Uncertainty, Risk, and Incentives: Theory and Evidence

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Uncertainty, Risk, and Incentives: Theory and Evidence Zhiguo He, Si Li, Bin Wei, and Jianfeng Yu 2013-18 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Uncertainty, Risk, and Incentives: Theory and Evidence* Zhiguo He Si Li Bin Wei Jianfeng Yu February 2013 Abstract Uncertaintyhasqualitativelydifferentimplicationsthanriskinstudyingexecutiveincentives. We study the interplay between profitability uncertainty and moral hazard, where profitability is multiplicative with managerial effort. Investors who face greater uncertainty desire faster learning, and consequently offer higher managerial incentives to induce higher effort from the manager. In contrast to the standard negative risk-incentive trade-off, this “learning-bydoing” effect generates a positive relation between profitability uncertainty and incentives. We document empirical support for this prediction. Keywords: Executive Compensation, Optimal Contracting, Learning, Uncertainty, Risk-Incentive Trade-off. * This work does not necessarily reflect the views of the Federal Reserve System or its staff. We thank seminar participants at the Third Annual Triple Crown Conference, the Northern Finance Association Annual Meeting, the SixthSingaporeInternationalConferenceonFinance,FinancialManagementAssociationMeetings,andTCFABest Paper Symposium. All errors are our own. Author affiliations/contact information: Zhiguo He: University of Chicago, Booth School of Business, 5807 South Woodlawn Ave., Chicago 60637. Phone: 773-834-3769, Email: zhiguo.he@chicagobooth.edu. SiLi: WilfridLaurierUniversity,SchoolofBusinessandEconomics,75UniversityAvenueWest,Waterloo,ONN2L 3C5, Canada. Phone: 519-884-0710×2395, Email: sli@wlu.ca. Bin Wei: Board of Governors of the Federal Reserve System, Washington, DC, 20551. Phone: 202-452-2693, Email: bin.wei@frb.gov. Jianfeng Yu: University of Minnesota, Carlson School of Management, CSOM 3-122, 321 19th Avenue South, Minneapolis, MN 55455. Phone: 612-625-5498. Email: jianfeng@umn.edu.

1 Introduction A central prediction of the principal-agent theory is the negative trade-off between risk and incentives (Holmstrom and Milgrom, 1987). Higher performance pay induces greater effort from the agent but increases risk, which, in turn, raises risk compensation. The greater the output risk, the higher the risk compensation, leading to a lower performance pay to the risk-averse agent in the optimal contract. Yet, numerous studies over the past two decades find mixed empirical evidence on such a negative relation between risk and incentives. After reviewing more than two dozen empirical studies and concluding that evidence on the risk-incentive trade-off is inconclusive, Prendergast(2002)arguesthatinamoreuncertainenvironment,theprincipalmaywanttodelegate responsibilitiestotheagent,leadingtoapositive risk-incentiverelation. Otherleadingexplanations for this puzzle includes the idea of endogenous firm risk, where firms offer high powered incentives to induce the agent to take risk (e.g., Core and Guay, 1999; Edmans and Gabaix, 2011a), or the view that risk does not affect incentives because, from the principal’s perspective, the cost of risk bearingisoutweighedbythebenefitsofefforts,andthusriskissecondorder(e.g.,Edmans,Gabaix, and Landier, 2009; Edmans and Gabaix, 2011b). In this paper we offer another plausible theory to explain why the negative risk-incentive tradeoff has received mixed empirical support. Empirically measured risk, which is essentially output performance variance, can come from either cash flow risk or project profitability uncertainty, or both. Specifically, in many types of economic environments with agency relationships, output performance not only consists of the agent’s effort plus some transitory random noise (i.e., cash flow risk), but also the project’s unobserved long-run profitability (i.e., profitability uncertainty).1 We incorporate endogenous learning about the firm’s profitability uncertainty into the standard Holmstrom and Milgrom (1987) setting, and show that a potentially positive relation between uncertainty and incentives emerges. In a nutshell, besides the traditional risk channel, the learning channel implies that greater effort, induced by high-powered incentives, leads to more-informative signals about uncertain project profitability, improving the firm’s future investment decisions. Moreover, somewhat surprisingly, even if one can perfectly separate risk from uncertainty, this learning channel may also overturn the traditional negative risk-incentive relation. Based on several widely used proxies for firm profitability uncertainty, we find empirical support for the positive uncertainty-incentive relation. This suggests that prior mixed empirical results in testing the negative risk-incentive trade-off may be attributable to a positive bias caused by omitting variables that are proxies for profitability uncertainty. In this paper we develop a two-period investment model, in which the firm hires a manager to manage a project at the beginning of period 1. The project generates an output of y = 1 1Most of the existing principal-agent literature assumes that the productivity of managerial input is known. Our paperintroducestheuncertaintyontheproductivityparametersinasimpletwo-periodsettingtostudytherelation between incentives, risk, and uncertainty. For other papers with learning in short-term contracting, see Murphy (1986) and Gibbons and Murphy (1992). Long-term optimal contracting with learning is much more technically challengingbecauseofthehidden-stateproblem;seeDeMarzoandSannikov(2010),PratandJovanovic(2011),and He, Wei, and Yu (2012). 1

θK1−λLλ + (cid:15) , where K is capital, L is managerial labor (effort) input, and (cid:15) is exogenous 1 1 1 1 1 1 cash flow shock. The parameter θ is the project’s marginal productivity or profitability. The key departure of our model from standard agency models is that profitability θ is unknown. Investors learn θ and then make future investment decisions. Both multiplicative labor with θ and additive cash flow noise (cid:15) are the drivers of our mechanism; they imply that a greater labor input can 1 increase the information-to-noise ratio of the output signal y based on Bayes’ rule. At period 2, 1 the firm with a posterior belief of θ adjusts capital K through investment, and resets labor input 2 L . 2 To optimize over period-2 investment, investors desire faster learning about θ from period- 1 output signal y . As a result, for a more informative signal y , high powered incentives that 1 1 induce greater effort from the manager are more preferable. Moreover, the higher the degree of uncertainty, thegreaterthereductionoftheposteriorvarianceofθ, andthusthegreaterthebenefit in inducing a higher period-1 effort. In other words, firms with uncertain profitability offer highpoweredincentivestotheirmanagersformoreinformativesignalstoguidetheirinvestmentpolicies. This mechanism is similar in spirit to the learning-by-doing literature (e.g., Jovanovic and Lach, 1989; Jovanovic and Nyarko, 1996; and Johnson, 2007). Because uncertainty in θ also increases the total volatility of output y on the risk-averse manager, when the manager’s risk aversion is 1 relatively high, the traditional negative risk-incentive effect dominates and leads to a standard negative uncertainty-incentive relation. However, when the manager’s risk aversion is relatively low, the learning-by-doing effect dominates and leads to a positive uncertainty-incentive relation.2 Moreover,thelearningmechanismmayalsooverturnthetraditionalnegativerisk-incentiverelation. Thehighertherisk,thesmallertheinformation-to-noiseratio,andthemoretheroomtolearnabout the unknown profitability uncertainty. Thus offering high-powered incentives might be desirable. Weempiricallytestwhethertheuncertainty-incentiverelationispositiveinSection3. Following Pastor and Veronesi (2003) and Korteweg and Polson (2009) we use firm age as our first proxy as older firms usually have lower uncertainty. We also use stock price reaction to earnings announcements (i.e., earnings response coefficient, or ERC) as another proxy for profitability uncertainty (Pastor et al., 2009). Intuitively, investors who are more uncertain about a company’s profitability should be more responsive to earnings surprises. Our other proxies for profitability uncertainty are tangibility and market-to-book ratio (Korteweg and Polson, 2009), and analyst forecast error (Lang and Lundholm, 1996). We then run panel regressions of pay-performance sensitivities (PPS henceforth) on these uncertainty proxies and the risk proxy while controlling for other factors known to affect PPS. We find that firm age and tangibility are negatively related to 2Our paper, with the inclusion of learning, is different from Prendergast (2002) and some other papers (see, e.g., Zabojnik, 1996; Baker and Jorgensen, 2003; and Peng and Roell, 2012) that predict a possible positive relation between uncertainty and incentives. For example, Prendergast (2002) argues that in a more uncertain environment that the agent knows more than the principal, the positive value of delegating responsibilities to the agent may dominatethenegativeeffectofriskonincentives,resultinginapositiverelationbetweenuncertaintyandincentives. By contrast, our model has symmetric information along the equilibrium path, and learning is the key mechanism. Peng and Roell (2012) study optimal contracting when managers can manipulate firm performance. They find that uncertainty in managerial manipulation propensity may also lead to a positive uncertainty-incentive relation. Based on a different type of uncertainty, the mechanism in their paper is complementary to ours. 2

PPS; ERC, market-to-book ratio, and analyst forecast error are positively related to PPS. Several remarks are worth highlighting in interpreting our empirical results. First, we acknowledge that each individual proxy for uncertainty is imperfect; these proxies may reflect firm characteristics such as growth opportunities. For example, firms with more growth opportunities are often younger, have higher market-to-book ratios, and have more intangible assets. These firms are also harder to analyze and thus are associated with larger ERC and analyst forecast errors. Hence, in all regressions, we control for firm growth using analysts’ long-term earnings growth forecast. This is not a perfect solution to remove the effect of growth from the uncertainty proxies, and the results in the paper need to be interpreted with this caveat in mind. Second, for our analysis, it is important (to try) to separate uncertainty from risk. Fortunately, some uncertainty variables we use are positively correlated with firm volatility, while others (e.g., ERC) are negatively correlated with volatility. Examining all of the different uncertainty variables will help us separate the role of uncertainty from that of volatility. We do, however, acknowledge that the separation of uncertainty from risk in the paper is not perfect. Third,inourmodel,profitabilityuncertaintyistakenasexogenous,andfirmsdesignendogenous optimal incentive contracts as a response to uncertainty. It could well be possible that the causality goes the other way in practice; that is, incentive contracts affect managers’ choices of project uncertainty. This reverse causality problem exists even if we can measure uncertainty perfectly. Althoughweusefixedeffectsregressionsintherobustnesssectiontoaddressthepotential endogeneity problem due to time invariant omitted variables, fixed effects can address neither the problem of time-variant omitted variables nor the reverse causality problem. In this paper we do not claim identification of causality, although we lag our uncertainty proxies by one year in our regression analysis in an attempt to mitigate the reserve causality issue. Because the incentive variables are persistent and some of the uncertainty proxies are forward looking, this treatment is far from perfect. The contribution of this paper is to propose a new explanation for mixed empirical evidence on the negative risk-incentive trade-off.3 Our learning-based model suggests two reasons: first, the effectofriskonincentivesmaybeconfoundedbytheuncertaintyeffectifuncertaintyisnotcaptured in the model; and second, under learning, the risk-incentive relation becomes ambiguous. On the empiricalside,weprovidepreliminaryanalysistoseewhetherthedataisconsistentwithourmodel. Our analysis suggests that controlling for profitability uncertainty helps partially (if not fully) to 3On the mixed evidence of risk-incentive relation, Aggarwal and Samwick (1999, 2002, 2003) find that the rank of dollar return volatility is negatively associated with pay performance sensitivities. Other papers supporting this negative relation include Garvey and Milbourn (2003), Jin (2002), Core, et al. (2003), Lambert and Larcker (1987), Bitler et al. (2005), Himmelberg et al. (1999), etc. In contrast, Becker (2006), Bushman et al. (1996), and Yermack (1995), do not find any significant impact of percentage stock return volatility on incentives, and Core and Guay (1999) obtain a positive effect of idiosyncratic risk on incentives. Other papers in this camp include Garen (1994), Conyon and Murphy (2000), Bizjak, Brickley, and Coles (1993), Coles, Daniel and Naveen (2006), etc. Prendergast (2002) reviews some mixed evidence for risk-incentive relationship in the areas other than executive compensation. Our theory is complimentary to other explanations for the mixed evidence of risk-incentive relation, e.g., Core and Guay (1999), Prendergast (2002), Edmans and Gabaix (2011a, 2011b), and Edmans, Gabaix, and Landier (2009); see the first paragraph in the introduction. 3

