RECURSIVE VALUATION AND SENTIMENTS
|
|
- Jeremy Atkins
- 6 years ago
- Views:
Transcription
1 1 / 32 RECURSIVE VALUATION AND SENTIMENTS Lars Peter Hansen Bendheim Lectures, Princeton University
2 2 / 32 RECURSIVE VALUATION AND SENTIMENTS ABSTRACT Expectations and uncertainty about growth rates that might prevail in the future influence the valuation of risky claims to consumption. I explore this mechanism using the recursive utility model pioneered by Koopmans, Kreps and Porteus, Epstein and Zin and others. This model gives a structured way to investigate how beliefs about the future are reflected in current-period assessments, including in the continuation values of prospective consumption processes and in the stochastic discount factors used to represent prices over alternative investment horizons. Thus the forward-looking nature of the recursive utility model provides an additional channel for which sentiments about the future matter. Using some recently developed methods for studying stochastic processes with uncertain growth, I provide revealing characterizations by exploring some limiting cases and suggesting alternative interpretations for sentiments.
3 3 / 32 MOTIVATION Explore ways in which expectations and uncertainty about future growth rates influence risky claims to consumption. Use a recursive utility model pioneered by Koopmans, Kreps and Porteus and others that, by design, can make beliefs about uncertain events figure prominently in asset valuation. Provide novel and revealing characterizations that will help us understand better how this mechanism operates.
4 4 / 32 RECURSIVE PREFERENCES Koopmans initiated an important line of research on recursive preferences that pushed beyond the additive discounted utility framework. Some References: Stationary Ordinal Utility and Impatience - Koopmans, Econometrica 1960 Koopmans, Diamond and Williamson - Econometrica 1964 Kreps and Porteus - Econometrica 1978 Epstein and Zin - Econometrica 1989
5 5 / 32 KOOPMANS AND RECURSIVE UTILITY Utility representation: V t = Φ[U(C t ), V t+1 ] as a generalization of V t = U(C t ) + exp( δ)v t+1 where C t is the current period consumption vector, V t is the continuation value or what Koopmans called the prospective utility.
6 Kreps-Porteus representation V t = Φ [U(C t ), E (V t+1 F t )] UNCERTAINTY as a generalization of expected utility V t = U(C t ) + exp( δ)e (V t+1 F t ). K-P does not reduce intertemporal compound consumption lotteries. Intertemporal composition of risk matters. I will feature a convenient special case V t = [(ζc t ) 1 ρ + exp( δ) [R t (V t+1 )] 1 ρ] 1 1 ρ where R t (V t+1 ) = ( E [ (V t+1 ) 1 γ F t ]) 1 1 γ where 1 ρ is the elasticity of intertemporal substitution and γ is a risk aversion parameter. Epstein and Zin. 6 / 32
7 Continuation values RECURSIVE UTILITY where V t = [(ζc t ) 1 ρ + exp( δ) [R t (V t+1 )] 1 ρ] 1 1 ρ R t (V t+1 ) = ( E [ (V t+1 ) 1 γ F t ]) 1 1 γ Induces some interesting nonlinearities in valuation. Intertemporal marginal rate of substitution S t+1 S t = exp( δ) ( Ct+1 C t ) ρ [ ] ρ γ Vt+1. R t (V t+1 ) Depends on continuation values, which gives a channel for sentiments to matter. Used to represent asset prices. 7 / 32
8 8 / 32 TALK OUTLINE Mathematical setup Two related applications Continuation values for infinite-horizon problems Asset pricing over alternative investment horizons Perron-Frobenius theory and martingales Applications revisited Estimated long-run risk model Robustness and beliefs
9 9 / 32 APPROACH Use Markov formulations and martingale methods to study compounding in stochastic environments Allow for nonlinear time series models including models of stochastic volatility and stochastic regime shifts. Use the long-term as a frame of reference. Explore the implications of state dependent compounding when we alter the forecast or consumption horizon. Study continuation values, asset values and growth.
10 DISCRETE-TIME FORMULATION Markov process X. Additive functional Y t = t κ(x j, X j 1 ) j=1 Multiplicative functional M t = exp(y t ) = t exp[κ(x j, X j 1 )] j=1 The product of two multiplicative functions is a multiplicative functional. Use multiplicative functionals to model state dependent growth and discounting. 10 / 32
11 11 / 32 EXAMPLE: MIXTURE OF NORMALS MODEL Let {W t+1 } be a multivariate iid sequence of standard normals. Construct an additive functional: Y t+1 Y t = Z t µ + Z t ΛW t+1 where Z is a component of X and evolves as a finite state Markov chain Y 0 = 0. Construct the multiplicative functional: M t = exp(y t ) and study conditional expectations. Recursive structure to the compounding over time.
