Content uploaded by Zhiting Wu
Author content
All content in this area was uploaded by Zhiting Wu on May 12, 2025
Content may be subject to copyright.
Content uploaded by Zhiting Wu
Author content
All content in this area was uploaded by Zhiting Wu on Jan 06, 2025
Content may be subject to copyright.
Content uploaded by Zhiting Wu
Author content
All content in this area was uploaded by Zhiting Wu on Dec 29, 2024
Content may be subject to copyright.
Content uploaded by Zhiting Wu
Author content
All content in this area was uploaded by Zhiting Wu on Nov 19, 2024
Content may be subject to copyright.
Leverage Volatility Risk∗
Liansheng Shentu †
Xiamen University
Yuqi Wang ‡
Xiamen University
Zhiting Wu §
Xiamen University
Current version: May 12, 2025
∗We are largely indebted to insightful comments from Andrew Davis, Oliver de Groot, Ji Huang, Zibin Huang,
Jiangyuan Li, Kai Li, Shusen Qi, Chi-Yang Tsou, Paul Whalen, Zhige Harry Yu, and Haonan Zhou, and the audience
and participants in several conferences and seminars. All errors are our own.
†Contact: guderian2020@gmail.com.; Xiamen, China
‡Contact: wangyuqi1996@stu.xmu.edu.cn.; Xiamen, China
§Corresponding Author, Contact: wuzhiting@xmu.edu.cn.; Xiamen, China
1
Conict-of-interest disclosure statement
Statement:
Liansheng Shentu:
I have nothing to disclose.
Yuqi Wang:
I have nothing to disclose.
Zhiting Wu:
I have nothing to disclose.
2
Leverage Volatility Risk
Current version: May 12, 2025
Abstract
We show that time variations in intermediary leverage volatility aect asset prices. Lever-
age volatility is a negatively priced source of risk. At the rm level, exposure to leverage
volatility risk inversely predicts future returns, yielding a spread across decile portfolios of
5.2 (4.4) percent annually when measuring intermediary leverage with security broker-dealer
(primary-dealer) data. When leverage volatility is high in recessions, equity term premiums
increase, and the intermediary expects low returns in the future, leading to negatively ex-
pected changes in equity yields. Thus, the slope of the equity yield curve is potentially neg-
ative in busts and vice versa.
Keywords: Financial Intermediary, Uncertainty Shock, Financial Leverage.
EFM Classication Codes: 150
JEL Classication: E44, G10, G20
1 Introduction
The nancial intermediary plays a crucial role in both the real economy and the capital market.
1Financial leverage is a key determinant of asset returns in an intermediary asset pricing model.
The prior literature emphasizes time variations of leverage or its rst moment. 2Moving one
step further, this article documents that the intermediary leverage volatility is high during reces-
sions, whose cyclical pattern is similar to nancial uncertainty proposed in Adrian, Boyarchenko,
and Giannone (2019). Our contribution is to show that exposure to time variations in leverage
volatility leads to a negative price of risk and a atter curve of the equity yield, both of which are
explained in an intermediary asset pricing model.
Built on Gertler and Karadi (2011), Christiano, Motto, and Rostagno (2014), and Li and Xu
(2024), we introduce a moral hazard problem associated with a nancial sector within an other-
wise standard intermediary asset pricing model. Financial intermediaries are marginal investors
in all assets. They combine their equity capital (i.e., net worth) with household deposits in their
balance sheet and invest in asset markets. However, nancial intermediaries can also divert a
fraction of assets, leading to a moral hazard problem. The severity of the moral hazard prob-
lem limits how much households are willing to lend to nancial intermediaries, thereby creating
an incentive constraint on equity nancing. Specically, when nancial intermediaries divert
more funds, household depositors are less inclined to supply funds, and the equity nancing
constraint becomes binding. This binding creates an endogenous nancial leverage constraint,
subsequently resulting in lower equilibrium leverage ratios, where endogenous nancial leverage
moves inversely to the fraction of diverted assets.
1For business cycles, see Bernanke and Gertler (1989), Kiyotak and Moore (1997), Bernanke, Gertler, and Gilchrist
(1999), among others; for asset prices, see He and Krishnamurthy (2013), Adrian, Etula, and Muir (2014), He, Kelly,
and Manela (2017), He and Krishnamurthy (2019), among others.
2For example, Adrian, Etula, and Muir (2014) argue that the market-based leverage is high in booms. In contrast,
He, Kelly, and Manela (2017) suggest that the enterprise value-based leverage is high in recessions, motivating Kargar
(2021) to build a heterogeneous intermediary asset pricing model. Fontaine, Garcia, and Gungor (2024) decompose
the rst moment of the leverage shock into demand- and supply-side, while the inconsistency remains in their setting.
1
In our benchmark model, we model the uctuation over time in the severity of the moral
hazard problem in a stochastic process, in which we refer to its magnitude as nancial uncertainty.
As nancial uncertainty increases, agency problems intensify, leading to a decrease in nancial
intermediaries’ net worth. Then the marginal utility of equity capital must rise, making it more
likely that the equity constraint will be binding in the future. Consequently, the present and
future borrowing capacity of nancial intermediaries diminishes, resulting in decreased demand
for risky assets and lower equity prices. This drop in equity prices further reduces intermediary
net worth and current stock returns but increases future stock returns. As a result, investors
demand a negative price of risk associated with nancial uncertainty. Although the fraction
of diverted funds moves inversely to the endogenous nancial leverage ratio, time variations
in leverage volatility resonate positively with nancial uncertainty due to Itô’s lemma within
stochastic processes, leading leverage volatility to carry a negative price of risk.
The uctuation over time of the severity of agency problems is unobservable directly, for
which we estimate observed leverage volatility instead. The observed leverage volatility shares
similar patterns due to Itô’s lemma and is solved from an intermediary asset pricing model. We
rst assume that both the rst and second moments of nancial leverage follow a hidden Markov
chain. We follow Adrian, Etula, and Muir (2014) (AEM) and collect data from the security broker-
dealer to measure the enterprise-value-based leverage and utilize the inverse of He, Kelly, and
Manela (2017) (HKM) proxy from the primary dealer to measure the market-based leverage. We
then follow Hamilton (1990) and use the expectation-maximization (EM) approach to estimate the
rst and second moments for both AEM and HKM leverage. For leverage growth, we nd that
transition probabilities for both states are close to 50 percent in the AEM leverage, implying that
the AEM leverage is barely cyclical. Instead, for the HKM leverage, the estimated probabilities
of remaining in the "Low" and "High" growth states are 0.335and 0.671and vary over time. For
leverage volatility, both the AEM and HKM measures move inversely to the NBER recession index
and are countercyclical.
2
Second, evidence on the pricing of leverage volatility is built on factor loadings obtained from
rolling window time series regressions of stock returns on leverage growth, as well as changes
in its perceived mean and volatility. In the cross-section, we take two approaches to test these
loadings: the Fama and MacBeth (1973) regression and classifying stocks into imitation portfo-
lios. Both approaches nd that loadings on the rst moment of leverage do not help explain
stock returns. Instead, loadings on leverage volatility signicantly and negatively forecast cross-
sectional dierences in stock returns. Specically, the return of holding a long position in the
value-weighted (VW) deciles of stocks with high leverage volatility risk and a short position of
stocks with low leverage volatility risk, which we call the leverage volatility risk () factor,
yields an annual average return of −5.2(−4.4)percent for the AEM (HKM) measure.
We inspect two mechanisms by which leverage volatility risk aects asset prices: the discount
rate and cash ow channels. We document that the AEM leverage volatility risk is negatively
associated with the weighted average of costs of equity capital proposed in Frank and Shen (2016),
but we do not nd similar associations for the HKM leverage volatility risk. Instead, we nd that
the HKM leverage volatility risk is inversely associated with the cash ow channel proposed
in Doyle, Lundholm, and Soliman (2003) to a large extent, while the association between the
AEM leverage volatility risk and cash ows is much weaker. The inverse association between
the HKM leverage volatility risk and cash ows motivates us to augment our benchmark model
with the assumption that time variations in leverage volatility conversely aect dividend growth,
which has a smaller eect on the discount rate. Both models feature time variations in nancial
uncertainty, such that we solve them globally using the projection algorithm.
Third, we extend the impact of time variations in leverage volatility to the slope of the equity
yield curve. We follow Li and Xu (2024) and decompose the slope of the equity yield curve into
equity term premiums and expected changes in equity yields, which move in inverse relation.
Since their data on the dividend strips are from Bansal, Miller, Song, and Yaron (2021) and not
publicly available, we employ the public data and methodology proposed in Giglio, Kelly, and
3
Kozak (2024) to recover the term structure of the equity premiums. For both the AEM and HKM
measurements, time variations in leverage volatility inversely aect expected changes in equity
yields. Specically, we nd that time variations in leverage volatility are positively associated
with equity term premiums using the AEM measure. As both the equity term premiums and
expected changes in equity yields determine the slope of the equity yield curve, the AEM leverage
volatility risk predicts a atter slope of the equity yield curve. 3
A potential concern with our empirical design is that the volatility patterns in leverage are
a byproduct of the hidden Markov Model (HMM) and quarterly frequency data. We address
this concern in two ways. First, we estimate quarterly leverage volatility patterns utilizing the
stochastic volatility (SV) model, which is an approximated HMM with a continuous innite state
space. Second, we utilize the inverse of the daily intermediary equity capital ratio from He, Kelly,
and Manela (2017) to measure the high-frequency HKM leverage. Our estimates suggest that
both the estimates of the SV model and daily HKM leverage volatility risk carry negative prices
of risk. Another concern is that leverage volatility is not the only observed proxy attributed to
nancial uncertainty. Since the intermediary equity capital ratio is, by denition, the inverse of
intermediary leverage, its volatility could also be a candidate proxy for estimating the unobserved
shocks to nancial uncertainty due to the severity of moral hazard problems. We thus utilize the
inverse of the Adrian, Etula, and Muir (2014) proxy and the He, Kelly, and Manela (2017) proxy
to measure the intermediary equity capital ratio, both of whose volatilities also carry a negative
price of risk. All estimates suggest that our empirical nding, in which leverage volatility risk is
a negatively priced source, could be robust.
We conrm our asset pricing predictions in a theoretical model by testing articial data. We
solve for our intermediary asset pricing models and simulate a panel of 1000 economies and
1000 years, each solving for an equilibrium leverage ratio. We nd the average of these 1000
equilibrium leverage ratios and compute its volatility as endogenous leverage volatility at the
3We also examine how leverage volatility aects convenience yields. If you are interested in convenience yields,
please see our Appendix C.
4
aggregate level, and each economy has a rm where investors can claim a return on its risky asset.
We nd that equilibrium leverage volatility is a negatively priced source of risk for articial stock
returns, and it aects the equity yield curve via equity term premiums and expected changes in
equity yield, aligning with observed data.
This research rst contributes to the literature on intermediary frictions about balance sheet
constraints. Both the leverage-constraint theories and equity nancing constraint theories high-
light the nonlinear implications of the marginal value of intermediary net worth for understand-
ing asset price variability. 4We contribute to this strand of literature in two ways. First, de-
spite the importance of the marginal value of net worth, there is an ongoing question about
which specic state variable serves as the optimal indicator of nancial intermediaries (Li and
Xu,2024). Our approach answers this question by utilizing the second moments of intermediary
leverage. Although the intermediary equity capital ratio is the inverse of intermediary leverage,
their volatilities share similar patterns due to Itô’s lemma. Whatever how we identify shocks
to nancial uncertainty caused by the severity of agency problems—the volatilities of either the
intermediary equity capital ratio or the intermediary leverage—we nd similar patterns. Second,
evidence documents seemingly dierent cyclical patterns between the AEM and HKM interme-
diary leverage, resulting in a question of whether the leverage is procyclical or countercyclical.
Fontaine, Garcia, and Gungor (2024) decompose the rst moment of leverage into the demand
side and the supply side to reconcile their dierent behaviors. Beyond a reduced-form approach,
Kargar (2021) unies the rst moment of AEM and HKM leverage in a heterogeneous nancial
intermediaries model. We answer this question by showing that the second moments of both
leverage measures are countercyclical. We thus unify the AEM and HKM leverage volatility in a
simple representative agent model and reduce the complexity of modeling. 5
4For leverage-constraint theories, see Fostel and Geanakoplos (2008), Adrian and Shin (2010), Geanakoplos (2010),
Adrian and Boyarchenko (2012), and Fontaine, Garcia, and Gungor (2024) among others. For equity nancing con-
straint theories, please see Brunnermeier and Pedersen (2009), Gertler and Karadi (2011), He and Krishnamurthy
(2013), Brunnermeier and Sannikov (2014), He and Krishnamurthy (2019) among others.
5Despite dierent patterns in the rst moments of dierent leverage measurements, their rst moments are
not priced. Without loss of generality, we only include the second moments of intermediary leverage in our model.
5
Second, our paper extends the literature on time variations in nancial uncertainty. Early
research into nancial frictions utilizes log-linearized solutions around the steady state, which
oer limited insights into nonlinear eects (Bernanke and Gertler,1989;Kiyotak and Moore,
1997;Bernanke, Gertler, and Gilchrist,1999;Gertler and Karadi,2011). The subsequent macro lit-
erature incorporates nancial frictions as propagations amplifying uncertainty shocks (Gilchrist,
Sim, and Zakrajšek,2014;Arellano, Bai, and Kehoe,2019;Fang and Liu,2021;Alfaro, Bloom, and
Lin,2024). Many of these approaches study how real economic uncertainty shocks aect nan-
cial constraints, but not nancial uncertainty shocks. Instead, we model nancial uncertainty
shocks due to the severity of agency problems similar to Christiano, Motto, and Rostagno (2014).
