Leverage Restrictions in a Business Cycle Model. Lawrence J. Christiano Daisuke Ikeda

Leverage Restrictions in a Business Cycle Model Lawrence J. Christiano Daisuke Ikeda

Background Increasing interest in the following sorts of questions: What restrictions should be placed on bank leverage? How should those restrictions be varied over the business cycle?

What We Do Modify a standard medium-sized DSGE model to include a banking sector. Assets Loans and other securities Liabilities Deposits Banker net worth Job of bankers is to identify and finance good investment projects. doing this requires exerting costly effort. Agency problem between bank and its creditors: banker effort is not observable. Consequence: leverage restrictions on banks generate a very substantial welfare gain in steady state. Desirable to encourage low leverage in good times, so that banks in better position to absorb bad shocks to net worth.

Outline Model first, without leverage restriction observable effort benchmark unobservable case then, with leverage restriction Steady state properties of leverage restrictions Dynamics

Standard Model L Firms Labor market household

Standard Model L Firms K Labor market Market for Physical Capital household

Standard Model L Firms K Labor market C I Market for Physical Capital household

Standard Model with Banking L Firms K K, ~F, t Labor market Entrepreneurs household

Standard Model with Banking L Firms Labor market C I Capital Producers 1 K Entrepreneurs household Entrepreneur pays everything to the bank and has nothing.

Standard Model with Banking Firms Labor market Capital Producers Entrepreneurs household banks Mutual funds

Entrepreneurs After goods production in period t : Purchase raw capital from capital producers, for price P k,t. entrepreneurs have no resources of their own and must obtain financing from banks. Entrepreneurs convert raw capital into effective capital. Some are good at it and some are bad. In period t + 1 : entrepreneurs rent capital to goods-producers in competitive markets, at rental rate, r t+1. after production, sell undepreciated capital back to capital producers at price, P k,t+1. entrepreneurs pay all earnings to bank at end of t + 1, keeping nothing. no agency problems between entrepreneurs and banks.

Earnings of Entrepreneurs there are good entrepreneurs and bad entrepreneurs. bad: 1 unit, raw capital e b t units, effective capital good: 1 unit, raw capital e g t > e b t units, effective capital return to capital enjoyed by entrepreneurs: R g t+1 = eg t R k t+1, Rb t+1 = eb t R k t+1 R k t+1 rk t+1 P t+1 + (1 δ) P k,t+1 P k t

Bankers each has net worth, N t. a banker can only invest in one entrepreneur (asset side of banker balance sheet is risky). by exerting effort, e t, a banker finds a good entrepreneur with probability p : p (e t ) = ā + be t in t, bankers seek to optimize: [ ] E t λ t+1 {p (e t ) R g t+1 (N t + d t ) R g d,t+1 d t [ ] + (1 p (e t )) R b t+1 (N t + d t ) R b d,t+1 d t } 1 2 e2 t Bankers have a cash constraint: R b t+1 (N t + d t ) R b d,t+1 d t

Bankers and their Creditors Bankers and Mutual Funds interact in competitive markets for loan contracts: ( ) d t, e t, R g d,t+1, Rb d,t+1 Free entry and competition among mutual funds implies: p (e t ) R g d,t+1 + (1 p (e t)) R b d,t+1 = R t Two scenarios: banker effort, e t, is observed by mutual fund banker effort, e t, is unobserved.

Observed Effort Benchmark Set ( of contracts available ) to bankers is the d t, e t, R g d,t+1, Rb d,t+1 s that satisfy MF zero profits: p (e t ) R g d,t+1 + (1 p (e t)) R b d,t+1 = R t, cash constraint: R b t+1 (N t + d t ) R b d,t+1 d t Each banker chooses the most preferred contract from the menu. Key feature of observed effort equilibrium: ( ) e t = E t λ t+1 p (e t ) R g t+1 Rb t+1 (N t + d t )

Unobserved Effort In this case, banker always sets e t to its privately optimal level, whatever e t is specified in the loan contract: ( ) incentive: e t = E t λ t+1 p (e t ) [ R g t+1 Rb t+1 (N t + d t ) ( ) R g d,t+1 Rb d,t+1 d t ]. Set ( of contracts available ) to bankers is the d t, e t, R g d,t+1, Rb d,t+1 s that satisfy incentive in addition to: MF zero profits: p (e t ) R g d,t+1 + (1 p (e t)) R b d,t+1 = R t, cash constraint: R b t+1 (N t + d t ) R b d,t+1 d t One factor that can make e t ineffi ciently low: R g d,t+1 > Rb d,t+1.

