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Optimism bias and incentive contracts in portfolio delegation
Economic Modelling 33 (2013) 493–499
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Economic Modelling journal homepage: www.elsevier.com/locate/ecmod
Optimism bias and incentive contracts in portfolio delegation☆ Jian Wang a, Jiliang Sheng b, Jun Yang c,⁎ a b c
School of Business Administration, Northeastern University, Shenyang, 110819, China School of Information Technology, Jiangxi University of Finance and Economics, Nanchang, 330013, China School of Business Administration, Acadia University, Wolfville, NS, B4P 2R6, Canada
a r t i c l e
i n f o
Article history: Accepted 22 April 2013 Available online xxxx JEL classification: D82 D86 J33
a b s t r a c t This paper incorporates the well-documented managerial optimism bias into a standard portfolio delegation problem to study its impact on investment strategies and the optimal incentive contract offered by the investor to the manager. It is shown that the optimistic manager trades a larger quantity of the risky asset and thus takes more risk than the rational manager. Managerial optimism bias can offset her risk aversion and increase the investor's wealth by reducing moral hazard between the investor and the manager. Furthermore, a pronounced optimism bias reduces the incentive component of the incentive contract, suggesting that an optimistic manager requires fewer incentives to align her decisions with the interests of the investor. © 2013 Elsevier B.V. All rights reserved.
Keywords: Optimism bias Incentive contract Portfolio delegation Investment strategy
1. Introduction The past few decades has seen tremendous growth in delegated portfolio management. Increasingly the financial markets are dominated by professional money managers, who work for institutions such as mutual funds, pension funds, and hedge funds, and manage investment for others. Investors trust the money managers' expertise to collect and apply information to make investment decisions. However, the investors cannot observe managerial actions by the money managers, who may not work for the best interest of investors. Therefore moral hazard is a concern in delegated portfolio management industry. An extensive literature on portfolio delegation that follows the early works of Bhattacharya and Pfleiderer (1985) has demonstrated that to mitigate the moral hazard problem a manager's compensation should be linked to investor's wealth. Linear performance-adjusted compensation contracts are a popular avenue to achieve this (e.g., Admati and Pfleiderer, 1997; Cohen and Starks, 1988; Diamond, 1998; Palomino and Prat, 2003; Sheng et al., 2012). These papers on resolving the
☆ This study is supported by the National Natural Science Foundation of China (#71101024, #71161013, #71171042), the Social Science Foundation of Ministry of Education of China (#10YJC790253, #10YJC630203), the Fundamental Research Funds for the Central Universities in China (#N110406008), and the China Postdoctoral Science Foundation (#2012M511450). Financial support from the Jiangxi Postdoctoral Science Foundation of China (2012) is gratefully acknowledged. ⁎ Corresponding author. Tel.: +1 902 585 1791. E-mail address: [email protected] (J. Yang). 0264-9993/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econmod.2013.04.042
principal–agent problem in portfolio delegation are usually tied to classical rationality assumptions thereby in contradiction with the reality, as an extensive and growing literature on human psychology and behavior shows that most people tend to develop behavioral biases that can significantly influence their decisions. This issue is first addressed by Wiseman and Gomez-Mejia (1998), who bridges agency and prospect theories and proposes a behavioral agency model of executive risk taking. The impact of overconfidence on incentive contracts has been studied by Keiber (2002), who finds that the more overconfident are both the principal and the agent the lower are the agency costs. Palomino and Sadrieh (2011) derive the optimal contract, compare the performance of financial institutions hiring overconfident managers relative to institutions hiring rational agents, and examine the impact of overconfidence on asset prices. This paper studies the influence of a cognitive bias that has been extensively documented in behavioral research, namely optimism, on investment strategies in portfolio delegation and the optimal incentive contract. The bias of optimism is closely related to overconfidence, but there are clear distinctions between them. As discussed in Gervais et al. (2003), optimism can make one overestimate the probability that favorable events will occur. In contrast, overconfidence can make one think that he is more competent and skilled than others. Following Weinstein (1980) and Kunda (1987), who note that individuals expect good things to happen to themselves more often than to their peers, many studies relate that optimism bias in decision making is among the most robust findings in research on social perceptions and cognitions (Glaser et al., 2008; Helweg-Larsen and Shepperd, 2001; Simmons and Massey, 2012).
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Prior research has demonstrated the impact of managerial optimism bias on many economic phenomena, including corporate financial and accounting decisions (Hackbarth, 2008; Heaton, 2002; March and Shapira, 1987), entrepreneurial activities (Åstebro, 2003; de Meza and Southey, 1996; Landier and Thesmar, 2009), stock investments (Barberis et al., 1998; Puri and Robinson, 2007), and financial analysts' forecasts and recommendations (Carleton et al., 1998; Easterwood and Nutt, 1999; Paleari and Vismara, 2007). However, in financial economics, there has been very little research on the role that optimism bias plays in portfolio delegation. For example, are optimistic managers more willing to take risks than similarly risk-averse rational managers? If so, this should lead to predictable relations among managerial optimism, trading strategies, and optimal incentive contracts between the investor and the manager. For a number of reasons we may expect that money managers are subject to optimism bias. First, optimistic people are more willing to take risk in financial decisions (Puri and Robinson, 2007). This selfselection makes optimistic individuals more likely to pursue careers in wealth management. Second, while the success in wealth management can be attributed to many factors such as manager's ability, market condition, and luck, optimistic managers are usually more willing to take risks and thus are often awarded for better performance. Therefore optimistic managers are more likely to survive in such a highly competitive industry. Third, institutions often hire optimistic money managers either because optimism and confidence are often perceived as signs of greater ability, or because, as will be argued in this paper, an optimistic manager can better serve the interest of investors (Gervais et al., 2003). In this paper we define optimism bias as the belief that favorable future events are more likely than they actually are. This definition is motivated by Heaton (2002), who explores managerial optimism and its relation to the benefits and costs of corporate free cash flows. We show that compared to a rational manager, an optimistic manager attaches less importance to the loss stemming from risks and invests more in the risky asset, suggesting that managerial optimism bias to some extent mitigates the moral hazard between the investor and the manager. Furthermore, the optimal incentive component of the contract offered by the investor decreases with the level of managerial optimism bias, indicating that the optimistic manager is willing to accept “cheaper” contracts. Therefore we conclude that the optimistic manager is more attractive than the rational manager to the investor. However, if the investor compensates the optimistic manager as if she was rational it hurts the investor by unnecessarily transferring the investor's wealth to the manager. 2. Related literature This paper bridges the literatures on portfolio delegation and those on optimism bias in financial markets. The seminal contribution to the literature on portfolio delegation is due to Bhattacharya and Pfleiderer (1985). However, their model is more one of hidden information rather than hidden action as the principal is able to verify the level of risk taken by the agent. The related literatures that have evolved since then often propose models in which there is also hidden action and study the incentive impact of linear contracts. Diamond (1998) studies a hidden-action moral hazard problem in which the agent controls both effort and the distribution of the outcome, and proves that if the control space of the agent has full dimensionality, the optimal contract converges to a linear contract as the cost of effort shrinks. Gomez and Sharma (2006) explore the incentive impact of short-sell constraints in portfolio delegation, and show that under moral hazard, linear performance-adjusted contracts can provide managers with incentives to gather information. More recently, Li and Tiwari (2009) demonstrate that the option-type incentive helps overcome the effort-underinvestment problem that undermines linear contracts. Dybvig et al. (2010) show that trading restrictions are essential because they prevent the portfolio manager from undoing the incentive effects of performance-based fees. Kyle et al. (2011) set up a strategic trading model in portfolio delegation, and find that a higher-powered
linear contract induces the manager to exert more effort for information acquisition. Several studies in the literature are related to ours. Gervais et al. (2003) investigate the impact of overconfidence and optimism bias on executive compensation. They consider a capital budgeting problem faced by a risk-averse manager who may be overconfident and optimistic, and examine the influence of these managerial biases on executive stock options. While their definition of optimism is the same as ours, they define overconfidence as the belief that the precision of one's information is greater than it actually is. They find that overconfidence and optimism provide an alternative solution to the agency problem between shareholders and the manager. Moreover, overconfidence motivates a manager to give more effort, but optimism reduces effort. In another investigation of incentive contracts in a moral-hazard framework, de la Rosa (2011) also carefully distinguishes overconfidence and optimism. While it is found that generally there are efficiency gains stemming from the agent's overconfidence, the impact on incentive contracts depends both on the overall level of overconfidence and on the particular type of bias (optimism or overconfidence). It is clear that higher optimism or overconfidence implies a higher implemented effort level, but different kinds of overconfidence can have conflicting effects in terms of risk taking in equilibrium. Palomino and Sadrieh (2011) also investigate the impact of overconfidence in delegated portfolio management. In particular the authors examine the mechanism (over-acquisition of information) through which overconfidence generates higher trading volumes. Solely focused on the optimism bias, this paper studies the impact of optimism in a portfolio delegation context in which the manager has the choice of portfolio risk. We show how optimism bias affects the manager's investment strategy, and the closed-form optimal incentive contract for portfolio delegation is derived and discussed. Recent theoretical behavioral corporate finance literature examines optimism in corporate settings and suggests that optimism bias can substantially influence the investment and financing decisions made by business managers. Hackbarth (2008) investigates the impact of optimism bias on corporate financial policy and firm value. He proves that mildly optimistic managers ameliorate manager–shareholder conflicts. Campbell et al. (2011) suggest that high managerial optimism should cause greater levels of firm investments. Chen and Lin (2012) find that an under-invested firm with a CEO that has a high level of managerial optimism can improve the firm's investment efficiency by reducing the degree of underinvestment, further increasing the value of a firm. However, the optimal contract is not derived in these studies. The remainder of the paper proceeds as follows. Section 3 introduces the basic model underlying portfolio delegation. Section 4 introduces equilibrium in the absence of optimism bias that serves as a benchmark to study the effects of behavioral bias. Section 5 formally introduces the concept of optimism bias, and shows the influence of this individual trait on the manager's investment strategy. The optimal incentive contract based on managerial optimism bias is derived in Section 6. The conclusions are in Section 7. All proofs are collected in the Appendix A.