restore the negative risk-incentive relation predicted by standard agency theories. Although the coefficients of the risk variable often become less positive or more negative after the uncertainty variables are incorporated in the empirical model, we acknowledge that our analysis cannot fully restore the negative risk-incentive tradeoff and thus is far from resolving Prendagast’s statement (2002) that the evidence on the risk-incentive trade-off is inconclusive. We further reiterate that our empirical methodology has several other limitations: our uncertainty proxies are not perfect, the separation of uncertainty from risk is not ideal, and our method does not allow us to establish causality. The attempt to rule out alternative explanations in the robustness section is suggestive rather than conclusive; we await future research on this topic. The rest of this paper is organized as follows. Section 2 presents the model and its prediction of the positive relation between profitability uncertainty and incentives. Section 3 conducts empirical analysis and Section 4 concludes the paper. All proofs are in the Appendix. 2 The Model 2.1 The Setting We consider a two-period investment model, where investment consists of capital and (managerial) laborinputs. Therisk-freerateiszero. Investorsareriskneutral,andmanagersareriskaversewith exponential (CARA) preference. We interpret labor input as the manager’s effort. For simplicity, we assume that moral hazard only exists in the first period; the firm matures in the second period and therefore is no longer subject to agency issues. The output in each period, before investment cost, is modeled as (similar to the standard Cobb–Douglas technology with constant returns to scale) y = θK1−λLλ+(cid:15) , (1) t t t t where K is capital level, L is managerial labor input, λ ∈ (0,1) and 1−λ are output elasticities t t of labor and capital, respectively, and (cid:15) ∼ N (cid:0) 0,σ2(cid:1) is i.i.d. normally distributed. Importantly, t (cid:15) θ, which can be interpreted as project profitability or marginal productivity, is uncertain. Neither the firm nor the manager observes profitability θ directly, and they will learn θ from the realized output. At time 0, the common prior about profitability is θ ∼ N (θ ,γ ), where θ > 0 and γ > 0 0 0 0 0 are prior mean and variance, respectively. At the beginning of period 1, the firm with a zero outside option decides whether or not to invest K . Given K , investors hire a manager to provide labor input L , which is unobservable. 1 1 1 We interpret L as managerial effort, and investors offer the manager a compensation contract for 1 proper incentives. We focus on the space of linear contracts. The contract w (y;α,β) takes the 1 following form with fixed salary α and incentive β: (cid:16) (cid:17) w (y;α,β) ≡ α+βy = α+β θK1−λLλ+(cid:15) . 1 1 1 1 1 4

At time 0, the firm At time 1, the agent At time 2, the firm decides whether to chooses effort and learns about θ and take the project. If output is realized. then adjusts the so, the firm offers a investment level. linear contract to the agent. t=0 t=1 t=2 Figure 1: Timeline of the model. Here, the monetary cost for managerial labor L is lL2, where l > 0 is a positive constant. 1 2 1 Therefore, the manager’s utility from accepting the contract w (y;α,β) and working L is given 1 1 by (cid:18) (cid:18) (cid:19)(cid:19) l U (L ,w ) = −exp −a α+βy − L2 , (2) 1 1 1 2 1 wherea > 0isthemanager’srisk-aversioncoefficient. Finally, themanagerhasareservationutility of U(cid:98) at time 0, which is normalized to −1 without loss of generality. Suppose that the firm induces a labor input of L∗ from the period-1 manager. At the second 1 period the firm makes capital investment and labor investment based on the updated posterior of profitability θ . For period-2 labor investment L , the firm hires another manager with the same 1 2 costfunction lL2, andforsimplicity, weassumeawayanyagencyproblematperiod2(asthefirm’s 2 2 operation becomes more routine). Capital investment is subject to standard (constant-return-toscale) quadratic adjustment cost; given initial capital K , a (gross) investment of I + κ I2 leads 1 2K1 to a new capital level of K +I, where κ > 0 is a positive constant. As a result, investors at the 1 beginning of period 2 will solve the following problem: (cid:20) (cid:12) (cid:21) m I,L a 2 xE θ(K 1 +I)1−λLλ 2 +(cid:15) 2 −I − 2K κ 1 I2− 2 l L2 2 (cid:12) (cid:12) (cid:12) y 1 ,L∗ 1 . We provide a summary of the model timeline as follows; see Figure 1. 1. At the beginning of t = 1, the firm is deciding whether to take a project. Its outside option is normalized to zero. Thus (θ ,γ ) must be sufficiently favorable for the project to be adopted. 0 0 Thisstageplaystheonlyroletoensurethatθ > 0(somaximizingexpectedoutputθK1−λLλ 0 t t in (1) makes sense), an assumption that holds throughout the paper.4 4For purely technical convenience, we follow Gaussian-learning framework where θ can be negative. Our results gothroughifweassumethatθ islognormal. However,duetotheprincipal’soptiontoabandontheproject,θ must 0 be reasonably high for the project to be taken. 5

2. If the firm decides to take this project, investors hire one manager and offer him a linear contract w = α + βy , where y = θK1−λLλ + (cid:15) is the project’s output in period 1. 1 1 1 1 1 1 Investors’ period-1 payoff is y −w −K = θK1−λLλ+(cid:15) −α−βy −K . 1 1 1 1 1 1 1 1 3. Given the outcome y , investors update their belief about θ based on the prior θ ∼ N (θ ,γ ). 1 0 0 4. At t = 2, the firm makes capital investment I and labor investment L , so that y = 2 2 θ(K +I)1−λLλ+(cid:15) . The period-2 payoff is 1 2 2 κ l θ(K +I)1−λLλ+(cid:15) −I − I2− L2. 1 2 2 2K 2 2 1 2.2 Discussion on Modeling Assumptions Before solving the model backwards, we briefly discuss the key assumptions of the model. In particular, we highlight the necessary assumptions for the key model mechanism and discuss the assumptions made for technical convenience as well. First, two features of production technology in Eq. (1) are important: multiplicative specification between productivity θ and managerial labor input L, and additive cash flow noise (cid:15) . Under this setting, a greater labor input can increase the information-to-noise ratio when 1 investorslearntheproject’sprofitabilityθ fromtheoutputsignaly usingBayes’rule,resultingina 1 potentially positive uncertainty-incentive relation due to the learning-by-doing effect. If instead we assume that output is additive in profitability and labor so that y = θ+K1−λLλ+(cid:15), the learningby-doing effect disappears. Our learning-by-doing effect also vanishes if we assume a multiplicative cash flow noise, i.e. y = θK1−λLλ(cid:15). This disappearance occurs because increasing effort does not reduce the posterior variance of the unknown parameter θ in these two alternative settings. Second, the common prior on the unknown parameter θ indicates that the agent and the principal have the same information regarding θ. It is possible that the agent knows θ more than the principal. This is especially true if θ captures the manager’s productivity type. Two questions arise under this asymmetric information scenario. The first question is whether the learning-bydoing effect remains. Typically, the mechanism design approach will first solicit information from the agent in an incentive-compatible manner, and then offer the agent some (potentially different) contract based on the agent’s truthful report. If the agent knows θ perfectly, then the principal will learn θ immediately, annihilating our learning-by-doing effect. Away from this extreme scenario, as long as there is uncertainty in θ (either because the agent does not know θ perfectly or the true θ varies over time), the principal’s learning-by-doing effect (that is orthogonal to soliciting the agent’s truthful report) remains. Another question is whether information asymmetry leads to an ambiguous uncertaintyincentive relation. A thorough analysis of this question is unavailable. However, from another 6

related angle, Sung (2005) allows for information asymmetry and endogenous project volatility in a setting similar to Holmstrom and Milgrom (1987), and finds that “sometimes the higher the volatility, the higher the sensitivity of the contract.” This effect may be complementary to our mechanism. Third, the assumption of no agency issue in the second period is innocuous and for convenience only. As long as the period-2 managerial labor input has impact on the learning of profitability of period3,period-2incentives(ifamoralhazardproblemstillpersists)willsharethesamequalitative feature as period-1 incentives. The important assumption is that the old period 1 manager is replaced by a new manager in period 2, so that the incentive contract is short-term. With longterm employment relationship and endogenous learning, the manager can enjoy some endogenous information rent (as the manager who shirks at period 1 knows that the project actually is better thanwhatinvestorsbelieve),whichmakesanalysiscomplicated. SeeDeMarzoandSannikov(2010), Prat and Jovanovic (2011), or He, Wei and Yu (2012). In sum, our main mechanism goes through as long as 1) unknown profitability enters marginal labor productivity, and 2) there is strictly positive cash flow noise that is not scaled with expected output. To highlight the insight, we have chosen to push these two assumptions to extremes so that y = θK1−λLλ+(cid:15). 2.3 Learning and Investing in Period 2 Immediately after observing y at period 1, investors update their belief about θ. Given the 1 optimal labor input L∗ implemented by the incentive contract at period 1, Bayes’ rule implies 1 that the posterior of the project’s profitability is characterized by the posterior mean and posterior variance: γ K1−λ(L∗)λ (cid:104) (cid:105) θ ≡ E[θ|y ,L∗] = θ + 1 1 1 y −θ K1−λ(L∗)λ , (3) 1 1 1 0 σ2 1 0 1 1 (cid:15) γ σ2 γ ≡ Var[θ|y ,L∗] = 0 (cid:15) . (4) 1 1 1 (cid:16) (cid:17)2 σ2+γ K1−λ(L∗)λ (cid:15) 0 1 1 Intuitively, y −θ K1−λ(L∗)λ represents an unexpected shock from the output. If investors observe 1 0 1 1 a positive unexpected shock y −θ K1−λ(L∗)λ > 0, which serves a positive signal to the project 1 0 1 1 profitability θ, then Eq. (3) says that they should update θ upwards. As we will see shortly, given period-1 output information, profitability estimate θ guides the firm’s investment decision 1 atperiod2; moreover, posteriorvarianceγ inEq. (4), whichmeasurestheprecisionofprofitability 1 estimate θ , determines investment efficiency at period 2. Finally, posterior variance γ negatively 1 1 depends on L∗, thanks to the structure in Eq. (1). 1 Without loss of generality, we set κ = 1 to simplify exposition. Solving the model backwards, 7

