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Agency and nominal frictions in financial markets Morales Veas, Gonzalo Andrés 2015

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Agency and Nominal Frictions in Financial MarketsbyGonzalo Andre´s Morales VeasB.Sc., Pontificia Universidad Cato´lica de Chile, 2005M.Sc., Pontificia Universidad Cato´lica de Chile, 2007A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Business Administration)The University of British Columbia(Vancouver)July 2015c© Gonzalo Andre´s Morales Veas, 2015AbstractThis dissertation examines the effects of nominal and agency frictions in three different environ-ments. The first essay builds a New Keynesian model to analyze the effects of changes in the ma-turity structure of nominal government debt on the real economy and inflation. The model includesnominal frictions, a time-varying maturity structure of nominal debt and allows for variations in theinteraction between the monetary and the fiscal authorities. This essay shows that when the slopeof the term structure of interest rates is nonzero in a fiscally-led policy regime the irrelevance ofopen market operations, changing the duration of government liabilities, is violated. Furthermore,maturity restructuring policies, conditional on the slope of the term structure of interest rates, cansmooth macroeconomic fluctuations and offer substantial welfare benefits.The second essay studies how agency frictions between spouses affect their consumption, assetallocation and marital decisions. To examine this nexus, a life cycle model with limited commitmentbetween spouses is built. The model is able to endogenously produce time-varying risk aversion atthe household-level through changes in the relative income between spouses that alter their relativebargaining power. Consistent with the data, changes in relative income are associated with signif-icant shifts in the portfolios of households. Also, the model can rationalize the empirical patternsrelating marital transitions to changes in portfolio allocations. Furthermore, the risk-sharing bene-fits of marriage in the model imply a positive link between wealth and risky asset holdings acrosshouseholds, which is observed in the data.The final essay presents a dynamic agency model to study the effects of firms’ exposure toaggregate risk on CEOs’ contracts. The model features a risk-averse representative shareholder,risk-averse managers that exert unobservable effort and firms that are heterogeneously exposed toaggregate risk. In the model, managers are incentivized by a mix of short- and long-term compen-sation, and the threat of being fired. The contract differs between firms with different exposure toaggregate risk. The model can explain salient features in the data; namely, the negative relationbetween aggregate risk and long-term compensation and the procyclicality of aggregate turnover.iiPrefaceThe research project in Chapter 2 was identified by Howard Kung and designed by AlexandreCorhay and by the author. The theoretical analysis was developed by Alexandre Corhay, HowardKung and the author. The numerical calculations and data analysis were performed by AlexandreCorhay and the author. A manuscript was written by Howard Kung in collaboration with AlexandreCorhay and the author.The research project in Chapter 3 was identified by Howard Kung and Jawad Addoum. Thetheoretical analysis was developed by Howard Kung and the author. The numerical calculationswere performed by the author. Data analysis was performed by Jawad Addoum. A manuscript waswritten by Howard Kung in collaboration with Jawad Addoum and the author.The research project in Chapter 4 was identified and designed by the author. The theoreticalanalysis, numerical calculations and data analysis were performed by the author. A manuscript waswritten by the author.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Government Maturity Structure Twists . . . . . . . . . . . . . . . . . . . . . . . . . 32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1.1 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Fixed Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.3 Regime-Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Limited Marital Commitment and Household Portfolios . . . . . . . . . . . . . . . . 333.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.2.1 Panel Study of Income Dynamics . . . . . . . . . . . . . . . . . . . . . . 363.2.2 Risky Portfolio Share Regressions . . . . . . . . . . . . . . . . . . . . . . 363.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37iv3.4 Model Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4.2 Implications for Portfolio Choice . . . . . . . . . . . . . . . . . . . . . . 413.4.3 Intra-Household Risk Sharing . . . . . . . . . . . . . . . . . . . . . . . . 424 Risk-Sensitive CEO Contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.2.1 CEO’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.2.2 Representative Shareholder’s Preferences . . . . . . . . . . . . . . . . . . 564.2.3 Representative Shareholder’s Problem . . . . . . . . . . . . . . . . . . . . 574.2.4 Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.2.5 Optimal Dynamic Contract . . . . . . . . . . . . . . . . . . . . . . . . . . 584.3 Empirical Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.3.2 Data Regressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3.3 Turnover Probability and Firm’s Exposure to Aggregate Risk . . . . . . . . 614.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.4.1 CEO’s Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624.4.2 Turnover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.4.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.4.4 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.1 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82A Appendix to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93A.1 Numerical Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93A.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95A.3 Household Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95A.4 Monopolistic Firm Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96A.5 Present Value of Surpluses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96B Appendix to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98B.1 Data Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98B.2 Numerical Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99B.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99vC Appendix to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100C.1 Data Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100C.1.1 Data Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100C.2 Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101C.2.1 Cash Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101C.2.2 Equity Portfolio Compensation . . . . . . . . . . . . . . . . . . . . . . . 102C.3 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104C.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105C.4.1 Grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107viList of TablesTable 2.1 Quarterly Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Table 2.2 Macroeconomic Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Table 2.3 Term Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Table 2.4 Forecasts with the Yield Spread . . . . . . . . . . . . . . . . . . . . . . . . . . 20Table 3.1 Bargaining Power, Marital Transitions, and Portfolio Share . . . . . . . . . . . 44Table 3.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Table 3.3 Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Table 3.4 Portfolios and Bargaining Power . . . . . . . . . . . . . . . . . . . . . . . . . 47Table 3.5 Portfolios and Marital Transitions . . . . . . . . . . . . . . . . . . . . . . . . . 48Table 3.6 Risk-Sharing within Households . . . . . . . . . . . . . . . . . . . . . . . . . 49Table 4.1 CEO Compensation and Risk Premia: Benchmark Specification . . . . . . . . 67Table 4.2 CEO Compensation and Risk Premia: Extended Specification I . . . . . . . . . 68Table 4.3 CEO Compensation and Risk Premia: Extended Specification II . . . . . . . . . 69Table 4.4 CEO Compensation and Risk Premia: Benchmark Specification, Factors Influ-ence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70Table 4.5 CEO Compensation and Risk Premia: Extended Specification I, Factors Influ-ence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Table 4.6 CEO Compensation and Risk Premia: Extended Specification II, Factors Influence 72Table 4.7 CEO Turnover and Risk Premia . . . . . . . . . . . . . . . . . . . . . . . . . 73Table 4.8 Monthly Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Table 4.9 Contract and Returns Moments . . . . . . . . . . . . . . . . . . . . . . . . . . 74Table 4.10 Comparison Data and Model: Compensation . . . . . . . . . . . . . . . . . . . 74Table 4.11 Comparison Data and Model: Turnover . . . . . . . . . . . . . . . . . . . . . 74Table 4.12 Comparative Statics: Contract . . . . . . . . . . . . . . . . . . . . . . . . . . 75Table 4.13 CEO Compensation and Risk Premia: Benchmark Specification (Robustness) . 76Table 4.14 CEO Compensation and Risk Premia: Extended Specification I (Robustness) . 77Table 4.15 CEO Compensation and Risk Premia: Extended Specification II (Robustness) . 78Table 4.16 CEO Turnover and Risk Premia (Robustness) . . . . . . . . . . . . . . . . . . 79viiTable C.1 Grid Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107viiiList of FiguresFigure 2.1 Average Maturity of Public Debt . . . . . . . . . . . . . . . . . . . . . . . . . 21Figure 2.2 Maturity Restructuring Shocks with Different Slopes . . . . . . . . . . . . . . 22Figure 2.3 Maturity Restructuring Shocks with High Uncertainty . . . . . . . . . . . . . 23Figure 2.4 Maturity Restructuring Shocks with High Debt . . . . . . . . . . . . . . . . . 24Figure 2.5 Maturity Restructuring Shocks During a Recession . . . . . . . . . . . . . . . 25Figure 2.6 Market Timing Restructuring . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 2.7 Surplus Shocks with Regime Shifts . . . . . . . . . . . . . . . . . . . . . . . 27Figure 2.8 Productivity Shocks with Regime Shifts . . . . . . . . . . . . . . . . . . . . . 28Figure 2.9 Maturity Restructuring with Regime Shifts . . . . . . . . . . . . . . . . . . . 29Figure 2.10 Maturity Restructuring with High Debt and Regime Shifts . . . . . . . . . . . 30Figure 2.11 Maturity Restructuring with Regime Shifts at the ZLB: Decrease in AverageMaturity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Figure 2.12 Maturity Restructuring with Regime Shifts at the ZLB: Increase in AverageMaturity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Figure 3.1 Policy Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Figure 3.2 Quintiles Correlation between Incomes . . . . . . . . . . . . . . . . . . . . . 51Figure 4.1 Timing of Events in a Period . . . . . . . . . . . . . . . . . . . . . . . . . . . 66ixAcknowledgmentsEver tried. Ever failed. No matter. Try again. Fail again. Fail better. — SamuelBeckett (1983)Do... or do not. There is no try. — YodaI would like to express my deepest gratitude to my two co-advisors, Ron Giammarino andHoward Kung. Their mentoring and inspiring discussions have helped me to grow as a person andresearcher.I am most grateful of Kai Li for all her help and constructive comments. I am thankful to theSauder School of Business faculty at UBC and in particular to Professors Jan Bena, Adlai Fisher,Lorenzo Garlappi, Russell Lundholm and Hernan Ortiz-Molina for their valuable suggestions.I would like to thank my friends and colleagues Pablo Moran, Jose Pizarro, Alberto Romeroand Ercos Valdivieso for their support.I thank the University of British Columbia and the Sauder School of Business for providingfinancial support to complete the PhD program.My parents, Williams and Maria Angelica and my big brother Juan Eduardo, never gave up onme. I deeply appreciate all their efforts. They raised me in a nurturing environment that fosteredmy love for knowledge. My parents-in-law, Julio and Valerie, have been a source of support andconstant encouragement.Finally, I would like to thank my wife Vero´nica for her infinite support, patience, love and foralways believing in me.xDedicationTo Vero´nicaxiChapter 1IntroductionTo understand the financial decisions of economic agents in the economy it is important to takeinto account economic frictions. Economic frictions are costs that do not allow the economy toefficiently adjust in the presence of disturbances. The work in the present thesis focuses in twoimportant types of frictions: nominal frictions and agency frictions. Nominal frictions correspondto the fact that firms only adjust prices infrequently (see Nakamura and Steinsson [2013]). Agencyfrictions correspond, among others, to imperfect coordination between economic agents with dif-ferent interests due to asymmetric information or limited commitment (see Bolton and Dewatripont[2005]).In this thesis the effects of nominal and agency frictions are studied in three different environ-ments. Chapter 2 examines the impact of government debt maturity restructuring on inflation andthe real economy using a model that features nominal frictions, a time-varying maturity structureof nominal debt and allows for changes in the monetary/fiscal policy mix. The irrelevance of openmarket operations changing the duration of government liabilities (holding the total market valueof debt constant) is violated when the slope of the yield curve is nonzero in a fiscally-led policyregime. When the yield curve is downward-sloping, shortening the maturity structure increases thegovernment discount rate, which generates fiscal inflation and an expansion in output. The oppo-site results are obtained when the yield curve is upward-sloping. Conditional maturity restructuringpolicies depending on the slope of the yield curve can smooth macroeconomic fluctuations and offersubstantial welfare benefits. In short, this chapter highlights the importance of bond risk premia, inconjunction with the government debt valuation equation, as a transmission channel for open marketoperations.Chapter 3 examines the link between marital decisions, consumption, and portfolio choice ina life cycle model with agency friction between the spouses due to limited marital commitment.Without commitment, income shocks to individuals alter relative bargaining power between spousesthrough renegotiation and endogenously generates time-varying risk aversion at the household-level.Consistent with the data, changes in relative income are associated with significant shifts in house-hold portfolios. The model can also rationalize the empirical patterns relating endogenous maritaltransitions to portfolio allocations. Also, the risk-sharing benefits of marriage imply a positive link1between wealth and risky asset holdings across households.Chapter 4 studies the compensation structure and firing policy of contracts for risk-averse man-agers implied by a dynamic agency model with sizeable risk premia. Due to the unobservabilityof effort and the dynamic contracting framework, managers are incentivized by a mix of short- andlong-term compensation, and the threat of being fired. Since firms are heterogeneously exposedto aggregate risk, the timing of compensation and termination policy differs between firms. Whenshareholders are averse to uncertainty about long-term growth prospects and the managers receivedisutility from effort, the model can explain the negative relation between aggregate risk and long-term compensation. Also, the model generates procyclical aggregate turnover. In short, this essayhighlights the importance of risk premia in understanding the dynamics of CEO contracts.2Chapter 2Government Maturity Structure Twists2.1 IntroductionDuring the global financial crisis, central banks, constrained by the zero lower bound (ZLB) on nom-inal interest rates, conducted open market operations on an unprecedented scale that significantlyaltered the maturity structure of government debt. For example, in the second round of QuantitativeEasing (QE) in 2010, the Federal Reserve announced the purchase of $600 billion of long-termTreasuries financed by selling an equivalent amount of short-term assets. The series of open mar-ket operations between 2008 and 2014 and the expansion in excess reserves reduced the averageduration of U.S. government liabilities by over 20% (from 4.6 years to 3.6 years).1 While there issome evidence for the effectiveness of these policies in flattening the yield curve in short-run,2 thelong-term effects on yields are uncertain.3 Further, the effects of these operations on the real econ-omy are even more controversial.4 This chapter contributes to this debate by demonstrating howbond risk premia, in conjunction with the equilibrium restrictions imposed by the government debtvaluation equation, provide a potentially important transmission channel for maturity restructuringpolicies to the macroeconomy.To quantitatively examine the impact of government debt restructuring policies, we construct aclosed New Keynesian model with several distinguishing features.5 First, households have recur-sive preferences (e.g., Epstein and Zin [1989]) which allows the model to generate realistic bondrisk premia (e.g., Piazzesi and Schneider [2007], Bansal and Shaliastovich [2013], Rudebusch andSwanson [2012], and Kung [2014]). Second, the supply of nominal government bonds over var-ious maturities is time-varying (e.g., Cochrane [2001], Vayanos and Vila [2009], Guibaud et al.1The Fed has been paying interest on reserves since 2008, so that reserves are effectively the same as short maturitytreasuries (i.e., Cochrane [2014]).2See, for example, Gagnon et al. [2011] Krishnamurthy and Vissing-Jorgensen [2011], Swanson [2011], DAmicoet al. [2012], Hamilton and Wu [2012], Joyce et al. [2012], Greenwood and Vayanos [2010], Greenwood and Vayanos[2014]3See, for example, the comments from Cochrane [2011a].4See Williams [2014] for an survey on the effects of QE.5The economy in the model is a closed economy. In the case of small open economies the fiscal theory of the pricelevel is important to explain several events (e.g., Daniel [2001], Daniel [2010], and Daniel and Shiamptanis [2012])3[2013], and Greenwood and Vayanos [2014]). Third, the monetary/fiscal policy mix are subject torecurrent stochastic changes between monetary- and fiscally-led regimes (e.g., Davig and Leeper[2007a], Davig and Leeper [2007b], Bianchi and Ilut [2013], and Bianchi and Melosi [2013]). Inthe monetary-led regime, the Taylor principle is satisfied and the monetary authority controls in-flation while the fiscal authority is committed to stabilizing the value of debt by adjusting primarysurpluses. In the fiscally-led regime, the fiscal authority determines the price level through thegovernment budget constraint while the monetary authority passively stabilizes debt and anchorsexpected inflation. Leeper [1991] refers to the monetary-led regime as Active Monetary/PassiveFiscal (AM/PF) and the fiscally-led regime as Passive Monetary/Active Fiscal (PM/AF). Lastly, wesolve the model using global projection methods to capture bond risk premia and account for theZLB constraint jointly with rational expectations in an extension of the benchmark model.In this framework, zero cost shocks to the maturity structure of nominal government debt affectinflation dynamics breaks Wallace [1981] neutrality when the nominal yield curve is nonzero andmonetary/fiscal policy is characterized by the PM/AF regime.6 In this policy mix, the fiscal authorityis not committed to adjusting surpluses to stabilize changes in the value of debt.7 For example, anincrease in the government discount rate reduces the market value of debt (without expectations ofhigher future taxes). Consequently, households reduce their debt holdings by increasing demandfor consumption goods, which translates into a rise in the price level (e.g., Cochrane [2011b]). Inparticular, the price level is determined by the ratio of nominal debt to the present value of surpluses,which is an equilibrium condition in this regime rather than a constraint that Cochrane [2005] refersto as the government debt valuation equation. Further, with sticky prices, restructuring policesimpact the real economy.We find that when the average yield curve is downward-sloping in the PM/AF regime, shorten-ing the maturity structure, while holding the total market value of debt fixed initially, in the PM/AFregime generates fiscal inflation and an expansion in output. Increasing the relative weight on shortterm debt when the yield curve is downward sloping raises the return on the government bond port-folio. A higher government discount rate implies that aggregate demand increases and the pricelevel increases via the government debt valuation equation. The effect on inflation is persistent dueto a positive supply of longer maturity debt, which spreads the rise in the price level over several pe-riods (Cochrane [2001]). With sticky prices, an increase in inflation expectations decreases the realrate and stimulates higher output. With a similar logic, shortening the maturity structure, when theaverage yield curve is upward-sloping, decreases the government discount rate, generates deflationand a contraction in output.When the nominal yield curve is flat (e.g., risk neutrality holds or an interest rate peg) in thePM/AF regime, maturity restructuring operations, holding market value constant, leave the govern-ment discount rate unchanged, and, therefore, inflation and real variables are unaffected. Further,when the yield curve is nonzero in the AM/PF (monetary-led regime) without policy regime shifts,6Given concerns of advanced economies in reaching their fiscal limits during the global recession, expectations ofentering a PM/AF regime appears to be a stronger possibility (e.g., Leeper [2013]).7See Leeper and Walker [2012] for a survey on monetary/fiscal policy interactions.4maturity restructuring affects government discount rates as in the PM/AF regime, which alters debtvalues. However, in the AM/PF regime, the fiscal authority is committed to adjusting future sur-pluses to absorb fluctuations in debt values, which leaves aggregate demand unchanged. Morebroadly, Ricardian equivalence holds in the AM/PF regime, so that households are insulated fromany fiscal disturbances. In short, the slope of the nominal yield curve, in conjunction with thePM/AF policy regime, plays a key role in determining the effects of debt maturity restructuring.To quantitatively evaluate the impact of the maturity restructuring shocks, it is important thatthe benchmark model generates a realistic term structure. The benchmark model produces sizeablebond risk premia through a similar mechanism as in Kung [2014] which generates countercyclicalreal marginal costs, and, therefore, a negative relation between expected consumption growth andinflation.8 With recursive preferences, these consumption and inflation dynamics lead to a positiveand sizeable average nominal term spread (e.g., Piazzesi and Schneider [2007] and Bansal andShaliastovich [2013]). The model can explain the level and persistence of nominal yields for variousmaturities. Also, the slope of the yield curve can forecast output growth and inflation.Even when the yield curve is upward-sloping, on average, shortening the maturity structure inthe PM/AF regime conditional on a temporarily downward-sloping yield curve has similar infla-tionary and stimulative effects as unconditionally reducing the maturity structure when the averageyield curve is downward-sloping. For example, high levels of nominal debt generate persistent fis-cal inflation to satisfy the government debt valuation equation. An increase in inflation leads themonetary authority to increase the short rate. A temporary increase in the short rate reduces theslope of the yield curve by the expectations hypothesis. If the increase in inflation is sharp enough,the yield curve slopes downward. Additionally, the onset of a deep recession (e.g., due to a very badproductivity shock) is associated with a downward sloping yield curve due to the negative inflation-growth link. Thus, in the PM/AF regime, shortening the maturity structure, in deep recessions orunder fiscal stress (high debt), can stimulate the economy and raise inflation expectations. Fur-ther, conditional maturity restructuring policies that shorten the maturity structure when the slopeis downward sloping and lengthen the maturity structure when the yield curve is upward slopingsmooth macroeconomic fluctuations and enhance welfare.When there are regime shifts, then fiscal disturbances are non-neutral in the AM/PF regimedue to expectations of possibly entering the PM/AF regime.9 In particular, shortening the maturitystructure has similar qualitative effects on inflation and macroeconomic quantities as in the PF/AMregime. Quantitatively, if regimes are very persistent (i.e., with small probability of switching), thenthe responses to fiscal shocks in the AM/PF regime are significantly weaker than in the PF/AMregime. Similarly, if the PM/AF regime is persistent, then, both quantitatively and qualitatively, theimpacts of maturity structuring shocks are similar to a fixed PM/AF regime. Bianchi and Ilut [2013]find that the AM/PF and PM/AF regimes are indeed very persistent via structural estimation.8Kung [2014] endogenizes the low-frequency component in productivity, which in the present model is assumed tobe exogenous.9In other words, in the long-run the Ricardian equivalence is not reestablished (see Davig and Leeper [2007a]), yetdebt is stable in the long-run (see Davig [2005]).5To examine the impact of maturity restructuring shocks in a liquidity trap, we augment thebenchmark model with preference shocks and a zero lower bound constraint on nominal