3. The model We consider an economy populated with a risk-neutral investor whose initial wealth for investment is one and a money manager (or just a manager for short). The manager has no initial wealth but some investment skills. The investor delegates the investment decisionmaking power to the manager. There are two types of assets on the market, a risk-free asset with constant gross return equal to one and a risky asset with gross return equal to v~ , which is either vH > 1 or vL b 1 with equal probabilities, so that Eðv~ Þ ¼ 1. Consequently, the return of the portfolio that is to be shared between the investor and the manager at ~ ðx; ~v Þ ¼ 1 þ ðv~ −1Þx, where x and 1 − x are the end of the period is w the amounts (or proportions, since the total amount is one) invested in the risky and the risk-free assets respectively.
J. Wang et al. / Economic Modelling 33 (2013) 493–499
Similar to Palomino and Sadrieh (2011), we assume that if exerting effort at a cost c~ ¼ c, c ∈ (0,1), the manager receives a private signal ~s about the return of the risky asset, but cannot credibly communicate it to the investor. The signal is either ~s ¼ sH or ~s ¼ sL , where sH and sL denote the good or bad information, corresponding to vH and vL respectively. Conditional on the signal, the distribution of the risky asset's return is Prðv~ ¼ vH j~s ¼ sH Þ ¼ Prðv~ ¼ vL j~s ¼ sL Þ ¼ ð1 þ aÞ=2
ð1Þ
Prðv~ ¼ vH j~s ¼ sL Þ ¼ Prðv~ ¼ vL j~s ¼ sH Þ ¼ ð1−aÞ=2
ð2Þ
where a ∈ (0,1) represents the manager's professional ability, such that the higher a, the better her ability to interpret the information in the signal received (i.e., to estimate the probabilities of the market conditions). The manager decides on the investment strategy according to the signal she receives, that is, she will buy on the good news (x(sH) > 0) and sell on the bad news (x(sL) b 0). In the same spirit as in Gomez and Sharma (2006), we assume that x belongs to ½−x; x , where x is finite. This means that there are some trading constraints (i.e., limits on leverage and short selling) for the agent. Thus the return of the portfolio depends on both the manager's ability and the market condition. The investor cannot observe the manager's effort level or her investment strategy. Therefore, the compensation contract that the investor offers the manager depends solely on the realized outcome ~ ðx; ~v Þ. Suppose Sðw ~ Þ represents such a contract that is assumed to w be a linear function of the portfolio return: ~ ðx; ~v ÞÞ ¼ γ þ βw ~ ðx; ~v Þ Sðw
ð3Þ
where γ > 0 and 0 b β b 1 represent the fixed and the incentive components of the manager's compensation contract respectively. Hence the final wealth of the manager depends on both the compensation ~ Þ and her effort cost c~. For not losing generality, we assume contract Sðw that the manager is assured a minimum certainty equivalent R ≥ 0. 4. Equilibrium in the absence of optimism bias 4.1. The manager's first-best strategy We assume that the investor cannot acquire information at any cost. This implies that if he does not hire a manager, his expected wealth will not exceed the return of the risk-free asset. The investor claims the residual return that remains after the manager is paid. The investor is assumed to be risk neutral and his utility function is represented by V, which depends on the portfolio return: ~ ðx; ~v Þ−Sðw ~ ÞÞ ¼ ð1−βÞ½1 þ ðv~ −1Þx−γ: V ðw
ð4Þ
We first investigate the optimal investment strategy from the investor's perspective. That is, if the investor could receive the signal as the manager, then his expected utility evolves as different information is gathered. The constraint Eðv~ Þ ¼ vH =2 þ vL =2 ¼ 1 is equivalent to vH − vL = 2(vH − 1) = 2(1 − vL). So the investor's expected utility is:
where x(sH) > 0 and x(sL) b 0 represent the optimal trading quantities of the risky asset bought and sold on good and bad news respectively. And the investor's optimization problem is ~ ðx; ~v Þ−Sðw ~ ÞÞj~s max E½V ðw x
where x is the decision variable. Eq. (5) indicates that the investor's expected utility increases with the absolute value of x(sH) and x(sL). Thus, it is easy to show that the optimal strategy is xFB ðsH Þ ¼ −xFB ðsL Þ ¼ x, which is the wealth-maximizing strategy – namely the first-best strategy – for the investor, and we use a subscript “FB” to denote it. The investor's expected utility under the optimal strategy is: ~ ðx; ~v Þ−Sðw ~ ÞÞj~s ¼ ð1−βÞ½1 þ aðvH −1Þx −γ EFB ½V ðw
4.2. The effect of risk aversion The first-best outcome maximizes the monetary outcome of the investment to the risk-neutral investor. To attain it, the manager must not care about risk when making her investment decision. However, the investment strategy has to be carried out by the manager whose sole income is tied to the investment. In the spirit of Jensen and Meckling (1976), we assume that, because the manager's human capital is less diversified than that of the investor, she is risk-averse about the outcome of her compensation. Incentive contracts can be used to reduce these agency costs by realigning the objective of the agent with that of the principle. We will discuss such contracts in Section 6. For now, we concentrate on how the risk aversion of the manager affects her investment strategy and in turn the payoff of the investment. The risk-averse manager has the utility function U as preference representation over her end-of-period wealth. Specifically we assume that the manager has a constant absolute risk aversion (CARA) utility function with an absolute risk aversion coefficient ρ: ~ ¼ − exp −ρW ~ U W
ð7Þ
~ is the wealth of the manager at the end of the period. where W According to the signal observed, the manager decides on the optimal investment strategy, which is given in Proposition 1. Proposition 1. Suppose that the investment is managed by a risk-averse manager with risk aversion ρ > 0 and ability a. The optimal investment strategy for the manager is
1þa ln 1−a ≤ xFB ðsH Þ ρβðvH −vL Þ
x ðsL Þ ¼ −x ðsH Þ
ð5Þ
ð6Þ
Eq. (6) implies that under the condition that the investor is risk-neutral (i.e., his utility is independent of risk taking), the optimal strategy is to take (either buy or sell) the largest position allowed in the risky asset given the signal received regarding the upcoming market condition. This strategy can also be reached by assuming that the manager is risk-neutral, as the personal objective of this agent is then precisely to maximize the final payoff of the portfolio.
x ðsH Þ ¼
11 þ a ~ ðx; ~v Þ−Sðw ~ ÞÞj~s ¼ E½V ðw fð1−βÞ½1 þ ðvH −1ÞxðsH Þ−γ g 2 2 1 1−a þ fð1−βÞ½1 þ ðvL −1ÞxðsH Þ−γ g 2 2 1 1−a þ fð1−βÞ½1 þ ðvH −1ÞxðsL Þ−γ g 2 2 11þ a þ fð1−βÞ½1 þ ðvL −1ÞxðsL Þ−γg 2 2 ð1−βÞ ¼ f2 þ aðvH −1Þ½xðsH Þ−xðsL Þg−γ 2
495
ð8Þ ð9Þ
where x ∗(sH) > 0 and x∗(sL) b 0 represent the optimal trading quantities of the risky asset bought and sold on good and bad news respectively. By inspection of Proposition 1 we realize that the absolute value of both x*(sH) and x*(sL) decreases with ρ, which implies that risk aversion makes the manager cut down the trading quantity of the risky asset. Then the derivation of Proposition 1 delivers the insight of Corollary 1 immediately.
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Corollary 1. Under the manager's optimal investment strategy, the investor's expected utility is a 1þa ~ ðx; ~v Þ−Sðw ~ ÞÞj~s ¼ ð1−βÞ 1 þ ~ ðx; ~v Þ−Sðw ~ ÞÞj~s E V ðw ln −γ ≤ EFB ½V ðw 2ρβ 1−a
He does revise his beliefs upwards (downwards) after receiving a positive (negative) signal, but the resulting posterior is still more optimistic than it should be. 5.2. Investment strategies with optimism bias
ð10Þ ~ ðx; v~ Þ−Sðw ~ ÞÞj~s represents the investor's utility of the where EFB ½V ðw first-best strategy. As we can see in Eqs. (5) and (10), the investor's wealth is increasing in his agent's ability a. However, the investor's wealth is strictly decreasing in his agent's risk aversion ρ. The loss of investor's wealth results from the fact that the trading quantity of the risky asset is reduced. The intuition of this result is fairly straightforward. Due to the imperfect signal, for a risk-averse manager, the shock that results from making a wrong decision is more severe than that for a risk-neutral manager. Consequently, the risk-averse manager cuts down the trading quantity of the risky asset. The manager's utility gained from reducing risk is not transferred to the investor. Thus the investor suffers a loss, namely the agency cost. As shown in Eq. (10), the more risk-averse the manager, the lower the investor's utility, and so the larger the agency cost. This is a well-known result in principal–agent theories. The moral hazard problem causes disutility for the investor. 5. The role of optimism bias We have shown that the final payoff to the risk-neutral investor is negatively affected by the manager's risk aversion. In this section, we show how managerial optimism bias may help to restore the wealth of the investor. 5.1. Definitions It is common for people to be subjected to optimism bias when making decisions. And the manager's psychology bias, which refers to ex ante view of the risky asset's return in our model, is doomed to affect the investor. Following Heaton (2002), we define optimism to be the ex-ante belief that the risky asset's return is, on average, more than it actually is. More specifically, we assume that the optimistic manager thinks that the probability of good outcome for the risky asset (v = vH) is not 1/2, but B∈ 12 ; 1 , that is Prðv~ ¼ vH > 1Þ ¼ B
ð11Þ
Prðv~ ¼ vL b1Þ ¼ 1−B
ð12Þ
and d≡B− 12 ∈ 0; 12 measures the level of managerial optimism bias. As d → 0, the manager approaches rationality; as d → 1/2, she thinks the good state is much more likely to happen than the bad state and effectively becomes overoptimistic. The following lemma shows how this bias will affect the way that imperfect information is interpreted by the manager. Lemma 1. A manager with optimism bias thinks that Prb ðv~ ¼ vH j~s ¼ sH Þ ¼ a þ ð1−aÞB≡ ϕ1 ða; BÞ
ð13Þ
Prb ðv~ ¼ vH j~s ¼ sL Þ ¼ ð1−aÞB≡ ϕ2 ða; BÞ
ð14Þ
Prb ðv~ ¼ vL j~s ¼ sH Þ ¼ ð1−aÞð1−BÞ≡ ϕ3 ða; BÞ
ð15Þ
Prb ðv~ ¼ vL j~s ¼ sL Þ ¼ a þ ð1−aÞð1−BÞ≡ϕ4 ða; BÞ
ð16Þ
where the subscript “b” refers to the fact that the manager is biased. Apparently ϕ1(a,B) and ϕ2(a,B) are increasing in B, whereas ϕ3(a,B) and ϕ4(a,B) are decreasing in B. This is very intuitive — the optimistic manager always thinks that the investment is better than it really is.
