uncertainty about government policy and stock prices

THE JOURNAL OF FINANCE • VOL. LXVII, NO. 4 • AUGUST 2012

Uncertainty about Government Policy and StockPrices

LUBOS PASTOR and PIETRO VERONESI∗

ABSTRACT

We analyze how changes in government policy affect stock prices. Our general equilib-rium model features uncertainty about government policy and a government whosedecisions have both economic and noneconomic motives. The model makes numerousempirical predictions. Stock prices should fall at the announcement of a policy change,on average. The price decline should be large if uncertainty about government policyis large, and also if the policy change is preceded by a short or shallow economic down-turn. Policy changes should increase volatilities and correlations among stocks. Thejump risk premium associated with policy decisions should be positive, on average.

GOVERNMENTS SHAPE THE ENVIRONMENT in which the private sector operates.They affect firms in many ways: for instance, by levying taxes, providing subsi-dies, enforcing laws, regulating competition, and defining environmental poli-cies. In short, governments set the rules of the game.

Governments change these rules from time to time, eliciting price reactionsin financial markets. These reactions are weak if the change is widely antic-ipated, but they can be strong if the markets are caught by surprise. Thispaper analyzes the effects of changes in government policy on stock prices in atheoretical setting. Naturally, policy changes are not exogenous; they are deter-mined by various economic and political forces. To account for this endogeneity,we develop a simple asset pricing model in which the government plays anactive decision-making role.

We interpret policy changes broadly as government actions that change theeconomic environment. Recent examples include the reforms in health careand financial regulation. Another example is the U.S. government’s decision to

∗Both authors are at the University of Chicago Booth School of Business, NBER, and CEPR.We are grateful for comments from Fernando Alvarez, Frederico Belo, Jaroslav Borovicka, PierreChaigneau, David Chapman, Josh Coval, Max Croce, Steve Davis, Allan Drazen, Francisco Gomes,John Graham (Co-editor), Cam Harvey (Editor), John Heaton, Monika Piazzesi, Lukas Schmid,Amit Seru, Rob Stambaugh, Milan Svolık, Francesco Trebbi, Rob Vishny, and an anonymousreferee, as well as from conference participants at the 2010 NBER AP, 2010 NBER EFG, 2011WFA, 2011 EFA, 2011 SED, 2011 SITE, 2011 Prague Economic Meeting, 2011 European SummerSymposium in Financial Markets (Gerzensee), and 2011 Chicago Booth CFO Forum and workshopparticipants at the Bank of England, Chicago, Einaudi Institute (Rome), Erasmus, ECB, Harvard,LBS, LSE, McGill, Northwestern, Notre Dame, Oxford, Vanderbilt, Warwick, and WU Vienna. Thisresearch was funded in part by the Initiative on Global Markets at the University of Chicago BoothSchool of Business.

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let Lehman Brothers go bankrupt in 2008, which was perceived by many assignaling a shift in the government’s implicit too-big-to-fail policy. Numerousother policy changes took place during the financial crisis of 2007 to 2008. Manyof those changes were unprecedented, with long-term effects that were difficultto predict in advance.

A key role in our analysis belongs to uncertainty about government policy,which is an inevitable by-product of policymaking. We consider two types ofuncertainty. The first type, which we call political uncertainty, relates to uncer-tainty about whether the current government policy will change. The secondtype, which we call impact uncertainty, corresponds to uncertainty about theimpact a new government policy will have on the profitability of the privatesector. In other words, there is uncertainty about what the government is goingto do, as well as uncertainty about what the effect of its action is going to be.We find that both types of uncertainty affect stock prices in important ways.

We develop a general equilibrium model in which firm profitability followsa stochastic process whose mean is affected by the prevailing government pol-icy. The policy’s impact on the mean is uncertain. Both the government andinvestors (firm owners) learn about this impact in a Bayesian fashion by ob-serving realized profitability. All agents have the same prior beliefs about thecurrent policy’s impact, and the same prior beliefs apply to any other poten-tial future government policy. The prior standard deviation is labeled impactuncertainty.

At a given point in time, the government decides whether to change its policy.If a policy change occurs, the agents’ beliefs are reset: the posterior beliefsabout the old policy’s impact are replaced by the prior beliefs about the newpolicy’s impact. When making its policy decision, the government is motivatedby both economic and noneconomic objectives: it maximizes investors’ welfare,as a social planner would, but it also takes into account the political cost (orbenefit) incurred by changing the policy. This cost is unknown to investors,who therefore cannot fully anticipate whether the policy change will occur. Thestandard deviation of the political cost is labeled political uncertainty.

We find that it is optimal for the government to replace its current policyif the policy’s impact on profitability is perceived as sufficiently unfavorable,that is, if the posterior mean of the impact is below a given threshold. Thisthreshold decreases with impact uncertainty as well as with the political cost.If the government derives an unexpectedly large political benefit from changingits policy, the policy will be replaced even if it worked well in the past. Inexpectation, however, the threshold is below the prior mean of the policy’simpact. To push the posterior mean below the prior mean, Bayes’s rule dictatesthat the signal—the realized profitability in the private sector—must be lowerthan expected. As a result, policy changes are expected to occur after periodsof unexpectedly low realized profitability, which we refer to as downturns.

We derive the conditions under which stock prices fall at the announcement ofa policy change. There are two effects. First, a policy change typically increasesfirms’ expected profitability, as a result of the government’s decision rule de-scribed in the previous paragraph. This cash flow effect generally pushes stock

Uncertainty about Government Policy and Stock Prices 1221

prices up. Second, a policy change increases discount rates because the newpolicy’s impact on profitability is more uncertain; adopting a new policy undoesthe gains from learning about the old policy. This discount rate effect pushesstock prices down. We find that the discount rate effect is stronger than thecash flow effect unless the old policy’s impact is perceived as sufficiently nega-tive. That is, stock prices fall at the announcement of a policy change unless theposterior mean of the old policy’s impact is below a negative cutoff. This cut-off is generally below the policy-change-triggering threshold discussed in theprevious paragraph. Stock prices rise only if the outgoing policy is perceived assufficiently harmful to profitability.

We find that, on average, stock prices fall at the announcement of a policychange. Positive announcement returns are typically small because they tendto occur in states of the world in which the policy change is largely anticipatedby investors. Given the government’s economic motive, any policy change thatlifts stock prices is mostly expected, so that much of its effect is priced in beforethe announcement. In contrast, negative announcement returns tend to belarger because they occur when the announcement of a policy change containsa bigger element of surprise. The probability distribution of the announcementreturns is left-skewed and, most interesting, its mean is below zero—we proveanalytically that the expected value of the stock return at the announcementof a policy change is negative. We also find that this expected announcementreturn is more negative when there is more uncertainty about governmentpolicy. When impact uncertainty is larger, so is the risk associated with a newpolicy, and thus also the discount rate effect that pushes stock prices downwhen the new policy is announced. When political uncertainty is larger, so isthe element of surprise in the announcement of a policy change.

We relate the stock return at the announcement of a policy change to thelength and depth of the preceding downturn. We find that announcement re-turns are negative, especially after downturns that are short or shallow. An-nouncement returns can also be positive, mostly after downturns that are longor deep. However, such positive returns tend to be small because, after long ordeep downturns, policy changes are largely anticipated.

Before the announcement of a policy decision, investors are uncertain aboutwhether the policy will change. If it does change, stock prices tend to jumpdown; if it does not change, they tend to jump up. The expected jump in stockprices at the announcement of a policy decision captures the risk premiumdemanded by investors for holding stocks while the decision is announced. Weshow that the conditional jump risk premium can be positive or negative, butthe unconditional premium is always positive and increasing in both impactuncertainty and political uncertainty.

The volatilities and correlations of stock returns are also affected by changesin government policy. By introducing new policies whose impact is more un-certain, policy changes raise the volatility of the stochastic discount factor. Asa result, risk premia go up and stock returns become more volatile and morehighly correlated across firms. Policy uncertainty also affects stock volatil-ity before a policy change through fluctuations in investors’ beliefs about the


current policy’s impact as well as in investors’ assessment of the probability ofa policy change.

In the benchmark version of our model, discussed above, the government canchange its policy only at a predetermined point in time. In the first extensionof the model, we endogenize the timing of the policy change by letting thegovernment choose the optimal time to change its policy. We find numericallythat the key results from our benchmark model continue to hold.

In the benchmark model, firms’ investment decisions are not modeled ex-plicitly. In the second extension, we allow firms to disinvest in response touncertainty about government policy. Firms’ investment decisions and the gov-ernment’s policy decision are made simultaneously: the government takes intoaccount firms’ anticipated response, and each firm considers the actions of theother firms as well as the government. In the resulting Nash equilibrium, afraction of firms remain invested while the rest hold cash. We show numericallythat firms respond to policy uncertainty by cutting investment. We also showthat our key results continue to hold.

In the third and final extension of our model, we introduce heterogeneity infirms’ exposures to government policy. We find that firms with more exposuretypically have higher expected returns, though the premium for governmentpolicy risk is state dependent and can be negative. The main results from ourbenchmark model continue to hold.

Prior studies analyze the effects of uncertainty, broadly defined, on variousaspects of economic activity. For example, it is well known that uncertaintygenerally reduces firm investment when this investment is at least partiallyirreversible.1 The impact of uncertainty about government policy on invest-ment has been analyzed both theoretically and empirically.2 The literaturealso analyzes the effects of policy uncertainty on inflation, capital flows, andwelfare.3 However, the literature does not have much to say about how this un-certainty affects asset prices, which is the subject of our study. A rare exceptionis Sialm (2006), who analyzes the effects of stochastic taxes on asset prices.4

1 See, for example, Bernanke (1983), McDonald and Siegel (1986), Pindyck (1988), and Dixit(1989). Bloom (2009) provides a structural analysis of various real effects of macroeconomic uncer-tainty shocks.

2 For example, Rodrik (1991) shows that even moderate policy uncertainty can impose a heftytax on investment. Hassett and Metcalf (1999) find that the impact of tax policy uncertainty oninvestment depends on the process followed by the tax policy. Julio and Yook (2012) and Yonce(2009) find that firms reduce their investment in years leading up to major elections. Durnev(2011) finds that corporate investment is less sensitive to stock prices during election years.

3 For example, Drazen and Helpman (1990) study how uncertainty about a future fiscal adjust-ment affects the dynamics of inflation. Hermes and Lensink (2001) show that uncertainty aboutbudget deficits, tax payments, government consumption, and inflation is positively related to capi-tal outflows at the country level. Gomes, Kotlikoff, and Viceira (2008) calibrate a life-cycle model tomeasure the welfare losses resulting from uncertainty about government policies regarding taxesand Social Security. They find that policy uncertainty materially affects agents’ consumption,saving, labor supply, and portfolio decisions.

4 Other studies, such as McGrattan and Prescott (2005), Sialm (2009), and Gomes, Michaelides,and Polkovnichenko (2009), relate stock prices to tax rates without emphasizing tax-related


Tax uncertainty also features in Croce et al. (2011), who explore its asset pric-ing implications in a production economy with recursive preferences. Finally,Ulrich (2011) analyzes how bond yields are affected by Knightian uncertaintyabout both Ricardian equivalence and the size of the government multiplier.All of these studies are quite different from ours. They analyze fiscal policy,whereas we consider a broader set of government actions. We focus on stockmarket reactions to government policy announcements, in contrast to any ofthese studies. These studies also use very different modeling techniques, andthey do not model the government’s policy decision explicitly as we do. Further,none of these studies feature Bayesian learning, which plays an important rolehere.

Our model is also different from the learning models recently proposed in thepolitical economy literature, such as Callander (2008) and Strulovici (2010). InCallander’s model, voters learn about the effects of government policies throughrepeated elections. In Strulovici’s model, voters learn about their preferencesthrough policy experimentation. Neither study analyzes the asset pricing im-plications of learning.

The paper is organized as follows. Section I presents the model. Section IIanalyzes the equilibrium stock prices. Section III extends the model by endog-enizing the time of the policy change. Section IV extends the model by allowingfirms to disinvest in response to policy uncertainty. Section V introduces het-erogeneity in firms’ exposures to government policy. Section VI summarizesthe empirical predictions of our model. Section VII concludes. The Appendixcontains some technical details, and the Internet Appendix provides all theproofs.5

I. The Model

We consider an economy with a finite horizon[0, T

]and a continuum of firms

i ∈ [0, 1]. Let Bi

t denote firm i’s capital at time t. Firms are financed entirelyby equity, so Bi

t can also be viewed as book value of equity. At time 0, all firmsemploy an equal amount of capital, which we normalize to Bi

0 = 1. Firm i’scapital is invested in a linear technology whose rate of return is stochasticand denoted by d�i

t. All profits are reinvested, so that firm i’s capital evolvesaccording to dBi

t = Bitd�i

t. Since d�it equals profits over book value, we refer to

it as the profitability of firm i. For all t ∈ [0, T ], profitability follows the process

d�it = (μ + gt) dt + σdZt + σ1dZi

t , (1)

where (μ, σ, σ1) are observable constants, Zt is a Brownian motion, and Zit is an

independent Brownian motion that is specific to firm i. The variable gt denotesthe impact of the prevailing government policy on the mean of the profitability

uncertainty. In empirical work, Erb, Harvey, and Viskanta (1996) relate political risk to futurecountry stock returns, and Boutchkova et al. (2012) relate political uncertainty to stock volatility.