Law of Motion of Net Worth Bankers live in a large representative household, with workers (as in Gertler-Karadi, Gertler-Kiyotaki). Bankers pool their net worth at the end of each period (we avoid worrying about banker heterogeneity) Law of motion of banker net worth profits when bank assets good {[ }} ]{ N t+1 = γ t+1 {p (e t ) R g t+1 (N t + d t ) R g d,t+1 d t profits when bank assets are bad [{}} ]{ + (1 p (e t )) R b t+1 (N t + d t ) R b d,t+1 d t } + lump sum transfer, households to their bankers {}}{ T t+1

Model Assumption that Banks Don t Systematically Rely on Equity Issues to Finance Assets Evidence from two sources provide support for this assumption as a description of the data. Adrian and Shin s examination of the assets and liabilities of two large French financial firms. US flow of funds data on assets and liabilities of financial corporations. Adrian and Shin, Procyclical Leverage and Value-at-Risk Changes in financial firm equity not systematically related to their assets. Changes in financial firm debt moves one-for-one with changes in assets.

Material taken from the work of Adrian Shin. Displays a scatter plot change in equity and debt on the horizontal axis against change in assets on the horizontal axis. Note that the slope of changes in debt against changes in assets is essentially unity, while the slope of changes in equity against changes in assets has a slope of zero. The results are consistent with the notion that this financial company headquartered in Paris finances changes in assets with changes in debt and not changes in equity. 5 BNP Paribas: annual change in assets, equity and debt (1999-21) 4 y = 1.51x - 6.2 R 2 =.9987 Change in equity and debt (billion euros) 3 2 1-1 Debt Change Equity Change -2-2 -1 1 2 3 4 5 Asset change (billion euros) Figure 3. BNP Paribas: annual change in assets, equity and debt (1999-21) (Source: Bankscope)

Discussion of Acharya and Seru 7 Societe Generale: annual changes in assets, equity and debt (1999-21) 3 Annual change in equity and debt (billion euros) 25 2 15 1 5-5 -1 y =.996x - 3.15 R 2 =.9985 Debt change Equity change -15-2 -1 1 2 3 Annual asset change (billion euros) Figure 4. Société Générale: annual change in assets, equity and debt (1999-21) (Source: Bankscope)

The model assumes that when bankers want funds, issuing equity is not an option. 8 Borrowing by Private Depository Institutions (Table F.19, Flow of Funds) billions of dollars 7 6 5 4 3 2 1 open market paper, bonds, other loans, deposits 26 27 28 29 21 211 12 11 1 This shows how major debt instruments were used at 9 private 8 depository institutions in the wake of the crisis. billions of dollars 7 Equity as a source of funds, Private Depository Institutions (F.19, F of F) 26 27 28 29 21 211

The model assumes that when bankers want funds, issuing equity is not an option. 8 Borrowing by Private Depository Institutions (Table F.19, Flow of Funds) billions of dollars 7 6 5 4 3 2 1 open market paper, bonds, other loans, deposits 26 27 28 29 21 211 12 Equity as a source of funds, Private Depository Institutions (F.19, F of F) billions of dollars 11 1 9 8 7 26 27 28 29 21 211

Crisis Suppose something makes banker net worth, N t, drop. For given d t, bank cash constraint gets tighter: R b t+1 (N t + d t ) R b d,t+1 d t. So, R b d,t+1 has to be low when N t is low, banks with bad assets cannot cover their own losses and creditors must share in losses. then, creditors require R g d,t+1 high So, interest rate spread, R g d,t+1 R t, high, banker effort low. Banks get riskier (cross sectional mean return down, standard deviation up).

Endogenous Risk Rate of return on equity, good banks and bad banks: p (e t ) good banks : 1 p (e t ) bad banks : R g t+1 (N t + d t ) R g d,t+1 d t, N t R b t+1 (N t + d t ) R b d,t+1 d t = N t Mean, E b t+1, and cross sectional standard deviation, sb t+1, of return on equity across banks: [p (e t ) (1 p (e t ))] 1/2 Rg t+1 (N t + d t ) R g d,t+1 d t N t E b t+1 = p (e t ) Rg t+1 (N t + d t ) R g d,t+1 d t N t In a crisis, risk rises and mean return falls.