The optimistic agent operates under the assumption that the return of the risky asset is intrinsically better than it actually is. Now we will investigate how the manager's psychological bias affects the investor's final wealth. More precisely, given the manager's risk aversion ρ, we would like to know the difference of the investor's utility between hiring a rational manager and hiring a biased manager. The following proposition derives the manager's optimal strategy and resultant investor's utility when the manager is optimistically biased. Proposition 2. Suppose that the investment is managed by a manager with risk aversion ρ > 0, ability a, and optimism bias B. Then the optimal investment strategy for the manager is
xb ðsH Þ ¼
xb ðsL Þ ¼
ln ϕϕ1 3
ρβðvH −vL Þ ln ϕϕ2 4
ρβðvH −vL Þ
¼
aþð1−aÞB ln ð1−a Þð1−BÞ
ρβðvH −vL Þ
¼−
Þð1−BÞ ln aþð1−a ð1−aÞB
ρβðvH −vL Þ
ð17Þ
ð18Þ
where xb∗ (sH) > 0 and xb∗ (sL) b 0 represent the optimal quantities of the risky asset bought and sold upon good and bad news respectively.With the optimal strategy under optimism bias, the investor's expected utility is a ϕ ϕ ~ ðx; ~v Þ−Sðw ~ ÞÞj~s ¼ ð1−β Þ 1 þ Eb ½V ðw ln 1 4 −γ 4ρβ ϕ2 ϕ3 1 ~ ðx; ~v Þ−Sðw ~ ÞÞj~s ∀B∈ ; 1 : ≥ E V ðw 2
ð19Þ
In the first part of Proposition 2, we can see that for the optimistic manager, xb∗ (sH) is bigger than x*(sH) whereas the absolute value of xb∗ (sL) is smaller than that of x*(sL). However, the increase of the former is larger than the decrease of the latter, so the optimism bias is useful in enhancing the investor's wealth. The intuition behind this result is that the biased manager, who is subject to optimism, thinks the return of the risky asset is larger than it really is, so attaches less importance to the loss stemming from risk, and thus enhances the trading quantity of the risky asset. In other words, the manager's optimism bias partially offsets her risk aversion. The second part of Proposition 2 tells us that managerial optimism bias somewhat alleviates conflicts between the investor and the manager. 6. Optimal incentive contracts based on optimism bias As is standard in the contract theory, we assume that the principal has all the bargaining power and makes a take-it-or-leave-it offer to the manager. The main assumption which makes this model differ from the standard principal-agent model is that the agent is subject to optimism bias with respect to the return of the risky asset. Similar to de la Rosa (2011), we assume that the investor knows that the manager is biased, and the manager knows that the investor thinks that way, but disagrees with him. Therefore, the investor and the manager “agree to disagree”. The investor's problem is to choose the incentive contract ~ Þ that maximizes his final wealth. Sb ðw The determination of the optimal incentive contract boils down to the investor's choice of γb∗ and βb∗ , which represent the contract's parameters. The timing of the players' actions and events is depicted in Fig. 1. At t = 0 the investor observes the manager's bias and makes a delegation contract offer (i.e., choosing γb∗ and βb∗ ). The manager decides whether or not to accept the investor's offer at t = 1. If the
J. Wang et al. / Economic Modelling 33 (2013) 493–499
The investor observes
The manager
If the manager
the manager’s bias and
accepts or
accepts offer she
offers the contract
rejects offer
expends effort
t=0
t=1
497
The manager
The manager
receives
chooses investment
signal
t=2
t=3
strategy
The investor observes and the manager is compensated
t=4
t=5
Fig. 1. Timing of the players' actions and events.
manager accepts the offer she expends effort at t = 2, and receives a private signal ~s at t = 3. At t = 4 the manager chooses the investment strategy xb ð~s Þ. The resolution of uncertainty occurs at t = 5 and the manager is compensated. The optimal incentive contract is the solution to the following constrained problem: ~ ðx; ~v Þ−Sb ðw ~ ÞÞj~s max Eb ½V ðw fSb ðw~ Þg ~ Þ ¼ γb þ βb w ~ s:t: Sb ðw xb ð~s Þ∈½−x; x
ð20Þ ð21Þ
~ ðx; ~v ÞÞ−cj~s g xb ð~s Þ∈ arg maxEb fU ½Sb ðw
ð22Þ
~ ðx; ~v ÞÞ−cjc~ ¼ cg ≥ Eb fU ½Sb ðw ~ Þjc~ ¼ 0g Eb fU ½Sb ðw
ð23Þ
~ ðx; ~v ÞÞ−cjc~ ¼ cg ≥ U ðRÞ Eb fU ½Sb ðw
ð24Þ
where γb and βb, the parameters of the contract offered to the biased manager, are the decision variables. Eq. (20) is the end-of-period compensation of the manager. Eq. (21) is the optimal asset selection constraint for the manager. Eqs. (22) and (23) are the incentive compatibility constraints guaranteeing that the manager maximizes her expected utility by exerting effort and allocating assets. Eq. (24) represents the participation constraint ensuring that it is rational for the manager to enter the principal–agent relationship because the engagement produces at least her reservation utility. The optimal incentive contract is given in Proposition 3. Proposition 3. Suppose that the manager hired by the investor is characterized by risk aversion ρ, ability a, and optimism bias B. Then the optimal incentive contract that realigns her incentives with that of the risk-neutral investor is
γb ¼ R
ð25Þ
and
βb ¼
i 1 1 1 h ln ðϕ1 ϕ3 Þ2 þ ðϕ2 ϕ4 Þ2 þ c: ρ
ð26Þ
By inspection of this proposition we realize that the manager's optimal fixed compensation depends on his reservation utility only, while the optimal wealth-sharing parameter βb∗ is a function of the manager's risk aversion, ability, and optimism bias. The sensitivity of the variable compensation βb∗ with respect to the level of managerial optimism bias d is gathered in Proposition 4. Proposition 4. Suppose that the manager hired by the investor is characterized by risk aversion ρ, ability a, and optimism bias d. Then the optimal incentive component of the contract βb∗ proposed by the risk-neutral investor is decreasing in the level of the managerial optimism bias d. In order to interpret this result with respect to the level of managerial bias d properly, we here emphasize again that d is used to measure how optimistic the manager is — the higher the level of bias is, the more optimistic the future is judged by the manager. Thus, Proposition 4 shows that a manager subject to optimism bias believes that she will realize a good
performance with a higher probability than the true one, so she accepts a “cheaper” contract than a rational manager would. This insight lets us conclude that the optimism bias makes an optimistic manager more attractive than a rational manager for the investor. This result is consistent with those of Gervais et al. (2003) in the corporate environment: moderate optimism makes managers reduce the need for option compensation. However, if the investor cannot observe the managerial optimism bias, he would offer the manager higher incentive compensation than actually needed, and this is virtually detrimental for the investor in terms of his expected utility. Another implication of this result is that incentive contracts can be used to screen prospective managers — those willing to accept a low-incentive contract may be optimistic. When considering Propositions 2 and 4 together we realize that managerial optimism bias can actually serve well the interest of the investor in at least two ways. First, it can reduce the agency cost caused by the different levels of risk aversion between the investor and the manager. Second, it costs less to hire an optimistic investor manager. This result can be added to the list of benefits of optimism as identified in Puri and Robinson (2007), which includes working harder, retiring later, saving more, and investing more in stocks. While psychological biases such as optimism may sound negative and are often treated as complications in traditional models with rational agents, they can actually have some positive impacts to decision making and social welfare. 7. Conclusion Within a principal–agent framework, this paper explores the investment strategy and the optimal incentive contract in portfolio delegation based on managerial optimism bias. Specifically, we assume that the manager is optimistic, and thus biased, when assessing the return of the risky asset. We find that the optimistic manager behaves as if she perceives less risk when allocating assets and show that such a biased manager can be beneficial to the investor because it takes less in compensation to align her decisions with the investor's preference. However, compensating the optimistic manager as if she was unbiased reduces the investor's wealth due to overpayment. These results differ significantly from those of portfolio delegation models under rationality assumptions. Future research may explore the incentive impact of optimism bias when the investor is not well diversified and therefore concerned with the portfolio risk. Appendix A Proof of Proposition 1 Assuming that after paying a cost c, the manager receives a signal ~s ¼ sH . 1 From Eq. (7) the manager maximizes h 1 þ a −ρfγþβ½1þxðvH −1Þ−cg ~ w ~ ðx; ~v ÞÞ−c sH ¼ E U Sð −e 2 1−a −ρfγþβ½1þxðvL −1Þ−cg þ −e : 2 1
The derivation for the case ~s ¼ sL is identical.
