exercise price becomes large (see the remark in footnote 25). Figure 1 illustrates the overall solution for α = 0.5 but is also very similar to the figures we could draw from the other parameter values displayed in Tables 2 and 3 as well as to other values tested but not displayed in the tables.26 The black line in the su )), figure represents the value of the shareholders’ equity if they adjust s to su (ΠS ( the grey line ΠS (sd ). The dashed black line finally represents the firm value net of compensation in the option case (ΠN (nsb u )). The figure shows that for small values of b u  (close to bM ) the optimal solution still consists in giving out su shares to the manager and for large values of b (close to bSH ) in letting the manager diversify the firm and ∗ issuing ssb d shares. However, for values of b in the environment of b it is now optimal to give out options to the manager instead of restricted stock. As can be seen from the figure, the issue of options leads to less diversification than the concentration on the two stock solutions. Moreover, Table 3 also shows that there are cases where b > b, i.e. the shareholders cannot make the manager forego diversification by issuing stock, but giving out options instead can avoid diversification. These results show that options not only have advantages in solving risk taking problems between managers and shareholders relative to restricted stock but that these advantages can more than compensate potential disadvantages in motivating working effort and can therefore make options enter optimal compensation contracts. Furthermore, the examples for α = 0.5 in Table 2 reveal that two compensation contracts inducing the same incentive strength (i.e. the same effort level) need not induce the same diversification incentives for the manager: Issuing options makes the manager forego diversification whereas giving out restricted stock implies diversification. Thus, stock and options can induce opposite diversification incentives even if the incentive strength is identical.27 This directly implies that it is not feasible to neglect options in empirical analyses or to combine them with the manager’s stock holdings as not only the strength of incentives (usually measured as pay performance sensitivity) influences the manager’s diversification incentives but also the shape of his incentive contract.28 26 For risk aversion parameters up to 15 we obtain qualitatively the same results as those displayed. Note however, that when the manager’s risk aversion becomes large there are also some cases where the option solution is always dominated by one of the two stock solutions. 27 See e.g. Guay (1999) for an example of two executives taken from his empirical sample whose compensation contracts exhibit the same pay performance sensitivity but differ very strongly with respect to their convexity. 28 Consistent with this intuition Coles, Daniel, and Naveen (2003) find in their empirical study that the diversification level and the manager’s risk taking incentives (measured as the increase in the option value from an increase in stock volatility) are significantly negatively correlated. Berger and Ofek (1999) find that the probability to refocus increases when the manager’s option holdings increase but not when his stock holdings increase.

18

5

Conclusion

Diversification is often seen as a typical example for agency conflicts between manager and shareholders: Although empirical contributions reveal that diversification negatively affects the value of the shareholders’ equity managers may have (strong) incentives to realize it. The literature usually suggests two motives for diversification: The manager may diversify his firm in order to reduce the firm specific risk inherent to his stock and option holdings or other firm risk-related wealth, and he may derive a private benefit from diversification. However, most empirical studies exploring diversification incentives do not consider the manager’s entire portfolio of compensation instruments and the incentives it provides since they either completely neglect the managers’ option holdings or only analyze the aggregated incentive strength of the stock and option holdings (measured as pay performance sensitivity). This is all the more surprising as the risk taking incentives of options are intensely discussed in the compensation literature and the different risk taking incentives of stock and options are regularly emphasized. The model presented in this paper is based on the basic model of Feltham and Wu (2001) and extends their setting in order to incorporate diversification decisions of the manager. Neither the options’ risk taking incentives nor the manager’s diversification incentives are treated as exogenously given but enter the shareholders’ optimization problem in this model. As in the basic model of Feltham and Wu, the benchmark case in which shareholders are able to make the diversification decision themselves shows that options are inefficient to motivate managerial working effort, a result often derived in recent contributions analyzing the efficiency of different compensation instruments. Thus, this paper also refers to the question how additional risk taking problems between manager and shareholders affect the design of optimal contracts originally excluding the use of options. The model shows that if the risk averse manager decides about diversification and is compensated in the same way as in the benchmark case he overdiversifies from the shareholders’ perspective. The reduction in firm risk from the diversification makes it more attractive to him to diversify than to the shareholders. In this case, the shareholders’ alternatives are twofold: First, they can let the manager diversify and issue the corresponding optimal number of shares and second, they can change his diversification decision by either adjusting the number of shares or giving out options to him. The following analysis reveals that giving out options to the manager instead of restricted stock helps to overcome the manager’s risk aversion more effectively than restricted stock and therefore makes it easier for the shareholders to avoid undesired diversification. In some cases the options’ advantage in solving the risk taking problem between manager and shareholders even more than compensates the options’ disadvantages in motivating working effort and consequently options become part of the optimal compensation contract in these cases. The results further show that even if two compensation contracts induce the same incentive strength they can imply opposite diversification incentives. While the manager foregoes diversification under an option contract he diversifies the firm when compensated with stock. Thus, we can draw the following conclusions: First, the findings prove that even in