12 12 / 32 Recall UTILITY RECURSION RECONSIDERED where V t = [(ζc t ) 1 ρ + exp( δ) [R t (V t+1 )] 1 ρ] 1 1 ρ. R t (V t+1 ) = ( E [ (V t+1 ) 1 γ F t ]) 1 1 γ. To adjust for growth, exploit homogeneity and divide by C t to obtain: V t C t = [ ζ 1 ρ + exp( δ) [ ( )] ] 1 Vt+1 C 1 ρ 1 ρ t+1 R t, C t+1 C t When consumption is a multiplicative functional, Vt C t function of the Markov state X t. will be a
13 RESTATING THE RECURSION Explore the parameter region γ > ρ and define 1 ν = 1 γ 1 ρ Restrict ρ 1 in what follows. ( ρ = 1 requires a separate argument.) Consumption dynamics log C t+1 log C t = κ(x t+1, X t ), and solution form: ( ) 1 ρ Vt = h(x t ). C t Formalism: Restated recursion Vf (x) = E (exp[(1 γ)κ(x t+1, X t )]f (X t+1 ) X t = x), h(x) = ζ 1 ρ + exp( δ) [ Vh 1 ν (x) ] 1 1 ν 13 / 32
14 14 / 32 ASSET PRICING OVER ALTERNATIVE Multi-period return: G t E (S t G t X 0 = x) INVESTMENT HORIZONS Logarithm of the expected return adjusted for horizon: 1 t log E (G t X 0 = x) 1 t log E (S tg t X 0 = x) In valuation problems there are two forces at work - stochastic growth G and stochastic discounting S. Study product SG. Term structure of risk and shock prices - look at value implications of marginal changes in growth exposure as represented by changes in G. Build recursions that exploit the Markov structure.
15 Normal mixture model: WARMUP Y t+1 Y t = Z t µ + Z t ΛW t+1 where Z evolves a finite state Markov chain. Realized value of Z t is a coordinate vector. Let f (X t+1 ) = g Z t : [ ] z Mt+1 Mg = E f (X t+1 ) X t = x M t = E [exp(y t+1 Y t )g Z t+1 Z t = z] where M is a matrix with nonnegative entries. Characterize M j. Grows or decays geometrically with the rate given by logarithm of the dominant eigenvalue. Perron-Frobenius theory - dominant eigenvalue has an eigenvector with positive entries. Adjust for growth or decay using the eigenvalue; construct a new probability matrix using the eigenvector. 15 / 32
16 16 / 32 PERRON-FROBENIUS THEORY/ MARTINGALES Solve, E [M 1 e(x 1 ) X 0 = x] = exp(η)e(x) where e is strictly positive. Eigenvalue problem. Construct martingale [ ] e(xt ) ˆM t = exp( ηt)m t. e(x 0 ) Invert to obtain factorization M t = exp(ηt) ˆM t [ e(x0 ) e(x t ) exp(ηt) is the eigenvalue for horizon t and e the eigenfunction. ].
17 17 / 32 MULTIPLICATIVE MARTINGALES Factorization: M t = exp(ηt) ˆM t [ e(x0 ) e(x t ) ]. Change of probability measure: ( ) Ê [f (X t ) X 0 = x] = E ˆM t f (X t ) X 0 = x preserves Markov structure at most one is stochastically stable - Hansen-Scheinkman
18 18 / 32 STOCHASTIC STABILITY [ ] f (Xt ) exp( ηt)e [M t f (X t ) X 0 = x] = e(x)ê e(x t ) X 0 = x Under stochastic stability and the moment restriction: [ ] f (Xt ) Ê <, e(x t ) the right-hand side converges to: [ ] f (Xt ) e(x)ê. e(x t ) Common state dependence independent of f.
19 Recall, CHANGE OF MEASURE REVISITED M t = exp(ηt) ˆM t [ e(x0 ) e(x t ) ]. Consider the stochastic discount factor S = M. Alternative factorization - risk neutral dynamics. Risk-neutral adjustment is a local or one-period adjustment whereas our adjustment features the long-term valuation. Short-term interest rates are typically state dependent whereas η is not. The state dependence of short-term interest rates adjusts for risks over multi-period horizons. Our change of measure features risk adjustments over multiple investment horizons with direct characterizations of limiting behavior. We are interested in other applications as well that include adjustments for stochastic growth. 19 / 32
20 20 / 32 Define CONTINUATION-VALUE RECURSION 1 ν = 1 γ 1 ρ where ν > 0. We presume that ρ 1 in what follows. Consumption dynamics log C t+1 log C t = κ(x t+1, X t ), Solution of the form: ( ) 1 ρ Vt = h(x t ). C t Formalism: Restated recursion Vf (x) = E (exp[(1 γ)κ(x t+1, X t )]f (X t+1 ) X t = x), h(x) = ζ 1 ρ + exp( δ) [ Vh 1 ν (x) ] 1 1 ν
21 21 / 32 ITERATING ON THE RECURSION Construct log M t+1 log M t = (1 γ)κ(x t+1, X t ) = (1 γ)(log C t+1 log C t ) which depends only on γ and not ρ. Factor M t = exp(ηt) ˆM t [ e(x0 ) e(x t ) Use change of measure: ]. Vf (x) = E (exp[(1 γ)κ(x t+1, X t )]f (X t+1 ) X t = x), [ ] f (Xt+1 ) = exp(η)ê e(x t+1 ) X t = x e(x)
22 Recall BOUNDS h(x) = ζ 1 ρ + exp( δ) [ Vh 1 ν (x) ] 1 1 ν The parameter restriction that is a necessary condition for finite values for an infinite horizon: δ η 1 ρ 1 γ Moment inequality ] Ê [e(x t ) 1 ν 1 < implies a bound on the infinite-horizon continuation value. Deduced by applying Jensen s Inequality to Ê [ f (X t+1 ) 1 ν X t = x ] for an appropriately defined f. In parametric examples these inequalities imply parameter restrictions that are typically ignored in practice. 22 / 32
23 WHAT HAPPENS AS WE APPROACH THE DISCOUNT FACTOR LIMIT? Recursive utility mechanism features beliefs about the future or sentiments. Makes these as potent as possible. Recall h(x) = ζ 1 ρ + exp( δ) [ Vh 1 ν (x) ] 1 1 ν Drive ζ to zero (scale doesn t matter) and δ to its bound. Then the equation simplifies to h 1 ν (x) = exp( η)vh 1 ν (x) Continuation value function (relative to consumption) converges to e(x) 1 1 γ up to scale. 23 / 32
24 24 / 32 LIMITING REPRESENTATION OF THE STOCHASTIC DISCOUNT FACTOR S t+1 S t S t+1 S t ( ) Ct+1 ρ [ Vt+1 ρ γ = exp( δ) C t R t(v t+1 )] in general. ( ) Ct+1 γ [ ] ρ γ e(xt+1 ) 1 γ = exp( η) C t e(x t) in the limit. Stochastic discount factors for power utility (constructed with γ) and recursive utility share the same martingale components. Changing ρ does not alter the martingale component.
