Download Deterioration of Bank Balance Sheets and

Housing Crisis, Deterioration of Bank Balance Sheets, and
Macroprudential Policies
Yavuz Arslan‡
Bulent Guler§
⇤†
Burhanettin Kuruscu¶
Very preliminary and incomplete. Please do not circulate.
Abstract
One of the most prominent features of the 2007-2009 US financial crisis was the central
role played by mortgages and mortgage-related assets. After the crisis, regulators have implemented several macroprudential policies targeting both the mortgage market and bank lending
practices in order to stabilize the financial system. Despite the key role played by mortgages
in the 2007-2009 financial crisis and the fact that many of the macroprudential policies implemented to stabilize the banking system are related to mortgage lending, the literature lacks a
rich and realistic quantitative analysis of such policies with explicit treatment of housing and
mortgage markets, and bank balance sheets. The purpose of this paper is to fill this gap by
developing a quantitative general equilibrium model that combines two key features: 1) a rich
heterogeneous-agent overlapping-generations structure of households who make housing tenure
decisions borrowing through long-term mortgages and 2) banks that issue long-term mortgages
and whose ability to intermediate funds depend on their capital. Using this framework, we
analyze stabilization e↵ects of macroprudential policies during financial crisis as well as their
aggregate and distributional implications. As part of our policy analysis, we consider minimum down payment requirements, limits on loan-to-income ratio, debt-service-to-income ratio
and mortgage maturities, and stricter capital requirements and countercyclical capital bu↵ers
(CCyB’s) for banks.
⇤
The views expressed here are ours and do not reflect those of the Bank of International Settlements.
We thank seminar participants at the University of Toronto and The Bank of Canada.
‡
The Bank of International Settlements; [email protected].
§
Indiana University; [email protected].
¶
University of Toronto; [email protected].
†
1
1
Introduction
One of the most prominent features of the 2007-2009 US financial crisis was the central role played
by mortgages and mortgage-related assets. Leading up to the financial crisis, a relaxation of credit
conditions led to increase in demand for housing, a higher leverage by households, and an increase
in house prices. Real house prices almost doubled between 1995 and 2006. Total mortgage debt
outstanding as a fraction of disposable income increased from about 60% to 100%. Once credit
conditions started tightening however, house prices started to decline. House prices experienced a
drop of around 40% between 2006 and 2011, which led to increase in foreclosures. The fraction of
loans in foreclosure process in a given quarter increased from 0.5% to 4% between 2006 and 2011.
Financial institutions experienced losses from residential mortgages (and related assets) due to the
unexpected increases in foreclosures. Depletion of capital forced financial institutions to contract
lending, which raised the cost of credit they o↵ered. The ratio of mortgage-backed securities (MBS)
held by banks in the US to the net-worth of these banks decreased by almost 30%. Parallel to these
developments, mortgage premium defined as the di↵erence between 30-year fixed mortgage interest
rate and 10-year Treasury bond increased by almost 1.5 percentage points. Meanwhile, the volume
of mortgage originations dropped by almost 50%. Higher cost of credit led to further contraction in
economic activity, lowering income of households and raising the cost of buying a house. Demand
for housing declined, which led to further declines in house prices and increases in foreclosures,
which fed back to the cycle.
After the crisis, regulators have implemented several macroprudential policies targeting both
the mortgage market and bank lending practices in general in order to stabilize the financial system
and to prevent any future crisis happening due to banks’ and households’ having leverage ratios that
are not sustainable. For example, Basel III (the third Basel Accord) was intended to strengthen
bank capital adequacy by decreasing bank leverage. The Canadian federal government has changed
mortgage regulations since 2008. Maximum amortization for insured mortgages has been reduced
from 40 to 25. Down payment requirement has been increased or equivalently maximum loan
to value (LTV) ratio has been reduced. Maximum gross debt service to income ratio (the share
of borrower’s gross household income needed to pay for mortgage, property taxes, and heating
expenses) has been limited to 39%. Other policies such as restrictions on loan to income ratio
for mortgages and countercyclical capital bu↵ers (CCyB’s) have also been have become part of
policy tools by some countries. Based on an IMF survey, Cerutti et al. (2015) report wide usage
of macroprudential policies (including the policies considered in this paper) for a large and diverse
set of 119 countries. They report that borrower based policies such as caps on loan to value and
debt to income ratios are especially used in advanced economies.
Despite the key role played by mortgages in the 2007-2009 financial crisis and their impact on
bank balance sheets, the literature lacks a rich and realistic quantitative framework with explicit
treatment of housing and mortgage markets, and bank balance sheets that can be used for analyzing
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macroprudential policies. The purpose of this paper is to develop such a framework, and, using this
framework, analyze feedback mechanisms between house prices, foreclosures, and household and
bank balance sheets in response to financial shocks and compare e↵ectiveness of macroprudential
policies in stabilizing the economy during financial crisis. To our knowledge, this paper is the
first to combine a rich household structure with heterogeneous long-term mortgages with a model
of bank balance sheet, which allows us to compare e↵ectiveness of policies across household and
banking sectors. Interactions between household and bank balance sheets might have important
consequences in evaluating policies as well. We consider the e↵ects of minimum down payment
requirements, restrictions on loan-to-income ratio, debt-service-to-income ratio, and maturity limits
on mortgages, as well as stricter capital requirements and CCyB’s. Our modeling elements make it
possible to study these policies in a unified framework. For example, long-term nature of mortgages
and house being used as collateral are two important features that help generate debt-to-income
ratios as high as those observed in the data. Thus, these are essential features needed to analyze
e↵ects of minimum down payment requirements and restrictions on loan-to-income ratio, debtservice-to-income ratio, and maturity limits on mortgages. A quantitative analysis of these policies
also requires realistic income, wealth, and age heterogeneity as we have in our framework. In the
absence of such heterogeneity, some of these regulations (for example, down payment requirement)
would either not a↵ect any household or would bind for all of them.
We build an overlapping generations model with idiosyncratic shocks and incomplete markets.
We explicitly model housing tenure choices of households, by allowing them to choose between owning and renting. The model features heterogeneity along income, wealth, and age, which are important determinants of housing and financial choices of individuals. Households make consumptionsaving decision and purchase houses by using long-term mortgages. We assume that households
have limited commitment to the mortgage, i.e. they can default on the mortgage in any period
throughout the life of the mortgage. Banks price mortgages by internalizing the default probability
of households. So, each mortgage is individual specific, and borrowing limits endogenously arise
due to limited commitment by households.
The model also features firms as in standard macroeconomic models as agents maximizing
profits. Since firms finance their production through loans from financial sector, the model also
enables us to study the impact of the deterioration of the bank balance sheets on the supply of the
goods by the firms and employment.
The key contribution of our paper is to incorporate the rich mortgage structure arising in the
household sector into the balance sheet of the banks while allowing individual pricing of each
mortgage. For this purpose, we assume a competitive banking industry with a continuum of
identical banks. Banks accept deposits from creditors, issue new mortgages, and invest in existing
mortgages. No arbitrage condition implies that prices of all mortgages will be such that banks
are indi↵erent between investing in capital or any mortgage, i.e. the expected return from all
3
mortgages will the same. Given this result, the distribution of mortgages becomes irrelevant for
the bank. As a result, even though each bank holds a rich portfolio of mortgages issued to di↵erent
types of households, we show that the bank’s problem can be reduced into a very simple portfolio
problem, where banks only choose total amount of lending and deposits regardless of the type of
the mortgage. Such an aggregation result allows us to analyze the interaction between banks and
households by still preserving the rich heterogeneity in the household sector.
Banks face capital requirement constraints, which limit their capacity to issue loans. This
constraint arises endogenously due to the possibility that a bank can walk away with a fraction of
its assets and not pay back creditors. More leveraged banks have a higher incentive to walk away
and thus, creditors lend to banks up to a limit so that banks do not default. When the capital
requirement constraint is slack, perfect competition among banks implies that the expected return
on mortgages is equal to the deposit rate, and banks make zero expected profit from each mortgage
(as in Chatterjee, Corbae, Nakajima, and J. V. Rios-Rull (2007), Livshits, MacGee, and Tertilt
(2007, 2010) etc.). When the capital requirement constraint binds however, the expected return
on mortgages becomes higher than the deposit rate. Even though banks make positive profits
from newly originated mortgages, no bank can reduce the interest rate on mortgages and attract
more households since they cannot issue additional mortgages due to binding capital requirement
constraint. Thus, during financial crisis, as banks’ balance sheet deteriorates and their equity
declines, banks’ ability to provide funding becomes limited. Due to higher cost of credit, households’
demand for housing and firms’ demand for capital go down, reducing house prices and wages. Lower
wages and house prices further increase default, which further reduce bank equity squeezing bank
lending further. Thus, a crisis is amplified due to this interaction between household and bank
balance sheets.
