Download Bank Bailouts and Moral Hazard?

Bank Bailouts and Moral Hazard?:
Evidence from Banks’ Investment and Financing
Decisions
Yunjeen Kim
∗
2016
ABSTRACT
The main goal of this paper is to estimate a dynamic model of banks with endogenous choice
of risks, specifically investments and financing decisions, to explain how the bank bailouts
exacerbate the moral hazards. The simulation results show that if the bank anticipates that it
will be rescued with a higher probability when it runs into a trouble, the bank deliberately fail
to prepare for bankruptcy by decreasing investment and increasing borrowings. Moreover,
when the bank is close to default it tends to take more risky investments and borrow more
risky debt beyond what it would take otherwise. Six model parameters that characterize the
banks’ behavior are estimated using simulated method of moments. Of particular interest
is the expected bailout probability conditional on default, which is estimated at 52.44% for
the full sample. By splitting the sample by size, the conditional bailout probability for the
small banks and the large banks are 35.69% and 76.20%, respectively. Lastly, the model
predicts the rescue funding would be 4.53% of the total assets, which is very close to the
actual capital injection made by the U.S. government under TARP in 2008, which is 4.39%.
Yunjeen Kim is at Boston University Questrom School of Business. Email: [email protected]. I would
like to thank Toni M. Whited, Ron Kaniel, Robert Ready, Jerold B. Warner, and Yongsung Chang for
their great comments and encouragement. I thank seminar participants at University of Rochester, Federal
Reserve Bank of Richmond, Moody’s Analytics, Southern Methodist University, University of Connecticut,
SUNY Buffalo, Johns Hopkins University, University of Oklahoma, University of Warwick, and Federal
Reserve Bank of Boston. Additional thanks to Candace Jens, Matthew Gustafson, and Ruoyan Huang for
their feedback.
∗
This paper aims to examine the effects of the government bailout policy on the moral
hazards of banks. Because failure of financial institutions are generally considered to play
a critical role in triggering the collapse of the entire economy, the U.S. government has
intervened in the financial markets to prevent the crises. The rescue loan approach favored
in the financial crises has been blamed because it increased not only the direct costs to the
tax payers, but also increased incentives for banks to behave badly. A common critique would
be the banks who expect the government to protect them take on excessively risky projects
and generally act less responsibly than they would if they had to take the full burden of their
behavior. Although the bailouts in the financial markets are in need of careful examination,
they have not been fully investigated in the literature due to the limitation of data.
To evaluate the bank bailouts more systematically and quantitatively, I construct a dynamic model of banks in which the choice of risks, specifically the investment and financing
policies, are endogenous and estimate the model parameters. Without this kind of model,
it is challenging to show the moral hazards of banks for the following reasons. Firstly, since
the risk taking is not directly measurable, empirical analyses often use proxies, which cause
measurement errors.1 The endogenous choice of risks in the model, on the other hand, would
be the direct measure of risk taking. Secondly, evaluating the moral hazards requires to observe the bank’s behavior under a hypothetical situation in which data cannot be available.
The structural model based approach allows one to explore counterfactuals. Thirdly, there is
no consensus on the beliefs about the government bailouts and the beliefs are not observable
(Dam and Koetter (2012)). Instead, the model is able to estimate the expectation of the
bailout probability. Thus, the model in this paper is able to cope with these challenges.
1
Such proxy indicators of risk taking include profit-volatility, debt-to-asset ratios, nonperforming loan
ratios, market capital-to-asset ratios, actual interest costs on large CDs, charged-off losses, loan loss provisions, standard deviation of return on assets or equity, Z-scores, etc. For example, see Shrieves and Dahl
(1992), Lepetit et al. (2008), etc.
2
Moreover, it is necessary to construct a new model that captures the distinctive properties
of banks. Financial institutions are very different from other non-financial firms in the sense
that their capital structure and the profit making mechanism are unique.2 The key features
of the model are summarized as follows: (i) Banks maximize shareholders’ value by choosing
the investment allocation and financing decisions subject to a budget constraint; (ii) On the
assets side, there are some cash reserves and two types of investments, one-period risk-free
bonds and risky loans; (iii) On the liabilities side, there are fully insured deposits and risky
debt; (iv) Banks can default, but there is a possibility that they can get bailed out.3 (v)
Time is discrete and the horizon is infinite; and (v) Shareholders are risk-neutral.
The model parameters governing the banks’ behavior are estimated by simulated method
of moments (SMM). This procedure is basically to find an optimal set of parameters to make
the simulated moments generated from the model close enough to the real moments. Picking
up the moments is a very important step in the SMM procedure. The moments should be
informative about the parameters that need to be pinned down. Nine moments that are
matched in this paper are: the first moment of leverage, the autocorrelation of leverage, the
standard deviation of the shock to leverage, the first moment of profits, the first moment of
dividends, the first and the second moment of charge-offs, the ratio of insured deposits to
the total liabilities, and the frequency of default.
Of particular interest is the belief on the government bailouts conditional on default.
2
The leverage of banks is on average much higher than that of non-financial firms. Also, most of the
creditors of a bank is depositors. The banks have only little control on the deposits. Moreover, a bank makes
profits by borrowing money at a low interest rate and lending money at a high interest rate. Such activities
provide liquidity in the markets.
3
The goal of this paper is not to find an optimal bailout policy. There can be several ways to rescue the
troubled banks, such as purchase of preferred stock, purchase of common stock, or buying toxic assets, etc.
The goal is not to solve the government’s problem in this paper. It is assumed that, from the shareholders’
point of view, the type of the government bailout is exogenously given and random, which in turn irrelevant
to banks’ decision. Thus, the differences between the methods used to bail out troubled banks are not
distinguishable. If you are interested in which kind of bailout is optimal, see Wilson and Wu (2010) and
Bernardo et al. (2011), for example.
3
The estimation results show that the banks believe that the government will bail them out
with a probability of 52.44% given that they are in trouble. The model parameters are also
estimated for subsamples by size of banks. The bailout probability conditional on default
of the small banks and the large banks are 35.69% and 76.20%, respectively. That is, the
“Too-Big-to-Fail” is prevalent in the financial markets; the large banks believe more strongly
about the bailouts than the small banks do. Moreover, the other 5 estimated values of the
model parameters show that the large banks have riskier loans, lower fire-sales price, shorter
maturity of loans, lower rate of return on loans, and smaller adjustment cost of loans than
the small banks.
The model reveals that the banks intentionally move closer to default threshold if they
expect the government bailouts to be more likely. The model in the present paper allows
one to directly measure the risk taking behavior even in a hypothetical situation, which is
not possible in the real world. Simulating the model while varying the bailout probability
parameter but keeping the other parameters fixed generates hypothetical data sets of banks.
As the expected bailout probability conditional on the banks being in distress increases,
the banks increase the leverage and decrease the loan investment. These changes shift the
distribution of banks closer to the default threshold and thus the default probability becomes
much higher; a 1% increase in the belief in the bailouts increases the default frequency from
4.5% to 16.6%.
The model is capable of predicting the banks’ dynamic behavior as they move toward the
default threshold. The distance to the default is defined as the net worth plus the existing
loans after fire-sales, named ex post net worth. Note that if the net worth of a bank is less
than 0, the bank defaults and is forced to sell the existing loans. On the one hand, the plot
of loan investments as a function of ex post net worth (the distance to the default) is a hump
shape; in the area where the ex post net worth is large enough (the banks are far from the
4
default), the banks increase the loan investment as the ex post net worth is getting smaller;
when the banks’ ex post net worth is below a certain level, they decrease the amount of
loans as they are closer to the default. On the other hand, the risky debt amount shows a
monotonically decreasing pattern as the banks are closer to the default threshold.
The same experiment is done with a 1% higher expectation of the government bailouts.
The 1% increase in the belief result in an upward shift of the borrowings, keeping the
downward slope. The plot of the loan investment is now shaped like a lightning bolt; those
who are far way from the default slightly decrease the loan amount as the ex post net worth
decreases. That is, they are willing to be closer to the default threshold. But then, they
start increasing the loan amount for a while and decrease again when they are very close to
the default threshold. In turn, the relative amount of loan of banks who are close to default
is not that smaller than that of those who are far from the default. Since the debt holders
require a higher rate of return on the debt due to the banks’ bad behavior, the price of debt
goes up to 140% higher. Lastly, the 1% change in the belief allows banks to enjoy higher
payout ratios and to issue less equity.
The last exercise tests if the predicted amount of the cash injection that would be made
by the government under the bailout policy in the model is reasonable or comparable to the
actual data in the crisis of 2008-2009. In the model, the bailout policy is designed that the
government will give just enough money to a troubled bank so as to the bank can pay back
the outstanding debt due today, only if the bank defaults and the government decides to
rescue it. Under the TARP, 736 financial firms were rescued, and on average they received
4.39% of their total assets as the rescue funds. In the simulated data using the parameter
estimates of the sample before the recent crisis, the expected amount of the government
rescue funds is about 4.53% of their total assets. The moments are surprisingly close. That
is, the banks’ expectation about the government’s rescue funds before the crisis is very similar
5
to what they actually received during the crisis.
The paper is organized as follows. Section 1 contains the motivation and the related
papers in the literature. Section 2 describes the model. Section 3 describes the data and the
characterization of the model. Section 4 presents the estimation results. Section 5 discusses
some counterfactuals. Section 6 describes the government bailout plan in the recent financial
crisis of 2008-2009. Section 7 concludes. The Appendix explains how to find the model
solutions.