interestrates (i.e., Eggertsson and Woodford [2003], Christiano et al. [2011], and Bianchi and Melosi[2013]). Further, we fully capture the nonlinear effects of the ZLB and rational expectations us-ing global projection methods (e.g., Ferna´ndez-Villaverde et al. [2012] and Aruoba and Schorfheide[2013]). To connect to the extant literature, we focus on the AM/PF regime for conducting the pol-icy experiments at the ZLB. We find that lengthening the maturity structure at the zero lower boundcreates inflationary pressure and stimulates output. A large preference shock drives the short termnominal rate to zero, which makes the yield curve steeper and upward-sloping. Lengthening thematurity structure therefore increases the government discount rate, and, in turn, creates inflationpressure and stimulates output. In contrast, shortening the maturity structure at the ZLB exacerbatesthe deflationary pressures and output losses from the liquidity trap.2.1.1 Related LiteratureThis chapter relates to the literature examining how the interactions between monetary and fiscalpolicy determine the price level. This literature begins with Sargent and Wallace [1981] who showthat permanent fiscal deficits have to eventually be financed by seignorage when the governmentonly issues real debt. Further, the money creation leads to inflation. Building on this paper, the fiscaltheory of the price level (FTPL) shows that when the government issues nominal debt and does notprovide the necessary fiscal backing, deficits are linked to current and expected inflation throughthe government debt valuation equation, without necessarily relying on seignorage revenues (e.g.,Leeper [1991], Sims [1994], Woodford [1994], Woodford [1995], Woodford [2001], Schmitt-Grohe´and Uribe [2000], Cochrane [1999], Cochrane [2001], and Cochrane [2005]).The current chapter connects to the literature examining the role of long-term public debt forpolicy. Angeletos [2002] and Buera and Nicolini [2004] solve for the optimal maturity structure ofgovernment debt and demonstrate how a portfolio of non-state-contingent debt of different matu-rities can replicate state-contingent debt. Leeper and Zhou [2013] highlight the importance of thematurity structure in determining optimal monetary/fiscal policy mix. The current chapter is mostclosely related to Cochrane [2001] in emphasizing the importance of the maturity structure for deter-mining inflation dynamics in the FTPL. We build on Cochrane [2001] by quantitatively examiningdebt maturity structure shocks on both inflation and the real economy in a New Keynesian frame-work featuring endogenous bond risk premia and stochastic shifts between the AM/PF and PM/AFregimes. A distinguishing feature of the current chapter is that it highlights the importance of thesign and magnitude of the yield curve slope for determining the effects of debt maturity twists.The Markov-switching Dynamic Stochastic General Equilibrium (DSGE) framework builds onDavig and Leeper [2007a], Davig and Leeper [2007b], Farmer et al. [2009], Bianchi and Ilut [2013],and Bianchi and Melosi [2013]. In particular, Bianchi and Ilut [2013] and Bianchi and Melosi [2013]also consider stochastic shifts between AM/PF and PM/AF policy regimes. However, our focus ison how the PM/AF policy regime (or expectations of entering the regime) propagates maturity6restructuring shocks.The current chapter relates to the theoretical literature studying the effects of unconventionalmonetary policy and transmission channels. Curdia and Woodford [2010], Gertler and Karadi[2011], and Arau´jo et al. [2013] analyze the role of financial frictions for central bank purchasesof risky assets. Correia et al. [2014] demonstrate how distortionary tax policy can deliver an eco-nomic stimulus when monetary policy is constrained at the zero lower bound. Gomes et al. [2013b]illustrate how incorporating nominal corporate debt in a DSGE framework provides an importantsource of monetary non-neutrality. The present work is most closely related to Chen et al. [2012]who also examine government debt maturity restructuring in a DSGE model. To break Wallaceneutrality, Chen et al. [2012] rely on market segmentation. In contrast, we introduce an alternativeand complementary channel that breaks Wallace neutrality by incorporating bond risk premia in afiscally-led policy regime.More broadly, this chapter relates to general equilibrium models that link policy to risk premia.Vayanos and Vila [2009], Guibaud et al. [2013], and Greenwood and Vayanos [2014] link bond sup-ply changes to the term structure of interest rates via investor clienteles with preferences for bondsof different maturities. In contrast, we link bond supply to interest rates through the intertemporalgovernment bond constraint (i.e., fiscal theory of the price level). More broadly, Rudebusch andSwanson [2008], Rudebusch and Swanson [2012], Palomino [2012], Dew-Becker [2014] Campbellet al. [2014], and Kung [2014] link asset prices to monetary policy. Croce et al. [2012], Pastor andVeronesi [2012], Gomes et al. [2013a], and Belo et al. [2013] look at fiscal policy and asset prices.This chapter looks at monetary/fiscal policy interactions, and focuses on how risk premia provide atransmission channel for open market operations.The rest of this chapter is structured as follows: Section 2.2 outlines the benchmark model.Section 2.3 explains the economic mechanisms and explores the quantitative implications of themodel.2.2 ModelThis section presents the benchmark model.2.2.1 HouseholdsThe representative household is assumed to have Epstein-Zin preferences over streams of consump-tion Ct and labor Lt :Ut ={(1−β )(C?t )1−1/ψ +β(Et[U1−γt+1]) 1θ} θ1−γC?t = Ct(L¯Lt)τ7where γ is the coefficient of risk aversion, ψ is the elasticity of intertemporal substitution, θ ≡1−γ1−1/psi is a parameter defined for convenience, β is the subjective discount rate, and L¯ is the agent’stime endowment. The time t budget constraint of the household isPtCt +Bt+1 =PtDt +WtLt +RtBt −Tt ,where Pt is the aggregate price level, Bt is the nominal market value of a portfolio of governmentbonds, Dt represents real dividends received from the intermediate firms, Rt is the gross nominalinterest rate on the bond portfolio, Wt is the nominal competitive wage, and Tt are lump sum taxesfrom the government. The household chooses sequences of Ct , Lt , and Bt to maximize lifetimeutility subject to the budget constraints.The household’s intertemporal condition isEt[Mt+1Πt+1Rt+1]= 1where Πt+1 is the inflation rate between t and t +1, andMt+1 = β(C?t+1C?t)1− 1ψ(Ct+1Ct)−1U1−γt+1Et[U1−γt+1]1− 1θis the real stochastic discount factor. The intratemporal labor condition is,WtPt=τCtLt.2.2.2 FirmsProduction in our economy is comprised of two sectors: the final goods sector and the intermediategoods sector.Final Goods A representative firm produces the final consumption goods Yt in a perfectly com-petitive market. The firm uses a continuum of differentiated intermediate goods Xit as input in aconstant elasticity of substitution (CES) production technology:Yt =(∫ 10(Xi,t)ν−1ν) νν−1where ν is the elasticity of substitution between intermediate goods. The profit maximization prob-lem of the final goods firm yields the following isoelastic demand schedule10 with price elasticity10See Appendix A for derivations.8ν :Xi,t = Yt(Pi,tPt)−νwhere Pt is the nominal price of the final goods and Pi,t is the nominal price of the intermediategoods i. The inverse demand schedule isPi,t =PtY1νt X−1νi,tIntermediate Goods The intermediate goods sector is characterized by a continuum of monopo-listic firms. Each intermediate goods firm produces Xi,t using labor Li,t :Xi,t = ZtLi,t −ΦZt ,where Zt represents an aggregate productivity shock common across firms, and is composed of bothtransitory and permanent components (e.g., Croce [2014] and Kung and Schmid [2014]):ln(Zt) = z?+at +ntat = ρaat−1 +σaεat∆nt = ρn∆nt−1 +σnεntwhere z? is the unconditional mean of log(Zt), ∆nt = nt − nt−1 is the permanent component, at isthe transitory component, εat and εnt are standard normal shocks with a contemporaneous correla-tion equal to ρan. The low-frequency component in productivity, ∆nt , is used to generate long-runrisks and sizeable risk premia (i.e., Bansal and Yaron [2004]).11 The fixed cost of production Φ ismultiplied by Zt to ensure that it does not become trivially small along the balanced growth path.Using the inverse demand function from the final goods sector, nominal revenues for intermedi-ate firm i can be expressed asPi,tXi,t = PtY1νt [ZtLi,t −ΦZt ]1− 1νThe intermediate firms face a cost of adjusting the nominal price a` la Rotemberg [1982], mea-sured in terms of the final good asG(Pi,t ,Pi,t−1;Pt ,Yt) =φR2(Pi,tΠssPi,t−1−1)2Ytwhere Πss ≥ 1 is the steady-state inflation rate and φR is the magnitude of the costs.11To better understand that nt+1 is a permanent shock, it can be re-written as nt+1 ≈ nt−K +∑Kj=0 σnεn,t+1− j +∑Kj=0∑ji=1 ρinσnεn,t+1− j. It is possible to see that the effects of the shocks εn,t+1− j are permanent.9The source of funds constraint isPtDi,t =Pi,tXi,t −WtLi,t −PtG(Pi,t ,Pi,t−1;Pt ,Yt)where Di,t is the real dividend paid by the firm. The objective of the firm is to maximize share-holder’s value V (i)t =V(i)(·) taking the pricing kernel Mt , the competitive nominal wage Wt , and thevector of aggregate state variables ϒt = (Pt ,Zt ,Yt) as given:V (i)t (Pi,t−1;ϒt) = maxPi,t ,Li,t{Di,t +Et[Mt+1 V(i) (Pi,t ;ϒt+1)]}subject to:Di,t =Pi,tPtXi,t −WtLt −G(Pi,t ,Pi,t−1;Pt ,Yt)Pi,tPt=(Xi,tYt)− 1νThe corresponding first order conditions are derived in Appendix A.Government The flow budget constraint of the government is given by:N∑i=1B(i)t+1 =N∑i=1R(i)t B(i)t −StwhereB(i)t+1 is the nominal debt of maturity i issued at the end of period t,R(i)t is the nominal interestpaid on debt of maturity i, St denotes the nominal value of primary surpluses. Following Bianchiand Melosi [2013], we assume that the government only levies lump-sum taxes and governmentexpenditures are excluded. Thus, the primary surplus equals lump-sum taxes. Denoting the totalmarket value of public debt by Bt and scaling the budget constraint by nominal output PtYt ,bt+1 =RgtΠt∆Ytbt − st (2.1)where bt+1 ≡Bt+1/(PtYt), st ≡St/(PtYt) and Rgt = ∑Ni=1 w(i)t R(i)t is the nominal gross interestpaid on the portfolio of government debt. The government issues nominal debt at N different matu-rities and we assume that each period the government retires outstanding debt and issues new debtover the N maturities. The proportion of the debt financed with bonds of maturity i is given by:w(i)t = w¯(i)+β (i)xmt , (2.2)where the constants w¯(i)’s determine the steady state maturity structure of debt and the β (i)’s deter-mine the sensitivity to xmt , a stochastic process that drives the dynamics of the maturity structure.10The evolution of xmt is given by:xmt = ρmxmt−1 +σmεmt , (2.3)subject to ∑Ni=1 w¯(i) = 1 and ∑Ni=1β(i)t = 0, ∀t. The latter condition ensures that the restructuringshocks do not change the total market value of debt initially so as to isolate the effects of change inmaturity.Monetary and Fiscal Rules The central bank follows an interest rate feedback rule:ln(R(1)tR(1))= ρr ln(R(1)t−1R(1))+(1−ρr)(ρpi,ρt ln(ΠtΠ)+ρy ln(ŶtŶ))+σrεrt , (2.4)where R(1)t+1 is the gross one-period nominal interest rate, Πt is inflation, Ŷt is detrended output, andεrt is a normal i.i.d. shock. Note that the coefficient ρpi,ρt is indexed by ρt , which determines thepolicy mix at time t.The fiscal authority adjusts primary surpluses according to the following rule:st − s = ρs(st−1− s)+(1−ρs)δb,ρt (bt −b)+σsεst .The coefficient δb,ρt is also indexed by ρt and therefore depends on the policy mix at time t.Monetary/Fiscal Policy Mix Leeper [1991] distinguishes four policy regions in a model with fixedpolicy parameters. Two of the parameter regions admit a unique bounded solution for inflation.One of the determinacy regions is the Active Monetary/Passive Fiscal (AM/PF) regime, which isthe familiar textbook case (e.g., Woodford [2003] and Galı´ [2008]). The Taylor principle is satisfied(ρpi > 1) and the fiscal authority adjusts taxes to stabilize debt (δb >(β∆Y 1−1ψ)−1− 1). In thispolicy mix, monetary policy determines inflation while fiscal policy passively provides the fiscal-backing to accommodate the inflation targeting objectives of the monetary authority.The other determinacy region is the Passive Monetary/Active Fiscal (PM/AF) regime. The fiscalauthority is not committed to stabilizing debt (δb <(β∆Y 1−1ψ)−1− 1), but instead the monetaryauthority passively accommodates fiscal policy (ρpi < 1) by allowing the price level to adjust (tosatisfy the government budget constraint). In this setting, fiscal policy determines inflation whilemonetary policy stabilizes debt and anchors expected inflation. Importantly, in this regime, fiscaldisturbances, including non-distortionary taxation, have a direct impact on the price level via thegovernment budget constraint because households know that changes in taxes will not be offset byfuture tax changes.12When both the fiscal and monetary authorities are active (AM/AF), no stationary equilibrium12In this regime, the government budget constraint is an equilibrium condition (rather than a constraint that has to holdfor any price path), which Cochrane [2005] refers to as the government debt valuation equation.11exists. When both authorities are passive, there exist multiple equilibria. Thus, in our regime-switching specification, we follow Bianchi and Melosi [2013] and assume that the policy mixalternates between AM/PF and PM/AF regimes according to a two-state Markov chain with thefollowing transition matrix:M =(pMM 1− pFF1− pMM pFF)where pi j ≡ Pr(ρt+1 = i|ρt = j) and M denotes the monetary-led (AM/PF) regime and F denotesthe fiscally-led (PM/AF) regime.2.3 ResultsThis section presents the results from the model. First, the calibration of the model is discussed andis followed by description of both the qualitative and quantitative implications.2.3.1 CalibrationTable 2.1 presents the quarterly calibration. Panel A reports the values for the preference parameters.The elasticity of intertemporal substitution ψ is set to 1.5 and the coefficient of relative risk aversionγ is set to 10.0, which are standard values in the long-run risks literature (e.g., Bansal and Yaron[2004]). The subjective discount factor β is calibrated to 0.9945 to be consistent with the averagereturn on the government bond portfolio (see Panel A of Table 2.2). The relative preference forleisure τ is set so that the household works one-third of the time in the steady-state.Panel B reports the calibration of the technological parameters. The price elasticity of demand νis set to 2. The fixed cost of production Φ is set such that dividend is zero in the deterministic steadystate. The price adjustment cost parameter φR is set to 10.13 The mean growth rate of productivityz? is set to obtain a mean growth rate of output of 2%. The parameters ρa and σa are set to beconsistent with the standard deviation and persistence of output growth, respectively (see Panel Bof Table 2.2). The parameters ρn and σn are set to match the standard deviation and persistence ofexpected productivity growth.For parsimony, we assume that shocks to the short-run and long-run components of productivityare perfectly correlated (εa,t = εn,t). Indeed, Kung and Schmid [2014] and Kung [2014] show that astochastic endogenous growth framework produces a very strong positive correlation between thesecomponents (i.e., around 0.98). Kung [2014] illustrates that these productivity dynamics help togenerate countercyclical real marginal costs, which implies a negative relation between inflation andexpected growth. Further, these inflation and growth dynamics imply an upward sloping nominalyield curve (see Table 2.3). Overall, the model does a good job in matching the level and persistenceof nominal yields. The volatility of yields falls a bit short of the empirical moments, however,Kung [2014] shows that incorporating conditional heteroscedastic productivity shocks helps to fit13For example, in a log-linear approximation, the parameter φR can be mapped directly to a parameter that governs theaverage price duration in a Calvo pricing framework. In this calibration, φR = 10 corresponds to an average price durationof 3.7 quarters, a standard value in the macroeconomics literature (e.g. Ferna´ndez-Villaverde et al. [2012]).12the second moments better. Further, the model can explain the joint dynamics between bond yieldsand real variables. Table 2.4 shows that the slope of the nominal yield curve can positively forecastconsumption and output growth while negatively forecast inflation as in the data. Generating arealistic term structure is important given that it plays a central role for the propagation of thematurity restructuring shocks.Panel C reports the calibration of the policy rule parameters. We set the steady state debt-to-GDP ratio to match the empirical average. The persistence and volatility parameters, ρs and σs,are chosen to match primary surplus dynamics. The surplus rule parameter, δb, is set to 0.05 and0.00 in the AM/PF and PM/AF regimes, respectively. The interest rate rule parameter, ρpi , is setto 1.5 and 0.4 in the AM/PF and PF/AM regimes, respectively. The calibration of these policyparameters, conditional on regime, are consistent with structural estimation evidence from Bianchiand Ilut [2013]. The persistence of the interest rate rule ρR is calibrated to 0.5. For parsimony, weabstract from monetary policy shock and output smoothing. Steady-state inflation Πss is calibratedto match the average level of inflation. Following Bianchi and Melosi [2013], we assume that thetransition matrix governing the dynamics of the policy/mix is symmetric: pMM = pFF ≡ p is set to0.9875, implying that the economy stays on average 20 years in a given regime.Panel D reports the calibration of the government bond supply dynamics, which are computedin the data following Greenwood and Vayanos [2014]. Fig. 2.1 plots the average maturity of netgovernment liabilities held by households from Q1:2005 to Q3:2013. Note that the three QE opera-tions and the Maturity Extension Program (MEP), show up quite visibly as each of these operationssignificantly shortened the maturity structure of debt.14. We calibrate the bond supply process tocapture salient features of maturity structure dynamics. We set N = 40, so that we include bondsup to a maturity of 10 years. The steady state maturity structure {w¯(i)} is set to match the sampleaverage. To calibrate the dynamics of the process driving the duration of government liabilities, xmt ,we proceed as follows. First, we run a principal component analysis on the panel data of maturitystructure. Next, we extract the first principal component (PC1) and fit the time series to an AR(1)process15. The estimates for ρm and σm are 0.9513 and 1.28%, respectively. The loadings {β (i)} forbonds of each maturity are also obtained from the first principal component.2.3.2 Fixed RegimesTo illustrate the economic mechanisms more clearly, we begin the analysis by first assuming fixedpolicy parameters (i.e., pMM = pFF = 1) before moving to the benchmark model with regime shifts.In the PM/AF regime, zero cost maturity structure shocks impact inflation when the slope of theyield curve is non-zero at the time of the shock. A nonzero slope implies that the changes in thecomposition of the government bond portfolio affect the portfolio return. Changes in the govern-ment discount rate, in turn, affect inflation through the government debt valuation equation, whichcan be seen by iterating on Eq. (2.1) and rewriting in present value form (see Appendix A for14Details on data construction are in Appendix A.15The first principal component explains about 62% of the cross-sectional dynamics of the debt maturity structure13derivations):bt = Et[∞∑i=0st+i∏ij=0Π−1t+ j∆Y−1t+ jRgt+ j], (2.5)and note that total debt bt is chosen at time t − 1. Importantly, Eq. (2.5) illustrates that any netchanges to current or expected surpluses, discount rates, growth rates are absorbed by inflation inthis regime. Since there is long-term debt outstanding, changes in inflation are spread out overmultiple periods, and the timing is determined by the relative proportion of bonds at each maturity(i.e., Cochrane [2001]). This inflation timing channel can be seen more clearly by rearrangingEq. (2.5):Et[∞∑i=0Mt,t+iYt+iYt−1st+i]=(b(1)tP(1)t−1)1Πt+(b(2)tP(2)t−1)Et[Mt+1ΠtΠt+1]+ ...+(b(n)tP(n)t−1)Et[∏n−1j=1 Mt+ j∏n−1j=0 Πt+ j](2.6)where Mt,t+i is the discount factor between t and t + i and P(i)t is the price at time t of a nominalbond of maturity i. Note that a shorter maturity structure implies less inflation smoothing.Fig. 2.2 plots impulse response functions to a positive maturity restructuring shock (εmt > 0),which shortens the maturity structure, when the average nominal yield curve is upward-sloping(solid line), flat (thin line with squares), and downward-sloping (dashed line). It should be recalledthat the maturity structure variable, xmt , follows an independent stochastic process. Thus, whenthe average yield curve is upward-sloping (the benchmark case), shortening the maturity structure,on average, decreases the government discount rate. A lower discount rate drives down the pricelevel and inflation through the government debt valuation equation. The intuition is that a lowerdiscount rate raises the value of debt, so households increase demand for bonds and reduce demandfor consumption goods. A fall in aggregate demand for consumption goods drives down prices andinflation. When prices are flexible, the prices will fall such that the households will be satisfied withtheir original consumption plan. However, when prices are sticky, the price adjustment is sluggishso that prices are too high relative to the flexible price case. Consequently, production falls and thereal rate rises. Also, a fall in inflation lowers the short nominal rate (FFR) due to the interest raterule, which increases the slope of the yield curve.When the average slope of the yield curve is downward-sloping the responses to a positive ma-turity restructuring shock are reversed from the upward-sloping case. Shortening the maturity in thiscase raises the government discount rate, which lowers the value of debt. Consequently, householdswant to unload their bond positions by increasing demand for consumption goods. Higher aggregatedemand creates inflationary pressure. The presence of sticky prices implies that prices are too lowrelative to flexible prices, so production increases and the real rate falls. A rise in inflation increasesthe short nominal rate, which decreases the slope of the yield curve.14When the average yield curve is flat, zero cost maturity restructuring shocks have no averageeffect on inflation or real economic variables.16 Changes in the maturity structure in this case donot affect the return on the government bond portfolio. Also, the maturity restructuring shocks areneutral in the AM/PF regime, even when the slope is nonzero. In this regime, any changes to thepresent value of surpluses (or equivalently to the value of debt) are offset by adjustments in futuretaxes. For example, if the government discount rate falls, the value of debt increases and the fiscalauthority increases taxes to stabilize debt.Fig. 2.3 plots the responses to debt restructuring for high (solid line) and low (dashed line)uncertainty cases in the benchmark model. Higher uncertainty raises risk premia and steepens theslope of the yield curve. A steeper slope implies that shortening the maturity structure reduces thegovernment discount rate more.Even when the average nominal yield curve is upward-sloping, conditional maturity restruc-turing can produce similar responses to unconditional restructuring shocks when the average yieldcurve is downward-sloping. Fig. 2.4 plots impulse response functions to a positive maturity struc-ture shock for high debt (dashed line) and low debt (solid line) from the benchmark model withan upward-sloping average yield curve. In the PM/AF regime, high debt (without offsetting futuretaxes) makes agents feel wealthier. This wealth effect drives up aggregate demand and creates in-flationary pressure. Higher inflation increases the short nominal rate, which lowers the slope of theyield curve. If the level of debt is sufficiently high, the yield curve is downward sloping. Thus,shortening the maturity structure when debt is high (e.g., during fiscal stress) increases the discountrate of the government. An increase in the discount rate generates more inflation and an expansionin output, as with unconditional restructuring shocks when the average yield curve is downward-sloping.As in the case with high debt, at the onset of a large recession, the nominal yield curve isdownward-sloping. In the model, the negative inflation-growth link implies that low expectedgrowth is associated with a rise in inflation. An increase in inflation leads to a rise in the shortterm nominal rate, and the yield curve is downward-sloping if the recession is large enough. Thus,shortening the maturity structure when expected growth is inflationary and expansionary. Fig. 2.5plots the impulse response functions to a positive maturity shock at the onset of a recession (dashedline) and at the steady-state (solid line). In short, shortening the maturity structure can be an effec-tive tool in stimulating the economy during large recessions and in times of fiscal stress.Motivated by the conditional restructuring examples relating to high debt and low expectedgrowth, we also consider a maturity restructuring policy that depends directly on the slope of thenominal yield curve:xm,t = ρmxm,t−1 +ρm,ys(y5Yt − y1Qt)+σmεm,t (2.7)16A flat average yield curve can be obtained, for example, by assuming that agents are risk neutral. For the flat yieldcurve case, we