Recent studies measure nancial volatility via stock returns (Berger, Dew-Becker, and Giglio,
2019), the National Financial Conditions Index (Adrian, Boyarchenko, and Giannone,2019), and
the nancial uncertainty index (Ludvigson, Ma, and Ng,2021). Instead, we estimate uncertainty
on intermediary leverage, utilizing the time variations of an observable state variable to reect
the patterns of unobserved uncertainty in the severity of agency problems. Much research builds
models in a continuous-time environment (He and Krishnamurthy,2013;Brunnermeier and San-
nikov,2014) and relies on Brownian motion to model time variations in nonlinear uctuation
eects but lacks empirical support, while our leverage volatility is empirically grounded.
Finally, this research also belongs to the intermediary asset pricing literature. Models of nan-
cial sectors have succeeded in explaining equity premium variability and have recently been ex-
tended to explain the equity yield curve. 6We build a new risk factor based on leverage volatility,
which carries a negative price of risk. Empirical ndings reach no agreement on the unconditional
slope of equity yields. Arguments have been made that it is downward- or upward-sloping, but
Although the volatility measured by AEM and HKM shows dierent associations with dividend growth, we augment
the benchmark model with predictable dividend growth to address this issue without rm heterogeneity.
6For domestic nancial markets, see He and Krishnamurthy (2013), Adrian, Etula, and Muir (2014), He, Kelly, and
Manela (2017), He and Krishnamurthy (2019), Kargar (2021), and Ma (2023) among others; for international markets,
see Maggiori (2017),Fang (2021), and Fang and Liu (2021) among others; and recently, Li and Xu (2024) extend models
of nancial sectors to examine equity yields.
6
there is consensus on its procyclical behavior. 7In a model of nancial sectors, Li and Xu (2024)
decomposes the equity yield curve into equity term premiums and changes in equity yields. How-
ever, Li and Xu (2024) relies on private data from Bansal, Miller, Song, and Yaron (2021), which
restricts the popularization of their methodology. We unify approaches introduced in Li and Xu
(2024) and Giglio, Kelly, and Kozak (2024), which can be broadly used in future research with pub-
licly available data. Based on our empirical methodology, we show that leverage volatility could
raise equity term premiums but reduce expected changes in equity yields. As both determinants
of the slope of equity yields depend on leverage volatility, we leave space for those who believe
in a downward-sloping and upward-sloping equity yield curve.
2 Empirical Evidence
Empirically, we rst estimate the rst and second moments of the leverage in a hidden Markov
model. At the rm level, we show that leverage volatility is a negatively priced source of risk.
Finally, we nd that changes in leverage volatility decrease expected changes in equity yields and
potentially raise the equity term premium, both of which determine the equity yield curve.
2.1 Data and Measurements
2.1.1 Data
The data sources are as follows:
1. Leverage. The quarterly total nancial assets and liabilities of broker-dealers in the United
States come from the Federal Reserve Flow of Funds.8The quarterly and daily HKM inter-
7See van Binsbergen, Brandt, and Koijen (2012), van Binsbergen, Hueskes, Koijen, and Vrugt (2013), van Bins-
bergen and Koijen (2017), Gormsen (2021) for downward-sloping; see Bansal, Miller, Song, and Yaron (2021), Giglio,
Kelly, and Kozak (2024), Li and Xu (2024) for upward-sloping.
8The data in Flow of Funds are also available from the Federal Reserve Economic website in St. Louis. Fed.
7
mediary equity capital ratios come from the personal website of Zhiguo He. 9The quarterly
data cover 1970Q1 to 2023Q4, and the daily samples cover January 1st 2000 to December
11th, 2018. The inverse of the intermediary equity capital ratio is the HKM leverage. 10
2. Stock Returns and Firm Characteristics. The monthly returns and prices of all common
stocks traded on the NYSE, AMEX, and NASDAQ come from the Center for Research in
Security Prices (CRSP). The annual rm characteristics come from Compustat. The sample
is from 1970Q1 to 2023Q4 to match the leverage series.
3. Other Asset Pricing Factors. The monthly size, value, and momentum factors come from
Ken French’s website. 11 The monthly investment, return on equity (ROE), and expected
growth (EG) factors in the factor model come from Zhang Lu’s website. 12 The sample is
still from 1970Q1 to 2023Q4.
4. Equity Term Structure and Relative Tightness Index. We obtain monthly equity yields with
maturities from one to ve years from Serhiy Kozak’s website. 13 The monthly relative
tightness index (RT) is from Li and Xu (2024). Data on equity yields are from 1974Q4 to
2020Q2, and data on the RT index are from 1970Q1 to 2020Q4.
2.1.2 Financial Leverage
In this paper, we use two widely accepted measures of nancial leverage: Adrian, Etula, and Muir
(2014) (AEM) computed from the broker-dealer leverage and the inverse of He, Kelly, and Manela
(2017) intermediary equity capital ratio computed from primary dealer leverage (HKM). Some
9More details can be found at https://zhiguohe.net/data-and-empirical-patterns/intermediary-capital-ratio-and-
risk-factor/
10Following Adrian, Etula, and Muir (2014), the raw data from the Flow of Funds cover from 1968Q1 to 2023Q4.
The leverage ratio and its growth rate for the AEM measure are from 1968Q4 to 2023Q4 after seasonal adjustment.
To keep the series the same length as the HKM measure, we take all leverage series beginning at 1970Q1.
11More details can be found at http://mba.tuck.dartmouth.edu/pages/faculty/ ken.french/data library.html.
12More details can be found at http://global-q.org/factors.html
13More details can be found at https://www.serhiykozak.com/data.
8
might argue that the primary dealer is a subset of the broker-dealer, and these two measurements
dier because the former is computed in book equity while the latter is computed in market
value. Without loss of generality, we do not distinguish the "AEM leverage," "security broker-
dealer leverage," and "enterprise-value based leverage," or the "HKM leverage," "primary-dealer
leverage," and "market-value based leverage" in this article. We rst follow Adrian, Etula, and
Muir (2014) (AEM) and measure the leverage obtained from the data of security broker-dealers
as: ≡Total Financial Assets
Total Financial Assets
−Total Liabilities
Here, Total Financial Assets..
and Total Liabilities..
are the total nancial assets and total -
nancial liabilities of security broker-dealers at time , which are available from Table L.130 of the
Federal Reserve Flow of Funds. Table L.130 oers an aggregate balance sheet of broker-dealers
in the United States. The broker-dealers dened in the Flow of Funds often hold company sub-
sidiaries of banks, such that AEM leverage captures broker-dealer subsidiary-level information.
Alternatively, we also follow He, Kelly, and Manela (2017) (HKM) and measure the interme-
diary leverage of primary dealers as the inverse of their intermediary equity capital ratio:
1
=∑Market Equity,
∑(Market Equity, +Book Debt,)
Here, ∑Market Equity, and ∑Book Debt, are, respectively, the aggregate value of market eq-
uity and the aggregate book debt of primary dealers who are active at time . HKM leverage uses
the market value instead of book equity, which captures the information contained in the current
market conditions. HKM leverage also regards primary dealers as a more diverse set of marginal
investors subject to more types of companies, which leads to a dierent leverage pattern.
We then construct the growth rate of the leverage ratio, which contains more statistical in-
formation than the level of data, but remains similar to cyclical patterns. The AEM leverage
growth rate, denoted Δ, is dened as the log changes in after seasonal adjustment.
9
We follow Adrian, Etula, and Muir (2014) and run an expanding-window regression of on
seasonal dummies at each date to obtain the deseasonalized growth rate. We also follow He, Kelly,
and Manela (2017) and compute the HKM leverage growth rate, which is denoted as Δ. It
is dened as shocks to leverage at each date divided by the lagged . Shocks to leverage
are innovations in the autoregression process of .
2.2 Identication Methodology
Estimating Leverage Dynamics. We assume that the growth rate of the leverage ratio, Δ, ∈
{AEM,HKM}, follows a Markov regime-switching model, which is given by:
Δ=
+
+1,
+1 ∼(0,1).(1)
where
is the conditional mean and
is the conditional standard deviation for Δ. To avoid
the under-identication issue in the estimation, for each leverage growth rate series, we assume
there are two states for the mean, which are denoted by
∈ {
,
}, and two states for the
standard deviation, which are denoted by
∈{
,
}with
>
,
>
>0. Hereafter, we
omit the superscript, , for convenience. Further, we assume the transition probability matrices
14 for parameters of the mean and volatility are given by:
Pr(𝝁+1|𝝁) =
1−
1−
,Pr(𝝈+1|𝝈) =
1−
1−
.(2)
where = {,},= {,}, and the elements of
(or
,∈ {,}) in the above matrix
denote the probability of (or ) remaining in its current state next period. The transition matrix
in Equation (2) implies that the transition probabilities of and are independent. There are two
14The (m,n)-th element of the transition matrix is the transition probability from state m to state n, where , ∈
{,}.
10
good reasons to make such an assumption: First, it can greatly reduce the number of parameters
that need to be estimated. Second, most papers show evidence that a Markov model with two
states, each for mean and volatility, can capture enough information to reect aggregate series,
such as consumption or TFP series.
We follow Hamilton (1990) and use the expectation maximization (EM) algorithm to estimate
the parameters of the conditional mean and volatility, as well as the elements in the transition
matrices. This algorithm recursively uses the "E" step and the "M" step to estimate the state
process and parameters, respectively. In the "E" step, we use a particle lter with 1000 particles
to calculate the expectation of a hidden state instead of using traditional numerical integration.
In the "M" step, we use the coordinate ascent algorithm to solve the likelihood maximization. 15
Perceived Moments. We follow Boguth and Kuehn (2013) and assume that a representative
agent in the economy would estimate the subsequent state in leverage using his prior beliefs.
Prior beliefs regarding the "Low" state of the mean and volatility are respectively given by:
≡Pr(+1 =|)and ≡Pr(+1 =|)(3)
Here, represents all the information that the agent can obtain at time .16 We can then dene
the perceived rst and second moments of leverage growth as belief-weighted averages:
=+(1−)and =+(1−)(4)
The innovations of both perceived moments dened in Equation (4) are given by,
Δ=−−1 and Δ=−−1 (5)
15There is no dierence to using a simpler EM algorithm. We use this method simply because it can be easily
extended to a nonlinear and high-dimensional case.
16The information lter includes the posterior beliefs over current states and the estimated parameters in
Equations (1) and (2).
11
Both changes in beliefs and economic states potentially aect asset prices. Thus, we dene inno-
vations to both perceived moments in Equation (5) as characteristic-based pricing factors. 17
Fama-MacBeth Regressions. We use Fama and MacBeth (1973) regression to estimate risk
premiums regarding the rst and second moments of leverage. First, we obtain the factor loadings
for each stock by running a rolling-window regression at a quarterly frequency at time ∗:
−
=
∗+
,∗Δ+
,∗Δ+𝕀=HKM
,∗Δ+
,∗Mkt+∗(6)
Here, and
are, respectively, the quarterly returns of rm and the risk-free rate; Δ is the
leverage growth dened in section 2.1.2 and its risk loading is
,∗;Δis the rst moment of
Δwith risk loading
,;Δis the second moment (leverage volatility) of Δwith associated
risk loading
,∗; Mkt is the market factor with risk loading,
,∗. The rolling window is chosen
from ∗−19to ∗, 20 quarters or 5 years in total. The indicator, 𝕀=HKM, denotes whether we include
the market factor and rst moment of Δin our regressions. 18
Second, we regress monthly returns on their latest available risk loadings and obtain the risk
price series of leverage volatility factors for each stock, 19 at every month, which is denoted by 20
=, +
,, +
,, +
, , +
(7)
where
, and
, are the risk loadings estimated in Equation (6), and represents other
commonly used pricing factors. The price of risk on a factor is then obtained by running regres-
17Note that the innovation in the mean is another potential risk factor. However, our empirical results show that
the magnitude of innovations in the mean is too small to be considered a risk factor.
18Note that the denition of involves the market value. Therefore, in our empirical specication for the
HKM measurement, we include the market factor. Furthermore, since only the HKM exhibits a signicant state in
mean, we also incorporate a time-varying leverage mean for the HKM.
19Using the latest available risk loadings does not mean we must update them quarterly. Most papers update their
factor loadings on an annual frequency. See Fama and French (1993) and Boguth and Kuehn (2013).
20We have controlled for market risk in the rst pass of the Fama-MacBeth regression. Thus, we drop it in the
second pass cross-section regression. There are a few changes in the empirical results if we reinclude this factor in
the cross-section regression.
12
sion
, = +with Newey-West standard errors with 12 lags. We follow Fama and
French (1993) and Boguth and Kuehn (2013) and update the factor loadings on Δ and Δin
July of every calendar year for AEM and in June for HKM.