25 ESTIMATED EXAMPLE Motivated by the work of Bansal and Yaron, I fit the following model to the aggregate time series data and specified in continuous time. dx [1] t = A 11 X [1] t dt + dx [2] t = A 22 (X [2] X [2] dy t = µdt + H 1 X [1] t dt + t B 1 dw t, t 1)dt + X [2] t B 2 dw t X [2] t FdW t. The continuous time autogregressive coefficients are: [ ] A 11 = A =.07. The shock loadings for the components of the state vector are: [ ] B 1 = B = [ ] F = [ ] H 1 = [ ] 25 / 32
26 SHOCK-PRICE TRAJECTORIES FOR POWER AND RECURSIVE UTILITY Temporary Shock Price Elasticity Permanent Shock Price Elasticity Volatility Shock Price Elasticity quarters See second lecture for interpretation and construction details. 26 / 32
27 SHOCK-PRICE TRAJECTORIES FOR ALTERNATIVE VALUES OF THE EIS 6 x 10 3 Temporary Shock Price Elasticity Permanent Shock Price Elasticity Volatility Shock Price Elasticity rho= quarters See second lecture for interpretation and construction details. 27 / 32
28 28 / 32 IMPACT OF STOCHASTIC VOLATILITY FIGURE: Densities for shock price elasticities for exposure to the growth rate shock. Green volatility shock and blue growth shock.
29 IMPACT OF PARAMETER ESTIMATION FIGURE: Densities for shock price elasticities for exposure to the growth rate shock. Green volatility shock an blue growth shock. See also Hansen, Heaton and Li (JPE). 29 / 32
30 30 / 32 ROBUSTNESS AND DISTORTED BELIEFS Lack of investor confidence in the models they use. Investors explore alternative specifications for probability laws subject to penalization. Martingale is the implied worst case model. Parameter θ > 0 determines the magnitude of the penalization where γ > 1 and θ = 1 ρ 1 γ. The parameters are now ρ and θ where ρ continues to measure the elasticity of intertemporal substitution. Related methods have a long history in robust control theory and statistics. Axiomatic treatments in recent decision theory papers.
31 ROBUSTNESS AND DISTORTED BELIEFS In empirical applications it is common to assume a large value of the risk aversion parameter γ. Instead appeal to a concern about robustness. For v t+1 = 1 ρ log V t+1 and solve min E [v t+1 F t ] + 1 m 0,E[m F t]=1 θ E [m log m F t] [ ( = θ log E exp 1 ) ] θ v t+1 F t. Minimizing m: exp ( 1 θ m t+1 = v ) t+1 E [ exp ( 1 θ v ) ] = t+1 Ft (V t+1 ) 1 γ E [(V t+1 ) 1 γ F t ]. This gives rise to the exponential tilting solution as m tilts the density in directions that have the largest adverse consequences for the continuation value. Altered beliefs. See third lecture for more details. 31 / 32
32 WHERE DOES THIS LEAVE US? The flat term structure for recursive utility shows the potential importance of macro growth components on asset pricing. Typical rational expectations modeling assumes investor confidence and uses the cross equation restrictions to identify long-term growth components from asset prices. Instead do asset prices identify subjective beliefs of investors and risk aversion? Predictable components of macroeconomic growth and volatility are hard for an econometrician to measure from macroeconomic data. Questions What are the interesting shocks? Where does investor confidence come from when confronted by weak sample evidence? Motivates my interest in modeling investors who have a concern for model specification. What about learning? Concerns about model specification of the type I described abstract from learning. 32 / 32
MODELING THE LONG RUN:
MODELING THE LONG RUN: VALUATION IN DYNAMIC STOCHASTIC ECONOMIES 1 Lars Peter Hansen Valencia 1 Related papers:hansen,heaton and Li, JPE, 2008; Hansen and Scheinkman, Econometrica, 2009 1 / 45 2 / 45 SOME
More informationLONG-TERM COMPONENTS OF RISK PRICES 1
LONG-TERM COMPONENTS OF RISK PRICES 1 Lars Peter Hansen Tjalling C. Koopmans Lectures, September 2008 1 Related papers:hansen,heaton and Li, JPE, 2008; Hansen and Scheinkman, forthcoming Econometrica;
More informationEXAMINING MACROECONOMIC MODELS
1 / 24 EXAMINING MACROECONOMIC MODELS WITH FINANCE CONSTRAINTS THROUGH THE LENS OF ASSET PRICING Lars Peter Hansen Benheim Lectures, Princeton University EXAMINING MACROECONOMIC MODELS WITH FINANCING CONSTRAINTS
More informationToward A Term Structure of Macroeconomic Risk
Toward A Term Structure of Macroeconomic Risk Pricing Unexpected Growth Fluctuations Lars Peter Hansen 1 2007 Nemmers Lecture, Northwestern University 1 Based in part joint work with John Heaton, Nan Li,
More informationStochastic Compounding and Uncertain Valuation
Stochastic Compounding and Uncertain Valuation Lars Peter Hansen University of Chicago and NBER José A. Scheinkman Columbia University, Princeton University and NBER January 31, 2016 Abstract Exploring
More informationUNCERTAINTY AND VALUATION
1 / 29 UNCERTAINTY AND VALUATION MODELING CHALLENGES Lars Peter Hansen University of Chicago June 1, 2013 Address to the Macro-Finance Society Lord Kelvin s dictum: I often say that when you can measure
More informationRisk Pricing over Alternative Investment Horizons
Risk Pricing over Alternative Investment Horizons Lars Peter Hansen University of Chicago and the NBER June 19, 2012 Abstract I explore methods that characterize model-based valuation of stochastically
More informationNotes on Epstein-Zin Asset Pricing (Draft: October 30, 2004; Revised: June 12, 2008)
Backus, Routledge, & Zin Notes on Epstein-Zin Asset Pricing (Draft: October 30, 2004; Revised: June 12, 2008) Asset pricing with Kreps-Porteus preferences, starting with theoretical results from Epstein