ð27Þ
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J. Wang et al. / Economic Modelling 33 (2013) 493–499
The first-order condition of utility maximization yields
The first-order condition of utility maximization yields
x ðsH Þ ¼
1þa ln 1−a > 0: ρβðvH −vL Þ
xb ðsH Þ ¼
Proof of Corollary 1
ln ϕϕ1 3
ρβðvH −vL Þ
¼
aþð1−aÞB ln ð1−a Þð1−BÞ
ρβðvH −vL Þ
:
Then the investor's expected utility based on managerial bias is:
Given that Eðv~ Þ ¼ v2H þ v2L ¼ 1, we have vH − vL = 2(vH − 1) = 2(1 − vL), and x*(sL) = − x*(sH). From Proposition 1, the investor's expected utility is:
11 þ a ð1−βÞ 1 þ x ðsH ÞðvH −1Þ −γ 2 2
1 1−a ð1−βÞ 1 þ x ðsH ÞðvL −1Þ −γ þ 2 2
1 1−a ð1−βÞ 1 þ x ðsL ÞðvH −1Þ −γ þ 2 2
11 þ a ð1−βÞ 1 þ x ðsL ÞðvL −1Þ −γ þ 2 2 ¼ ð1−βÞ 1 þ ax ðsH ÞðvH −1Þ −γ a 1þa ln −γ ¼ ð1−βÞ 1 þ 2ρβ 1−a
11 þ a ð1−βÞ 1 þ xb ðsH ÞðvH −1Þ −γ 2 2
1 1−a ð1−βÞ 1 þ xb ðsH ÞðvL −1Þ −γ þ 2 2
1 1−a ð1−βÞ 1 þ xb ðsL ÞðvH −1Þ −γ þ 2 2
11 þ a ð1−βÞ 1 þ xb ðsL ÞðvL −1Þ −γ þ 2 2 a ϕ ϕ ln 1 4 −γ ¼ ð1−βÞ 1 þ 4ρβ ϕ2 ϕ3
~ ðx; ~v Þ−Sðw ~ ÞÞj~s ¼ Eb ½V ðw
~ ðx; ~v Þ−Sðw ~ ÞÞj~s ¼ E ½V ðw
ð30Þ and from Eq. (10) ~ ðx; ~v Þ−Sðw ~ ÞÞj~s −E ½V ðw ~ ðx; ~v Þ−Sðw ~ ÞÞj~s Eb ½V ðw
ð28Þ Comparing Eq. (28) with Eq. (6), due to x ðsH Þ ≤ x, we can get ~ ðx; ~v Þ−Sðw ~ ÞÞj~s ≤ EFB ½V ðw ~ ðx; ~v Þ−Sðw ~ ÞÞj~s . E ½V ðw
ð1−aÞ2 Bð1−BÞ
≥0:
Proof of Proposition 3 From Proposition 2, Eq. (22) is equivalent to
Proof of Lemma 1
xb ðsH Þ ¼
Using Bayes' rule, we have Prb ðv~ ¼ vH j~s ¼ sH Þ ¼ ¼
Prb sH jvH ÞPrb ðvH Þ Prb ðsH jvH ÞPrb ðvH Þ þ Prb ðsH jvL ÞPrb ðvL Þ
xb ðsL Þ ¼
½a þ ð1−aÞBB ½a þ ð1−aÞBB þ ð1−aÞBð1−BÞ
ð1−aÞð1−BÞB ð1−aÞð1−BÞB þ ½a þ ð1−aÞð1−BÞð1−BÞ
¼
¼ ð1−aÞB and
ρβðvH −vL Þ ln ϕϕ2 4
ρβðvH −vL Þ
¼
aþð1−aÞB ln ð1−a Þð1−BÞ
ð31Þ
ρβðvH −vL Þ
¼−
Þð1−BÞ ln aþð1−a ð1−aÞB
ρβðvH −vL Þ
:
ð1−aÞBð1−BÞ ð1−aÞBð1−BÞ þ ½a þ ð1−aÞBB
Prb ðv~ ¼ vL j~s ¼ sH Þ ¼
ð32Þ
ð33Þ
¼ ð1−aÞð1−BÞ Prb ðv~ ¼ vL j~s ¼ sL Þ ¼
3
h i 1 ~ ðx; ~v Þ−cÞc~ ¼ c ¼ ϕ1 −e−ρfγþβ½1þxb ðsH ÞðvH −1Þ−cg Eb U S~b ðw 2 1 −ρ γþβ 1þx ðs Þðv −1Þ −c þ ϕ2 −e f ½ b L H g 2 1 h −ρ γþβ 1þx ðs Þðv −1Þ −c i þ ϕ3 −e f ½ b H L g 2 1 −ρ γþβ 1þx ðs Þðv −1Þ −c þ ϕ4 −e f ½ b L L g 2 h ih i 1 1 −ρðγþβ−cÞ : ¼ ðϕ1 ϕ3 Þ2 þ ðϕ2 ϕ4 Þ2 −e
Prb ðsL jvH ÞPrb ðvH Þ Prb ðsL jvH ÞPrb ðvH Þ þ Prb ðsL jvL ÞPrb ðvL Þ
Prbðv~ ¼ vH j~s ¼ sL Þ ¼
ln ϕϕ1
The expect utility of the optimistic manager when she exerts effort is
¼ a þ ð1−aÞB
The expect utility of the optimistic manager when she exerts no effort is
½a þ ð1−aÞð1−BÞð1−BÞ ½a þ ð1−aÞð1−BÞð1−BÞ þ ð1−aÞB
h i ~ Þ c~ ¼ 0 ¼ −e−ργ : Eb U S~b ðw
¼ a þ ð1−aÞð1−BÞ:
ð34Þ
Using these equations in the constrained program, we can get γb∗ and βb∗ .
Proof of Proposition 2 Assuming that after paying a cost c, the biased manager receives a signal sH. 2 From Lemma 1, she maximizes ~ ðx; ~v ÞÞ−cÞjsH ¼ ϕ1 ð−e Eb ½UðS~ðw
−ρfγþβ ½1þxb ðvH −1Þ−cg
Þ
−ρfγþβ½1þxb ðvL −1Þ−cg
þϕ3 ð−e
2
¼
h i 2 2 a þ að1−aÞ 2 B− 12 þ 12
The proof for the case s = sL is identical.
Þ:
ð29Þ
Proof of Proposition 4 Due to d ¼ B− 12, Eq. (26) is equivalent to ( 1
1 2 2 1 1 1 ð1 þ aÞ þ ð1−aÞd ð1−aÞ −d βb ¼ ln ρ 2 2
1 1 ) 2 1 2 1 þ c: þ ð1−aÞ þd ð1 þ aÞ−ð1−aÞd 2 2
ð35Þ
J. Wang et al. / Economic Modelling 33 (2013) 493–499
We define ϕ′ 1 ≡ 12 ð1 þ aÞ þ ð1−aÞd, ϕ′ 3 ≡ð1−aÞ 12−d , ϕ′ 2 ≡ð1− 12 ϕ′ 4 ≡ 12 ð1 þ aÞ−ð1−aÞd, then βb ¼ ρ1 ln ϕ′ 1 ϕ′ 3 þ aÞ 12 þ d , 12 ϕ′ 2 ϕ′ 4 g þ c. ∂ðϕ′ 1 ϕ′ 3 Þ ∂ðϕ′ 2 ϕ′ 4 Þ ¼ −ð1−aÞ½a þ 2ð1−aÞd and n≡ ∂d ¼ ð1− Let m≡ ∂d aÞ½a−2ð1−aÞd, then −1 −1 ∂βb 1 1 ′ ′ ′ ′ ϕ 1ϕ 3 2m þ ϕ 2ϕ 4 2n : ¼ 1 1 2ρ ϕ′ ϕ′ 2 þ ϕ′ ϕ′ 2 ∂d 1 3 2 4
ð36Þ
h i 2 m2 ϕ′ 2 ϕ′ 4 −n2 ϕ′ 1 ϕ′ 3 ¼ 2að1−aÞd a2 þ 4ð1−aÞ2 d þ h i −12 b0, we can get 4að1−aÞd ϕ′ 1 ϕ′ 3 þ ϕ′ 2 ϕ′ 4 > 0, and m ϕ′ 1 ϕ′ 3 As
∂β b ∂d
b0, which arrives at Proposition 4.
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Heaton, J.B., 2002. Managerial optimism and corporate finance. Financial Management 31 (2), 33–45. Helweg-Larsen, M., Shepperd, J.A., 2001. Do moderators of the optimistic bias affect personal or target risk estimates? A review of the literature. Personality and Social Psychology Review 5 (1), 74–95. Jensen, M.C., Meckling, W.H., 1976. Theory of the firm: managerial behavior, agency costs and ownership structure. Journal of Financial Economics 3 (4), 305–360. Keiber, K.L., 2002. Managerial compensation contracts and overconfidence. Working paper available at SSRN: http://ssrn.com/abstract=302416. Kunda, Z., 1987. Motivated inference: self-serving generation and evaluation of causal theories. Journal of Personality and Social Psychology 53 (4), 636–647. Kyle, A.S., Ou-Yang, H., Wei, B., 2011. A model of portfolio delegation and strategic trading. Review of Financial Studies 24 (11), 3778–3812. Landier, A., Thesmar, D., 2009. Financial contracting with optimistic entrepreneurs. Review of Financial Studies 22 (1), 117–150. Li, W., Tiwari, A., 2009. Incentive contracts in delegated portfolio management. Review of Financial Studies 22 (12), 4681–4714. March, J.G., Shapira, Z., 1987. Managerial perspectives on risk and risk taking. Management Science 33 (11), 1404–1418. Paleari, S., Vismara, S., 2007. Over-optimism when pricing IPOs. Managerial Finance 33 (6), 352–367. Palomino, F., Prat, A., 2003. Risk taking and optimal contracts for money managers. The RAND Journal of Economics 34 (1), 113–137. Palomino, F., Sadrieh, A., 2011. Overconfidence and delegated portfolio management. Journal of Financial Intermediation 20 (2), 159–177. Puri, M., Robinson, D.T., 2007. Optimism and economic choice. Journal of Financial Economics 86 (1), 71–99. Sheng, J., Wang, X., Yang, J., 2012. Incentive contracts in delegated portfolio management under VaR constraint. Economic Modelling 29 (5), 1679–1685. Simmons, J.P., Massey, C., 2012. Is optimism real? Journal of Experimental Psychology. General 141 (4), 630–634. Weinstein, N.D., 1980. Unrealistic optimism about future life events. Journal of Personality and Social Psychology 39 (5), 806–820. Wiseman, R., Gomez-Mejia, L., 1998. A behavioral agency model of managerial risk taking. Academy of Management Review 23 (1), 133–152. Dr. Jian Wang is an Associate Professor of finance at the School of Business Administration, Northeastern University, China. She received her Ph.D. in management science and engineering from Northeastern University of China. Her research interests include behavioral finance, portfolio management, and agency problems. Dr. Jiliang Sheng is an Associate Professor of finance at the School of Information Technology, Jiangxi University of Finance and Economics, China. He received his Ph.D. in finance from University of Electronic Science and Technology of China. His research interests include portfolio management, asset pricing, and risk management.
