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 HISTORY The manner in which risk operates upon time preference will differ, among other things, according to the particular periods in the future to which the risk applies. Irving Fisher (Theory of Interest, 1930)

EMPIRICAL MACROECONOMICS Identify macroeconomic shocks: w t Quantify responses to those shocks. How does the future macro or financial economic vector y t+j depend on the current shock w t? Compare models with alternative mechanism by which these shocks are transmitted to macro time series 3 / 45

4 / 45 HOW CAN ASSET PRICING CONTRIBUTE? The macroeconomic shocks are the ones that cannot be diversified. Investors that are exposed to such shocks require compensation for bearing this risk. Add (shadow) pricing counterparts to these impulse responses. Pricing dual to an impulse response: What is the current period price of an exposure to future macroeconomic growth rate shocks? Compare how alternative economic models assign prices to exposures even when these exposures are not in the center of the support of the historical time series. Structural model in the sense of Hurwicz.

5 / 45 UNCERTAINTY AND RECURSIVE PREFERENCES Kreps-Porteus representation V t = Φ [U(C t ), E (V t+1 F t )] as a generalization of expected utility V t = U(C t ) + βe (V t+1 F t ). Do not reduce intertemporal compound consumption lotteries. Intertemporal composition of risk matters. I will feature a convenient special case Vt = (1 β) log C t + β 1 γ log E ( exp [ (1 γ)vt+1 ] ) Ft where I have taken a monotone transformation of the continuation value. Links risk- sensitive control and recursive utility.

RESPONSE AND RISK PRICE TRAJECTORIES.020 Impulse Response to a Growth Rate Shock.015.010.005 0 0 20 40 60 80 100 120 140 160 180 200.20 Trajectory for the Growth Rate Risk Price.15.10.05 0 0 20 40 60 80 100 120 140 160 180 200 investment horizon (quarters) FIGURE: The horizontal axis is given in quarterly time units. The top panel gives the impulse-responses of the logarithm of consumption and the bottom panel gives corresponding risk prices for two alternative models. Blue assumes recursive utility and red assumes power utility. 6 / 45

7 / 45 REMAINDER OF THE TALK 1. My approach to characterizing risk-price dynamics. 2. Mathematical support for value decompositions. 3. Model comparisons and long-run components to value. 4. Recursive utility versus power utility: a comparison. 5. Recursive utility: a robust interpretation.

8 / 45 1. CHARACTERIZING RISK-PRICE DYNAMICS Use Markov formulations and martingale methods to produce decompositions of model implications. Allow for nonlinear time series models - stochastic volatility, stochastic regime shifts. Use the long-term as a frame of reference. Stochastic growth and discount factors will be state dependent. Explore the implications of this dependence when we alter the forecast or payoff horizon.

WHY USE THE LONG TERM AS A FRAME OF REFERENCE? 9 / 45

9 / 45 WHY USE THE LONG TERM AS A FRAME OF REFERENCE? growth uncertainty has important consequences for welfare

9 / 45 WHY USE THE LONG TERM AS A FRAME OF REFERENCE? growth uncertainty has important consequences for welfare stochastic component growth can have a potent impact on asset values

9 / 45 WHY USE THE LONG TERM AS A FRAME OF REFERENCE? growth uncertainty has important consequences for welfare stochastic component growth can have a potent impact on asset values economics more revealing for modeling long-run phenomenon

ASSET VALUATION AND STOCHASTIC DISCOUNTING π = E [S t G t x 0 ] j=0 where π is the date zero price of a cash flow process {G t } that grows stochastically over time. {S t } is a stochastic discount factor process. Encodes both discounting and adjustments for risk. Satisfies consistency constraints - Law of Iterated Values. Economic models imply stochastic discount factor S t Intertemporal investors MRS Work of Koopmans, Kreps and Porteus and others expanded the array of models of investor preferences. Dynamics of pricing are captured by the time series behavior of the stochastic discount factor. 10 / 45

11 / 45 CHALLENGES Extract dynamic pricing implications in a revealing way. Compare models and model ingredients.