5 The Internet Appendix is located on the Journal of Finance website at http://www.afajof.org/supplements.asp.


process of each firm.6 When gt = 0, the government policy is “neutral” in thatit has no impact on profitability.

The government policy’s impact, gt, is constant while the same policy is ineffect. At an exogenously given time τ , 0 < τ < T , the government makes anirreversible decision as to whether or not to change its policy.7 As a result, gt isa simple step function of time:

gt =

⎧⎪⎨⎪⎩gold for t ≤ τ

gold for t > τ if there is no policy changegnew for t > τ if there is a policy change,

(2)

where gold denotes the impact of the government policy prevailing at time 0.A policy change replaces gold by gnew, thereby inducing a permanent shift inaverage profitability. A policy decision becomes effective immediately after itsannouncement at time τ .

The values of gold and gnew are unknown. This key assumption captures theidea that government policies have an uncertain impact on firm profitability.The prior distributions of both gold and gnew at time 0 are normal with meanzero and known variance σ 2

g :

g ∼ N(0, σ 2

g

), (3)

for both g ≡ gold and g ≡ gnew. That is, both government policies, old and new,are expected to be neutral a priori.8 We refer to σg as impact uncertainty. Thevalue of gt is unknown for all t ∈ [0, T ] to all agents—the government as wellas the investors who own the firms.

The firms are owned by a continuum of identical investors who maximizeexpected utility derived from terminal wealth. For all j ∈ [0, 1

], investor j’s

utility function is given by

u(W j

T

) =(W j

T

)1−γ

1 − γ, (4)

where W jT is investor j’s wealth at time T and γ > 1 is the coefficient of relative

risk aversion. At time 0, all investors are equally endowed with shares of firmstock. Stocks pay liquidating dividends at time T .9 Investors observe whethera policy change takes place at time τ .

When making its policy decision at time τ , the government maximizes thesame objective function as investors, except that it also faces a nonpecuniary

6 The assumption that government policy affects all firms equally is relaxed in Section V.7 The assumption that τ is exogenous allows us to obtain analytical results. Section III shows

numerically that all of our key results survive when the government chooses the optimal time tochange its policy.

8 Any government effect on profitability that is stable across policy changes is included in μ.9 No dividends are paid before time T because investors’ preferences do not involve intermediate

consumption. Firms in our model reinvest all of their earnings, as mentioned earlier.


cost (or benefit) associated with a policy change. The government changes itspolicy if the expected utility under the new policy is higher than under the oldpolicy. Specifically, the government solves

max

{Eτ

[W1−γ

T

1 − γ

∣∣∣∣ no policy change

], Eτ

[CW1−γ

T

1 − γ

∣∣∣∣ policy change

]}, (5)

where WT = BT = ∫ 10 Bi

T di is the final value of aggregate capital and C is the“political cost” incurred by the government if a new policy is introduced. Valuesof C > 1 represent a cost (e.g., the government must exert effort or burn politicalcapital to implement a new policy), whereas C < 1 represents a benefit (e.g.,the government makes a transfer to a favored constituency, or it simply wantsto be seen doing something).10 The value of C is randomly drawn at time τ froma lognormal distribution centered at C = 1:

c ≡ log (C) ∼ N(

−12

σ 2c , σ 2

c

), (6)

where C is independent of the Brownian motions in equation (1). As soon as thevalue of C is revealed to the agents at time τ , the government uses it to makethe policy decision. We refer to σc as political uncertainty. Political uncertaintyintroduces an element of surprise into policy changes, resulting in stock pricereactions at time τ .

Given its objective function in equation (5), the government is “quasi-benevolent”: it is expected to maximize investors’ welfare (because E

[C] = 1),

but also to deviate from this objective in a random fashion. The assumptionthat governments do not behave as fully benevolent social planners is widelyaccepted in the political economy literature.11 This literature presents variousreasons why governments might not maximize aggregate welfare. For example,governments often redistribute wealth.12 Governments tend to be influencedby special interest groups.13 They might also be susceptible to corruption.14

Instead of modeling these political forces explicitly, we adopt a simple reduced-form approach to capturing departures from benevolence. In our model, allaspects of politics—redistribution, corruption, special interests, etc.—are bun-dled together in the political cost C. The randomness of C reflects the difficultyin predicting the outcome of a political process, which can be complex and non-transparent. For example, it can be hard to predict the outcome of a battlebetween special interest groups. By modeling politics in such a reduced-form

10 We refer to C as a cost because, given γ > 1, higher values of C translate into lower utility(since W1−γ

T / (1 − γ ) < 0).11 Drazen (2000) provides a useful overview of this literature.12 Redistribution is a major theme in political economy—see, for example, Alesina and Rodrik

(1994) and Persson and Tabellini (1994). Our model is not well suited for analyzing redistributioneffects because all of our investors are identical ex ante, for simplicity.

13 See, for example, Grossman and Helpman (1994) and Coate and Morris (1995).14 See, for example, Shleifer and Vishny (1993) and Rose-Ackerman (1999).


fashion, we can focus on the asset pricing implications of government policychanges.

Our modeling of government policies also merits some discussion. We imposethe same prior distribution in equation (3) on all policies. In reality, the priorsmight differ across policies. For example, if a potential new policy has alreadybeen tried in the past, the agents may believe they have more prior informationabout that policy, so they may assign a lower prior variance to it. The lowervariance might not be warranted though, because the same policy implementedat a different time in a different environment may have a different impact onprofitability. Our assumption that all policies are viewed as identical ex anteseems like a natural starting point in an asset-pricing analysis. The simplicityof our setup allows us to derive interesting analytical results.

A. Learning

The value of gt is unknown to all agents, that is, to investors and the gov-ernment alike. At time 0, all agents share the same prior beliefs about gt,summarized by the distribution in equation (3). All agents learn about gt inthe same Bayesian fashion by observing the realized profitabilities of all firms.The learning process is described in the following proposition.

PROPOSITION 1: Observing the continuum of signals d�it in equation (1) across

all firms i ∈ [0, 1]

is equivalent to observing a single aggregate signal about gt:

dst = (μ + gt) dt + σdZt. (7)

Under the prior in equation (3), the posterior for gt at any time t ∈ [0, T ] is givenby

gt ∼ N(gt, σ

2t

). (8)

For all t ≤ τ , the mean and variance of this posterior distribution evolve as

dgt = σ 2t σ−1dZt (9)

σ 2t = 1

1σ 2

g+ 1

σ 2 t,

(10)

where the “expectation error” dZt is given by dZt = (dst − Et (dst)) /σ for all t ∈[0, T ]. If there is no policy change at time τ , then the processes (9) and (10) alsohold for t > τ . If there is a policy change at τ , then gt jumps from gτ to zero rightafter the policy change, and for t > τ , gt follows the process in equation (9). Inaddition, for t > τ , σt follows

σ 2t = 1

1σ 2

g+ 1

σ 2 (t − τ ). (11)


Comparing equations (1) and (7), idiosyncratic shocks dZit wash out upon ag-

gregating infinitely many independent signals. The aggregate signal in equa-tion (7) is the average profitability across firms. When this signal is higherthan expected, the agents revise their beliefs about gt upward, and vice versa(see equation (9)). Uncertainty about gt declines deterministically over timedue to learning, except for a discrete jump up at time τ in the case of a pol-icy change (see equations (10) and (11)). A policy change resets agents’ be-liefs from the posterior N(gτ , σ

2τ ) to the prior N(0, σ 2

g ), where στ < σg due tolearning between times 0 and τ . Before time τ , agents learn about gold; af-ter τ , they learn about gold or gnew, depending on whether a policy changeoccurs.

B. Optimal Changes in Government Policy

After a period of learning about gold, the government decides whether tochange its policy at time τ . If a change occurs, the value of gt changes fromgold to gnew and the perceived distribution of gt changes from the posteriorin equation (8) to the prior in equation (3). According to equation (5), thegovernment changes its policy if and only if

Eτ

[CB1−γ

T

1 − γ

∣∣∣∣ policy change

]> Eτ

[B1−γ

T

1 − γ

∣∣∣∣ no policy change

]. (12)

Since government policy affects future profitability, the two expectations inequation (12) are computed under different stochastic processes for the aggre-gate capital Bt = ∫ 1

0 Bitdi. We show in the Internet Appendix that the aggregate

capital at time T is given by

BT = Bτ e(μ+g− σ22 )(T −τ )+σ (ZT −Zτ ), (13)

where g ≡ gold if there is no policy change and g ≡ gnew if there is one. Evaluat-ing the expectations in equation (12) based on equation (13) yields the followingproposition.

PROPOSITION 2: The government changes its policy at time τ if and only if

gτ < g(c), (14)

where

g(c) = −(σ 2

g − σ 2τ

)(γ − 1) (T − τ )

2− c

(T − τ ) (γ − 1). (15)

The government follows a simple cutoff rule: it changes its policy if gτ , theposterior mean of gold, is below a given threshold. That is, a policy is replaced ifits effect on firm profitability is perceived as sufficiently unfavorable. The twoterms on the right-hand side of equation (15) reflect the government’s economicand political motives. The first term reflects the increase in risk associated with


Table IParameter Choices

This table reports the parameter values used in the simulations. The parameters are: impactuncertainty σg (equation (3)), political uncertainty σc (equation (6)), μ, σ , and σ1 from equation (1),final date T , time τ of the policy decision, and risk aversion γ . All variables except for γ arereported on an annual basis.

σg σc μ σ σ1 T τ γ

0.02 0.10 0.10 0.05 0.10 20 10 5

adopting a new policy (σg > στ ). Since γ > 1, higher impact uncertainty reducesg(c), making a policy change less likely. The second term reflects the politicalcost or benefit incurred by the government if the new policy is adopted. If c > 0,the government incurs a cost, the second component is negative, and the newpolicy is less likely to be adopted. If c < 0, the government benefits from apolicy change and the new policy is more likely to be adopted. Since investorsdo not know c, they cannot fully anticipate whether a policy change will occur.The only exception occurs if σc = 0; in that case, investors know g(c), so theyknow the outcome of the government’s policy decision immediately before itsannouncement at time τ .

Investors expect c to be close to zero, so their expectation of g(c) is closeto g(0).15 Since σg > στ , we have g(0) < 0. That is, in expectation, a policy isreplaced if its impact is perceived as sufficiently negative. It is not enough forgτ to be negative; it must be sufficiently negative for the expected gain froma policy change to outweigh the higher risk associated with a new policy.16

For the posterior mean of gold to be negative at time τ while the prior meanis zero, Bayes’s rule implies that the signal received by investors before timeτ—realized profitability—must be unexpectedly low. Therefore, Proposition 2implies that policy changes are expected to occur after periods of unexpectedlylow realized profitability. We refer to such periods as “downturns.”

To illustrate this implication, we plot the expected dynamics of profitabilityconditional on a policy change. We choose a set of plausible parameter values,which are summarized in Table I. We simulate many samples of shocks in oureconomy and record the paths of gt and realized profitability in each sample.Realized profitability is the average profitability across all firms, reported inexcess of μ so that its unconditional mean is zero. We split the samples into twogroups, depending on whether the government changes its policy at time τ .17

We then plot the average paths of gt and realized profitability across all sampleswithin both groups. Panel A of Figure 1 plots gt (solid line) and profitability

15 Recall from equation (6) that E(C) = 1 implies E(c) = −σ 2c /2 for c = log (C).

16 The political economy literature recognizes that uncertainty associated with a policy changemay lead to a bias toward the status quo, but the literature focuses mostly on the uncertaintyabout how the gains and losses from reform are distributed across individuals (e.g., Fernandez andRodrik (1991)). Instead, we focus on the uncertainty about the aggregate effects of a policy change.

