Expectations and Credit Slumps*
Antonio Falato†

Jasmine Xiao‡

March 31, 2022

Abstract
Since the 2008-09 financial crisis, U.S. bank lending has been slow to recover, despite theperiod of very low interest rates. We show that banks do not process information efficiently, and this is a quantitatively important explanation for credit slumps after 2008.
Using a new dataset of bank expectations, we find that banks over-extrapolate the past,and their lending decisions are sensitive to beliefs. The behavioral bias is stronger forlarge banks, whose loan portfolios are also more sensitive to beliefs. To quantify the
economic significance of imperfect expectations, we build a dynamic model with heterogeneous banks that are over-extrapolative and face a small risk of economic disaster. We
show that a realistic degree of over-extrapolation estimated from the data generates thepace of credit and real recovery observed after the crisis. Banks’ distorted beliefs hamper
the effectiveness of policy tools, such as quantitative easing (QE) programs.
JEL codes: E7, G01, G21
Keywords: Financial Crises, Financial Intermediaries, Belief Distortions* We thank Jacob Faber for excellent research assistance. We thank George-Marios Angeletos, Nicola
Gennaioli, and Francois Gourio for helpful suggestions. The views expressed in this paper are those of theauthors and do not necessarily reflect the views of the Federal Reserve Board of Governors or the Federal
Reserve System.
† Federal Reserve Board, Division of Financial Stability. Email: antonio.falato@frb.gov.
‡ University of Notre Dame, Department of Economics. Email: jasmine.xiao@nd.edu.

11Introduction

The Great Recession associated with the 2008-09 financial crisis was one of the worst economic downturns in U.S. history. Bank lending declined significantly during the crisis,
and the Federal Reserve responded quickly by cutting interest rates and aggressively expanding the monetary base. One puzzling aspect of the recovery is why bank lending
failed to recover, even after most measures of economic activity have improved.1
We propose one potential explanation for the puzzle: banks over-extrapolate the past,and their lending decisions are sensitive to beliefs. As banks have been over-pessimistic
about the economic outlook and their loan performance since 2008, the growth in lending
remained subdued long after the crisis.
We offer the explanation in two steps. First and foremost, we construct a new data
set to study bank expectations, using the recently available bank-level responses on expected changes in loan performance over the next year, as reported on the Federal Reserve Board’s Senior Loan Officer Opinion Survey on Bank Lending Practices (SLOOS).
Since 2004, the survey began to include a list of special questions that inquire, among others, about banks’ expectations for changes in lending standards, loan demand, and loan
performance as measured by delinquencies and charge-offs, assuming that economic activity would evolve in line with consensus forecasts. We merge this sample with banklevel data on actual loan performance and loan growth, and the resulting data set allows
us to analyze the properties of bank expectations and their connection with actual lending
decisions. We find that banks’ forecast errors are positively and significantly autocorrelated, and that banks are more likely to reduce lending if they had been over-pessimistic
in past years. Second, to quantify the economic significance of imperfect expectations, webuild a dynamic model with heterogeneous banks that are over-extrapolative and face asmall risk of economic disaster. We show that a realistic degree of over-extrapolation
estimated from the data generates the pace of credit and real recovery observed after
the crisis. Banks’ distorted beliefs also hamper the effectiveness of policy tools, such as
quantitative easing (QE) programs.
Our analysis starts with the reduced-form evidence of systematic biases in banks’
forecasts. In a similar approach to recent contributions in the macroeconomics and finance literature using survey data, we show that forecast errors from banks are highly
persistent, with both the first- and second-order autocorrelation coefficients positively
1 Figure A.1 shows the total loans in the U.S. banking system from 2000 to 2020. Intermediated credit

grows steadily through the early 2000s, peaks in mid-2008 and declines through 2009. Growth resumes
around 2012 but at a slower pace than in the early 2000s. The dotted line shows the trend from 2000 to 2008
projected out through the rest of the period.

2and significantly significant.2 This result rejects the rational expectations hypothesis, under which banks’ forecast errors should not be predictable using variables in their information set, which includes past forecasts and past loans. Moreover, we show that forecast
errors have significant predictive power for future loan growth, after controlling for loan
demand, regulatory requirements, and other factors that can potentially influence banks’lending decisions. If a bank had been too pessimistic about its loan performance in thepast, its current lending is also likely to be lower than the pre-crisis average. Notably,
the persistence of forecast errors is stronger for large banks than small banks, and ceterisparibus, large banks’ loan portfolios are more sensitive to their forecast errors in the past.
Taking together, large banks appear to be slower at incorporating new information into
their lending decisions and their loan portfolio is plausibly more susceptible to fluctuations in beliefs.
We then develop a framework to quantify the impact of imperfect expectations on theslow recovery in bank credit after 2008. There is a continuum of heterogeneous banks
and an infinitely-lived representative investor, who owns all banks. A bank is defined as
a company that finances risky loan portfolios by equity and deposits. All agents share
a common exposure to an extreme economic adverse event (such as the financial crisis),that occurs with a time-varying probability (Barro, 2006; Gourio, 2012). Banks have limited liability, and face a regulatory capital requirement. Defaults occur endogenously, if abank’s continuation value becomes too low. Importantly, all agents have full information,
but they are not fully rational. Instead, they over-extrapolate past information when they
make forecasts, which is consistent with the micro evidence.
Our modeling choice for over-extrapolation is disciplined by the empirical evidence:in matching the dynamics of forecast errors, we show that agents perceive the fundamentals to evolve according to AR(2), while the actual processes follow AR(1). If beliefs
follow AR(1), only the current state the economy matters for expectation formation, even
if the perceived degree of persistence is higher than the actual persistence. However, thecurrent state does not necessarily indicate the trajectory of the economy, which is captured by AR(2). We show that by incorporating information on the trajectory in forming
expectations, agents tend to be overly pessimistic as the economy recovers from a crisis,and overly optimistic at the end of a boom.
For a realistic parametrization that is calibrated to match the mean and standard deviation of leverage, the mean profit-to-equity, bank default rate and the dynamics of bankforecast errors, we show that the model with over-extrapolation outperforms the rational
2 See, for example, the recent works by Coibion and Gorodnichenko (2015), Bordalo, Gennaioli, andShleifer (2017), Ma, Ropele, Sraer, and Thesmar (2020), which find the persistence in forecast errors using
firm survey data.

3

expectations benchmark in terms of matching the time-series properties of bank performance, expectations, and asset prices simultaneously.
We model the 2008-09 financial crisis as two consecutive positive shocks to the disaster probability. In terms of matching the post-crisis dynamics, the model with overextrapolation significantly outperforms its rational expectations counterpart: the OE model
generates a much slower recovery in the level of lending and hence a hump-shaped response in the loan growth rate. There is some persistence in banks’ asset growth in the
RE model since net worth takes time to build, but for realistic parameter values this is
not sufficient to match the slow credit growth after the crisis. The additional persistencein the OE model comes from two sources. First, lending is not only decreasing in thecurrent disaster probability, but also last period’s probability, so even when the disasterprobability starts to revert back to the long-run mean, bank lending continues to decline.
Second, the realized loan return increases more slowly in the OE model as disaster probability decreases. Since both lending and realized loan return recover more slowly, banks’current profit and hence net worth also recover more slowly. This in turn exacerbates
the slow recovery in lending, which is a function of net worth. In other words, the slow
recovery in the OE model comes both the over-extrapolative beliefs matched to the dataand its interactions with the evolution of net worth. Finally, we use the framework as alaboratory to study the impact of policy, such as quantitative easing. By subsidizing bankequity holders, these policies have some positive effects on lending and bank value, butthey are significantly hampered by the presence of forecasting bias.
Related Literature We build upon an influential and expanding literature on expectations and real outcomes, which has focused on professional forecasters, firms, households, and financial market investors (see, for example, Greenwood and Schleifer, 2014;
Coibion and Gorodnichenko, 2015; Bordalo, Gennaioli, and Shleifer, 2017 Bordalo, Gennaioli, Ma, and Shleifer, 2020; Angeletos, Huo, and Sastry, 2020; Rozsypal and Schlafmann, 2020; Kohlhas and Walther, 2021; Giglio, Maggiori, Stroebel, and Utkus, 2021,
Farmer, Nakamura, and Steinsson, 2021). Until recently, there has been little data thatmeasures banks’ expectations. Ma, Paligorova, and Peydró (2021) use a new dataset onbanks’ economic projections about all metropolitan statistical areas in the U.S. to study
the impact of lenders’ expectations on credit supply. We offer a complementary angle on
credit slumps after the 2007-09 crisis using a different dataset and quantify the economic
impact in a structural model.
Our paper also speaks to a growing theoretical literature on the role of expectations in
explaining credit cycles. For instance, Bordalo, Gennaioli, Shleifer, and Terry (2021) quantify the business cycle implications of diagnostic expectations in a heterogeneous-firm
4model; Krishnamurthy and Li (2020) and Maxted (2022) consider the impact of imperfect
expectations in models with financial intermediation. We complement this theoretical
literature in two ways. First, we build a heterogeneous-bank model and examine how
imperfect expectations propagate to the economy via changes in bank balance sheets.
Second, while models of diagnostic expectations focus on explaining the frothy pre-crisis
behavior and sudden reversals, our model focuses primarily on the recovery from a crisis.
Finally, our paper is also related to recent theories of banking, in particular thosewhich are quantitative in nature (see, for example, Egan, Hortaçsu, and Matvos, 2017;
Gourio, Kashyap, and Sim, 2018; Mankart, Michaelides, and Pagratis, 2018; Gomes, Grotteria, and Wachter, 2019; Corbae and D’Erasmo, 2021; Begenau and Landvoigt, 2021;
Elenev, Landvoigt, and Van Nieuwerburgh, 2021).3 Among these, our modeling choiceof the financial crisis as a rare disaster with time-varying probability follows Gourio,
Kashyap, and Sim (2018), and Gomes, Grotteria, and Wachter (2019). The key differencein our paper is that we introduce belief distortions, and we characterize the post-crisis
dynamics generated by the interaction of behavioral and financial frictions.
The rest of the paper is organized as follows. Section 2 presents reduced-form evidence of systematic biases in banks’ forecasts and their predictability for future lending decisions. Section 3 builds a dynamic model with heterogeneous banks and overexpectations. Section 4 provides an explanation for the calibration. Section 5 compares
the model fit under rational expectations versus over-extrapolation, and studies the impact of the 2008 financial crisis. Section 6 conducts policy analysis. Section 7 concludes.

2

Stylized Facts

2.1

Data Sources

Bank Expectations The key source of data used in the analysis comes from the Federal
Reserve’s Senior Loan Officer Opinion Survey of Bank Lending Practices. Since the early
1990s, the survey has queried banks about changes in their lending standards for the
major categories of loans to households and businesses and about changes in demand
for most of those types of loans. The survey is usually conducted four times per yearby the Federal Reserve Board, and the current reporting panel consists of up to 80 large
3 The financial crisis of 2008 renewed interest in banking theories, particularly the role of intermedi-aries in financial crises (see, for example, Farhi and Tirole, 2012; Adrian and Shin, 2013; He and Krishnamurthy, 2013; Brunnermeier and Sannikov, 2014; Sarin and Summers, 2016; Dang, Gorton, Hölmstrom,and Ordoñez, 2017; Rampini and Viswanathan, 2019.)

