WORKING PAPER SERIES

Back to the Real Economy: The Effects of Risk 

Mispricing on the Term Premium and Bank Lending

Kristina Bluwstein

Bank of England

Julieta Yung

Federal Deposit Insurance Corporation

February 2022

This version: August 2023

FDIC CFR WP 2022-03

fdic.gov/cfr

The Center for Financial Research (CFR) Working Paper Series allows CFR staff and their coauthors to circulate 

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revision.
Back to the Real Economy: The Eects of Risk Mispricing on the

Term Premium and Bank Lending*

Kristina Bluwstein Julieta Yung

Abstract

Bond markets can plummet or rally on the back of sentiment-driven reactions which are unrelated

to fundamentals. Therefore, changes in bond prices can not only be interpreted as reecting risk but

also mispricing of long-term assets. These perceived risks can often feed back into the economy by

aecting the supply of credit. We construct a DSGE model with heterogeneous banks, asset pricing

rules that generate a time-varying term premium, and introduce bond risk mispricing shocks to

study their eects on the real economy. A risk mispricing shock, in which agents overprice perceived

risk, increases the term premium and lowers output by reducing the availability of credit, as banks

increase rates and tighten lending standards. However, when investors underprice risk, a compressed

term premium leads to a `bad' credit boom that results in a more severe recession once the snapback

occurs.

JEL Codes: E43; E44; E58; G12

Keywords: Stochastic discount factor; risk mispricing; term premium; bank lending

*We would like to thank Rosalind Bennett, Roberto Billi, Ricardo Caballero, Fabio Canova, Martin Ellison, Mike Joyce,

Christian King, Luisa Lambertini, Chris Martin, Ian Martin, Fred Malherbe, Jordan Pandolfo, Evi Pappa, Ricardo Reis,

Glenn Rudebusch, and Mathias Trabandt, for their insightful comments and feedback, and Chi Hyun Kim and Gabriel

Madeira for their helpful discussions. We also beneted from comments made by participants of the 2021 Center for

Financial Research at the FDIC Seminar; 2021 University St. Andrews seminar, 2021 University of Hamburg Quantitative

Economics seminar, 2020 Macro Dynamics Workshop at Colby College; the ASSA 2020 CeMENT Workshop; the Financial

Fragility Session at the 2019 Midwest Macroeconomic Meetings in Michigan State University; the 2019 Women in Macro,

Finance and Economic History Workshop at DIW Berlin; the 2nd Annual NuCamp Conference at Nueld College, Oxford;

the XIII Annual Conference on Financial Stability and Banking in Brazil; the 50th Money, Macro, and Finance Conference

in Edinburgh; the 2018 European Economic Association Congress in Cologne; and the Bank of England Seminar series.

The opinions expressed in this article are the sole responsibility of the authors and should not be interpreted as reecting

the views of the Federal Deposit Insurance Corporation, the United States, the Bank of England or its Committees.

Bank of England, London EC2R 6DA, UK. Email: kristina.bluwstein@bankofengland.co.uk.

Center for Financial Research, FDIC, Washington, DC, 20429, USA. Email: jyung@fdic.gov.

1
1 Introduction

Bond markets are generally thought to be driven by fundamentals, with prices reecting the expected

future path of short-term assets adjusted for risk. However, markets often plummet or rally on the back

of changes to risk perceptions that do not necessarily stem from underlying actual risks in the economy,

suggesting an occasional mispricing of risks in asset markets (e.g. Pueger et al., 2020; Lewis et al.,

2021). If market participants overprice or underprice long-term risk, can these `mispricing shocks' aect

credit markets and feed back to the real economy, potentially threatening nancial stability?

In this paper, we develop a Dynamic Stochastic General Equilibrium (DSGE) model with a rich

nancial sector and bank lending that allows us to investigate the transmission mechanisms through

which bond mispricing leads banks to reassess their lending behavior and ultimately impact economic

activity. To this end, we introduce risk mispricing shocks, which inuence the compensation for bearing

long-term risk (the term premium) but are unrelated to economic fundamentals. An overpricing of

risk prompts banks to pass on the impact of the shock to the private sector in the form of higher

loan rates and reduce the amount of loans to entrepreneurs, as banks protect their protability and

capital-asset ratios. We nd that in addition to loans becoming more expensive, a risk mispricing

shock also alters the conditions for bank credit approval, increasing the de-facto required collateral

necessary to take out a loan, making access to credit yet more dicult. Even without any initial

change in underlying economic fundamentals, bond mispricing can lead to a pronounced contraction

in investment and economic activity, linking risk mispricing shocks to the real economy via the term

premium, through changes in bank lending decisions.

Our contribution to the literature is three-fold: (1) We combine a banking sector that is subject

to macroprudential regulation with asset pricing rules for long-term bonds in order to investigate how

changes in nancial markets aect credit allocation to the real economy. (2) The rich structure of our

general equilibrium model and non-linear solution technique enable us to match both macroeconomic

and term premium moments to the data. (3) By introducing a wedge to the stochastic discount factor

of nancial agents, we model a shock that mimics how nancial markets and the real economy respond

to the mispricing of long-term risk, dierentiating our shock from a more standard preference shock.

Our framework allows us to investigate lending conditions by dierentiating between a `good' credit

boom driven by economic fundamentals versus a `bad' credit boom driven by agents underpricing risk

2
in the economy. We nd that a bad boom that leads to a compressed term premium despite unchanged

fundamentals, induces excessive lending relative to a good boom driven by improvements in technology.

Moreover, once agents correct their mistake and price risk accordingly, the term premium snaps back

and output falls more sharply during the subsequent bust. A bad credit boom also has stronger eects

on nancial markets than a good credit boom. Nonetheless, while a bad credit boom still drives

investment, it is less supportive of consumption with wages remaining constant, as fundamentals and

therefore productivity, remain unchanged. We conclude by highlighting the implications of dierent

macroprudential policies targeted at reducing risk-taking behavior and promoting nancial stability

and nd that higher target capital-asset ratios help mitigate the eects of risk mispricing shocks.

The paper is structured as follows. Section 2 describes the main assumptions and equations of the

DSGE model that links the real economy and the nancial sector via the term premium. Section 3

outlines our solution method and calibration, compares the model-implied responses of risk mispricing

in the bond markets relative to a standard preference shock, and explores the role of dierent channels

in amplifying the eects of risk mispricing. It also includes a detailed robustness section to dierent

parametrization and shock specication, and the responses to other classic macroeconomic and nancial

shocks in our model. Section 4 explores the richness of our setup by simulating dierent credit boombust scenarios to analyze the eectiveness of macroprudential policies in mitigating nancial instability.

Section 5 concludes.

2 A DSGE model with bank lending and risk mispricing

First, our model features patient households and impatient entrepreneurs to introduce lenders and

borrowers with dierent sets of preferences. Households choose consumption, labor, and savings to

maximize their recursive utility with Epstein and Zin (1989) preferences as in Rudebusch and Swanson

(2012) and Van Binsbergen et al. (2012). Epstein-Zin preferences have the advantage that risk aversion can be modeled independently from the intertemporal elasticity of substitution, allowing for bond

pricing dynamics to be amplied, while retaining the desirable features of standard preferences. As

in Gambacorta and Signoretti (2014) and Bonciani and van Roye (2016), households own the banks

that collect their deposits and the monopolistically competitive nal-good rms (i.e., retail rms) for

which they receive a prot. They also pay taxes to the government. Perfectly competitive entrepreneurs

3
(that are credit constrained) need to borrow from the banks by providing capital as collateral in the

spirit of Iacoviello (2005) in order to produce intermediate goods that are sold to retailers. This feature

introduces a nancial friction via the borrowing constraint referred to as the collateral channel, in which

changes in the quantity and the value of capital impact consumption and investment decisions by entrepreneurs, generating the Bernanke and Gertler (1995) nancial-accelerator eect on the economy. We

follow Gerali et al. (2010) by allowing the borrowing constraint to depend on the LTV (loan-to-value)

ratio modeled as an exogenous stochastic process. Therefore, independent of bank choices, the LTV

ratio varies relative to a steady-state value set by the monetary policy authority and is subject to shocks

that directly impact the entrepreneur's ability to borrow.

In addition, entrepreneurs own perfectly competitive capital producing rms who buy undepreciated

capital stock and re-sell it to entrepreneurs at a new price taking into account a quadratic adjustment

cost, which helps provide the equilibrium condition for the price of capital. As in Kiyotaki and Moore

(1997), capital has many functions in this model and thus establishes an important feedback mechanism

between the economy and the nancial sector. Capital is used (i) in the production of intermediate

goods, (ii) as collateral for entrepreneurs' loans, and (iii) as a source of funds for investment. Therefore

capital, along with the LTV ratio and the bank lending rate, has a crucial impact on decisions made by

entrepreneurs. Finally, monopolistically competitive retail rms owned by the households dierentiate

the intermediate goods and sell the nal goods at a markup price subject to shocks, yielding a New

Keynesian Phillips curve that incorporates the recursive preferences of households.

Second, we model the banking sector to intermediate credit in the economy. Similar to Gerali et al.

(2010), Gambacorta and Signoretti (2014), and Bonciani and van Roye (2016), banks have branches that

perform specic functions under dierent degrees of market power. As in Gambacorta and Signoretti

(2014) and Bonciani and van Roye (2016), bank's demand for deposits is elastic, with the amount of

deposits being determined by the households, such that the deposit rate is the same as the risk-free rate.

On the other hand, the bank's wholesale branch manages its balance sheet by collecting the deposits

from households and issuing wholesale loans under monopolistic competition, such that the interest rate

on wholesale loans features a markup over the risk-free rate. The size of the markup, indicating the

degree of market power, is governed by the bank's penalty cost for deviating from its target capital-asset

ratio as in Gerali et al. (2010) and Gambacorta and Signoretti (2014). This represents the bank-lending

channel through which the monetary policy pass-through to banks' funding costs becomes incomplete.

4
Similar to Gerali et al. (2010) and Gambacorta and Signoretti (2014), the bank's retail branch operates

under monopolistic competition in the loan market for entrepreneurs. Market power is modeled with a

Dixit and Stiglitz (1977) framework such that the loan rate is chosen relative to the wholesale-loan rate,

amplied by a constant markup over the marginal cost. Dierent from their setup, is that our retail

branch can also purchase long-term government bonds. The bank, however, takes the rate on long-term

government bonds as given, which ultimately inuences the loan rate set by the bank. In our model,

government bond rates are subject to mispricing; therefore, our extension exposes bank loan rates to

shocks that make access to credit more dicult by increasing the loan rate faced by entrepreneurs, a

nancial friction that we identify as the loan-price channel.

Third, we introduce long-term government bonds as a source of investment, similar to Gertler and

Karadi (2013), Carlstrom et al. (2017), and Sims and Wu (2021). Government spending is modeled as

an exogenous process subject to shocks that can be nanced through taxes on households and by issuing

government bonds. The supply of government bonds is also modeled exogenously as a stochastic process

subject to shocks, but the rate of return on these bonds is determined in the nancial markets via asset

pricing conditions tied to the future state of the economy. An important extension of the modeland

our key contribution to understanding macro-nancial linkagesis the introduction of asset pricing

rules where the stochastic discount factor of households is subject to risk mispricing shocks right before

pricing long-term government bonds that endogenously elevate or compress the term premium, above

and beyond economic fundamentals. We hence allow nancial markets to misprice risk by introducing

a wedge between the actual price of long-term government bonds determined by the level of risk in

the economy versus the eective price that is driven by the mispricing of risk. A nal and important

point is that all debt in the economy is indexed to ination in order to focus on the real eects of risk

mispricing on the economy, rather than on the interaction between ination risk and asset pricing.

To close the model, the monetary policy authority follows a conventional monetary policy rule

that reacts to output and ination deviations from their steady states similar to Bonciani and van

Roye (2016). In addition, the real policy rate is subject to shocks, and the monetary policy authority

determines the target capital-asset ratio for banks and sets the steady-state LTV ratio for entrepreneurs,

which are then taken as given.

5
2.1 Households

Household i of unit mass chooses consumption (ch,t), labor (`h,t) and quantity of deposits (dt) to

maximize the recursive utility function

Vt (i) = U (ch,t (i), `h,t (i)) + βh



EtV

1−ξ

t+1 (i)

 1

1−ξ

, (1)

where βh is the patience discount factor and ξ is the Epstein-Zin parameter that captures risk aversion.

If ξ = 0, then Vt (i) collapses to standard preferences. The intra-period utility function is given by

U (ch,t (i), `h,t (i)) =

c

1−ψ

h,t (i)

1 − ψ

−

`

1+φ

h,t (i)

1 + φ

, (2)

with φ

−1

representing the Frisch elasticity of labor and ψ

−1

, the intertemporal elasticity of substitution

(IES). Households receive wt as (real) compensation for their labor, deposit their savings at the bank

for which they receive a risk-free gross return 1 + rtindexed to ination, pay taxes to the government

(Tt), and receive prots in real terms from the monopolistically competitive retail rms they own (JR,t),

such that their budget constraint is given by

ch,t (i) + dt (i) ≤ wt`h,t (i) + (1 + rt−1)dt−1 (i) − Tt (i) + JR,t (i). (3)

The rst order conditions yield a standard labor supply schedule, `

φ

h,t =

wt

c

ψ

h,t

, and Euler equation,

1 = Et

h

mh

t,t+1 (1 + rt)

i

, for mh

t,t+1 = βh

c

ψ

h,t

c

ψ

h,t+1







EtV

1−ξ

t+1  1

1−ξ

Vt+1





ξ

, (4)

where mh

t,t+1 represents the real stochastic discount factor of the households. Rudebusch and Swanson

(2012) show that with Epstein-Zin preferences, the consumption risk faced by households rises as the

state of the economy worsens, increasing the model risk premium via the ξ parameter. Intuitively,

these preferences imply that households are not just concerned with smoothing their consumption once

sudden shocks are realized in the short term, but are also concerned with medium- and longer-term

changes, allowing long-term risk to play a role in their decision-making process.

6
2.2 Entrepreneurs

Entrepreneurs operate under perfect competition to produce a homogeneous intermediate good, employ

households, and consume, but also need to borrow from banks by providing capital as collateral. They

form the link between the real economy and the banking sector and are thus important for generating a

feedback loop between the nancial and macroeconomic sides of the model. For simplicity, we assume

that entrepreneurs have constant relative risk aversion, as they, unlike households, do not invest or price

nancial assets. Therefore, entrepreneur j of unit mass chooses consumption (ce,t), labor demand (`d,t),

amount of bank loans (be,t), and capital stock (kt) in order to maximize utility subject to budget and

borrowing constraints,

max

{ce,t(j),`d,t(j),be,t(j),kt(j)}

E0

X∞

t=0

β

t

e

log (ce,t (j)) (5)

s.t. ce,t (j) + (1 + rb,t−1)be,t−1 (j) + wt`d,t (j) + qk,tkt (j)

≤

ye,t (j)

xt

+ be,t (j) + qk,t(1 − δk)kt−1 (j)

be,t (j) ≤

ΩtEt[qk,t+1kt (j) (1 − δk)]

1 + rb,t

.

βe represents the entrepreneurs' patience discount factor applied to their stream of future utility.