Dr. Jun Yang is a Professor of finance at the F.C. Manning School of Business Administration, Acadia University, Canada. He received his Ph.D. in finance from Queen's University, Canada. His research interests include corporate governance, asset pricing, and risk management.
Contents lists available at SciVerse ScienceDirect
Economic Modelling journal homepage: www.elsevier.com/locate/ecmod
Optimism bias and incentive contracts in portfolio delegation☆ Jian Wang a, Jiliang Sheng b, Jun Yang c,⁎ a b c
School of Business Administration, Northeastern University, Shenyang, 110819, China School of Information Technology, Jiangxi University of Finance and Economics, Nanchang, 330013, China School of Business Administration, Acadia University, Wolfville, NS, B4P 2R6, Canada
a r t i c l e
i n f o
Article history: Accepted 22 April 2013 Available online xxxx JEL classification: D82 D86 J33
a b s t r a c t This paper incorporates the well-documented managerial optimism bias into a standard portfolio delegation problem to study its impact on investment strategies and the optimal incentive contract offered by the investor to the manager. It is shown that the optimistic manager trades a larger quantity of the risky asset and thus takes more risk than the rational manager. Managerial optimism bias can offset her risk aversion and increase the investor's wealth by reducing moral hazard between the investor and the manager. Furthermore, a pronounced optimism bias reduces the incentive component of the incentive contract, suggesting that an optimistic manager requires fewer incentives to align her decisions with the interests of the investor. © 2013 Elsevier B.V. All rights reserved.
Keywords: Optimism bias Incentive contract Portfolio delegation Investment strategy
1. Introduction The past few decades has seen tremendous growth in delegated portfolio management. Increasingly the financial markets are dominated by professional money managers, who work for institutions such as mutual funds, pension funds, and hedge funds, and manage investment for others. Investors trust the money managers' expertise to collect and apply information to make investment decisions. However, the investors cannot observe managerial actions by the money managers, who may not work for the best interest of investors. Therefore moral hazard is a concern in delegated portfolio management industry. An extensive literature on portfolio delegation that follows the early works of Bhattacharya and Pfleiderer (1985) has demonstrated that to mitigate the moral hazard problem a manager's compensation should be linked to investor's wealth. Linear performance-adjusted compensation contracts are a popular avenue to achieve this (e.g., Admati and Pfleiderer, 1997; Cohen and Starks, 1988; Diamond, 1998; Palomino and Prat, 2003; Sheng et al., 2012). These papers on resolving the
☆ This study is supported by the National Natural Science Foundation of China (#71101024, #71161013, #71171042), the Social Science Foundation of Ministry of Education of China (#10YJC790253, #10YJC630203), the Fundamental Research Funds for the Central Universities in China (#N110406008), and the China Postdoctoral Science Foundation (#2012M511450). Financial support from the Jiangxi Postdoctoral Science Foundation of China (2012) is gratefully acknowledged. ⁎ Corresponding author. Tel.: +1 902 585 1791. E-mail address: [email protected] (J. Yang). 0264-9993/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.econmod.2013.04.042
principal–agent problem in portfolio delegation are usually tied to classical rationality assumptions thereby in contradiction with the reality, as an extensive and growing literature on human psychology and behavior shows that most people tend to develop behavioral biases that can significantly influence their decisions. This issue is first addressed by Wiseman and Gomez-Mejia (1998), who bridges agency and prospect theories and proposes a behavioral agency model of executive risk taking. The impact of overconfidence on incentive contracts has been studied by Keiber (2002), who finds that the more overconfident are both the principal and the agent the lower are the agency costs. Palomino and Sadrieh (2011) derive the optimal contract, compare the performance of financial institutions hiring overconfident managers relative to institutions hiring rational agents, and examine the impact of overconfidence on asset prices. This paper studies the influence of a cognitive bias that has been extensively documented in behavioral research, namely optimism, on investment strategies in portfolio delegation and the optimal incentive contract. The bias of optimism is closely related to overconfidence, but there are clear distinctions between them. As discussed in Gervais et al. (2003), optimism can make one overestimate the probability that favorable events will occur. In contrast, overconfidence can make one think that he is more competent and skilled than others. Following Weinstein (1980) and Kunda (1987), who note that individuals expect good things to happen to themselves more often than to their peers, many studies relate that optimism bias in decision making is among the most robust findings in research on social perceptions and cognitions (Glaser et al., 2008; Helweg-Larsen and Shepperd, 2001; Simmons and Massey, 2012).
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J. Wang et al. / Economic Modelling 33 (2013) 493–499
Prior research has demonstrated the impact of managerial optimism bias on many economic phenomena, including corporate financial and accounting decisions (Hackbarth, 2008; Heaton, 2002; March and Shapira, 1987), entrepreneurial activities (Åstebro, 2003; de Meza and Southey, 1996; Landier and Thesmar, 2009), stock investments (Barberis et al., 1998; Puri and Robinson, 2007), and financial analysts' forecasts and recommendations (Carleton et al., 1998; Easterwood and Nutt, 1999; Paleari and Vismara, 2007). However, in financial economics, there has been very little research on the role that optimism bias plays in portfolio delegation. For example, are optimistic managers more willing to take risks than similarly risk-averse rational managers? If so, this should lead to predictable relations among managerial optimism, trading strategies, and optimal incentive contracts between the investor and the manager. For a number of reasons we may expect that money managers are subject to optimism bias. First, optimistic people are more willing to take risk in financial decisions (Puri and Robinson, 2007). This selfselection makes optimistic individuals more likely to pursue careers in wealth management. Second, while the success in wealth management can be attributed to many factors such as manager's ability, market condition, and luck, optimistic managers are usually more willing to take risks and thus are often awarded for better performance. Therefore optimistic managers are more likely to survive in such a highly competitive industry. Third, institutions often hire optimistic money managers either because optimism and confidence are often perceived as signs of greater ability, or because, as will be argued in this paper, an optimistic manager can better serve the interest of investors (Gervais et al., 2003). In this paper we define optimism bias as the belief that favorable future events are more likely than they actually are. This definition is motivated by Heaton (2002), who explores managerial optimism and its relation to the benefits and costs of corporate free cash flows. We show that compared to a rational manager, an optimistic manager attaches less importance to the loss stemming from risks and invests more in the risky asset, suggesting that managerial optimism bias to some extent mitigates the moral hazard between the investor and the manager. Furthermore, the optimal incentive component of the contract offered by the investor decreases with the level of managerial optimism bias, indicating that the optimistic manager is willing to accept “cheaper” contracts. Therefore we conclude that the optimistic manager is more attractive than the rational manager to the investor. However, if the investor compensates the optimistic manager as if she was rational it hurts the investor by unnecessarily transferring the investor's wealth to the manager. 2. Related literature This paper bridges the literatures on portfolio delegation and those on optimism bias in financial markets. The seminal contribution to the literature on portfolio delegation is due to Bhattacharya and Pfleiderer (1985). However, their model is more one of hidden information rather than hidden action as the principal is able to verify the level of risk taken by the agent. The related literatures that have evolved since then often propose models in which there is also hidden action and study the incentive impact of linear contracts. Diamond (1998) studies a hidden-action moral hazard problem in which the agent controls both effort and the distribution of the outcome, and proves that if the control space of the agent has full dimensionality, the optimal contract converges to a linear contract as the cost of effort shrinks. Gomez and Sharma (2006) explore the incentive impact of short-sell constraints in portfolio delegation, and show that under moral hazard, linear performance-adjusted contracts can provide managers with incentives to gather information. More recently, Li and Tiwari (2009) demonstrate that the option-type incentive helps overcome the effort-underinvestment problem that undermines linear contracts. Dybvig et al. (2010) show that trading restrictions are essential because they prevent the portfolio manager from undoing the incentive effects of performance-based fees. Kyle et al. (2011) set up a strategic trading model in portfolio delegation, and find that a higher-powered
linear contract induces the manager to exert more effort for information acquisition. Several studies in the literature are related to ours. Gervais et al. (2003) investigate the impact of overconfidence and optimism bias on executive compensation. They consider a capital budgeting problem faced by a risk-averse manager who may be overconfident and optimistic, and examine the influence of these managerial biases on executive stock options. While their definition of optimism is the same as ours, they define overconfidence as the belief that the precision of one's information is greater than it actually is. They find that overconfidence and optimism provide an alternative solution to the agency problem between shareholders and the manager. Moreover, overconfidence motivates a manager to give more effort, but optimism reduces effort. In another investigation of incentive contracts in a moral-hazard framework, de la Rosa (2011) also carefully distinguishes overconfidence and optimism. While it is found that generally there are efficiency gains stemming from the agent's overconfidence, the impact on incentive contracts depends both on the overall level of overconfidence and on the particular type of bias (optimism or overconfidence). It is clear that higher optimism or overconfidence implies a higher implemented effort level, but different kinds of overconfidence can have conflicting effects in terms of risk taking in equilibrium. Palomino and Sadrieh (2011) also investigate the impact of overconfidence in delegated portfolio management. In particular the authors examine the mechanism (over-acquisition of information) through which overconfidence generates higher trading volumes. Solely focused on the optimism bias, this paper studies the impact of optimism in a portfolio delegation context in which the manager has the choice of portfolio risk. We show how optimism bias affects the manager's investment strategy, and the closed-form optimal incentive contract for portfolio delegation is derived and discussed. Recent theoretical behavioral corporate finance literature examines optimism in corporate settings and suggests that optimism bias can substantially influence the investment and financing decisions made by business managers. Hackbarth (2008) investigates the impact of optimism bias on corporate financial policy and firm value. He proves that mildly optimistic managers ameliorate manager–shareholder conflicts. Campbell et al. (2011) suggest that high managerial optimism should cause greater levels of firm investments. Chen and Lin (2012) find that an under-invested firm with a CEO that has a high level of managerial optimism can improve the firm's investment efficiency by reducing the degree of underinvestment, further increasing the value of a firm. However, the optimal contract is not derived in these studies. The remainder of the paper proceeds as follows. Section 3 introduces the basic model underlying portfolio delegation. Section 4 introduces equilibrium in the absence of optimism bias that serves as a benchmark to study the effects of behavioral bias. Section 5 formally introduces the concept of optimism bias, and shows the influence of this individual trait on the manager's investment strategy. The optimal incentive contract based on managerial optimism bias is derived in Section 6. The conclusions are in Section 7. All proofs are collected in the Appendix A.