Our framework combines key elements from two important literatures. On the one hand, an
active literature has studied pricing of default risk in the context of sovereign debt, unsecured debt,
or mortgage debt. Prominent examples for sovereign default are Aguiar and Gopinath (2006) and
Arellano (2008); for unsecured credit are Chatterjee et al. (2007), Livshits et al. (2007, 2010), and
for mortgage debt are Corbae and Quintin (2014), Chatterjee and Eyigungor (2015), Guler (2015),
Hatchondo et al. (2015). In this literature, banks are modeled as simple risk-neutral and zero-profit
making competitive financial intermediaries. Even though this literature studies environments with
rich heterogeneity, they ignore the consequences of default on bank balance sheets. On the other
hand, the literature on bank balance sheets studies how depletion of capital reduces a bank’s ability
to intermediate funds (See among others, Mendoza and Quadrini (2010), Gertler and Karadi (2011),
and Gertler and Kiyotaki (2010, 2015)). However, in this literature, banks’ asset structure typically
takes a simple form such as one-period bonds or lacks the rich heterogeneity observed in banks’
portfolios. Our contribution to this literature is to incorporate heterogenous long-term mortgages
into banks’ portfolio choice.
4
In terms of modeling, our framework shares similarities with some recent papers, which have
introduced heterogeneity or long-term assets into the model of bank balance sheets of Gertler
and Kiyotaki (2010) and Gertler and Karadi (2011). Gunn and Johri (2015) and Bocola (2016)
introduce long-term bonds into banks’ balance sheet. We have long-term defaultable debt as well
in banks’ portfolios similar to these papers. However, banks’ portfolios in our framework contains
heterogeneous mortgages from di↵erent households, which is essential to study policies such as
minimum down payment requirements and restrictions on loan-to-income ratio and debt-serviceto-income ratio. Navarro (2016) introduces heterogeneity by modeling banks’ lending decision to
firms which face productivity risk and can default on their debt. The debts are one-period default
loans in his framework while they are long-term and houses are used as collateral to support debt in
our case. Both long-term feature of mortgages and house being used as collateral are important in
obtaining the ratio of mortgage debt to GDP as high as in the data. Finally, another key di↵erence
of our paper from these papers is our focus on stabilization e↵ects of macroprudential policies,
which these papers do not address.
Garriga and Schlagenhauf (2009), Jeske et al. (2013), Corbae and Quintin (2015), Chatterjee and Eyigungor (2015), Jeske et al. (2013), Mitman (2016), and Kaplan et al. (2016) build
equilibrium models of housing, endogenous leverage choice, and foreclosure as in our framework.
Some of these papers have also studied mortgage market policies as in our paper, however generally
focusing on di↵erent policies than the ones we analyze here. For example, Jeske et al. (2013),
Mitman (2016), and Kaplan et al. (2016) analyze debt relief policies. The closest paper in terms
of policy analysis is Hatchondo et al. (2015). They analyze the implications of recourse mortgages
and loan-to-value ratios in the long-run as well as in response to exogenous declines in house prices.
The common di↵erence of our paper from all of these papers is that they ignore the consequences
of household default on bank balance sheets. Thus, they are silent about any implications of policies such as higher capital requirements for banks. Second, our preliminary analysis indicates that
interaction between bank and household balance sheets amplify impacts of shocks. This interaction
is likely to have important consequences for policy analysis too. Finally, one of the main targets of
the macroprudential policies is to make bank balance sheets more safe and thus more resilient to
shocks. These papers are silent about the e↵ects of macroprudential regulations on bank balance
sheets.
Several empirical studies have analyzed the relation between macroprudential policies and financial stability measures such growth in bank credit and leverage using cross-country data (see among
others, Lim et al. (2011), Dell’Ariccia et al. (2012), and Cerutti et al. (2015)). General conclusion
from these papers is that use of macroprudential policies are associated with slower credit growth
and leverage. Even though these papers provide evidence for stabilizing e↵ects of macroprudential
policies, they do not provide a framework to conduct counterfactual experiments with respect to
these policies. Our paper provides a quantitative framework that can be used to conduct such
5
experiments and analyze aggregate and distributional implications of policies in question as well as
their e↵ects on economic welfare.
To be continued...
2
Quantitative Model
Our economy consists of production, household, and the banking sector.
2.1
Production
A perfectly competitive firm produces final output by combining capital Kt and labour Nt . Denoting
the rental rate of capital as rt⇤ and the wage rate per efficiency unit of labor as wt , the firm’s problem
is given by
max At Kt↵ Nt1
↵
Kt ,Nt
2.2
w t Nt
(rt⇤ + )Kt .
Households
The household sector is closely related to the one in Guler (2016) and Arslan et al. (2015). The
economy is populated by a continuum of households who live up to age J. For each age j, there are
a continuum of households. Agents are forced to retire at the age Jr < J. After retirement, agents
receive retirement income, which depends on both the personal income level at the retirement age.
Before retirement, log labor income consists of deterministic component f (j) which only depends
on the age, and stochastic component zj which is an AR(1) process. Thus the income structure for
the households y(j, zj ) can be summarized by
y(j, zj ) =
zj = ⇢zj
8
<w exp(f (j) + zj ),
:wy (z ),
R Jr
1
+ "j ,
"j ⇠ i.i.d.
if j  Jr
(1)
if j > Jr
N (0,
2
" ),
where w is the wage rate per efficiency units of labor and yR (zJr ) is a function that approximates
the US retirement system.
Households receive utility from consumption and housing services and can choose between
renting and purchasing a house. All houses are of the same size, but households receive greater
utility when they own the house. The preferences of a household has the following form,
E0 [
J
X
j 1
j=1
6
uk (cj , hj )]
where
< 1 is the discount factor, cj is the consumption, and hj is the housing services at age
j. It is assumed that owner receives higher utility from consumption, that is, uo (c, h)
ur (c, h),
where “r” denotes renter and or “o” denotes owner.
Households start the economy as active renters, and can purchase a house and become homeowners at any time. They can access mortgage loans to finance their housing purchase and at the
same time save through capital at interest rate r⇤ . Terms of mortgage contracts, down payment and
mortgage interest rate, are endogenous. Homeowners can become renters again, by either selling
their houses or defaulting on mortgage loans if they have any. Defaulted households are classified
as inactive renters who are banned from accessing housing market until the red flag is removed. Inactive renters face a fixed probability of becoming active renters. Each period, homeowners receive
a moving shock, such as job or family-related move, with a fixed probability and they are forced to
sell their houses or default on the mortgage if they have any. Therefore, there are three status for
agents regarding their housing decision, homeowner, active renter, or inactive renter. The supply
of rental and owner-occupied units is constant and targeted to match the average ownership rate
in US.
Households are subject to idiosyncratic uncertainty, but there is no aggregate uncertainty in the
model. The source of the idiosyncratic shock is three folds. First of all, the stochastic component
of their income follows an AR(1) process as described in equation (1). Secondly, homeowners are
subject to idiosyncratic moving shocks, which force them to become renters in the next period.
Finally, homeowners also receive a house maintenance shock every period with some probability.
2.3
Household’s problem
In the household’s problem, the decision rules and value functions will be time dependent since
the interest rates and mortgage prices will be time dependent when we analyze the transition of
our economy after it is hit by a financial shock. However, we will ignore the time subscript in the
household’s problem as it does not create any confusion.
2.3.1
Active Renters
An active renter has two choices: to continue to rent or purchase a house, i.e. V r = max {V rr , V ro }
where V rr is the value function if she decides to continue renting and V ro is the value function if
she decides to purchase a house. If she decides to continue to rent, she pays the rental price, makes
her consumption and saving choices, and remains the next period as an active renter. If she decides
to purchase a house, she chooses a mortgage contract o↵ered by the bank. She is free to choose
any down payment level. After purchasing a house, she begins the next period as a homeowner.
The value function of an active renter who decide to remain as a renter is given by
7
Vjrr (!, z) = max
0
c 0,a
0
r
ur (c) + EVj+1
(! 0 , z 0 )
(2)
subject to
c + a0 + pr = y(j, z) + !
! 0 = a0 (1 + r⇤ ) .
where x0 denotes the next period value of any variable x, a is the end-of-period financial wealth
and ! is the beginning-of-period financial wealth, pr is the rental payment, and r is the risk-free
interest rate. The expectation operator is over the income shock z 0 .
Active renter can purchase a house, and access mortgage contract to finance the purchase. An
active renter chooses a mortgage debt level d given a mortgage contract schedule ⌦(d; !, z, j) =
{m(d; !, z, j), rm (d; !, z, j)} where m(d; !, z, j) is the mortgage payment and rm (d; !, z, j) is the
mortgage interest rate, which will be functions of the current state of the household and the amount
of debt. That is, the price of the mortgage depends on the mortgage debt level, the current wealth
and income realization and the age. The value function of an active renter who buys a house is
given by
Vjro (!, z, q) =
max
c 0,d 0,a0 0
o
uh (c) + EVj+1
(! 0 , ⌦(d; !, z, j), z 0 )
(3)
subject to
c + ph (1 + 'b ) + m(d; !, z, q, j) = y(j, z) + ! + d,
! 0 = a0 (1 + r⇤ )
d  (1
⇠0,
#) ph ,
(4)
where
⌦(d; !, z, q, j) = {m(d; !, z, q, j), rm (d; !, z, q, j)} ,
and ph is housing price and ⇠ 0 is the maintenance shock that a homeowner faces. Equation (4) is
the down payment requirement which implies that the household has to put # fraction of house
price ph as down payment. We assume that households cannot make any savings at the time of the
purchase if she chooses to obtain mortgage from banks, i.e. d > 0 and a0 > 0 cannot happen at the
same time. This assumption saves us from the complexity of two dimensional portfolio problem.