5

domestically chartered commercial banks and up to 24 large U.S. branches and agencies
of foreign banks. In 2004, the survey began to include a list of special questions thatinquire, among others, about banks’ expectations for changes in lending standards, loan
demand, and loan performance as measured by delinquencies and charge-offs, assumingthat economic activity would evolve in line with consensus forecasts. Our sample period
is from 2005 and 2020.
Specifically, these annual special questions ask about banks’ expectations for the behavior of loan delinquencies and charge-offs on selected categories of C&I, commercial
real estate, residential real estate, and consumer loans in the coming year. These questions follow the general pattern ofAssuming that economic activity progresses in line with consensus forecasts, what
is your outlook for delinquencies and charge-offs on your bank’s type X loans in thefollowing categories in the coming year?
Banks answer each question using a qualitative scale ranging from 1 to 5. The possible answers are: 1 =improve substantially; 2 =improve somewhat; 3 =remain aroundcurrent levels; 4 =deteriorate somewhat; 5 =deteriorate substantially. In a similar fashion to Bassett, Chosak, Driscoll, and Zakrajšek (2014), we use these responses to createkkcategorical variables, Et [ Ii,t+1 ], defined as Et [ Ii,t+1 ] =


1




0





if bank i in year t expects an improvement in type-k loan performance in t + 1
if bank i in year t expects no change in type-k loan performance in t + 1−1 if bank i in year t expects a worsening in type-k loan performance in t + 1
In our baseline analysis, we construct an index of expectations of future loan perfor-

mance, aggregated across loan types, for each bank i in year t. To this end, we use datafrom the Call Reports about the amount of outstanding loans each respondent bank has
in each loan category and we compute the following weighted average for each bank:k
Et [ Ii,t+1 ] = ∑ ωitk × Et [ Ii,t
+1 ] ,kwhere ωitk is the fraction of bank i’s core loan portfolio that is accounted for by loans in
category k, as reported on bank i’s Call Report at the end of year t. The higher the index,
the more pessimistic a bank is about loan performance in the following year.
Loan Performance

To maintain comparability with the measure of expectations, wemeasure the actual loan performance for each bank using the reported charge-off and6

delinquency rates from the Call Reports for our sample period. We compute, for eachbank, the annual change in charge-offs and delinquency rates on a particular type ofloans, and define a categorical variable, Iitk , such that:1
if bank i experiences an improvement in type-k loan performance in year t
Iitk = 0
if bank i experiences no change in type-k loan performance in year t−1 if bank i experiences a worsening in type-k loan performance in year t
where an “improvement” is defined as a decline in the sum of charge-offs and delinquency rates compared to the previous year, and a “worsening” is an increase in these
rates. Then we compute the weighted average across loan types:
Iit = ∑ ωitk × Iitk ,kwhere ωitk is the same weight that is used to construct the expectation measure, Et [ Ii,t+1 ].
Finally, we use the two categorical variables to compute each bank’s forecast errors
in each year, RitFE . Specifically, the forecast error is defined as the difference between the
expected and the actual change in loan performance:
RitFE = Et [ Ii,t+1 ] − Ii,t+1 .
For instance, if RitFE < 0, this indicates that the respondent bank is too pessimistic in yeart about its loan performance in the coming year.
Bank Lending and Other Bank-Specific Information We use the bank-level data on
outstanding loan balances in each year from the Call Reports to construct our measureof dependent variable, ∆Loanit , the logarithm change of total bank loans relative to thepre-crisis level. In addition, we also obtain other bank balance sheet data from the Call
Reports, and use them to control bank characteristic variables, including size (the logarithm of total assets), the share of loans in assets, the ratio of common equity Tier 1 (CET1)
capital to risk-weighted assets, and the cash ratio.

2.2

Persistence of Forecast Errors

Under rational expectations, banks’ forecast errors should not be predictable using variables in their information set, which includes past forecasts and past loans. We document
instead that banks’ forecast errors are persistent, and positively predictable by lagged
forecast errors.
7

We show this finding by estimating the following model:
RitFE = α +K∑

β k RitFE−k +

k =1

K

∑ gk Xit−k + τt + uit ,k = 1, 2, 3

(1)

k =1

where α is a constant, τt denotes year fixed effects. The vector Xit includes bank variables
such as size, the share of loans in assets, the ratio of core deposits to assets, loan loss
reserves, and loan performance. Standard errors are clustered at the bank level.
Table 1 reports the estimates of coefficients β k , for k = 1, 2, 3. The estimated β k is highly
significant for the first- and second-year lags: on average, forecast errors exhibit a positive autocorrelation for two lags. Intuitively, a positive news at date t about date t loan
performance – for example, a shock that increases house prices and hence the collateralvalue of a loan – implies a negative forecast error at date t, as the bank was too pessimistic
at date t − 1 about date t loan performance (i.e. expected delinquencies are higher than
the actual rate). If the shock is persistent, and the bank is slow to incorporate the information from the latest shock into their forecast for date t + 1, we end up with negative
forecast errors at t + 1 and beyond. Thus, the positive autocorrelation of forecast errors
are consistent with a hypothesis that banks, on average, under-react to loan performance
related news.
A possible concern about the relevance of this finding is that the persistence in forecast error might mostly be a phenomenon among small banks. To address this concern,
we split the sample of banks into two groups, small and large, based on the bottom andtop quartiles of bank total loans respectively. For each of the two groups, we re-run regression (1) separately. We report results in Table A.1. Across both groups, the estimatesfor β 1 and β 2 are strongly significant, and the degree of persistence increases with banksize. Overall, the results in Tables 1 and A.1 reject the hypothesis that banks process
information efficiently.

2.3

Expectations and Lending Dynamics

We now show that banks’ expectations are linked to their lending decisions. To this end,
we regress the logarithmic change of total loans (relative to the pre-crisis level) in year ton their forecast errors in the past three years:
K

K∆Loansit = α + ∑ β k RitFE−k + ∑ gk Xit−k + τt + uit , k = 1, 2, 3
k =1

(2)

k =1

where α is a constant, τt denotes year fixed effects, and Xit is a vector of bank controls
including past loan performance. Standard errors are clustered at the bank level. Ta8

ble 2 reports estimates of coefficients on lagged forecast errors (columns (1)–(3)) and on
past loan performance (columns (4)–(6)), respectively. The sample period is the recovery
after the Great Recession (2010-2020). We find that forecast errors in year t − 2 are positively and significantly related to banks’ lending decisions in year t: if banks were too
pessimistic in past (RitFE−2 < 0), their lending in year t is likely to stay below the pre-crisis
level (column (2)). We also find that the proxy for loan performance in years t − 2 andt − 3, the fraction of loan defaults, is negatively and significantly related to loan growthin year t: if loan default rates were high in the past, bank lending is still likely to stay
below the pre-crisis level today (columns (5)-(6)).
Alternative Hypotheses

Our results so far indicate that there is a significant degree of“stickiness” in the post-crisis lending dynamics, and that bank expectations appear toplay an important role in shaping the slow recovery in loan growth. Nonetheless, one
possible concern with the findings in Table 2 is that there could be alternative hypotheses
for the slow recovery in intermediated credit. For instance, one other potential explanation is the low level of economic growth and employment following the crisis. These
factors could indicate that low rates of lending were due to a lack of loan demand. Another potential explanation is the tightened bank regulation in response to the financial
crisis. To address these concerns, we repeat the same analysis as in Table 2 and add
controls for loan demand, loan performance in the crisis, the ratio of CET1 capital to riskweighted assets, and bank liquidity (measured by the cash ratio) in turn. We present
the additional results in Table A.2. The key takeaway is that the estimate of β 2 – thecoefficient on forecast errors in t − 2 – is robustly and positively significant. The magnitude of the coefficient also remains stable across specifications. In other words, the bankexpectation channel is still present, after controlling for alternative explanations for thepersistent decline in bank lending in the post-crisis period.

2.4

Bank and Loan Heterogeneity

Next, we examine if the relation between bank expectations and their lending dynamics
varies by bank size or loan types. To do so, we repeat the same analysis as in Table 2
and split the full sample according to bank size and loan types, in turn. As in Table
A.1, we define “small” and “large” banks based on the bottom and top quartiles of banktotal loans, respectively. Moreover, we also split the sample into four types of loans:commercial and industrial (C&I) loans, residential real estate (RRE) loans, commercial
real estate (CRE) loans, and consumer Loans. For each loan type, we regress the current
loan growth on the respective forecast error as well as on the average forecast error for9other loan types, in turn.
Table A.1 reports the estimates of coefficient β k , for k = 1, 2, 3. We find that the estimated β 2 is positive and statistically significant for both small and large banks, with theestimate significantly higher for large banks. Moreover, bank lending in all but consumer
loans is positively related to their forecast errors on that particular type of loans in thepast, and unrelated to forecast errors for other loan types. In particular, the estimates
are four times higher for real estate loans (both residential and commercial) than for C&I
loans. This evidence suggests that on average, past beliefs play a more prominent role in
influencing the current lending decisions of larger banks, and for real estate loans.

2.5

Key Takeaways

In this section, we document a set of new facts about bank expectations and lending
dynamics in the post-crisis period (2010-2020) using a unique bank-level dataset:
1. Banks’ forecast errors are positively and significantly autocorrelated.
2. Forecast errors have significant predictive power for future loan growth, even after
controlling for loan demand, regulatory requirements, and other factors that canpotentially influence banks’ lending decisions. If a bank had been too pessimistic
about its loan performance in the past, its current lending is also likely to be lower
than the pre-crisis average.
3. The persistence of forecast errors is stronger for large banks than small banks, andceteris paribus, large banks’ loan portfolios are more sensitive to their forecast errors in the past. Taking together, large banks appear to be slower at incorporating
new information into their lending decisions and their loan portfolio is plausibly
more susceptible to fluctuations in beliefs.
4. The growth of real estate loans is more sensitive to fluctuations in bank beliefs thanother loan types.
Figure 2 provides further demonstrative evidence from the micro data on the relevance of forecasting biases for bank lending in the post-crisis period. In particular, ourgoal here is to highlight the disconnect of loan growth from actual loan performance, andcontrast it with the link between loan growth and expected loan performance. In Panel(a), we plot the net fraction of banks that experience an actual worsening of loan performance: it is the number of banks that experience worsening (defined by an increasein expected the loan default rate) minus the number of banks that experience improving, divided by the total number of banks. Panel (b) shows the net fraction of banks
10that expect a worsening of loan performance. While actual loan performance appears
to start improving, on average, from 2010 (with more banks experiencing an improvement in loan performance than worsening), most banks remained over-pessimistic about
their loan portfolios well beyond 2010. This corroborates our evidence above that banks
adjust their expectations slowly. Furthermore, not only is expectation slow to adjust, so
is the rate of low growth. As shown in Panel (c), the average rate of loan growth only
returned to the pre-crisis level approximately seven years after the crisis. In other words,
the speed of recovery in bank lending appears to be much more correlated with the speedof recovery in expectations than in actual loan performance.
Taking together, the empirical evidence suggests that inefficient information processing by banks plays a potentially important role in the post-crisis decline in bank lending.
In particular, the evidence suggests that banks overextrapolate the past in forming their
expectations. Our next goal is to understand how overextrapolation affects banks’ lending decisions, and quantify the economic significance of such behavioral bias. We do sothrough the lens of a structural model, which allows us to “shut down” the behavioral
biases that we observe in the data and construct appropriate policy counterfactuals.

