Minsky s Financial Instability Hypothesis and the Leverage Cycle 1 Sudipto Bhattacharya London School of Economics Charles A.E. Goodhart London School of Economics Dimitrios P. Tsomocos University of Oxford Alexandros P. Vardoulakis Banque de France 1 The views expressed in this paper are those of the authors and do not necessarily represent those of the Banque de France. Minsky & Leverage Cycle 1 / 42
Outline 1 Approaches to financial crises 2 Baseline model 3 Minsky s Financial Instability Hypothesis and the Leverage Cycle 4 Policy Responses 5 Empirical Implications 6 Conclusions Minsky & Leverage Cycle 2 / 42
Approaches to financial crises 1 Shiller s (2000) irrational exuberance : a view that financial crisis results from a black-out of human reason 2 Diamond and Dybvig (1983), Morris and Shin (2004): a crisis results from a coordination failure, usually associated with a financial structure that has a built-in first-mover advantage, like demand deposits, or loss limits 3 Fisher (1933): The debt-deflation theory of Great Depressions Over indebtedness and deflation The main mechanism through which crises occur is the drop in the relative price of capital goods and industrial output relative to the value of corporate debt 4 Minsky s Financial Instability Hypothesis (1984) Minsky & Leverage Cycle 3 / 42
Externalities 1 Bank runs: Coordination problem 2 Fire sales: Market liquidity and funding liquidity / marginal buyer 3 Unduly pessimistic expectations due to portfolio opacity 4 Optimism and procyclicality of price based risk measures: This presentation 5 Network externalities: Portfolio commonality and chain reaction of default Minsky & Leverage Cycle 4 / 42
Tools i. Leverage requirements ii. Countercyclical capital buffers iii. Systemic surcharge iv. Haircuts regulation v. LTV regulation vi. Liquidity requirements vii. Creditors Association to Losses/Contingent capital Each tool is trying to correct for the externalities above in a different way Thus, its implementation may correct one externality and make another one more severe A combination of tools may be needed (Kashyap, Berner & Goodhart, 2011) Minsky & Leverage Cycle 5 / 42
Financial Instability Hypothesis by Hyman Minsky: over periods of prolonged prosperity and optimism about future prospects, financial institutions invest in riskier assets, which can make the economic system more vulnerable in the case that default materializes Expectations formation varies across the economic cycle giving rise to a leverage cycle and inevitable harsher default This paper: Question 1a What are the sources of excessive leverage: Optimizing behaviour of creditors and debtors? Question 1b How do portfolio choice and risk taking vary over the leverage cycle? Question 2 Is controlling leverage the way to ensure financial stability? Question 3 Can we predict the leverage cycle? Minsky & Leverage Cycle 6 / 42
Expectations formation Multiperiod economy. Two states possible at any point in time, u and d Set of all states: s t S = {0, u, d,..., uu, ud, du, dd,..., s t u, s t d,...} The probability that a good state occurs is constant at any point in time and denoted by θ Nature decides at t=0 whether θ = θ 1 or θ = θ 2, θ 1 > θ 2 Agents do not know this probability and try to infer it by observing past realizations No asymmetry of information among agents Cogley and Sargent (2008) Agents become more optimistic after they observe good outcomes in the past Proof. Minsky & Leverage Cycle 7 / 42
Portfolio choice Two financial institutions i I = {Γ, } with endowed capital of 1 at every t It can invest either in a safer project, denoted by L, or in riskier one, denoted by H For now, we make the assumption that the two projects are mutually exclusive, indivisible and that every bank can only invest in one unit of a project in each period X H u > X L u > 1 > X L d > X H d > 0 Ũ i s t = Π i s t γ i ( Π i s t ) 2 + ω i ( Ri st R i s t ) We introduce a strategic consideration by including a linear term in each financial institution s objective function, which is proportional to the difference between the institution s own return on its investments and the return on the investments of its competitors (following Bhattacharya, Goodhart, Sunirand and Tsomocos (2007)) More.. Minsky & Leverage Cycle 8 / 42