1
Motivation
This paper is motivated by the fact that the 20 largest corporate bankruptcies have occurred
in the last three decades in the U.S., and 9 out of 20 occurred in 2008 and 2009. Especially,
2009 marked the highest number of billion-dollar bankruptcies ever recorded. The number of
business bankruptcy filings for one year from July 2009 to June 2010 is about twice as many
as it is for three years from July 2006 to June 2008. Not only has the number of bankrupt
companies increased, but the number of bailouts has also increased. However, the effects of
the government bailout have not been fully investigated in the literature.
The recent global depression began with the largely unexpected U.S. financial crisis in
2007. In April of 2008, starting with the bailout of Bear Stearns Companies, Inc., which
costed $29 billion, the U.S. government bailed out big troubled firms and banks on an adhoc basis by buying toxic assets in the hope that it would rescue the economy. It was
strongly believed that the bailout of Bear Stearns sent a strong signal to the markets that
the government would bail out other large financial institutions in trouble. In the late
summer of 2008, the U.S. government injected $200 billion into Fannie Mae and Freddie
Mac to prevent their bankruptcies. In mid-September 2008, the U.S. government also rescued
6
American International Group, Inc. (AIG), one of the world’s largest insurance companies,
by committing up to $85 billion. On the other hand, on September 15, 2008, the U.S.
government let Lehman Brothers Holdings Inc. go under, so the firm eventually filed for
Chapter 11 bankruptcy. Some argue that since Lehman was expected to get bailed out as
well, Lehman was not fully prepared for the bankruptcy. The filing has recorded the largest
bankruptcy in U.S. history, and the failure of Lehman Bothers was thought to have played
a critical role in triggering the late-2000s global financial crisis. On October 3, 2008, after
some hesitations, the U.S. government announced its bailout plan of $700 billion, known as
the Troubled Asset Relief Program (TARP). It would buy the troubled assets, especially
mortgage-backed securities, of domestic financial institutions as well as equity positions in
the U.S. largest banks using taxpayer funds. The U.S. Treasury had primarily used preferred
stock to recapitalize troubled banks. These actions were supposed to stabilize the financial
markets, help them from going into bankruptcy, and prevent any further credit freeze.
However, the above recent examples including bailouts of non-financial firms, such as GM
and Chrysler, in 2008 remain very controversial. Even economists and regulators disagree
on how governments or central banks should react to financial crises. On the one hand, it
seems that the intervention prevents the imminent financial crisis and stops the spread of
failure before it ever begins, and helps preserve the firm value or save the costs that might
be caused by the failed firm. On the other hand, it is evident that it involves substantial
costs, especially to the taxpayers of rescue funding. Moreover, all seem to agree that the
unintended byproduct of the government bailouts is the moral hazard problem. If the bank
anticipates that it will be rescued when in distress, it may take risky investments beyond
what it would take otherwise. It is also possible that the troubled bank may deliberately
fail to prepare for bankruptcy since the government’s help becomes more likely. Thus, the
economic crisis of 2008-2009 requires the researchers to explore more systematically if the
7
bailouts can be justified, or at least what the consequences are quantitatively. The main
goal of this paper is to quantify the effects of the bank bailouts on their risk taking behavior.
There have been few papers that try to tackle a relationship between the government
bailouts and the moral hazards by constructing models. Cordella and Yeyati (2003) show
that a bailout has two effects. Letting the troubled banks survive preserves their values as
well as creates moral hazards. They find that the former is greater than the later. However,
the model is not suitable to explain the financial crises because their analysis does not involve
the real data.
To my knowledge, there is one paper which constructs a structural model to study the
relationship between bank bailouts and moral hazards. Cheng and Milbradt (2012) investigate the trade-off between risk-shifting problems and rollover risks of short-term debt. The
biggest different aspect of the present model from theirs is the onset of the disaster. Their
model is similar to He and Xiong (2009) in terms of the fact that the crisis comes from the
maturity mismatch between the long-term investment and the short-term debt. In their
model, the creditors have a choice to roll over the debt or run. Unlike these papers, the
model in the present paper assumes that the loans can default and that the deposits are
exogenously given. That is to say, there are two channels to lead a bank into trouble; the
existing loans default or the depositors withdraw their deposits all of a sudden. In the event
of the recent crisis, the turmoil started with the subprime mortgage crisis. Back then, the
subprime mortgages played important roles in the portfolios of banks, finance firms, insurance companies, pension funds, and sovereign wealth funds.4 Huge declines in the price of
housing, consequently, had major effects on the balance sheets and portfolios of financial
4
The value of American subprime mortgages was estimated at $1.3 trillion as of March 2007. The ratio
of subprime mortgages to total originations was 7-8% in 2001-2003, 18-21% in 2004-2006, and less than 10%
again during 2007 (3% by the end of fourth quarter of 2007). See more details in an article, “How severe is
subprime mess?”, at msnbc.com.
8
institutions throughout the world. Meanwhile, the insured deposit level of the U.S. commercial banks is constantly decreasing over the last decade. The model in this paper is more
pertinent to capture these two features of the recent crisis.
Some use the reduced-form approach, which requires data on the expected probability
of bailout as well as the risk taking. Unfortunately, both necessary ingredients are not
directly observable in the data. For instance, Dam and Koetter (2012) show that the bailouts
exacerbate moral hazards using German data. Even if they state that they construct a
structural model, they actually run a two-stage regression model. In the first stage, they
estimate the expected bailout probability conditional on a bank being in distress. In this
step, they use political factors as an instrumental variable, arguing that the political factors
are related to the likelihood of a bank bailout, but are not related to the risk taking of banks.
Using the estimates, in the second stage, they identify the causal effect of bank bailouts on
additional risk taking. Since the riskiness is unobservable, they first define risks using the
records of distress defined by regulatory authorities. They also define risk taking by the
likelihood of distress. Then, the moral hazard is estimated as the sensitivity of distress
probabilities with respect to an increase in the expected bailout probability. These proxies
cause serious measurement errors. Unlike to their paper, the structural model used in this
paper does not require any proxy measures of risks. Moreover, the risk taking behavior
in this paper is not simply the likelihood of default, but the choices made by the banks
endogenously, which are directly observable even in a hypothetical situation. It, in turn,
allows to investigate a dynamics of the banks.
Attempts to find the optimal type of bailout also have been made in the literature.
Wilson and Wu (2010) have a two-period model with two banks. One always chooses a safe
investment and the other may choose a risky investment. They interpret it inefficient if the
bank chooses the bad type investment. They conclude that buying up common stocks is
9
more efficient than preferred stocks. Bernardo, Talley, and Welch (2011) constructs a model
for more general firms (e.g., a car company). They suggest that the bailouts should punish
the managers to improve ex-ante incentives and prevent them from taking risks. They also
argue that the funds for bailouts should come from productive firms instead of from the
recipients. In the present paper, the main goal is not to find the optimal bailout policy,
but to quantify the consequences of the government’s announcement and commitment of the
bailout plan.
2
Model
Banks are different from other non-financial firms in terms of their capital structure. For
example, commercial banks borrow money on a short-term basis, such as consumer deposits,
which can be withdrawn from the banks at any time. In contrast, the major assets for most
banks are mortgages (real estate loans), credit card, auto loan receivables, and business
loans, which are illiquid and have longer maturities. The flow of money that banks make
creates liquidity in the markets. Such a bank can be represented by its balance sheet as in
Figure 1. Like any balance sheet it is divided into two sides: assets on the left, and liabilities
and net worth on the right. The percentage numbers in Figure 1 are the actual average of
commercial banks in the U.S. from 1987 to 2008. The data are from the Bank Regulatory
Database, from the Federal Reserve Bank of Chicago.
There are four main categories of assets. More than half of the assets are loans, which are
the primary source of interest revenue. This asset includes loans to consumers (home loans,
personal loans, auto loans, credit card loans) and business (real estate development loans,
capital investment loans). About 30% of assets are investment securities. Securities are safer
than loans but not as safe as reserves, while they pay more interest than reserves but not
10
as much as loans. Two important items in this category are U.S. Treasury securities and
Federal funds. The third asset category is reserves. While it is only about 10% of assets, it is
extremely important. In this category, there are vault cash and Federal Reserve deposits to
ensure the security of deposits. In the United States, the Board of Governors of the Federal
Reserve System sets a reserve requirement. It applies to some deposits held at depository
institutions, such as commercial bank including U.S. branch of a foreign bank, savings and
loan association, savings bank, credit union, etc. Lastly, the fourth type of assets is physical
assets, which include buildings, land, equipment that are owned by the bank. Unlike to
other types of business, this is relatively small for most banks.
On the other side of the balance sheet are net worth and liabilities. The average leverage
of commercial banks is 90%, which is extremely high compared to what we usually observe
in non-financial industries. The liabilities mostly consist of deposits. There are two ways to
separate the deposits into two categories. The first way is by the length or the accessibility
of the deposits: demand deposits (or transaction deposits) and time & saving deposits.
The demand deposits include all deposits in depository institutions that can be withdrawn
without prior notice, while the saving deposits are locked up for a certain length of time.
The second one divides deposits into insured or non-insured deposits. The Federal Deposit
Insurance Corporation (FDIC) provides deposit insurance of a depositor’s accounts up to
$250,000 for each deposit ownership category in each insured bank since October 3, 2008.