The above formulation also allows the household to purchase a house without any mortgage. In
that case, we have a0
0 and d = 0.
8
2.3.2
Inactive Renters
Inactive renters are households who defaulted in previous periods. They cannot access housing or
mortgage market until they exit the status and become active renter with probability . The value
function of an inactive renter is given by
Vji (!, z) =
max
0
c 0,a
0
ur (c) +
subject to
⇥
r
EVj+1
(! 0 , z 0 , q 0 ) + (1
i
)EVj+1
(! 0 , z 0 )
⇤
(5)
c + a0 + pr = y(j, z) + !
! 0 = a0 (1 + r⇤ ) .
2.3.3
Homeowners
A homeowner who does not receive a moving shock has three options: stay a homeowner, switch to
being a renter, or default. The value function of an owner is given as V o = max V oo , V or , V oi ,
where V oo is the value of staying as homeowner, V or is the value of switching to renter, and V oi is
the value of defaulting. The problem of the stayer is as follows:
Vjoo (!, ⌦, z) = max
0
c 0,a
0
uh (c) + EVj+1 ! 0 , ⌦, z 0
(6)
subject to
c + a0 + m = y (j, z) + !
! 0 = a0 (1 + r⇤ )
⇠ 0 ph
where
o
m
)Vj+1
+ Vj+1
Vj+1 = (1
and m is the mortgage payment specified in the mortgage contract ⌦ = (m, rm ). The stayer will
face a moving shock in the next period with probability , in which case she has to choose between
m = max{V or , V oi }.
selling or defaulting, i.e. Vj+1
t+1 t+1
The second possible choice for a homeowner is to sell the house and become a renter. The seller
incurs some transaction cost. The cost is assumed to be fraction 'h of the selling price. Moreover, a
˜ in full to the lender. The recursive formulation
seller has to pay the outstanding mortgage debt, d,
of her problem is the following:
Vjor (!, ⌦, z) = max
0
c,a
0
⇥
r
ur (c) + EVj+1
!0, z0
9
⇤
(7)
subject to
c + a0 + pr + de = y (j, z) + ! + ph (1
!
0
0
's )
⇤
= a (1 + r )
where
rm
m = de
1 + rm
(1 + rm )j
Jr
.
The third possible choice for a homeowner is to default on the mortgage, if she has any. A
defaulter has no obligation to the lender. The lender seizes the house, sells it in the market, and
pays any positive amount from the sale of the house, net of the outstanding mortgage debt and
transaction costs, back to the defaulter. For the lender,
n the sale price ofo the house is assumed
˜ 0 from the lender. The
to be (1 'i ) ph . Therefore, the defaulter receives max (1 'i ) ph d,
defaulter starts the next period as an active renter with probability . With probability (1
),
she becomes an inactive renter. The problem of a defaulter becomes the following:
Vjoi (!, ⌦, z) = max
ur (c) + E
0
c,a
0
subject to
⇥
r
Vj+1
(! 0 , z 0 , q 0 ) + (1
n
c + a0 + pr = ! + y (j, z) + max ph (1
! 0 = a0 (1 + r⇤ ) .
'i )
i
) Vj+1
(! 0 , z 0 )
⇤
(8)
o
e0
d,
Notice that the problemnof a defaulter is di↵erent
than the problem of a seller in two ways. First, the
o
˜
defaulter receives max ph (1 'i ) d, 0 from the housing transaction whereas a seller receives
˜ We assume that the default cost is higher than the sale transaction cost, i.e.
d.
'i > 's , the defaulter receives less than the seller as long as ph (1 's ) d˜ 0 (i.e., the home
ph (1
's )
equity net of the transaction costs for the homeowner is positive). Second, a defaulter does not
have access to the mortgage in the next period with some probability. Such an exclusion lowers the
continuation utility for a defaulter. In sum, since defaulting is costly, a homeowner will choose to
sell the house instead of defaulting as long as ph (1 's ) d˜ 0, i.e., net home equity is positive.
Hence, negative equity is a necessary condition for default in the model. Therefore, in equilibrium,
a defaulter gets nothing from the lender.
10
2.4
Banks
We assume a competitive banking industry with a continuum of identical banks that are risk-averse
and maximize the discounted lifetime utility
1
X
t 1
log cB
t
t=0
where cB
t is the banker’s consumption. There is no entry to the banking sector. Banks collect
deposits Bt+1 from the international market at a risk-free interest rate rt+1 , lend to the firm Lkt+1
⇤ , and issue mortgages and purchase existing mortgages. We will first present the problem of
at rt+1
banks, which face an exogenous capital requirement constraints, where the total amount of loans a
bank can issue is limited to a fraction
1 of its net worth net of consumption. Such exogenous
constraints can arise due to regulations and is also a good benchmark to present since it is simpler.
Then we will present the version where banks face endogenous capital constraints because they can
walk away with some of their assets without paying back their creditors. In our numerical analysis,
we will solve the version of the model with endogenous capital requirement constraint but impose
the exogenous constraint as a regulation. Thus, the full model will be a hybrid of the exogenous
and endogenous constraint models since in reality banks face both constraints. Depending on how
strict is the capital requirement regulation and the state of the economy, the exogenous capital
requirement constraint might be stricter than the endogenous capital requirement constraint or
vice versa.
2.4.1
Exogenous capital requirement constraint
Formulation of Bank’s Problem:
Letting ✓ = (!, ⌦, z, q, j) define the type of a mortgage,
Nt be the bank’s net worth and `t+1 (✓) be the amount of investment in mortgage type ✓ (which
includes any newly issued as well as existing mortgages), the bank’s problem is given by
t (Nt )
=
log cB
t +
max
Bt+1 ,Lkt+1 ,cB
t ,{`t+1 (✓)}
L
t+1 (Nt+1 )
s.t.
k
cB
t + Lt+1 +
Lkt+1 +
Z
pt (✓) `t+1 (✓) = Nt + Bt+1
Z✓
✓
pt (✓) `t+1 (✓) 
Nt+1 =
+
l
vt+1
✓0
Nt
Z Z
✓ ✓0
Lkt+1
cB
t
l
vt+1
✓0 ⇧ ✓0 |✓ `t+1 (✓)
⇤
1 + rt+1
Bt+1 (1 + rt+1 )
= mt+1 ✓0 + pt+1 ✓0 .
11
An important point to notice here is that the bank does not face any uncertainty in its net worth
even though each mortgage is a risky investment. This is because we assume a continuum within
each household type, that will translate into continuum within each mortgage type ✓. Thus, even
if a bank invests in a particular type of mortgage ✓ by a tiny amount, its return is deterministic
since a known fraction of ✓-type households defaults and the remainder continues to pay their
mortgages with certainty. The continuum assumption grants us tractability while keeping the rich
heterogeneity in the household sector.
Letting ⇤t
0 be the Lagrange multiplier on the capital requirement constraint in period t,
the first order conditions for the bank’s problem are gives as
Bt+1 :
d
1
= B
dBt+1
ct
Lkt+1 :
d
=
dLkt+1
L (1
+ rt+1 )
cB
t+1
⇤t = 0
1
L
⇤
+ B 1 + rt+1
+(
cB
c
t
t+1
1) ⇤t = 0
`t+1 (✓):
d
d`t+1 (✓)
=
1
L
+ B
cB
c
t
t+1
R
✓0
l
vt+1
(✓0 ) ⇧ (✓0 |✓)
+(
pt (✓)
1) ⇤t = 0
The last two equations give the price of the mortgage after mortgage payment has been made:
1
pt (✓) =
⇤
1 + rt+1
Z
✓0
l
vt+1
✓0 ⇧ ✓0 |✓ for all ✓.
The condition above is essentially
the no-arbitrage condition. To see this, note that the gross return
R
on a mortgage of type ✓ is
✓0
l
vt+1
(✓ 0 )⇧(✓ 0 |✓)
pt (✓)
⇤ .
and has to be equal to the return on capital 1 + rt+1
As we will illustrate, the no-arbitrage condition greatly simplifies the problem of the bank. Since
the bank is indi↵erent between investing in any asset, we do not have to keep track of its asset
distribution while we are solving the bank’s problem.
Perfect competition among banks imply that at the time of the mortgage initiation, mortgage
value should be equal to the loan amount
1
d=m+
⇤
1 + rt+1
Z
✓0
where
m=d
l
vt+1
✓0 ⇧ ✓0 |✓
rm
(1 + rm )j
1 + rm
Jr
,
and ✓ = (!, ⌦, z, q, j). The expressions above gives mortgage contract terms ⌦(d; !, z, q, j) =
{m(d; !, z, q, j), rm (d; !, z, q, j)} for a household in state (!, z, q, j) who borrows d from the bank.
