Uncertainty,Valuation, andPolicyLars Peter HansenUniversity of Chicago

IGIER

September 15, 2014

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Uncertainty in economic analysis• Outside a model

Given a dynamic economic model, researchers:▷ estimate unknown parameters▷ assess model implications

• Inside a modelWhen constructing a dynamic economic model, researchers:▷ depict economic actors (consumers, enterprises, policy

makers) as they cope with uncertainty▷ deduce the consequences for market outcomes and

resource allocations

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Investors inside an economicmodelWhen constructing a dynamic economic model, researchers:

• depict private sector economic actors as they cope withuncertainty

• deduce the consequences for market outcomes andresource allocations

Outcome: New sources for fluctuations in uncertainty pricesemerge in models of financial markets.

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Models of asset valuationTwo channels:

• Stochastic growth modeled as a process G = {Gt} where Gtcaptures growth between dates zero and t.

• Stochastic discounting modeled as a process S = {St} whereSt assigns risk-adjusted prices to cash flows at date t.

Date zero prices of a payoff Gt are

π = E (StGt|X0)

where X0 captures current period information.

Stochastic discounting reflects investor preferences throughintertemporal marginal rate of substitution for marginalinvestors.

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Recursive valuation andperceptions• Use a recursive utility model (see Koopmans, Kreps &

Porteus, Epstein & Zin, ...) to highlight how uncertaintyabout future events affects asset valuation.

• Explore ways in which expectations and uncertainty aboutfuture growth rates influence risky claims to consumption.

Investigate how beliefs about the future are reflected incurrent-period assessments through continuation values. Theforward-looking nature of the recursive utility model providesan additional channel through which perceptions about thefuture matter. (Bansal-Yaron and many others.)

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Recursive utilityConsider the aggregator specified in terms of Ct the currentperiod consumption and Vt the continuation value:

Vt =[(Ct)

1−ρ + exp(−δ) [Rt(Vt+1)]1−ρ

] 11−ρ

.

whereRt (Vt+1) =

(E[(Vt+)

1−γ |Ft]) 1

1−γ

adjusts the continuation value Vt+1 for risk. 1ρ is the elasticity

of intertemporal substitution and δ is a subjective discountrate.

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Stochastic discount factorStochastic discount factor:

St+1

St= exp(−δ)

(Ct+1

Ct

)−ρ [ Vt+1

Rt(Vt+1)

]ρ−γ

whereRt (Vt+1) =

(E[(Vt+1)

1−γ |Ft]) 1

1−γ

adjusts the continuation value Vt+1 for risk.

• Continuation values give a structured way to captureperceptions about the future.

• Special case: power utility sets ρ = γ.• When ρ = 1 [

Vt+1

Rt(Vt+1)

]1−γ

has conditional expectation equal to unity. Equivalentinterpretation as distorted beliefs.

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Impulse problem and valuationRagnar Frisch (1933):

There are several alternative ways in which one mayapproach the impulse problem .... One way which Ibelieve is particularly fruitful and promising is tostudy what would become of the solution of adeterminate dynamic system if it were exposed to astream of erratic shocks that constantly upsets thecontinuous evolution, and by so doing introducesinto the system the energy necessary to maintain theswings.

Irving Fisher (1930):

The manner in which risk operates upon timepreference will differ, among other things, accordingto the particular periods in the future to which therisk applies.

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Impulses and risk pricingDynamic asset pricing through altering cash flow exposure toshocks.

• Study implication on the price today of changing theexposure tomorrow on a cash flow at some future date.

• Represent shock price elasticities by normalizing theexposure and studying the impact on the logarithms of theexpected returns.

• Construct pricing counterpart to impulse responsefunctions.

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ElasticitiesCounterparts to impulse response functions pertinent tovaluation:

• shock-exposure elasticities

• shock-price elasticities

These are the ingredients to risk premia, and they have a termstructure induced by the changes in the investment horizons.

Hansen-Scheinkman (Finance and Stochastics),Borovicka-Hansen-Hendricks-Scheinkman (Journal of FinancialEconometrics), Hansen (Fisher-Schultz, Econometrica),Borovicka, Hansen and Schienkman ((Mathematical andFinancial Economics).

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Quantitative example(Bansal-Yaron)

dX[1]t = −.021X[1]

t dt +√

X[2]t

[.031 −.015 0

]dWt

dX[2]t = −013

(X[2]

t − 1)

dt +√

X[2]t

[0 0 −.038

]dWt

d log Ct = (.01)(.15 + X[1]t )dt + (.01)

√X[2]

t

[.34 .7 0

]dWt

X[1] captures predictability in growth rates;

X[2] captures stochastic volatility;

Components of dWt:

Permanent shock;Transitory shock;Stochastic volatility shock.

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Shock-price elasticities

0 20 40 60 80 100 1200

.2

.4

.6Permanent price elasticity

0 20 40 60 80 100 1200

.2

.4

.6Temporary price elasticity

0 20 40 60 80 100 1200

0.05

0.1

0.15

0.2

quarters

Volatility price elasticity

Recursive utility and Power utility. Bands depict quartiles.12 / 28

Success?• The mechanism relies on endowing investors with

knowledge of statistically subtle components of the macrotime series. Where does this confidence come from?