3

Model

3.1

Environment

Time and agents

Time is discrete. There is a continuum of heterogeneous banks andan infinitely-lived representative investor, who owns all banks. A bank is defined as acompany that finances risky projects by equity and deposits. Both entities share a common exposure to an extreme economic adverse event (“crisis”), that occurs with a timevarying probability. In order to focus solely and squarely on the impact of forecasting
biases, we do not fully integrate these sectors into a general equilibrium setting.
Aggregate shocks

There are two aggregate shocks in the economy, denoted by ε ct andxt . ε ct is a standard normal random variable that is i.i.d. over time. xt+1 is a Bernoulli
random variable which takes on the value 1 with probability pt and 0 otherwise. If xt+1 =1, the economy experiences a “crisis” in period t + 1, and the investor’s endowment falls
by a large fraction ξ (see, for example, Rietz, 1988; Barro, 2006; Gourio, 2012, 2013; Gourio,
Kashyap, and Sim, 2018). The probability pt follows a Markov process:log pt+1 = (1 − ρ p ) log p̃ + ρ p log pt + ε p,t+1 ,

11

(3)

where ε p,t+1 ∼ iidN (0, σp ). ε pt , ε ct and xt are independent. Following Gomes et al. (2019),
we assume the following process for the endowment of the investor:
Ct+1 = Ct eµc +σc ε c,t+1 +ξxt+1 ,(4)

where µc represents the mean growth in consumption in normal times.
Preferences

The representative investor who consumes endowment Ct has Epstein

and Zin (1989) preferences with time preference β ∈ (0, 1), relative risk aversion γ, andelasticity of intertemporal substitution ψ. Hence the stochastic discount factor of theinvestor is given by:

Mt,t+1 = βθ

Ct+1
Ct

!−γ

S t +1 + 1
St! −1+ θ(5)1− γwhere θ = 1−
1/ψ and St denotes the ratio of wealth to consumption as determined by:

"
θ
EP
t β

Ct+1
Ct

!1− γ



S t +1 + 1θ#= Stθ .

(6)

where P denotes the subjective belief of the investor, which we specify below. To avoiding clouding on our key mechanism, we do not fully integrate the agents in a generalequilibrium setting.

3.2

Loan Portfolios and Uncertainty

Each bank has a portfolio of risky private sector loans (j = 1, 2, ..., n) within a local economy. There is uncertainty about the collateral value for loan j of bank i, Wijt , which ismodeled as a random variable:
Wijt = eσc ε ct +ξxt +ωit +σj ε jt .

(7)

The collateral value is subject to three sources of uncertainty: (ε ct , xt ) are aggregate shocks;
ωit is a bank-specific (or local-economy specific) shock; ε jt is a borrower-specific shock.
ε ct and ε jt are standard normal random variables that are i.i.d. over time. ωit follows theMarkov process:ωi,t+1 = ρω ωit + ε ωi,t+1 ,

12

(8)

where ε ωi,t+1 ∼ iidN (0, σω ). ε ct , ε jt , ε ωit and ε pt are independent of each other and alsoof xt . Each of the four shocks can change the value of an individual loan and its default
probability.
We assume that borrower j defaults at time t if Wijt < κ, and κ is common acrossall borrowers. When a borrower defaults, the bank can recover a fraction 1 − L of thecollateral value, where L is the loss given default. The ex-post return on the portfolio ofloans equalsLri,t+1 ( sit , ε c,t+1 , xt+1 , ωi,t+1 ) =Lπi,t+1 ( ε c,t+1 , xt+1 , ωi,t+1 )PitL (sit )

− 1,

(9)

L
L
where πi,t+1 ( ε c,t+1 , xt+1 , ωi,t+1 ) is the payoff of the loan portfolio for bank i, and Pit ( sit )is the price of the loan. sit denotes the exogenous state variables for bank i, which we
specify below.
Assuming that each bank i holds an equal-weighted portfolio of an arbitrarily large
number of loans, then as shown in Gomes et al. (2019), the payoff of the loan portfolio
can be expressed as:
Lπi,t+1 ( ε c,t+1 , xt+1 , ωi,t+1 )hi= κProb Wij,t+1 ≥ κ ε c,t+1 , xt+1 , ωi,t+1 + (1 − L )E Wij,t+1 1Wij,t+1 <κ ε c,t+1 , xt+1 , ωi,t+1
|
{z
} |
{z
}
Repay
Default
 


1

κ
1
−
Φ
(
log
(
κ
)
−
σ
ε
−
ω
)

c c,t+1
i,t+1
σj



1

σ2
(z−σj )2jσj (log(κ )− σc ε c,t+1 − ωi,t+1 )1σc ε c,t+1 +ωi,t+1 + 2− 2 dz if x

√
+(
1
−
L
)
e
e

t +1 = 0

2π
−∞

=





κ 1 − Φ 1 (log(κ ) − σ ε

−
ξ
−
ω
)

c
c,t
+
1
i,t
+
1
σj



1
σ2

(z−σj )2j
σj (log(κ )− σc ε c,t+1 − ξ − ωi,t+1 )

1
σ
ε
+
ω
+
ξ
+
−
c
c,t
+
1
i,t
+
1
√ e
 +(1 − L )e22
dz if xt+1 = 1
2π

R

R

−∞

(10)
where Φ(·) denotes the standard normal cdf. The price of the loan portfolio equals
PitL (sit ) = EP
t

h

i

L
Mt,t+1 πi,t+1 ( ε c,t+1 , xt+1 , ωi,t+1 ) .

13

(11)

3.3

The Bank’s Balance Sheet

Each bank i enters period t with loan portfolio Li,t−1 , deposits Di,t−1 , and book value ofequity BEi,t−1 . Then all aggregate and idiosyncratic shocks to loan returns are realized
and the bank decides whether to continue or default. If the bank continues, its net profit
Πit depends on the rates of return on the bank’s assets ritL and liabilities (deposits) r D , any
non-interest income c L , and operating expenses c D , which captures overhead costs andthe FDIC surcharge to fund deposit insurance:Πit = ritL + c L Li,t−1 − (r D + c D ) Di,t−1 .
| {z }

(12)

≡r̃ D

The bank chooses dividends Divit and new loans Lit . Equity is accumulated retained
profits over time, i.e. after dividends and adjustment costs have been paid. Thus at theend of period t, each bank has a new equity level BEit , which is equal to the equity at the
beginning of the period t net of current dividends Divit and adjustment costs Φit , plus
current profit Πt :BEit = BEit−1 − Divit − Φit + Πit .

(13)

In choosing its investment in loans, each bank faces the balance sheet equation:
Lit = Dit + BEit ,(14)

as well as a quadratic loan adjustment cost:Φit = η L Li,t−1

Lit − Li,t−1
Li,t−1!2.

(15)

Banks also face a regulatory capital requirement, consisting of a maximum ratio of total
assets to equity captured by λ:
Lit
≤ λ.
BEit

(16)

Following Merton (1978), we assume Dit = e g Di,t−1 ; in other words, deposits grow at aconstant rate g.4
4 We calibrate g to equal expected consumption growth:



σc2
σc2
g = log (1 − Ept )eµc + 2 + Ept eµc + 2 +ξ .

14

3.4

Belief Distortions

Agents (both bank managers and investors) have distorted expectations about futureloan performances. First, whereas the true processes of pt and ωit follow (3) and (8),agents perceive the processes to be:ωi,t+1 = ρ̂1ω ωit + ρ̂2ω ωi,t−1 + ε ωi,t+1 ,log pt+1 = (1 − ρ̂1p − ρ̂2p ) log p̃ + ρ̂1p log pt + ρ̂2p log pt−1 + ε p,t+1 .

(17)
(18)

This formulation is motivated by the empirical observation in Table 1: forecast errors are
positively correlated up to a two-year lag. Secondly, while the true probability of xt+1 = 1
is pt , the subjective probability of the occurrence of a disaster in t + 1 depends on both pt
and pt−1 :χ 1− χ
ProbP xt+1 = 1 = pt pt−1 .

(19)

As explained below (see Section 4), we calibrate the belief parameters {ρ̂1p , ρ̂2p , ρ̂1ω , ρ̂2ω , χ}to match the dynamics of bank forecast errors in Table 1.

3.5

The Bank’s Problem

Banks are run by managers with limited liability that maximize the present discounted
value of investor utility from dividends Divit , and discount the future with the investor’s
SDF Mt,t+1 . Dividends represent equity payouts if Divit > 0 and equity issuance if
Divit < 0. Equity issuance is costly due to asymmetric information and incentive issues.
Following Bolton et al. (2021), we adopt a reduced-form approach by choosing a proportional equity issuance cost:Λ( Divit ) = 1Divit <0 η E Divit(20)

where 1Divit <0 is an indicator variable that is equal to 1 if Divit < 0 and 0 otherwise.
We define a composite state variable Nit , that is the sum of the beginning-of-period
equity and the current net profit:L
L
D
Ni,t+1 = BEit + ri,t
+1 + c Lit − r̃ Dit .

(21)

Conditional on not defaulting at time t, bank managers solve the following continuation15

problem:"(
ViC ( Li,t−1 , Di,t−1 , Nit , sit ) = max

Divit ,Lit#)

Divit + Λ( Divit ) + EPt Mt,t+1 Vi ( Lit , Dit , Ni,t+1 , si,t+1 ) sit(22)

subject to the return on risky loans (9), the current profit (12), the evolution of equity (13),the balance sheet constraint (14), asset adjustment costs (15), the capital requirement constraint (16), equity issuance costs (20), and the definition of Ni,t+1 (21). EPt (·) captures the
expectation formation process, which follows (17)-(19) with extrapolative beliefs. Limited liability implies that bank managers may choose an outside option V D , which we
normalize to zero, and the expected value in (22) is defined as the upper envelope:hi
Vi ( Lit , Dit , Ni,t+1 , si,t+1 ) = max ViC ( Lit , Dit , Ni,t+1 , si,t+1 ), V D .

(23)Default happens if a bank’s continuation value V C falls below the threshold level V D .5
A bank’s states include last period’s loan portfolio Li,t−1 , deposits Di,t−1 , the current
net worth Nit , and exogenous states sit , which depend on the expectation formation process.6 With rational expectations, sit includes the persistent aggregate and idiosyncratic
shocks sit = [ pt , ωit ]; with overextrapolative expectations, sit must also include last period’s aggregate and idiosyncratic shocks, i.e. sit = [ pt , pt−1 , ωit , ωi,t−1 ].

