Macro, Money and Finance: A Continuous Time Approach

Macro, Money and Finance: A Continuous Time Approach Markus K. Brunnermeier & Yuliy Sannikov Princeton University International Credit Flows, Trinity of Stability Conference Princeton, Nov. 6 th, 2015

Brunnermeier, Eisenbach & Sannikov & Sannikov Price stability Monetary policy Financial stability Macroprudential policy Fiscal debt sustainability Fiscal policy Short-term interest Policy rule (terms structure) Reserve requirements Capital/liquidity requirements Collateral policy Margins/haircuts Capital controls interaction interaction 2

Macro, Money and Finance Endogenous level Persistence & amplification Net worth trap Endogenous risk dynamics Tail risk Crisis probability Volatility Paradox Illiquidity and liquidity mismatch Undercapitalized sectors Time varying risk premia (dynamics) External funding premium Value of money Welfare Interaction: regulatory, monetary and other policies 3

timeline History: Macro & Finance Verbal Reasoning (qualitative) Fisher, Keynes, Macro Finance Growth theory Dynamic (cts. time) Determinisitc Portfolio theory Static Stochastic Introduce stochastic Discrete time Brock-Mirman Stokey-Lucas Kydland-Prescott DSGE models Introduce dynamics Continuous time Options Black Scholes Term structure CIR Agency theory Sannikov Cts. time macro with financial frictions 4

Amplification & Persistence Bernanke & Gertler (1989), Carlstrom & Fuerst (1997) Perfect (technological) liquidity, but persistence Bad shocks erode net worth, cut back on investments, leading to low productivity & low net worth of in the next period Kiyotaki & Moore (1997), BGG (1999) Technological/market illiquidity KM: Leverage bounded by margins; BGG: Verification cost (CSV) Stronger amplification effects through prices (low net worth reduces leveraged institutions demand for assets, lowering prices and further depressing net worth) 6

Impulse response vs. Volatility dynamics once and for all shock = no uncertainty about length of slump Sequence of adverse shock 8

Why continuous time modeling? Characterization for volatility and amplification Discrete: only impulse response functions Only for shocks starting at the steady state Only expected path fan charts help somewhat More analytical steps Return equations Next instant returns are essentially log normal (easy to take expectations) Explicit net worth and state variable dynamics Continuous: Discrete: only slope of price function determines amplification need whole price function (as jump size can vary) Numerically simple solve differential equations Discrete: IES/RA within period =, across periods 1/γ 9

Cts. time: special features of diffusions Continuous path fast enough deleveraging Never jumps over a specific point, e.g. insolvency point Implicit assumption: can react to small price changes Can continuously delever as wealth goes down Makes them more bold ex-ante 10

Recent macro literature (in cts time) Core BrunSan (2014), Basak & Cuoco (1998) He & Krishnamurthy (2012,13), DiTella (2013), Isohätälä et al. (2014) Intermediation/shadow banking Phelan (2014), Adrian & Boyarchenko (2012,13), Huang (2014), Moreira & Savov (2014), Klimenko & Rochet (2015) Quantification He & Krishnamurthy (2014), Mittnik & Semmler (2013) International BruSan (2015), Maggiori (2013) Monetary The I Theory of Money (2012), Drechsler et al. (2014). 11

Financial frictions Costly state verification (BGG) Leverage constraints Exogenous limit (Bewley/Ayagari) Collateral constraints Next period s price (KM) state 2 Rb t q t+1 k t Next periods volatility Current price Incomplete markets Endogenous leverage (VaR) state 1 Debt limit can depend on prices/volatility 12

Roadmap Why continuous time? Literature Simple model With undesirable features Add portfolio choice with general utility function Full model With all desired features Add equity issuance 13

A simple model Basak & Cuoco (1998) Experts Output: y t = ak t Households No output: a = 0 Consumption rate: c t Investment rate: ι t dk t = Φ ι k t δ dt+σdz t t Consumption rate: c t E 0 [ 0 e ρt log c t dt] E 0 [ 0 e ρt log c t dt] Can only issue risk-free debt 14