at period 2 the firm makes capital investment and labor investment so that (cid:20) (cid:12) (cid:21) m I,L a 2 xE θ(K 1 +I)1−λLλ 2 +(cid:15) 2 −I − 2K κ 1 I2− 2 l L2 2 (cid:12) (cid:12) (cid:12) y 1 ,L∗ 1 = Mθ 1 + K 2 1 , where the constant M ≡ 1 (λ/l)λ(1−λ)1−λK1−λ > 0. The investors’ period 2 value 2 1 K 1 V (θ ) = Mθ + 2 1 1 2 is a function of the period 1 posterior mean θ . For instance, had the investors perfectly known θ, 1 they would have chosen I∗ = (1−λ) 2− 2 λ (λ/l) λ 2 K 1 2− 2 λ θ−K 1 = (2M (1−λ)K 1 ) 1 2 θ−K 1 . (5) However, due to imperfect information, they choose I∗ = (2M (1−λ)K 1 ) 1 2 θ 1 −K 1 which deviates from the full-information benchmark (5). Standing at time 0, the time-0 expected payoff from period 2 is given by K E[V (θ )] = M (γ −γ )+Mθ2+ 1 , (6) 2 1 0 1 0 2 which is decreasing in γ , the posterior variance of the unobserved profitability θ. Intuitively, the 1 lowertheposteriorvarianceγ , themoreprecisetheestimateofθ, andthemoreefficientthesecond 1 period investment. Moreover, from Eq. (4), γ decreases with effort L∗. This decrease implies that, 1 1 raising incentive β in period 1 improves the information content of period-1 output y , and, hence, 1 1 investors learn more about θ. 2.4 Optimal Contracting in Period 1 We now solve for the optimal linear contract in period 1. Here, investors offer a linear contract w = α+βy to implement the optimal labor (effort) L∗, and the optimal contract maximizes their 1 1 1 expected total value (including both periods’ payoffs): max E[y −w −K +V (θ )], (7) 1 1 1 2 1 α,β,L∗ 1 subject to the manager’s incentive compatibility and participation constraints: (cid:20) (cid:18) (cid:18) (cid:19)(cid:19)(cid:21) (cid:20) (cid:18) (cid:18) (cid:19)(cid:19)(cid:21) l l L∗ 1 = argm L a 1 xE −exp −a w 1 − 2 L2 1 , and E −exp −a w 1 − 2 L2 1 ≥ U(cid:98). The following lemma gives the manager’s optimal labor (effort) input. Lemma 1 A contract w = α+βy implements labor L∗ and satisfies the manager’s participation 1 1 1 8

constraint, if and only if L∗ uniquely solves 1 λβθ K1−λ−lL2−λ−aγ λβ2K 2(1−λ) Lλ = 0, (8) 0 1 1 0 1 1 and l 1 (cid:16) (cid:17) α = −βθ K1−λ(L∗)λ+ L∗2+ aβ2 γ K 2(1−λ) (L∗)2λ+σ2 . (9) 0 1 1 2 1 2 0 1 1 (cid:15) Essentially,Lemma1establishesanimportantlinkbetweenimplementedlaborL∗ andincentive 1 loadings β in any incentive-compatible contracts, which allows the firm to choose implemented L∗ 1 to maximize its value function. In light of Lemma 1, we can replace the incentive compatibility and participation constraints in the investors’ problem by Eq. (8) and Eq. (9). Together with Eqs. (3), (4), and (6), we can rewrite the investors’ problem in Eq. (7) (for details, see the proof of Lemma 1 in Appendix A) as: (cid:34) (cid:35) lL2 a (cid:16) (cid:17) γ2K 2(1−λ) L2λ L∗ ∈ argmax θ K1−λLλ− 1 − β2 γ K 2(1−λ) L2λ+σ2 +M 0 1 1 1 L1 0 1 1 2 2 0 1 1 (cid:15) σ (cid:15) 2+γ 0 K 1 2(1−λ) L2 1 λ s.t. 0 = λβθ K1−λ−lL2−λ−aγ λβ2K 2(1−λ) Lλ. 0 1 1 0 1 1 The first term in the investors’ value function is expected period-1 output, the second term is labor cost, the third term is the manager’s risk compensation, and the last term is the firm’s period 2 payoff. Once we derive the optimal effort level L∗, the optimal contract (i.e., α∗ and β∗) is fully 1 determined by Eq. (8) and Eq. (9). 2.5 Positive Incentive-Uncertainty Relation In our model, learning could induce a positive relation between incentives and uncertainty. This result is rooted in the fact that investors’ expected value of period 2 value, E [V (θ )], 0 2 1 depends on learning about profitability θ from period-1 output y . As indicated by Eq. (6), 1 maximizing E [V (θ )] is equivalent to minimizing the posterior variance of θ, i.e., γ . Because 0 2 1 1 Lλ is multiplicative with θ in signal y as in Eq. (1), implementing a higher effort L raises 1 1 1 the informativeness of the period 1 signal y , or equivalently, reduces the posterior variance γ . 1 1 Essentially, this mechanism shares a spirit similar to the learning-by-doing literature. For example, Johnson (2007) shows that when return-to-scale in firm’s production function is unknown in advance, overinvestment relative to the full-information case becomes optimal, as overinvestment expedites learning about the unknown production function. Presumably, this learning-by-doing effect is stronger in a more uncertain environment (i.e., a larger γ ). The effect is stronger because starting with a larger initial uncertainty γ , the reduction 0 0 of the posterior variance will be more significant, which results in a greater benefit of inducing a higher effort. That is, based on Eq. (4), we have ∂2(−γ ) 1 > 0. ∂L∗∂γ 1 0 9

0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35 -0.4 -0.45 -0.50 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Effort L1 *- 1     0 0 0 0 = = = = 0 0 0 0 . . . . 0 1 2 5 1 5 Figure2: Thenegativeposteriorvariance−γ asafunctionofeffortinperiod1fordifferentvaluesofγ . Parameters: 1 0 l=1.6, κ=1, θ =1, λ=0.67, K =0.28, a=0.5, and σ =0.2. 0 1 (cid:15) In Figure 2, we plot −γ as a function of effort L for different levels of γ . As we can see, when 1 1 0 γ increases, the marginal benefit of raising effort L becomes greater. To implement a higher 0 1 effort, a greater incentive β∗ is needed, which results in a positive relation between uncertainty and incentives. InProposition2weformallyprovetheexistenceofsuchapositiveuncertainty-incentiverelation when the manager is sufficiently risk tolerant. Note that higher uncertainty also implies that the manager is bearing larger output volatility, hence a higher incentive provision cost. Therefore, for thepositiveuncertainty-incentiverelationtohold, themanagerneedstobesufficientlyrisktolerant so that the learning-by-doing effect is dominant. Proposition 2 For sufficiently small risk aversion coefficient a, a positive relation exists between β∗ and γ , i.e., dβ∗ > 0. 0 dγ0 Figure 3 plots the incentive β∗ as a function of both uncertainty γ and risk σ2. Here, we vary 1 0 (cid:15) profitability uncertainty γ from 0.2 to 0.3 in the left panels (Panels A and C) and cash flow risk σ 0 (cid:15) from 0.05 to 0.15 in the right panels (Panels B and D). We set the absolute risk aversion coefficient to a = 0.5 for the top two panels,5 and a = 5 for the bottom two panels. Figure 3 indicates that our simple model cannot quantitatively match the very low pay-performance sensitivity observed in the data. However, our focus is the qualitative implications of our model on the relationship between uncertainty and incentives under realistic parameterizations. 5GiventhatCEOsarerelativelywealthy,itisreasonabletochooseasmallabsoluteriskaversioncoefficientbecause a×Wealth is the relative risk aversion coefficient. We follow Haubrich (1994) to set absolute risk aversion to be relative risk aversion/(CEO wealth in millions). According to http://people.few.eur.nl/dittmann/data.htm, which is used in Dittmann and Maug (2007), the mean CEO non-firm wealth is about 4.4 million; then a = 0.5 implies a relative risk aversion of 2.2, a number that lies in the range widely used in the literature. In addition, Haubrich (1994)considerstherangeofabsoluteriskaversiontobefrom0.125to1.125. Ourvaluea=0.5isaroundthemiddle point of his range. 10

Panel A: Incentive * 1 Panel B: Incentive * 1 0.8985 0.95 0.948 0.898 0.946 0.944 0.8975 0.942 0.897 0.94 0.938 0.8965 0.936 0.934 0.8960.2 0.22 0.24 0.26 0.28 0.3 0.9302.05 0.1 0.15 Uncertainty 0 Cash flow risk  0.47 Panel C: Incentive * 1 0.64 Panel D: Incentive * 1 0.465 0.62 0.46 0.6 0.455 0.58 0.45 0.56 0.445 0.54 0.440.2 0.22 0.24 0.26 0.28 0.3 0.502.05 0.1 0.15 Uncertainty 0 Cash flow risk  Figure 3: Incentives β∗ as functions of γ (left panels A and C) and σ (right panels B and D). Parameters: l=1.6, 0 (cid:15) κ=1, θ =1, λ=0.67, and K =0.28, In Panel A, we set a=0.5, σ =0.2, and γ ∈[0.2,0.3]. In Panel B, we set 0 1 (cid:15) 0 a=0.5, γ =0.25, and σ ∈[0.05,0.15]. In Panel C, we set a=5, σ =0.2, and γ ∈[0.2,0.3]. In Panel B, we set 0 (cid:15) (cid:15) 0 a=5, γ =0.25, and σ ∈[0.05,0.15]. 0 (cid:15) Panel D shows the traditional negative trade-off between risk σ2 and incentives β∗. In contrast, (cid:15) as predicted by Proposition 2, Panel A shows a positive relation between profitability uncertainty γ and incentive β∗ when the manager is relatively risk tolerant. Of course, uncertainty also 0 raises the perceived volatility of output. When risk aversion is relatively high as in Panel C, the traditionalnegativerisk-incentiveeffectsdominate,leadingtoanegativerelationbetweenincentives and uncertainty. We observe another interesting result in Panel B with a = 0.5. Here, because of the learningby-doing effect, even the traditional risk-incentive relation becomes hump shaped. Notice that investors would like to reduce the posterior variance γ in Eq. (4), and ∂(−γ )/∂L∗ can be viewed 1 1 1 as the marginal benefit of expediting learning through raising effort. The higher ∂(−γ )/∂L∗, the 1 1 greater the incentive β∗ that investors would like to offer. Linking this benefit to output risk σ2, in 1 (cid:15) AppendixAweshowthat∂ ∂(−γ1) /∂σ2 ≤ 0ifandonlyifσ2 ≥ γ K 2(1−λ) (L∗)2λ, whichexplainsthe ∂L∗ (cid:15) (cid:15) 0 1 1 1 nonmonotone incentive-risk relation in Panel B. This intuition is rooted in the fact that a higher σ2 implies a lower information-noise ratio. When σ2 ≥ γ K 2(1−λ) (L∗)2λ so that we are on the (cid:15) (cid:15) 0 1 1 right-hand side of the hump shape in Panel B, the information-noise ratio is low and there is plenty of room for learning. Here, the marginal benefit of expediting learning is positively related to the information-to-noise ratio. Hence, a greater σ2 lowers the marginal benefit of learning ∂(−γ1) , and (cid:15) ∂L∗ 1 consequently investors offer a lower-powered incentive contract. On the left-hand side of the hump shape where σ2 < γ K 2(1−λ) (L∗)2λ, the opposite holds. This is because the information-to-noise (cid:15) 0 1 1 ratio is already high and investors have learned a great deal about θ, and a higher σ2 lowers the (cid:15) information-to-noise ratio. This increases the room to learn, leading to a greater marginal benefit from learning. Taken together, Panel B shows that a potential positive risk-incentive relation due to learning may overturn the traditional negative risk-incentive trade-off when the manager is 11