The deposit insurance limit was $100,000 from 1980 to 2008. Finally, the rest is net worth,
which can be thought of as what the bank owes the owners, equity holders. A negative net
worth would put the bank in default.
[Insert Figure 1: A representative, hypothetical balance sheet]
As seen above, the bank can be represented by its balance sheet. Accordingly the key
11
features of the model are summarized as follows: (i) On the asset side, there are cash reserves
and two types of investments, one-period risk-free bond, bt and a portfolio of risky loans, lt ;
(ii) On the liabilities side, there are fully insured deposits, dt , and risky debt, qt ; (iii) Banks
can default if the net worth is less than 0, but there is a possibility that they can get bailed
out; (iv) Banks maximize shareholders’ value by choosing the investment and financing
decisions; (v) shareholders are risk-neutral; and (vi) Time is discrete and the horizon is
infinite. Also, the bank’s balance should be balanced:
lt + bt = kt + dt + pt qt ,
(1)
where kt denotes the net worth at time t and pt the price of the risky debt. From now on,
variables with primes denote next period’s values. The more details about each ingredient
are in the following subsections.
2.1
Asset Side
A bank invests in a risk-free bond, b, which yields a risk-free rate, rf . The bank also invests
in risky loans, l, which yields a rate of return, rl . A fraction of the loans, δ, is due in each
period. δ < 1 means that the maturity of the loans is on average longer than 1 year. There
is a risk that the loan can default. Let 1 − z is the default rate of the loan, where z denotes
the survival rate of the loans. The default is independent with the maturity of the loans.
Thus, the income from the loan of l consists of two components: a stochastic survival rate
z and a deterministic profitability rate rl ; i.e., investing an amount of l in the portfolio of
loans yields (rl + δ)zl in the following period. And, the rest (1 − δ)zl remains and is rolled
over to the next period. The law of motion of l is then given by
l′ = (1 − δ)zl + i,
12
(2)
where i is the investment in new loans chosen at time t for the next period t+1. The survival
rate z of the loans is assumed to have a truncated-normal distribution:
iid
z ∼ N(µz , σz2 ), z ∈ [0, 1].
(3)
Lastly, the bank also holds reserves required by the Board of Governors of the Federal Reserve
System. Let’s denote the reserve requirement by α. The available capital for investments
at time t is then At − αdt , where At is the total assets and dt is deposits held by the bank.
That is, the reserves are determined by the deposits.
2.2
Liability Side
A bank borrows money from depositors, called “deposits” d. It is assumed that the deposits
are randomly given in each period; that is, the bank cannot choose how much it wants to
borrow from depositors. The deposits d follows an AR(1) process, that is,
d′ = µd + ρd d + ǫd ,
(4)
where ǫd ∼ N(0, σd2 ). All deposits are fully insured; that is, d is corresponding to the insured
deposits in Figure 1. The depositors require a fixed rate of interest at each period, rd , which
is lower than the risk-free rate rf ; rd < rf . This is because the difference between the two
rates includes the costs of the intermediary’s service as well as the costs of the insurance.
The bank can also issue debt q ′ which is risky since the bank can default on the debt.
The bank defaults if its net worth is less than zero after the credit shock is realized but
before the new deposits are given. The net worth in the next period is given by
w ′ = (rl + δ)z ′ l′ + (1 + rf )b′ − (1 + rd )d′ − q ′ .
13
(5)
The first two terms are the incomes from the investments in the loan portfolio and from
the risk-free bond investment, respectively. The last two terms are the payments to the
depositors and to the debt holders, respectively. A bank defaults on the debt when w ′ < 0.
One can define a default threshold of the survival rate z. The bank defaults when z < zd
where
−(1 + rf )b′ + (1 + rd )d′ + q ′
zd (b , q , l , d ) ≡
.
(rl + δ)l′
′
′
′
′
(6)
The recovery of the debt holders in the event of default is equal to
x(b′ , q ′ , l′ , d′ , z ′ ) = max{min{(rl + δ)z ′ l′ + ξ(1 − δ)z ′ l′ + (1 + rf )b′ − (1 + rd )d′, q ′ }, 0}, (7)
where ξ is fire-sales price of outstanding loans. Then, the debt-pricing formula, p(·), is given
by
Z zd h ′ ′ ′ ′ ′
!
i
x(b
,
q
,
l
,
d
,
z
)
1
1+E
− 1 Γ(z ′ |zt )d .
p(b′ , q ′ , l′ , d′ ; z, d) =
′
1 + rd
q
z
2.3
(8)
Default and Government Intervention
In the event of default, banks sell the existing loans at a discount price. As discussed in the
previous section, a bank defaults if the net worth w is less than 0. The default does not
mean that the bank exits the market. Note that the default occurs when the bank does not
have enough cash to pay back the outstanding risky debt. When it is in distress, the bank is
forced to sell the remaining loans at a fire-sales price ξ so as to meet the obligation as much
as possible. That is, the amount of the loans that has to be sold to pay back the debt is the
absolute value of
w
.
ξ
After the fire-sales, the remaining loans are ld ≡ (1 − δ)zl + wξ . If this
amount is less than 0, the bank finally goes bankrupt.
In the model, the government can intervene in the market when a bank is bankrupt.
If the existing loans are not enough to meet the debt obligation even after selling them,
14
the bank files for bankruptcy. Then, the government comes in and decides to rescue the
troubled bank or not. For simplicity, it is assumed that the bailout decision is random from
the bank’s perspective, and the probability of the government intervention conditional on
the bank being bankrupt is denoted by η. Note that the fire-sales are obviously inefficient. It
is especially so in the financial markets since the loans are very illiquid. So, the government
will intervene and inject direct money into the troubled banks so that the banks do not need
to immediately sell all the existing loans at a lower price and that they can continue their
business.
2.4
Banks’ Problem
[Insert Figure 2: Time-Line between time t and t + 1]
Figure 2 summarizes a bank’ problem from time t to time t + 1. The bank has chosen
the risk-free bond b, loans l, and risky debt q in the previous period t − 1. Given them,
at the beginning of time t, the bank observes the loan survival rate z and the deposit level
d. These shocks determine whether the bank defaults or not (see Equation (6)). If the net
worth is positive, the bank can continue by choosing a new financing decision q ′ , and new
investment decisions b′ and l′ . On the other hand, if the net worth is negative meaning that
the bank defaults, it sells the existing loans in order to meet the debt obligation. However,
if it is not feasible to pay back the outstanding debt even after it sells all existing loans, the
bank files for bankruptcy. Then, the government comes in and bails the bank out with a
probability of η. Otherwise, the bank is reorganized. Reorganization occurs costs levied to
shareholders.
Let ei be the cash flows to shareholders after choosing the next period’s decisions as well
as realizing the new deposits for each case i ∈ {c ≡ continuation, d ≡ default, b ≡ bailout}.
15
When the net worth w is positive, the cash flows to the shareholders are given by
ec = w + (1 − δ)zl + d′ − l′ − b′ + p(b′ , q ′ , l′ ; z, d)q ′ .
(9)
When the net worth w is negative, the bank starts selling the outstanding loans at a discount
price ξ until it can meet the debt obligation. Since the loans are illiquid the sales price of
the loans is less than 1. If w + ξ(1 − δ)zl is positive, it means that the bank does not need
to sell all the loans. In this case, the bank needs to sell the existing loans of | wξ |, which is
just enough to meet the obligation, and is then left with
w
ξ
+ (1 − δ)zl. Thus, the cash flows
to the shareholders are given by
ed =
w
+ (1 − δ)zl + d′ − l′ − b′ + p(b′ , q ′ , l′ ; z, d)q ′ .
ξ
(10)
It is also possible that the bank cannot still meet the debt obligation after selling all the loans
that it currently has. In such a case, the bank files for bankruptcy, and the debtors enter
into reorganization process and verify the net worth. The debtors have all bargaining power
ex post and extract all bilateral surplus. All the costs occurring in this process is imposed to
shareholders. On the other hand, in the event of bankruptcy, the government intervenes in
the market and bails out troubled banks with a probability of η. If the government decides
to rescue the troubled banks, it injects some cash into the banks.5 The cash injection is
denoted by τ and the cash flows to the shareholder are:
eb =
w+τ
+ (1 − δ)zl + d′ − l′ − b′ + p(b′ , q ′ , l′ ; z, d)q ′.
ξ
5
(11)
Under the Troubled Asset Relief Program (TARP), the government would buy the troubled assets,
especially mortgage-backed securities, of domestic financial institutions as well as equity positions in the
U.S. largest banks using taxpayer funds. These actions can be interpreted as cash inflows to the troubled
banks.
16
2.5
Model Simplification
Notice that the deposits are exogenously given and that the bank chooses the investment
decisions, l′ (or i) and b′ , and the financing decision, q ′ , after observing the loans income,
but before the deposits for the next period are realized. That is, the bank makes a decision
based on the expected value of the next deposits. There are two advantages of doing so.
First of all, it is computationally easier since the number of state variables is reduced by
1. Otherwise, d′ would also be either a decision variable or a state variable. Secondly, the
assumption makes the model more realistic since the uncertainty about the level of deposits
can be interpreted as a rollover risk. According to the deposits process, the expected value
of d′ conditional on d is given by
E[d′ |d] = µd + ρd d,
(12)
which is a function of the current level of deposits.
