Dynamic Robust Utility P Beissner* F Maccheroni # M Marinacci # S - PowerPoint PPT Presentation

Dynamic Robust Utility Dynamic Robust Utility P Beissner* F Maccheroni # M Marinacci # S Mukerji @ * ANU # U Bocconi and @ QMU Robustness in Economics and Econometrics Conference BFI - University of Chicago April 2019

Dynamic Robust Utility Robust Dynamic Utilities Robust Dynamic Utilities We study utility processes V ( c ) = ( V t ( c )) that evaluate consumption processes c = ( c t ) They are recursively defined as maximal bounded solutions of a forward recursion: � � T � V t ( c ) = E t f ( s , c s , V s ( c ) , σ s ( V ( c ))) d s t They are called robust because they feature σ s ( V ( c )) , interpreted as a variability index for the valuation process V ( c )

Dynamic Robust Utility Two noteworthy special cases Two noteworthy special cases Let f ( s , c , y , z ) = g ( s , c , y ) − κ t | z | where κ t accounts for “maxmin” ambiguity aversion a la Gilboa and Schmeidler (1989) We have a specification a la Chen and Epstein (2002), in their leading example of κ -ignorance (more generally, under rectangularity)

Dynamic Robust Utility Two noteworthy special cases Two noteworthy special cases Let f ( t , c , y , z ) = g ( t , c , y ) − υ ( y ) ς 2 t z 2 2 where ς t and υ account for model ambiguity perception and attitude a la Klibanoff et al. (2005) We have a specification a la Hansen and Sargent (2011) and Hansen and Miao (2018) This is the specification upon which we will delve deeper

Dynamic Robust Utility Further special cases Further special cases Here the recursion becomes � � T � � � g ( s , c s , V s ( c )) − 1 2 υ ( V t ( c )) ς 2 t σ 2 V t ( c ) = E t s ( V ( c )) d s t If either υ or ς are zero, it reduces to a stochastic differential utility recursion a la Duffie and Epstein (1992): � � T � V t ( c ) = E t g ( s , c s , V s ( c )) d s t

Dynamic Robust Utility Further special cases Further special cases If, in addition, g ( t , c , y ) = u ( c ) − β y , it further reduces to a traditional expected utility recursion � � T � V t ( c ) = E t ( u ( c s ) − β s V s ( c )) d s t that is, � � T � e − β ( s − t ) u ( c s ) d s V t ( c ) = E t t

Dynamic Robust Utility Heuristic interpretation (sketch) Heuristic interpretation (sketch) DM may posit a finite set � � Q θ : θ ∈ Θ of probability measures equivalent to P , each interpreted as an alternative possible true model We can regard Θ as a collection of time-varying parameters θ ∈ L 0 that parametrize the models Q θ via the relation � d Q θ � � � � t � t − 1 0 θ 2 E t = exp s d s + 0 θ s d B s d P 2

Dynamic Robust Utility Heuristic interpretation (sketch) Heuristic interpretation (sketch) A belief process π = ( π t ) ⊆ ∆ ( Θ ) describes the evolution of DM’s beliefs about parameters The expected parameter process ϑ = ( ϑ t ) is given by ϑ t = ∑ θ θ t π t ( θ ) Process ς = ( ς t ) ∈ L 2 of standard deviation is given by � θ 2 t π t ( θ ) − ϑ 2 ∑ ς t = t θ Process ς accounts for the DM perception of model ambiguity, a subjective feature that depends on the belief process

Dynamic Robust Utility Heuristic interpretation (sketch) Heuristic interpretation (sketch) Let φ : R → R be a function that describes attitudes toward model ambiguity (as in the smooth ambiguity model) In the robust aggregator f ( t , c , y , z ) = u ( c t ) − β y − 1 2 ς 2 t υ ( y ) z 2 one reads υ = λ φ ≡ − φ �� φ � A key caveat detailed in the “real” heuristic

Dynamic Robust Utility Summing up Summing up Interpretation of the separable specification: f ( t , c , y , z ) = u ( c t ) − β y − 1 2 ς 2 t υ ( y ) z 2 ς indexes the degree of (perceived) model ambiguity over time: the higher ς is, the higher such ambiguity is, with ς = 0 iff it is absent υ indexes attitudes toward model ambiguity: the (pointwise) higher υ is, the higher is the aversion to such uncertainty, with neutrality iff υ = 0 z 2 describes the impact of the volatility of uncertain streams’ valuations

Dynamic Robust Utility Summing up Summing up If υ is a constant coefficient (of aversion to model uncertainty), we can write the recursion as V t ( c ) = U t ( c ) − υ Σ t ( V ( c )) � � T � and Σ : L ∞ → L 0 t e − β ( s − t ) u ( c s ) d s where U t ( c ) = E t is given by � � T � e − β ( s − t ) ς 2 s σ 2 Σ t ( V ( c )) = E t s ( V ( c )) d s t Σ can be viewed as a variability index for the dynamic utility operator V A mean-variance flavor

