When Micro Prudence increases Macro Risk: The Destabilizing Effects of Financial Innovation, Leverage, and Diversification

Transcription

1 When Micro Prudence increases Macro Risk: The Destabilizing Effects of Financial Innovation, Leverage, and Diversification Fabrizio Lillo Scuola Normale Superiore di Pisa, University of Palermo (Italy) and Santa Fe Institute (USA) Cambridge - September 22, 214

2 Research questions Credit growth is driven by the dynamics of balance sheet size. What drives the balance sheet dynamics of FIs and what are the systemic risk implications? What is the effect of financial innovations allowing a more efficient diversification of risk? What are the systemic risk consequences of a change in capital requirements and in the maximum leverage allowed to banks? How does contagion propagate when financial players change their level of diversification? Is it always beneficial for the system to have more diversified portfolios? Can actions to reduce risk of a single financial institution increase the risk of a systemic event i.e. are micro-prudential policies always coherent with macro-prudential objectives? F. Corsi, S. Marmi, F. Lillo When Micro Prudence Increases Macro Risk: The Destabilizing Effects of Financial Innovation, Leverage, and Diversification. Available at SSRN: F. Lillo and D. Pirino, The Impact of Systemic and Illiquidity Risk on Financing with Risky Collateral. Journal of Economic Dynamics and Control (in press, 214). P. Mazzarisi, S. Marmi, F. Lillo, in preparation

3 Related literature Our paper tries to combine several strands of literature: on the impact of capital requirements on the behavior of FIs (Danielsson et al., 24,29; Adrian & Shin,29; Adrian et al., 211); on the effects of diversification and overlapping portfolios on systemic risk (Tasca & Battiston, 212; Caccioli et al., 212) on the risks of financial innovation (Brock et al. 29, Haldane & May, 211) on distressed selling and its impact on the market price dynamics (Kyle & Xiong, 21; Cont & Wagalath, 211, Thurner et al., 212; Caccioli et al., 212) on the determinants of balance sheet dynamics of FIs and credit supply (Stein 1998; Bernanke & Gertler 1989; Bernanke, Gertler, & Gilchrist, 1996,1999; Kiyotaki & Moore, 1997) Contribution: propose a solvable model that, by combining these different streams of literature, provides full analytical quantification of the links between micro prudential rules and macro prudential outcomes

4 The portfolio choice N financial institutions (FIs or banks), i = 1,..., N For simplicity, we assume that FIs adopt a simple investment strategy: equally weighted portfolio of m randomly selected investments (out of M) We consider the existence of costs of diversification" c reflecting the presence of transaction costs, firms specialization and other types of frictions. With r L the avg interest rate on liabilities the portfolio expected return is µ r L, FI maximizes portfolio returns under VaR constraints. VaR = ασ pa E. with σ p the holding period volatility, A asset of bank i, and α a constant,

5 The investment set The investment set: collection of risky investment j = 1,..., M FIs, correctly perceive that each risky investment entails both an idiosyncratic (diversifiable) risk component and a systematic (undiversifiable) risk component, σi 2 σ 2 s is the systematic risk σ 2 d is the diversifiable risk component. where = σ 2 s + σ 2 d Hence, the expected mean and volatility per dollar invested in the portfolio chosen by a given institution are µ and σ p = σ 2 s + σ2 d m

6 The portfolio optimization Then, facing cost of diversification and VaR constraints, FI chooses the total asset A (E is sticky) and diversification m which max their portfolio returns. max A,m A(µ r L ) cm s.t. αa σ 2 s + σ2 d m E. Dividing by E and with c = c, the max can be written in terms of the leverage E λ = A, E max λ,m λ(µ r L ) cm s.t. αλ σ 2 s + σ2 d m 1. chooses the optimal leverage λ = A E VaR and m which max ROE under the Squaring the constraint the Lagrangian can be written as L = λ(µ r L ) cm 1 ( ( ) ) 2 γ α 2 λ 2 σs 2 + σ2 d 1. m where γ is the Lagrange multiplier for the VaR constraint.