Next, the total assets are fixed and constant for simplicity. It is denoted by A. The
constant and fixed total capital assumption can be justified since of interest is not the total
size of investment, but the allocation between two investment options. As we discussed in
section 2.1, the available assets for investment are A − αd due to the reserve requirement.
Then, choosing the loan amount automatically determines the amount invested in the riskfree bond, and vice versa; i.e., b = A − αd − l. This again reduces the number of state
variables by 1. Therefore, there are 4 state variables {b, l, d, z} and 2 choice variables {q ′ , l′ }.
Lastly, the cash injection by the government τ is equal to the amount of money that
the troubled bank is short of. When the net worth after fire-sales
w
ξ
+ (1 − δ)zl is negative,
the bank files for bankruptcy. The government makes an amount of cash injections of w so
that the bank can survive and the creditors can still get the money from the bank. This
17
simplification assumption is necessary since the expectation about the government parameter
η and the cash injection parameter τ cannot be identified simultaneously given the limitation
of the data. From the bank’s point of view, they solve the model based on their expectation,
which is a multiplication between η and τ .
2.6
Bellman Equation
At each time t, a bank chooses qt+1 , and lt+1 that maximize the discounted life-time expected equity value given the set of state, {qt , lt , dt , zt }. The objective function of the bank’s
shareholders is given by
max
{qj+1 ,lj+1 ,j=t,··· ,T }
Et
∞
hX
β j−teit
j=t
i
,
i ∈ {c, d, b}
(13)
where β is a discount factor.
Furthermore, the model assumes that there are asymmetric adjustment costs for the
loans; when a bank increases the loan amount, it is costly. Let Λi (·) denote the adjustment
costs for each case.
Λi = λ(ld − l′ ))2 1{l′ > ld },
i ∈ {c, d, b}
(14)
in which λ is a coefficient of the adjustment cost function of the loans and ld is the remaining
loan amount.
Thus, the Bellman equation in each case is given by
i
i
′ ′ ′ ′
V i (q, l, d, z) = max
e
−
Λ
+
βE[V
(q
,
l
,
d
,
z
)|d,
z]
,
′ ′
{q ,l }
18
i ∈ {c, d, b}.
(15)



V c (q, l, d, z)
if w ≥ 0,


V (q, l, d, z) = V d (q, l, d, z)
if w < 0 and ld ≥ 0,



ηV b (q, l, d, z) + (1 − η)V d (q, l, d, z) if w < 0 and l < 0.
d
(16)
V d denotes the value function when the bank gets bailed out by the government.
3
Estimation
3.1
Estimated outside the Model
Most of the parameters of the model are estimated using simulated method of moments
(SMM). However, some of the parameters are estimated separately. The fixed and constant
capital A is set to be 1. The market discount factor β is equal to
1
,
1+rf
where rf is the
risk-free rate of return. I estimate the risk free rate of return as the annualized 1-month
T-Bill return and set equal to 3%.6 The rate of return on deposits rd is set to 1%.7 The depository institution’s reserve requirements is currently 10%8 of the net transaction accounts.
Accordingly, α is set to be 10% for simplicity. The survival rate of loan µz is the average
of the sum of charge-off and delinquency rates on loans and leases of all commercial banks
in the U.S. provided by the Board of Governors of the Federal Reserve System. It is 96%
over the sample period. The set of parameter values are summarized in the panel A and B
of Table I.
[Insert Table I: Model Parameters]
6
The source is from http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/Data_Library/
The source for rd are the 1-year CD national rate, the money market accounts national rate, the checking
account interest rate over the sample period.
8
Shrieves and Dahl (1992) show that most banks operate at leverage levels slightly above the regulatory
minimum.
7
19
3.2
Estimation by SMM
There are 9 parameters to pin down: the drift, serial correlation and standard deviation
of the deposits process, µd , ρd and σd ; the standard deviation of the loan survival rate, σz ;
the percentage of loans that matures in each period, δ; the rate of return on loans, rl ; the
conditional belief on the government bailout, η; the fire-sales price of existing loans, ξ; and
the loan adjustment cost coefficient, λ. In this paper, the simulated method of moments
estimates the model parameters, which is basically to choose model parameters that the
distance between the moments of the real data and the moments of the simulated data from
the model is as small as possible. It is necessary to choose moments that are informative
about the unknown parameters. The data used for the SMM are described below in Section
3.3.
The deposit process can be estimated separately outside the model. However, I adopt a
two-step procedure as in Cooper and Haltiwanger (2006). They suggest to split the parameters into two sets; in this paper, they are the parameters generating the deposit process
(Θ1 ≡ {µd , ρd , σd }) and the other 6 parameters (Θ2 ≡ {σz , δ, rl , η, ξ, λ}). First, given an
initial second set of parameter values in Θ02 , the 3 parameters related to deposits in Θ11 are
estimated to match the following three moments: the average, serial correlation and standard
deviation of the deposits process. Next, the second set of 6 parameters in Θ12 is estimated.
To pin down this set of parameters, 9 moments are chosen to be matched: the average of
leverage; the first moment of leverage, the autocorrelation of leverage, the standard deviation of the shock to leverage, the first moment of profits, the first moment of dividends,
the first and the second moment of charge-offs, the ratio of insured deposits to the total
liabilities, and the frequency of default. When the 9 moments are matched, the 3 moments
of deposits are calculated using the parameters that are found so far. If these moments are
close enough to the moments that are found in the first step, the procedure stops here. If
20
they are not, the first set of parameters Θ1 is re-estimated given Θ12 . Repeat the process
until the moments related to deposits are matched. However, in this paper, since the deposit
process is exogenously given and independent from other parameters, one iteration is enough
to find the parameters.
All variables are deflated by total assets, and the profitability is measured by operating
income. The default frequency is calculated by dividing the number of banks that are delisted
from the FDIC for a given year by the total number of banks that are listed in the FDIC in
the previous year: that is,
Default Frequency =
(# of Banks in t) − (# of Banks in t − 1)
.
(# of Banks in t − 1)
(17)
Note that (# of Banks in t) does not take into account the banks newly entering the markets
in year t. That is, the definition of default probability includes only delisted banks in each
year.
Note that the hypothetical banks generated by the model are heterogeneous only in terms
of their shocks and that the SMM attempts to estimate the parameters of an average bank.
On the other hand, the real data are heterogeneous in many dimensions. Thus, when using
the SMM, it is critical to soak up as much heterogeneity from the data as possible. I use
firm fixed effects in the estimates of variance and use the double-differencing method in
Han and Phillips (2010) to get the AR(1) coefficients. Moreover, I split the sample by the
bank size since small banks are different from large banks.9
According to Erickson and Whited (2000), the weighting matrix is the inverse of the
sample covariance matrix of the moments, which is the inner product of stacked influence
functions of the moments. When minimizing the objective function I use the weighting
9
Another reason why to estimate the subsamples by size is to see if the large banks believe more strongly
the government bailouts.
21
matrix with the firm fixed effects. On the other hand, the standard errors are calculated
using the clustered weighting matrix. More details about how to implement the SMM are
described in the Appendix and also see Strebulaev and Whited (2012).
3.3
Data
The data are from the Bank Regulatory Database, from the Federal Reserve Bank of Chicago,
which provides quarterly accounting data for commercial banks. The sample period is from
1994 to 2007. The data end in 2007 due to the lack of data. The database provides the
amount of deposits greater than the deposit insurance limit $100,000, while the deposit
insurance limit is increased to $250,000 as of October 3, 2008. They have not changed the
data variable definition since then. Data variables are defined as follows: total assets is
RCFD2170; debt is RCFD2950; operating income is RIAD4000; distributions is RIAD4460
(common stock) plus RIAD4470 (preferred stock) minus negative RIAD4346 (net sale of
stock); insured deposits is RCFD2200 (total deposits) minus RCON2710 or RCONF051
(deposits greater than $100,000, which are not insured by the FDIC before 2008).
First step is to delete observations of which total assets are less than one million dollars,
or operating income or deposits is non-positive. Since some variables are only available
on annual basis, the quarterly data are converted to annual data. To do so, I accumulate
the flow variables from first quarter to fourth quarter. As for the stock variables, I pick
the fourth quarter’s values. If there is a missing value in any quarters, the observation is
dropped. Otherwise, the flow variables are biased below. Then, all variables are deflated
by total assets of each year. Lastly, observations are included only when they have at least
three consecutive years. After winsorizing the top and bottom 1% of the variables, I end
up with an unbalanced panel of banks from 1994 to 2007 with between 7,597 and 10,948
observations per year, and 123,159 bank-year observations. It is found that the number of
22
banks in the market is monotonically decreasing.
3.4
Identification
The objective of SMM is to find the model parameters that make the simulated moments
as close to the actual moments as possible. Since model identification is very critical in
this process, it is necessary to choose the moments seriously and with care. I compute the
mean, variance, and the autocorrelation of all possible variables and find the moments that
are sensitive to variations in the model parameters. The three parameters related to the
deposits process are just identified by the actual process of deposits. The other 6 parameters
are pinned down by matching 9 moments described below.
Of particular interest is to identify the expectation about the government bailouts, η.