Macro Model Sticky wages and prices Investment adjustment costs Habit persistence in consumption Monetary policy rule

Calibration targets Table 2: Steady state calibration targets for baseline model Variable meaning variable name magnitude Cross-sectional standard deviation of quarterly non-financial firm equity returns s b.2 Fnancial firm interest rate spreads (APR) 4 R d g R.6 Financial firm leverage L 2. Profits of intermediate good producers (controled by fixed cost, ) Government consumption relative to GDP (controlled by g ).2 Growth rate of per capita GDP (APR) 4 z 1 1.65 Rate of decline in real price of capital (APR) 4 1 1.69

Data behind calibration targets.4 Figure 1: Cross-section standard deviation financial firm quarterly return on equity, HP-filtered US real GDP Cross section volatility (left scale).1.3.5.2.1 -.5 -.1 Q3 1962 Q1 1968 Q2 1973 Q4 1978 Q1 1984 Q2 1989 Q4 1994 Q1 2 Q3 25 Q4 21 quarterly data

Data behind calibration targets.4 Figure 1: Cross-section standard deviation financial firm quarterly return on equity, HP-filtered US real GDP Cross section volatility (left scale).1.3.5.2.1 -.5 HP filtered GDP (right scale) -.1 Q3 1962 Q1 1968 Q2 1973 Q4 1978 Q1 1984 Q2 1989 Q4 1994 Q1 2 Q3 25 Q4 21 quarterly data

Parameter Values Table 1: Baseline Model Parameter Values Meaning Name Value Panel A: financial parameters return parameter, bad entrepreneur b -.9 return parameter, good entrepreneur g. constant, effort function ā.83 slope, effort function b.3 lump-sum transfer from households to bankers T.38 fraction of banker net worth that stays with bankers.85 Panel B: Parameters that do not affect steady state steady state inflation (APR) 4 1 2.4 Taylor rule weight on inflation 1.5 Taylor rule weight on output growth Δy.5 smoothing parameter in Taylor rule p.8 curvature on investment adjustment costs S 5. Calvo sticky price parameter p.75 Calvo sticky wage parameter w.75 Panel C: Nonfinancial parameters steady state gdp growth (APR) z 1.65 steady state rate of decline in investment good price (APR) 1.69 capital depreciation rate.3 production fixed cost.89 capital share.4 steady state markup, intermediate good producers f 1.2 habit parameter bu.74 household discount rate 1 4 1.52 steady state markup, workers w 1.5 Frisch labor supply elasticity 1/ L 1. weight on labor disutility L 1. steady state scaled government spending g.89

Impact of Loss of Bank Net Worth 3.5 Bank net worth (N) levels 3 2.5 no leverage restrictions leverage restrictions

levels Bank net worth (N) 3.5 3 2.5 Impact of Loss of Bank Net Worth levels Deposit rate, bad (failed) banks (APR) 2 4 6 8 1 12 14 no leverage restrictions leverage restrictions

levels 3.5 3 2.5 Bank net worth (N) Impact of Loss of Bank Net Worth levels Deposit rate, bad (failed) banks (APR) Interest rate spread (APR) 2 1 4 6 8 1 12 14 levels.8.6.4.2 no leverage restrictions leverage restrictions

levels levels 3.5 3 2.5 Bank net worth (N) levels Std dev, in cross section, financial firm equity returns.25.2.15.1 Impact of Loss of Bank Net Worth Deposit rate, bad (failed) banks (APR) Interest rate spread (APR) 2 1 4 6 8 1 12 14 levels.8.6.4.2.5 no leverage restrictions leverage restrictions

levels 3.5 3 2.5 Bank net worth (N) levels Std dev, in cross section, financial firm equity returns.25 Impact of Loss of Bank Net Worth Deposit rate, bad (failed) banks (APR) Interest rate spread (APR) 2 1 4 6 8 1 12 14.2 GDP levels.8.6.4.2 levels.2.15.1 % dev, ss.4.6.8.5 no leverage restrictions leverage restrictions

levels levels 3.5 3 2.5 Bank net worth (N) levels Std dev, in cross section, financial firm equity returns.25.2.15.1.5 Impact of Loss of Bank Net Worth % dev, ss Deposit rate, bad (failed) banks (APR) Interest rate spread (APR) 2 1 4 6 8 1 12 14.2.4.6.8 GDP levels % dev, ss.8.6.4.2.2.4.6.8 1 Consumption no leverage restrictions leverage restrictions