Equilibrium An equilibrium consists of functions that for each history of macro shocks {Z s, s 0, t } specify q t the price of capital k t, k t = 0 capital holdings c t 0, c t = 0 consumption of representative expert and households ι t,ι t = 0 rate of internal investment, per unit of capital r the risk-free rate such that intermediaries and households maximize their utility, taking prices q t as given and markets for capital and consumption goods clear 15

Equilibrium Equilibrium is a map Histories of shocks {Z s, s t} prices, allocations q t, ψ t, ι t, ι, c t, c t wealth distribution η t = N t q t K t 0,1 experts wealth share Experts, HH solve optimal investment, portfolio, consumption Markets clear 16

Solution steps 1. Postulate endogenous processes dq t /q t = μ t q dt + σ t q dz t Returns from holding capital 2. Equilibrium conditions Agents optimization Internal investment (new capital formation) Optimal portfolio choice Optimal consumption Market clearing conditions 3. Law of motion of state variable wealth (share) distribution η t 4. Express in ODEs of state variable 17

1. Postulate endogenous process Postulate Recall dq t /q t = μ t q dt + σ t q dz t dk t /k t = Φ ι t δ + σdz t Return on capital dr t k = a ι t q t dividend yield dt + d(k t q t ) k t q t capital gains d(k tq t ) k t q t = Φ ι t δ + μ t q + σσ t q dt + σ + σ t q dz t by Ito s product rule In this simple model it will turn out that q is constant, i.e. μ t q = σ t q = 0. 18

2. Equilibrium optimality conditions a. Investment rate (capital formation) Static problem max Φ ι t ι t /q t ι FOC: Φ ι t = 1 (marginal Tobin s q) q t a ι t δ Φ ι t δ b. Consumption choice c t = ρn t due to log utility c. Portfolio choice Volatility of wealth = Sharpe ratio of risky investment 19

2. Equilibrium market clearing conditions Goods market price of capital q t = q = ρq t K t = a ι t q t Special case: Φ ι = log(κι+1) κ Risk free rate dr t k = a ι t q t ρ,dividend yield Sharpe ratio: Volatility of net worth: + Φ ι t δ dt+σdz t capital gains Sharpe ratio = volatility of N t K t ι = q 1 κ q = a+1/κ r+1/κ ρ+φ ι δ r t σ q t K t N t σ= σ η t r t = ρ + Φ ι δ σ2 η t 20

3. Law of motion of η t dn t N t =r t dt+ σ η t risk d(q tk t ) q t K t =.. σ dt+ σ dz η t η t ρdt t consumption Sharpe Use Ito ratio rule for η t = N t (q t K t ) dη t = 1 η t 2 σ η 2 dt+ 1 η t σdz t t η t 2 21

Observations dη t = 1 η t 2 η t η 2 t σ 2 dt+ 1 η t σdz t Wealth share η moves with macro shock dz t In the long run experts save their way out, η 1 Sharpe ratio ρ+φ ι δ r t σ Increases as η goes down, (to as η 0) Achieved through a lower risk free rate q is constant No endogenous risk No amplification No volatility effects 22

Generalizing preference: portfolio choice 1. Also postulate process for marginal utility dθ t /θ t = μ θ t dt + σ θ t dz t SDF: e ρs θ t+s /θ t 2. Portfolio choice: Optimality condition For asset A with payoff process dr A t = μ A t dt + σ A t dz t Intuition: 0 = μ t θ ρ + μ t A + σ t A σ t θ i. Discrete time analog: Take log of 1 = E t SDF t,t+s R t,t+s A ii. Consider wealth n t invested in A, so that dn t /n t = dr t n t+s e ρs θ t+s is a martingale θ t 23

Desired model properties Normal regime: stable around steady state Experts are adequately capitalized Experts can absorb macro shock Net worth trap Endogenous risk Fat tails Assets are more correlated SDF vs. cash-flow news Volatility paradox Financial innovation look at stationary distribution less stable economy 27

Full model Experts Output: y t = ak t Consumption rate: c t Investment rate: ι t dk t = Φ ι k t δ dt+σdz t t a a δ δ Households Output: y t = ak t Consumption rate: c t Investment rate: ι t dk t = Φ ι k t δ dt+σdz t t E 0 [ 0 e ρt c 1 γ t dt] 1 γ ρ ρ E 0 [ 0 e ρtc t 1 γ 1 γ dt] Can issue Risk-free debt Equity, but most hold χ t χ 28