The most informative moments for this parameter are the ratio of the insured deposits to
the total liabilities and the default frequency. In the model, it is assumed that the deposits
are given exogenously and that banks can choose extra borrowings by paying appropriate
prices for that. As in Figure 6, the insured deposits are monotonically decreasing over time,
while the total liabilities are stable around 90% of total assets. That is, the banks are
borrowing more risky debt to maintain the high level of leverage. Thus, the belief in the
bailouts parameter can be pinned down by this moment. The default frequency is also useful
for identifying the bailout belief parameter. Intuitively, one would expect that more banks
default if they believe that the government bailouts are more likely.
The mean of operating income and the mean of dividends help pin down the rate of
return on loans, rl . As the return on the loans go up, the banks are more profitable. The
more profitable are the loans, the more dividends the banks can pay out. The process of
leverage pins down the average maturity, δ. If the maturity of the loans are too short, the
leverage would not be stable. The adjustment coefficient affects the charge-offs as well as the
23
profits of the banks. The variance of the charge-offs are directly affected by the variance of
the survival rate of the loans, σz . Lastly, the fire-sales price, ξ, affects the default probability.
Even if a bank cannot meet its debt obligation, if it can sell the existing loans at a higher
price, it might be able to escape the distress. Otherwise, it is very likely to go under.
4
Estimation Results
4.1
Full Sample
Table II contains estimation results for the full sample ad shows that the model fits the
data pretty well. The top panel reports the actual moments and the simulated moments
with t-statistics. The moments related to deposit process are almost perfectly matched since
the deposit process is perfectly independent with the banks’ behavior. Fewer than half of
the simulated moments are statistically significantly different from their counterparts. The
model fits the first moments particularly well, such as average of leverage, profitability, and
charge-offs, deposit ratios, and default frequency. The model fails to match the moment of
dividends due to the absence of taxes in the model.
[Insert Table II: Simulated Moments Estimation for Full Sample]
The belief in the bailout probability conditional on default is estimated by 52.44%, which
is lower than the estimate in Dam and Koetter (2012), 69%, for German data. Note that
their estimation method requires a proxy measure of the risks since their estimation is based
on a reduced-form approach.10 In their paper, risk taking is measured as the probability of
10
Even if they define their model a structural model, they actually run a two-stage regression. The
instrumental variable in the first stage is the political factors, such as the election indicator, vote share
difference in state parliament elections, and political similarity between the federal and state prime minister.
They insist that the political factors are directly related to the bailouts, but not related to the risk taking
behavior of banks.
24
distress and the distress is defined as the event in which the regulator officially declares that
the bank is seriously in trouble. In their setup it is not feasible to distinguish what drives
the higher distress probability. On the other hand, the structural model of this paper does
not require any indirect measures of risk. The choice variables, the amount of risky loans
and the risky debt, are the direct measures of risk taking behavior. Moreover, the default
probability can be calculated by simulating the model.
The fire-sales price is estimated 46.42% which is lower than non-financial firms. (e.g.,
Hennessy and Whited (2005) estimate it 59.2%.) It can be interpreted that the investments
of loans are generally illiquid. The average amount of loans that mature in each period is
about 69%. It implies that the the average maturity of loans is about 525 days. The weighted
average maturity for all commercial and industrial (C&I) loans of all U.S. commercial banks
in the same sample period is about 469 days. Also, the weighted average maturity of new
car loans by U.S. commercial banks is about 58 months. The estimated value properly lies
in between the two values. The model predicts that the rate of return on the loan is about
10%, which is also consistent with the actual data. Since there are several types of loans,
one can think of the rate rl as a required rate of return on a portfolio of loans. For instance,
over the same sample period, the 30-year fixed mortgage rate is on average 7%; the finance
rate on consumer installment loans at commercial banks is 8%; the finance rate on personal
loans at commercial banks is 13%; and the interest rate on credit card plans of commercial
banks is 14%. Thus, the rate of return of the loans held by commercial banks should be the
weighted average of these rates. Finally, the parameter λ measures how difficult to adjust
the loan amount upwards and is estimated 6%. In order to increase an additional 1% of
loans it occurs 1.2% of costs.
25
4.2
Subsamples
The model assumes that the banks believe that the government would bail them out with a
constant probability given that they default. The assumption is fine in a sense that the model
explains an average bank’s behavior and the model parameters are estimated for an average
bank. Furthermore, according to the past events, the government rescue plan seems to be
done on an ad-hoc basis with varying degrees of taxpayer support. The U.S. government
rescued Bear Sterns Companies, Inc. by helping the merger with JP Morgan Chase. The U.S.
Treasury took over Fannie Mae and Freddie Mac. The Federal Reserve injected direct capital
into American International Group, Inc. However, the government declined to help Lehman
Brothers Holdings Inc. and the company eventually filed for Chapter 11 bankruptcy.11
On the other hand, the perception of “Too-Big-to-Fail” seems to be prevalent. Since a
big bank is connected to many other financial institutions as well as non-financial firms in
an economy, the failure of the big bank may have a domino effect on the entire economy and
eventually result in a global recession.12 It is likely that the government bailout probability is
dependent upon the bank size. To my knowledge, it has not been proved yet in the literature
the positive relationship between the bailout probability and the bank size. To mitigate this
concern, the sample is split by bank size, and then the expectation of bailout parameter is
estimated for each subsample.
[Insert Table III: Simulated Moments Estimation for Small Banks]
[Insert Table IV: Simulated Moments Estimation for Large Banks]
Table III and Table IV show the estimated results for subsamples of small banks and
large banks, respectively. The bank size is defined by total assets. Large (small) banks are
11
12
See Ayotte and Skeel Jr (2010).
For example, see Aharony and Swary (1983).
26
defined as the ones whose total assets are in the lower (higher) third of the distribution of
each year. The estimation method is identical to the one for the full sample described in
Section 4.1. Note that the moments of the deposit process are different for each subsample.
For example, the mean of insured deposits over the total liabilities is larger for the large
banks (64.97%) than the small banks (53.10%). Thus, the parameters of the deposit process
are re-estimated. The drift and the serial correlation of the deposit process is slightly higher
for the small banks than those of the large banks. The standard deviation of the shock to
the deposit process is higher for the large banks than that of the small banks.
First of all, of particular interest among the parameters governing the banks’ behavior
is the belief on the bailouts. It is estimated by 76.20% and 35.69% for the large banks and
for the small banks, respectively. This is a consistent finding with the “Too-Big-to-Fail”;
the large banks believe more strongly that the government would bail them out conditional
on that they are in distress than the small banks do. Secondly, the fire-sales price indicates
that it is much easier for the small banks to sell the existing loans than the large banks.
However, according to the adjustment cost coefficient, it is easier for the large banks to raise
loan amounts than the small banks. Thirdly, the average maturity of loans is a little bit
longer for small banks than the large banks. It implies that the large banks hold more C&I
loans and less personal loans than the small banks do, because the average maturity of C&I
loans is longer than the personal loans. Lastly, the large banks’ rate of return on the loans
are surprisingly lower than that of the small banks, whereas the standard deviation of the
loan survival rate is slightly higher for the small banks than that of the large banks. Since
the large banks believe that the bailouts are more likely, they are willing to invest in riskier
loans with lower returns.
27
5
Counterfactuals
Of particular use is comparative statics to quantify the responses of to changes in the bailout
belief parameter η. The results from this exercise are in Figure 3. Let η̂ denote the estimated
value of the belief in the government bailout using the full sample. To explore the counterfactuals, I simulate the model 20 times with 20 different values in [η̂ − 10%, η̂ + 10%] for η,
holding the other model parameters fixed as in Table II. Each time, the model is solved and
the simulated model generates hypothetical sets of data. Then, some interesting variables
are computed.
[Insert figure 3: Sensitivity to η]
The top panel of Figure 3 shows the policy functions as a function of the belief in the
government bailout η. The blue solid line and the red dotted line are the loan investment
and the debt borrowing, respectively. As the belief is stronger, banks decrease the allocation
in the risky loans and increase the borrowings. This can be explained by the defaulting
mechanism defined by Equation (5). Recall that the equation basically means that when
the net worth is less than 0, the bank defaults; i.e., when the net worth is smaller, the bank
is more likely to default. Notice also that the net worth w ′ is increasing in the loan amount
l′ and decreasing in the debt q ′ . That is, by investing less in the risky loans and borrowing
more the risky debt, the bank is closer to the default threshold. Thus, when a bank believes
that it would be bailed out more likely conditional on default, it does not try to avoid being
close to the default threshold. That is, the bank deliberately prepares for bankruptcy.
The bottom panel plots the dividends payout in response to the 20 different values for the
belief parameter η ∈ [η̂ − 10%, η̂ + 10%]. The dividends and equity issuance decrease as the
belief is stronger. This means that the banks are shirking as their beliefs in the government
bailouts are stronger. Since they are not afraid of being in default, they borrow more and
28
invest less, and in turn, it allows them to pay out less dividends and issue less equity. In the
end, the profitability is also lower.
While the above exercise explains the average bank’s behavior, it is not enough to show
the dynamics of the bank. So, the second exercise is done to investigate banks’ behavior
depending on how far away they are from the default threshold. First, I simulate the model
using the parameter values that are reported in Table II, except that the parameter of the
expectation of the government’s bailout probability η takes two different values: η1 = η̂ (as in
Table II) and η2 = η̂+1%. Next, let x denote the ex post net worth, defined as the current net
worth divided by the fire-sales price plus the existing loans of a bank, x ≡
w
ξ
+ (1 − δ)zl; i.e.,
x is the remaining loans, if any, after paying back the debt obligations but before choosing
the next period’s decisions in the event of default. This variable measures the distance to
the default threshold of each bank. Then, I split the simulated data into 20 bins according
to the level of x. Note that the banks whose x is less than 0 are not included, which means
that they are already bankrupt. Lastly, the average bank variables in each bin is computed
and plotted in Figure 4. As we move to the left side of each graph, x is smaller, meaning that
the bank is closer to the default threshold. The plots are smoothed by the spline method.