19

model structures that are extremely disadvantageous for options – as option compensation cannot be efficient from a purely motivational perspective – additional risk taking problems between manager and shareholders can make options enter optimal contracts. This implies that especially when analyzing compensation contracts of executives that usually make important investment decisions for their firm but are simultaneously not allowed to diversify the firm specific risk inherent to their stock and option holdings, risk taking problems between manager and shareholders should be considered, as they might play an important role when designing compensation contracts.29 Second, the results of this paper reveal that it is neither feasible to neglect options in empirical studies nor to analyze only the aggregated incentive strength of stock and options as the manager’s diversification incentives are not only influenced by the incentive strength but also by the shape of the compensation contract.

29

See e.g. Nohel and Todd (2002) who analyze the use of options to govern investment decisions of executives.

20

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Hanlon, M., S. Rajgopal, and T. Shevlin (2003): “Large Sample Evidence on the Relation Between Stock Option Compensation and Risk Taking,” Working Paper, University of Michigan und University of Washington. Henderson, V. (2002): “Stock Based Compensation: Firm-Specific Risk, Efficiency and Incentives,” Working Paper, University of Oxford. Hermalin, B. E., and M. L. Katz (2000): “Corporate Diversification and Agency,” in Incentives, Organization, and Public Economics: Papers in Honor of Sir James Mirrlees, ed. by P. J. Hammond, and G. D. Myles, pp. 17–39. Oxford. Jenter, D. (2001): “Understanding High-Powered Incentives,” Working Paper, Harvard Business School. John, K., and E. Ofek (1995): “Asset Sales and Increase in Focus,” Journal of Financial Economics, 37, 105–126. Ju, N., H. Leland, and L. W. Senbet (2002): “Options, Option Repricing and Severance Packages in Managerial Compensation: Their Effects on Corporate Risk,” Working Paper, University of Maryland und University of California at Berkeley. Lamont, O. A., and C. Polk (2002): “Does Diversification Destroy Value? Evidence from the Industry Shocks,” Journal of Financial Economics, 63, 51–77. Lang, L. H. P., and R. M. Stulz (1994): “Tobin’s Q, Corporate Diversification, and Firm Performance,” Journal of Political Economy, 102, 1248–1281. Lewellen, W., C. Loderer, and A. Rosenfeld (1985): “Merger Decisions and Executive Stock Ownership in Aquiring Firms,” Journal of Accounting and Economics, 7, 209–231. Lins, K., and H. Servaes (1999): “International Evidence on the Value of Corporate Diversification,” The Journal of Finance, 54, 2215–2239. May, D. O. (1995): “Do Managerial Motives Influence Firm Risk Reduction Strategies?,” The Journal of Finance, 50, 1291–1308. Meulbroek, L. K. (2001): “The Efficiency of Equity-Linked Compensation: Understanding the Full Cost of Awarding Executive Stock Options,” Financial Management, 30, 5–44. Morck, R., A. Shleifer, and R. W. Vishny (1990): “Do Managerial Objectives Drive Bad Acquisitions?,” The Journal of Finance, 45, 31–48. Nohel, T., and S. Todd (2002): “Compensation for Managers with Career Concerns: The Role of Stock Options in Optimal Contracts,” Working Paper, erscheint 2005 in: Journal of Corporate Finance. Nohel, T., and S. Todd (2004): “Stock Options and Managerial Incentives to Invest,” Journal of Derivatives Accounting, 1, 29–46. Rajgopal, S., and T. Shevlin (2002): “Empirical Evidence on the Relation Between Stock Option Compensation and Risk Taking,” Journal of Accounting and Economics, 33, 145–171. Schwetzler, B., and C. Reimund (2003): “Conglomerate Discount and Cash Distortion: New Evidence from Germany,” HHL Working Paper. Servaes, H. (1996): “The Value of Diversification During the Conglomerate Merger Wave,” The Journal of Finance, 51, 1201–1225. Shleifer, A., and R. W. Vishny (1989): “Management Entrenchment: The Case of ManagerSpecific Investments,” Journal of Financial Economics, 25, 123–139.

