Essays on Money, Banking, and Finance

A dissertation submitted in partial fulfillment of the requirements for the degree ofDoctor of Philosophy at George Mason University

By

Thomas L. HoganMaster of Business Administration

The University of Texas at Austin, 2007Bachelor of Business Administration

The University of Texas at Austin, 2000

Director: Lawrence H. White, ProfessorDepartment of Economics

Spring Semester 2011George Mason University

Fairfax, VA

Copyright c© 2011 by Thomas L. HoganAll Rights Reserved

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Dedication

I dedicate this dissertation to my parents. Thank you for your love, your integrity, and yourunending patience. You made me the man I am today. I love you both.

I would also like to make a special dedication to Douglas B. Rogers. Doug was a greatfriend and innovative economist whose time with us ended far too soon.

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Acknowledgments

I would like to thank my dissertation committee chaired by Lawrence H. White joined byTyler Cowen and Gerald A. Hanweck. Dr. White, thank you for your invaluable guidanceboth through the dissertation process and in my efforts to become a scholar. I aspire toyour example in teaching, writing, and research. Dr. Hanweck, I am privileged to havehad the opportunity to work with you on your research and appreciate your assistance andencouragement in my own research. Dr. Cowen, thank you for your insightful commentson my dissertation and other projects.

While at George Mason, I have had the great privilege to study with several gifted and in-spiring faculty members. Peter J. Boettke eats, sleeps, and breathes economics and infuseshis students with a passion for learning. Peter T. Leeson is a simply brilliant economist anda model that all Mason students would be proud to emulate. My other amazing instructorsinclude Bryan D. Caplan, Garett B. Jones, John V. C. Nye, Richard E. Wagner, and WalterE. Williams whose classes continue to permeate my understanding of economics. Virgil H.Storr, Chris J. Coyne, and Omar Al-Ubaydli greatly helped to further my understandingof economics and my career in the field. Thanks also to Peter Lipsey, Lane Conaway, andMary Jackson who provided invaluable assistance in my dissertation and throughout thePh.D. process.

Outside of George Mason I have been privileged to work with Andy Young of West VirginiaUniversity who has guided me in my writing and research. Thanks to Travis Wisemanof West Virginia University who has been a valuable co-author. Thanks also to professorsMichael W. Brandl, Eric Hirst, Greg F. Hallman, and Stephen P. Magee of the University ofTexas who encouraged me to pursue my dream of becoming a college professor and helpedmake that dream a reality.

Thank you to my friends in the graduate economics program at George Mason University.Daniel J. Smith and Alexander Fink are already great economists and will soon be joinedby Nicholas A. Curott, Stewart Dompe, Tom K. Duncan, David J. Hebert, Kyle Jackson,William Luther, and Nicholas A. Snow. Thanks to the students who preceded me in theGMU graduate economics program and helped guide my way, especially Dan J. D’Amico,Adam G. Martin, Jeremy M. Horpedahl, David Skarbek, Emily C. Skarbek, Adam C. Smith,Diana W. Thomas, Michael D. Thomas, and Tyler Watts.

One name that deserves to be on the list of great economists is Douglas B. Rogers. Dougwas an innovative economist, a gifted writer, an exciting teacher, an amazing athlete, anda loving friend. Doug challenged me, encouraged me, corrected me, and inspired me. Dougis dearly missed and will always be lovingly remembered by his friends, family, and all whoknew him.

For financial support, I would like to thank the Mercatus Center at George Mason Univer-sity, the Richard E. Fox Foundation Fellowship, the Lynde and Harry Bradley Foundation

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Fellowship, the Charles G. Koch Charitable Foundation, and the Institute for HumaneStudies. Their support made this dissertation possible. These papers also benefited fromcomments and criticisms in presentations at the 2011 Association of Private Enterprise Ed-ucation international conference and the GMU Monetary Research Cluster.

Thanks to my friends back home in Texas, across the country, and around the world. Mostof all, let me thank my best friends James Gill, Bill McMains, Phillip Olcese, Matt Steven-son, and Leo Welder. You have each been an inspiration to me. Thank you for putting upwith my stubbornness through all the years and with my lack of communication over therecent few. The many other people deserving of recognition are too numerous to name, solet me simply give one big “Thank you” to all the friends I grew up with in Corpus Christior made along the way at the University of Texas, the McCombs MBA program, GeorgeMason University, and in Vienna, Frankfurt, and London.

My family too deserves thanks and praise. My father Roland Hogan has always been myhero and a model for me in my hopes, goals, and dreams. My mother Kathy Hogan keptme grounded so that my goals and dreams could actually be accomplished. My sister LauraBrady has, despite my efforts to hide it, always been a role model to me in her academic,professional, and personal endeavors. My brother Kevin Hogan, who is now pursuing doc-torate of his own, is smarter than he knows and more creative and kind than I could hopeto be. Thanks to my bother-in-law Corbin who I feel is like a brother to me. Thanks tomy loving and wonderful grandmother Margaret Greene and her children, grandchildren,sisters, and cousins. I must fail here to thank my extended family by name since doingso would require adding a further appendix to this paper, but I love you all and appreci-ate your help and support. However, I would like to specifically thank my uncles ThomasGreene, Mark Hulings, and John Greene who kept me interested in business, politics, andeconomics and who in many ways inspired me to pursue my Ph.D. in economics.

My greatest thanks go to Jesica Tomlinson. Thank you, Jesica. You most of all deserve myappreciation. You most of all bore the burden of my overloaded schedule and frantic lifestyleover the past two years. You put up with my stressed-out phases and absent-minded antics.I appreciate everything you have given me and all you have done for me. I love you so much.

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Table of Contents

Page

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 Gresham’s Law Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Gresham’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 English Coinage, 1344-1815 . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Stability and Exchange in a Generalized Diamond-Dybvig Model . . . . . . . . . 24

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2 The Diamond-Dybvig Model . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.1 The All-Production Economy . . . . . . . . . . . . . . . . . . . . . . 26

3.2.2 Optimal Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2.3 The All-Banking Economy . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.4 Bank Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.5 Actual Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2.6 Do Banks Exist? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 The Multi-Sector Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3.1 Welfare in the Multi-Sector Model . . . . . . . . . . . . . . . . . . . 36

3.3.2 Is Trade Allowed? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3.3 Bank Runs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.4 Multi-Sector Consumption . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Historical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4.1 Trading at a Discount . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4.2 Fundamentals or Sunspots? . . . . . . . . . . . . . . . . . . . . . . . 45

3.4.3 Information and Exchange . . . . . . . . . . . . . . . . . . . . . . . . 46

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Capital and Risk in Commercial Banking . . . . . . . . . . . . . . . . . . . . . . 48

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2 Capital and Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Empirical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3.1 Equity Price Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . 51

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4.3.2 Bond Yield Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3.3 Regression Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.5.1 Absolute Value of Stock Returns . . . . . . . . . . . . . . . . . . . . 58

4.5.2 Standard Deviation of Stock Returns . . . . . . . . . . . . . . . . . . 58

4.5.3 Bond Yield Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

A Gresham’s Law Alternative Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

B The Generalized Diamond-Dybvig Model . . . . . . . . . . . . . . . . . . . . . . 63

B.1 Optimal Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

B.2 Actual Consumption in the All-Bank Economy . . . . . . . . . . . . . . . . 66

B.3 Welfare in the Multi-Sector Economy . . . . . . . . . . . . . . . . . . . . . . 70

B.4 Consumption in the Multi-Sector Economy . . . . . . . . . . . . . . . . . . 72

C Leverage and Capital Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

C.1 Federal Reserve Y9-C Regulatory Capital Schedule . . . . . . . . . . . . . . 75

C.2 Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

C.3 Regression Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

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List of Figures

Figure Page

2.1 Short and long-term effects of a price ceiling. . . . . . . . . . . . . . . . . . 5

2.2 Legal tender laws as a price floor or ceiling. . . . . . . . . . . . . . . . . . . 6

2.3 Price ratio of gold to silver, 1344-1815. . . . . . . . . . . . . . . . . . . . . . 13

2.4 Coinage of gold vs. silver when gold is overvalued, 1344-1815. . . . . . . . . 13

2.5 Regression of gold coinage on the overvaluation of gold. . . . . . . . . . . . 14

2.6 Coinage regression on overvaluation and dummy variables. . . . . . . . . . . 16

3.1 Consumption in the all-production economy. . . . . . . . . . . . . . . . . . . 28

3.2 Value of deposits at T=1 in the all-bank model. . . . . . . . . . . . . . . . . 31

3.3 Consumption in the all-bank model. . . . . . . . . . . . . . . . . . . . . . . 34

3.4 Probability-weighted expected value of social welfare. . . . . . . . . . . . . . 37

3.5 Value of deposits at T=1 in the multi-sector model. . . . . . . . . . . . . . 40

3.6 Consumption in the multi-sector model. . . . . . . . . . . . . . . . . . . . . 43

4.1 Absolute value of stock returns . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.2 Standard deviation of stock returns . . . . . . . . . . . . . . . . . . . . . . . 59

4.3 Bond yield spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

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Abstract

ESSAYS ON MONEY, BANKING, AND FINANCE

Thomas L. Hogan, PhD

George Mason University, 2011

Dissertation Director: Lawrence H. White

This dissertation will address three significant topics in money, banking, and finance.

The first chapter contributes to the current debate over is Gresham’s Law. This “law” had

long been considered a basic principle of monetary economics, yet over the last few decades

it has become commonly criticized as a failure. I clarify the theory of Gresham’s Law as a

description of price controls on the exchange of currencies and provide historical evidence

of the influence of Gresham’s Law on English coinage from 1344 to 1815.

The second chapter presents a generalized version of the Diamond-Dybvig model. The

recent literature on bank runs, following Diamond and Dybvig (1983), studies the banking

sector in isolation from the greater economy. Here I model an economy that includes not only

DD type bank depositors but also producers of goods. When consumers can exchange goods

for deposits, trade provides a welfare improvement, and bank runs are not an equilibrium

unless the bank is fundamentally insolvent.

The final chapter examines the influences of capital and risk-based capital (RBC) on

the stock prices and bond yield spreads of US bank holding companies from 2000 to 2010.

Both capital and RBC are significantly related to these risk indicators in several quarters.

However, there does not appear to be a significant difference between the influences of

capital and RBC in any quarter, indicating that RBC does not improve upon the standard

capital ratio.

Chapter 1: Introduction

This dissertation will address three significant topics in money, banking, and finance. Sev-

eral contemporary works have revived the debate over Gresham’s Law. The next chapter

on Gresham’s Law seeks to quell that discussion. Chapter 3 addresses the Diamond-Dybvig

model of deposit banking. I propose a previously unknown equilibrium in the famous model

of Diamond and Dybvig (1983). The final chapter analyzes the influence of capital and risk-

based capital (RBC) on perceptions of bank risk. I find that bank risk is related to RBC

but that RBC provides no new information compared to the standard capital ratio.

In chapter 2, I address the recent debate over Gresham’s Law. Gresham’s Law states

that bad (legally overvalued) money drives out good (legally undervalued) money. This

“law” had long been considered a basic principle of monetary economics, yet over the last

few decades it has become commonly criticized as a failure. This paper clarifies the theory of

Gresham’s Law as a description of price controls on the exchange of currencies. It provides

historical evidence of the influence of Gresham’s Law on English coinage from 1344 to 1815.

Chapter 3 presents a generalized version of the Diamond-Dybvig model of banking. The

recent literature on bank runs, following Diamond and Dybvig (1983), studies the banking

sector in isolation from the greater economy. Here I model an economy that includes not only

DD type bank depositors but also producers of goods. When consumers can exchange goods

for deposits, trade provides a welfare improvement, and bank runs are not an equilibrium

unless the bank is fundamentally insolvent. Instability is, thus, not inherent to banking but

results from restrictions on information and exchange. I show that this is consistent with

historical evidence from the US banking system.

Chapter 4 examines RBC as a measure of bank risk. Recent changes in US banking

regulation have emphasized RBC as a buffer against bank insolvency. This paper compares

RBC to the standard capital ratio of equity over assets. For each quarter from 2000 to

2010, I regress both capital and RBC against three indicators of risk: the absolute value

of stock returns, standard deviation of stock returns, and bond yield spreads. Both capital

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and RBC are significantly related to these indicators in several quarters. However, I am

unable to find a significant difference between the influences of capital and RBC in any

quarter, indicating that RBC does not improve upon the standard capital ratio.

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Chapter 2: Gresham’s Law Revisited

2.1 Introduction

Gresham’s Law was once considered a “verified fact” of monetary economics (Fisher 1894,

p.527 n.2). Yet over the last few decades, this “law” has come under dispute. This paper

seeks to clarify Gresham’s Law and respond to its critics. It provides evidence of the

influence of Gresham’s Law during England’s bimetallic period from 1344 to 1815.

A number of contemporary economists dispute Gresham’s Law. Rolnick and Weber

(1986) declares Gresham’s Law a failure and a “fallacy.” Velde, Weber, and Wright (1999,

p.1) states that “Despite it being one of the most generally accepted and frequently cited

propositions in economics, we think that existing theoretical analyses of Gresham’s Law

are lacking” and “its empirical validity is questionable, or at least seems to depend on

circumstances.” Sargent and Smith (1997, p.199), Velde and Weber (2000), and Li (2002) all

contend that Gresham’s Law sometimes holds but other times fails. Even Friedman (1990b,

p.1162 n.4) confesses that “For precision, the “law” must be stated far more specifically, as

Rolnick and Weber (1986) point out.” This paper attempts to restore Gresham’s Law by

providing the specificity which Friedman requested.

After clarifying the theory behind Gresham’s Law, its empirical validity is tested. The

history of English coinage provides strong evidence in favor of Gresham’s Law. From 1344

to 1815 the Royal Mint in London minted currencies from both silver and gold.1 The

metal content of each currency when combined with the legal tender rate that relates the

currencies creates an implied price ratio of silver to gold. Gresham’s Law predicts that

when the implied ratio between the prices of gold and silver in England overvalues gold

relative to international prices, then gold will be imported and minted while silver will be

exported. When silver is overvalued, the opposite occurs. Regression analysis on English

1Although England left the bimetallic standard in 1815 and moved to a mono-metallic gold standard,both gold and silver continued to be coined until 1917 (Craig 1953, p.422).

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coinage shows that overvaluation significantly influence mint production at the 1% level.

This paper contends that Gresham’s Law is both logically sound and empirically ob-

servable. The next section outlines the theory of Gresham’s Law and the claims against it.

Section 2.3 describes the effects of Gresham’s Law with the historical example of English

coinage from 1344 to 1815 and verifies the strength of this relationship using regression

analysis. Section 2.4 concludes.

2.2 Gresham’s Law

Gresham’s Law is often summarized as the tendency for bad (legally overvalued) money to

drive good (legally undervalued) money out of circulation. When two monies have different

intrinsic or international values but may only be exchanged at equal value, traders benefit

by paying out the less valuable coins and hoarding, melting, or exporting the more valuable

ones. Over time, more of the good money leaves circulation until only the bad money

remains in use.

Hayek concisely explains, “The essential condition for Gresham’s Law to operate is that

there must be two (or more) kinds of money which are of equivalent value for some purposes

and of different value for others” (Hayek 1967, p.318). A disparity between prices of the

two monies persists only when subject to some non-market restriction such as a legal tender

law. As White (2000, p.xxx ) describes, “When a legal tender law sets a fixed exchange rate

between two monies of unequal market value, the legally overvalued currency drives the

legally undervalued currency out of circulation.”

A legal tender law is a price control on the exchange of currency. The law may be

imposed as either a price ceiling or a floor or both. For example, in 1696 the local exchange

rate in England of silver shillings to gold guineas shot up from 22 to 30. The Crown

responded by instituting a ceiling price of 22 shillings per gold guinea in all market exchange

of shillings and guineas (Jenkinson 1805, p.161-162). What would an economist predict as

the result of this imposition?

Figure 1 illustrates the price ceiling and its effects. The price is drawn as gold in terms

of silver, and the quantity is the amount of gold exchanged. If suppliers can only sell

gold guineas at the price of 22 shillings or below, then the quantity of gold supplied will

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(a) Short-term shortage. (b) Long-term adjustment.

Figure 2.1: Short and long-term effects of a price ceiling.

fall. Additionally, the suppliers of silver (demanders of gold) will find the price to their

advantage and the quantity of silver supplied will rise (and the quantity of gold demanded).

The short-term effect will be a shortage of gold and a surplus of silver as in figure 1.a. But

if the international price ratio of gold to silver is above the price ceiling, then traders can

benefit by importing gold and exporting silver. The long-term effect will be an increase in

the supply of gold and a decrease in the supply of silver as in figure 1.b. These predictable

results are exactly what occurred following the imposition of the English price ceiling in

1696 (Fay 1935; Breckinridge 1969[1903], p.45).

Suppose instead that the legal tender law had been created as a price floor on the price

of gold. Then when the international price fell below the price floor, silver would have been

imported and coined and gold would have been melted and exported. Since a price floor on

gold is equivalent to a price ceiling on silver, any price control generally creates a shortage

of one metal currency and a surplus of the other. The relative magnitudes of these effects

will be determined by the elasticities of each currency.

When the government dictates that currencies may only be exchanged at the legal rate,

the ratio acts as both a ceiling and a floor. The market price of the currencies will converge

to the legal rate and create a shortage or surplus in the quantity. Figure 2.2 illustrates

the effects of price controls on the supply and demand of silver and gold currencies. When

the market rate of gold rises above the legal rate as in figure 2.2.a, the legal rate acts as a

price ceiling. To return the market to the legal rate, the supply of gold must increase, or

the supply of silver must decrease, or both. When the market rate of gold falls below the

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(a) Price ceiling. (b) Price floor.

Figure 2.2: Legal tender laws as a price floor or ceiling.

legal rate as in figure 2.2.b, the legal rate acts as a price floor. To return the market to the

legal rate, the supply of silver must increase, or the supply of gold must decrease, or both.

This is the essence of Gresham’s Law, that a legal tender law will act as a price control and

create a surplus or shortage.

When legal tender laws equate coins of two different metals such as gold and silver,

Gresham’s Law takes the form of arbitrage through the import and export of precious

metal. If the local exchange rate between gold and silver deviates from the international

rate, arbitragers can profit by importing the overvalued metal and minting it into coins while

melting and exporting coins made from the undervalued currency. Thus, bad (overvalued)

money drives out good (undervalued) money as described by Gresham’s Law. Alternatively,

legal tender laws can also create a fixed rate of exchange between two currencies of the same

metal, such as when a government debases a currency by lowering its weight. In this case, the

more valuable, full-bodied coins are usually hoarded while the less valuable, underweight

coins are passed on through exchange. Again, bad (overvalued) money drives out good

(undervalued) money as described by Gresham’s Law.2

The formulation of Gresham’s Law in terms of price controls on currency is consis-

tent with historical descriptions. Thomas Gresham was the founder of the London Royal

Exchange and opposed its regulation (Fetter 1932, p.481-483). Buckley (1924, p.593-4)

describes Gresham’s opinion that “The Rate of Exchange... is fixed on the market by the

plenty of Deliverers and Takers (i.e. by Supply and Demand).” When it was suggested that

2This paper deals mainly with the operation of Gresham’s Law through international channels. Forexamples of hoarding, see Craig (1946, p.7) and Li (1963, p.47), and Kelly (1991, p.45).

6

the Crown might stabilize English currency by regulating or even closing the market for

currency exchange, “Sir Thomas Gresham... discredited all idea of direct interference with

the exchange.” Mcleod (1855) was the first to interpret Gresham’s ideas as a singular law.3

’When two sorts of coin are current in the same nation of like value by denom-

ination, but not intrinsically, that which has the least value will be current,

and the other as much as possible will be hoarded,’ or exported, we may add.

(Mcleod (1903[1855], pp.119-120) quoted in Fetter (1932, p.488))

Through the twentieth century, economists described the shortage (Fetter 1933, p.821) or

surplus (Giffen 1891, p.304) created whenever the implied ratio of gold to silver in England

was over- or undervalued relative to international rates. These descriptions are consistent

with more modern terminologies of price ceiling and price floor.

Gresham’s Law has long been considered a true and fundamental law of monetary eco-

nomics. Academic works on Gresham’s Law include older articles such as Giffen (1891),

Fisher (1894), Daniels (1895), Buckley(1924), and Fetter (1932, 1933). Contemporary works

such as Selgin (1996) and Oppers (1996, 2000) buttress the theoretical foundations of Gre-

sham’s Law, while others such as Greenfield and Rockoff (1995), Mundell (1998), and Fried-

man (1990b) provide empirical support. Friedman (1990a) and Flandreau (2002) describe

how bad money drives out good without specific reference to Gresham. Historical works

which refer to Gresham’s Law including Giffen (1891), Farrer (1968[1898]), Spooner (1972),

Friedman and Schwartz (1963), Hayek (1967, 1976), Galbraith (1975), and Selgin (2008).

Yet despite the ample historical evidence, many contemporary works proclaim the failure

of Gresham’s Law. These criticisms fall into three categories. Most common is the empirical

conjecture that Gresham’s Law sometimes works and sometimes fails. Second is the claim

that legal tender laws were often unenforceable, so Gresham’s Law could not have operated.

As will be discussed, these first two critiques are compatible with the characterization of

Gresham’s Law describing a price control on currency exchange. Last, a few critics make

the extreme case that legal tender laws have no effect on commodity money, and therefore,

Gresham’s Law always fails. It is only this final claim, that Gresham’s Law always fails,

3Giffen (1891) and (Daniels 1895) challenge whether this phrasing is consistent Thomas Gresham’s orig-inal description.

7

which this paper seeks to rebut.

Several works dispute Gresham’s Law by providing empirical cases where the law appears

to have failed. Rolnick and Weber (1986, p.193) provides examples of times when bad

money did not appear to drive out good money.4 Sargent and Smith (1997) creates a

theoretical model of commodity money and cites empirical examples from Cipolla (1956,

p.17) and Rolnick and Weber (1986) of times when the Gresham’s Law appears to have

failed. Li (2002) describes theoretical cases which “violate Gresham’s Law,” again referring

to Rolnick and Weber (1986) for specific examples. Redish (2000) examines Gresham’s Law

but finds the evidence ambiguous.

However, these historical cases are not inconsistent with Gresham’s Law as a description

of price controls. As previously described, Gresham’s Law simply asserts that a price ceiling

on the exchange of currencies creates a shortage of the undervalued currency. A case where

bad money fails to drive out the good should hardly be cited as a failure of Gresham’s

Law. It is simply an instance in which the legal tender rate was not a binding constraint,

and therefore, Gresham’s Law did not apply. If some authors chose to describe the non-

application of a law as a “failure” of that law, then the distinction is purely semantic.

Other critics of Gresham’s Law accept the influence of legal tender laws but argue

that such laws were not enforceable. Velde, Weber, and Wright (1999, p.293) cite Miskimin

(1963, p.84) that “[Gresham’s Law] assumes that the government possesses enough political

force to insist upon the legal tender value of the coinage and to decree circulation at par.

There is, however, substantial evidence that neither the French nor the English monarchies

gained this power until the end of the middle age”. Sargent and Velde (2002, p.31) claims

that “During shortages, the laws were often disobeyed.”

Clearly the critics are correct that enforcement of legal tender restrictions was not

consistent through time. During some periods of English monetary history the exchange

rates of gold and silver currencies floated freely in the market (Jenkinson 1805, p.180),

while at other times the Crown attempted to control the rate by assigning values for tax

collection (Craig 1953, p.185) or enforcing a price ceiling or floor on the exchange of currency

(Jenkinson 1805, p.161-162; Shirras and Craig 1946, p.228). Yet, this line of argument

4Most of these examples have since been disputed by Greenfield and Rockoff (1997).