[Insert figure 4: Dynamics As Close to Default]
The blue solid line is when the belief in the government bailout is the estimated value
using the full sample by the SMM, denoted by η̂, and the red dotted line is when the value of
the belief parameter is increased by 1%, that is η̂ + 1%. Panel (A) and (B) plot the decision
variables, the risky loan
l and the risky debt q, respectively. Each of the panels is scaled by the average values in
order to understand the relative dynamics instead of the absolute levels. Under the original
belief η̂, when x is large enough, the bank increases the loan amount as the bank’s ex post
29
net worth x gets smaller. When x passes a certain level toward to the default threshold, the
bank decreases the loan amount. However, when the belief about the government bailout
is 1% higher, the bank decreases the loan amount even when it has enough ex post net
worth. By decreasing the loan amount, the bank is closer to the default threshold, but it
can enjoy high dividends payout (as in panel (E)). This is because the bank does not mind
being near the default threshold. As x goes beyond a certain point, the bank increases the
loan amount in order to escape from the low net worth. This is the evidence of risk taking
near the default threshold. When it comes to the borrowing decisions, as the bank is closer
to the default threshold, it monotonically decreases the risky debt regardless of the beliefs.
The 1% increase in the belief causes an upward shift of the curve, meaning that the bank is
overall willing to take more risks from the borrowing side.
By having these adjustments of decisions in response to the 1% increase in the belief,
the bank has to pay much higher prices for the debt (see panel (C)). The distribution of
the banks shifts to the left as shown in panel (D); the banks are willing to be near the
default threshold by having the lower ex post net worth. Moreover, the higher belief in the
government bailouts causes higher default probability, there are fewer banks survived in the
market; the probability of default increases from 4.50% to 16.60%. Notice also that the
changes in the banks’ behavior allow them to enjoy more dividends when their ex post net
worth is high and to issue less equity even if they are on the verge of default.
Lastly, banks’ reaction is investigated when an unexpected huge one-time shock is introduced. In the model, there are two shock processes: deposits and loans. First, simulating
the model many times lead the banks at a steady state. After the banks arrive at the steady
state, an unexpected huge shock is introduced at time 1. The deposit level is set to 34%, the
lowest value in the deposit space, and the loan survival rate is set to 50%. Figure 7 show the
impulse responses for 50 time periods. In order to compare the dynamics after the shocks,
30
all values are deflated by the initial value at time 0. The experiment is done twice; one with
the original belief (blue solid line) and the other with a 1% higher belief (red dotted line).
[Insert figure 7: Responses to Unexpected Shocks]
The left-side of Figure 7 shows the impulse responses to an unexpected huge drop in
deposits. Under the original belief, after a huge drop in the deposits level, banks increase
the risky debt to substitute the low level of deposits. They initially decrease the investment
in loans, increase the risky investment for a while, and then go back to the steady state level.
By doing so, they can keep the price stable. On the other hand, when their belief in the
government bailout is higher, the increase in debt is not as much as the original belief and
the speed of going back to the original state is slower. Also, the banks decrease the loan
amount, but they keep the less risky investment for a while until they go back to the steady
state. These lackadaisical reactions cause higher price for risky debt.
The right-side of Figure 7 shows the impulse responses to an unexpected huge drop in
loan survival rate. When compared to the impulse response to the deposit shock, the size
of reactions is relatively small, the speed of recovering is much faster, and the difference
between the reactions of two belief is small. The banks decrease the risky debt due to the
fear of inability of repayment and increase the loan amounts to recover the bad situation.
The price of the risky debt is rather lower than before the shock is introduced.
6
What Happened?
To estimate the model presented in this paper, the data before the recent crisis are used. This
section describes what actually happened during the crisis. After the subprime mortgage
crisis, the U.S. government decided to purchase assets and equity from financial institutions
in October of 2008. Originally the total amount of costs was about $700 billion. As part
31
of TARP, the Bank Capital Purchase (CPP) program was designed to stabilize the financial
system by providing capital to financial institutions of all sizes throughout the nation and
it was conducted by the U.S. Treasury’s Office of Financial Stability. Under the CPP, the
Treasury provided capital to 736 financial institutions and the total amount paid is about
$200 billion.
I collect all banks, thrifts, or bank holding companies that received the government help
under CPP, identify their RSSD IDs, and find out the total assets of the quarter that they
received the money. If it is a bank holding company, the total assets of all related banks are
summed up. I end up with 585 financial institutions. The average size of the banks is $10.9
billion, while the median is only $0.3 billion. It implies that the distribution is left skewed.
Figure 5 shows the histogram of the ratio of the rescue funding to the total assets under the
TARP. The money that they received from the Treasury is on average 4.39% of their total
assets. Using the same criterion to split the sample by size (total assets) in Section 4.2, the
amount of the capital injection is on average 6.75% and 3.92% of their total assets for small
institutions and for large institutions, respectively. Although the percentage is higher for the
small banks than that of the large banks, in terms of the dollar amount, it is much larger for
the large banks than for the small banks: on average, it is $3.1 million and $533.9 million
for the small banks and for the large banks, respectively.
[Insert figure 5: The histogram of TARP]
These numbers are very comparable with the model. In the model, the amount of the
rescue loan is denoted by τ , which is equivalent to | wξ +(1−δ)zl|. This is the expected amount
of money that banks would get if they would default and get bailed out. By simulating the
model using the set of parameters in Table II, this variable is on average 4.53% of total
assets. For the small banks and the large banks, the expected rescue loan amount is 3.56%
32
and 9.03%, respectively. Thus, the large banks expected more than what they actually
received, while the small banks received more than what they expected.
7
Conclusion
This paper quantifies the effects of the government bailouts on banks’ risk-taking behavior by
estimating the model parameters governing the banks’ investment and financing decisions.
The government bank bailouts have been criticized since they would create incentives for
banks to further engage in risk-taking behavior. Although this topic has been regarded very
important, the literature has not fully tackled the research question due to the unobservable
variables in the data. To cope with the challenges, this paper adopts a new way – structural
estimation – which allows one to directly observe banks’ behavior, such as investment and
financing decisions, even in a hypothetical scenario.
The estimation results show banks predict that the government would bail them out with
a probability of 52.44% conditional on default. It is also found that large banks believe more
strongly in the government bailouts than small banks do. The counterfactual exercises show
that banks deliberately move toward the default threshold by borrowing more and investing
less when they expect that the government bailout is more likely. The higher belief in the
government bailout affects the banks’ dynamics in the sense that they tend to take more risky
investments and borrow more risky debt beyond what they would take otherwise when they
are close to default. Lastly, the model with the parameters estimated using the pre-crisis
data predicts well the actual amount of the rescue loans injected by the U.S. government in
2008 (known as TARP).
Unfortunately, the current model is not capable of explaining different types of capital
injections, such as common stock repurchase or preferred stock repurchase. However, one can
33
think of another government intervention. For instance, the government would rescue not
only the banks but the debt holders of banks. If the debt holders believe that the government
would bail them out as well, it would directly affect the price of debt. The model can also
be developed in a way to incorporate the macroeconomic shock. By having this component,
it would be able to explain the domino effect or contagion effect in the financial markets,
which are usually observed during financial crises.
34
Appendix
A. Model Solution
Here I describe how to solve the model and the simulation procedure. First, to find a
numerical solution, I discretize a finite state space for the four state variables, {q, l, d, z}.
The loan amount q and the risk debt q lie between 0 and A(≡ 1). Both spaces are equally
discretized. As for the deposit process, I transform the AR(1) process into discrete-state
spaces using quadrature method following Tauchen and Hussey (1991). The survival rate of
loan is truncated-normal distributed between 0 and 1 with mean µz and standard deviation
σz .
The model is then solved via iterations on the Bellman equation. This yields the policy
functions, {q ′ , l′ } = h(q, l, d, z). To generate an artificial data set, I first take random draws
of the survival rates of loans z and the deposits d. Then, I simulate each bank to generate q
and l using the policy functions, while updating the shocks. I simulate each bank 200 time
periods, and keep the last 100 time periods, corresponding to the sample period of data,
1994-2007. I drop some number of simulations in order to reach an optimal point.
B. Truncated Normal Distribution
I adopt and modify the method proposed by Ada and Cooper (2003). Let nz be the number
nz −1
of grids on z. First of all, I construct {mi }i=1
such that
mi −µz
− Φ(α)
Φ σz
Φ(β) − Φ(α)
35
=
i
,
nz
(B.1)
where Φ(·) is the cumulative density function (CDF) of N(0, 1), α =
0−µz
σz
and β =
1−µz
.