We check the robustness of the leverage volatility factor using dierent controls in the re-
gression in Equation (7). These controls include: market equity, ,, book-to-market ratio ,,
12-month momentum, ,, 5-year idiosyncratic volatility, ,, investment-to-assets, ,,
and expected investment growth, ,. We follow Fama and French (1993) and construct size and
value factors, and follow Hou, Mo, Xue, and Zhang (2021) and formulate the and expected in-
vestment growth factors. The momentum factor is the sum of the prior 11-month returns (except
the latest month), and the idiosyncratic volatility (IVOL) denotes the standard error of residuals
from the 5-year rolling Fama-French 3-factor model.
Mimicking Portfolios. Alternatively, we sort stocks into mimicking portfolios based on
estimated factor loadings. We dene the leverage volatility risk () factor as the return of
holding a long position in the decile portfolio of stocks with high leverage volatility loadings and
a short position in the decile portfolio of stocks with low leverage volatility loadings. The sign
of the leverage volatility mimicking factor should be consistent with the estimated obtained
from the second-pass cross-sectional regression.
We obtain the risk-adjusted (i.e., abnormal returns) for our mimicking factor via re-
gressions on widely recognized factor pricing models, consisting of the CAPM, the Fama and
French (1993) 3-factor model (FF3), the Carhart (1997) 4-factor model (Carhart4), the Fama and
French (2015) 5-factor model (FF5), the Fama and French (2018) 6-factor model (FF6), the Hou,
Xue, and Zhang (2015)-factor model, and its augmented model extended in Hou, Mo, Xue, and
Zhang (2021) (5). We conjecture that the estimated risk-adjusted resulting from various factor
pricing models with dierent controls should remain statistically signicant and keep the same
sign as the estimated prices of risk.
Decomposing the Equity Term Structure. In our third empirical part, we study the eect
13
of leverage volatility on equity term structure. Using a denition from van Binsbergen, Hueskes,
Koijen, and Vrugt (2013), a stock price at time is denoted as and can be written as the sum of
a series of claims on all its future dividends, {+}∞
=1, yielding
≡∞
=1 𝔼[,+]= ∞
=1 , (8)
where , represents the maturity-n pricing kernel at time ;, is dened from , ≡𝔼[,+]
and represents the price of a claim on maturity-(+)dividends at time ., is also known as
the "dividend strip" or "zero-coupon equity." Given the concept of dividend strips, the equity yield
at time with maturity can be dened as
, ≡1
ln(
,).(9)
In practice, the forward price, denoted as ,, is more widely recognized than the "spot" price,
,. As such, we obtain the forward equity yield, which is dened as follows:
, ≡1
ln(
,).(10)
In an arbitrage-free environment, the spot price is identical to the forward price, such that
, =, exp(,),(11)
which yields
, =, −,.(12)
where , denotes the nominal bond yield at time with the same maturity . Given the denition
of (or forward) equity yields, the conditional and unconditional slopes of (or forward) equity
14
yields are denoted by
Conditional: , −,1(or
, −
,1)
Unconditional: 𝔼[, −,1] (or 𝔼[
, −
,1]) (13)
The sign of the (conditional or unconditional) slope of (or forward) equity yields is key in the
literature on equity term structure (van Binsbergen, Hueskes, Koijen, and Vrugt,2013;van Bins-
bergen and Koijen,2017;Bansal, Miller, Song, and Yaron,2021;Gormsen,2021). We follow Li and
Xu (2024) and emphasize the slope of forward equity yields, excluding the nominal bond.
Accordingly, we decompose the slope of equity yields into two components: (i) the equity term
premium and (ii) the expected changes in yields. As proved in Li and Xu (2024), the (conditional)
slope of forward equity yields is given by:
, −
,1=1
=1(, −,1)
equity term premium
+1
=1(−1)𝔼(
+1,−1)−
.−1
expected change in forward equity yields
(14)
where , represents the maturity-n equity premium at time . The maturity-n equity premium
is equivalent to the expected one-period holding log return on the forward of the dividend strip,
which is denoted by: , ≡𝔼ln(+1,−1
, ) =𝔼[
,].(15)
where
, denotes the one-period holding log return on dividend futures. However, the dividend
forward data used in the Li and Xu (2024) are not publicly available. Hence, we obtain publicly
available equity yield data from Giglio, Kelly, and Kozak (2024) and then use their approach to
transform the equity yields into the equity premium. We provide more details in Appendix A.
Li and Xu (2024) state that their decomposition framework is an algebraically convenient way
to investigate the determinants of the slope of equity yield. In line with their framework, we test
15
the eect of leverage volatility on the slope of the equity yield curve in the following steps.
1. In the rst stage, we estimate the overall eect of leverage volatility on equity yield slopes.
To this end, we regress forward equity yields on leverage volatility as follows: 21
, −
,1=1
0+1
+1.(16)
Note that the changes in leverage volatility are small, but for better interpretability, we use
leverage volatility as the regressor instead of the leverage volatility factor.
2. In our second stage, we regress the equity term premium, dened in Equation (14), on
leverage volatility. Since the expected log returns on dividend futures, ,, are unobservable,
we follow Li and Xu (2024) and adopt the ex-post log return on dividend futures as its ex-
ante proxy, which yields the following regression:
1
=1(
+1, −
+1,) =2
0+2
+2.(17)
3. In terms of the expected changes in yields, we regress expected changes in equity yields,
dened in Equation(14), on leverage volatility. We follow Li and Xu (2024) and employ
the ex-post forward equity yields to approximate their ex-ante measurement. The keynote
regression is then given by
1
=1(−1)(
+1,−1 −
,−1)= 3
0+3
+3.(18)
In addition, Li and Xu (2024) consider the RT index as a measure of the tightness of nancial con-
straints and as being able to capture variations in the slope of equity yields. It is straightforward
21Theoretically, both maturity and time are of the same frequency. In reality, maturity is often annual, but time
is monthly. We follow Li and Xu (2024) and maintain the consistency of this frequency in which we replace
+1,−1
with
+12,−1.
16
to conjecture that leverage volatility can interact with nancial constraints and have an impact
on the equity yield slope. We intend to understand their interaction through testing regressions
from Equations (16) to (18) again while incorporating an additional RT index term.
2.3 Empirical Findings
2.3.1 Estimates of Leverage Dynamics
The estimates of the Markov switching model are displayed in Table 1. In Panel A, the conditional
volatility in the "H" state for AEM is around 0.99 percent, with a standard error of 0.058 percent.
For the "L" state, the conditional volatility is 0.45 percent, with a standard error of 0.035 percent.
In Panel B, the conditional volatility in the "H" state for HKM is 1.92 percent, with a standard error
of 0.17 percent, and for the "L" state, it is 0.69 percent, with a standard error of 0.058 percent. Our
estimates illustrate the existence of leverage volatility dynamics for each leverage factor. Table 1
also shows that the probability of remaining in the "L" state is 0.60for both the AEM and HKM.
The likelihood of staying in the "H" state is 0.43for AEM and 0.41for HKM. As a result, the low
state is more persistent for both leverage measures.
However, the cyclical patterns of the rst moment of leverage growth are quite dierent. As
shown in Table 1, for the AEM measure, the conditional mean in the "High" state is 0.072 percent
with a standard error of 0.084 percent, and in the "Low" state, it becomes −0.03percent with a
standard error of 0.084 percent. Also note that the values of both
and
are very close to 0.5.
These results show that the rst moment of AEM leverage in both states is statistically close to
zero, which is barely cyclical. However, for the HKM measure, the conditional mean in the "High"
state is 0.75 percent with a standard error of 0.2 percent, and in the "Low" state, it becomes −0.53
percent with a standard error of 0.087 percent. The probability of remaining in the "Low" state is
0.65, and that of remaining in the "High" state is 0.34. These results imply time variations in the
rst moment of the HKM leverage.
17
Next, we construct the leverage growth factor and leverage volatility factor via changes in the
rst and second moments of leverage, respectively. 22 For each measurement, Figure 1presents
changes in the rst and second moments of leverage as characteristic-based factors. Table 2
presents their correlation matrix, in addition to the NBER recession indicator, 𝕀, and the
volatility index, VIX.
Figure 1and Table 2show detailed patterns for each estimated leverage volatility growth
and its innovations. First, the coecients of correlation of the NBER indicator to ,Δ,
, and Δ are 0.074, 0.0091, 0.31, and 0.095, respectively. As such, the volatility in both
measurements is countercyclical. Second, the coecients of correlation of VIX to Δ and
Δ are −0.006and 0.11, which are much lower than the correlation of leverage volatility with
VIX. As such, innovations to the volatility of leverage are less related to economic uncertainty.
Third, the correlation coecient between and is 0.16, and that between Δ and
Δ is 0.13. Despite both leverage volatilities being countercyclical, their correlation is lower.
2.3.2 Volatility Factor Pricing
Table 3shows risk premiums estimated from the second pass of the Fama and MacBeth (1973)
regression. Model specications I to II show the estimated for each leverage pricing factor,
and model specications III to VI show estimated risk premiums when we control for additional
pricing factors. Model III combines both the leverage growth and volatility factors; Model IV
additionally controls for size and value factors; Model V adds the and momentum factors;
Model VI extends Model V with the and expected investment growth factors. We document
that the exposure to leverage volatility risk signicantly commands a negative price of risk (the
single -values for AEM and HKM are −2.84 and −1.87, respectively). This negative risk price
remains largely impacted when controlling for other pricing factors.
22Since we nd no evidence of time variations in the rst moment of AEM leverage, a 3-factor model, including
leverage growth, as well as changes in its rst and second moments, does not improve our empirical performance.
18
2.3.3 Cross-Sectional Stock Returns
Table 4presents mimicking portfolios sorted on the leverage volatility risk. We report the average
return of the value-weighted (VW) deciles as well as the long-short spread return of the rst and
last decile (WML). For the AEM measure, the return of the high-minus-low value-weighted LVR
portfolio is 0.42percent (with a Newey-West -value of −2.20, henceforth "") per month. For the
HKM measure, the return of the high-minus-low value-weighted LVR portfolio is −0.36percent
(=−1.79). As such, a long-short investment strategy based on the factor loadings obtained from
the leverage volatility risk can yield a statistically signicant negative premium of around 5.2
percent for AEM leverage and 4.4 percent for HKM leverage per annum.
Table 4also documents abnormal returns (i.e., risk-adjusted ) accruing to the factor
when adjusted by commonly used factor pricing models. For the AEM measure, the abnormal
returns of high-minus-low value-weighted LVR portfolios range from −0.42 to −0.87 percent
(with = −2.19 and = −4.21, respectively). For the HKM measure, the adjusted abnormal
returns of the factor are smaller in magnitude than those of the AEM measure. Second, we
nd that seven value-weighted portfolios have monthly excess returns higher than −0.24percent
(=−1.66). In sum, this factor survives in most empirical tests and seems robust.
2.3.4 Inspecting the Mechanism
Our previous analysis documents a negative price of risk led by leverage volatility. This section
investigates how leverage volatility works through two channels: the discount rate and the cash
ow.
Discount Rates. For cross-sectional stock returns, we test the discount rate and cash ow
channels utilizing rm-level weighted average cost of capital and future cash ows. We measure
the discount rate via the weighted average cost of capital (WACC), a weighted average of the
19
implied cost of capital (ICC) and the cost of debt, as follows:
WACC_ICC, ≡,, ∗(1−
, )+,, ∗
, ∗(1−,)(19)
Here,
, denotes the leverage of rm , a quotient of the value of equity over the total value
of the rm. We calculated
, from Compustat items (Item DLTT +Item DLC)/[Item AT +
(Item PRCC×Item CSHO)−Item SEQ−Item TXDB]. Tax is the corporate average tax rate, which
is calculated by Compustat items (Item TXT/Item PI). is the average cost of debt calculated by
[Item XINT/(Item DLTT+Item DLC)]. Specically, for the ICC, we follow Frank and Shen (2016)
and compute the average of four widely used measures as our proxy for the implied cost of capital,
which is calculated by: =1
4 + + +(20)
Here, ,,, and are the ICC proxies introduced in Claus and Thomas (2001), Geb-
hardt, Lee, and Swaminathan (2001), Ohlson and Juettner-Nauroth (2005), and Easton (2004),
respectively. To test the discount rate channel, we regress the weighted average cost of capital
on factor loadings associated with leverage volatility:
WACC_ICC, =
, +controls, ++1
+2
+, (21)
Here,
, are the factor loadings, is the coecient that represents the associated eect of
the risk factors on the discount rate, controls denotes the set of control variables, represents
the time xed eects, 1
denes the industry xed eect based on the Fama-French 48 industrial
classication, 2
denotes the rm-level xed eects, and , represents the residuals. The selected
control variables consist of the quarterly market leverage (lev), log rm size (lsize), log book-to-
market ratio (lbm), return on assets (roa), idiosyncratic volatility (IVOL), and risk loadings of the
market portfolio obtained from CAPM ().
Table 5presents our test results. We nd that loadings on the AEM leverage volatility in-
20
versely move to the rm-level weighted average cost of capital, indicating a negative price of
risk, in which all estimates are statistically signicant at the 1% level. This eect is explained
by the cross-sectional sensitivity of the discount rate to leverage volatility. Stocks with volatile
AEM measured intermediary leverage in uncertain aggregate times require lower expected re-
turns. Instead, we do not nd the association between loadings on the HKM leverage volatility
and the rm-level weighted average cost of capital.