For small enough shocks, Iacoviello (2005) shows that βh>βe makes entrepreneurs more impatient than

households, ensuring that the borrowing constraint is binding around the steady state and credit is

constrained in the economy. Therefore, households in this model are considered net lenders, while

entrepreneurs are net borrowers. In the budget constraint, 1 + rb,t is the gross interest rate on bank

loans indexed to ination; qk,t is the real price of capital, which depreciates at rate δk; and xt represents

the gross markup price of the nal good over the intermediate good. ye,t (j) is the intermediate output

produced by entrepreneur j, which follows a standard Cobb-Douglas form ye,t (j) = Atk

α

t−1

(j) `

1−α

d,t (j),

where α denotes the capital share and Atis an exogenous stochastic AR(1) process with persistence ρa,

and i.i.d. technology shock εa,t with variance σ

2

a

.

The entrepreneurs are also subject to a borrowing constraint that limits how much can be borrowed

from banks, depending on (i) the stock of physical capital they posses and can thus be oered as

collateral, (ii) the expected price of capital stock, (iii) the lending rate set by the bank, and (iv) the

LTV ratio Ωt, modeled as an AR(1) process with persistence ρΩ, and i.i.d. shock εΩ,t with variance

7
σ

2

Ω. A high target LTV ratio, which can be interpreted as a high amount of loans relative to the future

value of assets, implies that banks can lend more for the same amount of collateral in the steady state.

The rst order conditions yield the labor demand schedule wt =

(1−α)ye,t

`d,txt

and the entrepreneurs' Euler

equation

1 = Et



me

t,t+1 (1 + rb,t)



, for me

t,t+1 = βe



1

ce,t

− λe,t−1

Et

1

ce,t+1

, (6)

where me

t,t+1 is the entrepreneur's real stochastic discount factor, in which the Lagrange multiplier

on the borrowing constraint, λe,t, represents the discounted marginal value of one unit of additional

borrowing.

Finally, the investment Euler equation equalizes the marginal benet to the marginal cost of saving capital. Since capital serves as collateral, the equation also depends on the discounted Lagrange

multiplier of the entrepreneur and the stochastic LTV ratio,

qk,t

ce,t

= Etβe

1

ce,t+1

(rk,t+1 + qk,t+1(1 − δk)) + λe,t 

ΩtEt[qk,t+1(1 − δk)]

1 + rb,t 

, (7)

where rk,t is the return to capital dened as rk,t ≡ α

Atk

α−1

t−1

`

1−α

d,t

xt

.

2.3 Capital good producers

Capital good producers are perfectly competitive and their main task is to transform the old, undepreciated capital from entrepreneurs to new capital without any additional costs. In addition, capital

producers `invest' in the nal goods bought from retailers, which are not consumed by households, and

also transform these into new capital. They then resell the new capital to entrepreneurs in the next

period. Taking the entrepreneurs' stochastic discount factor from Eq. (6) as given, capital producers maximize their expected discounted prots accounting for the law of motion of physical capital,

kt = (1 − δk)kt−1 +



1 −

κi

2



It

It−1

− 1

2



It, such that

1 = qk,t 1 −

κi

2



It

It−1

− 1

2

− κi



It

It−1

− 1



It

It−1

!

+ βeEt

"

ce,t

ce,t+1

qk,t+1κi



It+1

It

− 1

 It+1

It

2

#

, (8)

where It represents aggregate investment and κi governs the importance of the adjustment cost associated with the production of the nal good.

8
2.4 Retailers

Let yt (v) be the nal quantity of output sold by retailer v of unit mass and Pt (v) the associated price.

Given the nal-good production technology: yt =



R 1

0

yt (v)

y,t−1

y,t dv y,t

y,t−1

, where y,t represents the

elasticity of substitution between dierentiated nal goods, the retailer's demand for consumption goods

is yt (v) = 

Pt(v)

Pt

−y,t

yt, with aggregate price level Pt =

R 1

0

Pt (v)

1−y,t dv 1

1−y,t . The monopolistically competitive retailers dierentiate the intermediate goods from entrepreneurs sold at wholesale

price PW,t at no cost and sell them with a markup in order to maximize the expected discount value of

prots, subject to the demand for consumption goods:

max

{Pt(v)}

JR,t (v) = Et−1

X∞

t=1

mh

t−1,t "

Pt (v) yt (v) − PW,tyt (v) −

κπ

2



Pt (v)

Pt−1 (v)

− 1

2

Ptyt

#

(9)

s.t. yt (v) = 

Pt (v)

Pt

−y,t

yt.

Since retail rms are owned by the households, who keep their prots, retailers take their stochastic

discount factor as given and face quadratic price adjustment costs parametrized by κπ, which cause

prices to be sticky (Rotemberg, 1982), and yield the New Keynesian Phillips curve,

0 = 1 −

μy,t

μy,t − 1

+

μy,t

μy,t − 1

1

xt

− κπ (πt + 1) πt + Et

"

mh

t,t+1

mh

t−1,t

κπ (πt+1 + 1) πt+1

yt+1

yt

#

, (10)

where Pt

Pt−1

= πt + 1 is the gross ination rate, 1

xt

=

PW,t

Pt

, and y,t =

μy,t

μy,t−1

. The rm's price markup,

μy,t, is assumed to follow a stochastic AR(1) process with an autocorrelation coecient ρμy and an i.i.d.

price markup shock, εμy,t, with variance σ

2

μy.

2.5 Banks

The banking sector consists of a wholesale and a retail branch. The wholesale branch manages the

overall balance sheet of the bank, such that bank loans are nanced through deposits and bank capital.

This branch therefore chooses the amount of deposits and total bank lending (bt) in order to maximize

9
the discounted sum of cashows subject to a standard balance sheet constraint:

max

{bt}

Et−1

X∞

t=1

mh

t−1,t "

(1 + rw,t) bt − bt+1 + dt+1 + ∆kb,t+1 − (1 + rt) dt −

θ

2



kb,t

bt

− ν

2

kb,t#(11)

s.t. bt = dt + kb,t,

where 1 + rw,t is the gross interest rate on total bank lending (or wholesale loans) and kb,t is bank

capital, which accumulates out of reinvested prots. The bank's optimal choices are subject to a

quadratic adjustment cost of deviating from a target capital-asset ratio, ν, which is exogenously set by

the monetary policy authority and is parametrized by θ. This setup yields the following condition for

the net interest rate on wholesale loans,

rw,t = rt − θ



kb,t

bt

− ν

 kb,t

bt

2

, (12)

that relates rw,t to the deposit rate and the capital position of the bank relative to its target. When

kb,t

bt

= ν, the wholesale rate equals the deposit rate, as in the case of no market power. Therefore, the

dierence between the wholesale and the risk-free rates widens the more over- or under-capitalized a

bank is, with θ regulating the magnitude of the level shift relative to the risk-free rate; the bank-lending

channel.

The bank's retail branch buys wholesale loans from its wholesale branch at the given rate rw,t,

dierentiates them at no cost, and either lends to entrepreneurs to earn rb,t interest rate or purchases

bl,t government bonds to earn rl,t interest rate. While the bank chooses the interest rate on loans subject

to the entrepreneurs' Dixit-Stiglitz demand for debt, the interest rate on long-term government bonds is

determined by asset pricing relationships and thus taken as given. The retail branch therefore sets the

loan rate that maximizes prots for bank n subject to the total assets condition, i.e., bank assets are

comprised of loans to entrepreneurs and government bonds, and the demand for loans to entrepreneurs,

max

{rb,t(n)}

Et−1

X∞

t=1

mh

t−1,t [rb,t (n) be,t (n) + rl,tbl,t (n) − rw,tbt (n)] (13)

s.t. bt (n) = be,t (n) + bl,t (n)

be,t (n) = 

rb,t (n)

rb,t −b

be,t,

10
where be,t (n) depends on overall loan volumes and on the interest rates charged by bank n relative

to the average rate prevailing in the economy, with constant elasticity of substitution equal to b > 1.

The equilibrium condition for the bank leads to the net loan rate depending on the wholesale-loan rate,

the interest rate on long-term government bonds, and a constant markup, after imposing a symmetric

equilibrium condition:

rb,t = (rl,t + rw,t)

b

b − 1

. (14)

Notice that, everything else held constant, a shock that increases the long-term government bond

rate would also increase the loan rate charged by retail banks, making credit more expensive for entrepreneurs. This is eectively the loan-price channel through which increases in long-term government

bond rates via risk mispricing make borrowing less attractive.

Finally, bank prots (Jb,t) from both branches,

Jb,t = rb,tbe,t + rl,tbl,t − rtdt −

θ

2



kb,t

bt

− ν

2

kb,t, (15)

get reinvested as bank capital every period, which is used by the bank at rate δb under the following

accumulation equation:

kb,t = (1 − δb) kb,t−1 + Jb,t−1. (16)

2.6 Government sector

For simplicity, we assume that government spending, Gt, follows a stationary AR(1) process

Gt = (1 − ρg)G + ρgGt−1 + εg,t, (17)

where G is the steady state value of government spending, ρg captures the degree of persistence in scal

policy, and εg,t is a government spending i.i.d. shock with variance σ

2

g

. Government spending is nanced

through taxes on households and through debt held by banks, modeled as an exogenous AR(1) process

around the bond supply steady state 

bl



with autocorrelation coecient ρbl,

bl,t = (1 − ρbl) bl + ρblbl,t−1 + εbl,t, (18)

11
which is subject to i.i.d. shocks εbl,t with variance σ

2

bl. Finally, government spending is subject to the

standard budget constraint:

Gt + (1 + rl,t−1) bl,t−1 = Tt + bl,t. (19)

2.7 Asset pricing and risk mispricing

Long-term government bonds are default-free securities issued by the scal authority that pay a geometrically declining coupon every period in perpetuity, normalized to 1.1 Let pl,t be the price of the

long-term government bond at time t, where δc is the coupon decay rate that controls the duration of

the bond, discounted by the household's stochastic discount factor subject to risk mispricing:

pl,t = Et

hmh

t,t+1 + risk,t

(1 + δcpl,t+1)

i

, (20)

where risk mispricing, risk,t, is an exogenous AR(1) process with autocorrelation coecient ρrisk and

i.i.d. shock εrisk,t with variance σ

2

risk,

risk,t = ρriskrisk,t−1 − εrisk,t. (21)

Notice that our risk mispricing shock is not a preference shock, as it only changes the level of the

stochastic discount factor when pricing bonds, acting more like a wedge when it comes to asset prices

rather than a fundamental change in agents' beliefs.2 This setup is intuitive in that one can imagine how

events or news that trigger a higher perception of risk might cause investors to change their portfolio

allocations immediately yet not alter their consumption patterns on impact.

The rate that equates the price of the long-term bond today to the present discounted value of

cashows on perpetuity is given by

rl,t = ln 

δcpl,t + 1

pl,t 

, (22)

1This is equivalent to assuming that government bonds are innitely-lived consol-style bonds as in Chin et al. (2015).

The purpose of this assumption is to reduce the pricing relationship to just one recursive equation in the model, rather

than having to solve for each maturity level. As shown in Rudebusch and Swanson (2012), this simplication still generates

equivalent results to using ten-year zero-coupon bonds, while signicantly reducing the computational burden.

2The idea that there is a wedge between the actual household's stochastic discount factor and the discount factor used

to price risk premia has been micro-founded for example by Barillas and Nimark (2017), who priced bonds subject to

speculative behavior of individual traders, and Ellison and Tischbirek (2021), who decomposed the real term premium at

any maturity into covariances of realized stochastic discount factors and covariances of expectations of stochastic discount

factors that dier due to informational assumptions.

12
and the risk-neutral present value of the bond, pbl,t (i.e., discounted at the risk-free rate) is

pbl,t = Et



1 + δcpbl,t+1

1 + rt



, (23)

with the standard risk-neutral rate, rbl,t, given by

rbl,t = ln 

δcpbl,t + 1

pbl,t 

. (24)

As guaranteed by the absence of arbitrage in the bond markets, we compute the term premium (tpl,t)

as the dierence between the yield on the long-term bond and the yield on the equivalent risk-neutral

bond:

tpl,t = rl,t − rbl,t, (25)

reecting the compensation that risk-averse investors require in order to be exposed to long-term risk.3

We model risk mispricing as an AR(1) process following the empirical evidence supporting the idea

that investors' perceived changes in the term premium are persistent (see Adrian et al., 2013).4 The

risk mispricing shock in Eq. (21) enters the stochastic discount factor that prices government bonds

negatively, so that a positive shock lowers the marginal utility growth rate, as prospects for the future

valuation of bonds worsen. Since the demand (and hence price) of long-term bonds goes down, long-term

yields increase via the term premium, while the risk-neutral yield remains unaected by the shock.

2.8 Monetary policy

Monetary policy follows a standard Taylor rule, in which the central bank sets the one-period real rate

according to

1 + rt

1 + r

=



1 + rt−1

1 + r

ρr

 

1 + πt

1 + π

φπ 

yt

y

φy

!1−ρr

(1 + εr,t), (26)

where ρr is the interest-rate smoothing coecient, {r, π, y} are the real interest rate, ination and

3An important assumption for a positive, time-varying term premium is that the expectations hypothesis therefore

does not hold and households are allowed to be risk averse.

4Li et al. (2017b) empirically estimated the eect of changes in U.S. term premia, concurrent with changes in implied

U.S. equity volatility and the broad dollar exchange rate index, and found that the eect of a U.S. term premium shock

is persistent with a signicant estimate of the autocorrelation coecient of 0.78. Using a term structure model of interest

rates, Osterrieder and Schotman (2017) found that when allowing for stronger persistence of interest rate shocks, e.g.

with fractional integration I(0.89), the correlation between risk prices and the spot rate becomes negative, matching the

volatility observed in the data. This assumption is also consistent with the literature on uncertainty or credit risk shocks

being persistent, as in Christiano et al. (2014) and Sims and Wu (2021).

13
output steady states, respectively, and φπ and φy are the ination and output monetary policy parameters.5 We allow rt to be subject to shocks, where εr,t is the i.i.d. monetary policy shock with variance

σ

2

r

.

The monetary policy authority is responsible for setting a target capital-asset ratio for banks to

avoid an over-leveraging of the economy, similar to the Basel Tier 1 leverage ratios. ν in Eq. (11)

can therefore be interpreted as a capital adequacy/leverage constraint, ensuring all loans are backed by

sucient bank capital and deposits at the beginning of the period, inuencing bank prots. Moreover,

the central bank also sets the steady state LTV ratio for entrepreneurs that governs their borrowing

constraint and therefore the entrepreneur's equilibrium condition for the price of capital in Eq. (7).

2.9 Market clearing and aggregation

We add the budget constraints of the households, entrepreneurs, and the governmentEqs. (3), (5),

and (19)to derive the aggregate resource constraint in the economy,

yt = ct + qk,t (kt − (1 − δk)kt−1) + Gt +

κπ

2

π

2

t

yt + δbkb,t−1 +

θ

2



kb,t−1

bt−1

− ν

2

kb,t−1, (27)

where the last three terms represent the adjustment costs for prices and bank capital-asset ratios in real

terms, the clearing condition for the labor market is `h,t = `d,t, the aggregate condition for output is

yt = ye,t (j), aggregate consumption is dened as ct = ch,t + ce,t, and banks' symmetry condition holds

in equilibrium.6 We end up with 37 variables and 37 equations to solve, including 7 stochastic shocks.

3 Results

We describe the solution methods and baseline calibration for our model in Section 3.1 and compare

the t of the simulated model moments to the data in Section 3.2, along with several variations of our

5Fuerst and Mau (2019) pointed out that the exact monetary policy rule specication is important to generate variability

in the term premium in response to macroeconomic shocks. In order to achieve greater variability in the term premium,

the monetary authority should respond to the level of output relative to the steady state rather than the output gap (see

Rudebusch and Swanson, 2012). As an output level rule means the central bank is committing to a contractionary policy

for longer, thus reducing ination by more, the term premium is more aected than in the case of an output gap rule.