No competitive considerations Proposition 1: Given that banks do not compete with each other (i.e. ω i = 0 i I), bank i will choose to invest in the riskier project if the number of good realization before the time of the decision is n with π st (n ) > Āi Xd L π st > X d H ( γi Xd L ) 2 + γ i ( Xd H ) 2 Xu H Xu L + Xd L X d H + ( γi Xu L ) 2 ( γ i Xu H ) 2 ( + γ i Xd H ) 2 ( γ i Xd L ) 2 = Āi Proof. Minsky & Leverage Cycle 9 / 42
Competitive considerations Proposition 2: For a given risk-aversion and distribution of returns, there exists an ω i after which bank i chooses to invest in the riskier project Proposition 3: For a given distribution of returns, the more risk averse a bank is the higher its relative performance coefficient has to be for it to invest in the riskier project Proposition 4: For a given risk-aversion, when expectations about project returns become more optimistic, it is more profitable for a bank to deviate to the riskier project Proposition 5: Given that banks compete with each other, bank i will choose to invest in the riskier project after a lower number of good realizations compared to the case when there is no competition, i.e. n < ω i n where n ω i is the minimum number of good realizations to deviate to the riskier project when ω i > 0 Proof. Minsky & Leverage Cycle 10 / 42
Richer model Banks can raise additional funds from the credit markets Generalized portfolio problem: Projects not mutually exclusive and infinitely divisible Endogenous default in credit markets: Banks weigh the marginal utility with the marginal disutility from defaulting Non pecuniary default penalties guarantee commitment to repay Alternative way to model default: Collateralized loans More.. Geanakoplos (2010) Minsky & Leverage Cycle 11 / 42
Bank s Optimization Problem 1 max E 0 Ũ w s i t,j,vi s,π i t+1 s,t i t+1 u,t d i s i t+1 λ st+1 E 0 max [(1 ṽs i t+1 ) w st (1 + r st ), 0] = t=0 ]] [ ]] = E 0 [Ũi s1 + E s1 [Ũi s2 λ st+1 E 0 [max (1 ṽs i )w 1 0 (1 + r 0 ) [ [ ]]] λ st+1 E 0 E s1 [max (1 ṽs i 2 ) w s1 (1 + r s1 ) w0,l i + w 0,H i w 0 i + w0 i (ψ0) i i.e. investment in the safer and the riskier assets initial capital + leverage at t=0 ws i t,l + w s i t,h T s i t + ws i t (ψs i t ) s t {u, d} i.e. investment in the safer and the riskier projects reinvested profits + leverage Π i s t + Ts i t w0,l i X g L + w0,h i X g H w0 i v s i t (1 + r 0 ) (φ st ) s t {u, d} i.e. distributed + retained profits safer and riskier investments payoff - loan repayment Π i s t ws i t 1,L X g L + ws i t 1,H X g H ws i v i t 1 s t (1 + r st 1 ) (φ st ) s t {uu, ud, du, dd} i.e. distributed profits safer and riskier investments payoff - loan repayment Minsky & Leverage Cycle 12 / 42
Creditors Optimization Problem max E 0 c cs c,w c s c t = c0 c + π 0cu c + (1 π 0 )cd c + π 0π u cuu c + π 0 (1 π u )cud c t s t s + (1 π 0 )π d cdu c + (1 π 0)(1 π d )cdd c s.t. c0 c w 0 c w 0 c i.e. consumption initial endowment - credit extension at t=0 cs c t w s c t + vs i t (1 + r 0 )w0 c w s c t s t {u, d} consumption endowment + loan repayment - credit extension in s t {u, d} cs c t vs i t (1 + r st 1 )ws c t 1 s t {uu, ud, du, dd} consumption loan repayment in s t {uu, ud, du, dd} E st [ v i s t+1 ] (1 + r st ) = 1, s t {0, u, d} Minsky & Leverage Cycle 13 / 42