12
R
Rewriting the bank’s problem and the solution: Letting Lm
pt (✓) `t+1 (✓)and using
t+1 =
R l
1
0
0
pt (✓) = 1+r⇤
✓ 0 vt+1 (✓ ) ⇧ (✓ |✓), we can write the bank’s budget constraint and evolution of its
t+1
net worth as:
k
m
cB
t + Lt+1 + Lt+1 = Nt + Bt+1
⇤
Nt+1 = Lm
t+1 1 + rt+1
⇤
+ Lkt+1 1 + rt+1
Bt+1 (1 + rt+1 ) .
Letting Lt+1 = Lkt+1 + Lm
t+1 , the bank’s problem becomes
t (Nt )
=
max
Bt+1 ,Lt+1 ,cB
t
log cB
t +
L
t+1 (Nt+1 )
s.t.
cB
t + Lt+1 = Nt + Bt+1
Lt+1 
Nt
cB
t
⇤
Nt+1 = Lt+1 1 + rt+1
Bt+1 (1 + rt+1 ) .
Note that we can alternatively write the capital requirement constraint as (
1)Lt+1
Bt+1 ,
which will be useful for comparison when we analyze the endogenous version of this constraint in
the next section. The following proposition characterizes the bank’s optimal decisions.
Proposition 1. The solution to the bank’s problem can be represented by the following three cases:
⇤
1. If rt+1
> rt+1 ,
Lt+1 =
L
Bt+1 =
L(
Nt
1) Nt .
⇤
2. If rt+1
= rt+1 ,
Lt+1
Bt+1 =
L Nt
Lt+1 
L
Bt+1 
L(
Nt
⇤
3. If rt+1
< rt+1 ,
Lt+1 =
L Nt
Bt+1 = 0.
13
1) Nt .
2.4.2
Endogenous capital requirement constraint
Similar to Gertler and Kiyotaki (2010) and Gertler and Karadi (2011), we assume that banks can
walk away at the beginning of a period without paying back their creditors. In that case, the bank
can steal a fraction ' of its assets but is excluded from banking operations in the future and can
invest those assets at rate rtD . Knowing this, creditors lend to the bank so that the bank does not
walk away. Since the bank’s outside option depends on its assets in this case, we need to keep track
of assets and debt separately. Thus, the bank’s problem can be written as:
t (Lt , Bt )
=
max
Bt+1 ,Lt+1 ,cB
t
log cB
t +
L
t+1 (Lt+1 , Bt+1 )
s.t.
⇤
cB
t + Lt+1 = (1 + rt ) Lt
D
t+1
t+1 (Lt+1 , Bt+1 )
(1 + rt ) Bt + Bt+1
⇤
' 1 + rt+1
Lt+1
where
D
t+1 (s)
⇤
= max
log ' 1 + rt+1
Lt+1
0
s
s0 +
L
D
t+2
D
(1 + rt+2
)s0 .
We can show that the capital requirement constraint of the bank can be written as
⇤
1 + rt+1
(1
'⌫t+1 ) Lt+1
(1 + rt+1 ) Bt+1 ,
where ⌫t is defined recursively as follows:
⌫t =
D ) 1+r
⇤ )
(1 + rt+1
(1 '⌫t+1 )(1 + rt+1
t+1
⇤ )(1 + r
'(1 + rt+1
t+1 )
!
L
.
Note that the capital requirement constraint has the same format as the exogenous one but with endogenous coefficients. Given this constraint, we can solve the bank’s problem, which is summarized
in the following proposition.
⇤
Proposition 2. The decision rules when the no-default constraint is binding (if rt+1
> rt+1 ) are:
Lt+1 =
Bt+1 =
where Nt = (1 + rt⇤ )Lt
1 + rt+1
(1
1 + rt+1
(1 + rt+1 )
⇤ )
(1 '⌫t+1 )(1 + rt+1
⇤ )
'⌫t+1 )(1 + rt+1
⇤ )
(1 '⌫t+1 )(1 + rt+1
(1 + rt )Bt .
14
L Nt
L Nt ,
⇤
The decision rules when the no-default constraint is not binding (if rt+1
 rt+1 ) are:
Bt+1 =
and
8 h
<2 0,
L (1
1+rt+1
⇤ )
'⌫t+1 )(1+rt+1
⇤ ) Nt
(1 '⌫t+1 )(1+rt+1
:0
Lt+1 = Bt+1 +
L ((1
+ rt⇤ )Lt
i
⇤
if rt+1
= rt+1
⇤
if rt+1
< rt+1
(1 + rt )Bt ) .
Note that the main di↵erence between this solution and the solution from the exogenous constraint case is that
(1+rt+1 )
⇤ )
1+rt+1 (1 '⌫t+1 )(1+rt+1
in the equations above is replaced with exogenous .
Thus, in what follows we will write the solution of the bank with bt where bt can be either or
(1+rt+1 )
⇤ )
1+rt+1 (1 '⌫t+1 )(1+rt+1
2.5
depending on the model version we analyze.
Symmetric equilibrium
We focus on a symmetric equilibrium where each bank holds the market portfolio of mortgages.
Thus, we have a representative bank. The symmetric equilibrium assumption is without loss of
generality in the deterministic equilibrium since all mortgages give the same return. Even if different banks invested in di↵erent assets, the stationary equilibrium of that economy would be the
same as the symmetric stationary equilibrium. However, if di↵erent banks had di↵erent mortgages
portfolios, their balance sheets would be a↵ected at di↵erentially when the economy is hit by an
unexpected shock.
Letting
t (✓)
be the distribution of available mortgage’s after HH’s make their decisions at time
t, the credit market clearing conditions can be summarized by the following two conditions:
1. The representative bank holds the mortgage portfolio
`t+1 (✓) =
t (✓).
2. Total supply of loans by the bank and households are equal to total capital demand by the
firm and mortgage demand by HH’s
At+1 + Lt+1 = Kt+1 +
Z
pt (✓)
t (✓)
✓
⇤ .
which gives equilibrium rt+1
Remember that total owner occupied housing supply is fixed. Thus, total demand for owner
occupied housing should be equal to the supply, which determines house price ph (t).
15
2.5.1
Characterization of the Bank’s Problem in Stationary equilibrium
We can further characterize the bank’s problem under stationarity. Throughout the paper, we will
focus on stationary equilibria where the capital requirement constraint is binding. If it was not,
then bank balance sheets would not have any impact on the economy. However, we do not rule out
the case that there might be some periods where this constraint becomes slack in transitions. Using
the general formula capturing both the exogenous and endogenous capital requirement constraint,
we have the following decision rules when the constraint is binding:
b
⇣
Lt+1 =
L t Nt
Bt+1 =
L
⌘
1 Nt ,
bt
and the law of motion for net worth is given as
⇤
Nt+1 = Lt+1 1 + rt+1
Bt+1 (1 + rt+1 ) .
Then, we can obtain the next period’s net worth as
Nt+1 =
L
⇣
bt 1 + r⇤
t+1
Imposing steady state Nt+1 = Nt and bt = b gives
r⇤ = r +
Note that if
L (1
1
⇣
bt
L (1
b
⌘
⌘
1 (1 + rt+1 ) Nt .
+ r)
.
L
+ r) < 1 and b < 1 then r⇤ > r . Thus, capital requirement constraint will be
binding in the stationary equilibrium. To understand this point, assume that
L (1 + r)
< 1 but the
bank starts with a high net worth so that the capital requirement constraint is not binding. Since
⇤
the constraint is not binding rt+1
= r. Using the decision rule when the constraint is not binding,
i.e. Lt+1
Bt+1 =
L Nt ,
we can show that Nt+1 = (1 + r)
L Nt
< Nt . Thus, the bank eats up its
net worth until the capital requirement constraint starts to bind. Thus, the economy will converge
to a stationary equilibrium where it actually binds.
3
Calibration and Quantitative Results
This part is in progress. However, we attach slides at the end of the document that contain
preliminary results and conclusions from a previous version of the model with exogenous capital
requirement and without retirement.
16
References
[1] Aguiar, M. and G. Gopinath, 2006. “Defaultable Debt, Interest Rates, and the Current Account,” Journal of International Economics, 69 (1), 64-83.
[2] Arellano, C., 2008. “Default Risk and Income Fluctuations in Emerging Economies,” American
Economic Review, 98(3), 690-712.
[3] Arslan, Yavuz, Bulent Guler and Temel Taskin (2015): “Joint Dynamics of House Prices and
Foreclosures,” Journal of Money, Credit and Banking, 47(S1), pp. 133-169.
[4] Bocola Luigi (2015): “The Pass-Through of Sovereign Risk”, Journal of Political Economy,
124 (4), 879-926.
[5] Cerutti, Eugenio, Stijn Claessens, and Luc Laeven (2015): “The Use and E↵ectiveness of
Macroprudential Policies: New Evidence” IMF Working paper.
[6] Chatterjee, S. and B. Eyigungor (2012): “Maturity, Indebtedness, and Default Risk,” American
Economic Review, Vol. 102, No. 6, October.
[7] Chatterjee, S. and B. Eyigungor (2015): “Quantitative Analysis of the US Housing and Mortgage Markets and the Foreclosure Crisis,” Review of Economic Dynamics, 18(2), pp.165-184.
