Balance Sheet Capacity and Endogenous Risk
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Balance Sheet Capacity and Endogenous Risk Jon Danielsson London School of Economics Hyun Song Shin Princeton University Jean—Pierre Zigrand London School of Economics December 2010 Balance Sheet Capacity • Balance sheet capacity is capacity of banking (intermediary) sector to channel credit • Balance sheet capacity is impaired during crises — Erosion of bank capital due to credit losses — Deleveraging • One rationale for central bank purchase of risky securities is to restore lost balance sheet capacity 1 Total Asset Growth 0 10 20 Procyclical Leverage 2008q1 2007q3 -10 2008q2 2007q4 2008q3 1987q4 1998q4 -20 2008q4 -40 -20 0 20 Leverage Growth Adrian and Shin (2010) 2 Balance Sheet Capacity Decomposed Asset growth = log ( + 1) − log () Leverage Growth = log ( + 1) − log () − (log ( + 1) − log ()) 45 degree line represents constant equity line: log ( + 1) − log () = 0 Above 45 degree line, equity increasing. Below 45 degree line, equity decreasing. Straight line with slope 1 is constant equity growth line, with intercept being growth rate 3 Balance Sheet Capacity Decomposed • Equity seems to play role of exogenous variable — Contrast to corporate finance textbooks extending MM • Balance sheet capacity is Equity × Permitted Leverage • Our approach: intermediaries are subject to Value-at-Risk constraint Equity = Total Assets × VaR per dollar of assets Leverage = Assets 1 = Equity Unit VaR 4 Endogeneity of Risk • True risks impacting financial markets are attributable (at least in part) to actions of economic agents. • Endogenous risk is fixed point of mapping: perceived risk ⇒ actual risk • Solve for volatility and risk premium − in closed form, where — Banking sector capital (equity) is state variable — Fundamental uncertainty is constant 5 Some Themes • Volatility depends on balance sheet capacity, even though fundamental uncertainty is constant (“Endogenous Risk” Danielsson and Shin (2003)) • Risk premium depends on balance sheet capacity — Lagrange multiplier of VaR constraint enters like a risk aversion parameter — “As if” preferences + endogenous risk generate risk premiums that depend on balance sheet capacity • Implication: intermediary balance sheet size forecasts asset returns — Adrian, Moench and Shin (2009) for asset returns in US — Etula (2009) for commodities — Adrian, Etula and Shin (2009) for FX returns for USD cross rates 6 Simplified Financial System Intermediated Credit end-user borrowers Banks (Active Investors) Directly granted credit Debt Claims Households (Passive Investors) Intermediated and Directly Granted Credit 7 Model • Time indexed by ∈ [0 ∞). • 0 non-dividend paying risky securities (date price of th risky security is ) • One risk-free bond (0 = 1 = , with constant) • Two types of traders — Active (risk-neutral, with VaR constraints) - banks — Passive (residual demand/supply curves) - households, value investors 8 Posit equilibrium of form: ⎡ ⎣ 11 ⎤ ⎡ 1 ⎤ ⎡ − 1 .. ⎦ = ⎣ .. ⎦ + ⎣ − − − ⎤⎡ ⎦⎣ 1 ⎤ .. ⎦ © ª independent Brownian motions (fundamental shocks enter via passive traders’ demands) © ª © ª Scalars and 1 × vectors are as yet undetermined coefficients to be solved in equilibrium Solve for rational expectations equilibrium (REE) with respect to active traders’ beliefs 9 Portfolio Choice of Active Traders • Risk-neutral traders • (Ultra) short horizons • Value-at-Risk (VaR) constraint: enough to meet VaR capital (i.e. equity) should be large is dollar holding of th security at is trader’s capital (notice no superscript for trader, due to aggregation result, to follow) Balance sheet identity = − P 10 Evolution of capital £ ¤ > = + ( − ) + > > is transpose of , is the × diffusion matrix, = ( )>. Expected capital gain: [] = [ + >( − )] (1) Var() = > > (2) Variance of capital: Trader maximizes (1) subject to VaR constraint, where VaR is times forward-looking standard deviation of return on equity. 11 Assuming trader is solvent ( 0) maximization problem is max + >( − ) subject to q > > ≤ First-order condition − = (>Σ)−12 Σ where is Lagrange multiplier for VaR constraint, and Σ := > . 1 −1 Σ ( − ) = (>Σ)−12 12 Constraint binds due to risk-neutrality q = >Σ Therefore (3) −1 = 2 Σ ( − ) “As if” preferences. Optimal portfolio is similar to mean-variance optimal portfolio where the Lagrange multiplier appears like a risk-aversion coefficient. Substitute into (3) to solve for Lagrange multiplier p = 13 where := ( − )>Σ−1 ( − ) Lagrange multiplier is √ • proportional to generalized Sharpe ratio • does not depend directly on equity Interpretation. Additional unit of capital relaxes VaR constraint by multiple of standard deviation, raising expected return by risk-premium on the portfolio per unit of standard deviation 14 Finally, solve for optimal portfolio: = p Σ−1 ( − ) Optimal portfolio is homogeneous of degree one in equity Aggregation result. Portfolio depends on , total capital of active trading (banking?) sector, not on profile of individual equity capital. ⇒ Take aggregate capital, , as state variable 