4

Calibration

We solve and calibrate two variants of the model at an annual frequency:
1. Model with rational expectations (RE): Agents believe that the aggregate shock pt
and idiosyncratic shock wt follow (3) and (8), respectively, and that the probabilityof a disaster in t + 1 is pt .
2. Model with overextrapolative expectations (OE): Agents believe that pt and wt follow (18) and (17), and that the probability of a disaster in t + 1 is (19).
There are two groups of parameters: the first group is exogenously calibrated; the second
group is calibrated in a moment-matching exercise. We calibrate the model parameters
5 When a bank defaults, an identical bank is created with the same state variables, so we maintain astationary distribution of banks.
6 We simplify the computation of the problem by scaling the value of a bank by deposits. See Appendix

A.1 for details.

16,

to best match moments for each version of the model, thus giving each model the best
chance to represent the data. When we simulate the model with overextrapolation, allshocks and distributional dynamics are determined according to their true processes,
even though the asset prices and bank polices involve overextrapolative beliefs.
A list of the exogenously calibrated parameters are shown in Table 4. For most ofthem, we take their values from the literature, in order to facilitate comparison of models
with and without belief distortions. More specifically, the values for β, µc and σc follow
the standard values in the business cycle literature (e.g. Cooley and Prescott, 1995), whilethe values for ψ and γ follow the literature on asset pricing with rare events (e.g. Gourio,2012). We follow the estimates of an average probability of disaster on OECD countries
by Barro and Ursúa (2008) and set the average probability of an economic collapse to be
2 percent per year and an associated drop in consumption of ξ to be 30%. For parameters
governing the disaster probability process, we set ρ p to be 0.8 and σp to be 0.42 (Gourio, 2012). For parameters governing the bank-specific shock process, we follow Gomes
et al. (2019) and set ρω to be 0.9 and σω to be 0.02. We fix the loan-to-value ratio at loan
origination at 0.66, following Nagel and Purnanandam (2020). The loss given default on
loans, L, is set to match the observed average recovery rate on loans. The regulatory
capital requirement parameter λ is set to 12.5, corresponding to an 8% equity to asset
ratio in accordance with the Basel rules. Lastly, we follow Bolton et al. (2021) and set the
proportional equity issuance cost η E to 5%.
The rest of the parameters are calibrated in a moment-matching exercise. All targeted moments are computed using our sample from Section 2, for the period 2005 and2020. We calibrate the volatility of loan-specific shock σj and the asset adjustment cost parameter η L to match the unconditional mean and the cross-sectional dispersion in bankleverage. Bank equity is defined as total bank assets minus total bank liabilities. Bank
leverage is calculated as the sum of bank equity and liabilities to bank equity. The noninterest income parameter c L and the cost of liabilities (including both interest expenses
and operating expenses) r̃ D are chosen to target the mean profit-to-equity ratio and thebank default rate.
Lastly, we calibrate the parameters governing distorted beliefs {ρ̂1p , ρ̂2p , ρ̂1ω , ρ̂2ω , χ}to match the dynamics of bank forecast errors in the data. We remain agnostic about
whether it is more difficult to forecast aggregate or local conditions, and assume thatbanks have similar belief distortions when they are forecasting both the aggregate ( pt+1 )ρ̂and the local (ωi,t+1 ) conditions, i.e. ρ̂1p = ρ̂1ω and ρ̂2p = ρ̂2ω . Moreover, we set χ = ρ̂1p +1pρ̂1p
and calibrate ρ̂1p and ρ̂2p to target the dynamics of bank forecast errors shown in Table 1.
We measure bank forecast errors as the difference between the expected and the realized
loan performance over the next year or two, and loan performance is defined as the loan
17

default rate.
We obtain model-implied moments by simulating 1,000 banks for 3,000 periods, discarding the first 300 observations. All shocks and distributional dynamics are determinedaccording to the true process (i.e. the rational expectations representation), even though
the asset prices and bank polices may involve distorted expectations. Table 5.A presents
the targeted moments in each model.

5

Quantitative Results

5.1

Model Fit

Before using the model to study the impact of a credit crisis, we first examine the timeseries statistical properties of the key variables on both bank performance and expectations. As we seek to explain the stylized facts in Figure 2, our main variables of interest are loan growth, changes in expected loan performance, changes in realized loan
performance, and the growth of loan rates. For untargeted moments, we examine theirautocorrelations and correlations with the aggregate economic condition, both contemporaneously and with lags.7
Table 5.B reports the moments in each model. First, in the data, the annual loan growthis positively correlated with current and lagged GDP growth, and both the first and second order autocorrelations are positive. In the RE model, only the level of loans is positively correlated with lagged GDP growth and has a positive second order autocorrelation, but not the growth rate; in other words, the RE model does not generate enoughpersistence in the bank lending series that we observe in the data. As shown in the table, the OE model outperforms the RE model on the time-series statistics for loan growth
rates.
Next, we turn to the annual change in expected loan performance, where loan performance is measured by loan defaults, so we look at the properties of changes in expectedloan defaults. This is our main measure for bank expectations. These are highly countercyclical in the data: if the current GDP growth is low, banks’ expected loan default rates
increase at a faster rate. Moreover, if last year’s GDP growth was low, banks’ expectations
today also worsen at a faster rate. While both models match the contemporaneous correlations with aggregate conditions, only the OE model can match the negative correlation
with lagged GDP growth. In addition, the OE model also outperforms the RE model on
7 While the obvious candidate for measuring aggregate economic conditions is GDP growth, which isused to compute the data moments, we use one minus the disaster probability (i.e. the probability of nothaving a disaster in the next period) as a proxy for the aggregate condition.

18

the autocorrelations of bank expectations. Both AR(1) and AR(2) correlation coefficientsare positive for changes in expectations in the data; in other words, bank expectations
adjust slowly. We find positive correlations for the counterparts in the OE model, butthey are counterfactually negative in the OE model.
We also examine the time-series properties of realized loan performance, or realized
loan default rates. These have similar cyclical properties as the expected loan performance: changes in realized default rates are negatively correlated with current and oneyear lagged GDP growth, but the autocorrelations are negative. Notably, both models
generate the correct signs, and the magnitudes of coefficients are very similar in the twomodels. This is unsurprising since they only differ in terms of how expectations are computed, and loan default decisions depend on the true shock processes, which are drawn
according to the RE representation in both models.
To examine the model prediction on asset prices, we look at changes in loan rates.
The model counterparts are the changes in ex-post returns on loan portfolios ritL , which
are affected by how expectations are computed, as shown in (9). In the data, changes
in loan rates are positively correlated with GDP growth up to a two-year lag. In the RE
model, the positive correlation only holds for contemporaneous GDP growth and notfor either lag. The OE model outperforms the RE model on the cyclical properties, aschanges in loan rates are also positively correlated with last year’s aggregate conditions.
Furthermore, the OE model generates positive first-order autocorrelation and negative
second-order autocorrelation as shown in the data, whereas both autocorrelations arenegative in the RE model.
Lastly, we compare the model-implied regression results with our findings in Tables
1 and 2. The RE model predicts that bank forecast errors follow an AR(1), with a strong
reversal in expectations after one year: the coefficient estimate on the second lag is negative and strongly significant, contrary to the evidence. Moreover, forecast errors only
predict loan growth for one year in the RE model, but the predictability remains positively significant beyond that in the data and the OE model.
Main Takeaways The OE model outperforms the RE model on the time-series properties of key indicators of bank performance, expectations, and asset prices. Specifically,
the OE model is successful on two fronts that the RE model fails at: one is the autocorrelations of both bank performance and expectation variables, and the other is theircorrelations with 1- and 2-year lagged aggregate economic conditions. The data counterparts indicate that banks exhibit strong sluggishness in adjusting their expectations,and therefore, performance. We show that the OE model can simultaneously account for
both.
19

5.2

A Temporary Increase in Disaster Probability

The main experiment of the paper is an increase in disaster probability pt . We mimic
the financial crisis in 2008 and 2009 with two consecutive years of increase in disasterprobability. Figures 3 and 4 present the impulse response functions of bank variables
when the probability of disaster increases from the long-run average of 2.61 percent per
year to 4.03 percent and 8.11 percent, respectively. Then the probability of disaster meanreverts to its long-run average according to equation (3).
Impact on Bank Lending

Figure 3 shows that an increase in disaster risk leads to acredit crisis. To inspect the mechanism, we plot the policy functions in Figures A.7–A.9.
In both models, two related forces lead to a reduction in lending upon impact of theshock. First, lending is decreasing in the disaster probability, until it hits the capital requirement constraint (see Figure A.7). This is because as at a higher pt , investment inassets becomes less profitable, as shown in Panel B of Figure A.6. Secondly, lending isincreasing in the current net worth of a bank (nit ). Recall that the current net worth includes banks’ beginning-of-period equity and the current net profit. As a higher disasterprobability reduces realized loan returns, banks’ current profits take a hit and hence their
net worth decreases. Since external financing (both debt and equity) is costly for banks,
they decrease their loan portfolios in response to a reduction in net worth.
The key difference between the two models is that the OE model generates a muchslower recovery than the RE model. There is some persistence in banks’ asset growthin the RE model, but, for realistic parameter values, this is not sufficient to match theslow credit growth after the crisis. The persistence in the RE model comes from twosources: asset growth is costly and the net worth takes time to build. Nonetheless, it only
takes three years for the annual loan growth rate to return to the pre-crisis level, and theimpulse response of loan growth does not exhibit any hump-shape. By contrast, it takes
seven years for the loan growth rate to recover in the OE model, which resembles theslow recovery in the data, as the annual growth rate only returned to the pre-crisis level
after 2014.
There is additional persistence in the OE model because lending is not only decreasingin Pt , but also in Pt−1 , so even when the disaster probability starts to revert back to thelong-run mean in t = 3, banks’ lending decisions respond with a lag. Moreover, realized
loan return increases more slowly in the OE model than in the RE model, as disasterprobability returns to the steady state level (see Figure A.8). As both lending and realized
loan return recover more slowly, banks’ current profit and hence net worth also recover
more slowly. This in turn exacerbates the slow recovery: lending policies of banks with20

lower net worth increase more slowly as disaster probability decreases. As we illustratein Panel B of Figure A.9, the slope of the policy function is flatter for banks with low
net worth, at each level of Pt . In particular, when Pt−1 is high, the capital requirement
only becomes non-binding when Pt falls to the lowest possible level (see the blue line in
the bottom right corner of Figure A.9). Intuitively, since external financing is costly andbanks want to avoid default, banks are more cautious about lending risky loans when
their net worth is low and the disaster probability in the last period was high. As aresult, we get a hump-shaped impulse response function for the loan growth rate in the
OE model, which more closely resembles the data (Panel (c) of Figure 2).
It is worth noting that the OE model generates a milder decline in lending upon impact of the shock to disaster probability. This is because lending at the (stochastic) steadystate is lower in the OE economy than in the RE economy, which means that the capitalrequirement constraint becomes binding more quickly as banks reduce their loan portfolios. Hence, on average, aggregate lending falls by less in percentage deviation terms in
the OE model.
Impact on Bank Expectations