7 Optimal leverage and diversification F.O.C. λ = 1 γ 1 α 2 µ r L σ 2 p with Lagrange multiplier γ = 1 µ r L α σ p The optimal leverage is λ 1 = α σs 2 + σ2 d m = 1 ασ p The optimal level of diversification is m = γαλσd 2c = λσ d α 2c µ r L σ p Bottom line: leverage λ is an inverse function of the portfolio volatility σ p portfolio size m is an inverse function of diversification costs c

8 Diversification cost and optimal leverage 8 leverage λ diversification cost parameters are: M = 2, α =.5, µ r L =.8, σ d = 1. We then choose σ s equal to (solid line),.3 (dashed line), and.6 (dotted line).

9 Portfolio overlap between two financial institutions

10 Diversification cost and portfolio overlap 1 1 portfolio overlap portfolio overlap diversification cost α VaR parameters are: M = 2, α =.5, µ r L =.8, σ d = 1. We then choose σ s equal to (solid line),.3 (dashed line), and.6 (dotted line).

11 Leverage targeting and balance sheet adjustments Initial position Assets Liabilities Asset 1 Debt 9 Equity 1 Asset growth Assets Liabilities Asset 11 Debt 9 Equity 11 Leverage adjustment Assets Liabilities Asset 11 Debt 99 Equity 11 from Adrian and Shin (21)

12 Balance sheet adjustments: empirical evidence from Adrian, Colla, and Shin (21)

13 Dynamics of asset with portfolio rebalancing At the beginning of each investment period FIs rebalance their portfolio by the difference between the desired amount of asset A j,t = λe j,t and the actual one A j,t R j,t A j,t A j,t = λe j,t A j,t, By defining realized portfolio return r p j,t, can be rewritten as R j,t = (λ 1)r p j,t A j,t 1 any P&L from the investments portfolio r p j,t A j,t 1 results in a change of FI asset value amplified by the target leverage (for λ > 1). VaR induces a perverse demand function: buy if r p j,t >, sell if r p j,t < positive feedback

14 Dynamics of investments demand The aggregate demand of asset i will be simply the sum of the individual demands of the FIs who picked asset i in their portfolio. D i,t = N j=1 I {i j} 1 m R j,t N j=1 I {i j} (λ 1)r p A j,t 1 j,t m where I {i j} is 1 if asset i is in the portfolio of institution j and zero otherwise. Considering total assets approximately the same across FIs, A j,t 1 A t 1, demand of investment i can be approximated as D i,t (λ 1) A t 1 N r i,t + m 1 r k,t m M M 1 Note: it can be shown that demand correlation between two assets ρ(d i, D k ) 1 m M k i

15 Portfolio overlap and demand variance & correlation Var[D] portfolio overlap Corr[D i,d j ] portfolio overlap Parameters are M = 2, N = 1, and σ d = 1. We then choose σ s equal to (solid line),.3 (dashed line), and.6 (dotted line).

16 Risky asset dynamics with endogenous feedbacks I With rebalancing feedbacks, the return process is now made of 2 components r i,t = e i,t 1 }{{} endogenous + ε i,t }{{} exogenous We assume that the exogenous component has a multivariate factor structure ε i,t = f }{{} t + ɛ i,t }{{} factor idiosyncratic uncorrelated and distributed with mean and constant volatility, σ f and σ ɛ (the same for all investments). Thus, the variance of the exogenous component of the risky investment i is V (ε i ) = σf 2 + σɛ 2

17 Risky asset dynamics with endogenous feedbacks II Assuming a linear price impact function the endogenous component becomes where - γ i is the market liquidity of asset i D i,t e i,t = γ i C i,t - C i,t = N j=1 I A j,t 1 {i j} N m M A t 1 is a proxy for market cap Substituting D, r, and C, we obtain the following VAR(1) for e t e t = Φ (e t 1 + ε t) where Φ (λ 1) Γ 1 Ψ with γ 1... Γ = γ 2... M M ,... γ M Ψ = M M 1 m 1 m m 1 M 1. 1 m 1 m M 1 1 m 1 m 1 1 m 1 M 1 m 1 M 1... M 1 m 1... m m m 1... m M 1 1 m.

