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Notes on Financial Frictions Under Asymmetric Information and Costly State Verification Incorporating Financial Frictions into a Business Cycle Model • General idea: – Standard model assumes borrowers and lenders are the same people..no conflict of interest – Financial friction models suppose borrowers and lenders are different people, with conflicting interests – Financial frictions: features of the relationship between borrowers and lenders adopted to mitigate conflict of interest. Discussion of Financial Frictions • Simple model to illustrate the basic costly state verification (csv) model. – Original analysis of Townsend (1978), BernankeGertler. • Integrating the csv model into a full-blown dsge model. – Follows the lead of Bernanke, Gertler and Gilchrist (1999). – Empirical analysis of Christiano, Motto and Rostagno (2003,2009). Simple Model • There are entrepreneurs with all different levels of wealth, N. – Entrepreneur have different levels of wealth because they experienced different idiosyncratic shocks in the past. • For each value of N, there are many entrepreneurs. • In what follows, we will consider the interaction between entrepreneurs with a specific amount of N with competitive banks. • Later, will consider the whole population of entrepreneurs, with every possible level of N. Simple Model, cont’d • Each entrepreneur has access to a project with rate of return, k 1 R • Here, is a unit mean, idiosyncratic shock experienced by the individual entrepreneur after the project has been started, 0 dF1 • The shock, , is privately observed by the entrepreneur. • F is lognormal cumulative distribution function. Banks, Households, Entrepreneurs ~ F , dF 1 0 entrepreneur entrepreneur entrepreneur Households Bank entrepreneur entrepreneur Standard debt contract • Entrepreneur receives a contract from a bank, which specifies a rate of interest, Z, and a loan amount, B. – If entrepreneur cannot make the interest payments, the bank pays a monitoring cost and takes everything. • Total assets acquired by the entrepreneur: total assets net worth loans A N B • Entrepreneur who experiences sufficiently bad , loses everything. luck, • Cutoff, gross rate of return experience by entrepreneur with ‘luck’, total assets A 1 R k interest and principle owed by the entrepreneur ZB 1 R k A ZB Z 1 Rk B N A N leverage L • Cutoff higher with: – higher leverage, L – higher Z/1 R k Z 1 Rk 1 A N A N Z L1 1 Rk L • Cutoff, gross rate of return experience by entrepreneur with ‘luck’, total assets A 1 R k interest and principle owed by the entrepreneur ZB 1 R k A ZB Z 1 Rk B N A N leverage L • Cutoff higher with: – higher leverage, L – higher Z/1 R k Z 1 Rk 1 A N A N Z L1 1 Rk L • Cutoff, gross rate of return experience by entrepreneur with ‘luck’, total assets A 1 R k interest and principle owed by the entrepreneur ZB 1 R k A ZB Z 1 Rk B N A N leverage L • Cutoff higher with: – higher leverage, L – higher Z/1 R k Z 1 Rk 1 A N A N Z L1 1 Rk L • Cutoff, gross rate of return experience by entrepreneur with ‘luck’, total assets A 1 R k interest and principle owed by the entrepreneur ZB 1 R k A ZB Z 1 Rk B N A N leverage L • Cutoff higher with: – higher leverage, L – higher Z/1 R k Z 1 Rk 1 A N A N Z L1 1 Rk L • Cutoff, gross rate of return experience by entrepreneur with ‘luck’, total assets A 1 R k interest and principle owed by the entrepreneur ZB 1 R k A ZB Z 1 Rk B N A N leverage L • Cutoff higher with: – higher leverage, L – higher Z/1 R k Z 1 Rk 1 A N A N Z L1 1 Rk L • Cutoff, gross rate of return experience by entrepreneur with ‘luck’, total assets A 1 