17 A policy change occurs in about 38% of all simulated samples.


0 2 4 6 8 10 12 14 16 18 20−3

−2

−1

0

1

2

3

Time

Pro

fitab

ility

(%)

Panel A. Policy change

Realized profitabilityExpected profitabilityThreshold

0 2 4 6 8 10 12 14 16 18 20−3

−2

−1

0

1

2

3

Time

Pro

fitab

ility

(%)

Panel B. No policy change

Realized profitabilityExpected profitabilityThreshold

Figure 1. Profitability dynamics around the policy decision. This figure plots the expecteddynamics of gt (solid line) and realized profitability (dashed line) under the parameter values fromTable I. Realized profitability is the average profitability across all firms, plotted in excess of μ

so that its unconditional mean is zero. The figure also plots the threshold g(c) (dotted line). Allthree lines are average paths across many simulated samples. The top panel averages across thesamples in which a policy change occurred at time τ = 10, whereas the bottom panel conditions onno policy change.

(dashed line) conditional on a policy change at time τ . Panel B plots the samequantities conditional on no policy change. Both panels also plot the averagevalue of the cutoff g(c) (dotted line).

Panel A of Figure 1 shows that policy changes tend to be preceded by periodsof low realized profitability. Between times zero and τ = 10 years, profitabilityaverages −2.5% per year. The profitability is negative due to ex post condition-ing on gτ < g(c). The average value of g(c) is −0.5% per year, and the averagevalue of gt gradually falls from zero to −1.5% per year. For the posterior meangt to decline in this manner, realized profitability must be below gt, as discussedearlier. When the policy changes at time τ , both gt and profitability jump up tozero, on average, and stay there until time T = 20 since there is no more expost conditioning after time τ .


Panel B of Figure 1 shows the opposite but milder patterns when there isno policy change. Due to ex post conditioning on gτ > g(c), the average value ofgt rises from zero to 1% per year at time τ , and average realized profitabilityis 1.5% until time τ . After time τ , the average gt remains unchanged at 1%because the policy remains the same. Profitability therefore drops from 1.5%to 1% at time τ , and it stays there until time T .

Having established the government’s optimal policy rule and its key impli-cations for profitability, we are ready to analyze the asset pricing implications.The following section presents our main results on stock price dynamics aroundgovernment policy changes.

II. Stock Prices

Firm i’s stock represents a claim on the firm’s liquidating dividend at time T ,which is equal to Bi

T . Investors’ total wealth at time T is equal to BT = ∫ 10 Bi

T di.Stock prices adjust to make investors hold all of the firms’ stock. In addition tostocks, there is also a zero-coupon bond in zero net supply, which makes a unitpayoff at time T with certainty. We use this risk-free bond as the numeraire.18

Under the assumption of market completeness, standard arguments imply thatthe state price density is uniquely given by

πt = 1λ

Et[B−γ

T

], (16)

where λ is the Lagrange multiplier from the utility maximization problem ofthe representative investor. The market value of stock i is given by the standardpricing formula

Mit = Et

[πT

πtBi

T

]. (17)

A. Stock Price Reaction to the Announcement of a Policy Change

When the government announces its policy decision at time τ , stock pricesjump. The direction and size of the jump depend on whether the governmentdecides to change or maintain its policy, as well as on the extent to whichthis decision is unexpected. We derive a closed-form solution for each firm’s“announcement return,” defined as the instantaneous stock return at time τ

conditional on the announcement of a change in government policy.19

18 This assumption is equivalent to assuming a risk-free rate of zero. Such an assumption isinnocuous because, without intermediate consumption, there is no intertemporal consumptionchoice that would pin down the interest rate. This modeling choice ensures that interest ratefluctuations do not drive our results.

19 The stock return conditional on the announcement of no change in policy is analyzed in SectionII.B.


PROPOSITION 3: Each firm’s stock return at the announcement of a policy changeis given by

R (gτ ) = (1 − p (gτ )) F (gτ ) (1 − G (gτ ))p (gτ ) + (1 − p (gτ )) F (gτ ) G (gτ )

, (18)

where

F (gτ ) = e−γ gτ (T −τ )− 12 γ 2(T −τ )2(σ 2

g −σ 2τ ) (19)

G(gτ ) = egτ (T −τ )− 12 (1−2γ )(T −τ )2(σ 2

g −σ 2τ ) (20)

p(gτ ) = N

(gτ (1 − γ ) (T − τ ) − (1 − γ )2

2(T − τ )2 (

σ 2g − σ 2

τ

); −σ 2

c

2, σ 2

c

), (21)

and N(x; a, b) denotes the c.d.f. of a normal distribution with mean a andvariance b.

The function in equation (21) is the probability of a policy change perceived byinvestors just before time τ . According to Proposition 2, p(gτ ) = Prob(gτ < g(c)).Since investors observe gτ but not c, p(gτ ) can be expressed as the probabilitythat c is below a given threshold, implying equation (21). The rest of Proposition3 follows from the comparison of stock prices right before and right after thepolicy decision at time τ . Right after time τ , at time τ+, the market value ofeach firm i takes one of two values:

Miτ+ =

⎧⎨⎩Mi,yesτ+ = Bi

τ+ e(μ−γ σ 2)(T −τ )+ 1−2γ

2 (T −τ )2σ 2g if policy changes

Mi,noτ+ = Bi

τ+ e(μ−γ σ 2+gτ )(T −τ )+ 1−2γ

2 (T −τ )2σ 2τ if policy does not change.

(22)

The values of the market-to-book (M/B) ratios, Mit /Bi

t, are equal across firmsfor all t because all firms are identical ex ante. We show that right before timeτ , the market value of firm i is given by

Miτ = ωMi,yes

τ+ + (1 − ω) Mi,noτ+ , (23)

where the weight ω, which is always between zero and one, is given by

ω = pτ

pτ + (1 − pτ ) F (gτ ), (24)

using the abbreviated notation pτ ≡ p (gτ ). As one would expect, ω increaseswith pτ , with pτ → 0 implying ω → 0 and pτ → 1 implying ω → 1. The an-nouncement return R (gτ ) in equation (18) is given by the ratio of the quantitiesin equations (22) and (23):

R(gτ ) = Mi,yesτ+Mi

τ

− 1. (25)


To gain some insight into the announcement return R(gτ ), recall that a policychange replaces a policy whose impact is perceived to be distributed as N(gτ , σ

2τ )

by a policy whose impact is perceived as N(0, σ 2g ). Relative to the old policy,

the new policy typically increases firms’ expected future cash flows (because gτ

must be below a threshold that is typically negative).20 However, the new policyalso increases discount rates due to its higher uncertainty (because σg > στ dueto learning). The cash flow effect pushes stock prices up, whereas the discountrate effect pushes them down. Either effect can win, depending on gτ andthe parameter values. The following corollaries describe the behavior of R(gτ ),when risk aversion γ > 1 takes extreme values.

COROLLARY 1: As risk aversion γ → ∞, the announcement return R (gτ ) → −1for any gτ .

COROLLARY 2: As risk aversion γ → 1, the expected value of the announcementreturn goes to zero (E {R (gτ )} → 0), where the expectation is computed withrespect to gτ as of time 0.

Both corollaries follow quickly from Proposition 3. When γ → ∞, the discountrate effect discussed above dwarfs the cash flow effect and stocks lose all oftheir value when the more uncertain policy is installed. In this limiting case,the probability of a policy change goes to zero. By continuity, Corollary 1 impliesthat if risk aversion is large, policy changes are unlikely but if they do occur,stock prices fall dramatically. When γ → 1, the discount rate effect and thecash flow effect cancel out, on average.

A.1. When Is the Announcement Return Negative?

In this subsection, we derive the conditions under which the announcementreturn R (gτ ) < 0.

PROPOSITION 4: The market value of each firm drops at the announcement of apolicy change (i.e., R (gτ ) < 0) if and only if

gτ > g∗, (26)

where

g∗ = −(σ 2g − σ 2

τ

)(T − τ )

(γ − 1

2

). (27)

Proposition 4 shows that stock prices drop at the announcement of a newpolicy unless the old policy is perceived as having a sufficiently negative impacton profitability. The relative importance of the cash flow and discount rateeffects depends on gτ . Under (26), the discount rate effect is stronger and the

20 To clarify the word “typically,” exceptions occur if the government derives an unexpectedlylarge political benefit from changing its policy. If c is sufficiently negative, the cutoff g(c) in equa-tion (15) can be positive, and the old policy can be replaced even if its impact on profitability isperceived to be positive.


announcement return is negative. Formally, (26) implies G (gτ ) > 1, resultingin R (gτ ) < 0 (see equations (18) and (20)).

Combining the results in Propositions 2 and 4, stock prices drop at the an-nouncement of a policy change if and only if

g∗ < gτ < g(c). (28)

That is, gτ must be sufficiently low for the policy change to occur (Proposition2), but it must be sufficiently high for the discount rate effect to overcome thecash flow effect (Proposition 4). Comparing the definitions of g(c) and g∗ inequations (15) and (27),

g(c) − g∗ = c(T − τ ) (1 − γ )

+ (σ 2

g − σ 2τ

)(T − τ )

γ

2. (29)

The first term on the right-hand side of equation (29) is expected to be close tozero. The second term is always positive, so g(c) − g∗ is generally positive, andits magnitude (and thus also the size of the interval in which condition (28)holds) increases with σg.

Since g∗ in equation (27) can also be expressed as g∗ = g(0) − (σ 2g − σ 2

τ )(T −τ )γ /2, we have g∗ < g(0) < 0. Figure 2 illustrates four possible locations of gτ

relative to g∗, g(0), and zero. Also plotted is the probability distribution of g(c),as perceived by investors (who do not observe c) just before time τ . This normaldistribution is centered at a value just above g(0) (see equations (6) and (15)).This distribution, along with the values of g∗ and g(0), are computed basedon the parameters in Table I. The shaded area represents the probability of apolicy change, as perceived by investors just before time τ .

In Panel A of Figure 2, gτ is very low, and the probability of a policy changeis nearly one. Since gτ < g∗, stock prices rise at the announcement of a policychange. Given the high probability of such a change, the price increase will besmall because most of it is already priced in. In contrast, stock prices plunge inthe unlikely event of no policy change, which occurs if such a change imposesa very large political cost on the government.

In Panels B through D, gτ > g∗, so stock prices fall if the policy is changed.In Panel B, g∗ < gτ < g(0), and the policy change reduces stock prices eventhough it increases investors’ expected utility. Stock prices are lower due tohigher discount rates, but the expected utility is higher due to higher ex-pected wealth. Expected utility and stock prices need not move in the samedirection because stock prices are related to marginal utility rather thanthe level of utility. In Panel D, gτ > 0, indicating that the prevailing policyboosts profitability. A policy change is then unlikely, but should it occur, thestock price reaction will be strongly negative. If the government derives anunexpectedly large political benefit from changing its policy, it replaces evena policy that appears to work well, and stock prices exhibit a large drop as aresult.

Figure 2 captures uncertainty about c while conditioning on gτ . Such a per-spective is relevant at time τ when gτ is known, but less so before time τ while


Figure 2. Probability of a policy change. The shaded area represents the probability of apolicy change, as perceived by investors just before time τ . The bell curve represents the normaldistribution of the random threshold g(c). The four panels illustrate four possible locations of theposterior mean gτ relative to g∗, g(0), and zero. The vertical dotted lines are drawn at g∗, g(0),and zero. The normal distribution as well as the values of g∗ and g(0) are computed based on theparameter values in Table I.

gτ is uncertain. As of time 0, the perceived distribution of gτ is N(0, σ 2g − σ 2

τ ).Therefore, the values of gτ considered in Panel A of Figure 2, for which stockprices rise at the announcement of a policy change, are less likely than thosein Panels B through D, for which stock prices fall. The distribution of R (gτ )that is relevant from the perspective of an econometrician (or an investor attime 0) must incorporate uncertainty about both c and gτ . We integrate out allof that uncertainty in computing the expected value of R (gτ ) in the followingsubsection.


A.2. The Expected Announcement Return

We define the expected announcement return (EAR) as the expected value ofR (gτ ) after conditioning on gτ < g(c) and integrating out all uncertainty aboutc and gτ as of time 0:

EAR ≡ E{R(gτ )|Policy Change}. (30)

EAR is the return that we expect to see at the announcement of a policychange—the expected value of the instantaneous return at time τ conditionalon a policy change at the same time τ . Given the conditioning on a contem-poraneous event, EAR does not represent the more traditional expectationof a future return based on information available today. (The latter expectedreturns are analyzed in Section II.B.) Instead, EAR is the expected return rel-evant from the perspective of an empiricist conducting an event study of howstock prices respond to the announcements of policy changes. The followingproposition presents our main result.

