EXAMINING MACROECONOMIC MODELS

1 / 24 EXAMINING MACROECONOMIC MODELS WITH FINANCE CONSTRAINTS THROUGH THE LENS OF ASSET PRICING Lars Peter Hansen Benheim Lectures, Princeton University

EXAMINING MACROECONOMIC MODELS WITH FINANCING CONSTRAINTS THROUGH THE LENS OF ASSET PRICING Dynamic stochastic equilibrium models of macro economy are designed to match transient time series properties including impulse response functions. Gertler, Kiyotaki and others have extended this literature by introducing financing constraints that alter investment opportunities. Since these models aim to be structural, they have implications for asset pricing. To assess these implications, we consider asset pricing counterparts to impulse response functions. We quantify the exposures of alternative macroeconomic cash flows to shocks over alternative investment horizons and the corresponding prices or compensations that investors must receive because of their exposure to such shocks. We build on the continuous-time methods developed in Hansen and Scheinkman and our earlier work and construct discrete-time shock elasticities that measure the sensitivity of cash flows and their prices to economic shocks including economic shocks featured in the empirical macroeconomics literature. 2 / 24

OUTLINE Construct shock price and shock exposure elasticities as inputs into valuation accounting. Go beyond log-linearization methods to accommodate stochastic volatility and other sources of nonlinearity in the dynamic evolution. Previously - long-run cash flow predictability and time-variation in expected returns. Our aim - dynamic risk or shock exposure elasticities for cash-flows and dynamic risk or shock price elasticities. References: Pricing Growth Rate Risk, Hansen and Schienkman, Finance and Stochastics Risk Price Dynamics, Borovička, Hansen, Hendricks and Scheinkman, Journal of Financial Econometrics Explore these elasticities in the context of a recent macroeconomic model with financial frictions. Ongoing work with Borovička. 3 / 24

4 / 24 TALK OUTLINE Elasticities and valuation accounting Log-exponential parameterization Recursive utility revisited Financial market wedges

5 / 24 SETUP Suppose X is first-order Markov, and W is an iid sequence of multivariate, standard normally distributed random vectors. Conditional Gaussian model in logarithms: t 1 Y t = [β (X s ) + α (X s ) W s+1 ]. s=0 Levels M t = exp(y t ). Examples of M include a macroeconomic growth functional G such as consumption or capital and a stochastic discount factor functional S used to price assets.

SINGLE-PERIOD ASSET PRICING Suppose that log G 1 = β g (X 0 ) + α g (X 0 ) W 1 log S 1 = β s (X 0 ) + α s (X 0 ) W 1 R 1 = G 1 E(S 1 G 1 X 0 ) Logarithm of the expected return is: log E(G 1 X 0 = x) log E(S 1 G 1 X 0 = x) = β s (x) α g (x) α s (x) α s(x) 2 2 Then α s is the risk price vector for exposure to the components of W 1. Asset pricing puzzle: Modeled versions of α s are too small. Recursive utility gives one way to address using seemingly large values of γ. 6 / 24

ALTERNATIVE APPROACH THAT EXTENDS TO Compute elasticities. OTHER INVESTMENT HORIZONS Consider a parameterized family of payoffs. where H 1 (r) = rα h (X 0 ) W 1 r2 2 α h(x 0 ) 2 E[ α h (X 0 ) 2 ] = 1. Then α h gives an exposure direction and H 1 (r) has conditional expectation equal to one. Form G 1 H 1 (r) where log G 1 +log H 1 (r) = [α g (X 0 ) + rα h (X 0 )] W 1 +β g (X 0 ) (r)2 2 α h(x 0 ) 2 Parameterized family of asset payoffs to be priced. 7 / 24

8 / 24 Compute expected return: ELASTICITIES log E[G 1 H 1 (r) X 0 = x] log E[S 1 G 1 H 1 (r) X 0 = x] Differentiate: d dr log E[G 1H 1 (r) X 0 = x] r=0 d dr log E[S 1G 1 H 1 (r) X 0 = x] r=0 Component elasticities: 1. shock-exposure elasticity: ε g (x) = d dr log E[G 1H 1 (r) X 0 = x] r=0 = α g (x) α h (x) 2. shock-value elasticity: ε v (x) = d dr log E[S 1G 1 H 1 (r) X 0 = x] r=0 = α s (x) α h (x)+α g (x) α h (x) 3. shock-price elasticity: ε p (x) = ε g (x) ε v (x) = α s (x) α h (x)

