Why Are Big Banks Getting Bigger?

Why Are Big Banks Getting Bigger? or Dynamic Power Laws and the Rise of Big Banks Ricardo T. Fernholz Christoffer Koch Claremont McKenna College Federal Reserve Bank of Dallas ACPR Conference, Banque de France Paris, Monday 11. December 2017 The views expressed in this presentation are those of the authors and are not necessarily reflective of views at the Federal Reserve Bank of Dallas or the Federal Reserve System. Any errors or omissions are the sole responsibility of the authors.

The Changing U.S. Bank Size Distribution 70 Top 10 Top 11-100 60 Share of Total Assets (%) 50 40 30 20 1985 1990 1995 2000 2005 2010 2015 Year Figure: The share of total assets held by the largest bank-holding companies.

The Changing U.S. Bank Size Distribution 0.10 0.08 Herfindahl Index 0.06 0.04 0.02 0.00 1985 1990 1995 2000 2005 2010 2015 Year Figure: The Herfindahl Index (HHI) for U.S. BHC s over time.

Consolidation Processes...

This is an empirical paper with a novel perspective

Thinking about a very specific source of risk

Changing Distributions Cross-Sectional Mean Reversion Two Different Bank Asset Concentrations Number of Banks Mean Idiosyncratic Volatility Smallest Banks Biggest Banks Total Assets Per Bank

Thick-Tailed Distributions (Pareto) High Asset Concentration Shares of Total Assets θ(k) Medium Asset Concentration Low Asset Concentration Rank k

Our Approach Illustration Illustration for Intuition (log) Asset Shares (log) Size Rank

Typical approach: a i,t = X i,t β + ε i,t A more general rank-based approach: a i,t a (k,t) 1 2 3 4 5 6 7 8 9 10 1 3 5 7 9 2 4 6 8 10 Size Rank Odd Ranks Even Ranks

The Changing U.S. Bank Size Distribution Growing concentration of U.S. BHC assets starting in the 1990s Why did big banks get bigger?

Three Literatures 1. Changes in the banking industry and in bank size Janicki and Prescott (2006) Wheelock and Wilson (2012) Lucas (2013) 2. Idiosyncratic risk/random growth and power laws Gabaix (1999, 2009) 3. Idiosyncratic risk as a potential source of aggregate volatility, especially when combined with complex and opaque interlinkages Gabaix (2011), Acemoglu et al. (2012), Caballero and Simsek (2013) One of the main contributions is to unify and extend these literatures via a purely empirical investigation of the changing U.S. bank size distribution Idiosyncratic volatility as a shaping force of power law distributions

Empirical Methods for Dynamic Power Law Distributions Nonparametric approach applied to distribution of bank assets Provides simple description of stable distribution: bank asset concentration = idiosyncratic asset volatilities reversion rates of assets Reversion rates measure cross-sectional mean reversion By estimating the changing values of the idiosyncratic volatilities and reversion rates, can answer the two questions: 1. What caused these changes in the past? 2. What can we learn from the drivers of the size distribution?

Findings U.S. Bank Holding Company asset concentration: Increased asset concentration a result of lower reversion rates Idiosyncratic volatilities are actually lower for BHCs Bigger banks are not necessarily riskier banks Even though BHCs are bigger, one source of risk has declined Acemoglu et al. (2012), Carvalho and Gabaix (2013)

Basics Economy is populated by N banks, time t [0, ) is continuous Total assets of each bank given by process a i : M d log a i (t) = µ i (t) dt + δ iz (t) db z (t) z=1 B 1,..., B M are independent Brownian motions (M N) Nonparametric approach with little structure imposed on µi and δ iz More general than previous random growth literature based on equal growth rates and volatilities of Gibrat s Law (Gabaix, 1999, 2009)

Rank-Based Asset Dynamics and Local Times Let a (k) (t) be the total assets of the k-th largest bank: M d log a (k) (t) = µ pt(k)(t) dt + δ pt(k)z(t) db z (t) z=1 + 1 2 dλ log a (k) log a (k+1) (t) 1 2 dλ log a (k 1) log a (k) (t) p t (k) = i when bank i is the k-th largest bank Λ x is the local time at 0 for the process x Measures amount of time x spends near 0 (Karatzas and Shreve, 1991)

