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A Macroeconomic Model with a Financial Sector Markus K. Brunnermeier and Yuliy Sannikov

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Motivation Financial crises happen every now and then, spill over into real economy –Once about every 10 years in the 400 years in Western Europe - Kindleberger (1993) –Great Depression and the failure of the financial system Fisher (1933), Keynes (1936) –Current financial turmoil Current macro approach –Most DSGE models use representative agents, ignore financing frictions –Models with a financial sector (e.g. Bernanke-Gertler-Gilchrist) log-linearize near steady state, miss instability below steady state 2

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Motivation Current financial turmoil –Spirals and adverse feedback loops –Spillovers Across financial institutions To real economy Current macro approach –Representative agent analysis ignores wealth distribution (including leveraged lending) spillovers (externalities) and hence, financial stability –Steady-state log-linearization ignore many dynamic effects (like precautionary hoarding due to higher volatility). 3

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Modeling financial frictions Central idea: heterogeneous agents have special skills/more productive Bernanke-Gertler-Gilchrist, He-Krishnamurthy, Kiyotaki-Moore less risk-averse Geanakoplos, Adrian-Shin, Garleanu-Pedersen more optimistic Geanakoplos type 1: financially constrained type 2: unlevered 4

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Modeling financial frictions Central idea: heterogeneous agents have special skills/more productive Bernanke-Gertler-Gilchrist, He-Krishnamurthy, Kiyotaki-Moore less risk-averse Geanakoplos, Adrian-Shin, Garleanu-Pedersen more optimistic Geanakoplos type 1: financially constrained type 2: unlevered Diamond (1984) 5 Debt Equity capital inside equity financing capital inside equity financing type 3: intermediary

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Modeling financial frictions Central idea: heterogeneous agents have special skills/more productive Bernanke-Gertler-Gilchrist, He-Krishnamurthy, Kiyotaki-Moore less risk-averse Geanakoplos, Adrian-Shin, Garleanu-Pedersen more optimistic Geanakoplos type 1: financially constrained type 2: unlevered Amplification (e.g. Brunnermeier-Pedersen) Shocks affect the distribution of assets between agents of types 1 and 2, total output 6 Loss of capital Loss of capital shock to net worth shock to net worth Precaution + tighter margins Precaution + tighter margins volatility price volatility price Fire sales Fire sales

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Modeling financial frictions Central idea: heterogeneous agents have special skills/more productive Bernanke-Gertler-Gilchrist, He-Krishnamurthy, Kiyotaki-Moore less risk-averse Geanakoplos, Adrian-Shin, Garleanu-Pedersen more optimistic Geanakoplos type 1: financially constrained type 2: unlevered Amplification (e.g. Brunnermeier-Pedersen) Shocks affect the distribution of assets between agents of types 1 and 2, total output But… Existing models study only steady-state dynamics or local dynamics two-three period models 7 Loss of capital Loss of capital shock to net worth shock to net worth Precaution + tighter margins Precaution + tighter margins volatility price volatility price Fire sales Fire sales

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Preview of results 1.Unified model, replicates dynamics near steady state (e.g. Bernanke-Gertler-Gilchrist) … but unstable dynamics away from steady state due to (nonlinear) liquidity spirals 2.Welfare: Fire-sale externalities within financial sector, externalities b/w financial sector and real economy… –When levering up, institutions ignore that their fire-sales depress prices for others –Inefficient pecuniary externality in incomplete market setting 3.Securitization can lead to excessive leverage pricesexpert net worth volatility

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Overview Start with simple model Show non-linearities in equilibrium Add households, capital producers and direct investment to show externalities Add idiosyncratic shocks to show effects of securitization 9

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Model: production of output and capital Experts hold capital, which produces output y t = a k t Investment of ι(g) k t makes capital grow at rate g δ: expert depreciationι(-δ) = 0 Liquidation value: ι’(-∞) = a/(r + δ * ), δ * > δ 10 Household discount rateHousehold depreciation