More informationSteven Heston: Recovering the Variance Premium. Discussion by Jaroslav Borovička November 2017
Steven Heston: Recovering the Variance Premium Discussion by Jaroslav Borovička November 2017 WHAT IS THE RECOVERY PROBLEM? Using observed cross-section(s) of prices (of Arrow Debreu securities), infer
More information1 Consumption and saving under uncertainty
1 Consumption and saving under uncertainty 1.1 Modelling uncertainty As in the deterministic case, we keep assuming that agents live for two periods. The novelty here is that their earnings in the second
More informationCONSUMPTION-BASED MACROECONOMIC MODELS OF ASSET PRICING THEORY
ECONOMIC ANNALS, Volume LXI, No. 211 / October December 2016 UDC: 3.33 ISSN: 0013-3264 DOI:10.2298/EKA1611007D Marija Đorđević* CONSUMPTION-BASED MACROECONOMIC MODELS OF ASSET PRICING THEORY ABSTRACT:
More informationLECTURE NOTES 10 ARIEL M. VIALE
LECTURE NOTES 10 ARIEL M VIALE 1 Behavioral Asset Pricing 11 Prospect theory based asset pricing model Barberis, Huang, and Santos (2001) assume a Lucas pure-exchange economy with three types of assets:
More informationRisks for the Long Run: A Potential Resolution of Asset Pricing Puzzles
: A Potential Resolution of Asset Pricing Puzzles, JF (2004) Presented by: Esben Hedegaard NYUStern October 12, 2009 Outline 1 Introduction 2 The Long-Run Risk Solving the 3 Data and Calibration Results
More informationConsumption and Portfolio Decisions When Expected Returns A
Consumption and Portfolio Decisions When Expected Returns Are Time Varying September 10, 2007 Introduction In the recent literature of empirical asset pricing there has been considerable evidence of time-varying
More informationBirkbeck MSc/Phd Economics. Advanced Macroeconomics, Spring Lecture 2: The Consumption CAPM and the Equity Premium Puzzle
Birkbeck MSc/Phd Economics Advanced Macroeconomics, Spring 2006 Lecture 2: The Consumption CAPM and the Equity Premium Puzzle 1 Overview This lecture derives the consumption-based capital asset pricing
More informationCHOICE THEORY, UTILITY FUNCTIONS AND RISK AVERSION
CHOICE THEORY, UTILITY FUNCTIONS AND RISK AVERSION Szabolcs Sebestyén szabolcs.sebestyen@iscte.pt Master in Finance INVESTMENTS Sebestyén (ISCTE-IUL) Choice Theory Investments 1 / 65 Outline 1 An Introduction
More informationNon-Time-Separable Utility: Habit Formation
Finance 400 A. Penati - G. Pennacchi Non-Time-Separable Utility: Habit Formation I. Introduction Thus far, we have considered time-separable lifetime utility specifications such as E t Z T t U[C(s), s]
More informationLong-Run Risk, the Wealth-Consumption Ratio, and the Temporal Pricing of Risk
Long-Run Risk, the Wealth-Consumption Ratio, and the Temporal Pricing of Risk By Ralph S.J. Koijen, Hanno Lustig, Stijn Van Nieuwerburgh and Adrien Verdelhan Representative agent consumption-based asset
More informationAppendix to: Long-Run Asset Pricing Implications of Housing Collateral Constraints
Appendix to: Long-Run Asset Pricing Implications of Housing Collateral Constraints Hanno Lustig UCLA and NBER Stijn Van Nieuwerburgh June 27, 2006 Additional Figures and Tables Calibration of Expenditure
More informationMacroeconomics Sequence, Block I. Introduction to Consumption Asset Pricing
Macroeconomics Sequence, Block I Introduction to Consumption Asset Pricing Nicola Pavoni October 21, 2016 The Lucas Tree Model This is a general equilibrium model where instead of deriving properties of
More informationSkewness in Expected Macro Fundamentals and the Predictability of Equity Returns: Evidence and Theory
Skewness in Expected Macro Fundamentals and the Predictability of Equity Returns: Evidence and Theory Ric Colacito, Eric Ghysels, Jinghan Meng, and Wasin Siwasarit 1 / 26 Introduction Long-Run Risks Model:
More informationImplementing an Agent-Based General Equilibrium Model
Implementing an Agent-Based General Equilibrium Model 1 2 3 Pure Exchange General Equilibrium We shall take N dividend processes δ n (t) as exogenous with a distribution which is known to all agents There
More informationIdentifying Long-Run Risks: A Bayesian Mixed-Frequency Approach
Identifying : A Bayesian Mixed-Frequency Approach Frank Schorfheide University of Pennsylvania CEPR and NBER Dongho Song University of Pennsylvania Amir Yaron University of Pennsylvania NBER February 12,
More informationLong-term Risk: An Operator Approach 1
Long-term Risk: An Operator Approach 1 Lars Peter Hansen University of Chicago and NBER lhansen@uchicago.edu José Scheinkman Princeton University and NBER joses@princeton.edu June 18, 8 1 Comments from
More informationThe stochastic calculus
Gdansk A schedule of the lecture Stochastic differential equations Ito calculus, Ito process Ornstein - Uhlenbeck (OU) process Heston model Stopping time for OU process Stochastic differential equations
More informationA Note on the Economics and Statistics of Predictability: A Long Run Risks Perspective
A Note on the Economics and Statistics of Predictability: A Long Run Risks Perspective Ravi Bansal Dana Kiku Amir Yaron November 14, 2007 Abstract Asset return and cash flow predictability is of considerable
More informationProblem set 5. Asset pricing. Markus Roth. Chair for Macroeconomics Johannes Gutenberg Universität Mainz. Juli 5, 2010
Problem set 5 Asset pricing Markus Roth Chair for Macroeconomics Johannes Gutenberg Universität Mainz Juli 5, 200 Markus Roth (Macroeconomics 2) Problem set 5 Juli 5, 200 / 40 Contents Problem 5 of problem
More informationThe Life Cycle Model with Recursive Utility: Defined benefit vs defined contribution.
