Monetary Policy and the Equity Premium

Monetary Policy and the Equity Premium Christopher Gust David López-Salido Federal Reserve Board Bank of Spain Workshop on Monetary Policy Madrid February 26, 29 GLS () Equity Premium Madrid February 26, 29 / 34

Motivation Monetary policy affects the macroeconomy primarily through financial markets. Two possible channels: Monetary policy affects the (unconditional mean of the) real rate, which in turn affects real economy. Monetary policy affects economic risk (conditional variances). Standard models (i.e. CIA and NK) abstract from the second channel. Evidence that economic risk is important in accounting for asset prices. We develop a DSGE model with both channels to explain the response of equity prices to monetary shocks. GLS (Bank of Spain) Equity Premium Madrid February 26, 29 2 / 34

Sluggish Portfolio Rebalancing We extend the neoclassical CIA model to allow for segmented goods and asset markets. Fixed costs of transferring funds from a brokerage to a checking account. Households only infrequently update their desired allocation of cash across these two accounts. Households are heterogenous in their fixed cost of transferring funds. Recent micro evidence on household finance provides strong support for infrequent portfolio rebalancing with considerable heterogeneity across households. Brunnermeier & Nagel (28) and Haliassos et al. (28) Calvet, Campbell, and Sodini (28, 29) GLS (Bank of Spain) Equity Premium Madrid February 26, 29 3 / 34

Monetary Policy and Stock Prices Using high frequency data Bernanke & Kuttner (25) show that: Stock prices rise percent to a 25 bp surprise in federal funds rate. 2 An important part of the increase is due to changes in equity premia. Standard models can not capture the second fact. We show that a model with infrequent portfolio rebalancing can account for this evidence. GLS (Bank of Spain) Equity Premium Madrid February 26, 29 4 / 34

Related Literature Alvarez, Atkeson, Kehoe (27) emphasize endogenous participation in financial markets to account for time-varying risk in exchange rates. We differ by emphasizing the endogenous asset segmentation along an intensive margin (i.e. rebalancing) rather than the extensive margin (i.e. participation). Abel, Eberly, and Panageas (27) study infrequent adjustment of funds between assets and goods markets in a partial equilibrium setting with rational inattention. We study monetary policy in a general equilibrium framework. GLS (Bank of Spain) Equity Premium Madrid February 26, 29 5 / 34

Model Overview Agents:(i) households, (ii) firms, and (iii) the government. In t =, there is trade in asset markets, and no trade in goods markets. In t, In the asset markets, households trade a complete set of state-contingent claims and equity in the firms. In the goods markets, households use money to buy goods subject to a CIA constraint. There are two sources of uncertainty: aggregate shocks to technology, z t, and to money growth, µ t. We index the states at date t by s t = (z t, µ t ), and s t = (s,..., s t ) denote history through period t. GLS (Bank of Spain) Equity Premium Madrid February 26, 29 6 / 34

Market Segmentation At date t =, households decide on a costless, non-state contingent amount of cash to transfer from assets to goods markets in periods t. The initial non-state-contingent allocation plan (i.e., the annuity) ensures that all agents participate in financial markets. At t, make state-contingent transfers between these markets, households pay a fixed cost. This cost is constant over time but varies across households. If pays this cost, HH actively rebalances portfolio. If not, then inactive. GLS (Bank of Spain) Equity Premium Madrid February 26, 29 7 / 34

Firms Technology: Y(s t ) = K(s t ) α exp(z t )L(s t ) α, with z t = ρ z z t + ɛ zt, and ɛ zt N(, σ 2 z). Firms have a one-period planning horizon. To operate capital in period t +, a firm must purchase it by issuing equity. In equilibrium: w(s t+ ) = ( α) Y(st+ ) L(s t+ ), h i α Y(st+ ) + p + r e (s t+ K(s ) = t ) k (s t+ ) p k (s t ) Fixed factor supply: K(s t ) =, L(s t ) = GLS (Bank of Spain) Equity Premium Madrid February 26, 29 8 / 34