Hördahl et al. (2008) showed that the degree of interest rate smoothing in the monetary policy rule is also important for

matching bond and macroeconomic moments in their microfounded DSGE framework. Refer to Palomino (2012) for an

analysis of the role of monetary policy regimes and central bank credibility in inuencing bond risk premia in long-term

bonds.

6Refer to the Online Appendix for all the derivations and further details of the model.

14
baseline specication. Section 3.3 showcases the impact of our risk mispricing shock on the economy,

which we compare to a more traditional preference shock. We then study the role of dierent channels

in propagating the risk mispricing shock in Section 3.4 and evaluate the robustness of our results to

dierent calibration and shock specications in Section 3.5. Finally, we describe the impact of other

shocks in our model in Section 3.6.

3.1 Solution and calibration

Since the dimensions of the model are relatively high with 16 state variables, the most feasible option

to solve the model is through perturbation methods.7 We use third-order solutions and apply pruning

to cut out unstable higher-order explosive terms. The advantage of using third-order solutions is that

the macroeconomic responses remain mostly unchanged, and thus correspond to results in the previous

literature, while the responses for the bond markets can be rendered more realistically.8 A potential

disadvantage of this methodology is that the solution method is inherently local and is only valid around

the steady state, so that larger shock variances might lead to more inaccurate results. As estimation of

larger-scale, non-linear models is still dicult, we follow the literature (e.g., Rudebusch and Swanson,

2012) and calibrate our model based on standard assumptions about parameter values and to t specic

moments for key macroeconomic variables and the term premium.

Table 1 reports the values of the calibrated parameters for our baseline model. For households and

entrepreneurs, their discount factors are set such that βh implies an annual interest rate of 1.4 percent

to approximate the mean of the real policy rate in the data, and βh > βe ensures that entrepreneurs

are more impatient. The values of βh = 0.9965 and βe = 0.9787 are similar to those reported in Gerali

et al. (2010), Bonciani and van Roye (2016) and Gambacorta and Signoretti (2014)0.99, 0.9943, and

0.996, respectivelyfor the households, and within the 0.975-to-0.98 range reported in Iacoviello (2005)

and Iacoviello and Neri (2010) for the entrepreneurs. For household preferences, φ is based on the

Frisch elasticity being 2.5 and ψ is based on the IES being 0.95, in line with previous micro-founded

studies which have found the IES to be smaller than one (e.g. Vissing-Jørgensen, 2002) so that the

7Caldara et al. (2012) showed that perturbation methods provide equally accurate solutions to models with recursive

preferences than Chebychev polynomials and value function iterations, but are considerably faster.

8This feature occurs because rst-order solutions imply that the expectation hypothesis holds and the term premium

is zero, with second-order solutions generating a positive, yet constant term premium, both results inconsistent with

empirical evidence on risk pricing dynamics (e.g. Shiller, 1979; Campbell, 1987; Longsta, 1990; Cuthbertson, 1996;

Piazzesi and Schneider, 2007).

15
utility function is everywhere positive and the certainty equivalence is well dened. We set the EpsteinZin parameter ξ = 2 to match two term premium moments (mean and standard deviation). Using

the constant relative risk aversion (CRRA) formula in Swanson (2010), this number implies an overall

CRRA of 4. This is a low value relative to typical numbers found in DSGE models and more consistent

with the estimates found in the empirical macro-nance literature (see Havranek et al., 2015 for a

meta-study).

Table 1: Calibrated parameters for baseline specication

households, entrepreneurs, production banks, bonds, monetary policy

household patience (βh) 0.9965 bank capital adj. cost (θ) 11

IES (1/ψ) 0.95 target capital ratio (ν) 0.09

Frisch (1/φ) 2.5 loan elasticity (b) 9

EZ risk aversion (ξ) 2 bank management cost (δb) 0.06

entrepreneur patience (βe) 0.9787 consol bond decay rate (δc) 0.982

price adj. cost (κπ) 40 MP rule ination weight (φπ) 1.5

investment adj. cost (κi) 0.55 MP rule output weight (φy) 0.25

capital depreciation rate (δk) 0.05 shocks: persistence and standard deviation

steady states productivity: ρa = 0.9 σa = 0.003

ination (π) 0.00855 government spending: ρg = 0.9 σg = 0.003

price elasticity (y) 6 monetary policy: ρr = 0.5 σr = 0.005

target LTV ratio Ω



0.35 bond supply: ρbl = 0.9 σbl = 0.003

gov. spending-to-GDP ratio 

G

y



0.20 price markup: ρμy = 0.9 σμy = 0.003

bond supply-to-GDP ratio 

bl

y



0.10 LTV ratio: ρΩ = 0.9 σΩ = 0.003

bank capital-asset ratio 

kb

b



ν =0.09 risk mispricing: ρrisk = 0.9 σrisk = 0.005

For production, we set the adjustment cost for prices (κπ) to 40 and for investment (κi) to 0.55, and

later explore robustness to dierent values within our setup. The capital share α is assumed to be 0.3

and δk = 0.05 implies an annual capital depreciation rate of 20 percent. The bank parameter ν is set

to match the Basel target capital-asset ratio of 0.09 and the bank capital adjustment cost is θ = 11 as

in Gambacorta and Signoretti (2014). We set the loan elasticity coecient to 9, such that the loan rate

is 12.5 percent higher than the long-term and wholesale-loan rates combined, bank management costs

δb to 0.06, and the decay rate for consol bonds, δc = 0.982, is set to match the average real rate of a

10-year government bond of 2.87 percent. The monetary policy rule parameters φπ = 1.5 and φy = 0.25

reect that the central bank targets both ination and output with a stronger weight on ination, in

line with the literature (e.g., Gertler et al., 2020; Sims and Wu, 2021).

16
We calibrate the steady state of ination to match the annual rate of 3.42 percent we observe in

the data, and the price elasticity of demand is assumed to be 6 in steady state, implying a 20 percent

markup in the goods market steady-state price as in, for example, Bonciani and van Roye (2016). The

steady-state or target LTV ratio of 0.35 follows Gerali et al. (2010), bank capital-asset ratios are equal

to their target in the steady state, the ratio of government spending to output is 20 percent in the

steady state as in Gertler et al. (2020), and bond supply is calibrated to be 10 percent of output in the

steady state. The parameters governing our stochastic processes are also set to standard values, with

the persistence of the autoregressive processes ρ being 0.9 for all but the monetary policy smoothing

coecient, which is set to 0.5. The standard deviations of the shocks σ are calibrated to 0.005 for the

monetary policy and risk mispricing shocks and 0.003 for all other shocks.

3.2 Model t

To evaluate the t of the model, we compare key macroeconomic as well as term premium moments

implied by the DSGE model to the data. We use the Hodrick-Prescott lter to compute the business

cycle component of log quarterly U.S. data for chained GDP, consumption, investment, and labor.

The annualized real policy rate moments are calculated from the ination-adjusted Federal Funds rate

and annual ination is calculated using the GDP deator. The term premium is the Adrian et al.

(2013) nominal ten-year Treasury term premium from the Federal Reserve Bank of New York. Details,

summary statistics, and sources can be found in the Online Appendix.

Table 2 shows the results for the moments observed in the U.S. data in the rst row and the simulated

moments using the baseline calibration from Table 1 in the second row. The model performs very well

in matching key macroeconomic moments: the standard deviations of output, investment, and labor

are under 2 percent of the data variation, and consumption deviates within less than 7 percent from its

data moment. Our model ination rate, however, has a lower standard deviation from what we observe

in the data. This is a trade-o that arises when it comes to matching term premium moments. If we

set the Epstein-Zin (EZ) parameter to zero, the standard deviation of ination increases signicantly,

as can be observed in the no EZ preferences row, at the expense of not being able to match the term

premium moments, particularly its variability. When EZ= 0, we can increase the exibility of capital in

the model to match the standard deviation of ination (and investment) closer to what we observe in the

data (κi = 0.0255 yields σ (πt) = 2.44 and σ (It) = 4.13), yet term premium moments remain virtually

17
unaected. This trade-o occurs because in our model retail rms are owned by the households and

therefore take their stochastic discount factor as given when maximizing the discounted value of their

prots. The EZ parameter that governs the stochastic discount factor of the households with recursive

preferences derived in Eq. (4), therefore inuences the New Keynesian Phillips curve in Eq. (10) and

regulates relative price dynamics in the model via the markup price over intermediate goods.

Table 2: Comparing data with simulated model moments

rt tpl,t

σ (yt) σ (ct) σ (It) σ (`t) σ (πt) mean σ mean σ

data 1.46 0.85 4.08 2.10 2.45 1.54 2.47 1.64 1.19

baseline 1.46 0.80 4.12 2.14 1.08 1.52 1.06 1.64 1.19

no TFP shock 1.85 1.10 5.73 2.49 1.12 1.58 1.22 0.50 1.29

no risk mispricing shock 1.37 0.84 3.77 2.05 1.11 1.46 1.06 0.48 1.11

no bond supply shock 1.48 0.84 4.35 2.04 1.19 1.35 1.05 0.48 1.66

no gov. spending shock 1.47 0.89 4.37 2.12 1.16 1.42 1.07 0.40 1.65

no price markup shock 1.60 1.02 4.97 2.32 1.22 1.49 1.23 0.11 1.26

no monetary policy shock 1.06 0.97 3.52 1.54 0.55 1.47 0.75 0.10 0.91

no LTV ratio shock 1.22 0.92 3.35 1.79 1.06 1.53 0.89 -0.02 1.33

only risk mispricing shock 0.23 0.23 1.24 0.30 0.06 1.43 0.17 0.18 0.65

no EZ preferences 1.51 1.13 0.52 2.14 10.5 3.65 1.38 0.58 0.01

xed capital (κi = 1, 000) 1.63 1.02 0.06 2.31 1.06 1.44 0.93 1.43 1.17

Notes: Moments for real aggregate output (yt), consumption (ct), investment (It), and labor (`t) are reported in

quarterly percentage points; whereas ination (πt), the real policy rate (rt), and nominal term premium (tpl,t) are

reported in annual frequency. The model moments are computed by simulating the data 224 times to be consistent with

the duration of the time series. Refer to the Online Appendix for data sources and calculations. Mean and standard

deviations (σ) correspond to the model with the baseline calibration from Table 1 in the row labeled baseline, and

the same model without each shock at a time in subsequent rows. The row titled no EZ preferences corresponds to a

version of the model with CRRA preferences for the households. The nal row proxies for a model with xed capital

by setting κi = 1, 000.

When it comes to nancial variables, our baseline calibration allows us to perfectly match the mean

and standard deviation of the nominal term premium, and obtain a real policy rate that is on average

similar to the data yet with lower variability. Again, this is a trade-o that occurs as we prioritize

matching term premium moments, since under EZ= 0 a more exible capital calibration increases the

standard deviation of the policy rate to 2.07. The term premium is the compensation that investors

require in order to hold a long-term bond instead of a series of short-term bonds during the same horizon.

A high term premium therefore reects a perceived increase in nancial risk over the life of a bond.

Although unobservable, and therefore dicult to measure, it has been established in the literature that

this compensation for risk varies throughout time as investors update their beliefs about the future path

18
of the economy (e.g. Campbell and Shiller, 1991), hence the model term premium variability we obtain

is crucial for the purpose of our study.9

In order to understand the components of our model that help us match the data moments, we turn

o one shock at a time to observe their contribution to modeling the term premium. With regards to

matching the level of the term premium, all shocks at least partially contribute to delivering a high term

premium, particularly the LTV ratio shock. In a model without LTV ratio shocks, the mean of the term

premium is −2 basis points, far from the 164 basis points observed in the data. The variance of the

term premium, on the other hand, does not seem to be driven by any specic shock: risk mispricing and

monetary policy shocks are the most important drivers, yet without them, term premium variability

is still present due to risk aversion. Under no EZ preferences the variance of the term premium is

practically zero. This is consistent with the literature identifying that the additional parameter modeling

households' risk aversion is crucial to matching the empirical features of bond moments. Fuerst and Mau

(2019) explored business cycle moments for dierent model specications with and without segmented

markets and Epstein-Zin preferences, and found these features to also help deliver a counter-cyclical term

premium. We nd, consistent with their results and empirical evidence (e.g. Campbell and Cochrane,

1999; Wachter, 2006; Bauer et al., 2012; Lustig et al., 2014), that the model-implied correlation between

future consumption and nominal term premia is 0.30. This counter-cyclicality suggests that we expect

the term premium to rise along with the deterioration of future consumption prospects during economic

downturns and fall during economic upswings, as compensation for risk increases during bad times and

vice versa. Finally, we consider a model with only risk mispricing shocks and proxy for a version of the

model with xed capital by setting the investment adjustment cost parameter to a very high number,

the xed capital (κi = 1, 000) row. This implies that retailers nd it prohibitively costly to change

investment levels relative to the previous period. We conclude that risk mispricing shocks alone cannot

generate the variability we observe in the term premium data. Furthermore, while dynamic capital in

our model is critical for matching the standard deviation of investment, it is not crucial for delivering

a positive term premium with high variability.

9The term premium is typically estimated or inferred from the term structure of interest rates, forecasts, or surveys of

market participants. Swanson (2007), Rudebusch et al. (2007), and Li et al. (2017a) compared dierent estimates of the

term premium and provide excellent overviews of the challenges faced when measuring the long-term expectation of short

rates. Importantly, despite the disagreement on what the level of the term premium is (e.g. Rudebusch and Swanson,

2012 report a mean of 106 b.p., whereas Adrian et al., 2013 report a mean of 169 b.p. for the 1961/22007/8 period), all

measures broadly agree on the general movements of the term premium and are hence highly correlated.

19
3.3 The impact of a risk mispricing shock: When investors panic

We begin by analyzing how a risk mispricing shock aects the macroeconomy as a temporary, exogenous

wedge to the real price of long-term government bonds. Figure 1 reports the results for a shock that

underprices long-term bonds in solid blue by generating a 90 basis point increase in the annual nominal

term premium, a magnitude comparable to the increase experienced from September to October 2008,

during the initial stages of the Global Financial Crisis. This shock lowers the real price of long-term

government bonds by 11 percent on impact and increases the real long-term annualized interest rate by

0.24 percentage points, as investors demand higher compensation due to the perceived elevated risk.

Risk mispricing has a negative eect on the real economy with output declining by 0.42 percent on

impact, returning to the steady state only after 6 years, and has a small eect on ination, a short-lived

2 basis point decline, much as if our risk mispricing shock behaved like a demand shock. The central

bank reacts to ease economic conditions by decreasing the real policy rate by 7 basis points, lowering

the cost of funding for banks with an imperfect pass-through. The decline in the policy rate brings

expectations of the future policy rate down, incentivizing households to increase their consumption and

lower their supply of labor on impact, as well as reduce their savings. We nd that, overall, the results

of our theoretical model are in line with the previous empirical literature identifying sizable eects on

economic activity resulting from uncertainty shocks (e.g., 1.2 percent decline in output in Baker et al.,

2016; 1 percent change in output during expansions and 2 percent during contractions in Caggiano

et al., 2022; a persistent eect on industrial production Carriero et al., 2018) and that our risk mispricing

shock seems to correspond with a demand shock (e.g. Leduc and Liu, 2016).10

With the long-term interest rate on government bonds increasing, banks pass on the impact of the

risk mispricing shock to the private sector in the form of higher loan rates, around 200 basis points

on impact, and reduce the amount of loans to entrepreneurs, as banks protect their protability and

capital-asset ratios. The higher loan rates lead to higher bank prots and eventually higher bank net

worth as banks accumulate capital out of reinvested prots in Eq. (16). Since the amount of government

bonds in the bank portfolio remains the same, while loan volumes decline, total bank assets decrease.