Leverage Cycle Once expectations become more optimistic, banks will reallocate their portfolios towards the riskier asset In order to fund their position, they will increase their leverage, since they cannot go short in the safer asset Once uncertainty is resolved, banks need to repay their loans and they are confronted with the decision to default If realisations turn out to be bad after a period of previously good news, they will default more on their loans, since they would have invested more in the riskier asset One might have expected that creditors would reduce their credit extension and leverage would go down, since loss given default would be higher However, this is not the case since the probability of a good outcome has increased and consequently the interest rate creditors charge is lower Minsky & Leverage Cycle 14 / 42
Quantitative Analysis The choice of exogenous parameters is in line with our assumptions about the evolution of beliefs and the riskiness of the two projects Exogenous variables. Table: Portfolio weight of the riskier project Portfolio weight on the risky project at t = 0 w0,h i =0 Portfolio weight on the risky project after bad news wd,h i =0 Portfolio weight on the risky project after good news wu,h i =65.97% Risky-to-safe project ratio of weights at t=0 w0,h i /w 0,L i =0 Risky-to-safe asset ratio of weights after bad news wd,h i /w d,l i =0 Risky-to-safe asset ratio of weights after good news wu,h i /w u,l i =1.94 Minsky & Leverage Cycle 15 / 42
Quantitative Analysis ctd. Table: Interest rates, leverage and default Increase in leverage after good news 182% Interest rate change after good news -10.36% Decrease in leverage after bad news 14.51% Interest rate change after bad news 9.14% Expected default at s t = 0 9.18% Realized default at s t = d 52.08% Expected default at s t = u 8.24% Realized default at s t = ud 64.78% Expected default at s t = d 9.86% Realized default at s t = d 24.29% Loss given default at s t = d 2.33 Loss given default at s t = ud 8.10 Loss given default at s t = dd 0.94 Endogenous variables. Minsky & Leverage Cycle 16 / 42
Shock in relative performance Table: Change in equilibrium variables under 1% shock in ω i in state u Equilibrium variable Change Holdings of riskier project in s t = u 0.15% Holdings of safer project in s t = u -1.33% Riskier holdings over leverage in s t = u 0.70% Leverage in s t = u -0.54% Interest rate in s t = u 0.39% Expected default in s t = u 0.36% Loss given default in s t = ud -0.15% The relative performance channel seems to have shut down in state s t = u, since higher risk-taking if fully anticipated by creditors Aggregate holding of the riskier project do not increase much and loss given default decreases due to lower borrowing Minsky & Leverage Cycle 17 / 42
Shock in relative performance Aggregate holdings are much lower for a significantly lower coefficient Risk-taking increases as the relative performance coefficient becomes higher A measure of risk taking could be the volatility of the bank s portfolio We construct another measure: the difference between riskier and safer holdings in aggregate terms per unit of leverage Minsky & Leverage Cycle 18 / 42
Shock in relative performance Figure: Risk-taking and default for low and high relative performance coefficient Minsky & Leverage Cycle 19 / 42
Shock in relative performance Figure: Portfolio holdings under low and high ω s in state u Minsky & Leverage Cycle 20 / 42
Stricter Default Penalties after good news The fundamental reason that allows banks to invest largely on the riskier project after expectations have been updated upwards is that interest rates go down Although loss given default is higher in bad outcomes, optimism dominates making expected default lower Thus, banks are able to leverage up to invest in the risker project without having to reduce their investment in the safer one, the holdings of which increase as well It looks like that banks are not penalized ex-ante with a higher interest rate due to optimistic expectations We examine: 1 Stricter Default Penalties 2 Stricter Leverage Requirements 3 Another type of requirement Minsky & Leverage Cycle 21 / 42
