UNCERTAINTY AND VALUATIONMODELING CHALLENGES

Lars Peter HansenUniversity of Chicago

June 1, 2013Address to the Macro-Finance Society

1 / 29

Lord Kelvin’s dictum: “I often say that when you can measuresomething that you are speaking about, express it in numbers, youknow something about it; but when you cannot measure it, when youcannot express it in numbers, your knowledge is of the meagre andunsatisfactory kind: it may be the beginning of knowledge, but youhave scarcely, in your thoughts advanced to the stage of science,whatever the matter might be.”

2 / 29

CHICAGO SCHOLARS RESPONSES TO THEKELVIN DICTUM

I KnightIf you cannot measure a thing, go ahead and measureit anyway.

I Viner’s proposed amendment to the Kelvin Dictum... and even when we can measure a thing, ourknowledge will be meager and unsatisfactory.

3 / 29

USING DECISION THEORY TO CONFRONTUNCERTAINTY

I Wald approach - Consider a family of models without resortingto averages.

I Robust Bayesian - Consider a family of priors over models anduse historical data to update each one.

I Smooth ambiguity models - Consider a family of models inconjunction with a posterior from Bayes rule to average overmodels. Risk and ambiguity are two stages of a compoundlottery. Responses to these two sources are governed by distinctparameters.

4 / 29

WHAT ABOUT SUBJECTIVE PROBABILITY?

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 neighborhood theassumed opinion.

Savage:

No matter how neat modern operational definitions ofpersonal probability may look, it is usually possible todetermine the personal probabilities of events only verycrudely.

5 / 29

MAKING ROBUSTNESS OPERATIONAL

I Explore a family of alternative potential models or a class ofperturbations to a benchmark model subject to constraints orpenalization. Future perturbations may not be tied to the pastmaking learning about them impossible. (Control theory origins.)

I Explore a family of “posteriors/priors” used to weight modelspossibly relative to a benchmark specification. Dynamic learningplays a central role.

I Unbundle attitudes towards risk and ambiguity in preferences.

Use the decision problem to target the member of the family that hasthe largest utility consequences.

6 / 29

MODERN DECISION THEORY AND CONTROLTHEORY

What is available:

I Axiomatic foundationsI Tractable representationsI Recursive construction

What is not available:

I A fully fleshed out theoretical justification for the sensible use ofmisspecified models.

I A clear discussion of the best source of “new” parameters.I A clear discussion of parameter invariance across alternative

environments.

7 / 29

WORST-CASE MODELS

I The analysis often yields a so-called (restrained) worst-casemodel for comparison.

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

I Explore how close the original model is to the worst-case modelusing statistical measures such as “Chernoff” entropy.

8 / 29

MODELS OF ASSET VALUATION

Two channels:

I Stochastic growth modeled as a process G = {Gt} where Gt

captures growth between dates zero and t.I Stochastic discounting modeled as a process S = {St} where St

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 marginal investors.

9 / 29

RECURSIVE VALUATION AND SENTIMENTS

I Use a recursive utility model (see Koopmans, Kreps & Porteus,Epstein & Zin, ...) to highlight how uncertainty about futureevents affects asset valuation.

I Explore ways in which expectations and uncertainty about futuregrowth rates influence risky claims to consumption.

Investigate how beliefs about the future are reflected in current-periodassessments, through continuation values of prospective consumptionprocesses and through stochastic discount factors which representprices over alternative investment horizons. The forward-lookingnature of the recursive utility model provides an additional channelthrough which sentiments about the future matter. (Bansal-Yaron andmany others.)

10 / 29

FAMILIAR ASSET PRICING WITH RECURSIVEUTILITY

Consider the homogeneous-of-degree-one aggregator specified interms of Ct the current period consumption and Vt the continuationvalue :

Vt =[(Ct)

1−ρ + exp(−δ) [Rt(Vt+1)]1−ρ] 1

1−ρ.

where

Rt (Vt+1) =(E[(Vt+1)

1−γ |Ft]) 1

1−γ

adjusts the continuation value Vt+1 for risk. With these preferences, 1ρ

is the elasticity of intertemporal substitution and δ is a subjectivediscount rate.

11 / 29

STOCHASTIC DISCOUNT FACTOR

Stochastic discount factor:

St+1

St= exp(−δ)

(Ct+1

Ct

)−ρ [ Vt+1

Rt(Vt+1)

]ρ−γ

I Continuation value gives a structured way to introducesentiments.

I Special case: Power utility sets ρ = γ.I Multiply to compound over multiple periods.

12 / 29

IMPACTS ON RISK-RISK RETURN TRADEOFFS

Dynamic asset pricing through altering cash flow exposure to shocks.

I Alter cash flow exposure to shocks.I Study implication on the price today of changing the exposure

tomorrow on a cash flow at some future date.I Represent shock price elasticities by normalizing the exposure

and studying the impact on the logarithms of the expectedreturns.

I Pricing counterpart to impulse response functions.

