剩余消费比率、风险溢酬和经济波动——基于外部习惯的连续时间一般均衡模型
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摘要
2008年的金融危机再次引发了人们对现有宏观经济模型的反思,大量文献试图将金融部门纳入宏观模型。近年来,连续时间框架下的金融中介模型具有高度非线性、内生波动、时变风险溢酬和稳态随机等特点。这不仅使之在刻画金融市场与实体经济的互动关系和拟合经济现实等方面更具优势,同时也为融合宏观经济与资产定价提供了新的思路。以剩余消费比率作为度量投资者风险容忍度的状态变量,在连续时间下将基于消费的传统资产定价模型推广到了包含宏观变量的一般均衡模型,并利用机器学习求解了该模型。在边际消费倾向递减的假设下,谨慎性储蓄使得家庭消费在状态较差时对负向冲击更敏感,从而导致逆周期的消费波动、低风险容忍度和高风险溢酬。结果表明,该模型除了可以刻画消费波动、谨慎性储蓄和风险溢酬与状态变量剩余消费比率的非线性关系外,还能较好地匹配负向冲击发生后风险溢酬、无风险利率、夏普比率等资产价格变量和投资、经济产出等宏观经济变量的变化。
When the surplus consumption ratio is taken as the state variable measuring investors' tolerance of risk, the traditional consumption-based asset pricing model can be extended to a general equilibrium model with macro variables in continuous time. Under the assumption of decrease law in marginal consumption, precautionary savings render household consumption more sensitive to shocks when the state is poor, leading to countercyclical consumption fluctuations, lower risk tolerance and higher risk premiums. The results show that the model can not only describe the non-linear relationships between consumption fluctuation, precautionary savings, risk premium and state variable, but also better match the changes over time of both asset price variables and macroeconomic variables following the negative shock of the state variable. These asset price variables and macroeconomic variables include risk premium, risk-free interest rate, Sharpe ratio, investment and economic output.
When the surplus consumption ratio is taken as the state variable measuring investors' tolerance of risk, the traditional consumption-based asset pricing model can be extended to a general equilibrium model with macro variables in continuous time. Under the assumption of decrease law in marginal consumption, precautionary savings render household consumption more sensitive to shocks when the state is poor, leading to countercyclical consumption fluctuations, lower risk tolerance and higher risk premiums. The results show that the model can not only describe the non-linear relationships between consumption fluctuation, precautionary savings, risk premium and state variable, but also better match the changes over time of both asset price variables and macroeconomic variables following the negative shock of the state variable. These asset price variables and macroeconomic variables include risk premium, risk-free interest rate, Sharpe ratio, investment and economic output.
引文
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[3]明雷、王天轶、杨胜刚:《宏观经济条件下的一个资产定价模型》,《数学的实践与认识》2015年第11期;Brunnermeier,M.K.,Sannikov,Y.,“A Macroeconomic Model with a Financial Sector”,The American Economic Review,2014,104,pp.379-421;Brunnermeier,M.K.,Sannikov,Y.,“Macro,Money and Finance:A Continuous Time Approach”,Handbook of Macroeconomics,2016,Vol.2B,pp.1497-1546,North-Holland,Amsterdam;He,Z.,Krishnamurthy,A.,“A Macroeconomic Framework for Quantifying Systemic Risk”,Working Paper,No.19885,National Bureau of Economic Research,2014;Silva H.,“The Risk Channel of Unconventional Moentary Policy”,Unpublished Working paper,2015;Tella,S.D.,“Uncertainty shocks and balance sheet recessions”,Journal of Political Economy,2017,125,pp.2038-2081;Drechsler,Itamar,Alexi Savov,and Philipp Schnabl,“A Model of Monetary Policy and Risk Premia”,the Journal of Finance,2018,73,pp.317-373.
[6]Mian,A.,and A.Sufi,“Who Bears the Cost of Recessions?The Role of House Prices and Household Debt”,Chapter 5 in John B.Taylor and in Harald Uhlig ed.,Handbook of Macroeconomics,2016,Vol.2.pp.255-296,North-Holland,Amsterdam.
[7][13][16][17][22][23][30][33][50]Campbell,J.Y.,and Cochrane,J.H.,“By force of habit:a consumption-based explanation of aggregate stock market behavior”,Journal of Political Economy,1999,2(107),pp.205-251.
[8][9][14][18][19][21][24][31][48][51]Chen,Andrew Y.,“External Habit in a Production Economy:A Model of Asset Prices and Consumption Volatility Risk”,The Review of Financial Studies,2017,30,pp.2890-2932.
[10][15][26][27][28] [47][54]Brunnermeier,M.K.,Sannikov,Y.,“A Macroeconomic Model with a Financial Sector”,The American Economic Review,2014,104,pp.379-421.
[11]Silva H.,“The Risk Channel of Unconventional Moentary Policy”,Unpublished Working paper,2015.
[12]Tella,S.D.,“Uncertainty shocks and balance sheet recessions”,Journal of Political Economy,2017,125,pp.2038-2081.