Stricter Default Penalties after good news The only penalty is due to default Policy cannot set the credit spreads, since they are market based and depend on expectations which follow an exogenous process But, what policy can affect is the propensity to default by changing the default penalty Our main consideration is that policy cannot affect default penalties in the straightforward way described here They rely a lot on market reaction and market discipline Their presence is implicit and necessary, but their level is not directly observable Minsky & Leverage Cycle 22 / 42
Stricter Default Penalties after good news Figure: Loss given default (top), borrowing (middle) and interest rates (bottom) under various default penalties in state ud However, banking profits fall as expected. Profits. Minsky & Leverage Cycle 23 / 42
Leverage Requirements A leverage requirement can take the form of a maximum ratio of borrowing over the total investment in projects We show that such a requirement achieves the exact opposite result than hoped: it results in increased loss given default in state ud instead of bringing it down Intuition Banks will divert their own funds away from the safer asset and put them to the riskier one Although borrowing goes down, they will invest even more in the riskier asset to compensate for the loss in gearing, since expectation are optimistic Minsky & Leverage Cycle 24 / 42
Leverage Requirements Figure: Loss given default (top) and portfolio holdings (bottom) under various leverage requirement in state u The x-axis denotes the leverage requirement (the ratio of borrowing over total asset portfolio value) in the up-turn of the cycle, i.e. state u. The requirement gets tighter to the right. Banks divert funds from the safer to the riskier project to make up for the loss in gearing. Increased risk-taking leads to higher loss given default. Minsky & Leverage Cycle 25 / 42
An alternative requirement Our analysis highlights that the adverse consequences of the leverage cycle depend on financial institutions shifting their portfolios towards previously riskier projects due to the fact that beliefs have been updated upwards Policy could restrict relative portfolio holdings A requirement on the difference between riskier and safer holdings per unit of leverage results in higher financial stability Intuition It is the shift towards riskier projects in combination with high leverage that creates the problem, which is something that leverage requirements by themselves cannot handle It is leverage that goes directly to risky investment which is the appropriate variable to control Minsky & Leverage Cycle 26 / 42
An alternative requirement Figure: Percentage default (top), loss given default (middle) and portfolio holdings in state u The requirement is defined as riskier minus safer investment per unit of borrowing. It gets tighter to the right. Given that it targets directly risk-taking behaviour, riskier holdings in state u go down, as do loss given default and percentage default in state ud. Minsky & Leverage Cycle 27 / 42
Measuring the time-dimension in riskiness of banking portfolios or of the financial sector as a whole over the leverage cycle is not an easy task Commonly used measures to capture risk building up, such as the volatility of banking assets or credit spreads, fail to do so due to the fact that they are biased by optimistic expectations 5 TED spread (lhs) VIX (rhs) 1 0 0.5 5 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 0 Figure: VIX and TED spread evolution over time Minsky & Leverage Cycle 28 / 42
The index we propose is the difference between riskier and safer portfolio holdings per unit of leverage Although absolute riskiness goes down for both types, their ranking is preserved (assuming bank are beta long..and that risk is captured in the cross-section) We normalize by leverage, because it is default on debt that causes a financial crisis, tightening in credit and forced liquidations that lead to fire sales externalities Figure: Optimism and riskiness Minsky & Leverage Cycle 29 / 42