10Our results also support the reduced-form evidence found in the macro-nance literature that identies persistent

eects in macroeconomic variables in response to term premia shocks (e.g. Gil-Alana and Moreno, 2012; Jardet et al.,

2013). Joslin et al. (2014), for example, identied that both economic activity and ination decline when a canonical term

structure model of interest rates incorporates macroeconomic fundamentals beyond the information spanned by the yield

curve. Although not a general equilibrium framework, their model allows future bond prices to be inuenced by yield

curve factors as well as macroeconomic risks, which in turn account for variation in the term premium.

20
Figure 1: Impulse responses to risk mispricing shock

Notes: The blue solid line represents the impulse responses to a risk mispricing shock, whereas the blue dashed line

represents the responses to a preference shock. Both shocks are scaled to raise the nominal term premium by 90 basis

points for comparison. All variablesexcept for the term premiumare in real terms. Responses are deviations relative

to the steady state in the corresponding units indicated in the vertical axis. All variables are quarterly, except for the

term premium and the interest rate, which are annualized.

Higher bank capital and lower bank assets lead to a further reduction in demand for deposits through

the balance sheet structure of the bank. The decline in deposits occurs as households increase their

21
consumption and reduce their supply of labor on impact, before the eects of the economic downturn

bring household (and aggregate) consumption down.

As a response to the reduction in the amount of loans to entrepreneurs, whose consumption falls by

more than 1 percent and are less willing to take out a loan at a higher rate (the loan-price channel in

our model), investment falls by 1.65 percent. The contraction in investment is what drives the decline

in output, as capital decreases by 0.1 percent on impact, reaching a 0.4 percent decline 3 years later

and staying persistently below the steady state for 10 years. The fall in the stock and price of capital

reduces the collateral value of the entrepreneurs, further restricting their ability to take out a loan,

and amplifying the impact of the risk mispricing shock on bank lending and the real economy. This

mechanism is consistent with anecdotal evidence on bank behavior: The Euro area bank lending survey

providing qualitative information on bank loan demand and supply across Euro area enterprises and

households, for example, identied `risk perceptions' as one of the most important factors in periods

of net tightening of credit standards on housing loans and loans to enterprises (Köhler Ulbrich et al.,

2016).

To further understand our shock, Figure 1 also displays the results of a preference shock in blue

dashed lines. In order to transform our shock into a preference shock, we incorporate risk mispricing into

the household value function Eq. (1), such that it multiplies the consumption term in the intra-period

utility function Eq. (2).11 This leads to a dierent equation for the stochastic discount factor of the

households featuring a preference shock:

1

1 + rt

= Etβh

risk,t+1

risk,t

| {z }

preference shock

c

ψ

h,t

c

ψ

h,t+1







EtV

1−ξ

t+1  1

1−ξ

Vt+1





ξ

| {z }

mh

t,t+1: stochastic discount factor

, (28)

where we make two additional modications. First, we change the exogenous process dening risk

mispricing to have a mean of 1 and make the shock enter Eq. (21) positively, such that a positive

shock reduces the stochastic discount factor on impact, lowers the real price of bonds, and elevates the

nominal term premium. Second, we set IES= 0.5 to avoid running into the asymptote that can arise

11Our version of a preference shock is similar in principle to the uncertainty shock of Basu and Bundick (2017) in that it

impacts the EZ preferences of the households, lowering their stochastic discount factor. Their uncertainty shock, however,

impacts the volatility or second moment of the exogenous process, as opposed to the rst moment.

22
when the IES is close to unity under this modied specication, as explained in De Groot et al. (2018).

Finally, we scale the preference shock to induce a 90 basis point increase in the nominal term premium

to be in line with our risk mispricing shock, although an apples-to-apples comparison is dicult given

the dierent nature of the shocks. The rst important dierence is that a preference shock lowers the

stochastic discount factor of the households on impact, a feature that is not present in our baseline

specication. Via Eq. (28), we can observe that the preference shock therefore inuences mh

t,t+1 as

much as rt, breaking the segmentation previously featured in our bond markets. Without the wedge

between 1

1+rt

pricing the risk-neutral bond in Eq. (23) and mh

t,t+1 pricing the risky bond in Eq. (20),

the real term premium as dened by the dierence between the risky and risk-neutral yields in Eq. (25)

is zero. The nominal term premium therefore elevates in this context driven solely by the increase in

ination, since the decline in the bond price of 2 percent is oset by the decrease in the risk-neutral

bond price of the same magnitude. The initial contraction in output is smaller, 0.25 percent below the

steady state, and the monetary policy increases the rate this time in response to higher ination. The

reason behind the more pronounced impact on ination is that retail rms, owned by the households

such that retailers take their stochastic discount factor as given in Eq. (9), are now exposed to the shock,

inuencing the New Keynesian Phillips curve in Eq. (10) that determines relative prices in the economy.

Banks reduce lending and increase the loan rate even more, since the tighter monetary policy increases

banks' funding costs. Although a preference shock delivers similar macroeconomic responsesoutput

and investment decline all the eect is transmitted via ination as opposed to the bond markets, a

very dierent mechanism from the risk mispricing shock we want to study.

3.4 The role of dierent channels in propagating the risk mispricing shock

There is recent evidence to support the importance of nancial frictions in restricting credit availability

during the early 2020 outbreak of the COVID-19 pandemic. Between March 9th and March 19th, the

10-year U.S. Treasury yield rose by 60 basis points, reecting a pronounced increase in the Kim and

Wright (2005) daily 10-year term premium of almost 40 basis points, as estimated by the Board of

Governors.12 During the following weeks, consumer lending by all commercial banks declined, a higher

percentage of banks tightened their standards for issuing all types of loans (e.g. commercial, consumer,

credit card, mortgage) as well as increased the spread of loan rates over the cost of funds, further

12See https://www.federalreserve.gov/pubs/feds/2005/200533/200533abs.html.

23
restricting the availability of credit between 2020-Q1 and 2020-Q2.13 While the COVID-19 shock was

not a risk mispricing shockand was followed by lock-down policies and other interventions that we do

not model in our frameworkit is a clear recent example of a disruption to the economy whose origins

were unrelated to fundamentals, was reected in a higher term premium elevating long-term interest

rates, and at least partially impacted the availability of credit via the types of channels we identify in

our model.

We hence asses the relative importance of dierent channels in allowing risk mispricing shocks to feed

back to the real economy by considering specications of our model in which we remove nancial frictions

one at a time. We begin by illustrating the main transmission mechanism in our baseline specication

by changing the retail bank problem in Eq. (13), such that banks no longer incorporate long-term

government bonds into their portfolios, eectively removing the loan-price channel. In this case, the

loan rate in Eq. (14) is dened as a constant markup over the wholesale-loan rate: rb,t = rw,t 

b

b−1



,

changing bank prots in Eq. (15) to Jb,t = rb,tbe,t − rtdt −

θ

2



kb,t

bt

− ν

2

kb,t, and rendering the risk

mispricing shock inconsequential for the banking system. Figure 2 illustrates that the presence of this

channel is of rst order importance for the transmission of the shock to the real economy (gold dashes).

With the risk mispricing shock no longer impacting the loan rate, loans to entrepreneurs, investment, and

therefore output, there is no longer a need for the monetary policy authority to intervene. Therefore, the

bank's cost of funding, capital, and prots remain unaected by risk mispricing. These results illustrate

that our risk mispricing shock is only transmitted via bond prices held by the banks, as opposed to the

case of a preference shock, for example, which can still impact the bank's balance sheet via ination. In

fact, the responses to a preference shock remain virtually unaltered, even after removing the loan-price

channel entirely.

We next investigate the role of the bank-lending channel in propagating our risk mispricing shock.

To this end we set the parameter that governs the importance of the capital-asset ratio deviations from

its target to zero, by letting θ = 0 in Eq. (9), such that the wholesale-loan rate in Eq. (12) now moves

one-to-one relative to the policy rate; yet the imperfect pass-through of monetary policy to the loan

rate in Eq. (14) remains. This modication changes the equation for bank prots and eliminates the

adjustment term for aggregate output in Eq. (27). Figure 2 (orange dashes) shows that without a bank13Board of Governors of the Federal Reserve System's April 2020 Senior Loan Ocer Opinion Survey on Bank Lending

Practices release, May 4, 2020, available online at http://www.federalreserve.gov/boarddocs/SnLoanSurvey/.

24
lending channel, banks are no longer concerned about deviations from their target capital-asset ratios,

hence when the risk mispricing shock hits and the loan rate increases, banks protect their protability by

contracting lending even more and charging persistently higher loan rates than under the baseline case.

The banks' more pronounced reduction in their balance sheet leads to higher bank prots on impact

and a more sustained and persistent accumulation of bank capital over time as those higher prots

get reinvested. With a stronger reduction of credit in the economy, investment's return to the steady

state after the initial decline is slower relative to the baseline case, making the eect of a risk mispricing

shock on output more persistent. The presence of this channel therefore suggests that actively managing

their capital-asset ratios to achieve a specic target can be benecial in mitigating the eects of risk

mispricing on the real economy, at the expense of reduced protability for banks.

Lastly, we evaluate the impact of the collateral channel on amplifying the eects of the risk mispricing

shock (orange dots). Similar to the no asset price channel scenario of Iacoviello (2005), we consider the

case in which the entrepreneur's borrowing limit is present but no longer tied to the value of the assets,

such that the borrowing constraint in Eq. (5) is set to be,t (j) ≤ z for z > 0. This removes the last term

in the investment Euler equation of the entrepreneur and eliminates the presence of the LTV ratio in

the model. When the risk mispricing shock increases the loan rate via the long-term interest rate, banks

raise rates by an additional 5 basis points relative to the baseline scenario to increase prots, leading

eventually to higher bank net worth. Since banks do not reduce their supply of credit in response to

the shock, the changes in capital, investment, and output are much more muted than in the baseline

case. We nd that in the absence of the collateral channel, changes in the stock and price of capital no

longer constrain the entrepreneurs' ability to borrow, hence the deterioration in their consumption and

investment opportunities is less pronounced: only the price of credit changes, not the quantity. This

result is independent from the presence/absence of the LTV shock; it is solely driven by bank lending

no longer being tied to the value of entrepreneurs' capital. Given the lack of change in the quantity of

credit availability, the overall eect in the macroeconomy is more modest, with the policy rate reacting

to the milder changes in output and ination. It may seem that banks requiring loans to be backed

by collateral amplies the transmission of risk mispricing as it acts as a nancial accelerator; however,

is important to recall that our model features no loan default and hence no upside to collateralized

debt. Incorporating credit risk in this type of model by allowing lenders to default on their payments

is outside the scope of this project but it would allow one to better assess these relative trade-os.

25
Figure 2: Impulse responses to risk mispricing shock: NO channels

Notes: The blue solid line represents the impulse responses to a risk mispricing shock (scaled to raise the nominal term

premium by 90 basis points) under the calibration in Table 1, which we label our baseline specication. The orange

dashes represent a no bank-lending channel scenario by setting θ = 0. The orange dots represent a no collateral channel

scenario by setting the entrepreneur's borrowing constraint to be,t (j) ≤ z, for z > 0. The gold dashes represent a no

loan-price channel scenario, which delivers a loan rate no longer tied to the long-term interest rate, deactivating the

transmission of the risk mispricing shock. All variables are quarterly and in real terms, except for the term premium,

which is annualized and in nominal terms. Responses are deviations relative to the steady state in the corresponding units

indicated in the vertical axis.

3.5 Robustness to parametrization and shock specication

We next evaluate the robustness of the responses to a risk mispricing shock when changing key preference,

production, and bank parameters. For reference, Figure 3 displays the impulse response functions for

the baseline model calibration in solid blue. In panel (a), we lower the household IES to 0.25 (light blue

dots), increasing the curvature of households' utility with respect to consumption from 1.05 to 4.

26
Figure 3: Impulse responses to risk mispricing shock: Robustness

(a) preferences

(b) production and banking

Notes: The blue solid line represents the impulse responses to a risk mispricing shock (scaled to raise the nominal term

premium by 90 basis points) under the calibration in Table 1, which we label our baseline specication. We change

dierent parameters regulating household preferences in panel (a) and production and banking in panel (b). All variables

are quarterly and in real terms, except for the interest rate, which is annualized. Responses are deviations relative to the

steady state in the corresponding units indicated in the vertical axis.

27
Responses to a risk mispricing shock are similar to the baseline scenario, with both ination (−0.3

percentage points) and output (−0.5 percent) decreasing more on impact and hence inducing a stronger

policy rate response (2 basis points lower than the baseline). Since a higher curvature makes consumption growth less sensitive to changes in the policy rate, household consumption raises by less, while

entrepreneur consumption goes down by just as much as in the baseline scenario. Aggregate consumption hence contracts more on impact, magnifying the response of output, with more muted eects in the

long run. The impact on bank loans is more subdued, as the change in the IES coecient aects the

quantity and not the price of capital, with all other responses relatively unchanged. Second, we lower

the elasticity of substitution of labor supply (light blue dashes), increasing households' utility curvature

with respect to labor from 0.4 to 1. While with a lower Frisch elasticity most responses remain relatively

the same, the risk mispricing shock lowers wages by more on impact (−0.2 percent), and generates a

stronger policy rate response stemming from a more pronounced decline in ination (5 additional basis

points). Impulse response functions are also robust to a model without EZ preferences (solid orange

line). While crucial for matching the term premium variability observed in the data, macroeconomic

responses to the risk mispricing shock are quantitatively the same, albeit slightly more persistent in

the long run, with the eect on credit being more pronounced over time. Analogous to the baseline

specication, the risk mispricing shock is transmitted to the economy via the loan-price channel and

amplied through the collateral channel under standard preferences.

In regards to production in panel (b), we make price adjustments more exible by lowering κπ from

40 to 10 (gold dots), making ination more sensitive to the risk mispricing shock (4 additional basis

points), yet all other responses to the shock remain the same as in the baseline scenario, since our shock

works through the real pricing of risky bonds. We also evaluate the importance of investment adjustment

costs by considering a fully exible capital model where κi = 0 and a quasi-xed capital model where

κi = 1000. When we set the investment adjustment cost parameter to zero (not shown), the capital

accumulation equation collapses to a standard version without adjustment costs and the price of capital

in Eq. (8) is xed. Investment is therefore more sensitive (2 additional percentage points) and hence

the response of output to the risk mispricing shock is stronger, with bank capital and loans exhibiting

a more pronounced decline via the change in the quantity of capital. On the other hand, we create a

quasi-xed capital model by setting the investment adjustment cost parameter to a very high value, such

that capital-good producers nd it prohibitively costly to change investment relative to the previous

28
period, making capital a xed rather than a dynamic variable that only changes due to depreciation

(red dashes). Since investment cannot adjust, the sharper contraction in output is driven by the decline

in entrepreneurs' consumption. In this case the quantity of capital cannot change in response to the

shock, yet the decline in the price of capital diminishes the ability of entrepreneurs to borrow (the value

of the collateral required to take out a loan plummets), exacerbating the decline in credit, which more

than doubles on impact. These eects, although more severe on impact, are shorter-lived relative to the

baseline specication we present. Lastly, we lower the loan elasticity coecient to 3, such that the loan

rate features a 50 percent markup over the long-term and wholesale-loan rates combined (gold solid

line). With a higher loan rate as a response to the risk mispricing shock, the banks contract lending by

the same amount on impact, but their balance sheet, lending, and deposits revert back to their steady

states faster.