[20][40][45][49][52]Duarte V.,“Sectoral Reallocation and Endogenous Risk-Aversion:Solving Macro-Finance Models with Machine Learning”,Unpublished Working Paper,2018.
[25][29][32][41]Brunnermeier,M.K.,Sannikov,Y.,“A Macroeconomic Model with a Financial Sector”,The American Economic Review,2014,104,pp.379-421;Brunnermeier,M.K.,Sannikov,Y.,“Macro,Money and Finance:A Continuous Time Approach”,Handbook of Macroeconomics,2016,vol.2B,pp.1497-1546,North-Holland Amsterdam.
[34]Duarte V.,“Sectoral Reallocation and Endogenous Risk-Aversion:Solving Macro-Finance Models with Machine Learning”,Unpublished Working Paper,2018;Duarte,Victor,“Machine Learning for Continuous-Time Economics”,Available at SSRN,https://ssrn.com/abstract=3012602,2018.
[35]郑泽宇、梁博文、顾思宇:《Tensorflow 实战Google深度学习框架》,北京:电子工业出版社,2018年。
[36][42] Brunnermeier,M.K.,Sannikov,Y.,“Macro,Money and Finance:A Continuous Time Approach”,Handbook of Macroeconomics,2016,vol.2B.pp.1497-1546,North-Houand Amsterdam.
[37]Drechsler,Itamar,Alexi Savov,and Philipp Schnabl,“A Model of Monetary Policy and Risk Premia”,the Journal of Finance,2018,73,pp.317-373.
[38]Achdou,Y.,Buera,F.J.,Lasry,J.-M.,Lions,P.L.,Moll,B.,“Partial differential equation models in macroeconomics”,Philosophical Transactions of the Royal Society of London A:Mathematical,Physical and Engineering Sciences,2014,372.
[39]Mnih,V.,Kavukcuoglu,K.,Silver,D.,Rusu,A.A.,Veness,J.,Bellemare,M.G.,Graves,A.,Riedmiller,M.,Fidjeland,A.K.,Ostrovski,G.,Petersen,S.,Beattie,C.,Sadik,A.,Antonoglou,I.,King,H.,Kumaran,D.,Wierstra,D.,Legg,S.,Hassabis,D.,“Human-level control through deep reinforcement learning”,Nature,2015,518,pp.529-533.
[43]Cochrane,J.H.,“Macro-finance”,Review of Finance,2017,21(3),pp.945-985;Campbell,J.Y.,and Cochrane,J.H.,“By force of habit:a consumption-based explanation of aggregate stock market behavior”,Journal of Political Economy,1999,2(107),pp.205-251.
[44]Norets.A.,“Estimation of dynamic discrete choice models using artificial neural network approximations”,Econometric Reviews,2012,31(1),pp.84-106.
[46]Duffie,D.,Singleton,K.J.,“Simulated moments estimation of markov models of asset prices”,Econometrica,1993,61,pp.929-952.
① 关于资产定价与宏观经济割裂的详述,见Brunnermeier,M.K.,Sannikov,Y.,“Macro,Money and Finance:A Continuous Time Approach”,Handbook of Macroeconomics,2016。
(1)Chen、Duarter的研究中剩余消费比率长期均值分别为-2.65和-2.86,外部习惯强度λ取值分别为13.28和6.25,详见Chen,Andrew Y.,“External Habit in a Production Economy:A Model of Asset Prices and Consumption Volatility Risk”,The Review of Financial Studies,2017,30,pp.2890-2932;Duarte V.,“Sectoral Reallocation and Endogenous Risk-Aversion:Solving Macro-Finance Models with Machine Learning”,Unpublished Working paper,2018。在本文研究过程中,我们用国内实际数据和模拟矩估计得到的剩余消费比率长期均值和外部习惯强度λ分别约为-2.5和1.5,而且在异质投资者和考虑货币政策的时两者的估计值依然在-2.5和1.5左右。
[2][4][5][53]Cochrane,J.H.,“Macro-finance”,Review of Finance,2017,21(3),pp.945-985.
[3]明雷、王天轶、杨胜刚:《宏观经济条件下的一个资产定价模型》,《数学的实践与认识》2015年第11期;Brunnermeier,M.K.,Sannikov,Y.,“A Macroeconomic Model with a Financial Sector”,The American Economic Review,2014,104,pp.379-421;Brunnermeier,M.K.,Sannikov,Y.,“Macro,Money and Finance:A Continuous Time Approach”,Handbook of Macroeconomics,2016,Vol.2B,pp.1497-1546,North-Holland,Amsterdam;He,Z.,Krishnamurthy,A.,“A Macroeconomic Framework for Quantifying Systemic Risk”,Working Paper,No.19885,National Bureau of Economic Research,2014;Silva H.,“The Risk Channel of Unconventional Moentary Policy”,Unpublished Working paper,2015;Tella,S.D.,“Uncertainty shocks and balance sheet recessions”,Journal of Political Economy,2017,125,pp.2038-2081;Drechsler,Itamar,Alexi Savov,and Philipp Schnabl,“A Model of Monetary Policy and Risk Premia”,the Journal of Finance,2018,73,pp.317-373.