[8] Corbae, Dean and Erwan Quintin (2014): “Leverage and the Foreclosure Crisis.” Journal of
Political Economy, 123(1): 1–65.
[9] Dell’Ariccia, Giovanni, Deniz Igan, Luc Laeven, and Hui Tong, with Bas Bakker and Jerome
Vandenbussche (2012): “Policies for Macrofinancial Stability: How to Deal with Credit
Booms,” IMF Sta↵ Discussion Note 12/06.
[10] Garriga, Carlos, and Don Schlagenhauf (2009): “Home Equity, Foreclosures, and Bailouts.”
Working Paper, Federal Reserve Bank of St. Louis
[11] Gertler, Mark and Nobuhiro Kiyotaki (2010): “Financial Intermediation and Credit Policy
in Business Cycle Analysis.” In Handbook of Monetary Economics, vol. 3A, edited by B.
Friedman and M. Woodford. Elsevier.
[12] Gertler, Mark and Nobuhiro Kiyotaki (2015): “Banking, Liquidity, and Bank Runs in an
Infinite Horizon Economy”, American Economic Review, 105(7): 2011–2043.
[13] Guler, Bulent (2016): “Innovations in Information Technology and the Mortgage Market,”
Review of Economic Dynamics.
[14] Gunn, Christopher and Alok Johri (2015): “Financial news, banks and business cycles,” Forthcoming at Macroeconomic Dynamics.
17
[15] Hatchondo, Juan Carlos, Leonardo Martinez and Juan M. Sanchez (2015): “Mortgage defaults”. Journal of Monetary Economics, vol. 76: 173-190.
[16] Jeske, Karsten, Dirk Krueger, and Kurt Mitman (2013): “Housing, mortgage bailout guarantees and the macro economy.” Journal of Monetary Economics, 60(8): 917–935.
[17] Kaplan, Greg, Kurt Mitman, and Gianluca Violante (2016): “Consumption and House Prices
in the Great Recession: Model meets Evidence,” work-in-progress.
[18] Lim, Cheng H., Francesco Columba, Alejo Costa, Piyabha Kongsamut, Akira Otani, Mustafa
Saiyid, Torsten Wezel, Xiaoyong Wu, 2011, “Macroprudential Policy: What Instruments and
How Are They Used? Lessons from Country Experiences”, IMF Working Paper 11/238.
[19] Livshits, Igor, James MacGee and Michele Tertilt (2007): “Consumer Bankruptcy: A Fresh
Start,” American Economic Review, 97/1, 402-418.
[20] Livshits, Igor, James MacGee and Michele Tertilt (2010): “Accounting for the Rise in Consumer Bankruptcies,” American Economic Journal: Macroeconomics, 2(2): 165-193.
[21] Mendoza, E. and V. Quadrini (2010): “Financial Globalization, Financial Crises and Contagion,” Journal of Monetary Economics, 57(1), 24-39.
[22] Mitman, Kurt (2016): “Macroeconomic E↵ects of Bankruptcy and Foreclosure Policies,” American Economic Review, Vol. 106(8): 2219-2255
[23] Navarro, Gaston (2016): “Financial Crises and Endogenous Volatility”, working paper.
18
D ETERIORATION OF H OUSEHOLD AND B ANK B ALANCE
S HEETS AND M ORTGAGE C RISIS1
Yavuz Arslan
Bulent Guler
Burhan Kuruscu
BIS
Indiana
Toronto
December 9, 2016
1 The views expressed here are ours, and do not reflect those of the BIS.
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
M OTIVATION : A N I NTERTWINED C IRCLE OF E VENTS
During the 2007-2009 crisis:
œ
House prices declined, which triggered foreclosures by households.
œ
Financial institutions experienced losses from residential
mortgages (or related assets).
œ
Depletion of capital forced financial institutions to contract
lending, which raised the cost of credit they offered.
œ
Higher cost of credit led to contraction in economic activity
œ
Demand for housing declined, which led to further declines in
house prices.
lowering income of households
raising the cost of buying a house
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
W HAT WE AIM TO DO
1
Provide a rich and realistic framework with explicit treatment of
household and bank balance sheets, housing and realistic mortgage
markets,
2
Analyze feedback mechanisms between house prices, foreclosures,
and household and bank balance sheet in response to unexpected
shocks.
3
Use the framework to compare effectiveness of government policies
during (and before) financial crisis.
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
W HAT WE AIM TO DO
œ
Our paper combines two literatures:
1
Mortgage:
œ
2
Hatchondo et al (2014), Chatterjee and Eyigungor (2015), Guler (2015),
Arslan, Guler and Taskin (2015)
Bank Balance Sheet Effects
œ
Mendoza and Quadrini (2009), Gertler and Karadi (2011), Gertler,
Kiyotaki and Queralto (2011), Gertler and Kiyotaki (2015)
which allows us to study the interaction between household
balance sheets and bank balance sheets in a financial crisis and in
response to government policy.
œ
To our knowledge, this is the first paper that incorporates a rich
household sector and heterogenous long-term mortgages into a
model of bank balance sheets.
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
O UTLINE OF THE TALK
œ
Model
HH sector
Banks
œ
Exogenous capital requirement
œ
Endogenous capital requirement
œ
Numerical Experiments
œ
Preliminary conclusions
With exogenous capital requirement
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
E NVIRONMENT: A GGREGATE T ECHNOLOGY
Representative firm’s problem is given by
max At KtÆ Nt1°Æ ° wt Nt ° (rt§ + ±)Kt
K t , Nt
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
E NVIRONMENT: H OUSEHOLDS
œ
Life-cycle model with deterministic time horizon.
œ
Fixed house supply and house size.
œ
Households either rent or own a house:
uo (c) ∏ ur (c)
where o : owner and r : renter.
œ
Utility from both consumption good and housing:
E0 [
JR
X
Øj °1 uk (cj ) + ØJr W (!JR )]
j =1
where JR is the retirement age and !JR is the wealth at retirement.
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
E NVIRONMENT: H OUSEHOLDS
œ
Household’s labor income at age j is
y (j , zj ) = w £ exp(f (j) + zj ) for j ∑ JR
where
zj = Ω zj °1 + "j , "j ª i.i.d. N(0, æ2" )
and w is the aggregate wage rate per unit of efficiency labor.
œ
HH is also subject to an idiosyncratic investment opportunity that follows
a two-state Markov process º(q 0 |q):
q = 0 ! HH saves through bank deposits
q = 1 ! HH can also invest in other assets (capital and mortgages)
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
E NVIRONMENT: H OUSEHOLDS
œ
Purchase of a house can be done through a mortgage
œ
Mortgage holders can default on the mortgage
œ
Terms of mortgage contracts are endogenous (down payment and
mortgage interest rate)
œ
Only fixed-rate mortgages (FRM) and maturity is determined by the
age of the individual
œ
No unsecured borrowing
œ
HH’s are subject to idiosyncratic moving shocks
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
VALUE F UNCTIONS
œ
Four possible housing status: inactive renter (i ), active renter (r ),
owner (o ) and mover (m)
Inactive Renter: Renter with default flag (cannot purchase a house):
Vi
Active Renter: Can stay as a renter or purchase a house:
©
™
V r = max V rr , V ro
Owner: Can stay in the house, sell the house or default on the
mortgage (if any):
n
o
V o = max V oo , V or , V oi
Mover: Can sell the house or default on the mortgage (if any):
n
o
V m = max V or , V oi
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
D EFAULTER-I NACTIVE R ENTER
œ
Inactive renter:
Vji (!, z , q) = max0
c ∏0,a ∏0
n
h
ur (c) + Ø ±EVjr+1 (!0 , z 0 , q 0 ) + (1 ° ±)EVji+1 (!0 , z 0 , q 0 )
subject to
c + a 0 + pr
0
!
= y (j , z) + !
= a0 (1 + r (q)) .
io
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
A CTIVE R ENTERS P ROBLEM : V r = max {V rr , V ro }
œ
Active renter can either stay as renter
n
o
Vjrr (!, z , q) = max0
ur (c) + ØEVjr+1 (!0 , z 0 , q 0 )
c ∏0,a ∏0
subject to
c + a 0 + pr
0
!
= y (j , z) + !
= a0 (1 + r (q)) .
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
A CTIVE R ENTERS P ROBLEM : V r = max {V rr , V ro }
œ
OR can buy a house
Vjro (!, z , q) = max
c ∏0,d ∏0
n
uh (c) + ØEVjo+1 (!0 , ≠(d; !, z , q , j), z 0 , q 0 )
subject to
c + ph (1 + 'b ) + m(d; !, z , q , j) = y (j , z) + ! + d ,
!0
d
= °ª0 ph ,
∑ ph ,
where
œ
©
™
≠(d; !, z , q , j) = m(d; !, z , q , j), rm (d; !, z , q , j) .
We assume that a0 = 0 at the time of the house purchase.
o
Introduction Model
Crisis Examples
H OMEOWNER :
œ
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
©
™
V o = max V oo , V or , V oi ;V m = max{V or , V oi }
Homeowner who decides to stay as homeowner
Vjoo (!, ≠, z , q) = max0
c ∏0,a ∏0
subject to
©
uh (c) + ØEVj +1 (!0 , ≠, z 0 , q 0 )
c + a0 + m = y (j , z) + !