Government Issues one-period state-contingent bonds and controls the economy s money stock, M t. At date t =, the government also issues an annuity at price, P A, which has a constant payoff A in units of consumption. Budget constraints at date t = : Z B = q(s )B(s )ds + P A A, s B is given, q(s ) is the price of the state-contingent bond, B(s ). At dates t, the government s budget constraint: Z B(s t ) + M t + P(s t )A = M t + q(s t, s t+ )B(s t, s t+ )ds t+, s t+ Finally, the government injects cash: M t M t = µ t = ( ρ µ ) µ + ρ µ µ t + ɛ µt, ɛ µt N(, σ 2 µ). GLS (Bank of Spain) Equity Premium Madrid February 26, 29 9 / 34

Households Fixed Costs A household can purchase and sell bonds and stocks in asset markets. Asset and goods markets are segmented, so that a household must pay a real fixed cost, γ, to transfer cash between them. Hence, households are indexed by γ. This cost is constant for a household but differs across households according to the distribution F(γ) with density f (γ). GLS (Bank of Spain) Equity Premium Madrid February 26, 29 / 34

Households and Cash in the Goods Market CIA in goods market for household γ: P(s t )c(s t, γ) = M(s t, γ) + P(s t )A(γ) + P(s t )x(s t, γ)z(s t, γ). At dates t, HH receives a non-state contingent transfer of cash, A(γ), from the annuity purchased at date t =. Households pay fixed cost γ of making state contingent transfer, x(s t, γ), between checking and brokerage accounts. If HH pays fixed cost γ, then z(s t, γ) =. z(s t, γ) =, otherwise. GLS (Bank of Spain) Equity Premium Madrid February 26, 29 / 34

Households and Cash in Asset Markets HH with different fixed costs of transferring x(s t, γ) will demand different A(γ). Through the choice of A(γ), all HHs participate in financial markets. If HH cannot make non-state contingent transfers (i.e., A(γ) = ), there is limited participation in asset markets (AAK (27)). Household cash constraint in asset markets at dates t : Z B(s t, γ)+(+r e (s t ))S(s t, γ)= q(s t, s t+ )B(s t, s t+, γ)ds t+ + s t+ S(s t, γ)+p(s t )[x(s t, γ)+γ]z(s t, γ). GLS (Bank of Spain) Equity Premium Madrid February 26, 29 2 / 34

Households Utility Households maximize: Z β t U(c(s t, γ))g(s t )ds t t= s t whith g(s t ) denotes the probability distribution over history s t, and: U(c) = c σ σ GLS (Bank of Spain) Equity Premium Madrid February 26, 29 3 / 34

Households Budget Constraint Household budget constraint simplifies to: M(s t, γ) = P(s t )w(s t ) Beginning of period cash is independent of γ. This requires: Z β U [c(s t+, γ)] s t+ U [c(s t, γ)] P(s t ) P(s t+ ) g(s t+)ds t+ < For HHs that always rebalance, this implies: R(s t ) > GLS (Bank of Spain) Equity Premium Madrid February 26, 29 4 / 34

Timing in the Two Markets Asset Markets s = (μ, z) are observed Starting Assets B(s)+(+R e )S - Asset Market Constraint If cash transferred: q(s') B(s') + S + P [x(s)+γ] If no transfer: q(s') B(s') + S If transfer x(s), pay fixed cost γ Ending Assets B(s') + S Goods Markets Cash-in-Advanced Constraint: Consumption If cash transferred: c = n + x(s) + A(γ) If no transfer: c = n + A(γ) Shopper Starting Cash: (P - w) Real Balances n = P - w/p Ending Cash: (Pw) Worker Wages sold for cash: P w GLS (Bank of Spain) Equity Premium Madrid February 26, 29 5 / 34