assume a nominal interest peg which implies a flat and constant term structure. Thus, restructuring shocksare always neutral (and not only average).15A positive coefficient (ρm,ys > 0) implies that the government shortens the maturity structure whenyield curve is upward-sloping and lengthens the maturity structure when the yield curve is downward-sloping. A negative coefficient implies the opposite policy. Fig. 2.6 plots comparative statics forvarying ρm,ys from -1 to 1. Note that more negative values of ρm,ys smooth macroeconomic fluctu-ations, reduce risk premia, and improve welfare. In contrast, more positive values of ρm,ys increaseconsumption and inflation volatility, and, in turn, increase welfare costs. Negative values of ρm,ysshorten the maturity structure when the yield curve is downward sloping, which stimulates the econ-omy and generates fiscal inflation exactly during low expected growth states. Using similar logic,positive values for ρm,ys deepen recessions and increase deflationary pressure.2.3.3 Regime-SwitchingWith stochastic and recurrent policy shifts, fiscal shocks are no longer neutral in the AM/PF regimedue to the possibility of entering the PM/AF regime. Fig. 2.7 plots impulse response functionto a positive surplus shock (εst > 0) conditional on being in the AM/PF (solid line) and PM/AF(dashed line) regimes. Note that the surplus shock is less persistent in the AM/PF regime due to thedependence of the surplus rule on debt that smoothes the shock. In PF/AM regime, a positive surplusshock (without the expectation of lower future taxes), makes agents feel poorer. This negative wealtheffect makes agents demand less consumption goods, which lowers prices. Due to sticky prices, theprice level is too high relative to the flexible price case, so production falls and the real rate rises.A fall in inflation also lowers the short-term nominal rate, which steepens the slope of the yieldcurve. The responses to the surplus shock in the AM/PF regime are qualitatively similar due toagents expectations of possibly entering in the PM/AF. However, since regimes are persistent andthe probability of switching is small, quantitatively, the responses are significantly smaller.As agents are less insulated from fiscal shocks in the PM/AF regime, macroeconomic volatilityis higher than in the AM/PF regime (see Panel A in Table 2.2). Since the fiscal shocks induce pos-itive correlation between inflation and macroeconomic quantities, the negative correlation (inducedby the productivity shocks) between inflation and consumption growth is weaker in the PM/AFregime. Fig. 2.8 plots impulse response functions to a positive productivity shock. A weaker corre-lation reduces bond risk premia (see Panel B in Table 2.2).Fig. 2.9 plots impulse response functions for a positive maturity shock in the AM/PF (solid line)and PM/AF (dashed line) regimes. The responses from the PM/AF with regime shifts are quitesimilar to the fixed regime cases given that the regimes are persistent. Although the responses aresomewhat smaller, the magnitudes are quantitatively significant. Shortening the maturity structureby 0.18 years (as in QE2), reduces output by just less than one percentage point. The responses inthe AM/PF regime are qualitatively similar to those in the PM/AF regime, but significantly smaller.Fig. 2.10 illustrates that shortening the maturity structure conditional on high debt in the PM/AFregime stimulates output as in the fixed regime case. The debt-to-GDP in the high debt case (dashedline) is set to be 50% higher than in the steady-state case (solid line), as in the onset of the financialcrisis in 2008 and reducing the maturity structure by 0.18 years increases output by around 1%.16To address the efficacy of maturity restructuring policies in a liquidity trap, we augment thebenchmark model with regime shifts to include preference shocks and a zero lower bound (ZLB)constraint on the short-term nominal rate. To capture preference shocks, the time discount factor ofthe agent is assumed to follow:ln(βt) = (1−ρβ ) ln(β )+ρβ ln(βt−1)+σβ εβ . (2.8)The ZLB constraint is given byln(R(1)tR(1))= max{0,ρr ln(R(1)t−1R(1))+(1−ρr)(ρpi,ρt ln(ΠtΠ)+ρy ln(ŶtŶ))+σrεrt}. (2.9)We focus on the AM/PF regime to relate to the literature on the zero lower bound and also becausethe ZLB rarely binds in the PM/AF regime.17 Fig. 2.11 plots impulse response functions to anegative maturity restructuring shock at the ZLB (dashed line) and away from the ZLB (solid line).Away from the ZLB (i.e., in steady state), the average yield curve is upward-sloping, so a maturityrestructuring shock reduces inflation and output as discussed above. To reach the ZLB, we assumea large preference shock that drives nominal rates to zero, which binds, on average, for around fourquarters. This shock steepens the slope of the yield curve, which amplifies the responses relative tobeing away from the ZLB, and further exacerbates the problems in a liquidity trapTo alleviate deflationary pressures and output losses associated with being at ZLB, requires alengthening of the maturity structure rather than a shortening of it. Fig. 2.12 plots impulse responsefunctions to a negative maturity restructuring shock at the ZLB (dashed line) and away from theZLB (solid line). At the ZLB, lengthening the maturity structure by 0.18 years raises expectedinflation by 4 basis points and output by over 20 basis points at the peak of the responses.17In the PM/AF regime, the interest rule is less sensitive to changes in inflation, so it is less likely that the deflationaryshock sends the short rate to the ZLB.17Table 2.1: Quarterly CalibrationParameter Description ModelA. Preferencesβ Subjective discount factor 0.9945ψ Elasticity of intertemporal substitution 1.5γ Risk aversion 10B. Productionν Price elasticity for intermediate goods 2φR Magnitude of price adjustment costs 10z? Unconditional mean growth rate 0.5%ρa Persistence of at 0.95σa Volatility of transitory shock eat 1.25%ρn Persistence of ∆nt 0.99σn Volatility of permanent shock ent 0.015%C. Policyρs Persistence of government surpluses 0.975σs Volatility of government surpluses est 0.0005%δb(M/F) Sensitivity of taxes to debt 0.05 / 0.0ρr Degree of monetary policy inertia 0.5 / 0.5ρpi (M/F) Sensitivity of interest rate to inflation 1.5 / 0.4p Switching probability 0.9875D. Bond Supplyb¯ Steady state Debt-to-GDP ratio 0.5ρm Persistence of xmt 0.958σm Volatility of emt 1.28%This table reports the parameter values used in the quarterly calibration of the model. The table is divided into fourcategories: Preferences, Production, Policy, and Bond Supply parameters.18Table 2.2: Macroeconomic MomentsData Model AM/PF PM/AFA. MeansE(∆y) (in %) 2.0 2.0 2.0 2.0E(y(20)− y(1))(in %) 1.02 1.01 1.88 0.14E(rg)(in %) 5.47 5.81 6.18 5.45B. Standard Deviationsσ(∆y) (in %) 2.22 2.53 2.34 2.72σ(pi) 1.64 1.06 0.36 1.40σ(∆l)/σ(∆y) 0.92 0.47 0.13 0.63σ(y(20)− y(1))(in %) 1.05 0.55 0.26 0.37σ(rg)(in %) 4.53 0.93 0.92 0.91C. Correlationscorr(pi,∆c) -0.56 -0.10 -0.17 -0.09This table presents the means, and standard deviations for key macroeconomic variables from the data and the model.The model is calibrated at a quarterly frequency and the reported statistics are annualized.Table 2.3: Term StructureMaturity1Q 1Y 2Y 3Y 4Y 5Y 5Y - 1QNominal yieldsMean (Model) (in %) 5.15 5.47 5.74 5.93 6.05 6.16 1.00Mean (Data) (in %) 5.03 5.29 5.48 5.66 5.80 5.89 1.02Std (Model) (in %) 0.51 0.47 0.42 0.42 0.44 0.48 0.55Std (Data) (in %) 2.97 2.96 2.91 2.83 2.78 2.72 1.05AC1 (Model) 0.96 0.91 0.92 0.94 0.96 0.97 0.96AC1 (Data) 0.93 0.94 0.95 0.95 0.96 0.96 0.74This table presents summary statistics for the term structure of interest rates: the annual mean, standard deviation, andfirst autocorrelation of the one-quarter, one-year, two-year, three-year, four-year, and five-year nominal yields and the5-year and one-quarter spread from the model and the data. The model is calibrated at a quarterly frequency and themoments are annualized.19Table 2.4: Forecasts with the Yield SpreadData ModelHorizon (in quarters)1 4 8 1 4 8A. Outputβ 1.023 0.987 0.750 0.723 0.492 0.310S.E. 0.306 0.249 0.189 0.257 0.219 0.196R2 0.067 0.148 0.147 0.025 0.057 0.060B. Consumptionβ 0.731 0.567 0.373 0.700 0.474 0.298S.E. 0.187 0.163 0.153 0.253 0.215 0.193R2 0.092 0.136 0.088 0.025 0.055 0.058C. Inflationβ -1.328 -1.030 -0.649 -0.977 -0.680 -0.412S.E. 0.227 0.315 0.330 0.118 0.171 0.181R2 0.180 0.157 0.071 0.276 0.175 0.094This table presents output growth, consumption growth, and inflation forecasts for horizons of one, four, and eightquarters using the five-year nominal yield spread from the data and the model. The n-quarter regressions, 1n (xt,t+1 +· · ·+ xt+n−1,t+n) = α+β (y(5)t − y(1Q))+ εt+1, are estimated using overlapping quarterly data and Newey-West standarderrors are used to correct for heteroscedasticity.20Figure 2.1: Average Maturity of Public Debt2006 2007 2008 2009 2010 2011 2012 20133.43.63.844.24.44.64.855.25.4Time (years)Average Maturity (years)QE1↓QE2↓MEP↓QE3↓This figure plots the average maturity structure of government held by the public from Q1-2005 to Q3-2013.21Figure 2.2: Maturity Restructuring Shocks with Different Slopes0 5 10 15 20−0.2−0.10Avg MS   slope > 0slope < 0slope = 00 5 10 15 20−0.200.2E[rg] 0 5 10 15 20−0.500.5E[pi] 0 5 10 15 20−0.100.1yield spread0 5 10 15 20−0.200.2FFR0 5 10 15 20−0.200.21Y−yield0 5 10 15 20−0.200.2r quarters0 5 10 15 20−1012yquartersThis figure plots impulse response functions of the expected government cost of capital, expected inflation, the 5 years to1 quarter nominal yield spread, the federal fed fund rate, the one-year nominal yield, the real interest rate, and output to adecrease in the average maturity of government debt of the size of QE2 (emt ) in the PM/AF regime. Results are reportedfor three shapes of the unconditional term structure: upward sloping (solid line), flat (squares), and downward sloping(dashed line). The units of the y-axis are annualized percentage deviations from the steady-state, except for the averageduration that is in years.22Figure 2.3: Maturity Restructuring Shocks with High Uncertainty0 5 10 15 20−0.2−0.10Avg MS   high uncertaintylow uncertainty0 5 10 15 20−0.200.2E[rg] 0 5 10 15 20−0.500.5E[pi] 0 5 10 15 20−0.100.1yield spread0 5 10 15 20−0.200.2FFR0 5 10 15 20−0.100.11Y−yield0 5 10 15 20−0.200.2r quarters0 5 10 15 20−1−0.500.5yquartersThis figure plots impulse response functions of the expected government cost of capital, expected inflation, the 5 years to1 quarter nominal yield spread, the federal fed fund rate, the one-year nominal yield, the real interest rate, and output to adecrease in the average maturity of government debt of the size of QE2 (emt ) in the PM/AF regime. Results are reportedfor the benchmark calibration (solid line), and when all shock volatilites, except for emt , are cut in half (dashed line). Theunits of the y-axis are annualized percentage deviations from the steady-state, except for the average duration that is inyears.23Figure 2.4: Maturity Restructuring Shocks with High Debt0 5 10 15 20−0.2−0.10Avg MS   low Bhigh B0 5 10 15 20−0.500.5E[rg] 0 5 10 15 20−0.500.51E[pi] 0 5 10 15 20−0.200.2yield spread0 5 10 15 20−0.500.5FFR0 5 10 15 20−0.500.51Y−yield0 5 10 15 20−0.500.5r quarters0 5 10 15 20−2024yquartersThis figure compares the impulse response functions of the expected government cost of capital, expected inflation, the5 years to 1 quarter nominal yield spread, the federal fed fund rate, the one-year nominal yield, the real interest rate, andoutput to a decrease in the average maturity of government debt of the size of QE2 (emt ) in the PM/AF regime whendebt-to-GDP is at the steady state (solid line) or higher than the steady state (dashed line). The units of the y-axis areannualized percentage deviations from the steady-state, except for the average duration that is in years.24Figure 2.5: Maturity Restructuring Shocks During a Recession0 5 10 15 20−0.2−0.10Avg MS   benchmarkrecession0 5 10 15 20−0.200.2E[rg] 0 5 10 15 20−0.500.5E[pi] 0 5 10 15 20−0.100.1yield spread0 5 10 15 20−0.200.2FFR0 5 10 15 20−0.100.11Y−yield0 5 10 15 20−0.200.2r quarters0 5 10 15 20−101yquartersThis figure reports the impulse response functions of the expected government cost of capital, expected inflation, the 5years to 1 quarter nominal yield spread, the federal fed fund rate, the one-year nominal yield, the real interest rate, andoutput to a decrease in the average maturity of government debt of the size of QE2 (emt ) in the PM/AF regime. Theresponse in a recession (dashed) is obtained by shocking the economy with a negative productivity shock (eat ) at the timeof the restructuring. The units of the y-axis are annualized percentage deviations from the steady-state, except for theaverage duration that is in years.25Figure 2.6: Market Timing Restructuring−1 −0.5 0 0.5 1−0.01−0.00500.0050.010.0150.020.025Welfare cost−1 −0.5 0 0.5 10.811.21.41.61.8Yield spread−1 −0.5 0 0.5 10.620.630.640.650.660.670.68std[E[cg]]−1 −0.5 0 0.5 11.081.11.121.141.161.181.2std[pi]This figure plots the welfare cost, the yield spread (annualized percentage), the standard deviation of consumption growthand the standard deviation of inflation for various debt restructuring policies. The y-axis is normalized to the benchmarkcase of exogenous debt management, i.e. ρm,yldsprd = 0 (differenced for first moments and divided by the standarddeviation for second moments)26Figure 2.7: Surplus Shocks with Regime Shifts0 5 10 15 20246 x 10−4S  AM/PFPM/AF0 5 10 15 20−10−505 x 10−3E[rg] 0 5 10 15 20−0.03−0.02−0.010E[pi] 0 5 10 15 20−0.01−0.00501Y−yield0 5 10 15 20−0.01−0.0050FFR0 5 10 15 200246 x 10−3yield spread0 5 10 15 20−0.0100.010.02r quarters0 5 10 15 20−0.100.1yquartersThis figure plots impulse response functions of the expected government cost of capital, expected inflation, the 5 years to1 quarter nominal yield spread, the federal fed fund rate, the one-year nominal yield, the real interest rate, and output to apositive one standard deviation surplus shock (est ), when the economy is initially in the active monetary (solid line) andfiscal (dashed line) regime. The units of the y-axis are annualized percentage deviations from the steady-state, except forthe surplus that is in levels.27Figure 2.8: Productivity Shocks with Regime Shifts0 5 10 15 200246a  AM/PFPM/AF0 5 10 15 20−0.500.5E[rg] 0 5 10 15 20−1−0.500.5E[pi] 0 5 10 15 20−0.500.51Y−yield0 5 10 15 20−0.4−0.20FFR0 5 10 15 2000.10.2yield spread0 5 10 15 20−0.500.51r quarters0 5 10 15 200246yquartersThis figure plots impulse response functions of the expected government cost of capital, expected inflation, the 5 years to1 quarter nominal yield spread, the federal fed fund rate, the one-year nominal yield, the real interest rate, and output to apositive one standard deviation technology shock (eat ), when the economy is initially in the active monetary (solid line)and fiscal (dashed line) regime. The units of the y-axis are annualized percentage deviations from the steady-state.28Figure 2.9: Maturity Restructuring with Regime Shifts0 5 10 15 20−0.2−0.10Avg MS   AM/PFPM/AF0 5 10 15 20−0.100.1E[rg] 0 5 10 15 20−0.200.2E[pi] 0 5 10 15 2000.050.1yield spread0 5 10 15 20−0.04−0.0200.02FFR0 5 10 15 2000.050.11Y−yield0 5 10 15 2000.050.1r quarters0 5 10 15 20−1−0.50yquartersThis figure reports the conditional impulse response functions of the expected government cost of capital, expectedinflation, the 5 years to 1 quarter nominal yield spread, the federal fed fund rate, the one-year nominal yield, the realinterest rate, and output to a decrease in the average maturity of government debt of the size of QE2 (emt ) in the AM/PFregime (solid line) and PM/AF regime (dashed line). The units of the y-axis are annualized percentage deviations fromthe steady-state, except for the average duration that is in years.29Figure 2.10: Maturity Restructuring with High Debt and Regime Shifts0 5 10 15 20−0.2−0.10Avg MS   low Bhigh B0 5 10 15 20−0.500.51E[rg] 0 5 10 15 20−0.500.5E[pi] 0 5 10 15 20−0.100.1yield spread0 5 10 15 20−0.100.1FFR0 5 10 15 2000.20.41Y−yield0 5 10 15 20−0.200.2r quarters0 5 10 15 20−1012yquartersThis figure reports the conditional impulse response functions of the expected government cost of capital, expectedinflation, the 5 years to 1 quarter nominal yield spread, the federal fed fund rate, the one-year nominal yield, the realinterest rate, and output to an increase in the average maturity of government debt of the size of QE2 (emt ) when theeconomy is initially in the PM/AF regime. The units of the y-axis are annualized percentage deviations from the steady-state, except for the average duration that is in years.30Figure 2.11: Maturity Restructuring with Regime Shifts at the ZLB: Decrease in Average Ma-turity0 5 10 15 20−0.2−0.10Avg MS   away ZLBat ZLB0 5 10 15 20−0.04−0.03−0.02−0.01E[rg] 0 5 10 15 20−0.06−0.04−0.020E[pi] 0 5 10 15 200.020.040.060.08yield spread0 5 10 15 200510FFR0 5 10 15 200.010.020.030.041Y−yield0 5 10 15 2000.010.020.03r quarters0 5 10 15 20−0.4−0.20yquartersThis figure reports the conditional impulse response functions of the expected government cost of capital, expectedinflation, the 5 years to 1 quarter nominal yield spread, the federal fed fund rate, the one-year nominal yield, the realinterest rate, and output to a decrease in the average maturity of government debt of the size of QE2 (emt ) when theeconomy starts initially in the AM/PF regime. The dashed line represents the response at the Zero Lower Bound and thesolid line the response away from the Zero Lower Bound. The units of the y-axis are annualized percentage deviationsfrom the steady-state, except for the average duration that is in years and the FFR that is in levels.31Figure 2.12: Maturity Restructuring with Regime Shifts at the ZLB: Increase in Average Ma-turity0 5 10 15 2000.10.2Avg MS   away ZLBat ZLB0 5 10 15 200.010.020.030.04E[rg] 0 5 10 15 2000.020.040.06E[pi] 0 5 10 15 20−0.08−0.06−0.04−0.02yield spread0 5 10 15 200510FFR0 5 10 15 20−0.04−0.03−0.02−0.011Y−yield0 5 10 15 20−0.03−0.02−0.010r quarters0 5 10 15 2000.20.4yquartersThis figure reports the conditional impulse response functions of the expected government cost of capital, expectedinflation, the 5 years to 1 quarter nominal yield spread, the federal fed fund rate, the one-year nominal yield, the realinterest rate, and output to an increase in the average maturity of government debt of the size of QE2 (emt ) when theeconomy starts initially in the AM/PF regime. The dashed line represents the response at the Zero Lower Bound and thesolid line the response away from the Zero Lower Bound. The units of the y-axis are annualized percentage deviationsfrom the steady-state, except for the average duration that is in years and the FFR that is in levels.32Chapter 3Limited Marital Commitment andHousehold Portfolios3.1 IntroductionThe household portfolio choice literature typically assumes that households act as single agents.An extensive literature in labor economics shows that accounting for intra-household interactionsin married households is important for explaining consumption, fertility, children’s nutrition, andlabor decisions.1 Also, consumption Euler equation tests provide empirical support for collectivehousehold models where marital decisions are endogenous (e.g., Mazzocco [2007] and Mazzocco[2008b]). This chapter studies the portfolio allocation of multiple-member households when house-holds’ members earn non-tradable labor income, they jointly make the investment decision and theycannot commit to future plans. More specifically, the members of the household are married andhave limited commitment to future plans due to divorce.The model embeds a life cycle portfolio choice model (e.g., Cocco et al. [2005] and Gomes andMichaelides [2005]) into a limited intra-household commitment framework (e.g., Mazzocco et al.[2013]). Households make marital status, consumption, and portfolio choice decisions (between ariskfree and risky asset). Single individuals meet potential spouses and have the option to marry.Married couples choose household allocations cooperatively but can divorce if both spouses arebetter off as singles. Due to the lack of commitment, income shocks to individuals alter relativebargaining power between spouses through renegotiation.In the model individuals get married because marriage has risk-sharing benefits since eachspouse earns non-tradable labour income. Thus, as long as the incomes are not perfectly corre-lated, marriage provides mutual insurance for smoothing individual consumption. This property ofmarriage is known as the marital surplus (gains from marriage). The cost of marriage is that eachspouse does not have the portfolio allocation and consumption path that he/she would have if he/she1For example, see Browning and Chiappori [1998], Rasul [2008], Thomas [1990], Gray [1998], and Gallipoli andTurner [2013] for evidence relating to household expenditures, fertility decisions, family health and nutrition, laborsupply, and retirement, respectively.33had all the bargaining power. The cost of marriage fluctuates with the relative bargaining power.When the bargaining power is high the portfolio allocation and consumption path reflect his/herpreferences more strongly. On the contrary, when the bargaining power is low the cost is highersince the portfolio allocation and consumption path marginally reflect his/her preferences. In themodel divorce occurs when marriage can no longer generate a positive marital surplus. This happensbecause the outside option for each of the spouses includes the possibility of getting married againafter the divorce. In other words the gains from marriage are not lost forever after a divorce, sinceeach of the spouses can marry again under better conditions (i.e. better bargaining power). Themodel allows for a renegotiation and this happens when only one of the spouses wants the divorceat the beginning of the renegotiation process. The renegotiation step is unsuccessful (i.e. divorce) ifafter the renegotiation the spouse who wanted to stay married is left worse off than his/her outsideoption.When the risk aversions of the spouses are different, renegotiation generates time-varying riskaversion at the household level. For example, suppose that the wife is more risk averse than the hus-band. If the wife receives a positive income shock, this increases her outside option (i.e., divorce)and therefore increases her bargaining power in the marriage if renegotiation is successful. Thehousehold-level preferences will then reflect her preferences more strongly and the effective house-hold risk aversion increases. Furthermore, an increase in risk aversion reduces demand for riskyassets. If renegotiation is unsuccessful and the couple divorces, then the wife’s risky asset demandfalls relative to the married portfolio while the husband’s demand increases. A similar logic followsfor when the wife and husband go from single to married.There is significant empirical and experimental evidence supporting the conclusion that, onaverage, women are more risk averse than men.2 This preference assumption is made along withassuming that individuals have recursive preferences (i.e., Epstein and Zin [1989]) to separate theeffects of risk aversion from the intertemporal elasticity of substitution (IES). The model predictsthat (i) an increase in the bargaining power of females (males) decreases (increases) risky assetholdings for the household, (ii) after marriage, risky asset holdings in the married portfolio increase(decrease) relative to the female’s (male’s) single portfolio (iii) after divorce, risky asset holdings inthe female’s (male’s) single portfolio decrease (increase) relative to the married portfolio.The model predictions are supported in household data from the Panel Study of Income Dynam-ics (PSID). Although bargaining power cannot be observed directly, the model implies that relativeincome is a key determinant of intra-household bargaining power. Longitudinal regressions fromthe data indicate that increases in the wife’s share of income are highly negatively correlated withhousehold-level allocations to risky assets, controlling for a variety of household characteristics.Marital transitions are associated with dichotomous changes in the portfolios of men and womenas predicted by the model.3 When calibrated to match key empirical moments, such as average2For example, Powell and Ansic [1997] provide experimental evidence, Barsky et al. [1997] use survey-based evi-dence, Barber and Odean [2001] show strong gender-based differences in investment preferences, and Mazzocco [2008a]provides evidence from Euler equation estimates.3In a companion paper, Addoum [2013] documents similar effects in the data during retirement. Love [2010] docu-34risky asset holdings, income dynamics, and divorce rates, the model can quantitatively rationalizethe empirical regressions.Wachter and Yogo [2010] document a positive relationship between wealth and risky asset hold-ings across households. The risk-sharing benefits of marriage in the present model provide a po-tential explanation for this stylized fact. Couples whose incomes are less correlated provide bettermutual insurance for smoothing individual consumption. With reduced background risks, marriedhouseholds increase their position in risky assets which raises expected returns and average wealth.Single households are less wealthy (due to only a single income source) and lack the mutual in-surance mechanism, so they invest less in risky assets than married households. This risk-sharingchannel is empirically supported.This paper relates to household portfolio choice models with nontradable labor income (e.g.,Bodie et al. [1992], Heaton and Lucas [2000], Viceira [2001], Campbell and Viceira [2002], Coccoet al. [2005], and Gomes and Michaelides [2005]). These papers assume a unitary household, andthe model here departs from this assumption