PROPOSITION 5: The expected announcement return is negative (EAR < 0).According to Proposition 5, stock prices are expected to fall at the an-

nouncement of a policy change. This result relies on the government’s quasi-benevolence. Since the government is expected to maximize investors’ welfare,the government’s value-enhancing policy decisions are mostly expected by mar-ket participants. Positive announcement returns tend to occur when the policychange is widely anticipated (gτ < g∗, see Panel A of Figure 2). The positiveeffect of the policy change is thus largely priced in before the announcement,resulting in a weak price reaction to the announcement. In contrast, nega-tive announcement returns occur when the policy change comes as a biggersurprise (Panels B through D), so the price reaction is stronger. In short, thepositive returns tend to be small and the negative returns tend to be large. Inaddition, the negative announcement returns are fairly likely since even someutility-increasing policy changes reduce market values, as explained earlierwhile discussing Panel B of Figure 2.

To understand Proposition 5, it also helps to realize that, upon observing apolicy change, investors revise their beliefs about the political cost C. Beforetime τ , they expect C to be one (see equation (6)), but conditional on a policychange at time τ , the expectation drops below one: E(C|policy change) < E(C) =1. This fact follows from condition (14) for the optimality of a policy change,which can be rewritten as c < ξ (gτ ), where ξ (gτ ) is a decreasing function of gτ .(The expected value of c conditional on c < ξ (gτ ) is less than the unconditionalexpected value of c.) As a result, investors expect the government to derive apolitical benefit from any policy change. Even though C does not affect stockprices directly, it adds noise to the policy decision. Observing a policy change,investors infer that the increase in risk is certain but the increase in cashflow is not, and stock prices typically fall. They fall more when gτ is higherbecause the policy change is then more likely to be politically motivated. Afterintegrating gτ out, we prove that EAR< 0.


0 2 4 6 8 10 12 14 16 18 20−2

−1.8

−1.6

−1.4

−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

σc (%)

Ret

urn

(%)

σg = 1%

σg = 2%

σg = 3%

Figure 3. Expected announcement return. This figure plots the expectation of the announce-ment return R(gτ ), which is the instantaneous stock return at the announcement of a policy changeat time τ . The expectation integrates out uncertainty about the posterior mean gτ as perceived attime 0, as well as uncertainty about the political cost c conditional on the policy being changed.We vary impact uncertainty (σg) and political uncertainty (σc). All other parameters have valuesassigned in Table I.

The magnitude of EAR depends on uncertainty about government policy.We relate EAR to σg and σc in Figure 3 . We compute EAR by averaging theannouncement returns R (gτ ) across all simulated paths for which a policychange occurs at time τ . The parameter values are from Table I, as before,except that we vary σg and σc.

Figure 3 shows that EAR is more negative for larger values of σg and σc.Fixing σc = 20%, EAR is only −0.3% for σg = 1%, but it is −2% for σg = 3%.When σg is larger, so is the risk associated with a new policy, and so is thediscount rate effect that pushes stock prices down. Fixing σg = 2%, EAR is−0.5% for σc = 10% and −1.1% for σc = 20%. When σc is larger, so is the ele-ment of surprise in the announcement of a new policy. Also note that, wheneither uncertainty is zero, so is the announcement return. When σc = 0, thepolicy change is fully anticipated, and all of its effect is priced in before the an-nouncement. When σg = 0, one neutral policy is replaced by another, makingno difference to investors.


A.3. The Determinants of the Announcement Return

In this subsection, we analyze two key determinants of the announcementreturn R (gτ ) from equation (18): the length and depth of the downturns thatprecede policy changes. Recall that we define a downturn as a period of unex-pectedly low realized profitability. We vary the downturns’ length and depth ina simple way. We measure the length of a downturn as

LENGTH = τ − t0, (31)

where we set gt0 = 0. At time t0, the posterior mean of the policy impact is thenequal to the prior mean, but after t0, gt generally falls, conditional on a policychange at time τ (Proposition 2). We refer to t0 as the beginning of a downturn.

We measure the depth of a downturn by the number of standard deviationsby which gt drops during the downturn. Conditional on gt0 = 0, the distribution

of gτ is normal with mean zero and standard deviation Std(gτ ) =√

σ 2t0 − σ 2

τ . Wedefine

DEPTH = gτ

Std(gτ ). (32)

Panel A of Figure 4 plots the announcement return as a function of LENGTHand DEPTH. Panel B plots the corresponding probability of a policy change,as perceived by investors just before time τ (see Figure 2). Whenever thisprobability is close to one, the announcement return is close to zero becausethe announcement is already priced in. When the probability is smaller thanone, a policy change contains an element of surprise, and the announcementreturn is nonzero. The parameter values are from Table I, as before.

Figure 4 shows that DEPTH has a large effect on the announcement return.When the downturn is shallow (i.e., DEPTH is a small negative value), the an-nouncement return is negative. This result makes sense given Proposition 4,since DEPTH is proportional to gτ . The magnitude of the announcement re-turn can be as large as −10% if the downturn is sufficiently shallow. Evenlarger returns can happen if the government derives a huge political benefitfrom changing its policy. In contrast, if the downturn is sufficiently deep, theannouncement return is essentially zero; it is positive (Proposition 4) but smallbecause the policy change is fully anticipated (Panel B).

The announcement return also depends on LENGTH. Figure 4 shows thatshorter downturns are followed by more negative announcement returns, hold-ing DEPTH at a constant negative value. Intuitively, shorter downturns areless likely to induce a policy change, so if such a change occurs, it is more likelyto be politically motivated, resulting in a more negative announcement return.

A.4. The Distribution of Stock Returns on the Announcement Day

In this subsection, we examine the probability distribution of stock returnson the day of the announcement of a policy change, without conditioning on


−2.5 −2 −1.5 −1 −0.5 0 0.5−20

−15

−10

−5

0

Depth

Ret

urn

(%)

Panel A. Announcement return

−2.5 −2 −1.5 −1 −0.5 0 0.50

0.2

0.4

0.6

0.8

1

Depth

Pro

babi

lity

Panel B. Probability of a policy change

Length = 10Length = 5Length = 1

Length = 10Length = 5Length = 1

Figure 4. Announcement return and downturn length and depth. Panel A plots the an-nouncement return R(gτ ), the instantaneous stock return at the announcement of a policy changeat time τ , as a function of the length and depth of the downturn that precedes the policy change.Downturn length is computed as τ − t0 > 0, where gt0 = 0. Downturn depth is given by the numberof standard deviations by which gt drops during the downturn. Panel B plots the correspondingprobability of a policy change, as perceived by investors just before time τ . The parameters are inTable I.

DEPTH. This distribution is relevant from the empirical perspective if theevent study of the announcement effects is conducted on daily returns, whichis commonly done.21 The announcement day returns have a jump compo-nent R(gτ ) pertaining to the instant of the announcement, as well as a dif-fusion component Mτ+day/Mτ − 1 covering the rest of the announcement day.Figure 5 plots the distribution of the announcement day returns across all sim-ulated paths for which a policy change occurs at time τ . The randomness in thesimulations comes from the randomness in gτ and c. The downturn length is

21 For example, Savor and Wilson (2010) use daily returns to analyze the announcement effectsof macroeconomic news announcements regarding employment, inflation, and interest rates. Ifthe event study is conducted on tick-by-tick returns instead, the distribution of the instantaneousreturn R (gτ ) is more relevant. That distribution is similar to the distribution of the announcementday returns plotted here, except that it looks even more left-skewed and has less mass above zero.


−20 −15 −10 −5 00

10

20

30

40

Return (%)

Freq

uenc

y di

strib

ution

Panel A. Political uncertainty σc = 10%

σg = 1 %

σg = 2 %

σg = 3 %

−20 −15 −10 −5 00

10

20

30

40

Return (%)

Freq

uenc

y di

strib

ution

Panel B. Political uncertainty σc = 20%

σg = 1 %

σg = 2 %

σg = 3 %

Figure 5. Probability distribution of stock returns on the day of a policy changeannouncement. This figure plots the probability distribution of stock returns on the day whena policy change is announced. These per-day stock returns have two components: the jump com-ponent R(gτ ) at the instant of the announcement, and the diffusion component Mτ /Mτ−τ − 1covering the rest of the announcement day (τ = 1/252 years). This distribution is comparablewith an empirical distribution of daily announcement returns. The downturn length is τ − t0 = 5years. We vary impact uncertainty (σg) and political uncertainty (σc). All other parameters are inTable I.

5 years. The parameter values are from Table I, as before, except that we varyσg and σc.

Figure 5 shows that the distribution of the announcement day returns isstrongly left-skewed. The mode is close to zero, returns larger than 2% areextremely rare, but returns below −5% are more common, especially when σgand σc are large. This skewness is due to the asymmetry discussed earlier.Positive announcement returns tend to occur when the policy change is widelyanticipated, whereas negative returns occur when the policy change comes as abigger surprise. We also see that, as we increase σg or σc, the probability massin Figure 5 shifts to the left, consistent with our earlier result that EAR is morenegative when σg or σc is larger.


B. Stock Price Dynamics

The dynamics of stock prices are closely related to the dynamics of thestochastic discount factor from equation (16), which are described by the fol-lowing proposition.

PROPOSITION 6: The stochastic discount factor (SDF) follows the process

dπt

πt= −σπ,tdZt + Jπ1{t=τ }, (33)

where dZt is the Brownian motion from Proposition 1, 1{t=τ } is an indicatorfunction equal to one for t = τ and zero otherwise, and the jump component Jπ

is given by

Jπ =

⎧⎪⎪⎪⎨⎪⎪⎪⎩Jyes

π = (1 − pτ )(1 − F (gτ ))pτ + (1 − pτ )F (gτ )

if policy changes

Jnoπ = pτ (F (gτ ) − 1)

pτ + (1 − pτ )F (gτ )if policy does not change.

(34)

For t > τ , σπ,t is given by

σπ,t = γ[σ + (T − t)σ 2

t σ−1], (35)

and for t ≤ τ , it is given in equation (A2) in the Appendix.

Proposition 6 shows that the SDF jumps when the policy decision is an-nounced. The jump magnitudes Jyes

π and Jnoπ always have opposite signs. The

expected value of the jump, as perceived just before time τ , is zero: Eτ (Jπ ) =pτ Jyes

π + (1 − pτ ) Jnoπ = 0. It is also straightforward to show that Jyes

π < 0 (andJno

π > 0) if and only if gτ < g∗∗, where

g∗∗ = −γ

2(T − τ )

(σ 2

g − σ 2τ

). (36)

PROPOSITION 7: The return process for stock i is given by

dMit

Mit

= μM,tdt + σM,tdZt + σ1dZit + JM1{t=τ }, (37)

where the jump component JM is given by

JM ={

JyesM = R(gτ ) if policy changes

JnoM = R (gτ ) G (gτ ) + G (gτ ) − 1 if policy does not change.

(38)

For t > τ , we have

μM,t = γ[σ + (T − t)σ 2

t σ−1]2 (39)

σM,t = σ + (T − t)σ 2t σ−1, (40)


and for t ≤ τ , σM,t and μM,t are given in equations (A4) and (A5) in theAppendix.

Proposition 7 shows that stock returns have two components: a diffusioncomponent and a jump component. We discuss these components in separatesubsections.

C. Stock Price Jump at the Announcement of a Policy Decision

Stock prices jump at time τ when the government announces its policy de-cision. If the decision is to change the prevailing policy, the jump Jyes

M is equalto R (gτ ), which is given in equation (18) and analyzed extensively in SectionII.A. If the decision is not to change the policy, the jump Jno

M in equation (38)also involves the function G (gτ ) from equation (20).

COROLLARY 3: The market value of each firm increases at the announcementof no policy change (i.e., Jno

M > 0) if and only if gτ > g∗, where g∗ is given inequation (27).

Comparing Corollary 3 with Proposition 4, we see that the jumps JyesM and

JnoM always have opposite signs: whenever one is positive, the other is negative.

When gτ > g∗, we have JyesM < 0 and Jno

M > 0. When gτ < g∗, we have JyesM > 0

and JnoM < 0.

In the remainder of this subsection, we focus on the expected value of JM,or the expected jump in stock prices at the announcement of a policy decision.This expectation captures the risk premium that investors demand for holdingstocks at time τ . We consider both conditional and unconditional expectations.The conditional expectation of JM, denoted by Eτ (JM), is perceived by agentsjust before time τ :

Eτ (JM) = pτ JyesM + (1 − pτ ) Jno

M . (41)

This conditional expectation conditions on gτ , which is observable just beforethe policy decision. We also compute the unconditional expectation E(JM) =E(Eτ (JM)) by integrating out uncertainty about gτ as of time 0. (Recall thatgτ ∼ N(0, σ 2

g − σ 2τ ).) This latter expectation matters to an econometrician who

averages jumps in stock prices across all announcements of policy decisions,without controlling for gτ . Neither expectation conditions on whether the deci-sion changes the policy or not, so that both Eτ (JM) and E(JM) can be viewed asexpected returns (or risk premia) in the traditional forward-looking sense.