σz
Then, taking the inverse function of Φ(·) yields
−1
mi = Φ
i
Φ(β) − Φ(α) + Φ(α) σz + µz ,
nz
(B.2)
which are the points to discretize the space [0, 1]. Next, define the abscissas {zi }ni=1 such
that zi is the expected value of each interval between the points mi ’s. That is,
zi = E z|z ∈ [mi−1 , mi ]
= µz − σz
−µz
z
φ( miσ−µ
) − φ( mi−1
)
σz
z
−µz
z
) − Φ( mi−1
)
Φ( miσ−µ
σz
z
i = 2, 3, · · · , n − 1,
(B.3)
where φ(·) is the probability density function (PDF) of N(0, 1). For the end points, m0 = 0
and mn = 1. By construction, notice that the probability of each abscissa pi =
1
n
∀i.
C. SMM Estimation
Let xit and yits (β) denote the data and the simulated data, respectively, i = 1, · · · , n, t =
1, · · · , T , and s = 1, · · · , S; T the sample period; and S the number of simulated data sets.
The artificial data sets are dependent upon a set of parameters β. The SMM is designed to
find the optimal β to minimize the distance between a set of simulated moments, m(yits (β)),
and a set of actual moments from the data m(xit ). The moment vector can be written as:
g(xit , β) =
n
T
S
i
1X
1 XXh
m(xit ) −
m(yits (β)) .
nT i=1 t=1
S s=1
(B.4)
The simulated moments estimator of β is the solution of
β̂ = arg min g(xit , β)′ Ŵ g(xit , β),
β
36
(B.5)
where Ŵ is a positive definite matrix that converges in probability to a deterministic positive
definite matrix W .
To find the weight matrix, Ŵ , I adopt the influence function method from Erickson and Whited
(2002). When I calculate the influence functions, I demean each of variables at the bank
level to take out the heterogeneities in the data. This is because the data are greatly heterogeneous, whereas the simulated data are heterogeneous only by the shocks and I estimate
the parameters of an average bank. The inverse of the covariance matrix of the moments is
Ŵ .
For the standard errors, I use a clustered weight matrix within time and bank, denoted
Ω. The asymptotic distribution of β is given by
√
d
n(β̂ − β) → N 0, avar(β̂)
(B.6)
in which
1 h ∂gn (β) ∂gn (β) i−1 h ∂gn (β)
∂gn (β) i−1 h ∂gn (β) ∂gn (β) i−1
avar(β̂) ≡ 1 +
W
W
ΩW
W
S
∂β
∂β ′
∂β
∂β ′
∂β
∂β ′
37
References
Ada, Jerome and Russell W Cooper (2003), “Dynamic economics.”
Aharony, Joseph and Itzhak Swary (1983), “Contagion effects of bank failures: Evidence
from capital markets.” Journal of Business, 305–322.
Ayotte, Kenneth and David A Skeel Jr (2010), “Bankruptcy or bailouts?” Iowa J. Corp. L.,
35, 469–849.
Bernardo, Antonio, Eric Talley, and Ivo Welch (2011), “A model of optimal government
bailouts.”
Cheng, Haw and Konstantin Milbradt (2012), “The hazards of debt: Rollover freezes, incentives, and bailouts.” Review of Financial Studies, 25, 1070–1110.
Cooper, Russell W and John C Haltiwanger (2006), “On the nature of capital adjustment
costs.” The Review of Economic Studies, 73, 611–633.
Cordella, Tito and Eduardo Levy Yeyati (2003), “Bank bailouts: Moral hazard vs. value
effect.” Journal of Financial Intermediation, 12, 300–330.
Dam, Lammertjan and Michael Koetter (2012), “Bank bailouts and moral hazard: Evidence
from Germany.” Review of Financial Studies, 25, 2343–2380.
Erickson, Timothy and Toni M Whited (2000), “Measurement error and the relationship
between investment and q.” Journal of Political Economy, 108, 1027–1057.
Erickson, Timothy and Toni M Whited (2002), “Two-step gmm estimation of the errors-invariables model using high-order moments.” Econometric Theory, 18, 776–799.
38
Han, Chirok and Peter CB Phillips (2010), “Gmm estimation for dynamic panels with fixed
effects and strong instruments at unity.” Econometric Theory, 12, 119.
He, Zhiguo and Wei Xiong (2009), “Dynamic bank runs.” Work. Pap., Univ. Chicago.
Hennessy, Christopher A and Toni M Whited (2005), “Debt dynamics.” The Journal of
Finance, 60, 1129–1165.
Lepetit, Laetitia, Emmanuelle Nys, Philippe Rous, and Amine Tarazi (2008), “Bank income
structure and risk: An empirical analysis of European banks.” Journal of Banking &
Finance, 32, 1452–1467.
Shrieves, Ronald E and Drew Dahl (1992), “The relationship between risk and capital in
commercial banks.” Journal of Banking & Finance, 16, 439–457.
Strebulaev, Ilya and Toni Whited (2012), “Dynamic models and structural estimation in
corporate finance.” Available at SSRN 2091854.
Tauchen, George and Robert Hussey (1991), “Quadrature-based methods for obtaining approximate solutions to nonlinear asset pricing models.” Econometrica: Journal of the
Econometric Society, 371–396.
Wilson, Linus and Yan Wendy Wu (2010), “Common (stock) sense about risk-shifting and
bank bailouts.” Financial Markets and Portfolio Management, 24, 3–29.
39
Table I: Model Parameters
Panel A: Estimated outside the model
Descriptions
Notation
Risk Free Rate
rf
Deposit Interest Rate
rd
Mean of Survived Loans
µz
Panel B: Automatically Determined
Discount Rate
Panel C: Estimated by SMM
Descriptions
Drift of Deposits
Serial Correlation of Deposits
Residual Std. Dev. of Deposits
Std. Dev. of Survived Loans
Bailout Probability
Fire-sale Price
Percentage of Maturing Loans
Loan Interest Rate
Loan Adjustment Cost
Values
3.0%
1.0%
0.96
β
1/(1 + rf )
Notation
µd
ρd
σd
σz
η
ξ
δ
rl
λ
Most Informative Moments
Mean of Deposits
Serial correlation of Deposits
Variance of Deposits
Variance of Charge-offs
Deposits/Total Liabilities
Default Frequency
Leverage Process
Mean Profits & Dividends
Mean Charge-offs
40
Table II: Simulated Moments Estimation for Full Sample
Actual moments are calculated using a sample of commercial banks from the Bank Regulatory
Database. The sample period is for 15 years from 1994 to 2007. Estimation is done by the
Simulated Method of Moments, which is designed to minimize the distance between the actual
moments from the data and the simulated moments from the model. The moments for the actual
data and the simulated data are constructed identically. The first panel reports the actual and
simulated moments and t-statistics. The second panel reports the estimated structural parameters
and the clustered standard errors in parentheses. σz is the standard deviation of the loan survival
rate; η the expectation of the government bailouts; ξ fire-sale price of loans; δ average portion of
maturing loans; rl rate of return of loans; and λ loan adjustment cost coefficient.
Descriptions
Mean Debt/Assets
Autocorrelation Debt/Assets
Std. Dev Shock to Debt/Assets
Mean Operating Income/Assets
Mean Dividends/Assets
Mean Charge-offs/Assets
Std. Dev. Charge-offs/Assets
Deposits/Total Liabilities
Default Frequency
Mean Deposits/Assets
Variance Deposits/Assets
Autocorrelation Deposits/Assets
Panel A: Moments
Actual
0.8962
0.8565
0.0056
0.2015
0.0108
0.0060
0.0075
0.6650
0.0487
0.5960
0.0074
0.8930
Simulated
0.9020
0.5270
0.0022
0.1888
0.0287
0.0100
0.0056
0.6586
0.0450
0.5959
0.0074
0.8869
t-stat.
-1.9382
0.1439
0.0611
2.1501
-9.0441
-1.1640
21.0600
0.1051
1.5235
<0.0001
<0.0001
<0.0001
Panel B: Parameter Estimates
µd
ρd
σd
σz
η
ξ
δ
rl
λ
0.0669
0.8878
0.0399
0.0278
0.5244
0.4642
0.6942
0.1061
0.0616
(0.0042) (0.0071) (0.0014) (0.0021) (0.1940) (0.2261) (0.0765) (0.0272) (0.0440)
41
Table III: Simulated Moments Estimation for Small Banks
Actual moments are calculated using a sample of commercial banks from the Bank Regulatory
Database. The sample period is for 15 years from 1994 to 2007. The size is defined by total assets.
Large (small) banks are defined as the ones whose total assets are in the lower (higher) third of
the distribution of each year. Estimation is done by the Simulated Method of Moments, which is
designed to minimize the distance between the actual moments from the data and the simulated
moments from the model. The moments for the actual data and the simulated data are constructed
identically. The first panel reports the actual and simulated moments and t-statistics. The second
panel reports the estimated structural parameters and the clustered standard errors in parentheses.
σz is the standard deviation of the loan survival rate; η the expectation of the government bailouts;
ξ fire-sale price of loans; δ average portion of maturing loans; rl rate of return of loans; λ loan
adjustment cost coefficient; and τ cash injection.
Panel A: Moments
Descriptions
Actual
Simulated
t-stat.