Cash Flows. We follow Doyle, Lundholm, and Soliman (2003) and denote the (quarterly) fu-
ture cash ow at time as the summation of (quarterly) cash ows from +1to +1+. We
measure future cash ows in two approaches: the operating cash ow (CFO) obtained from the
Compustat item OANCF and free cash ow (FCF) dened as the CFO minus the capital expendi-
tures (Compustat item CAPX).
The argument in Doyle, Lundholm, and Soliman (2003) suggests that it is challenging to de-
termine an exact value for since nobody knows when cash ow would respond to changes
in the risk factor. Hence, we adopt their approach and choose to be 13, 25, and 37, which
correspond to the next 1, 2, and 3 years, respectively. We dene the CFO and FCF for the next
years (∈{1,2,3}) as CFO_sumand FCF_sum, respectively. We then regress future cash ows
on factor loadings as follows:
CFO_sum(or FCF_sum)=Γ
, +Growth +Accurals +constant +, (22)
Here,
, are the factor loadings, Γis the coecient of the associated risk factor on future
cash ows. We follow Doyle, Lundholm, and Soliman (2003) and control for Accruals dened
as GAAP earnings (Compustat item EPSPI) minus CFO, and may predict future cash ows. We
also control for growth, which is the one-year change in quarterly sales (Compustat item SALE),
because growing rms tend to invest more in long-term capital and consequently lower their
short-term cash ows. As stated in Doyle, Lundholm, and Soliman (2003), using standard panel
21
regression faces a serious econometric issue of high autocorrelation because these dependent
variables overlap between consecutive observations. Instead, we use Fama-MacBeth regression.
Table 6reports our ndings. We nd a much weaker negative association between loadings
on the AEM leverage volatility and future cash ows, in which only the rst-year free cash ows
respond to the AEM leverage volatility. However, loadings on the HKM leverage volatility are
negatively associated with future cash ows to a large extent. We explain this nding via the
cross-sectional sensitivity of dividends to leverage volatility. Stocks with volatile HKM measured
intermediary leverage in uncertain aggregate times require lower cash ows.
2.3.5 Testing Equity Term Structure
To test the eects of leverage volatility on equity yields, we rst follow Li and Xu (2024) and
decompose equity yields into equity term premiums and expected changes in equity yields. We
then regress each component of equity yields on each leverage volatility measurement. Table 7
reports our regression results. The estimated coecient of volatility leverage on the equity term
premium is 0.42 (with a Newey-West -value of 3.37, henceforth "") for the AEM measure and
0.16 (= 1.16) for the HKM measure. Second, our estimated coecient of volatility leverage on
the changes in yields is −0.020(= −2.22) for the AEM measure and is −0.019(= −2.38) for
the HKM measure. As such, this leverage volatility largely aects the expected changes in equity
yields.
In Table 7, we also investigate how the nancial constraint measured by the relative tightness
(RT) index moderates the eects of leverage volatility on both equity term premiums and expected
changes in equity yields. For each panel, columns (2), (4), and (6) show estimates of the eects of
the RT index. Table 7shows that the RT index directly aects equity yields, even if they only play
a small determining role in equity term premiums. This RT index mainly aects expected changes
in equity yields. Since the RT index does not impact equity term premiums, the unconditional
slope of equity yield is often upward-sloping in Li and Xu (2024). By contrast, leverage volatility
22
not only has impacts on changes in equity yields, but the AEM measure has the potential to
aect equity term premiums. This additional eect on equity term premiums osets the eects
of expected changes in equity yields. Hence, leverage volatility could predict a atter equity yield
curve and leave additional space for both sides (i.e., upward vs. downward) in the discussion.
3 Robustness Check
In this section, we make three robustness checks, including (i) employing a dierent model and
estimation method for leverage volatility, (ii) utilizing high-frequency data, and (iii) using the
volatility of the intermediary equity capital ratio. The robustness checks indicate that our con-
clusions do not depend on the choice of the estimation model.
3.1 A Stochastic Volatility Model
Someone may argue that our results depend on the selection of the model used for estimating
leverage volatility. To address this concern, we employ a stochastic volatility (SV) model, which
is an approximated HMM with a continuous innite state space, instead of the hidden Markov
Model (HMM) to estimate leverage volatility. Let denote the leverage growth process, such as
Δ or Δ. To be consistent with the HMM, we assume that follows an SV model
with an AR(1) process for the mean process and another AR(1) process for the volatility.
=(1−)+−1 +,z,t
, =(1−)+,−1 +,t
(23)
Here, and denote the unconditional values of leverage growth and leverage volatility; and
are the coecients of the AR(1) process to measure their persistence; z,t and ,t respectively
capture the residuals of the leverage growth process and shocks to its conditional volatility. We
23
assume both shocks follow a standard normal distribution, (0,1). Obviously, the exponent of
represents the leverage volatility dynamic, which is what we need to estimate.
The estimate of leverage volatility is determined not only by the model but also by the estima-
tion method. To mitigate parameter uncertainty, we employ Bayesian estimation rather than the
Maximum Likelihood Estimation (MLE) for the SV model. Specically, we follow Bretscher, Hsu,
and Tamoni (2023) and choose Beta priors for the AR(1) parameters, an inverse-Gamma prior for
, and calibrate the unconditional mean of . We then use Markov Chain Monte Carlo (MCMC)
to obtain 5000 posterior samples. 23 Statistical summaries for the samples are displayed in Table
8. After that, we employ a particle lter with 1000 particles to obtain the volatility process, ,.
We display the plot of , ≡, in Figure 2.
Given our benchmark settings, we compute changes in leverage volatility with the rst-lag
dierence of , as the associated characteristic-based pricing factor:
Δ,,+1 ≡,,+1 −,, (24)
We also display the plot of Δ,,+1 in Figure 2. Clearly, the leverage volatility series (and its
underlying characteristic-based pricing factors) based on the SV model seems to have the same
pattern. 24 but is smoother than those based on the HMM. We should not be surprised by these
ndings because, under similar settings, the SV model is just a continuous-state version of the
HMM, allowing for smoother transitions between volatility states.
We then assess the performance of the new leverage volatility risk factors in relation to equity
premiums and the equity term structure. Tables 9,10, and 11 present the results for the SV-based
volatility factors of the Fama-MacBeth regression, the value-weighted decile portfolio sorted on
leverage volatility risk, and the equity term structure analysis, respectively. These results are
23We simulate 10000 samples and discard the rst 5,000 as burn-in.
24The correlation coecient between , and is 0.78, and that between , and is 0.82. The
correlation coecient between , and is 0.86, and that between , and is 0.86.
24
highly consistent with those obtained using the HMM, suggesting that our ndings are robust to
the choice of volatility model and estimation technique.
3.2 High-frequency Data
In this section, we demonstrate that leverage volatility exhibits a stable and signicantly negative
risk premium in daily-frequency data. This nding provides strong empirical evidence for the
existence of leverage volatility risk.
It is a challenging task to observe high-frequency data on nancial leverage. Luckily, Professor
Asaf Manela released a high-frequency dataset on HKM intermediary equity capital ratio, cov-
ering the period from 2000-01-01 to 2018-12-11. We retrieved the daily squared leverage data for
HKM 25 from his website 26 and calculated the rst-order log-dierence of HKM squared leverage
as the daily leverage growth rate. The plots of daily squared leverage data and leverage growth
rate are displayed in the top panel in Figure 3.
Due to the limitations of HMM and SV approaches for modeling high-frequency data, we
employ the GARCH family to estimate the daily leverage volatility process. Specically, we use
an AR (1)-EGARCH (1,1) model to capture the mean and volatility of the leverage growth process,
respectively. The advantage of using the EGARCH model is that it can capture the asymmetric
volatility patterns, as the observed leverage growth rate in Figure 3.27 Let denote the daily
leverage growth rate, then is assumed to follow
=+−1 +
log(2
)= +−1 +(|−1|−𝔼[|−1|])+log(2
−1)(25)
25Squared inverse of daily HKM leverage.
26More details, see https://asafmanela.github.io/data/.
27Specically, the frequency of positive spikes exceeds that of negative spikes, suggesting a leverage eect. Except
for the EGARCH model, the GJR-GARCH model can also capture asymmetric volatility dynamics. However, its
performance is very similar to that of the EGARCH model but slightly weaker; thus, we only report the results of
the EGARCH model.
25
Here, and represent the unconditional mean and variance of the leverage growth rate, respec-
tively. measures the rst-moment persistence. measures the second-moment persistence
and is also called the ARCH eect. The coecient captures the sign eect, and captures the
size eect. Both eects are the key features of the EGARCH model. Unlike HMM and SV mod-
els, the GARCH family shares a single standard normal shock, . We use maximum likelihood
estimation (MLE) to estimate the model and report the estimation results in Table 12.
We can directly obtain the estimated volatility process in the GARCH family because no ad-
ditional shock is introduced in the second-moment dynamics. Following our prior studies, we
dene the daily leverage volatility factor as the rst-order dierence of the daily leverage volatil-
ity. We report the daily leverage volatility and its associated factor in the middle panel in Figure
3. An interesting nding is that the estimated daily leverage volatility shares the same trend
as the VIX index. The correlation coecient between leverage volatility and VIX reaches 0.86.
However, after log-dierence adjustment for both indices, the correlation drops to −0.04. This
nding suggests that although leverage volatility is highly correlated with VIX, it originates from
a dierent risk source.
We now evaluate the impact of the daily volatility factor in the equity market. We use the
method described in Ang, Hodrick, Xing, and Zhang (2006) to construct the daily decile portfolios
of stocks. To do this, we rst obtain the factor loadings in each month for every rm by running
the daily two-factor model:
=
+
+,Δ, +
(26)
Here
is the excess return of rm at time .is the market excess return. Δ, is
the innovation to the daily HKM leverage volatility that we have constructed previously. The
factor associated with the rst moment of leverage is excluded since the GARCH model accounts
for only one shock term, and the information in the mean may overlap with that in volatility in
26
some sense. Then we sort all stocks into decile daily portfolios according to their previous month
factor loadings with respect to the leverage volatility. To be comparable with our prior analysis,
we aggregate all daily portfolios into monthly ones and then assess whether the return of the
high-minus-low portfolio of high-frequency leverage volatility is attributed to other risk factors.
Table 13 presents the average returns of value-weighted decile portfolios formed on daily
leverage volatility, along with the risk-adjusted estimated from dierent factor models for these
portfolios. This table clearly indicates that the daily leverage volatility exhibits a signicant neg-
ative premium of −0.69percent after being aggregated to monthly, which is comparable to the
monthly leverage volatility premium of −0.42percent. Furthermore, this table demonstrates that
the negative premium based on daily leverage volatility cannot be explained by models such as
CAPM, FF3, the 5model, and others. This nding strongly supports the notion that the leverage
risk factor constitutes a genuine pricing factor.
We also examine the predictive ability of daily leverage volatility with respect to the eq-
uity term structure. Because the daily leverage volatility is not a return measure, we cannot
directly aggregate it to a monthly or quarterly frequency. To address this issue, we employ the
method proposed by Lamont (2001) to construct the economic tracking portfolios for daily lever-
age volatility. We rst regress the daily leverage volatility risk factor on the daily excess returns to
the base assets. We follow Ang, Hodrick, Xing, and Zhang (2006) and Barinov (2012) and choose
the decile portfolios sorted on daily leverage volatility as the base assets. Thus, the regression
model is: Δ, =
0+10
=1
(
, −,)+, (27)
Here, , denotes the daily risk-free rate. We run the above regression for each month and obtain
the monthly series of coecients on all base portfolios, {
, }, = 1,2,..,10. Then the daily eco-
nomic tracking portfolio is dened as the tted part of the regression minus the constant, that
27
is: Δ, =10
=1
(
, −,).(28)
We have now successfully transformed the economic variable into a stock portfolio, and we can
obtain the portfolio-mimicking variable at any desired frequency through time aggregation. The
bottom panel in Table 3displays the mimic factors related to the leverage volatility and leverage
volatility growth derived from the economic tracking portfolio.
Table 14 presents the results of the predictive ability of daily leverage volatility regarding
the equity term structure. Aligning with the quarterly HKM leverage volatility, the daily HKM
leverage volatility is conversely associated with expected changes in equity yield after controlling
for the RT index. Since this eect is much weaker, it seems not to move conversely with the
unconditional slope of equity yield, in which daily HKM leverage volatility leads to a atter equity
yield curve than its quarterly measure. This weaker pattern might be due to measurement errors
in aggregating the economic tracking portfolio.
3.3 Intermediary Equity Capital Ratio Volatility
The intermediary capital ratio is dened as the inverse of the intermediary leverage. Mathe-
matically, if the logarithm of Y follows a normal distribution, 28 then the logarithm of 1/also
follows a normal distribution with the same variance. 29 This statement still holds according to
Itô’s lemma when the process is assumed to follow a geometric Brownian motion. Therefore,
in principle, the volatility of the growth rate of the intermediary equity capital ratio should be
identical to that of the growth rate of the intermediary leverage.
To examine this statement, we construct factors of the volatility of the intermediary capital
ratio for both AEM and HKM proxies, and then check their asset pricing implications. We begin
28This also means that Y follows log-normal. In some sense, Y is the level data, and log(Y) is the growth rate of .
29However, the sign of the mean value of variable log(Y) will reverse.
28
by dening the intermediary equity capital ratio. For the AEM proxy, the intermediary equity
capital ratio is dened as the inverse of the leverage of security broker-dealers:
≡1
=Total Financial Assets
−Total Liabilities
Total Financial Assets
Here, Δ denotes the log changes in the intermediary equity capital ratio after
seasonal adjustment. The seasonal adjustment method, which is described in Adrian, Etula, and
Muir (2014), is the same as the one we applied to Δ.