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Model: production of output and capital Experts hold capital, which produces output y t = a k t Investment of ι(g) k t makes capital grow at rate g δ: expert depreciationι(-δ) = 0 Liquidation value: ι’(-∞) = a/(r + δ * ), δ * > δ Holding capital is risky dk t = g k t dt + k t dZ t ρ ≥ r: expert discount rate (may “pay out” too much) 11 Household discount rateHousehold depreciation Brownian macro shock

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Financing through experts Agency problem: experts can increase depreciation/cause losses Get benefit b per unit of capital –Assume: effort, k t i not contractible, but market value of assets p t k t i is –Incentive constraint b - t p t ≤ 0 t b/p t -Solvency constraint: debt holders liquidate when k t p t drops to d t -Expert capital depends on aggregate risk, amplified through prices -Later: contracting on aggregate risk separately 12 Assets Liabilities d t = k t i p t – e t Inside n t = α t e t Outside (1-α t )e t ktiptktipt etet Experts borrow and may unload some risk

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Balance sheets 13 Assets Liabilities Equity = net worth n t = p t k t - d t Debt d t p t k t dp t = t p dt+ t p dZ t dk t = gk t dt+ k t dZ t d(k t p t ) = k t (g p t + t p + t p ) dt + k t ( p t + t p ) dZ t dd t = (r d t - a k t + ι(g) k t ) dt - dc t For simplicity, take α = 1 for most of this talk

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Evolution of balance sheets –Asset side – long maturity d(k t p t ) = k t (g p t + t p + t p )dt + k t ( p t + t p )dZ t –Liability side 1.Debt 1.Debt – overnight maturity dd t = (r d t - a k t + ι(g) k t ) dt - dc t 2.Equity d n t = d (k t p t ) - dd t = rn t + k t [(a-ι(g)-(r-g)p t + t p + t p )dt+( p t + t p )dZ t ] -dc t 14

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Equilibrium Aggregate: N t, K t State variable t = N t /K t Price p( t ) Expert value function f( t )n t Bellman equation f( t )n t = max k,g,c E[dc+d(f( t )n t )] = max k,g,c {dc t +μ t f n t + f( t ) (rn t + k t (a-ι(g)-(r-g)p t + t p + t p )) + σ t f k t ( p t + t p )} FOC : ι’(g) = p t, a-ι(g)-(r-g)p t + t p + t p = - σ t f /f( t ) ( p t + t p ) dc t = 0 unless f( t )=1 Bellman equation: ( - r) f( t ) = μ t f Ito’s lemma: t p = t η p’ + ½( t η ) 2 p’’ differential equations 15 (risk premium) * p t

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Stochastic Discount Factor Experts’ SDF: m 0,t = e -ρt f(η t )/f(η 0 ) Households’ SDF: m 0,t HH = e -rt –Note that m 0,t = m 0,t HH, since δ*> δ / time preference agency constraint

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Equilibrium 17

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Equilibrium 18

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Full vs. steady-state dynamics Log-linearized impulse response functions Stationary distribution around steady-state But.. below steady state system gets unstable, volatility Liquidity spiral prices expert net worth volatility ηtηt time

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System dynamics and instabilities Volatility effects –Macro shocks –Higher volatility exacerbates precautionary hoarding motive, amplifying price drops –Outside equity shrinks (+deleveraging) Dynamical system spends most time near steady state, but has occasional volatile destructive episodes 20 Loss of capital Z t -shock on k t Precaution + tighter margins tppttppt Fire sales amplification Sensitivity of price to η t

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Asset pricing (time-series) Predictability –For low η values price is (temporarily) depressed –Price-earnings ratio predicts future prices Current earnings: ak 0 Current price: p 0 k 0 =E 0 [e -ρt f(η t )/f(η 0 ) (p t k t +div t )] Excess volatility –Volatility of k t p t per dollar is σ + σ t p /p t –Cash flow volatility amplified by SDF movements Stochastic volatility –See σ η plot

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Asset pricing (cross section) Correlation increases with σ p –Extend model to many types j of capital dk j /k j = g dt + σ dz + σ' dz j –Experts hold diversified portfolios Equilibrium looks as before, but Volatility of p t k t is σ + σ p + σ’ For uncorrelated z j and z l correlation (p t j k t j, p t l k t l ) is (σ + σ p )/(σ + σ p + σ’) which is increasing in σ p –xxx aggregate shock uncorrelated shock

