A Model of Capital and Crises

Transcription

1 A Model of Capital and Crises Zhiguo He Booth School of Business, University of Chicago Arvind Krishnamurthy Northwestern University and NBER AFA, 2011

2 ntroduction ntermediary capital can a ect asset prices. We study the role of distressed intermediary capital in the crises of market with complex asset classes (e.g. MBS).

3 ntroduction ntermediary capital can a ect asset prices. We study the role of distressed intermediary capital in the crises of market with complex asset classes (e.g. MBS). A General Equilibrium (GE) model where intermediaries, rather than households, are marginal. Frictions are endogenously derived based on optimal contracting considerations. Mechanism: ntermediation capital a ects participation/risk-sharing. n normal times households participate through intermediation; When intermediaries su er losses, Distressed intermediary sector averse to hold risky positions, risk premium goes up. Households y to quality, drive down interest rate.

4 Model Structure (1) Unit supply of risky asset with dividend dd t D t = gdt + σdz t, and riskless asset in zero-net supply. Risky asset price Pt and interest rate r t are determined in GE. h R i Households E 0 e ρht ln ct hdt. Limited participation in risky asset market. They invest in intermediaries. Specialists E R 0 e ρt ln c t dt, ρ < ρ h. They run intermediaries. Only intermediaries/specialists can invest in the risky asset. They are marginal investors. Derive ntermediation Constraint from moral hazard primitives.

5 Model Structure (2) The economy. ntermediation: 1) Short-term contracting between agents; 2) Equilibrium in competitive intermediation market; Asset pricing: 3) Optimal consumption/portfolio decisions; 4) GE.

6 The Heart of the Model: (equity) Capital Constraint Say household with wealth Wt h, and specialist with wealth W t. Given specialist s contribution Wt in the intermediary, household contributes Tt h as equity investment (for risk sharing). Capital Constraint: T h t is capped at mw t so risk sharing is capped at 1 : m. ntermediation capacity mw t is increasing in the specialist s contribution W t, as re ection of agency friction.

7 The Heart of the Model: (equity) Capital Constraint Say household with wealth Wt h, and specialist with wealth W t. Given specialist s contribution Wt in the intermediary, household contributes Tt h as equity investment (for risk sharing). Capital Constraint: T h t is capped at mw t so risk sharing is capped at 1 : m. ntermediation capacity mw t is increasing in the specialist s contribution W t, as re ection of agency friction. How to interpret m? 1. ntermediary capital requirement: outside/inside contribution ratio; (Holmstrom-Tirole, QJE) O cers/directors inside holdings in nancial industry around 18%. 2. ncentive contract the performance share of hedge fund managers. Think of 2 and Mutual funds ow-performance sensitivity. Specialist s W t tracks his past gains and losses (Shleifer-Vishny, JF)

8 ntermediation Constraint: An Example (1) Say m = 1, W h t = 80. Comparing W h t to mw t. Unconstrained Region: W t = 100. Then Tt h = Wt h = 80; Zero net debt. Risky asset price Pt = W t + Wt h = 180. Fund s total equity 180. ntermediary holds risky asset without leverage, rst-best risk sharing.

9 ntermediation Constraint: An Example (2) Constrained Region: W t = 50. Then T h t = mw t = 50;

10 ntermediation Constraint: An Example (2) Constrained Region: W t = 50. Then T h t = mw t = 50; ntermediary s total equity is = 100. But P t = 130. n equilibrium, the intermediary borrows 30 from the debt market; t is supplied by households W h t T h t = 30. Specialist and household have equal shares in the intermediary; Specialist s leveraged position in risky asset: α = specialist s portion of asset specialist s equity = 130/2 50 = 130%. Risk premium has to adjust to make this high leverage optimal.

11 Risk Premium and nterest Rate 0.7 Risk Premium 0.05 nterest Rate m=4 m= w c (m=6)=9.07 w c (m=4)= w c (m=6)=9.07 w c (m=4)=13.02 m=4 m= Scaled Specialist's Wealth w Scaled Specialist's Wealth w

12 ntermediation Stage Game Short-term contracts only. At time t, contract from t to t + dt. Household with wealth W h t, and specialist with wealth W t. Household contributes T h t, specialist T t. T t = T h t + T t.