Figure 4 shows the evolution of beliefs during and after

a credit crisis. For each model, we plot the net fraction of banks that expect a worseningof loan performance and forecast errors, respectively, following two consecutive years ofshocks to disaster probability in 2008 and 2009. To facilitate comparison, we constructthese variables as closely as possible to the data definition. In Panel (a), the net fractionof banks that expect a worsening of loan performance is the fraction of banks that expect higher loan default rates in the coming year minus the fraction that expect lower
default rates. In Panel (b), forecast errors are the difference between the expected andthe actual change in loan performance, so positive forecast errors indicate that banks areover-optimistic (or the expected loan default rate is lower than the actual default rate)
while negative forecast errors indicate that banks are over-pessimistic. We compute forecast errors for each bank over time, and average them across banks for each year.
Following the positive shocks to disaster probability in 2008-9, more banks expect
loan performance to worsen than to improve in both models, i.e. the net fraction in Panel(a) becomes positive. However, in the post-crisis period, bank expectations follow thedata more closely in the OE model. While it starts declining in 2010 in the RE model as
the disaster probability begins to revert to the long-run mean, it rises further in the OE
model and matches the data almost exactly in 2010-11. In the data, it is not until 2014
– seven years after 2008 – for bank expectations to be “neutral”, i.e. to have the same
number of banks to expect worsening and improvement. The OE model matches thisfact, whereas in the RE model, more banks start to expect improvement than worsening
21

from 2012.
Turning to forecast errors, the main difference between the two models occurs in the
post-crisis period (see Panel (b) of Figure 4). In the RE model, banks’ forecast errors
remain above zero between 2007 and 2011. By contrast, in the data, banks turn pessimistic(with forecast errors below zero) from 2009 and remain as such until 2018. In the RE
model, banks’ expectations of loan defaults fall more quickly than the actual default rates,as the disaster probability starts to revert back after 2009. In the OE model, expectations
are more “sticky”, i.e. banks continue to over-predict the loan default rate after 2009. The
OE model closely matches the data on forecast errors from 2008 to 2018.
Bottom LineWe model the 2008-09 financial crisis as two consecutive positive shocks

to the disaster probability. In terms of matching the post-crisis dynamics, the OE outperforms the RE model on two fronts. First, it generates a much slower recovery in the levelof lending and hence a hump-shaped response in the loan growth rate. Secondly, thebelief process backed out from the OE model follows the data much more closely thanthe RE model: banks remain persistently pessimistic long after 2009.

5.3

Beliefs as AR(2) Processes

In the model, agents perceive the processes of pt and ωit to be AR(2) processes (equations
17 and 18), while the true processes follow AR(1). This modeling choice is motivatedby the empirical evidence in Table 1, as we calibrate the belief parameters to match the
dynamics of forecast errors in the data.
As a result, agents over-extrapolate, and this generates significantly more persistencein the model compared to the rational expectations counterpart due to a combination of
two forces. First, lending is not only decreasing in the current disaster probability, butalso last period’s probability, so even when the disaster probability starts to revert back to
the long-run mean, bank lending continues to decline. Second, with over-extrapolative
beliefs, the realized loan return increases more slowly as disaster probability decreases.
Since both lending and realized loan return recover more slowly, banks’ current profit
and hence net worth also recover more slowly. This in turn exacerbates the slow recovery
in lending, which is a function of net worth.
Now we ask the question: What if we model agents’ over-extrapolative beliefs asan AR(1) process, where the perceived degree of persistence is higher than the actual
persistence? To this end, we perform the following counterfactual exercise. While the22true processes of pt and ωit continue to be (3) and (8), we let agents’ beliefs follow:
ωi,t+1 = ρ̃ω ωit + ε ωi,t+1 ,log pt+1 = (1 − ρ̃ p ) log p̃ + ρ̃ p log pt + ε p,t+1 .
With over-extrapolation, ρ̃ω > ρω and ρ̃ p > ρ p . To maintain comparability with the AR(2)specification in the baseline model, we set ρ̃ω = ρ̂1ω ωit + ρ̂2ω ωit and ρ̃ p = ρ̂1p + ρ̂2p in the
counterfactual. In Figure A.2, we compare the impulse response functions to an increasein disaster probability under RE and two OE models: one with AR(2) (baseline) and one
with AR(1) (counterfactual).
To understand the difference between the two OE specifications, let Pt ≡ log pt andrewrite the belief process (18) as follows:
Pt+1 = (1 − ρ̂1p − ρ̂2p ) P̃ + ρ̂1p Pt + ρ̂2p Pt−1 + ε p,t+1= (1 − ρ̂1p − ρ̂2p ) P̃ + (ρ̂1p + ρ̂2p ) Pt − ρ̂2p ∆Pt + ε p,t+1= (1 − ρ̃ p ) P̃ + ρ̃ p Pt −ρ̂2p ∆Pt +ε p,t+1| {z }
momentumThe first term is the same as the AR(1) process above. The second term represents where
the economy has been – in other words, whether the economy is in a recovery phase(with ∆Pt < 0) or heading into a recession (with ∆Pt > 0) – and it is the key difference
between an AR(2) belief process and a corresponding AR(1) with ρ̃ p = ρ̂1p + ρ̂2p .
Figure 1 illustrates the role of momentum in agents’ expectations of future disasterprobability when ρ̂2p > 0, which is the case in our calibration disciplined by the forecast
error dynamics. The key observation is that if beliefs follow an AR(1), only the current
state of the economy (Pt ) matters for expectation formation, regardless of whether beliefs
are rational or over-extrapolative. However, the current state of the economy does nottell us where the economy has been, which is captured by ∆Pt . As a result, an AR(1)specification of agents beliefs does not capture the asymmetry illustrated in Figure 1: As
the economy recovers from a crisis (∆Pt < 0), agents with AR(2) beliefs are more likely
to be more pessimistic about the state of the economy next year than agents with overextrapolative AR(1) beliefs; by contrast, if the economy is at the end of a boom or the
beginning of a crisis (∆Pt > 0), agents are more likely to be more optimistic under AR(2).
The impulse response functions in Figure A.2 confirm the intuition above. When theeconomy is disturbed by an unexpected increase in disaster probability, the decline inlending is milder at the beginning but the recovery is slower in the OE-AR(2) model thanthe OE-AR(1) model (see Figure A.2). At the beginning of a crisis, since ∆Pt > 0, agents23

Figure 1: Role of Momentum in Expectations of Disaster Probability∆Pt > 0∆Pt < 0ρ̃Pt − ρ̂2p ∆Pt

ρ̃Pt

ρ̃Pt

ρ̃Pt − ρ̂2p ∆Pt

Over-extrapolation of Pt+1 under AR(1)Over-expectation of Pt+1 under AR(2), with ∆Pt > 0 and ρ̂2p > 0
Over-expectation of Pt+1 under AR(2), with ∆Pt < 0 and ρ̂2p > 0

with AR(2) expectations under predict the probability of disaster in the forthcoming year.
On the other hand, when disaster probability starts declining, agents with AR(2) expectations over-predict the probability of disaster in the forthcoming year. Quantitatively,
the degree of persistence in the OE-AR(1) model is not sufficient to match the slow lending recovery in the data post-2009. Specifically, the OE-AR(1) model does not generate a“hump-shape” in the loan growth rate, or match the banks’ expectations of loan performance after 2009.
Although the focus of the paper is on the post-crisis period, we also examine themodel predictions during a boom, which is modeled as a temporary decrease in disasterprobability. In the face of an unexpected decrease in disaster probability (see Figure A.3),
the increase in lending is milder immediately after the shock in the OE-AR(2) model thanin the OE-AR(1) since ∆Pt < 0. However, the lending boom continues well after the shockis over in the OE-AR(2) model as agents are over-optimistic when at the end of a boom
when disaster probability starts to increase (∆Pt > 0).8
8 The asymmetry is that a negative disaster probability generates a larger boom, whereas a positive
disaster probability generates a milder recession in the OE-AR(2) model compared to the OE-AR(1) model.
This is because the capital requirement constraint becomes binding more quickly in the OE-AR(2) model,
which has a lower stochastic steady state for lending.

24

5.4

Bank Heterogeneity

Figure 5 presents the impulse response functions from the OE model by bank size: smallbanks are those whose asset-to-deposit ratios (lit ) are below the sample median before
the crisis (i.e. before the increase in disaster probability), while large banks are thosewith total assets above the median. All banks are subject to the same sequential shocks
as in Figure 3.9 The main takeaway from Figure 5 is that disaster probability shocks havea deeper and longer-lasting impact on larger banks’ lending decisions.
In order to understand the different responses, we first examine loan returns for small
and large banks, respectively. It turns out that the decline in realized return is quantitatively similar for the two groups, and slightly greater for smaller banks (see the bottom
right panel of Figure 5). However, small banks are much closer to the capital requirement constraint than large banks. As shown in the top panel of Figure A.9.C, the lending
decision for banks with low lt−1 quickly becomes flat at the level at which the capitalrequirement constraint binds, as disaster probability increases.
For banks with high lt−1 , there is some asymmetry. Focusing on the policy function
between Pt = 0.02 and Pt = 0.0549 in the top panel of Figure A.9.C, we see that the slope
for the pink solid line on is steeper on the left (i.e. when Pt−1 is low) than on the right(when Pt−1 is high). In other words, for large banks, optimal lending falls steeply during
the crisis when Pt unexpected increases from the stochastic steady state level, and rises
slowly after the crisis. Intuitively, the rate at which banks change their lending policies
depends on the asset adjustment cost and their risk-taking incentives. If they believethat a disaster is coming, their primary concern is to reduce risk-taking in order to avoid
shutting down. If they believe that the probability of disaster is diminishing, the benefitof adjusting assets gradually over time outweighs the benefit of rebuilding their loan
portfolios quickly.

5.5

A Disaster Realization

Figure 6 illustrates the response of the economy to a typical disaster that lasts for two
years (xt = 1 for t = 0, 1), with the realizations of pt and ωit equal to the mean of theirdistributions. In both economies, ex-post returns on loan portfolios (rtL ) fall significantly– and slightly more in the OE model – during the disaster years, and recover when eacheconomy exits the disaster state. Aggregate lending, bank value and current net worth
fall in both economies. More specifically, in the OE economy, lending falls by less but
9 In our simulations, all banks have the same belief process (17)-(18), and have the same values for{ρ̂1p , ρ̂2p , ρ̂1ω , ρ̂2ω , χ}. Assigning a greater degree of over-extrapolation to large banks as observed in thedata would amplify the difference between small and large banks in Figure 5.