!0
= a0 (1 + r (q)) ° ª0 ph
where
Vj +1 (:) = (1 ° √)Vjo+1 (:) + √Vjm
+1 (:) .
™
Introduction Model
Crisis Examples
H OMEOWNER :
œ
©
™
V o = max V oo , V or , V oi ;V m = max{V or , V oi }
Homeowner who decides to become renter
nh
io
r
0
0
(c)
(a
(1
)
)
Vjor (!, ≠, z , q) = max
u
+
Ø
EV
+
r
,
z
r
j
+
1
0
c ,a ∏0
subject to
c + a0 + pr + de = y (j , z) + ! + ph (1 ° 's )
!0
where
œ
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
m = de
= a0 (1 + r (q))
rm
1 + rm ° (1 + rm )j °Jr
.
Homeowner who decides to default
n
h
Vjoi (!, ≠, z , q) = max
ur (c) + ØE ±Vjr+1 (!0 , z 0 , q 0 ) + (1 ° ±)Vji+1 (!0 , z 0 , q 0 )
0
subject to
c ,a ∏0
c + a 0 + pr
!0
©
™
= ! + y (j , z) + max ph (1 ° 'i ) ° de, 0
= a0 (1 + r (q)) .
io
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
B ANKS
œ
Competitive and identical bankers.
œ
Bankers maximize
≥ ¥
1
X
Øt °1 log ctB
t =0
where ctB
œ
is the banker’s consumption.
Bankers can
accept deposits at rate rt (exogenous) and
lend to the firm at rt§ (endogenous)
issue mortgages and purchase existing mortgages.
œ
Are subject to capital requirement constraint: the amount of assets
they can purchase cannot exceed a multiple of their net worth net of
consumption
exogenous
endogenous
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
B ANK (E XOGENOUS C APITAL R EQUIREMENT )
œ
Letting µ = (!, ≠, z , q , j), the bank’s problem:
n ≥ ¥
o
™t (Nt ) =
max
log ctB + ØL ™t +1 (Nt +1 )
Bt +1 ,Lkt+1 ,ctB ,{`t +1 (µ )}
s.t.
ctB + Lkt+1 +
Lkt+1 +
Z
Zµ
µ
pt (µ ) `t +1 (µ ) = Nt + Bt +1
≥
pt (µ ) `t +1 (µ ) ∑ ∏ Nt ° ctB
Nt +1 =
ZZ
µ
µ0
¥
vtl +1 (µ 0 ) ¶ (µ 0 |µ ) `t +1 (µ )
+ Lkt+1 (1 + rt§+1 ) ° Bt +1 (1 + rt +1 )
vtl +1 (µ 0 ) = mt +1 (µ 0 ) + pt +1 (µ 0 )
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
FOC’ S OF THE B ANK
œ
œ
Let §t ∏ 0 be the lagrange multiplier on the capital requirement
constraint in period t , then
Bt +1 :
d
1 ØL (1 + rt +1 )
= B°
° ∏§t = 0
B
dBt +1
œ Lk :
t +1
œ
`t +1 (µ ):
ct +1
d
1
ØL
= ° B + B (1 + rt§+1 ) + (∏ ° 1) §t = 0
k
dLt +1
ct
ct +1
d
d `t +1 (µ )
œ
ct
1
ØL
= ° B+ B
ct
ct +1
R
l
) ¶ (µ 0 |µ )
+ (∏ ° 1) §t = 0
pt ( µ )
µ 0 vt +1 (µ
0
The last two equations give the price of the mortgage:
Z
1
pt (µ ) =
v l (µ 0 ) ¶ (µ 0 |µ )
1 + rt§+1 µ0 t +1
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
M ORTGAGE P RICING AND I NITIATION
œ
Perfect competition among banks imply that at the time of the
mortgage initiation, mortgage value should be equal to the loan
amount
Z
d =m+
where
1
1 + rt§+1
m=d
œ
and µ = (!, ≠, z , q , j).
µ0
vtl +1 (µ 0 ) ¶ (µ 0 |µ )
rm
1 + r m ° (1 + r m )j °Jr
,
The expressions above gives mortgage contract terms
©
™
≠(d; !, z , q , j) = m(d; !, z , q , j), rm (d; !, z , q , j) for an individual in
state (!, z , q , j) who borrows d from the bank.
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
R EWRITING THE B ANK ’ S P ROBLEM
Letting
Lm
t +1 =
and using
pt (µ ) =
Z
pt (µ ) `t +1 (µ )
1
1 + rt§+1
Z
µ0
vtl +1 (µ 0 ) ¶ (µ 0 |µ )
we can write the bank’s budget constraint and evolution of its net worth
as:
ctB + Lkt+1 + Lm
t +1 = Nt + Bt +1
§
Nt +1 = Lm
t +1 (1 + rt +1 )
+ Lkt+1 (1 + rt§+1 ) ° Bt +1 (1 + rt +1 )
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
S IMPLIFIED B ANK ’ S P ROBLEM
Letting Lt +1 = Lkt+1 + Lm
t +1 , we can write the bank’s problem as
™t (Nt ) =
max
Bt +1 ,Lt +1 ,ctB
n
≥
¥
log ctB + ØL ™t +1 (Nt +1 )
o
s.t.
ctB + Lt +1 = Nt + Bt +1
≥
Lt +1 ∑ ∏ Nt ° ctB
¥
Nt +1 = Lt +1 (1 + rt§+1 ) ° Bt +1 (1 + rt +1 )
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
S OLUTION TO THE B ANK ’ S P ROBLEM
œ
If rt§+1 > rt +1 ,
Lt +1 = ØL ∏Nt
Bt +1 = ØL (∏ ° 1)Nt
œ
If rt§+1 = rt +1 ,
Lt +1 ° Bt +1 = ØL Nt
Lt +1 ∑ ØL ∏Nt
Bt +1 ∑ ØL (∏ ° 1)Nt
œ
If rt§+1 < rt +1 ,
Lt +1 = ØL Nt
Bt +1 = 0.
E NDOGENOUS C ASE
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
S YMMETRIC E QUILIBRIUM
œ
œ
œ
Focus on a symmetric equilibrium where each bank holds the
mortgage portfolio.
Symmetric equilibrium assumption is without loss of generality in a
stationary equilibrium since all mortgages give the same return.
Letting °t (µ ) be the distribution of available mortgage’s after HH’s
make their decisions at time t , the credit market clearing conditions
are:
The bank holds the mortgage portfolio
`t +1 (µ )
=
°t (µ )
Total supply of loans by the bank and investors are equal to total
capital demand by the firm and mortgage demand by HH’s
Z
Aqt +=11 + Lt +1 = Kt +1 + pt (µ ) °t (µ )
µ
which gives equilibrium rt§+1 .
House market clearing condition determines house price ph .
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
S TATIONARY E QUILIBRIUM WITH C APITAL R EQUIREMENT C ONSTRAINT
B INDING
œ
Using
Lt +1 = ØL ∏Nt
Bt +1 = ØL (∏ ° 1)Nt
and
Nt +1 = Lt +1 (1 + rt§+1 ) ° Bt +1 (1 + rt +1 )
we can obtain
Nt +1 = ØL (∏ (1 + rt§+1 ) ° (∏ ° 1)(1 + rt +1 ))Nt .
œ
In steady state Nt +1 = Nt , which gives
r§ = r +
œ
1 ° ØL (1 + r )
∏ØL
Note that r § > r if ØL (1 + r ) < 1 and ∏ < 1.
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
C RISIS E XAMPLES
œ
Illustrate mechanisms through some examples.
œ
No hard conclusions yet.
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
F INANCIAL C RISIS
œ
œ
Calibrate the stationary equilibrium of the economy with r § > r (i.e.
capital requirement constraint is binding)
Hit the economy with a
Shock to the bank balance sheet: Temporary rise in interest rate rt +1
Shock to the HH balance sheet: Temporary rise in maintance shock
Permanent increase in ∏ followed by a permanent decline in ∏ (in the
endogenous case, shock to the amount of assets that the banker can
steal)
Temporary negative productivity shock
Unfullfilled house price expectation
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
U NANTICAPATED I NTEREST R ATE S HOCK
Interest Rate
0.06
rt+1
0.05
0.04
0.03
0.02
0.01
0
5
10
t
15
20
25
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
S TATIONARY E QUILIBRIUM WITH C APITAL R EQUIREMENT C ONSTRAINT
B INDING
Nt
=
ZZ
µ
µ0
(mt (µ 0 ) + pt (µ 0 )) ¶ (µ 0 |µ ) °t °1 (µ )
+ Lkt (1 + rt§ ) ° Bt (1 + rt )
Lt +1 = ØL ∏Nt
Bt +1 = ØL (∏ ° 1)Nt
— Note that rt +1 does not affect Nt directly. However, with higher rt +1 ,
°
¢
Nt +1 = Lt +1 1 + rt§+1 ° Bt +1 (1 + rt +1 ) # and Lt +2 = ØL ∏Nt +1 # .