8

confuses the causes and effects of Gresham’s Law. If legal tender laws fail to create a

shortage of currency, then one possibility is that the laws were not effectively enforced.

However, if coinage of a currency is found to be dependent on its legal tender laws despite

the fact that legal tender restrictions were not fully enforced, then the evidence is in favor

of Gresham’s Law, not against it.

A few works go so far as to deny the influence of legal tender laws entirely. Rolnick

and Weber (1986, p.193) states that “legal tender laws provide no reason for good money

to disappear from circulation.” The authors “propose a more feasible qualification to the

popular version of Gresham’s Law, one that depends on fixed transaction costs rather than

a fixed rate of exchange.” They find that “History seems to support our new version of

Gresham’s Law” (p.186). Sargent and Velde (1999, 2002) proposes another transaction cost

model intended to solve the “big problem of small change.” The authors dismiss Gresham’s

Law, asserting that “In our model, there is no need to appeal to that ambiguous law”

(Sargent and Velde 2002, p.31). Neither of the models from Rolnick and Weber (1986) and

Sargent and Velde (1999, 2002) contain reference to the legal tender rates since they assume

that transaction costs will trump the influence of Gresham’s Law.

Yet the success of these transaction cost models do not preclude the operation of Gre-

sham’s Law. Selgin (1996) proposes that “Rolnick and Weber’s Law” is a complement to

Gresham’s Law rather than a substitute. The same can be said of “Sargent and Velde’s

Law” of small change. Evidence in favor of these models is not evidence against Gresham’s

Law since both forces may have operated simultaneously. For this reason, the effects of

Gresham’s must be tested independently. The next section analyzes the history of En-

glish coinage, a period discussed by both Rolnick and Weber (1986) and Sargent and Velde

(2002). This analysis indicates that coin production is indeed influenced by the legal ratio

as predicted by Gresham’s Law.5

5This evidence does not contradict the models of Rolnick and Weber (1986) and Sargent and Velde (1999,

2002) since the phenomena they describe are not mutually exclusive with Gresham’s Law. Indeed, it seemslikely that both price controls and transaction costs affected the rates of gold and silver coinage.

9

2.3 English Coinage, 1344-1815

This section analyzes influence of Gresham’s Law on the history of English coinage. Gre-

sham’s Law predicts that the government’s assignment of the relative values of the currencies

(which money is good and which is bad) will determine which is minted and which is driven

out. Therefore, the implied price ratio in England relative to the world price ratio should

influence the amounts of gold and silver that are coined each year. This prediction will be

tested using historical data on English and international prices and coinage records from

the Royal Mint in London.

The history of English coinage is replete with examples of Gresham’s Law. Since ancient

times, England employed the silver penny or shilling as the medium of exchange.6 In 1344

the mint began producing gold coins in addition to silver, and England employed a bimetallic

system until 1816.7 It was during this bimetallic period that Gresham’s Law had its greatest

effect.

Throughout the bimetallic period, England experienced successive waves of change in

its monetary base which oscillated between mostly gold and mostly silver. The Crown and

agents of the mint attempted to stabilize the monetary base by legally assigning the relative

values of gold and silver currencies. However, monetary authorities of the time were often

unaware that the stability of English currency was highly dependent on the price ratios of

gold and silver in Europe and the rest of the world (Shaw 1967[1896], pp.48-49; Sargent

and Velde 2002, p.5). When the legal values of gold and silver coins implied a price ratio

between gold and silver that overvalued gold, gold was imported and coined while silver was

exported. When the implied price ratio overvalued silver, silver was imported and coined

while gold was exported.

The implied price ratio of silver to gold in England can be calculated from the legal

tender ratio and the amount of precious metal used to produce each coin. For example, in

1699 the Royal Mint in London specified the guinea’s weight at 44.5 guineas per pound of

6The shilling was “First issued in 1504,” and was “Always proportional to the penny except during thedebasement period” (Feavearyear 1931, p.349).

7In 1816 England moved to a mono-metallic gold standard, but coins of both metals continued to beminted. The nation reverted to a silver standard during the first World War, and minting of gold coinsceased in 1917 (Challis 1992, p.557).

10

gold and coined shillings at 62 shillings per pound of silver.8 While Master of the Royal

Mint, Sir Isaac Newton described how the legal tender rate of 21.5 shillings per guinea

created an implied value of just over 15.5 pounds of silver per troy pound of gold:

A pound weight Troy of Gold, eleven ounces fine & one ounce allay, is cut into

4412 Guineas, & a pound weight of silver, 11 ounces, 2 pennyweight fine, &

eighteen pennyweight allay is cut into 62 shillings, & according to this rate, a

pound weight of fine gold is worth fifteen pounds weight six ounces seventeen

pennyweight & five grains of fine silver, recconing a Guinea at 1£, 1s. 6d. in

silver money. (Newton 1717, p.166)

Newton’s calculation of 15 pounds 6 ounces 17 pennyweight and 5 grains of silver per

pound of gold is a ratio of 15.58 in decimal terms. An international trader with access to

prices above this ratio could earn an arbitrage profit by importing and coining silver while

melting and exporting gold. If international prices fell below the implied ratio, the trader

could profit by importing and coining gold and exporting silver.9

An ideal test of Gresham’s Law would be to analyze the stocks of gold and silver currency

in England to see if they behave as predicted when the implied English ratio of gold to silver

differs from international rates. Although the quantities of currency in circulation are almost

impossible to measure, comprehensive records on the annual production of gold and silver

coins are available from the Royal Mint. If Gresham’s Law is correct, then coinage rates

of gold and silver should be largely determined by the international price ratio of gold and

silver. This hypothesis will be tested using English mint production from 1344 to 1815.10

Data for this analysis will be taken from 4 sources. The implied price ratio of gold to

silver in England is calculated from the official coinage weights and legal tender ratios in

Redish (2000, p.89-92). For the international price ratio of gold to silver, data for years

8English coinage was denominated in terms of pounds sterling (£), shillings (s), and pennies or pence

(d) where £1=20s and 1s=12d.9For an example, consider the market prices as of January 1697 of £4, 1.6s. per troy pound of gold, 5s.

2d. per pound of silver, and 22s. per guinea (Li 1963, p.10). Suppose a merchant begins with 100 shillingsand can exchange pure gold and silver at an international rate of 15 to 1. The merchant trades his shillingsfor 1.61 troy pounds of silver, then exchanges them for 0.11 troy pounds of gold, and then takes his gold tothe mint where it is coined into 4.78 guineas. He can now trade his guineas for 102.88 shillings.

10Another possibile approach would be to analyze the imports, exports, and coinage of each metal. Un-fortunately, records on the import and export of gold and silver are sparse and unreliable.

11

1344 to 1499 are taken as the European average from Shaw (1967[1896], p.40), and years

1500 to 1815 are from Farrer (1898, Appendix II). Mint production of gold and silver coins

is taken from Craig (1953, pp.408-422). These data sets can be found in tables 1 through

4 respectively.

Two measures of mint production will be used for the dependent variable. Since the

dependent variable is meant to capture the amount of mint production of gold coins relative

to silver, each measure is based on the annual percentage of gold coins produced out of the

total coinage of gold and silver in that year. Equation 2.3.1 specifies this relationship where

γt is the percentage of gold coined in year t calculated form annual gold coinage gt (by tale

value) and annual silver coinage st (by tale value).

γt = gt/(gt + st) (2.3.1)

For the second measure, the dependent variable is calculated as a lagging five-year moving

average γa as calculated in equation 2.3.2.

γt =

( t+4∑i=t

gi

)/

( t+4∑i=t

(gi + si)

)(2.3.2)

The lag is intended to account for the time required for any changes in the ratio to take

effect. The purchase and transportation of foreign metals was not instantaneous, and the

minting process alone often required several months.

Figure 2.3 shows the price ratios of gold to silver over the entire period. The solid line

represents the world price. The dotted line represents the English price ratio implied by

mint weights and the legal tender ratio. The background is shaded gray in time periods

when the English implied price ratio exceeds the international ratio. This indicates that

gold is likely to be imported and minted according to Gresham’s Law.

Figure 2.4 shows the relative annual amounts of gold and silver coinage. The solid line

is a five-year moving average of annual gold coinage as a percentage of total coinage. The

shaded areas, taken from figure 2.3, indicate that the English ratio exceeds the international

ratio, so gold was overvalued in England compared to international standards. Gresham’s

12

Figure 2.3: Price ratio of gold to silver, 1344-1815.

Law predicts that when the English ratio exceeds the international ratio, gold will be

imported and minted. Silver will be minted when the English ratio is below the international

ratio. Figure 2.4 shows this correspondence to be generally accurate.

The influence of Gresham’s Law on English coinage will be tested with regression anal-

ysis. The dependent variable is the annual percentage of gold coinage as described in

equations 2.3.1 and 2.3.2. The annual over- or undervaluation in England is calculated as δt

in equation 2.3.3 where Et is the gold to silver ratio in England, and Wt is the international

ratio.

δt = Et −Wt (2.3.3)

Therefore, the regression equation is given in terms of γt and δt.

γt = α+ βδt + ε (2.3.4)

Figure 2.4: Coinage of gold vs. silver when gold is overvalued, 1344-1815.

13

The regression results are shown in figure 2.5. The first column shows that the difference

between the local and international ratios of gold to silver does significantly affect annual

mint production at a significance level of 1% with an adjusted R-squared of 6.2%. The

second column shows the results for the lagging five-year average. The result is again

significant at the 1% level but with an adjusted R-squared of 15.7%. Since the regression uses

a five-year average, the first 4 years were excluded leaving the sample with 467 obseervations

from 1344 to 1811.

Annual coinage Five-year average

Overvaluation of gold 0.084∗∗∗ 0.659∗∗∗

(0.015) (0.013)

Constant 0.131∗∗∗ 0.668∗∗∗

(0.014) (0.012)

Observations 471 467Adj. R-squared 6.2% 15.7%

* Statistically significant the 1% level.

Figure 2.5: Regression of gold coinage on the overvaluation of gold.

The results of the regression analysis show that even a basic model can detect the effects

of Gresham’s Law. Of course, there are many factors omitted from the model which might

influence mint production. Most of these effects would either decrease the actual effect of

Gresham’s Law or make its influence less detectable in the data. If accounted for, they

would likely increase the significance of the regression.

For example, the model does not consider the transportation costs of importing and

exporting precious metal. These costs slowed the transmission of bullion between countries

and dulled the “knife’s edge” created by Gresham’s Law as described by Friedman (1990a)

and formalized by Flandreau (2002). Quinn (1996, p.475) uses an estimated transportation

cost of 3% based on Jones (1988, p.123). Transaction costs can be included by adding a

dummy variable with a value of 1 in any years where the difference between the implied

price ratio in England and the international price ratio differ by less than 1.5% indicating

that arbitrage may not have been profitable during these periods. The dummy has a value

of 1 in 85 of the 472 years.

14

Several historical events affected the normal process of mint production. For example,

there were several times when the English government interfered with minting such as the

recoinages of 1414-1417, 1559, and 1696-99 (Shaw 1967[1896], pp.55,129,225). Additionally,

the implied market ratio is based on the official mint weight of each coin and does not

account for underweight coins being traded in the market. Carlile (1901, p.14) describes

how Gresham’s Law may not operate when overvalued coins are heavily worn or depreciated.

There were clearly period of English history when clipping became so bad that they inhibited

local trade and therefore could not have feasibly been exported for profit. These include the

crises of the 1540’s (Feavearyear 1931, p.56) and 1694-1696 (Craig 1953, p.184). Separate

dummy variables are added to the regression for periods of forced recoinage and depreciated

coinage.

The results of regressions which include these dummy variables are presented in figure

2.6. Accounting for transaction costs increases the adjusted R-squared to 17.2% at a sig-

nificance level of 1%. Adjusting for forced recoinages and periods of depreciated coinage

increases the adjusted R-squared to 16.2% and maintains a significance level of 1%. Us-

ing all three dummies for transaction costs, recoinage, and underweight coins increases the

adjusted R-squared to 18.6% at the same level of significance.11

As with most empirical analysis, there are limitations to the data sets used for this

examination. As noted earlier, the world price of gold is a combined price series from Shaw

(1967[1896]) and Farrer (1898). Prices from Shaw are the simple average of mint prices from

France, Germany, Spain, and the Netherlands. Farrer is a single series but does not begin

until 1500. These data sets were combined for the main regressions but were also tested

separately. The results of are given in Appendix A. The implied English price ratio from

Reddish (2000) is also imperfect since it contains minting fees for both silver and gold. A

more accurate measure would include minting fees for either gold or silver but not both. As

an alternative, the implied ratio is calculated from Feavearyear (1931, pp.348-349) which

does not include minting fees. Results from the regressions on all data sets are given in

Appendix A. The overvaluation of gold was found to significantly influence mint production

at the 1% level in 21 of the 24 variatations.

11The dummy for underweignt coins is not significant in the regressions shown in figure 2.6 but is significantin many regressions shown in Appendix A.

15

(1) (2) (3)

Overvaluation of gold 0.125∗∗∗ 0.131∗∗∗ 0.124∗∗∗

Standard error (0.014) (0.014) (0.014)

Transportation cost dummy -0.092∗∗∗ -0.091∗∗∗

Standard error (0.029) (0.029)

Recoinage dummy -0.196∗∗ -0.193∗∗

Standard error (0.007) (0.081)

Underweight dummy 0.007 0.004Standard error (0.055) (0.054)

Constant 0.687∗∗∗ 0.671∗∗∗ 0.69∗∗∗

Standard error (0.013) (0.012) (0.014)

Observations 467 467 467Adj. R-squared 17.2% 16.2% 18.6%

*** Statistically significant the 1% level.

** Statistically significant the 5% level.

Figure 2.6: Coinage regression on overvaluation and dummy variables.

One last worry common to all regression analysis is the problem of endogeneity. There

is little room for reverse causation in the English price ratio since it is chosen by the King

and Council and only changed 16 times over the 472-year sample. There is some possibility

of feedback effects between mint production and the international price ratio. For example,

increases production of gold indicates that silver is likely to be exported. If exports were

large enough, they could increase the supply of silver outside of England and reduce the

international price ratio. However, this effect would reduce the spread between the implied

English ratio and the world ratio and thereby diminish the influence of Gresham’s Law.

The fact that Gresham’s Law is still observable and significant indicates that endogeneity

is not a problem.

These many short-comings of the data and model inhibit the detection Gresham’s Law.

Yet despite these mitigating factors, the difference between the implied price ratio in Eng-

land and the international price is consistently shown to significantly influence the mint’s

production of gold coins. These results indicate that Gresham’s Law did have a significant

effect on English coinage. The empirical evidence from the period 1344 to 1815 suggests

16

that bad money does indeed drive out good money.

2.4 Conclusion

This paper argues that Gresham’s Law is both logically sound and empirically valid. Gre-

sham’s Law that bad (legally overvalued) money drives out good (legally undervalued)

money can be understood as a price control on the exchange of currency. Historical evidence

from England’s experience with bimetallism from 1344 to 1815 testifies to the significant

influence of Gresham’s Law.

Gresham’s Law simply states that a binding price ceiling on the exchange of currencies

will lead to a shortage. Yet when a price ceiling does not result in a shortage, economists do

not describe the laws of economics as having “failed.” Rather, they seek out explanations

for why the ceiling was not binding. The same must be true of Gresham’s Law. Cases

where the law appears to have failed are simply situations to which the law did not actually

apply, often due to the absence of legal tender laws or lack of their enforcement. However,

the absence of legal tender enforcement does not invalidate Gresham’s Law. Evidence from

English monetary history demonstrates that Gresham’s Law can be effective even when

legal tender restrictions were not always fully enforced.

The past few decades have brought major advances in monetary theory. New models

abound of search theories and transaction costs. Yet evidence in favor of these models

does not negate existing evidence for the old, and while aspiring to more complex theories,

economists must take care not to overlook the simple ones. Despite these new and interesting

models, basic relative price economics should not be forgotten. Economists cannot dismiss

the simple reasoning and overwhelming evidence of Gresham’s Law.

17

Table 1. Implied Price Ratio of Gold to Silver in England, 1343-1815.

Year Gold Silver Implied Ratio Year Gold Silver Implied Ratio1343 14.76 1.23 12.02 1545 32.58 2.8 11.641344 13.6 1.21 11.21 1546 35.84 2.82 12.731345 13.6 1.22 11.12 1547 34.2 3 11.421346 14.31 1.23 11.6 1549 35.46 3.4 10.431349 14.39 1.23 11.67 1550 36.96 3.4 10.871351 15.58 1.37 11.35 1551 36.05 3.21 11.251355 15.73 1.4 11.27 1553 35.99 3.19 11.281361 15.81 1.39 11.36 1557 35.99 3.19 11.291412 17.6 1.68 10.46 1560 35.94 3.16 11.381413 17.56 1.67 10.5 1572 35.99 3.16 11.391423 17.6 1.67 10.53 1578 35.98 3.18 11.331445 17.56 1.67 10.5 1582 35.98 3.16 11.391464 19.66 1.9 10.34 1583 35.89 3.13 11.461465 23.02 1.9 12.11 1593 35.62 3.13 11.381466 23.02 1.91 12.05 1601 36 3.24 11.11467 23.02 2.01 11.47 1604 38.94 3.22 12.111469 23.35 2.01 11.64 1605 38.94 3.24 12.011470 23.57 2.05 11.52 1611 38.13 3.24 11.761472 23.73 2.07 11.44 1612 44.27 3.24 13.661492 24 2.1 11.42 1623 43.91 3.24 13.541526 27.01 2.38 11.37 1666 44.73 3.35 13.351533 27 2.38 11.36 1670 48.54 3.35 14.491542 29.91 2.64 11.34 1717a 50.97 3.35 15.211544 29.91 2.62 11.41

Source: Redish (2000, pp.89-92)

Notes:

a. This ratio remained until the minting of gold was discontinued in 1816.

b. Prices of gold and silver given as the mint prices including minting fees.

18

Table 2. Price Ratio of Gold to Silver in Europe, 1344-1499.

Year Ratio Year Ratio Year Ratio Year Ratio1350 9.69 1400 9.61 1450 9.721351 9.69 1401 9.61 1451 9.721352 9.69 1402 9.61 1452 9.721353 9.69 1403 9.61 1453 9.721354 9.69 1404 9.61 1454 9.721355 9.69 1405 9.61 1455 9.721356 9.69 1406 9.61 1456 9.81357 9.69 1407 9.61 1457 9.81358 9.69 1408 9.61 1458 9.81359 9.69 1409 9.61 1459 9.81360 9.69 1410 9.61 1460 9.81361 9.91 1411 9.61 1461 9.81362 9.91 1412 9.61 1462 9.81363 9.91 1413 9.61 1463 9.81364 9.91 1414 9.61 1464 10.81365 9.92 1415 9.61 1465 10.81366 9.92 1416 9.61 1466 10.81367 9.92 1417 9.59 1467 10.81368 9.92 1418 9.59 1468 10.81369 9.92 1419 9.59 1469 10.81370 9.92 1420 9.59 1470 10.81371 9.92 1421 9.5 1471 10.81372 9.92 1422 9.5 1472 10.81373 9.92 1423 9.5 1473 10.81374 9.92 1424 9.5 1474 10.611375 9.92 1425 9.5 1475 10.91376 9.92 1426 9.5 1476 10.91377 9.92 1427 9.17 1477 10.91378 9.92 1428 9.17 1478 10.91379 9.92 1429 9.17 1479 10.91380 9.92 1430 9.17 1480 11.041381 9.92 1431 9.17 1481 11.041382 9.92 1432 9.64 1482 11.041383 9.92 1433 9.64 1483 11.071384 9.92 1434 9.64 1484 11.141385 9.92 1435 10 1485 11.141386 9.92 1436 10 1486 11.141387 9.92 1437 10 1487 11.141388 9.92 1438 10 1488 11.341389 9.92 1439 10 1489 11.31390 9.92 1440 10 1490 11.31391 9.61 1441 9.94 1491 11.31392 9.61 1442 9.94 1492 11.31393 9.61 1443 9.94 1493 11.3

1344 10.07 1394 9.61 1444 9.94 1494 11.31345 10.07 1395 9.61 1445 9.94 1495 11.31346 9.69 1396 9.61 1446 9.94 1496 11.31347 9.69 1397 9.61 1447 9.72 1497 11.071348 9.69 1398 9.61 1448 9.72 1498 11.071349 9.69 1399 9.61 1449 9.72 1499 11.07

Source: Shaw (1896, p.40)

19

Table 3. Price Ratio of Gold to Silver in Europe, 1500-1819.

Year Price Ratio1501 10.751511 10.751521 11.251531 11.251541 11.31551 11.31561 11.51571 11.51581 11.81591 11.81601 12.251611 12.251621 141631 141641 14.51651 14.51661 151671 151681 14.981691 14.961701 15.271711 15.151721 15.091731 15.071741 14.931751 14.561761 14.811771 14.641781 14.761791 15.421801 15.611811 15.51

Source: Farrer (1898, Appendix II)

20

Table 3. Gold and silver coinage by reign (in £), 1334-1815.

Year Gold Silver Year Gold Silver Year Gold Silver1344 39,326 40,833 1396 8,050 212 1448 1,458 1321345 9,970 25,129 1397 17,175 735 1449 3,408 1,0521346 8,325 7,746 1398 17,175 735 1450 5,950 6,9531347 36,593 4,612 1399 16,635 1,434 14511348 43,300 8,235 1400 7,328 286 1452 6,918 16,1841349 11,001 4,255 1401 7,328 286 1453 4,367 6,1391350 36,298 9,409 1402 7,328 286 1454 2,058 5,4081351 95,577 24,972 1403 4,479 161 1455 1,244 4,1021352 72,609 87,896 1404 4,707 452 1456 1,244 4,1021353 53,857 112,314 1405 3,320 87 1457 2,136 9,9941354 124,508 46,736 1406 5,407 101 1458 1,408 5,4911355 84,750 48,294 1407 2,981 80 1459 321 4,6551356 8,278 28,330 1408 2,170 8 1460 1,885 10,5631357 77,243 18,137 1409 14611358 112,167 12,315 1410 14621359 97,909 10,422 1411 1463 2,444 8,9141360 62,909 6,144 1412 149,870 2,911 1464 2,444 8,9141361 207,870 6,020 1413 138,822 5,463 1465 139,376 51,8761362 131,140 14,164 1414 78,376 7,017 1466 139,376 51,8761363 37,746 3,148 1415 78,376 7,017 14671364 20,482 2,840 1416 78,376 7,017 14681365 15,651 1,485 1417 78,376 7,017 1469 45,997 15,2761366 16,519 0 1418 23,874 1,579 1470 51,509 15,1221367 11,115 0 1419 23,874 1,579 1471 0 01368 25,203 2,194 1420 43,313 2,220 1472 42,557 17,0061369 72,715 1,535 1421 98,472 3,326 1473 42,557 17,0061370 22,205 1,945 1422 109,703 3,462 1474 42,557 17,0061371 15,446 801 1423 109,703 3,462 1475 42,557 17,0061372 21,826 174 1424 109,703 3,462 1476 25,349 5,9011373 14,597 453 1425 57,760 2,418 1477 25,349 5,9011374 9,642 465 1426 51,307 4,054 1478 25,173 3,9231375 10,414 4,168 1427 12,703 898 1479 23,562 5,0911376 5,645 2,915 1428 28,192 1,745 1480 31,014 3,6571377 4,101 225 1429 12,740 2,171 1481 17,865 1,8291378 4,880 1,264 1430 12,740 2,171 1482 17,280 3,2081379 4,880 1,264 1431 21,681 4,222 1483 10,485 5,8191380 4,880 1,264 1432 10,890 2,200 1484 16,537 13,3261381 4,880 1,264 1433 8,163 1,292 1485 8,737 4,6411382 4,880 1,264 1434 10,575 855 1486 10,622 7,6931383 4,880 1,264 1435 6,711 666 1487 7,762 3,5711384 4,880 1,264 1436 6,711 666 1488 9,029 4,9711385 11,454 1,091 1437 5,658 820 1489 5,312 5,1071386 11,454 1,091 1438 5,251 2,216 14901387 11,454 1,091 1439 9,090 6,233 14911388 1440 4,986 1,230 14921389 27,366 354 1441 4,986 1,230 14931390 24,402 2,243 1442 4,986 1,230 14941391 23,031 2,736 1443 4,986 1,230 1495 18,422 9,1161392 25,420 410 1444 4,057 233 1496 18,422 9,1161393 13,040 222 1445 2,700 310 1497 18,422 9,1161394 13,554 184 1446 1498 18,422 9,1161395 13,554 184 1447 4,567 3,865 1499 20,104 22,456

21

Table 3 continued.