18 Multivariate return dynamics The VAR(1) dynamics e t = Φ (e t 1 + ε t) is dictated by the eigenvalues of the matrix Φ (λ 1)Γ 1 Ψ. Being the max eigenvalue of Ψ equal 1 m, we have: Λ max (λ 1) γ 1 where γ 1 is the average of all the γ 1 i. the max eig depends on leverage and on the average illiquidity of the assets. When Λ max > 1, the return processes become non-stationary and explosive

19 Max eigenvalue, diversification cost and portf overlap non-stationary 2 non-stationary Λ max Λ max stationary diversification cost.5 stationary portfolio overlap parameters are: M = 2, α =.5, µ r L =.8, γ = 4, and σ d = 1. We then choose σ s equal to (solid line),.3 (dashed line), and.6 (dotted line). The horizontal solid line shows the condition Λ max = 1,

20 Alternative representation of the endogenous dynamics We can write, m M 1 m M 1 mψ = m 1 m M 1 M 1 m 1 M 1 m 1 M 1 = (1 b)i + bιι with b = m 1 M 1. Thus, the endogenous component of an individual investment i becomes e i,t = (1 b) a i (e i,t 1 + ε i,t ) + b Ma i (ē t 1 + ε t) }{{}}{{} idiosyncratic comp common average comp with a i = λ 1 mγ i. Moreover, assuming all investments have the same liquidity, we can show: - the average process ē t is an AR(1) (systemic component) - the distance from the avg e i,t e i,t ē t is an AR(1) (idiosyncratic) the endogenous return dynamics can be seen as a multivariate ARs around AR"

21 Endogenous Variance & Covariance formulas Thanks to this representation we can explicitly compute the variance and covariances of endogenous components e i,t and show that: - leverage both var and cov of e i,t - diversification var and cov - Both correlations - Corr(e i,t, e j,t ) m M 1

22 Return Var-Cov with rebalancing feedbacks The feedback induced by portfolio rebalancing, introduces a new endogenous component in the variances and covariances of individual and portfolio returns individual returns: V (r i,t ) = V (e i,t ) }{{} endogenous + V (ε i,t ) }{{} exogenous Cov(r i,t, r j,t ) = Cov(e i,t, e j,t ) + }{{} σf 2 }{{} endogenous exogenous portfolio returns: with V (ε i,t ) = σ 2 f + σ 2 ɛ V (r p t ) = V (e i,t) m + m 1 m Cov(e i,t, e j,t ) + σf 2 + σ2 ɛ }{{}}{{ m } endogenous exogenous Cov(r p h,t, r p k,t ) = } V {{ (ē) } endogenous V (r M t ) = 1 1 Λ 2 max }{{} variance multiplier" + V (ε M,t ) }{{} exogenous V (ε M,t ) with V (ε M,t ) = σ 2 f + σ2 ɛ M

23 Portfolio variance & correlation vs diversification cost variance of portfolios diversification cost correlation of portfolios diversification cost Parameters are: M = 2, α =.5, µ r L =.8, γ = 4, and σ d = 1. We then choose σ s equal to (solid line),.3 (dashed line), and.6 (dotted line).

24 Systemic risk Correlation between assets and between institutions increases with overlaps. Diversification tends to increase the probability of a systemwide failures A negative realization of the factor f t, now triggers a sequence of portfolio rebalances causing the price of all risky assets to decay for several periods. Being r t = e t 1 + ιf t + ɛ t = Φr t 1 + ιf t + ɛ t, also a VAR(1) the h-period cumulative mean return conditioned on systematic shock ft shock is [ ] E r t:t+h f t = ft shock (I Φ) 1 ιft shock.

25 Including systemic risk in the haircuts on repo Analytical Monte Carlo No Sys. Risk Difference λ = 5, b 2 = 5% N=5, K=7, Turn Over=2 Haircut (%) Maturity (days) Haircut (%) Mean Scalar Product (%) F. Lillo and D. Pirino, The Impact of Systemic and Illiquidity Risk on Financing with Risky Collateral. Journal of Economic Dynamics and Control (in press, 214), ssrn.com/abstract=

26 Introduction of financial innovation: summary High costs of diversification c small diversification m heterog. portfolios and P&L individual feedbacks weak and uncoordinated. Introduction of financial innovation makes: c m σ p λ Hence we have: 1) Increase in leverage λ increases risk exposure 2) Increase in diversification m increases correlations 3) Increase in λ and m increases endogenous feedback var, cov & corr So, individual reaction more aggressive (due to higher leverage) and more coordinated (due to higher correlation) aggregate feedback between prices and total asset makes bank total asset A more erratic liquidity and funding booms and bursts

27 Simulation results: simulated structural break 14 Asset evolution of fin institutions Aggregat Asset Structural Break at 1: 1) low diversification and leverage 2) high diversification and levarage

28 Expectation formation and systemic risk In the model c m σ p λ A The model presented so far assumes that financial institutions do not consider their effect on prices and do not use past information to create future expectations of assets volatilities and correlation. This leads to a dynamics described by a VAR(1) model and predicts the existence of a transition between a stationary and a nonstationary phase when the control parameter (the cost of diversification or the parameter of the Value at Risk) crosses a given threshold What happens if one uses other (more realistic) expectation formation schemes?