13 ntermediation Stage Game Short-term contracts only. At time t, contract from t to t + dt. Household with wealth W h t, and specialist with wealth W t. Household contributes T h t, specialist T t. T t = T h t + T t. Specialist in charge of intermediary. Moral Hazard: 1. Unobserved due diligence action s t = f0, 1g. Shirking (s t = 1) reduce return by X t but brings private bene t B t. 2. Unobserved portfolio choice E t (dollar exposure to risky asset); Undoing activity. Not crucial. Fund s return E t (dr t r t dt) + T t r tdt s t X t dt, private bene t s t B t dt. Focus on implementing working.

14 ntermediation Contract A ne contracts for sharing returns. βt : specialist s share; ˆK t dt: transfer to specialist. Set K t β t Tt T t r t + ˆK t. Dynamic budget constraint 8 < dw t = r t W t dt c t dt + β t Et (dr t r t dt) + K t dt, : dw h t = r t W h t dt ch t dt + (1 β t )E t (dr t r t dt) K t dt. Reduce contract to (β t, K t ). Sharing rule and fee. Specialist chooses Et = β t Et. Household buys risk exposure Et h = (1 β t )Et from intermediary. n competitive intermediation market, the fee will take some simple linear form.

15 C Constraint and Maximum Household s Exposure C constraint: specialist bears at least a certain fraction of risk. ncentive provision. Skin in the game. No shirking: βt X t B t 0 ) β t B t X t 1+m 1. A lower bound on βt. Specialist always chooses β t Et = Et independent of β t. n the paper we show E t is independent of K.

16 C Constraint and Maximum Household s Exposure C constraint: specialist bears at least a certain fraction of risk. ncentive provision. Skin in the game. No shirking: βt X t B t 0 ) β t B t X t 1+m 1. A lower bound on βt. Specialist always chooses β t E t = E t independent of β t. n the paper we show E t is independent of K. E t fund s total risk exposure. S: E t = β t E t, H: E h t = (1 β t ) E t. Household exposure from the contract, or exposure supply: E h t = (1 β t )E t = 1 β t β t E t. As β t 1 1+m, households maximum exposure E h t me t.

17 Key ntuition and Equity mplementation The households exposure is capped due to agency frictions E h t me t. t caps a risk-sharing rule between households and specialists. ncentive provision implies that specialists have to bear su cient risk. n bad times this friction kicks in. Even if specialists wealth is low, they still have to bear disproportionally large risk.

18 Key ntuition and Equity mplementation The households exposure is capped due to agency frictions E h t me t. t caps a risk-sharing rule between households and specialists. ncentive provision implies that specialists have to bear su cient risk. n bad times this friction kicks in. Even if specialists wealth is low, they still have to bear disproportionally large risk. Equity implementation: Households (outsiders) cannot hold more than 1+m m (equity) shares. Equity capital constraint: Given specialist s equity W t, households can make at most mw t equity contributions. Recall contract (β t, K t ). We have derived equilibrium β t. What determines fee K t? Households pay competitive fees in the intermediation market.

19 Competitive ntermediation Market At time t, specialists make o ers (β t, K t ) to households who can accept/reject o ers. The intermediation market reaches equilibrium if: 1) β t is incentive compatible; 2) no pro table deviation coalitions. Lemma: n equilibrium, households face a per-unit-exposure price of k t 0: to purchase E h t, he has to pay K t = k t E h t. dea: equivalence between core and Walrasian equilibrium. Households and specialists form coalitions to chop o the exposure linearly. Now we start studying agents consumption/portfolio problems.

20 Households Consumption/Portfolio Rules Log investors. Simple consumption rule; myopic mean-variance portfolio choice. Risky asset excess return dr t r t dt = π R,t dt + σ R,t dz t. h Household max E R i fc t,e t g 0 e ρht ln ct hdt subject to dw h t = W h t r t dt c h t dt + E h t (dr t r t dt) k t E h t dt. Standard problem; households achieve exposure Et h by paying per-unit-cost of k t. Optimal consumption ct h = ρ h Wt h, optimal exposure Et h = π R,t k t Wt h. σ 2 R,t Optimal risk exposure is decreasing in exposure price kt.

21 Specialists Consumption/Portfolio Rules The specialist supplies an exposure 1 β t β E t t k t, he gets intermediation fees K t dt = k t 1 The specialist is solving: 1 dw t = E t (dr t r t dt) + max β 1 t 2[,1] 1+m. Given exposure price dt. βt β E t t max E R fc t,e t,β t g 0 e ρt ln c t dt subject to h i βt = 1+m 1 if k t > 0, otherwise βt m, 1 supply schedule. β t βt k t E t dt + W t r t dt if k t = 0. Exposure Et is the exposure expected by households, and must coincide with the specialist s actual optimal choice in REE. c t dt.