Experts Households A L A L Debt Loans Capital ψ t q t K t N t Outside equity Equity Capital 1 ψ t q t K t Net worth q t K t N t χ Experts must hold fraction χ t χ 29

Solution steps 1. Postulate endogenous processes dq t /q t =, dθ t /θ t =.., dθ t /θ t = μ t θ dt + σt θ dzt 2. Equilibrium conditions Agents optimization Internal investment (new capital formation) Optimal portfolio choice with equity issuance Optimal consumption Market clearing conditions 3. Law of motion of state variable wealth (share) distribution η t 4. Express in ODEs of state variable 30

2. Optimal portfolio condition Without equity issuance a ι t q t +Φ ι t δ+μ t q +σσt q rt σ+σ q = σ t θ χ t σ t θ + (1 χ t )( σ t θ ) a ι t q t +Φ ι t δ+μ t q +σσt q rt σ+σ q σ t θ with equality if ψ t < 1 31

2. Optimal portfolio condition Without equity issuance a ι t q t +Φ ι t δ+μ t q +σσt q rt σ+σ q = σ t θ χ t σ t θ + (1 χ t )( σ t θ ) a ι t q t +Φ ι t δ+μ t q +σσt q rt σ+σ q σ t θ with equality if ψ t < 1 If experts require higher returns than HH if σ θ > σ t θ χt = χ Otherwise σ θ = σ t θ (a a)/q t σ+σ t q χ with equality if ψ t < 1 32

3. Law of motion of η t dn t N t = r t dt + χ tψ t η t (σ+σ t q ) + χ t ψ t η t risk ( σ t θ ) risk premium dt σ + σ t q dz t C t N t dt Use Ito ratio rule for η t = N t (q t K t ) dη t η t =.. 33

4. Express in functions q η, θ η, ψ η, χ η Convert equilibrium conditions and law of motion Replace terms μ t q, μ t θ, σ t q, σ t θ, with expressions containing derivatives of q and θ using Ito s lemma A simple example: Leland (1994) dv t = rv t dt + σv t dz t (under Q) default at V t = V B to αv B 1. Postulate de t = μ E t E t dt + σ E t E t dz t 2. Equilibrium condition: r = μ E t E t Ito lemma on E(V): μ E t E t = E V t rv t + 1 2 σ2 V 2 t E (V t ) 37

Amplification closed form σ t η = 1 [ χ tψ t η t χ t ψ t 1 η t leverage 1] q (η t) q η t /η t σ Market illiquidity (price impact elasticity) Leverage effect Loss spiral χ t ψ t 1 η t 1/{1 [ χ t ψ t 1] q (η t ) } η t q η t /η t (infinite sum) Technological illiquidity (κ, δ) market illiquidity q η (dis)investment adjustment cost 39

5. Solving system of ODE numerically Matlab ODE solver, ode45 Boundary conditions θ 0 = M for large constant M θ η q 0 = (closed form for log utility and log Φ) 40

Monetary Models Money models without intermediaries Store of value: Money pays no dividend and is a bubble \Friction OLG Incomplete Markets + idiosyncratic risk Risk deterministic endowment risk borrowing constraint investment risk Only money Samuelson Bewley With capital Diamond Aiyagari, Krusell-Smith Basic I Theory

Inside equity Risky Claim Risky Claim Risky Claim Risky Claim Risky Claim Risky Claim HH Net worth Monetary Models The I Theory of Money Step 1: Postulate process for value of money p t K t vectors dp t = μ p p p t dt + σ t t dzt (money + bond) db t /B t = μ B t dt + σ B t dz t (part due to consul bond) Outside Money A L Outside Money Pass through A A A Money L L L Gov. bond Inside Money (deposits) A A A A L L L L Money A 1 Net worth B 1 45

Conclusion Manual for continuous time macro-finance models 4 step approach More tractable: explicit amplification terms Volatility dynamics characterization Precautionary motive Endogenous fat tails, crisis probability Undercapitalized sectors, liquidity mismatch, fire-sales, equity issuance cycles, fat tails, Revival of Money and Banking The I Theory of Money with short-term money and long-term bond 46