Year Gold Silver Year Gold Silver Year Gold Silver1500 20,104 22,456 1555 19,656 51,799 1610 77,513 22,7311501 27,272 19,661 1556 1611 38,585 20,6411502 27,272 19,661 1557 1612 73,752 20,1051503 28,622 28,622 1558 18,588 109,929 1613 302,801 6,4241504 36,277 25,896 1559 6,487 51,068 1614 326,537 15,3551505 47,522 45,148 1560 2,362 5,247 1615 168,978 1,8011506 95,546 38,514 1561 17,085 72,524 1616 252,839 12,3221507 85,601 30,302 1562 1,035 16,826 1617 263,656 1,4651508 122,683 23,987 1563 1,821 18,933 1618 153,684 2161509 119,241 9,241 1564 1,205 7,242 1619 149,343 4701510 69,147 3,303 1565 562 8,569 1620 186,953 34,0711511 50,479 1,248 1566 1,023 1,800 1621 520,658 1,461,0911512 26,919 10,383 1567 1,069 1,344 1622 12,086 63,8831513 73,138 13,562 1568 1,144 1,582 1623 21,250 102,7831514 31,934 4,464 1569 1,535 13,204 1624 321 3531515 41,987 1,024 1570 1,634 14,890 1625 740 40,6071516 53,933 180 1571 7,674 25,474 1626 130,165 365,2731517 0 0 1572 11,835 21,727 1627 3,021 15,9711518 48,110 1,000 1573 12,164 95,752 1628 5,313 25,6961519 5,462 14,237 1574 4,428 9,838 1629 80 881520 36,272 64 1575 32,034 135,982 1630 185 10,1521521 24,869 1,860 1576 1,940 31,548 1631 1,304 15,1831522 14,679 14,238 1577 3,414 35,500 1632 402 9,0201523 9,137 17,560 1578 2,260 13,579 1633 2,888 3,1401524 1579 1,054 16,067 1634 5,617 2,8151525 1580 1,918 3,375 1635 4,889 2,2241526 1581 2,004 2,519 1636 9,269 5,1131527 44,066 65,075 1582 35,424 78,705 1637 5,028 1,9331528 44,066 65,075 1583 23,297 98,896 1638 4,899 5,5241529 44,066 65,075 1584 1,411 22,944 1639 8,087 5,1741530 44,066 65,075 1585 2,483 25,818 1640 3,620 11,4161531 4,552 7,675 1586 1,643 9,875 1641 5,298 12,7051532 4,552 7,675 1587 766 11,685 1642 3,648 1,7161533 4,552 7,675 1588 1,395 2,455 1643 2,248 2401534 4,552 7,675 1589 1,458 1,832 1644 10,102 13,1151535 4,552 7,675 1590 1,560 2,157 1645 10,127 18,8221536 4,552 7,675 1591 2,094 18,005 1646 5,425 1,0311537 837 42,519 1592 2,228 20,304 1647 23,337 10,5421538 25,401 27,646 1593 10,465 34,738 1648 25,160 8,2541539 44,903 61,525 1594 15,518 252,384 1649 13,015 3,8411540 19,090 25,842 1595 27,314 283,998 1650 18,129 95,8241541 11,631 18,587 1596 18,077 108,630 1651 18,129 95,8241542 5,198 17,642 1597 8,430 128,538 1652 18,129 95,8241543 5,198 17,642 1598 15,347 27,003 1653 18,129 95,8241544 5,198 17,642 1599 16,035 20,154 1654 25,500 123,3401545 165,931 149,287 1600 17,165 23,726 1655 25,500 123,3401546 1601 7,676 66,020 1656 25,500 123,3401547 420,855 324,267 1602 7,676 66,020 1657 25,500 123,3401548 0 242,680 1603 7,676 66,020 1658 25,500 123,3401549 1604 24,506 223,346 1659 641 7061550 0 500,000 1605 115,110 382,115 1660 641 7061551 0 500,000 1606 177,531 325,906 1661 641 7061552 611 1,500 1607 152,709 176,346 1662 4,439 243,6401553 19,656 51,799 1608 138,169 153,082 1663 31,305 364,3911554 19,656 51,799 1609 128,778 74,453 1664 9,649 216,490

22

Table 3 continued.

Year Gold Silver Year Gold Silver Year Gold Silver1665 69,321 75,365 1715 1,826,480 5,093 1765 538,272 191666 134,814 67,555 1716 1,110,420 5,113 1766 820,725 2981667 117,344 53,386 1717 709,566 2,939 1767 1,271,808 01668 222,444 122,708 1718 140,642 7,114 1768 844,554 01669 120,667 46,398 1719 688,960 3,444 1769 626,582 01670 117,576 132,580 1720 885,859 24,279 1770 623,779 681671 194,078 124,171 1721 272,500 7,170 1771 637,796 01672 86,887 273,990 1722 594,716 6,147 1772 843,853 3351673 127,150 304,930 1723 388,098 149,106 1773 1,317,645 01674 87,540 41,185 1724 273,808 3,121 1774 4,685,624 01675 53,945 5,754 1725 58,360 7,734 1775 4,901,219 01676 242,442 314,765 1726 872,963 2,592 1776 5,006,350 3151677 243,036 451,729 1727 292,779 2,049 1777 3,680,996 01678 130,201 24,742 1728 539,874 2,644 1778 350,437 01679 560,076 253,001 1729 0 6,371 1779 1,696,118 2541680 603,836 198,103 1730 91,628 3,478 1780 0 01681 312,354 92,185 1731 305,768 2,182 1781 876,795 621682 186,517 29,589 1732 373,473 2,620 1782 698,074 01683 376,736 229,744 1733 833,947 3,580 1783 227,083 01684 319,220 53,660 1734 487,108 4,929 1784 822,128 2031685 564,204 94,773 1735 107,234 3,460 1785 2,488,106 01686 648,281 99,814 1736 330,579 5,310 1786 1,107,383 01687 421,370 250,630 1737 67,284 3,720 1787 2,849,057 55,4591688 589,375 76,231 1738 269,837 0 1788 3,664,174 01689 134,864 96,573 1739 283,854 10,528 1789 1,530,711 01690 51,159 1,995 1740 196,246 0 1790 2,660,522 01691 57,222 3,731 1741 25,232 9,486 1791 2,456,567 01692 120,223 4,160 1742 0 0 1792 1,171,863 2521693 54,094 1,995 1743 0 7,440 1793 2,747,430 01694 64,780 9,277 1744 9,812 7,836 1794 2,558,895 01695 753,079 160 1745 292,966 1,860 1795 493,416 2951696 145,548 2,511,915 1746 474,492 136,431 1796 464,680 01697 126,469 2,192,196 1747 37,146 4,650 1797 2,000,297 01698 495,145 326,628 1748 338,523 0 1798 2,967,505 01699 148,445 60,444 1749 710,687 0 1799 449,962 01700 126,223 14,898 1750 558,597 0 1800 198,936 911701 1,249,520 116,179 1751 450,663 8,103 1801 450,242 331702 170,172 359 1752 572,657 58 1802 437,019 621703 1,596 2,226 1753 364,876 59 1803 596,444 721704 0 12,422 1754 0 59 1804 718,397 781705 4,859 1,332 1755 224,690 59 1805 54,668 1831706 25,091 2,889 1756 492,983 121 1806 405,106 01707 28,362 3,639 1757 0 16,612 1807 0 1081708 48,192 11,628 1758 651,814 62,586 1808 371,744 01709 115,317 78,811 1759 2,429,010 105 1809 298,946 1151710 173,630 2,532 1760 676,231 133 1810 316,936 1211711 435,664 76,781 1761 550,888 31 1811 312,263 01712 133,400 5,532 1762 553,691 3,162 1812 0 531713 613,826 7,232 1763 513,041 2,629 1813 519,722 901714 1,379,602 4,855 1764 883,102 15 1814 0 161

Source: Craig (1953, pp.408-422)

Notes:

a. Multi-year values are divided evenly between years. For example, if Craig notes 3,000 for years

1-3, then 1000 is assigned for each of years 1, 2, and 3.

23

Chapter 3: Stability and Exchange in a Generalized

Diamond-Dybvig Model

3.1 Introduction

Diamond and Dybvig (1983) is by far the most commonly cited article on the theory of

deposit banking.1 The DD model illustrates the important role of banks as providers of

liquidity but also the danger they face due to the potential for runs. The authors discuss

suspension of convertibility in optimal deposit contracts and the potential role of government

in providing deposit insurance. Many citations of DD (1983) regard it as describing the

inherent instability of deposit banking. Here, by contrast, we show that instability is a

special case of the model rather than a general property.

Dozens of papers have extended the DD model in various ways. Optimal deposit con-

tracts have been discussed by Wallace (1990), Villamil (1991), Selgin (1993), Peck and Shell

(2003), and Green and Lin (2003). Many have debated the presence or absence of bank run

equilibria including Postlewaite and Vives (1987), Hazlett (1997), Green and Lin (2000),

Goldstein and Pauzer (2005), and Ennis and Keister (2009). Further extensions of the DD

model include Chari (1989), Dowd (1993), Wallace (1996), Cooper and Ross (2002), and

Cavalcanti (2004). Garratt and Keister (2005), Schotter and Yorulmazer (2003), and Duffy

(2008) describe experimental tests of the DD model. Yet none of these works consider the

banking system in the context of a larger economy. Each assumes that banking occurs in

isolation rather than interacting with other sectors as originally proposed in DD (1983).

As Wallace (1990) posited, “The model suggests that illiquid banking system portfolios

can be understood without complications like separate financial and business sectors.” All

progressive works appear to have followed on that assumption.

1DD (1983) has 861 citations according to the Social Science Citation Index as of March 5, 2011. It is

ranked as the 24th highest paper by impact factor by Research Papers in Economics (http://repec.org).

24

This paper demonstrates that the banking sector cannot be effectively studied in iso-

lation. It explores a state of the DD model which includes both banking and production

sectors. DD (1983, p.409) briefly discusses this scenario but does not explore it in detail.

There are two benefits to the multi-sector model. First, agents can hedge risk by investing

in multiple sectors which creates improvements in welfare. The same risk aversion that mo-

tivates agents to invest in a bank also motivates them to invest outside the bank. Second,

exchange can prevent bank runs because deposit contracts can be traded at a discount to

real goods. As the bank nears default, the uncertain payoff will cause deposits to be traded

at a price lower than the stated value but higher than the default value. The likelihood of

bank runs is thereby reduced and is not an equilibrium unless the bank is fundamentally

insolvent. This is consistent with historical experience of US banking before the Federal

Reserve. When a bank was suspected of insolvency, its notes continued to circulate but

were often traded at a discount to their redemption value.

The issue of bank stability is especially pertinent today due to the current turmoil in

the financial sector. Uhlig (2010) uses the DD model to describe the financial panic of

2008 as a modern-day bank run. Other works use the DD model to analyze instability

and contagion in the financial sector (Manz 2010, Ennis and Keister 2010, and Peck and

Shell 2010). Yet if analysis of the banking sector in isolation is inappropriate, as this

paper suggests, then the use of standard DD models to study these phenomena will draw

precisely the wrong conclusions about bank stability, especially regarding the degree to

which government intervention can stem financial panics.

In a multi-sector DD model solvent banks do not experience runs. The next section

describes the basics of the DD model and outlines the variables which influence stability

and exchange. Section 3.3 describes the incentives for trade in a multi-sector economy and

their effects on bank stability. Section 3.4 discusses historical evidence from the US banking

system. Section 3.5 concludes with implications for further research.

3.2 The Diamond-Dybvig Model

This section will describe the basic model described in DD (1983). Although the reader

may be familiar with the DD model, we discuss it here for two reasons. First, we outline the

25

incentives for both production and banking sectors so that they may be easily combined in

section 3.3. Second, we emphasize the importance of analyzing ex post consumption. While

most DD models focus on ex ante optimization in period 0, the possibility of a bank run

can be better analyzed ex post in period 1 after agents have learned their types. This is

also the period when exchange, if any, will occur.

We begin by describing the agent population and available technology in section 3.2.1.

Section 3.2.2 calculates ex ante optimal consumption which is shown to be beyond the

limit of the basic technology. The formation of banks is described in section 3.2.3 and the

potential for runs in 3.2.4. Section 3.2.5 calculates actual ex post consumption, and section

3.2.6 shows that banks may still be created despite the threat of a run.

3.2.1 The All-Production Economy

Consider an economy to be analyzed over 3 time periods (T = 0, 1, 2). Each agent of the

population is endowed with 1 unit of capital at time T = 0. The economy has only one

production technology with the following properties:

T=0 T=1 T=2

-1

0 R

1 0

A capital investment of 1 unit at T = 0 produces consumption goods of up to 1 unit at

T = 1, up to R > 1 units at T = 2, or any linear combination thereof. At T = 1 each

agent allocates consumption for periods 1 and 2. Individual consumption is not publicly

observable.

There are two types of agents, patient and impatient. As of T = 0, agent type is

unknown, and all agents identically prefer to patiently defer consumption until period T = 2.

However, each faces some positive probability of becoming impatient in period T = 1, in

which case he will chose to maximize consumption in that period. Impatient agents who

prefer consumption at T = 1 are described as type 1 with Θ = 1, while patient agents who

prefer consumption at T = 2 are type 2 with Θ = 2. Agent types are also not publicly

observable, however the proportion t ∈ (0, 1) of type 1 agents in the population of N agents

26

where t = ΣN1 (2−Θ)/N is known to all agents.

Let cΘT represent individual consumption by type Θ agents in period T .2 Impatient type

1 agents always maximize consumption in period 1. Patient type 2 agents can maximize

consumption in period 2 either by leaving their capital in the production process until

T = 2 or by withdrawing consumption goods in period 1 and storing them at no cost

for consumption at T = 2. All agents exhibit constant relative risk aversion in utility of

consumption with 0 < γ < 1. Type 2 agents face some discount factor 0 < ρ < 1. These

utilities are given by equation 3.2.1.

u(c1, c2,Θ) =

(c11)1−γ

1−γ if Θ = 1

ρ(c21+c22)1−γ

1−γ if Θ = 2(3.2.1)

Individual utility can be maximized subject to the resource constraint 3.2.2.

c2 = R(1− c1) (3.2.2)

These preferences indicate that each agent will invest his endowment of 1 unit at T = 0.

When agent types are revealed at T = 1, all type 1 agents will maximize consumption in

T = 1. All type 2 agents will defer consumption to maximize at T = 2. The price of c2 in

terms of c1 will be R−1. Agents may exchange promises of goods in different periods, but as

DD describe, “Given these prices, there is never any trade” (DD 1983, p.406). Therefore,

all type 1 agents will withdraw and consume 1 unit at T = 1, while all type 2 agents will

refrain until T = 2 when they receive and consume R units.

These choices are presented graphically in figure 3.1. The resource constraint, drawn in

solid black, is the same for both agent types. Gray lines represent the indifference curves for

each agent, the points at which the agent is indifferent between consumption at times 1 and

2. Type 1 agents have vertical indifference curves since they consume only c1. Indifference

curves are unitary for type 2 agents since c1 and c2 are perfect substitutes. Thus, type 1

agents maximize their consumption set c1 = (1, 0) of 1 unit at time T = 1 and 0 time T = 2.

2Notations of c1 and c2 refer to consumption in periods 1 and 2 regardless of agent type. Alternatively,

c1 and c2 represent consumption for agent types 1 and 2 regardless of time.

27

(a) Type 1 agents. (b) Type 2 agents.

Figure 3.1: Consumption in the all-production economy.

Type 2 agents leave their goods in production and consume c2 = (0, R) of nothing at time

T = 1 and R units at time T = 2.

3.2.2 Optimal Consumption

Expected individual consumption can be shown to be below the social optimum. Social

welfare in the all-production economy can be stated as the weighted sum of utilities by

agent type shown in equation 3.2.3.

WProd = tu(c11) + (1− t)ρu(c2

1 + c22) (3.2.3)

Social welfare can be maximized according to the constraints on resources and between

marginal production and consumption.

tc11 +

(1− t)c22

R= 1 (3.2.4)

u′(c11) = Ru′(c2

2), c12 = c2

1 = 0 (3.2.5)

Optimal individual consumption in each period can be calculated as equations 3.2.6 and

3.2.7.3

c11 =

R

tR+ (1− t)(ρR)1/γ(3.2.6)

3These calculations are shown in Appendix B.1.

28

c22 =

ρR1/γ

tR+ (1− t)(ρR)1/γ(3.2.7)

Because agents are risk averse, their optimal consumption is some combination of c1

and c2. However, agents acting independently are unable to achieve this socially optimal

consumption set. As DD (1983, p.407) describes “Optimal consumption levels satisfy c11 > 1

and c22 < R. Therefore, there is room for improvement on the competitive outcome.”

3.2.3 The All-Banking Economy

In the DD model, it is sometimes possible to improve upon the competitive equilibrium

through collective action.4 At T = 0, each agent can increase his expected future utility by

trading off some potential consumption at T = 2 for increased consumption at T = 1.

Suppose that all agents agree to pool their capital investments as a form of insurance

against the lower utility that comes with being impatient. Each agent receives a deposit

contract which he may redeem at t = 1 for an amount r1 ≥ 1. Any goods not withdrawn

at T = 1 will be used to produce consumption goods which will be divided equally among

the remaining agents at T = 2. Return r2 at time 2 is, therefore, a function of t.

r2(t) =(1− tr1)R

1− r1. (3.2.8)

To optimize expected utility as of T = 0, the group sets r1 = c1 from equation 3.2.6.

Expected time 2 consumption will be the optimal value r2(t) = c2 from equation 3.2.7.

This consumption set increases social welfare above the level achieved in the all-production

economy.

3.2.4 Bank Runs

The optimal levels of consumption described in the previous section will only result if

information about agent types is publicly observable. Banks can ensure optimal risk sharing

by verifying individual agent type Θ. Alternatively, they may use other mechanisms such as

4It is not always possible to improve upon the no-trade equilibrium since in some cases the potential forruns may preclude any gains, so no bank will be established. This possibility is acknowledged by DD (1983,

p.409), expanded by Huo and Yu (1994), and will be discussed further in section 3.2.6.

29

suspension clauses based on the portion t of type 1 agents (DD 1983, p.411). However, when

neither individual agent type nor the proportion of type 1 agents is observable, consumption

may be suboptimal since some type 2 agents may choose to consume at T = 1. When

this information is lacking, “No bank contract can attain the full-information optimal risk

sharing” (p.412).

Suppose now that t is random and stochastic. Agents have only some expectation t

which they use to calculate optimal consumption. If t 6= t, consumption is suboptimal. For

all t < t, first period consumption will be too low, and when t > t period 1 consumption

will be too high and consumption in period 2 too low. In fact, consumption in time T = 1

may be so high that there are no goods are left for consumption in T = 2. In this case, the

bank becomes insolvent before the second period. The threshold for insolvency is given in

equation 3.2.9.

r1(t+ c21) > 1. (3.2.9)

When period 1 consumption is expected to be above the threshold, all agents have the

incentive to redeem their deposits at T = 1 which constitutes a bank run.

When a run occurs, all agents attempt to redeem their deposits at T = 1, however,

not all will be successful. We follow DD (1983, p.408) in assuming a sequential service

constraint. All agents who choose to redeem at T = 1 form a line at the bank, and each

agent’s place in line is assigned at random. The line progresses with each agent redeeming

his deposit for r1 units until the bank is devoid of capital. Thus, each of the first 1/r1

redeemers will receive r1 units of consumption goods from the of 1 total unit of invested

capital (representing 100% of the capital in the economy). The remaining 1 − 1/r1 agents

are left with nothing. Therefore, each agent’s welfare in the case of a run is a probabilstic

outcome shown in equation 3.2.10.

WRun = (1/r1)u(r1) + (1− 1/r1)u(0) (3.2.10)

Welfare in the bank run equilibrium is clearly lower than the no-run equilibrium and even

lower than in the all-production economy. Total social welfare in the all-bank economy

can be calculated as the weighted expected value of run and no-run equilibria given some

30

(a) Fundamental values. (b) Expected values.

Figure 3.2: Value of deposits at T=1 in the all-bank model.

expected probability δ that a run will occur.

WBank = (1− δ)WNoRun + δWRun (3.2.11)

The incentive to run on the bank can be described in terms of the period 1 value V (c1)

of deposits to each type of depositor as illustrated in figure 3.2. These graphs show the

deposit value on the y-axis as a function of time 1 redemptions c1 = c11 + c2

1 on the x-

axis. Figure 3.2a. depicts the “fundamental” value of the deposit earned if the deposit is

redeemed in the period which matched the agent’s type. Type 1 depositors always redeem

in t = 1. They earn c11 = r1 up to the point where there c1 = 1 − t redemptions, at which

time the bank defaults. After the point of default, all agents have an expected payoff of r1

with probability 1/r1 and a payoff of 0 with probability 1− 1/r1. This probabilistic default

payoff, denoted here as r0, can range anywhere from −∞ < r0 < 1/r1 depending on the

values of ρ and γ but for convenience is shown in the range 0 < r0 < 1.

The fundamental value of deposits to type 2 agents r2 is a function of c1 given in equation

3.2.8. This is shown in figure 3.2a. as a line declining from R at c1 = 0 to 1 at c1 = 1−t, the

point where the bank defaults. When the bank goes into default, even type 2 agents choose

to redeem their deposits since nothing will be left at T = 2. All deposits will be worth r0

regardless of agent type. However, type 2 agents may decide to redeem their deposits at

T = 1 even before c1 = 1− t. Figure 3.2a. shows that after the point c1 = 1/r1− t, the time

T = 2 redemption value falls below the value at T = 1 to r2 < r1. Since type 2 agents value

31

c21 and c2

2 equally, they will choose to redeem their deposits early at T = 1. Since all agents

decide to redeem at T = 1, there is a run on the bank, and the value of deposits falls to r0.

This scenario is depicted in figure 3.2b. Whenever the expected value of c1 > 1/r1 − t, all

agents redeem at T = 1 causing a run on the bank, and the expected value of all deposits

falls to r0.

Runs can occur for two reasons. First, as previously described, the number of types 1

agents may be higher than expected to the point that they consume all goods at T = 1,

and none are left for consumption at T = 2. This will be referred to as a “fundamental

run” since the bank has more legitimate claims than it can pay out. The second type occurs

when type 2 agents choose to consume at T = 1. This “sunspot run” can occur if the agents

fear that the number of type 1 agents may be higher than previously expected (that t > t).