Relative Growth Rates and Volatilities M d log a (k) (t) = µ pt(k)(t) dt + δ pt(k)z(t) db z (t) + local time terms z=1 Let α k be the relative growth rate of the k-th largest bank, 1 T ( α k = lim µpt(k)(t) µ(t) ) dt, T T 0 where µ(t) is growth rate of total assets a(t) = a 1 (t) + + a N (t). Let σ k be the volatility of relative asset holdings log θ (k) log θ (k+1), σk 2 = lim 1 T T T 0 M ( δpt(k)z(t) δ ) 2 pt(k+1)z(t) dt. z=1

Reversion Rates and Idiosyncratic Volatilities Refer to α k as reversion rates of asset holdings Equal to minus the growth rate of assets for rank k bank relative to growth rate of total assets of all banks (cross-sectional mean reversion) Regulatory and competition policy, mergers and acquisitions (Kroszner and Strahan, 1999, 2014), and the preferences, constraints, and strategic choices that drive asset growth (Corbae and D Erasmo, 2013) Parameters σ k measure idiosyncratic bank asset volatility Unanticipated changes in liabilities and defaults cause by shocks to borrowers production technology (Corbae and D Erasmo, 2013) One potential source of contagion (Acemoglu et al., 2012)

Theorem (Bank Size Distribution) There is a stable distribution of bank assets in this economy if and only if α 1 + + α k < 0, for k = 1,..., N 1. Furthermore, if there is a stable distribution, then for k = 1,..., N 1, this distribution satisfies [ ] E log ˆθ (k) (t) log ˆθ (k+1) (t) = σ 2 k 4(α 1 + + α k ). Stable distribution entirely determined by two factors 1. Idiosyncratic asset volatilities: σ k 2. Reversion rates of asset holdings: α k Theorem describes behavior of stable versions of asset shares, ˆθ (k)

Theorem (Bank Size Distribution) There is a stable distribution of bank assets in this economy if and only if α 1 + + α k < 0, for k = 1,..., N 1. Furthermore, if there is a stable distribution, then for k = 1,..., N 1, this distribution satisfies [ ] E log ˆθ (k) (t) log ˆθ (k+1) (t) = σ 2 k 4(α 1 + + α k ). Stable distribution entirely determined by two factors 1. Idiosyncratic asset volatilities: σ k 2. Reversion rates of asset holdings: α k Only a change in these factors can alter the distribution Consider a transition from one stable distribution to another

Idiosyncratic Volatility, Reversion Rates, and Concentration Idiosyncratic Volatility σ k Low Asset Concentration Shares of Total Assets θ(k) Low Asset Concentration Sum of Reversion Rates (α 1 + + α k) Rank k [ ] E log ˆθ (k) (t) log ˆθ (k+1) (t) = σ 2 k 4(α 1 + + α k )

Idiosyncratic Volatility, Reversion Rates, and Concentration Idiosyncratic Volatility σ k Medium Asset Concentration Low Asset Concentration Shares of Total Assets θ(k) Medium Asset Concentration Low Asset Concentration Sum of Reversion Rates (α 1 + + α k) Rank k [ ] E log ˆθ (k) (t) log ˆθ (k+1) (t) = σ 2 k 4(α 1 + + α k )

Idiosyncratic Volatility, Reversion Rates, and Concentration High Asset Concentration High Asset Concentration Idiosyncratic Volatility σ k Medium Asset Concentration Low Asset Concentration Shares of Total Assets θ(k) Medium Asset Concentration Low Asset Concentration Sum of Reversion Rates (α 1 + + α k) Rank k [ ] E log ˆθ (k) (t) log ˆθ (k+1) (t) = σ 2 k 4(α 1 + + α k )

Relation to Previous Literature Rank-based, nonparametric approach nests much of previous literature Gibrat s law: Growth rates and volatilities equal for all agents Gabaix (2009) shows that Gibrat s law yields a Pareto distribution Gabaix (1999) shows that Gibrat s law sometimes yields Zipl s law Gibrat s law: α = α 1 = = α N 1 and σ = σ 1 = = σ N 1 This implies that [ E log ˆθ (k) (t) log ˆθ ] (k+1) (t) = σ 2 k 4(α 1 + + α k ) = σ2 4kα

Gibrat s Law and Pareto Distributions Rank-based, nonparametric approach nests much of previous literature Gibrat s law: Growth rates and volatilities equal for all agents Gabaix (2009) shows that Gibrat s law yields a Pareto distribution Gabaix (1999) shows that Gibrat s law sometimes yields Zipl s law Gibrat s law: α = α 1 = = α N 1 and σ = σ 1 = = σ N 1 Log-log plot of shares θ (k) vs. rank k has constant slope (Pareto) [ ] E log ˆθ (k) (t) log ˆθ (k+1) (t) log k log k + 1 = kσ2 4kα = σ2 4α