C.1. The Conditional Jump Risk Premium

PROPOSITION 8: The conditional expected jump in stock prices at time τ , asperceived by investors just before time τ , is given by

Eτ (JM) = − pτ (1 − pτ )(1 − F (gτ ))(1 − G(gτ ))pτ + (1 − pτ )F (gτ )G(gτ )

. (42)


COROLLARY 4: We have Eτ (JM) < 0 if and only if

g∗ < gτ < g∗∗, (43)

where g∗ is given in equation (27) and g∗∗ is given in equation (36).

Corollary 4 shows that Eτ (JM) can be positive or negative, depending on gτ .22

This expected jump, which captures the jump risk premium, is related to thecovariance between the jumps in prices and SDF, as perceived by the agentsjust before the policy decision:

Eτ (JM) = −Covτ (Jπ , JM) , (44)

where Jπ is given in equation (34). The sign of this covariance depends onwhether (43) is satisfied. If the covariance is positive, upward jumps in marginalutility at time τ tend to be accompanied by upward jumps in stock prices, whichmakes stocks an effective hedge against jumps in SDF. As a result, investors arewilling to accept a negative jump risk premium. In contrast, if the covarianceis negative, stocks are a poor hedge and the jump risk premium is positive.

COROLLARY 5: As risk aversion γ → ∞, Eτ (JM) → 0 from above for any valueof gτ .

COROLLARY 6: As risk aversion γ → 1, Eτ (JM) converges to a nonnegative valuefor any gτ . It converges to zero if and only if gτ = − 1

2 (T − τ )(σ 2g − σ 2

τ ).As γ → ∞, the probability of a policy change goes to zero. Given the di-

minishing element of surprise in the policy decision, the jump risk premiumdiminishes as well. In contrast, the jump risk premium remains positive almostsurely as γ → 1.

C.2. The Unconditional Jump Risk Premium

The expectation E (JM), which does not condition on gτ , represents the un-conditional jump risk premium. As noted earlier, E (JM) is the expected valueof the quantity in Proposition 8, where the expectation is taken with respect togτ as of time 0.

COROLLARY 7: As risk aversion γ → ∞, E (JM) → 0 from above.

COROLLARY 8: As risk aversion γ → 1, E (JM) converges to a positive value.According to Corollaries 7 and 8, there exist values γ and γ exceeding one

such that E (JM) > 0 for all γ ∈ (1, γ ) as well as all γ ∈ (γ ,∞). In fact, E (JM) > 0appears to hold for any γ . While we are unable to prove E (JM) > 0 in itsfull generality, our numerical investigation supports this statement. Havingevaluated E (JM) on a large multidimensional grid of parameter values, wehave not found any set of parameters for which E (JM) > 0 is violated. Thisresult is not surprising because the condition (43) is unlikely to hold (since

22 The size of the interval in (43) is always positive because g∗∗ − g∗ = −g(0) > 0.


0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

σc (%)

Ret

urn

(%)

Panel A. Expected return at announcement of a policy decision

0 2 4 6 8 10 12 14 16 18 20

−4

−2

0

σc (%)

Ret

urn

(%)

Panel B. Expected return at announcement of a policy change

σg = 1%

σg = 2%

σg = 3%

0 2 4 6 8 10 12 14 16 18 200

2

4

σc (%)

Ret

urn

(%)

Panel C. Expected return at announcement of no policy change

σg = 1%

σg = 2%

σg = 3%

σg = 1%

σg = 2%

σg = 3%

Figure 6. Expected return at the announcement of a policy decision. Panel A plots theexpected value of the jump in stock prices at the announcement of a policy decision, withoutconditioning on whether the decision is to change the policy or not. This value, E(JM), representsthe unconditional risk premium investors demand at the announcement. Panel B reports theexpected jump conditional on the announcement of a policy change, E(Jyes

M ). Panel C reports theexpected jump conditional on the announcement of no policy change, E(Jno

M ). All three expectationsintegrate out uncertainty about gτ as perceived at the beginning of the downturn. The downturnlength is τ − t0 = 5 years. We vary impact uncertainty (σg) and political uncertainty (σc). All otherparameters have values assigned in Table I.

g∗∗ < g(0) < 0, g0 = 0, and gt is a martingale). We thus conclude that our modelimplies a positive unconditional jump risk premium.

Panel A of Figure 6 plots E (JM) as a function of σg and σc for the parametervalues from Table I. We set the downturn length to 5 years, so that uncertaintyabout gτ is integrated out as of time t0 = 5 conditional on gt0 = 0. The figureshows that the jump risk premium E(JM) increases with both σg and σc. In thebenchmark case of σg = 2% and σc = 10%, E(JM) = 5 basis points; for σg = 3%,it is 10 basis points.

Panels B and C of Figure 6 plot the unconditional expected jumps E(JyesM )

and E(JnoM ), which are computed analogously to E (JM) by integrating out uncer-

tainty about gτ . Panel B shows E(JyesM ), which corresponds to the EAR analyzed

earlier. Panel B looks similar to Figure 3, except that the EAR plotted here ismore negative than in Figure 3 due to the shorter downturn length (i.e., 5 yearsin Figure 6 vs. 10 years in Figure 3). Panel C of Figure 6 shows that E(Jno

M ) is


positive and increasing in both σg and σc, but its magnitude is much smallerthan that of E(Jyes

M ). The reason is that the probability-weighted average of thejumps in Panels B and C is close to zero (see Panel A) and the unconditionalprobability of a policy change is less than 0.5. This probability, which is as-sessed at the beginning of the downturn, ranges from 11% to 49%, dependingon the values of σg and σc (it is 29% in the benchmark case of σg = 2% andσc = 10%).

D. The Diffusion Component of Stock Returns

Whereas the jump components are nonzero only at time τ , the diffusion com-ponents in equations (33) and (37) drive the dynamics of returns and SDF forall t ∈ (0, T ). Both components are affected by policy uncertainty—the jumpcomponent is due to political uncertainty, whereas the diffusion component isdriven by impact uncertainty. Equations (35), (39), and (40) show that uncer-tainty about gt increases the volatility of SDF as well as the mean and varianceof stock returns. After time τ , σπ,t, μM,t, and σM,t all vary over time as increasingfunctions of σt, and similar dependence is present before time τ . If the policychange occurs at time τ , uncertainty about gt jumps up (since σg > στ ), causingupward jumps in σπ,t, μM,t, and σM,t.

Figure 7 plots the expected dynamics of σπ,t, μM,t, and σM,t around a policychange. We simulate many samples of shocks in the model, keeping the down-turn length at 5 years. We plot the paths of σπ,t, μM,t, and σM,t averaged acrossthe subset of samples in which a policy change occurs. The parameters are fromTable I, except for σg, which takes three different values.

Figure 7 is dominated by the upward jumps in σπ,t, μM,t, and σM,t at the timeof a policy change. These jumps are induced by increases in uncertainty aboutgt, as discussed earlier. For example, for σg = 2%, σπ,t jumps from 28% to 65%per year, μM,t jumps from 2% to 9% per year, and σM,t jumps from 12% to 16%per year. The magnitudes of all three quantities increase in σg: when σg = 3%,both μM,t and σM,t jump to about 25% per year. After time τ , all quantitiesgradually fall as a result of the learning-induced gradual decline in σt.

Stock returns before time τ are affected by multiple forces. Let pt denote theprobability of a policy change at time τ , as perceived by investors at time t < τ .Fluctuations in pt contribute to volatility: stock prices fluctuate as investorschange their minds about what the government is going to do. Conditional ona policy change, pt grows towards pτ = 1, and the volatility induced by fluc-tuations in pt typically increases as time τ approaches. There are also twoopposite effects. First, σt declines over time due to learning, thereby pushingσπ,t, μM,t, and σM,t down. Second, when pt increases, the current value of gtbecomes less likely to matter after time τ , making stock prices less sensitiveto gt. The rising probability of a policy change makes the current policy lessrelevant, reducing the sensitivity of stock prices to time-varying beliefs aboutthis policy. Which of these effects prevails depends on the parameter values. InFigure 7, all three quantities fall before τ , suggesting that the effect of growingfluctuations in the probability of a policy change is weaker than the combined


9 9.5 10 10.5 1110

15

20

25

30

Time

Per

cent

per

yea

r

Panel C. Return volatility

σg=1%

σg=2%

σg=3%

9 9.5 10 10.5 1120

40

60

80

100

Time

Per

cent

Panel D. Correlation

σg=1%

σg=2%

σg=3%

9 9.5 10 10.5 1120

40

60

80

100

120

Time

Per

cent

per

yea

rPanel A. SDF volatility

σg=1%

σg=2%

σg=3%

9 9.5 10 10.5 110

5

10

15

20

25

30

Time

Per

cent

per

yea

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Panel B. Expected return

σg=1%

σg=2%

σg=3%

Figure 7. Properties of returns around policy changes. This figure plots the expected dy-namics of the volatility of the stochastic discount factor, σπ,t (Panel A), the conditional expectedstock market return, μM,t (Panel B), the conditional volatility of stock market returns, σM,t(Panel C), and the pairwise correlation between stocks, ρt (Panel D), all conditional on a pol-icy change at time τ = 10. The jump-related components at time τ are not plotted. The downturnlength is τ − t0 = 5 years, so that g5 = 0. The parameters are in Table I, except for σg (impactuncertainty), which takes three different values, namely, 1%, 2%, and 3% per year.

effects of learning and the diminished sensitivity of prices to gt. Also note that,when the political uncertainty is resolved and the government makes its deci-sion at time τ , pt stops fluctuating and this component of volatility disappears.Nonetheless, volatility jumps up at time τ due to a stronger opposing effect ofhigher uncertainty about gt.

COROLLARY 9: The correlation between the returns of any pair of stocks for t > τ

is given by

ρt =[σ + (T − t)σ 2

t σ−1]2[

σ + (T − t)σ 2t σ−1

]2 + σ 21

. (45)

For t < τ , the correlation is given in equation (A6) in the Appendix. For t = τ ,the instantaneous correlation is one.


Equation (45) shows that, after time τ , the correlation ρt increases withσt. Intuitively, uncertainty about gt increases each stock’s sensitivity to thecommon factor gt, making returns more correlated. Similar dependence of ρton σt is also present before time τ , when ρt also depends on the probability ofa policy change. At time τ , ρt = 1 due to the resolution of political uncertainty,which results in a common jump in stock prices.

Panel D of Figure 7 plots the expected dynamics of ρt around the policychange. The correlation jumps up at time τ , for reasons discussed earlier. Thisjump is substantial: from 30% to 62% when σg = 2%, and from 39% to 84%when σg = 3%. (We do not plot the instantaneous jump to ρt = 1 at time τ .)After time τ , the correlation falls due to learning; before time τ , it falls due tothe previously discussed effects of learning and the diminished price sensitivityto gt.

III. Extension: Endogenous Timing of Policy Change

Our benchmark model assumes that the government can change its policyonly at a given time τ . This assumption buys us closed-form solutions for manyquantities of interest, and it also allows us to formally prove our propositionsand corollaries. In this section, we extend the model by endogenizing the timingof the policy change. We solve the problem numerically. To give a preview, wefind that our main results continue to hold in this setting. In addition, wefind that policy changes that happen earlier tend to produce more negativeannouncement returns.

We assume that the government can change its policy at any time τ ∈[τ1, τ2, . . . , τn], where τi = i and n = 19, so the policy change can occur in anyyear (T = 20). Once the change is made, it is irreversible. At each time τi, anew value of the political cost Ci is drawn from the distribution in equation (6).The values of Ci are iid. After observing Ci, the government decides whetherto change its policy, maximizing the same objective as before. Let V (gt, Bt, t)denote the value function given no policy change at or before time t ∈ [Ti, Ti+1):

V (gt, Bt, t) = Et

{max

{V (gτi+1, Bτi+1 , τi+1), Eτi+1

[C B1−γ

T1−γ

∣∣∣∣Change at τi+1

]}}.