9 / 24 EXTENDING THE INVESTMENT HORIZON Construct payoff: G t H 1 (r). Compute price: E [S t G t H 1 (r) X 0 = x] Form elasticities: 1. shock-exposure elasticity for horizon t: ε g (x, t 1) = d 1 dr t log E[G th 1 (r) X 0 = x] r=0 2. shock-value elasticity for horizon t and (and shock date one): ε v (x, t 1) = d dr 3. shock-price elasticity for horizon t: 1 t log E[S tg t H 1 (r) X 0 = x] r=0 ε p (x, t 1) = ε g (x, t 1) ε v (x, t 1).

10 / 24 REPRESENTATIONS Let M be a multiplicative functional (either G or SG). Then ε m (x, t) = α h (x) E (M tw 1 X 0 = x) E (M t X 0 = x) Observations When M is log-linear, essentially recovers the impulse response function for log M in response to a shock α h W 1. Shock exposure elasticities reflect impulse response functions for log G, shock-price elasticities reflect impulse response functions for log S With stochastic volatility and other sources of nonlinearity, the choice of G matters for computing the shock-price elasticities. The elasticities are inputs into valuation accounting.

11 / 24 REPRESENTATION CONTINUED Recall the change of measure based on the factorization: M t = exp(ηt) ˆM t e(x 0 ) e(x t ) where ˆM is multiplicative martingale. Then ε m (x, t 1) = α h (x) Ê [ê(x t)w 1 X 0 = x]. Ê [ê(x t ) X 0 = x] where ê = 1 e. Observations: Under the change of measure, the expectation of W 1 is not zero and determines limiting value: α h (x) Ê [W 1 X 0 = x]. Reflects the state-dependent counterpart to the impulse response function for ê using Markov diffusions.

12 / 24 SHIFTING THE SHOCK EXPOSURE DATE Shock date is at τ and the impact date at t + τ. The resulting elasticity is: Ê [ê(x t+τ )ε(x τ 1, t) X 0 = x] Ê [ê(x t+τ ) X 0 = x] = Ê [ê(x t+τ )α h (X τ 1 ) W τ X 0 = x] Ê [ê(x t+τ ) X 0 = x] The shifted elasticity depends on ê and on the alternative probability distribution. Some limits: For large t and a fixed τ the limiting elasticity is: Ê [α h (X τ 1 ) W τ X 0 = x]. For t = 0 and large τ the limiting elasticity is: Ê [ê(x 0 )α h (X 0 ) W 1 ]. Ê [ê(x 0 )]

13 / 24 TALK OUTLINE Elasticities and valuation accounting Log-exponential parameterization Recursive utility revisited Financial market wedges

14 / 24 EXPONENTIAL-QUADRATIC FRAMEWORK Triangular state vector system: X 1,t+1 = 10 + 11 X 1,t + Λ 10 W t+1 X 2,t+1 = 20 + 21 X 1,t + 22 X 2,t + 23 (X 1,t X 1,t ) +Λ 20 W t+1 + Λ 21 (X 1,t W t+1 ) + Λ 23 (W t+1 W t+1 ) Stable dynamics if 11 and 22 have stable eigenvalues. Structure allows for stochastic volatility through X 1,t W t+1 Additive functionals Y t+1 Y t = Γ 0 + Γ 1 X t + Γ 2 (X 1,t X 1,t ) +Θ 0 W t+1 + Θ 1 (X 1,t W t+1 ) + Θ 2 (W t+1 W t+1 ) Use to model stochastic discount factor and growth functionals.

COMPONENT CALCULATIONS The framework allows for quasi-analytical formulas for conditional expectations of multiplicative functionals and for elasticities. Start with log f (x) = φ + Φx + 1 2 (x 1) Ψ(x 1 ) Then log E [( Mt+1 M t ) ] f (X t+1 ) X t = x = log E [exp (Y t+1 Y t ) f (X t+1 ) X t = x] = φ + Φ x + 1 2 (x 1) Ψ (x 1 ) = log f (x) Functional form for positive eigenfunctions and conditional expectations that accommodates growth and discounting and stochastic volatility. 15 / 24

16 / 24 APPROXIMATION Much of the existing macroeconomic literature uses perturbation methods. Our approach log-linearization/ first-order approach used to fit responses to macroeconomic quantities to small shocks not suitable for stochastic volatility log approximation of our one-period valuation operators second-order in x 1. include both zeroth order and first-order terms in shock exposure include zeroth and first-order terms in x 2 and zeroth, first and second-order terms in x 1 impose stochastic stability - follow Schmitt-Grohe and Uribe, Kim, Kim, Schaumburg and Sims, and Lombardo. Convenient, but...