sufficiently risk tolerant. In sum, in addition to the leading alternative explanations surveyed in the introduction, our model provides another plausible explanation for why it is difficult to identify a negative riskincentive trade-off in the data. According to our model, there could be two reasons. First, we mighthaveapositiverelationbetweenuncertaintyandincentivesforsmallriskaversioncoefficients (Panel A), and existing empirical analysis does not distinguish uncertainty from risk. Second, even if we can identify risk from uncertainty, with learning there is not necessarily a clear-cut relation between risk and incentives (Panel B). 3 Empirical Analysis In this section, we empirically test the prediction of a positive relation between uncertainty and incentives. We also investigate how this positive relation affects the traditional trade-off between risk and incentives. In Section 3.1, we describe our data, incentive and risk measures, and profitability uncertainty proxies. We then provide regression results in Section 3.2. 3.1 Data, Variables, and Summary Statistics 3.1.1 Data and Sample Selection Our sample consists of a manager-firm matched panel data set from 1992 to 2008. This data set allows us to track the highest paid executives of firms covered by ExecuComp through time. We merge the manager-level ExecuComp data with the firm-level annual accounting variables from Compustat, stock returns from CRSP, corporate board information from RiskMetrics, and analyst forecast information from IBES. We then remove the observations with incomplete data. We also winsorize the continuous variables that present obvious outliers by replacing the extreme values with the 1 and 99 percentile values. The main regressions are estimated based on our full sample, which includes 2,441 firms and 25,999 top executives. 3.1.2 Pay-Performance Sensitivity The dependent variable in the paper is pay-performance sensitivity (PPS), a standard variable used in the literature to measure managerial incentives. There are three PPS measures in the executive compensation literature. The first measure, dollar-to-dollar measure (PPS1) (Jensen and Murphy, 1990), is equal to the dollar change in stock and option holdings for a one dollar change in firm value (see also Demsetz and Lehn, 1985; Yermack, 1995; Schaefer, 1998; Palia, 2001; Jin,2002; andAggarwalandSamwick,2003). Thismeasureisessentially∂Wealth/∂(Firm Value) (where Wealth is the CEO’s wealth) and is also called value-sensitivity or share of the money in Becker (2006). The second measure, dollar-to-percentage measure (PPS2) (Hall and Liebman, 1998), is equal to the dollar change in stock and option holdings for a one percent change in 12

firm value (see also Holmstrom, 1992; and Core and Guay, 2002). The PPS2 measure is equal to ∂Wealth/∂ln(Firm Value) and is also referred to as return-sensitivity or money at stake in Becker (2006). The third measure, scaled wealth-performance sensitivity measure (PPS3) (Edmans et al., 2009), is equal to PPS2 divided by TDC1, where TDC1 is the total compensation of an executive.6 This incentive measure is similar to the percentage-to-percentage incentives (i.e., ∂(ln(Wealth))/∂(ln(Firm Value)) used (or advocated) by Murphy (1985), Gibbons and Murphy (1992), Rosen (1992), and Peng and R¨oell (2008), but replaces flow compensation in the numerator of the Murphy (1985) measure with the change in the executives’ wealth. 3.1.3 Empirical Proxies for Profitability Uncertainty Despitealargeliteraturestudyingtheeffectofparameteruncertaintyonassetpricesandinvestment (see Pastor and Veronesi, 2009, for a recent survey), separating uncertainty from risk is empirically challenging. Intheexistingliterature,mostofthestudies(e.g.,PastorandVeronesi,2003;Korteweg and Polson, 2009; and Pastor, Taylor, and Veronesi, 2009) use imperfect proxies to test model implications. Following their footsteps, we use five profitability uncertainty proxies in our study. These proxies have been used in the existing literature; for detailed definitions of these proxies, see Appendix B. We do not use firm size as an uncertainty proxy, although it is proposed by such literature as Korteweg and Polson (2009). There exists a strong empirical relation between size and PPS; that is, firm size is negatively correlated with PPS1 and positively correlated with PPS2 (e.g., Edmans et al., 2009).7 We do, however, include firm size and (size)2 as control variables in all of our regressions to capture the (potentially nonlinear) size effect.8 Natural log of firm age The first proxy that we employ is firm age. Previous studies such as PastorandVeronesi(2003)andKortewegandPolson(2009)usefirmageasaproxyforprofitability uncertainty. Uncertainty declines over a firm’s lifetime due to learning, and younger firms have higher uncertainty. Following Pastor and Veronesi (2003), we consider each firm as “born” in the year of its first appearance in the CRSP database. Specifically, we obtain the first occurrence 6The values of PPS3 for each individual executive are available from Alex Edmans’ website. We thank Alex Edmans for kindly sharing his data. 7The literature has proposed various explanations for this pattern, and therefore size may not be a clean profitability uncertainty variable for our purpose. For instance, in the Holmstrom and Milgrom’s CARA-Normal framework, risk is measured in dollar returns. Then dollar-to-dollar PPS1 should be lower for larger firms with greater dollar variances in output. For the dollar-to-percentage PPS2 measure, the matching model in Gabaix and Landier (2008) suggests that pay increases with firm size. Since part of compensation is in variable pay, it suggests that PPS2 is positively correlated with firm size. 8We also decide not to use some other uncertainty proxies found in the literature. Baker and Wurgler (2006) provide some proxies for hard-to-value stocks. Besides the variables we mention above, they mention that nondividend-paying stocks are harder to value than dividend-paying stocks because the value of a firm with stable dividends is less subjective. As a result, dividend-paying firms possibly have lower uncertainty and thus may be relatedtolowerincentives. Ourregressionscontrolfordividend-payingindicatoranddoobserveaconsistentnegative associationbetweenthedividend-payingindicatorandPPS.Analternativeexplanationofthenegativeassociationis that firms with cash constraints (such as non-dividend-paying companies) might prefer restricted stock and options over cash compensation. As a result, a higher PPS is more likely to be observed among non-dividend payers (Jin, 2002, and Yermack, 1995). 13

of a valid stock price on CRSP, as well as the first occurrence of a valid market value in the CRSP/COMPUSTAT database, and take the earlier of the two. The firm’s age is assigned the value of one in the year in which the firm is born and increases by one in each subsequent year. As in Pastor and Veronesi (2003), we take the natural log of firm age. Log(age) is concave in firm’s plain age, and captures the idea that regarding uncertainty, one year of age should matter more for young firms than for old firms. Earnings response coefficient (ERC) We follow Pastor et al. (2009) and Cremers and Yan (2010) to use the stock price reaction to earnings announcements (i.e., earnings response coefficient orbriefly,ERC).Morespecifically,ERCistheaverageofafirm’sprevious12stockpricereactionsto quarterlyearningssurprises.9 Intuitively,investorswhoaremoreuncertainabouttheprofitabilityof acompanyshouldrespondmorestronglytoearningssurprises. AsnotedinPastoretal. (2009),the ERCmeasureisidealtoseparateuncertaintyfromvolatilitybecauseERCishighwhenuncertainty is high and earnings volatility is low. When realized earnings are more precise, investors react more to earnings surprises, leading to a higher value of ERC. The shortcoming of the ERC measure is its measurement error. As a result, we also incorporate other empirical proxies of uncertainty in the analysis. Market-to-book ratio Thethirdproxyforprofitabilityuncertaintyisthemarket-to-bookratio, which equals market value of equity plus the book value of debt, divided by total assets. Pastor and Veronesi (2003) show that aging in the life of a firm is accompanied by a decrease in the market-to-book ratio. According to Korteweg and Polson (2009), the market-to-book ratio is a proxy for firm growth opportunities, and such opportunities are inherently more difficult to value than the assets in place. As a result, the market-to-book ratio increases with uncertainty about firm profitability. Tangibility The fourth proxy is tangibility. Korteweg and Polson (2009) mention that firms with more tangible assets (property, plant, and equipment) are easier to value and thus are related to lower profitability uncertainty. We use net property, plant, and equipment scaled by firm total assets to measure tangibility. Analyst forecast error We also construct an analyst forecast error variable as a proxy of profitability uncertainty. Based on Bae et al. (2008) and Lang and Lundholm (1996), for each specific company in each fiscal year, we first obtain the absolute value of the forecast error made by each analyst, where forecast errors are defined as the difference between the forecast value and the actual value of earnings per share. We then use the median value of these absolute forecast errors, 9Pastor et al. (2009) also use a second ERC measure that is the negative of the regression slope of the firm’s last20quarterlyearningssurprisesonitsabnormalstockreturnsaroundearningsannouncements. Wereportinthe paper the results from using the ERC1 variable. The results from the ERC2 variable are similar and available upon request. 14

scaled by the absolute value of the actual EPS. Using the mean value of the absolute forecast errors gives similar results.10 We end this section by pointing out that uncertainty is hard to measure and could be endogenous. We use five different proxies for uncertainty, hoping that establishing similar results for all of them can raise hurdles for other alternative explanations. Unfortunately, the five proxies we use can be all linked to firm growth. Fast-growing firms have higher marginal benefit of managerial effort and thus should have higher-powered incentives, which can also explain the positive uncertainty-incentive relation.11 To address this issue at least partially, our control variables include the long-term earnings growth forecast from analysts, which gives a more precise measure of firm growth (relative to our five uncertainty proxies). Indeed, in the regressions, the coefficient on long-term earnings growth forecast is always significantly positive, suggesting the validity of this alternative mechanism. 3.1.4 The Risk Variable Similar to the literature that tests the risk-incentive relation, we take stock return volatility as a measure of risk in our regression analysis. We measure stock return volatility as the standard deviation of daily log (percentage) returns over the past five years, which is then annualized by multiplying by the square root of 254 (Yermack, 1995, and Palia, 2001). We acknowledge that this proxy for firm risk may be imperfect and can also capture profitability uncertainty. We also use the percentage rank of stock dollar return variance (Aggarwal and Samwick, 1999, 2002, 2003, Garvey and Milbourn, 2003, and Jin, 2002) in the empirical analysis, but obtain essentially the same results. 3.1.5 Control Variables In the regressions, we include various control variables that could potentially affect the incentives a firmprovidestoitsmanagers; seedetaileddefinitionsofallofthefollowingvariablesinAppendixB. Thesecontrolvariableshavebeenusedintheempiricalliteratureonthedeterminantsofmanagerial incentives (Aggarwal and Samwick, 2003, Core et al., 1999, Jin, 2002, Palia, 2001, etc.). As mentioned at the beginning of Section 3.1.3, since there is a well-established empirical pattern between incentives and firm size, we first include firm size and the square of firm size as controls. Following the literature, we also include profitability, the ratio of capital expenditure to total assets, advertising expenses scaled by total assets, a dummy variable that is set to one whenever advertising expenses are missing, firm leverage, and dividend payout indicator. We further control 10Another widely used measure based on IBES data is analysts’ forecast dispersion, which usually proxies for potentialdisagreementinthemarket. Thedifferencebetweenforecastdispersionandforecasterroristhatthelatter considers the distance between EPS forecast and actual EPS, while the former considers the distance between EPS forecast and the mean forecast among analysts. The forecast error variable better captures profitability uncertainty studied in this paper. Consider the situation where two analysts issued the same EPS forecast of $5, and the actual EPSturnsouttobe$3. Then,inthisexampletheforecasterrorwillbe2(whichmightresultfromlargeuncertainty), but the forecast dispersion is just 0. 11We owe an anonymous referee for this excellent point. 15

