UNCERTAINTY 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 something that you are speaking about, express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of the meagre and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts advanced to the stage of science, whatever the matter might be. 2 / 29

CHICAGO SCHOLARS RESPONSES TO THE KELVIN DICTUM Knight If you cannot measure a thing, go ahead and measure it anyway. Viner s proposed amendment to the Kelvin Dictum... and even when we can measure a thing, our knowledge will be meager and unsatisfactory. 3 / 29

USING DECISION THEORY TO CONFRONT UNCERTAINTY Wald approach - Consider a family of models without resorting to averages. Robust Bayesian - Consider a family of priors over models and use historical data to update each one. Smooth ambiguity models - Consider a family of models in conjunction with a posterior from Bayes rule to average over models. Risk and ambiguity are two stages of a compound lottery. Responses to these two sources are governed by distinct parameters. 4 / 29

5 / 29 WHAT ABOUT SUBJECTIVE PROBABILITY? De Finetti: Subjectivists should feel obligated to recognize that any opinion (so much more the initial one) is only vaguely acceptable... So it is important not only to know the exact answer for an exactly specified initial problem, but what happens changing in a reasonable neighborhood the assumed opinion. Savage: No matter how neat modern operational definitions of personal probability may look, it is usually possible to determine the personal probabilities of events only very crudely.

6 / 29 MAKING ROBUSTNESS OPERATIONAL Explore a family of alternative potential models or a class of perturbations to a benchmark model subject to constraints or penalization. Future perturbations may not be tied to the past making learning about them impossible. (Control theory origins.) Explore a family of posteriors/priors used to weight models possibly relative to a benchmark specification. Dynamic learning plays a central role. Unbundle attitudes towards risk and ambiguity in preferences. Use the decision problem to target the member of the family that has the largest utility consequences.

7 / 29 MODERN DECISION THEORY AND CONTROL THEORY What is available: Axiomatic foundations Tractable representations Recursive construction What is not available: A fully fleshed out theoretical justification for the sensible use of misspecified models. A clear discussion of the best source of new parameters. A clear discussion of parameter invariance across alternative environments.

8 / 29 WORST-CASE MODELS The analysis often yields a so-called (restrained) worst-case model for comparison. Apply the theory of two-person games. The decision maker optimizes taking as given the worst-case model. Explore how close the original model is to the worst-case model using statistical measures such as Chernoff entropy.

9 / 29 MODELS OF ASSET VALUATION Two channels: Stochastic growth modeled as a process G = {G t } where G t captures growth between dates zero and t. Stochastic discounting modeled as a process S = {S t } where S t assigns risk-adjusted prices to cash flows at date t. Date zero prices of a payoff G t are π = E (S t G t X 0 ) where X 0 captures current period information. Stochastic discounting reflects investor preferences through intertemporal marginal rate of substitution for marginal investors.

10 / 29 RECURSIVE VALUATION AND SENTIMENTS Use a recursive utility model (see Koopmans, Kreps & Porteus, Epstein & Zin,...) to highlight how uncertainty about future events affects asset valuation. Explore ways in which expectations and uncertainty about future growth rates influence risky claims to consumption. Investigate how beliefs about the future are reflected in current-period assessments, through continuation values of prospective consumption processes and through stochastic discount factors which represent prices over alternative investment horizons. The forward-looking nature of the recursive utility model provides an additional channel through which sentiments about the future matter. (Bansal-Yaron and many others.)

11 / 29 FAMILIAR ASSET PRICING WITH RECURSIVE UTILITY Consider the homogeneous-of-degree-one aggregator specified in terms of C t the current period consumption and V t the continuation value : where V t = [(C 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 adjusts the continuation value V t+1 for risk. With these preferences, 1 ρ is the elasticity of intertemporal substitution and δ is a subjective discount rate.

12 / 29 STOCHASTIC DISCOUNT FACTOR Stochastic discount factor: S t+1 S t = exp( δ) ( Ct+1 C t ) ρ [ Vt+1 ] ρ γ R t (V t+1 ) Continuation value gives a structured way to introduce sentiments. Special case: Power utility sets ρ = γ. Multiply to compound over multiple periods.

13 / 29 IMPACTS ON RISK-RISK RETURN TRADEOFFS Dynamic asset pricing through altering cash flow exposure to shocks. Alter cash flow exposure to shocks. Study implication on the price today of changing the exposure tomorrow on a cash flow at some future date. Represent shock price elasticities by normalizing the exposure and studying the impact on the logarithms of the expected returns. Pricing counterpart to impulse response functions.

14 / 29 ELASTICITIES Counterparts to impulse response functions pertinent to valuation: shock-exposure elasticities shock-price elasticities These are the ingredients to risk premia, and they have a term structure induced by the changes in the investment horizons. Hansen-Scheinkman (Finance and Stochastics), Borovička-Hansen-Hendricks-Scheinkman (Journal of Financial Econometrics), Hansen (Fisher-Schultz, Econometrica)

15 / 29 SHOCK ELASTICITIES Let G be a stochastic growth process and S a stochastic discount factor process. Form perturbed payoff G t H ɛ where log H ɛ = 1 2 ɛ 0 α h (X t ) 2 dt + ɛ 0 α h (X t ) dw t and E α h 2 = 1. Construct the logarithm of the expected return: log E [G t H ɛ X 0 = x] log E [S t G τ H ɛ X 0 = x] and differentiate with respect to ɛ to form a shock price elasticity. Construct the logarithm of the expected growth: log E [G t H ɛ X 0 = x] and differentiate with respect to ɛ to form a shock exposure elasticity. Repeat for different t and α h. Elasticities are linear in α h.