R k interest and principle owed by the entrepreneur ZB 1 R k A ZB Z 1 Rk B N A N leverage L • Cutoff higher with: – higher leverage, L – higher Z/1 R k Z 1 Rk 1 A N A N Z L1 1 Rk L • Cutoff, gross rate of return experience by entrepreneur with ‘luck’, total assets A 1 R k interest and principle owed by the entrepreneur ZB 1 R k A ZB Z 1 Rk B N A N leverage L • Cutoff higher with: – higher leverage, L – higher Z/1 R k Z 1 Rk 1 A N A N Z L1 1 Rk L • Expected return to entrepreneur, over opportunity cost of funds: Expected payoff for entrepreneur 1 Rk AZB dF N 1 R For lower values of , entrepreneur receives nothing ‘limited liability’. opportunity cost of funds • Rewriting entrepreneur’s rate of return: 1 R A ZB dF k N 1 R 1 R k A 1 R k A dF N 1 R k 1 R dF 1 R Z L1 1 L k L R L Z 1 Rk • Entrepreneur’s return unbounded above – Risk neutral entrepreneur would always want to borrow an infinite amount (infinite leverage). • Rewriting entrepreneur’s rate of return: 1 R A ZB dF k N 1 R 1 R k A 1 R k A dF N 1 R k 1 R dF 1 R Z L1 1 L k L R L Z 1 Rk • Entrepreneur’s return unbounded above – Risk neutral entrepreneur would always want to borrow an infinite amount (infinite leverage). • Rewriting entrepreneur’s rate of return: 1 R A ZB dF k N 1 R 1 R k A 1 R k A dF N 1 R k 1 R dF 1 R Z L1 1 L k L R L Z 1 Rk • Entrepreneur’s return unbounded above – Risk neutral entrepreneur would always want to borrow an infinite amount (infinite leverage). • Rewriting entrepreneur’s rate of return: 1 R A ZB dF k N 1 R 1 R k A 1 R k A dF N 1 R k 1 R dF 1 R L Z L1 Z 1 with L LGets smaller Rk L 1 Rk Larger with L • Entrepreneur’s return unbounded above – Risk neutral entrepreneur would always want to borrow an infinite amount (infinite leverage). • Rewriting entrepreneur’s rate of return: 1 R A ZB dF k N 1 R 1 R k A 1 R k A dF N 1 R k 1 R dF 1 R Z L1 1 L k L R L Z 1 Rk • Entrepreneur’s return unbounded above – Risk neutral entrepreneur would always want to borrow an infinite amount (infinite leverage). Expected return for entrepreneur Expected entrepreneurial return, over opportunity cost, N(1+R) 2.2 In our baseline parameterization, risk spread = 1.0063, return is monotonically increasing in leverage 2 1.8 1.6 1.4 1.2 1 1.5 2 2.5 3 3.5 leverage 4 4.5 5 5.5 Expected return for entrepreneur Expected entrepreneurial return, over opportunity cost, N(1+R) 2.2 High leverage always preferred eventually linearly increasing Z/(1+R) = 1.0063 Z/(1+R) = 1.5 2 1.8 1.6 1.4 Baseline parameters 1.2 More leverage locally reduces expected return with high risk spread. 1 1.5 2 2.5 3 3.5 leverage 4 4.5 5 5.5 • If given a fixed interest rate, entrepreneur with risk neutral preferences would borrow an unbounded amount. • In equilibrium, bank can’t lend an infinite amount. • This is why a loan contract must specify both an interest rate, Z, and a loan amount, B. • Need to represent preferences of entrepreneurs over Z and B. – Problem, possibility of local decrease in utility with more leverage makes entrepreneur indifference curves ‘strange’ .. Indifference Curves Over Z and B Problematic Entrepreneurial indifference curves 1.4 1.35 Z/(1+R), risk spread 1.3 1.25 1.2 Utility increasing 1.15 1.1 1.05 1.5 2 2.5 3 3.5 Leverage (i.e., Assets/Net Worth) Downward-sloping indifference curves reflect local fall in net worth with rise in leverage when risk premium is high. Solution to Technical Problem Posed by Result in Previous Slide • Think of the loan contract in terms of the loan amount (or, leverage, (N+B)/N) and the cutoff, 1 Rk AZB dF N 1 R dF 1 Rk 1 R L Indifference curve, (leverage, - bar) space 7 B L NA N N 6 - bar 5 4 3 2 Utility increasing 