a macroeconomic model with a financial sector markus k. brunnermeier and yuliy sannikov

A Macroeconomic Model with a Financial Sector

Markus K. Brunnermeier and Yuliy Sannikov

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).

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Modeling financial frictions• Central idea: heterogeneous agents

• have special skills/more productiveBernanke-Gertler-Gilchrist, He-Krishnamurthy, Kiyotaki-Moore

• less risk-averse Geanakoplos, Adrian-Shin, Garleanu-Pedersen• more optimistic Geanakoplos

type 1: financially constrained type 2: unlevered

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Diamond (1984) 5

Debt

Equity

capital inside equity

financing

capital inside equity

financing

type 3: intermediary





• Amplification (e.g. Brunnermeier-Pedersen)• Shocks affect the distribution

of assets between agents of

types 1 and 2, total output

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Loss of capital

Loss of capital

shockto net worth

shockto net worth

Precaution+ tighter margins


volatilityprice

volatilityprice

FiresalesFire

sales




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

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Loss of capital

Loss of capital

shockto net worth

shockto net worth



volatilityprice

volatilityprice

FiresalesFire

sales

Preview of results1. Unified model, replicates dynamics near steady state (e.g.

Bernanke-Gertler-Gilchrist)

… but unstable dynamics away from steady statedue 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

prices expert net worth volatility

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

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Model: production of output and capital

• Experts hold capital, which produces output

yt = a kt

• Investment of ι(g) kt makes capital grow at rate g

• δ: expert depreciation ι(-δ) = 0• Liquidation value: ι’(-∞) = a/(r + δ*), δ* > δ

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Household discount rate Household depreciation

Model: production of output and capital

• Experts hold capital, which produces output

yt = a kt

• Investment of ι(g) kt makes capital grow at rate g

• δ: expert depreciation ι(-δ) = 0• Liquidation value: ι’(-∞) = a/(r + δ*), δ* > δ

• Holding capital is risky

dkt = g kt dt + kt dZt

• ρ ≥ r: expert discount rate (may “pay out” too much)11

Household discount rate Household depreciation

Brownian macro shock

Financing through experts• Agency problem: experts can increase depreciation/cause losses• Get benefit b per unit of capital

– Assume: effort, kti not contractible, but market value of assets ptkt

i is – Incentive constraint

b - t pt ≤ 0 t b/pt

- Solvency constraint: debt holders liquidate when ktpt drops to dt

- Expert capital depends on aggregate risk, amplified through prices- Later: contracting on aggregate risk separately

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Assets Liabilities

dt = ktipt – et

Insident = αtet

Outside(1-αt)et

ktipt

et

Experts borrow and may unload some risk

Balance sheets

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Assets Liabilities

Equity =net worth

nt = ptkt - dt

Debt dtptkt

dpt= tp dt+t

p dZt

dkt = gkt dt+kt dZt

d(ktpt) = kt (g pt + tp + t

p) dt + kt (pt + tp) dZt

ddt = (r dt - a kt + ι(g) kt) dt - dct

• For simplicity, take α = 1 for most of this talk

Evolution of balance sheets– Asset side – long maturity


p)dt + kt (pt + tp)dZt

– Liability side

1.1. DebtDebt – overnight maturity

ddt = (r dt - a kt + ι(g) kt) dt - dct

2.2. EquityEquity

dnt = d(ktpt) - ddt = rnt + kt [(a-ι(g)-(r-g)pt+t

p+tp)dt+(pt+t

p)dZt] -dct14

Equilibrium• Aggregate: Nt, Kt

• State variable t = Nt/Kt

• Price p(t)• Expert value function f(t)nt • Bellman equation

f(t)nt = maxk,g,c E[dc+d(f(t)nt)] = maxk,g,c {dct+μt

fnt + f(t) (rnt + kt(a-ι(g)-(r-g)pt+tp+t

p)) + σtfkt(pt+t

p)}

FOC: ι’(g) = pt, a-ι(g)-(r-g)pt+tp+t

p = - σtf/f(t) (pt+t

p)

dct = 0 unless f(t)=1

Bellman equation: ( - r) f(t) =μtf

Ito’s lemma: tp = t

η p’ + ½(tη)2p’’ differential equations

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(risk premium) * pt

Stochastic Discount Factor

• Experts’ SDF: m0,t = e-ρt f(ηt)/f(η0)

• Households’ SDF: m0,tHH= e-rt

– Note that m0,t = m0,tHH, since δ*> δ/

time preference agency constraint

Equilibrium

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Equilibrium

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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

ηt

time

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 episodes20

Loss of capital

Zt-shock on kt


tp

pt

Firesales

.)('1

)()('

t

tttpt p

pp

amplification

Sensitivity of price to ηt

Asset pricing (time-series)

• Predictability – For low η values price is (temporarily) depressed– Price-earnings ratio predicts future prices

• Current earnings: ak0

• Current price: p0k0 =E0[e-ρt f(ηt)/f(η0) (ptkt+divt)]

• Excess volatility

– Volatility of ktpt per dollar is σ + σtp/pt

– Cash flow volatility amplified by SDF movements

• Stochastic volatility– See ση plot

Asset pricing (cross section)

• Correlation increases with σp

– Extend model to many types j of capital dkj/kj = g dt + σ dz + σ' dzj

– Experts hold diversified portfolios• Equilibrium looks as before, but

• Volatility of ptkt is σ + σp + σ’

• For uncorrelated zj and zl

correlation (ptjkt

j, ptlkt

l) is (σ + σp)/(σ + σp + σ’)which is increasing in σp

– xxx

aggregateshock

uncorrelatedshock

Externalitiesso 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 pt = 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.