22 Specialists Consumption/Portfolio Rules The specialist supplies an exposure 1 β t β E t t k t, he gets intermediation fees K t dt = k t 1 The specialist is solving: 1 dw t = E t (dr t r t dt) + max β 1 t 2[,1] 1+m. Given exposure price dt. βt β E t t max E R fc t,e t,β t g 0 e ρt ln c t dt subject to h i βt = 1+m 1 if k t > 0, otherwise βt m, 1 supply schedule. β t βt k t E t dt + W t r t dt if k t = 0. Exposure Et is the exposure expected by households, and must coincide with the specialist s actual optimal choice in REE. Solution: ct = ρw t and Et = π R,t W σ 2 t, and specialists receive fee R,t 1 β of K t = q t W t where q t = t π βt k R,t t. σ 2 R,t c t dt.

23 Unconstrained vs. Constrained Regions (1)

24 Unconstrained vs. Constrained Regions (1)

25 Equilibrium Asset Prices: Solution We derive everything in closed form. State variables (D t, W t ). Scales with D t. Uni-dimensional state variable w t W t /D t captures wealth distribution.

26 Equilibrium Asset Prices: Solution We derive everything in closed form. State variables (D t, W t ). Scales with D t. Uni-dimensional state variable w t W t /D t captures wealth distribution. Consumption rules ct = ρwt h, ch t = ρ h Wt h. Zero net debt W t + W h t = P t, goods clearing c t + c h t P t = 1 D t ρ h + 1 ρ ρ h w t. Specialist s risky (percentage) position α t = constrained region. P t (1+m)W t = D t. So > 1 in

27 Asset Pricing (1) The economy is in constrained region whenever w t = W t /D t < w c 1 mρ h + ρ. n unconstrained region, w t increases deterministically toward w c. Perfect risk sharing rule. Relative patience level ρ < ρ h matters. n constrained region, specialists take a higher leverage than households. Therefore w t becomes stochastic and drops when fundamental falls. When (scaled) intermediary capital w t falls in constrained region, Risk premium rises; nterest rate falls; Volatility rises; Correlation endogenously rises.

28 Asset Pricing (2) Uncon. Region Con. Region E t W t 1 1+m P t ρ)w α t 1 t > 1 (1+m)ρ h w t σ (1+m)ρ h σ R,t σ > σ 1+(ρ h ρ)w t mρ h +ρ π R,t σ 2 σ2 (1+m)ρ h 1 w t (mρ h +ρ) mρ h +ρ 1+(ρ h ρ)w t k t 0 ρ h + g σ 2 r t +ρ ρ ρ h w t 1+(ρ h 1 (ρ+mρ h )w t (1 ρw t )(1+(ρ h ρ)w t) σ 2 ρ h + g + ρ ρ h ρ > σ 2 (1+m)ρ h σ 2 w t (mρ h +ρ) 2 > 0 ρ h w t (1+m) 1 w ρ m 2 ρ h +(mρ h ) 2i t (1 ρw t )(ρ+mρ h ) 2

29 Risk Premium and nterest Rate 0.7 Risk Premium 0.05 nterest Rate m=4 m= w c (m=6)=9.07 w c (m=4)= w c (m=6)=9.07 w c (m=4)=13.02 m=4 m= Scaled Specialist's Wealth w Scaled Specialist's Wealth w Asymmetry. Crisis like. When constraint binds w t < w c, specialist bears disproportionally large risk, causing more volatile pricing kernel. Flight to quality. 1) Specialists precautionary savings. 2) Household y to debt market.

30 Comovement Consider an in nitesimal asset with d ˆD t ˆD t = dd t D t + ˆσdẐ t. The correlation between dr t and corr(dr t, ˆ dr t ) = ˆ dr t is: 1 q1 + ( ˆσ/σ R,t ) 2. Unconstrained region, since σ R is constant, the correlation is constant. Constrained region, rising correlation. Market return volatility σr,t rises, magnifying the common component of returns.

31 Concluding Remarks (1) Canonical intermediation friction meets canonical GE asset pricing models. Calibratable, easy to quantify e ects. We have another paper where specialists have general CRRA power utility, with capital constraint as given. Calibrate the model to the MBS market; Add in labor income, debt households (create leverage in unconstrained region), and other necessary twists... Study the crisis dynamics (especially recovery), government liquidity injection policies, etc.

32 Concluding Remarks (2): Calibration result % % % % % w c /P=0.91 w / P Crisis Recovery Transit from 12% W/o Capital nfusion W. Capital nfusion (48bn) 10% % % %