This differs from the fundamental run since it is only the expected difference between t and

t that causes the run.

What causes this shift in expectations? DD do not specifically say. It may be any

economy-wide signal or event and need not be economic in nature. As described by DD

(1983, p.410), it “need not be anything fundamental about the bank’s condition” but could

be any “commonly observed random variable in the economy... even sunspots.” In this

case, even a solvent bank will experience a run and default. In fact, any widely observable

signal may cause all banks in the economy to be run upon simultaneously.

Although the information which can signal the run is not specifically included in the

DD model, other works have provided explicit mechanisms. Green and Lin (2000) assumes

that information is revealed by the agent’s order in line at the bank. Alonso (1993) uses

a signal from each agent that may or may not be truthful, while Andolatto, Nosal, and

Wallace (2007) assumes that agents reveal their true types when redeeming deposits. In

Samartin (2003), agents observe their return on investment from which they can deduct

the portion of impatient agents. Because this paper attempts to replicate DD (1983) as

originally written, we continue on the assumption that agents receive some new information

at T = 1 which is not included in the model.

32

3.2.5 Actual Consumption

We can study actual consumption (as opposed to ex ante expected consumption) by analyz-

ing agents’ consumption opportunities at T = 1. Because type 1 agents only get utility from

consumption at time 1, we assume that all t agents will maximize period 1 consumption

c11 = r1 with c1

2 = 0. Therefore, type 2 agents will receive all of the remaining 1− tr1 capital

which they will divide between c21 and c2

2 consumption at times 1 and 2.

The actual consumption in period 2 is a non-linear function of consumption in period

1. In equation 3.2.8, r2 is dependent on period 1 consumption by type 1 agents. However,

since some type 2 agents may also consume in period 1, actual period 2 consumption by

type 2 agents c22 is dependent on t+ c2

1 as shown in equation 3.2.12.

c22 =

R(1− r1(t+ c21))

1− t− c21

(3.2.12)

The more agents who redeem their deposits at T = 1, the lower the payoff to patient

depositors who withdraw at T = 2. The number of early redeemers can increase until the

point c21 = 1/r1 − t where all resources are consumed, the bank becomes bankrupt, and a

run occurs. Since deposits will be worth nothing at T = 2, all type 2 agents will choose to

redeem their deposits at T = 1. Again, c21 need not actually reach the level of c2

1 = 1/r1− t.

Merely the expectation that this will occur is sufficient to create a bank run.

We can optimize social welfare according to the bank’s resource constraint. The welfare

function in equation 3.2.3 includes only type 2 depositors. Type 1 agents care only for

consumption at T = 1 and uniformly choose c11 = r1. Therefore, the social welfare function

given in equation 3.2.13 depends solely on consumption by type 2 agents.

WBank = tu(r1) + (1− t)ρu(c21 + c2

2) (3.2.13)

Unlike consumption in the all-production economy shown in figure 3.1, the resource con-

straint is not a straight line. While total period 1 consumption is less than c1 = 1/r1 − t,

individual period 2 consumption is a function of c1 = t+c21. Once total period 1 consumption

33

(a) Type 1 agents. (b) Type 2 agents.

Figure 3.3: Consumption in the all-bank model.

reaches c1 = 1/r1 − t, the bank defaults, and c2 = 0 as shown in 3.2.14.

c22 =

R(1−r1(t+c21))

1−t−c21for c2

1 < 1/r1 − t

0 for c21 ≥ 1/r1 − t

(3.2.14)

Figure 3.3 shows the resource constraints between c21 and c2

2 and utility functions for

agent types 1 and 2. This analysis matches the proof in DD (1983) that banks are subject

to non-fundamental runs, however, it deviates from their analysis in another respect. DD

(1983) analyzes ex ante expected consumption as of T = 0. It is assumed that actual

consumption in the non-run equilibrium will match the optimal levels predicted at that

time. However, that prediction does not match the agents’ ex post decisions analyzed

here as of T = 1. As shown in figure 3.3, type 2 agents do not optimize consumption by

redeeming only at T = 2. Rather, they consume some combination of c21 and c2

2. This is

due to the nonlinear nature of r2 as a function of c1. By redeeming a small amount of their

deposits at T = 1 for r1, they reduce the number of depositors waiting to redeem at T = 2.

In the second period, goods are divided among as smaller group, so each agent gets a larger

share. The optimal consumption for type 2 agents is show below.5

c21 = 1− t−

√R(r1 − 1) (3.2.15)

5These calculations are shown in Appendix B.2.

34

c22 =

1− r(

1 +√R(r1 − 1)

)r − 1

(3.2.16)

At this consumption set, period 1 consumption is always below the bank run threshold

c21 < 1/r1 − t. (3.2.17)

Thus, there exists one stable non-run equilibrium. However, as described in section 3.2.4,

any informational shock which causes the expectation of a bank run can push c21 above the

threshold and into the bank run equilibrium.

3.2.6 Do Banks Exist?

Given that runs can occur at any time for any reason, one might wonder whether a bank will

ever be formed at all. As described in section 3.2.4, a run leaves a portion 1/r1 agents with

r1 units each and the remaining portion of 1− 1/r1 agents with 0 units each. This point is

especially pertinent since the DD model assumes relative risk aversion. Zero consumption

can lead to infinitely negative utility as discussed by Huo and Yu (1994) which shows that

for some combinations of γ, t, and R it may be impossible to establish a bank. The converse

is that in some cases banks are optimal despite positive probability of a run.

Let us compare social welfare of the all-production economy WProd given in equation

3.2.3 to that of the all-bank economy WBank in equation 3.2.11. As before, WBank is given

in terms of weighted expected value given some expected probability δ that a run will occur

as shown in 3.2.18.

WBank = (1− δ)WNoRun + δWRun

= (1− δ)[tu(c11) + (1− t)ρu(c2

1 + c22)] + δ[(1/r1)u(r1) + (1− 1/r1)u(0)]

≥ tu(c11) + (1− t)ρu(c2

1 + c22) = WProd (3.2.18)

Since a bank will only be formed if welfare for an all-bank economy is expected to be greater

than or equal to that of the all-production economy, there must be some value δ low enough

35

such that WBank > WProd in order for agents to invest in the bank. We can use this

inequality in equation 3.2.11 to solve for the threshold probability δ for which a bank will

be formed.

δ =WNoRun −WProd

WNoRun −WRun(3.2.19)

For all δ ≤ δ, forming a bank creates an ex ante improvement in social welfare. However,

this all-bank economy is only a special case of a more general model.

3.3 The Multi-Sector Model

DD (1983, p.405-410) describes a model which includes both banking and production sec-

tors. However, DD (1983) and all subsequent models have focused on the banking sector

alone. This section discusses a DD model with both banking and production. Allowing

multiple sectors creates a welfare improvement since agents can hedge their risk by invest-

ing in multiple sectors. Exchange between producers and bank depositors further increases

welfare and helps prevent bank runs since risky bank notes can be traded at a discount for

real goods.

The next section describes how allowing investment in multiple sectors creates a Pareto

improvement in social welfare. Section 3.3.2 discusses exchange in the DD model. Con-

sumption decisions are discussed in section 3.3.4. Section 3.3.3 explains the lack of bank

runs in equilibrium.

3.3.1 Welfare in the Multi-Sector Model

Optimal investment in the DD model occurs when capital is divided between production

and deposits. DD (1983) note that under an optimal investment strategy, “agents will invest

at least some of their wealth in banks even if they anticipate a positive probability of a run”

(p.409, emphasis added). This clearly implies that some capital will remain outside of banks

as well. However, DD (1983) does not discuss this scenario in detail. The article begins

with the all-production economy to demonstrate the risk diversification benefits provided

by banks. It then describes an all-banking economy stating that “for now we will assume

36

all agents are required to deposit initially” (p.409, emphasis added). The language implies

that the authors will also discuss the case in which not all agents are depositors, however

they never return to this scenario.

In the multi-sector model, capital is divided between banking and production. Each

agent invests a portion φ of his goods invested in the bank and (1−φ) in production. Payoffs

for each sector are the same as previously described, c1 = 1 and c2 = R for production and

c1 = r1 and c2 = r2(c1) for banking. In the case of a bank run, payoffs to the banking sector

are c1 = r1 with probability 1/r1 and c1 = 0 with probability 1− 1/r1.

Agent utility (WMS)

No run(1− δ)

Type 1(t)

φr1 + (1− φ)(1)

Type 2(1− t)

φr2 + (1− φ)R

Run(δ)

Deposit redeemed(1/r1)

Type 1(t)

φr1 + (1− φ)(1)

Type 2(1− t)

φr1 + (1− φ)R

Deposit not redeemed(1− 1/r1)

Type 1(t)

φ(0) + (1− φ)(1)

Type 2(1− t)

φ(0) + (1− φ)R

Figure 3.4: Probability-weighted expected value of social welfare.

We calculate ex ante social welfare as the weighted sum of expected utilities for each

agent type in three potential states. These are shown in figure 3.4 along with the probability

for each state. When there is no bank run, type 1 agents receive the expected value of

c11 = φr1+(1−φ)(1), and type 2 agents receives c2

2 = φr2+(1−φ)R. In the case of a bank run,

type 1 agents have probability 1/r1 chance of receiving c11 = φr1 +(1−φ)(1) and probability

(1−1/r1) of receiving c11 = φ(0)+(1−φ)(1). Type 2 agents have probability 1/r1 chance of

receiving c21 = φr1 + (1− φ)R and probability (1− 1/r1) of receiving c2

1 = φ(0) + (1− φ)R.

37

Social welfare is the weighted sum of the potential states as calculated in equation 3.3.1.

WMS =(1− δ)[tu(φr1 + (1− φ)(1)) + (1− t)ρu(φr2 + (1− φ)R)]

+ δt[(1/r1)u(φr1 + (1− φ)(1)) + (1− 1/r1)u(φ(0) + (1− φ)(1))]

+ δ(1− t)ρ[(1/r1)u(φr1 + (1− φ)R) + (1− 1/r1)u(φ(0) + (1− φ)R)] (3.3.1)

Let us compare this social welfare function to the other previously discussed welfare

functions using three values of δ. First, notice that if agents know there will be a bank run

(δ = 1), then no capital will be invested in the bank (φ = 0), so

WMS = tu(c11) + (1− t)ρu(c2

2) = WProd (3.3.2)

Similarly, if there are no bank runs (δ = 0), then all capital will be invested in the bank

(φ = 1), so

WMS = (1− δ)[tu(c11) + (1− t)ρu(c2

2)] + δ[(1/r1)u(r1) + (1− 1/r1)u(0)] = WBank (3.3.3)

However, for all values 0 < φ < 1 we can see that the multi-sector utility is not simply the

sum of utility in each sector. For example, the no-run type 1 utility is φu(r1) for banking

and (1 − φ)u(1) for production. However, the multi-sector value of u(φ(r1) + (1 − φ)(1))

is greater due to relative risk aversion in the utility function. This is true of all states and

agent types. Therefore, we know that for all 0 < φ < 1, WMS is greater than the linear

combination of WBank and WProd.

WMS > (1− φ)WProd + φWBank (3.3.4)

Considering these three cases, only the last is consistent with our prior assumptions.

The probability of a bank run was assumed to be positive (δ > 0), so equation 3.3.3 can be

ignored. Conversely, if the probability of a run is high, then no bank will ever be established,

so 3.3.2 is not relevant. Thus, we can see that social welfare is improved for any value of

38

0 < φ < 1 as shown in equation 3.3.4. The ability to invest in both banking and production

sectors creates an ex ante welfare improvement for all agents.6

Note that this ex ante welfare improvement is the result of diversification alone. Since

utility exhibits relative risk aversion, all agents prefer to invest in goods and production in

order to hedge the risk of becoming impatient. Further gains may be created through trade

in T = 1 once agents have discovered their utility preferences.

3.3.2 Is Trade Allowed?

Exchange between producers and depositors is clearly described in DD (1983). The authors

compare deposit contracts to market exchange and point out cases where trade will not be

advantageous. Thus, analysis of a multi-sector economy with exchange is not an extension

of the DD model but simply a case within the original model that has not been previously

examined.

DD assume tradability of goods and deposit contracts. Trade in consumption goods is

clearly allowed as DD predict that “There will be trade in claims on goods for consumption

at T = 1 and at T = 2” (p.406). Regarding deposit contracts, they write that “illiquidity is

a property of the financial assets in the economy in our model, even though they are traded

in a competitive market” (p.403, emphasis added). The original specification of the model

assumes “a competitive market in claims on future goods” (p.406), so even if bank notes

themselves were not tradable, agents could transact in forward contracts with the same

payoffs.

Some works such as Selgin (1993, p.348) assume that deposit contracts are “nontrans-

actable” in the standard DD model. However, it appears that the reason exchange is not

discussed by DD is not that such trade is prohibited but rather that it is non-optimal in the

scenarios analyzed. DD (1983, p.406) notes that no exchange will occur in an all-production

economy since each agent already holds his optimal consumption bundle. The same condi-

tions apply to the all-bank economy. Only in an economy of both producers and depositors

does an agent have the incentive to trade.

6A complete proof is given in Appendix B.3.

39

(a) Fundamental values. (b) Expected values.

Figure 3.5: Value of deposits at T=1 in the multi-sector model.

3.3.3 Bank Runs

In contrast to the all-banking model, bank runs do not form an equilibrium in the multi-

sector model. The lack of runs can be shown by describing the value V (c1) of redemption at

T = 1 to each agent type as illustrated in figure 3.5. The fundamental values are shown in

figure 3.5a as functions of total redemptions c1 = c11 +c2

1. The deposit value is constant at r1

for type 1 agents but is falling for type 2 agents since r2 is a decreasing function of c1. These

are the same lines depicted in figure 3.2a except for the addition of the new horizontal line

at V = r2/R. Since deposits represent r2 worth of goods to type 2 agents, these agents will

be willing to buy depositsat any price below r2/R. This new line represents the exchange

value of deposits.

Let us explore further the incentives for trade. A type 2 producers holds real goods which

are worth c22 = R to himself but only c1

1 = 1 to a type 1 agent. Type 1 depositors hold

deposits worth c11 = r1 to themselves but worth c2

2 = r2 to type 2 agents. At these prices,

there will be no exchange. However, if type 2 depositors suspect that c1 > 1/r1 − t, then

they will expect a bank run, and the expected value of deposits will fall to c1 = r0 < r2/R.

At this price, it become profitable for type 2 producers to trade some portion of their goods

for deposits. For each deposit purchased by a type 2 producer, another deposit can be

redeemed by a type 1 agent without fear of a bank run. Therefore, the average expected

value of type 2 deposits will lie somewhere between in r1 and r2/R. This value is represented

by the dashed line in figure 3.5b, although the actual expected value will depend on the

40

portions of agents t and deposits φ as well as the values used in the utility function.

Just as the existence of deposit insurance prevents bank runs (DD 1983, p.413-416), so

does the potential for exchange. If type 2 depositors know that type 2 producers stand ready

to purchase deposits, there is no incentive for an expectations-led run. Type 2 depositors

are not afraid of other type 2 depositors running on the bank because they have a more

profitable option available through exchange. There is still be a potential for a fundamental

bank run if c1 > 1− t, but the threat of sunspots becomes irrelevant.

Should we worry about fundamental bank runs? A bank becomes insolvent when its

commitments exceed its capital. This occurs in the model when the portion t of type 1

agents exceeds its expected value t by some amount. When agents realize that the bank is

insolvent, all type 2 depositors run on the bank since they know there will be no capital

left for redemption at T = 2. Yet the misjudgment of r1 based on t is no different than

any other business error that causes firm failure. A bank that over-promises should be held

responsible for its errors and its capital be reallocated to production or other banks. In

this sense, the fundamental run is an efficient mechanism for closing insolvent banks. As

White (1999, p.121) describes, “A run on an insolvent bank serves the same function as a

bankruptcy proceeding.” Therefore, the result of the multi-sector model, that fundamental

runs are allowed but sunspot runs are not, is the optimal set of responses to potential runs.

Exchange also provides information conveyed through the price system. The price of

goods in terms of deposits provides an indication to all agents of the portion of impatient

type 1 agents in the population. This public information may enable other efficient mech-

anisms for preventing runs. DD (1983, p.411-412) demonstrates that suspension of deposit

redemption can induce optimal levels of consumption only if the portion t of type 1 agents

is publicly available, which it is not. However, in the multi-sector model banks may be able

to implement optimal suspension contracts based on market prices since the portion of type

1 agents may be derived from the price.

3.3.4 Multi-Sector Consumption

Actual consumption in the multi-sector model can be optimized as of time T = 1. As in

section 3.2.5, c2 is a non-linear function of c1. Social welfare is the same as in the production

41

economy. Unlike the all-bank optimization, we include some type 1 consumers who may

wish to participate in trade. Specifically, we include the portion of t(1−φ) of type 1 agents

who invested in production. These agents seek to maximize period 1 consumption and may

be able to improve upon their endowment of 1 unit of goods from production. Conversely,

type 1 depositors are included as a constant term since these agents already hold their

optimum consumption bundle of r1 and are unlikely to improve upon that outcome.

WBank = tφu(r1) + t(1− φ)u(c11) + (1− t)ρu(c2

1 + c22) (3.3.5)

The resource constraint 3.3.6 is a combination of the production constraint from equation

3.2.2 and the banking constraint from equation 3.2.14 allocated according to φ.

c2 =

(1− φ)R[1− (1− φ)c21] + φ

R(1−r1(t+c21))

1−t−φc21for c2

1 ≥ 1/r1 − t

0 for c21 > 1− t

(3.3.6)

This equation describes consumption in period 2 from banking and production. The portion

φ of capital allocated to banking is depleted by 1 − r1(t + c21) redemptions in time T = 1,

then grows by R, and will be apportioned to 1− t−φc21 agents at T = 2. The portion 1−φ

of goods allocated to production are depleted only by the type 1 agents 1− (1−φ)c21. Bank

capital is fully depleted when c21 reaches 1 − t. This limit is the full amount of capital in

the economy excluding the portion tφ of type 1 depositors who did not engage in trade.

Optimizing social welfare in equation 3.3.5 according to the resource constraint 3.3.6,

we find the optimal consumption set in equations 3.3.7 and 3.3.8.7.

c21 = 1− t±

√φR(r1 − 1)(1− t)

1− (1− φ)R(3.3.7)

c22 =

1− r(

1 +√

φR(r1−1)(1−t)1−(1−φ)R

)r − 1

(3.3.8)

7These calculations are shown in Appendix B.4.

42

(a) Type 1 agents. (b) Type 2 agents.

Figure 3.6: Consumption in the multi-sector model.

These consumption levels can be seen graphically in figure 3.6. Like figure 3.3, the

tradeoff between consumption in times T = 1 and T = 2 is non-linear. In this case, the

y-intercept of c22 lies between r2 and R since the potion φ of capital deposited in the bank is

positive but less than 1. However, unlike figure 3.3, the constraint does not cross the x-axis

at c21 = 1/r − t but instead crosses at c2

1 = 1 − t just like the all-production constraint in

3.1. This is because deposits do not default prematurely but always retain some positive

value through trade until the point of consumption c21 = 1− t where the bank faces default.

Utility for type 1 depositors is maximized by consuming all of their goods in T = 1 as seen

in 3.6a, while type 2 depositors choose some optimal combination of c21 + c2

2 < 1/r1 − t, we

know that there is one stable equilibrium with no bank runs and no bank run equilibrium

before c21 = 1− t where the bank becomes insolvent.

3.4 Historical Evidence

The history of banking in the United States before the Federal Reserve indicates that the

multi-sector DD model is more analogous to the real world than the single-sector models.

Section 3.4.1 describes how risky bank notes and deposits were often traded at a discount.

Section 3.4.2 shows that bank runs are based on fundamental factors rather than expecta-

tions alone. Section 3.4.3 emphasizes the important role of information both in the model

and in the real world.

43

3.4.1 Trading at a Discount

There is ample historical evidence that the deposits, particularly bank notes, of risky banks

were traded at a discount to their face values. Gorton (1999, p.36) explains that “a bank

note is equivalent to risky debt with maturity equal to the time it takes to return from

the particular location of the note holder to the site of the issuing bank.” During the

US free banking era (1837-67), the notes issued by private banks were regularly traded

at a discount depending upon the expected probability that they would be successfully

redeemed. Rolnick and Weber (1988, p.33) provides evidence from this period, stating that

“Free bank notes were demanded because they were priced to reflect the expected value

of their backing.” Mullineaux (1987) and Calomiris and Kahn (1996) provide additional

evidence of note discounting in the Suffolk banking system.

Some bank notes carried an “option clause” which allowed the bank temporary suspen-

sion of convertibility. During periods of suspension, bank notes continued to circulate as

currency but were often traded at a discount. The possibility of optimal suspension is dis-

cussed by DD (1983, p.410-411), Dowd (1992), Diamond and Rajan (2001), and Peck and

Shell (2003). Commenting on the suspended convertability of the greenback from 1862 to

1878, Calomiris (1988, p.190) notes that expectations of future redemtpions“play a vital role

in determining the exchange rate” of greenbacks to gold. Calomiris and Schweikart (1991)

analyze the Panic of 1857 and find that note discounting varied widely across states depend-

ing on regulation and clearing house activities. Gorton (1985, p.280-282) and Calomiris and

Gorton (1991, p.117-119) describe how multiple banks sometimes coordinated suspensions

until the weaker banks could rebuild their capital bases. White (1984, p.26-30) and Dowd

(1988, p.325) provide further evidence from Scottish banking of discountounted note trading

in times of suspension.

During the pre-Fed period there was some a danger of “wildcat” banks opening long

enough to issue notes but closing again before their notes could be redeemed. Rockoff

(1985, p.887) shows that a discount was often applied to the notes of new banks entering

the market which reflected their level of risk. This system was generally effective at limiting

the danger of wildcat banking since “market participants could discipline banks by pricing

factors that affected risk and via the contractual redemption option” (Gorton 1999, p.61).

44

As a bank’s reputation was successfully established, the price of its notes increased to

approach is face value. Gorton (1996) confirms that reputation mechanism caused both

the bonds and notes of risky banks to be traded at a discount, noting that “Redemption

and reputation, combined with public and private restrictions on risk taking that limited

the degree of adverse selection, explain the success of the Free Banking Era” (Gorton 1996,

p.386).

3.4.2 Fundamentals or Sunspots?

In the DD model, economy-wide bank runs can occur for non-economic reasons which have

nothing to do with the bank’s fundamental solvency. However, in reality, sunspots do not

cause runs. Even during a financial panic, insolvent banks are more likely to be run upon

than solvent banks.

Articles such as Ennis and Keister (2010) cite the bank runs during the Great Depres-

sion and panics in the national banking era as evidence of economy-wide runs. However,

Gorton (1988) examines this period and finds that bank runs are not caused by sun spots

but by bank fundamentals. The study disputes the view that panics “are random events,

perhaps self-confirming equilibria in settings with multiple equilibria, caused by shifts in

the beliefs of agents which are unrelated to the real economy” and maintains instead that

“variables predicting deposit riskiness cause panics just as such movements would be used

to price such risk at all other times” (Gorton 1988, p.751). Champ, Smith, and Williamson

(1989) and Wicker (2000) show that pre-Fed bank runs tended to occur in heavily reg-

ulated regions, indicating that runs were related to fundamental issues rather than wide

spread panics. Champ, Freeman, and Weber (1999) suggests that bank runs were often

caused by variation in redemption costs, again indicating a fundamental issue rather than

a self-fulfilling prophecy.