2. MATHEMATICAL SETUP {X t : t 0} be a continuous time Markov process on a state space D. This process can be stationary and ergodic. X = X c + X d X c is the solution to dx c t = µ(x t )dt + σ(x t )dw t where W is an {F t } Brownian motion and X t = lim τ 0 X t τ. X d with a finite number of jumps in any finite interval. Simple distinction between small shocks and big shocks. Discrete time works too, but Markov structure is central. 12 / 45

13 / 45 ADDITIVE FUNCTIONAL - DEFINITION Construct a scalar process {Y t : t 0} as a function of X u for 0 u t. An additive functional is parameterized by (β, γ, κ) where: I) β : D R and t 0 β(x u)du < for every positive t; II) γ : D R m and t 0 γ(x u) 2 du < for every positive t; III) κ : D D R, κ(x, x) = 0. Y t = t 0 β(x u)du + t 0 γ(x u) dw u + 0 u t κ(x u, X u ) smooth small shocks big shocks Process Y is nonstationary and can grow linearly. Sums of additive functionals are additive. Add the parameters.

14 / 45 MULTIPLICATIVE FUNCTIONAL - DEFINITION Let {Y t : t 0} be an additive functional. Construct a multiplicative functional {M t : t 0} as M t = exp(y t ) Process M is nonstationary and can grow exponentially. products of multiplicative functionals are multiplicative. Multiply the parameters. Use multiplicative functionals to model state dependent growth and discounting.

15 / 45 DISCRETE-TIME COUNTERPART Additive functional Y t = t κ(x j, X j 1 ) j=1 Multiplicative functional t M t = exp[κ(x j, X j 1 )] j=1 Use multiplicative functionals to model state dependent growth and discounting.

16 / 45 ILLUSTRATION 0 Yo=0 Y1 =Y0 +/- 1 Y2 =Y1 +/- 1 Range of Y grows + or - A Simple Discrete-Time Additive Functional Y k(xt, Xt-1) =+/- 1 Note: X s can be temporally dependent exp(1) Or exp(-1) 1 0 A Simple Multiplicative Functional M made from exponentiating Y Yt = exp[k(x0,x1)]exp[k(x1, X2)] exp[k(xt-1,xt)] Y0 =0 M0 =exp(0) = 1 Y1= Y0 +/- 1 M1 =exp(0)exp(+/- 1) Y2 = Y1 +/- 1 M2 = exp(0)exp(+/-1)exp(+/- 1 ) Range of M is non-negative.

17 / 45 MULTIPLICATIVE DECOMPOSITION [ ] M t = exp (ρt) ˆM e(x0 ) t e(x t). (1) exponential trend martingale ratio ρ is a deterministic growth rate; ˆM t is a multiplicative martingale; e is a strictly positive function of the Markov state; Observations Reminiscent of a permanent-transitory decomposition from time series. Important differences! Not unique and co-dependence between components matters.

18 / 45 WHY? In valuation problems there are two forces at work - stochastic growth G and stochastic discounting S. Study product SG. Decompose pricing implications of a model as represented by a stochastic discount factor S. Term structure of risk prices - look at value implications of marginal changes in growth exposure as represented by changes in G.

19 / 45 FROBENIUS-PERRON THEORY/ MARTINGALES Solve, E [M t e(x t ) X 0 = x] = exp(ρt)e(x) where e is strictly positive. Eigenvalue problem. Construct martingale [ ] e(xt ) ˆM t = exp( ρt)m t. e(x 0 ) Invert M t = exp(ρt) ˆM t [ e(x0 ) e(x t ) ].

20 / 45 MULTIPLICATIVE MARTINGALES Decomposition: M t = exp(ρt) ˆM t [ e(x0 ) e(x t ) ]. Observations about ˆM. 1. Converge - often to zero - raise to powers for refined analysis - Chernoff 2. Change of measure preserves Markov structure at most one is stochastically stable - Hansen-Scheinkman

21 / 45 STOCHASTIC STABILITY [ ] f (Xt ) exp( ρt)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. Hyperbolic approximation in valuation horizon: 1 t log E [M tf (X t ) X 0 = x] ρ + 1 t ( log e(x) + log Ê [ ]) f (Xt ) e(x t )

22 / 45 LONG-TERM CASH FLOW RISK 1 ρ(m) = lim t t log E [M t X 0 = x]. Cash flow return over horizon t: E (G t X 0 = x) E (S t G t X 0 = x). long-term expected rate of return (risk adjusted): ρ(g) ρ(sg). long-term expected excess rate of return (risk adjusted): ρ(g) + ρ(s) ρ(sg) using G = 1 as a long run risk free reference.