Gibrat s Law and Zipf s Law Rank-based, nonparametric approach nests much of previous literature Gibrat s law: Growth rates and volatilities equal for all agents Gabaix (2009) shows that Gibrat s law yields a Pareto distribution Gabaix (1999) shows that Gibrat s law sometimes yields Zipl s law Gibrat s law: α = α 1 = = α N 1 and σ = σ 1 = = σ N 1 Log-log plot of shares θ (k) vs. rank k has slope -1 (Zipf s law) [ ] E log ˆθ (k) (t) log ˆθ (k+1) (t) log k log k + 1 = kσ2 4kα = σ2 4α = 1 iff σ2 = 4α

Top k Banks at time t θ (1) (t) θ (2) (t) Top k Banks at time t + 1 θ (1) (t + 1) θ (2) (t + 1).... θ (k) (t) θ (k) (t + 1) θ (k+1) (t) θ (k+1) (t + 1) θ (k+2) (t) θ (k+2) (t + 1).... θ (N) (t) θ (N) (t + 1)

. θ (k) (t)... θ (k) (t + 1)..

. θ pt(k)(t) = θ (k) (t).. θ pt(j)(t) = θ (j) (t).. θ pt(j)(t + 1) = θ (k) (t + 1). θ pt(k)(t + 1)...

Bank Holding Companies Entity Level Data Estimate volatility and reversion rates on bank level data set: Bank-holding companies (1986 2016) FR Y9-C: publically available Quarterly data on total assets 500 largest included BHCs own commercial banks and thrifts

Estimation: Reversion Rates It can be shown that for all k = 1,..., N 1, the reversion rates α k are increasing in the quantity log [ θ pt+1 (1)(t + 1) + + θ pt+1 (k)(t + 1) ] log [ θ pt(1)(t + 1) + + θ pt(k)(t + 1) ]. Reversion rates measure the intensity of mean reversion, since they are increasing in the difference between the time t + 1 assets of the largest banks at t + 1 and the time t + 1 assets of the largest banks at t.

Estimation: Idiosyncratic Volatilities Idiosyncratic volatilities measure variance of relative asset holdings for adjacent ranked banks, log θ (k) log θ (k+1) More general definition than is common in economics and finance Discrete-time approximation yields σ 2 k = 1 T T [( log θpt(k)(t + 1) log θ pt(k+1)(t + 1) ) t=1 ( log θ pt(k)(t) log θ pt(k+1)(t) )] 2

When Did the Bank Size Distribution Start Transitioning? No standard techniques for determining when a transition starts Estimate parameters α k and σ k for different transition start dates Find date that minimizes the distance between predicted shares (before and after the transition) and those observed in the data For each date, smooth estimated parameters αk and σ k to achieve best possible fit

When Did the Bank Size Distribution Start Transitioning? 0.46 Root Mean Squared Error 0.44 0.42 0.40 0.38 1998 Q3 1997 1998 1999 2000 2001 Year

Idiosyncratic Volatilities: Bank-Holding Companies 30 More Volatility 1986 Q2-1998 Q2 1998 Q3-2016 Q3 Sigma (%) 25 20 15 0 100 200 300 400 500 Rank Figure: Standard deviations of idiosyncratic asset volatilities (σ k ) for different ranked BHCs.

Reversion Rates: Bank-Holding Companies 0.5 More Mean Reversion 0.0 Alpha (%) -0.5-1.0-1.5-2.0 1986 Q2-1998 Q2 1998 Q3-2016 Q3 0 100 200 300 400 500 Rank Figure: Minus the reversion rates (α k ) for different ranked BHCs.

Interpreting the Results Idiosyncratic asset volatilities decreased for bank-holding companies Cross-sectional mean reversion also decreased...... and this is why BHC assets still grew more concentrated

Interpreting the Results Why did mean reversion decrease for bank-holding companies? Repeal of Glass-Steagall Act (Lucas, 2013), changes in scale economies (Wheelock & Wilson, 2012), end of inter-state branching restrictions Industry concentration, interlinkages, contagion, and aggregate risk Gabaix (2011), Acemoglu et al. (2012), Caballero and Simsek (2013) Bigger banks are not necessarily riskier banks One source of contagion idiosyncratic risk has diminished, even as another more obvious source concentration has intensified

Thank you.