The eects of the risk mispricing shock are not driven by dierent calibration choices nor its additive

specication. To this nal point, risk mispricing enters the stochastic discount factor while pricing longterm bonds in an additive fashion, which entails generating an up or downward parallel shift in the

term premium. We therefore allow this shock to act as a wedge between the stochastic discount factor

of the households and the price of the risky bond. We test whether this additive specication drives

our results by changing Eq. (20) such that risk mispricing multiplies the stochastic discount factor

instead.14 Unlike Kim (2000) additive vs. multiplicative TFP shock to rms' production function, for

example, we nd our results to be robust to either shock specication (not shown), since our shock only

operates via asset prices.

3.6 Other shocks

We analyze the impulse responses of traditional macroeconomic shocks, including their eects on the

term premium, to cross-check that their responses are consistent with economic theory and our model

reproduces standard results that are well understood in the literature. The responses to a positive,

one standard deviation productivity (a), government spending (b), and monetary policy (c) shocks are

14This modication generates two changes in the interpretation of our risk mispricing shock. First, the shock still aects

the level of the stochastic discount factor, but it also alters its slope, steepening or attening the rate of growth instead

of adding a wedge. Second, because a multiplicative shock is proportional to the stochastic discount factor, when the

pricing kernel is very high (the good times), then the eect of the risk mispricing shock is larger than during the bad

times, when the pricing kernel is smaller. Therefore, the impact of a risk mispricing shock on the term premium is no

longer independent from the level of the stochastic discount factor.

29
reported in Figure 4 with red solid lines.

As is standard in the literature, a productivity shock persistently increases output (0.3 percent) and

lowers ination (6 basis points on impact), inducing the monetary policy authority to marginally lower

the policy rate, given its stronger emphasis on managing ination. Driven by a boost in investment,

wages go up and consumption increases, and banks meet the higher demand for credit by issuing more

loans, expanding their balance sheet. Since the loan rate falls on impact due to easier monetary policy,

bank capital and prots go down, and investment is thus supported both by the productivity boost

and easier access to credit. Furthermore, higher capital induces a higher valuation of entrepreneurs'

collateral and thus higher demand for credit, amplifying the impact on investment. In fact, when we

turn the collateral channel o (light blue solid lines), investment expands by half the value under the

baseline model, leading to a more muted increase in output, since credit is no longer constrained. The

productivity shock also boosts bond prices, putting downward pressure on interest rates and compressing

the term premium, which declines by 24 basis points, consistent with the qualitative ndings outlined

in Rudebusch and Swanson (2012) revealing a negative, albeit weaker, relationship between output and

the term premium. This correlation is robust to removing the nancial frictions in our model, since

most macroeconomic and term premium responses remain unchanged, despite bank capital and prot

losses being somewhat amplied in the absence of the loan-price and bank-lending channels, similar to

the amplication eects observed in the case of a risk mispricing shock.

A one standard deviation positive shock to real government spending in Eq. (17) boosts real output

by 0.12 percent, with almost no eect on ination, leading the monetary policy authority to marginally

raise the policy rate, tightening economic conditions. The 1 percent increase in government spending is

nanced through higher taxes to the households, for which they lower their consumption (0.1 percent on

impact), reduce their deposits, and work more despite lower wages. The small decline in term premium

occurs as the average expected future policy rate from Eq. (24)due to the monetary policy response

to higher outputis higher than the increase in the long-term rate due to reduced bond prices from Eq.

(22). We therefore nd a small 4 basis point impact on the term premium from such a shock. Bank

lending initially declines and then increases and remains persistently above the steady state, while banks

increase their net worth and prots on the back of higher loan rates. Since investment goes up (0.06

percent), entrepreneurs' consumption increases 0.6 percent on impact and capital accumulates. Results

are again, robust to removing our nancial frictions.

30
Figure 4: Impulse responses to other shocks

(a) productivity shock (b) government spending shock (c) monetary policy shock

Notes: The red solid lines represent the impulse responses to a one standard deviation shock to (a) productivity, (b)

government spending, and (c) monetary policy, which we label our baseline specication. The red dots represent a no

bank-lending channel scenario by setting θ = 0. The light blue solid lines represent a no collateral channel scenario

by setting the entrepreneur's borrowing constraint to be,t (j) ≤ z, for z > 0. The gray dashes represent a no loan-price

channel scenario, which delivers a loan rate no longer tied to the long-term interest rate, deactivating the transmission of

the risk mispricing shock. All variables are quarterly and in real terms, except for the term premium, which is annualized

and in nominal terms. Responses are deviations relative to the steady state in the corresponding units indicated in the

vertical axis.

31
We consider a one-standard deviation monetary policy shock when σr = 0.003, such that the real

policy rate in Eq. (26) increases by 130 basis points. As expected, this shock has a contractionary eect

in the economy, with both output and ination declining on impact, yet returning back to the steady

state after only 3 quarters. In this scenario, the term premium goes down 50 basis points driven by the

increase in the expected future policy rate surpassing the increase in the long-term interest rate, and

returns back to the steady state as monetary policy quickly unwinds. The increase in the policy rate

gets directly passed onto the loan rates, such that bank loans decline on impact and remain persistently

below the steady state, while bank prots and bank capital benet from the higher loan rates. We

also nd that nancial frictions play no role in the transmission of the monetary policy shock to the

macroeconomy, as expected.

Three other shocks are featured in our model, not shown in Figure 4. While a bond supply shock in

Eq. (18) has almost no impact on output, ination, nor the term premium, a positive nal-good markup

price shock increases ination while lowering output, driving the term premium up by 10 basis points.

This is consistent with the eects of inationary supply-side shocks identied in Rudebusch and Swanson

(2012) that cause the term premium to rise at the same time that they generate declines in consumption,

investment and output. Since the policy rate remains relatively unchanged, so does the loan rate, and

entrepreneurs react to the higher relative prices by lowering their demand for credit, their consumption,

and investment. Finally, we consider a shock to the LTV ratio that relaxes the entrepreneurs' budget

constraint, increasing the demand for credit by 0.43 percent and their consumption by more than

2.5 percent on impact. This is in essence a short-lived demand-side shock, for which we see higher

investment drive an expansion in both production and ination simultaneously. The monetary policy

authority reacts by increasing the policy rate by 150 basis points, which increases the loan rate charged

by the banks, increasing bank prots and net worth, and putting upward pressure on the term premium,

which goes up by 26 basis points on impact.

4 Application: When investors underprice risk

With the unfolding of the Financial Crisis, there has been a surge in both empirical and theoretical

models trying to explain the underlying causes of nancial market uctuations. While recessions are a

normal feature of business cycle dynamics, a consensus emerged that recessions following a credit fueled

32
boom are particularly damaging to the economy (Minsky, 1986; Borio and Lowe, 2002; Kindleberger

and Aliber, 2011; Claessens et al., 2012; Jordà et al., 2013). Dell'Ariccia et al. (2016) found that one

third of all credit booms are followed by a nancial crisis and 60 percent are followed by lower economic

performance. As such, a distinction that is often made is between a `good' credit boom and a `bad'

credit boom. While both booms increase output and the availability of credit, a good boom is based on

fundamentals in the economy improving, such as higher productivity or technological innovation (e.g.,

Kydland and Prescott, 1982). In contrast, a bad boom can be dened as a credit boom that is solely

based on sentiment or `animal spirits' and is thus likely to mean revert, once agents realize their mistake

(e.g., Azariadis, 1981; De Grauwe, 2011; De Grauwe and Macchiarelli, 2015; Bianchi and Melosi, 2016).

An event that triggers the reversal in expectation, i.e., a Minsky moment, could likely set o a chain

reaction ultimately inducing a nancial crisis, a recession, or both. From a nancial stability perspective,

a bad boom could therefore have devastating consequences on nancial markets and the real economy.

Beaudry and Willems (2022), for example, found cross-country empirical evidence that over-optimism

about the economic prospects of a country that later on fail to materialize, lead to excessive borrowing

and is therefore associated with future economic recessions.

4.1 Good and bad boom-bust scenarios

We simulate dierent boom-bust scenarios with our baseline specication, labeling productivity shocks

as `good' booms and risk mispricing or preference shocks as `bad' booms. In all three cases, we generate

shocks every period scaled such that output increases by 1 percent every year for three consecutive years

resulting in an economic boom that induces banks to lend more. Figure 5 shows that all three shocks

are scaled to lead output to be 3.03 percent above the steady state by quarter 12. During this economic

boom, investment and credit expand, while ination, interest rates, and the term premium go down.

The magnitudes of the responses, yet, depend on the type of shocks driving the boom. The booms

generated by risk mispricing (solid blue lines) or preference shocks (dashed blue lines) lead to credit and

investment growth around three times higher than a boom generated by technological improvements

(solid green lines). These booms, however, are labeled `bad' booms since they are not driven by stronger

economic fundamentals, but misperceptions about the current and future state of the economy. In the

case of a risk-mispricing boom, agents underprice the actual risk in the economy because they perceive

risks, for some exogenous reason, to be lower; signicantly compressing the term premium and pushing

33
long-term interest rates 2 percentage points below their steady state. The eect of the preferenceshock boom is driven by changes in ination, generating a similarly-sized term premium compression

via the ination channel instead. Therefore, expansions fueled by credit growth that are accompanied

by pronounced changes in interest rates or ination can indicate that the nature of the credit boom is

driven by unjustied perceptions of the economy or nancial market returns.

Figure 5: Impulse responses to dierent boom-bust scenarios

Notes: Economic expansion (boom) is generated by risk mispricing shocks in solid blue lines, preference shocks in dashed

blue lines, and productivity shocks in solid green lines, scaled such that output increases by 1 percent every year for

3 consecutive years using continuous compounding. Each boom is then counteracted in period 13, labeled a bust. All

variablesexcept for the term premiumare in real terms. Responses are deviations relative to the steady state in the

corresponding units indicated in the vertical axis. All variables are quarterly, except for the term premium and the interest

rate, which are annualized.

In order to evaluate the potential consequences of credit fueled expansion via dierent shocks, we

generate a severe bust that counteracts the entire three-year boom on period 13. For the good boom

scenario, we model the bust as a negative productivity shock, and for the bad boom scenarios, we

assume that an exogenous event occurs making agents realize that they have not priced risk correctly

or their consumption preferences reverse back to the baseline. All busts lead to a contraction in output

and investment and an increase in interest rates and the term premium.15 The decline in output is,

15It should also be noted that the only non-linearity we assume in the model is the one arising due to the higher order

34
however, more pronounced for the scenario in which the economic booms are driven by risk mispricing,

justifying the notion of this being a particularly `bad' boom. Although the depth of the recession is

about halved by the following quarter, output does not revert back to the steady state for about four

additional years. This exercise suggests that the transmission from the nancial sector to the real

economy can be amplied if bank lending is driven by unfounded optimism or mispricing instead of

economic fundamentals.

4.2 Macroprudential policies

Financial stability policy makers are tasked with identifying bad credit booms in advance, and building

up resilience against a potential burst that could be harmful for the economy, such as experienced in

the 2008-2009 Great Recession. Policy makers are hence given powers over macroprudential policies

to ensure the resilience of the nancial system. We test two specic types of macroprudential policies

that the nancial stability authority can implement in our model: (a) a macroprudential policy that

increases banks' target capital-asset ratio, ν in Eq. (11), and (b) a macroprudential policy that decreases

the steady state of the LTV ratio of entrepreneurs, Ω in Eq. (5). While bank capital-asset ratios are

targeted in particular at strengthening lenders' solvency, the LTV ratio aims to improve borrowers'

solvency and should help to avoid unsustainable debt levels. As an illustrative example, we assume a 33

percent increase in the target bank capital-asset ratio from 0.09 to 0.12, so that banks are encouraged

to have a larger capital buer with respect to their assets. In a similar vein, we analyze a 25 percent

reduction of the LTV target ratio for entrepreneurs from 0.35 to 0.2625 implying that entrepreneurs

need to back up the same quantity of loans with more collateral than before, reasonable values within

those discussed in Gerali et al. (2010). Both measures are intended to make the nancial system more

resilient.

Figure 6 displays the boom-bust scenario driven by risk mispricing shocks under the baseline calibration (solid blue lines), compared to a scenario with higher target capital-asset ratios (dashed gray lines)

and lower target LTV ratios (solid gray lines). During the economic boom, higher bank capital-asset

ratios reduce the ability of banks to issue credit, leading to loan volume increases similar to those that

would occur under a good boom scenario driven by productivity shocks. During the bust, this policy

perturbation method with which we solve the model. The eects of a bust are likely to be larger when accounting for

other non-linearities like occasionally binding constraints (see e.g., Bluwstein, 2017).

35
also helps reduce the severity of the decline in output by almost half, attenuating the contractionary

eect on the economy and the volatility of credit growth. A similar result is found by Caballero and

Simsek (2020), who showed that macroprudential policies can be welfare improving by reducing the

risk-taking behavior of overly optimistic agents. In our simulation, the macroprudential policy targeting lender solvency is deemed more eective in reducing the depth of the recession during a bad-boom

reversal driven by risk mispricing of nancial markets. Which policy is preferred, of course, will depend

on the exact specication of the calibration and the policy maker's preference function which is beyond

the scope of this paper.

Figure 6: Impulse responses to bad boom-bust scenario under dierent macroprudential policies

Notes: Economic expansion (boom) is generated by risk mispricing shocks with the baseline calibration in solid blue lines,

under a higher target capital-asset ratio in dashed gray lines, and under a lower target LTV ratio in solid gray lines, scaled

such that output increases by 1 percent every year for 3 consecutive years using continuous compounding. The boom is

then counteracted in period 13, labeled a bust. All variables are quarterly, in real terms, and in percent deviations relative

to the steady state.

5 Conclusion

We construct a unifying model of the macroeconomy with a nancial sector to show that incorporating

a feedback loop via bank lending helps quantify the general equilibrium eects of risk mispricing shocks.

Our model generates a time-varying term premium, consistent with empirical evidence, which allows us

to match both macro and nance moments with standard calibrated parameters. We nd that shocks

that might occur during a panic and lead investors to overprice risk, reduce the supply of credit via the

loan price and collateral channels, and signicantly reduce the bank's balance sheet, investment, and

output. Our simulation also shows that a bad credit boom driven by agents underpricing risks is very

dierent from a good credit boom driven by economic fundamentals. Furthermore, we demonstrate how

36
our model can be used for macroprudential policy analysis to foster nancial stability over the business

cycle.

There are many avenues in which the model can be extended. In terms of the banking sector, one

useful addition would be to allow for private loans to default endogenously and thus generate another

source of risk in the model beyond duration risk. By introducing the possibility of corporate default, one

could endogenize the risk premium that is charged on top of private loans based on the relative riskiness

of private debt over government debt. An example could be modeling the empirical relationship between

credit spreads and economic activity found in Faust et al. (2013), in the spirit of Görtz and Tsoukalas

(2017). Note also that in our setup, the reason for risk mispricing, and thus the under- or over-valuation

of long-term bond prices via the term premium, is exogenous. In reality, these perceptions might have

an endogenous cyclical nature. Another interesting avenue to pursue would be to estimate the model

formally. Especially, the household parameters, which are crucial to pin down both macroeconomic as

well as asset price behavior, would benet from estimation. As methods that allow to estimate a model

to the third order (see e.g. Andreasen et al., 2018) are still dicult to implement for high-dimensional

models, we shall leave this possible extension for future investigation.