3. MODEL COMPARISON Factorization [ ] e(x0 ) M t = exp(ρt) ˆM t. e(x t ) [ ] f If Mt (Xt ) = M t, f (X 0 ) then M and M share the same martingale component. M = S G discount growth Observations: Applied to G - long-term components of consumption processes or cash flows - in the limit these dominate pricing. Applied to S - long-term model components of valuation - models with common martingale components share the same long-term value implications. 23 / 45

TRANSIENT MODEL COMPONENTS Bansal-Lehmann style decomposition: [ ] f St (Xt ) = S t. f (X 0 ) Same martingale component Examples Habit persistence models - big differences between empirical macro and empirical asset pricing models Solvency constraint models Recursive utility models, long-term risk prices coincide with those from a power utility model Preference shock models and social externalities, I) Santos-Veronesi - asset pricing - transient value implications relative to power utility model. II) Campbell-Cochrane - asset pricing - more subtle limiting analysis. 24 / 45

25 / 45 WHAT IS A RISK PRICE? Parameterize the risk exposure as a function of shocks, G(α), to be a martingale. Compute Abstract from dynamics of risk-exposure. Alternatively, extract martingale component from a consumption of cash flow. α log E [S tg t (α) X 0 = x] α=α0 for each horizon t. (Exposure is bad so we take the negative of the price and adjust for the horizon.) Term structure emphasizes the time dependence on the horizon t of the payoffs that are priced.

26 / 45 LIMITING BEHAVIOR Stochastic discount models with the same martingale components have the same limit prices. The hyperbolic (in the investment horizon) approximation extends to the risk prices.

27 / 45 0 RISK-PRICE 0 25 50 75FIGURE 100 125REVISITED 150 175 200.20 Trajectory for the Growth Rate Risk Price.15.10.05 0 0 25 50 75 100 125 150 175 200 FIGURE: The horizontal axis is given in quarterly time units. Blue assumes recursive utility and red assumes power utility. Grey line is the hyperbolic approximation.

4. RECURSIVE UTILITY INVESTORS Vt = (1 β) log C t + β 1 γ log E ( exp [ (1 γ)vt+1 ] ) Ft Risk sensitive control theory (Jacobson, Whittle) - linked to recursive utility theory (Kreps-Porteus and Epstein-Zin). Achieved by applying an exponential risk adjustment to continuation values. Hansen-Sargent. Intertemporal compound lotteries are no longer reduced. The intertemporal composition of risk matters. Macroeconomic/asset pricing implications originally studied by Tallarini - increases risk prices while having modest implications for stochastic growth models. Asset pricing success achieved by imposing high risk aversion (Tallarini) or a predictable growth component (Bansal and Yaron). 28 / 45

29 / 45 REMINDER The study of asset pricing implications typically focus on one-period risk prices, but not on the entire term-structure of risk prices.

30 / 45 ASSET PRICING In the discounted version of recursive risk-sensitive preferences, the stochastic discount factor is; ( ) St C0 = exp( δt) ˆV t C t where ˆV is a martingale component of {exp [(1 γ)(v t V 0 )] : t 0} and V is the stochastic process of continuation values. The process V and hence ˆV are constructed from the underlying consumption dynamics. δ continues to be the subjective rate of discount and the inverse ratio of consumption growth reflects a unitary intertemporal elasticity of substitution in the preferences of the investor.

31 / 45 LIMITING STOCHASTIC DISCOUNT FACTOR Martingale component ˆV for consumption and continuation values: ( ) 1 γ [ ] Ct e(x0 ) = exp(ρt)ˆv t. e(x t ) C 0 Limiting stochastic discount factor S t = ( C0 C t ) ( ) γ ( ) Ct e(xt ) ˆV t = exp( ρt). C 0 e(x 0 ) Different limiting risk-free interest rate but the same long-term risk prices.