for corporate governance variables, which include the CEO chair indicator and the proportion of inside directors on the board. Manager-level variables, such as log(tenure), the CEO indicator, and the female indicator, are also controlled in the regressions. Finally, year and industry effects are included to capture the time and industrial differences in the level of managerial incentives. 3.1.6 Summary Statistics and Correlations between Variables Table 1 contains summary statistics of the variables used in the regression analysis. For instance, theaverage(median)dollar-to-dollarmeasureofPPS1is1.13%(0.22%),suggestingthattheaverage (median)dollarchangeinthesampleexecutives’stockandoptionholdingsforaonethousanddollar change in firm value is $11.3 ($2.2). These summary numbers are consistent with those provided in the empirical literature such as Core and Guay (1999), Palia (2001), and Yermack (1995). The statistics also imply a positive skewness in PPS, with a few companies having very high incentives. The average, median, minimum, and maximum age of the sample firms are 26, 20, 1, and 84 years, respectively, similartothosereportedinPastorandVeronesi(2003). Thefirmsinthesample have an average (median) earnings response coefficient of 4.44 (2.88), market-to-book ratio of 2.08 (1.51), tangibility of 0.29 (0.23), and total assets of $6.6 ($1.3) billion. The average analyst forecast error relative to the actual value is about 16%. In addition, the average (median) annual stock return volatility is 44% (39%). Table 2 examines the pairwise correlations between the variables. Not surprisingly, the three PPSvariablesarepositivelycorrelated; thecorrelationcoefficientbetweenthedollar-to-dollarPPS1 and the dollar-to-return PPS2 is 0.55, and PPS1 (PPS2) is correlated with PPS3 at 0.21 (0.25). The PPS variables are in general negatively correlated with firm age and tangibility, and are positively correlated with the earnings response coefficient (ERC) and the market-to-book ratio. The correlations between PPS2 and firm age are very low. The low correlations may be due to the fact that PPS2 is PPS1 multiplied by market value of equity, and the negative relation between age and PPS1 is canceled out by the positive relation between age and market value. When we control for firm size in the model, the relation between PPS2 and firm age becomes negative and significant. PPS3 has a very low correlation (-0.03) with firm size, consistent with the property mentioned in Edmans et al. (2009) that the PPS3 measure is independent of firm size. Table 2 also shows that the uncertainty proxy variables are correlated with each other, with the correlation between firm age and market to book being -0.23 and the correlation between firm age and tangibility around 0.18. These correlations indicate that younger firms tend to be firms with more growth options and lower tangibility ratios. The table also reveals very low correlations between ERC and volatilities and between ERC and firm size, suggesting that ERC serves an ideal proxy variable that separates uncertainty from volatility and firm size. In contrast, the percentagereturn and dollar-return volatilities have opposite signs in correlations with other variables. This is perhaps due to the fact that the dollar return volatility, which equals percentage return volatility multiplied by firm market value, captures the firm size effect. 16

3.2 Empirical Results This section uses regression analysis to examine the effect of profitability uncertainty and risk on incentives. The main empirical model is as follows: PPS = α+β (Uncertainty proxies) +β (Risk) (10) ijt 1 j,t−1 2 j,t−1 +β (Firm characteristics) +β (Managerial characteristics) 3 j,t−1 4 i,t−1 +β (Year dummies) +β (Industry dummies) +(cid:15) . 5 t 6 j ijt In the equation, we use i to denote manager, j to denote firm, and t to denote year. The dependent variable is pay-performance sensitivities. In the OLS regressions, we control for industry effects using two-digit SIC indicator variables. In the firm-manager pair fixed effects regressions, we replace industry effects with firm-manager fixed effects in Eq. (10), as the latter absorbs the former. We lag all the explanatory variables by one year to mitigate potential reverse causality issues, and later use the fixed effects model in robustness analysis to deal with the endogeneity problem caused by time-invariant unobservable factors. We acknowledge that lagging may not fully resolve endogeneity because serial correlations may exist in some uncertainty proxies (some of our proxies may be forward-looking). We also note that the fixed effects model cannot deal with time-variant unobservable factors. 3.2.1 Main Results Tables 3-5 report the OLS regression results, with each table having different PPS dependent variables. The t-statistics in these regressions are heteroskedasticity robust and are adjusted for clustering within firms. In all three tables, Column (1) does not include any of the five uncertainty variables, Columns (2)-(6) include one of the five uncertainty variables, and Column (7) includes all five uncertainty variables. Positive uncertainty-incentive relation The results in Tables 3-5 show that firm age is negativelyrelatedtoincentives(Columns(2)and(7)),indicatingthatyoungerfirms,i.e.,firmswith higher uncertainty, are associated with greater managerial incentives. Both the earnings response coefficient(ERC)andthemarket-to-bookratioarepositivelyassociatedwiththeincentivevariables inmostregressions. TherelationbetweentangibilityandPPSisgenerallynegative,suggestingthat firmsthathavemoretangibleassetsareassociatedwithlowerincentives. Firmswithgreateranalyst forecast errors (that might be due to greater uncertainty) are weakly related to higher incentives. All of these results indicate a positive relation between profitability uncertainty and incentives, consistent with our model when the manager’s risk aversion is relatively low. This positive relation is not only statistically significant but also economically important. Take Column (7) in Tables 3-5 as examples. A one-standard-deviation decrease in log(firm age), which is about 0.97 (i.e., firm age reduces by about 3 years), is associated with an increase of approximately 0.23% (=0.97×0.24) in 17

PPS1, 34.09 (=0.97×35.14) in PPS2, and 11.72 (=0.97×12.08) in PPS3. These increases in PPS are of similar magnitude to that of the median values of PPS. Other uncertainty variables have similar economic significance. Reexamining the risk-incentive relation The negative risk-incentive relation is a key predictionfromstandardagencytheories,butwithmixedempiricalsupportfromexistingliterature. From the point of view of this paper, the risk proxies used in the previous literature, namely stock volatility and rank of dollar return volatility, could be contaminated by profitability uncertainty. If profitability uncertainty is positively related to incentives, then it is not surprising that previous research,inwhichtheriskproxycapturesboththecashflowriskσ2 andtheprofitabilityuncertainty (cid:15) γ , finds an ambiguous risk-incentive relation. 0 The above reasoning suggests that in revealing the negative risk-incentive relation, it is important to control for uncertainty, as it helps correct for the positive bias potentially caused by omitting relevant variables that are proxies for profitability uncertainty. Our empirical results offer evidence for this implication. Compared with the specification that does not include the uncertainty proxies (i.e., Columns(1) of Tables 3-5), when we include the uncertainty variables in the regressions (Columns (7) of Tables 3-5), the relation between volatility and incentives becomes less positive or more negative. This pattern generally holds in other specifications considered in Section 3.2.2 for robustness checks. Although our results do not fully restore the significantly negative risk-incentive relation from the data (possibly due to such reasons as endogenous matching between firm risk and CEO’s risk appetite, the learning-by-doing effect in Figure 3 Panel D of this paper, etc.), it should be safe to say that separating profitability uncertainty from cash flow risk is important when empirically examining the negative risk-incentive relation. Our results also indicate that it may be important to separate the effect of profitability uncertainty from that of risk in other empirical studies. 3.2.2 Robustness Analysis This section performs additional analysis to investigate the robustness of our empirical results. Risk measured as dollar return volatility In addition to measuring firm risk using the variance of stock percentage returns, we attempt to use a different measure of firm risk: volatility of stock dollar returns. Following Aggarwal and Samwick (1999, 2003) and Jin (2002), we use the percentage rank of the variance of dollar returns,12 and report results in Panel A of Table 6. In Column (1), we find that the rank of dollar return volatility is negative and significant, consistent with Aggarwal and Samwick (1999, 2002, 2003), Garvey and Milbourn (2003), and Jin (2002). In Column (2), we include the uncertainty variables, and find that greater profitability uncertainty is 12According to Aggarwal and Samwick (1999, 2003), the use of the percentage ranks deals with potential outliers inthedollarreturndataandalsoallowsthepay-performanceincentivesatdifferentpointsinthedistributionoffirm risk to be easily compared. In the regressions, we also use an alternative transformation of the raw dollar return variance, namely the logarithm of dollar return variance, and we find basically the same results. 18

related to higher incentives. Moreover, the dollar return volatility (i.e., the risk proxy) continues to benegativeandsignificantafterincludinguncertaintyvariables. InColumns(3)-(6),inwhichPPS2 and PPS3 are dependent variables, we continue to find that firms with greater uncertainty provide higher incentives to their executives. The effect of the risk variable is positive and significant when the uncertainty variables are excluded, but the effect becomes insignificant when the uncertainty variables are introduced to the model. Median regressions Following Aggarwal and Samwick (1999, 2003) and Jin (2002), we use median regressions to deal with outliers and right skewness in the compensation data. Results are reported in Panel B of Table 6 (with risk measured by the percentage return volatility) and Panel C of Table 6 (with risk measured by the rank of dollar return volatility). Both tables show that in general, uncertainty is positively related to incentives. The coefficient on the risk variable becomes less positive or more negative if profitability uncertainty is captured in the model. Fixed effect regressions In Panel D of Table 6, we deal with potential endogeneity issues by adding the firm-manager paired fixed effects in the regressions. For example, it is possible that some unobservable managerial attributes (e.g., risk aversions) are correlated with the explanatory variables, such as firm age and at the same time are correlated with the dependent variable, PPS. Thefirm-managerfixedeffectmayalsocapturetime-invariantunobservablefactorsthatpotentially affect endogenous matching between the firm and the manager (Graham et al., 2011). We can see from Panel D of Table 6 that the coefficients on the profitability uncertainty proxies continue to show a positive relation between profitability uncertainty and incentives. Admittedly,thefixedeffectspecificationcanonlyaddressthepotentialendogeneityproblemdue totimeinvariantomittedvariables. Fixedeffectscannotaddressthetime-variantomittedvariables, nor the reverse causality problem, where some of our proxies of uncertainty (e.g., Market-to-book ratio) are forward looking and thus respond to tomorrow’s pay-performance sensitivity (recall that we have lagged uncertainty proxies by one year in regression). Other robustness checks Finally, the tables reported so far examine each top executive’s incentives. In untabulated analysis, we also examine CEO incentives only, non-CEO incentives, and the average incentives for top executives in each individual company. We also examine the incentives from stock and options, separately. The results, omitted for brevity, provide the same implications as those reported here. In addition, Pastor and Veronesi (2003) find that the market-to-book ratio increases with uncertainty about average profitability, especially for firms that pay no dividends. We interact the dividend paying dummy with the uncertainty proxy variables, and run regressions with interaction variables. The coefficients of the interaction variables are not significant, suggesting that the positive relation between uncertainty and incentives does not vary significantly between firms that pay dividends and firms that do not. 19

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Table 1: Summary Statistics The table provides summary statistics of the variables. Detailed definitions of the variables are in Appendix B. Variables N Mean Stdev Median Min Max Dependent variables: Pay-performance sensitivity PPS1 179,930 1.13% 3.15% 0.22% 0% 22.66% PPS2 ($thousands) 179,930 168.61 476.91 27.97 0 3,519.93 PPS3 169,841 36.34 119.97 6.77 0 939.61 Profitability uncertainty variables Firm age 143,291 25.56 20.02 20 1 84 Earnings response coefficient (ERC) 117,263 4.44 10.40 2.88 -57.01 75.09 Market-to-book (M/B) 141,405 2.08 2.03 1.51 0.51 43.19 Tangibility 139,799 0.29 0.24 0.23 0 0.94 Analyst forecast error 131,689 0.16 0.55 0.03 0 6 Risk variables Stock return volatility 141,623 0.44 0.20 0.39 0.18 1.14 Dollar return volatility ($millions) 108,557 1497.20 2639.38 477.79 27.98 14382.63 Control variables Total assets ($millions) 143,182 6,589 14,022 1,343 0.07 72,282 Analysts’ long-term growth forecast 179,930 15.39 6.09 15.39 1.95 60 (%) Profitability 140,222 0.13 0.13 0.13 -5.09 0.45 Capital expenditure 134,919 0.06 0.06 0.05 0 0.48 Advertisement 143,195 0.01 0.03 0 0 0.19 Advertisement missing indicator 143,195 0.69 0.46 1 0 1 Leverage 142,528 0.23 0.20 0.21 0 3.09 Dividend paying indicator 143,195 0.57 0.50 1 0 1 CEO chair indicator 163,936 0.65 0.48 1 0 1 Fraction of inside directors 163,936 0.27 0.14 0.25 0 0.9 Tenure 173,383 9.23 5.85 8.86 0 40 CEO indicator 179,930 0.15 0.35 0 0 1 Female indicator 179,930 0.05 0.22 0 0 1 24