The Life Cycle Model with Recursive Utility: Defined benefit vs defined contribution. Knut K. Aase Norwegian School of Economics 5045 Bergen, Norway IACA/PBSS Colloquium Cancun 2017 June 6-7, 2017 1. Papers
More informationInternational Asset Pricing and Risk Sharing with Recursive Preferences
p. 1/3 International Asset Pricing and Risk Sharing with Recursive Preferences Riccardo Colacito Prepared for Tom Sargent s PhD class (Part 1) Roadmap p. 2/3 Today International asset pricing (exchange
More informationAsset Pricing and Equity Premium Puzzle. E. Young Lecture Notes Chapter 13
Asset Pricing and Equity Premium Puzzle 1 E. Young Lecture Notes Chapter 13 1 A Lucas Tree Model Consider a pure exchange, representative household economy. Suppose there exists an asset called a tree.
More informationConsumption and Portfolio Choice under Uncertainty
Chapter 8 Consumption and Portfolio Choice under Uncertainty In this chapter we examine dynamic models of consumer choice under uncertainty. We continue, as in the Ramsey model, to take the decision of
More informationHeterogeneous Firm, Financial Market Integration and International Risk Sharing
Heterogeneous Firm, Financial Market Integration and International Risk Sharing Ming-Jen Chang, Shikuan Chen and Yen-Chen Wu National DongHwa University Thursday 22 nd November 2018 Department of Economics,
More information1 Precautionary Savings: Prudence and Borrowing Constraints
1 Precautionary Savings: Prudence and Borrowing Constraints In this section we study conditions under which savings react to changes in income uncertainty. Recall that in the PIH, when you abstract from
More informationRoss Recovery theorem and its extension
Ross Recovery theorem and its extension Ho Man Tsui Kellogg College University of Oxford A thesis submitted in partial fulfillment of the MSc in Mathematical Finance April 22, 2013 Acknowledgements I am
More informationA Continuous-Time Asset Pricing Model with Habits and Durability
A Continuous-Time Asset Pricing Model with Habits and Durability John H. Cochrane June 14, 2012 Abstract I solve a continuous-time asset pricing economy with quadratic utility and complex temporal nonseparabilities.
More informationLong-term Risk: An Operator Approach 1
Long-term Risk: An Operator Approach 1 Lars Peter Hansen University of Chicago and NBER lhansen@uchicago.edu José Scheinkman Princeton University and NBER joses@princeton.edu November 7, 27 1 Comments
More informationDisagreement, Speculation, and Aggregate Investment
Disagreement, Speculation, and Aggregate Investment Steven D. Baker Burton Hollifield Emilio Osambela October 19, 213 We thank Elena N. Asparouhova, Tony Berrada, Jaroslav Borovička, Peter Bossaerts, David
More informationLong-Run Risks, the Macroeconomy, and Asset Prices
Long-Run Risks, the Macroeconomy, and Asset Prices By RAVI BANSAL, DANA KIKU AND AMIR YARON Ravi Bansal and Amir Yaron (2004) developed the Long-Run Risk (LRR) model which emphasizes the role of long-run
More informationSensitivity Analysis on Long-term Cash flows
Sensitivity Analysis on Long-term Cash flows Hyungbin Park Worcester Polytechnic Institute 19 March 2016 Eastern Conference on Mathematical Finance Worcester Polytechnic Institute, Worceseter, MA 1 / 49
More informationApplication of Stochastic Calculus to Price a Quanto Spread
Application of Stochastic Calculus to Price a Quanto Spread Christopher Ting http://www.mysmu.edu/faculty/christophert/ Algorithmic Quantitative Finance July 15, 2017 Christopher Ting July 15, 2017 1/33
More informationUNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS
UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Postponed exam: ECON4310 Macroeconomic Theory Date of exam: Monday, December 14, 2015 Time for exam: 09:00 a.m. 12:00 noon The problem set covers 13 pages (incl.