3. The model We consider an economy populated with a risk-neutral investor whose initial wealth for investment is one and a money manager (or just a manager for short). The manager has no initial wealth but some investment skills. The investor delegates the investment decisionmaking power to the manager. There are two types of assets on the market, a risk-free asset with constant gross return equal to one and a risky asset with gross return equal to v~ , which is either vH > 1 or vL b 1 with equal probabilities, so that Eðv~ Þ ¼ 1. Consequently, the return of the portfolio that is to be shared between the investor and the manager at ~ ðx; ~v Þ ¼ 1 þ ðv~ −1Þx, where x and 1 − x are the end of the period is w the amounts (or proportions, since the total amount is one) invested in the risky and the risk-free assets respectively.
J. Wang et al. / Economic Modelling 33 (2013) 493–499
Similar to Palomino and Sadrieh (2011), we assume that if exerting effort at a cost c~ ¼ c, c ∈ (0,1), the manager receives a private signal ~s about the return of the risky asset, but cannot credibly communicate it to the investor. The signal is either ~s ¼ sH or ~s ¼ sL , where sH and sL denote the good or bad information, corresponding to vH and vL respectively. Conditional on the signal, the distribution of the risky asset's return is Prðv~ ¼ vH j~s ¼ sH Þ ¼ Prðv~ ¼ vL j~s ¼ sL Þ ¼ ð1 þ aÞ=2
ð1Þ
Prðv~ ¼ vH j~s ¼ sL Þ ¼ Prðv~ ¼ vL j~s ¼ sH Þ ¼ ð1−aÞ=2
ð2Þ
where a ∈ (0,1) represents the manager's professional ability, such that the higher a, the better her ability to interpret the information in the signal received (i.e., to estimate the probabilities of the market conditions). The manager decides on the investment strategy according to the signal she receives, that is, she will buy on the good news (x(sH) > 0) and sell on the bad news (x(sL) b 0). In the same spirit as in Gomez and Sharma (2006), we assume that x belongs to ½−x; x , where x is finite. This means that there are some trading constraints (i.e., limits on leverage and short selling) for the agent. Thus the return of the portfolio depends on both the manager's ability and the market condition. The investor cannot observe the manager's effort level or her investment strategy. Therefore, the compensation contract that the investor offers the manager depends solely on the realized outcome ~ ðx; ~v Þ. Suppose Sðw ~ Þ represents such a contract that is assumed to w be a linear function of the portfolio return: ~ ðx; ~v ÞÞ ¼ γ þ βw ~ ðx; ~v Þ Sðw
ð3Þ
where γ > 0 and 0 b β b 1 represent the fixed and the incentive components of the manager's compensation contract respectively. Hence the final wealth of the manager depends on both the compensation ~ Þ and her effort cost c~. For not losing generality, we assume contract Sðw that the manager is assured a minimum certainty equivalent R ≥ 0. 4. Equilibrium in the absence of optimism bias 4.1. The manager's first-best strategy We assume that the investor cannot acquire information at any cost. This implies that if he does not hire a manager, his expected wealth will not exceed the return of the risk-free asset. The investor claims the residual return that remains after the manager is paid. The investor is assumed to be risk neutral and his utility function is represented by V, which depends on the portfolio return: ~ ðx; ~v Þ−Sðw ~ ÞÞ ¼ ð1−βÞ½1 þ ðv~ −1Þx−γ: V ðw
ð4Þ
We first investigate the optimal investment strategy from the investor's perspective. That is, if the investor could receive the signal as the manager, then his expected utility evolves as different information is gathered. The constraint Eðv~ Þ ¼ vH =2 þ vL =2 ¼ 1 is equivalent to vH − vL = 2(vH − 1) = 2(1 − vL). So the investor's expected utility is:
where x(sH) > 0 and x(sL) b 0 represent the optimal trading quantities of the risky asset bought and sold on good and bad news respectively. And the investor's optimization problem is ~ ðx; ~v Þ−Sðw ~ ÞÞj~s max E½V ðw x
where x is the decision variable. Eq. (5) indicates that the investor's expected utility increases with the absolute value of x(sH) and x(sL). Thus, it is easy to show that the optimal strategy is xFB ðsH Þ ¼ −xFB ðsL Þ ¼ x, which is the wealth-maximizing strategy – namely the first-best strategy – for the investor, and we use a subscript “FB” to denote it. The investor's expected utility under the optimal strategy is: ~ ðx; ~v Þ−Sðw ~ ÞÞj~s ¼ ð1−βÞ½1 þ aðvH −1Þx −γ EFB ½V ðw
4.2. The effect of risk aversion The first-best outcome maximizes the monetary outcome of the investment to the risk-neutral investor. To attain it, the manager must not care about risk when making her investment decision. However, the investment strategy has to be carried out by the manager whose sole income is tied to the investment. In the spirit of Jensen and Meckling (1976), we assume that, because the manager's human capital is less diversified than that of the investor, she is risk-averse about the outcome of her compensation. Incentive contracts can be used to reduce these agency costs by realigning the objective of the agent with that of the principle. We will discuss such contracts in Section 6. For now, we concentrate on how the risk aversion of the manager affects her investment strategy and in turn the payoff of the investment. The risk-averse manager has the utility function U as preference representation over her end-of-period wealth. Specifically we assume that the manager has a constant absolute risk aversion (CARA) utility function with an absolute risk aversion coefficient ρ: ~ ¼ − exp −ρW ~ U W
ð7Þ
~ is the wealth of the manager at the end of the period. where W According to the signal observed, the manager decides on the optimal investment strategy, which is given in Proposition 1. Proposition 1. Suppose that the investment is managed by a risk-averse manager with risk aversion ρ > 0 and ability a. The optimal investment strategy for the manager is
1þa ln 1−a ≤ xFB ðsH Þ ρβðvH −vL Þ
x ðsL Þ ¼ −x ðsH Þ
ð5Þ
ð6Þ
Eq. (6) implies that under the condition that the investor is risk-neutral (i.e., his utility is independent of risk taking), the optimal strategy is to take (either buy or sell) the largest position allowed in the risky asset given the signal received regarding the upcoming market condition. This strategy can also be reached by assuming that the manager is risk-neutral, as the personal objective of this agent is then precisely to maximize the final payoff of the portfolio.
x ðsH Þ ¼
11 þ a ~ ðx; ~v Þ−Sðw ~ ÞÞj~s ¼ E½V ðw fð1−βÞ½1 þ ðvH −1ÞxðsH Þ−γ g 2 2 1 1−a þ fð1−βÞ½1 þ ðvL −1ÞxðsH Þ−γ g 2 2 1 1−a þ fð1−βÞ½1 þ ðvH −1ÞxðsL Þ−γ g 2 2 11þ a þ fð1−βÞ½1 þ ðvL −1ÞxðsL Þ−γg 2 2 ð1−βÞ ¼ f2 þ aðvH −1Þ½xðsH Þ−xðsL Þg−γ 2
495
ð8Þ ð9Þ
where x ∗(sH) > 0 and x∗(sL) b 0 represent the optimal trading quantities of the risky asset bought and sold on good and bad news respectively. By inspection of Proposition 1 we realize that the absolute value of both x*(sH) and x*(sL) decreases with ρ, which implies that risk aversion makes the manager cut down the trading quantity of the risky asset. Then the derivation of Proposition 1 delivers the insight of Corollary 1 immediately.