1 2 3 4 leverage 5 6 7 Solution to Technical Problem Posed by Result in Previous Slide • Think of the loan contract in terms of the loan amount (or, leverage, (N+B)/N) and the cutoff, 1 Rk AZB dF N 1 R dF 1 Rk 1 R L Indifference curve, (leverage, - bar) space 7 B L NA N N 6 - bar 5 4 3 2 Utility increasing 1 2 3 4 leverage 5 6 7 Banks • Source of funds from households, at fixed rate, R • Bank borrows B units of currency, lends proceeds to entrepreneurs. • Provides entrepreneurs with standard debt contract, (Z,B) Banks, cont’d • Monitoring cost for bankrupt entrepreneur Bankruptcy cost parameter with 1 R k A • Bank zero profit condition fraction of entrepreneurs with quantity paid by each entrepreneur with 1 F ZB quantity recovered by bank from each bankrupt entrepreneur 1 dF 1 R k A 0 amount owed to households by bank 1 R B Banks, cont’d • Monitoring cost for bankrupt entrepreneur Bankruptcy cost parameter with 1 R k A • Bank zero profit condition fraction of entrepreneurs with quantity paid by each entrepreneur with 1 F ZB quantity recovered by bank from each bankrupt entrepreneur 1 dF 1 R k A 0 amount owed to households by bank 1 R B Banks, cont’d • Simplifying zero profit condition: 1 F ZB 1 dF 1 R k A 1 RB 0 1 F 1 R k A 1 dF 1 R k A 1 RB 0 B/N 1 R 1 F 1 dF 0 1 R k A/N 1 Rk L 1 L 1 R , L • Expressed naturally in terms of Banks, cont’d • Simplifying zero profit condition: 1 F ZB 1 dF 1 R k A 1 RB 0 1 F 1 R k A 1 dF 1 R k A 1 RB 0 B/N 1 R 1 F 1 dF 0 1 R k A/N 1 Rk L 1 L 1 R , L • Expressed naturally in terms of Bank zero profit condition, in (leverage, - bar) space 14 •Free entry of banks ensures zero profits 12 - bar 10 8 6 • zero profit curve represents a ‘menu’ of contracts, , L , that can be offered in equilibrium. •Only the upward-sloped portion of the curve is relevant, because entrepreneurs would never select a high value of if a lower one was available at the same leverage. 4 2 1.2 1.4 1.6 1.8 2 2.2 leverage 2.4 2.6 2.8 3 Bank zero profit condition, in (leverage, - bar) space 14 •Free entry of banks ensures zero profits 12 - bar 10 8 6 • zero profit curve represents a ‘menu’ of contracts, , L , that can be offered in equilibrium. •Only the upward-sloped portion of the curve is relevant, because entrepreneurs would never select a high value of if a lower one was available at the same leverage. 4 2 1.2 1.4 1.6 1.8 2 2.2 leverage 2.4 2.6 2.8 3 Bank zero profit condition, in (leverage, - bar) space 14 •Free entry of banks ensures zero profits 12 - bar 10 8 6 • zero profit curve represents a ‘menu’ of contracts, , L , that can be offered in equilibrium. •Only the upward-sloped portion of the curve is relevant, because entrepreneurs would never select a high value of if a lower one was available at the same leverage. 4 2 1.2 1.4 1.6 1.8 2 2.2 leverage 2.4 2.6 2.8 3 Bank zero profit condition, in (leverage, - bar) space 14 •Free entry of banks ensures zero profits 12 - bar 10 8 6 • zero profit curve represents a ‘menu’ of contracts, , L , that can be offered in equilibrium. •Only the upward-sloped portion of the curve is relevant, because entrepreneurs would never select a high value of if a lower one was available at the same leverage. 