APPENDIX

How Good is the Fit? Period 1 10 Share of Total Assets (%) 1 0.1 0.01 0.001 Predicted Average for 1986 Q2-1998 Q2 1 2 5 10 20 50 100 200 500 Rank

How Good is the Fit? Period 1 10 Share of Total Assets (%) 1 0.1 0.01 0.001 Predicted for 1986 Q2-1998 Q2 Average for 1986 Q2-1998 Q2 Average for 1998 Q3-2016 Q3 1 2 5 10 20 50 100 200 500 Rank

How Good is the Fit? Period 1 10 Share of Total Assets (%) 1 0.1 0.01 0.001 Predicted using Nonparametric Methods Predicted using Gibrat's Law Average for 1986 Q2-1997 Q4 1 2 5 10 20 50 100 200 500 Rank

How Good is the Fit? Period 2 10 Share of Total Assets (%) 1 0.1 0.01 0.001 Predicted Average for 1998 Q3-2016 Q3 1 2 5 10 20 50 100 200 500 Rank

How Good is the Fit? Period 2 10 Share of Total Assets (%) 1 0.1 0.01 0.001 Predicted for 1998 Q3-2016 Q3 Average for 1998 Q3-2016 Q3 Average for 1986 Q2-1998 Q2 1 2 5 10 20 50 100 200 500 Rank

How Good is the Fit? Period 2 10 Share of Total Assets (%) 1 0.1 0.01 0.001 Predicted using Nonparametric Methods Predicted using Gibrat's Law Average for 1998 Q3-2016 Q3 1 2 5 10 20 50 100 200 500 Rank

Confidence Bands σ s 35 1986 Q2-1998 Q2 1998 Q3-2016 Q3 95% Conf. Intervals 30 Sigma (%) 25 20 15 0 100 200 300 400 500 Rank

Confidence Bands α s 1 0 Alpha (%) -1-2 1986 Q2-1998 Q2 1998 Q3-2016 Q3 95% Conf. Intervals 0 100 200 300 400 500 Rank

Herfindahl Index Over Time 0.10 0.08 Herfindahl Index 0.06 0.04 0.02 0.00 1985 1990 1995 2000 2005 2010 2015 Year

Herfindahl Index Difference Frequency 0 200 400 600 800 1000 1200 1986 Q2-1998 Q2 1998 Q3-2016 Q3 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Herfindahl Index

Herfindahl Index Difference Frequency 0 100 200 300 400 500 600-0.5-0.4-0.3-0.2-0.1 0.0 0.1 0.2 0.3 0.4 Difference in Herfindahl Indexes

Herfindahl Index Difference Frequency 0 200 400 600 800 1000 1986 Q2-1998 Q2 1998 Q3-2016 Q3 10 20 30 40 50 60 70 80 90 100 Share of Assets Held by Top 10 Biggest Banks (%)

Herfindahl Index Difference Frequency 0 100 200 300 400 500 600 80 70 60 50 40 30 20 10 0 10 20 30 40 Difference in Shares of Assets Held by Top 10 Largest Banks (%)

Volatilities Over Time: Bank-Holding Companies 30 1986 Q2-1997 Q4 1998 Q1-2014 Q4 Sigma (%) 25 20 15 More Volatility 0 100 200 300 400 500 Rank

Volatilities Over Time: Bank-Holding Companies 30 Top Third Middle Third Bottom Third Sigma (%) 25 20 15 1990 1994 1998 2002 2006 2010 2014 Year Figure: Ten-quarter moving averages of σ k for different ranked BHCs.

Volatilities Over Time: Commercial Banks 30 1960 Q4-1998 Q1 1998 Q2-2014 Q4 Sigma (%) 25 20 15 More Volatility 0 500 1000 1500 2000 2500 3000 Rank

Volatilities Over Time: Commercial Banks 30 Top Third Middle Third Bottom Third Sigma (%) 25 20 15 1970 1980 1990 2000 2010 Figure: Ten-quarter moving averages of σ k for different ranked commercial banks. Year

Idiosyncratic Volatilities: Beyond Gibrat s Law 30 1986 Q2-1997 Q4 1998 Q1-2014 Q4 Sigma (%) 25 20 15 More Volatility 0 100 200 300 400 500 Rank Figure: Standard deviations of idiosyncratic asset volatilities (σ k ) for different ranked BHCs.

Idiosyncratic Volatilities: Beyond Gibrat s Law 30 1986 Q2-1997 Q4 1998 Q1-2014 Q4 Sigma (%) 25 20 15 More Volatility 0 100 200 300 400 500 Figure: Standard deviations of idiosyncratic asset volatilities (σ k ) when imposing Gibrat s Law. Rank

Bootstrap Resampling Previous figures suggest that at least some of these changes are statistically significant, especially for the volatilities σ k Underlying distribution of parameters α k and σ k is unknown Bootstrap resampling generates confidence intervals and estimates of probability that σ k is smaller in one time period versus another 10,000 replicate samples randomly generated with replacement Confidence intervals based on range of estimates in these resamples How often is σ k in time period one greater than in time period two?