22

α 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2

sb b ssb u = eu 0.235 0.95694 0.235 0.95694 0.235 0.95694 0.235 0.95694 0.237 0.95694 0.237 0.95694 0.237 0.95694 0.237 0.95694 0.239 0.95694 0.239 0.95694 0.239 0.95694 0.239 0.95694 0.220 0.91743 0.220 0.91743 0.220 0.91743 0.220 0.91743 0.225 0.91743 0.225 0.91743 0.225 0.91743 0.225 0.91743 0.230 0.91743 0.230 0.91743 0.230 0.91743 0.230 0.91743 0.205 0.84746 0.205 0.84746 0.205 0.84746 0.205 0.84746 0.210 0.84746 0.210 0.84746 0.210 0.84746 0.210 0.84746 0.215 0.84746 0.215 0.84746 0.215 0.84746 0.215 0.84746

su = eu 0.97763 0.97763 0.97763 0.97763 0.98630 0.98630 0.98630 0.98630 0.99498 0.99498 0.99498 0.99498 0.95126 0.95126 0.95126 0.95126 0.97484 0.97484 0.97484 0.97484 0.99852 0.99852 0.99852 0.99852 0.96740 0.96740 0.96740 0.96740 0.99635 0.99635 0.99635 0.99635 1.02570 1.02570 1.02570 1.02570

en 0.973 0.974 0.975 0.976 0.9805 0.981 0.983 0.985 0.989 0.990 0.992 0.994 0.9435 0.945 0.947 0.949 0.9675 0.969 0.971 0.973 0.9915 0.993 0.995 0.997 0.958 0.960 0.962 0.965 0.9875 0.990 0.992 0.994 1.0175 1.020 1.022 1.024

 n) X(e 0.181 0.150 0.112 0.061 0.220 0.207 0.146 0.045 0.231 0.205 0.141 0.024 0.196 0.164 0.114 0.042 0.205 0.173 0.120 0.040 0.217 0.184 0.130 0.041 0.180 0.146 0.106 0.017 0.193 0.149 0.104 0.039 0.204 0.159 0.110 0.031

nu 0.97863 0.97813 0.97779 0.97762 0.98814 0.98775 0.98667 0.98625 0.99691 0.99615 0.99523 0.99491 0.95389 0.95267 0.95169 0.95122 0.97691 0.97591 0.97507 0.97474 1.00023 0.99933 0.99866 0.99840 0.96956 0.96831 0.96756 0.96716 0.99794 0.99675 0.99624 0.99610 1.02668 1.02583 1.02548 1.02545

Table 1: Exercise price and number of options necessary to induce nondiversification for given effort levels.

24

α 0.5 0.5 0.5 0.5 0.5 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3

b 0.233 0.235 0.237 0.239894 0.240813 0.220 0.224 0.226 0.229451 0.232 0.195 0.200 0.205 0.208299 0.210 0.175 0.185 0.187486 0.190 0.200

ΠS (sd ) 0.470636 0.472656 0.474656 0.477551 0.478469∗ 0.445907 0.449907 0.451907 0.455359 0.457907∗ 0.399030 0.404030 0.409030 0.412329 0.414030∗ 0.359075 0.369075 0.371561 0.374075∗ 0.384075∗

su = eu 0.968969 0.977633 0.986304 0.998863 1.002851 0.951261 0.970113 0.979564 0.995914 1.008018 0.910597 0.938817 0.967398 0.986461 0.996353 0.885063 0.955849 0.974126 0.992913 1.070977 ΠS ( su ) 0.478438∗ 0.478245 0.478018 0.477551 0.477367 0.458092∗ 0.457203 0.456612 0.455357 0.454243 0.421377∗ 0.418804 0.415241 0.412329 0.410649 0.387644∗ 0.375683 0.371561 0.366881 0.342637