13 / 29

ELASTICITIES

Counterparts to impulse response functions pertinent to valuation:

I shock-exposure elasticitiesI 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)

14 / 29

SHOCK ELASTICITIESI Let G be a stochastic growth process and S a stochastic discount

factor process.I Form perturbed payoff GtHε where

log Hε = −12

∫ ε

0|αh(Xt)|2dt +

∫ ε

0αh(Xt) · dWt

and E|αh|2 = 1.I Construct the logarithm of the expected return:

log E [GtHε|X0 = x]− log E [StGτHε|X0 = x]

and differentiate with respect to ε to form a shock price elasticity.I Construct the logarithm of the expected growth:

log E [GtHε|X0 = x]

and differentiate with respect to ε to form a shock exposureelasticity.

I Repeat for different t and αh. Elasticities are linear in αh.15 / 29

SHOCK-PRICE TRAJECTORIES FOR POWER ANDRECURSIVE UTILITY

0 20 40 60 80 100 1200

.2

.4

.6Consumption price elasticity

0 20 40 60 80 100 1200

.2

.4

.6Growth−rate price elasticity

0 20 40 60 80 100 1200

0.05

0.1

0.15

0.2

quarters

Volatility price elasticity

Bands depict quartiles. Parameter values from Hansen (Econometrica,Fisher-Schultz Lecture).

16 / 29

SUCCESS?

I The mechanism relies on endowing investors with knowledge ofstatistically subtle components of the macro time series. Wheredoes this confidence come from?

I Stochastic volatility or fluctuations in the volatility of risk pricesis imposed from the “outside”.

I “Large” risk aversion is imposed.

17 / 29

IMITATING HIGH RISK AVERSION

From Jacobson and Whittle, there is a well known connectionbetween risk sensitivity and concern about robustness. Hansen andSargent (IEEE) and Maenhout (RFS) show how to adapt the Jacobsonand Whittle formulation to recursive utility. The risk aversionparameter γ from recursive utility is related to a relative entropypenalization parameter ξ used to discipline a concern about modelmisspecification via the formula:

ξ =1

γ − 1,

which is positive provided that γ > 1.

I Relative entropy is used as a discrepancy measure betweenprobabilities. Used extensively in the applied probabilityliterature. Convenient.

I Penalize the exploration of alternative probability specifications.

18 / 29

SKEPTICISM

Figure 2. Georges de la Tour’s circa 1635 painting “Le Tricheur” (or “The Cheat”). The painting currently exists in two versions. This version(also known as “The Ace of Diamonds”) is in the Louvre. The online version of this figure is in color.

was a hope for a Robust New World on a scale that would dis-place most of classical statistics, now there was no audible voic-ing of such a hope. The euphoria of the first decade provedto be unsustainable. Reasons for this change were not hardto find. Indeed, some were noted in the discussion of Bickel(1976), and many of them were summarized in a new chapterPeter Huber himself added to the 1996 second edition of his1977 SIAM monograph. As models grew more complicated, sotoo did the question of just what “robust” meant. The poten-tial model failures in a multivariate time series model are huge,with no consensus upon where to start. Even in such cases,important progress was made, but with lower expectations ofcompleteness. The crisp formulation and brilliant solutions forthe location parameter problem were not to be repeated. TheNeyman–Pearson Lemma extends effortlessly to Banach andHilbert spaces; Huber’s (1964) result was not available for themore complex worlds that modern statisticians live in.

I suspect all of you are familiar with these issues, and I knowsome of them that have been at least partially overcome—thisdiscussion has after all been going on for at least thirty years.There is a further issue that I suspect is perhaps as important,and I have not seen it brought up in this connection, as part ofthis discussion. It is the nature of the established methodology,the normal theory of linear models and multivariate analysis.Ever since the statistical world fully grasped the nature of whatFisher created in the 1920s with the analysis of multiple regres-sion models and the analysis of variance and covariance—eversince about 1950—we have seen what that analysis can do andseen the magic of the results it permits. The perfection of thatdistribution theory, the ease of assessing additional variablesand partitioning sums of squares as related to potential causes—no other set of methods can touch it in these regards. Leastsquares is still and will remain King for these reasons—these

magical properties—even if for no other reason. In the UnitedStates many consumers are entranced by the magic of the newiPhone, even though they can only use it with the AT&T system,a system noted for spotty coverage—even no receivable signalat all under some conditions. But the magic available when itdoes work overwhelms the very real shortcomings. Just so, leastsquares will remain the tool of choice unless someone concoctsa robust methodology that can perform the same magic, a stepthat would require the suspension of the laws of mathematics.

Where does this leave robustness? In very good condi-tion, thank you. I hasten to reiterate and extend my earlierdisclaimer—there was and is no shortage of important and ex-citing research on robustness. And a lot that was learned forsimpler problems contributed to the much greater understand-ing that has been achieved in the more complex problems. Ro-bust thinking is everywhere; it is just a different sort of think-ing than might have been expected 40 years ago. Modern re-searchers are seeking deeper understanding from the geometryof statistical models, looking at missing data problems, givingserious attention to the diagnostic use of robustness, and ad-dressing robustness as a point of view rather than the answer toall ills.