More informationAppendix to: Quantitative Asset Pricing Implications of Housing Collateral Constraints
Appendix to: Quantitative Asset Pricing Implications of Housing Collateral Constraints Hanno Lustig UCLA and NBER Stijn Van Nieuwerburgh December 5, 2005 1 Additional Figures and Tables Calibration of
More informationBeliefs, Doubts and Learning: Valuing Macroeconomic Risk
Beliefs, Doubts and Learning: Valuing Macroeconomic Risk Lars Peter Hansen Published in the May 2007 Issue of the American Economic Review Prepared for the 2007 Ely Lecture of for the American Economic
More informationAll Investors are Risk-averse Expected Utility Maximizers
All Investors are Risk-averse Expected Utility Maximizers Carole Bernard (UW), Jit Seng Chen (GGY) and Steven Vanduffel (Vrije Universiteit Brussel) AFFI, Lyon, May 2013. Carole Bernard All Investors are
More informationLeads, Lags, and Logs: Asset Prices in Business Cycle Analysis
Leads, Lags, and Logs: Asset Prices in Business Cycle Analysis David Backus (NYU), Bryan Routledge (CMU), and Stanley Zin (CMU) Society for Economic Dynamics, July 2006 This version: July 11, 2006 Backus,
More informationLong Run Labor Income Risk
Long Run Labor Income Risk Robert F. Dittmar Francisco Palomino November 00 Department of Finance, Stephen Ross School of Business, University of Michigan, Ann Arbor, MI 4809, email: rdittmar@umich.edu
More informationLecture 1: Traditional Open Macro Models and Monetary Policy
Lecture 1: Traditional Open Macro Models and Monetary Policy Isabelle Méjean isabelle.mejean@polytechnique.edu http://mejean.isabelle.googlepages.com/ Master Economics and Public Policy, International
More informationLocal Volatility Dynamic Models
René Carmona Bendheim Center for Finance Department of Operations Research & Financial Engineering Princeton University Columbia November 9, 27 Contents Joint work with Sergey Nadtochyi Motivation 1 Understanding
More informationRisks for the Long Run and the Real Exchange Rate
Risks for the Long Run and the Real Exchange Rate Riccardo Colacito - NYU and UNC Kenan-Flagler Mariano M. Croce - NYU Risks for the Long Run and the Real Exchange Rate, UCLA, 2.22.06 p. 1/29 Set the stage
More informationAll Investors are Risk-averse Expected Utility Maximizers. Carole Bernard (UW), Jit Seng Chen (GGY) and Steven Vanduffel (Vrije Universiteit Brussel)
All Investors are Risk-averse Expected Utility Maximizers Carole Bernard (UW), Jit Seng Chen (GGY) and Steven Vanduffel (Vrije Universiteit Brussel) First Name: Waterloo, April 2013. Last Name: UW ID #:
More informationIntertemporal Risk Attitude. Lecture 7. Kreps & Porteus Preference for Early or Late Resolution of Risk
Intertemporal Risk Attitude Lecture 7 Kreps & Porteus Preference for Early or Late Resolution of Risk is an intrinsic preference for the timing of risk resolution is a general characteristic of recursive
More informationImplications of Long-Run Risk for. Asset Allocation Decisions
Implications of Long-Run Risk for Asset Allocation Decisions Doron Avramov and Scott Cederburg March 1, 2012 Abstract This paper proposes a structural approach to long-horizon asset allocation. In particular,
More informationMisspecified Recovery
Misspecified Recovery Jaroslav Borovička New York University jaroslav.borovicka@nyu.edu Lars Peter Hansen University of Chicago and NBER lhansen@uchicago.edu José A. Scheinkman Columbia University, Princeton
More informationTangent Lévy Models. Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford.
Tangent Lévy Models Sergey Nadtochiy (joint work with René Carmona) Oxford-Man Institute of Quantitative Finance University of Oxford June 24, 2010 6th World Congress of the Bachelier Finance Society Sergey
More informationAMH4 - ADVANCED OPTION PRICING. Contents
AMH4 - ADVANCED OPTION PRICING ANDREW TULLOCH Contents 1. Theory of Option Pricing 2 2. Black-Scholes PDE Method 4 3. Martingale method 4 4. Monte Carlo methods 5 4.1. Method of antithetic variances 5
More informationNominal Exchange Rates Obstfeld and Rogoff, Chapter 8
Nominal Exchange Rates Obstfeld and Rogoff, Chapter 8 1 Cagan Model of Money Demand 1.1 Money Demand Demand for real money balances ( M P ) depends negatively on expected inflation In logs m d t p t =
More informationAsset pricing in the frequency domain: theory and empirics
Asset pricing in the frequency domain: theory and empirics Ian Dew-Becker and Stefano Giglio Duke Fuqua and Chicago Booth 11/27/13 Dew-Becker and Giglio (Duke and Chicago) Frequency-domain asset pricing
More informationComprehensive Exam. August 19, 2013
Comprehensive Exam August 19, 2013 You have a total of 180 minutes to complete the exam. If a question seems ambiguous, state why, sharpen it up and answer the sharpened-up question. Good luck! 1 1 Menu
More informationOnline Appendix (Not intended for Publication): Federal Reserve Credibility and the Term Structure of Interest Rates
Online Appendix Not intended for Publication): Federal Reserve Credibility and the Term Structure of Interest Rates Aeimit Lakdawala Michigan State University Shu Wu University of Kansas August 2017 1
More informationThe Role of Risk Aversion and Intertemporal Substitution in Dynamic Consumption-Portfolio Choice with Recursive Utility
The Role of Risk Aversion and Intertemporal Substitution in Dynamic Consumption-Portfolio Choice with Recursive Utility Harjoat S. Bhamra Sauder School of Business University of British Columbia Raman
More information1 Dynamic programming
1 Dynamic programming A country has just discovered a natural resource which yields an income per period R measured in terms of traded goods. The cost of exploitation is negligible. The government wants
More informationMarket Survival in the Economies with Heterogeneous Beliefs
Market Survival in the Economies with Heterogeneous Beliefs Viktor Tsyrennikov Preliminary and Incomplete February 28, 2006 Abstract This works aims analyzes market survival of agents with incorrect beliefs.