Table 2: Correlations between Variables The table includes pairwise correlations of the main variables used in the regressions. Detailed definitions of the variables are in Appendix B. Incentives Profitability Uncertainty Risk PPS1 PPS2 PPS3 age ERC M/B tang forerr vol dolvol size PPS2 0.55 1 Wealth-performance 0.21 0.25 1 sensitivity (PPS3) Log(firm age) (age) -0.16 0.003 -0.10 1 Earnings response 0.04 0.05 0.07 -0.06 1 coefficient (ERC) Market-to-book (M/B) 0.08 0.20 0.19 -0.23 0.07 1 Tangibility (tang) -0.05 -0.08 -0.03 0.18 -0.06 -0.12 1 Analyst forecast error 0.02 -0.04 -0.01 -0.04 -0.05 -0.04 0.01 1 (forerr) Stock return volatility 0.10 -0.03 0.03 -0.44 -0.04 0.23 -0.22 0.13 1 (vol) Rank of dollar return -0.14 0.29 0.09 0.23 0.01 0.16 -0.06 -0.12 -0.12 1 volatility (dolvol) Firm size (size) -0.20 0.19 -0.03 0.44 -0.01 -0.24 0.03 -0.10 -0.48 0.73 1 Long-term growth 0.11 0.06 0.12 -0.38 0.06 0.39 -0.17 0.02 0.45 -0.04 -0.37 forecast 25

Table 3: Effects of Profitability Uncertainty and Risk on Incentives (PPS1) The table presents the OLS regression results on the effects of profitability uncertainty and risk on incentives. The dependent variable is the dollar-to-dollar measure (PPS1) of pay-performance sensitivity. All explanatory variables are lagged by one year. The sample includes all companies in ExecuComp and covers the period from 1992 to 2008. Detailed definitions of all the variables are in Appendix B. Heteroskedasticity robust t-statistics adjusting for clustering within companies are in parentheses. Significance at the 10%, 5%, and 1% levels is indicated by *, **, and ***. Dependent variable = PPS1 (1) (2) (3) (4) (5) (6) (7) Profitability uncertainty variables -0.20*** -0.24*** Log(firm age) (-) -- -- -- -- -- (-6.02) (-6.80) 0.0071*** 0.007*** ERC (+) -- -- -- -- -- (4.32) (4.03) 0.007 -0.002 Market-to-book (+) -- -- -- -- -- (0.60) (-0.13) -0.29* -0.17 Tangibility (-) -- -- -- -- -- (-1.65) (-0.97) 0.03 0.04* Analyst forecast error (+) -- -- -- -- -- (1.61) (1.87) Risk variable 0.0023 -0.18 0.20 0.008 -0.007 0.08 -0.058 Stock return volatility (0.01) (-1.02) (1.05) (0.05) (-0.04) (0.47) (-0.31) Control variables -0.31*** -0.29*** -0.29*** -0.31*** -0.31*** -0.30*** -0.26*** Firm size (-16.20) (-13.90) (-14.91) (-16.08) (-15.95) (-16.04) (-12.75) 0.056*** 0.05*** 0.05*** 0.053*** 0.06*** 0.06*** 0.049*** Squared firm size (8.26) (8.15) (6.85) (7.93) (8.18) (8.29) (7.05) 0.014*** 0.01*** 0.01*** 0.013*** 0.013*** 0.01*** 0.008*** Long-term growth forecast (4.77) (3.68) (3.88) (4.69) (4.53) (4.92) (2.77) 0.52** 0.43** 0.49** 0.48** 0.54** 0.48** 0.43* Profitability (2.47) (2.06) (2.17) (2.25) (2.54) (2.32) (1.94) 0.90** 0.75* 1.00** 0.92** 1.33*** 1.18*** 1.13*** Capital expenditure (2.23) (1.86) (2.42) (2.28) (3.09) (2.90) (2.64) 1.14 1.23 1.30 1.14 1.08 1.43 1.36 Advertisement (0.98) (1.06) (1.09) (0.98) (0.93) (1.23) (1.16) Advertisement missing -0.01 -0.01 0.04 -0.02 -0.01 0.02 0.04 dummy (-0.25) (-0.20) (0.66) (-0.27) (-0.18) (0.31) (0.69) 0.03 0.02 -0.06 0.005 0.07 -0.06 -0.03 Leverage (0.20) (0.12) (-0.39) (0.03) (0.49) (-0.44) (-0.23) -0.28*** -0.21*** -0.26*** -0.27*** -0.27*** -0.27*** -0.17*** Dividend paying indicator (-4.65) (-3.41) (-4.23) (-4.65) (-4.51) (-4.54) (-2.78) 0.29*** 0.29*** 0.25*** 0.29*** 0.29*** 0.26*** 0.26*** CEO chair indicator (6.92) (6.97) (6.09) (6.83) (6.89) (6.43) (6.20) 2.60*** 2.57*** 2.37*** 2.57*** 2.57*** 2.46*** 2.26*** Fraction of inside directors (12.45) (12.38) (11.44) (12.42) (12.21) (12.02) (11.01) 0.59*** 0.60*** 0.59*** 0.59*** 0.60*** 0.58*** 0.60*** Log(tenure) (14.75) (14.99) (14.66) (14.82) (14.74) (15.13) (14.88) 2.80*** 2.79*** 2.71*** 2.80*** 2.80*** 2.76*** 2.71*** CEO indicator (30.68) (30.67) (28.91) (30.70) (30.52) (30.58) (28.63) -0.21*** -0.22*** -0.24*** -0.21*** -0.21*** -0.22*** -0.25*** Female indicator (-2.59) (-2.66) (-3.25) (-2.64) (-2.58) (-3.05) (-3.31) Year & 2-digit SIC dummies Yes Yes Yes Yes Yes Yes Yes Adjusted R2 0.22 0.22 0.22 0.22 0.22 0.22 0.22 Number of observations 119,281 119,281 102,537 119,079 118,149 113,496 100,760 26

Table 4: Effects of Profitability Uncertainty and Risk on Incentives (PPS2) The table presents the OLS regression results on the effects of profitability uncertainty and risk on incentives. The dependent variable is the dollar-to-percentage measure (PPS2) of pay-performance sensitivity. Other information is the same as that in Table 3. Dependent variable = PPS2 (1) (2) (3) (4) (5) (6) (7) Profitability uncertainty variables -33.36*** -35.14*** Log(firm age) (-) -- -- -- -- -- (-5.89) (-5.65) 1.31*** 1.01*** ERC (+) -- -- -- -- -- (4.25) (3.48) 53.53*** 59.52*** Market-to-book (+) -- -- -- -- -- (13.83) (13.65) -186.15*** -98.39*** Tangibility (-) -- -- -- -- -- (-6.12) (-3.27) -2.83 3.40 Analyst forecast error (+) -- -- -- -- -- (-0.71) (1.12) Risk variable -6.59 -36.69 35.88 -42.81 -11.30 19.98 -62.67* Stock return volatility (-0.22) (-1.23) (1.00) (-1.54) (-0.37) (0.63) (-1.90) Control variables 93.35*** 98.18*** 97.74*** 95.59*** 93.63*** 96.28*** 101.61*** Firm size (19.66) (19.86) (19.23) (21.72) (19.63) (19.69) (21.19) 11.04*** 10.83*** 9.21*** 7.23*** 10.32*** 10.31*** 5.67*** Squared firm size (6.66) (6.58) (4.94) (4.44) (6.36) (5.91) (3.25) 8.39*** 7.83*** 8.37*** 3.86*** 7.97*** 8.33*** 2.63*** Long-term growth forecast (11.74) (11.19) (11.38) (6.79) (11.21) (11.40) (4.25) 294.99*** 278.55*** 393.89*** 134.40*** 299.38*** 321.27*** 79.62** Profitability (6.05) (5.74) (8.69) (3.69) (6.14) (6.50) (2.24) 223.51*** 197.56*** 205.42*** 117.67* 500.55*** 207.27*** 286.90*** Capital expenditure (3.09) (2.78) (2.60) (1.76) (6.56) (2.82) (3.77) 584.96*** 599.36*** 573.61*** 446.43** 535.77** 614.50*** 426.87** Advertisement (2.99) (3.07) (2.73) (2.44) (2.78) (3.01) (2.21) Advertisement missing 3.14 3.64 6.53 6.83 3.62 6.23 7.21 dummy (0.30) (0.35) (0.59) (0.69) (0.35) (0.57) (0.71) -144.48*** -146.54*** -158.20*** -109.93*** -129.62*** -165.64*** -107.34*** Leverage (-5.22) (-5.36) (-5.09) (-4.56) (-4.92) (-5.57) (-3.91) -43.39*** -31.58*** -43.56*** -48.30*** -38.85*** -45.91*** -32.33*** Dividend paying indicator (-4.10) (-2.99) (-3.79) (-4.83) (-3.70) (-4.25) (-3.01) 20.04*** 20.18*** 20.05*** 19.43*** 19.10*** 19.75*** 18.29*** CEO chair indicator (2.84) (2.89) (2.75) (2.96) (2.72) (2.74) (2.71) 318.06*** 312.53*** 319.31*** 324.43*** 306.52*** 316.90*** 301.93*** Fraction of inside directors (9.79) (9.69) (9.15) (10.32) (9.52) (9.36) (9.07) 83.01*** 84.90*** 86.82*** 84.69*** 83.92*** 83.93*** 91.80*** Log(tenure) (14.11) (14.38) (13.62) (14.62) (14.19) (13.80) (14.33) 417.28*** 417.14*** 424.42*** 416.30*** 415.57*** 424.14*** 422.89*** CEO indicator (32.33) (32.33) (30.24) (32.36) (32.04) (31.95) (29.97) -52.13*** -53.03*** -54.06*** -53.90*** -52.23*** -53.81*** -57.52*** Female indicator (-4.83) (-4.88) (-4.51) (-5.05) (-4.84) (-4.84) (-4.84) Year & 2-digit SIC Yes Yes Yes Yes Yes Yes Yes dummies Adjusted R2 0.23 0.24 0.24 0.27 0.24 0.24 0.27 Number of observations 119,281 119,281 102,537 119,079 118,149 113,496 100,760 27