to analyze the relationship between intra-householddynamics and portfolio decisions. This paper also highlights a new channel for which labor incomerisk influences risky asset demands. Idiosyncratic shocks to labor income reallocate bargainingpower within married households through renegotiation, which alters the effective risk aversion ofthe household. Love [2010] examines how exogenous marital transitions impact portfolio deci-sions in a life cycle model. This paper endogenizes the marital transitions and the link to portfolioallocations. Furthermore, the focus of this paper is to highlight the importance of changes in intra-household bargaining power on portfolio choice.This paper also relates to models of limited commitment (e.g., Thomas and Worrall [1988],Kocherlakota [1996], Ligon et al. [2002], Mazzocco [2007], Gallipoli and Turner [2011], Maz-zocco et al. [2013]). These papers show that in contracts with limited commitment, bargainingpower changes with the value of outside options. In the present model, this channel generates en-dogenous time-varying risk aversion at the household level. The theoretical setting is most closelyrelated to Mazzocco et al. [2013] who also analyzes a collective household model with equilibriummarriage and divorce. This paper extends this literature to consider portfolio choice decisions. At abroader level, this paper provides additional theoretical and empirical support for models of limitedcommitment by considering household financial decisions.The paper is organized as follows. Section 3.2 presents the empirical relations between house-hold bargaining, marital transitions, and portfolio choice. Section 3.3 describes the life cycle model.Section 3.4 presents the quantitative model results.ments the effects from marital transitions in a model where divorce and marriage are exogenous.353.2 Empirical Evidence3.2.1 Panel Study of Income DynamicsThe data is from the Panel Study of Income Dynamics (PSID), a nationally representative longi-tudinal survey of nearly 9,000 U.S. families.4 The main variables of interest are those concernedwith household financial asset holdings. These include holdings in stocks, bonds, and cash, aswell as primary residential equity, the value of private business interests, equity in vehicles, andnon-primary real estate. The data also makes available a host of demographic and socioeconomicmeasures, including age, education, marital status, labor income, and total income. See Appendix Bfor additional details regarding use of the data and variable definitions. Table 3.2 presents summarystatistics for single and married households.3.2.2 Risky Portfolio Share RegressionsTo examine the relation between household portfolios and intra-household bargaining power, threedifferent measures of female’s bargaining power are constructed following the labor economicsliterature.5 These measures are the female’s income share, female’s non-wage income share, andfemale’s relative hourly wage rate (to the male’s). To examine the effect of variations in bargainingpower on households’ allocations to risky assets, consider regressions of the following form:αi,t = χi +φMi,t +ΓXi,t + εi,t , (3.1)where αi,t is family i’s allocation to risky assets at time t, Mi,t is the female’s relative bargainingpower, φ is the coefficient of interest on Mt , and Γ is a vector of coefficients on a set of controlvariables, Xi,t , for household i at time t. χi is a household-specific constant term, since the abovefunction with household-level fixed effects is estimated in order to control for unobserved hetero-geneity.Columns (1) to (3) in Table 3.1 present the results of running this regression using each ofthe proposed measures of bargaining power. Column (1) uses the wives’ total income share. Thecoefficient φ is negative with a point estimate of -0.18 and is statistically significant at the 5% level.Interpretation of this estimate indicates that, holding the full set of control variables constant, ahousehold within the estimation sample in which the female controls all non-wage income will havean equity portfolio allocation that is, on average, 18 percentage points lower than a household inwhich the male controls all non-wage income. Columns (2) and (3) use female’s non-wage incomeand relative hourly wage, respectively, and similar results are obtained.In similar spirit as Love [2010], Column (4) of Table 3.1 examines the link between maritaltransitions and risky portfolio shares. Consider regressions following males in the sample through4The collection of PSID data used in this study was partly supported by the National Institutes of Health under grantnumber R01 HD069609 and the National Science Foundation under award number 1157698.5See, for example, McElroy and Horney [1981], Thomas [1990], Browning et al. [1994], and Pollak [2005].36time:αi,t = χi +δ Marriedi,t +ΓXi,t + εi,t , (3.2)where αi,t is individual i’s allocation to risky assets at time t, δ is the coefficient of interest on theindicator variable Married (equal to one when individual i is married, and zero otherwise), and Γ isa vector of coefficients on a set of control variables, Xi,t , for individual i at time t. χi is an individual-specific constant term, since the above function with individual-level fixed effects in order to controlfor unobserved heterogeneity.The point estimate for δ is -0.067, and is statistically significant at the 1% level. Interpretationof the estimate indicates that, holding the full set of control variables constant, the average malein the estimation sample tilts his portfolio allocation toward stocks by almost 7 percentage pointswhen he is single relative to when he is married.3.3 ModelTime is discrete and each period t corresponds to a year. There are two types of individuals, male (m)and female ( f ). Following the convention in the literature, each type lives for 81 periods, startingfrom age 20. Each individual enters a period as either single or married. Following Mazzocco et al.[2013], if single, the individual draws a potential spouse of the opposite type, and the randomlymatched pair decides to marry or stay single. If married, the couple chooses to stay married ordivorce. Divorce entails a fixed cost κ . Also, a constant fraction x ∈ [0,1] of the couple’s wealth isallocated to the wife.Labor income process Labor income for an individual of type i ∈ {m, f}, Y it , is exogenously spec-ified as:log(Y it ) ={(1−ρy)µ iy(t)+ρy log(Y it−1)+σyε iy,t , if t ≤ 65, (3.3)λ × yi65 if t > 65.where ε i ∼ N(0,1) is independently and identically distributed (iid). Following the literature (e.g.,Cocco et al. [2005]), the trend of the income profile is a deterministic function of age, for agesbelow 66. For ages above 65, the income process is a constant fraction, λ , of the income receivedjust before retirement. The correlation between Y mt and Yft is given by ρm, f .Financial assets Households can invest in a risk-free bond with a constant gross return R f > 1 anda risky asset with a random gross return Rt . The law of motion for the risky asset is:log(Rt) = (1−ρr)µr +ρr log(Rt−1)+σrεr,t . (3.4)37where εr,t ∼N(0,1) is iid.6 The correlation between labor income (of either type) and the risky assetis given by ρr,y. Define αt as the risky asset portfolio weight. The gross portfolio return is given by:Rp,t ≡ αtRt +(1−αt)R f (3.5)Households face borrowing and short-sales constraints as in Cocco et al. [2005] and Gomes andMichaelides [2005] (i.e., αt ∈ [0,1]).Single household’s problem Consider the scenario when household enters period t as single. Indi-viduals have Epstein-Zin preferences defined over nondurable consumption Cit . These preferencesseparate the effects of heterogeneity in risk aversion across types from the IES. The value of beingsingle at period t for an individual of type i ∈ {m, f} is given by the following program:V 0,it = max{Cit ,α it }(1−β i)(Cit)1− 1ψi +β i(Et[(V it+1)1−γ i])1− 1ψi1−γi11− 1ψi(3.6)subject to the wealth accumulation equationW it+1 = Rip,t+1(W it +Yit −Cit), (3.7)where β i is the discount factor, γ i is the coefficient of relative risk aversion, and ψ i is the IES.Married household’s problem Consider the scenario when household enters period t married. Fol-lowing the limited commitment literature assume that married households solve a Pareto efficientproblem, which has a recursive representation.7 The married household’s problem is computed intwo steps. First, the value of being married is computed without taking into account the participa-tion constraint. Second, it is verified if the individual participation constraints are satisfied. Also,define Mt as the relative bargaining power of the female in the marriage.The value of being married ismax{Cmt ,Cft ,αt}V 1,mt +MtV1, fts.t.Wt+1 = Rp,t+1(Wt +Ymt +Yft −Cmt −Cft). (3.8)Define Ĉmt , Ĉft , and α̂t to be the optimal values from the optimization problem above. Then, the6The process for the risky asset allows autocorrelation in the returns. However, in practice this autocorrelation is smalland negligible since the risky asset is calibrated to the market portfolio.7See, for example, Thomas and Worrall [1988].38value of being married for individual i isV 1,it =(1−β i)(Ĉit)1− 1ψi +β i(Et[(V it+1)1−γ i])1− 1ψi1−γi11− 1ψi. (3.9)Next, verify the participation constraints:V 1,it ≥V0,it , i ∈ {m, f} (3.10)There are three cases to consider. First, if the constraints are satisfied for both individuals, then thecouple stays married. Second, if both constraints are violated, then the couple divorces.8 Third, ifonly one constraint is violated, say for individual m, then there is renegotiation. As shown in Ligonet al. [2002], it is optimal to allocate resources so that m is indifferent between being married andbeing single. If the renegotiated bargaining power and allocations are such that the participationconstraint for f is satisfied then the agents stay married. Otherwise the couple gets divorced. Thisthird case can be formalized in the following program:max{Cmt ,Cft ,αt ,Mt}V 1,mt +MtV1, ft (3.11)subject toWt+1 = Rp,t+1(Wt +Ymt +Yft −Cmt −Cft), (3.12)V 0,mt (St) = Um(Cmt )+βmEt[V mt+1(St+1)]. (3.13)Define ̂̂Cmt ,̂̂Cft ,̂̂α t , and ̂̂Mt as the optimal values from the program above. The value of stayingmarried for f after renegotiation isV 1, ft =(1−β f )(̂̂Cft )1− 1ψ f +β f(Et[(V ft+1)1−γ f])1− 1ψ f1−γ f11− 1ψ f(3.14)If the constraint for f is not satisfied, they divorce. After renegotiation, m is indifferent betweenbeing married and being single:V 1, ft =V0, ft . (3.15)8As explained in Mazzocco et al. [2013] both constraints can bind simultaneously since the outside options includethe value of reentering into matrimony, after divorce, under better conditions in the future. In other words, the gainsfrom marriage, in some cases, are not enough to replicate the outside options of the spouses since these options includethe probability of obtaining the risk-sharing benefits of marriage in the future, making divorce unavoidable under certainstates of the economy. In contrast, in Ligon et al. [2002] the outside option precludes the possibility of re-entering to therisk-sharing contract. Thus, making it possible to replicate the outside option alternative inside the risk-sharing contract.39The value of individual i at time t isV it (St) = max{V 1,it ,V0,it}. (3.16)3.4 Model Results3.4.1 CalibrationTable 3.2 presents the benchmark calibration. Panel A reports the values for the preference parame-ters. The discount factor is set to 0.97, similar to the value in Cocco et al. [2005]. The intertemporalelasticity of substitution parameters, ψm and ψ f , are both set to 1.1.9The coefficient of relative riskaversion for males γm and females γ f are set to 6.0 and 10.0, respectively, to match the averagerisky asset allocation by gender (reported in Panel A of Table 3.3). There is significant evidencesupporting that women, on average, are more risk averse than men. For example, Powell and Ansic[1997] provide experimental evidence, Barsky et al. [1997] use survey-based evidence, and Barberand Odean [2001] show strong gender-based differences in investment preferences. Also, the ra-tio γ f /γm = 1.67 is consistent with the estimated value from Mazzocco [2008a]. More generally,the preference configuration is within the range of values used in the long-run risks literature (e.g.,Bansal and Yaron [2004]).Panel B reports the parameters relating to the income process that are common across bothtypes. The parameters ρy and σy are set to match the median persistence and volatility in income,respectively, across all individuals. The fraction of income that individuals receive during retirementλ is set to the value in Cocco et al. [2005]. The average correlation between income processes ρm, fis calibrated to the value in the data. The average correlation between the income process and therisky asset ρy,r is set to 0.2, which is consistent with the empirical findings from Davis and Willen[2000]. The divorce cost κ is calibrated to be consistent with the mean divorce rate. The fraction ofhousehold wealth allocated to the female x is set to value in Mazzocco et al. [2013].Panels C and D report the male and female income process parameters, respectively, to accountfor heterogeneity across genders. The parameter values are taken from Love [2010], who fits thelog income processes, by gender, to a third-order polynomial. The use of a third-order polynomialfollows Cocco et al. [2005] to capture non-linearities in the life cycle income profile.Panel E reports the calibration of the parameters relating to financial assets. Following Coccoet al. [2005], the gross riskfree is set to 1.02 and the mean of the risky asset µr is set to 0.06.The parameters σr and ρr is set to match the volatility and persistence of the stock market return,respectively.10Table 3.3 reports key summary statistics from the model and data. A description of the numericalsolution and simulation are in Appendix B.9The IES parameters are kept the same to isolate the effects of heterogeneity in risk aversion.10The autocorrelation of the stock market return is not crucial to obtain the results in the model. The calibration showsthat the value is very small (0.05). The results do not change if the autocorrelation is set to zero.403.4.2 Implications for Portfolio ChoiceThe relation between portfolio choice, household bargaining, and marital transitions is exploredhere. In the model, an increase in the relative income of the female raises her outside option (i.e.,divorce and becoming single) and hence, given a successful renegotiation, increases her bargainingpower within the marriage. Since females have higher risk aversion than males, a higher bargainingweight for the female raises the effective risk aversion of the household. Higher risk aversiondecreases demand for risky assets. Fig. 3.1 shows policy functions from the model that illustratethis mechanism. Panel A depicts the positive relation between the relative income of the femaleand her bargaining power across various ages. Panel B depicts the negative relation between thehouseholds risky asset holdings and female bargaining power.Columns (1) and (2) of Table 3.4 run regressions of risky asset shares for married couples onfemale bargaining power (measured as the female’s relative income share). Panel A reports censoredregressions of the following form:αi,t = φMi,t +∑jγ jAge j,i,t + εi,t ,where αi,t is the family i’s allocation to risky assets at time t, Mi,t is the female’s relative bargainingpower, φ is the coefficient of interest on Mi,t , and {γ j}Jj=1 is a set of coefficients on a collectionof dummy variables for different age groups {Age j,i,t}Jj=1, for household i at time t. The omittedcategory is households whose head is married and his/her age is between 46 and 55 years. Theestimate of the coefficient φ is negative and of similar magnitude between the data and model (-0.22in the data and -0.19 in the model), which supports the key model mechanism.11 The interpretationof these estimates indicates that, holding the full set of control variables constant, a household inwhich the female controls all income will have an equity portfolio allocation that is, on average, 22(19) percentage points lower in the data (model) than a household in which the male controls allincome. Panel B reports the results of estimating Eq. 3.1. The point estimates for φ are similar withvalues of -0.11 for the model and -0.18 for the data.Next, the link between marital transitions and portfolio choice is examined. In the model, mar-riage occurs when for a randomly matched pair, expected lifetime utility is higher for both indi-viduals as married than as single. From the perspective of the male, effective household-level riskaversion is higher during marriage (which also reflects the female’s preferences) than being single.Hence, after marriage, risky asset demand decreases for the male relative to being single. Divorcearises if both individuals are better off as singles than being married or if only one individual isbetter off as single and renegotiation is unsuccessful. By a similar logic as above for marriage, riskyasset demand increases for the male relative to being married. The converse result holds for femalesduring marital transitions.Columns (1) and (2) of Table 3.5 run regressions of risky asset shares on a married indicator for11For the model, the regressions are calculated for ages between 20-65, because after retirement the relative income isconstant and does not affect the outside option of the spouses.41the male (1 if male is married and 0 otherwise). Column (3) runs regressions of risky asset shareson a married indicator for the female, and the coefficient on the married indicator is positive. Notethat the regression for the female individuals is unavailable for the data because in the PSID, withonly few exceptions, the male is defined as the head of the family unit. This rule makes it difficultto track female’s asset allocation in and out of marriage. Panel A reports censored regressions ofthe following form:αi,t = δ Marriedi,t +∑jγ jAge j,i,t + εi,t ,where αi,t is individual i’s allocation to risky assets at time t, δ is the coefficient of interest on theindicator variable Marriedi,t , and {γ j}Jj=1 is a set of coefficients on a collection of dummy variablesfor different age groups,{Age j,i,t}Jj=1, for individual i at time t. The omitted category is householdswhose head is 46-55 years old. The estimate of the coefficient δ for males is negative and of similarmagnitude between the data and the model (-0.02 in the data and -0.03 in the model). Interpretationof these estimates indicates that, holding the full set of control variables constant, the average malein the data (model) increases his portfolio allocation on stocks by 2 (3) percentage points when heis single relative to when he is married. The model’s result indicates that the average female willdecrease her portfolio allocation on stocks by 18 percentage points when she is single relative towhen she is married. Panel B reports the results of estimating Eq. 3.2. The estimates for male aresimilar with point estimates of -0.04 for the model and -0.07 for the data. The model’s result forfemale is positive with a point estimate of 0.03.3.4.3 Intra-Household Risk SharingThe risk sharing implications of marriage are explored in this section. When spouses’ incomestreams are not strongly correlated, marriage offers substantial benefits in hedging nontradable in-come risk (relative to being single). In the model, the hedging benefits of marriage provide aneconomic incentive for singles to get married. Further, by smoothing household-level income fluc-tuations, this channel can significantly influence risky asset demands.Fig. 3.2 plots average spouses’ income correlations sorted in quintiles from the data. There isconsiderable variation in the correlations, ranging from around -0.5 in the bottom quintile to 0.7 inthe top quintile. Motivated by these statistics and to explore the magnitude of the risk sharing bene-fits, an extension of the model with heterogeneity in income correlations is considered. Specifically,half of the male-female pairs are assumed to have an income correlation of 0.0 and the other half isassumed to have a correlation of 0.2.12Columns (1) and (2) of Table 3.6 run censored regressions of married household portfolio shareson spouses’ income correlations and a set of dummy variables for different age groups. The coeffi-cient on the income correlation is negative (-0.05 in the data and -0.14 in the model). In the model,12For the model the regressions are calculate for ages between 20-55, because in the model the risk-sharing benefits ofmarriage decrease with age. After retirement the income processes of the spouses are constant and marriage no longergenerates risk-sharing benefits. The disappearance of the benefits affect the households’ behavior before retirement. Forexample, a 60 year old couple have only 5 year of benefits but they have 35 years of life after retirement.42lower income correlations provide a better hedge and reduce background risks, which increasesrisky asset holdings. Also, a larger position in risky assets increases the expected return of the port-folio, and with an IES greater than one, this increases the average wealth of the household. Indeed,regressions of wealth on income correlation indicate a negative link both in the model and the data.Putting these two results together implies a positive relation between wealth and risky asset holdingsfor married households. Moreover, single households are less wealthy (due to only a single incomesource) and lack the mutual insurance mechanism, so they invest less in risky assets than marriedhouseholds. Thus, overall the model generates a positive link between wealth and risky asset al-location, as documented empirically in Wachter and Yogo [2010]. Columns (5) and (6) report theregression results for wealth and risky asset shares for the model and the data.43Table 3.1: Bargaining Power, Marital Transitions, and Portfolio ShareRisky portfolio share(1) (2) (3) (4)M -0.18 -0.14 -0.25 -(0.07) (0.06) (0.08)Married Indicator - - - -0.067(0.025)Income -7.10 0.04 -0.47 0.000(4.44) (0.23) (0.29) (0.001)Wealth 4.28 0.15 0.20 -0.011(2.03) (0.12) (0.13) (0.001)Child Indicator 5.18 3.53 4.40 0.019(4.66) (4.55) (4.58) (0.016)# Children -2.26 -1.94 -2.90 -0.002(2.35) (2.17) (2.33) (0.006)Age2 (head) -3.75 -2.55 -4.94 0.005(2.56) (1.92) (2.43) (0.002)Education (head) 9.72 4.20 11.72 0.029(4.11) (5.71) (4.21) (0.013)Age2 (wife) 3.68 2.84 3.26 -(2.18) (1.81) (2.20)Education (wife) 1.55 1.88 1.95 -(1.73) (2.07) (1.69)Fixed Effects Y Y Y YN 887 793 908 4,835R2 0.13 0.13 0.15 0.19This table presents the results of specifications with male head-of-household fixed effects, regressing the allocation toequity on different measures of bargaining power, αi,t = χi+φMi,t +ΓXi,t +εi,t (regressions 1 to 3), or marriage indicator,αi,t = χi+δ Marriedi,t +ΓXi,t +εi,t (regression 4). For the first specification bargaining power (M) is defined as the wife’sshare of total income. For the second specification bargaining power is the wife’s share of non-wage income. Finally, forthe third specification bargaining power is defined as the difference in hourly wages between wives and husbands. Forthe fourth regression the marriage indicator is set to one when the male is married, and zero otherwise (single, separated,divorced, or widowed). Regressions 1 to 3 contain controls for family labor income, wealth (net worth), an indicatorfor having children, a measure of the number of children in the household, quadratic age of the head and spouse, andtheir education levels in years. The fourth regression controls for family labor income, wealth (net worth), an indicatorfor having children, a measure of the number of children in the household, quadratic age of the household head, and theeducation level of the household head in years. Standard errors are heteroskedasticity robust and clustered by head-of-household.44Table 3.2: CalibrationParameter Description ModelA. Preferencesβ Subjective discount factor 0.97ψm,ψ f Intertemporal elasticity of substitution 1.1γm Risk aversion male 6.0γ f Risk aversion female 10.0B. Income Process (General)ρy Persistence of income process 0.50σy Volatility of income process 0.20λ Fraction received during retirement 0.68ρm, f Correlation between income types 0.18ρy,r Correlation between income and risky asset 0.20κ Fixed divorce cost 0.03x Fraction of wealth to female 0.50C. Income Process (Male)am Coefficient fitted polynomial order 0 -2.075b1,m Coefficient fitted polynomial order 1 0.163b2,m×102 Coefficient fitted polynomial order 2 -0.290b3,m×104 Coefficient fitted polynomial order 3 0.163D. Income Process (Female)a f Coefficient fitted polynomial order 0 -1.859b1, f Coefficient fitted polynomial order 1 0.163b2, f ×102 Coefficient fitted polynomial order 2 -0.303b3, f ×104 Coefficient fitted polynomial order 3 0.180E. Financial assetsR f Gross return risk-free asset 1.02µr Mean of risky asset 0.06ρr Persistence of risky asset 0.05σr Volatility of risky asset 0.19This table reports the parameter values used in the calibration of the model. The table is divided into six categories:preferences, general income process, male income process, female income process, and financial assets.45Table 3.3: Summary StatisticsData ModelA. Means (single)Risky asset weight, male 58.59% 88.73%Risky asset weight, female 55.74% 75.42%Income, male (in ’000s) 30.23 19.14Income, female (in ’000s) 22.99 16.00B. Standard deviations (single)Log income, male 0.32 0.21Log income, female 0.31 0.24E. Means (married)Divorce rate 1.58% 2.26%Risky asset weight 55.67% 81.95%Income, male (in ’000s) 35.69 22.85Income, female (in ’000s) 20.29 14.62F. Standard deviations (married)Log income, male 0.30 0.22Log income, female 0.35 0.24This table presents summary statistics for key variables by gender for single and married