Characterizing Allocations Consumption of inactive rebalancers: c I (s t, γ) = w(st ) µ t + A(γ) Complete risk-sharing among active rebalancers: c A (s t, γ) = c A (s t ), 8 γ HH annuity provides insurance for inactive types: β t U t= Zs (c A (s t )) U (c I (s t, γ)) ( z(s t, γ))g(s t )ds t =. t To get proceeds from equity, inactive types set A(γ). GLS (Bank of Spain) Equity Premium Madrid February 26, 29 6 / 34

The Marginal Rebalancer γ(s t ) is the fixed cost of the marginal rebalancer determined by: U[c A (s t )]-U[c I (s t, γ(s t ))]=U [c A (s t )] c A (s t )-c I (s t, γ(s t ))+ γ(s t ), where c I (s t, γ(s t )) = w(st ) µ t +A( γ(s t )). Consumption of rebalancers is given by: Z Z F( γ(s t ))c A (s t )+ [ w(st ) γ(s t +A(γ)]f (γ)dγ=y(s t ) )- γf (γ)dγ, γ(s t ) µ t where F(γ) and f (γ) are the cdf and pdf of γ: log γ N(eγ m, σ 2 γ). GLS (Bank of Spain) Equity Premium Madrid February 26, 29 7 / 34

State-dependent Rebalancing.364 The Demand Schedule for the Annuity.363 Annuity.362.36.5.5.25 Size of the Fixed Costs of Rebalacing (γ) Household Distribution (Percentile).2.5..5 Active Region Regions of Active and Inactive Rebalancing Inactive Region Active Region.2...2.3.4 Deterministic Level of Money Growth GLS (Bank of Spain) Equity Premium Madrid February 26, 29 8 / 34

Consumption of Rebalancers and Asset Pricing Pricing kernel depends on the consumption of rebalancers: m(s t ca (s, s t+ ) = t σ ) β c A (s t+ ) Risk-free rate: [ + r f t ] = E t m t,t+ Return on equity: The equity premium: + r e t+ = [α exp[( α)z t+] + p k,t+ ] p k,t + r ep t = E t[ + r e t+ ] + r f = cov t β t ca,t c A,t+ σ, + r e t+ GLS (Bank of Spain) Equity Premium Madrid February 26, 29 9 / 34

Global Model Solution Given A(γ), the resource constraint and equilibrium condition for the marginal rebalancer determine c A (s t ) and γ(s t ). Following Tauchen and Hussey (99) and Judd (998), we use the linear Fredholm integral equations (Type 2) and quadrature to determine the price of capital from the stochastic difference equation: Z p k (s t ) = m(s t, s t+ ) s t+ h i α exp[( α)z t+ ] + p k (s t+ ) g(s t+ js t )ds t+ Similar approach to determine A(γ) for a fixed value of γ. Then A(γ) is approximated using piecewise linear interpolation. GLS (Bank of Spain) Equity Premium Madrid February 26, 29 2 / 34

Calibration Parameter Values β.99 α.36 σ 3 γ m.2 σ γ.35 Parameter Values µ.4 ρ z.97 σ z (%).3 ρ µ.9 σ µ (%).7 GLS (Bank of Spain) Equity Premium Madrid February 26, 29 2 / 34

Deterministic Steady State The endogenous rebalancing model reduces to a representative agent model: c A = c I = ( α)µ + A The model only becomes interesting in the presence of uncertainty. The endogenous participation model (strong incentive to participate): c A > c I (= ( α)µ ) GLS (Bank of Spain) Equity Premium Madrid February 26, 29 22 / 34

Endogenous Rebalancing and the Equity Premium 2 Avg. Fraction of Rebalancers = % (γ m =3.5%) Equity Premium (AR%) 8 6 4 Avg. Fraction of Rebalancers = 5% (σ = 4) Avg. Fraction of Rebalancers = 3% (γ =2%, σ=3) m U.S. Data Avg. Fraction of Rebalancers = 24% (γ m =.5%) Avg. Fraction of Rebalancers = % (σ = 2) Standard CIA Model (σ = 5) 2 Standard CIA Model (σ = 3).5.5.5 2 2.5 3 3.5 4 Risk Free Rate (AR%) GLS (Bank of Spain) Equity Premium Madrid February 26, 29 23 / 34