Table 5: Effect of Profitability Uncertainty and Risk on Incentives (PPS3) The table presents the OLS regression results on the effects of profitability uncertainty and risk on incentives. The dependent variable is the percentage-to-percentage measure (i.e., wealth-performance sensitivity or PPS3) proposed in Edmans et al. (2009). In the regressions, PPS3 is winsorized at 99% to deal with outliers. Other information is the same as that in Table 3. Dependent variable = PPS3 (1) (2) (3) (4) (5) (6) (7) Profitability uncertainty variables -9.35*** -12.08*** Log(firm age) (-) -- -- -- -- -- (-3.80) (-4.04) 0.38*** 0.34** ERC (+) -- -- -- -- -- (2.59) (2.27) 9.81*** 11.39*** Market-to-book (+) -- -- -- -- -- (5.70) (4.72) -7.58 13.39 Tangibility (-) -- -- -- -- -- (-0.54) (0.85) 1.62 2.01 Analyst forecast error (+) -- -- -- -- -- (0.81) (1.30) Risk variable -6.65 -15.50 1.16 -13.33 -6.76 -2.50 -21.37 Stock return volatility (-0.45) (-1.02) (0.07) (-0.92) (-0.45) (-0.16) (-1.17) Control variables 5.55*** 6.89*** 5.97** 5.93*** 5.68*** 5.93*** 7.72*** Firm size (2.64) (3.11) (2.56) (2.88) (2.66) (2.72) (3.15) -0.14 -0.20 -0.21 -0.83 -0.13 -0.24 -0.70 Squared firm size (-0.25) (-0.35) (-0.33) (-1.46) (-0.22) (-0.41) (-1.04) 1.91*** 1.76*** 1.92*** 1.08*** 1.89*** 1.93*** 0.81** Long-term growth forecast (5.12) (4.75) (4.83) (3.21) (4.98) (5.04) (2.18) 71.03*** 66.26*** 94.49*** 40.79** 70.64*** 81.80*** 31.97 Profitability (3.51) (3.22) (3.84) (2.39) (3.49) (3.80) (1.51) 82.70** 75.48** 77.75** 63.68* 93.35** 79.71** 39.03 Capital expenditure (2.44) (2.26) (2.04) (1.90) (2.55) (2.29) (0.97) 225.72 230.51 241.75 201.76 224.88 244.70 222.15 Advertisement (1.42) (1.45) (1.40) (1.29) (1.42) (1.44) (1.29) Advertisement missing -0.87 -0.66 0.32 -0.21 -0.62 0.33 0.49 dummy (-0.15) (-0.12) (0.05) (-0.04) (-0.11) (0.05) (0.08) -50.53*** -51.22*** -55.62*** -43.79*** -50.66*** -53.09*** -50.13*** Leverage (-3.87) (-3.92) (-3.75) (-3.51) (-3.79) (-3.74) (-3.34) -5.20 -1.91 -4.49 -5.96 -5.12 -5.16 -1.17 Dividend paying indicator (-1.07) (-0.41) (-0.84) (-1.25) (-1.05) (-1.05) (-0.23) 17.78*** 17.81*** 18.41*** 17.63*** 18.02*** 17.52*** 18.54*** CEO chair indicator (4.79) (4.81) (4.54) (4.83) (4.82) (4.56) (4.59) 155.76*** 154.16*** 148.42*** 157.18*** 156.80*** 151.66*** 147.22*** Fraction of inside directors (8.76) (8.70) (7.87) (8.85) (8.68) (8.37) (7.77) 3.22** 3.75*** 3.41** 3.52** 3.28** 3.01** 4.50*** Log(tenure) (2.25) (2.59) (2.20) (2.49) (2.28) (2.03) (2.91) 0.05 -0.02 -0.04 -0.11 0.04 0.09 -0.44 CEO indicator (0.07) (-0.03) (-0.06) (-0.16) (0.06) (0.12) (-0.63) -3.48 -3.74 -2.28 -3.77 -3.60 -3.25 -3.11 Female indicator (-0.85) (-0.91) (-0.51) (-0.93) (-0.87) (-0.77) (-0.70) Year & 2-digit SIC dummies Yes Yes Yes Yes Yes Yes Yes Adjusted R2 0.10 0.10 0.10 0.12 0.10 0.10 0.12 Number of observations 117,238 117,238 101,449 117,130 116,115 112,050 99,730 28

Table 6: Robustness Analysis The table presents the robustness analysis results on the effects of profitability uncertainty and risk on incentives. Panel A contains OLS regressions with the volatility variable being dollar return volatility. Panel B contains median regression results. Panel C is median regressions with the volatility variable being dollar return volatility. Panel D is firm-manager paired fixed effect regression results, in which there is one fixed effect for each unique firm-manager combination. Unless mentioned, the return volatility is percentage return volatility. The dependent variable is the dollar-to-dollar measure (PPS1) of pay-performance sensitivity in Columns (1) and (2), the dollar-to-percentage measure (PPS2) in Columns (3) and (4), and the wealth-performance sensitivity (PPS3) in Columns (5) and (6). All the specifications include the same control variables as those in Table 3, but to save space, the coefficient estimates on these control variables are not reported. All explanatory variables are lagged by one year. The sample includes all companies in ExecuComp and covers the period from 1992 to 2008. Detailed definitions of all the variables are in Appendix B. For median regressions, t-statistics derived from the bootstrapped standard errors (based on 20 replications) are in parentheses. For OLS (firm-manager fixed effect) regressions, heteroskedasticity robust t-statistics adjusting for clustering within companies (firm-manager pairs) are in parentheses. Significance at the 10%, 5%, and 1% levels is indicated by *, **, and ***. Panel A: Dollar return volatility and OLS regressions Expected Dependent variable = PPS (1) PPS1 (2) PPS1 (3) PPS2 (4) PPS2 (5) PPS3 (6) PPS3 sign Profitability uncertainty variables -0.27*** -38.06*** -10.79*** Log(firm age) - -- -- -- (-6.72) (-5.03) (-3.53) 0.006*** 1.17*** 0.28* ERC + -- -- -- (3.08) (3.46) (1.89) 0.02 68.35*** 9.80*** Market-to-book + -- -- -- (1.34) (11.69) (3.71) -0.17 -105.74*** 20.04 Tangibility - -- -- -- (-0.83) (-2.93) (1.14) 0.05* 4.71 2.04 Analyst forecast error + -- -- -- (1.94) (1.23) (1.40) Risk variable Rank of dollar return -0.0081*** -0.0077*** 2.67*** -0.28 0.66*** 0.32 volatility (-5.12) (-5.10) (8.70) (-0.70) (4.05) (1.53) Control variables, year dummies, and two digit Yes Yes Yes Yes Yes Yes SIC dummies Adjusted R2 0.23 0.24 0.26 0.30 0.11 0.12 Number of observations 92,970 80,642 92,970 80,642 92,424 80,425 29

Panel B: Median regressions Expected Dependent variable = PPS (1) PPS1 (2) PPS1 (3) PPS2 (4) PPS2 (5) PPS3 (6) PPS3 sign Profitability uncertainty variables -0.03*** -4.98*** -1.00*** Log(firm age) - -- -- -- (-17.33) (-18.83) (-14.51) 0.0008*** 0.14*** 0.04*** ERC + -- -- -- (8.22) (5.24) (9.50) -0.006*** 21.55*** 1.22*** Market-to-book + -- -- -- (-10.17) (32.57) (42.91) -0.15*** -24.16*** -1.92*** Tangibility - -- -- -- (-17.50) (-14.12) (-4.88) -0.01*** -0.05 -0.25*** Analyst forecast error + -- -- -- (-3.42) (-0.20) (-3.15) Risk variable 0.06*** 0.06*** -9.50*** -12.84*** -2.75*** -3.50*** Stock return volatility - (5.20) (5.97) (-6.60) (-8.79) (-8.61) (-8.68) Control variables, year dummies, and two digit Yes Yes Yes Yes Yes Yes SIC dummies Pseudo R2 0.11 0.12 0.11 0.12 0.03 0.03 Number of observations 119,281 100,760 119,281 100,760 117,238 99,730 Panel C: Dollar return volatility and median regressions Expected Dependent variable = PPS (1) PPS1 (2) PPS1 (3) PPS2 (4) PPS2 (5) PPS3 (6) PPS3 sign Profitability uncertainty variables -0.04*** -6.46*** -0.88*** Log(firm age) - -- -- -- (-17.79) (-17.29) (-17.07) 0.0007*** 0.18*** 0.04*** ERC + -- -- -- (7.13) (4.79) (9.64) -0.002* 28.35*** 1.03*** Market-to-book + -- -- -- (-1.88) (34.02) (16.65) -0.19*** -31.00*** -1.50*** Tangibility - -- -- -- (-12.96) (-15.99) (-4.99) -0.010*** 0.55** -0.23*** Analyst forecast error + -- -- -- (-3.13) (2.02) (-2.95) Risk variable Rank of dollar return -0.001*** -0.002*** 0.67*** -0.16*** 0.05*** 0.02*** volatility (-11.08) (-11.96) (34.40) (-5.12) (30.41) (5.51) Control variables, year dummies, and two digit Yes Yes Yes Yes Yes Yes SIC dummies Pseudo R2 0.12 0.13 0.12 0.14 0.03 0.03 Number of observations 92,970 80,642 92,970 80,642 92,424 80,425 30

Panel D: Fixed effect regressions Expected Dependent variable = PPS (1) PPS1 (2) PPS1 (3) PPS2 (4) PPS2 (5) PPS3 (6) PPS3 sign Profitability uncertainty variables -0.42*** -71.50*** -29.06*** Log(firm age) - -- -- -- (-4.51) (-4.03) (-6.64) 0.003*** 0.20 0.25*** ERC + -- -- -- (3.39) (0.90) (5.04) -0.0001 54.85*** 4.36*** Market-to-book + -- -- -- (-0.02) (20.45) (6.50) -0.60*** 13.57 6.25 Tangibility - -- -- -- (-3.31) (0.37) (0.85) -0.01 -1.68 -0.49 Analyst forecast error + -- -- -- (-0.94) (-0.89) (-1.04) Risk variable -0.52*** -0.82*** -150.40*** -147.10*** -4.42 0.08 Stock return volatility - (-3.57) (-5.33) (-5.77) (-4.62) (-0.61) (0.01) Control variables, year dummies, and firm- Yes Yes Yes Yes Yes Yes manager paired fixed effects Adjusted R2 0.81 0.82 0.72 0.74 0.69 0.70 Number of observations 119,365 100,835 119,365 100,835 117,238 99,730 31

Appendix A: Proofs Proof of Lemma 1. Note that given α,β and L , the manager’s expected utility is: 1 (cid:20) (cid:18) (cid:18) (cid:19)(cid:19)(cid:21) (cid:104) (cid:105) l 1 (cid:16) (cid:17)2 E −e−a(w1− 2 lL2 1 ) = −exp −a α+βθ 0 K 1 1−λLλ 1 − 2 L2 1 − 2 aβ2 γ 0 K 1 1−λLλ 1 +σ (cid:15) 2 . Denote the above function by U (L ). Its first-order condition is 1 dU (L ) (cid:104) (cid:105) 1 = U (L )(−a) λβθ K1−λLλ−1−lL −aγ λβ2K 2(1−λ) L2λ−1 , dL 1 0 1 1 1 0 1 1 1 and its second-order condition is d2U (L ) (cid:104) (cid:105)2 1 = (−a)2U (L ) λβθ K1−λLλ−1−lL −aγ λβ2K 2(1−λ) L2λ−1 dL2 1 0 1 1 1 0 1 1 1 (cid:16) (cid:17) +U (L )a λ(1−λ)βθ K1−λLλ−2+l+(2λ−1)aγ λβ2K 2(1−λ) L2λ−2 < 0. 1 0 1 1 0 1 1 The optimal L∗ is determined by the first-order condition of the manager’s optimization problem. 1 That is, it is the unique solution of the following equation: λβθ K1−λ−lL2−λ−aγ λβ2K 2(1−λ) Lλ = 0. 0 1 1 0 1 1 The assumption θ > 0 ensures a unique positive solution for L∗. The fixed salary α is chosen to 0 1 satisfy the manager’s participation constraint: (cid:18) (cid:19) l 1 (cid:16) (cid:17)2 1 (cid:16) (cid:17) α+βθ 0 K 1 1−λ(L∗ 1 )λ− 2 (L∗ 1 )2− 2 aβ2 γ 0 K 1 1−λ(L∗ 1 )λ +σ (cid:15) 2 = − a log −U(cid:98) ; or, after substituting the expression of L∗ and U(cid:98) = −1, we have 1 (cid:18) (cid:19) l 1 (cid:16) (cid:17)2 α = −βθ K1−λ(L∗)λ+ (L∗)2+ aβ2 γ K1−λ(L∗)λ +σ2 . 0 1 1 2 1 2 0 1 1 (cid:15) Proof of Proposition 2. We first prove that dβ∗ > 0 holds when the manager is risk neutral dγ0 (i.e., a = 0); the statement in the proposition immediately follows in light of the continuity of the derivative dβ∗/dγ in a. We can view the maximization problem in terms of implemented effort 0 L∗. If the optimal effort increases with uncertainty γ , i.e. 1 0 dL∗ 1 > 0, (1) dγ 0 and if higher effort is linked to higher incentives, which requires that dL∗ 1 > 0, (2) dβ 1