Dynamic Robust Utility In the paper In the paper We prove that dynamic robust utilities exist, are monotone and dynamically consistent, and can be concave We study general decision problems based on dynamic robust utilities To illustrate, we study a portfolio problem and derive some asset pricing formulas featuring risk and ambiguity components For instance, excess return has the form E t [ d R t ] − r t d t ≈ λ u ( ^ c )) ς 2 c t ) σ t ( ^ + υσ t ( V ( ^ c ) σ t ( R ) d t t σ t ( R ) d t � �� � � �� � risk term ambiguity term

Dynamic Robust Utility At variance At variance A first study of aggregators that depend on variability is Lazrak and Quenez (2003). They require, however, Lipschitz conditions that are not germane to our “quadratic analysis” But, why including valuation variability is a non-trivial issue? Back to Principles of Economics

Dynamic Robust Utility Lotteries Lotteries X is a prize space ∆ ( X ) is the set of lotteries p Lotteries are simple prob. measures, with supp p = { x ∈ X : p ( x ) > 0 } u : X → R is a utility function, with Im u = ( a , b )

Dynamic Robust Utility Ranking lotteries Ranking lotteries Expected utility E u : ∆ ( X ) → R is given by ∑ E u ( p ) = u ( x ) p ( x ) x ∈ supp p Standard deviation σ u : ∆ ( X ) → R is given by � ( u ( x ) − E u ( p )) 2 p i ∑ σ u ( p ) = x ∈ supp p

Dynamic Robust Utility Expected Utility? Expected Utility? Expected utility E u is the standard (normative) criterion A good student might well ask: what about the standard deviation? Specifically, given a function λ : ( a , b ) → R , define M u ( p ) = E u ( p ) − λ ( E u ( p )) σ 2 u ( p ) 2 Why E u and not M u ?

Dynamic Robust Utility Expected Utility? Expected Utility? Standard answer: the von Neumann-Morgenstern axioms, in particular the Independence Axiom If you do not buy this axiom, note that M u is not even monotone, a basic rationality tenet

Dynamic Robust Utility Small Variance Small Variance Yet, if σ u is small enough, monotonicity holds Can M u then be resurrected as an “asymptotic” criterion?

Dynamic Robust Utility Small Variance Small Variance � � σ 2 A function r : ∆ ( X ) → R is o u ( p ) if ⇒ r ( p n ) ∀ { p n } ⊆ ∆ ( X ) σ u ( p n ) → 0 = u ( p n ) → 0 σ 2 A function v : X → R is ordinally equivalent to u if v = g ◦ u for some strictly increasing function g : ( a , b ) → R

Dynamic Robust Utility Expected Utility Redux Expected Utility Redux THM (e.g. de Finetti ’52) There exists v : X → R , ordinally equivalent to u , such that � � σ 2 ∀ p ∈ ∆ ( X ) M u ( p ) = E v ( p ) + o u ( p ) So, the relevance of the standard deviation σ u vanishes asymptotically faster than its magnitude g is such that λ = − g �� / g �

Dynamic Robust Utility Continuous Time Continuous Time Even if you do not buy the Independence Axiom, this casts some serious doubts on M u even as an asymptotic criterion That said, all this suggests that continuous time might be a key framework where to study monotone decision criteria where variability plays a role

Dynamic Robust Utility Continuous time setup Continuous time setup Time interval [ 0 , T ] Probability space ( Ω , F , P ) Brownian motion B = ( B t ) Brownian standard filtration F = ( F t ) Filtered probability space ( Ω , F , F , P )

Dynamic Robust Utility Continuous time setup Continuous time setup Basic primitive: atemporal utility function u : C → R over a material consequence c ∈ C , say a consumption good The utility function u is bounded, monotone and measurable

Dynamic Robust Utility Probabilities Probabilities DM faces uncertain consumption streams that depend on exogenous contingencies ω ∈ Ω Uncertainty is governed by a true generative mechanism or model DM has a subjective probability P on Ω that quantifies beliefs about the relative likelihood of the exogenous contingencies P is DM’s best guess of the true model

Dynamic Robust Utility Dynamic Utility Operators Dynamic Utility Operators Consumption streams are adapted processes, denoted by c = ( c t ) ∈ C They are evaluated by the DM through the following operator: DEF A dynamic utility operator is a map V : C → L 0 that associates to each consumption plan c = ( c t ) a utility process V ( c ) = ( V t ( c )) such that � � �� = � c � � c � ( u ( c t )) = u ⇒ V ( c ) = V t for all consumption plans c , c � ∈ C

Dynamic Robust Utility Dynamic Utility Operators Dynamic Utility Operators Each V t : C → L ∞ t is an intertemporal utility function that, at time t , evaluates consumption streams based on his information F t at t . From the standpoint of t = 0, we can interpret V t ( c ) as the continuation value of stream c at time t . A standard separable example is � � T � e − β ( s − t ) u ( c s ) d s U t ( c ) = E t ∀ c ∈ C t