Let me end by returning to my colleague’s book, Robustness(Hansen and Sargent 2008). It is a tour de force by two topeconometricians, who summarize a large body of work, includ-ing their own, to develop the use of a robust control theory foranalyzing economic decision making, a theory that acknowl-edges the real potential for model misspecification. The pic-ture on the cover represents an evil agent, one that can justify ause of minimax criteria in certain realms of game theory, evenwhen in most ordinary statistical problems we have discardedit as too pessimistic. But the designer of their cover has donemore than cast De La Tour’s scene in mottled shades: it has

280 History Corner

The Cheat by La Tour

19 / 29

FORMALIZATION

I Construct a specification of preferences as inHansen-Sargent(AER) andMaccheroni-Marinacci-Rustichini(Econometrica, JET)

I Relative entropy penalization gives a rationale for exponentialtilting using the value function as the penalized worst-casemodel:

exp(− log Vt+1

ξ

)E[exp

(− log Vt+1

ξ

)|Ft

] =(Vt+1)

1−γ

E [(Vt+1)1−γ |Ft]

where Vt+1 is the next-period continuation value.

No endogenous source for fluctuations in uncertainty prices.

20 / 29

ENRICHING THE UNCERTAINTY PRICINGDYNAMICS

Two approaches:

I Structural MisspecificationI Robust Learning under Misspecification - Fragile Beliefs

I will report results based on the first approach. Results for the secondcan be found in Hansen and Sargent (QE,2010) and Hansen ElyLecture (AER, 2007).

21 / 29

ROBUSTNESS CONCERNS RECONSIDERED

I A representative consumer has instantaneous utility log Ct andthe following approximating model for the dynamics ofconsumption Ct

d log Ct = (.01) (µ+ Xt) dt + (.01)α · dWt

dXt = −κXtdt + σ · dWt

where µ+ Xt is the date t growth rate expressed as a percent andW is a two-dimensional Brownian motion.

I Disguise drift distortions inside Brownian motions.I Little reliable historical information on the parameters µ and κ.

Other distortions are allowable but the decision problem featuresthese.

I A concern for robustness is reflected in the implied “risk return”tradeoff over alternative investment horizons.

22 / 29

MISSPECIFICATION

I Change the evolution of W:

dWt = htdt + dWt

where W is a Brownian motion.I Interpreting a quadradic penalty:

I Conditional relative entropy: |ht|22 .

I Conditional Chernoff entropy: |ht|28 .

I Worst-case drift distortion is constant.

23 / 29

MISSPECIFICATION RECONSIDERED

Build on approach in Petersen, James, Dupuis (IEEE, 2000).

I Offset the penalty with a quadratic term in Xt guaranteeing thatchanging the AR coefficient is always permissible.

I Permit

ht = ±[

0ηXt

],

but do not impose it. Thus

dXt = −κXtdt ± σ ·[

01

]ηXtdt + σ · dWt

becomes an allowable misspecification.

24 / 29

LONG-TERM UNCERTAINTY

I Large “uncertainty prices” without large risk aversion.I Endogenous fluctuations: uncertainty prices are larger in bad

times than good times.

25 / 29

WORST-CASE MODEL

0 20 40 60 80 100 120 1400

0.005

0.01

0.015

0.02

0.025

0.03

0.035

shock−exposure elasticity to WX

horizon (quarters)

worst case

original

FIGURE : Shock-exposure elasticities or a growth rate shock.

26 / 29

UNCERTAINTY PRICES

horizon (quarters)

ela

sticity

shock−price elasticity

0 20 40 60 80 100 120 1400

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

WC

WX

FIGURE : Shock-price elasticities. The shaded region picks the quartiles forthe elasticities.

27 / 29

SYSTEMIC UNCERTAINTY

I Recall Viner’s proposed amendment to the Kelvin Dictum... and even when we can measure a thing, ourknowledge will be meager and unsatisfactory.

I Financial crisis has led to calls for better and morecomprehensive oversight of the financial system because ofsystemic risk. Our knowledge of this concept is “meager”.

I How do we best express skepticism in our probabilisticmeasurement of systemic risk?

I This skepticism when expressed appropriately can have importantconsequences for policy design.

28 / 29

SOME EXTENSIONS

I Robust Ramsey planner: Ramsey planner is a stand-in for thebenevolent policy maker. The introduction of this entity is usedas a device to determine “optimal” policies. Robust concernsfrom private and/or public sectors alter the design of goodpolicies. Policy maker engages in “managing” or “monitoring”expectations and may be cautious because of model uncertainty.

I Heterogeneous beliefs: This framework “endogenizes”expectations through the computation of “worst-case models.”Such models depend on the decision problem, and hencegenerates ex post heterogeneous beliefs.

29 / 29