households.46Table 3.4: Portfolios and Bargaining PowerPortfolio ShareData Model(1) (2)A.M -0.22 -0.19(0.05)Age:25 -0.71 0.14(0.05)26-35 -0.29 0.10(0.03)36-45 -0.15 0.06(0.03)56-65 0.06 -0.60(0.04)B.All Ages -0.18 -0.11(0.07)This table presents the comparison between the data and the model for the results of regressing the risky asset share onbargaining power. Panel A reports the results of estimating a censored regression for the portfolio share on bargainingpower and a set of dummy variables for different age groups: αi,t = φMi,t +∑ j γ jAge j,i,t + εi,t . The omitted category ishouseholds whose head is married and his/her age is between 46 and 55 years. Panel B reports the results of regressingrisky asset share on bargaining power and a set of controls: αi,t = χi + φMi,t +ΓXi,t + εi,t ,. For the panel A the dataregression includes an age dummy for 66-75 age group. For panel B both specifications contain controls for family laborincome, age, quadratic age, and wealth (net worth). Data regression includes male head-of-household fixed effects. Also,they include controls for the age of the spouse, quadratic age of the spouse, the spouses education levels in years, anindicator for having children in the household, and a control for the number of such children. For the data regression thestandard errors are reported in parentheses. The standard errors are heteroskedasticity robust and clustered by head-of-household.47Table 3.5: Portfolios and Marital TransitionsMarital StatusData (Male) Model (Male) Model (Female)(1) (2) (3)A.Married -0.02 -0.03 0.18(0.007)25 -0.11 0.13 0.13(0.02)26-35 -0.06 0.09 0.09(0.01)36-45 -0.03 0.05 0.05(0.01)56-65 0.004 -0.57 -0.63(0.01)B.All Ages -0.07 -0.04 0.03(0.03)This table presents the comparison between the data and the model for the results of regressing the portfolio share of therisky asset on a marital status indicator. The indicator is 1 if the head is married and 0 otherwise. Panel A reports theresults of estimating a censored regression for the portfolio share on a marriage indicator and a set of dummy variablesfor different age groups: αi,t = δ Marriedi,t +∑ j γ jAge j,i,t + εi,t . The omitted category is households whose head is46-55 years old. Panel B reports the results of regressing risky asset share on a marriage indicator and a set of controls:αi,t = χi + δMarriedi,t + ΓXi,t + εi,t ,. For panel A the data regression includes an age dummy for 66-75 age group.For panel B both specifications contain controls for family labor income, quadratic age, and wealth (net worth). Dataregression includes male head-of-household fixed effects. Also, they include controls for the education level in years, anindicator for having children in the household, and a control for the number of such children. For the data regression thestandard errors are reported in parentheses. The standard errors are heteroskedasticity robust and clustered by head-of-household.48Table 3.6: Risk-Sharing within HouseholdsPortfolio Share Wealth WealthData Model Data Model Data Model(1) (2) (3) (4) (5) (6)ρm, f -0.05 -0.14 -0.07 -2.92 - -(0.02) (0.03)Port. Share - - - - 0.01 0.13(0.00)25 -0.63 0.08 -2.31 -0.40 -2.15 -0.41(0.05) (0.07) (0.07)26-35 -0.29 0.02 -1.35 -1.17 -1.22 -1.17(0.02) (0.03) (0.03)36-45 -0.14 0.02 -0.56 -0.66 -0.48 -0.66(0.02) (0.03) (0.03)This table presents the comparison between the data and the model for 3 different regressions. Columns 1 and 2 displaythe results of estimating a censored regression for the portfolio share of the risky asset on the correlation between theincomes of the spouses and a set of dummy variables for different age groups: αi,t = φ [ρm, f ]i,t +∑ j γ jAge j,i,t + εi,t .Columns 3 and 4 display the estimation results of regressing the household wealth on the correlation between the incomesof the spouses and a set of dummy variables for different age groups: Wealthi,t = φ [ρm, f ]i,t +∑ j γ jAge j,i,t +εi,t . Finally,columns 5 and 6 display the estimation results of regressing the wealth of the household on the risky asset share and aset of dummy variables for different age groups: Wealthi,t = ϕαi,t +∑ j γ jAge j,i,t + εi,t . For columns 1 to 4 the omittedcategory is households whose head is married and his/her age is between 46 and 55 years. For columns 5 and 6 theomitted category is households whose head is 46-55 years old. The data regression includes age dummies for 56-65and 66-75 age groups. For the data regressions the standard errors are reported in parentheses. The standard errors areheteroskedasticity robust and clustered by head-of-household.49Figure 3.1: Policy Functions(a) Bargaining Power (b) Portfolio Share(c) Relative ConsumptionThis figure shows three policy functions, conditional on being married. Sub-figure A displays the bargaining power asa function of age and relative income (Y f /Y m). The wealth level, the risky asset return, and the relative level of themale’s income are fixed. Sub-figure B displays the portfolio share as a function of age and bargaining power. The wealthlevel, the risky asset return, and the relative level of the spouses’ incomes are fixed . Sub-figure C displays the relativeconsumption (C f /Cm) as a function of age and bargaining power. The wealth level, the risky asset return, and the relativelevel of the spouses’ incomes are fixed.50Figure 3.2: Quintiles Correlation between IncomesThis figure shows the quintiles for the correlation between the income processes of the spouses in the data.51Chapter 4Risk-Sensitive CEO Contracts4.1 IntroductionIn the last decades a vigorous debate has risen among shareholders, stakeholders, regulators, andnews media. They have questioned whether the high level of compensation can be justified as thereward needed to incentivize the managers, or instead the compensation corresponds to rent ex-traction by the managers. The empirical literature has found (Bertrand and Mullainathan [2001],Garvey and Milbourn [2006]) that their remunerations increase with positive events that are out oftheir control (e.g. positive aggregate shocks). Furthermore, Garvey and Milbourn [2003], and Jin[2002] find a negative effect of aggregate risk on stock-based compensation for young managersand for managers with short-selling constraints, respectively. Also, Bushman et al. [2010] find thatforced turnover is negatively related with firms’ exposure to aggregate risk. This evidence is diffi-cult to rationalize in a static principal-agent model, which shows that exogenous shocks should befiltered out while analyzing the remuneration of executives (Ho¨lmstrom [1979]). The current chap-ter proposes a dynamic agency model to rationalize remuneration and turnover, and their relationwith risk premia.To study the effects of risk premia on executive compensation and turnover I develop a dy-namic moral hazard model with several key features. First, the economy is subject to aggregateshocks, which are observable for all agents in the economy. Second, firms are heterogeneous intheir exposure to aggregate risk. Third, the representative shareholder is risk-averse and sensitiveto uncertainty about long-term growth prospects. Fourth, CEOs are risk-averse and their effort isunobservable. Fifth, an incumbent CEO can be replaced, if it is optimal.The model is calibrated to match salient moments of contracts and returns: (i) the ratio ofexpected cash compensation to expected change in equity compensation, (ii) the ratio of the standarddeviation of cash compensation to the standard deviation of the change in equity compensation, (iii)the correlation between cash compensation and (iv) the change in equity-based compensation andthe value premium.1 The calibrated model explains in the cross-section the negative relation of1Along this chapter I refer to short-term compensation as cash compensation and long-term compensation as equity52risk premia with equity-based compensation and forced turnover. Also, the model explains theprocyclicality of aggregate turnover.In the present framework, each firm needs the effort exerted by its CEO to produce, which is un-observable. The representative shareholder (“she”) and the CEO (“he”) enter into a long-term con-tract to incentivize him. The contract specifies: short-term compensation, long-term compensation,and a replacing policy of the manager. The manager is incentivized by linking his compensation torealized cash flows. Thus, the manager’s compensation increases with high profits and decreaseswith low profits. This means that with high profits the incentives problem is relaxed but with lowprofits the incentives problem is more severe.In contrast to a static setting where exogenous observable shocks should not affect the man-ager’s remuneration (Ho¨lmstrom [1979]), aggregate shocks will affect manager’s compensation aslong as they persistently affect the profitability of the firm (DeMarzo et al. [2012], Hoffmann andPfeil [2010], Piskorski and Tchistyi [2010]). This happens because positive aggregate shocks in-crease the profits of the firm and the marginal return of the effort of the manager is higher. Thus,incentivizing the manager (i.e. increasing his/her remuneration) to exert a high level of effort afterpositive shocks will increase firm’s profits. An important point is that the aggregate shock has to bepersistent. If the aggregate shock is not persistent, then the aggregate shock is going to increase theprofitably of the firm for just a period and, therefore, the marginal return of effort of the manageris negligible and it is not efficient to expose the manager to aggregate risk. In other words, whenthe aggregate shock is not persistent we are back to the static setting result. The dynamic settinggenerates that during expansions, when profits are high, the incentives problem is relaxed. How-ever, during recessions the incentives problem is more severe. For the representative shareholder itis costly that the incentives problem is more severe during recessions, because her marginal utilityincreases with negative aggregate shocks. Thus, agency costs are high precisely when the marginalutility of the representative shareholder is high.Since firms are heterogeneously exposed to aggregate risk, they have different compensationstructures and replacement policies. Firms with high exposure to aggregate shocks have higherrevenue volatility and need to increase the exposure to aggregate risk of their managers (relatively tomanagers at low exposure firms) to incentivize them, increasing the volatility of their compensation.However, firms with high exposure to aggregate risk will reduce their managers’ long-termcompensation (relatively to managers at firms with low exposure) if managers get strong disutilityfrom effort. This happens because incentivizing managers at high exposure firms is more costly.The contract will reduce the cost by decreasing their equity portfolio compensation, generating anegative relation between risk premia and equity-based compensation.When the economy is subject to a series of positive aggregate shocks firms’ profitability willincrease. Consequently, managers’ compensations will increase (since they are rewarded for highprofits). In the model the cost to incentivize managers increases with their income level (incomeeffect). Thus, managers at firms with low exposure to aggregate risk are more difficult to incentivize,portfolio compensation, indistinctively.53since the managers in this type of firms have, ex-ante, higher long-term compensation than managersat high exposure firms. If the increase in compensation is high enough it will render the managerstoo costly to be incentivized. Thus, low exposure firms will be better-off replacing them, generatinga negative relation between forced turnover and exposure to aggregate risk.Also, in the model aggregate turnover is procyclical. Since firms have a positive exposure toaggregate risk, for all firms it is more expensive to incentivize incumbent CEOs during booms. Thebetter the state of the economy is, the harder it is to incentivize the incumbent CEO. Thus, duringbooms a high fraction of firms will replace their CEOs.Empirically, the current chapter extends in key aspects previous works that study the relations ofaggregate risk with compensation and forced turnover. First, I use a long sample that includes boomsand recessions that allows a better understanding of the effects of firms’ exposure to aggregaterisk. Second, following the asset pricing literature, aggregate risk is measured using the Fama andFrench three factor model and the Carhart four factor model. Third, to calculate the aggregateand idiosyncratic components of returns, I use daily returns for the previous fiscal year. Usingthis empirical approach I find a stronger relation between firms’ exposure to aggregate risk andcompensation than in the previous literature; the relation is strongly negative for all managers. Forturnover, I find similar results than previous literature but stronger.This chapter relates to the dynamic agency literature. DeMarzo and Sannikov [2006] study theoptimal dynamic contract implemented using the firm’s capital structure. DeMarzo and Fishman[2007] derive the investment, capital structure and dividend dynamics under agency conflicts. He[2009] finds the optimal compensation of the manager when the firm’s size follows a geometricBrownian process. Biais et al. [2010] study the optimal contract when the risk-neutral manager haslimited liability and needs to exert effort to reduce the probability of disaster shocks. He [2011]studies the optimal dynamic compensation of the CEO and the optimal capital structure of the firm.Edmans and Gabaix [2011] develop a framework that allows closed-form solutions of dynamicagency models. Edmans et al. [2012] and He [2012] study the optimal compensation of a CEOwhen he can privately save. Zhu [2012] studies a dynamic contract when the agent optimally shirksunder certain circumstances. Szydlowski [2014] studies the optimal contract when the agent needsto allocate effort between different risky projects. Varas [2014] studies a dual moral hazard modelwith unobservable effort and imperfectly observable project’s quality. Nikolov and Schmid [2012]estimate a dynamic agency model to quantify the effects of agency conflicts.Inderst and Mueller [2010], Spear and Wang [2005], Sannikov [2008], Wang [2011], and Garrettand Pavan [2012] study the effects of firing policy on firm’s value and the agent’s compensation.DeMarzo et al. [2012], Piskorski and Tchistyi [2010], and Hoffmann and Pfeil [2010] show howthe agent’s compensation is affected by aggregate shocks that are related to future profitability ofthe project but are not under the control of the agent. Axelson and Baliga [2009] also study thecircumstances under which the compensation of the CEO should depend on variables out of hiscontrol. Eisfeldt and Rampini [2008] find that aggregate turnover is procyclical. Lustig et al. [2011]study the effects of technological changes on the cross-section of CEOs’ compensation. Ai and54Li [2014] study the compensation of managers and firms’ investment decision in a neoclassicalinvestment model when there is limited commitment on contracts. Ai et al. [2013] and Ai and Li[2012] study firms’ investment policies and the compensation of managers in general equilibriummodels with two-sided limited commitment model and moral hazard, respectively.In general, the current chapter belongs to the literature that studies the effects of agency conflictson firms’ decisions. In a dynamic moral hazard model Clementi et al. [2010] study the drivers offirm’s decline. Quadrini [2004] studies the liquidation and investment dynamics of firms in a dy-namic moral hazard framework, with renegotiation-proof contracts. Albuquerque and Hopenhayn[2004] and Schmid [2012] study endogenous borrowing constraints under limited commitment,while Clementi and Hopenhayn [2006] study them under moral hazard. Li et al. [2014], Rampiniand Viswanathan [2010], and Rampini and Viswanathan [2013] study the effects of collateral con-straints on optimal leverage and taxes, risk management, and capital structure, respectively. Boltonet al. [2014] study the optimal contract between a risk-averse entrepreneur and a risk-neutral in-vestor under limited commitment with inalienable risky human capital. Zhang [2014] studies theoptimal wage contract between the firm and its workers.Also, this chapter is related to dynamic models that study managers’ compensation and turnover,Taylor [2010] studies firms’ firing policy when the board of directors learns about the ability of themanager and firing is costly. Taylor [2013] studies wage dynamics when the board of directorslearns about the ability of the CEO.Empirically, this chapter is close to several strands of literature. This chapter is related to therelative performance evaluation literature (Aggarwal and Samwick [1999a,b], Gibbons and Murphy[1990], Murphy [1985, 1999] and Himmelberg and Hubbard [2000]). Also, this chapter is linkedwith studies that find a relation between compensation structure and firms’ characteristics (Gopalanet al. [2013]). Further, this essay is related to the turnover literature (Gao et al. [2012], Jenter andKanaan [2014], Kaplan and Minton [2012] and Peters and Wagner [2014]).This essay is linked to the asset pricing literature that studies the effects of aggregate risk onfirms’ decisions. Zhang [2005], Carlson et al. [2004], and Bai et al. [2015] study the effects onfirms’ investment decisions. Bhamra et al. [2010] analyze the effects on firms’ capital structuredecisions. Finally, this chapter is related to Tallarini Jr [2000], which highlights the importance ofaccounting for asset pricing data while understanding the effects of business cycle fluctuations.The chapter is organized as follows. Section 4.2 presents the model and calibration. Section 4.3presents the empirical strategy and data. Section 4.4 discusses the results for the data and the model.Appendix C describes in detail the data and calculations of the different types of compensations,together with the model’s solution procedure and normalization.4.2 ModelThe model consists of heterogeneous firms and an infinite-lived risk-averse representative share-holder. The production technology requires labor, i.e. effort, to produce. The production technologyis subjected to idiosyncratic and aggregate shocks. The representative shareholder cannot operate55the production technology and hires managers to do so (one for each firm). Managers are risk-averseand receive utility from consumption and disutility from effort. Effort is unobservable and managersneed to be incentivized to exert effort. Each manager and the representative shareholder enter in along-term contract. The contract defines CEO’s consumption, effort, and next period life-time util-ity level. Firms take the stochastic discount factor as given when taking production decisions. Thestochastic discount factor is derived from the representative shareholder’s preferences.4.2.1 CEO’s ProblemThe executive’s life-time utility Wt isWt = Et[∞∑i=tδ i−tu(Ht)−ν(zt)], (4.1)where u(Ht) is the per-period utility over consumption Ht , ν(zt) is the per-period disutility ofeffort, and δ is the subjective discount factor of the CEO. The executive per-period utility preferenceis of the CRRA form:u(Ht) =H1−σt1−σ , (4.2)where σ is the reciprocal of the elasticity of intertemporal substitution. The functional form forthe disutility of effort is:ν(zt) = ζX1−σt−1 zεt , (4.3)where ζ is a parameter that scales the marginal cost of effort, ε is the effort curvature, and X1−σt−1is scaling the effort so it does not become negligible along the balance growth path.4.2.2 Representative Shareholder’s PreferencesThe representative shareholder has Epstein-Zin preferences over the aggregate consumption CtstreamUt ={(1−β )C1−1/ψt +βEt [U1−γt+1 ]1θ} θ1−γ(4.4)where β is the subjective discount factor, , ψ is the elasticity of intertemporal substitution, γ is therisk aversion, θ ≡ (1− γ)/(1−1/ψ) is defined for convenience.The stochastic discount factor isMt+1 = β(Ct+1Ct) 1ψ(U1−γt+1Et [U1−γt+1 ])1ψ −γ1−γ(4.5)Following Bansal and Yaron [2004], the log consumption growth (∆ct = log(Ct/Ct−1)) is ex-56ogenously specified as:log(CtCt−1)= g¯+gct , (4.6)where gct is an AR(1) process:gct = ρcgct−1 +σcεct , (4.7)where ρc is the persistence, σc is the volatility, and εct is a standard normally distributed randomvariable.The model is a partial equilibrium one. The fact that the representative shareholder’s consump-tion process is exogenous and not endogenous eliminates feedback effects from the firms profitsthat would be taken into account in a general equilibrium model. In this case the importance of thepreferences of the representative shareholder and the exogenous consumption process is to allowthe aggregate risk to be priced in the economy, so firms with different exposure to the aggregate riskcan have realistic risk premia. This allows the model to quantitatively study the effects of firms’ riskpremia on the contracts of the managers.4.2.3 Representative Shareholder’s ProblemEquation 4.8 shows that firm’s profit depends on an aggregate component Xt , an idiosyncratic com-ponent ξt and CEO’s current consumption Ht .pit = ξtXt −Ht . (4.8)The aggregate risk component exposes the firm to consumption growth risk:log(XtXt−1)= g¯+φgct . (4.9)As in Bansal et al. [2005] firm’s exposure heterogeneity is given by φ , a higher φ increases theexposure to consumption growth. The executive’s consumption, as well as his effort, are endogenousand depend on the optimal contract between the firm and the executive.Following Spear and Wang [2005] and Wang [2011], the firm can fire the CEO after paying acost C and can hire a new one. The representative shareholder has to compare between the value ofthe firm retaining the current CEO (V rt ) and the value of the firm replacing the CEO (Vft ), the valueof the firm Vt is given by:Vt = max{Vrt ,Vft }, (4.10)the value of the firm retaining the CEO V rt is given by:22For exposition purposes in this section the idiosyncratic shock and the effort exerted by the manager take only adiscrete set of values.57V rt = maxzt ,Ht ,Wt+1∑ξp(ξ |zt)ECt [ξ (zt)Xt −Ht +Mt+1Vt+1] , (4.11)where Vt+1 is the value of the firm in the next period, EC[.] is the expected value with respect to theaggregate shock, and p(ξ |zt) is the probability of the idiosyncratic shock that is a function of theeffort exerted by the manager. The probability of a positive idiosyncratic shock increases with theeffort level exerted by the manager. 3The value of the firm when it fires the CEO V ft is given by:V ft (W, .) = maxŴ{V rt (Ŵ , .)