Other works show that the danger of economy-wide bank runs is often attributable

to the central bank rather than being inherent to deposit banking. Gorton (1985, p.222)

states that “The number of banks that failed during the thirties was roughly twenty-five

times what it would have been had the pre-Federal Reserve System institutions been in

place.” Rolnick and Weber (1985, p.4) notes that pre-Fed bank runs tended not to cause

45

contgion. Selgin, Lastrapes, and White (forthcoming, p.22-25) describes how bank runs

actually increased in the early years of the Federal Reserve. It was FDIC deposit insurance

rather than the Fed that ended the incidence of banking panics and runs.

3.4.3 Information and Exchange

The primary source of instability in the DD model is the lack of information. Agents run

on the bank when they have insufficient information about the safety of their deposits, and

banks cannot prevent runs because they lack information regarding the preferences of their

depositors. However, in the real world there are practical solutions to these information

asymmetries. Banks provide information about their financial condition, customers can

receive verification for their claims, and the price system provides information on both

bank condition and consumer preference.

Sunspot bank runs occur in the DD model because agents lack information about the

safety of their deposits. However, banks developed mechanisms to share information so

that this issue was not a problem. Some banks published their capital levels while others

announced their relationships with clearing houses and other lenders. Similarly, customers

can have their personal and financial information verified through third parties.

Markets reveal information about the bank through the price system. As described

in section 3.4.1, there are many historical example of notes from risky banks trading at a

discount. Discounted note trading was especially important during times of suspension. DD

(1983, p.410-411) examines suspension of convertibility as one method for preventing runs.

In the all-bank version of the model there is insufficient information upon which banks can

base their decision to suspend. However, in the multi-sector version, the price can fulfill

this function.

It is often claimed that the DD model illustrates the inherent instability of deposit

banking. However, instability in the model originates in restrictions on information and

exchange that occur only in certain cases of the model. When studied as part of a larger

economy, banking provides a welfare improvement without creating instability. This con-

clusion is consistent with historical evidence. The multi-sector version of the DD model can

be used to study the effectiveness of information and exchange in preventing runs.

46

3.5 Conclusions

The generalized Diamond-Dybvig model shows that banking is not inherently unstable when

considered as part of a larger economy. The exchange of goods and bank deposit contracts

can prevent bank runs on solvent banks. It is curious that the dozens of articles extending

the DD model have failed to discuss this scenario. If exchange alone is sufficient to prevent

bank runs, the conclusions of some of these works may need to be reconsidered.

Exchange in the DD model opens the door to other preventative solutions to bank

runs. First, exchange creates prices which may reveal information about a bank’s financial

position. DD (1983, pp.403, 406) states that the concealment of private information pre-

vents competitive markets from providing optimal liquidity. However, information conveyed

through the price system might be used as a basis for contingent contracts that allow com-

petitive markets to achieve this optimum. Second, the simultaneous existence of goods and

deposit contracts enables trade of deposit contracts in times of suspension. This is consis-

tent with historical evidence and can help prevent runs as described in Selgin (1993). Third,

banks may be able to provide the same payoffs as government deposit insurance since risky

deposit contracts will trade at a discount in the market. These are but a few examples of

ways in which exchange between producers and depositors enables more realistic solutions

to be incorporated into the DD model.

This version of the DD model can also be used to study the importance of exchange in

inhibiting bank runs. One implication of the model is that restricting the exchange of deposit

contracts may actually cause bank runs (or at least handicap the market’s ability to deter

them). Historical evidence from the nineteenth and early twentieth centuries supports this

notion. The DD model can be used to analyze the degree to which government intervention

may have been the cause rather than deterrent of banks runs.

47

Chapter 4: Capital and Risk in Commercial Banking

4.1 Introduction

The US Federal Reserve is currently in the process of implementing stricter standards of

risk-based capital (RBC) for commercial banks based on the Basel II accords.1 These

measures are intended to improve upon the standard capital ratio of equity over assets, yet

there is serious disagreement on whether RBC requirements provide any new and useful

information. Here we examine the correlations in the levels of capital and RBC held by

bank holding companies to each bank’s bond yield spreads and the volatility of its stock

returns. Although both capital and RBC are correlated to these measures of risk, we fail

to find a significant difference between the coefficients of capital and RBC.

Basel II (2004, p.2) intends to “strengthen the soundness and stability of the interna-

tional banking system” through international RBC requirements. Bernanke (2007) noted

that “Strong capital levels and sound risk management are important for maintaining bank

safety and soundness and, thus, promoting financial stability more generally.” However,

many works have disputed the effectiveness of these statutes (DanIelsson et al. 2001, Van-

Hoose 2007, Blundell-Wingnall and Atkinson 2010).

We offer some evidence to this debate by examining the relationship between a bank’s

capital holdings and its perceived level of risk. To do this, we estimate the correlations of

bank capital and RBC to three indicators of a bank’s perceived risk of future insolvency: the

absolute value of stock returns, standard deviation of stock returns, and yield spreads on the

bank’s bonds. Capital and RBC are found to be significantly related to all three measures

of risk. However, the coefficient estimates for capital and RBC are not significantly different

from each other indicating that RBC does not provide an improvement over the standard

capital ratio.

1“Basel II” refers to the policies recommended by the Basel Committee on Banking Supervision (2004).

48

This paper contends that capital and RBC are both important indicators of bank sol-

vency but that RBC fails to provide new information about bank risk. The next section

describes the literature on bank capital and risk of insolvency. Section 4.3 outlines the

empirical model. Section 4.4 describes our sources and methods of data collection. Results

of our analysis are presented in section 4.5, and section 4.6 concludes.

4.2 Capital and Risk

Capital standards have a long history in US banking. Ronn and Verma (1989, p.21) claims

that capital regulation “is as old as the banking industry itself, since it directly affects

the safety of deposits.” Berger et al. (1995, p.401-403) describes the evolution of capital

standards in the US noting that the National Banking Act of 1863 created an “implicit 10%

regulatory capital ratio.” Formalized capital requirements were adopted in 1981, and RBC

requirements were introduced in 1992.

The US is currently in the process of implementing stricter RBC standards. Adoption

of the Basel Accords has required the Federal Reserve to “completely overhaul bank capital

requirements” (Furfine 2000, p.1). Every major BHC is required to report its RBC ratio

in quarterly reports to the Federal Reserve. The RBC ratio is calculated as risk-based

capital over risk-based assets. “Risk-based capital” is the sum of Tier 1 and Tier 2 capital

adjusted for illiquid items such as intangible assets and unrealized gains or losses. “Risk-

based assets” is the sum of all asset categories multiplied by their designated risk weight.

Risky assets receive a high risk weight which lowers the RBC ratio while safer assets are

assigned a lower weight which raises the RBC ratio. For example, holdings of subordinated

debt with less than 1 year to maturity receive a rating of 0% while subordinated debt with

five years or more to maturity receives a rating of 100%. All AAA rated securites (including

mortgage-backed securities) receive a risk weight of 0%. The Regulatory Capital Schedule

used to calculate a bank’s RBC ratio is provided in appendix C.1.

Bank capital has long been known to reduce default risk. Capital acts as a buffer

against liquidity shocks (Diamond and Rajan 2000, p.2431) and against portfolio losses

(Avery and Berger 1991, p.848; Cordell and King 1995, p.532). “Virtually every bank

failure model finds that higher equity-to-asset ratio is associated with lower probability of

49

default” (Berger et al. 1995, p.409). However, models of risk-based capital show mixed

results. “The theoretical banking literature is sharply divided about the effects of capital

requirements on bank behavior and, hence, on the risks faced by individual institutions and

the banking system as a whole” (VanHoose 2007, p.3681). Shrieves and Dahl (1992) claims

that RBC regulation can effectively limit bank volatility and risk while Hellman, Murdock,

and Stiglitz (2000) finds that RBC alone is not sufficient.

The empirical evidence on RBC is also mixed. Avery and Berger (1991, p.872) claims

that “risk weights provide an improvement over the old capital standards,” while Hirtle

(2003, p.38) states that “the market risk capital figures provide little additional information

about the extent of an institution’s market risk exposure.” These points are further debated

by DanIelsson et al. (2001), Gambacorta and Mistrulli (2004), Van den Heuvel (2008),

Berger et al. (2008) and Blundell-Wingnall and Atkinson (2010).

This paper studies the effectiveness of capital and RBC requirements by using equity

return volatility and bond yield spreads as proxies for bank risk. “Almost all asset pricing

theories rest on a specification of the way in which first moments (expected returns and

risk premia) depend on second moments (variances and covariances)” (Engle et al. 1990,

213). Engle (2004) describes how various ARCH models have been used to demonstrate the

relationship between firm risk and equity price volatility. Resti and Sironi (2007) shows a

relationship between firm risk and bond yield spreads. We follow this literature by testing

the influence of capital and RBC on bank risk by using a bank’s stock returns, volatility of

stock returns, and bond yield spread as proxies for the risk of future default.

Options theory provides insights into how a bank’s stock and bond prices influence its

probability of default. A firm’s bond prices, stock prices, and stock price volatility are used

to calculate the the value of the firm’s assets and the probability that the asset value will fall

to zero. Merton (1977) employed the options pricing model to analyze deposit insurance.

Based on the models of Black and Cox (1973) and Merton (1974), Hanweck (2001) explains

how a bank’s debt and equity prices can be used to calculate its probability of default with

the example of the Bank of New England. Hanweck and Spellman (2005) applies this model

to the Prompt Corrective Action standards for insolvent banks. Ronn and Verma (1989)

reverses this process by using market values of bank stock and bond prices to derive optimal

50

book-value capital standards. Cordell and King (1995) uses bank stock and bond prices

to estimate the market’s perception of bank capital and finds that it is highly related to

RBC. These studies confirm the relationship between a bank’s stock and bond prices and

its probability of default.

4.3 Empirical Model

This section describes a few simple econometric models intended to test the correlation

between the perceived risk of a BHC and its levels of capital and RBC. For each quarter we

sum the absolute values of daily returns and calculate the standard deviation of daily returns

on the stock prices of each BHC. We also calculate the average yield spread on the bonds of

each BHC. Each of these three quarterly cross-sections (absolute value of return on stocks,

standard deviation of return on stocks, and bond yield spreads) is then regressed against

the quarterly cross-section of BHC capital and RBC ratios to determine the influence of

these ratios on bank risk. Bank holdings of mortgage-backed securities (MBS) and real

estate assets are used as controls since these assets may also influence perceptions of the

bank’s level of risk.

4.3.1 Equity Price Volatility

We use two measures of equity price volatility: the sum of the absolute values of daily

returns and the standard deviation of daily returns. In order to calculate these measures,

we first compute the daily returns on equity. For each BHCs, daily return is calculated as

in equation 4.3.1 where pit is the closing price of bank i’s stock on day t.

rit =pit − pi(t−1)

pi(t−1)(4.3.1)

The absolute value of daily returns per quarter for each BHC are denoted αi and calculated

as in equation 4.3.2 where T is the number of trading days over the quarter and t is a

51

particular trading day.

αi =T∑t=1

|rit| (4.3.2)

The quarterly standard deviation of daily returns is computed as in equations 4.3.3 and

4.3.4.

ri =T∑t=1

rit/T (4.3.3)

σi =T∑t=1

(ri − rit)2 (4.3.4)

The median number of days is 62 per quarter. Companies with prices for fewer than 30

trading days in the quarter are excluded for that particular quarter. Equity price volatility

is a common indicator of firm risk. These two measures of price volatility will be used as

dependent variables in regressions on capital and RBC.

4.3.2 Bond Yield Spreads

For bonds, we calculate the average yield premium above US treasuries for each BHC in

each quarter. Each bond is matched with a US treasury bond index of similar maturity. The

indexes used are constant-maturities of 1, 2, 3, 5, 10, 20, and 30 years. Bonds are matched

to the index of the closest maturity. For example, a bond with 1.4 years to maturity is

matched to the 1-year maturity index while a bond with 1.5 or 1.6 years to maturity would

be matched to the 2-year maturity index. Once the proper index is identified, the spread

calculated as the yield on the bonds of bank i less the yield on treasury j. The average

spread is calculated for each BHC in each quarter as in equation 4.3.5 where T is the number

of trading days over the quarter and t is a particular trading day.

δi =T∑t=1

(yit − yjt)/T (4.3.5)

52

This quarterly average yield premium above the treasury rate indicates the perceived risk-

iness of the bond over the period. It will be used as the dependent variable in regressions

against BHC capital and RBC.

4.3.3 Regression Equations

The three measures of price volatility (absolute value of return on stocks, standard deviation

of return on stocks, and bond yield spreads) are used as the dependent variables in our

regression analysis. Independent variables are taken from each bank’s balance sheet, the

most important of which are capital and RBC. We include other risky asset categories of

real estate asset holdings and MBS as controls since they may influence market participants’

perception of the bank’s level of risk. Each independent variable (capital, RBC, real estate

loans, and MBS) is listed as a percentage of total assets. The regression equation for capital

on the absolute value of stock returns is shown in equation 4.3.6.

αi = βcap(capitali) + β0(reali) + β1(mbsi) + β2 + ε (4.3.6)

We repeat this test with the bank’s risk-based capital ratio replacing the capital ratio as

the main independent variable. This regression is given in equation 4.3.7.

αi = βrisk(RBCi) + β0(reali) + β1(mbsi) + β2 + ε (4.3.7)

We then repeat these regressions using standard deviation of equity prices σi as the depen-

dent variable and then again using bond yield spreads δi as the dependent variable.

These regression give us a series of quarterly cross-sectional beta estimates representing

the influence of capital and RBC on the riskiness of a BHC’s debt and equity. These

quarterly data are used to evaluate three testable hypotheses for each of our quarterly

cross-sections (absolute value of return on stocks, standard deviation of return on stocks,

and bond yield spreads). H1: Bank capital has no influence on stock returns or bond yields

(βcap = 0). H2: Bank risk-based capital has no influence on stock returns or bond yields

(βrisk = 0). H3: Bank capital and risk-based capital have the same influence on stock

returns and bond yields (βcap = βrisk).

53

The use of quarterly cross-sections has two advantages. First, we can test the significance

of βcap and βrisk in each quarter to find out not only when each β was significant but also how

their significance has changed over time. Second, using quarterly cross-sections excludes

time-dependent factors which may have affected the entire banking industry such as changes

in Fed policy or trends in the composition of bank balance sheets.

4.4 Data

The data for this analysis are taken from multiple sources. A list of BHC ID numbers

and corresponding CUSIPs as of 2008 is available from the Federal Reserve Bank of New

York (FRBNY).2 For this list of banks, we obtain stock prices from Wharton Research

Data Services (WRDS)3 and bond yield data from Thomson Reuters Datastream4 which

are matched with treasury yields to calculate the spread on each bond. These data sets are

then combined with quarterly balance sheet data from the Federal Reserve.

Daily stock prices for each bank holding company are downloaded from WRDS. Banks

are selected by PERMCO according to the bank list from the FRBNY. The fields selected

are date, PERCMO, CUSIP, and daily closing price for the period January 1, 1999 to

December 31, 2010. Any entries with characters or missing data in the price field are

dropped. A list of distinct CUSIP numbers is created from this data set to be used for

downloading bond data.

Daily bond yields for each bank holding company are downloaded from Datastream.

Bonds are selected if their CUSIP matches any CUSIP from the stocks data set. Fields

selected are date, CUSIP, book value, yield, and bond life (years to maturity) for the period

January 1, 1999 to December 31, 2010. Firms with less than one year of data are dropped.

For each firm, we calculate a daily average yield of all bonds outstanding weighted by book

value. This weighted average yield is used as the single bond yield for the firm. Each

bond is then matched to a constant-maturity index of US treasuries or similar maturity as

described in section 4.3.2. Treasury index data are downloaded from the Federal Reserve

2Available at http://www.newyorkfed.org/research/banking research/datasets.html.3Available at https://wrds-web.wharton.upenn.edu/wrds/.4Available at http://online.thomsonreuters.com/datastream/.

54

Bank of St. Louis.5

Data on the balance sheets and capital is obtained from the Federal Reserve Bank of

Chicago (FRBC) in their Consolidated Financial Statements for Bank Holding Companies

(Y-9C) reports.6 These quarterly reports contain full financial statements from all large

BHCs.7 The fields taken from these reports are report date, BHC ID number, total equity,

holdings of real estate assets, MBS, and risk-based capital ratio. New fields are calculated

for the ratios of capital, real estate, and MBS as percentages of total equity.

These three data sets are then matched using the FRBNY bank list. Stock prices are

joined to bond yield spreads using one-to-many mapping using date and CUSIP as the key

fields (for each firm, dates with a stock price but not bond spread are kept while dates with

a bonds spread but no stock price are dropped). For these daily yields and prices, a new

field is created to identify the year and quarter. This set of stock prices and bond yield

spreads is then matched to the balance sheet data using the quarter and BHC ID number

as the key fields. Daily yields and prices are matched to the firm’s balance sheet from the

beginning of the quarter (the end of the previous quarter).

Summary statistics for the final combined data set are given in appendix C.2. Table

C.2.1 contains means of balance sheet data and daily returns for the entire sample. There

are an average of 301 stock prices and 23 bond yields per quarter. Figure C.2.1 shows a

scatter plot of all quarterly BHC capital and risk-based capital ratios. Quarterly averages

for capital and RBC ratios are shown in figures C.2.2. Figure C.2.3 shows the quarterly

average yield on treasury indexes of 1, 5, and 10 year maturities along with the average BHC

bond yield in each quarter. Figure C.2.4 calculates the average yield spread per quarter.

C.2.5 shows quarterly standard deviation of stock prices and C.2.6 lists the number of stock

price observations per quarter. The number of observations per quarter declines slightly

over the sample period, likely due to the industry trend towards consolidation during this

time. Figures C.2.7 and C.2.8 show quarterly standard deviations of bond yields and the

number of bond yield observations per quarter. The number of observations is low in early

5Available at http://research.stlouisfed.org/fred2/categories/115.6Available at http://www.chicagofed.org/webpages/banking/financial institution reports/bhc data.cfm.7Until 2006, BHCs were required to complete the quarterly Y-9C report if their total assets exceeded

$150 million. Since 2006, the limit has been raised to $500 million, but some firms below this level continueto report.

55

years due to the fewer number of banks issuing bonds at that time but also due to the

difficulty of matching bond data to banks in those years, many of which are no longer in

existence. The year 2000 is not included in bond yield analysis since there were too few

observations to conduct the regression analysis.

4.5 Results

The results from these regressions show that both capital and RBC are related to bank risk

measured by bond yields and stock price volatility. Capital and RBC are most significantly

related to bond yields around the recession of 2001 and to stock returns since 2008. The

coefficients of capital and RBC are very similar in most regressions.

As described in section 4.3, our intent is to analyze the significance of βcap and βrisk

representing the influence of capital or RBC in each quarter and examine their changes

over time. We display the results of this analysis in a series of charts which plot time in

terms of quarters on the x-axis, and β coefficient on the y-axis. The solid line in each figure

represents the quarterly β estimate while the thin dotted lines represent upper and lower

confidence intervals at the 10% level. When the lower CI is above zero or the higher line

below zero, β is significant at the 10% level. These periods of significanct are shaded grey

in each figure. As mentioned in the previous section, analysis on bond yields does not begin

until 2001 due to the low number of observations before that time.

Several alternative specifications were used to test the robustness of these regressions

on leverage and capital. The bank asset categories of real estate assets and MBS were

included in regression equations because they were found to be significantly related to price

volatility. Coefficient estimates and standard errors for real estate assets and MBS are given

in appendix C.3. Other asset categories tested were subordinate debt, treasuries, trading

assets, and cash, all as percentages of equity. Most of these asset categories did have some

periods of significance but less often than real estate or MBS. All regressions yielded similar

overall results.

56

Figure 4.1: Absolute value of stock returns

Figure 4.1.a. Beta of absolute stock returns on capital

Figure 4.1.b. Beta of absolute stock returns on risk-based capital

Figure 4.1.c. Betas of absolute stock returns on capital and risk-based capital

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57

4.5.1 Absolute Value of Stock Returns

Figure 1 shows the results of the quarterly regressions of the absolute values of stock returns

on capital and RBC. Figure 1.a shows the quarterly estimates of βcap which are used to test

H1. The chart is shaded gray in 10 of the 43 quarters indicating that the βcap coefficient

is significant in these periods. We find significance for 2 quarters in 2001 and all quarters

since Q4 of 2008. Figure 1.b tests H2 with quarterly estimates of βrisk which are significant

in 8 of 43 quarters. These are Q3 of 2001 and since Q1 of 2009, all of which were also

significant

Figure 1.c tests H3 that βcap = βrisk. Quarterly estimates of βcap are shown as a dotted

black line while estimates of βrisk are shown in solid black. We can see that the patterns

of these coefficients are quite similar. Periods where these coefficients are significantly

different at the 10% level would be shaded grey except that there are no such periods. In

every quarter we fail to reject the null hypothesis that βcap = βrisk.

4.5.2 Standard Deviation of Stock Returns

Figure 2 shows the results of quarterly regressions of the standard deviation of stock returns

on capital and RBC. Figure 2.a shows the quarterly estimates of βcap. Grey shading indi-

cates that we reject the null hypothesis H1 at the 10% level. The coefficient is significant in

10 of 43 quarters: Q4 of 2001, Q2 of 2004, and since Q4 of 2008. Figure 2.b gives the βrisk

estimates which are used to test H2. We find that βrisk is significant in 8 of 43 quarters.

These are Q3 of 2005 and all quarters since Q1 of 2009.

Figure 2.c tests hypothesis H3 that βcap = βrisk. It shows the estimates of βcap as

a dotted line and βrisk as a solid line. The trends are again similar. Periods when the

estimates are significantly different would be shaded gray, but again there are none. In all

43 quarters we fail to reject the null hypothesis.

4.5.3 Bond Yield Spreads

Figure 2 shows the results of quarterly regressions of bond yield spreads on capital and

RBC. H1 is tested in figure 2.a which shows the quarterly estimates of βcap. We can reject

H1 in 6 quarters in which the coefficient is found to be significant: Q2 of 2002 and Q4 of

58

Figure 4.2: Standard deviation of stock returns

Figure 4.2.a. Beta of standard deviation of stock returns on capital

Figure 4.2.b. Beta of standard deviation of stock returns on risk-based capital

Figure 4.2.c. Beta of standard deviation of stock returns on capital and risk-based capital

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2005 to Q4 of 2006. Figure 2.b gives the βrisk estimates in each quarter which are used to

test H2. The coefficient is significant in 7 quarters from Q2 of 2001 to Q2 of 2002, Q4 of

2002, and Q2 of 2003.

Figure 3.c compares the quarterly coefficient estimates of βcap and βrisk. Like figures

1.c and 2.c, βcap is shown as a dotted black line, and βrisk is shown in solid black. However,

unlike 1.c and 2.c, there appears to be more variation between these estimates. Both begin

below 0 and decline in 2001, although βrisk reaches a minimum in Q4 of 2001 while βcap

declines until Q2 of 2002. βcap then becomes positive in 2003 and stays mostly positive

until 2005, while βrisk remains negative over the period. Despite these modest variations,

the trends of βcap and βrisk are generally similar. They both start off negative, first decline,

then begin to rise. Both are close to 0 from 2005 through 2008 and become slightly negative

in 2009. The lack of shading in figure 3.c indicates that in every quarter we fail to reject

the null hypothesis H3 that βcap = βrisk.

Overall, the results from figures 1-3 are very similar. Estimates of the β coefficients tend

to be significant around the recessions of 2001, especially for bonds, and of 2008-2009 for

stocks. One unusual period is the significance of βcap for bond yields in 2005 and 2006 as

shown in figure 3.a. However the coefficient is small and is not significantly different from

βrisk. The trends of βcap and βrisk are remarkably similar in figures 1.c, 2.c, and 3.c, and

these coefficient estimates are not found to be significantly different from each other in any

quarter. Capital and RBC are found to be significantly related to bank risk but are not

significantly different from each other.