(46)

This value represents a solution to a partial differential equation whose finalcondition is given by V (gT , BT , T ) = B1−γ

T /(1 − γ ), as detailed in the Appendix.Conditional on a policy change at a given time τ , the value function at that timeis available in closed form:

V (Bτ , τ, c) = B1−γτ

1 − γec+μ(1−γ )(T −τ )+σ 2

g (T −τ )2−γ (1−γ ) σ22 (T −τ ). (47)

The government changes its policy at time τ = τi if and only if

V (Bτi , τi, c) > V (gτ , Bτi , τi). (48)


0 5 10 15 20

−4

−3

−2

−1

0Panel A. Announcement return

σc (%)

Per

cent

σg = 1%

σg = 2%

σg = 3%

5 10 15 20−10

−8

−6

−4

−2

0Panel B. Announcement return

Policy announcement date

Per

cent

σg = 1%

σg = 2%

σg = 3%

−1 −0.5 0 0.5 110

12

14

16

18

20Panel C. Return volatility

Time relative to policy announcement date

Per

cent

per

yea

r

σg = 1%

σg = 2%

σg = 3%

−1 −0.5 0 0.5 120

30

40

50

60

70Panel D. Correlation

Time relative to policy announcement date

Per

cent

σg = 1%

σg = 2%

σg = 3%

Figure 8. Endogenous timing of a policy change. Panel A plots the expected stock return,EAR, at the endogenously timed announcement of a policy change. The timing of the policy changeis optimally chosen by the government from the set τ ∈ [1, 2, . . . , 19

]. Whereas Panel A averages

returns across all τ , Panel B plots EAR as a function of τ . Panels C and D plot the expecteddynamics of σM,t and ρt, respectively, around a policy change. The results are plotted in eventtime, with time 0 marking the policy change (time τ ). The parameters are in Table I, except for σg(impact uncertainty), which takes three different values, namely, 1%, 2%, and 3% per year.

Figure 8 shows that our key results continue to hold when τ is endogenous.For the parameter values from Table I, Panel A plots EAR as a function ofσg and σc. This panel is analogous to Figure 3, in which τ is exogenous. Asin Figure 3, EAR is negative and its magnitude is increasing in both σg andσc. The magnitudes are generally larger than in Figure 3; for example, inthe benchmark case with σg = 2% and σc = 10%, EAR is −1.9%, compared to−0.5% in Figure 3. The reason the magnitudes are larger when τ is endogenousis difficult to convey precisely because the results are obtained numerically.However, some intuition emerges when we consider Propositions 2 and 4, whichapply when τ is exogenous. These propositions imply that a policy changeoccurs at time τ if gτ < g(c), and that stock prices drop at the announcementif gτ > g∗. Assuming that these results are also somewhat relevant when τ

is endogenous, a policy change occurs at time τi if gτi < g(ci) but gτi−1 > g(ci−1).


Since gt has only just fallen below the threshold at time τi (but not at time τi−1),it seems unlikely that gτi is so low that it is also below g∗, in which case theannouncement return would be positive. This intuition is only approximate,not only because Propositions 2 and 4 rely on exogenous τ but also becauseg(ci) varies due to variation in ci. Nonetheless, it is comforting to see that ourmain result strengthens when we endogenize τ .

Panel B of Figure 8 plots the same quantity as Panel A, the EAR conditionalon a policy change, but this time as a function of the time τ at which thepolicy is changed. Recall that τ is optimally chosen by the government from theset τ ∈ [1, 2, . . . , 19

]. Whereas Panel A averages the announcement returns

across all τ , Panel B averages them only across those simulations in whichthe policy change occurs at the given τ . We keep σc = 10% as in Table I andvary σg.23 There are two basic results: (i) the announcement return is negativefor all τ , and (ii) its magnitude is larger when τ is smaller. For example,holding σg at 2%, the announcement return is −1% for τ = 15 but almost−5% for τ = 5.24 This result is easy to understand. At each point in time, thegovernment weighs the costs and benefits of changing the policy. One importantcost is that the irreversible policy change destroys the option to wait (e.g., thepolitical benefit from changing the policy might be higher in the future). Thisoption value of waiting declines as time passes. Early on, while this optionvalue is still high, it takes a large political benefit for the policy change tooccur. As a result, policy changes occurring at low τi tend to be politicallymotivated and highly unexpected, producing larger negative announcementreturns.

Panels C and D of Figure 8 plot the expected dynamics of σM,t and ρt, re-spectively, around a policy change. These dynamics are computed by aver-aging across those simulated paths in which a policy change occurs at anyτ ∈ [1, 2, . . . , 19

]. The results are plotted in event time, with time 0 marking

the policy change (time τ ). We see that both volatility and correlation jump upat the time of a policy change and decline afterwards, just like in Figure 7,in which τ is exogenous. We also find that policy changes tend to occur afterperiods of low profitability, as they do when τ is exogenous. To summarize, allof our key results remain unchanged when τ is endogenous.

IV. Extension: Investment Adjustment

In the benchmark model, firms’ decision-making is not modeled explicitly.Any investment decisions taken by firms are assumed to be reflected in theprofitability process in equation (1). That process might not adequately capture

23 We vary σg because the effect of σc is predictable and weaker than the effect of σg.24 The magnitudes are even more negative for τ < 5 but the simulation results then become less

precise due to the small probability of policy changes for small values of τ . To keep the number ofsimulations manageable and the plot smooth rather than jagged, we do not plot the announcementreturns for τ < 5.


all investment decisions. For example, prompted by policy uncertainty, a firmmight shut down its operations. In this section, we extend the model by allowingeach firm to make a major investment decision at time τ . To preview our results,we find that firms often cut their investment in response to government-induceduncertainty. We also find that EAR remains negative, as in the benchmarkmodel.

Between times 0 and τ , each firm is fully invested in its risky technology, asin the benchmark model (equation (1)). At time τ , each firm has the option todisinvest and shift all of its capital into a risk-free storage technology whoseconstant return is normalized to zero. If the firm disinvests, its capital earnsthe zero rate of return from time τ until time T ; otherwise, its profitabilitycontinues to follow equation (1). Partial investment in the two technologies isruled out for simplicity.

All firms make their investment decisions at the same time τ at which thegovernment makes its policy decision. This simplifying assumption capturesthe idea that firms decide on their investment while facing policy uncertainty,and the government decides on its policy while taking into account its im-pact on investment. The government and firms have the same prior beliefs(equations (3) and (8)). We solve for the Nash equilibrium in which the govern-ment’s policy choice is optimal given the firms’ investment decisions, and eachfirm’s investment decision is optimal given the decisions of all other firms aswell as the government’s decision rule.

PROPOSITION 9: In equilibrium at time τ , a randomly selected fraction ατ ∈[0, 1] of firms continue investing in their risky technologies, while the remainingfirms disinvest and park their capital in the risk-free technology. The govern-ment changes its policy if and only if

gτ < g(c, ατ ), (49)

where the threshold g (c, ατ ) and the equilibrium value of ατ are describedbelow.

Similar to Proposition 2, Proposition 9 shows that a policy is replaced if itis perceived to have a sufficiently unfavorable impact on profitability. A keydifference from Proposition 2 is that the threshold g (c, ατ ) depends on thefraction of investing firms, ατ , which in turn depends on gτ .

To shed light on Proposition 9, we first take the government’s perspective. Thegovernment makes its policy decision by taking firms’ investment decisions, ατ ,as given. Recall that a policy change occurs if and only if condition (12) holds.This condition involves functions of the aggregate capital BT . Given ατ , thevalue of BT is given by

BT

Bτ

= ατ e(μ+g− σ22 )(T −τ )+σ (ZT −Zτ ) + (1 − ατ ), (50)


where g ≡ gold if there is no policy change and g ≡ gnew if there is a policychange. Using equation (50), we find that condition (12) holds if and only if

ec Eτ

{[ατ eεyes(τ,T ) + (1 − ατ )

]1−γ }< Eτ

{[ατ eεno(τ,T ) + (1 − ατ )

]1−γ }, (51)

where

εi(τ, T ) ∼ N((μ + gi − σ 2/2)(T − τ ), (T − τ )2σ 2i + σ 2(T − τ )), (52)

for i = yes, no and gno = gτ , σ 2no = σ 2

τ , gyes = 0, and σ 2yes = σ 2

g . The right-handside of (51) decreases with gτ , whereas the left-hand side does not depend ongτ . Therefore, condition (51) implies the cutoff rule in Proposition 9, where thecutoff g(c, ατ ) is the value of gτ for which the left-hand side of (51) equals theright-hand side.

Next, we take the firms’ perspective. At the time of their investment deci-sions, firms do not know whether the government’s policy will change. Eventhough firms observe gτ , they do not observe g(c, ατ ) (because they do notobserve c), so they cannot fully anticipate the government’s action. Firms max-imize their market value, which obeys equation (17), where the state price den-sity πt is computed based on equations (16) and (50). If firm i decides to switchto the risk-free technology at time τ , its market value is Mi

τ = Biτ (because

BiT = Bi

τ , so that Eτ [πTπτ

BiT ] = Eτ [πT

πτ]Bi

τ = Biτ ). Therefore, each firm chooses to

remain invested in the risky technology if and only if doing so results in amarket-to-book ratio greater than one at time τ .

If firm i decides at time τ to remain invested, its market value right aftertime τ is given by

Miτ+ =

⎧⎪⎨⎪⎩Mi,yes

τ+ = Biτ+

Eτ+{eεyes (τ,T )[ατ eεyes (τ,T )+(1−ατ )]−γ }Eτ+{[ατ eεyes (τ,T )+(1−ατ )]−γ } if policy changes

Mi,noτ+ = Bi

τ+Eτ+{eεno (τ,T )[ατ eεno (τ,T )+(1−ατ )]−γ }

Eτ+{[ατ eεno (τ,T )+(1−ατ )]−γ } if policy does not change.

(53)

Right before time τ , the market value of firm i is a weighted average of Mi,yesτ+

and Mi,noτ+ as in equation (23), but the weights ωτ are in equation (A9) in the

Appendix. The values of Miτ /Bi

τ are equal across i because all firms are iden-tical ex ante. Therefore, if Mi

τ /Biτ > 1, all firms remain invested, so ατ = 1 in

equilibrium. For an interior solution 0 < ατ < 1 to occur, firms must be indif-ferent between the risky and risk-free technologies, so that Mi

τ /Biτ = 1 for all

i. Combining this condition with equation (23), we obtain the condition forequilibrium with 0 < ατ < 1:25

1 = ωτ

(Mi

τ+Bi

τ+

)yes

+ (1 − ωτ )(

Miτ+

Biτ+

)no

. (54)

25 The same condition is obtained as a first-order condition in an alternative formulation ofthe problem in which a social planner chooses ατ to maximize investors’ expected utility. See theInternet Appendix.


The right-hand side of equation (54) depends on ατ , which affects both ωτ andthe M/B ratios. If there exists a value of ατ strictly between zero and one forwhich equation (54) holds, then this is the equilibrium value of ατ in Proposition9. If instead the right-hand side of equation (54) exceeds one for all ατ ∈ [0, 1],then ατ = 1 in equilibrium, and all the results from Sections I and II hold. Ifthe right-hand side of equation (54) is smaller than one for all ατ ∈ [0, 1], thenthe equilibrium features ατ = 0, an uninteresting case of no investment.

For the parameter values in Table I, the equilibrium solution is ατ = 1 (nodisinvestment), so that all the results from the benchmark model also apply tothe problem analyzed here. To analyze the effect of disinvestment, we reducethe value of μ from its benchmark value of 10% to 2%, which is the largestinteger percentage value for which ατ < 1 in equilibrium (i.e., for all μ ≥ 3%,the equilibrium has ατ = 1). All other parameter values are in Table I. We setthe downturn length to 5 years, as before. We solve the problem numericallyand plot the results in Figure 9 .

Panel A of Figure 9 reports the equilibrium value of ατ as a function of σcand σg. Higher values of σc or σg imply lower values of ατ . Fixing σc = 10%and raising σg from 1% to 2% to 3%, the fraction of firms that remain investeddecreases from one to 0.87 to 0.77, respectively. The effect of σc is weaker: fixingσg = 2%, ατ decreases from 0.88 to 0.85 as we raise σc from zero to 20%. In short,both types of policy uncertainty reduce aggregate investment.

Panel B of Figure 9 plots EAR as a function of σc and σg. The plot is similarto that in Panel B of Figure 6, which corresponds to the benchmark model (inwhich we force ατ = 1). For σg = 1%, EAR is the same as in Figure 6 becauseατ = 1 in equilibrium (see Panel A of Figure 9). For σg = 2% and 3%, EARis more negative than for σg = 1% but less negative than its counterparts inFigure 6. For example, for σg = 2% and σc = 10%, EAR is −0.6% comparedto −1.1% in Figure 6, and the difference from Figure 6 is even larger forσg = 3%. The reason is that a substantial fraction of firms disinvest in equilib-rium (ατ < 1), thereby reducing aggregate risk. By cutting their investment,firms essentially undo some of the uncertainty associated with governmentpolicy. Nonetheless, EAR remains negative for all values of σc and σg, as in thebenchmark model.