29 Model setting The portfolio dynamics between t 1 and t is: at time t 1, financial institutions choose m t 1 and λ t 1 between t 1 and t, financial institutions adopt a rebalancing strategy in order to maintain the target leverage λ t 1 at time t, financial institutions estimate their actual risk position ( σ p,t) 2 at time t, financial institutions deduce ( σ d,t ) 2 and they form the expectation ( σ e d,t) 2 about the future (t + 1) risk at time t, according to their expectation ( σ e d,t) 2, financial institutions solve the problem of portfolio optimization choosing m t and λ t

30 Model setting We assume that the exogenous systematic risk component σ s is equal to zero. Hence the the risk of the portfolio, realized at time t, is ( σ p,t) 2 = ( σ d,t) 2 m t 1 (1) where ( σ d,t ) 2 is the diversifiable risk of an asset as a consequence of the impact of rebalancing feedback. When banks maintain the optimal target leverage λ t 1 rebalancing their portfolio in the time period between t 1 and t, the portfolio risk realized at time t is ( σ p,t) 2 = σ2 d + V (e j,t 1) + m t 1 1 Cov(e j,t 1, e k,t 1 ) (2) m t 1 m t 1 m t 1 In the previous slides we have derived the equations for the variance and covariance of the different assets when investors have a certain leverage λ t and a degree of diversification m t

31 The dynamical system At each time step banks have an expectation σ d,t e about the future value of the volatility. The optimization of the Lagrangian leads to the choice of leverage λ t and a degree of diversification m t given by m t = λ t σ d,t e α 2c µ r L ( σ e d,t )2 m t (3) 1 λ t = (4) ( σ d,t α e )2 m t σ e d,t σ e d (m t 1, λ t 1 ) (5) The system is closed when we set the expectation scheme. We assume that banks use past observations of volatility to create expectations This is a deterministic dynamical system for the variables {m, λ, σ e d} in the domain {m, λ, σ e d} [1, M] [1, γ + 1) R + (6)

32 Naive expectations Under naive expectations, last period volatility of the risky investments will coincide to the future one ( σ e d,t) 2 = ( σ d,t ) 2 (7) When the systematic component σ s =, m t is proportional to λ t. We will show the dynamics of the latter The dynamical system has as control parameters the diversification cost c and the Value at Risk parameter α

33 Dynamics of the leverage Λt t Λt t Figure: Dynamics of the leverage λ t changing the value of diversification cost, c =.4 (left) and c =.175 (right). The other parameters are fixed at M = 3, α = 1.64, µ r L =.8, the exogenous volatility σ d = , the liquidity γ = 1. Similar dynamics for the diversification m t; when leverage is high (low), diversification is high (low) For large diversification costs (or conservative VaR), the dynamics converges quickly to a stable state For small diversification costs (or aggressive VaR), a 2-period orbit appears. High values of leverage are alternated by low leverage periods

34 Bifurcation diagram Λt endogenous risk c c 1. c c Ρt Λt 4 2 c c Α Α Figure: Orbit plots as function of the cost parameter c. Top left shows the orbit plot for the leverage λ t, top right for the endogenous risk, and bottom left for the Pearson Correlation Coefficient between two endogenous components, e j and e k. The bottom right panel is the orbit plot for the leverage, using the VaR parameter α as controlling variable. The used parameters are: M = 3, α = 1.64, µ r L =.8, σ d = , γ = 1.