More informationOnline Appendix: Extensions
B Online Appendix: Extensions In this online appendix we demonstrate that many important variations of the exact cost-basis LUL framework remain tractable. In particular, dual problem instances corresponding
More informationStability in geometric & functional inequalities
Stability in geometric & functional inequalities A. Figalli The University of Texas at Austin www.ma.utexas.edu/users/figalli/ Alessio Figalli (UT Austin) Stability in geom. & funct. ineq. Krakow, July
More informationFinancial Risk Management
Financial Risk Management Professor: Thierry Roncalli Evry University Assistant: Enareta Kurtbegu Evry University Tutorial exercices #4 1 Correlation and copulas 1. The bivariate Gaussian copula is given
More informationAsset Pricing under Information-processing Constraints
The University of Hong Kong From the SelectedWorks of Yulei Luo 00 Asset Pricing under Information-processing Constraints Yulei Luo, The University of Hong Kong Eric Young, University of Virginia Available
More informationINTERTEMPORAL ASSET ALLOCATION: THEORY
INTERTEMPORAL ASSET ALLOCATION: THEORY Multi-Period Model The agent acts as a price-taker in asset markets and then chooses today s consumption and asset shares to maximise lifetime utility. This multi-period
More informationInfluence of Real Interest Rate Volatilities on Long-term Asset Allocation
200 2 Ó Ó 4 4 Dec., 200 OR Transactions Vol.4 No.4 Influence of Real Interest Rate Volatilities on Long-term Asset Allocation Xie Yao Liang Zhi An 2 Abstract For one-period investors, fixed income securities
More informationMacroeconomics and finance
Macroeconomics and finance 1 1. Temporary equilibrium and the price level [Lectures 11 and 12] 2. Overlapping generations and learning [Lectures 13 and 14] 2.1 The overlapping generations model 2.2 Expectations
More informationConsumption- Savings, Portfolio Choice, and Asset Pricing
Finance 400 A. Penati - G. Pennacchi Consumption- Savings, Portfolio Choice, and Asset Pricing I. The Consumption - Portfolio Choice Problem We have studied the portfolio choice problem of an individual
More informationIMPA Commodities Course : Forward Price Models
IMPA Commodities Course : Forward Price Models Sebastian Jaimungal sebastian.jaimungal@utoronto.ca Department of Statistics and Mathematical Finance Program, University of Toronto, Toronto, Canada http://www.utstat.utoronto.ca/sjaimung
More informationLinearity-Generating Processes, Unspanned Stochastic Volatility, and Interest-Rate Option Pricing
Linearity-Generating Processes, Unspanned Stochastic Volatility, and Interest-Rate Option Pricing Liuren Wu, Baruch College Joint work with Peter Carr and Xavier Gabaix at New York University Board of
More informationRobustness, Model Uncertainty and Pricing
Robustness, Model Uncertainty and Pricing Antoon Pelsser 1 1 Maastricht University & Netspar Email: a.pelsser@maastrichtuniversity.nl 29 October 2010 Swissquote Conference Lausanne A. Pelsser (Maastricht
More informationFINANCIAL OPTIMIZATION. Lecture 5: Dynamic Programming and a Visit to the Soft Side
FINANCIAL OPTIMIZATION Lecture 5: Dynamic Programming and a Visit to the Soft Side Copyright c Philip H. Dybvig 2008 Dynamic Programming All situations in practice are more complex than the simple examples
More informationThe Paradox of Asset Pricing. Introductory Remarks
The Paradox of Asset Pricing Introductory Remarks 1 On the predictive power of modern finance: It is a very beautiful line of reasoning. The only problem is that perhaps it is not true. (After all, nature
More informationOptimal Portfolio Composition for Sovereign Wealth Funds
Optimal Portfolio Composition for Sovereign Wealth Funds Diaa Noureldin* (joint work with Khouzeima Moutanabbir) *Department of Economics The American University in Cairo Oil, Middle East, and the Global
More informationRisk Price Dynamics. Lars Peter Hansen. University of Chicago. José A. Scheinkman. Princeton University. July 13, Abstract
Risk Price Dynamics Jaroslav Borovička University of Chicago Lars Peter Hansen University of Chicago José A. Scheinkman Princeton University July 13, 21 Mark Hendricks University of Chicago Abstract We
More informationCombining Real Options and game theory in incomplete markets.