25

takes much longer to recover; bank value and current net worth fall by a greater extent
and also take longer to recover than in the RE economy.
Since the disaster realization is not a state variable, the current net worth of banks nit
plays an important role in the propagation of a realized disaster. Recall that the current
net worth includes banks’ beginning-of-period equity and the current net profit. Whenthe returns on loan portfolios fall, banks’ current profits take a hit and hence their net
worth decreases. Since external financing is costly, banks decrease their loan portfolios
in response to a reduction in net worth. As a result, both the current and future values ofa bank decrease. Due to the asset adjustment cost, lending gradually reverts back to thesteady state when the disaster is over.
As explained above, the capital requirement constraint becomes binding more quickly
in the OE economy, which has a lower stochastic steady state level of lending. Thus,
lending decreases by less in terms of percentage deviation from the steady state in the
OE model. Moreover, the optimal lending policy is less responsive to changes in bank
net worth in the OE economy, especially if the disaster probability was high in the lastperiod (see Appendix A.9).10 As a result, in the OE model, lending falls by less upon
impact of a disaster, but also takes much longer to return to the stochastic steady state,
and the loan growth rate exhibits a hump-shape recovery.

6

Policy Evaluation

In this section, we conduct two policy experiments in turn: first, we increase the minimum equity to asset ratio from 8% (λ = 12.5) to 10% (λ = 10); secondly, we examine
the impact of quantitative easing by considering the effects of an increase in the discountfactor β.

6.1

Tighter Capital Adequacy Requirements

In this experiment, we solve the two models again, but this time we set λ = 10, whichcorresponds to a minimum equity-to-asset ratio of 10%. Then we compute the impulse
response functions to an increase in disaster probability under the baseline scenario (witha minimum equity-to-asset ratio of 8%) with those in the policy experiment. In doing so,
we ask the following question: If banks face tighter capital adequacy requirements innormal times, how does this affect the variables of interest in a crisis?
We first examine the stochastic steady state. Higher capital adequacy requirements
increase the overall loan portfolio ait in both models’ steady states, with a higher ait cor10 This is also the case if we set the same asset adjustment cost η B in the two models.

26

responding to a lower leverage. Banks respond to tighter capital requirements by accumulating more equity, and quantitatively, such incentives are stronger in the OE model.
In the (stochastic) steady state, book equity (scaled by deposits) beit increases by 23%
in the OE model, and by 10% in the RE model; similarly, current net worth (scaled bydeposits) nit increases by 24% in the OE model, and by 9% in the RE model.
If agents are rational (Panel A), tighter capital requirements can, to some extent, mitigate the adverse impact of a disaster risk: lending and current net worth fall by less thanthe baseline, though the mitigating effect on bank values is more limited. However, in
the OE model (Panel B), a higher minimum equity-to-asset ratio has little impact on the
dynamic impact of a disaster risk: the impulse responses for lending and bank value aresimilar to those in the baseline, though net worth falls by less compared to the baseline.
Overall, the model suggests that tighter capital requirements raise aggregate lending and bank values in normal times, but if the disaster risk increases unexpectedly, its
stabilization role is limited in the presence of belief distortions.

6.2

Quantitative Easing

During the financial crisis and the subsequent recovery, many central banks around theworld turned to quantitative easing (QE) as a monetary policy tool. These policies effectively subsidized the banking sector by providing banks with funding at favorable terms.
We use our model to examine the effectiveness of this policy tool during and after a crisisin the presence of belief distortions. Formally, for the first 2,979 periods in our simulation, we use the r̃ D specified in Table 4. Then from t = 2, 980 (i.e. when the disaster riskincreases unexpectedly and thereafter), we lower r̃ D by 10 basis points in each model.
In our model, this policy affects the shock propagation through the accumulation ofbanks’ net worth nit , which is a function of r̃ D . If banks face a lower cost of debt throughgovernment interventions, their profits fall by less during the crisis; hence their net worth
also falls by less and recovers faster when shocks are over. As shown in Figure 8, this istrue in both RE and OE models. Bank values also fall by less, and as a result, bank default
rates are lower in the presence of policy interventions.
By subsidizing bank equity holders, these policies have some positive effects on banklending: in the RE model, lending falls by 15 percent less and recovers more quickly with
QE (Panel A). However, the effect is much milder in the OE model (Panel B). As shownin Appendix A.9, lending policies are less responsive to changes in bank net worth in theOE economy, especially when last period’s disaster probability Pt−1 was high. In otherwords, the success of unconventional monetary policies in stimulating bank credit to theprivate sector after a financial crisis may be limited in the presence of belief distortions.
27

7Conclusion

This paper uses a unique survey data on banks’ forecasts to explore the transmission of
forecasting bias to bank lending, and offers one potential explanation for the slow recovery in bank credit after the 2008-09 financial crisis. We make two contributions. First, we
document significant persistence in forecast errors, suggesting that banks process information inefficiently. Moreover, we find that over-pessimistic banks are more likely to cut
their lending in the future. The behavioral bias is stronger for large banks, whose loan
portfolios are also more sensitive to beliefs. Second, we build a quantitative model thatjointly explains: (a) the dynamics of beliefs, (b) the dynamics of loan growth, and (c) the
link between beliefs and lending in the post-crisis period. Finally, our policy counterfactuals suggest that biased beliefs dampen the effectiveness of stimulative policies.

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31

Table 1: Dynamics of Bank Forecast Errors
Panel A of this table summarizes results of time-series regressions of bank expectation errors:

RitFE = α +K∑

β k RitFE−k +

k =1

K

∑ gk Xit−k + τt + uit

k =1

FE , as the difference between the expected and the actual
We measure bank expectation errors, Ritchange in loan performance over the next year at time t. We include loan performance in thecontrol set, Xit−k . Information on bank expectations is from the Senior Loan Officer Opinion
Survey (SLOOS). In Panel A, the number of lags is 1-, 2-, and 3-years. For reference, in Columns
4-6 we report results for repeating the analysis for actual loan performance. t-statistics are basedon standard errors clustered at the bank level, with ***, **, and * denoting significance at the 1%,5%, and 10% level, respectively.

Panel A: Bank-Level Estimates
Forecast Errors
k=1 year
k=2 year
k=3 year
(1)
(2)
(3)

βk
[t]
Time FE

R2

0.233∗∗∗[5.57]

0.153∗∗∗[4.33]

0.024[0.68]

Yes
0.59

Loan Performance
k=1 year
k=2 year
k=3 year
(4)
(5)
(6)
0.064
[0.70]-0.070∗∗-0.086∗∗∗[-2.16][-3.36]

Yes
0.42

32

Table 2: Bank Expectations and Lending Dynamics
This table summarizes results of bank-level regressions of bank loans on bank expectations of loan perfornance:
K∆Loansit = α + ∑β k RitFE−k +

k =1

K

∑ gk Xit−k + τt + uit−1

k =1

We measure the change in bank loans, ∆Loansit , as the logarithmic change of total bank loans
relative to pre-crisis, and bank expectation errors, RitFE , as the difference between the expected
and the actual change in loan performance over the next year at time t. We include loan performance in the control set, Xit−k . Information on bank expectations is from the Senior Loan Officer
Opinion Survey (SLOOS). In Panel A, the number of lags is 1-, 2-, and 3-years. t-statistics are
based on standard errors clustered at the bank level, with ***, **, and * denoting significance atthe 1%, 5%, and 10% level, respectively.

Panel A: Bank-Level Estimates
k=1 year(1)

βk-0.071[t][-0.52]

Forecast Errors
k=2 year
k=3 year(2)(3)
0.239∗∗∗[2.96]

k=1 year(4)

Loan Performance
k=2 year
k=3 year
(5)
(6)

0.048
[0.55]

gk-0.161-0.199∗∗∗-0.170∗∗[t][-0.89][-2.49][-3.03]Time FE

Yes
1,702
0.14

N
R2

33

Table 3: Heterogeneity and Analysis by Loan Type
This table repeats the analysis of Table 2 by sub-sample based on bank size (Panel A) and by subgroups of loan types (Panels B-C). In Panel A, we split the sample into two groups of banks, small
and large based on the bottom and top quartiles of bank total loans respectively. We report theestimates for repeating the analysis of lending dynamics of Table 2 separately for each of the twosize sub-samples, in turn. In Panels B-C, we repeat the analysis of Table 2 separately for each loan
type regressed on its respective forecast error as well as on the (average) forecast error for otherloan types, in turn. Information on bank expectations is from the Senior Loan Officer Opinion
Survey (SLOOS). t-statistics are based on standard errors clustered at the bank level, with ***, **,and * denoting significance at the 1%, 5%, and 10% level, respectively.

Panel A: Bank Expectations and Lending Dynamics by Bank Sizeβk[t]

k=1 year(1)

Small Banks
k=2 year(2)

k=3 year(3)

0.021[0.15]

0.258∗[1.81]-0.079[-0.86]k=1 year(4)Large Banks
k=2 year(5)k=3 year(6)0.246[1.46]

0.313∗∗[2.36]

0.121[0.82]

Panel B: Bank Expectations and Lending Dynamics by Loan Type

βown
k
[t]

βotherk[t]

k=1 year(1)

C&I Loans
k=2 year(2)

k=3 year(3)

k=1 year(4)

RRE Loans
k=2 year(5)k=3 year(6)

0.023[1.48]

0.025∗[1.88]

0.032[1.52]-0.021[-0.30]0.092∗∗[2.37]-0.045[-0.39]0.021[0.89]

0.015[0.64]

0.021[1.17]

0.050[0.35]

0.036[0.50]

0.037[0.39]

Panel C: Bank Expectations and Lending Dynamics by Loan Type

βown
k
[t]

βotherk[t]

k=1 year(1)

CRE Loans
k=2 year(2)

k=3 year(3)

0.091[1.47]

0.111∗[2.59]

0.121[1.40]-0.212[-1.25]0.005[0.03]

0.125[-0.65]-0.121[1.27]

0.065[0.83]-0.010[-0.14]0.042[0.29]

0.215[1.40]

0.142[1.15]

34

Consumer Loans
k=1 year
k=2 year
k=3 year
(4)
(5)
(6)

Table 4: Parameterization
A. Exogenous calibration
Parameter

Description

Value

β
ψ
γ
p̄
σp
ρp
σω
ρω
ξ
µc
σc
L
κ
λ
ηE

Rate of time preference
Elasticity of intertemporal substitution
Relative risk aversion
Average probability of crisis
Volatility of crisis probability
Persistence in crisis probability
Volatility of bank-specific shock
Persistence in bank-specific shock
Impact of crisis on endowment
Mean growth in consumption (normal times)
Volatility of aggregate shock (normal times)
Loss given default on loans
Loan-to-value ratio
Capital requirement
Equity issuance cost

0.987
2
3
0.02
0.42
0.80
0.02
0.90
log(1 − 0.3)
0.01
0.015
0.40
0.66
12.5
0.05

B. Moment-matching exercise
Parameter

Description

OE

RE

σj
ηL
cL
r̃ D
ρ̂1p
ρ̂2p
ρ̂1ω
ρ̂2ω
χ

Volatility of loan-specific shock
Asset adjustment cost
Non-interest income
Interest & operating expenses
Overextrapolation of crisis probability
Overextrapolation of crisis probability
Overextrapolation of bank-specific shock
Overextrapolation of bank-specific shock
Weight of pt in subjective probability of disaster

0.09
0.51
0.005
0.0066
0.329
0.613
0.329
0.613
0.349

0.1
0.5
0.006
0.0073−−−−−

Note: The model is calibrated at annual frequency. “−” indicates that the parameter is absent in the model.