32 / 45 CONSUMPTION DYNAMICS Suppose that X and Y evolve according to: dy t = ν + H 1 X [1] t dt + X [2] t FdW t dx [1] t = A 1 X [1] t dt + X [2] t B 1 dw t, dx [2] t = A 2 (X [2] t 1)dt + X [2] t B 2 dw t Variables Y is the logarithm of consumption. X [1] governs the predictable growth rate in consumption. X [2] governs the macro volatility. Shocks dw F 1 dw t is the consumption shock. B 1 dw t is the consumption growth shock. B 2 dw t is the consumption volatility shock.

GROWTH-RATE STATE VARIABLE 0.02 0.015 0.01 0.005 0 0.005 0.01 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Year FIGURE: Consumption growth rate and growth-rate state variable. 33 / 45

34 / 45 VOLATILITY STATE VARIABLE 2.5 2 1.5 1 0.5 0 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Year FIGURE: Smoothed volatility estimates and quartiles.

35 / 45 RISK PRICE VECTORS Recall that a risk price for horizon t is: 1 t Power utility log E [S t G t (α) X 0 = x] α ( ) γ Ct S t = exp( δt) C 0 Parameterized multiplicative martingale d log G t (α) = X [2] t α dw t X [2] α 2 t 2

36 / 45 RISK PRICE LIMITS Local risk price vector (Breeden): X [2] 0 γf. Long-term risk price vector: ( ) Ê [ γf (B 1 ) r 1 (B 2 ) r 2 ] where X [2] t X [2] t (r 1 ) B 1 dw t is the surprise movement in γh 1 0 local growth volatility ( ) E X [1] t+τ X[1] t dτ

37 / 45 RISK PRICES.20.15.10.05 Consumption Risk Price 0 0 50 100 150 200 250 300 350 400 investment horizon (quarters) Recursive Utility Model Expected Utility Model

38 / 45 RISK PRICES.20.15.10.05 Consumption Risk Price 0 0 50 100 150 200 250 300 350 400.20.15.10.05 Growth Rate Risk Price 0 0 50 100 150 200 250 300 350 400 investment horizon (quarters) Recursive Utility Model Expected Utility Model

39 / 45 RISK PRICES.20.15.10.05 Consumption Risk Price 0 0 50 100 150 200 250 300 350 400.20.15.10.05 Growth Rate Risk Price 0 0 50 100 150 200 250 300 350 400.10 Volatility Risk Price.05 0 0 50 100 150 200 250 300 350 400 investment horizon (quarters) Recursive Utility Model Expected Utility Model

40 / 45 QUARTILES FOR GROWTH-RATE RISK PRICES 0.2 Recursive Utility 0.15 0.1 0.05 0 0 50 100 150 200 250 300 350 400 0.2 Expected Utility 0.15 0.1 0.05 0 0 50 100 150 200 250 300 350 400 investment horizon (quarters)

41 / 45 SPECIFICATION UNCERTAINTY.20 Consumption Risk Price.15.10.05 0 0 50 100 150 200 250 300 350 400.20 Growth Rate Risk Price.15.10.05 0 0 50 100 150 200 250 300 350 400.10 Volatility Risk Price.05 0 0 50 100 150 200 250 300 350 400 investment horizon (quarters) Recursive Utility Model Expected Utility Model

42 / 45 OTHER EMPIRICAL SPECIFICATIONS Explicit production with long-term uncertainty about technological growth. Regime shift models of volatility and growth - great moderation.

ESTIMATION ACCURACY OF RISK PRICES Fig. 4. A, Posterior histogram for the magnitude Fl 0 F of the immediate response of consumption to shocks. B, Posterior histogram for the magnitude Fl(1)F of the long-run response of consumption to the permanent shock. The vertical axis in each case is constructed so that the histograms integrate to unity. Vertical lines are located at the posterior medians. Source:Hansen, Heaton, Li (JPE) 43 / 45

5. HIGH RISK AVERSION OR A CONCERN FOR MODEL MISSPECIFICATION? The stochastic discount factor is; ( ) St C0 = exp( δt) ˆV t C t where ˆV is a martingale component of the transformed continuation value process. Martingale component in the stochastic discount factor implies a change of probability measure and manifests the alternative robust interpretation of risk-sensitive preferences. 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 γ determines the penalization. Related methods have a long history in robust control theory and statistics. 44 / 45

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 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 make reference to a single benchmark model and as a consequence abstract from learning. 45 / 45