37
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43
Back to the Real Economy: The Eects of Risk Mispricing on the

Term Premium and Bank Lending

- Appendix -

Kristina Bluwstein* Julieta Yung

Contents

1 A DSGE model with bank lending and risk mispricing 2

1.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Entrepreneurs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Capital good producers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Retailers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Banks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5.1 Wholesale branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5.2 Retail branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.5.3 Prots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6 Government sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 Asset pricing equations and risk mispricing . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.8 Monetary policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.9 Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.10 Equilibrium conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Data 16

3 Robustness 17

3.1 Dierent model specications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Additional impulse response functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

*Bank of England, London EC2R 6DA, UK. Email: kristina.bluwstein@bankofengland.co.uk.

Center for Financial Research, Federal Deposit Insurance Corporation, Washington, DC, 20429, USA. Email:

jyung@fdic.gov.

1
1 A DSGE model with bank lending and risk mispricing

1.1 Households

Equations

Recursive utility function for household i, each of unit mass:

Vt (i) = U(ch,t (i), `h,t (i)) + βh



EtV

1−ξ

t+1 (i)

 1

1−ξ

. (1)

Intra-period utility function:

U(ch,t (i), `h,t (i)) =

c

1−ψ

h,t (i)

1 − ψ

−

`

1+φ

h,t (i)

1 + φ

. (2)

Budget constraint:

ch,t (i) + dt (i) ≤ wt`h,t (i) + (1 + rt−1)dt−1 (i) − Tt (i) + JR,t (i). (3)

Solution

Households choose consumption, labor, and deposits to maximize V0 subject to the utility function (1)

and the budget constraint (3), with λ

h

1,t (i) and λ

h

2,t (i) being the respective multipliers of the Lagrangean

function (L) and substituting Eq. (2) for the intra-period utility function:

L = V0 (i) − E0

X∞

t=0

λ

h

1,t (i)



Vt (i) − U(ch,t (i), `h,t (i)) − βh



EtV

1−ξ

t+1 (i)

 1

1−ξ



− E0

X∞

t=0

β

t

hλ

h

2,t (i) [ch,t (i) + dt (i) − wt`h,t (i) − (1 + rt−1)dt−1 (i) − JR,t (i) + Tt (i)]

L = V0 (i) − E0

X∞

t=0

λ

h

1,t (i)

"

Vt (i) −

c

1−ψ

h,t (i)

1 − ψ

+

`

1+φ

h,t (i)

1 + φ

− βh



EtV

1−ξ

t+1 (i)

 1

1−ξ

#

− E0

X∞

t=0

β

t

hλ

h

2,t (i) [ch,t (i) + dt (i) − wt`h,t (i) − (1 + rt−1)dt−1 (i) − JR,t (i) + Tt (i)] .

2
First order necessary conditions are:

∂L

∂ct (i)

= λ

h

1,t (i)

1 − ψ

1 − ψ

c

−ψ

h,t (i) − β

t

hλ

h

2,t (i) = 0 =⇒ λ

h

2,t (i) =

λ

h

1,t (i)

β

t

h

c

ψ

h,t (i)

(4)

∂L

∂`t (i)

= −λ

h

1,t (i)

1 + φ

1 + φ

`

φ

h,t (i) + β

t

hλ

h

2,t (i) wt = 0 =⇒ λ

h

2,t (i) =

λ

h

1,t (i) `

φ

h,t (i)

β

t

hwt

(5)

∂L

∂dt (i)

= −β

t

hλ

h

2,t (i) + Etβ

t+1

h

λ

h

2,t+1 (i) (1 + rt) = 0 =⇒ λ

h

2,t (i) = βhEtλ

h

2,t+1 (i) (1 + rt) (6)

∂L

∂Vt (i)

= −λ

h

1,t (i) + λ

h

1,t−1

(i)



1

1 − ξ

βh



Et−1V

1−ξ

t

(i)

 ξ

1−ξ

(1 − ξ) V

−ξ

t

(i)



= 0

=⇒

λ

h

1,t (i)

λ

h

1,t−1

(i)

= βh







Et−1V

1−ξ

t

(i)

 1

1−ξ

Vt (i)





ξ

(7)

Obtain the labor supply decision from Eq. (4) = (5):

λ

h

1,t (i) `

φ

h,t (i)

β

t

hwt

=

λ

h

1,t (i)

β

t

h

c

ψ

h,t (i)

=⇒ `

φ

h,t (i) = wt

c

ψ

h,t (i)

Set Eq. (4) = (6), iterate Eqs. (4) and (7) forward, and substitute to obtain the consumption Euler

equation:

λ

h

1,t (i)

β

t

h

c

ψ

h,t (i)

= βhEtλ

h

2,t+1 (i) (1 + rt)

1

c

ψ

h,t (i)

= Et

β

t+1

h

λ

h

2,t+1 (i) (1 + rt)

λ

h

1,t (i)

1

c

ψ

h,t (i)

= Et

β

t+1

h

λ

h

1,t+1 (i) (1 + rt)

λ

h

1,t (i) β

t+1

h

c

ψ

h,t+1 (i)

1

c

ψ

h,t (i)

= Et









βh

(1 + rt)

c

ψ

h,t+1 (i)







EtV

1−ξ

t+1 (i)

 1

1−ξ

Vt+1 (i)





ξ









1 = Et

h

mh

t,t+1 (i) (1 + rt)

i

,

where mh

t,t+1 (i) = βh

c

ψ

h,t (i)

c

ψ

h,t+1 (i)







EtV

1−ξ

t+1 (i)

 1

1−ξ

Vt+1 (i)





ξ

.

3
1.2 Entrepreneurs

Problem

Entrepreneur j of unit mass chooses consumption, labor demand, loan amount, and the stock of physical

capital in order to maximize utility subject to budget and borrowing constraints, where ye,t (j) =

Atk

α

t−1

(j) `

1−α

d,t (j)

max

{ce,t(j),`d,t(j),be,t(j),kt(j)}

E0

X∞

t=0

β

t

e

log (ce,t (j))

s.t. ce,t (j) + (1 + rb,t−1)be,t−1 (j) + wt`d,t (j) + qk,tkt (j)

≤

ye,t (j)

xt

+ be,t (j) + qk,t(1 − δk)kt−1 (j) (8)

be,t (j) ≤

ΩtEt[qk,t+1kt (j) (1 − δk)]

1 + rb,t

Solution

L = E0

X∞

t=0

β

t

e

log (ce,t (j)) − λ

e

1,t (j) [ce,t (j) + (1 + rb,t−1)be,t−1 (j) + wt`d,t (j)

+qk,tkt (j) −

Atk

α

t−1

(j) `

1−α

d,t (j)

xt

− be,t (j) − qk,t(1 − δk)kt−1 (j)

#

− λ

e

2,t (j)



be,t (j) −

ΩtEt[qk,t+1kt (j) (1 − δk)]

1 + rb,t 

∂L

∂ce,t (j)

= β

t

e

1

ce,t (j)

− λ

e

1,t (j) = 0 =⇒ λ

e

1,t (j) = β

t

e

1

ce,t (j)

(9)

∂L

∂`d,t (j)

= −λ

e

1,t (j) wt + λ

e

1,t (j) (1 − α)

"

Atk

α

t−1

(j) `

−α

d,t (j)

xt

#

= 0

=⇒ wt = (1 − α)

"

Atk

α

t−1

(j) `

−α

d,t (j)

xt

#

(10)

∂L

∂be,t (j)

= λ

e

1,t (j) − Etλ

e

1,t+1 (j) (1 + rb,t) − λ

e

2,t (j) = 0

=⇒ λ

e

2,t (j) = λ

e

1,t (j) − Etλ

e

1,t+1 (j) (1 + rb,t) (11)

4
∂L

∂kt (j)

= −λ

e

1,t (j) qk,t + Etλ

e

1,t+1 (j)

"

α

At+1k

α−1

t

(j) `

1−α

d,t+1 (j)

xt+1

+ qk,t+1(1 − δk)

#

+λ

e

2,t (j)



ΩtEt[qk,t+1(1 − δk)]

1 + rb,t 

= 0 (12)

From Eq. (10), the labor demand schedule is

wt = (1 − α)

"

Atk

α

t−1

(j) `

−α

d,t (i)

xt

#

=⇒ wt =

(1 − α)ye,t (j)

`d,t (j) xt

From substituting Eq. (9) into Eq. (11), and dening λe,t (j) ≡

λ

e

2,t(j)

βt

e

, the consumption Euler equation

is

λ

e

2,t (j) =λ

e

1,t (j) − Etλ

e

1,t+1 (j) (1 + rb,t)

λ

e

2,t (j) =β

t

e

1

ce,t (j)

− Etβ

t+1

e

1

ce,t+1 (j)

(1 + rb,t)

λ

e

2,t (j) =β

t

e



1

ce,t (j)

− Etβe

1

ce,t+1 (j)

(1 + rb,t)



1 =Et



me

t,t+1 (j) (1 + rb,t)



where me

t,t+1 (j) = βe



1

ce,t (j)

− λe,t (j)

−1

Et

1

ce,t+1 (j)

Substitute Eq. (9) into Eq. (12) to obtain the investment Euler equation:

λ

e

1,t (j) qk,t = Etλ

e

1,t+1 (j)

"

α

At+1k

α−1

t

(j) `

1−α

d,t+1 (j)

xt+1

+ qk,t+1(1 − δk)

#

+ λ

e

2,t (j)



ΩtEt[qk,t+1(1 − δk)]

1 + rb,t 

β

t

e

1

ce,t (j)

qk,t = Etβ

t+1

e

1

ce,t+1 (j)

"

α

At+1k

α−1

t

(j) `

1−α

d,t+1 (j)

xt+1

+ qk,t+1(1 − δk)

#

+ λ

e

2,t (j)



ΩtEt[qk,t+1(1 − δk)]

1 + rb,t 

qk,t

ce,t (j)

= Etβe

1

ce,t+1 (j)











α

At+1k

α−1

t

(j) `

1−α

d,t+1 (j)

xt+1

| {z }

rk,t+1(j)

+ qk,t+1(1 − δk)











+ λe,t (j)



ΩtEt[qk,t+1(1 − δk)]

1 + rb,t 

5
1.3 Capital good producers

Problem

Capital producers maximize their expected discounted prots:

max

{It}

Et−1

X∞

t=1

me

t−1,t (qk,t (kt − (1 − δk)kt−1) − It)

s.t. kt = (1 − δk)kt−1 +

"

1 −

κi

2



It

It−1

− 1

2

#

It

me

t−1,t =



1

ce,t−1

− λe,t−1

−1

βe

1

ce,t

Solution

Let the change in the marginal value of one unit increase in the budget be equal to the change in the

marginal value of one unit of additional borrowing.

L =Et−1

X∞

t=1

me

t−1,t 

qk,t "1 −

κi

2



It

It−1

− 1

2

#

It − It

!

∂L

∂It

= me

t−1,tqk,t − met−1,t − met−1,tqk,t

κi

2



It

It−1

− 1

2

− me

t−1,tqk,tItκi



It

It−1

− 1



1

It−1

+me

t,t+1qk,t+1κi



It+1

It

− 1



It+1

I

2

t

It+1 = 0

0 = me

t−1,t 

qk,t − 1 − qk,t

κi

2



It

It−1

− 1

2

− qk,tItκi



It

It−1

− 1



1

It−1

+Et

me

t,t+1

me

t−1,t

qk,t+1κi



It+1

It

− 1



It+1

I

2

t

It+1!

1 = qk,t "

1 −

κi

2



It

It−1

− 1

2

− κi



It

It−1

− 1



It

It−1

#

+ Etβe

ce,t

ce,t+1

qk,t+1κi



It+1

It

− 1

 It+1

It

2

1.4 Retailers

Let yt (v) be the quantity of output sold by retailer v and Pt (v) the associated price. Given the nalgoods production technology: yt =



R 1

0

yt (v)

y,t−1

y,t dv y,t

y,t−1

, where y,t represents the elasticity of

substitution between dierentiated nal goods, we derive the retailers demand function

6
max

{yt(v)}

Ptyt −

Z 1

i=0

Pt (v) yt (v) dv

s.t. yt =

Z 1

0

yt (v)

y,t−1

y,t dv y,t

y,t−1

L = Pt

Z 1

0

yt (v)

y,t−1

y,t dv y,t

y,t−1

−

Z 1

i=0

Pt (v) yt (v) dv

∂L

∂yt (v)

= Pt

y,t

y,t − 1

Z 1

0

yt (v)

y,t−1

y,t dv y,t

y,t−1 −1

y,t − 1

y,t

yt (v)

y,t−1

y,t

−1

− Pt (v) = 0

Pt (v) = Pt

Z 1

0

yt (v)

y,t−1

y,t dv 1

y,t−1

yt (v)

−1

y,t

Pt (v) = Pt





Z 1

0

yt (v)

y,t−1

y,t dv y,t

y,t−1





1

y,t

yt (v)

−1

y,t

Pt (v) = Pty

1

y,t

t

yt (v)

−1

y,t

yt (v)

−1

y,t =

Pt (v)

Pt

y

− 1

y,t

t

yt (v) = 

Pt (v)

Pt

−y,t

yt.

We then substitute the demand function into the nal-goods production technology to obtain the aggregate price level Pt

yt =





Z 1

0

 

Pt (v)

Pt

−y,t

yt

!y,t−1

y,t

dv





y,t

y,t−1

1 = Z 1

0



Pt (v)

Pt

1−y,t

dv! y,t

y,t−1

P

y,t

t =

Z 1

0

Pt (v)

1−y,t dv y,t

1−y,t

Pt =

Z 1

0

Pt (v)

1−y,t dv 1

1−y,t

.

7
Problem

Each retailer v chooses its price to maximize the expected discounted value of prots subject to the

demand for consumption goods, which we express in real dollars:

max

{Pt(v)}

Et−1

X∞

t=1

mh

t−1,t "

Pt (v) yt (v) − PW,tyt (v) −

κπ

2



Pt (v)

Pt−1 (v)

− 1

2

Ptyt

#

(13)

s.t. yt (v) = 

Pt (v)

Pt

−y,t

yt

Solution

L = Et−1

X∞

t=1

mh

t−1,t "

Pt (v)



Pt (v)

Pt

−y,t yt

Pt

− PW,t 

Pt (v)

Pt

−y,t yt

Pt

−

κπ

2



Pt (v)

Pt−1 (v)

− 1

2

Pt

yt

Pt

#

∂L

∂Pt (v)

= mh

t−1,t "

Pt (v)

Pt

−y,t yt

Pt

− y,tPt (v)

yt

Pt

P

y,t

t Pt (v)

−y,t−1 + y,tPW,tP

y,t

t

yt

Pt

Pt (v)

−y,t−1

−κπyt



Pt (v)

Pt−1 (v)

− 1



1

Pt−1 (v)



+ Etmh

t,t+1 

κπyt+1 

Pt+1 (v)

Pt (v)

− 1



Pt+1 (v)

P

2

t

(v)



= 0

0 = 

Pt (v)

Pt

−y,t

(1 − y,t) + y,tPW,tP

y,t

t Pt (v)

−y,t−1 − κπPt



Pt (v)

Pt−1 (v)

− 1



1

Pt−1 (v)

+Et

"

mh

t,t+1

mh

t−1,t

κπ



Pt+1 (v)

Pt (v)

− 1

 Pt+1 (v)

P

2

t

(v)



yt+1

yt

Pt

#

0 = 

Pt (v)

Pt

−y,t

(1 − y,t) + y,tPW,tP

y,t

t Pt (v)

−y,t−1 − κπ



Pt (v)

Pt−1 (v)

− 1



Pt

Pt−1 (v)

+Et

"

mh

t,t+1

mh

t−1,t

κπ



Pt+1 (v)

Pt (v)

− 1



Pt+1 (v)

Pt (v)

yt+1

yt

#

.