17 / 24 TALK OUTLINE Elasticities and valuation accounting Log-exponential parameterization Recursive utility revisited Financial market wedges

Continuation values RECURSIVE UTILITY where V t = [(ζc 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 Intertemporal marginal rate of substitution S t+1 S t = exp( δ) ( Ct+1 C t ) ρ [ ] ρ γ Vt+1. R t (V t+1 ) Depends on continuation values, which gives a channel for sentiments to matter. Used to represent asset prices. 18 / 24

19 / 24 A LOGNORMAL EXAMPLE WITH ρ = 1 Use a specification from Hansen-Heaton-Li (JPE). State dynamics: X t+1 = 0 + 1 X t + ΛW t+1 Consumption dynamics: Y t+1 Y t = Γ 0 + Γ 1 X t + ΘW t+1 where log C t = Y t. Impulse response/shock exposure elasticity for shock α h W 1 and G = C: [ ] ε g (0) = Θα h ε g (t) = Θ + Γ t 1 1 j=0 ( 1) j Λ α h No state dependence.

20 / 24 RECURSIVE UTILITY EXAMPLE CONTINUED Shock-exposure elasticity for shock α h W 1 and G = C: [ ] ε c (0) = Θα h ε g (t) = Θ + Γ t 1 1 j=0 ( 1) j Λ α h Shock-price elasticity for shock α h W 1 [ ] ε p (j) = ε g (j) + (γ 1) Θ + Γ 1 (I β 1 ) 1 Λ α h When γ is large the price elasticity is dominated by ( ) (γ 1) Θ + Γ 1 [I exp( δ) 1 ] 1 Λ α h, which does not depend on the investment horizon. When exp( δ) 1, this term coincides with the limiting consumption exposure elasticity scaled by γ 1. Recursive utility adds a forward-looking component to valuation.

SHOCK-PRICE TRAJECTORIES FOR POWER AND RECURSIVE UTILITY 0.04 0.03 0.02 0.01 0.3 0.2 0.1 Temporary Shock Price Elasticity 0 0 10 20 30 40 50 60 70 80 Permanent Shock Price Elasticity 0 0 10 20 30 40 50 60 70 80 Volatility Shock Price Elasticity 0.04 0.03 0.02 0.01 0 0 10 20 30 40 50 60 70 80 quarters Revisited from lecture one. Stochastic volatility is incorporated. Volatility state set at its unconditional mean. 21 / 24

22 / 24 TALK OUTLINE Elasticities and valuation accounting Log-exponential parameterization Recursive utility revisited Financial market wedges (more discussion will be added)

23 / 24 MODEL INGREDIENTS Gertler-Kiotaki with recursive utility. Communication friction - firms that produce output return proceeds to consumers at random dates in the future. Wedge between internal and external financing of new capital - external financing recognizes that producers could confiscate a fraction of the capital after respecting commitments to internal financiers. Represent this wedge with two stochastic discount factors. No financial distortions for the production of new investment goods. Four shocks - neutral technology shock, investment specific shock, capital quality shock, and a financing constraint shock that represents the potency of the threat to confiscate. Stochastic volatility.

Pricing differences across stochastic discount factors SHOCK-PRICE ELASTICITIES Shock-price elasticities for S (household) and S 1 (bank) 0.25 0.2 neutral technology shock household sdf bank sdf 0.15 0.1 investment specific shock 0.15 0.1 0.05 0.05 0 0 0 20 40 60 80 100 maturity (quarters) 0.05 0 20 40 60 80 100 maturity (quarters) 0.8 capital quality shock 0.01 financing constraint shock 0.6 0 0.01 0.4 0.02 0.2 0.03 0.04 0 0 20 40 60 80 100 maturity (quarters) 0.05 0 20 40 60 80 100 maturity (quarters) Borovička, Hansen, University of Chicago Financial constraints 30/46 Banks are the internal financiers. Shock-price trajectories are depicted at the median state volatility state and at the upper and lower quartiles. 24 / 24