The characteristic-based pricing factor formed on the HKM intermediary capital ratio exhibits
a dierent pattern from Adrian, Etula, and Muir (2014). For this HKM proxy, we directly use the
intermediary equity capital ratio from He, Kelly, and Manela (2017):
=∑Market Equity,
∑(Market Equity, +Book Debt,)
As stated in He, Kelly, and Manela (2017), the innovations to the HKM intermediary capital ratio,
denoted Δ, are dened as shocks to leverage at each date divided by the lagged ,
which is captured in an autoregression process of .
Similar to the estimates of leverage volatility, we use a hidden Markov Model (HMM) to esti-
mate the dierent states of both the AEM and HKM intermediary equity capital ratio series and
then construct their volatility by a simulation-based ltering method. The estimate results of
HMM for innovations to the intermediary capital ratio are shown in Table 15. Not surprisingly,
due to Itô’s lemma, the absolute values of ,,, and for the growth of the intermediary
capital ratio are very close to those of ,,and for the leverage growth, respectively. We
again use the method in Boguth and Kuehn (2013) to calculate the volatility of the intermediary
equity capital ratio. The volatility of the intermediary equity capital ratio and its associated inno-
vations are shown in Figure 4.30 It is noted that the volatilities of the intermediary equity capital
30We employ the symbol ’cap’ to denote that the volatility and risk factors are derived from the intermediary
29
ratio and its innovations show similar trends to the leverage volatility and its innovations. 31
We now evaluate the impact of the factor related to the volatility of the intermediary eq-
uity capital ratio on the equity market. We again use the Fama-MacBeth regression and Fama-
Mimicking portfolio to assess the predictive power for cross-sectional returns and the decompo-
sition framework on the equity term structure. Tables 16,17, and 18 present the results of the
Fama-Macbeth regression, the value-weighted decile portfolios of stocks, and the decomposition
of the equity term structure, respectively. We document that the estimates based on the volatility
of the intermediary equity capital ratio are very similar to those based on leverage volatility. Ta-
bles 16 and 17 indicate that volatilities of the intermediary equity capital ratio for both measures
yield a stable negative premium. Table 18 reveals that a risk factor associated with the volatility
of the AEM intermediary equity capital ratio has signicant negative impacts on changes in yield
and positive impacts on term premiums. A similar factor based on the HKM proxy only generates
signicant changes in yield, aligning with our ndings before.
Overall, we show that the volatility factor derived from the intermediary equity capital ra-
tio exhibits almost the same pattern as factors originating from leverage. This nding strongly
suggests that the capital ratio volatility shares substantial similarities with leverage volatility.
4 Dynamic Model
Model Environment. We follow Li and Xu (2024) and assume an endowment economy is pop-
ulated by a representative household and a representative nancial intermediary, both of which
live innitely in a discrete time environment. The Li and Xu (2024) endowment economy diers
from the business cycle model introduced in Gertler and Karadi (2011), where nancial interme-
equity capital ratio.
31The correlation coecient between , and is 0.99, and that between , and can still reach
0.99. While the correlation coecient between , and is only 0.78, and that between , and
is 0.77. The correlation coecient for HKM is lower because He and Krishnamurthy (2013) employ AR(1) innovations
as shocks in their model, which can be aected by the estimate of the AR(1) process.
30
diaries do not supply funds for physical investment or production. This setting does not aect
asset pricing implications because an endowment economy could be a special case for production
economies with linear technology (Campbell and Cochrane,1999). Without loss of generality, we
assume an endowment economy and fold the physical investment or production, where nan-
cial intermediaries mainly supply funds to capital markets. Then, we introduce a moral hazard
problem into an otherwise Li and Xu (2024) intermediary asset pricing model, and we model
the uctuation over time in the severity of the moral hazard problem in aggregate nancial un-
certainty. Our endowment economy begins with a Lucas (1978) tree with random walk output
growth: +1 =+1
=exp(+,+1)(29)
in which ,+1 follows an i.i.d. standard normal distribution, and denotes the volatility of output
growth. The price of the Lucas tree at period is and the return on the Lucas tree is dened
as ,+1 =+1++1
.
4.1 Household
Built on Bansal and Yaron (2004) and Li and Xu (2024), we assume the representative household
features Epstein and Zin (1989) preferences:
={(1−)1−1
+([1−
+1 ])1− 1
1−}1
1−1
(30)
where is consumption, is the risk aversion parameter, and is the intertemporal elasticity of
substitution. Each period, households save only via deposits with nancial intermediaries, and
households own the whole nancial sector. The budget constraint of households is:
+=,−1 +(31)
31
where {}∞
=0 is the stream of income that households receive from the nancial intermediaries,
which we specify below. , denotes the return on deposits. The household receives income
from nancial intermediaries and then makes consumption and saving decisions. The sequence
{}∞
=0 guarantees that =satises the budget constraint.
4.2 Financial Intermediaries
The benchmark model generally includes banks, mutual funds, or hedge funds as nancial in-
stitutions. We assume nancial institutions employ their net worth to fund rm equities in
the capital markets and borrow deposits from households. At every period , households also
hold
shares of stock, such that the balance sheet of nancial intermediaries can be written as
follows:
=+(32)
Then, the law of motion for the bank’s net worth is given by:
+1 =
(+1 ++1)−(33)
Here, nancial intermediaries receive +1 ++1 from each share of the Lucas tree and repay
deposits at a xed deposit rate at period + 1. This deposit rate is determined by the
household Euler equation: (+1)=1 (34)
Here, +1 denotes the household stochastic discount factor (SDF). As in Gertler and Karadi (2011),
nancial intermediaries face a participation constraint. Each period, they can divert fraction
of equity in order to consume it themselves, which creates a moral hazard problem. Specically,
nancial intermediaries can divert: ≥
(35)
32
where is subject to the severity of a moral hazard problem. The severity of the moral haz-
ard problem uctuates over time, which is characterized by a stochastic process. We model the
magnitude of this stochastic process in a time-varying nancial uncertainty:
=+(36)
Here, and denote the conditional mean and standard deviation of the process, respectively.
Since we nd no evidence that the rst moment of leverage is priced, we assume only shocks to
leverage volatility. We assume follows a two-state Markov chain ∈ (,). The transition
matrix for the volatility states is given by:
Pr(𝝈+1|𝝈) =
1−
1−
.(37)
where denotes the discounted value of payos. Financial intermediaries maximize their values
subject to Equations (32) to (35), which yields:
=max
{+1[+1 +(1−)+1]} (38)
where denotes the fraction of nancial intermediaries that exit at period . Equation (38) implies
that every period fraction of nancial intermediaries exit and receive their net worth (i.e.,
fraction 1−of nancial intermediaries survive). Based on the homogeneity properties of the
nancial intermediaries’ value function, one can rewrite as:
=(39)
Here, is the marginal value of net worth, and its growth rate is log(+1
).
Aggregate Net Worth. We now aggregate net worth across nancial intermediaries. In
33
equilibrium, the aggregate share of the Lucas tree is
= 1. We aggregate the balance sheet
constraint in Equation (32) and obtain =+, in which is aggregate net worth and
denotes aggregate deposits. As noted in Gertler and Karadi (2011), households use a fraction of
the Lucas tree to set up new nancial intermediaries, whose net worth is given by:
=(1−)[(+1 ++1)−(−)]++1 (40)
4.3 Recursive Equilibrium
The recursive equilibrium consists of prices {,}∞
=0, and quantities {
,,}∞
=0 that satisfy: (i)
household utility maximization; (ii) nancial intermediaries’ optimizations; (iii) market clearing
conditions; and (iv) a set of consistency conditions. In this economy, we dene the endogenous
state variable as aggregate net worth divided by total asset value, =
. We demonstrate that
a Markov equilibrium exists with state variables {,+1,}. We use a global method introduced
in Li and Xu (2024) to solve this model.
Asset Markets. As noted in Li and Xu (2024), we model aggregate consumption and aggre-
gate dividends as separate processes in which nancial intermediaries can trade Arrow–Debreu
securities among themselves. The aggregate dividend growth process is as follows:
log(+1
)= +,+1 +,+1 (41)
Here, ,+1 and ,+1 denote shocks to output growth and shocks to dividend growth, respectively.
Both are IID shocks and uncorrelated with each other. The parameters >1and >1are used
to match the overall volatility of dividends. We use , to represent the price of the dividend
claim, and the market return is dened as ,+1 =,+1++1
, .
34
4.4 Leverage Constraint
In this section, we show how a Li and Xu (2024) endowment economy creates a leverage constraint
via a participation constraint in Equation (35). As we have shown in Equation (39), the franchise
value of the nancial sector can be expressed as =. The participation constraint can be
expressed as: ≥
(42)
This participation constraint provides a microfoundation for a leverage constraint:
≤
(43)
Here, is our nancial uncertainty shock, and the left-hand side of this equation is a bank’s
leverage ratio. In a symmetric equilibrium where all banks choose the same leverage, we can also
sum across all individual banks to obtain the aggregate leverage:
≤
(44)
The maximum leverage ratio depends on the aggregate state variable ≡/. Suppose that a
nancial intermediary purchases stock with loans from the household, and the household needs
a collateral constraint against the loans. Therefore, the participation constraint links to the col-
lateral constraint as follows: ≤(
−1)(45)
Here, the right-hand side of Equation (45) is equal to the aggregate net worth of the nancial
sector with a multiplier.
35
4.5 Asset Pricing Implications
Equity Market. In this section, we inspect the mechanism that determines asset price variabil-
ity. We price the assets traded among nancial intermediaries using the augmented stochastic
discount factor described in Lemma 4.1.
Lemma 4.1 The returns, +1, for any assets held by nancial intermediaries must satisfy the Euler
equation: [+1{+(1−)+1}+1]=Ω(46)
in which Ω=[+1{+(1−)+1}]
[+1]+(47)
We denote ..
+1 as the "augmented stochastic discount factor" and write it as:
+1 =+1+(1−)+1
Ω(48)
This SDF can price all assets traded among nancial intermediaries. Beyond the household
stochastic discount factor +1,
+1 also depends on Φ+1:
Φ+1 =+(1−)+1
Ω(49)
In Equation (49), the term +(1−)+1 is the shadow price of nancial intermediaries’ net worth
in period +1.Ωrepresents the risk-adjusted present value of investing one unit of net worth for
period . Hence, Φ+1 is the shadow price appreciation from period to +1such that ..
+1 cap-
tures both the household marginal utility of consumption and nancial intermediaries’ marginal
utility of equity capital, which we interpret as the intertemporal marginal rate of substitution
with respect to an additional unit of net worth.
Our model is similar to Gertler and Karadi (2011) and generates nancial accelerator eects.
36
Figure 5shows these accelerator eects through the implications of the transition probability
for the marginal value of net worth, the conditional price of risk, conditional expected returns,
and the conditional volatility of returns. In Figure 5, Panels A and B show positive relationships
for marginal values of net worth and the price of risk conditional on ; Panels C and D predict
positive associations between the mean and volatility of equity returns conditional on .
Beyond the stylized equity premium pattern, Figure 6shows how slopes of equity yields, eq-
uity term premiums, and expected changes in equity yields change conditionally on . Shocks
to nancial uncertainty are assumed to transfer from a high regime to a high regime. Panel A
shows that one-period equity yields increase when nancial uncertainty stays in the high volatil-
ity regime. Panels B, C, and D depict (i) the conditional slope of equity yields, (ii) equity term
premiums, and (iii) expected changes in equity yields with respect to the transition probability
, respectively. Since the state indicates high persistence for nancial uncertainty, the model
economy falls into recession. Panel B states that the model-implied conditional slope of equity
yields is procyclical, resulting from countercyclical model-implied equity premiums in Panel C
and procyclical model-implied expected changes in Panel D. This is in line with our empirical es-
timates that expected changes in equity yields move inversely to equity term premiums, yielding
a atter equity yield curve.
Impulse Response Functions. Figure 7shows impulse responses of nancial intermedi-
aries’ stochastic discount factor, the widely recognized patterns in equity premiums, and the
cyclical patterns on equity yields in response to shocks to nancial uncertainty that enter in the
fth period. The rst moment of nancial leverage is not priced, and the rst moment of shocks
to nancial uncertainty appears as a horizontal line. A rise in nancial uncertainty causes a
slight decline in stock returns, followed by a subsequent increase. This mean-reversion behav-
ior predicts a negative price of risk in the present and a procyclical expected change in equity
yields in the future. A rise in nancial intermediaries’ SDF moves inversely to stock returns and
commands considerable countercyclical equity term premiums. A procyclical expected change
37
in equity yield osets countercyclical equity term premiums and predicts a procyclical behavior
of equity yields. Both the benchmark and a model of predictable growth behave similarly, but
the latter is less likely to aect the discount rate than the former. Our additional result is that
the conditional variance of equity returns is also countercyclical, in line with other volatilities of
aggregate variables (e.g., consumption volatility and production volatility). Our model-implied
results largely align with the empirical evidence.