In equilibrium all retailers behave identically, charging the same price, such that the optimality condition

for prices can be written in terms of ination rates, where Pt

Pt−1

= πt + 1 is the gross ination rate,

1

xt

=

PW,t

Pt

and y,t =

μy,t

μy,t−1

:

0 = 1 −

μy,t

μy,t − 1



+

μy,t

μy,t − 1

1

xt

− κπ (πt + 1) πt + Et

"

mh

t,t+1

mh

t−1,t

κπ (πt+1 + 1) πt+1

yt+1

yt

#

.

8
1.5 Banks

1.5.1 Wholesale branch

The wholesale branch collects deposits dt from the households that pay rtinterest rate and issue wholesale loans bt that receive net interest rate rw,t. The branch manages bank capital kb,t, accumulated out

of reinvested prots, subject to a quadratic cost function that penalizes capital-asset ratio deviations

from their target ν. The optimization problem is to choose the amount of wholesale loans that maximize

the discounted sum of cashows subject to the balance sheet constraint:

max

{bt}

Et−1

X∞

t=1

mh

t−1,t "

(1 + rw,t) bt − bt+1 + dt+1 + ∆kb,t+1 − (1 + rt) dt −

θ

2



kb,t

bt

− ν

2

kb,t#

s.t. bt = dt + kb,t. (14)

Prots are discounted with the stochastic discount factor of the households that own the banks, but our

setup collapses to a per-period static problem, leading to the equilibrium condition for the wholesale

(real) rate in net terms:

L = rw,tbt − rt (bt − kb,t) −

θ

2



kb,t

bt

− ν

2

kb,t

∂L

∂bt

= rw,t − rt + θ



kb,t

bt

− ν



kb,t

b

2

t

kb,t = 0

rw,t = rt − θ



kb,t

bt

− ν

 kb,t

bt

2

.

1.5.2 Retail branch

To model market power, we assume a Dixit-Stiglitz framework for the entrepreneur loans market, such

that units of loan contracts bought by entrepreneurs are a composite CES basket of slightly dierentiated

products supplied by a branch of a bank n with constant elasticities of substitution equal to b > 1.

The demand function for entrepreneur j seeking an amount of real loans equal to be,t (j) can be derived

from minimizing the due total repayment:

9
min

{be,t(j,n)}

Z 1

n=0

rb,t (n) be,t (j, n) dn s.t. Z 1

n=0

be,t (j, n)

b−1

b dn

b

b−1

≥ be,t (j)

L =

Z 1

n=0

rb,t (n) be,t (j, n) dn + λb,t "be,t (j) −

Z 1

n=0

be,t (j, n)

b−1

b dn

b

b−1

#

∂L

∂be,t (j, n)

= rb,t (n) − λb,t

b

b − 1

Z 1

n=0

be,t (j, n)

b−1

b dn

b

b−1 −1

b − 1

b

be,t (j, n)

b−1

b

−1

= 0

rb,t (n) = λb,t Z 1

n=0

be,t (j, n)

b−1

b dn 1

b−1

be,t (j, n)

−1

b

be,t (j, n)

−1

b =

rb,t (n)

λb,t 

be,t (j)

b−1

b

− 1

b−1

be,t (j, n) = 

rb,t (n)

λb,t −b

be,t (j).

This equation can also be rewritten as: rb,t (n) = 

be,t(j,n)

be,t(j)

 1

−b λb,t. Let rb,t =

hR 1

n=0 rb,t (n)

1−b dni 1

1−b

to nd that

rb,t =

Z 1

n=0

rb,t (n)

1−b dn 1

1−b

rb,t =





Z 1

n=0 

be,t (j, n)

be,t (j)

 1

−b

λb,t!1−b

dn





1

1−b

rb,t = λb,t,

such that the nal demand condition for loans to entrepreneurs be,t (n) at bank n depends on overall

volumes and on the interest rates charged by bank n relative to the average rates in the economy:

be,t (n) = 

rb,t (n)

rb,t −b

be,t.

The retail branch problem is then dened as choosing the loan interest rate subject to the demand

for loan rates in a monopolistically competitive environment:

10
max

{rb,t(n)}

Et−1

X∞

t=1

mh

t−1,t [rb,t (n) be,t (n) + rl,tbl,t (n) − rw,tbt (n)]

s.t. bt (n) = be,t (n) + bl,t (n) (15)

be,t (n) = 

rb,t (n)

rb,t −b

be,t

L = rb,t (n) be,t (n) + rl,t (bt (n) − be,t (n)) − rw,t (be,t (n) + bl,t (n))

L = rb,t (n)



rb,t (n)

rb,t −b

be,t + rl,t bt (n) −



rb,t (n)

rb,t −b

be,t!− rw,t 

rb,t (n)

rb,t −b

be,t + bl,t (n)

!

L = r

1−b

b,t (n)



1

rb,t −b

be,t + rl,t 

bt (n) −



rb,t (n)

rb,t −b

be,t!

− rw,t 

rb,t (n)

rb,t −b

be,t + bl,t (n)

!

∂L

∂rb,t (n)

= (1 − b)



rb,t (n)

rb,t −b

be,t + brl,tr

−b−1

b,t (n)



1

rb,t −b

be,t + brw,tbe,tr

−b−1

b,t (n)



1

rb,t −b

0 = (1 − b) + br

−1

b,t (n) (rl,t + rw,t)

rb,t = (rl,t + rw,t)

b

b − 1

.

1.5.3 Prots

Bank prots Jb,t from both branches,

Jb,t = rw,tbt − rt (bt − kb,t) −

θ

2



kb,t

bt

− ν

2

kb,t

+rb,tbe,t + rl,tbl,t − rw,tbt

Jb,t = rb,tbe,t + rl,tbl,t − rtdt −

θ

2



kb,t

bt

− ν

2

kb,t, (16)

get reinvested as bank capital, which depreciates at rate δb in the following accumulation equation:

kb,t = (1 − δb) kb,t−1 + Jb,t−1. (17)

11
1.6 Government sector

Government budget constraint:

Gt + (1 + rl,t−1) bl,t−1 = Tt + bl,t (18)

Government spending (exogenous):

Gt = (1 − ρg) G + ρgGt−1 + εg,t

Supply of government bonds (exogenous):

bl,t = (1 − ρbl) bl + ρblbl,t−1 + εbl,t

1.7 Asset pricing equations and risk mispricing

Long-term bond price formula (including risk mispricing shocks):

pl,t = Et

hmh

t,t+1 + risk,t

(1 + δcpl,t+1)

i

.

In equilibrium, this collapses to the standard asset pricing equation for long-term bonds:

pl =

βh

1 − βhδc

.

The rate of return on the long-term bond (rl,t) equates the price of the long-term bond today to the

present discounted value of cashows on perpetuity paying a coupon normalized to 1 that decays at

rate δc. Using the formula for the innite geometric series and ln (1 + rl,t) ≈ rl,t:

pl,t =

1

1 + rl,t

+

δc

(1 + rl,t)

2 +

δ

2

c

(1 + rl,t)

3 + ...

pl,t =

1

1 + rl,t 

1 + rl,t

1 + rl,t − δc



rl,t = ln 

δcpl,t + 1

pl,t 

.

12
Similarly for the risk-neutral price of the long-term bond (discounted by the risk-free rate) and its

rate of return:

pbl,t = Et



1 + δcpbl,t+1

1 + rt



,

rbl,t = ln 

δcpbl,t + 1

pbl,t 

.

The real term premium is then dened as:

tpl,t = rl,t − rbl,t.

Let risk mispricing be an exogenous process:

risk,t = ρriskrisk,t−1 − εrisk,t.

Finally, to calibrate the decay rate for the long-term government bond, we set the duration of a 10-year

bond (40 quarters) equal to the discounted present value of cashows of a perpetual bond, and solve

for δc by setting rl = 0.007175, such that we match the average of a 10-year real rate on a government

bond of 2.87 percent:

δc = 1 + rl −

1 + rl

40

' 0.982.

1.8 Monetary policy

1 + rt

1 + r

=



1 + rt−1

1 + r

ρ

 

1 + πt

1 + π

φπ 

yt

y

φy

!1−ρ

(1 + εr,t)

1.9 Aggregation

For the aggregate resource constraint, we combine the budget constraint of the households Eq. (3), the

budget constraint of the entrepreneurs Eq. (8), and the government budget constraint Eq. (18). We

substitute retailers' prots JR,t = yt −

yt

xt

−

κπ

2



Pt

Pt−1

− 1

2

yt from Eq. (13), the bank balance sheet

constraint bt − kb,t = dt from Eq. (14), and the bank assets denition bt = be,t + bl,t from Eq. (15).

The clearing condition for the labor market is `h,t = `d,t and the aggregate condition for output is

13
yt = ye,t (j). We use Eq. (17) and the iterated version of Eq. (16) to obtain the aggregate condition:

ch,t + dt + ce,t + (1 + rb,t−1)be,t−1 + wt`d,t + qk,tkt + Gt + (1 + rl,t−1) bl,t−1 = wt`h,t + (1 + rt−1)dt−1

−Tt + JR,t +

ye,t

xt

+ be,t + qk,t(1 − δk)kt−1 + Tt + bl,t

ch,t + ce,t + Gt +

κπ

2

π

2

t

yt = yt + qk,t(1 − δk)kt−1 − qk,tkt + kb,t + (1 + rt−1)dt−1

−(1 + rb,t−1)be,t−1 − (1 + rl,t−1) bl,t−1

ch,t + ce,t + Gt +

κπ

2

π

2

t

yt = yt − qk,tkt + qk,t(1 − δk)kt−1 + (1 − δb) kb,t−1 + rb,t−1be,t−1

+rl,t−1bl,t−1 − rt−1dt−1 −

θ

2



kb,t−1

bt−1

− ν

2

kb,t−1

+(1 + rt−1)dt−1 − (1 + rb,t−1)be,t−1 − (1 + rl,t−1) bl,t−1

ch,t + ce,t + Gt +

κπ

2

π

2

t

yt = yt − qk,tkt + qk,t(1 − δk)kt−1 − δbkb,t−1 −

θ

2



kb,t−1

bt−1

− ν

2

kb,t−1

+kb,t−1 + dt−1 − be,t−1 − bl,t−1

yt = ch,t + ce,t

| {z }

ct

+ qk,t (kt − (1 − δk)kt−1) + Gt +

κπ

2

π

2

t

yt + δbkb,t−1 +

θ

2



kb,t−1

bt−1

− ν

2

kb,t−1

1.10 Equilibrium conditions

(HH) Vt =

c

1−ψ

h,t

1 − ψ

−

`

1+φ

h,t

1 + φ

+ βh



EtV

1−ξ

t+1  1

1−ξ

(HH)

1

1 + rt

= Et









βh

c

ψ

h,t

c

ψ

h,t+1







EtV

1−ξ

t+1  1

1−ξ

Vt+1





ξ









(HH) c

ψ

h,t =

wt

`

φ

h,t

(HH) JR,t = ch,t + dt − wt`h,t − (1 + rt−1)dt−1 + Tt

(Ent) 1 = Et

"

βe



1

ce,t

− λe,t−1

1

ce,t+1

(1 + rb,t)

#

(Ent) ye,t = Atk

α

t−1

`

1−α

d,t

(Ent) rk,t = α

Atk

α−1

t−1

`

1−α

d,t

xt

(Ent) wt =

(1 − α)ye,t

`d,txt

14
(Ent)

qk,t

ce,t

= Et



βe

1

ce,t+1

(rk,t+1 + qk,t+1(1 − δk))+ λe,t 

ΩtEt[qk,t+1(1 − δk)]

1 + rb,t 

(Ent) be,t =

ΩtEt[qk,t+1kt(1 − δk)]

1 + rb,t

(Ent) ce,t =

yt

xt

+ be,t − (1 + rb,t−1)be,t−1 − wt`d,t − qk,t (kt − (1 − δk)kt−1)

(Cap) kt = (1 − δk)kt−1 +

 

1 −

κi

2



It

It−1

− 1

2

!

It

(Cap) 1 = qk,t 

1 −

κi

2



It

It−1

− 1

2

− κi



It

It−1

− 1



It

It−1

!

+ Et

"

βe

ce,t

ce,t+1

qk,t+1κi



It+1

It

− 1

 It+1

It

2

#

(Ret) 0 = 1 −

μy,t

μy,t − 1

+

μy,t

μy,t − 1

1

xt

− κπ (πt + 1) πt + Et

"

mh

t,t+1

mh

t−1,t

κπ (πt+1 + 1) πt+1

yt+1

yt

#

(Bank) bt = be,t + bl,t

(Bank) dt = bt − kb,t

(Bank) kb,t = (1 − δb) kb,t−1 + Jb,t−1

(Bank) rw,t = rt − θ



kb,t

bt

− ν

 kb,t

bt

2

(Bank) rb,t = (rl,t + rw,t)

b

b − 1

(Bank) Jb,t = rb,tbe,t + rl,tbl,t − rtdt −

θ

2



kb,t

bt

− ν

2

kb,t

(MP)

1 + rt

1 + r

=



1 + rt−1

1 + r

ρ

 

1 + πt

1 + π

φπ 

yt

y

φy

!1−ρ

(1 + εr,t)

(Bond) pl,t = Et

hmh

t,t+1 + risk,t

(1 + δcpl,t+1)

i

(Bond) pbl,t = Et



1 + δcpbl,t+1

1 + rt



(Bond) rl,t = ln 

δcpl,t + 1

pl,t 

(Bond) rbl,t = ln 

δcpbl,t + 1

pbl,t 

(Bond) tpl,t = rl,t − rbl,t

(Gov) Tt = Gt − bl,t + (1 + rl,t−1) bl,t−1

(Agg) ct = ch,t + ce,t

(Agg) ye,t = yt

(Agg) `h,t = `d,t

(Agg) yt = ct + qk,t (kt − (1 − δk)kt−1) + Gt +

κπ

2

π

2

t

yt + δbkb,t−1 +

θ

2



kb,t−1

bt−1

− ν

2

kb,t−1

15
Exogenous Processes

(T F P) At = (1 − ρa) A + ρaAt−1 + εa,t

(LT V ) Ωt = (1 − ρΩ) Ω + ρΩΩt−1 + εΩ,t

(MARKUP) μy,t = (1 − ρμy) μy + ρμyμy,t−1 + εμy,t

(RISK) risk,t = ρriskrisk,t−1 − εrisk,t

(BOND) bl,t = (1 − ρbl) bl + ρblbl,t−1 + εbl,t

(GOV ) Gl,t = (1 − ρg) G + ρgGt−1 + εg,t

2 Data

Quarterly from 1961-Q1 to 2016-Q4, expressed in annual terms.