Combining Real Options and game theory in incomplete markets. M. R. Grasselli Mathematics and Statistics McMaster University Further Developments in Quantitative Finance Edinburgh, July 11, 2007 Successes
More informationTopic 11: Disability Insurance
Topic 11: Disability Insurance Nathaniel Hendren Harvard Spring, 2018 Nathaniel Hendren (Harvard) Disability Insurance Spring, 2018 1 / 63 Disability Insurance Disability insurance in the US is one of
More informationThe Risky Steady State and the Interest Rate Lower Bound
The Risky Steady State and the Interest Rate Lower Bound Timothy Hills Taisuke Nakata Sebastian Schmidt New York University Federal Reserve Board European Central Bank 1 September 2016 1 The views expressed
More informationNBER WORKING PAPER SERIES CONSUMPTION STRIKES BACK?: MEASURING LONG-RUN RISK. Lars Peter Hansen John Heaton Nan Li
NBER WORKING PAPER SERIES CONSUMPTION STRIKES BACK?: MEASURING LONG-RUN RISK Lars Peter Hansen John Heaton Nan Li Working Paper 11476 http://www.nber.org/papers/w11476 NATIONAL BUREAU OF ECONOMIC RESEARCH
More informationM.I.T Fall Practice Problems
M.I.T. 15.450-Fall 2010 Sloan School of Management Professor Leonid Kogan Practice Problems 1. Consider a 3-period model with t = 0, 1, 2, 3. There are a stock and a risk-free asset. The initial stock
More informationRisks For The Long Run And The Real Exchange Rate
Riccardo Colacito, Mariano M. Croce Overview International Equity Premium Puzzle Model with long-run risks Calibration Exercises Estimation Attempts & Proposed Extensions Discussion International Equity
More informationExtended Libor Models and Their Calibration
Extended Libor Models and Their Calibration Denis Belomestny Weierstraß Institute Berlin Vienna, 16 November 2007 Denis Belomestny (WIAS) Extended Libor Models and Their Calibration Vienna, 16 November
More informationReturn to Capital in a Real Business Cycle Model
Return to Capital in a Real Business Cycle Model Paul Gomme, B. Ravikumar, and Peter Rupert Can the neoclassical growth model generate fluctuations in the return to capital similar to those observed in
More informationHousehold Finance in China
Household Finance in China Russell Cooper 1 and Guozhong Zhu 2 October 22, 2016 1 Department of Economics, the Pennsylvania State University and NBER, russellcoop@gmail.com 2 School of Business, University
More informationGeneral Examination in Macroeconomic Theory SPRING 2016
HARVARD UNIVERSITY DEPARTMENT OF ECONOMICS General Examination in Macroeconomic Theory SPRING 2016 You have FOUR hours. Answer all questions Part A (Prof. Laibson): 60 minutes Part B (Prof. Barro): 60
More informationChapter 5 Macroeconomics and Finance
Macro II Chapter 5 Macro and Finance 1 Chapter 5 Macroeconomics and Finance Main references : - L. Ljundqvist and T. Sargent, Chapter 7 - Mehra and Prescott 1985 JME paper - Jerman 1998 JME paper - J.
More informationMaking Decisions. CS 3793 Artificial Intelligence Making Decisions 1
Making Decisions CS 3793 Artificial Intelligence Making Decisions 1 Planning under uncertainty should address: The world is nondeterministic. Actions are not certain to succeed. Many events are outside
More informationThe Equity Premium Puzzle, the consumption puzzle and the investment puzzle with Recursive Utility: Implications for optimal pensions.
The Equity Premium Puzzle, the consumption puzzle and the investment puzzle with Recursive Utility: Implications for optimal pensions. Knut K. Aase Norwegian School of Economics 5045 Bergen, Norway IAALS
More information1 Asset Pricing: Bonds vs Stocks
Asset Pricing: Bonds vs Stocks The historical data on financial asset returns show that one dollar invested in the Dow- Jones yields 6 times more than one dollar invested in U.S. Treasury bonds. The return
More informationChapter 9 Dynamic Models of Investment
George Alogoskoufis, Dynamic Macroeconomic Theory, 2015 Chapter 9 Dynamic Models of Investment In this chapter we present the main neoclassical model of investment, under convex adjustment costs. This
More informationDisaster risk and its implications for asset pricing Online appendix
Disaster risk and its implications for asset pricing Online appendix Jerry Tsai University of Oxford Jessica A. Wachter University of Pennsylvania December 12, 2014 and NBER A The iid model This section
More informationPricing Dynamic Solvency Insurance and Investment Fund Protection
Pricing Dynamic Solvency Insurance and Investment Fund Protection Hans U. Gerber and Gérard Pafumi Switzerland Abstract In the first part of the paper the surplus of a company is modelled by a Wiener process.
More informationExplaining basic asset pricing facts with models that are consistent with basic macroeconomic facts
Aggregate Asset Pricing Explaining basic asset pricing facts with models that are consistent with basic macroeconomic facts Models with quantitative implications Starting point: Mehra and Precott (1985),
More informationGeneral Examination in Macroeconomic Theory. Fall 2010
HARVARD UNIVERSITY DEPARTMENT OF ECONOMICS General Examination in Macroeconomic Theory Fall 2010 ----------------------------------------------------------------------------------------------------------------
More informationA Note on the Relation between Risk Aversion, Intertemporal Substitution and Timing of the Resolution of Uncertainty
ANNALS OF ECONOMICS AND FINANCE 2, 251 256 (2006) A Note on the Relation between Risk Aversion, Intertemporal Substitution and Timing of the Resolution of Uncertainty Johanna Etner GAINS, Université du
More informationLeads, Lags, and Logs: Asset Prices in Business Cycle Analysis
Leads, Lags, and Logs: Asset Prices in Business Cycle Analysis David Backus (NYU), Bryan Routledge (CMU), and Stanley Zin (CMU) NYU Macro Lunch December 7, 2006 This version: December 7, 2006 Backus, Routledge,
More informationRecursive Preferences
Recursive Preferences David K. Backus, Bryan R. Routledge, and Stanley E. Zin Revised: December 5, 2005 Abstract We summarize the class of recursive preferences. These preferences fit naturally with recursive
More informationAmbiguity, Learning, and Asset Returns
Ambiguity, Learning, and Asset Returns Nengjiu Ju and Jianjun Miao September 2007 Abstract We develop a consumption-based asset-pricing model in which the representative agent is ambiguous about the hidden
More information