Financial agents are Bayesian learners and update their beliefs about future good realisations by observing the sequence of past ones After a prolonged period of good news, expectations are boosted and financial institutions find it profitable to shift their portfolios towards projects that are on average riskier, but promise higher expected returns Creditors are willing to provide them with funds, since their expectations have improved as well When bad news realise, default is higher and the consequences for financial stability are more severe Minsky & Leverage Cycle 30 / 42
Banks will shift their portfolios towards riskier assets faster when they compete more for expected returns: This channel is mitigated when there is no asymmetry of information through higher borrowing costs Making default most costly can stabilise the leverage cycle Leverage requirements do not yield the desired outcome Price based measures, such as VIX or TED spread, may not be adequate to predict the leverage cycle Minsky & Leverage Cycle 31 / 42
Agents subjective belief is given by: π st = Pr st (θ = θ 1 s t ) θ 1 + Pr st (θ = θ 2 s t ) θ 2 Their conditional probability given past realizations is: Pr t (θ = θ 1 s t ) = Pr t (s t θ = θ 1 ) Pr(θ = θ 1 ) Pr(s t ) = = Pr t (s t θ = θ 1 ) Pr(θ = θ 1 ) Pr t (s t θ = θ 1 ) Pr t (θ = θ 1 ) + Pr t (s t θ = θ 2 ) Pr(θ = θ 2 ) θ n 1 (1 θ 1) t n Pr(θ = θ 1 ) θ n 1 (1 θ 1) t n Pr(θ = θ 1 ) + θ n 2 (1 θ 2) t n Pr(θ = θ 2 ) where n is the number of good realization up to time t Minsky & Leverage Cycle 32 / 42
θ1 n π st = (1 θ 1) t n Pr(θ = θ 1 ) θ1 n(1 θ 1) t n Pr(θ = θ 1 ) + θ2 n(1 θ 2) t n Pr(θ = θ 2 ) θ 1 + θ n 2 (1 θ 2) t n Pr(θ = θ 2 ) θ n 1 (1 θ 1) t n Pr(θ = θ 1 ) + θ n 2 (1 θ 2) t n Pr(θ = θ 2 ) θ 2 Proof. Pr st (θ = θ 1 s t ) > Pr st 1 (θ = θ 1 s t 1 ) θ n+1 1 (1 θ 1 ) t+1 (n+1) θ n+1 1 (1 θ 1 ) t+1 (n+1) + θ n+1 2 (1 θ 2 ) ( ) θ2 1 θ n ( ) 1 1 t θ2 > 1 θ 2 θ 1 1 θ 1 1 > θ 2 1 θ 2 1 θ 1 θ 1 1 θ 2 1 θ 1 1 > θ 2 θ 1 t+1 (n+1) > θn 1 (1 θ 1) t n ( θ2 1 θ 2 1 θ 1 θ 1 θ1 n(1 θ 1) t n + θ2 n(1 θ 2) t n ) n+1 ( ) 1 t+1 θ2 1 θ 1 Return Minsky & Leverage Cycle 33 / 42
The expected payoff at date t and state s t is: E st Ũ i s k+1 = k=t k=t = k=t E st Π i s k+1 γ i E st ( Π i s k+1 ) 2 + ω i ( E st Ri sk+1 E sk+1 Ri s k+1 ) = E st Π ( ) 2 ( ) i s k+1 γ i E st Π ( ) i s k+1 γ i Var Π i sk+1 + ω i E st Ri sk+1 E st Ri s k+1 Upper limit for a bank s risk-aversion, such that it chooses to invest in one of the two projects and not hold onto its capital. The individual rationality constraint for bank b is: ( ) 1 + µ j s t γ i 1 + µ j 2 ( ) ( s t γ i σs j 2 ) t + ω i µ j s t E st Ri s t > 1 γ i ω i E st Ri s t γ i < µj s t (ω i + 1) ) 1 + µ j s t + (σ s j 2 t Return Minsky & Leverage Cycle 34 / 42
A bank will prefer to invest in the risker project in s t if its expected utility at s t is higher compared to that coming for the safer investment, i.e. E st ( X L) γ i ( E st ( X L)) 2 < Est ( X H ) γ i ( E st ( X H )) 2 ( ) 2 π st Xu L + (1 π st )Xd L π s t Xu H (1 π st )Xd H γi π st (X u L ( ) 2 ( ) 2 ( ) ) 2 +(1 π st ) Xd L πst Xu H (1 πst ) Xd H < 0 Return Minsky & Leverage Cycle 35 / 42
For any choice of project j by bank i, bank i will choose to invest in the safer project and will not have an incentive to deviate if: ) (1 ( ) ( 2 + µ L st γ i 1 + µ L s t γ i (σs L t ) 2 + ω i µ L s t E st Ri ) s t ) (1 ( ) ( 2 + µ H st γ i 1 + µ H s t γ i (σs H t ) 2 + ω i µ H s t E st Ri ) s t ω i ω i = 1 + 2γ i + γ i ( µ H s t + µ L s t + (σh s t ) 2 (σ L s t ) 2 µ H s t µ L s t ) The derivative of ω i in above expression with respect to the risk aversion coefficient, γ i, is positive. Hence, the more risk averse banks are the higher ω i is. This means that if banks are very risk averse the competition among them for higher returns has to be very high for them to invest in the riskier project Minsky & Leverage Cycle 36 / 42