then we obtain our desired result dβ∗ > 0. Below we proceed to show that both Eq. (1) and Eq. dγ0 (2) hold. From the incentive compatibility condition λβθ K1−λ −l(L∗)2−λ −aγ λβ2K 2(1−λ) (L∗)λ = 0, 0 1 1 0 1 1 we have dL∗ dL∗ λθ K (1−λ) −l(2−λ)(L∗)1−λ 1 −aγ λ2β2K 2(1−λ) (L∗)λ−1 1 −2aγ λβK 2(1−λ) (L∗)λ = 0. 0 1 1 dβ 0 1 1 dβ 0 1 1 Simplifying the above equation, we have dL∗ λθ K1−λ−2aγ λβK 2(1−λ) (L∗)λ 1 = 0 1 0 1 1 . dβ l(2−λ)(L∗)1−λ+aγ λ2β2K 2(1−λ) (L∗)λ−1 1 0 1 1 Setting a = 0 and noticing that θ > 0, we have 0 dL∗ 1 (cid:12) (cid:12) (cid:12) = λθ 0 K 1 1−λ > 0. dβ (cid:12) l(2−λ)(L∗)1−λ a=0 1 dL∗ dL∗ Now we use the supermodularity property to prove 1 > 0. To prove 1 > 0, it suffices to show dγ0 dγ0 that ∂2V (L,γ 0 ) (cid:12) (cid:12) (cid:12) > 0. ∂L∂γ (cid:12) 0 a=0 Recall that the time-0 expected payoff function is given by l 1 γ2K 2(1−λ) L2λ V (L,γ ) = θ K1−λLλ− L2− aβ2γ K 2(1−λ) L2λ+M 0 1 0 0 1 2 2 0 1 σ2+γ K 2(1−λ) L2λ (cid:15) 0 1 1 1 − aσ2β2+Mθ2− K . 2 (cid:15) 0 2 1 We thus have ∂V (L,γ ) γ2K (1−λ) L2λ−1 0 = λθ K1−λLλ−1−lL−λaβ2γ K 2(1−λ) L2λ−1+2λMσ2 0 1 , ∂L 0 1 0 1 (cid:15)(cid:16) σ2+γ K 2(1−λ) L2λ (cid:17)2 (cid:15) 0 1 and (cid:18) (cid:19)2 ∂ γ0 ∂2V (L,γ 0 ) = −λaβ2K2ηL2λ−1+2λMσ2K2ηL2λ−1 σ (cid:15) 2+γ0K 1 2ηL2λ ∂L∂γ 1 (cid:15) 1 ∂γ 0 0 γ σ2 = −λaβ2K 2(1−λ) L2λ−1+4λMσ2K 2(1−λ) L2λ−1 0 (cid:15) . 1 (cid:15) 1 (cid:16) (cid:17)3 σ2+γ K 2(1−λ) L2λ (cid:15) 0 1 2

Therefore, when a = 0 (and hence when a is sufficiently small), ∂2V (L,γ ) 4λMσ2K 2(1−λ) L2λ−1γ σ2 0 = (cid:15) 1 0 (cid:15) > 0. ∂L∂γ 0 (cid:16) σ2+γ K 2(1−λ) L2λ (cid:17)3 (cid:15) 0 1 This completes the proof. Appendix of Section 2.4. We have (cid:16) (cid:17)2 ∂(−γ 1 ) = − γ 0 2 K 1 1−λ(L∗ 1 )λ = − γ 0 2 , ∂σ (cid:15) 2 (cid:18) σ2+γ (cid:16) K1−λ(L∗)λ (cid:17)2 (cid:19)2 (cid:18) σ (cid:15) 2 +γ (cid:16) K1−λ(L∗)λ (cid:17) (cid:19)2 (cid:15) 0 1 1 K1−λ(L∗)λ 0 1 1 1 1 therefore ∂2(−γ1) depends on the sign of ∂L∗∂σ2 1 (cid:15) (cid:20) (cid:21) ∂ K1−λ σ ( (cid:15) 2 L∗)λ +γ 0 K 1 1−λ(L∗ 1 )λ λ (cid:104) γ 0 K 1 2(1−λ) (L∗ 1 )2λ−σ (cid:15) 2 (cid:105) 1 1 = . ∂L∗ K1−λ(L∗)λ+1 1 1 1 3

Appendix B: Definition of Variables Firm-Level Variables Firm Age: Based on Pastor and Veronesi (2003), we consider each firm as “born” in the year of its first appearance in the CRSP database. Specifically, we look for the first occurrence of a valid stock price on CRSP, as well as the first occurrence of a valid market value in the CRSP / COMPUSTAT database, and take the earlier of the two. The firm’s plain age is assigned the value of one in the year in which the firm is born and increases by one in each subsequent year. We use natural log of firm’s plain age as the proxy for uncertainty. Earnings Response Coefficient (ERC): This variable is the ERC1 as defined in Pastor, et al. (2009) and is equal to the average of the firm’s previous 12 stock price reactions to quarterly earnings surprises. Specifically, we first obtain RC, which is the abnormal return due to a quarterly earnings announcement divided by the unexpected quarterly earnings. The abnormal return is measured as the cumulative return of stock i in excess of stock i’s industry’s return starting one trading day before the firm’s earnings announcement and ending one trading day after the same announcement. Quarterly earnings announcement dates are from IBES. The industry returns are the daily returns of 49 value-weighted industry portfolios from Ken French’s website. The unexpected quarterly earnings are equal to the difference between the actual quarterly earnings per share (obtained from the IBES unadjusted actuals file) and the mean of all analyst forecasts of EPS using IBES’s last preannouncement set of forecasts for the given fiscal quarter, deflated by bookequitypershareofthecompany. WewinsorizeRCat5%and95%andaveragethewinsorized quarterly RCs over the rolling three-year window to obtain ERC1. Pastor et al. (2009) contain more detailed information on constructing the ERC variables. Market to Book: (Market value of equity plus the book value of debt)/total assets = (CSHO×PRCC F + AT - CEQ)/AT = (data25×data199+data6-data60)/data6. Tangibility: Net property, plant, and equipment/total assets = PPENT/AT = data8/data6. Analyst Forecast Error: For each individual company in each fiscal year, we first obtain the absolute value of the forecast error (equal to the difference between the forecast and the actual EPS values) made by each analyst, and then we use the median value of these absolute forecast errors scaled by the absolute value of the actual EPS. Using the mean value of the absolute forecast errors or scaling by stock price per share gives similar results. The analyst forecast error variable is constructed from the I/B/E/S details database. Stock Return Volatility: First, we obtain the standard deviation of daily log returns over the past five years, and then annualize the standard deviation by multiplying by the square root of 254. This is the percentage return volatility. Rank of Dollar Return Volatility: Dollar return volatility is equal to stock percentage returnvolatilitymultipliedbythebeginning-of-yearfirmmarketvalue. Thisvariableismeasuredin 1

$millions. ConsistentwithAggarwalandSamwick(1999)andJin(2002), weemploythepercentage ranks of dollar return variance in our tests and these percentage ranks range from 0 (lowest risk) to 100 (highest risk). Firm Size: Natural log of total assets = log(AT) = log(data6). Assets are measured in $millions. Analysts’ Long-Term Growth Forecast: This variable comes from I/B/E/S analysts’ forecast of long-term earnings growth (LTG in I/B/E/S). When multiple analysts give LTG forecasts about the same company during the same period, the median forecast is used. Profitability: Operating income before depreciation and amortization/total assets = OIBDP/AT = data13/data6. Capital expenditure: Capital expenditures/total assets = CAPX/AT = data128/data6. Advertisement: Advertising expense/total assets = XAD/AT = data45/data6. This variable issettozeroifitismissingandanadvertisementmissingindicatoristhusincludedintheregressions to deal with the missing advertisement issue. Advertisement Missing Indicator: A dummy variable equal to one if the advertisement variable is missing. Leverage: (Long term debt + debt in current liabilities)/total assets = (DLTT+DLC)/AT = (data9+data34)/data6. Dividend-Paying Indicator: A dummy variable equal to one if dividends on common stock (data21 or DVC) are strictly positive, and zero otherwise. CEO Chair Indicator: A dummy variable equal to one if the CEO of the company is also the board chairman, and zero otherwise. Fraction of Inside Directors: Number of inside board directors divided by board size, where an inside director is defined as a director who is a current or former firm manager or one of his or her family members is a current or former firm manager.. Manager Level Variables PPS1: Dollar-to-dollar measure of pay-performance sensitivity. This variable measures the dollar change in stock and option holdings for a one dollar change in firm value. To estimate PPS1, first calculate a variable named totaldelta, which is obtained from multiplying the Black-Scholes hedge ratio by the shares in options owned by the executive, and then adding the shares in stock owned by the executive. PPS1 in year t is equal to an executive’s totaldelta over fiscal year t divided by total number of shares outstanding (Compustat data item CSHO) of the company at thebeginningofyeart. Theconstructionoftotaldeltainvolveslotsofdetails(e.g.,howtoconstruct Black-Scholeshedgeratio,howtodealwithpreviouslygrantedoptions,whattoassumeforexpected 2

life on the options, etc.), and we follow Appendix B in Edmans, et al. (2009) in estimating the totaldelta variable. In the regressions, PPS1 is in percentages. PPS2: Dollar-to-percentage measure of pay-performance sensitivity. This variable measures the dollar change in stock and option holdings for a one percent change in firm value. PPS2 in year t is equal to PPS1 in year t × share price at the beginning of fiscal year t × total number of shares outstanding at the beginning of t / 100, where share price is Compustat data item PRCC F and total number of shares outstanding is Compustat data item CSHO. In the regressions, PPS2 is in $thousands. PPS3: The scaled wealth-performance sensitivity proposed in Edmans et al. (2009). It is available from Alex Edmans’ website. Specifically, this sensitivity measure equals the dollar change in executive wealth for a 100 percentage point change in firm value, divided by annual flow compensation (TDC1). This incentive measure is a variant of the percentage-to-percentage incentivesusedinMurphy(1985),GibbonsandMurphy(1992),andRosen(1992),andreplacesflow compensationinthenumeratorofthemeasureinMurphy(1985)withthechangeintheexecutives’ wealth. By considering the change in wealth, the scaled wealth-performance sensitivity captures the important incentives from changes in the value of previously granted stock and options. See Edmans et al. (2009) for details. Log(tenure): Natural log of the number of years the manager has been with the company, which equals the difference between the year of the observation and the year when the individual joined the company. CEO Indicator: A dummy variable that equals one if the manager is the CEO in a particular year and zero if the manager is a non-CEO top executive. This dummy variable is time variant for a given individual because a specific manager could be a CEO in some years and a non-CEO in other years. Female Indicator: A dummy variable that equals one if the manager is a female and zero otherwise. 3