• Imposes stochastic volatility exogenously

• Imposes large risk aversion.

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Multiple components touncertainty• Model risk – what probabilities does a model assign to

events in the future?

• Model ambiguity – how much confidence do we place ineach model?

• Model misspecification – how do we use models that are notperfect?

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What about subjectiveprobability?• De Finetti:

“Subjectivists should feel obligated to recognize that anyopinion (so much more the initial one) is only vaguelyacceptable. . . So it is important not only to know the exactanswer for an exactly specified initial problem, but whathappens changing in a reasonable neighbourhood theassumed opinion.”

• Savage:“No matter how neat modern operational definitions of

personal probability may look, it is usually possible todetermine the personal probabilities of events only verycrudely.”

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Skepticism

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Making robustness operational• Explore a family of “posteriors/priors” used to weight

models possibly relative to a benchmark specification.Dynamic learning plays a central role. (Robust Bayesiananalysis)

• Explore a family of alternative potential models or a classof perturbations to a benchmark model subject toconstraints or penalization. Future perturbations may notbe tied to the past making learning about them impossible.(Control theory and statistical origins.)

Use the decision problem to target the member of the familythat has the largest utility consequences.

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Asset pricing under a beliefdistortion

π = E[StGt|X0

](1)

where E is the distorted expectation operator and S is thecorresponding stochastic discount factor.

• Convenient to represent distorted beliefs using a positivemartingale M with a unit expectation via the formula:

E [Yt+ℓ|Ft] = E[(

Mt+ℓ

Mt

)Yt+ℓ

∣∣∣∣Ft

].

• Rewrite (1) as:

π = E[StGt|X0

]= E

[MtStGt|X0

]= E [StGt|X0]

where S = MS.18 / 28

Asset pricing under a beliefdistortionSDF representation

S = M × Sdistorted riskbeliefs preferences

• S constructed from data and model parameters.

• M is a likelihood ratio.

• When M is close to one, the distortion is small.

• Statistical criteria provide interpretable measures of themagnitude of the distortion.

When the distortion is small, a statistician with a largenumber of observations will struggle to tell the differencebetween two models.

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Worst-case model of a beliefdistortion• The analysis often yields a so-called (restrained) worst-case

model for comparison.

• Apply the theory of two-person games. The decision makeroptimizes taking as given the worst-case model.

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Enriching the uncertainty pricingdynamicsTwo approaches:

• Structural Misspecification

• Robust Learning under Misspecification - Fragile Beliefs

Results for the first are part of current research. Results forthe second can be found in Hansen and Sargent (QE,2010) andHansen Ely Lecture (AER, 2007).

Outcome: New sources for fluctuations in uncertainty pricesemerge in models of financial markets.

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Uncertainty outside the modelGiven a dynamic economic model, researchers:

• estimate unknown parameters

• assess model implications

• use as a guide for the policy

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Friedrich Hayek (1974)

Even if true scientists shouldrecognize the limits of studyinghuman behaviour, as long as thepublic has expectations, therewill be people who pretend orbelieve that they can do more tomeet popular demand than whatis really in their power.(From Hayek’s Nobel address)

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Milton Friedman (1965)

“As Josh Billings wrote manyyears ago, ‘The trouble with mostfolks isn’t so much theirignorance, as knowing so manythings that ain’t so.’ Pertinent asthis remark is to economics ingeneral, it is especially so inmonetary economics.”

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Uncertainty and policy guidance• Systemic risk: A grab bag of scenarios rationalizing

interventions in financial markets

• Challenges: Limited understanding of systemic riskchallenges its value as a guiding principle for financialoversight (Haldane, Bank of England and Tarullo, Board ofGovernors)

• Systemic uncertainty: Complicated problems do notnecessarily require complicated solutions

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Uncertainty and policy guidance• Friedman (1961): advocated simple rules because of long

and variable lags in the monetary transmission mechanism.

• Current monetary policy: forward guidance with limitedknowledge of the the impact of monetary policy on labormarkets.

• Simple and transparent?

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Private and Public Sectors Insidethe Model

◦ Robust Ramsey planner: Ramsey planner is a stand-in forthe benevolent policy maker. Robustness concerns fromprivate and/or public sectors alter the design of goodpolicies. Policy maker engages in managing or monitoringexpectations and may be cautious because of modeluncertainty.

◦ Heterogeneous beliefs: This framework endogenizesexpectations through the computation of “worst-casemodels.” Such models depend on the decision problem,and hence generates ex post heterogeneous beliefs.Gives a way to introduce overconfidence by some agentsinto a model.

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Summary• Two perspectives on uncertainty: outside the model and

inside the model

• Multiple components to uncertainty

▷ Model risk - what probabilities does a model assign toevents in the future

▷ Model ambiguity - how much confidence do we place ineach model

▷ Model misspecification - how do we use models that arenot perfect

• Given limits to our understanding, complicated problemsmay be best addressed with simple solutions

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