— Due to capital market clearing condition in period t + 2, rt§+2 " .
— With higher rt§+2 reduces pt # =) Nt # =) Lt +1 # =) rt§+1 ".
— With higher future interest rates, housing demand in period t #, house
prices #, foreclosures ", mt #, Nt #.
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
I NTEREST R ATE S HOCK
Value of Mortgage Portfolio
Bank Net Worth (Nt )
1.05
0.19
1.04
0.18
1.03
0.17
1.02
0.16
1.01
0.15
1
0.14
0.99
0.13
0
5
10
t
15
20
25
0.12
0
5
10
t
15
20
25
Interest Rates
Bank Assets (Lt+1 )
1.5
0.06
rt+1
$
rt+1
1.45
1.4
0.05
1.35
0.04
1.3
1.25
0.03
1.2
1.15
0.02
1.1
0
5
10
t
15
20
25
0.01
0
5
10
t
15
20
25
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
I NTEREST R ATE SHOCK
Consumption (ct )
Output (Yt)
1.39
0.94
1.38
0.935
1.37
0.93
1.36
0.925
1.35
0.92
1.34
0.915
1.33
0.91
0
5
10
t
15
20
25
0
5
Foreclosure rate
10
t
15
20
25
15
20
25
House prices
3.4
0.03
3.35
0.025
3.3
0.02
3.25
0.015
3.2
0.01
3.15
0
5
10
t
15
20
25
0
5
10
t
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
I NTEREST R ATE S HOCK : W ELFARE (A LL AND N EW B ORN )
Welfare (All)
Welfare (New Born)
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
-1
-1
-1.2
-1.2
-1.4
-1.6
-1.4
0
5
10
t
15
20
25
0
5
10
t
15
20
25
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
C OMPARE THE RESPONSES OF THE ECONOMY TO TWO SHOCKS
œ
Shock to the bank balance sheet: Temporary rise in interest rate rt +1
œ
Shock to the HH balance sheet: Temporary rise in maintanence
shock (5 period increase in shock probability form 3% to 6%)
œ
A normalization (needs more thinking!): Choose the level of shocks
so that on impact the effect on rt§ and Yt are the same.
Some insights from this experiment:
œ
Decline in bank net worth and loan supply is key in interest rate
shock
Foreclosures and decline in housing demand are key in maintenance
shock
While interest rate and output recovery are lower with the interest
rate shock, the recovery of house prices is lower with maintanence
shock,
œ
which has implications for the distributional effects of crisis.
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
B ANK N ET W ORTH (%¢) AND ¢rt§+1
Interest rate shock
Maintanence shock
1
1
0.95
0.95
0.9
Nt
Nt
0.9
0.85
0.85
0.8
0.8
0.75
0.75
0.7
0.7
0
5
10
15
20
25
0
0.02
0.02
0.015
0.015
0.01
0.01
0.005
0
-0.005
-0.005
0
5
10
15
15
20
25
20
25
0.005
0
-0.01
10
Maintanence shock
rt$
rt$
Interest rate shock
5
20
25
-0.01
0
5
10
15
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
O UTPUT AND CONSUMPTION (%¢)
Interest rate shock
Maintanence shock
1.01
1
1
0.99
0.99
Yt
Yt
1.01
0.98
0.98
0.97
0.97
0.96
0.96
0
5
10
15
20
25
0
Interest rate shock
10
15
20
25
20
25
Maintanence shock
1.015
1.015
1.01
1.01
1.005
1.005
1
1
0.995
0.995
0.99
Ct
0.99
Ct
5
0.985
0.985
0.98
0.98
0.975
0.975
0.97
0.97
0.965
0.965
0
5
10
15
20
25
0
5
10
15
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
F ORECLOSURE R ATE (¢) AND H OUSE P RICE (%¢)
Maintanence shock
0.03
0.025
0.025
Foreclosure Rate
Foreclosure Rate
Interest rate shock
0.03
0.02
0.015
0.01
0.005
0.02
0.015
0.01
0.005
0
0
-0.005
-0.005
-0.01
0
5
10
15
20
-0.01
25
0
1
0.98
0.98
0.96
0.94
0.92
0.9
15
25
20
25
0.92
0.88
10
20
0.94
0.88
5
15
0.96
0.9
0
10
Maintanence shock
1
house price
house price
Interest rate shock
5
20
25
0
5
10
15
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
W ELFARE (A LL AND N EW B ORN ) (¢)
Maintanence shock
0
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
-0.8
Welfare
Welfare
Interest rate shock
0
-0.2
-1
-1.2
-1
-1.2
-1.4
-1.4
-1.6
-1.6
-1.8
-1.8
-2
-2
0
5
10
15
20
25
0
Interest rate shock
10
15
20
25
20
25
Maintanence shock
1
Welfare New Born
1
Welfare New Born
5
0.5
0
-0.5
-1
-1.5
0.5
0
-0.5
-1
-1.5
0
5
10
15
20
25
0
5
10
15
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
Analyze Bank and HH Balance Sheet
Interaction
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
B ANK AND HH B ALANCE S HEET I NTERACTION
1
To eliminate HH balance sheet deterioration, keep wages (+ house
prices) constant at the initial steady state
*Shock to the bank balance sheet
2
To eliminate bank balance sheet effect, keep rt§ at the before crisis
level:
Shock to the bank balance sheet: No crisis since crisis is created by
the changes in rt§ .
*Shock to the HH balance sheet.
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
I NTEREST R ATE S HOCK : WAGES (+H OUSE P RICE F IXED )
rt$
Nt
1
Benchmark
Fix wt
Fix wt and ph
0.02
0.95
0.015
0.9
0.01
0.85
0.005
0.8
Benchmark
Fix wt
Fix wt and ph
0.75
0.7
0
5
10
15
20
25
0
-0.005
-0.01
0
5
Foreclosure Rate
Benchmark
Fix wt
Fix wt and ph
0.015
0.01
10
15
20
25
house price
1
0.98
0.005
0.96
0
0.94
-0.005
Benchmark
Fix wt
Fix wt and ph
0.92
-0.01
0
5
10
15
20
25
0
5
10
15
20
25
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
I NTEREST R ATE S HOCK : WAGES (+H OUSE P RICE F IXED )
Ct
Yt
1.01
Benchmark
Fix wt
Fix wt and ph
1.03
1
1.02
0.99
1.01
0.98
1
0.97
Benchmark
Fix wt
Fix wt and ph
0.96
0
5
10
15
20
0.99
0.98
25
0
Welfare
5
10
15
20
25
Welfare New Born
0
0.5
-0.2
-0.4
0
-0.6
-0.8
-0.5
-1
-1.2
Benchmark
Fix wt
Fix wt and ph
-1
0
5
10
15
20
25
Benchmark
Fix wt
Fix wt and ph
-1.4
-1.6
0
5
10
15
20
25
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
B ANK AND HH B ALANCE S HEET I NTERACTION
1
To eliminate HH balance sheet deterioration, keep wages (+ house
prices) constant at the initial steady state
*Shock to the bank balance sheet
2
To eliminate bank balance sheet effect, keep rt§ at the before crisis
level:
Shock to the bank balance sheet: No crisis since crisis is created by
the changes in rt§ .
*Shock to the HH balance sheet.
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
M AINTENANCE S HOCK : rt§ FIXED
rt$
Benchmark
r$ Fixed
0.07
0.065
0.06
0.055
0.05
0.045
0.04
0.035
0.03
0
5
10
15
20
25
Yt
1.01
Ct
1.01
Benchmark
r$ Fixed
1
Benchmark
r$ Fixed
1
0.99
0.99
0.98
0.98
0.97
0.97
0.96
0
5
10
15
20
25
0
5
10
15
20
25
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
M AINTENANCE S HOCK : rt§ FIXED
Foreclosure Rate
house price
Benchmark
r$ Fixed
0.03
Benchmark
r$ Fixed
1
0.025
0.98
0.02
0.96
0.015
0.94
0.01
0.005
0.92
0
0.9
-0.005
-0.01
0
5
10
15
20
25
0.88
0
Welfare
5
10
15
20
25
Welfare New Born
1.5
Benchmark
r$ Fixed
0
-0.5
Benchmark
r$ Fixed
1
-1
0.5
-1.5
0
-2
0
5
10
15
20
25
0
5
10
15
20
25
Policy Analysis
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
P OLICY A NALYSIS
Analyze effectiveness of different government policies.
œ
œ
œ
Stricter capital requirement: ∏ = 5 (instead of 10).
Higher down payment requirement of 20% (instead of 0%).
Asset purchases: government buys assets (mortgages) in period t ,
Z
q =1
LG
+
A
+
L
=
K
+
pt (µ ) °t (µ ).
t
+
1
t
+
1
t +1
t +1
µ
œ
œ
œ
Equity injection: government injects ¢Nt in period t and buys a
share ¢Nt /(Nt + ¢Nt ) fraction of the bank, which entitles the
government a payment of csB £ ¢Nt /(Nt + ¢Nt ) for s ∏ t .
Bailout HH’s
Social insurance.