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J. Wang et al. / Economic Modelling 33 (2013) 493–499
Corollary 1. Under the manager's optimal investment strategy, the investor's expected utility is a 1þa ~ ðx; ~v Þ−Sðw ~ ÞÞj~s ¼ ð1−βÞ 1 þ ~ ðx; ~v Þ−Sðw ~ ÞÞj~s E V ðw ln −γ ≤ EFB ½V ðw 2ρβ 1−a
He does revise his beliefs upwards (downwards) after receiving a positive (negative) signal, but the resulting posterior is still more optimistic than it should be. 5.2. Investment strategies with optimism bias
ð10Þ ~ ðx; v~ Þ−Sðw ~ ÞÞj~s represents the investor's utility of the where EFB ½V ðw first-best strategy. As we can see in Eqs. (5) and (10), the investor's wealth is increasing in his agent's ability a. However, the investor's wealth is strictly decreasing in his agent's risk aversion ρ. The loss of investor's wealth results from the fact that the trading quantity of the risky asset is reduced. The intuition of this result is fairly straightforward. Due to the imperfect signal, for a risk-averse manager, the shock that results from making a wrong decision is more severe than that for a risk-neutral manager. Consequently, the risk-averse manager cuts down the trading quantity of the risky asset. The manager's utility gained from reducing risk is not transferred to the investor. Thus the investor suffers a loss, namely the agency cost. As shown in Eq. (10), the more risk-averse the manager, the lower the investor's utility, and so the larger the agency cost. This is a well-known result in principal–agent theories. The moral hazard problem causes disutility for the investor. 5. The role of optimism bias We have shown that the final payoff to the risk-neutral investor is negatively affected by the manager's risk aversion. In this section, we show how managerial optimism bias may help to restore the wealth of the investor. 5.1. Definitions It is common for people to be subjected to optimism bias when making decisions. And the manager's psychology bias, which refers to ex ante view of the risky asset's return in our model, is doomed to affect the investor. Following Heaton (2002), we define optimism to be the ex-ante belief that the risky asset's return is, on average, more than it actually is. More specifically, we assume that the optimistic manager thinks that the probability of good outcome for the risky asset (v = vH) is not 1/2, but B∈ 12 ; 1 , that is Prðv~ ¼ vH > 1Þ ¼ B
ð11Þ
Prðv~ ¼ vL b1Þ ¼ 1−B
ð12Þ
and d≡B− 12 ∈ 0; 12 measures the level of managerial optimism bias. As d → 0, the manager approaches rationality; as d → 1/2, she thinks the good state is much more likely to happen than the bad state and effectively becomes overoptimistic. The following lemma shows how this bias will affect the way that imperfect information is interpreted by the manager. Lemma 1. A manager with optimism bias thinks that Prb ðv~ ¼ vH j~s ¼ sH Þ ¼ a þ ð1−aÞB≡ ϕ1 ða; BÞ
ð13Þ
Prb ðv~ ¼ vH j~s ¼ sL Þ ¼ ð1−aÞB≡ ϕ2 ða; BÞ
ð14Þ
Prb ðv~ ¼ vL j~s ¼ sH Þ ¼ ð1−aÞð1−BÞ≡ ϕ3 ða; BÞ
ð15Þ
Prb ðv~ ¼ vL j~s ¼ sL Þ ¼ a þ ð1−aÞð1−BÞ≡ϕ4 ða; BÞ
ð16Þ
where the subscript “b” refers to the fact that the manager is biased. Apparently ϕ1(a,B) and ϕ2(a,B) are increasing in B, whereas ϕ3(a,B) and ϕ4(a,B) are decreasing in B. This is very intuitive — the optimistic manager always thinks that the investment is better than it really is.
The optimistic agent operates under the assumption that the return of the risky asset is intrinsically better than it actually is. Now we will investigate how the manager's psychological bias affects the investor's final wealth. More precisely, given the manager's risk aversion ρ, we would like to know the difference of the investor's utility between hiring a rational manager and hiring a biased manager. The following proposition derives the manager's optimal strategy and resultant investor's utility when the manager is optimistically biased. Proposition 2. Suppose that the investment is managed by a manager with risk aversion ρ > 0, ability a, and optimism bias B. Then the optimal investment strategy for the manager is
xb ðsH Þ ¼
xb ðsL Þ ¼
ln ϕϕ1 3
ρβðvH −vL Þ ln ϕϕ2 4
ρβðvH −vL Þ
¼
aþð1−aÞB ln ð1−a Þð1−BÞ
ρβðvH −vL Þ
¼−
Þð1−BÞ ln aþð1−a ð1−aÞB
ρβðvH −vL Þ
ð17Þ
ð18Þ
where xb∗ (sH) > 0 and xb∗ (sL) b 0 represent the optimal quantities of the risky asset bought and sold upon good and bad news respectively.With the optimal strategy under optimism bias, the investor's expected utility is a ϕ ϕ ~ ðx; ~v Þ−Sðw ~ ÞÞj~s ¼ ð1−β Þ 1 þ Eb ½V ðw ln 1 4 −γ 4ρβ ϕ2 ϕ3 1 ~ ðx; ~v Þ−Sðw ~ ÞÞj~s ∀B∈ ; 1 : ≥ E V ðw 2
ð19Þ
In the first part of Proposition 2, we can see that for the optimistic manager, xb∗ (sH) is bigger than x*(sH) whereas the absolute value of xb∗ (sL) is smaller than that of x*(sL). However, the increase of the former is larger than the decrease of the latter, so the optimism bias is useful in enhancing the investor's wealth. The intuition behind this result is that the biased manager, who is subject to optimism, thinks the return of the risky asset is larger than it really is, so attaches less importance to the loss stemming from risk, and thus enhances the trading quantity of the risky asset. In other words, the manager's optimism bias partially offsets her risk aversion. The second part of Proposition 2 tells us that managerial optimism bias somewhat alleviates conflicts between the investor and the manager. 6. Optimal incentive contracts based on optimism bias As is standard in the contract theory, we assume that the principal has all the bargaining power and makes a take-it-or-leave-it offer to the manager. The main assumption which makes this model differ from the standard principal-agent model is that the agent is subject to optimism bias with respect to the return of the risky asset. Similar to de la Rosa (2011), we assume that the investor knows that the manager is biased, and the manager knows that the investor thinks that way, but disagrees with him. Therefore, the investor and the manager “agree to disagree”. The investor's problem is to choose the incentive contract ~ Þ that maximizes his final wealth. Sb ðw The determination of the optimal incentive contract boils down to the investor's choice of γb∗ and βb∗ , which represent the contract's parameters. The timing of the players' actions and events is depicted in Fig. 1. At t = 0 the investor observes the manager's bias and makes a delegation contract offer (i.e., choosing γb∗ and βb∗ ). The manager decides whether or not to accept the investor's offer at t = 1. If the
J. Wang et al. / Economic Modelling 33 (2013) 493–499
The investor observes
The manager
If the manager
the manager’s bias and
accepts or
accepts offer she
offers the contract
rejects offer
expends effort
t=0
t=1
497
The manager
The manager
receives
chooses investment
signal
t=2
t=3
strategy
The investor observes and the manager is compensated
t=4
t=5
Fig. 1. Timing of the players' actions and events.
manager accepts the offer she expends effort at t = 2, and receives a private signal ~s at t = 3. At t = 4 the manager chooses the investment strategy xb ð~s Þ. The resolution of uncertainty occurs at t = 5 and the manager is compensated. The optimal incentive contract is the solution to the following constrained problem: ~ ðx; ~v Þ−Sb ðw ~ ÞÞj~s max Eb ½V ðw fSb ðw~ Þg ~ Þ ¼ γb þ βb w ~ s:t: Sb ðw xb ð~s Þ∈½−x; x
ð20Þ ð21Þ
~ ðx; ~v ÞÞ−cj~s g xb ð~s Þ∈ arg maxEb fU ½Sb ðw
ð22Þ
~ ðx; ~v ÞÞ−cjc~ ¼ cg ≥ Eb fU ½Sb ðw ~ Þjc~ ¼ 0g Eb fU ½Sb ðw
ð23Þ
~ ðx; ~v ÞÞ−cjc~ ¼ cg ≥ U ðRÞ Eb fU ½Sb ðw
ð24Þ
where γb and βb, the parameters of the contract offered to the biased manager, are the decision variables. Eq. (20) is the end-of-period compensation of the manager. Eq. (21) is the optimal asset selection constraint for the manager. Eqs. (22) and (23) are the incentive compatibility constraints guaranteeing that the manager maximizes her expected utility by exerting effort and allocating assets. Eq. (24) represents the participation constraint ensuring that it is rational for the manager to enter the principal–agent relationship because the engagement produces at least her reservation utility. The optimal incentive contract is given in Proposition 3. Proposition 3. Suppose that the manager hired by the investor is characterized by risk aversion ρ, ability a, and optimism bias B. Then the optimal incentive contract that realigns her incentives with that of the risk-neutral investor is
γb ¼ R
ð25Þ
and
βb ¼
i 1 1 1 h ln ðϕ1 ϕ3 Þ2 þ ðϕ2 ϕ4 Þ2 þ c: ρ
ð26Þ
By inspection of this proposition we realize that the manager's optimal fixed compensation depends on his reservation utility only, while the optimal wealth-sharing parameter βb∗ is a function of the manager's risk aversion, ability, and optimism bias. The sensitivity of the variable compensation βb∗ with respect to the level of managerial optimism bias d is gathered in Proposition 4. Proposition 4. Suppose that the manager hired by the investor is characterized by risk aversion ρ, ability a, and optimism bias d. Then the optimal incentive component of the contract βb∗ proposed by the risk-neutral investor is decreasing in the level of the managerial optimism bias d. In order to interpret this result with respect to the level of managerial bias d properly, we here emphasize again that d is used to measure how optimistic the manager is — the higher the level of bias is, the more optimistic the future is judged by the manager. Thus, Proposition 4 shows that a manager subject to optimism bias believes that she will realize a good
performance with a higher probability than the true one, so she accepts a “cheaper” contract than a rational manager would. This insight lets us conclude that the optimism bias makes an optimistic manager more attractive than a rational manager for the investor. This result is consistent with those of Gervais et al. (2003) in the corporate environment: moderate optimism makes managers reduce the need for option compensation. However, if the investor cannot observe the managerial optimism bias, he would offer the manager higher incentive compensation than actually needed, and this is virtually detrimental for the investor in terms of his expected utility. Another implication of this result is that incentive contracts can be used to screen prospective managers — those willing to accept a low-incentive contract may be optimistic. When considering Propositions 2 and 4 together we realize that managerial optimism bias can actually serve well the interest of the investor in at least two ways. First, it can reduce the agency cost caused by the different levels of risk aversion between the investor and the manager. Second, it costs less to hire an optimistic investor manager. This result can be added to the list of benefits of optimism as identified in Puri and Robinson (2007), which includes working harder, retiring later, saving more, and investing more in stocks. While psychological biases such as optimism may sound negative and are often treated as complications in traditional models with rational agents, they can actually have some positive impacts to decision making and social welfare. 7. Conclusion Within a principal–agent framework, this paper explores the investment strategy and the optimal incentive contract in portfolio delegation based on managerial optimism bias. Specifically, we assume that the manager is optimistic, and thus biased, when assessing the return of the risky asset. We find that the optimistic manager behaves as if she perceives less risk when allocating assets and show that such a biased manager can be beneficial to the investor because it takes less in compensation to align her decisions with the investor's preference. However, compensating the optimistic manager as if she was unbiased reduces the investor's wealth due to overpayment. These results differ significantly from those of portfolio delegation models under rationality assumptions. Future research may explore the incentive impact of optimism bias when the investor is not well diversified and therefore concerned with the portfolio risk. Appendix A Proof of Proposition 1 Assuming that after paying a cost c, the manager receives a signal ~s ¼ sH . 1 From Eq. (7) the manager maximizes h 1 þ a −ρfγþβ½1þxðvH −1Þ−cg ~ w ~ ðx; ~v ÞÞ−c sH ¼ E U Sð −e 2 1−a −ρfγþβ½1þxðvL −1Þ−cg þ −e : 2 1
The derivation for the case ~s ¼ sL is identical.