levels levels % dev, ss 3.5 3 2.5 Bank net worth (N) levels Std dev, in cross section, financial firm equity returns.25.2.15.1.5.4.6.8 1 1.2 Investment Impact of Loss of Bank Net Worth % dev, ss Deposit rate, bad (failed) banks (APR) Interest rate spread (APR) 2 1 4 6 8 1 12 14.2.4.6.8 GDP levels % dev, ss.8.6.4.2.2.4.6.8 1 Consumption no leverage restrictions leverage restrictions

levels levels % dev, ss 3.5 3 2.5 Bank net worth (N) levels Std dev, in cross section, financial firm equity returns.25.2.15.1.5.4.6.8 1 1.2 Investment Impact of Loss of Bank Net Worth % dev, ss level Deposit rate, bad (failed) banks (APR) Interest rate spread (APR) 2 1 4 6 8 1 12 14.2.4.6.8 2.5 2.4 2.3 2.2 2.1 2 GDP Inflation (APR) no leverage restrictions leverage restrictions levels % dev, ss.8.6.4.2.2.4.6.8 1 Consumption

levels levels % dev, ss 3.5 3 2.5 Bank net worth (N) levels Std dev, in cross section, financial firm equity returns.25.2.15.1.5.4.6.8 1 1.2 Investment Impact of Loss of Bank Net Worth % dev, ss level Deposit rate, bad (failed) banks (APR) Interest rate spread (APR) 2 1 4 6 8 1 12 14.2.4.6.8 2.5 2.4 2.3 2.2 2.1 2 GDP Inflation (APR) no leverage restrictions leverage restrictions levels % dev, ss levels.8.6.4.2.2.4.6.8 1 24 22 2 18 16 14 Consumption Bank leverage

Leverage Restrictions Banks taxed for issuing deposits d t 1.2% AR (versus 3% AR on the risk free nominal rate). revenues redistributed back to banks in lump-sum form. What is the consequence of this restriction? With less d t, banks with bad assets more able to cover losses interest rate spread falls, so banker effort rises. Second effect of leverage restriction, leverage restriction in effect implements collusion among bankers allows them to behave as monopsonists make profits on demand deposits...lots of profits: big {}}{ [ ( p (e t ) R g ) ( t+1 Rg + (1 p (e d,t+1 t )) R b )] t+1 Rb d t d,t+1 N t makes N t grow, offseting incentive effects of decline in d t.

levels levels % dev, ss 3.5 3 2.5 Bank net worth (N) levels Std dev, in cross section, financial firm equity returns.25.2.15.1.5.4.6.8 1 1.2 Investment Impact of Loss of Bank Net Worth % dev, ss level Deposit rate, bad (failed) banks (APR) Interest rate spread (APR) 2 1 4 6 8 1 12 14.2.4.6.8 2.5 2.4 2.3 2.2 2.1 2 GDP Inflation (APR) no leverage restrictions leverage restrictions levels % dev, ss levels.8.6.4.2.2.4.6.8 1 24 22 2 18 16 14 Consumption Bank leverage

Conclusion Described a model in which there is a problem that is mitigated by the introduction of leverage restrictions. Currently exploring what are the optimal dynamic properties of leverage. the cyclical behavior of the tax on leverage depends on which shock drives the cycle. if driven by permanent technology shocks, then act to discourage debt in a boom.

Steady State Calculations Next study steady state impact of leverage Quantify role of hidden effort in the analysis (essential!)

Table 3: Steady State Properties of the Model Variable meaning Variable name Unobserved Effort Observed Effort Leverage Restriction Leverage Restriction non-binding binding non-binding binding Spread 4 R d g R.6.211 NA NA scaled consumption c 1.84 1.88 2.1 1.95 labor h 1.18 1.16 1.15 1.14 scaled capital stock k 51.52 51.4 59.75 53.86 bank assets N d 51.52 51.31 59.55 53.68 bank net worth N 2.58 3.2 2.58 3.16 bank deposits d 48.94 48.29 56.98 5.52 bank leverage N d /N 2. 17. 23.12 17. g p et R bank return on equity (APR) 4 t 1 1 p et Rb t 1 Nt dt Rtdt 1 4.59 14.96 4.59 17.63 Nt fraction of firms with good balance sheets p e.962.982 1. 1. Benefit of leverage (in c units) 1 NA 1.19 NA -2.7 Benefitofmakingeffortobservable(inc units) 1 NA NA 6.11 2.3 Note: (i) NA, not applicable, indicates that the number is not defined. (ii) All calculations based on a single set of parameter values, reported in Table 1.