4 2 1.2 1.4 1.6 1.8 2 2.2 leverage 2.4 2.6 2.8 3 Some Notation and Results • Let expected value of , conditional on 0 dF G , 1 F G , • Results: Leibniz’s rule G d dF d 0 F F 1 F 1 F G Moving Towards Equilibrium Contract • Entrepreneurial utility: k 1 R dF L 1 R k 1 R 1 G 1 F L 1 R share of entrepreneur return going to entrepreneur 1 1 R k L 1 R Moving Towards Equilibrium Contract, cn’t • Bank profits: share of entrepreneurial profits (net of monitoring costs) given to bank 1 F 1 dF 0 1 Rk L 1 L 1 R G 1 Rk L 1 L 1 R L k R 1 1 1 R 1 G Equilibrium Contract • Entrepreneur selects the contract is optimal, given the available menu of contracts. • The solution to the entrepreneur problem is that solves: the profits, per unit of leverage, earned by entrepreneur, given log dF 1 R 1 R k higer drives share of profits to entrepreneur down (bad!) log 1 leverage offered by bank, conditional on 1 1 Rk 1 R 1 G higher drives leverage up (good!) k k log 1 R log 1 1 R G 1 R 1 R Equilibrium Contracting Problem Not Globally Concave, But Has Unique Solution Characterized by First Order Condition entrepreneurial utility as a function of - bar only 1.0138 1.0136 utility 1.0134 1.0132 1.013 Close up of the objective, in neighborhood of optimum. Locally concave. 1.0128 0.02 0.04 0.06 0.08 - bar 0.1 0.12 0.14 0.16 1 0.9 0.8 utility 0.7 0.6 0.5 Non-concave part 0.4 0.3 0.2 0.1 1 2 3 4 5 - bar 6 7 8 9 Equilibrium Contracting Problem Not Globally Concave, But Has Unique Solution Characterized by First Order Condition entrepreneurial utility as a function of - bar only 1.0138 1.0136 utility 1.0134 1.0132 1.013 Close up of the objective, in neighborhood of optimum 1.0128 0.02 0.04 0.06 0.08 - bar 0.1 0.12 0.14 0.16 1 0.9 0.8 utility 0.7 0.6 0.5 Non-concave part 0.4 0.3 0.2 0.1 1 2 3 4 5 - bar 6 7 8 9 Computing the Equilibrium Contract • Solve first order optimality condition uniquely for the cutoff, : elasticity of leverage w.r.t. elasticity of entrepreneur’s expected return w.r.t. 1 F 1 1 Rk 1 R 1 1 F F 1 Rk 1R G • Given the cutoff, solve for leverage: L 1 k R 11 G 1 R • Given leverage and cutoff, solve for risk spread: risk spread Z 1 R k R L 1 1 R L1 Result • Leverage, L, and entrepreneurial rate of interest, Z, not a function of net worth, N. • Quantity of loans proportional to net worth: L A N B 1 B N N N B L 1 N • To compute L, Z/(1+R), must make assumptions about F and parameters. 1 R k , , F 1 R The Distribution, F Log normal density function, E = 1, = 0.82155 0.9 0.8 0.7 density 0.6 0.5 0.4 0.3 0.2 0.1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Results for log-normal • Need: G dF , F 0 Can get these from the pdf and the cdf of the standard normal distribution. These are available in most computational software, like MATLAB. Also, they have simple analytic representations. Results for log-normal • Need: G dF , F 0 change of variables, xlog 0 dF E1 requires Ex12 2x prob v 1 e x e x 2 log 1 e x e x 2 combine powers of e and rearrange change of variables, log x1 2 2 x v x x log 1 e x 2 log 1 2 2 x x 1 x 2 log 12 2x 2 x1 2 2 x 22 x x dx dx 2 x1 2 x 2 22 x x xEx 2 22 x exp dx v 2 2 x dv cdf for standard normal Results for log-normal • Need: G dF , F 0 change of variables, xlog 0 dF E1 requires Ex12 2x prob v 1 e x e x 2 log 1 e x e x 2 combine powers of e and rearrange change of variables, log x1 2 2 x v x x log 1 e x 2 log 1 2 2 x x 1 x 2 log 12 2x 2 x1 2 2 x 22 x x dx dx 2 x1 2 x 2 22 x x xEx 2 22 x exp dx v 2 2 x dv cdf for standard normal Results for log-normal • Need: G dF , F 0 change of variables, xlog 0 dF E1 requires Ex12 2x prob v 1 e x e x 2 log 1 e x e x 2 combine powers of e and rearrange change of variables, log x1 2 2 x v x x log 1 e x 2 log 1 2 2 x x 1 x 2 log 12 2x 2 x1 2 2 x 22 x x dx dx 2 x1 2 x 2 22 x x xEx 2 22 x exp dx v 2 2 x dv cdf for standard normal Results for log-normal • Need: G dF , F 0 change of variables, xlog 0 dF E1 requires Ex12 2x prob v 1 e x e x 2 log 1 e x e x 2 combine powers of e and rearrange change of variables, log x1 2 2 x v x x log 1 e x 2 log 1 2 2 x x 1 x 2 log 12 2x 2 x1 2 2 x 22 x x dx dx 2 x1 