Endogenous Participation 2 Endogenous Rebalancing 8 Equity Premium (AR%) 6 U.S. Data Avg. Participation Rate = 8% (α =.) 4 Endogenous Participation 2 Avg. Participation Rate = % (α =.36).5.5.5 2 2.5 3 3.5 4 Risk Free Rate (AR%) GLS (Bank of Spain) Equity Premium Madrid February 26, 29 24 / 34

Understanding the Mechanism Annuity.364.363.362.36.36 The Demand Schedule for the Annuity.5.5.25 Size of the Fixed Costs of Rebalacing (γ) Endogenous Rebalancing Model Higher γ )less frequency)higher demand A Fraction of Rebalancers.8.6.4.2 MR Schedule GM Schedule.5.7.2 Consumption of Rebalancers.5 Endogenous Participation Model Fraction of Rebalancers.5 MR Schedule GM Schedule.5.64.2 Consumption of Rebalancers GLS (Bank of Spain) Equity Premium Madrid February 26, 29 25 / 34

Understanding the Mechanism Annuity.364.363.362.36.36 The Demand Schedule for the Annuity.5.5.25 Size of the Fixed Costs of Rebalacing (γ) Endogenous Rebalancing Model Higher γ )less frequency)higher demand A Insurance against consumption losses Fraction of Rebalancers.8.6.4.2 MR Schedule GM Schedule.5.7.2 Consumption of Rebalancers.5 Endogenous Participation Model Fraction of Rebalancers.5 MR Schedule GM Schedule.5.64.2 Consumption of Rebalancers GLS (Bank of Spain) Equity Premium Madrid February 26, 29 25 / 34

Understanding the Mechanism Annuity.364.363.362.36.36 Fraction of Rebalancers.8.6.4.2 The Demand Schedule for the Annuity.5.5.25 Size of the Fixed Costs of Rebalacing (γ) Endogenous Rebalancing Model GM Schedule MR Schedule Higher γ )less frequency)higher demand A Insurance against consumption losses Understanding the mechanism (A(γ) = A).5.7.2 Consumption of Rebalancers.5 Endogenous Participation Model Fraction of Rebalancers.5 MR Schedule GM Schedule.5.64.2 Consumption of Rebalancers GLS (Bank of Spain) Equity Premium Madrid February 26, 29 25 / 34

Understanding the Mechanism Annuity.364.363.362.36 Fraction of Rebalancers.36.8.6.4.2 The Demand Schedule for the Annuity.5.5.25 Size of the Fixed Costs of Rebalacing (γ) Endogenous Rebalancing Model.5.7.2 Consumption of Rebalancers.5 MR Schedule Endogenous Participation Model GM Schedule Higher γ )less frequency)higher demand A Insurance against consumption losses Understanding the mechanism (A(γ) = A) GM schedule: γc A γ u +(- γ γ u )c I =e [(-α)z] - γ2 2γ u Fraction of Rebalancers.5 MR Schedule GM Schedule.5.64.2 Consumption of Rebalancers GLS (Bank of Spain) Equity Premium Madrid February 26, 29 25 / 34

Understanding the Mechanism Annuity.364.363.362.36 Fraction of Rebalancers.36.8.6.4.2 The Demand Schedule for the Annuity.5.5.25 Size of the Fixed Costs of Rebalacing (γ) Endogenous Rebalancing Model.5.7.2 Consumption of Rebalancers.5 MR Schedule Endogenous Participation Model GM Schedule Higher γ )less frequency)higher demand A Insurance against consumption losses Understanding the mechanism (A(γ) = A) GM schedule: γc A γ u +(- γ γ u )c I =e [(-α)z] - γ2 2γ u Fraction of Rebalancers.5 MR Schedule GM Schedule.5.64.2 Consumption of Rebalancers MR schedule: (c A -c I ) 2 =c I γ, c I =(-α)µ +A GLS (Bank of Spain) Equity Premium Madrid February 26, 29 25 / 34