15 Closing the Model Passive traders in aggregate have vector-valued exogenous demand schedule for the risky assets, = (1 ) where ¡ 1 ¢ ⎤ 1 − ln .. ⎦ ⎣ = Σ−1 ¡ ¢ − ln ⎡ 1 is scaling parameter for slope of the demand curve is positive demand shock for asset = ∗ + 16 The market-clearing condition + = 0 gives ¡ 1 ¢ ⎤ 1 − ln .. ⎦=0 p ( − ) + ⎣ ¡ ¢ − ln ⎡ 1 Equilibrium prices are = exp à ! p ( − ) + ; = 1 17 Single Risky Asset Look for equilibrium of form: = + (4) and are undetermined coefficients to be solved in equilibrium, is standard (scalar) Brownian motion. Equivalently, ¶ µ 1 2 (5) ln = − + 2 “Seeds” of fundamental shocks given by exogenous shocks to passive traders’ demands: = ∗ + , for known constants 0 0 18 Start with market-clearing price with VaR-constrained traders ¶ µ = exp + Take logs and apply Ito’s Lemma ¶ µ ln = + = ∗ + + 1 ( + + ) (6) • Unpack and , and substitute back into (6) • Compare coefficients with (5) 19 Step 1. Unpack as an Ito process = [ + ( − )] + ¸ ∙ ( − ) + = + Key step is the simplification allowed by binding VaR constraint: = 20 Step 2. Unpack as an Ito process From Itô’s Lemma on (), 1 2 2 + ( ) = 2 ()2 ) ( ∙ ¸ µ ¶ 2 1 2 ( − ) + + (7) + = 2 2 () 2 where we substutute in and where () = ¡ ¢2 21 Substitute everything back into (6) and re-arrange: µ ¶¸ ∙ 1 ln = (drift term) + + + By hypothesis, ¶ µ 1 ln = − 2 + 2 (*) (**) Comparing coefficients between (*) and (**), µ ¶ 1 + () = + 22 giving ordinary differential equation: 2 = 2( − ) − Generic solution: " 1 − 2 () = − 2 Z ∞ 2 − − # where is a constant of integration, and Ei () function (“exponential R ∞ − integral” function) − − is defined provided 6= 0. 23 Constant of integration can be tied down as follows. Consider limit case → 0 where “fundamental shocks” are shut down As → 0 () → Restriction. In the absence of fundamental shocks, volatility is zero This implies = 0 so that ³ 2 ´ n 2 o () = exp − × Ei 2 24 Equilibrium drift = + 2 ½ ∙ ¸¾ 2 2 (∗ − ) + 2 − + ( − ) 22 + −2 Sharpe ratio: ½ ∙ ¸¾ 2 1 − ∗ 2 2 2 ( − ) + − + ( − ) 2 + = −2 2 25 Example = 001 ∗ = 0047 = 6 = 27 = 03 = 15 26 Risk and Return σ µ 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 20 40 Equity 60 80 Risk and Return σ µ 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 20 40 Equity 60 80 Risk and Return σ µ 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 20 40 Equity 60 80 Leverage under P Asset Growth 0.4 0.2 0.0 −0.2 −0.4 −0.6 −0.6 −0.4 −0.2 0.0 Leverage growth 0.2 0.4 Risk and Return σ µ γ 0.6 0.5 0.20 0.15 0.4 0.10 0.3 0.2 0.05 0.1 0.0 0 20 40 Equity 60 80 Many Risky Assets Special case of risky securities case can be solved using ODE solution from the single risky asset case. Assumption (Symmetry) 1. Diffusion matrix for is ̃ where ̃ 0 is a scalar and is the × identity matrix. 2. = for all . Let be coefficient giving effect of change in demand shock of th security on price of th security. 27 From assumption of symmetry, we only need to solve for one diffusion 11 variable, = , since for 6= the cross effects are tied down by 12 11 = = − ̃ . Define ≡ () the solution to the ODE for single risky asset with replaced by . Proposition 1. Assume (S). The following is a REE. −1 The REE diffusion coefficients are = + ̃ , and for 6= , 1 2 2 = − ̃ . Also, Σ = Var (return on security ) = ̃ + ¡ ¢ 1 2 2 2 2 , and for = 6 , Σ − ̃ = Cov(return on security ¡ ¢ return on security ) = 1 22 − 2̃ 2 and Corr(return on security return on security ) = Risky holdings are = 1 2 ̃ 2 2 − −1 2 2 2 + ̃ . . 32 28 The risk-reward relationship is given by ( µ ¶2 − 1 −1 √ ̃ = + − ̃ + 2 ̃ ∙ ¸¾ 2 √ ³ ´ − ̃ 22 + −2 (8) The intuition and form of the drift term is very similar to the = 1 case and reduces to it if is set equal to 1. 29 Related Literature • Two strands coming together — Competitive equilibrium models of crashes: Leland (1990), Genakoplos (1997) and Geanakoplos and Zame (2003) — Corporate finance elements: Shleifer and Vishny’s (1997), Holmström and Tirole (2001), He and Krishnamurthy (2007)... • Portfolio constraints: Basak and Croitoru (2000), Chabakauri (2008), Aiyagari and Gertler (1999), Gromb and Vayanos (2002), Brunnermeier and Pedersen (2009), Rytchkov (2008), Pavlova et al (2008) • Wealth effects: Kyle and Xiong (2001). Xiong (2001) solves for fixed point numerically. 30 • Lagrange multipliers associated with VaR constraints: Danielsson, Shin and Zigrand (2004), Brunnermeier and Pedersen (2009), Danielsson and Zigrand (2008), Oehmke (2008). • Asset pricing taking account of balance sheet constraints: Adrian, Etula and Shin (2009) [for exchange rates], Etula (2009) [commodities], Adrian, Moench and Shin (2009) 31 Avenues for Further Research • Having as sole state variable is limiting — Single state variable cannot take account of time dependence — “Long period of tranquility builds up vulnerabilites...” • Backward-looking learning (e.g. RiskMetrics) — In practice, banks use historical data to forecast volatility — ... also encouraged by regulators (Basel II) — This is one way to tie down beliefs, and overcome multiplicity of equilibrium — Encompasses history-dependence 32