Panels C and D of Figure 9 plot the expected dynamics of return volatility,σM,t, and correlation, ρt, respectively, around a policy change. Both quantitiesexhibit patterns very similar to those in Panels C and D of Figure 7, whichcorrespond to the benchmark model (ατ = 1). The post-τ increases in σM,t andρt are slightly smaller than in Figure 7, as one would expect as a result of areduction in aggregate risk, but the conclusions are the same. In short, the keyasset pricing results from the benchmark model hold also when we allow fordisinvestment.

V. Extension: Different Policy Exposures

Government policies typically affect all firms, but they may affect differentfirms differently. In this section, we extend our benchmark model by allowing


0 5 10 15 20−1.5

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Ret

urn

(%)

Panel B. Announcement return

σg = 1%

σg = 2%

σg = 3%

0 5 10 15 2070

75

80

85

90

95

100

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α (%

)Panel A. Equilibrium α

σg = 1%

σg = 2%

σg = 3%

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Panel C. Return volatility

σg=1%

σg=2%

σg=3%

9 9.5 10 10.5 1120

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60

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100

Time

Per

cent

Panel D. Correlation

σg=1%

σg=2%

σg=3%

Figure 9. Investment adjustment. This figure plots the results from the extension of the bench-mark model in which firms can disinvest at time τ . Panel A plots the equilibrium fraction of firmsthat choose to remain invested in the risky technology after the policy change. Panel B plots thecorresponding stock return expected at the announcement of a policy change, or EAR. Panels C andD plot the expected dynamics of σM,t and ρt, respectively, around a policy change at time τ = 10.The parameters are in Table I, except that μ = 0.02 (we reduce μ from its benchmark value toobtain α < 1 in equilibrium) and σg takes three different values, namely, 1%, 2%, and 3% per year.The downturn length is set to 5 years in all four panels.

firms to have different exposures to government policy. We find that firms withlarger government exposures tend to have higher expected returns. The gov-ernment risk premium can be negative though, and it is highly state dependent.We also find that EAR remains negative, as in the benchmark model.

Firms are divided into N sectors, n = 1, . . . , N, where N is finite. Each sectorn is characterized by a nonnegative “government beta” βn, which representsa loading on the policy impact. Specifically, the profitability of each firm i insector n is given by

d�it = (μ + βngt) dt + σdZn

t + σ1dZit , (55)


where dZnt is a sector-specific shock, dZi

t is a firm-specific shock, and all Brown-ian motions are independent of each other.26 Equation (55) replaces equation (1)in the benchmark model from Section I; the rest of the model remains un-changed. The benchmark model is thus a special case of this model with βn = 1and N = 1. Similar to Proposition 1, by observing each firm’s profitability, allagents effectively receive N signals

dsnt = (μ + βngt) dt + σdZn

t , n = 1, . . . , N. (56)

Let Bt denote the total capital of all firms and Bnt denote the total capital of all

firms in sector n. We denote sector n’s share of total capital by wnt , so that

wnt = Bn

t

Bt, n = 1, . . . , N. (57)

The N × 1 vector of these shares, which we denote by wt, represents additionalstate variables introduced by the presence of multiple sectors.

PROPOSITION 10: The government changes its policy at time τ if and only if

gτ < g (c, wτ ) , (58)

where the threshold g (c, wτ ) is described in the Internet Appendix.Proposition 10 shows that a policy is replaced if its impact on profitability

is perceived as sufficiently unfavorable. This result is similar to Proposition 2,except that the threshold for gτ now also depends on the vector wt of the sectorshares.

We solve for stock prices and their dynamics before time τ by using numericalmethods. The pricing results after time τ are in closed form. The new statevariable wt affects both the level and the volatility of stock prices, as well asthe expected rates of return. All of the asset pricing results, along with theresults on learning, are presented and proved in the Internet Appendix.

Figure 10 plots various results for a two-sector framework (i.e., N = 2). Wedenote the two sectors by H and L. The government betas of the two sectors areβH and βL, where βH ≥ βL. We vary these betas while holding their averageequal to one. In the baseline case, explored in Panel A, we set βH = 1.5 andβL = 0.5. We also vary the relative size of the high-beta sector, which we denoteby wt = BH

t /(BHt + BL

t ). In the baseline case, we set w0 = 0.5, so that, as of time0, half of the firms have high betas. Unless specified otherwise, the parametersare from Table I.

A. The Expected Announcement Return

Panel A of Figure 10 plots EAR, obtained by simulations for the baselinecase, as a function of σg and σc. This panel is the counterpart of Figure 3, which

26 We could make the dZnt shocks correlated at the cost of additional complexity. Given our

interest in the cross-sectional differences in stock returns as opposed to their level, such cross-correlation seems to be of second-order importance, so we simply assume no correlation.


Figure 10. Different government policy exposures. This figure plots the results from theextension of the benchmark model in which there are two types of firms with different exposuresto government policy, βH and βL. Panel A plots the stock return expected at the announcement ofa policy change, or EAR, when half of the firms as of time 0 have βH = 1.5 and the other half haveβL = 0.5. Panel B plots EAR as a function of βH − βL while varying w0, the fraction of high-betafirms as of time 0. Panels C and D plot the difference in the conditional expected returns of thehigh-beta and low-beta firms, μH − μL, as a function of βH − βL while varying wt, the fraction ofhigh-beta firms as of time t = 5 years. In Panel C, gt = 2%, whereas in Panel D, gt = −2%. In PanelsB, C, and D, we vary βH − βL while holding the average of the two betas constant, (βH + βL)/2 = 1.Unless specified otherwise, the parameters are in Table I.

corresponds to the benchmark model. As in Figure 3, EAR is always negativeand its magnitude is increasing in both σg and σc. The magnitudes of EARs arealso close to those from Figure 3.

Panel B of Figure 10 plots EAR as a function of βH − βL, for three differentvalues of w0. The values of σg and σc are fixed at their baseline values from


Table I. When βH − βL = 0, both sectors have the same government beta ofone. EAR is then −0.6%, which is close to the corresponding value of −0.5% inthe benchmark model (Figure 3). These two values are different even thoughall firms have a beta of one in both models. The reason is that the speeds oflearning in the two models are different. The two-sector model has two signalsabout gt (equation (56)), whereas there is only one signal in the benchmarkmodel (equation (7)). As a result, learning in the two-sector model is faster.27

Therefore, the difference between the prior and posterior variances is larger,the discount rate effect is stronger, and EAR is more negative. Panel B alsoshows that EAR is more negative for larger values of w0. For example, forβH − βL = 1, EAR is −0.4% for w0 = 0.2, but it is −0.8% for w0 = 0.8. Whenw0 is larger, the high-beta firms account for a larger fraction of the market.As a result, the risk associated with learning about gt is harder to diversifyand shocks to gt are more highly correlated with the stochastic discount factor.The increase in uncertainty at time τ thus generates a larger discount rateeffect, leading to a more negative EAR when w0 is larger. Finally, consider thecase of βH − βL = 2, in which βH = 2 and βL = 0. In the limiting case w0 → 0,all firms have zero beta; as a result, policy changes become irrelevant andEAR→ 0.

B. Government Betas and Expected Stock Returns

Given the government’s pervasive effect on the private sector, policy uncer-tainty is unlikely to be fully diversifiable. Motivated by this observation, werelate government betas to expected stock returns. The instantaneous expectedstock returns are given by the drifts of the stocks’ return processes. We denotethese drifts for sectors H and L at time t by μH

t and μLt , respectively.

Panels C and D of Figure 10 plot μHt − μL

t as a function of βH − βL. Weconsider three different values of wt and set t = 5 years (which is half-waybetween time 0 and time τ ). We set wt rather than w0 as in the previousparagraph because our focus is on the conditional expected return at time trather than EAR as of time 0.

Panels C and D show that, as the share of sector H increases, so does μHt − μL

t ,holding βH − βL constant. There are two reasons for this result. First, a largerwt makes the gt shocks more powerful, as explained above. As a result, ahigher government beta has a larger effect on the expected return. Second,there is an interesting size effect even when both sectors have equal govern-ment betas (βH − βL = 0). Both panels show that equal betas imply equal riskpremia (i.e., μH

t − μLt = 0) only when both sectors are equally large (wt = 0.5).

When wt �= 0.5, the risk premia are different. For example, wt = 0.8 impliesμH

t − μLt = 0.7% per year, and wt = 0.2 implies μH

t − μLt = −0.7%. Firms in the

larger sector command a higher risk premium because their capital Bit covaries

27 Specifically, learning in the multiple-sector model is faster than in the benchmark modelunder the condition β ′β > 1, where β is an N × 1 vector of βn’s. This condition is trivially satisfiedhere.


more closely with aggregate capital Bt (by construction, since the larger sectoraccounts for a larger share of aggregate capital). The higher covariance withaggregate capital makes the larger-sector firms riskier. This size effect canmake μH

t − μLt negative even when βH − βL is positive if wt is low; we observe

such instances in both panels when wt = 0.2.To abstract from the size effect, suppose both sectors are equally large (wt =

0.5). In that case, Panels C and D always show μHt − μL

t > 0, indicating thatthe higher-beta firms earn higher expected returns. In addition, μH

t − μLt is

generally increasing in βH − βL. The only exception occurs in Panel D for largevalues of βH − βL. In Panel D, gt < 0. Since the H sector loads more heavily ongt, it is expected to shrink relative to the L sector, especially when βH − βL islarge. As a result, the expected future impact of gt is diminished, reducing theimpact of the gt shocks on stock prices when gt < 0 and βH − βL is large. Withthis caveat, we can state that larger differences in government betas generallytranslate into larger differences in expected returns.

Finally, a comparison of Panels C and D shows that μHt − μL

t is substantiallylarger for gt = 2% (Panel C) than for gt = −2% (Panel D). The gt shocks are lesspowerful in Panel D for two reasons. First, the H sector is expected to shrink inrelative terms, as explained in the previous paragraph. Second, the probabilityof a policy change is higher in Panel D. When gt = 2%, the prevailing policyis likely to be retained, so the gt shocks are likely to be permanent. The sameshocks have a shorter expected duration when gt = −2% because the policychange is then more likely. Since the gt shocks are expected to be longer-lastingwhen gt is higher, the risk premium associated with exposure to those shocksis larger as well. Overall, we see that the premium for government beta risk istypically positive but also highly state dependent.

VI. Empirical Predictions

In this section, we summarize the empirical predictions of our model. The firstprediction is that policy changes should be more likely to occur after downturns,or periods of unexpectedly low profitability (see Proposition 2 and Figure 1).Similar results have been discussed in the political economy literature. Forexample, Rodrik (1996, p. 27) writes: “Reform naturally becomes an issue onlywhen current policies are perceived to be not working.” According to Drazen(2000, p. 449), “it is striking how little formal empirical testing there has beenof the view that a crisis is necessary for significant policy change.” An earlyexception is Bruno and Easterly (1996), who find that inflation crises tend tobe followed by reforms. Drazen and Easterly (2001) and Alesina, Ardagna, andTrebbi (2006) also find evidence supporting the hypothesis that crises inducereforms, although the former study finds such evidence only for a subset of thecrisis indicators.

Our most interesting predictions describe stock market reactions to policychanges. First, stock prices should fall at the announcement of a policy changeunless the policy being replaced is perceived as sufficiently harmful to prof-itability (see Proposition 4). Since the policy’s perceived impact is determined byrealized profitability, this prediction can also be stated differently: stock prices


should rise at the announcement of a policy change preceded by a sufficientlydeep downturn, but they should fall otherwise. Second, and perhaps most in-teresting, stock prices should fall at the announcements of policy changes, onaverage (see Proposition 5). That is, averaging across all policy changes, thestock market return at the announcement of a policy change should be nega-tive. Third, the average announcement return should be more negative if eitherimpact uncertainty or political uncertainty is large (see Figure 3). Fourth, theannouncement return should be more negative if the policy change is precededby a short or shallow downturn (see Figure 4). Fifth, the empirical distributionof stock market returns at the announcements of policy changes should be left-skewed (see Figure 5). Finally, policy changes should make stock returns morevolatile and more highly correlated across firms (see Figure 7).28

One challenge for empirical work on the announcement returns is the tim-ing of the policy change. In some cases, this timing is clear.29 However, mostreforms occur gradually, clearing multiple hurdles. For example, the reformof U.S. financial regulation has involved separate passages of related bills bythe House and the Senate, a subsequent reconciliation, and a continuing emer-gence of specific regulatory measures. According to our model, stock pricesshould respond at each step of the way, with bigger responses following biggerincreases in the probability of a policy change.30 Identifying the pivotal stepsof the key reforms is a nontrivial task. For example, Cutler (1988) examinesthe stock market reaction to the Tax Reform Act of 1986. He identifies two keyevents in the bill’s progress—the vote by the House for the bill in December1985, and the vote for a similar bill by the Senate Finance Committee in May1986—and argues that both events surprised the market. He reports that theday after the late-night favorable House vote, the S&P 500 index fell by 0.40%,and the day after the late-night favorable Senate Finance Committee vote, theindex fell by 0.49%. It would be useful to carefully examine other policy changesas well.