We need to calculate the derivative of ω i with respect to the probability of the good state occurring. For ease of notation, we will denote this probability by π. Expanding the expression for ω i we get: ( ) 1 + 2γ i + γ i µ H s t + µ L s t + (σh s t ) 2 (σs L t ) 2 µ H s t µ L ( s t ) 1 + 2γ i + γ i πxu H + (1 π)xd H + πx u L + (1 π)xd L + + γ i π ( Xu H πxu H (1 π)xd H ) 2 ( + (1 π) X H d πxu H (1 π)xd H ) 2 πxu H + (1 π)xd H πx u L (1 π)xd L γ i π ( Xu L πxu L (1 π)xd L ) 2 ( + (1 π) X L d πxu L (1 π)xd L ) 2 πxu H + (1 π)xd H πx u L (1 π)xd L The derivative of the above expression with respect to π is: ( X γ i H u Xu L ) ( X H d Xd L ) ( X H u Xd H + X u L Xd L ) ( πx H u ( 1 + π)xd H πx u L + ( 1 + π)xd L ) 2 Given that X H u > X L u > X L d > X H d > 0 the derivative is negative, which means that ω i decreases when expectations become more optimistic. Minsky & Leverage Cycle 37 / 42
The next step is to examine whether relative performance and competition for higher expected returns result in banks choosing the riskier project even if expectations are revised upwards less than in the case that there were no strategic considerations. When ω i > 0 expression we get: (1 + ω i ) ( Xd L π st > X d H ) γ i ( Xd L ) 2 + γ i ( Xd H ) 2 (1 + ω i ) ( Xu H Xu L + Xd L X d H ) ( + γ i Xu L ) 2 ( γ i Xu H ) 2 ( + γ i Xd H ) 2 ( γ i Xd L ) 2 = A i ω After some algebra we find that Ai ω ω i < 0 Return Minsky & Leverage Cycle 38 / 42
Note on endogenous default Non-pecuniary default penalties Penalties where the entrepreneur s loss is not enjoyed by the lenders This allows the agent s utility function to be defined over negative values of its domain without allowing negative consumption of goods Shubik & Wilson (Journal of Economics 1977), Diamond (RES 1984), Zame (AER 1993), Dubey et al. (Econometrica 2005) Diamond states that: projects which could not be undertaken at all without the (default) penalties can be operated using the penalties Agents default completely when the marginal utility for zero delivery of the asset they sell is higher than the marginal disutility (default penalty) from defaulting, i.e. u (c s ) > λ If not then they will default up to the level that the marginal utility is equal to the marginal disutility, i.e. u (c s ) = λ Agents will deliver fully when their marginal utility for full delivery is lower than the marginal disutility, i.e. u (c s ) < λ Return Minsky & Leverage Cycle 39 / 42
Table: Exogenous variables Probability of good outcome at t = 0 π 0 =0.82 Relative performance coefficient ω i =0.1 Probability of good outcome at s t = u π u =0.87 Risk-aversion coefficient γ i =0.035 Probability of good outcome at s t = d π d =0.59 Default penalty λ=0.94 Safer s project payoff in good state Xg L =1.37 Initial banking capital w 0 i =1.5 Safer s project payoff in bad state Xb L=0.78 Riskier s project payoff in good state Xg H =1.89 Riskier s project payoff in good state Xb H =0.19 Return Minsky & Leverage Cycle 40 / 42
Table: Initial equilibrium variables Interest rate in s t = 0 r 0 =10.01% Interest rate in s t = u r u=8.98% Interest rate in s t = d r d =10.93% Profits (reinvested) in s t = u T u=3.15 Profits (reinvested) in s t = d T d =2.20 Profits (distributed) in s t = uu Π uu=12.55 Profits (distributed) in s t = ud Π ud =1.31 Profits (distributed) in s t = du Π du =3.92 Profits (distributed) in s t = dd Π dd =1.51 Investment in safer asset in s t = 0 w 0,L =5.56 Investment in riskier asset in s t = 0 w 0,H =0 Loan amount in s t = 0 w 0 =4.06 Investment in safer asset in s t = u w u,l =4.98 Investment in riskier asset in s t = u w u,h =9.65 Loan amount in s t = u w u=11.47 Investment in safer asset in s t = d w d,l =5.67 Investment in riskier asset in s t = d w d,h =0 Loan amount in s t = d w d =3.47 Percentage delivery in state s t = u v u=100% Percentage delivery in state s t = d v d =48.92% Percentage delivery in state s t = uu v uu=100% Percentage delivery in state s t = ud v ud =35.22% Percentage delivery in state s t = du v du =100% Percentage delivery in state s t = dd v dd =75.71% Return Minsky & Leverage Cycle 41 / 42
Stricter Default Penalties after good news Figure: Banking profits under various default penalties in state ud Return Minsky & Leverage Cycle 42 / 42