4.6 A Model of Predictable Growth
Since the HKM leverage volatility aects asset prices via cash ows, we augmented our model
with predictable dividend and output growth that is dependent on net worth and nancial uncer-
tainty, where dividend growth is often assumed to be positively associated with output growth
in an endowment economy. 32
In our augmented model, the output growth process is rewritten as follows:
+1 =+1
=exp((,
)+,+1)(50)
In this setup, output growth increases with net worth. This positive correlation implies that
intermediaries with higher nancial uncertainty may struggle with ecient capital management,
which can result in economic deadweight losses. The dividend growth process is then given by:
log(+1
)= +((,
)− )+,+1 +,+1 (51)
in which is the average of output growth.
32We have already shown that the HKM leverage volatility aects dividend growth. In Appendix B.1, we document
that the HKM leverage volatility is also conversely related to output growth.
38
5 Benchmark Calibration
In this section, we simulate moments of aggregate quantities, asset prices, and equity yields in a
model of nancial sectors. Section 5.1 presents details regarding the choices of calibrated param-
eters. Sections 5.2 and 5.3 report simulated aggregate and conditional moments, respectively.
5.1 Structural Parameters
Our calibrated parameters mainly rely on Li and Xu (2024). We rst choose the discount factor ,
risk aversion , and intertemporal elasticity of substitution to be 0.9985, 10, and 1.5, respectively.
For consumption growth, we select the average and volatility to be 0.5 and 1.25, respectively.
For dividend dynamics, we select the average dividend growth to be 0.35, and its loadings on
shocks to consumption growth and shocks to dividend growth to be 3.5 and 4, respectively.
For nancial intermediaries, we choose the fraction of divertible assets to be 0.4, the fraction
of wealth transferred to new banks to be 0.01, and the exit probability of banks to be 0.08,
while and are 0.14 and 0.1, respectively. Finally, we select parameters from our empirical
ndings in Table 1for leverage volatility, which is assumed to have the same regime of nancial
shock. For the benchmark model, the high volatility is 0.00987 and low volatility is 0.00447;
the transition probabilities, and , are 0.434 and 0.595; for the augmented model, the high
volatility is 0.01476 and low volatility is 0.00528; the transition probabilities, and , are
0.455 and 0.552.
5.2 Aggregate Moments
Table 20 reports calibrated aggregate quantities and asset prices for the benchmark and aug-
mented models, respectively. For the benchmark model, Panel A of Table 20 shows that our
benchmark model closely matches consumption growth (1.98 percent vs. 1.90 percent in the
39
data) and its volatility (2.51 percent vs. 2.25 percent) and replicates dividend dynamics well with
a model-implied average dividend growth rate of 1.29 percent, which has a volatility of 13.34
percent. The benchmark model also generates a plausible correlation between consumption and
investment (0.66 vs. 0.47). We approximately match the capital ratio (0.24 vs. 0.21), book-to-
market ratio (0.57 vs. 0.65), and relative volatility of marginal net work growth /(Δlog())
(Δlog()) (1.88
vs. 1.67) in nancial sectors.
Regarding asset pricing variability (Panel B), our model precisely matches the equity premium
(6.28 percent vs. 6.09 percent in the data) and approximately captures equity volatility (16.28
percent vs. 19.12 percent). For the price-dividend (PD) ratio, this baseline model closely replicates
the average log PD ratio (2.75 percent vs. 3.41 percent) and generates plausible volatility for the
log price-dividend ratio (0.09 percent vs. 0.47 percent). The model-implied average equity yield
slope is 1.18 percent with a volatility of 1.79 percent, which is lower than the volatility of the
equity yield slope in the data (6.69 percent). Our model-implied equity yield curve is 0.02 percent
atter than Li and Xu (2024) of 1.20 percent. Our model perfectly replicates the 2-year bond
premium of 0.60 percent.
The column "Predictable" of Table 20 presents the quantitative results for the model aug-
mented with predictable output growth. Panel A indicates that our extended model reproduces
nearly the same consumption moments as the baseline model and precisely matches the aver-
age dividend growth (1.37 percent vs. 1.39 percent in the data) and its volatility (11.21 percent
vs. 12.54 percent). This extended model overestimates the correlation between consumption and
investment (0.78), as well as the intermediary capital ratio, book-to-market ratio, and relative
volatility of marginal growth compared to the baseline model. This model largely matches the
sensitivity of 1-year dividend growth on (−1.34vs. −1.15).
In Panel B, our augmented model generates reasonable equity premiums (6.98 percent vs. 6.09
percent in the data) and their volatility (18.45 percent vs. 19.12 percent), both of which are close
to the observed data. For the price-dividend (PD) ratio, this extended model largely matches
40
the mean of the log price-dividend ratio (2.65 percent vs. 3.41 percent). Although the model-
implied volatility of the log PD ratio is still low (0.13 vs. 0.47), this is slightly improved relative
to the quantitative performance of the benchmark model (0.13 vs. 0.09 in the baseline model).
This model produces a plausible average equity yield slope (2.27 percent vs. 0.37 percent) and its
volatility (3.93 percent vs. 6.69 percent). Our extended model predicts a plausible average 2-year
bond premium (2.07 percent).
5.3 Conditional Moments
For each model, Panel A of Table 21 reports model-implied moments conditional on expansions
and recessions. Quantitatively, we dene periods where the marginal value of net worth is in the
top 15 percent of its distribution as recessions.
For the benchmark model, Panel A of Table 21 demonstrates model-implied aggregate quan-
tities and asset pricing moments in dierent phases of the business cycle. In expansions, this
model largely matches the observed average consumption growth (2.93 percent vs. 2.48 percent
in the data), average market excess return (11.33 percent vs. 9.28 percent), integrated volatility
(17.19 percent vs. 15.57 percent), and relative volatility in expansions (0.66 vs. 0.79). During the
recession, the model tends to amplify the downturn, with predicted average consumption growth
of −3.36percent (compared to −1.14percent in the data) and the market excess returns of −22.32
percent (compared to −12.99percent in the data). The model also matches integrated volatility
and relative volatility well with the data. For equity yields, this model reasonably replicates 1-
year equity yields in recessions (8.62 percent vs. 10.89 percent in the data) and matches the equity
yield slope well both in expansions (1.79 percent vs. 1.59 percent) and recessions (−2.33percent
vs. −5.74percent). Analogous to Li and Xu (2024), we inversely predict the 1-year equity yield
during an expansion (2.56 percent vs. −3.22 percent in the data). In summary, our benchmark
model mostly replicates observed aggregate quantities and asset prices. It additionally captures
the fact that persistent leverage volatility raises equity (term) premiums, as we empirically ob-
41
served.
Panel A of Table 21 also demonstrates the same conditional moments for the model augmented
with predictable dividend growth. In booms, the augmented model-implied average consumption
growth (2.95 percent vs. 2.48 percent in the data), average market excess return (13.28 percent
vs. 9.28 percent), integrated volatility (19.25 percent vs. 15.57 percent), and the relative volatility
of consumption growth (0.47 vs. 0.79) are not far from the data. During recessions, the aug-
mented model also amplies the downturn, with model-generated average consumption growth
of −3.48 percent (compared to −1.14 percent in the data) and market excess returns of −28.76
percent (compared to −12.99percent in the data). The augmented model generates a more accu-
rate downward-sloping equity yield in recessions (−4.91) but a steeper upward-sloping yield in
expansions (3.54) compared to the baseline model. Panel B of Table 21 reports the discount rate
slope and dividend growth slope in expansions, recessions, and their dierence. In a recession,
the downward-sloping discount rate slope and upward-sloping expected dividend growth slope
both generate a downward-sloping equity yield term structure and vice versa.
5.4 Testing Leverage Volatility with Simulated Data
This section tests how model-simulated leverage volatility aects articial cross-sectional stock
returns. Although we assume a representative agent model, we simulate a panel of 1000 economies
over 1000 years for cross-sectional tests. Each panel has a rm that owns the risky asset, and its
return is considered the stock return. We then compute the average of equilibrium leverage ratios
as the aggregate nancial leverage ratio and nd its volatility. Then, we test the articial data in
two ways: (i) sorting stocks into decile portfolios based on factor loadings of leverage volatility;
(ii) testing how simulated leverage volatility aects the equity yield curve.
Table 22 states that the return of the value-weighted high-minus-low portfolio associated
with loadings on simulated leverage volatility is −0.28 percent per month (with Newey-West
42
=−11.37), whose magnitude is comparable to the mimicking factors associated with AEM and
HKM leverage volatility. Our simulated leverage volatility is also a negatively priced source of
risk.
Table 23 presents that the simulated leverage volatility is positively associated with the eq-
uity term premium but negatively associated with expected changes in equity yield. Overall,
their tradeo suggests a negative association between leverage volatility and equity yield. Since
simulated leverage volatility determines both components of the equity yield, it leaves a space
for the debate between those who believe in an upward-sloping equity term structure and those
who believe in a downward-sloping equity term structure.
6 Conclusion
This paper models the severity of a moral hazard problem in a stochastic process, which we refer
to as its magnitude of nancial uncertainty. We introduce shocks to nancial uncertainty into a Li
and Xu (2024) style intermediary asset pricing model. While shocks to nancial uncertainty are
unobservable, the proposed model solves for an equilibrium leverage ratio observed from data.
We then examine how time variations in leverage volatility aect asset prices empirically
and theoretically. The key nding in our research is that exposure to the second moment of
intermediary leverage leads to negative risk premiums. Second, we nd that leverage volatility
is negatively associated with expected changes in equity yields. Moreover, the leverage volatility
obtained from the security broker-dealer is positively associated with equity term premiums,
indicating an additional mechanism for understanding equity yields.
In theory, we solve for our intermediary asset pricing model and simulate a panel of 1000
economies and 1000 years. We test the model-implied data as we did in our previous empirical
tests. We document that leverage volatility is a negatively priced source of risk, and it is neg-
atively (positively) associated with expected changes in equity yields (equity term premiums).
43
This nding results in a atter equity yield curve, aligning with our ndings in observed data.
References
Adrian, Tobias and Nina Boyarchenko. 2012. Intermediary Leverage Cycles and Financial Stabil-
ity. New York Fed Working Paper.
Adrian, Tobias, Nina Boyarchenko, and Domenico Giannone. 2019. Vulnerable Growth. American
Economic Review 109 (4):1263—1289.
Adrian, Tobias, Erkko Etula, and Tyler Muir. 2014. Financial Intermediaries and the Cross-Section
of Asset Returns. Journal of Finance 69 (6):2557—2596.
Adrian, Tobias and Hyun Song Shin. 2010. Liquidity and Leverage. Journal of Financial Interme-
diation 19 (3):418–437.
Alfaro, Iván, Nicholas Bloom, and Xiaoji Lin. 2024. The Finance Uncertainty Multiplier. Journal
of Political Economy 132 (2):577—615.
Ang, Andrew, Robert J Hodrick, Yuhang Xing, and Xiaoyan Zhang. 2006. The cross-section of
volatility and expected returns. Journal of Finance 61 (1):259–299.
Arellano, Cristina, Yan Bai, and Patrick J. Kehoe. 2019. Financial Frictions and Fluctuations in
Volatility. Journal of Political Economy 127 (5):577—615.
Bansal, Ravi, Shane Miller, Dongho Song, and Amir Yaron. 2021. The Term Structure of Equity
Risk Premia. Journal of Financial Economics 142 (3):1209–1228.
Bansal, Ravi and Amir Yaron. 2004. Risk for the Long Run: A Potential Resolution of Asset Pricing
Puzzles. Journal of Finance 59 (4):1481–1509.
44
Barinov, Alexander. 2012. Aggregate volatility risk: Explaining the small growth anomaly and
the new issues puzzle. Journal of Corporate Finance 18 (4):763–781.
Berger, David, Ian Dew-Becker, and Stefano Giglio. 2019. Uncertainty Shocks as Second-Moment
News Shocks. Review of Economic Studies 87 (1):40–76.
Bernanke, Ben S. and Mark Gertler. 1989. Agency Costs, Net Worth, and Business Fluctuations.
American Economic Review 79 (1):14–31.
Bernanke, Ben S., Mark Gertler, and Simon Gilchrist. 1999. The Financial Accelerator in a Quanti-
tative Business Cycle Framework. In Handbook of Macroeconomics, vol. 1. Elsevier, 1341—1393.
Boguth, Oliver and Lars-Alexander Kuehn. 2013. Consumption Volatility Risk. Journal of Finance
68 (6):2589–2615.
Bretscher, Lorenzo, Alex Hsu, and Andrea Tamoni. 2023. The Real Response to Uncertainty
Shocks: The Risk Premium Channel. Management Science 69 (1):119–140.
Brunnermeier, Markus K. and Lasse Heje Pedersen. 2009. Market Liquidity and Funding Liquidity.
Review of Financial Studies 22 (6):2201–2238.
Brunnermeier, Markus K. and Yuliy Sannikov. 2014. A Macroeconomic Model with a Financial
Sector. American Economic Review 104 (2):379–421.
Campbell, John Y. and John H. Cochrane. 1999. By Force of Habit: A Consumption-Based Expla-
nation of Aggregate Stock Market Behavior. Journal of Political Economy 107 (2):205–251.
Carhart, Mark M. 1997. On Persistence in Mutual Fund Performance. Journal of Finance 52 (1):57–
82.
Christiano, Lawrence J., Roberto Motto, and Massimo Rostagno. 2014. Risk Shocks. American
Economic Review 104 (1):27–65.