28 When interpreting these model predictions, recall from Section II that we normalize stockprices by the price of a zero-coupon bond that pays one dollar at time T with certainty. Forexample, a zero stock market reaction combined with an increase in the value of a long-termzero-coupon Treasury bond would therefore qualify as a stock market drop in our model.

29 One example is the abrupt shift in the too-big-to-fail policy mentioned in the introduction.The U.S. government’s decision to let Lehman Brothers fail was followed by a 4.7% drop in the S&P500 index on September 15, 2008. Another example, fresh as of this writing, is Germany’s surpriseannouncement of a partial ban on naked short selling late on May 18, 2010. This announcementwas followed by a roughly 2% drop in the European share indices the following morning (e.g.,Germany’s DAX dropped 1.9%). This drop is consistent with our model, but not with theoriespredicting that short-sale constraints lift asset prices.

30 A key step in the financial regulation reform took place on Thursday, May 20, 2010. At 2:31pm, the Senate voted by the narrowest possible margin (60 to 40) to overcome filibusters and sendthe bill to a final vote. A CNN Money article on the same afternoon read: “Wall Street reformcleared a crucial test vote on Thursday, all but assuring final Senate passage of the most sweepingregulatory overhaul since the New Deal.” By the end of the day, the S&P 500 index fell 3.9%, withabout half of the decline occurring in the last hour. The Senate passed the reform bill at 8:25 pmon the same day. The following morning, the U.S. stock market fell about 1% in the first minute oftrade, followed by a rebound later in the day.


Another prediction of our model is that investors demand a positive uncon-ditional premium for the jump risk associated with announcements of govern-ment policy decisions (see Panel A of Figure 6). That is, averaging across allpolicy decisions, without conditioning on whether these decisions result in apolicy change or not, the stock market return at the announcement should bepositive. The magnitudes of the jump risk premium in Figure 6 are comparableto the 9.6 basis point jump risk premium estimated by Savor and Wilson (2010).These authors find empirically that stock market returns tend to be higher ondays when news about inflation, unemployment, or interest rates is announced.We focus on a different kind of announcements, so Savor and Wilson’s estimatesare not directly comparable to the magnitudes in our Figure 6. Nonetheless, itseems interesting that the magnitude of the jump risk premium produced byour model is similar to the premium estimated empirically for a different kindof market-wide announcements.

The extensions of our model provide two additional empirical implications.First, firms should often cut their investment in response to policy uncertainty(see Figure 9). Supporting empirical evidence is reported by studies cited ear-lier in footnote 2 (e.g., Rodrik (1991), Yonce (2009), and Julio and Yook (2012)).Second, firms that are more exposed to government policy risk should typicallyhave higher expected returns (see Figure 10). The government risk premiumcan be negative though, and it is highly state dependent. Related empirical ev-idence is presented by Belo, Gala, and Li (2011). These authors find that firmswith higher exposures to the government sector (measured by the fraction ofsales generated by government spending) earn higher average stock returns,but only during Democratic presidential terms; the relation flips during Repub-lican terms.31 The match between their analysis and ours is not perfect. Theymeasure government spending, whereas we focus on government policy moregenerally. Further, they measure return exposure to the government, whereasour government betas are profitability exposures. They also study Democratsversus Republicans, whereas our model does not distinguish between differenttypes of government. Despite these differences, it seems interesting that theyestimate a state dependent government risk premium. More research into thisstate dependence is clearly warranted.

The effects of policy changes on asset prices have been analyzed in variousspecific contexts. For example, Thorbecke (1997), Rigobon and Sack (2004),and Bernanke and Kuttner (2005) examine the effects of changes in monetarypolicy on stock prices. Klingebiel et al. (2001) examine stock market responsesto announcements of bank restructuring policies by East Asian governmentsduring the 1997 to 1998 crisis. And Aıt-Sahalia et al. (2010) investigate theeffects of policy changes during the recent financial crisis on interbank creditand liquidity risk premia. A broader analysis of major policy changes would bebeneficial. Our model can be viewed as a benchmark for such analysis.

31 In another study of political cycles, Santa-Clara and Valkanov (2003) find that the averagestock market return is higher under Democratic presidencies than under Republican presidencies.


VII. Conclusions

We develop a simple asset pricing model in which the government playsthe role of an active decision maker. This model makes numerous testablepredictions about the effects of changes in government policy on stock prices.For example, the model predicts that stock market returns at the announce-ments of policy changes should be negative, on average. The magnitude ofthese negative returns should be large if uncertainty about government policyis large, as well as if the policy change is preceded by a short or shallow down-turn. The distribution of stock returns at the announcements of policy changesshould be left-skewed. Policy changes should make stock returns more volatileand more highly correlated across firms. The average stock market return atthe announcements of policy decisions, without conditioning on whether thesedecisions result in a policy change or not, should be positive. Future researchcan test these predictions empirically.

The model’s predictions follow from three key assumptions: that the gov-ernment is quasi-benevolent, that there is uncertainty about the government’sactions as well as their impact, and that the impacts of all policy choices lookidentical a priori. The assumption of the government’s quasi-benevolence seemscrucial for our results. Due to that assumption, policy changes that lift stockprices are largely anticipated, producing small positive announcement returns,whereas policy changes that reduce prices are less expected, eliciting largernegative announcement returns. The results might flip if the government wereinstead malevolent because policy changes that benefit investors would then beunexpected, and thus associated with large positive announcement returns. Inreality, the government’s objectives are surely more complex than benevolenceor malevolence, but they are likely to have a benevolent tilt due to the mechan-ics of a democratic process. For example, the government might maximize theprobability of reelection, which is imperfectly but positively related to agents’material well-being.32 Alternative objective functions for the government canbe analyzed in future research.

As noted above, we also assume that the prevailing government policy canonly be replaced by a policy that is perceived as identical a priori. The govern-ment can choose from multiple new policies, but as long as all of those policieslook identical a priori, knowing which policy is chosen is irrelevant and themultiple-policy setting collapses to a single-policy setting. The convergence toa single-policy setting also obtains at the opposite extreme, namely, if the po-tential new policies are so different from each other that one of them dominatesa priori. Pastor and Veronesi (2011) analyze the intermediate scenarios in amultiple-policy setting and find that policy heterogeneity affects stock prices in

32 See Downs (1957) for an early model in which the government maximizes the probabilityof reelection. After surveying the empirical literature, Lewis-Beck and Stegmaier (2000) concludethat economic indicators explain most of the variance in government support. A more recent surveyby Anderson (2007) finds that the relation between economic outcomes and election outcomes isimperfect and complicated. Brender and Drazen (2008) find that this relation is significant onlyamong the less-developed countries.


interesting ways. These authors also allow investors to learn about the politicalcosts of the various policies. Instead of focusing on announcement returns, aswe do, they focus on the risk premium induced by political uncertainty.

It would also be interesting to extend our model by endogenizing the politicalcost. We treat this cost as exogenous for simplicity, but it would be useful tobuild the microfoundations for it based on the insights from the political econ-omy literature. Such an extension could establish new links between financialmarkets and various political economy variables. More generally, we hope ourpaper will spur more interest in building bridges between asset pricing andpolitical economy. We still have much to learn about the role of the governmentin asset pricing.

Appendix

The Appendix contains the formulas that are mentioned in the text butomitted for the sake of brevity. The proofs of all results are available in thecompanion Internet Appendix.

PROPOSITION 11: In the benchmark model for t ≤ τ , the state price density isgiven by

πt = B−γt � (gt, t) , (A1)

where

�(gt, t) = pyest Gyes

t + (1 − pno

t

)Gno

t

Gyest = e−γμ(T −t)−γ gt(τ−t)+ γ 2

2 ((T −τ )2σ 2g +(τ−t)2σ 2

t )+γ (1+γ ) σ22 (T −t)

Gnot = e−γ (μ+gt)(T −t)+ γ 2

2 (T −t)2σ 2t +γ (1+γ ) σ2

2 (T −t)

and

pyest = N

(g(0); gt − γ σ 2

t (τ − t) + σ 2c /2

(T − τ )(1 − γ ), σ 2

t − σ 2τ + σ 2

c

(T − τ )2(1 − γ )2

)pno

t = N(

g(0); gt − γ[σ 2

t (T − t) − (T − τ )σ 2τ

]+ σ 2c /2

(T − τ )(1 − γ ), σ 2

t − σ 2τ

+ σ 2c

(T − τ )2(1 − γ )2

).

COROLLARY 10 (used in Proposition 6): In the benchmark model for t ≤ τ , thevolatility of the stochastic discount factor is given by

σπ,t = γ σ − 1� (gt, t)

∂� (gt, t)∂ gt

σ 2t σ−1. (A2)


PROPOSITION 12: In the benchmark model for t ≤ τ , the stock price for firm i isgiven by

Mit = Bi

t� (gt, t)� (gt, t)

, (A3)

where

� (gt, t) = pyest Kyes

t + (1 − pno

t

)Kno

t

Kyest = e(1−γ )μ(T −t)+(1−γ )gt(τ−t)+ (1−γ )2

2 ((T −τ )2σ 2g +(τ−t)2σ 2

t )−(1−γ )γ σ22 (T −t)

Knot = e(1−γ )μ(T −t)+(1−γ )gt(T −t)+ (1−γ )2

2 σ 2t (T −t)2−(1−γ )γ σ2

2 (T −t)

and

pyest = N

(g(0); gt + (1 − γ )σ 2

t (τ−t) + σ 2c /2

(T −τ )(1−γ ), σ 2

t −σ 2τ + σ 2

c

(T − τ )2(1 − γ )2

)pno

t = N(

g(0); gt + (1 − γ )[σ 2

t (T − t) − (T − τ )σ 2τ

]+ σ 2c /2

(T − τ ) (1 − γ ), σ 2

t − σ 2τ

+ σ 2c

(T − τ )2(1 − γ )2

).

COROLLARY 11 (used in Proposition 7): In the benchmark model for t ≤ τ , themean and volatility parameters of the return process in Proposition 7 are givenby

σM,t = σ +(

∂� (gt, t) /∂ gt

� (gt, t)− ∂� (gt, t) /∂ gt

� (gt, t)

)σ 2

t σ−1 (A4)

μM,t = σπ,tσM,t. (A5)

COROLLARY 12 (used in Corollary 9): In the benchmark model for t < τ , thecorrelation between the returns on any pair of stocks is given by

ρt = σ 2M,t

σ 21 + σ 2

M,t

. (A6)

PROPOSITION 13: Let the timing of the policy change be endogenous as in Sec-tion III. For every t ∈ [τi, τi+1), the indirect utility function V (gt, Bt, t) fromequation (46) is given by

V (gt, Bt, t) = B1−γt � (gt, t) , (A7)


where � (gt, t) satisfies the partial differential equation

0 = ∂� (gt, t)∂t

+{

(1 − γ ) (μ + gt) − 12

γ (1 − γ ) σ 2}

� (gt, t)

+ 12

∂2� (gt, t)∂ g2

t

(σ 2

t

)2σ−2 + (1 − γ )

∂� (gt, t)∂ gt

σ 2t .

(A8)

The boundary conditions at time τi are given by

�(gτi− , τi−

)=Eτi−

[max

{�(gτi , τi),

11−γ

ec+(1−γ )μ(T −τi )+ 12 (1−γ )2σ 2

g (T −τi )2−γ (1−γ ) σ22 (T −τi )

}],

where the expectation is taken with respect to c just before the policy decision attime τi . The final condition at time T is � (gT , T ) = 1

1−γ.

Note on Section IV:

Right before time τ , the market value of firm i is given by a weighted averageof Mi,yes

τ+ and Mi,noτ+ as in equation (23), except that the weights ωτ are given by

ωτ = pτ

pτ + (1 − pτ )Hτ

, (A9)

where

Hτ = Eτ

{[ατ eεno(τ,T ) + (1 − ατ )

]−γ }Eτ

{[ατ eεyes(τ,T ) + (1 − ατ )

]−γ }and pτ is the probability of a policy change as perceived just before time τ

(i.e., the probability that condition (49) holds). This probability is computednumerically.

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