4.6 Conclusion

This paper compares the influences of capital and risk-based capital on bank default risk. We

analyze the correlations of capital and RBC to three indicators of bank risk: the absolute

value of stock returns, the standard deviation of stock returns, and bond yield spreads.

Both capital and RBC are found to be significantly related to all three measures of risk in

several quarters, especially around the recessions of 2001 and 2008-2009. However, the β

coefficients of capital and RBC are not found to be significantly different from each other

in any quarter.

60

Figure 4.3: Bond yield spreads

Figure 4.3.a. Beta of bond yield spreads on capital

Figure 4.3.b. Beta of bond yield spreads on risk-based capital

Figure 4.3.c. Beta of bond yield spreads on capital and risk-based capital

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61

Appendix A: Gresham’s Law Alternative Data

England World Over Trans Recoinage Weight const N R2Reddish Shaw/Farrer 0.084∗∗∗ 0.659∗∗∗ 445 6.2%

(0.015) (0.013)0.131∗∗∗ 0.668∗∗∗ 467 15.7%(0.014) (0.012)0.086∗∗∗ -0.053 -0.158∗ 0.144∗∗ 0.664∗∗∗ 445 8.5%(0.016) (0.033) (0.088) (0.061) (0.015)0.124∗∗∗ -0.091∗∗∗ -0.193 0.004 0.69∗∗∗ 467 18.6%(0.014) (0.029) (0.081) (0.054) (0.014)

Reddish Farrer 0.189∗∗∗ 0.707∗∗∗ 305 14.4%(0.027) (0.016)0.223∗∗∗ 0.714∗∗∗ 312 21.8%(0.024) (0.014)0.209∗∗∗ -0.124∗∗∗ -0.059 0.125∗∗ 0.733∗∗∗ 305 19%(0.026) (0.035) (0.119) (0.061) (0.019)0.239∗∗∗ -0.155∗∗∗ -0.289∗∗∗ -0.023 0.762∗∗∗ 312 29.5%(0.023) (0.031) (0.107) (0.054) (0.017)

Reddish Shaw 0.003 0.679∗∗∗ 445 0%(0.018) (0.014)0.047∗∗∗ 0.688∗∗∗ 467 1.7%(0.017) (0.013)0.059∗∗∗ 0.215∗∗∗ -0.098 0.202∗∗∗ 0.595∗∗∗ 445 11.7%(0.018) (0.03) (0.087) (0.062) (0.018)0.104∗∗∗ 0.26∗∗∗ -0.120 0.090 0.598∗∗∗ 467 18.9%(0.017) (0.027) (0.081) (0.056) (0.016)

Feavearyear Shaw/Farrer 0.129∗∗∗ 0.518∗∗∗ 445 11.8%(0.017) (0.024)0.204∗∗∗ 0.445∗∗∗ 467 31.2%(0.014) (0.02)0.144∗∗∗ 0.015 -0.174∗∗ 0.202∗∗∗ 0.492∗∗∗ 445 14.7%(0.019) (0.06) (0.085) (0.059) (0.028)0.215∗∗∗ 0.031 -0.218∗∗∗ 0.101∗∗ 0.43∗∗∗ 467 33.1%(0.016) (0.051) (0.073) (0.05) (0.024)

Feavearyear Farrer 0.237∗∗∗ 0.455∗∗∗ 305 23.2%(0.025) (0.028)0.326∗∗∗ 0.376∗∗∗ 312 47.4%(0.019) (0.022)0.282∗∗∗ 0.122∗∗ -0.126 0.219∗∗∗ 0.391∗∗∗ 305 27.4%(0.028) (0.062) (0.112) (0.059) (0.034)0.366∗∗∗ 0.141∗∗∗ -0.376∗∗∗ 0.123∗∗∗ 0.325∗∗∗ 312 52.5%(0.022) (0.048) (0.088) (0.045) (0.026)

Feavearyear Shaw 0.037∗ 0.631∗∗∗ 445 0.7%(0.021) (0.03)0.123∗∗∗ 0.542∗∗∗ 467 8.5%(0.019) (0.027)0.047∗∗ -0.098 -0.164∗ 0.162∗∗ 0.617∗∗∗ 445 2.8%(0.023) (0.082) (0.091) (0.067) (0.034)0.128∗∗∗ -0.036 -0.201∗∗ 0.074 0.537∗∗∗ 467 9.8%(0.02) (0.076) (0.085) (0.061) (0.031)

*** Significant at the 1% level.** Significant at the 5% level.* Significant at the 10% level.

62

Appendix B: The Generalized Diamond-Dybvig Model

B.1 Optimal Consumption

Let us use the same model set forth in section 3.2. The production technology has payoffs:

T=0 T=1 T=2

-1

{0 R1 0

Each agent is endowed with 1 unit of capital which he is required to invest at T = 0. Payoffs

are a nonnegative linear combination of 1 at T = 1 and R at T = 2. The tradeoff between

consumption in periods T = 1 and T = 2 forms the resource constraint.

c2 = R(1− c1) (B.1)

Agents are of two types, either impatient Θ = 1 or patient Θ = 2. The notation cΘt

represents consumption by agent type Θ in time t. The utility function given in equation

B.2 is dependent on agent type Θ with 0 < ρ < 1, 0 < γ < 1, and cΘT ≥ 0.

u(c1, c2,Θ) =

{(c11)1−γ

1−γ if Θ = 1ρ(c21+c22)1−γ

1−γ if Θ = 2(B.2)

Additionally, we assume that agents would always prefer to be patient and consume at

T = 2. Therefore, we know that u(c2) > u(c1).

The proportion t ∈ (0, 1) of type 1 agents in the population of N agents is calculated in

equation B.3 and is assumed (for now) to be known to all agents.

t = ΣN1 (2−Θ)/N (B.3)

Total social utility is the weighted sum of expected utilities for all agents.

V = tu(c11) + (1− t)ρu(c2

1 + c22) (B.4)

63

Social utility is subject to the budget constraint from B.1 which can be re-written to solve

for c22 in terms of the variables c1

1 and c21 and constants t and R.

c22 =

R(1− tc11 + (1− t)c2

2)

(1− t)(B.5)

Proposition: The socially optimum levels of consumption c11, c

12, c

21, and c2

2 for agent types

1 and 2 in periods T = 1 and T = 2 (calculated ex ante as of time T = 0) are not equal to

the consumption levels of independent agents c11 = 1, c2

2 = R, c12 = c2

1 = 0.

First, we can see from the utility functions in equation B.2 that c12 = 0 since type 1 agents

get no utility from time 2 consumption. The other optimum consumption values can be

found according to the Lagrangian function in B.6.

L = tu(c11) + (1− t)ρu(c2

1 + c22) + λ

(R[1− tc1

1 − (1− t)c21]− (1− t)c2

2) (B.6)

First order conditions are:

∂L

∂c11= tu′(c1

1) + λRt = 0 (B.7)

∂L

∂c21= (1− t)ρu′(c2

1 + c22) + λR(1− t) = 0 (B.8)

∂L

∂c22= (1− t)ρu′(c2

1 + c22) + λ(1− t) = 0 (B.9)

∂L

∂λ= R[1− tc1

1 − (1− t)c21]− (1− t)c2

2 = 0 (B.10)

Comparing ∂L∂c21

and ∂L∂c22

we find that they have equal marginal benefit ρu′(c21 + c2

2), however

∂L∂c21

has a higher marginal cost R(1 − t) > (1 − t). Agents will, therefore, choose the

minimum amount of c21 = 0. Equations A.4 and A.6 can then be rearranged into B.11 and

B.12 respectively.

−λ =u′(c1

1)

R(B.11)

64

−λ = ρu′(0 + c22) (B.12)

These equations can be combined into equation B.13 and rearranged to form B.14.

u′(c22) =

u′(c11)

ρR(B.13)

c22 = (ρR)1/γc1

1 (B.14)

This version of c22 given in B.14 can now be inserted back into ∂L

∂λ into B.10.

0 = R[1− tc11 − (1− t)(0)]− (1− t)(ρR)1/γc1

1 (B.15)

By rearranging B.15, we solve for optimal consumption of type 1 agents in T = 1.

c11 =

R

tR+ (1− t)(ρR)1/γ(B.16)

Now inserting equation B.16 back into equation B.14, we can solve for optimal consumption

at T = 2.

c22 =

ρR1/γ

tR+ (1− t)(ρR)1/γ(B.17)

The final step of this proof is to show that the socially optimal levels of consumption are

not equal to the private consumption levels c11 6= 1 and c2

2 6= R. To prove this for c11, we

recall the assumptions that 0 < ρ < 1 and 0 < γ < 1 imply the following inequalities.

R > ρR > (ρR)1/γ (B.18)

Therefore the numerator of B.16 must be larger than the denominator.

R = tR+ (1− t)R > tR+ (1− t)(ρR)1/γ (B.19)

Comparing this to equation B.16, we see that c11 > 1.

c11 =

R

tR+ (1− t)(ρR)1/γ>

R

tR+ (1− t)R= 1 (B.20)

65

To get c22 from c1

1, we multiply by (ρR)1/γ as in equation B.14. Performing this on equation

B.20 and again using the fact that (ρR)1/γ < R, we see that c22 < R.

c22 =

R(ρR)1/γ

tR+ (1− t)(ρR)1/γ< R

((ρR)1/γ

t(ρR)1/γ + (1− t)(ρR)1/γ

)= R(1) = R (B.21)

B.2 Actual Consumption in the All-Bank Economy

This section will calculate the actual ex post incentives and consumption as of period T = 1

as described in section 3.2.3. We show that optimal ex post consumption values do not

match the ex ante expected optima.

We assume the same technology and preferences described in appendix B.1 but with a few

changes. First, the portion t of type 1 agents was known to all agents. It is now assumed to

be random and stochastic. Instead, there is some expectation t that is shared by all agents

at T = 0. Agents discover their own type at T = 1, but this information is not publicly

observable. Therefore, agents cannot tell whether the actual value of t is higher, lower, or

equal to the expected value t.

As demonstrated in appendix B.1, agents cannot achieve the socially optimal levels of

consumption through private action alone. Let us now assume that agents engage in the

group action of creating a bank at time T = 0. All agents deposit their 1 unit of capital

into the bank. They can withdraw some amount r1 at time T = 1. The remaining capital

will be divided among the patient agents who receive r2 at T = 2 with 1 < r1 < r2 < R.

The values of r1 and r2 are set at the expected optima of c1 and c2 calculated from t.

r1 = c11 =

R

tR+ (1− t)(ρR)1/γ(B.1)

66

r2 = c22 =

R(ρR)1/γ

tR+ (1− t)(ρR)1/γ(B.2)

These new potential payoffs change the agents’ resource constraint. They now have the

option of receiving r1 at T = 1 or some portion of the remaining capital r2 at T = 2.

Although the expected value of r2 is the optimal consumption level r2 = c22, the actual

value of r2 is a function of c11 + c2

1 as shown in equation B.3.

r2 =R(1− r1(c1

1 + c21))

1− c11 − c2

1

(B.3)

The important difference between this constraint and equation B.1 used in the previous

section is that B.3 is nonlinear. We can see that not all values of c22 will fit the constraint

that c22 > c1

1. In fact, r2 can be negative or undefined for some values of c11 + c2

1, but these

values are not feasible since no type 2 agent will choose negative consumption. Since type

2 agents have the option of consuming at time T = 1 or T = 2, type 2 agents will choose

whichever value is higher between r1 and r2. For any values r2 < r1, c22 goes to zero. In

order to satisfy c22 > c1

1, we assume that equation B.3 holds only for values of c11 +c2

1 < 1/r1.

Otherwise, r2 = 0.

The social utility function is the same as before.

WBank = tu(c11) + (1− t)ρu(c2

1 + c22) (B.4)

Proposition: When agents invest all capital in the bank, their actual ex post levels of

consumption c11, c

12, c

21, and c2

2 (calculated at T = 1) are not equal to the ex ante expected

optima c11, c

12, c

21, and c2

2 (calculated at T = 0).

First, we can see that as before that c12 is not included in the utility function, so c1

2 = 0. We

can therefore conclude that each type 1 agent consumes c11 = r1, and the total consumption

by all t type 1 agents is tr1. We can, thus, rewrite our utility equation to solve only for c21

67

and c22.

WBank = tu(r1) + (1− t)ρu(c21 + c2

2) (B.5)

The resource constraint is given in B.6.

c22 =

{R(1−r1(t+c21))

1−t−c21for c2

1 < 1/r1 − t0 for c2

1 ≥ 1/r1 − t(B.6)

To optimize WBank, we use the Lagrangian function in B.7.

L = tu(r1) + (1− t)ρu(c21 + c2

2) + λ

(R(1− r1(t+ c2

1))

1− t− c21

− c22

)(B.7)

First order conditions are:

∂L

∂c21= (1− t)ρu′(c2

1 + c22) + λ

(−Rr1(1− t− c2

1) +R(1− r1(t− c21)

(1− t− c21)2

)= 0 (B.8)

∂L

∂c22= (1− t)ρu′(c2

1 + c22)− λ = 0 (B.9)

∂L

∂λ=R(1− r1(t+ c2

1))

1− t− c21

− c22 = 0 (B.10)

Equations B.8 and B.10 can then be combined into B.11.

0 = λ+ λ

(−Rr1(1− t− c2

1) +R(1− r1(t− c21)

(1− t− c21)2

)(B.11)

Dividing both sides by λ and multiplying by (1− t− c21)2, we get B.12.

0 = (1− t− c21)2 −Rr1(1− t− c2

1) +R(1− r1(t− c21)) (B.12)

This can be expanded to B.13 and simplified to B.14.

0 =(1− t− c21 − t− t2 + tc2

1 − c21 + tc2

1 + (c21)2)

− (Rr1 +Rr1t+Rr1c21) + (R−Rr1t−Rr1c

21) (B.13)

68

0 = (c21)2 + (2t− 2)c2

1 + (1− 2t+ t2 −Rr1 +R) (B.14)

Using the quadratic formula to solve for c21, we find B.15 which can be simplified to B.16.

c21 =−(2t− 2)±

√(2t− 2)2 − 4(1− 2t+ t2 −Rr1 +R)

2(B.15)

c21 = 1− t±

√R(r1 − 1) (B.16)

This gives us two possible cases for c21. Inserting these potential values into our resource

constraint from equation B.6, we find that the value of c21 = 1− t+

√R(r1 − 1) violates the

assumption of c22 ≤ R as shown in equation B.17.

c22 = R

(1− r1t− r1(1− t+

√R(r1 − 1))

1− t− (1− t+√R(r1 − 1))

)= R

(1− r1t− r1 + r1t− r1

√R(r1 − 1))

1− t− 1 + t−√R(r1 − 1)

)

= R

(1− r1 − r1

√R(r1 − 1)

−√R(r1 − 1)

)= R

(r1 − 1

−√R(r1 − 1)

+−r1

√R(r1 − 1)

−√R(r1 − 1)

)

= R

(−(r1 − 1)

−√R(r1 − 1)

+ r1

)= R

(r1 +

(r1 − 1)√R(r1 − 1)

)

> R(r1) > R(1) = R (B.17)

Alternatively, we see that c21 = 1 − t −

√R(r1 − 1) in equation B.18 fits the assumption

that c22 ≤ R.

c22 = R

(1− r1t− r1(1− t−

√R(r1 − 1))

1− t− (1− t+√R(r1 − 1))

)= R

(1− r1t− r1 + r1t− r1

√R(r1 − 1))

1− t− 1 + t+√R(r1 − 1)

)

= R

(1− r1 + r1

√R(r1 − 1)√

R(r1 − 1)

)= R

(1− r1√R(r1 − 1)

+r1

√R(r1 − 1)√R(r1 − 1)

)

= R

(−(r1 − 1)√R(r1 − 1)

+ r1

)= R

(r1 −

(r1 − 1)√R(r1 − 1)

)= R

(r1 −

√(r1 − 1)

R

)

< R(r1 − (r1 − 1)) = R(1) = R (B.18)

69

We can therefore conclude that the optimal levels of consumption c21 andc2

2 are given in

equations B.19 and B.20.

c21 = 1− t−

√R(r1 − 1) (B.19)

c22 =

R−Rr1

(1−

√R(r1 − 1)

)√R(r1 − 1)

(B.20)

We can see that these ex post optimal consumption levels do not match the ex ante expected

optima since c21 > 0, and c2

2 < r2.

B.3 Welfare in the Multi-Sector Economy

This section will show that investment in both banking and production sectors creates a

welfare improvement. We begin by restating the social welfare function of the multi-sector

economy as described in figure 3.4 and equation 3.3.1.

WMS =(1− δ)[tu(φr1 + (1− φ)(1)) + (1− t)ρu(φr2 + (1− φ)R)]

+ δt[(1/r1)u(φr1 + (1− φ)(1)) + (1− 1/r1)u(φ(0) + (1− φ)(1))]

+ δ(1− t)ρ[(1/r1)u(φr1 + (1− φ)R) + (1− 1/r1)u(φ(0) + (1− φ)R)] (B.1)

We can see that when the there is zero probability of a bank run (δ = 0), then all capital

will be invested in the bank (φ = 1), so

WMS = (1− δ)[tu(c11) + (1− t)ρu(c2

2)] + δ[(1/r1)u(r1) + (1− 1/r1)u(0)] = WBank (B.2)

Alternatively, when a bank run is expected (δ = 1), then no capital will be invested in the

bank (φ = 0).

WMS = tu(c11) + (1− t)ρu(c2

2) = WProd (B.3)

However, these scenarios have been ruled out by prior assumptions. Section 3.2.1 assumed

70

that the probability of a run is positive. Section 3.2.4 assumed that the probability of a run

is small enough that a bank will be formed. Therefore, we need only consider a the case

0 < φ < 1 where positive amounts of capital are invested in both banking and production.

Proposition: For an economy with positive probability of a bank run δ > 0, dividing

capital between production and banking sectors such that 0 < φ < 1 creates higher welfare

than investing in either sector alone.

The social utility function B.1 takes advantage of the agents’ relative risk aversion as shown

in equation B.2. Therefore, he prefers u(φx+ (1−φ)y) over φu(x) + (1−φ)u(y). As shown

in figure 3.4, there are six potential utility outcomes which depend on the agent’s type,

his holdings of goods and deposits, and some random probability that the agent will be

able to redeem his deposits in the case of a bank run. In each of these cases, the agent is

partly hedged since against the possibility of a bank run as demonstrated by the following

inequalities:

u(φr1 + (1− φ)(1)) > φu(r1) + (1− φ)u(1) (B.4)

u(φr2 + (1− φ)R) > φu(r2) + (1− φ)u(R) (B.5)

u(φr1 + (1− φ)(1)) > φu(r1 + (1− φ)u(1) (B.6)

u(φ(0) + (1− φ)(1)) > φu(0) + (1− φ)u(1) (B.7)

u(φr1 + (1− φ)R) > φu(r1 + (1− φ)u(R) (B.8)

u(φ(0) + (1− φ)R) > φu(0) + (1− φ)u(R) (B.9)

Since each of the hedged values is lower then the unhedged values, we know that substituting

these into B.1 will give us something lower than WMS . We then rearrange these and simplify

71

to get equation B.13.

WMS >(1− δ)[t(φu(r1) + (1− φ)u(1)) + (1− t)ρ(φu(r2) + (1− φ)u(R))]

+ δt[(1/r1)(φu(r1 + (1− φ)u(1)) + (1− 1/r1)(φu(0) + (1− φ)u(1))]

+ δ(1− t)ρ[(1/r1)(φu(r1 + (1− φ)u(R)) + (1− 1/r1)(φu(0) + (1− φ)u(R))]

(B.10)

=δ(1− φ)t(1/r1)u(1) + δ(1− φ)t(1− 1/r1)u(1) + (1− δ)(1− φ)tu(1)

+ δ(1− φ)(1/r1)tρu(R) + δ(1− φ)(1− 1/r1)tρu(R) + (1− δ)(1− φ)(1− t)ρu(R)

+ (1− δ)φtu(r1) + (1− δ)φ(1− t)ρu(r2)

+ δφ(1/r1)tu(r1) + δφ(1/r1)tρu(r1) + δφ(1− 1/r1)tu(0) + δφ(1− 1/r1)(1− t)ρu(0)

(B.11)

=(1− φ)[tu(1) + (1− t)ρu(R)]

+ φ(1− δ)[tu(r1) + (1− t)ρu(r2)]

+ φδ[(1/r1)tu(r1) + (1− t)ρ(u(r1) + (1− 1/r1)tu(0) + (1− 1/r1)(1− t)ρu(0)]

(B.12)

=(1− φ)WProd + φWBank (B.13)

B.4 Consumption in the Multi-Sector Economy

This section will calculate the actual ex post incentives and consumption as of period T = 1

as described in section 3.3. We assume that preferences described in appendix B.1 and that

resource constraints are a combination of the regular technological constraint from equation

B.1 in section B.1 and the banking constraint from equation B.3 of section B.2. These

72

constraints are combined according to the portions of capital invested in the production

and banking sectors as shown in equation B.1.

c2 = (1− φ)R[1− (1− φ)c21] + φ

R(1− r1(t+ c21))

1− t− φc21

(B.1)

We assume that each type 1 depositors consume c11 = r1 and type 2 depositors divide

consumption between c21 and c2

2. The social utility function is the same as before.

WBank = tu(r1) + (1− t)ρu(c21 + c2

2) (B.2)

To optimize c21 and c2

2, we use the Lagrangian function in B.3.

L = tu(r1)+(1−t)ρu(c21 +c2

2)+λ

((1−φ)R[1−(1−φ)c2

1]+φR(1− r1(t+ c2

1))

1− t− φc21

−c22

)(B.3)

First order conditions are:

∂L

∂c21= (1− t)ρu′(c2

1 + c22) + λφR

(−r1(1− t− c2

1) + (1− r1(t− c21))

(1− t− c21)2

)+ λ(1− φ)R

−1

1− t= 0

(B.4)

∂L

∂c22= (1− t)ρu′(c2

1 + c22)− λ = 0 (B.5)

∂L

∂λ= (1− φ)R[1− (1− φ)c2

1] + φR(1− r1(t+ c2

1))

1− t− φc21

− c22 = 0 (B.6)

Equations B.4 and B.6 can then be combined into B.7.

−λ = λφR−(r − 1)

(1− t− c21)2

+ λ(1− φ)R−1

1− t(B.7)

Dividing both sides by −λ, we get B.8.

1 = φRr − 1

(1− t− c21)2

+R1− φ1− t

(B.8)

73

Re-arranging and multiplying by (1− t− c21)2, we get B.9

(1−R1− φ

1− t)(1− t− c2

1)2 = φR(r − 1) (B.9)

Then dividing both sides by (1−R 1−φ1−t ), we get equation B.10.

(1− t− c21)2 =

φR(r − 1)(1− t)1−R(1− φ)

(B.10)

We take the square root of both sides to get B.11 and re-arranged into B.12.