|Ŵ > W0}−C(W ). (4.12)Equation 4.12 shows that the value of the firm when it fires the CEO has two terms. The firstterm is the value of the firm under the new management. The firm hires the new CEO and promiseshim a utility level that maximizes the value of the firm V rt .4 The second term is the cost of firingthe current CEO, which is the cost of the severance package that has to be paid to the CEO. Thepayment is the amount that is the certainty equivalent, in consumption units, of the life-time utilitylevel, W , promised to the CEO.4.2.4 TimingTo understand the firm’s problem is important to understand the timing of the model. Figure 4.1shows the timing of events within a period. Before time t the promised utility and the aggregateshock for period t are determined. At the beginning of period t the firm value is calculated. Fol-lowing Sannikov [2008] and Sannikov [2012] effort and consumption are executed at the beginningof the period. The idiosyncratic shock is realized in the middle of the period after effort is exerted.After the realization of the idiosyncratic shock the continuation value for the next period Wt+1 isdetermined.4.2.5 Optimal Dynamic ContractIn the model all the state variables and policy functions can be observed by the representative share-holder, except for the effort level exerted by the CEO. Since effort is unobservable, this causes amoral hazard problem. The contract has to incentivize the CEO to exert effort and not to deviatefrom the contract. Following Spear and Srivastava [1987] the optimal contract can be expressedrecursively if the state space is extended to include the promised utility to the CEO as a new statevariable. Thus, the state variables of the model are aggregate consumption (C) and CEO’s promisedutility (W ).The optimal contract problem with moral hazard, written in a recursive form, is3The functional form of p(ξ |zt) is given in Appendix C4Implicitly this assumption means that a new CEO is hired and retained for at least one period.58V r(Wt ,Ct) = maxzt ,Ht ,Wt+1∑ξtp(ξt |zt)(ξ (zt)Xt −Ht +ECt [Mt,t+1V (Wt+1,Ct+1)]), (4.13)subject to the incentive compatibility constraints (IC):∑ξtp(ξt |zt)(H1−σt1−σ −ζX1−σt−1 zεt +δECt [Wt+1(ξt ,Ct+1)])≥∑ξtp(ξt |zˆt)(H1−σt1−σ −ζX1−σt−1 zˆεt +δECt [Wt+1(ξt ,Ct+1)])∀z, zˆ (4.14)and the promise keeping constraint (PK):Wt =∑ξp(ξ |zt)(H1−σt1−σ −ζX1−σt−1 zεt +δECt [Wt+1(ξt ,Ct+1)]). (4.15)The IC constraint (Equation 4.14) is needed since the CEO effort is unobservable. The IC saysthat the promised utility offered by the contract, given the state of the economy, suggested effort z,and CEO’s consumption, has to be greater or equal than the life-time utility obtained by exerting adifferent effort level zˆ. Thus, the contract incentivizes the CEO to exert the optimal effort by givinghim a life-time utility level that it is at least equal to any other. The promise keeping constraint saysthat the utility promised to the CEO has to be delivered. The promise keeping constrain is neededto be able to write the problem in a recursive form.4.3 Empirical Methodology4.3.1 DataThe data is obtained from different sources. Firms’ information is from COMPUSTAT, executiveinformation is from EXECUCOMP, and returns information is from CRSP. CEO compensation isdivided into 2 categories: cash compensation and equity portfolio compensation. Cash compensa-tion represents cash payments received by the executive in the current fiscal year. Equity portfoliocompensation is the market value of compensations that the executive will receive in the future.Total compensation is defined as cash compensation plus equity portfolio compensation. AppendixC includes a detailed definition of the different types of compensations.The market value of future compensation depends on the market value of the firm’s sharesowned by the executive, and the market value of the stock options granted to the executive. Themost demanding task is to calculate the market value of the different stock options. Stock optionsdiffer in their maturity, vested/unvested condition, and vesting period. In the literature, differentways are used to calculate the value of the stock options. In the current chapter I follow Coles et al.59[2006]. This method also takes into account different maturities, vesting periods, and vested/un-vested conditions. Appendix C explains in detail the different methods.The firm’s aggregate risk is measured in two different ways. The Fama and French three factormodel, and the Carhart four factor model are used. The aggregate exposure is calculated using oneyear of daily returns for the preceding fiscal year. Only firms with more than six months of stockreturns are used.4.3.2 Data RegressionsFollowing Aggarwal and Samwick [1999a], and Garvey and Milbourn [2003] the executive’s com-pensation is regressed on the change in shareholder’s wealth, the interactions between the change inshareholder’s wealth and the variance of stock returns (decomposed into aggregate and idiosyncraticcomponents), and a set of controls:Cashit = α0 +η ∆ShrWealthit +α1 ∆ShrWealthit ×CDF(σ2Agg)it+α2 ∆ShrWealthit ×CDF(σ2Idio)it + γ ′Xit + εC,it , (4.16)∆Equityit = β0 +ν ∆ShrWealthit +β1 ∆ShrWealthit ×CDF(σ2Agg)it+β2 ∆ShrWealthit ×CDF(σ2Idio)it +ρ ′Xit + εE,it , (4.17)∆Totalit = ϕ0 +δ ∆ShrWealthit +ϕ1 ∆ShrWealthit ×CDF(σ2Agg)it+ϕ2 ∆ShrWealthit ×CDF(σ2Idio)it + ι ′Xit + εT,it , (4.18)where CDF stands for the cross sectional cumulative density function, and Xit is the set of controls.The controls are the interaction between the change in shareholders’ wealth and the CDF of Tobin’sQ, CDF of Tobin’s Q, and the CDFs of the firm’s aggregate and idiosyncratic variances. Instead ofdirectly using the aggregate and idiosyncratic variance I use the cumulative distribution function ofthe variables, as in Aggarwal and Samwick [1999a]. The CDF is used since it allows to directlycalculate the elasticity of the executive’s compensation to a change in the stock returns variance.Firm’s aggregate variance and idiosyncratic variance are measured in dollar terms.In multi-factor models it is also important to analyze the relative contribution of each of thefactors. For this purpose, firm’s aggregate variance is decomposed into the contribution of each ofthe factors. Regressions 4.16-4.18 are substituted by60Cashit = α0 +η ∆ShrWealthit +N∑j=1α j ∆ShrWealthit ×CDF(σ2f j)it+αN+1 ∆ShrWealthit ×CDF(σ2Idio)it + γ ′Xit + εC,it ,∆Equityit = β0 +ν ∆ShrWealthit +N∑j=1β j ∆ShrWealthit ×CDF(σ2f j)it+βN+1 ∆ShrWealthit ×CDF(σ2Idio)it +ρ ′Xit + εE,it ,∆Totalit = ϕ0 +δ ∆ShrWealthit +N∑j=1ϕ j ∆ShrWealthit ×CDF(σ2f j)it+ϕN+1 ∆ShrWealthit ×CDF(σ2Idio)it + ι ′Xit + εT,it ,where { f j}Nj=1 are the factors.4.3.3 Turnover Probability and Firm’s Exposure to Aggregate RiskI follow Bushman et al. [2010] to study the effects of firm’s exposure to aggregate risk on forcedturnover probability. I extend their study in two important ways. First, I extend their sample toinclude the financial crisis (1993-2010). Second, to calculate the aggregate risk of the firm, I usethe Fama and French 3 factor model and the Carhart 4 factor model, instead of only the marketreturn. Empirically, the Fama and French and the Carhart models fit returns better than just usingthe market return as regressor.Following Bushman et al. [2010] I use a two step-approach. In the first step firm’s stock returnis regressed on a factor model (Fama and French three factor model or Carhart four factor model).The first regression is calculated using daily data for the previous fiscal year. Thus, the processcalculates different exposure for each year. In the second step a forced turnover dummy is regressedon the firm’s aggregate and idiosyncratic risks, which are, respectively, the standard deviation of thepredicted and the error term of the first step regression:Pr(Forcedit) = F(α+ γσAggit +ρσIdioit + ς ′Xit + εit), (4.19)where Forcedit is a dummy variable that measures forced turnovers and is equal to one when theturnover is forced and zero otherwise, σAggit is the firm’s aggregate risk, σIdioit is the idiosyncraticrisk, and Xit is a set of controls.In Equation 4.19 the probability is calculated using a logit regression. To classify turnovers asforced or voluntary I follow Parrino [1997].5 The data is winsorized at the 1th and 99th percentiles.Variables denominated in dollars are measured in real terms; they are deflated using the CPI.5I thank Kai Li for providing me the data614.4 Results4.4.1 CEO’s CompensationThe relation between aggregate risk and CEO compensation is explored here. Table 4.1 shows theresult for the Fama and French 3 factor model. Column (1) shows that in the data there is positiverelation between pay-performance sensitivity to aggregate risk (∆Wealth×CDF(σ2Agg)) and cashcompensation, although the point estimate is not statistically significant. In contrast, column (2)shows that there is a statistically significant negative relation between the change in equity portfoliocompensation and firm’s exposure to aggregate risk. For total compensation (column (3)) there isalso a negative relation.To study the robustness of this result Tables 4.2 and 4.3 present specifications with several con-trols. Table 4.2 shows specifications with the interaction between change in shareholders’ wealthand the CDF of Tobin’s Q, CDF of Tobin’s Q, the CDFs of firm’s aggregate and idiosyncratic vari-ances. For equity portfolio and total compensation the results are consistent along the differentspecifications. The pay-performance sensitivity to aggregate risk (∆Wealth×CDF(σ2Agg)) is neg-ative and stable; it slightly decreases in magnitude when more controls are included. For columns(8) and (9) the effect of firm’s aggregate risk (CDF(σ2Agg)) is negative for equity portfolio and totalcompensation. The results suggest a strong negative relation between aggregate risk and equity-based compensation. For cash compensation, the results are not stable. For columns (1) and (4)∆Wealth×CDF(σ2Agg) is positive and statistically significant. For column (7) the coefficient is neg-ative and statistically insignificant, but the coefficient for CDF(σ2Agg) is positive and statisticallysignificant. This suggests that there is a weak positive link between cash compensation needs andfirm’s exposure to aggregate risk.Table 4.3 includes the expected return, in dollars, as explanatory variable (Bench(Agg)). Anegative coefficient indicates that the firm is reducing aggregate risk from compensation. The datashows that in fact firms are decreasing the exposure of CEOs to aggregate risk. The results are stableand in line with the other specifications.To analyze in more detail the importance of each risk factor Table 4.4 shows the results spec-ifying the contribution of each factor. The estimates for the market factor and the SMB factor arenegative; the market factor is the only statistically significant. For the HML factor the coefficient ispositive and statistically significant. Economically, the risk market factor is, in magnitude, the mostimportant of the three factors. This suggests that the pay-performance sensitivity decreases for themarket and the SMB factors.In Table 4.5 and 4.6 I run several specifications with different controls. Table 4.5 shows that theinteractions of the change in shareholders’ wealth with the market factor (∆Wealth×CDF(σ2RM))and the SMB factor (∆Wealth×CDF(σ2Size)) are negative for equity portfolio compensation andtotal compensation. Only the interaction with the market factor is statistically significant. In Table4.6 I include the direct effect of each of the factors (column (2)) I find that for the market factor andthe SMB factor the coefficients are negative and statistically significant. In short, there is a strong62negative relation between equity-based compensation and the aggregate risk generated by each ofthe factors, with the exception of the HML factor.The current results are in line with the previous literature (Garvey and Milbourn [2003], and Jin[2002] ). Current results are statistically stronger since I use a longer series that takes into accountthe effects of the great recession, and better reflects the effects of aggregate risk on the compensationof managers.4.4.2 TurnoverThe relation between aggregate risk and turnover is explored here. Table 4.7 shows the effectsof aggregate risk on CEOs’ turnover. The probability decreases with firm’s exposure to aggregaterisk and increases with idiosyncratic risk. The higher the exposure to aggregate risk the lower isthe probability of turnover since they have less equity portfolio compensation. Thus, firms withlow exposure have a higher turnover rate. The results presented here are in line with the results ofBushman et al. [2010]. Current results are statistically stronger, since, as in the previous case, alonger sample better captures the effects of good and bad times.4.4.3 ModelIn this section simulations from the model are explored. A panel of 5,000 firms is simulated for 520periods; the first 400 periods are dropped to eliminate the influence of the initial distribution on theresults.Table 4.8 presents the monthly calibration. Panel A presents the preference parameters for therepresentative shareholder. The subjective discount factor β is set to 0.99, the risk aversion γ is setto 5, and the elasticity of intertemporal substitution ψ is set to 2. The values are in line with thelong-run risks literature.Panel B reports the preference parameters of the managers. The subjective discount factorδ is set to 0.95. The risk aversion parameter σ , and the effort curvature ε are fixed at 1.2 and2, respectively. These parameters are calibrated to standard values in the dynamic agency models(Karaivanov and Townsend [2014], Moll et al. [2014]). Finally, the marginal cost of effort parameterζ is set to 2.5. The parameter is calibrated to help match the ratio of expected cash compensation toexpected equity compensation (reported in Table 4.9 ).Panel C reports the parameters for the aggregate consumption process. The average consump-tion growth rate g¯ is set to 0.019/12, the persistence of consumption growth ρg and the volatility ofconsumption growth σg are fixed at 0.92 and 0.0015, respectively. These values are standard in thelong-run risks literature. They help to match the value premium (reported in Table 4.9 ).Table 4.10 presents the results of the influence of aggregate risk on executive compensation.The model is able to replicate the data results. In the model, managers’ compensation structureswill differ for firms with different exposure to aggregate risk, because firms’ riskiness will differ.A firm with high exposure to aggregate risk has more volatile revenues than a low exposure firm.Its manager would also have a high exposure to incentivize him. It is more costly to incentivize the63manager in high exposure firms; the contract will reduce the cost by decreasing his equity portfoliocompensation (relative to low exposure firms).The model is also able to replicate the relation between turnover and aggregate risk, as shownin Table 4.11. When the economy is subject to a series of positive aggregate shocks the firms’profitability and managers’ total compensation will increase. In the model the cost to incentivizemanagers increases in their income level (income effect). Thus, managers at firms with low exposureto aggregate risk are more costly to incentivize, since the managers in this type of firms have, ex-ante, higher equity-based compensation than managers at high exposure firms. Thus, low exposurefirms are better-off firing them, generating a negative relation between turnover and exposure toaggregate risk.The model can replicate the procyclicality of aggregate turnover. Eisfeldt and Rampini [2008]find that the correlation between aggregate turnover and output growth is 0.54; aggregate turnoverin the model has a correlation with aggregate consumption growth of 0.828. In the model, all typesof firms are positively exposed to aggregate risk. As explained before, during booms firms need toincrease total compensation to incentivize the managers. When the aggregate shock is positive andlarge enough, a big fraction of firms will fire the incumbents CEOs because they are too expensive.Table 4.12 presents comparative statistics for cash compensation, equity-based compensationand turnover. In Column 3, the subjective discount factor of the managers is lowered from thebenchmark calibration of 0.95 to 0.9. A lower discount factor makes the manager more impatientand he prefers consumption now rather than later. This can be seen in the relation between cashcompensation and risk premia, the sensitivity is higher than on the benchmark case. Furthermore,for the equity portfolio compensation the parameter is lower (in absolute value) than in the bench-mark case, decreasing the sensitivity to equity portfolio compensation. The probability of turnoverdecreases less with an increase in exposure to aggregate risk, this is in line with the higher sensitivityto cash payment and lower sensitivity to equity portfolio compensation.In column 4, the representative shareholder has CRRA preferences. In this case the risk averseshareholder does not care about the uncertainty of long term growth prospects. This can be seen inthe low sensitivity of equity compensation to firm’s exposure to aggregate risk. Cash compensationsensitivity is similar between the CRRA case and the benchmark case; this means that the sensitivityfor short term compensation is similar in both cases. This also can be seen in the sensitivity of theturnover probability to aggregate risk which decreases less with an increase in exposure to aggregaterisk.In column 5, the representative shareholder is risk neutral. The representative shareholder has nolonger a strong negative sensitivity to recessions (marginal utility is constant). Thus, the incentivesproblem is, in this case, much less costly for her. Therefore, managers at high exposure firmscan be incentivized using more long-term compensation and less short-term compensation. This isreflected in the positive sensitivity to equity portfolio compensation and the negative sensitivity tocash compensation. Now, managers at high exposure firms have more equity portfolio compensationthan managers at low exposure firms, generating a positive relation between turnover and exposure64to aggregate risk.In column 6, the risk aversion parameter of the manager is increased from 1.2 to 2. In thiscase the manager is more risk-averse and also his elasticity of intertemporal substitution is smaller(from 0.8 to 0.5). The manager prefers a smoother consumption path; this can be seen in the lowsensitivity of equity compensation to risk premia. This means that equity compensation barelychanges when there is a change in exposure to aggregate risk. Cash compensation changes to keepthe manager incentivized since equity portfolio compensation is not changing. In contrast to the basecase, turnover probability increases with firm’s exposure to aggregate risk. In this case managersat firms with high exposure to aggregate risk have similar levels of equity portfolio compensationthan managers at low exposure firms; this means that during expansions managers at high exposurefirms have higher levels of equity portfolio compensation. Thus, they are the first to be fired duringexpansion, generating a positive relation between exposure to aggregate risk and turnover.In column 7, the marginal cost of effort parameter is decreased from 2.5 to 0.5. In this casethe effort exerted by the manager is less costly for him, and he is easier to be incentivized. Thisis reflected in the negative sensitivity to cash compensation and the positive sensitivity to equity-based compensation. In this case, it is much less costly for the representative shareholder that theincentives problem is more severe during recessions. Therefore, the representative shareholder canincrease the equity portfolio compensation for managers at firms with high exposure (relatively tomanagers at low exposure firms) to incentivize them without suffering the negative consequencesduring downturns. Turnover decreases more in this case than in the benchmark; this is due to thenegative sensitivity to cash compensation, which generates that total compensation is negativelyrelated to aggregate risk exposure.4.4.4 RobustnessIn this section the data regressions are calculated using the Carhart four factor model to calculatethe aggregate component of returns. Tables 4.13, 4.14, and 4.15 replicate the results presented inTables 4.1, 4.2, and 4.3, respectively. The negative relation between firm’s exposure to aggregaterisk and equity-based compensation is stronger economically and statistically. Also, the positiverelation between cash compensation and firm’s exposure to aggregate risk is stronger.In the case of turnover Table 4.16 replicates the results in Table 4.7. The results are strongereconomically and statistically.65Figure 4.1: Timing of Events in a Periodt t +1 TimePeriodCt Wi,tValue of the firm V (Wi,t ,Ct) is calculatedCt+1 Wi,t+1(ξi,t ,Ct+1)ξi,t(zi,t)Idiosyncratic shock[Hi,t ,zi,t]This figure shows the timing of events in the model within a period.66Table 4.1: CEO Compensation and Risk Premia: Benchmark Specification(1) (2) (3)Cash Equity Total∆Wealth -0.0000886 0.0135∗∗∗ 0.0135∗∗∗(-1.53) (30.64) (30.58)∆Wealth×CDF(σ2Agg) 0.000168 -0.00720∗∗∗ -0.00721∗∗∗(1.64) (-9.76) (-9.76)∆Wealth×CDF(σ2Idio) -0.0000839 -0.00546∗∗∗ -0.00547∗∗∗(-0.75) (-7.41) (-7.43)Constant 2.215∗∗∗ -0.579∗∗∗ -0.573∗∗∗(401.63) (-11.91) (-11.73)Coperol FE yes yes yesN 126407 100806 100628t statistics in parentheses∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01This table presents data results for different specifications of 3 different regressions. Column (1) shows the resultsof: Cashit = α0 + η ∆ShrWealthit + α1 ∆ShrWealthit ×CDF(σ2Agg)it + α2 ∆ShrWealthit ×CDF(σ2Idio)it + εC,it ; it re-gresses cash compensation on the change in shareholders’ wealth, the interaction between shareholders’ wealth andfirm’s aggregate variance, and the interaction between shareholders’ wealth and idiosyncratic variance. Firm’s aggre-gate variance is calculated using the Fama and French 3 factor model. Column (2) shows the results for ∆Equityit =β0+ν ∆ShrWealthit +β1 ∆ShrWealthit×CDF(σ2Agg)it +β2 ∆ShrWealthit×CDF(σ2Idio)it +εE,it ; it regresses equity port-folio compensation on the same exogenous variables as the previous regression. Column (3) shows the results for∆Totalit = ϕ0 +δ ∆ShrWealthit +ϕ1 ∆ShrWealthit ×CDF(σ2Agg)it +ϕ2 ∆ShrWealthit ×CDF(σ2Idio)it + εT,it ; it regressestotal compensation on the same exogenous variables as the previous regressions. The standard errors are clustered bycoperol (executive-firm unit).67Table 4.2: CEO Compensation and Risk Premia: Extended Specification I(1) (2) (3) (4) (5) (6) (7) (8) (9)Cash Equity Total Cash Equity Total Cash Equity Total∆Wealth -0.0000755 0.0137∗∗∗ 0.0137∗∗∗ -0.000259∗∗∗ 0.0121∗∗∗ 0.0121∗∗∗ -0.0000385 0.0116∗∗∗ 0.0116∗∗∗(-1.32) (31.20) (31.14) (-4.16) (26.92) (26.83) (-0.61) (25.85) (25.75)∆Wealth×CDF(σ2Agg) 0.000197∗ -0.00688∗∗∗ -0.00689∗∗∗ 0.000245∗∗ -0.00644∗∗∗ -0.00644∗∗∗ -0.000154 -0.00606∗∗∗ -0.00606∗∗∗(1.90) (-9.45) (-9.45) (2.39) (-9.01) (-9.01) (-1.45) (-8.36) (-8.35)∆Wealth×CDF(σ2Idio) -0.000153 -0.00639∗∗∗ -0.00639∗∗∗ -0.0000147 -0.00517∗∗∗ -0.00518∗∗∗ 0.000157 -0.00505∗∗∗ -0.00506∗∗∗(-1.31) (-8.64) (-8.65) (-0.12) (-7.00) (-7.02) (1.30) (-6.72) (-6.74)∆Wealth×TobinQCDF 0.0000362 0.000466∗∗∗ 0.000463∗∗∗ 0.0000345 0.000448∗∗∗ 0.000445∗∗∗ 0.0000322 0.000439∗∗∗ 0.000436∗∗∗(1.53) (3.55) (3.54) (1.46) (3.43) (3.42) (1.37) (3.35) (3.34)TobinQCDF 1.258∗∗∗ 11.39∗∗∗ 11.30∗∗∗ 1.240∗∗∗ 11.68∗∗∗ 11.59∗∗∗(9.60) (15.90) (15.77) (9.74) (16.17) (16.04)CDF(σ2Agg) 3.565∗∗∗ -2.286∗∗∗ -2.290∗∗∗(23.13) (-3.64) (-3.65)CDF(σ2Idio) -0.394∗∗∗ -3.790∗∗∗ -3.747∗∗∗(-2.70) (-5.89) (-5.83)Constant 2.207∗∗∗ -0.671∗∗∗ -0.664∗∗∗ 1.603∗∗∗ -6.117∗∗∗ -6.066∗∗∗ -0.0242 -3.116∗∗∗ -3.085∗∗∗(314.55) (-12.46) (-12.30) (25.56) (-17.78) (-17.64) (-0.25) (-7.67) (-7.61)Coperol FE yes yes yes yes yes yes yes yes yesN 126341 100767 100589 126341 100767 100589 126341 100767 100589t statistics in parentheses∗ p< 0.10, ∗∗ p< 0.05, ∗∗∗ p< 0.01This table presents data results for different specifications of 3 different regressions. Column (1) shows the results of: Cashit = α0 + η ∆ShrWealthit + α1 ∆ShrWealthit ×CDF(σ2Agg)it +α2 ∆ShrWealthit ×CDF(σ2Idio)it + γ1 ∆ShrWealthit ×TobinQCDFit + γ2TobinQCDFit + γ3CDF(σ2Agg)it + γ4CDF(σ2Idio)it + εC,it ; it regresses cash compensation on thechange in shareholders’ wealth, the interaction between shareholders’ wealth and firm’s aggregate variance, the interaction between shareholders’ wealth and idiosyncratic vari-ance, the interaction between shareholders’ wealth and Tobin’s Q, Tobin’s Q, firm’s aggregate variance, and idiosyncratic variance. Firm’s aggregate variance is calculated usingthe Fama and French 3 factor model. Column (2) shows the results for ∆Equityit = β0+ ν ∆ShrWealthit + β1 ∆ShrWealthit ×CDF(σ2Agg)it + β2 ∆ShrWealthit ×CDF(σ2Idio)it +ρ1 ∆ShrWealthit ×TobinQCDFit +ρ2TobinQCDFit +ρ3CDF(σ2Agg)it +ρ4CDF(σ2Idio)it + εE,it ; it regresses equity-based compensation on the same exogenous variables as the previousregression. Column (3) shows the results for ∆Totalit = ϕ0+δ ∆ShrWealthit+ϕ1 ∆ShrWealthit×CDF(σ2Agg)it+ϕ2 ∆ShrWealthit×CDF(σ2Idio)it+ ι1 ∆ShrWealthit×TobinQCDFit +ι2TobinQCDFit +ι3CDF(σ2Agg)it+ι4CDF(σ2Idio)it+εT,it ; it regresses total compensation on the same exogenous variables as the previous regressions. The standard errors are clusteredby coperol (executive-firm unit).68Table 4.3: CEO Compensation and Risk Premia: Extended Specification II(1) (2) (3)Cash Equity Total∆Wealth -0.0000130 0.0115∗∗∗ 0.0115∗∗∗(-0.21) (25.53) (25.43)∆Wealth×CDF(σ2Agg) -0.000174 -0.00598∗∗∗ -0.00597∗∗∗(-1.64) (-8.26) (-8.25)∆Wealth×CDF(σ2Idio) 0.000156 -0.00503∗∗∗ -0.00504∗∗∗(1.29) (-6.67) (-6.70)∆Wealth×TobinQCDF 0.0000298 0.000447∗∗∗ 0.000445∗∗∗(1.26) (3.40) (3.40)TobinQCDF 1.114∗∗∗ 12.25∗∗∗ 12.16∗∗∗(8.75) (16.57) (16.44)CDF(σ2Agg) 3.535∗∗∗ -2.151∗∗∗ -2.157∗∗∗(22.97) (-3.41) (-3.43)CDF(σ2Idio) -0.425∗∗∗ -3.654∗∗∗ -3.613∗∗∗(-2.91) (-5.67) (-5.61)Bench(Agg) 0.00648∗∗∗ -0.0290∗∗∗ -0.0288∗∗∗(3.54) (-4.54) (-4.52)Constant 0.0341 -3.376∗∗∗ -3.344∗∗∗(0.35) (-8.19) (-8.12)Coperol FE yes yes yesN 126341 100767 100589t statistics in parentheses∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01This table presents data results for different specifications of 3 different regressions. Column (1) shows the results of:Cashit = α0 +η ∆ShrWealthit +α1 ∆ShrWealthit×CDF(σ2Agg)it +α2 ∆ShrWealthit×CDF(σ2Idio)it +γ1 ∆ShrWealthit×TobinQCDFit +γ2TobinQCDFit +γ3CDF(σ2Agg)it +γ4CDF(σ2Idio)it +γ5Benchit +εC,it ; it regresses cash compensation on thechange in shareholders’ wealth, the interaction between shareholders’ wealth and firm’s aggregate variance, the interac-tion between shareholders’ wealth and idiosyncratic variance, the interaction between shareholders’ wealth and Tobin’sQ, Tobin’s Q, firm’s aggregate variance, idiosyncratic variance, and the expected return in dollars. Firm’s aggregatevariance is calculated using the Fama and French 3 factor model. Column (2) shows the results for ∆Equityit = β0 +ν ∆ShrWealthit +β1 ∆ShrWealthit ×CDF(σ2Agg)it +β2 ∆ShrWealthit ×CDF(σ2Idio)it +ρ1 ∆ShrWealthit ×TobinQCDFit +ρ2TobinQCDFit +ρ3CDF(σ2Agg)it +ρ4CDF(σ2Idio)it +ρ5Benchit + εE,it ; it regresses equity portfolio compensation on thesame exogenous variables as the previous regression. Column (3) shows the results for ∆Totalit = ϕ0 +δ ∆ShrWealthit +ϕ1 ∆ShrWealthit ×CDF(σ2Agg)it + ϕ2 ∆ShrWealthit ×CDF(σ2Idio)it + ι1 ∆ShrWealthit × TobinQCDFit + ι2TobinQCDFit +ι3CDF(σ2Agg)it + ι4CDF(σ2Idio)it + ι5Benchit + εT,it ; it regresses total compensation on the same exogenous variables asthe previous regressions. The standard errors are clustered by coperol (executive-firm unit).69Table 4.4: CEO Compensation and Risk Premia: Benchmark Specification, Factors Influence(1) (2) (3)Cash Equity Total∆Wealth -0.000103∗ 0.0133∗∗∗ 0.0133∗∗∗(-1.90) (30.77) (30.73)∆Wealth×CDF(σ2RM) 0.000104 -0.00163∗∗∗ -0.00163∗∗∗(1.56) (-4.02) (-4.02)∆Wealth×CDF(σ2BM) -0.0000351 0.000494∗∗∗ 0.000491∗∗∗(-1.30) (3.22) (3.21)∆Wealth×CDF(σ2Size) 0.0000469∗ -0.0000458 -0.0000459(1.77) (-0.39) (-0.39)∆Wealth×CDF(σ2Idio) -0.00000946 -0.0113∗∗∗ -0.0113∗∗∗(-0.10) (-19.37) (-19.39)Constant 2.216∗∗∗ -0.554∗∗∗ -0.546∗∗∗(375.48) (-11.65) (-11.45)Coperol FE yes yes yesN 126407 100806 100628t statistics in parentheses∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01This table presents data results for different specifications of 3 different regressions. Column (1) shows the resultsof: Cashit = α0 +η ∆ShrWealthit +∑3j=1 α j ∆ShrWealthit ×CDF(σ2f j )it +α4 ∆ShrWealthit ×CDF(σ2Idio)it + εC,it ; it re-gresses cash compensation on the change in shareholders’ wealth, the interaction between shareholders’ wealth andthe contribution to aggregate variance of each of the factors, and the interaction between shareholders’ wealth and id-iosyncratic variance. The factor model used is the Fama and French 3 factor model. Column (2) shows the results for∆Equityit = β0 + ν ∆ShrWealthit +∑3j=1 β j ∆ShrWealthit ×CDF(σ2f j )it +β4 ∆ShrWealthit ×CDF(σ2Idio)it + εE,it ; it re-gresses equity-based compensation on the same exogenous