1. Consumption*. Real personal consumption is computed as the period-to-period log growth rates

of real expenditures of non-durable goods and services (SAAR, Bil.$), averaged using their shares

in nominal expenditures. The weighted average growth rate is applied to the sum of nominal

expenditures in both categories in 1961-Q1 to produce chained real consumption with a base of

1961-Q1.

2. Investment*. SSAR, Chn.2009$ log growth of the private domestic investment of chained real

GDP.

3. Labor*. Computed as the amount of aggregate weekly hours of total private production and nonsupervisory employees (SA, Thous.), multiplied by number of weeks in the quarter to produce

quarterly hours of labor. Since the data start in 1964-Q1, business sector compensation per hour

(SA) from the Bureau of Labor Statistics is used to extend the series backwards to the start of

the dataset.

4. Ination. Ination is annualized log growth rate of the chain price index of GDP.

5. Output*. Seasonally adjusted annual log growth rate of chained real GDP.

16
6. Policy Rate. The short-term nominal interest rate is computed as the average discount rate from

1961-Q1 to 1961-Q4; the end-of-period discount rate at Federal Reserve Bank of New York from

1962-Q1 to 1982-Q2; and the Federal Funds target rate from 1982-Q3 to 2016-Q4; converted to

real terms using the ination rate described above.

7. Term Premium. The nominal ten-year Treasury average term premium from the Federal Reserve

Bank of New York, developed by Tobias Adrian, Richard Crump, and Emanuel Moench, which can

be downloaded at https://www.newyorkfed.org/research/data_indicators/term_premia.html.

For details on the methodology refer to Adrian et al. (2013). Data from January 1961 to May

1961 are extended back using the growth rate of the ten-year Treasury note yield at constant

maturity from the Federal Reserve Board.

* HP ltered to extract the cyclical component.

Table: Data sources and summary statistics (19612016)

Variable Name Obs. Mean St. Dev. Min Max Source

Consumption (ct) 224 0.00 0.85 −2.00 2.92 Bureau of Economic Analysis

Investment (It) 224 0.00 4.08 −12.62 9.50 FRB St. Louis

Labor (`t) 224 0.00 2.10 −6.71 4.71 Bureau of Labor Statistics

Ination (πt) 224 3.42 2.45 −0.62 12.77 Bureau of Economic Analysis

Output (yt) 224 0.00 1.46 −4.78 3.75 FRB St. Louis

Policy Rate (rt) 224 1.37 2.73 −5.06 7.66 FRB, FRB Atlanta

Term Premium (tpl,t) 224 1.64 1.19 −0.59 4.94 Federal Reserve Board

3 Robustness

3.1 Dierent model specications

 No loan price channel: the following equilibrium conditions change:

(Bank) bt = be,t

(Bank) rb,t = rw,t

b

b − 1

(Bank) Jb,t = rb,tbe,t − rtdt −

θ

2



kb,t

bt

− ν

2

kb,t

17
 No bank-lending channel: the following equilibrium conditions change when θ = 0:

(Bank) rw,t = rt

(Bank) rb,t = (rl,t + rt)

b

b − 1

(Bank) Jb,t = rb,tbe,t + rl,tbl,t − rtdt

(Agg) yt = ct + qk,t (kt − (1 − δk)kt−1) + Gt +

κπ

2

π

2

t

yt + δbkb,t−1

 No collateral channel: We now have 36 variables and 36 equations to solve for, including 6 stochastic shocks, after removing the exogenous process for the LTV ratio, we set z = 0.64. The following

equilibrium conditions change:

(Ent)

qk,t

ce,t

= Et



βe

1

ce,t+1

(rk,t+1 + qk,t+1(1 − δk))

(Ent) be,t = z

 Preference shock:

Equations

Recursive utility function for household i, each of unit mass:

Vt (i) = risk,tU(ch,t (i), `h,t (i)) + βh



EtV

1−ξ

t+1 (i)

 1

1−ξ

. (19)

Intra-period utility function:

U(ch,t (i), `h,t (i)) =

c

1−ψ

h,t (i)

1 − ψ

−

`

1+φ

h,t (i)

1 + φ

. (20)

Budget constraint:

ch,t (i) + dt (i) ≤ wt`h,t (i) + (1 + rt−1)dt−1 (i) − Tt (i) + JR,t (i). (21)

Solution

Households choose consumption, labor, and deposits to maximize V0 subject to the utility function

18
(19) and the budget constraint (21), with λ

h

1,t (i) and λ

h

2,t (i) being the respective multipliers of

the Lagrangean function (L) and substituting Eq. (20) for the intra-period utility function:

L = V0 (i) − E0

X∞

t=0

λ

h

1,t (i)



Vt (i) − risk,tU(ch,t (i), `h,t (i)) − βh



EtV

1−ξ

t+1 (i)

 1

1−ξ



− E0

X∞

t=0

β

t

hλ

h

2,t (i) [ch,t (i) + dt (i) − wt`h,t (i) − (1 + rt−1)dt−1 (i) − JR,t (i) + Tt (i)]

L = V0 (i) − E0

X∞

t=0

λ

h

1,t (i)

"

Vt (i) − risk,t 

c

1−ψ

h,t (i)

1 − ψ

+

`

1+φ

h,t (i)

1 + φ

!

− βh



EtV

1−ξ

t+1 (i)

 1

1−ξ

#

− E0

X∞

t=0

β

t

hλ

h

2,t (i) [ch,t (i) + dt (i) − wt`h,t (i) − (1 + rt−1)dt−1 (i) − JR,t (i) + Tt (i)] .

First order necessary conditions are:

∂L

∂ct (i)

= λ

h

1,t (i)

1 − ψ

1 − ψ

risk,tc

−ψ

h,t (i) − β

t

hλ

h

2,t (i) = 0 =⇒ λ

h

2,t (i) =

λ

h

1,t (i) risk,t

β

t

h

c

ψ

h,t (i)

(22)

∂L

∂`t (i)

= −λ

h

1,t (i)

1 + φ

1 + φ

risk,t`

φ

h,t (i) + β

t

hλ

h

2,t (i) wt = 0

=⇒ λ

h

2,t (i) =

λ

h

1,t (i) `

φ

h,t (i) risk,t

β

t

hwt

(23)

∂L

∂dt (i)

= −β

t

hλ

h

2,t (i) + Etβ

t+1

h

λ

h

2,t+1 (i) (1 + rt) = 0

=⇒ λ

h

2,t (i) = βhEtλ

h

2,t+1 (i) (1 + rt) (24)

∂L

∂Vt (i)

= −λ

h

1,t (i) + λ

h

1,t−1

(i)



1

1 − ξ

βh



Et−1V

1−ξ

t

(i)

 ξ

1−ξ

(1 − ξ) V

−ξ

t

(i)



= 0

=⇒

λ

h

1,t (i)

λ

h

1,t−1

(i)

= βh







Et−1V

1−ξ

t

(i)

 1

1−ξ

Vt (i)





ξ

(25)

Obtain the labor supply decision from Eq. (22) = (23):

λ

h

1,t (i) `

φ

h,t (i) risk,t

β

t

hwt

=

λ

h

1,t (i) risk,t

β

t

h

c

ψ

h,t (i)

=⇒ `

φ

h,t (i) = wt

c

ψ

h,t (i)

Set Eq. (22) = (24), iterate Eqs. (22) and (25) forward, and substitute to obtain the consumption

19
Euler equation:

λ

h

1,t (i) risk,t

β

t

h

c

ψ

h,t (i)

= βhEtλ

h

2,t+1 (i) (1 + rt)

1

c

ψ

h,t (i)

= Et

β

t+1

h

λ

h

2,t+1 (i) (1 + rt)

λ

h

1,t (i) risk,t

1

c

ψ

h,t (i)

= Et

β

t+1

h

λ

h

1,t+1 (i) risk,t+1(1 + rt)

λ

h

1,t (i) risk,tβ

t+1

h

c

ψ

h,t+1 (i)

1

c

ψ

h,t (i)

= Et









βh

risk,t+1(1 + rt)

risk,tc

ψ

h,t+1 (i)







EtV

1−ξ

t+1 (i)

 1

1−ξ

Vt+1 (i)





ξ









1 = Et

h

mh

t,t+1 (i) (1 + rt)

i

,

where mh

t,t+1 (i) = βh

risk,t+1c

ψ

h,t (i)

risk,tc

ψ

h,t+1 (i)







EtV

1−ξ

t+1 (i)

 1

1−ξ

Vt+1 (i)





ξ

.

Final conditions that change

 (IES) ψ = 0.50

 risk = 1

 εrisk,t enters the exogenous process as a plus such that a positive shock reduces the SDF:

(HH) Vt = risk,t

c

1−ψ

h,t

1 − ψ

−

`

1+φ

h,t

1 + φ

+ βh



EtV

1−ξ

t+1  1

1−ξ

(HH)

1

1 + rt

= Et









βh

risk,t+1

risk,t

c

ψ

h,t

c

ψ

h,t+1







EtV

1−ξ

t+1  1

1−ξ

Vt+1





ξ









(RISK) risk,t = (1 − ρrisk) risk + ρriskrisk,t−1 + εrisk,t

(Bond) pl,t = Et

h

mh

t,t+1 (1 + δcpl,t+1)

i

20
3.2 Additional impulse response functions

Figure 1: Impulse responses to preference shock

Notes: The blue solid line represents the impulse responses to a positive one standard deviation preference shock

under the calibration in Table 1 of the main draft, which we label our baseline specication. The orange dashes

represent a no bank-lending channel scenario by setting θ = 0. The orange dots represent a no collateral channel

scenario by setting the entrepreneur's borrowing constraint to be,t (j) ≤ z, for z > 0. The gold dashes represent a no

loan-price channel scenario, which delivers a loan rate no longer tied to the long-term interest rate, deactivating the

transmission of risk mispricing. All variables are quarterly and in real terms, except for the term premium, which

is annualized and in nominal terms, and the interest rate, which is annualized. Responses are deviations relative to

the steady state in the corresponding units indicated in the vertical axis.

21
Figure 2: Impulse responses to risk mispricing shock with standard preferences

Notes: The blue solid line represents the impulse responses to a risk mispricing shock when (EZ preferences) ξ = 0

(scaled to raise the nominal term premium by 90 basis points) under the calibration in Table 1 of the main draft,

which we label our baseline specication. The orange dashes represent a no bank-lending channel scenario by

setting θ = 0. The orange dots represent a no collateral channel scenario by setting the entrepreneur's borrowing

constraint to be,t (j) ≤ z, for z > 0. The gold dashes represent a no loan-price channel scenario, which delivers a

loan rate no longer tied to the long-term interest rate, deactivating the transmission of risk mispricing. All variables

are quarterly and in real terms, except for the term premium, which is annualized and in nominal terms, and the

interest rate, which is annualized. Responses are deviations relative to the steady state in the corresponding units

indicated in the vertical axis.

22
Figure 3: Impulse responses to productivity shock

Notes: The blue solid line represents the impulse responses to a positive one standard deviation productivity shock

under the calibration in Table 1 of the main draft, which we label our baseline specication. The orange dashes

represent a no bank-lending channel scenario by setting θ = 0. The orange dots represent a no collateral channel

scenario by setting the entrepreneur's borrowing constraint to be,t (j) ≤ z, for z > 0. The gold dashes represent a no

loan-price channel scenario, which delivers a loan rate no longer tied to the long-term interest rate, deactivating the

transmission of risk mispricing. All variables are quarterly and in real terms, except for the term premium, which

is annualized and in nominal terms, and the interest rate, which is annualized. Responses are deviations relative to

the steady state in the corresponding units indicated in the vertical axis.

23
Figure 4: Impulse responses to government spending shock

Notes: The blue solid line represents the impulse responses to a positive one standard deviation government

spending shock under the calibration in Table 1 of the main draft, which we label our baseline specication. The

orange dashes represent a no bank-lending channel scenario by setting θ = 0. The orange dots represent a no

collateral channel scenario by setting the entrepreneur's borrowing constraint to be,t (j) ≤ z, for z > 0. The gold

dashes represent a no loan-price channel scenario, which delivers a loan rate no longer tied to the long-term interest

rate, deactivating the transmission of risk mispricing. All variables are quarterly and in real terms, except for the

term premium, which is annualized and in nominal terms, and the interest rate, which is annualized. Responses are

deviations relative to the steady state in the corresponding units indicated in the vertical axis.

24
Figure 5: Impulse responses to monetary policy shock

Notes: The blue solid line represents the impulse responses to a positive one standard deviation monetary policy

shock under the calibration in Table 1 of the main draft, which we label our baseline specication. The orange

dashes represent a no bank-lending channel scenario by setting θ = 0. The orange dots represent a no collateral

channel scenario by setting the entrepreneur's borrowing constraint to be,t (j) ≤ z, for z > 0. The gold dashes

represent a no loan-price channel scenario, which delivers a loan rate no longer tied to the long-term interest rate,

deactivating the transmission of risk mispricing. All variables are quarterly and in real terms, except for the term

premium, which is annualized and in nominal terms, and the interest rate, which is annualized. Responses are

deviations relative to the steady state in the corresponding units indicated in the vertical axis.

25
Figure 6: Impulse responses to bond supply shock

Notes: The blue solid line represents the impulse responses to a positive one standard deviation government bond

supply shock under the calibration in Table 1 of the main draft, which we label our baseline specication. The

orange dashes represent a no bank-lending channel scenario by setting θ = 0. The orange dots represent a no

collateral channel scenario by setting the entrepreneur's borrowing constraint to be,t (j) ≤ z, for z > 0. The gold

dashes represent a no loan-price channel scenario, which delivers a loan rate no longer tied to the long-term interest

rate, deactivating the transmission of risk mispricing. All variables are quarterly and in real terms, except for the

term premium, which is annualized and in nominal terms, and the interest rate, which is annualized. Responses are

deviations relative to the steady state in the corresponding units indicated in the vertical axis.

26
Figure 7: Impulse responses to price markup shock

Notes: The blue solid line represents the impulse responses to a positive one standard deviation price markup shock

under the calibration in Table 1 of the main draft, which we label our baseline specication. The orange dashes

represent a no bank-lending channel scenario by setting θ = 0. The orange dots represent a no collateral channel

scenario by setting the entrepreneur's borrowing constraint to be,t (j) ≤ z, for z > 0. The gold dashes represent a no

loan-price channel scenario, which delivers a loan rate no longer tied to the long-term interest rate, deactivating the

transmission of risk mispricing. All variables are quarterly and in real terms, except for the term premium, which

is annualized and in nominal terms, and the interest rate, which is annualized. Responses are deviations relative to

the steady state in the corresponding units indicated in the vertical axis.

27
Figure 8: Impulse responses to LTV ratio shock

Notes: The blue solid line represents the impulse responses to a positive one standard deviation LTV ratio shock

under the calibration in Table 1 of the main draft, which we label our baseline specication. The orange dashes

represent a no bank-lending channel scenario by setting θ = 0. The orange dots represent a no collateral channel

scenario by setting the entrepreneur's borrowing constraint to be,t (j) ≤ z, for z > 0. The gold dashes represent a no

loan-price channel scenario, which delivers a loan rate no longer tied to the long-term interest rate, deactivating the

transmission of risk mispricing. All variables are quarterly and in real terms, except for the term premium, which

is annualized and in nominal terms, and the interest rate, which is annualized. Responses are deviations relative to

the steady state in the corresponding units indicated in the vertical axis.

28