[6]Mian,A.,and A.Sufi,“Who Bears the Cost of Recessions?The Role of House Prices and Household Debt”,Chapter 5 in John B.Taylor and in Harald Uhlig ed.,Handbook of Macroeconomics,2016,Vol.2.pp.255-296,North-Holland,Amsterdam.
[7][13][16][17][22][23][30][33][50]Campbell,J.Y.,and Cochrane,J.H.,“By force of habit:a consumption-based explanation of aggregate stock market behavior”,Journal of Political Economy,1999,2(107),pp.205-251.
[8][9][14][18][19][21][24][31][48][51]Chen,Andrew Y.,“External Habit in a Production Economy:A Model of Asset Prices and Consumption Volatility Risk”,The Review of Financial Studies,2017,30,pp.2890-2932.
[10][15][26][27][28] [47][54]Brunnermeier,M.K.,Sannikov,Y.,“A Macroeconomic Model with a Financial Sector”,The American Economic Review,2014,104,pp.379-421.
[11]Silva H.,“The Risk Channel of Unconventional Moentary Policy”,Unpublished Working paper,2015.
[12]Tella,S.D.,“Uncertainty shocks and balance sheet recessions”,Journal of Political Economy,2017,125,pp.2038-2081.
[20][40][45][49][52]Duarte V.,“Sectoral Reallocation and Endogenous Risk-Aversion:Solving Macro-Finance Models with Machine Learning”,Unpublished Working Paper,2018.
[25][29][32][41]Brunnermeier,M.K.,Sannikov,Y.,“A Macroeconomic Model with a Financial Sector”,The American Economic Review,2014,104,pp.379-421;Brunnermeier,M.K.,Sannikov,Y.,“Macro,Money and Finance:A Continuous Time Approach”,Handbook of Macroeconomics,2016,vol.2B,pp.1497-1546,North-Holland Amsterdam.
[34]Duarte V.,“Sectoral Reallocation and Endogenous Risk-Aversion:Solving Macro-Finance Models with Machine Learning”,Unpublished Working Paper,2018;Duarte,Victor,“Machine Learning for Continuous-Time Economics”,Available at SSRN,https://ssrn.com/abstract=3012602,2018.
[35]郑泽宇、梁博文、顾思宇:《Tensorflow 实战Google深度学习框架》,北京:电子工业出版社,2018年。
[36][42] Brunnermeier,M.K.,Sannikov,Y.,“Macro,Money and Finance:A Continuous Time Approach”,Handbook of Macroeconomics,2016,vol.2B.pp.1497-1546,North-Houand Amsterdam.
[37]Drechsler,Itamar,Alexi Savov,and Philipp Schnabl,“A Model of Monetary Policy and Risk Premia”,the Journal of Finance,2018,73,pp.317-373.
[38]Achdou,Y.,Buera,F.J.,Lasry,J.-M.,Lions,P.L.,Moll,B.,“Partial differential equation models in macroeconomics”,Philosophical Transactions of the Royal Society of London A:Mathematical,Physical and Engineering Sciences,2014,372.
[39]Mnih,V.,Kavukcuoglu,K.,Silver,D.,Rusu,A.A.,Veness,J.,Bellemare,M.G.,Graves,A.,Riedmiller,M.,Fidjeland,A.K.,Ostrovski,G.,Petersen,S.,Beattie,C.,Sadik,A.,Antonoglou,I.,King,H.,Kumaran,D.,Wierstra,D.,Legg,S.,Hassabis,D.,“Human-level control through deep reinforcement learning”,Nature,2015,518,pp.529-533.
[43]Cochrane,J.H.,“Macro-finance”,Review of Finance,2017,21(3),pp.945-985;Campbell,J.Y.,and Cochrane,J.H.,“By force of habit:a consumption-based explanation of aggregate stock market behavior”,Journal of Political Economy,1999,2(107),pp.205-251.
[44]Norets.A.,“Estimation of dynamic discrete choice models using artificial neural network approximations”,Econometric Reviews,2012,31(1),pp.84-106.
[46]Duffie,D.,Singleton,K.J.,“Simulated moments estimation of markov models of asset prices”,Econometrica,1993,61,pp.929-952.
① 关于资产定价与宏观经济割裂的详述,见Brunnermeier,M.K.,Sannikov,Y.,“Macro,Money and Finance:A Continuous Time Approach”,Handbook of Macroeconomics,2016。
(1)Chen、Duarter的研究中剩余消费比率长期均值分别为-2.65和-2.86,外部习惯强度λ取值分别为13.28和6.25,详见Chen,Andrew Y.,“External Habit in a Production Economy:A Model of Asset Prices and Consumption Volatility Risk”,The Review of Financial Studies,2017,30,pp.2890-2932;Duarte V.,“Sectoral Reallocation and Endogenous Risk-Aversion:Solving Macro-Finance Models with Machine Learning”,Unpublished Working paper,2018。在本文研究过程中,我们用国内实际数据和模拟矩估计得到的剩余消费比率长期均值和外部习惯强度λ分别约为-2.5和1.5,而且在异质投资者和考虑货币政策的时两者的估计值依然在-2.5和1.5左右。