variables as the previous regression. Column (3) shows the re-sults for ∆Totalit = ϕ0 +δ ∆ShrWealthit +∑3j=1 ϕ j ∆ShrWealthit×CDF(σ2f j )it +ϕ4 ∆ShrWealthit×CDF(σ2Idio)it +εT,it ;it regresses total compensation on the same exogenous variables as the previous regressions. The standard errors areclustered by coperol (executive-firm unit).70Table 4.5: CEO Compensation and Risk Premia: Extended Specification I, Factors Influence(1) (2) (3) (4) (5) (6)Cash Equity Total Cash Equity Total∆Wealth -0.0000916∗ 0.0134∗∗∗ 0.0134∗∗∗ -0.000274∗∗∗ 0.0117∗∗∗ 0.0117∗∗∗(-1.67) (31.15) (31.10) (-4.58) (26.72) (26.65)∆Wealth×CDF(σ2RM) 0.000133∗∗ -0.00144∗∗∗ -0.00144∗∗∗ 0.000141∗∗ -0.00136∗∗∗ -0.00136∗∗∗(2.00) (-3.52) (-3.53) (2.13) (-3.41) (-3.42)∆Wealth×CDF(σ2BM) -0.0000504∗ 0.000355∗∗ 0.000353∗∗ -0.0000556∗∗ 0.000304∗∗ 0.000304∗∗(-1.86) (2.35) (2.34) (-2.05) (2.06) (2.06)∆Wealth×CDF(σ2Size) 0.0000465∗ -0.0000597 -0.0000598 0.0000457∗ -0.0000656 -0.0000652(1.76) (-0.47) (-0.47) (1.72) (-0.52) (-0.52)∆Wealth×CDF(σ2Idio) -0.0000724 -0.0117∗∗∗ -0.0117∗∗∗ 0.000110 -0.0101∗∗∗ -0.0101∗∗∗(-0.78) (-19.62) (-19.64) (1.16) (-16.71) (-16.73)∆Wealth×TobinQCDF 0.0000512∗∗ 0.000388∗∗∗ 0.000386∗∗∗ 0.0000503∗∗ 0.000377∗∗∗ 0.000375∗∗∗(2.30) (2.78) (2.77) (2.26) (2.73) (2.72)TobinQCDF 1.256∗∗∗ 11.86∗∗∗ 11.78∗∗∗(9.59) (16.51) (16.40)Constant 2.206∗∗∗ -0.621∗∗∗ -0.612∗∗∗ 1.603∗∗∗ -6.296∗∗∗ -6.248∗∗∗(301.20) (-12.03) (-11.83) (25.62) (-18.15) (-18.03)Coperol FE yes yes yes yes yes yesN 126341 100767 100589 126341 100767 100589t statistics in parentheses∗ p< 0.10, ∗∗ p< 0.05, ∗∗∗ p< 0.01This table presents data results for different specifications of 3 different regressions. Column (1) and (4) show the results of: Cashit = α0+η ∆ShrWealthit+∑3j=1α j ∆ShrWealthit×CDF(σ2f j )it+α4 ∆ShrWealthit×CDF(σ2Idio)it+γ1 ∆ShrWealthit×TobinQCDFit +γ2TobinQCDFit +εC,it ; it regresses cash compensation on the change in shareholders’ wealth, the inter-action between shareholders’ wealth and the contribution to aggregate variance of each of the factors, the interaction between shareholders’ wealth and idiosyncratic variance, the in-teraction between shareholders’ wealth and Tobin’s Q, Tobin’s Q, the contribution to aggregate variance of each of the factors, and idiosyncratic variance. The factor model used is theFama and French 3 factor model. Column (2) and (5) show the results for ∆Equityit = β0+ν ∆ShrWealthit+∑3j=1 β j ∆ShrWealthit×CDF(σ2f j )it+β4 ∆ShrWealthit×CDF(σ2Idio)it+ρ1 ∆ShrWealthit×TobinQCDFit +ρ2TobinQCDFit +εE,it ; it regresses equity-based compensation on the same exogenous variables as the previous regression. Column (3) and (6) showsthe results for ∆Totalit = ϕ0+ δ ∆ShrWealthit +∑3j=1ϕ j ∆ShrWealthit ×CDF(σ2f j )it +ϕ4 ∆ShrWealthit ×CDF(σ2Idio)it + ι1 ∆ShrWealthit ×TobinQCDFit + ι2TobinQCDFit + εT,it ; itregresses total compensation on the same exogenous variables as the previous regressions. The standard errors are clustered by coperol (executive-firm unit).71Table 4.6: CEO Compensation and Risk Premia: Extended Specification II, Factors Influence(1) (2) (3)Cash Equity Total∆Wealth -0.000118∗ 0.0120∗∗∗ 0.0120∗∗∗(-1.90) (26.56) (26.47)∆Wealth×CDF(σ2Idio) 0.000120 -0.00530∗∗∗ -0.00531∗∗∗(1.01) (-7.14) (-7.16)∆Wealth×TobinQCDF 0.0000306 0.000449∗∗∗ 0.000446∗∗∗(1.30) (3.43) (3.42)TobinQCDF 1.296∗∗∗ 11.43∗∗∗ 11.34∗∗∗(10.07) (15.97) (15.85)∆Wealth×CDF(σ2Agg) -0.0000325 -0.00615∗∗∗ -0.00615∗∗∗(-0.31) (-8.51) (-8.50)CDF(σ2RM) 2.792∗∗∗ -2.041∗∗∗ -2.015∗∗∗(21.19) (-3.87) (-3.83)CDF(σ2BM) -0.552∗∗∗ 0.659∗ 0.634∗(-6.83) (1.81) (1.75)CDF(σ2Size) -0.261∗∗∗ -1.135∗∗∗ -1.126∗∗∗(-3.28) (-3.08) (-3.06)Constant 0.562∗∗∗ -4.827∗∗∗ -4.783∗∗∗(6.37) (-11.72) (-11.62)Coperol FE yes yes yesN 126341 100767 100589t statistics in parentheses∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01This table presents data results for different specifications of 3 different regressions. Column (1) shows the results of:Cashit = α0 +η ∆ShrWealthit +α1 ∆ShrWealthit ×CDF(σ2Agg)it +α2 ∆ShrWealthit ×CDF(σ2Idio)it + γ1 ∆ShrWealthit ×TobinQCDFit + γ2TobinQCDFit +∑3j=1 γ j+2CDF(σ2f j )it + εC,it ; it regresses cash compensation on the change in sharehold-ers’ wealth, the interaction between shareholders’ wealth and the contribution to aggregate variance of each of the factors,the interaction between shareholders’ wealth and idiosyncratic variance, the interaction between shareholders’ wealth andTobin’s Q, Tobin’s Q, the contribution to aggregate variance of each of the factors, and idiosyncratic variance. The factormodel used is the Fama and French 3 factor model. Column (2) shows the results for ∆Equityit = β0 +ν ∆ShrWealthit +β1 ∆ShrWealthit ×CDF(σ2Agg)it + β2 ∆ShrWealthit ×CDF(σ2Idio)it + ρ1 ∆ShrWealthit × TobinQCDFit + ρ2TobinQCDFit +∑3j=1 ρ j+2CDF(σ2f j )it + εE,it ; it regresses equity-based compensation on the same exogenous variables as the previ-ous regression. Column (3) shows the results for ∆Totalit = ϕ0 + δ ∆ShrWealthit +ϕ1 ∆ShrWealthit ×CDF(σ2Agg)it +ϕ2 ∆ShrWealthit ×CDF(σ2Idio)it + ι1 ∆ShrWealthit × TobinQCDFit + ι2TobinQCDFit +∑3j=1 ι j+2CDF(σ2f j )it + εT,it ; it re-gresses total compensation on the same exogenous variables as the previous regressions. The standard errors are clusteredby coperol (executive-firm unit).72Table 4.7: CEO Turnover and Risk Premia(1) (2)turnover turnoverσAgg -13.76∗∗∗ -13.15∗∗(-2.70) (-2.57)σIdio 31.95∗∗∗ 31.42∗∗∗(10.43) (10.20)Ret(Agg) -0.289∗ -0.286(-1.66) (-1.63)Ret(Idio) -0.612∗∗∗ -0.611∗∗∗(-6.32) (-6.32)Equity−based pay > 0 -1.822∗∗∗(-3.62)Constant -3.897∗∗∗ -2.078∗∗∗(-42.97) (-4.08)Firm FE yes yesN 20821 20821t statistics in parentheses∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01This table presents data results for different specifications of: Pr(Forcedit) = F(α + γσAggit + ρσIdioit + ς ′Xit + εit ),where Forcedit is a dummy variable that is equal to one if the CEO was fired in year t and zero otherwise (based on anews analysis), σAggit is a variable that measures firm’s aggregate risk, σIdioit is a variable that measures idiosyncraticrisk, and Xit is a set of controls. For column (1) the controls are: The stock’s expected return, the mean idiosyncraticreturn in the past year. Column (2) includes as control the variable Equity-based pay>0, it is a dummy variable that isequal to one if equity-based pay is greater than zero and zero otherwise. The Fama and French 3 factor model is used toobtain systematic and idiosyncratic measures. The standard errors are clustered at the firm level.Table 4.8: Monthly CalibrationParameter Description ModelA. Preferences Shareholderβ Subjective discount factor 0.99ψ Elasticity of intertemporal substitution 2.0γ Risk aversion 5.0B. Preferences CEOδ Subjective discount factor 0.95σ Risk aversion 1.2ε Effort curvature 2.0ζ Scaling parameter of the marginal cost of effort 2.5C. Consumptiong¯ Average consumption growth rate 0.019/12ρg Persistence of consumption growth 0.92σg Conditional volatility of consumption growth 0.0015This table reports the monthly calibration. Panel A reports the preferences parameters for the representative shareholder.Panel B presents the preference parameters for the managers. Finally, panel C reports the parameters of aggregateconsumption process73Table 4.9: Contract and Returns MomentsStatistic Data ModelE[Cash]/E[∆Equity] 1.1 0.9σ [Cash]/σ [∆Equity] 0.23 0.2corr(Cash,∆Equity) -0.12 -0.25Value Premium (basis points monthly) 40 30This table reports the moments matched by the model and their data counterpart.Table 4.10: Comparison Data and Model: CompensationCash EquityData Model Data Model(1) (2) (3) (4)∆Wealth -0.0000886 0.0103993 0.0135 0.017356(-1.53) (30.64)∆Wealth×CDF(σ2Agg) 0.000168 0.0067455 -0.00720 -0.011941(1.64) (-9.76)∆Wealth×CDF(σ2Idio) -0.0000839 -0.0165998 -0.00546 -0.0069235(-0.75) (-7.41)This table presents data results for different specifications of 2 different regressions. Column (1) and (2) show theresults of: Cashit = α0 +η ∆ShrWealthit +α1 ∆ShrWealthit ×CDF(σ2Agg)it +α2 ∆ShrWealthit ×CDF(σ2Idio)it + εC,it ; itregresses cash compensation for the data and the model on the change in shareholders’ wealth, the interaction betweenshareholders’ wealth and firm’s aggregate variance, and the interaction between shareholders’ wealth and idiosyncraticvariance. Firm’s aggregate variance is calculated using the Fama and French 3 factor model for the data. Column (3)and (4) show the results for ∆Equityit = β0 + ν ∆ShrWealthit + β1 ∆ShrWealthit ×CDF(σ2Agg)it + β2 ∆ShrWealthit ×CDF(σ2Idio)it + εE,it ; it regresses equity-based compensation for the data and the model on the same exogenous variablesas the previous regression.Table 4.11: Comparison Data and Model: TurnoverTurnoverData Model(1) (2)σAgg -13.1536 -74.823(-2.57)σIdio 31.4177 2.17342(10.20)Ret(Agg) -0.286363 0.02242(-1.63)Ret(Idio) -0.6134 2.86964(-6.32)This table presents data results for the data and the model of: Pr(Forcedit) = F(α + γσAggit + ρσIdioit + εit ), whereForcedit is a dummy variable that is equal to one if the CEO was fired in year t and zero otherwise (based on a newsanalysis), σAggit is the firm’s aggregate risk, and σIdioit is the idiosyncratic risk. The Fama and French 3 factor model isused to obtain firm’s aggregate and idiosyncratic measures for the data. The standard errors are clustered at the firm level.74Table 4.12: Comparative Statics: ContractModelA. Cash compensation Data Benchmark δ = 0.9 CRRA Risk Neutral σ = 2 ζ = 0.5(1) (2) (3) (4) (5) (6) (7)∆Wealth×CDF(σ2Agg)it 0.000168 0.0067455 0.015257 0.0067815 -0.0023201 0.0496758 -0.0008742∆Wealth×CDF(σ2Idio)it -0.0000839 -0.0165998 -0.0284832 -0.0164617 -0.0023489 -0.1520065 0.0076301B. Equity compensation∆Wealth×CDF(σ2Agg)it -0.0072 -0.011941 -0.0032341 -0.0023296 0.0014339 -0.000335 0.0059709∆Wealth×CDF(σ2Idio)it -0.00546 -0.0069235 -0.0398544 -0.0265555 -0.0193543 -0.0464216 -0.0479778C. TurnoverσAgg -13.1536 -74.8223 -63.81425 -67.3666 43.34675 7.0762 -85.3593σIdio 31.4177 2.1734 1.57597 2.15754 4.438744 -0.49074 2.2637This table compares alternative calibrations of the benchmark model for cash compensation, equity-based compensation and turnover. Model benchmark is the benchmark model.Model δ = 0.9 lowers the subjective discount rate of the CEO from the benchmark value of 0.95 to 0.9. Model CRRA lowers the elasticity of intertemporal substitution from thebenchmark value of 2 to 1/5 (the reciprocal of the risk aversion) for the representative shareholder. In column 5 the representative shareholder is risk neutral. Model σ = 2 increasesthe risk aversion parameters of the managers from 1 to 2. Model ζ = 0.5 decreases the marginal cost of effort from 2.5 to 0.5.75Table 4.13: CEO Compensation and Risk Premia: Benchmark Specification (Robustness)(1) (2) (3)Cash Equity Total∆Wealth -0.0000967∗ 0.0135∗∗∗ 0.0135∗∗∗(-1.67) (30.58) (30.53)∆Wealth×CDF(σ2Agg) 0.000160 -0.00809∗∗∗ -0.00809∗∗∗(1.50) (-10.53) (-10.53)∆Wealth×CDF(σ2Idio) -0.0000675 -0.00458∗∗∗ -0.00458∗∗∗(-0.58) (-6.01) (-6.03)Constant 2.215∗∗∗ -0.584∗∗∗ -0.579∗∗∗(402.37) (-11.96) (-11.78)Coperol FE yes yes yesN 126407 100806 100628t statistics in parentheses∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01This table presents data results for different specifications of 3 different regressions. Column (1) shows the re-sults of: Cashit = α0 + η ∆ShrWealthit + α1 ∆ShrWealthit × CDF(σ2Agg)it + α2 ∆ShrWealthit × CDF(σ2Idio)it + εC,it ;it regresses cash compensation on the change in shareholders’ wealth, the interaction between shareholders’ wealthand firm’s aggregate variance, and the interaction between shareholders’ wealth and idiosyncratic variance. Firm’saggregate variance is calculated using the 4 factor model. Column (2) shows the results for ∆Equityit = β0 +ν ∆ShrWealthit + β1 ∆ShrWealthit × CDF(σ2Agg)it + β2 ∆ShrWealthit × CDF(σ2Idio)it + εE,it ; it regresses equity port-folio compensation on the same exogenous variables as the previous regression. Column (3) shows the results for∆Totalit = ϕ0 +δ ∆ShrWealthit +ϕ1 ∆ShrWealthit ×CDF(σ2Agg)it +ϕ2 ∆ShrWealthit ×CDF(σ2Idio)it + εT,it ; it regressestotal compensation on the same exogenous variables as the previous regressions. The standard errors are clustered bycoperol (executive-firm unit).76Table 4.14: CEO Compensation and Risk Premia: Extended Specification I (Robustness)(1) (2) (3) (4) (5) (6) (7) (8) (9)Cash Equity Total Cash Equity Total Cash Equity Total∆Wealth -0.0000826 0.0137∗∗∗ 0.0138∗∗∗ -0.000268∗∗∗ 0.0121∗∗∗ 0.0121∗∗∗ -0.0000466 0.0116∗∗∗ 0.0116∗∗∗(-1.45) (31.19) (31.13) (-4.31) (26.90) (26.82) (-0.74) (25.84) (25.75)∆Wealth×CDF(σ2Agg) 0.000197∗ -0.00769∗∗∗ -0.00769∗∗∗ 0.000251∗∗ -0.00722∗∗∗ -0.00722∗∗∗ -0.000168 -0.00686∗∗∗ -0.00686∗∗∗(1.84) (-10.17) (-10.18) (2.35) (-9.73) (-9.73) (-1.52) (-9.14) (-9.12)∆Wealth×CDF(σ2Idio) -0.000146 -0.00558∗∗∗ -0.00558∗∗∗ -0.0000113 -0.00440∗∗∗ -0.00441∗∗∗ 0.000181 -0.00424∗∗∗ -0.00426∗∗∗(-1.21) (-7.34) (-7.35) (-0.09) (-5.79) (-5.81) (1.46) (-5.50) (-5.53)∆Wealth×TobinQCDF 0.0000364 0.000455∗∗∗ 0.000453∗∗∗ 0.0000344 0.000435∗∗∗ 0.000432∗∗∗ 0.0000308 0.000425∗∗∗ 0.000423∗∗∗(1.53) (3.44) (3.43) (1.45) (3.31) (3.30) (1.31) (3.22) (3.21)TobinQCDF 1.265∗∗∗ 11.36∗∗∗ 11.27∗∗∗ 1.268∗∗∗ 11.68∗∗∗ 11.59∗∗∗(9.65) (15.85) (15.73) (9.96) (16.14) (16.01)CDF(σ2Agg) 3.671∗∗∗ -1.736∗∗∗ -1.738∗∗∗(23.54) (-2.74) (-2.74)CDF(σ2Idio) -0.523∗∗∗ -4.227∗∗∗ -4.184∗∗∗(-3.59) (-6.51) (-6.45)Constant 2.208∗∗∗ -0.673∗∗∗ -0.667∗∗∗ 1.600∗∗∗ -6.104∗∗∗ -6.053∗∗∗ -0.0260 -3.183∗∗∗ -3.152∗∗∗(316.65) (-12.47) (-12.32) (25.50) (-17.74) (-17.61) (-0.26) (-7.82) (-7.76)Coperol FE yes yes yes yes yes yes yes yes yesN 126341 100767 100589 126341 100767 100589 126341 100767 100589t statistics in parentheses∗ p< 0.10, ∗∗ p< 0.05, ∗∗∗ p< 0.01This table presents data results for different specifications of 3 different regressions. Column (1) shows the results of: Cashit = α0 + η ∆ShrWealthit + α1 ∆ShrWealthit ×CDF(σ2Agg)it +α2 ∆ShrWealthit ×CDF(σ2Idio)it + γ1 ∆ShrWealthit ×TobinQCDFit + γ2TobinQCDFit + γ3CDF(σ2Agg)it + γ4CDF(σ2Idio)it + εC,it ; it regresses cash compensation on thechange in shareholders’ wealth, the interaction between shareholders’ wealth and firm’s aggregate variance, the interaction between shareholders’ wealth and idiosyncratic vari-ance, the interaction between shareholders’ wealth and Tobin’s Q, Tobin’s Q, firm’s aggregate variance, and idiosyncratic variance. Firm’s aggregate variance is calculated usingthe 4 factor model. Column (2) shows the results for ∆Equityit = β0+ ν ∆ShrWealthit + β1 ∆ShrWealthit ×CDF(σ2Agg)it + β2 ∆ShrWealthit ×CDF(σ2Idio)it + ρ1 ∆ShrWealthit ×TobinQCDFit +ρ2TobinQCDFit +ρ3CDF(σ2Agg)it +ρ4CDF(σ2Idio)it + εE,it ; it regresses equity portfolio compensation on the same exogenous variables as the previous regression. Col-umn (3) shows the results for ∆Totalit = ϕ0+δ ∆ShrWealthit+ϕ1 ∆ShrWealthit×CDF(σ2Agg)it+ϕ2 ∆ShrWealthit×CDF(σ2Idio)it+ ι1 ∆ShrWealthit×TobinQCDFit + ι2TobinQCDFit +ι3CDF(σ2Agg)it + ι4CDF(σ2Idio)it + εT,it ; it regresses total compensation on the same exogenous variables as the previous regressions. The standard errors are clustered by coperol(executive-firm unit).77Table 4.15: CEO Compensation and Risk Premia: Extended Specification II (Robustness)(1) (2) (3)Cash Equity Total∆Wealth -0.0000214 0.0115∗∗∗ 0.0115∗∗∗(-0.34) (25.53) (25.43)∆Wealth×CDF(σ2Agg) -0.000194∗ -0.00677∗∗∗ -0.00676∗∗∗(-1.75) (-9.05) (-9.03)∆Wealth×CDF(σ2Idio) 0.000185 -0.00424∗∗∗ -0.00425∗∗∗(1.49) (-5.50) (-5.52)∆Wealth×TobinQCDF 0.0000282 0.000434∗∗∗ 0.000432∗∗∗(1.19) (3.28) (3.27)TobinQCDF 1.142∗∗∗ 12.24∗∗∗ 12.16∗∗∗(8.96) (16.54) (16.41)CDF(σ2Agg) 3.647∗∗∗ -1.631∗∗ -1.634∗∗(23.43) (-2.56) (-2.57)CDF(σ2Idio) -0.561∗∗∗ -4.064∗∗∗ -4.023∗∗∗(-3.85) (-6.26) (-6.20)Bench(Agg) 0.00656∗∗∗ -0.0290∗∗∗ -0.0288∗∗∗(3.58) (-4.54) (-4.53)Constant 0.0325 -3.440∗∗∗ -3.408∗∗∗(0.33) (-8.33) (-8.27)Coperol FE yes yes yesN 126341 100767 100589t statistics in parentheses∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01This table presents data results for different specifications of 3 different regressions. Column (1) shows the results of:Cashit = α0 +η ∆ShrWealthit +α1 ∆ShrWealthit×CDF(σ2Agg)it +α2 ∆ShrWealthit×CDF(σ2Idio)it +γ1 ∆ShrWealthit×TobinQCDFit + γ2TobinQCDFit + γ3CDF(σ2Agg)it + γ4CDF(σ2Idio)it + γ5Benchit + εC,it ; it regresses cash compensation onthe change in shareholders’ wealth, the interaction between shareholders’ wealth and firm’s aggregate variance, the in-teraction between shareholders’ wealth and idiosyncratic variance, the interaction between shareholders’ wealth andTobin’s Q, Tobin’s Q, firm’s aggregate variance, idiosyncratic variance, and the expected return in dollars. Firm’saggregate variance is calculated using the 4 factor model. Column (2) shows the results for ∆Equityit = β0 +ν ∆ShrWealthit +β1 ∆ShrWealthit ×CDF(σ2Agg)it +β2 ∆ShrWealthit ×CDF(σ2Idio)it +ρ1 ∆ShrWealthit ×TobinQCDFit +ρ2TobinQCDFit +ρ3CDF(σ2Agg)it +ρ4CDF(σ2Idio)it +ρ5Benchit + εE,it ; it regresses equity portfolio compensation on thesame exogenous variables as the previous regression. Column (3) shows the results for ∆Totalit = ϕ0 +δ ∆ShrWealthit +ϕ1 ∆ShrWealthit ×CDF(σ2Agg)it + ϕ2 ∆ShrWealthit ×CDF(σ2Idio)it + ι1 ∆ShrWealthit × TobinQCDFit + ι2TobinQCDFit +ι3CDF(σ2Agg)it + ι4CDF(σ2Idio)it + ι5Benchit + εT,it ; it regresses total compensation on the same exogenous variables asthe previous regressions. The standard errors are clustered by coperol (executive-firm unit).78Table 4.16: CEO Turnover and Risk Premia (Robustness)(1) (2)turnover turnoverσAgg -13.75∗∗∗ -13.15∗∗(-2.71) (-2.57)σIdio 32.33∗∗∗ 31.79∗∗∗(10.40) (10.17)Ret(Agg) -0.358∗∗ -0.355∗∗(-2.10) (-2.08)Ret(Idio) -0.620∗∗∗ -0.619∗∗∗(-6.34) (-6.33)Equity−based pay > 0 -1.825∗∗∗(-3.62)Constant -3.884∗∗∗ -2.062∗∗∗(-43.35) (-4.03)Firm FE yes yesN 20821 20821t statistics in parentheses∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01This table presents data results for different specifications of: Pr(Forcedit) = F(α + γσAggit + ρσIdioit + ς ′Xit + εit),where Forcedit is a dummy variable that is equal to one if the CEO was fired in year t and zero otherwise (based on anews analysis), σAggit is a variable that measures firm’s aggregate risk, σIdioit is a variable that measures idiosyncraticrisk, and Xit is a set of controls. For column (1) the controls are: The stock’s expected return, the mean idiosyncraticreturn in the past year. Column (2) includes as control the variable Equity-based pay>0, it is a dummy variable that isequal to one if equity-based pay is greater than zero and zero otherwise. The 4 factor model is used to obtain aggregateand idiosyncratic measures. The standard errors are clustered at the firm level.79Chapter 5ConclusionsThis thesis studies the effects of nominal and agency frictions on the decisions of economic agents inthree different environments. The essay in Chapter 2 studies how changes in the maturity structureof government nominal debt affects the firms’ and households’ decisions in a model with nominalfrictions and changes in the interaction between the monetary and the fiscal authority. In the model,the effects of maturity structure changes crucially depending on bond risk premia. When bond riskpremia decrease with maturity (i.e. a negative slope in the term structure of interest rates) a decreasein the average maturity of government debt decreases real output and inflation. In contrast, whenthe bond risk premia increase with maturity (i.e. a positive slope in the term structure of interestrates) a decrease in the average maturity of government debt increases real output and inflation.When the slope is nonzero, maturity restructuring changes the government discount rate, whichaffects the price level through the government debt valuation equation. These effects can be seenin the benchmark model when the economy is in the fiscally-led regime. Also, they can be seen inboth regimes (i.e. fiscally-led and monetary-led regime) when the policy mix can switch betweenregimes, which is the case of the extended model. Conditional restructuring policies that shortenthe maturity structure when the yield curve is downward-sloping, such as during times of fiscalstress or at the onset of a recession, create fiscal inflation and an expansion in output. Market timingpolicies that shorten the maturity structure when the yield curve is downward-sloping and lengthenthe maturity structure when the yield curve is upward-sloping smooth macroeconomic fluctuationsand improve welfare. In a liquidity trap, lengthening the maturity structure can be effective inattenuating deflationary pressure and output losses.The essay in Chapter 3 studies how changes in the relative income between spouses affect theasset allocation, consumption and marital decision in a model with agency friction (i.e. divorce).In the model, a change in the relative income induces a change in the relative bargaining powerdue to the lack of marital commitment between the spouses. This mechanism allows the modelto rationalize empirical patterns relating fluctuations in spouses’ relative income to portfolio deci-sions. The model can also explain the link between marital transitions and portfolio allocations.Interestingly, the risk-sharing benefits of marriage imply a positive link between wealth and riskyasset holdings across households. In short, this chapter highlights the importance of intra-householdfrictions to understand the portfolio allocation of multiple-member households when households’members cannot commit to future plans.The essay in Chapter 4 relates firms’ risk premia with the compensation structure and firingpolicy of managers, by studying a dynamic moral hazard model with aggregate risk. When man-agers receive high levels of disutility from effort and shareholders are averse to uncertainty aboutlong-term growth prospects, firms will link compensation to realized cash flows to be able to in-80centivize the managers. Managers in firms with high exposure will reduce the use of long-termcompensation. Consequently, the model explains the negative relation between aggregate risk andequity-based compensation. The model can also explain the negative relation between risk premiaand forced turnover, and the procyclicality of turnover at the aggregate level. In short, this chapterhighlights the importance of risk premia in understanding the dynamics of CEO contracts.5.1 LimitationsEach of the essays presented in this thesis have limitations and can be improved. The model inChapter 2 assumes that changes in the monetary/fiscal policy mix are driven by an exogenous pro-cess. In the same line, the maturity structure of government debt is also driven by an exogenousprocess. While these limitations do not affect the understanding of the direct effects of these changeson the real economy and inflation, they do leave open the question of what are the feedback effectsof the real economy on the monetary/fiscal policy mix and the maturity structure.Also, the essay in Chapter 3 focuses only in some of the financial decisions made by households.Including other financial decisions would allow a better understanding, quantitatively, of the effectsof intra-household frictions in the financial decisions of households over the life cycle. For example,the model generates counter-factually high asset allocation to the risky asset for young households,which impacts the effects of the agency friction for young households.Lastly, Chapter 4 concentrates in understanding how the CEO’s compensation and firing policyis affected by the firm’s exposure to aggregate risk. However, since the exposure of the firm isfixed, the model cannot address the issue of how CEO’s decisions affects the firm’s exposure tothe aggregate risk. This would require a more complex model making it computationally infeasiblegiven the resources available to me.5.2 Future WorkSeveral interesting questions related to the effects of nominal and agency frictions remain open.In the case of nominal frictions an interesting research area is to study the effects on the maturitystructure of firms’ debt and the firms’ investment policies. Also, this extension can be used to assesthe effects of fiscal and monetary policies on firms’ decisions.As for Chapter 3, the model can be extended to take into account other important decisions madeby households. For example, incorporating housing and mortgage decisions. These are decisionswhere, I think, agency frictions can have an important effect. 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The Review of Economic Studies, page rds038, 2012.