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Optimal payout policy of monopolistDebt dDt = (rDt - a Kt + ι(g) Kt) dt – dCt, take ρ > rSolvency constraint: Dt ≤ LKt Value function h(ωt)Kt where ωt = -Dt/Kt Bellman equation…

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Modification 1: add labor sector• Fixed labor supply L• Production function

a’ Kt kt

1- lt

• Workers get wages wt = a’ Kt L-1

• Experts get ak = (1 - ) a’ L k • Worker welfare depends on Kt

• Experts do not take that into account when choosing leverage (investment and payout decisions)

• Household value function…

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Externalities with households

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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- rpt - δ*pt + tp+ t

p ≤ 0,equality when experts hold fraction ψt < 1

of assets – depreciation rate is δ* > δ – pt ≥ a/(r + δ*)

• In equilibrium households pick up assets when financial sector suffers losses, i.e. ηt 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

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Capital gains/losses, E[d(ktpt)]

earnings

financing cost

Equilibriumwhen

households provide liquidity support

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Modification 3: Idiosyncratic losses

dkti = g kt

i dt + kti dZt + kt

i dJti

Jti is an idiosyncratic compensated Poisson loss process, recovery distribution F and intensity λ(σt

p)

Vt = ktpt drops below Dt, costly state verification by debt

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 Vt for private benefit); debtholders verify if Vt falls below Dt

Modification 3: Idiosyncratic losses

dkti = g kt

i dt + kti dZt + kt

i dJti

Jti is an idiosyncratic compensated Poisson loss process, recovery distribution F and intensity λ(σt

p)

V = ktpt drops below Dt, costly state verification by debt

• Debtholders’ loss rate

• Verification cost rate

• Leverage bounded not only by

precautionary motive, but also

the cost of borrowing

( p )V (0

DV

DV x)dF(x)

Asset Liabilities

Dt = ktpt – Et

Equity

Vt = ktpt

)(

0

)()(

VD

VD

C

p xcxdFV

Equilibrium• Experts borrow at rate larger than r • Rate depends on leverage, price volatility

• dt = diffusion process (without jumps) because losses cancel out in aggregate

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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 tf (pt + t

p)

– with securitization, experts lever up more (as a function of t) and pay themselves sooner

– financial system becomes less stable

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Contracting• Agency problem: experts can lower growth rate/cause losses• Get benefit b per unit of capital

– Assume: effort, kti not contractible, but market value of assets ptkt

i is – Incentive constraint

b - t pt ≤ 0 t b/pt

- Expert capital depends on aggregate risk, amplified through prices- Contracting on aggregate risk separately: consider that in anextension, but only within financial sector

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Assets Liabilities

dt = ktipt – et

Insident = αtet

Outside(1-αt)et

ktipt

et

Evolution of balance sheets– Asset side – long maturity


p)dt + kt (pt + tp)dZt

– Liability side1.1. DebtDebt – overnight maturity2.2. Outside equityOutside equity: 1-t

3.3. Inside equityInside equitydnt = nt + kt [(a - pt +g pt + t

p+ tp)dt

+ t (pt+tp) dZt]

All results (nonlinear dynamics, externalities, excessive leverage under securitization) hold in this expanded model

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Capital appreciation, E[d(ktpt)]

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)

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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

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Thank you! ☺

Differences to Bernanke-Gertler-Gilchrist

BGG1. “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

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Brunnermeier-Sannikov1. 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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Differences to Kiyotaki-MooreKM – (Kiyotaki version)1. Zero-prob. temporary shock

– Persistent (dynamic loss spiral)– Amplified through collateral

value2. Non- vs. productive

(leveraged) sector3. Dual role of durable asset

1. Production2. Collateral

4. Exogenous contract– One period contract– Debt is limited by collateral value

5. Durable asset doesn’t depreciates (capital, fully)

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BruSan1. Permanent TFP shocks

Margin/haircut spiral (leverage)

Loss spiral

2. Investment through leveraged financial sector

3. Dual role of durable asset1. Production2. Securitization

4. Optimal contract Dynamic contract Debt is limited due

idiosyncratic risk and costly state verification

5. δ-depreciation rate

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Differences to He-KrishnamurthyHe-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 experts5. Procyclical Leverage6. 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 lDt

4. Non-log utility function

BruSan1. 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

Graphs: leverage

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a macroeconomic model with a financial sector markus k. brunnermeier and yuliy sannikov

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