2 x 2 22 x x xEx 2 22 x exp dx v 2 2 x dv cdf for standard normal Results for log-normal • Need: G dF , F 0 change of variables, xlog 0 dF E1 requires Ex12 2x prob v 1 e x e x 2 log 1 e x e x 2 combine powers of e and rearrange change of variables, log x1 2 2 x v x x log 1 e x 2 log 1 2 2 x x 1 x 2 log 12 2x 2 x1 2 2 x 22 x x dx dx 2 x1 2 x 2 22 x x xEx 2 22 x exp dx v 2 2 x dv cdf for standard normal Results for log-normal, cnt’d • The log-normal cumulative density: log 1 F dF e 0 x 2 2 x1 2 x 2 22 x dx • Differentiating (using Leibniz’s rule): log 1 2 2 1 1 exp F ; 2 2 2 1 Standard Normal pdf log 12 2 ‘Test’ of the Model • Obtain the following for each firm from a micro dataset: probability of default (from rating agency) F firm leverage , L interest rate , Z • Using definition of F, risk spread, first order condition associated with optimal contract and zero profit condition of banks, can compute: ex ante mean return on firm investment project Rk ex ante idiosyncratic uncertainty , cutoff productivity monitoring costs , , • Test the model: do the results look sensible? • Levin, Natalucci, Zakrajsek, ‘The Magnitude and Cyclical Behavior of Financial Market Frictions’, Finance and Economics Discussion Series, Federal Reserve Board, 2004-70. A jump in spreads occurred here, interpreted by the model as a jump (in part) of bankruptcy costs. Changes in idiosyncratic volatility not very important k 1 R 400 1 1 R High values consistent with high bankruptcy costs. k 400 1 R 1 1 R Effect of Increase in Risk, • Keep 0 dF1 • But, double standard deviation of Normal underlying F. Impact on lognormal cdf of doubling standard deviation 0.9 0.8 0.7 Doubled standard deviation density 0.6 0.5 0.4 0.3 Increasing standard deviation raises density in the tails. 0.2 0.1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 Effect of a 5% jump in 9 Risk spread = 400 8 Z 1 , Leverage = (B+N)/N 1 R risk spread (APR) 7 Entrepreneur Indifference curve 6 Risk spread= 2.67 Leverage = 1.12 5 4 3 Risk spread=2.52 Leverage = 1.13 2 Zero profit curve 1 1.08 1.1 1.12 1.14 leverage, qK/N 1.16 1.18 Issues With the Model • Strictly speaking, applies only to ‘mom and pop grocery stores’: entities run by entrepreneurs who are bank dependent for outside finance. – Not clear how to apply this to actual firms with access to equity markets. • Assume no long-run connections with banks. • Entrepreneurial returns independent of scale. • Overly simple representation of entrepreneurial utility function. • Ignores alternative sources of risk spread (risk aversion, liquidity) • Seems not to allow for bankruptcies in banks. Incorporating BGG Financial Friction into Neoclassical Model Model with Financial Frictions Firms L Labor market K C I Capital Producers household Entrepreneurs Model with Financial Frictions Firms Labor market Capital Producers K’ Entrepreneurs Loans household banks Equations of the Model • Aggregate resource constraint: bought by households ct monitoring costs of banks bought by capital producers It t dF 1 R kt k t y t 0 • Households: t uct , ct b t1 1 R t1 b t wt l t t0 • Capital producers: u c,t u c,t1 1 R t , l t 1 k t1 1 k t I t Equations, ctn’d • Entrepreneurs: – First order condition associated with optimization problem. – Zero profit condition of banks. – Law of motion of aggregate net worth. N t1 time t earnings of entrepreneurs net of interest on previous period’s bank loans T t 1 Tt, ~fraction of entrepreneurs that survive, 1 ~fraction of entrepreneurs born T t ~small transfer made to all entrepreneurs Conclusion • We’ve reviewed one interesting model of financial frictions. • Needs a lot more work!