SHOCK-PRICE TRAJECTORIES FOR POWER AND RECURSIVE UTILITY.6 Consumption price elasticity.4.2 0 0 20 40 60 80 100 120.6 Growth rate price elasticity.4.2 0 0 20 40 60 80 100 120 0.2 Volatility price elasticity 0.15 0.1 0.05 0 0 20 40 60 80 100 120 quarters Bands depict quartiles. Parameter values from Hansen (Econometrica, Fisher-Schultz Lecture). 16 / 29

17 / 29 SUCCESS? The mechanism relies on endowing investors with knowledge of statistically subtle components of the macro time series. Where does this confidence come from? Stochastic volatility or fluctuations in the volatility of risk prices is imposed from the outside. Large risk aversion is imposed.

18 / 29 IMITATING HIGH RISK AVERSION From Jacobson and Whittle, there is a well known connection between risk sensitivity and concern about robustness. Hansen and Sargent (IEEE) and Maenhout (RFS) show how to adapt the Jacobson and Whittle formulation to recursive utility. The risk aversion parameter γ from recursive utility is related to a relative entropy penalization parameter ξ used to discipline a concern about model misspecification via the formula: ξ = 1 γ 1, which is positive provided that γ > 1. Relative entropy is used as a discrepancy measure between probabilities. Used extensively in the applied probability literature. Convenient. Penalize the exploration of alternative probability specifications.

S KEPTICISM ure 2. Georges de la Tour s circa 1635 painting Le Tricheur (or The Cheat ). The painting currently exists in two versions. This version o known as The Ace of Diamonds ) is in the Louvre. The online version of this figure is in color. The Cheat by La Tour a hope for a Robust New World on a scale that would disce most of classical statistics, now there was no audible voic- magical properties even if for no other reason. In the United States many consumers are entranced by the magic of the new 19 / 29

20 / 29 FORMALIZATION Construct a specification of preferences as in Hansen-Sargent(AER) and Maccheroni-Marinacci-Rustichini(Econometrica, JET) Relative entropy penalization gives a rationale for exponential tilting using the value function as the penalized worst-case model: ( exp ( [ E exp log V t+1 ξ log V t+1 ξ ) ) F t ] = (V t+1 ) 1 γ E [(V t+1 ) 1 γ F t ] where V t+1 is the next-period continuation value. No endogenous source for fluctuations in uncertainty prices.

21 / 29 ENRICHING THE UNCERTAINTY PRICING DYNAMICS Two approaches: Structural Misspecification Robust Learning under Misspecification - Fragile Beliefs I will report results based on the first approach. Results for the second can be found in Hansen and Sargent (QE,2010) and Hansen Ely Lecture (AER, 2007).

ROBUSTNESS CONCERNS RECONSIDERED A representative consumer has instantaneous utility log C t and the following approximating model for the dynamics of consumption C t d log C t = (.01) (µ + X t ) dt + (.01)α dw t dx t = κx t dt + σ dw t where µ + X t is the date t growth rate expressed as a percent and W is a two-dimensional Brownian motion. Disguise drift distortions inside Brownian motions. Little reliable historical information on the parameters µ and κ. Other distortions are allowable but the decision problem features these. A concern for robustness is reflected in the implied risk return tradeoff over alternative investment horizons. 22 / 29

23 / 29 MISSPECIFICATION Change the evolution of W: dw t = h t dt + d W t where W is a Brownian motion. Interpreting a quadradic penalty: Conditional relative entropy: ht 2 2. Conditional Chernoff entropy: h t 2 8. Worst-case drift distortion is constant.

24 / 29 MISSPECIFICATION RECONSIDERED Build on approach in Petersen, James, Dupuis (IEEE, 2000). Offset the penalty with a quadratic term in X t guaranteeing that changing the AR coefficient is always permissible. Permit [ ] 0 h t = ±, ηx t but do not impose it. Thus [ ] 0 dx t = κx t dt ± σ ηx 1 t dt + σ dw t becomes an allowable misspecification.

25 / 29 LONG-TERM UNCERTAINTY Large uncertainty prices without large risk aversion. Endogenous fluctuations: uncertainty prices are larger in bad times than good times.

26 / 29 WORST-CASE MODEL 0.035 shock exposure elasticity to W X 0.03 0.025 0.02 0.015 0.01 0.005 worst case original 0 0 20 40 60 80 100 120 140 horizon (quarters) FIGURE : Shock-exposure elasticities or a growth rate shock.

27 / 29 UNCERTAINTY PRICES 0.45 shock price elasticity 0.4 0.35 0.3 elasticity 0.25 0.2 0.15 0.1 W 0.05 C W X 0 0 20 40 60 80 100 120 140 horizon (quarters) FIGURE : Shock-price elasticities. The shaded region picks the quartiles for the elasticities.

28 / 29 SYSTEMIC UNCERTAINTY Recall Viner s proposed amendment to the Kelvin Dictum... and even when we can measure a thing, our knowledge will be meager and unsatisfactory. Financial crisis has led to calls for better and more comprehensive oversight of the financial system because of systemic risk. Our knowledge of this concept is meager. How do we best express skepticism in our probabilistic measurement of systemic risk? This skepticism when expressed appropriately can have important consequences for policy design.

29 / 29 SOME EXTENSIONS Robust Ramsey planner: Ramsey planner is a stand-in for the benevolent policy maker. The introduction of this entity is used as a device to determine optimal policies. Robust concerns from private and/or public sectors alter the design of good policies. Policy maker engages in managing or monitoring expectations and may be cautious because of model uncertainty. Heterogeneous beliefs: This framework endogenizes expectations through the computation of worst-case models. Such models depend on the decision problem, and hence generates ex post heterogeneous beliefs.