Idiosyncratic Volatilities: Bank-Holding Companies 35 1986 Q2-1997 Q4 95% Conf. Int. 1998 Q1-2014 Q4 30 Sigma (%) 25 20 15 0 100 200 300 400 500 Rank Figure: Standard deviations of idiosyncratic asset volatilities (σ k ) for different ranked BHCs.

Idiosyncratic Volatilities: Bank-Holding Companies 35 1986 Q2-1997 Q4 1998 Q1-2014 Q4 95% Conf. Int. 30 Sigma (%) 25 20 15 0 100 200 300 400 500 Rank Figure: Standard deviations of idiosyncratic asset volatilities (σ k ) for different ranked BHCs.

P-Values 0.20 0.15 Probability 0.10 0.05 0.05 0.00 0.01 1 2 5 10 20 50 100 200 500 Rank Figure: Probability that σ k in time period 1 is greater (less) than or equal to σ k in time period 2 for different ranked U.S. BHCs.

Break Point 0.46 Root Mean Squared Error 0.44 0.42 0.40 0.38 1998 Q3 1997 1998 1999 2000 2001 Year Figure: Likely breakpoint betweene time period 1 and 2 for U.S. BHCs.

Break Point 1 0 Alpha (%) -1-2 1986 Q2-1998 Q2 1998 Q3-2016 Q3 95% Conf. Intervals 0 100 200 300 400 500 Rank Figure: Likely breakpoint betweene time period 1 and 2 for U.S. BHCs.

Break Point 35 1986 Q2-1998 Q2 1998 Q3-2016 Q3 95% Conf. Intervals 30 Sigma (%) 25 20 15 0 100 200 300 400 500 Rank Figure: Likely breakpoint betweene time period 1 and 2 for U.S. BHCs.

The Future of the U.S. Bank Size Distribution Empirical approach allows for an analysis of the future U.S. bank size distribution as well Compare predicted shares to observed shares after transition start date If predicted shares show more concentration than is observed, then transition is likely not complete expect further asset concentration If predicted shares match observed, then transition is likely complete How well do these nonparametric empirical methods match the data? Log-log plots in which straight lines correspond to Pareto distributions

Predicted vs. Data: Bank-Holding Companies 10 Share of Total Assets (%) 1 0.1 0.01 0.001 Predicted Average for 1986 Q2-1997 Q4 Maximum/Minimum for 1986 Q2-1997 Q4 1 2 5 10 20 50 100 200 500 Rank Figure: Shares of total assets held by the 500 largest U.S. BHCs for 1986 Q2-1997 Q4 as compared to the predicted shares.

Predicted vs. Data: Bank-Holding Companies 10 Share of Total Assets (%) 1 0.1 0.01 0.001 Predicted Average for 1998 Q1-2014 Q4 Maximum/Minimum for 1998 Q2-2014 Q4 2014 Q4 1 2 5 10 20 50 100 200 500 Rank Figure: Shares of total assets held by the 500 largest U.S. BHCs for 1998 Q1-2014 Q4 as compared to the predicted shares.

Predicted vs. Data: Beyond Gibrat s Law 10 Share of Total Assets (%) 1 0.1 0.01 0.001 Predicted Average for 1986 Q2-1997 Q4 Maximum/Minimum for 1986 Q2-1997 Q4 1 2 5 10 20 50 100 200 500 Rank Figure: Shares of total assets held by the 500 largest U.S. BHCs for 1986 Q2-1997 Q4 as compared to the predicted shares.

Predicted vs. Data: Beyond Gibrat s Law 10 Share of Total Assets (%) 1 0.1 0.01 0.001 Predicted Average for 1998 Q1-2014 Q4 Maximum/Minimum for 1998 Q2-2014 Q4 2014 Q4 1 2 5 10 20 50 100 200 500 Rank Figure: Shares of total assets held by the 500 largest U.S. BHCs for 1998 Q1-2014 Q4 as compared to the predicted shares.

Extensions and Applications Empirical methods for dynamic power law distributions Model can be applied to power law distributions other than bank size Nonparametric techniques are flexible and robust Some possible applications World income distribution: Are we converging, and if so, to what? Wealth and income City size: Like Gabaix (1999), but with more flexibility Historical bank size distribution (extend data back to 1800s) Firm size