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Externalities so far there are no externalities… Proposition. The competitive equilibrium in this economy is equivalent to the optimal policy by a monopolist expert. Sketch of proof. (1) Write Bellman equation for monopolist. (2) Define price p t = ι’(g). (3) Show that prices etc. are as in competitive eq. Intuition: In competitive equilibrium experts do affect prices by their choices (compensation and investment), but they are isolated from prices because they don’t trade given equilibrium prices. 23

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Optimal payout policy of monopolist Debt dD t = (rD t - a K t + ι(g) K t ) dt – dC t, take ρ > r Solvency constraint:D t ≤ LK t Value function h(ω t )K t where ω t = -D t /K t Bellman equation… 24

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Modification 1: add labor sector Fixed labor supply L Production function a’ K t k t 1- l t Workers get wages w t = a’ K t L -1 Experts get ak = (1 - ) a’ L k Worker welfare depends on K t Experts do not take that into account when choosing leverage (investment and payout decisions) Household value function… 25

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Externalities with households 26

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Modification 2: speculative households So far fixed liquidation value at a/(r + δ * )… now households can sell back to experts –Break even for HH a- rp t - δ * p t + t p + t p ≤ 0, equality when experts hold fraction ψ t < 1 of assets –depreciation rate is δ * > δ –p t ≥ a/(r + δ * ) In equilibrium households pick up assets when financial sector suffers losses, i.e. η t becomes small Fire sale externalities (within financial sector) – when levering up, experts hurt prices that other experts can sell to households in the event of a crisis 27 Capital gains/losses, E[d(k t p t )] earnings financing cost

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Equilibrium when households provide liquidity support 28

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Modification 3: Idiosyncratic losses dk t i = g k t i dt + k t i dZ t + k t i dJ t i J t i is an idiosyncratic compensated Poisson loss process, recovery distribution F and intensity λ(σ t p ) V t = k t p t drops below D t, costly state verification by debt

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Review: costly state verification Developed by Townsend (1979), used in Diamond (1984), Bernanke-Gertler-Gilchrist Time 0: principal provides funding I to agent Time 1: agent’s profit y ~ F[0, y * ] is his private information but principal can verify y at cost Optimal contract (with deterministic verification) is debt with face value D: agent reports y truthfully and pays D if y ≥ D, triggers default and pays y if y < D In our context: expert can cause losses (reduce V t for private benefit); debtholders verify if V t falls below D t

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Modification 3: Idiosyncratic losses dk t i = g k t i dt + k t i dZ t + k t i dJ t i J t i is an idiosyncratic compensated Poisson loss process, recovery distribution F and intensity λ(σ t p ) V = k t p t drops below D t, costly state verification by debt Debtholders’ loss rate Verification cost rate Leverage bounded not only by precautionary motive, but also the cost of borrowing Asset Liabilities D t = k t p t – E t Equity V t = k t p t

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Equilibrium Experts borrow at rate larger than r Rate depends on leverage, price volatility d t = diffusion process (without jumps) because losses cancel out in aggregate 32

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Securitization Experts can contract on shocks Z and L directly among each other, contracting costs are zero In principle, good thing (avoid verification costs) Equilibrium –experts fully hedge idiosyncratic risks –experts hold their share (do not hedge) aggregate risk Z, market price of risk depends on t f ( p t + t p ) –with securitization, experts lever up more (as a function of t ) and pay themselves sooner –financial system becomes less stable 33

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Contracting Agency problem: experts can lower growth rate/cause losses Get benefit b per unit of capital –Assume: effort, k t i not contractible, but market value of assets p t k t i is –Incentive constraint b - t p t ≤ 0 t b/p t -Expert capital depends on aggregate risk, amplified through prices -Contracting on aggregate risk separately: consider that in an extension, but only within financial sector 34 Assets Liabilities d t = k t i p t – e t Inside n t = α t e t Outside (1-α t )e t ktiptktipt etet