35

Table 5: Moments
A. Targeted moments
Description

Data

OE

RE

Leverage (mean)

8.50

8.72

8.69

Leverage (std)

2.95

3.10

2.50

Profit-to-equity (mean)

0.169

0.137

0.149

Bank default rate (mean)

0.041

0.062

0.053

1-year

0.233

0.243

−

2-year

0.153

0.169

−

Dynamics of bank forecast errors

Note: The model is calibrated at annual frequency.

36

B. Untargeted Moments
Description

Data

OE

RE

Std dev

0.019

0.018

0.014

Corr(∆lt , ∆GDPt )

0.134

0.219

0.327

Corr(∆lt , ∆GDPt−1 )

0.239

0.135

−0.015Corr(∆lt , ∆GDPt−2 )

0.218

0.056

−0.136Corr(∆lt , ∆lt−1 )

0.207

0.597

0.123

Corr(∆lt , ∆lt−2 )

0.118

0.160

−0.036−0.225−0.192−0.023−0.077

0.192

0.295

0.230

0.134

0.465

0.197

−0.152

0.153

0.082

−0.094−0.013−0.011−0.064−0.015−0.015

0.023

0.017

0.016

−0.195−0.499−0.499−0.083−0.010−0.009Corr(∆Et [rtL+1 ], ∆GDPt )

0.075

0.375

0.306

Corr(∆Et [rtL+1 ], ∆GDPt−1 )Corr(∆Et [rtL+1 ], ∆GDPt−2 )Corr(∆Et [rtL+1 ], ∆Et−1 [rtL ])Corr(∆Et [rtL+1 ], ∆Et−2 [rtL−1 ])

0.071

0.110

−0.301

0.022

0.009

−0.270

0.017

0.177

−0.141−0.013−0.072−0.109Annual loan growth (∆lt )Annual change in expected loan performance (∆Et [LoanDefaultt+1 ])Corr ∆Et [LoanDefaultt+1 ], ∆GDPt−0.354Corr(∆Et [LoanDefaultt+1 ], ∆GDPt−1 )Corr(∆Et [LoanDefaultt+1 ], ∆GDPt−2 )Corr ∆Et [LoanDefaultt+1 ], ∆Et−1 [LoanDefaultt ]Corr ∆Et [LoanDefaultt+1 ], ∆Et−2 [LoanDefaultt−1 ]Annual change in realized loan performance (∆LoanDefaultt+1 )Corr ∆LoanDefaultt+1 , ∆GDPt−0.110Corr(∆LoanDefaultt+1 , ∆GDPt−1 )Corr(∆LoanDefaultt+1 , ∆GDPt−2 )Corr ∆LoanDefaultt+1 , ∆LoanDefaulttCorr ∆LoanDefaultt+1 , ∆LoanDefaultt−1Annual loan rate growth

Post crisis dynamics: years taken for the recovery of the following variables
Net fraction of banks experience worsening

3

5

5

Net fraction of banks expect worsening

8

8

5

Loan growth

7

7

3

Note: The model is calibrated at annual frequency.

37

Table 6: Dynamics of Bank Expectations and Lending in the Model
Panel A of this table summarizes results of time-series regressions of bank expectation errors:

RitFE = α +K∑

β k RitFE−k +

k =1

K

∑ gk Xit−k + τt + uit

k =1

FE , as the difference between the expected and the actual
We measure bank expectation errors, Ritchange in loan performance over the next year at time t. Panel B summarizes results of bank-level
regressions of bank loans on bank expectations of loan perfornance:
K

K

k =1

k =1

∆Loansit = α + ∑ β k RitFE−k + ∑ gk Xit−k + τt + uit−1
We measure the change in bank loans, ∆Loansit , as the logarithmic change of total bank loans
relative to pre-crisis, and We include loan performance in the control set, Xit−k . The number oflags is 1- and 2-years. t-statistics are based on standard errors clustered at the bank level, with ***,**, and * denoting significance at the 1%, 5%, and 10% level, respectively.

A. Dynamics of Bank Expectation Errors

βk[t]

RE Modelk=1 year
k=2 year(1)(2)

OE Modelk=1 year
k=2 year(3)(4)

0.920∗∗∗[3.64]

0.243∗∗[2.42]

Time FE−0.599∗∗[−2.53]

Yes
0.78

R2

0.169∗[2.16]Yes
0.80

B. Bank Expectations and Lending Dynamics

βk[t]Time FE

R2

RE Modelk=1 year
k=2 year(1)(2)

OE Modelk=1 year
k=2 year(3)(4)

0.417∗∗[2.45]

0.218∗∗[2.79]−0.082[−0.43]Yes
0.56

0.187∗[2.10]

Yes
0.52

38

Figure 2: Stylized Facts from Bank-Level Data(a) Net fraction of banks that experience an actual worsening of loan performance(b) Net fraction of banks that expect a worsening of loan performance(c) Annual loan growthNote: Panel A plots the net fraction of banks that experience an actual worsening of loan performance: it is the number of banks
that experience worsening (defined by an increase in expected the loan default rate) minus the number of banks that experience
improving, divided by the total number of banks. Panel (b) demonstrates the net fraction of banks that expect a worsening of loan
performance. Panel C shows the actual loan growth (aggregated across banks). The shaded area indicates the NBER recession dates.

39

Figure 3: Impulse Response Functions to A Temporary Increase in Disaster Probability

40
Note: The figure shows the response to a temporary increase in the annual disaster probability. We simulate the model for 3,000 years, and at t = 2, 980 and t = 2, 981 we perturb the model with twoconsecutive shocks to the probability of disaster. We repeat the procedure 1,000 times and calculate the average across simulations. All shocks and distributional dynamics are determined according
to the true process (i.e. the rational expectations representation), even though the asset prices and bank polices may involve distorted expectations. OE: Overextrapolation. RE: Rational expectation.

Figure 4: Belief Process: Data vs. Model(a) Net fraction of banks that expect a worsening of loan performance(b) Forecast Errors

Note: Panel A plots the net fraction of banks that expect a worsening of loan performance in each year, i.e. it is the number of banks
that expect worsening minus number of banks that expect improving, divided by the total number of banks. In Panel B, forecast errors
are the difference between the expected and the actual change in loan performance, so positive forecast errors indicate that banks areover-optimistic (or the expected loan default rate is lower than the actual default rate) while negative forecast errors indicate thatbanks are over-pessimistic.

41

Figure 5: Impulse Response Functions By Bank Size (OE Model)

Note: The figure shows the responses to a temporary increase in the annual disaster probability in the overextrapolative model. We
simulate the model for 3,000 years, and at t = 2, 980 and t = 2, 981 we perturb the model with two consecutive shocks to the probabilityof disaster. We repeat the procedure 1,000 times and calculate the average across simulations. Small banks are those whose total assets
are below the median level before the increase in disaster probability, and large banks are those with total assets above the median.
All shocks and distributional dynamics are determined according to the true process (i.e. the rational expectations representation),even though the asset prices and bank polices have over-extrapolative expectations.

42

Figure 6: Disaster in the Model

43
Note: The figure shows the response of the economy to a typical disaster, that lasts for 2 years. We simulate the model for 3,000 years, and at t = 2, 981 − 2, 982 we perturb the model with a disaster(xt = 1). pt and ωit are randomly drawn according to their true processes. We repeat the procedure 1,000 times and calculate the average across simulations. OE: Overextrapolation. RE: Rational
expectation.

Figure 7: Policy Counterfactual: Higher Capital Requirement
A. Rational Expectations

44

B. Overextrapolation

Note: The figure shows the response to a temporary increase in the annual disaster probability in terms of percentage deviation from the stochastic steady state. We simulate the model for 3,000 years,
and at t = 2, 981 and t = 2, 982 we perturb the model with two consecutive shocks to the probability of disaster. We repeat the procedure 1,000 times and calculate the average across simulations. All
shocks and distributional dynamics are determined according to the true process, even though the asset prices and bank polices may involve distorted expectations.

Figure 8: Policy Counterfactual: Quantitative Easing
A. Rational Expectations

45

B. Overextrapolation

Note: The figure shows the response to a temporary increase in the annual disaster probability in terms of percentage deviation from the stochastic steady state. We simulate the model for 3,000 years,
and at t = 2, 981 and t = 2, 982 we perturb the model with two consecutive shocks to the probability of disaster. We repeat the procedure 1,000 times and calculate the average across simulations. All
shocks and distributional dynamics are determined according to the true process, even though the asset prices and bank polices may involve distorted expectations.

Expectations and Credit Slumps
Online Appendix
Antonio Falato
Federal Reserve Board

Jasmine Xiao
University of Notre Dame

March 2022

A

Model Solution

A.1. Normalization
We scale the market value of a bank by deposits, and conjecture that it is a function ofli,t−1 , nit , and sit :viC (li,t−1 , nit , sit ) =ViC ( Li,t−1 , Di,t−1 , Nit , sit )
Dit

L

Nit
it
. We further define divit ≡ Div
where li,t−1 ≡ Di,t−1 and nit ≡ D
D =
i,t−1

it

it


1 
BEi,t−1 + (rtL + c) Li,t−1 − r̃ D Di,t−1 −( Lit − Dit ) − Φ( Li,t−1 , Lit )
Dit |
{z
}
≡ Nit

=nit + 1 − lit − ϕ(li,t−1 , lit ),which uses the evolution of equity (13), the balance sheet constraint (14), andϕ(li,t−1 , lit ) = ηli,t−1 e− glit − li,t−1 e− gli,t−1 e− g!2.

where ϕ(li,t−1 , lit ) ≡. Recursively define the bank’s problem as(viC (li,t−1 , nit , sit ) = max nit + 1 − lit − ϕ(li,t−1 , lit ) + Λ(divit )
lit

"
g

+ Et Mt,t+1 e max

1nviC (lit , ni,t+1 , si,t+1 ), 0

o

#)
sit

,

(A.1)

subject to the evolution of ni,t+1 :
ni,t+1 = e− g lit − 1 + rtL+1 (sit , ε c,t+1 , xt+1 , ωi,t+1 ) + c lit − r̃ D

(A.2)

the capital requirement constraint:lit≤λlit − 1

(A.3)

Λ(divit ) = 1divit <0 η E divit ,(A.4)

and the equity issuance cost:thus verifying the conjecture.