45
Claus, James and Jacob Thomas. 2001. Equity Premia as Low as Three Percent? Evidence from
Analysts Earnings Forecasts for Domestic and International Stock Markets. Journal of Finance
56 (5):1629–1666.
Doyle, Jerey T., Russell J. Lundholm, and Mark T. Soliman. 2003. The Predictive Value of Ex-
penses Excluded from Pro Forma Earnings. Review of Accounting Studies 8:145–174.
Easton, Peter D. 2004. PE Ratios, PEG Ratios, and Estimating the Implied Expected Rate of Return
on Equity Capital. The Accounting Review 79 (1):73–95.
Epstein, Larry G. and Stanley E. Zin. 1989. Substitution, Risk aversion, and the Temporal Behavior
of Consumption and Asset Returns: A Theoretical Framework. Econometrica 57 (4):937–969.
Fama, Eugene F. and Ken R. French. 1993. Common Risk Factors in the Returns on Stocks and
Bonds. Journal of Financial Economics 33 (1):3–56.
———. 2015. A Five-Factor Asset Pricing Model. Journal of Financial Economics 116 (1):1–22.
———. 2018. Choosing Factors. Journal of Financial Economics 128 (2):234–252.
Fama, Eugene F. and James D. MacBeth. 1973. Risk, Return, and Equilibrium: Empirical Tests.
Journal of Political Economy 81 (3):607—636.
Fang, Xiang. 2021. Intermediary Leverage and the Currency Risk Premium. Job Market Paper.
Fang, Xiang and Yang Liu. 2021. Volatility, Intermediaries, and Exchange Rates. Journal of Finan-
cial Economics 141 (1):217—233.
Fontaine, Jean-Sebastien, Rene Garcia, and Sermin Gungor. 2024. Intermediary Leverage Shocks
and Funding Conditions. Journal of Finance, forthcoming .
Fostel, Ana and John Geanakoplos. 2008. Leverage Cycles and the Anxious Economy. American
Economic Review 98 (4):1211—1244.
46
Frank, Murray Z. and Tao Shen. 2016. Investment and the Weighted Average Cost of Capital.
Journal of Financial Economics 119 (2):300–315.
Geanakoplos, John. 2010. The Leverage Cycle. NBER Macroeconomics Annual. 24:1–65.
Gebhardt, William R., Charles M.C. Lee, and Bhaskaran Swaminathan. 2001. Toward an Implied
Cost of Capital. Journal of Accounting Research 39 (1):135–176.
Gertler, Mark and Peter Karadi. 2011. A Model of Unconventional Monetary Policy. Journal of
Monetary Economics 58 (1):17–34.
Giglio, Stefano, Bryan T. Kelly, and Serhiy Kozak. 2024. Equity Term Structures without Dividend
Strips Data. Journal of Finance, forthcoming .
Gilchrist, Simon, Jae W. Sim, and Egon Zakrajšek. 2014. Uncertainty, Financial Frictions, and
Investment Dynamics. National Bureau of Economic Research Working Paper.
Gormsen, Niels Joachim. 2021. Time Variation of the Equity Term Structure. Journal of Finance
76 (4):1959–1999.
Gürkaynak, Refet S., Brian Sack, and Jonathan H. Wright. 2007. The US Treasury Yield Curve:
1961 to the Present. Journal of Monetary Economics 54 (8):2291–2304.
Hamilton, James D. 1990. Analysis of Time Series Subject to Changes in Regime. Journal of
Econometrics 45 (1):39–70.
He, Zhiguo, Brian Kelly, and Asaf Manela. 2017. Intermediary Asset Pricing: New Evidence from
Many Asset Classes. Journal of Financial Economics 126 (1):1–35.
He, Zhiguo and Arvind Krishnamurthy. 2013. Intermediary Asset Pricing. American Economic
Review 103 (2):732—770.
47
———. 2019. A Macroeconomic Framework for Quantifying Systemic Risk. American Economic
Journal: Macroeconomics 11 (4):1—37.
Hou, Kewei, Haitao Mo, Chen Xue, and Lu Zhang. 2021. An Augmented q-Factor Model with
Expected Growth. Review of Finance 25 (1):1–41.
Hou, Kewei, Chen Xue, and Lu Zhang. 2015. Digesting Anomalies: An Investment Approach.
Review of Financial Studies 28 (3):650–705.
Kargar, Mahyar. 2021. Heterogeneous Intermediary Asset Pricing. Journal of Financial Economics
141 (2):505—532.
Kiyotak, Nobuhiro and John Moore. 1997. Credit Cycles. Journal of Political Economy 105 (2):211–
248.
Koenker, Roger and Gilbert Bassett. 1978. Regression Quantiles. Econometrica 46 (1):33–50.
Lamont, Owen A. 2001. Economic tracking portfolios. Journal of Econometrics 105 (1):161–184.
Li, Kai and Chenjie Xu. 2024. Intermediary-Based Equity Term Structure. Journal of Financial
Economics 157:103856.
Lucas, Robert E. 1978. Asset Pricing in an Exchange Economy. Econometrica 46 (6):1429–1445.
Ludvigson, Sydney C., Sai Ma, and Serena Ng. 2021. COVID-19 and the Costs of Deadly Disasters.
American Economic Association Papers and Proceedings 111 (2021):366–370.
Ma, Sai. 2023. Heterogeneous Intermediaries and Asset Prices: A Semiparametric Approach. Job
Market Paper.
Maggiori, Matteo. 2017. Financial Intermediation, International Risk Sharing, and Reserve Cur-
rencies. American Economic Review 107 (10):3038–3071.
48
Ohlson, James A. and Beate E. Juettner-Nauroth. 2005. Expected EPS and EPS Growth as Deter-
minants of Value. Review of Accounting Studies 10:349–365.
van Binsbergen, Jules H., Michael Brandt, and Ralph Koijen. 2012. On Timing and Pricing of
Dividends. American Economic Review 102 (4):1596–1618.
van Binsbergen, Jules H., William F. Diamond, and Marco Grotteria. 2022. Risk-Free Interest
Rates. Journal of Financial Economics 143 (1):1–29.
van Binsbergen, Jules H., Wouter Hueskes, Ralph Koijen, and Evert Vrugt. 2013. Equity Yields.
Journal of Financial Economics 110 (2013):503–519.
van Binsbergen, Jules H. and Ralph Koijen. 2017. The Term Structure of Returns: Facts and the
Theory. Journal of Financial Economics 102 (2017):1–21.
49
Time
1970 1980 1990 2000 2010 2020
−4 −2 0 2 4
lev
lev growth
(a) and Δ
Time
1970 1980 1990 2000 2010 2020
−2 −1 0 1 2 3
sigma
delta sigma
(b) and Δ
Time
1970 1980 1990 2000 2010 2020
−2 0 2 4
lev
lev growth
(c) and Δ
Time
1970 1980 1990 2000 2010 2020
−3 −2 −1 0 1 2 3
sigma
delta sigma
(d) and Δ
Figure 1: Financial Leverage and Risk Factors
This gure shows several quarterly time series related to nancial leverage, including nancial
leverage and and their growth Δ and Δ, as well as leverage volatility
and and their growth Δ and Δ. In the top panel (subgures (a) and (b)), we
show the series constructed using AEM leverage. In the bottom panel (subgures (c) and (d)),
we show the series constructed using HKM. The blue curves in subgures (a) and (c) represent
nancial leverage and ; using their growth, the red curves in subgures (a) and (c)
represent the leverage factor Δ and Δ. Analogously, the green curves in subgures
(b) and (d) represent the leverage volatility and ; using their growth, the orange curves
in subgures (b) and (d) represent the leverage volatility factor Δ and Δ. All series are
normalized with zero mean and unit variance. Shaded areas represent NBER recessions. The data
is from 1970Q1 to 2023Q4.
50
Time
1970 1980 1990 2000 2010 2020
−2 0 2 4 6
sigma
delta sigma
(a) , and Δ,
Time
1970 1980 1990 2000 2010 2020
−2 0 2 4
sigma
delta sigma
(b) , and Δ,
Figure 2: Financial Leverage Risk Factors: The Stochastic Volatility Model
This gure shows several quarterly time series related to nancial leverage, including nancial
leverage volatility and and their innovations Δ and Δ. In the top panel
(subgure (a)), we show the series constructed using AEM leverage. In the bottom panel (sub-
gure (b)), we show the series constructed using HKM. The green curves in subgures (a) and
(b) represent the leverage volatility and ; the orange curves in subgures (a) and (b)
represent the innovations of leverage volatility Δ and Δ. All series are normalized with
zero mean and unit variance. Shaded areas represent NBER recessions. The data is from 1970Q1
to 2023Q4.
51
2000 2005 2010 2015
0 1000 2000 3000 4000
Date
leverage square
(a) Squared Leverage
2000 2005 2010 2015
−0.3 −0.1 0.0 0.1 0.2 0.3
Date
leveage growth
(b) Leverage growth
2000 2005 2010 2015
0.2 0.4 0.6 0.8 1.0
Date
sigma
(c) Leverage volatility
2000 2005 2010 2015
0.00 0.05 0.10 0.15 0.20
Date
delta sigma
(d) Leverage volatility growth
2000 2005 2010 2015
−0.06 −0.04 −0.02 0.00
Date
hat sigma
(e) Leverage volatility mimic factor
2000 2005 2010 2015
0.00 0.05 0.10 0.15
Date
hat delta sigma
(f) Leverage volatility growth mimic factor
Figure 3: High-Frequency Financial Leverage and Risk Factors
This gure shows daily time series of the HKM leverage, including the squared leverage, leverage
growth, leverage volatility, and innovations to leverage volatility (i.e., leverage volatility growth),
as well as the mimick factors related to the leverage volatility and leverage volatility growth. In
the top panel (subgures (a) and (b)), we show the series constructed using daily HKM leverage
and its growth. In the middle panel (subgures (c) and (d)), we display the series constructed
using the daily HKM leverage volatility and its growth. In the bottom panel (subgures (e) and
(f)), we show mimic factors associated with the leverage volatility and leverage volatility growth
derived from the economic tracking portfolio. All series are normalized with zero mean and unit
variance. The data is from January 1st, 2000, to December 11th, 2018.
52
Time
1970 1980 1990 2000 2010 2020
−4 −2 0 2 4
lev
lev growth
(a) and Δ
Time
1970 1980 1990 2000 2010 2020
−2 −1 0 1 2
sigma
delta sigma
(b) , and Δ,
Time
1970 1980 1990 2000 2010 2020
−3 −2 −1 0 1 2 3
lev
lev growth
(c) and Δ
Time
1970 1980 1990 2000 2010 2020
−2 −1 0 1 2
sigma
delta sigma
(d) , and Δ,
Figure 4: Intermediary Capital Ratio and Risk Factors
This gure shows several quarterly time series related to the intermediary equity capital ratio,
including the ratio and and its innovations Δ and Δ, as well as
its volatility , and , and the associated innovations Δ, and Δ,. In
the top panel (subgures (a) and (b)), we show the series constructed using the inverse of the
AEM proxy. In the bottom panel (subgures (c) and (d)), we show the series constructed using
HKM. The blue curves in subgures (a) and (c) represent the intermediary equity capital ratio
for both the AEM and HKM proxies; using their growth, the red curves in subgures (a) and (c)
represent innovations to the intermediary equity capital ratio for both proxies. Analogously, the
green curves in subgures (b) and (d) represent the volatilities of the intermediary equity capital
ratio; using their growth, the orange curves in subgures (b) and (d) represent innovations to the
intermediary equity capital ratio. All series are normalized with zero mean and unit variance.
Shaded areas represent NBER recessions. The data is from 1970Q1 to 2023Q4.
53
0.0 0.2 0.4 0.6 0.8 1.0
4.4890
4.4892
4.4894
4.4896
4.4898
4.4900
4.4902
4.4904
A: Marginal Values of Net Worth
0.0 0.2 0.4 0.6 0.8 1.0
0.673
0.674
0.675
0.676
0.677
0.678
B: Conditional price of risk
0.0 0.2 0.4 0.6 0.8 1.0
0.218
0.219
0.220
0.221
0.222
0.223
0.224
0.225
0.226
C: Conditional Expected Returns
0.0 0.2 0.4 0.6 0.8 1.0
0.0906
0.0908
0.0910
0.0912
0.0914
0.0916
D: Conditional Volatility of Returns
Figure 5: Equity Prices in the Intermediary Model
Figure 5shows the equity price patterns as functions of : transition probabilities (from high regime to high
regime) under benchmark calibration in Table 19. Panel A displays the marginal value of net worth, Panel B displays
the conditional prices of risk, Panel C displays the conditional expected returns on risky assets, and Panel D displays
the conditional volatility of risky assets.
54
0.0 0.2 0.4 0.6 0.8 1.0
0.0555
0.0560
0.0565
0.0570
0.0575
A: One-Period Equity Yield
0.0 0.2 0.4 0.6 0.8 1.0
0.02835
0.02830
0.02825
0.02820
0.02815
B: Conditional slope of equity yield
0.0 0.2 0.4 0.6 0.8 1.0
0.022700
0.022725
0.022750
0.022775
0.022800
0.022825
0.022850
0.022875
C: Equity term premium
0.0 0.2 0.4 0.6 0.8 1.0
0.05110
0.05108
0.05106
0.05104
D: Expected change in yield
Figure 6: Equity Yields in the Intermediary Model
Figure 6shows equity yields in the intermediary model as functions of the : transition probability (from high
regime to high regime) under the benchmark calibration in Table 19. Panel A displays the one-period equity yield,
and panels B, C, and D display the decomposition of the term structure slope of equity yields into the conditional
slope of equity yields, the equity term premium, and the expected change in equity yields, respectively.