→ pages 5492Appendix AAppendix to Chapter 2A.1 Numerical ProcedureSingle Regime For each of the regimes the model is solved using a global method following Juddet al. [2012] and Judd et al. [2013]. A subset of the policy functions are approximated by standardordinary polynomials of the state variables. The state variables are:St =(rt−1,at−1,st−1,bt−1,xmt ,Yt−1,∆nt−1,βt ,+εat ,εnt ,εst ,{P(n)t−1}Ni=2),where rt−1 is the nominal one-period risk-free rate, at−1 is the transitory productivity shock, st−1 isthe governments’ surplus, bt−1 is the total debt of the government, xmt is the stochastic process driv-ing the maturity structure, Yt−1 is the final consumption goods, ∆nt−1 is the permanent productivityshock, βt is the subjective discount rate, eat is the innovation to the transitory productivity shock,ent is the innovation to the permanent productivity shock, est is the innovation to the government’ssurplus, and {P(n)t−1}Ni=2 are the nominal bond bond prices. The approximated policy functions are:G=(Ft ,Ct ,Ut ,{P(n)t−1}Ni=2),whereFt =(ΠtΠss−1)ΠtΠss,Ct is the household’s consumption, Ut is the household’s utility, and {Pt−1}Ni=2 are the nominalbond prices.The model is solved by finding the set of polynomial coefficients Θ that minimizes the meansquared residuals for the approximated decision rules over a fixed grid. For each point j on the grid93the residuals are calculated as:R j1 = φRFjt Yjt −Ejt [Mt+1 φRFt+1Yt+1]−Λt ,R j2 = Ejt[Mt+1RWt+1]−1,R j3 = Ujt −{(1−β )(C j?t)1−1/ψ+β(E jt[U1−γt+1]) 1θ} 11−1/ψ,R ji+2 = Ejt[Mt+1Πt+1P(i−1)t+1]−P j,(i)t ∀i = 2..N.R j1 is calculated using the first order condition (in equilibrium) of the firms in the intermediategoods sector, R j2 is calculated using the Euler equation for the return on the wealth portfolio, Rj3is computed using the value function equation. Finally, {R ji }Ni=4 are computed using the Eulerequations for the nominal bonds.The grid on the state variables space is calculated in 4 steps. First, the model is solved using asecond order perturbation approximation. The solution is used to find an initial guess for the set ofcoefficients Θ. Second, the model is simulated and the principal components of the state variablesare calculated. Third, an auxiliary grid on the principal components space is calculated using theSmolyak algorithm. Finally, the grid on the state variables space is calculated by performing a lineartransformation of the auxiliary grid calculated in the previous step.The Smolyak algorithm is used for the auxiliary grid because it is a highly efficient methodto calculate a sparse grid in a hypercube. The drawback of the Smolyak algorithm is that thepoints are not chosen to maximize the number of points on the region of the state space wherethe model’s ergodic distribution is located. The Smolyak algorithm is improved by adapting it tothe characteristics of the model using the principal components transformation.Second order polynomials are used for each of the regimes. The minimization is done using anumerical optimizer. To improve the speed of the code analytical gradient and monomial integrationare used. The mean square error is of the order of 10−7 for the monetary-led regime and 10−6 forthe fiscally-led regime.Regime Switching The regime switching model is also solved using the global approximationmethod. In this case the decision rules are approximated by a piece-wise polynomial, as in Aruobaand Schorfheide [2013]. Let L be a policy function, the policy function is approximated as:L̂ = 1F pF +1M pM, (A.1)where 1 j is an indicator function that takes value one in regime j and zero otherwise, and p j is apolynomial. Equation A.1 shows that for each regime a different polynomial is used, pF for thefiscally-led regime and pM for the monetary-led regime. The use of piece-wise polynomials allowsfor a more flexible structure to fit the model. The initial guesses for the polynomial coefficients arethe solutions for each of the single regime models. The grid is calculated as the union of the twosingle regime grids (fiscally-led regime and monetary-led regime). As in the single regime case,second order standard ordinary polynomials are used for each of the regime-specific polynomials.A numerical optimizer with analytical gradient is used in this case as well. The mean square erroris of the order of 10−6, in line with the mean square errors for the single regime models.94A.2 DataWe obtain quarterly data for consumption, and output from the Bureau of Economic Analysis(BEA). Consumption is measured as real personal consumption expenditures (DPCERX1A020NBEA).Output is measured as real gross domestic product (GDPC1). Inflation is computed by taking thelog return on the Consumer Price Index for All Urban Consumers (CPIAUCSL), obtained fromthe Bureau of Labor Statistics (BLS). Monthly yield data are from CRSP. Nominal yield data formaturities of 4, 8, 12, 16, and 20 quarters are from the CRSP Fama-Bliss discount bond file. Theone-quarter nominal yield is from the the CRSP Fama risk-free rate file. Finally we build our bondsupply maturity structure data using the same methodology as Doepke and Schneider [2006] andGreenwood and Vayanos [2014]. In particular each month we collect the complete history of U.S.government bonds issued from the CRSP historical bond database. We then break the stream of eachbonds cash flows into principal and coupon payments. Summing the streams from each outstandingbond vintage over their respective maturity give us the monthly maturity structure of governmentdebt. The sample period runs from Q1-1964 to Q3-2013.A.3 Household ProblemThe time-t Lagrangian writesUt ={(1−β )(C?t )1−1/ψ +β(Et[U1−γt+1]) 1θ} 11−1/ψ+λt[Dt +WtPtLt +RtBtPt+Tt −Ct −Bt+1Pt]The first order conditions are[Ct ] : (1−β )U1/ψt (C?t )−1/ψ(L¯Lt)τ= λt (A.2)[Bt+1] : U1/ψt βEt[U1−γt+1] 1θ −1Et[U−γt+1λt+1PtPt+1Rt+1]= λt (A.3)[Lt ] : (1−β )τU1/ψt (C?t )−1/ψCt(L¯Lt)τ−1L¯ = λtWtPt(A.4)Dividing both sides of (A.3) by(λt/U1/ψt)and using (A.2) to replace λt+1, we getβEt[U1−γt+1] 1θ −1Et[(Ct+1Ct)− 1ψU1/ψ−γt+1Rt+1Πt+1]= 1orEt[Mt+1Πt+1Rt+1]= 1whereMt+1 = β(C?t+1C?t)1− 1ψ(Ct+1Ct)−1U1−γt+1Et[U1−γt+1]1− 1θThe intratemporal decision is obtained by plugging (A.2) into (A.4).95A.4 Monopolistic Firm ProblemThe maximization problem of the individual firm isV (i)t (Pi,t−1;ϒt) = maxPi,t ,Li,t{Di,t +Et[Mt+1 V(i) (Pi,t ;ϒt+1)]}subject to:Di,t =Pi,tPtXi,t −WtPtLi,t −G(Pi,t ,Pi,t−1;Pt ,Yt)Pi,tPt=(Xi,tYt)− 1νAfter plugging the definition of Di,t , the Lagrangian of the problem isLt = Y1νt [ZtLi,t −ΦZt ]1− 1ν −WtPtLi,t −φR2(Pi,tΠssPi,t−1−1)2Yt+Λi,t(Pi,tPt−Y1νt [ZtLi,t −ΦZt ]− 1ν)+Et[Mt+1 V(i)t+1 (Pi,t ;ϒt+1)]The first order conditions areΛi,tPt= φR(Pi,tΠssPi,t−1−1)YtΠssPi,t−1−Et[Mt+1φR(Pi,t+1ΠssPi,t−1)Yt+1Pi,t+1ΠssP2i,t]WtPt=(1−1ν)Y1νt X− 1νi,t Zt +Λi,t(1ν)Y1νt X− 1ν−1i,t ZtThis specification yields a symmetric equilibrium in whichPi,t =Pt , Xi,t = Xt , Li,t = Lt , Di,t =Dt , and V(i)t =Vt . The equilibrium condition for the economy are:WtPt=(1−1ν)Zt +Λi,t(1ν)ZtYtΛt = φR(ΠtΠss−1)ΠtYtΠss−Et[Mt+1 φR(Πt+1Πss−1)Yt+1Πt+1Πss]where Λt is the Lagrange multipliers on the inverse demand constraint.A.5 Present Value of SurplusesThe flow budget constraint of the government (2.1) can be written asbt =Πt∆YtRgt(bt+1 + st) (A.5)96where st = τt − gt is the government surplus. Leading expression (A.5) for one period and takingthe conditional expectation at time t,Et [bt+1] = Et[Πt+1∆Yt+1Rgt+1bt+2]+Et[Πt+1∆Yt+1Rgt+1st+1](A.6)Iterating forward on Et [bt+1] using the law of iterated expectation and assuming a transversalitycondition on real debt,Et [bt+1] = Et[∞∑i=1Pt+i Yt+iPt−1Yt−11∏ij=1Rgt+ jst+i](A.7)Using equations (A.5) and (A.7) together, the present value of government surpluses isbt = Et[∞∑i=0st+i∏ij=0Π−1t+ j∆Y−1t+ jRgt+ j](A.8)97Appendix BAppendix to Chapter 3B.1 Data DescriptionAttention is restricted to eight waves of the PSID from 1984 through 2007, since the main variablesof interest are available only for waves from 1984 onward.1In constructing the final data set, a series of filters are applied to the raw data. As highlighted inthe existing literature on household portfolio choice, wealthy households are willing to take greaterrisk in their portfolios, the result of not only higher participation rates in risky asset classes, but alsogreater portfolio weights conditional on participation (e.g., Campbell [2006]). A measure of networth is defined as the sum of holdings in stocks, bonds, cash accounts, primary residential equity,private business interests, non-residential real estate holdings, and equity in vehicles, less additionaloutstanding debts (e.g., credit cards, student loans). Only those observations for which this measureof net worth is available are kept. Further, observations are required to have non-missing valuesfor labor income, total income, age, education, and number of children, all variables that have beenshown in the literature to be important determinants of household portfolio choice (e.g., Campbell[2006] and Curcuru et al. [2004]).Financial wealth is defined as the sum of holdings in stocks, bonds, and cash accounts. Holdingsin cash and bonds are classified as non-risky, as is standard in the extant empirical portfolio choiceliterature (e.g., Gomes and Michaelides [2005] and Cocco et al. [2005]). Risky asset holdings aredefined as the value of stocks, and the risky asset portfolio share is defined as the fraction of financialwealth held in stocks.We collect demographic variables such as age, marital status, number of children, and education.Education is measured in years, with 12 representing a high school diploma and 16 representing abachelors degree. Respondents with education at the graduate and professional levels are top-codedat 17 years.The PSID collects labor income data for both partners in married households. We calculateincome growth for each individual. As in Vissing-Jørgensen [2002] and Angerer and Lam [2009],we discard observations where income growth for either the husband or wife is less than -70% orexceeds 300%. We calculate the within-household correlation between spouses incomes using allavailable data for the household. This is consistent with the calculation of the correlation betweenhousehold labor income growth and stock market returns in the literature examining householdsincome hedging (e.g., Vissing-Jørgensen [2002], Massa and Simonov [2006], and Bonaparte et al.1PSID waves including household asset allocations were released every five years until 1999, and every two yearsthereafter. Hence, we make use of eight waves of data: 1984, 1989, 1994, 1999, 2001, 2003, 2005, and 2007.98[2013]).B.2 Numerical SolutionThe model is solved by backward induction, over a discretized state space . In each period 3 valuefunctions are calculated: the value functions of being single today (one for each sex), and the valuefunction of being married today. Finally, the renegotiation step is calculated. The maximizationstep was done using search grid. Thus, avoiding the possibility of local maximums. Consumptiongrids are equally spaced in the logarithm of the variable. The grid used for the asset allocation isequally spaced in level.The grid for wealth and bargaining power are equally spaced in the logarithm and in the levelof the variable, respectively. The grids’ boundaries of the endogenous variables and policies wereselected to be non-biding at all time. Linear interpolation is used to evaluate the value functionfor levels of wealth and bargaining power outside their grids. Since the income processes and therisky asset process are correlated, they are treated, for the numerical procedure, as a single VAR(1)with time-varying mean. The VAR(1) is discretized in two steps. First, the fitted polynomials arediscretized. The fitted polynomials are approximated by a step function that changes every 5 years.The value of the step function is set to the average for the period. Second, the VAR(1) is discretized.The discretization process is an adapted version of the method used by Caldara et al. [2012] forDSGE models with stochastic volatility. The VAR(1) with time-varying mean is approximated by aset of grids each of them calculated using the Terry and Knotek II [2011] procedure with a differentmean. The mean for each grid is the corresponding value of the step function. When there is achange in the value of the step function the transition probabilities for the relative positions on thegrids remain the same, but the values of the grid-points change. So, when there is a change in thestep function we always stay on the grid, thus avoiding interpolation. In the current implementationthe VAR(1) is approximated with 6 points to cover 1 standard deviation in each direction for eachof the 3 variables.B.3 SimulationA panel of 10,000 individuals is simulated, the sex of the individuals is drawn from a discreteuniform distribution. Each individual is tracked from age 20 to 100. Prospective spouses are drawnfrom outside the set of tracked individuals. Spouses of the tracked individuals are follow as longas they stay married to the tracked individuals. After divorce the spouses are dropped from thesimulation. At age 20 tracked individuals are randomly selected to be married or single. The initialdistribution of single and married households is set to match the empirical distribution calculatedfrom the PSID for all households whose head is 20 years old. Initial wealth distributions for singlesand married households are calculated from the PSID for all households whose head is 20 years old.Initial bargaining power for married couples is set to 1. Incomes for individuals are drawn from thesteady state distributions of the income processes. Each period single individuals are matched withan individual of different sex. For the prospective spouses income is drawn from the steady statedistribution of the process, initial bargaining power of the possible couple is set to one.22This condition assumes that each single individual is matched with a prospective spouse that has the same wealthas him/her. Mazzocco et al. [2013] argue that this assumption is in-line with the insight that individuals meet potentialspouses with similar socio-economic backgrounds.99Appendix CAppendix to Chapter 4C.1 Data ImplementationC.1.1 Data DefinitionExecutive compensation data is from EXECUCOMP, firms’ balance sheet data is from COMPUS-TAT, stock returns data is from CRSP.• ALLOTHPD: The portion of “All Other Compensation” that is paid or payable in cash duringthe indicated fiscal year. The indicated amount of a portion of the item “All Other Total”.• ALLOTHTOT: This is the amount listed under“All Other Compensation” in the SummaryCompensation Table.• Bonus: The dollar value of a bonus earned by the named executive officer during the fiscalyear.• DIVYIELD: The Dividends per Share by Ex-Date divided by Close Price for the fiscal year.This quotient is then multiplied by 100.• OPT UNEX EXER NUM: The aggregate number of unexercised options held by the execu-tive at fiscal year end that were vested.• OPT UNEX EXER EST VAL: Value of in-the-money vested options at fiscal year end asreported by the company (pre-2006).• OPT UNEX UNEXER NUM: The aggregate number of unexercised options held by the ex-ecutive at fiscal year end that were not yet vested.• OPT UNEX UNEXER EST VAL: Value of in-the-money unvested options at fiscal year endas reported by the company (pre-2006).• OPTION AWARDS NUM: Total number of options awarded during the year as detailed inthe Plan Based Awards table.• OPTION AWARDS RPT VALUE: The aggregate value of all options granted to the execu-tive during the year as valued by the company.100• OPTION AWARDS FV : Fair value of all options awarded during the year as detailed in thePlan Based Awards table. Valuation is based upon the grant-date fair value as detailed in FAS123R.• OPT EXER NUM: Number of options exercised by the executive during the year.• OTHANN: The dollar value of other annual compensation not properly categorized as salaryor bonus.• PRCCF: The Close Price of the company’s stock for the fiscal year.• RSTKGRNT: The value of restricted stock granted during the year (determined as of the dateof the grant).• Salary: The dollar value of the base salary earned by the named executive officer during thefiscal year.• SHROWN EXCL OPTS: Shares owned by the executive, excluding options that are exercis-able or will become exercisable within 60 days. Share ownership is generally reported as ofa date between the fiscal year end and proxy publication.• STOCK AWARDS FV: Fair value of all stock awards during the year as detailed in the PlanBased Awards table. Valuation is based upon the grant-date fair value as detailed in FAS123R.• STOCK UNVEST NUM: Aggregate shares of restricted stock held by the executive that hadnot yet vested as of fiscal year end.C.2 CompensationC.2.1 Cash Compensation• Salary.• Bonus.• Net revenue from trade in stock.• Dividends• Long-term incentives payouts (ALLOTHTOT-ALLOTHPD).• Others (ALLOTHPD+OTHANN)Net revenue from trade in stockThe net revenue from trading (Net Rev) is calculated as:Net Rev = max{Net Sold ·PRCCF−CEX,0},where Net Sold is the net number of shares sold (including shares received by exercising stockoptions), PRCCF is the price at the end of the fiscal year, and CEX is the cost of exercising stockoptions. Net Sold and CEX are unknown.101First, CEX is calculated from Equation C.1. It shows that the cost of exercising the stock optionsis equal to the number of options exercised (OPT EXER NUM) times the strike price (STRIKE)CEX = OPT EXER NUM×STRIKE. (C.1)The strike price is unknown. To find the strike price Clementi and Cooley (2009) assume thatthe options are exercised at the maximum stock price during the fiscal year. The strike price is foundby solving:OPT EXER VAL = (MAX PRICE−STRIKE)OPT EXER NUM, (C.2)where OPT EXER VAL is the value of options exercised and MAX PRICE is the maximumstock price.Second, the net number of shares sold can be calculated from the law of motion for the numbersof shares own by the executive (SHROWN):SHROWNt = SHROWNt−1 +VESTt +OPT EXER NUMt −Net Sold. (C.3)Equation C.3 shows that the number of shares own today (SHROWNt), is equal to the numberof shares own last period (SHROWNt−1) plus the number of restricted shares that vested (VESTt)plus the number of shares received by exercising stock options (OPT EXER NUMt) minus the netnumber of shares sold (Net Sold).Thus, the net value of shares sold is:Net Sold = SHROWNt−1−SHROWNt +VESTt +OPT EXER NUMt .The number of stocks that vested VESTt is unknown. It can be obtained from the law of motionfor the total number of restricted stocks (STOCK UNVEST NUM):STOCK UNVEST NUMt = STOCK UNVEST NUMt−1 +GRNTt −VESTt ,where the number of granted restricted stocks GRNTt can be calculated as the value of restrictedstocks granted during the current fiscal year divided by the closing price for the fiscal year:GRNTt =RST KGRNTPRCCF.C.2.2 Equity Portfolio CompensationEquity-based compensation is the summation of:• Executive’s shares value.• Executive’s options value.The executive’s shares value is:S = SHROWNt ·PRCCFt .102OptionsTo calculate the options value there are different methods. The present paper follows Core and Guay[2002]; for a detailed explanation of the method see Coles et al. [2013]. The options portfolio isdivided in three different groups: options granted during the current fiscal year (assumed unvested),unvested options and vested options. Options values, for the three different portfolios, are calculatedat the end of the fiscal year using the Black and Scholes formula.For the options granted during the current fiscal year is possible to obtain: the strike price(EXPRIC), expiration (EXDATE), and number of options (OPTION AWARDS NUM) from EXE-CUCOMP.The unvested options portfolio excludes current option grants. Thus, the net number of unvestedoptions (NET OPT UNEX UNEXER NUM) is calculated as:NET OPT UNEX UNEXER NUMt =max{OPT UNEX UNEXER NUMt −OPTION AWARDS NUMt ,0}. (C.4)Equation C.4 states that if the difference between unvested options and current option grants isnegative then the net number of unvested options is zero. The time to maturity is calculated as theaverage time to maturity for current grants minus 1 year. If the number of current grants is zero, thematurity is set to nine years. The exercise price is obtained as if the options were to be exercisedimmediately, i.e., the value of the options today is equal to the difference between the current spotprice and the strike price times the number of options:NET OPT UNEX UNEXER VALt =(PRCCFt−STRIKEt)·NET OPT UNEX UNEXER NUMt .Thus, the strike price can be calculated as:STRIKE UNVESTt = PRCCFt −NET OPT UNEX UNEXER VALtNET OPT UNEX UNEXER NUMt.For vested options the time to maturity is calculated as the time to maturity for unvested optionsminus 3 year. The number of vested options in the portfolio is equal to the total number of vestedoptions minus the net number of unvested options (only if it is negative):NET OPT UNEX EXER NUMt = OPT UNEX EXER NUMt+max{−(OPT UNEX UNEXER NUMt −OPTION AWARDS NUMt),0}.The exercise price is also obtained as if the options were to be exercised immediately:STRIKE VESTt = PRCCFt −NET OPT UNEX EXER VALtNET OPT UNEX EXER NUMt.The Black and Scholes formula for options written on dividend paying stock, is given by:optt = Se−qT N(d1)−Xe−rT N(d2),where S is the spot price of the underlying asset, q is the dividend yield, X is the strike price, T isthe time to maturity, and N is the standard normal cumulative distribution function. d1 and d2 are103given by:d1 =log(SX)+(r−q+ 12σ2)Tσ√T,d2 = d1−σ√T ,where σ denotes the stock return volatility.q is calculated using the DIVYIELD variable from EXECUCOMP, r is obtained from the Fed-eral Reserve, specifically, the Constant Maturity Treasuries (annual frequency). The interest rateused corresponds to the CMT with the same maturity as the option. 1 Also, S is the stock price atthe end of the fiscal year PRCCF .C.3 NormalizationSince the model is non-stationary it has to be normalized to be able to solve it. Equation 4.13 isnormalized by Xt−1:V (Wt ,Xt)Xt−1=maxzt ,Ht ,Wt+1∑ξp(ξ |zt)(ξ (zt)XtXt−1−HtXt−1+∑CXtXt−1Q(Ct ,Ct+1)[MV (Wt+1,Xt+1)Xt]), (C.5)where Q(., .) is the transition matrix of the Markov-chain approximation of the aggregate riskprocess. 2 The normalized value of the firm is:V̂t = maxzt ,Ĥt ,Wˆt+1∑ξp(ξ |zt)(ξ (zt)eg¯+φigct − Ĥt + eg¯+φigct ∑∆cQ(∆ct ,∆ct+1)[M(∆ct+1)V̂t+1]), (C.6)The IC (Equation 4.14) and the PK (Equation 4.15) constraints are normalized by X1−σt−1 :∑ξp(ξ |zt)(HtXt−1)1−σ1−σ −ζ zεt +δXtXt−1∑∆cQ(∆ct ,∆ct+1)Wt+1Xt≥∑ξp(ξ |ẑt)(HtXt−1)1−σ1−σ −ζ zˆεt +δXtXt−1∑∆cQ(∆ct ,∆ct+1)Wt+1Xt∀z, zˆ (C.7)and the promise keeping constraint (PK):1i.e. if the maturity of the option is one year, the interest rate used is the one year CMT yield. For options withmaturity greater or equal to 10 years, the ten year yield is used.2In this section the aggregate risk has been discretized, which is inline with the implementation process (for moredetails see the next section).104WtX1−σt−1=∑ξp(ξ |zt)(HtXt−1)1−σ1−σ −ζ zεt +δXtXt−1∑∆cQ(∆ct ,∆ct+1)Wt+1Xt , (C.8)getting:∑ξp(ξ |zt)(Ĥt1−σ1−σ −ζ zεt +δe(1−σ)(g¯+φigct)∑∆cQ(∆ct ,∆ct+1)Ŵt+1)≥∑ξp(ξ |zˆt)(Ĥ1−σt1−σ −ζ zˆεt +δe(1−σ)(g¯+φigct)∑∆cQ(∆ct ,∆ct+1)Ŵt+1)∀z, zˆ (C.9)and the promise keeping constraint (PK):Ŵt =∑ξp(ξ |zt)(Ĥ1−σt1−σ −ζ zεt +δe(1−σ)(g¯+φigct)∑∆cQ(∆ct ,∆ct+1)Ŵt+1), (C.10)C.4 ImplementationThe model is solved following the method developed by Phelan and Townsend [1991]. The methodhas three steps. First, the model is discretized and re-written in term of lotteries over the policyfunctions and shocks, which are the new decision variables. Second, the value of the firm is foundusing value function iteration. Third, at each iteration and at each point of the state space the lotteriesare found by solving a linear programming problem. The advantage of this method is twofold. First,by using probabilities it convexifies the constraint set, allowing to solve problems that do not havea convex constraint set. Second, there are highly optimized computational routines to solve linearprogramming problems with accuracy and speed. The drawback is the curse of dimensionality, itis necessary to understand the characteristic of the model to choose the state variables and policiesgrids in an optimal manner.In the discretized model the state variables as well as the policy functions can take only a discreteset of values (grids). LetZ ,H ,W , G , and Ξ be the grids for effort, CEO’s consumption, promisedutility, consumption growth 3, and idiosyncratic shock, respectively. Then, the probabilities aredefined over the cartesian products of the grids, given the current state of the economy:pi(.|gct ,Ŵt) :H ×Ξ×Z ×W ×G → [0,1] (C.11)The model expressed in terms of the probabilities pi(.) is:V̂t = maxpi(.)∑Ĥt ,ξt ,Ŵt+1,zt ,gc t+1pi(Ĥt ,ξt ,zt ,Ŵt+1,gc t+1|gct ,Ŵt )[ξte(g¯+φigct)− Ĥt + e(g¯+φigct)Mt+1V̂t+1], (C.12)subject to:3In the implementation I write the grid in terms of gc instead of ∆c, remember that ∆ct = g¯+gct .105Ŵt = ∑Ĥt ,ξt ,Ŵt+1,zt ,gc t+1pi(Ĥt ,ξ ,zt ,Ŵt+1,gc t+1|gct ,Ŵt)[Ĥ1−σt1−σ −ζ zεt +δe(1−σ)(g¯+φgc,t)Ŵt+1], (C.13)∑Ĥt ,ξt ,Ŵt+1,zt ,gc t+1pi(Ĥt ,ξ ,zt ,Ŵt+1,gc t+1|gct ,Ŵt)(Ĥt1−σ1−σ −ζ zεt +δe(1−σ)(g¯+φigct)Ŵt+1)≥∑Ĥt ,ξt ,Ŵt+1,zt ,gc t+1pi(Ĥt ,ξ ,zt ,Ŵt+1,gc t+1|gct ,Ŵt)(Ĥ1−σt1−σ −ζ zˆεt +δe(1−σ)(g¯+φigct)Ŵt+1)∀(z, zˆ) ∈ Z×Z (C.14)pi(Ĥt ,ξ ,zt ,Ŵt+1,gc t+1|gct ,Ŵt)≥ 0, (C.15)∑Ĥt ,ξt ,Ŵt+1,zt ,gc t+1pi(Ĥt ,ξ ,zt ,Ŵt+1,gc t+1|gct ,Ŵt) = 1, (C.16)∑Ŵt+1pi( ¯̂Ht , ξ¯t , z¯t ,Ŵt+1,gc t+1|gct ,Ŵt) =P(ξ¯ |z¯t ,gc,t ,Ŵt) ∑ξt ,Ŵt+1,gc t+1pi( ¯̂Ht ,ξt , z¯t ,Ŵt+1,gc t+1|gct ,Ŵt)Q(gct ,gc t+1)∀(ξ¯ , z¯, ¯̂H,gc t+1) ∈ Ξ×Z×H×G. (C.17)Equation C.12 is the maximization of the firm’s value. The maximization is over the proba-bilities. Equations C.13 and C.14 are the promise keeping and incentive compatibility constraints.Equations C.15- C.17 are included to insure that the pis are true probabilities. Equation C.15 statesthat the probabilities are positive, Equation C.16 states that the sum of the probabilities must beequal to one, Equation C.17 shows that the Bayes’ rule has to hold. To see more clearly the relationbetween condition probabilities Equation C.17 can be written as:P( ¯̂Ht , ξ¯t , z¯t ,gc t+1|gct ,Ŵt) = P(ξ¯t |z¯t , ¯̂Ht ,gct ,Ŵt)P(z¯t , ¯̂Ht |gc,t ,W˜t)Q(gct ,gc t+1). (C.18)In the model the idiosyncratic shock depends only on the effort of the executive (P(ξ¯t |z¯t) =P(ξ¯t |z¯t , ¯̂Ht ,gct ,Ŵt)). Thus, the previous equation can be simplified even further:P( ¯̂Ht , ξ¯t , z¯t ,gc t+1|gc,t ,Ŵt) = P(ξ¯t |z¯t)P(z¯t , ¯̂Ht , |gc,t ,W˜t)Q(gct ,gc t+1). (C.19)106Following Karaivanov and Townsend [2014] the functional form of P(ξ¯t |z¯t) is:P(ξ¯ = ξ1|z¯) = 1−η z¯ (C.20)P(ξ¯ = ξi|z¯) =η z¯#Ξ−1∀i = 2, ...,#Ξ ∈ Ξ, (C.21)where #Ξ is the number of points in the grid Ξ.4 It is important to highlight that Equation C.16shows that effort and consumption are established at the beginning of each period, as in Sannikov[2012]. In other words, the consumption received by the executive in the current period does notdepend on the value of the current idiosyncratic shock, which is realized in the middle of the period.C.4.1 GridsThe number of points used for each grid is given in Table C.1.Table C.1: Grid PointsNumber of pointsZ 5H 10W 15G 7Ξ 5This table presents the number points for each grid used in the implementation of the model.4In the implementation η is set to 1/2.107

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