A.2. Computation
Discretization and Asset Prices We discretize the shock processes for the probabilityof crisis pt , the collateral value ωit using the method by Tauchen (1986). We solve thefixed-point problem (6) to find the equilibrium wealth-consumption ratio St . Then thestochastic discount factor follows from (5), and the price of the loan portfolio PtL (st ) andrtL+1 (ε c,t+1 , xt+1 , ωt+1 , st ) are derived from (11) and (9), respectively. Note that st include:• ( pt , ωt ) for rational expectations;• ( pt , pt−1 , ωt , ωt−1 ) for overextrapolation;
Value Function Iteration

The bank takes prices as given, and decides on its loan port-

folio to maximize the sum of dividends and continuation value, subject to the regulation
constraint. We solve the problem by iterating on the Bellman equation.
1. Compute nt+1 (lt , rtL+1 ) that satisfies (A.2), given rtL+1 (ε c,t+1 , xt+1 , ωt+1 , st ).
2. Guess vC (lt−1 , nt , st ), and denote it as v0C .
3. Evaluate vC (lt , nt+1 , st+1 ) using piecewise interpolation of our guess v0C . Then solve
the bank’s problem (A.1) on the discretized state space. If the candidate lt violates
the capital requirement constraint (A.3), the default value v D = 0 is assigned. Denote the maximum bank value as v1C .
4. Check for convergence:• If |v1C (lt−1 , nt , st ) − v0C (lt−1 , nt , st )| < ε, the value function has converged;2• Otherwise update v0C (lt−1 , nt , st ) to v1C (lt−1 , nt , st ) and repeat step 3.
Simulation We obtain model-implied moments by simulating 1,000 banks for 3,000 periods, discarding the first 300 years. Each bank starts with some specific initial values forli,t−1 and nit . We simulate the series for the exogenous state variables ωit , pt , ωi,t−1 , and
pt−1 , the endogenous state variables ai,t−1 and nt , and shocks ε c,t+1 that determine theex-post return on the bank investments and the ex post output of the firm. Importantly,
when we simulate the model, all shocks and distributional dynamics are determined according to the true process (i.e. the rational expectations representation), even though
the asset prices and bank polices may involve distorted expectations.
Recall that a bank defaults if its continuation value V C falls below the threshold level
V D , which we normalize to zero. For simplicity, we assume that when a bank defaults, anidentical bank is created with the same state variables. Hence we maintain a stationary
distribution of banks.

3

Table A.1: Additional Heterogeneity
This table repeats the analysis of Table 1 by sub-sample based on bank size. In Panel A, we split
the sample into two groups of banks, small and large, based on the bottom and top quartiles ofbank total loans respectively. We report the estimates for repeating the analysis of forecast error
dynamics of Table 2 separately for each of the two size sub-samples, in turn. Information on bankexpectations is from the Senior Loan Officer Opinion Survey (SLOOS). t-statistics are based onstandard errors clustered at the bank level, with ***, **, and * denoting significance at the 1%, 5%,
and 10% level, respectively.

Panel A: Dynamics of Bank Forecast Errors by Bank Sizeβk[t]

k=1 year(1)

Small Banks
k=2 year(2)

k=3 year(3)0.254∗∗[2.15]

0.121∗[1.73]

0.107[1.28]

4

k=1 year(4)Large Banks
k=2 year(5)k=3 year(6)0.308∗∗[4.64]

0.138∗∗[2.14]-0.046[-0.70]Table A.2: Robustness to Controlling for Alternatives
This table repeats the analysis of Table 2 using a specification that adds controls for alternatives.
In Panel A, we add controls for loan demand and loan performance in the crisis, in turn. In
Panel B, we add controls for bank capital (tier 1 capital ratio) and liquidity (cash ratio), in turn.
Information on bank expectations is from the Senior Loan Officer Opinion Survey (SLOOS). tstatistics are based on standard errors clustered at the bank level, with ***, **, and * denoting
significance at the 1%, 5%, and 10% level, respectively.

Panel A: Bank Expectations and Lending Dynamics, Controlling for Alternatives
Loan Demand
k=1 year
k=2 year
k=3 year
(1)
(2)
(3)

βk
[t]

Controlk[t]Crisis Loan Performance
k=1 year
k=2 year
k=3 year
(4)
(5)
(6)

-0.141[-0.97]0.220∗∗∗[2.64]

0.053[0.58]-0.169[-0.95]0.236∗∗[2.36]

0.039[0.45]

0.053[0.60]

0.092[1.39]

0.120∗∗[2.42]-0.200[-0.95]0.465∗[1.93]-0.187[-0.62]

Panel B: Bank Expectations and Lending Dynamics, Controlling for Alternatives

βk[t]Controlk[t]

k=1 year(1)

Bank Capital
k=2 year(2)

k=3 year(3)-0.067[-0.48]0.239∗∗∗[2.96]

0.048[0.55]-0.075[-0.52]0.245∗∗∗[2.87]

0.043[0.46]

0.114[0.77]-0.182[-1.48]-0.053[-0.49]0.116∗∗[2.07]-0.085∗∗[-2.09]-0.041[-0.57]

5

Bank Liquidity
k=1 year
k=2 year
k=3 year
(4)
(5)
(6)

Figure A.1: Aggregate Data

Note: This plots total bank loans (in trillions of dollars) by all commercial banks between 2000 and 2020, deflated by the GDP deflator,
and the pre-crisis trend of the series. Source: H.8 Assets and Liabilities of Commercial Banks in the U.S.

6

Figure A.2: Impulse Response Functions to A Temporary Increase in Disaster
Probability

Note: The figure shows the response to a temporary decrease in the annual disaster probability. We simulate the model for 3,000
years, and at t = 2, 980 and t = 2, 981 we perturb the model with two consecutive shocks to the probability of disaster. We repeat theprocedure 1,000 times and calculate the average across simulations. All shocks and distributional dynamics are determined according
to the true process (i.e. the rational expectations representation), even though the asset prices and bank polices may involve distorted
expectations. OE with AR(2): Overextrapolation with AR(2) expectations. OE with AR(1): Overextrapolation with AR(1) expectations.
RE: Rational expectation.

7

Figure A.3: Impulse Response Functions to A Temporary Decrease in Disaster
Probability

Note: The figure shows the response to a temporary decrease in the annual disaster probability. We simulate the model for 3,000
years, and at t = 2, 980 and t = 2, 981 we perturb the model with two consecutive shocks to the probability of disaster. We repeat theprocedure 1,000 times and calculate the average across simulations. All shocks and distributional dynamics are determined according
to the true process (i.e. the rational expectations representation), even though the asset prices and bank polices may involve distorted
expectations. OE with AR(2): Overextrapolation with AR(2) expectations. OE with AR(1): Overextrapolation with AR(1) expectations.
RE: Rational expectation.

8

Figure A.4: Robustness: No Asset Adjustment Cost

Note: The figure shows the response to a temporary increase in the annual disaster probability. We simulate the model for 3,000 years,
and at t = 2, 981 we perturb the model assuming a change in the probability of disaster to a new value. We repeat the procedure 1,000
times and calculate the average across simulations. All shocks and distributional dynamics are determined according to the true
process, even though the asset prices and bank polices may involve distorted expectations.

9Figure A.5: Autocorrelation Functions
A. Disaster Risk Probability (log Pit )B. Bank-Specific Shock (ωit )Note: This figure plots the autocorrelation functions for the persistent processes in the model, Pit and ωit . The left panels give the
ACF for the true processes, according to (3) and (8). The right panels give the ACF for the belief processes, according to (18) and (17)for the parameters given in Table 4.

10

Figure A.6: Asset Prices
A. Risk Premia

B. Realized Loan Return

Note: This figure illustrates the asset prices in the OE model. Panel A shows – for each level of the probability of crisis pt – theffex-ante expected rate of return on loans relative to the rate of return on a one-year safe asset, i.e. Et [rtL+1 − rt+1 ], where rt+1 satisfiesfff
rt+1 = 1/Pt − 1 and Pt = Et [ Mt,t+1 ]. The expected return and the probability are in annual terms. The risk premium is a function of( pt , pt−1 , ωit , ωi,t−1 ). Different lines represent alternative levels of pt−1 . Panel B shows – for each level of pt – the realized return onloans in t + 1, depending on whether a disaster occurs in t + 1. The realized return is a function of ( pt , pt−1 , ωit , ωi,t−1 , ϵt+1 , ωi,t+1 ).
Different lines represent alternative levels of pt−1 , and the other states (ωit , ωi,t−1 , ϵt+1 , ωi,t+1 ) are fixed.

11Figure A.7: Policy Functions in the OE Model
A. Lending

B. Bank value

Note: This figure illustrates the policy functions in the OE model. Panel A shows the optimal amount of bank lending scaled bydeposits lit = Lit /Dit for each level of the probability of crisis pt . Panel B shows the value of the bank scaled by deposits vitC = VCit /Dit .
In the left panels, different lines represent alternative levels of lagged asset-to-debt ratio li,t−1 = Li,t−1 /Dit , holding the other states(nit , Pt−1 , ωit , ωi,t−1 ) fixed. In the right panels, different lines represent alternative levels of lagged disaster probability Pt−1 , holding
the other states (li,t−1 , nit , ωit , ωi,t−1 ) fixed.

12Figure A.8: Comparison of Asset Prices in OE and RE Models
A. Asset prices as a function of disaster probability (for different levels of lt−1 )B. Asset prices as a function of disaster probability (for different levels of nt )Note: This figure compares asset prices as a function of disaster probability Pt in the two models. OE: Overextrapolation. RE: Rational
expectations. The risk premium is the ex-ante expected rate of return on loans relative to the rate of return on a one-year safe asset,
fi.e. Et [rtL+1 − rt+1 ]. The risk premium a function of ( Pt , Pt−1 , ωit , ωi,t−1 ) in the OE model, and a function of ( Pt , ωit ) in the RE model.
The realized return on loans depends on whether a disaster occurs. It is a function of ( Pt , Pt−1 , ωit , ωi,t−1 , ϵt+1 , ωi,t+1 ) in the OE model,
and a function of ( Pt , ωit , ϵt+1 , ωi,t+1 ) in the RE model. We fix ωit , ϵt+1 , ωi,t+1 ) at the same level in both models, and we set ωi,t−1 = ωitin the OE model.

13

Figure A.9: Comparison of Policy Functions in OE and RE Models
C. Lending as a function of disaster probability

Note: This figure compares the optimal lending lit = Lit /Dit as a function of disaster probability Pt in the two models. OE: Overextrapolation. RE: Rational expectations. In the RE model, lit is a function of (li,t−1 , nit , Pt , ωit ); in the OE model, lit is a functionof (li,t−1 , nit , Pt , ωit , Pt−1 , ωi,t−1 ). We fix li,t−1 , nt , ωit at the same levels in both models, and in the OE model, we set Pt−1 = Pt andωi,t−1 = ωit .

14

B. Lending as a function of current net worth

Note: This figure compares the optimal lending lit = Lit /Dit as a function of current net worth nit in the two models. OE: Overextrapolation. RE: Rational expectations. In the RE model, lit is a function of (li,t−1 , nit , Pt , ωit ); in the OE model, lit is a function of(li,t−1 , nit , Pt , ωit , Pt−1 , ωi,t−1 ). We fix li,t−1 , Pt , ωit at the same levels in both models.

15