X − 0.128 0.222 0.237 0.240 − 0.117 0.165 0.224 0.229 − 0.008 0.096 0.145 0.158 − 0.049 0.095 0.127 0.203

nsb u − 0.977900 0.988198 1.000874 1.004823 − 0.970359 0.980345 0.997945 1.009870 − 0.938596 0.967443 0.986907 0.996864 − 0.955482 0.973969 0.992944 1.070972

esb n − 0.97460 0.98039 0.99276 0.99678 − 0.96625 0.97420 0.98846 1.00080 − 0.93605 0.96245 0.97988 0.98951 − 0.95095 0.96770 0.98550 1.06256

Table 2: Optimal compensation contracts for σ = 0.3 and c = 1.

sb ssb d = ed 0.968969 0.975312 0.975312 0.975312 0.975312 0.951261 0.951814 0.951814 0.951814 0.951814 0.908059 0.908059 0.908059 0.908059 0.908059 0.868150 0.868150 0.868150 0.868150 0.868150

ΠN (nsb u) − 0.478255∗ 0.478064∗ 0.477674∗ 0.477516 − 0.457239∗ 0.456695∗ 0.455556∗ 0.454527 − 0.418814∗ 0.415359∗ 0.412571∗ 0.410963 − 0.375851∗ 0.371843∗ 0.367309 0.343795

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α 10 10 10 10 10 11 11 11 11 11 12 12 12 12 12 13 13 13 13 13

b 0.0790 0.0791 0.0792 0.07937 0.0800 0.0710 0.0718 0.0720 0.072165 0.0730 0.0654 0.0655 0.0660 0.06614 0.0670 0.0600 0.0605 0.0607 0.060878 0.0610

ΠS (sd ) 0.868501 0.868601 0.868701 0.868871 0.869501∗ 0.843286 0.844086 0.844286 0.844450 0.845286∗ 0.821032 0.821132 0.821632 0.821772 0.822632∗ 0.799511 0.800011 0.800211 0.800389 0.800511∗

eu 1.694958 1.679036 1.659694 no sol. no sol. 1.625186 1.543516 1.508812 no sol. no sol. 1.462450 1.452624 1.383078 no sol. no sol. 1.381134 1.336900 1.313459 1.285771 1.255965 ΠS ( su ) 0.913534∗ 0.912224 0.910447 no sol. no sol. 0.887755∗ 0.878338 0.873209 no sol. no sol. 0.849907∗ 0.848287 0.835222 no sol. no sol. 0.821083∗ 0.812148 0.806949 0.800389 0.792826

X − 0.079 0.412 0.387 no sol. − 0.349 0.337 0.314 no sol. − 0.156 0.275 0.260 no sol. − 0.150 0.232 0.225 0.216

nsb u − 0.671944 0.664357 0.639162 no sol. − 0.618000 0.604643 0.581159 no sol. − 0.581155 0.555251 0.539716 no sol. − 0.535189 0.526983 0.516861 0.507091

esb n − 1.679650 1.660650 1.597525 no sol. − 1.544525 1.511000 1.452000 no sol. − 1.452775 1.386900 1.347725 no sol. − 1.337503 1.315775 1.290125 1.265375

Table 3: Optimal compensation contracts for σ = 0.3 and c = 0.4.

esb d 2.079002 2.079002 2.079002 2.079002 2.079002 2.044572 2.044572 2.044572 2.044572 2.044572 2.011263 2.011263 2.011263 2.011263 2.011263 1.979022 1.979022 1.979022 1.979022 1.979022

ΠN (nsb u) − 0.912231∗ 0.910487∗ 0.903279∗ no sol. − 0.878373∗ 0.873423∗ 0.863179∗ no sol. − 0.848287∗ 0.835751∗ 0.827197∗ no sol. − 0.812172∗ 0.807108∗ 0.801017∗ 0.794779

Π 0.4784

0.4782

0.478

0.4778

0.4776

0.4774

0.4772

0.477

0.232

0.234

0.236

0.238

0.24

b

Figure 1: Firm values net of compensation costs for α = 0.5, σ = 0.3 and c = 1. Grey su ). Dashed black line: ΠN (nsb line: ΠS (sd ). Solid black line: ΠS ( u ).

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