A Technology Improvement.9.8 Fraction of Rebalancers.7.6.5.4.3.2. MR MR GM GM Initial (z ) Final (z ).5.5 Consumption GLS (Bank of Spain) Equity Premium Madrid February 26, 29 26 / 34

Individual Heterogeneity Individual Probability of Rebalancing.8.6.4.2..2.3 Fixed Cost of Rebalancing (γ) Average Consumption Heterogeneity in portfolio rebalancing, with a large fraction showing inertia..2..99..2.3 Fixed Cost of Rebalancing (γ) 5 Standard Deviation of Individual Consumption Relative to Aggregate 4 3 2..2.3 Fixed Cost of Rebalancing (γ) GLS (Bank of Spain) Equity Premium Madrid February 26, 29 27 / 34

Individual Heterogeneity Individual Probability of Rebalancing.8.6.4.2..2.3 Fixed Cost of Rebalancing (γ).2. Average Consumption.99..2.3 Fixed Cost of Rebalancing (γ) 5 Standard Deviation of Individual Consumption Relative to Aggregate Heterogeneity in portfolio rebalancing, with a large fraction showing inertia. HH are trading off consumption volatility against higher average consumption. 4 3 2..2.3 Fixed Cost of Rebalancing (γ) GLS (Bank of Spain) Equity Premium Madrid February 26, 29 27 / 34

Individual Heterogeneity Individual Probability of Rebalancing.8.6.4.2..2.3 Fixed Cost of Rebalancing (γ).2. Average Consumption.99..2.3 Fixed Cost of Rebalancing (γ) 5 4 3 2 Standard Deviation of Individual Consumption Relative to Aggregate..2.3 Fixed Cost of Rebalancing (γ) Heterogeneity in portfolio rebalancing, with a large fraction showing inertia. HH are trading off consumption volatility against higher average consumption. Higher volatility of c A translates into high return on equity. GLS (Bank of Spain) Equity Premium Madrid February 26, 29 27 / 34

Monetary Policy and Equity Prices: Bernanke-Kuttner They find that a broad index of stock prices registers a one-day gain of percent in reaction to a 25 basis point easing of the federal funds rate. They decompose the response of stock prices into three components: Current and expected changes in the real rate Expected future excess equity returns or equity premia Current and expected changes in dividends, They conclude that an important channel by which stock prices increase occurs through changes in the equity premium. GLS (Bank of Spain) Equity Premium Madrid February 26, 29 28 / 34

Impulse Response to a Monetary Policy Shock An IRF of y(s t ) to µ is defined as the revision in expectations from a variable s conditional mean (Hamilton (994)): E[log y(s t ) j µ, z ] E[log y(s t ) j µ, z ] where µ = µ and z = z. We use Monte Carlo integration to compute the conditional expectation, which involves multidimensional integrals. GLS (Bank of Spain) Equity Premium Madrid February 26, 29 29 / 34

IRFs to a Money Growth Shock Nominal Interest Rate 3 Stock Price Percentage Points.5.5 Percent 2.5 2.5.5.5 5 9 3.5 5 9 3.5 Real Interest Rate (r ) t Equity Premium Percentage Points.5.5 Percentage Points.5 2 5 9 3.5 5 9 3.2 Consumption of Active Types.7 Fraction of Active Types.6 Percent.8.6.4.2 Percentage Points.5.4.3.2..2 5 9 3 5 9 3 Endogenous Rebalancing Exogenous Rebalancing Endogenous Participation (42%) GLS (Bank of Spain) Equity Premium Madrid February 26, 29 3 / 34