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Evolution of balance sheets –Asset side – long maturity d(k t p t ) = k t (g p t + t p + t p )dt + k t ( p t + t p )dZ t –Liability side 1.Debt 1.Debt – overnight maturity 2.Outside equity 2.Outside equity: 1- t 3.Inside equity 3.Inside equity dn t = n t + k t [(a - p t +g p t + t p + t p )dt + t ( p t + t p ) dZ t ] All results (nonlinear dynamics, externalities, excessive leverage under securitization) hold in this expanded model 35 Capital appreciation, E[d(k t p t )]

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Conclusion Incorporate financial sector in macromodel –Higher growth –Exhibits instability – due to non-linear liquidity spirals (away from steady state) Leverage/payouts chosen without taking into account externalities –Towards households (labor provision) –Within financial sector: possible fire sales compromise others’ balance sheets Securitization avoids inefficiencies, but can increase instabilities (higher leverage/payout rate) 36

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To do list Introducing risk aversion –Asset pricing implication – time-varying risk premia –Time-varying risk-free interest rate Incorporating instabilities into general DSGE model Optimal regulation Introducing money/assets with different liquidity Exploring maturity mismatch 37

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Thank you!☺

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Differences to Bernanke-Gertler-Gilchrist BGG 1. “small” aggregate shocks, log- linearization around steady state 2.Price dynamics driven by idiosyncratic shocks and default risk –Higher state verification costs when expert capital goes down 3.With small aggregate noise, expert incentives to keep “dry powder” (liquidity) are negligible 4.Countercyclical leverage –Experts take on same position after drop in net worth –Leverage increases after drop in net- worth 5.Debt vs. Equity 6.No fire-sale externality 39 Brunnermeier-Sannikov 1. Focus on (large) aggregate shocks (idiosyncratic shocks not essential), explore nonlinearities using Bellman equation 2. Asset price drops also due to fire sales 3. Expert’s rent depends on state t Incentive to keep “dry powder” (liquidity) 4. Procyclical leverage: Experts reduce position after drop in net worth Liquidity spirals 5. Securitization (debt, inside + outside equity) 6. Fire-sale externality (rationale for regulation)

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40 Differences to Kiyotaki-Moore KM – (Kiyotaki version) 1.Zero-prob. temporary shock –Persistent (dynamic loss spiral) –Amplified through collateral value 2.Non- vs. productive (leveraged) sector 3.Dual role of durable asset 1.Production 2.Collateral 4.Exogenous contract –One period contract –Debt is limited by collateral value 5.Durable asset doesn’t depreciates (capital, fully) 40 BruSan 1. Permanent TFP shocks Margin/haircut spiral (leverage) Loss spiral 2. Investment through leveraged financial sector 3. Dual role of durable asset 1. Production 2. Securitization 4. Optimal contract Dynamic contract Debt is limited due idiosyncratic risk and costly state verification 5. δ-depreciation rate

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41 Differences to He-Krishnamurthy He-Krishnamurthy 1.Endowment economy –GDP growth is exogenously fixed –No physical investment 2.No direct investment in risky asset by households –Limited participation model 3.Contracting –Only short-run relationship (t to t+dt) –Fraction of return, fee –Asset composition (risky vs. risk-free) is not contractable –Non-effort lowers return by xdt x is exogenous,not linked to fundamental –Private benefit from shirking –No benchmarking 4.Pricing Implications –When experts wealth declines, their market power increases, and so does their fee –Price impact depends on assumption that household have larger discount rate than experts 5.Procyclical Leverage 6.In H-K calibration paper 1.No fee, households are rationed in their investment 2.As expert wealth approaches 0, interest rate can go to –∞ 3.Heterogeneous labor income for newborns of lD t 4.Non-log utility function BruSan 1. Production economy GDP growth depends on net-wealth Physical investment 2. Direct investments by all households 3. Contracting (Potential) long-run relationship Fraction of return, fee, size of asset pool Effort increases fundamental growth to gdt Monetary benefit from shirking No benchmarking 4. Pricing Implication Price drop with state variable 5. Countercyclcial Leverage Entrepreneur take on same position after drop in networth Leverage increases after drop in net- worth

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Graphs: leverage 42

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