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
B ANK BALANCE SHEET AND r §
Interest rate shock
Maintanence shock
1
1
0.95
0.95
0.9
Nt
Nt
0.9
0.85
0.8
0.85
0.8
Benchmark
Low 6
min down payment =20%
0.75
0.75
0.7
Benchmark
Low 6
min down payment =20%
0.7
0
5
10
15
20
25
0
Interest rate shock
0.01
rt$
rt$
0.015
0.01
0.005
0
-0.005
-0.005
5
10
15
20
25
0.005
0
0
15
20
Benchmark
Low 6
min down payment =20%
0.02
0.015
-0.01
10
Maintanence shock
Benchmark
Low 6
min down payment =20%
0.02
5
25
-0.01
0
5
10
15
20
25
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
O UTPUT AND C ONSUMPTION
Interest rate shock
Maintanence shock
1.01
1
1
0.99
0.99
Yt
Yt
1.01
0.98
0.97
0.98
0.97
0.96
0.96
0
5
10
Benchmark
Low 6
min down payment =20%
15
20
25
Benchmark
Low 6
min down payment =20%
0
Interest rate shock
10
15
20
25
Maintanence shock
1.015
1.015
1.01
1.01
1.005
1.005
1
1
0.995
0.995
0.99
Ct
0.99
Ct
5
0.985
0.985
0.98
0.98
0.975
0.975
0.97
Benchmark
Low 6
min down payment =20%
0.965
0
5
10
15
20
25
0.97
Benchmark
Low 6
min down payment =20%
0.965
0
5
10
15
20
25
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
F ORECLOSURE R ATE AND H OUSE P RICES
Interest rate shock
Maintanence shock
Benchmark
Low 6
min down payment =20%
0.03
0.025
Foreclosure Rate
0.025
Foreclosure Rate
Benchmark
Low 6
min down payment =20%
0.03
0.02
0.015
0.01
0.005
0.02
0.015
0.01
0.005
0
0
-0.005
-0.005
-0.01
-0.01
0
5
10
15
20
25
0
10
15
1
0.98
0.98
0.96
0.94
0.92
0.9
0
5
10
15
25
0.96
0.94
0.92
0.9
Benchmark
Low 6
min down payment =20%
0.88
20
Maintanence shock
1
house price
house price
Interest rate shock
5
20
25
Benchmark
Low 6
min down payment =20%
0.88
0
5
10
15
20
25
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
W ELFARE (A LL & N EW B ORN )
Maintanence shock
0
-0.5
-0.5
Welfare
Welfare
Interest rate shock
0
-1
-1.5
-1
-1.5
Benchmark
Low 6
min down payment =20%
Benchmark
Low 6
min down payment =20%
-2
0
5
10
15
20
-2
25
0
Interest rate shock
15
0.5
0
-0.5
-1
-1.5
5
10
15
25
0.5
0
-0.5
-1
Benchmark
Low 6
min down payment =20%
-1.5
0
20
1
Welfare New Born
Welfare New Born
10
Maintanence shock
Benchmark
Low 6
min down payment =20%
1
5
20
25
0
5
10
15
20
25
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
T ENTATIVE P OLICY C ONCLUSIONS
Stricter capital requirement:
œ
œ
Reduces the impact of the shock but also leads to slower recovery.
Results in SS welfare losses, typically larger than the gains during
crisis.
S.S. welfare (all) is lower by 0.8% lower but the welfare loss during
crisis is reduced by 1.2% (0.5%) under interest rate shock
(maintenance shock).
S.S. welfare (new born) is lower by 6% lower but the welfare loss
during crisis is reduced by 0.75% under interest rate shock. Welfare
gain is lower by about 0.9% under maintenance shock.
Higher down payment requirement:
œ Results in SS welfare gains for all! but SS losses larger than the crisis
gains during crisis.
S.S. welfare (all) is higher and welfare loss from crisis is reduced by
0.7% (0.5%) under interest rate shock (maintenance shock).
S.S. welfare (new born) is lower 0.8% the welfare loss during crisis is
reduced by 0.4% under interest rate shock. Welfare gain is lower by
about 0.7% under maintenance shock.
Tentative Conclusions
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
T ENTATIVE C ONCLUSIONS AND C URRENT W ORK
œ
We have constructed a rich framework to analyze HH and bank
balance sheets jointly.
œ
The model generated realistic comovements between macro and
financial variables.
œ
There seems to be significant feedback between household and
bank balance sheet deterioration.
œ
Effectiveness of macro prudential policies depends on the type of
the shock.
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
T ENTATIVE C ONCLUSIONS AND C URRENT W ORK
œ
Stricter capital requirement:
œ
Higher down payment requirement:
Reduces the impact of the shock on output and consumption but
also leads to slower recovery.
Results in SS welfare losses, typically larger than the gains from
policy during crisis.
More effective in mitigating output decline when the shock hits HH
balance sheet.
Implies SS and transitional welfare gain for all.
Implies SS and transitional welfare losses for new born.
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
C URRENT WORK
œ
Analyze the model with endogenous capital requirement.
œ
Quantify the importance of long-term mortgages in magnifying
shocks.
œ
Explore heterogeneity dimension.
œ
Evaluate other policies:
Asset purchases
Equity injection
Household bailout
Social insurance.
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
PARAMETERS
Parameter
æ
Æ
Ω
æ"
'b
'h
'i
r
∏
ph
±
Explanation
risk aversion
capital share
persistence of income
std of innovation to AR(1)
buying cost for a household
selling cost for a household
selling cost for a defaulter
risk-free interest rate
capital requirement ratio
house price to income ratio
prob. of being an active renter
Value
2
0.3
0.97
0.13
3%
7%
27%
2%
10
3.5
0.14
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
PARAMETERS
Parameter
Ø
∞h /∞r
≤
Prob(≤)
√
≥
¥
ØL
Explanation
discount factor
utility advantage of ownership
house depreciation cost
probability of depreciation
moving probability
utility parameter-terminal period
fraction of capital holders
bank discount factor
Value
0.91
1.15
50%
0.03
5.7%
19.4
0.95
0.82
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
M OMENTS
Statistic
Wealth-income ratio
Homeownership rate
Aggregate house depreciation
Foreclosure rate
Moving rate-owners
Wealth-income-retirees
Ratio of C-I loans to mortgage loans
Risk-free mortgage premium
Data
4
66%
1.5%
1.5%
8%
6.5
0.5
2%
Model
4
66%
1.5%
1.5%
8%
6.5
0.5
2%
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
B ANK ’ S PROBLEM (E NDOGENOUS C APITAL R EQUIREMENT )
™t (Lt , Bt ) =
s.t.
max
Bt +1 ,Lt +1 ,ctB
n
≥
¥
log ctB + ØL ™t +1 (Lt +1 , Bt +1 )
o
ctB + Lt +1 = (1 + rt§ )Lt ° (1 + rt )Bt + Bt +1
§
™t +1 (Lt +1 , Bt +1 ) ∏ ™D
t +1 (' (1 + rt +1 )Lt +1 )
where
≥
¥
§
0
D
D
0
(s)
(
(1
)L
)
™D
=
max
log
'
+
r
°
s
+
Ø
™
(1
+
r
)s
.
t
+
1
L t +2
t +1
t +1
t +2
0
s
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
S OLUTION TO THE B ANK ’ S P ROBLEM
œ
The no-default constraint in period t can be written as
(1 + rt§+1 )(1 ° '∫t +1 )Lt +1 ∏ (1 + rt +1 )Bt +1 ,
where
√
°
(1 + rtD+1 ) 1 + rt +1 ° (1 ° '∫t +1 )(1 + rt§+1 )
∫t =
'(1 + rt§+1 )(1 + rt +1 )
œ
¢ !Ø L
.
The bank’s solution satisfies the following expression regardless of
whether the no-default constraint binds:
Lt +1 ° Bt +1 = ØL ((1 + rt§ )Lt ° (1 + rt )Bt ) .
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
S OLUTION TO THE B ANK ’ S P ROBLEM
œ
The decision rules when the no-default constraint is binding (if
rt§+1 > rt +1 ):
Lt +1 =
Bt +1 =
œ
(1 + rt +1 )
Ø L Nt
1 + rt +1 ° (1 ° '∫t +1 )(1 + rt§+1 )
(1 ° '∫t +1 )(1 + rt§+1 )
Ø L Nt ,
1 + rt +1 ° (1 ° '∫t +1 )(1 + rt§+1 )
where Nt = (1 + rt§ )Lt ° (1 + rt )Bt .
The decision rules when the no-default constraint is not binding (if
rt§+1 ∑ rt +1 ):
8 h
i
§
<2 0, ØL (1°'∫t +1 )(1+rt +1 )§ N
if rt§+1 = rt +1
1+rt +1 °(1°'∫t +1 )(1+rt +1 ) t
Bt +1 =
:0
if r § < r
and
t +1
Lt +1 = Bt +1 + ØL ((1 + rt§ )Lt ° (1 + rt )Bt ) .
E XOGENOUS C ASE
t +1
Introduction Model
Crisis Examples
A Decomposition Exercise Policy Analysis Tentative Conclusions Miscella
W ELFARE (A LL & N EW B ORN )
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