IRFs to a Money Growth Shock (cont.) Nominal Interest Rate 3.5 Stock Price Percentage Points 2 3 4 Percent 3 2.5 2.5.5 5 5 9 3 5 9 3 Real Interest Rate (r ) t Equity Premium Percentage Points 2 3 4 Percentage Points.5 5 6 5 9 3.5 5 9 3.4 Consumption of Active Types.7 Fraction of Active Types.2.6 Percent.8.6.4.2 Percentage Points.5.4.3.2. 5 9 3 5 9 3 ρ μ =.9 ρ μ =.68 GLS (Bank of Spain) Equity Premium Madrid February 26, 29 3 / 34

Why Does a Monetary Expansion Lower Risk? A Small Monetary Expansion.8 Fraction of Rebalancers.6.4.2 Initial (μ ) Final (μ >μ ) Higher µ )higher c A and γ γ u..9.95.5. Consumption of Rebalancers First Derivative of Consumption With Respect to Money Growth 3.5 3 2.5 2.5.5.7..2.4 Deterministic Level of Money Growth GLS (Bank of Spain) Equity Premium Madrid February 26, 29 32 / 34

Why Does a Monetary Expansion Lower Risk? A Small Monetary Expansion Fraction of Rebalancers.8.6.4.2 Initial (μ ) Final (μ >μ ).9.95.5. Consumption of Rebalancers Higher µ )higher c A and γ γ u. Final )less volatile c A. Translates into lower equity premium. First Derivative of Consumption With Respect to Money Growth 3.5 3 2.5 2.5.5.7..2.4 Deterministic Level of Money Growth GLS (Bank of Spain) Equity Premium Madrid February 26, 29 32 / 34

Why Does a Monetary Expansion Lower Risk? A Small Monetary Expansion Fraction of Rebalancers.8.6.4.2.9.95.5. Consumption of Rebalancers 3 2.5 2.5 Initial (μ ) Final (μ >μ ) First Derivative of Consumption With Respect to Money Growth 3.5 Higher µ )higher c A and γ γ u. Final )less volatile c A. Translates into lower equity premium. Lower volatility because γ γ u is higher. In the limit, as γ γ u!, c A is unaffected by µ..5.7..2.4 Deterministic Level of Money Growth GLS (Bank of Spain) Equity Premium Madrid February 26, 29 32 / 34

Why Does a Monetary Expansion Lower Risk? A Small Monetary Expansion Fraction of Rebalancers.8.6.4.2.9.95.5. Consumption of Rebalancers First Derivative of Consumption With Respect to Money Growth 3.5 3 2.5 2.5.5 Initial (μ ) Final (μ >μ ).7..2.4 Deterministic Level of Money Growth Higher µ )higher c A and γ γ u. Final )less volatile c A. Translates into lower equity premium. Lower volatility because γ γ u is higher. In the limit, as γ γ u!, c A is unaffected by µ. Changes in risk reflect that c A is increasing and concave in µ. GLS (Bank of Spain) Equity Premium Madrid February 26, 29 32 / 34

A Reduction in Money Growth A Small Monetary Contraction Fraction of Rebalancers.8.6.4.2 Initial (μ ) Final (μ <μ ).9.95.5. Consumption of Rebalancers A Big Monetary Contraction.8 Fraction of Rebalancers.6.4 Final (μ <μ ) Initial (μ ).2.9.95.5. Consumption of Rebalancers GLS (Bank of Spain) Equity Premium Madrid February 26, 29 33 / 34

Conclusion and Further Research We have developed a DSGE model with infrequent portfolio rebalancing where monetary policy affects the economy through changes in risk. The model is helpful in accounting for the average equity premium and the response of the equity prices to monetary policy shocks. Future research: Feedback from changes in risk to the policy instrument. Endogenous capital and labor supply to jointly analyze asset prices and business cycles. Address how endogenous movements in risk affect optimal monetary policy. GLS (Bank of Spain) Equity Premium Madrid February 26, 29 34 / 34