1− t− c21 =

√φR(r − 1)(1− t)

1−R(1− φ)(B.11)

c21 = 1− t±

√φR(r − 1)(1− t)

1−R(1− φ)(B.12)

We know from equation B.18 that in order to satisfy our prior assumptions the square root

in equation B.12 must be subtracted rather than added. We can therefore conclude that

the optimal levels of consumption c21 andc2

2 are given in equations B.13 and B.14.

c21 = 1− t−

√φR(r − 1)(1− t)

1−R(1− φ)(B.13)

c22 =

R−Rr(

1 +√

φR(r−1)(1−t)1−R(1−φ)

)√

φR(r−1)(1−t)1−R(1−φ)

(B.14)

74

Appendix C: Leverage and Capital Data

C.1 Federal Reserve Y9-C Regulatory Capital Schedule

Tier 1 capital 1. Total bank holding company equity capital (from Schedule HC, item 27.a) ....................................... 1. 2. LESS: Net unrealized gains (losses) on available-for-sale securities1 (if a gain, report as a

positive value; if a loss, report as a negative value)........................................................................... 2. 3. LESS: Net unrealized loss on available-for-sale equity securities1 (report loss as a positive value) .. 3. 4. LESS: Accumulated net gains (losses) on cash flow hedges1 (if a gain, report as a positive value;

if a loss, report as a negative value) .................................................................................................. 4. 5. LESS: Nonqualifying perpetual preferred stock ................................................................................. 5. 6. a. Qualifying Class A noncontrolling (minority) interests in consolidated subsidiaries ...................... 6.a.

b. Qualifying restricted core capital elements (other than cumulative perpetual preferred stock)2 ... 6.b. c. Qualifying mandatory convertible preferred securities of internationally active bank holding

companies .................................................................................................................................... 6.c. 7. a. LESS: Disallowed goodwill and other disallowed intangible assets .............................................. 7.a.

b. LESS: Cumulative change in fair value of all financial liabilities accounted for under a fair value option that is included in retained earnings and is attributable to changes in the bank holding company's own creditworthiness (if a net gain, report as a positive value; if a net loss, report as a negative value) ............................................................................................................ 7.b.

8. Subtotal (sum of items 1, 6.a., 6.b., and 6.c., less items 2, 3, 4, 5, 7.a, and 7.b) .............................. 8. 9. a. LESS: Disallowed servicing assets and purchased credit card relationships ............................... 9.a.

b. LESS: Disallowed deferred tax assets .......................................................................................... 9.b.10. Other additions to (deductions from) Tier 1 capital ............................................................................ 10.11. Tier 1 capital (sum of items 8 and 10, less items 9.a and 9.b) ........................................................... 11.

Tier 2 capital12. Qualifying subordinated debt, redeemable preferred stock, and restricted core capital elements2

(except Class B noncontrolling (minority) interest) not includible in items 6.b. or 6.c. ..................... . 12.13. Cumulative perpetual preferred stock included in item 5 and Class B noncontrolling (minority)

interest not included in 6.b., but includible in Tier 2 capital ................................................................ 13.14. Allowance for loan and lease losses includible in Tier 2 capital ......................................................... 14.15. Unrealized gains on available-for-sale equity securities includible in Tier 2 capital ........................... 15.16. Other Tier 2 capital components ........................................................................................................ 16.17. Tier 2 capital (sum of items 12 through 16) ........................................................................................ 17.18. Allowable Tier 2 capital (lesser of item 11 or 17) ................................................................................ 18.

19. Tier 3 capital allocated for market risk ............................................................................................... 19.20. LESS: Deductions for total risk-based capital .................................................................................... 20.21. Total risk-based capital (sum of items 11, 18, and 19, less item 20) .................................................. 21.

Total assets for leverage ratio22. Average total assets (from Schedule HC-K, item 5) .......................................................................... 22.23. LESS: Disallowed goodwill and other disallowed intangible assets (from item 7.a above) ............... 23.24. LESS: Disallowed servicing assets and purchased credit card relationships (from item 9.a above) . 24.25. LESS: Disallowed deferred tax assets (from item 9.b above) ............................................................ 25.

26. LESS: Other deductions from assets for leverage capital purposes .................................................. 26.27. Average total assets for leverage capital purposes (item 22 less items 23 through 26) .................... 27.28.–30. Not applicable

3210 BHCK

8434 A221 4336 B588 G214 G215 G216 B590 F264 C227 B591 5610 B592 8274

G217 G218 5310 2221 B594 5311 8275 1395 B595 3792

bhct

3368 B590 B591 5610 BHCK

B596 A224

FR Y–9CPage 41Schedule HC-R—Regulatory Capital

This schedule is to be submitted on a consolidated basis. For Federal Reserve Bank Use Only

C.I.

Dollar Amounts in Thousands

BHCX Bil Mil Thou

3/09

1. Report amount included in Schedule HC, item 26.b, "Accumulated other comprehensive income."2. Includes subordinated notes payable to unconsolidated trusts issuing trust preferred securities net of the bank holding company's investment

in the trust, trust preferred securities issued by consolidated special purpose entities, and Class B and Class C noncontrolling (minority) interests that qualify as Tier 1 capital.

7204 7206 7205

BHCK Percentage

. % . % . %

Capital ratios31. Tier 1 leverage ratio (item 11 divided by item 27) .................................................................... 31.32. Tier 1 risk-based capital ratio (item 11 divided by item 62) ...................................................... 32.33. Total risk-based capital ratio (item 21 divided by item 62) ....................................................... 33.

75

(Col

umn

A)

Tota

ls(fr

omS

ched

ule

HC

)

FR Y

–9C

Pag

e 42

Sche

dule

HC

-R—

Con

tinue

d

Dol

lar A

mou

nts

in T

hous

ands

C00

0

Bal

ance

She

et A

sset

Cat

egor

ies

34.

Cas

h an

d du

e fro

m d

epos

itory

inst

itutio

ns (c

olum

n A

equa

ls th

e su

m o

f Sch

edul

e H

C, i

tem

s 1.

a, 1

.b.(1

) and

1.

b.(2

)) ...

......

......

......

......

......

......

......

......

......

......

......

......

......

.....

34

.

35.

Hel

d-to

-mat

urity

sec

uriti

es ..

......

......

......

......

......

......

......

......

.....

35

.

36.

Avai

labl

e-fo

r-sa

le s

ecur

ities

.....

......

......

......

......

......

......

......

......

36

.37

. Fe

dera

l fun

ds s

old

and

secu

ritie

s pu

rcha

sed

unde

r ag

reem

ents

to re

sell.

......

......

......

......

......

......

......

......

......

......

....

37.

38.

Loan

s an

d le

ases

hel

d fo

r sal

e ...

......

......

......

......

......

......

......

...

38.

39.

Loan

s an

d le

ases

, net

of u

near

ned

inco

me

......

......

......

......

.....

39

.

40.

LES

S: A

llow

ance

for l

oan

and

leas

e lo

sses

......

......

......

......

.....

40

.

41.

Trad

ing

asse

ts ...

......

......

......

......

......

......

......

......

......

......

......

.....

41

.

42.

All

othe

r ass

ets1 .

......

......

......

......

......

......

......

......

......

......

......

....

42.

43.

Tota

l ass

ets

(sum

of i

tem

s 34

thro

ugh

42) .

......

......

......

......

......

43

.

◄

B

il M

il Th

ou

Bil

Mil

Thou

B

il M

il Th

ou

Bil

Mil

Thou

B

il M

il Th

ou

Bil

Mil

Thou

(Col

umn

B)

Item

s N

otS

ubje

ct to

Ris

k-W

eigh

ting

(Col

umn

C)

(Col

umn

D)

(Col

umn

E)

(Col

umn

F)

0%20

%50

%10

0%

1. In

clud

es p

rem

ises

and

fi xe

d as

sets

, oth

er re

al e

stat

e ow

ned,

inve

stm

ents

in u

ncon

solid

ated

sub

sidi

arie

s an

d as

soci

ated

com

pani

es, d

irect

and

indi

rect

inve

stm

ents

in re

al e

stat

e ve

ntur

es, i

ntan

-gi

ble

asse

ts, a

nd o

ther

ass

ets.

9/05

B

HC

E

BH

C0

BH

C2

BH

C5

BH

C9

Ban

k ho

ldin

g co

mpa

nies

are

not

req

uire

d to

ris

k-w

eigh

t eac

h on

-bal

ance

she

et a

sset

and

the

cred

it eq

uiva

lent

am

ount

of e

ach

off-b

alan

ce s

heet

item

that

qua

lifi e

s fo

r a

risk

wei

ght o

f les

s th

an 1

00 p

erce

nt (5

0 pe

rcen

t for

der

ivat

ives

) at i

ts lo

wer

risk

wei

ght.

Whe

n co

mpl

etin

g ite

ms

34 th

roug

h 54

of S

ched

ule

HC

-R, e

ach

bank

hol

ding

com

pany

sho

uld

deci

de fo

r its

elf h

ow d

etai

led

a ris

k-w

eigh

t ana

lysi

s it

wis

hes

to p

erfo

rm. I

n ot

her

wor

ds, a

ban

k ho

ldin

g co

mpa

ny c

an c

hoos

e fro

m a

mon

g its

ass

ets

and

off-b

alan

ce s

heet

ite

ms

that

hav

e a

risk

wei

ght o

f les

s th

an 1

00 p

erce

nt w

hich

one

s to

risk

-wei

ght a

t an

appr

opria

te lo

wer

risk

wei

ght,

or it

can

sim

ply

risk-

wei

ght s

ome

or a

ll of

thes

e ite

ms

at a

10

0 pe

rcen

t ris

k w

eigh

t (50

per

cent

for d

eriv

ativ

es).

B

HC

K 0

010

bh

cx 1

754

bh

cx 1

773

B

HC

K C

225

bh

ct 5

369

bh

ct B

528

bh

cx 3

123

bh

cx 3

545

B

HC

K B

639

bh

ct 2

170

Allo

catio

n by

Ris

k W

eigh

t Cat

egor

y

76

Sche

dule

HC

-R—

Con

tinue

d

Dol

lar A

mou

nts

in T

hous

ands

B

il M

il Th

ou

(Col

umn

A)

Face

Val

ue

or N

otio

nal

Am

ount

(Col

umn

C)

(Col

umn

D)

(Col

umn

E)

(Col

umn

F)

0%20

%50

%10

0%

Cre

dit

Con

vers

ion

Fact

or

(Col

umn

B)

Cre

dit

Equ

ival

ent

Am

ount

1

B

il M

il Th

ou

Bil

Mil

Thou

B

il M

il Th

ou

Bil

Mil

Thou

B

il M

il Th

ou

1. C

olum

n A

mul

tiplie

d by

cre

dit c

onve

rsio

n fa

ctor

.2.

For

fi na

ncia

l sta

ndby

lette

rs o

f cre

dit t

o w

hich

the

low

-leve

l exp

osur

e ru

le a

pplie

s, u

se a

cre

dit c

onve

rsio

n fa

ctor

of 1

2.5

or a

n in

stitu

tion

spec

ifi c

fact

or. F

or o

ther

fi na

ncia

l sta

ndby

lette

rs o

f cre

dit,

use

a cr

edit

conv

ersi

on fa

ctor

of 1

.00.

See

inst

ruct

ions

for f

urth

er in

form

atio

n.3.

Or i

nstit

utio

n-sp

ecifi

c fa

ctor

.

FR Y

–9C

Pag

e 43 6/09

BH

C0

BH

C2

BH

C5

BH

C9

BH

CE

1.00

or 1

2.52

.50

.20

1.00

1.00

1.00

12.5

3

1.00

1.00 .50

.10

Der

ivat

ives

and

Off-

Bal

ance

She

et It

ems

44.

Fina

ncia

l sta

ndby

lette

rs o

f cre

dit..

......

......

...

44.

45.

Per

form

ance

sta

ndby

lette

rs o

f cr

edit

......

......

......

......

......

......

......

......

......

......

45

.46

. C

omm

erci

al a

nd s

imila

r let

ters

of

cred

it ...

......

......

......

......

......

......

......

......

......

...

46.

47.

Ris

k pa

rtici

patio

ns in

ban

kers

ac

cept

ance

s ac

quire

d by

the

repo

rting

in

stitu

tion

......

......

......

......

......

......

......

......

......

47

.

48.

Sec

uriti

es le

nt...

......

......

......

......

......

......

......

.. 48

.49

. R

etai

ned

reco

urse

on

smal

l bus

ines

s ob

ligat

ions

sol

d w

ith re

cour

se ...

......

......

......

. 49

. 50

. R

ecou

rse

and

dire

ct c

redi

t sub

stitu

tes

(oth

er th

an fi

nanc

ial s

tand

by le

tters

of

cred

it) s

ubje

ct to

the

low

-leve

l ex

posu

re ru

le a

nd re

sidu

al in

tere

sts

subj

ect t

o a

dolla

r-fo

r-do

llar c

apita

l re

quire

men

t ....

......

......

......

......

......

......

......

....

50.

51.

All

othe

r fi n

anci

al a

sset

s so

ld w

ith

reco

urse

.....

......

......

......

......

......

......

......

......

.. 51

.52

. A

ll ot

her o

ff-ba

lanc

e sh

eet

liabi

litie

s....

......

......

......

......

......

......

......

......

....

52.

53.

Unu

sed

com

mitm

ents

:a.

With

an

orig

inal

mat

urity

ex

ceed

ing

one

year

......

......

......

......

......

...

53.a

.b.

With

an

orig

inal

mat

urity

of o

neye

ar o

r les

s to

ass

et-b

acke

d co

mm

erci

al p

aper

con

duits

.....

......

......

....

53.b

.

54.

Der

ivat

ive

cont

ract

s....

......

......

......

......

......

....

54.

B

HC

K B

546

bh

ct 6

570

bh

ct 3

411

B

HC

K 3

429

bh

ct 3

433

bh

ct A

250

B

HC

K B

541

B

HC

K B

675

B

HC

K B

681

B

HC

K 6

572

B

HC

K G

591

B

HC

E A

167

Allo

catio

n by

Ris

k W

eigh

t Cat

egor

y

77

Sche

dule

HC

-R—

Con

tinue

d

Dol

lar A

mou

nts

in T

hous

ands

B

il M

il Th

ou

(Col

umn

A)

Face

Val

ue

or N

otio

nal

Am

ount

(Col

umn

C)

(Col

umn

D)

(Col

umn

E)

(Col

umn

F)

0%20

%50

%10

0%

Cre

dit

Con

vers

ion

Fact

or

(Col

umn

B)

Cre

dit

Equ

ival

ent

Am

ount

1

B

il M

il Th

ou

Bil

Mil

Thou

B

il M

il Th

ou

Bil

Mil

Thou

B

il M

il Th

ou

1. C

olum

n A

mul

tiplie

d by

cre

dit c

onve

rsio

n fa

ctor

.2.

For

fi na

ncia

l sta

ndby

lette

rs o

f cre

dit t

o w

hich

the

low

-leve

l exp

osur

e ru

le a

pplie

s, u

se a

cre

dit c

onve

rsio

n fa

ctor

of 1

2.5

or a

n in

stitu

tion

spec

ifi c

fact

or. F

or o

ther

fi na

ncia

l sta

ndby

lette

rs o

f cre

dit,

use

a cr

edit

conv

ersi

on fa

ctor

of 1

.00.

See

inst

ruct

ions

for f

urth

er in

form

atio

n.3.

Or i

nstit

utio

n-sp

ecifi

c fa

ctor

.

FR Y

–9C

Pag

e 43 6/09

BH

C0

BH

C2

BH

C5

BH

C9

BH

CE

1.00

or 1

2.52

.50

.20

1.00

1.00

1.00

12.5

3

1.00

1.00 .50

.10

Der

ivat

ives

and

Off-

Bal

ance

She

et It

ems

44.

Fina

ncia

l sta

ndby

lette

rs o

f cre

dit..

......

......

...

44.

45.

Per

form

ance

sta

ndby

lette

rs o

f cr

edit

......

......

......

......

......

......

......

......

......

......

45

.46

. C

omm

erci

al a

nd s

imila

r let

ters

of

cred

it ...

......

......

......

......

......

......

......

......

......

...

46.

47.

Ris

k pa

rtici

patio

ns in

ban

kers

ac

cept

ance

s ac

quire

d by

the

repo

rting

in

stitu

tion

......

......

......

......

......

......

......

......

......

47

.

48.

Sec

uriti

es le

nt...

......

......

......

......

......

......

......

.. 48

.49

. R

etai

ned

reco

urse

on

smal

l bus

ines

s ob

ligat

ions

sol

d w

ith re

cour

se ...

......

......

......

. 49

. 50

. R

ecou

rse

and

dire

ct c

redi

t sub

stitu

tes

(oth

er th

an fi

nanc

ial s

tand

by le

tters

of

cred

it) s

ubje

ct to

the

low

-leve

l ex

posu

re ru

le a

nd re

sidu

al in

tere

sts

subj

ect t

o a

dolla

r-fo

r-do

llar c

apita

l re

quire

men

t ....

......

......

......

......

......

......

......

....

50.

51.

All

othe

r fi n

anci

al a

sset

s so

ld w

ith

reco

urse

.....

......

......

......

......

......

......

......

......

.. 51

.52

. A

ll ot

her o

ff-ba

lanc

e sh

eet

liabi

litie

s....

......

......

......

......

......

......

......

......

....

52.

53.

Unu

sed

com

mitm

ents

:a.

With

an

orig

inal

mat

urity

ex

ceed

ing

one

year

......

......

......

......

......

...

53.a

.b.

With

an

orig

inal

mat

urity

of o

neye

ar o

r les

s to

ass

et-b

acke

d co

mm

erci

al p

aper

con

duits

.....

......

......

....

53.b

.

54.

Der

ivat

ive

cont

ract

s....

......

......

......

......

......

....

54.

B

HC

K B

546

bh

ct 6

570

bh

ct 3

411

B

HC

K 3

429

bh

ct 3

433

bh

ct A

250

B

HC

K B

541

B

HC

K B

675

B

HC

K B

681

B

HC

K 6

572

B

HC

K G

591

B

HC

E A

167

Allo

catio

n by

Ris

k W

eigh

t Cat

egor

y

78

C.2 Summary Statistics

Mean St. Dev. Min. Max. Num. of Obs.

Total Assets 6,632 64,348 2,366,000 101,751Total Liabilities 6,060 59,064 2,225,000 101,751Total Equity 553 5,310 -1,375 257,700 101,751

Real estate loans 561 5,718 -15,002 258,600 1,731,825Mortgage-backed securities 1,732 14,161 573,900 77,367

Stock returns 0 0.03 -.92 4.00 2,042,371Bond returns 0 1 -46.45 41.91 121,802

Table C.2.1. Summary statistics for entire sample (dollars in millions)

Figure C.2.1. Scatter plot of capital and risk-based capital ratios

Sheet1

Page 1

2000

-03

2000

-06

2000

-09

2000

-12

2001

-03

2001

-06

2001

-09

2001

-12

2002

-03

2002

-06

2002

-09

2002

-12

2003

-03

2003

-06

2003

-09

2003

-12

2004

-03

2004

-06

2004

-09

2004

-12

2005

-03

2005

-06

2005

-09

2005

-12

2006

-03

2006

-06

2006

-09

2006

-12

2007

-03

2007

-06

2007

-09

2007

-12

2008

-03

2008

-06

2008

-09

2008

-12

2009

-03

2009

-06

2009

-09

2009

-12

2010

-03

2010

-06

2010

-09

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

riskcap

Figure C.2.2. Average quarterly capital risk-based capital ratios

79

Figure C.2.3. Average quarterly BHC bond yields and yields on 1, 5, and 10-year treasuries

Figure C.2.4. Average quarterly BHC bond yield spread

80

Figure C.2.5. Quarterly standard deviation of stock returns

Figure C.2.6. Quarterly number of stock observations

81

Figure C.2.7. Quarterly standard deviation of bond returns

Figure C.2.8. Quarterly number of bond observations

82

C.3 Regression Results

Table C.3.1. Quarterly regressions of absolute stock returns on capital

Quarter Beta(debt) Beta(real) Beta(mbs) N2000-03 -2.9699* -0.2347 -0.2208 190

(1.7546) (0.2197) (0.1633)2000-06 -1.4373 -0.001 0.1515 192

(1.3853) (0.168) (0.1288)2000-09 -0.3625 -0.7166*** 0.0323 192

(1.6201) (0.2065) (0.1566)2000-12 0.5173 -0.5546*** 0.1627 194

(1.3542) (0.1787) (0.1325)2001-03 -1.4436 0.1209 -0.0484 378

(1.033) (0.1613) (0.1226)2001-06 -1.0915* 0.1601 0.0986 370

(0.5909) (0.16) (0.1186)2001-09 -1.2998*** -0.1109 -0.148 374

(0.6457) (0.1723) (0.124)2001-12 -2.0041*** 0.1419 0.0114 340

(0.8272) (0.1153) (0.0892)2002-03 -0.2251 0.4251*** 0.0174 362

(0.4536) (0.1239) (0.0865)2002-06 -0.0733 -0.3434* 0.1208 362

(0.6405) (0.1808) (0.1294)2002-09 -0.5005 -0.6629*** -0.0286 363

(0.5419) (0.1508) (0.1073)2002-12 -0.2838 -0.1485 -0.0329 361

(0.4118) (0.1134) (0.0805)2003-03 -0.1736 -0.2108* -0.1909*** 356

(0.4563) (0.1263) (0.0856)2003-06 -0.2119 0.0288 -0.0873 364

(0.4606) (0.1205) (0.0832)2003-09 -0.0997 0.0543 -0.0716 359

(0.5194) (0.133) (0.0938)2003-12 -0.0601 -0.0215 -0.1023 357

(0.371) (0.096) (0.066)2004-03 -0.0097 0.1428 0.0707 353

(0.4578) (0.1172) (0.0807)2004-06 -0.8118 0.0313 -0.0284 345

(0.5175) (0.1329) (0.0951)2004-09 -0.7783* 0.0771 -0.0245 339

(0.4423) (0.1153) (0.0855)2004-12 -0.1195 0.1491 0.0073 340

(0.3532) (0.0917) (0.0706)2005-03 -0.045 0.3089*** 0.0549 337

(0.4643) (0.1174) (0.091)2005-06 0.2718 0.2983*** 0.0485 331

(0.4382) (0.1163) (0.0926)2005-09 -0.138 0.1884* 0.0637 329

(0.3804) (0.1043) (0.0872)2005-12 0.0479 0.1082 0.0259 322

(0.2878) (0.088) (0.074)

Quarter Beta(debt) Beta(real) Beta(mbs) N2006-03 -0.0238 -0.0073 -0.0852 295

(0.323) (0.101) (0.0865)2006-06 -0.0444 0.2031* 0.0287 294

(0.3839) (0.1208) (0.1047)2006-09 -0.1632 0.1521 -0.0363 290

(0.317) (0.1009) (0.089)2006-12 -0.0856 0.1913 0.0736 280

(0.4292) (0.1442) (0.126)2007-03 -0.1779 0.1646 -0.0673 274

(0.4165) (0.1397) (0.1313)2007-06 -0.0444 0.1028 0.2109 268

(0.58) (0.1984) (0.2035)2007-09 0.7375 0.3617* 0.1953 261

(0.6318) (0.2131) (0.2223)2007-12 0.0119 0.5287*** -0.0488 252

(0.8151) (0.271) (0.2803)2008-03 -0.1046 0.6891*** -0.3821 246

(0.7717) (0.2467) (0.2446)2008-06 -1.338 0.0081 -0.8993* 243

(1.4774) (0.52) (0.4893)2008-09 -0.5441 0.2749 -1.0679* 239

(1.7107) (0.6083) (0.5673)2008-12 -3.9581*** -0.0642 -1.6943*** 236

(1.9082) (0.706) (0.6393)2009-03 -3.5327*** 1.1884*** -1.1862*** 237

(1.4288) (0.5466) (0.4688)2009-06 -3.1213*** 1.552*** -0.5751 232

(1.1164) (0.4289) (0.3669)2009-09 -4.6064*** 1.9484*** -0.3711 231

(1.1278) (0.455) (0.3715)2009-12 -4.7056*** 1.82*** -0.3855 227

(1.0465) (0.4306) (0.3346)2010-03 -4.6284*** 1.2953*** -0.3747 222

(1.0529) (0.4273) (0.3364)2010-06 -3.3379*** 1.2356*** -0.346 221

(0.9348) (0.3604) (0.284)2010-09 -3.9168*** 1.2037*** -0.4046 220

(0.9266) (0.3552) (0.2714)

*** Significant at the 1% level.

** Significant at the 5% level.

* Significant at the 10% level.