Dynamics Here, we consider the dynamic effects of two shocks shock to monetary policy lump sum shock to net worth

R t.8r t 1 1.8 1.5 t 1.5g y,t t p p 25 annual basis points % dev from ss -.15 -.2 -.25 -.3 GDP % dev from ss -.15 -.2 -.25 -.3 -.35 Consumption % dev from ss -.3 -.4 -.5 Investment 2.44 2.42 2.4 2.38 2.36 2.34 2.32 Inflation (APR) level % points dev from ss 4.75 4.7 4.65 4.6-2 -4-6 Risk free rate (APR) Bank net worth (N) level Interest rate spread (APR).2.1 -.1 21.5 21 2.5 2 Leverage (N+d)/N level.962.96.958.956 Prob of success quarterly, level.22.215.21.25 Contractionary M policy shock: fall in: c, i, y, bank net worth, N, inflation.2 Cross-section std dev, Bank return on equity Rise in: leverage cross sectional dispersion of bank performance

Bankers and their Creditors Assets Liabilities Loans and other securities Deposits, d t N t + d t Banker net worth, N t No agency problems on asset side of bank balance sheet. Problems are on liability side. Bankers receive credit, d t, from mutual funds. Mutual funds deal with households.

Risky Bankers Funded By Mutual Funds Household banker Household Household Diversified, competitive mutual funds banker banker banker

% dev from ss -.4 -.6 -.8 GDP log % dev from ss -.2 -.4 -.6 -.8-1 T t.95log T T.1 Consumption % dev from ss -.4 -.6 -.8-1 -1.2 T t 1 T Investment t T 2.4 2.3 2.2 2.1 Inflation (APR) % points dev from ss level 4.5 4.4 4.3 4.2-7 -8-9 -1-11 Risk free rate (APR) Bank net worth (N) level.35.3.25.2.15 22 21.5 21 2.5 2 Interest rate spread (APR) Leverage (N+d)/N level.962.96.958.956.954.952 Prob of success quarterly, level.23.22.21.2 Cross-section std dev, Bank return on equity Negative shock to bank net worth: fall in: c, i, y, bank net worth, N, inflation Rise in: leverage cross sectional dispersion of bank performance

L e t = dl e t = = a f t a f t lf t da f t a f ( ) t a f l f ( a f l f ) da f 2 t dlf t a f a f l f âf t a f ( t ( a f l f ) 2 ˆL e t = â f t 1 ( a f l f a f â f t lfˆlf t l = f ) (ˆlf t âf t f f a f â f t lfˆlf t ) )

Cyclicality of Leverage The model appears to imply countercyclical leverage. We took data from the Flow of Funds accounts to measure leverage. Problem: only report financial assets (a f ) and liabilities (l f ) L f = af a f l f This measure of leverage can be negative or gigantic. We took measures of L f for three components of financial business, over a period for which L f does not behave strangely, the 2s.

1975 198 1985 199 1995 2 25 21 Holding Companies (L.128) liability growth - asset growth (yoy) 15 1 5-5 -1-15 Private Depository Institutions (L.19) liability growth - asset growth (yoy) 3 2.5 2 1.5 1.5 -.5-1 -1.5 1975 198 1985 199 1995 2 25 21 Security Brokers and Dealers (L.127) liability growth - asset growth (yoy) 1 8 6 4 2-2 -4-6 -8 1975 198 1985 199 1995 2 25 21 liability growth - S&P5 growth (yoy) 1 8 6 4 2-2 1975 198 1985 199 1995 2 25 21 liability growth - S&P5 growth (yoy) 1 8 6 4 2-2 1975 198 1985 199 1995 2 25 21 liability growth - S&P5 growth (yoy) 5 4 3 2 1-1 -2-3 -4 1975 198 1985 199 1995 2 25 21

L t = LˆL t = ˆL t = = = a nf t a nf t + a f t + a f t lf t a nf a nf + a f l f ânf t + a nf + a f ( a nf + a f l f ) 2 a nf a nf + a f ânf [ a nf a nf + a f [ a nf a nf + a f t + af a f a nf + a f l f âf t ( a nf â nf t + a f â f t lfˆlf t a nf + a f âf t 1 ( ) a nf + a f l f a nf â nf t + a f â f t lfˆlf t ] [ ] a nf a nf + a f l f â nf a t + f a a nf + a f f a nf + a f l f â a nf a nf + a f l f ] â nf t ) l f ( a nf + a f l f ) ( a nf + a f )af â l = f a ( nf a nf + a f ) ( l a nf + a f l f )ânf t f a ( f a nf + a f ) ( a nf + a f l f )âf t