Mean Debt/Assets
0.8870
0.9122
-9.1987
Autocorrelation Debt/Assets
0.9061
0.7270
0.2952
Std. Dev Shock to Debt/Assets
0.0023
0.0017
0.0662
Mean Operating Income/Assets
0.1919
0.1860
0.8142
Mean Dividends/Assets
0.0098
0.0283
-15.3295
Mean Charge-offs/Assets
0.0053
0.0094
-3.1898
Std. Dev. Charge-offs/Assets
0.0069
0.0040
68.5929
Deposits/Total Liabilities
0.7319
0.7118
0.3453
Default Frequency
0.0970
0.0934
18.2714
Mean Deposits/Assets
0.5310
0.5306
<0.0001
Variance Deposits/Assets
0.0070
0.0070
<0.0001
Autocorrelation Deposits/Assets
0.8847
0.8738
<0.0001
Panel B: Parameter Estimates
µd
ρd
σd
σz
η
ξ
δ
rl
λ
0.0673
0.8964
0.0321
0.0151
0.3569
0.6946
0.6423
0.1290
0.1873
(0.0031) (0.0047) (0.0010) (0.0031) (0.0721) (0.1341) (0.2682) (0.0116) (0.0112)
42
Table IV: Simulated Moments Estimation for Large Banks
Actual moments are calculated using a sample of commercial banks from the Bank Regulatory
Database. The sample period is for 15 years from 1994 to 2007. The size is defined by total assets.
Large (small) banks are defined as the ones whose total assets are in the lower (higher) third of
the distribution of each year. Estimation is done by the Simulated Method of Moments, which is
designed to minimize the distance between the actual moments from the data and the simulated
moments from the model. The moments for the actual data and the simulated data are constructed
identically. The first panel reports the actual and simulated moments and t-statistics. The second
panel reports the estimated structural parameters and the clustered standard errors in parentheses.
σz is the standard deviation of the loan survival rate; η the expectation of the government bailouts;
ξ fire-sale price of loans; δ average portion of maturing loans; rl rate of return of loans; λ loan
adjustment cost coefficient; and τ cash injection.
Descriptions
Mean Debt/Assets
Autocorrelation Debt/Assets
Std. Dev Shock to Debt/Assets
Mean Operating Income/Assets
Mean Dividends/Assets
Mean Charge-offs/Assets
Std. Dev. Charge-offs/Assets
Deposits/Total Liabilities
Default Frequency
Mean Deposits/Assets
Variance Deposits/Assets
Autocorrelation Deposits/Assets
Panel A: Moments
Actual
0.9035
0.8309
0.0015
0.2097
0.0147
0.0068
0.0076
0.5876
0.0868
0.6497
0.0053
0.8990
Simulated
0.9066
0.2928
0.0023
0.2044
0.0301
0.0106
0.0054
0.5856
0.0841
0.6499
0.0052
0.8956
t-stat.
-1.1161
0.4834
-0.1758
0.5513
-9.1061
-5.8384
25.0367
0.0254
0.4515
<0.0001
<0.0001
<0.0001
Panel B: Parameter Estimates
µd
ρd
σd
σz
η
ξ
δ
rl
λ
0.0665
0.8747
0.0406
0.0253
0.7620
0.2232
0.7078
0.0995
0.0393
(0.0031) (0.0059) (0.0013) (0.0504) (0.1113) (0.0781) (0.0356) (0.0189) (0.0933)
43
Figure 1: A Representative, Hypothetical Balance Sheet
Assets
Liabilities & Net Worth
Equity (10%)
Other (5%)
Demand
Not Insured
(20%)
(22%)
Loans
(53%)
Liabilities
Deposits
(90%)
(95%)
Securities
(29%)
Time &
Savings
(80%)
Insured
(78%)
Reserves (9%)
Physical Assets & etc (9%)
The percentages are based on the Bank Regulatory Database, from the Federal Reserve Bank of
Chicago, which provides quarterly accounting data for commercial banks. Sample period is from
1987 to 2008. Data variables are defined as follows: Total Assets is RCFD2170; Total Liabilities
RCFD2950; Total Deposits RCFD2200; Demand Deposits RCON2210; Time and Savings Deposits
RCON2350; Insured Deposits RCON2710 or RCONF051; Loans RCFD2170; Securities RCFD0390;
Cash RCFD0010; Reserves RCFD3260.
44
Figure 2: Time-Line between time t and t + 1
✛
t
b, l, q
t+1 ✲
✲
z, d
✲✂
✂
✂
Default
Criterion ✂
✯
✂ ✟
✟
✂❍❍
❥
✲
✲
Continue
✯
✟
Default ✟
❍
❍
❥
❍
b′ , l′ , q ′
✻✻
Fire-sale
✯ Bailout
✟✟
Bankruptcy❍
❍
❥Reorganization
45
Debt
0.35
0.30
0.20
0.50
0.45
0.50
↑
0.55
^
η
Bailout Belief η
0.60
0.024
Dividends
0.030
0.028
0.026
0.004
0.002
0.000
Equity Issuance
0.006
0.45
0.25
0.10
0.15
Loan
0.25
0.40
0.30
Figure 3: Sensitivity to η
↑
0.55
^
η
Bailout Belief η
0.60
Let η̂ denote the parameter estimate for the belief in the government bailout η using the full sample
(Table II). First, I simulate the model 20 times while varying the bailout belief parameter η from
η̂ − 10% to η̂ + 10%, with the rest of the parameters are as in Table II. The top panel shows the
policy functions as a function of the belief in the government bailout. The blue solid line and the
red dotted line are the loan investment and the financing decisions, respectively. In the bottom
panel, I plot the equity issuance (blue solid line) and the dividends payout (red dotted line) in
response to the 20 different belief parameters η ∈ [η̂ − 10%, η̂ + 10%].
46
0.7
1.0
(B) Risky Debt
1.2
1.4
(A) Riksy Loans
0.9
1.1
1.6
1.3
Figure 4: Dynamics As Close to Default
(D) Distribution
0.04
0.08
0.12
0.00
Close to Default
Far from Default
x (ex post networth)
(F) Equity Issuance
0.010
0.020
0.030
Close to Default
Far from Default
x (ex post networth)
The Original Belief
0.000
(E) Dividends
0.10
0.20
0.00
Close to Default
Far from Default
x (ex post networth)
Close to Default
Far from Default
x (ex post networth)
0.30
0.010
(C) Debt Price
0.012
0.014
Close to Default
Far from Default
x (ex post networth)
Close to Default
Far from Default
x (ex post networth)
The Original Belief + 1%
First, I simulate the model using the parameter values that are reported in Table II. However, I
use two different values for the belief in the government’s bailout probability: η1 = η̂ (as in Table
II) and η2 = η̂ + 1%. Let x denote the ex post net worth; that is, the current net worth plus the
existing loans of a bank, wξ + (1 − δ)zl. As we move to the left side of each graph, x is smaller,
which means the bank is closer to the default. The plots are smoothed by the spline method. Also,
note that panel (A) and (B) are scaled by the average loans and debts, respectively, in order to
understand the relative dynamics instead of the absolute levels. The blue solid line is when the
belief in the government bailout is the estimated value using the SMM, denoted by η̂, and the red
dotted line is when the belief parameter is increased by 1%, η̂ + 1%.
47
100
0
50
Frequency
150
200
Figure 5: The Histogram of TARP
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Capital/Total Assets
In 2008, the U.S government purchased assets and equity from financial institutions to stabilize the
financial markets in the recent financial crisis, which is known as TARP. There were 736 financial
institutions that received the government help and only 585 were identified for this graph. I plot
the histogram of the government rescue funds scaled by the total assets of the financial institutions.
48
Figure 6: Time Series Patterns
0.8
Ratios
0.6
0.4
0.2
07
20
06
20
05
20
04
20
03
20
02
20
01
20
00
20
99
19
98
19
97
19
96
19
95
19
19
94
0.0
(A) Full Sample
0.8
Ratios
0.6
0.4
0.2
07
20
06
20
05
20
04
20
03
20
02
20
01
20
00
20
99
19
98
19
97
19
96
19
95
19
19
94
0.0
(B) Small Banks
0.8
Ratios
0.6
0.4
0.2
07
20
06
20
05
20
04
20
03
20
02
20
01
20
00
20
99
19
98
19
97
19
96
19
95
19
19
94
0.0
(C) Large Banks
Total
Liabilities
Insured
Deposits
Loans
Profits
The sample includes U.S. commercial banks from the Bank Regulatory Database. The sample
period is from 1994 to 2007. Each variable is scaled by the end of year total assets. The size is
defined by total assets. Large (small) banks are defined as the ones whose total assets are in the
lower (higher) third of the distribution of each year.
49
Figure 7: Responses to Unexpected Shocks
(A) Shock to Deposits
(B) Shock to Loans
1.00
1.8
0.99
0.98
Debt
Debt
1.6
1.4
0.97
0.96
1.2
0.95
0.94
Time
50
45
40
35
30
25
20
15
5
10
0
50
45
40
35
30
25
20
15
5
10
0
1.0
Time
1.20
1.25
1.15
1.20
1.05
Loan
Loan
1.10
1.00
1.15
1.10
0.95
1.05
0.90
1.00
35
40
45
50
40
45
50
30
35
Time
25
20
15
10
5
0
50
45
40
35
30
25
20
15
10
5
0
0.85
Time
1.002
1.15
1.000
1.10
0.998
Price
Price
0.996
1.05
0.994
0.992
1.00
0.990
0.95
Time
30
25
20
15
10
5
0
50
45
40
35
30
25
20
15
10
5
0
0.988
Time
The Original Belief
The Original Belief + 1%
The left-side of the figure shows the impulse responses to an unexpected huge drop in deposits and
the right-side of the figure shows the impulse responses to an unexpected huge drop in loan survival
rate.
50