Table C.3.2. Quarterly regressions of the standard deviation of stock returns on capital

83

Quarter Beta(debt) Beta(real) Beta(mbs) N2000-03 -5.1119* 4.9159*** 0.1618 376

(2.8916) (1.5496) (0.5703)2000-06 -4.7613* 4.8229*** 0.7715 376

(2.8273) (1.4525) (0.532)2000-09 -2.6071 4.3932*** -0.1136 377

(2.8113) (1.658) (0.5356)2000-12 -0.3366 3.8913*** 0.1327 370

(2.9066) (1.5479) (0.5638)2001-03 -2.7257 3.1862*** 1.1652*** 365

(2.7878) (1.3035) (0.5307)2001-06 -2.7969* 4.8404*** 0.4538 362

(1.5506) (1.1328) (0.4807)2001-09 -1.6304 2.5641** 1.0962*** 368

(1.641) (1.3374) (0.4989)2001-12 -4.0185*** 3.462*** 0.3384 335

(1.8916) (0.6501) (0.3186)2002-03 -0.7106 4.2314*** 0.8294*** 356

(1.0442) (0.8102) (0.3203)2002-06 -1.5689 3.7086*** -0.9807*** 362

(1.4312) (1.2763) (0.4605)2002-09 -1.5268 1.7539*** -1.3928*** 363

(1.1347) (0.8026) (0.363)2002-12 -0.8984 2.0896*** -0.0309 356

(1.165) (0.7467) (0.3738)2003-03 0.4491 1.5244*** 0.5976* 356

(1.1429) (0.7667) (0.3619)2003-06 0.2357 1.4259*** 0.7932*** 364

(0.9732) (0.6148) (0.2959)2003-09 0.3794 0.9793 1.0462*** 359

(1.1041) (0.9046) (0.3276)2003-12 -0.5971 1.2193 0.4331 357

(0.8691) (0.8867) (0.2655)2004-03 -0.1485 0.2663 0.4098 353

(1.0806) (1.2123) (0.3234)2004-06 -2.4337*** -1.1123 0.0886 345

(1.1919) (1.3726) (0.3617)2004-09 -1.9891* 0.5762 0.4296 339

(1.0756) (1.2666) (0.3314)2004-12 -0.7145 1.0272 -0.0442 340

(0.8363) (1.0154) (0.2664)2005-03 -0.0909 0.3345 0.3571 337

(1.293) (1.6805) (0.3918)2005-06 0.7204 -0.4573 0.5503** 331

(0.9011) (1.1764) (0.2875)2005-09 -1.2456 -0.3525 -0.2026 329

(0.9893) (1.2388) (0.3321)2005-12 -0.0549 0.3539 0.1764 322

(0.694) (0.9302) (0.2619)

Quarter Beta(debt) Beta(real) Beta(mbs) N2006-03 0.1458 1.3054 0.3512 295

(0.7353) (1.0973) (0.2938)2006-06 0.2553 2.2293 1.1425*** 294

(1.0015) (1.7966) (0.4073)2006-09 -0.205 0.3769 0.2524 290

(0.7441) (1.4579) (0.3072)2006-12 -1.383 10.9427*** -0.1276 280

(1.2779) (2.5083) (0.5401)2007-03 -0.5013 14.9306*** 0.5296 274

(1.1861) (3.1416) (0.5145)2007-06 -2.1653 -0.3267 -0.9812 268

(1.5845) (4.566) (0.6904)2007-09 0.4436 6.1944 0.1095 261

(1.5569) (5.493) (0.6816)2007-12 -3.3728 13.2335* -1.9988*** 252

(2.206) (7.8897) (1.0127)2008-03 -1.4671 10.2057*** 0.4823 246

(2.1404) (5.037) (0.9212)2008-06 -3.1214 39.6293*** -0.7146 243

(4.4214) (12.3399) (2.0155)2008-09 -2.7854 40.532*** -1.276 239

(3.9354) (9.9434) (1.8368)2008-12 -11.2633*** 16.9114 -0.4099 236

(4.6107) (11.4396) (2.2862)2009-03 -9.3847*** 25.9002*** 3.0028 237

(4.129) (11.2488) (2.0759)2009-06 -7.8051*** 9.5474 3.2184*** 232

(2.9907) (8.4616) (1.502)2009-09 -10.8666*** 5.9644 4.0795*** 231

(3.0517) (10.9038) (1.6253)2009-12 -13.0399*** 3.1911 3.9328*** 227

(3.6972) (10.6445) (1.9783)2010-03 -12.1302*** 5.49 2.104 222

(2.8253) (10.7496) (1.5205)2010-06 -8.4495*** 1.4755 2.8198*** 221

(2.5535) (6.8711) (1.2676)2010-09 -11.7612*** 3.19 1.089 220

(2.8543) (7.6118) (1.4232)

*** Significant at the 1% level.

** Significant at the 5% level.

* Significant at the 10% level.

84

Table C.3.3. Quarterly regressions of absolute stock returns on risk-based capital

Quarter Beta(debt) Beta(real) Beta(mbs) N2000-03 -0.3868 -0.2277 -0.1287 190

(1.2757) (0.2228) (0.1561)2000-06 0.3905 0.0244 0.1945 192

(0.9449) (0.1704) (0.1215)2000-09 1.4029 -0.6617*** 0.019 191

(1.1149) (0.2072) (0.1484)2000-12 0.6884 -0.5258*** 0.1318 193

(0.9979) (0.1799) (0.1262)2001-03 -1.2364* 0.1096 0.0002 377

(0.688) (0.1614) (0.1174)2001-06 -0.7851* 0.1628 0.1392 369

(0.4517) (0.1605) (0.115)2001-09 -1.1096*** -0.1201 -0.1092 373

(0.5256) (0.173) (0.1203)2001-12 0.0723 0.1819 0.0761 339

(0.5668) (0.1167) (0.086)2002-03 0.2183 0.4633*** 0.0324 361

(0.3463) (0.1248) (0.0834)2002-06 -0.179 -0.35** 0.1198 361

(0.5013) (0.1821) (0.125)2002-09 -0.6268 -0.6747*** -0.0127 362

(0.4903) (0.1508) (0.1033)2002-12 0.0835 -0.1246 -0.0189 361

(0.3698) (0.114) (0.0783)2003-03 0.027 -0.1981 -0.1825*** 356

(0.4164) (0.1275) (0.0831)2003-06 0.2455 0.0593 -0.0757 364

(0.3936) (0.1219) (0.0808)2003-09 0.4455 0.093 -0.0659 359

(0.4382) (0.1341) (0.0904)2003-12 0.4638 0.0183 -0.1006 357

(0.3118) (0.0971) (0.0638)2004-03 0.2974 0.1675 0.0674 353

(0.3705) (0.1183) (0.0787)2004-06 -0.1722 0.0588 0.0087 345

(0.406) (0.1355) (0.0925)2004-09 -0.2184 0.1028 0.0088 339

(0.3394) (0.1177) (0.0837)2004-12 0.0931 0.1685* 0.0146 340

(0.2608) (0.0931) (0.0683)2005-03 0.537 0.3732*** 0.0652 337

(0.3537) (0.1192) (0.0872)2005-06 0.1704 0.297*** 0.0367 331

(0.3216) (0.1174) (0.0896)2005-09 0.3105 0.2344*** 0.0798 329

(0.285) (0.1049) (0.0842)2005-12 0.1793 0.1248 0.0272 322

(0.2156) (0.0886) (0.072)

Quarter Beta(debt) Beta(real) Beta(mbs) N2006-03 0.1252 0.0107 -0.0792 295

(0.2391) (0.1014) (0.084)2006-06 0.117 0.2224* 0.0369 294

(0.2697) (0.1208) (0.1015)2006-09 0.2625 0.195** -0.0231 290

(0.2565) (0.1014) (0.0873)2006-12 0.3961 0.2455* 0.0893 280

(0.3354) (0.1443) (0.1233)2007-03 0.1616 0.2023 -0.0469 274

(0.331) (0.1406) (0.1289)2007-06 0.1094 0.1199 0.2197 268

(0.5091) (0.1999) (0.1992)2007-09 0.4734 0.3497 0.1534 261

(0.5391) (0.2149) (0.2174)2007-12 0.0497 0.5335*** -0.0475 252

(0.6816) (0.2734) (0.2746)2008-03 -0.119 0.683*** -0.3805 246

(0.6278) (0.2507) (0.2408)2008-06 -1.7592 -0.1182 -0.8808* 243

(1.0941) (0.5217) (0.4784)2008-09 -0.7844 0.218 -1.0645** 239

(1.3862) (0.6163) (0.5584)2008-12 -2.723* -0.0568 -1.5893*** 236

(1.6319) (0.7271) (0.6358)2009-03 -2.8248*** 1.1665*** -1.0933*** 237

(1.2252) (0.5544) (0.4649)2009-06 -2.1172*** 1.6067*** -0.4772 232

(0.9601) (0.4356) (0.3654)2009-09 -3.2866*** 2.012*** -0.2513 231

(1.0073) (0.4671) (0.3733)2009-12 -3.4304*** 1.8599*** -0.2362 227

(0.9011) (0.4415) (0.3347)2010-03 -3.1818*** 1.333*** -0.2183 222

(0.9162) (0.4425) (0.3377)2010-06 -4.5337*** 1.0606*** -0.2181 221

(1.0363) (0.3624) (0.2773)2010-09 -5.2637*** 1.0083*** -0.223 220

(0.9789) (0.3519) (0.2624)

*** Significant at the 1% level.

** Significant at the 5% level.

* Significant at the 10% level.

85

Table C.3.4. Quarterly regressions of the standard deviation of stock returns on risk-basedcapital

Quarter Beta(debt) Beta(real) Beta(mbs) N2000-03 0.7298 0.0365 -0.3529 190

(2.9195) (0.51) (0.3573)2000-06 1.6988 0.3805 0.324 192

(2.3917) (0.4312) (0.3075)2000-09 4.4855* -1.3504*** -0.1056 191

(2.6035) (0.4839) (0.3467)2000-12 2.7995 -0.9068*** 0.1476 193

(2.2378) (0.4036) (0.2831)2001-03 -2.6433 0.2404 -0.2292 377

(1.9185) (0.4501) (0.3274)2001-06 -2.1551* 0.5155 0.0754 369

(1.236) (0.439) (0.3147)2001-09 -2.3313* -0.1199 -0.4511 373

(1.36) (0.4477) (0.3113)2001-12 0.9868 0.8069*** 0.1249 339

(1.3988) (0.2879) (0.2121)2002-03 0.3291 1.2449*** 0.0032 361

(0.8208) (0.2957) (0.1978)2002-06 -0.3653 -0.4528 0.3123 361

(1.1657) (0.4233) (0.2907)2002-09 -1.3449 -1.2592*** -0.1771 362

(1.0475) (0.3222) (0.2208)2002-12 0.8544 0.0838 0.0178 361

(1.0655) (0.3285) (0.2256)2003-03 -0.0072 -0.024 -0.4302*** 356

(1.0562) (0.3234) (0.2107)2003-06 0.8609 0.4189 -0.1223 364

(0.8443) (0.2615) (0.1732)2003-09 1.1244 0.3913 -0.1752 359

(0.953) (0.2917) (0.1967)2003-12 0.6743 0.0579 -0.2285 357

(0.7402) (0.2304) (0.1515)2004-03 0.5394 0.3059 0.0614 353

(0.8762) (0.2798) (0.1862)2004-06 -0.7478 0.0831 -0.0387 345

(0.9419) (0.3144) (0.2147)2004-09 -0.6956 0.1741 -0.0581 339

(0.828) (0.2872) (0.2041)2004-12 0.5276 0.5262*** 0.088 340

(0.6167) (0.22) (0.1615)2005-03 3.2062*** 0.9518*** 0.2326 337

(0.9799) (0.3303) (0.2416)2005-06 0.7989 0.5272*** 0.0956 331

(0.667) (0.2435) (0.1859)2005-09 1.2947* 0.7805*** 0.3731* 329

(0.7351) (0.2706) (0.2173)2005-12 0.3035 0.2474 0.2249 322

(0.5102) (0.2097) (0.1704)

Quarter Beta(debt) Beta(real) Beta(mbs) N2006-03 0.2141 0.0237 -0.1942 295

(0.5338) (0.2265) (0.1874)2006-06 0.2645 0.5127 -0.0546 294

(0.7028) (0.3147) (0.2645)2006-09 0.5673 0.3506 -0.0548 290

(0.5854) (0.2315) (0.1993)2006-12 1.1659 0.9687*** 0.5653 280

(0.9948) (0.4279) (0.3657)2007-03 0.2412 0.4254 -0.2777 274

(0.9414) (0.3999) (0.3665)2007-06 -0.361 0.3255 0.3224 268

(1.358) (0.5332) (0.5313)2007-09 1.5793 1.2335*** 0.4425 261

(1.2918) (0.5149) (0.5208)2007-12 0.1796 1.5531*** -0.1382 252

(1.806) (0.7243) (0.7276)2008-03 -0.0177 1.6937*** -1.1071* 246

(1.6452) (0.657) (0.6311)2008-06 -4.5668 -0.224 -2.4033* 243

(3.2319) (1.541) (1.4131)2008-09 -2.0954 1.0085 -2.4397** 239

(3.175) (1.4116) (1.279)2008-12 -7.2206** 0.6066 -3.4341*** 236

(3.7616) (1.676) (1.4656)2009-03 -7.992*** 3.0312*** -2.6871*** 237

(3.2743) (1.4817) (1.2424)2009-06 -5.502*** 3.8679*** -1.3249 232

(2.3216) (1.0533) (0.8834)2009-09 -7.6723*** 4.8825*** -0.4975 231

(2.429) (1.1264) (0.9002)2009-12 -8.9989*** 5.3762*** -0.8337 227

(2.8283) (1.3857) (1.0506)2010-03 -7.8691*** 3.4445*** -0.4316 222

(2.2233) (1.0739) (0.8195)2010-06 -11.626*** 2.6238*** -0.4432 221

(2.5384) (0.8878) (0.6792)2010-09 -13.4271*** 2.5099*** -0.5906 220

(2.7037) (0.9719) (0.7249)

*** Significant at the 1% level.

** Significant at the 5% level.

* Significant at the 10% level.

86

Table C.3.5. Quarterly regressions of bond yield spreads on capital

Quarter Beta(debt) Beta(real) Beta(mbs) N2001-03 -43.1549 0.7957 -11.2978*** 9

(31.1088) (2.8984) (5.7911)2001-06 -50.4799 1.9431 -12.163 8

(51.1048) (4.6166) (9.2258)2001-09 -19.5735 -0.6459 -12.558 8

(73.7491) (6.4236) (11.194)2001-12 3.6632 6.088 -16.2425* 7

(68.5147) (6.5234) (9.547)2002-03 -2.5645 -1.7193 -11.7747 8

(84.1943) (8.3106) (12.9858)2002-06 -176.85*** -7.64*** -31.9473*** 5

(39.5565) (2.1702) (4.6674)2002-09 -128.0512 -7.19 -23.8462*** 5

(86.9827) (5.5794) (10.0286)2002-12 -22.1429 -7.7805 -9.7838 7

(45.4512) (5.0065) (6.8886)2003-03 -10.5498 -6.6242 -5.9312 7

(54.962) (5.8606) (7.1129)2003-06 3.4378 -0.7071 1.8192 11

(55.1853) (5.4851) (4.0295)2003-09 37.0473 -0.9562 3.4488 15

(47.3164) (4.8903) (4.3042)2003-12 55.265 -1.9146 3.1706 16

(43.5301) (4.156) (3.8714)2004-03 65.2534* -2.7598 1.3649 16

(37.4446) (3.7089) (3.3954)2004-06 -6.5996 1.3385 2.6876 17

(50.2612) (4.5098) (4.1423)2004-09 40.1146 0.0415 3.0977 18

(43.0519) (2.8672) (3.1581)2004-12 21.3872 1.2161 1.8692 19

(19.0544) (1.8829) (2.6794)2005-03 15.9919 1.3458 2.6288 21

(13.9686) (1.4657) (1.9582)2005-06 -11.5891 0.4477 -1.0623 22

(12.9256) (1.8542) (2.1284)2005-09 -14.5503 -0.0065 -1.8822 22

(10.3189) (1.5343) (1.8668)2005-12 -21.2361*** -0.1104 -2.079 23

(9.7878) (1.3525) (1.6428)

Quarter Beta(debt) Beta(real) Beta(mbs) N2006-03 -17.8174*** 0.0463 -2.6248* 26

(7.8419) (1.1678) (1.4189)2006-06 -14.6793*** 0.1478 -2.3535* 27

(6.6985) (1.0429) (1.3188)2006-09 -14.4033*** 0.6789 -1.4689 28

(7.2791) (1.0624) (1.3928)2006-12 -16.2876*** 1.2091 -1.1412 29

(7.7976) (1.1527) (1.4995)2007-03 -13.7151** 1.1405 -0.4136 29

(7.102) (1.1378) (1.471)2007-06 -12.1949* 1.0392 -0.5131 30

(6.6816) (1.0056) (1.3885)2007-09 -6.1542 0.7889 -0.1948 30

(6.1024) (1.3015) (1.9648)2007-12 -2.0448 0.8352 -1.1348 29

(6.9739) (1.5783) (2.4214)2008-03 3.2981 2.3449 -0.5708 29

(10.0316) (2.2089) (3.2039)2008-06 1.7738 2.9541 -1.0816 32

(10.4953) (2.0206) (3.2702)2008-09 7.3517 3.947 -0.846 33

(12.7857) (2.5022) (4.0468)2008-12 -5.4642 3.1838 1.3132 34

(25.3796) (4.6034) (6.8954)2009-03 -47.0606 6.2913 15.6307*** 37

(30.0592) (5.2699) (6.9018)2009-06 -20.1836 7.4945 7.5852 37

(23.3574) (4.6571) (5.5699)2009-09 -4.8208 9.2032*** 6.3852 40

(22.0208) (4.3188) (5.2093)2009-12 15.0696 9.3786*** 5.0968 41

(20.5275) (3.7965) (4.8132)2010-03 4.7832 11.1557*** 4.8535 40

(21.3591) (3.7408) (4.3358)2010-06 3.6272 9.9792*** 4.5538 41

(5.2934) (3.3012) (3.9447)2010-09 1.353 8.9115*** 3.38 42

(5.854) (3.5684) (4.0357)

*** Significant at the 1% level.

** Significant at the 5% level.

* Significant at the 10% level.

87

Table C.3.6. Quarterly regressions bond yield spreads on risk-based capital

Quarter Beta(debt) Beta(real) Beta(mbs) N2001-03 -32.8087 -1.5083 -14.167*** 9

(21.8005) (2.0965) (5.5728)2001-06 -61.0421*** -1.0208 -16.8752*** 8

(22.4663) (2.1597) (6.2205)2001-09 -97.7899*** 1.0119 -13.5202** 8

(40.5734) (3.3104) (7.108)2001-12 -154.6541*** 1.7298 -23.7314*** 7

(72.0116) (4.2567) (6.9109)2002-03 -129.8197*** -1.5957 -21.2475*** 8

(33.6672) (3.1577) (6.454)2002-06 -118.1711*** -3.3741 -11.4243*** 5

(23.2709) (2.3318) (2.5475)2002-09 -88.4667 -3.3367 -11.0371 5

(60.8588) (6.7199) (6.8546)2002-12 -97.2741*** -2.829 -5.191 7

(40.652) (3.6654) (4.5762)2003-03 -80.621* -2.6218 -3.9735 7

(48.2919) (4.492) (5.2684)2003-06 -75.9329*** 1.359 1.9746 11

(26.392) (2.4573) (2.6202)2003-09 -44.6937 2.0424 4.293 15

(51.9621) (3.6516) (4.259)2003-12 -61.5921 2.1145 4.3792 16

(38.3118) (3.1222) (3.6617)2004-03 -57.5971 1.517 1.6327 16

(41.9045) (3.1152) (3.5235)2004-06 -67.0188* 1.5322 2.3377 17

(37.3037) (3.0444) (3.6841)2004-09 -47.8042* 1.9543 1.0218 18

(28.7352) (2.4518) (3.0298)2004-12 -7.1106 0.6527 0.6305 19

(17.7106) (2.037) (2.8168)2005-03 6.9254 1.1537 1.5872 21

(11.0978) (1.4941) (1.7323)2005-06 4.3447 0.9791 -0.2968 22

(11.264) (1.8662) (2.0566)2005-09 0.7081 0.5525 -0.9014 22

(9.2075) (1.596) (1.8688)2005-12 -0.6936 0.1653 -1.0635 23

(11.4948) (1.5044) (1.7591)

Quarter Beta(debt) Beta(real) Beta(mbs) N2006-03 -2.9027 0.3313 -2.076 26

(10.2941) (1.2879) (1.5508)2006-06 -5.1662 0.3863 -1.8809 27

(8.6711) (1.1356) (1.4177)2006-09 0.5603 0.7728 -1.2593 28

(8.1852) (1.1614) (1.4986)2006-12 9.5542 0.6416 -1.6521 29

(11.3397) (1.2205) (1.6654)2007-03 4.2032 0.6302 -1.104 29

(11.0932) (1.2121) (1.5987)2007-06 1.9898 0.5217 -1.1155 30

(8.7736) (1.0468) (1.4751)2007-09 6.855 0.2099 -1.2911 30

(10.0405) (1.27) (1.9112)2007-12 11.9271 0.7225 -1.7062 29

(11.2572) (1.4948) (2.2262)2008-03 15.2626 2.5614 -1.3222 29

(17.9773) (2.1257) (3.2761)2008-06 -17.171 2.6311 0.5219 32

(17.4014) (2.0118) (3.3563)2008-09 -7.8485 4.1021* 0.3295 33

(19.873) (2.4825) (3.8513)2008-12 -4.2947 2.8071 1.0382 34

(36.5261) (5.098) (6.7813)2009-03 -33.7913 4.0879 13.9422*** 37

(51.8577) (6.1059) (7.0023)2009-06 -28.1705 5.7689 7.7707 37

(37.0692) (5.4118) (5.6262)2009-09 -15.2812 7.986 6.5064 40

(29.645) (4.9737) (5.1895)2009-12 -2.3721 8.9759*** 5.7588 41

(27.7904) (4.4539) (4.8525)2010-03 -16.2028 10.6118*** 4.6706 40

(29.8166) (3.8853) (4.3344)2010-06 4.233 10.0488*** 4.4978 41

(7.2582) (3.4695) (3.9796)2010-09 -1.3565 8.2935*** 2.967 42

(7.525) (3.6148) (4.0067)

*** Significant at the 1% level.

** Significant at the 5% level.

* Significant at the 10% level.

88

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Curriculum Vitae

Thomas Hogan holds bachelors and masters degrees in business administration from theUniversity of Texas at Austin. He is an adjunct instructor in the Department of Financeat George Mason University and has taught classes on financial management and moneyand banking. Thomas has worked for Merrill Lynch’s commodity trading group and as aderivatives trader for hedge funds in the U.S. and Europe. He is also a former consultantto the World Bank.

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