ð27Þ
498
J. Wang et al. / Economic Modelling 33 (2013) 493–499
The first-order condition of utility maximization yields
The first-order condition of utility maximization yields
x ðsH Þ ¼
1þa ln 1−a > 0: ρβðvH −vL Þ
xb ðsH Þ ¼
Proof of Corollary 1
ln ϕϕ1 3
ρβðvH −vL Þ
¼
aþð1−aÞB ln ð1−a Þð1−BÞ
ρβðvH −vL Þ
:
Then the investor's expected utility based on managerial bias is:
Given that Eðv~ Þ ¼ v2H þ v2L ¼ 1, we have vH − vL = 2(vH − 1) = 2(1 − vL), and x*(sL) = − x*(sH). From Proposition 1, the investor's expected utility is:
11 þ a ð1−βÞ 1 þ x ðsH ÞðvH −1Þ −γ 2 2
1 1−a ð1−βÞ 1 þ x ðsH ÞðvL −1Þ −γ þ 2 2
1 1−a ð1−βÞ 1 þ x ðsL ÞðvH −1Þ −γ þ 2 2
11 þ a ð1−βÞ 1 þ x ðsL ÞðvL −1Þ −γ þ 2 2 ¼ ð1−βÞ 1 þ ax ðsH ÞðvH −1Þ −γ a 1þa ln −γ ¼ ð1−βÞ 1 þ 2ρβ 1−a
11 þ a ð1−βÞ 1 þ xb ðsH ÞðvH −1Þ −γ 2 2
1 1−a ð1−βÞ 1 þ xb ðsH ÞðvL −1Þ −γ þ 2 2
1 1−a ð1−βÞ 1 þ xb ðsL ÞðvH −1Þ −γ þ 2 2
11 þ a ð1−βÞ 1 þ xb ðsL ÞðvL −1Þ −γ þ 2 2 a ϕ ϕ ln 1 4 −γ ¼ ð1−βÞ 1 þ 4ρβ ϕ2 ϕ3
~ ðx; ~v Þ−Sðw ~ ÞÞj~s ¼ Eb ½V ðw
~ ðx; ~v Þ−Sðw ~ ÞÞj~s ¼ E ½V ðw
ð30Þ and from Eq. (10) ~ ðx; ~v Þ−Sðw ~ ÞÞj~s −E ½V ðw ~ ðx; ~v Þ−Sðw ~ ÞÞj~s Eb ½V ðw
ð28Þ Comparing Eq. (28) with Eq. (6), due to x ðsH Þ ≤ x, we can get ~ ðx; ~v Þ−Sðw ~ ÞÞj~s ≤ EFB ½V ðw ~ ðx; ~v Þ−Sðw ~ ÞÞj~s . E ½V ðw
ð1−aÞ2 Bð1−BÞ
≥0:
Proof of Proposition 3 From Proposition 2, Eq. (22) is equivalent to
Proof of Lemma 1
xb ðsH Þ ¼
Using Bayes' rule, we have Prb ðv~ ¼ vH j~s ¼ sH Þ ¼ ¼
Prb sH jvH ÞPrb ðvH Þ Prb ðsH jvH ÞPrb ðvH Þ þ Prb ðsH jvL ÞPrb ðvL Þ
xb ðsL Þ ¼
½a þ ð1−aÞBB ½a þ ð1−aÞBB þ ð1−aÞBð1−BÞ
ð1−aÞð1−BÞB ð1−aÞð1−BÞB þ ½a þ ð1−aÞð1−BÞð1−BÞ
¼
¼ ð1−aÞB and
ρβðvH −vL Þ ln ϕϕ2 4
ρβðvH −vL Þ
¼
aþð1−aÞB ln ð1−a Þð1−BÞ
ð31Þ
ρβðvH −vL Þ
¼−
Þð1−BÞ ln aþð1−a ð1−aÞB
ρβðvH −vL Þ
:
ð1−aÞBð1−BÞ ð1−aÞBð1−BÞ þ ½a þ ð1−aÞBB
Prb ðv~ ¼ vL j~s ¼ sH Þ ¼
ð32Þ
ð33Þ
¼ ð1−aÞð1−BÞ Prb ðv~ ¼ vL j~s ¼ sL Þ ¼
3
h i 1 ~ ðx; ~v Þ−cÞc~ ¼ c ¼ ϕ1 −e−ρfγþβ½1þxb ðsH ÞðvH −1Þ−cg Eb U S~b ðw 2 1 −ρ γþβ 1þx ðs Þðv −1Þ −c þ ϕ2 −e f ½ b L H g 2 1 h −ρ γþβ 1þx ðs Þðv −1Þ −c i þ ϕ3 −e f ½ b H L g 2 1 −ρ γþβ 1þx ðs Þðv −1Þ −c þ ϕ4 −e f ½ b L L g 2 h ih i 1 1 −ρðγþβ−cÞ : ¼ ðϕ1 ϕ3 Þ2 þ ðϕ2 ϕ4 Þ2 −e
Prb ðsL jvH ÞPrb ðvH Þ Prb ðsL jvH ÞPrb ðvH Þ þ Prb ðsL jvL ÞPrb ðvL Þ
Prbðv~ ¼ vH j~s ¼ sL Þ ¼
ln ϕϕ1
The expect utility of the optimistic manager when she exerts effort is
¼ a þ ð1−aÞB
The expect utility of the optimistic manager when she exerts no effort is
½a þ ð1−aÞð1−BÞð1−BÞ ½a þ ð1−aÞð1−BÞð1−BÞ þ ð1−aÞB
h i ~ Þ c~ ¼ 0 ¼ −e−ργ : Eb U S~b ðw
¼ a þ ð1−aÞð1−BÞ:
ð34Þ
Using these equations in the constrained program, we can get γb∗ and βb∗ .
Proof of Proposition 2 Assuming that after paying a cost c, the biased manager receives a signal sH. 2 From Lemma 1, she maximizes ~ ðx; ~v ÞÞ−cÞjsH ¼ ϕ1 ð−e Eb ½UðS~ðw
−ρfγþβ ½1þxb ðvH −1Þ−cg
Þ
−ρfγþβ½1þxb ðvL −1Þ−cg
þϕ3 ð−e
2
¼
h i 2 2 a þ að1−aÞ 2 B− 12 þ 12
The proof for the case s = sL is identical.
Þ:
ð29Þ
Proof of Proposition 4 Due to d ¼ B− 12, Eq. (26) is equivalent to ( 1
1 2 2 1 1 1 ð1 þ aÞ þ ð1−aÞd ð1−aÞ −d βb ¼ ln ρ 2 2
1 1 ) 2 1 2 1 þ c: þ ð1−aÞ þd ð1 þ aÞ−ð1−aÞd 2 2
ð35Þ
J. Wang et al. / Economic Modelling 33 (2013) 493–499
We define ϕ′ 1 ≡ 12 ð1 þ aÞ þ ð1−aÞd, ϕ′ 3 ≡ð1−aÞ 12−d , ϕ′ 2 ≡ð1− 12 ϕ′ 4 ≡ 12 ð1 þ aÞ−ð1−aÞd, then βb ¼ ρ1 ln ϕ′ 1 ϕ′ 3 þ aÞ 12 þ d , 12 ϕ′ 2 ϕ′ 4 g þ c. ∂ðϕ′ 1 ϕ′ 3 Þ ∂ðϕ′ 2 ϕ′ 4 Þ ¼ −ð1−aÞ½a þ 2ð1−aÞd and n≡ ∂d ¼ ð1− Let m≡ ∂d aÞ½a−2ð1−aÞd, then −1 −1 ∂βb 1 1 ′ ′ ′ ′ ϕ 1ϕ 3 2m þ ϕ 2ϕ 4 2n : ¼ 1 1 2ρ ϕ′ ϕ′ 2 þ ϕ′ ϕ′ 2 ∂d 1 3 2 4
ð36Þ
h i 2 m2 ϕ′ 2 ϕ′ 4 −n2 ϕ′ 1 ϕ′ 3 ¼ 2að1−aÞd a2 þ 4ð1−aÞ2 d þ h i −12 b0, we can get 4að1−aÞd ϕ′ 1 ϕ′ 3 þ ϕ′ 2 ϕ′ 4 > 0, and m ϕ′ 1 ϕ′ 3 As
∂β b ∂d
b0, which arrives at Proposition 4.
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Dr. Jun Yang is a Professor of finance at the F.C. Manning School of Business Administration, Acadia University, Canada. He received his Ph.D. in finance from Queen's University, Canada. His research interests include corporate governance, asset pricing, and risk management.