35 Comments There is a threshold c 2 for the cost (or α 2 for the VaR) under which financial cycles of period 2 appear. These are due to the feedback effect induced by the target leverage strategy adopted by financial institutions. There is also a second threshold c < c 2 corresponding to the transition from stationarity to non-stationarity return dynamics Technically, at c 2 we have a period-doubling (flip) bifurcation The value of c 2 grows with the illiquidity. For a more illiquid market (higher γ 1 ) the transition occurs for larger transaction costs Near criticality (c c 2 ) the system converges to the stable state very slowly By using the Poincaré-Birkhoff normal form it is possible to estimate the flip bifurcation s amplitude for c c 2

36 Forecasting errors ACF. ACF Time Lag Time Lag ACF. ACF Time Lag Time Lag Figure: The linear autocorrelation function of the errors time series for different values of diversification costs for c =.24 (top left),.22 (top right),.18 (bottom left),.16 (bottom right). The amplitude of the gaussian noise is σ noise = σ d 1. Below the bifurcation the autocorrelation function of the forecasting errors E t = σ e p,t σ p,t+1 (8) converges quickly to zero Above the bifurcation the autocorrelation oscillates remaining different from zero and financial institutions might recognize this structure in the autocorrelation function and they might improve their forecast. Adaptive expectations

37 Adaptive expectations The adaptive expectation about the volatility of a typical asset is ( σ e d,t) 2 = ( σ e d,t 1) 2 + ω[ ( σ d,t ) 2 ( σ e d,t 1) 2 ] (9) where ω 1 Naive expectations corresponds to ω = 1. An equivalent form for adaptive expectations is ( σ e d,t) 2 = (1 ω) ( σ e d,t 1) 2 + ω ( σ d,t ) 2 (1) that is, expected risk at time t is a weighted average of the most recently observed risk and the most recently expected one. Using recursion ( σ d,t) e 2 = ω(1 ω) i ( σ d,t i ) 2 (11) i= i.e. adaptive expectation is an Exponential Weighted Moving Average of all past observed values of the realized volatility. The weights with which the past realizations of risk are considered, depend on the memory parameter, ω. In financial markets volatility is strongly correlated and moving averages are frequently used to forecast future volatility (e.g. RiskMetrics by JP Morgan)

38 Dynamics of leverage t m t t Λ t t m t t Λ t Figure: Dynamics of the diversification m t and the financial leverage λ t for a fixed value of ω equal to.95 and for two "small" values of the diversification cost, c =.15 (top) and c =.14 (bottom). The other parameters are fixed at M = 3, α = 1.64, µ r L =.8, the exogenous σ d = , the liquidity γ = 1.

39 Bifurcation diagram: chaos appears! Λt Λt c8 c4 c c Α4 1.4 Α Α Figure: Bifurcation diagrams of the leverage λt as function of the parameter c (left) or of the parameter α (right). We have fixed the value of ω equal to.95. The other parameters are fixed at M = 3, α = 1.64, µ rl =.8, the exogenous σd = , the liquidity γ = 1.

40 The role of memory and of noise Λt Λt Ω2.6 Ω4 Ω Ω Ω Figure: Bifurcation diagrams of the leverage λt for a fixed value of c equal to.14, varying the value of the expectations weight factor ω. The other parameters are fixed at M = 3, α = 1.64,.5 µ rl =.8, the exogenous σd = 1.64, the liquidity γ = 1. A longer memory stabilizes the market

41 Entropies Ω Figure: Largest Lyapunov Exponent (LLE, black circle), the Kolmogorov-Sinai entropy (KS, red circle), the order-two Renyi entropy (K 2,blue circle) as a function of the parameter ω. The parameters are fixed at c =.14, M = 3, α = 1.64, µ r L =.8, the exogenous σ d = , the liquidity γ = 1. The dynamical behavior of the system is truly chaotic (deterministic chaos)

42 Conclusions An analytical model of systemic risk where the bipartite network of investments, illiquidity, and leverage management play a crucial role In the baseline model the feedback between investment prices and bank asset induced by portfolio rebalancing leads to a multivariate VAR process whose max eigenvalue depends on the degree of leverage and average illiquidity of the assets; both the variance and correlation of individual investments monotonically increase with reduction in diversification costs; reduction in diversification costs, by increasing the strength and coordination of individual feedbacks, increases the variability of bank total asset. A transition between a stationary and a non stationary VAR is predicted In the model with naive expectations A new transition corresponding to a flip bifurcation appears, when the costs or the VaR parameter are small enough 2 period cycles of high and low leverage (diversification) is predicted in the unstable phase In the model with adaptive expectations A full bifurcation cascade is observed, as well as a transition to a chaotic phase of the dynamics of leverage or diversification Small changes in the control parameters (c and α) can trigger complex dynamics