55
0246810
0.0
0.1
0.2
0.3
Conditional leverage volatility
0246810
0.050
0.025
0.000
0.025
0.050
Conditional leverage mean
0246810
0.005
0.000
0.005
Stock Return
0246810
0.00
0.01
0.02
0.03
Conditional variance of equity returns
0246810
0.0000
0.0001
0.0002
0.0003
Financial intermediary SDF
0246810
0.003
0.002
0.001
0.000
Equity yield slope
0246810
0.03
0.02
0.01
0.00
Expected change in Equity yield
0246810
0.000
0.002
0.004
Equity term premium
predictable growth model
benchmark model
Figure 7: Impulse Response Functions: Benchmark Model vs. Predictable Growth Model
Figure 7displays the impulse response functions when the conditional volatility of leverage rises from the low
regime to the high regime.
56
Table 1: The Markov Model for Leverage Volatility
This table reports the estimates of the Markov model for leverage volatility. The results in Panel A use the leverage
factor from AEM, and those in Panel B use HKM. The standard errors are also reported in parentheses. The full
sample period is from January 1970 to December 2023. The parameters in mean and volatility are standardized by
multiplying by 10.
Panel A: AEM
Parameters ×10
0.072 −0.030 0.987 0.447
(0.084) (0.084) (0.058) (0.035)
Transition Probabilies
0.501 0.498 0.434 0.595
(0.705) (0.576) (0.134) (0.113)
Panel B: HKM
Parameters ×10
0.751 −0.529 1.922 0.685
(0.200) (0.087) (0.169) (0.058)
Transition Probabilies
0.335 0.650 0.409 0.602
(0.124) (0.087) (0.121) (0.182)
57
Table 2: Correlation Matrix of Variables
This table shows the correlation matrix of variables, which includes the nancial leverage, lever-
age factor, leverage volatility, and the leverage volatility factor for both AEM and HKM. In addi-
tion, we consider the factors correlated with the NBER indicator, 𝕀, and the volatility index,
VIX. 𝕀 indicates the regime whether the U.S. economy is in recession, and VIX measures
the overall uncertainty in the nancial markets. Our data covers 1970Q1 to 2023Q4 (except VIX,
which begins at 1991).
Δ Δ Δ Δ 𝕀 VIX
1 0.0726 0.1793 0.0373 −0.2625 0.1085 0.0134 0.0220 0.1456 0.0367
Δ 0.0726 1 0.1747 0.3020 −0.2234 0.0208 −0.2353 −0.1832 −0.1905 −0.1385
0.1793 0.1747 1 0.6742 0.0057 −0.0784 0.1630 0.0198 0.0740 0.1162
Δ 0.0373 0.3020 0.6742 1 −0.0098 −0.0664 0.1549 0.1250 0.0091 −0.0059
−0.2625 −0.2234 0.0057 −0.0098 1 0.2679 0.3022 0.0898 0.4053 0.3538
Δ 0.1085 0.0208 −0.0784 −0.0664 0.2679 1 −0.1168 0.0546 0.1944 0.4983
0.0134 −0.2353 0.1630 0.1549 0.3022 −0.1168 1 0.6967 0.3062 0.2364
Δ 0.0220 −0.1832 0.0198 0.1250 0.0898 0.0546 0.6967 1 0.0946 0.1085
𝕀 0.1456 −0.1905 0.0740 0.0091 0.4053 0.1944 0.3062 0.0946 1 0.4742
VIX 0.0367 −0.1385 0.1162 −0.0059 0.3538 0.4983 0.2364 0.1085 0.4742 1
58
Table 3: Fama-MacBeth Regressions: Leverage Volatility from Hidden Markov Chain.
This table shows the results of Fama-MacBeth regressions of monthly returns on lagged estimated risk loadings and
other control variables. The factor loadings on the rst and second moments of intermediary leverage come from
the 5-year quarterly rolling time series regression. We estimate a hidden Markov model (HMM) for both the rst
and second moments of leverage. The control variable includes market equity, ME, Book-to-Market ratio, BM, 12-
month momentum, MoM, 5-year idiosyncratic volatility, IVOL, investment-to-asset, IA, and expected growth on IA,
EG. Panels A and B respectively report the results of AEM and HKM leverage. All coecients are in percent. The
Newey-West t-statistics using 12 lags are reported in parentheses. The period of the full sample is from January 1970
to December 2023.
ME BM MoM IVOL IA EG
Panel A: AEM
I 0.395
(0.446)
II −0.323
(−2.841)
III −0.044 −0.323
(−0.033) (−2.289)
IV 0.117 −0.362 −6.687 23.611
(0.074) (−2.220) (−1.292) (3.183)
V−1.303 −0.39 17.524 42.523 0.316 15.493
(−0.650) (−1.929) (4.470) (7.226) (2.127) (5.546)
VI -1.019 -0.346 15.909 42.383 0.147 14.837 29.146 39.763
(−0.539) (−1.743) (4.285) (7.474) (1.049) (5.354) (4.627) (6.120)
Panel B: HKM
I 0.009
(0.019)
II −0.054
(−1.874)
III −0.868 −0.077
(−0.790) (−2.290)
IV 0.258 −0.062 −7.547 21.465
(0.235) (−1.827) (−1.482) (2.864)
V−2.261 −0.055 17.484 42.532 0.262 15.222
(−0.982) (−1.191) (4.421) (7.200) (1.700) (5.388)
VI −2.138 −0.041 15.841 42.238 0.106 14.614 30.218 40.262
(−0.899) (−0.865) (4.273) (7.507) (0.718) (5.257) (4.763) (6.066)
59
Table 4: Value-Weighted Portfolios and Risk-Adjusted Univariately Sorted on Leverage
Volatility Risk from Hidden Markov Chain.
This table reports the average returns and risk-adjusted of decile value-weighted portfolios based on . We
estimate the leverage volatility in a hidden Markov model (HMM). The factor loading of Leverage volatility, ,
is from the 5-year quarterly rolling time series regression. Factor models for calulating alphas include the CAPM,
the Fama-French three factor model (FF3), the Carhart four factor model (Carhart4), the Fama-French ve factor
model (FF5), the Fama-French ve factor model with a momentum factor (FF5+MoM), the -factor model (), and the
augmented -factor model(5). The last column shows the returns and risk-adjusted of a zero investment portfolio
that is long the "H" portfolio and short the "L" portfolio. The results in Panel A are using the leverage factor from
AEM, and those in Panel B are from HKM. All returns and alphas are in percent. The Newey-West t-statistics using
12 lags are also reported in parentheses. The period of the full sample is from January 1970 to December 2023.
L 2 3 4 5 6 7 8 9 H H-L
Panel A: AEM
Average 2.499 1.716 1.606 1.517 1.661 1.477 1.467 1.54 1.757 2.075 −0.423
(8.292) (8.019) (8.675) (8.489) (8.574) (7.634) (8.268) (7.979) (8.350) (7.691) (−2.198)
CAPM alpha 1.593 1.009 1.001 0.86 1.011 0.806 0.786 0.813 1 1.174 −0.419
(8.834) (7.348) (8.090) (7.707) (8.182) (8.212) (7.668) (11.945) (9.572) (7.602) (−2.177)
FF3 alpha 1.69 0.968 0.923 0.813 0.957 0.757 0.752 0.794 1.003 1.22 −0.47
(9.433) (7.787) (9.377) (7.892) (9.537) (9.812) (8.138) (12.280) (9.947) (9.170) (−2.381)
Carhart4 alpha 1.699 1.001 0.938 0.823 0.961 0.826 0.766 0.834 0.976 1.272 −0.427
(10.213) (8.139) (9.515) (8.145) (9.225) (9.921) (8.149) (11.152) (9.540) (8.646) (−2.148)
FF5 alpha 2.016 0.983 0.846 0.757 0.886 0.681 0.687 0.754 1.005 1.258 −0.758
(10.436) (7.459) (9.502) (7.911) (10.666) (9.811) (8.418) (12.654) (8.617) (8.584) (−3.508)
FF5+MoM alpha 2.005 1.01 0.866 0.768 0.894 0.741 0.702 0.787 0.981 1.294 −0.71
(10.710) (7.573) (9.428) (8.091) (10.225) (10.589) (8.377) (12.008) (8.557) (8.305) (−3.262)
alpha 1.731 0.926 0.848 0.789 0.91 0.717 0.719 0.782 1.024 1.24 −0.491
(8.641) (7.112) (8.522) (7.433) (9.677) (8.975) (8.219) (11.843) (10.073) (8.512) (−2.348)
5alpha 2.128 1.076 0.928 0.8 0.801 0.656 0.649 0.716 0.824 1.254 −0.874
(10.201) (8.373) (9.443) (8.091) (10.364) (8.573) (7.785) (11.383) (7.706) (8.442) (−4.207)
Panel B:HKM
Average 2.493 1.898 1.749 1.635 1.441 1.418 1.341 1.576 1.656 2.135 −0.358
(8.304) (8.792) (9.284) (7.715) (8.251) (7.577) (6.554) (8.442) (7.996) (7.298) (−1.787)
CAPM alpha 1.561 1.127 1.028 0.933 0.807 0.765 0.680 0.905 0.935 1.252 −0.310
(7.421) (8.646) (9.715) (8.523) (9.180) (7.304) (7.362) (8.539) (8.073) (9.246) (−1.712)
FF3 alpha 1.679 1.149 1.040 0.920 0.792 0.715 0.621 0.861 0.894 1.285 −0.394
(8.111) (8.882) (10.102) (8.431) (9.290) (8.310) (7.871) (8.330) (8.288) (10.143) (−2.090)
Carhart4 alpha 1.697 1.173 1.026 0.925 0.816 0.731 0.669 0.834 0.891 1.313 −0.384
(9.008) (9.703) (9.853) (8.452) (8.390) (8.696) (8.345) (8.651) (7.990) (9.809) (−2.215)
FF5 alpha 1.961 1.222 1.077 0.888 0.769 0.639 0.553 0.806 0.805 1.300 −0.660
(8.872) (8.581) (9.247) (7.549) (8.699) (8.591) (7.108) (7.959) (8.739) (12.087) (−3.273)
FF5+MoM alpha 1.957 1.236 1.064 0.895 0.791 0.656 0.597 0.790 0.811 1.324 −0.632
(9.403) (9.142) (9.127) (7.585) (8.071) (8.734) (7.722) (8.343) (8.392) (11.542) (−3.405)
alpha 1.780 1.175 1.058 0.903 0.784 0.683 0.552 0.813 0.821 1.265 −0.515
(7.963) (8.863) (9.627) (8.079) (8.276) (7.651) (6.919) (8.263) (8.395) (10.094) (−2.691)
5alpha 1.862 1.136 1.030 0.847 0.765 0.586 0.579 0.819 0.802 1.480 −0.381
(8.039) (8.388) (8.929) (8.077) (7.346) (8.706) (6.501) (9.001) (8.720) (9.611) (−1.494)
60
Table 5: Testing the Discount Rate Channel for Leverage Volatility
This table reports the test results of leverage volatility for the discount rate channel. Panels A and B show the results
for the AEM and HKM leverage, respectively. The dependent variable, WACC_ICC, is the weighted average cost of
capital measured by the implied cost of capital.
, and
, are the loadings on the AEM and HKM leverage
volatility, respectively. We estimate for both loadings. The selected control variables include: the quarterly market
leverage (lev), log rm size (lsize), log book-to-market ratio (lbm), return on asset (roa), idiosyncratic volatility (IVOL),
and betas of the market obtained from CAPM (). The standard errors clustered at the rm level are reported in
parentheses. All variables, except factor loadings, are winsorized at the 1% and 99% levels. All available samples
are from 1977 to 2023. "*","**", and "***" represent the statistical signicance under the level of 10%, 5%, and 1%,
respectively.
Panel A: AEM
(1) (2) (3) (4) (5) (6)
−0.116∗∗∗ −0.107∗∗∗ −0.046∗∗ −0.142∗∗∗ −0.112∗∗∗ −0.038∗∗∗
(0.018) (0.019) (0.020) (0.013) (0.013) (0.014)
constant 7.965*** 14.073*** 15.197***
(−0.057) (−0.214) (−0.574)
controls No Yes Yes No Yes Yes
Year xed eect No No Yes No No Yes
industry xed eect No No Yes No No No
rm xed eect No No No Yes Yes Yes
obs 177 841 169 785 169 785 177 841 169 785 169 785
20.001 0.167 0.215 0.003 0.042 0.068
adj 20.001 0.167 0.214 −0.044 −0.004 0.023
Panel B: HKM
(1) (2) (3) (4) (5) (6)
0.015 0.025 −0.001 0.008 −0.003 −0.006
(0.017) (0.018) (0.018) (0.012) (0.012) (0.012)
constant 7.944*** 13.881*** 15.046***
(0.057) (0.223) (0.563)
controls No Yes Yes No Yes Yes
Year xed eect No No Yes No No Yes
industry xed eect No No Yes No No No
rm xed eect No No No Yes Yes Yes
obs 175 635 167 640 167 640 175 635 167 640 167 640
20.000 0.166 0.215 0.000 0.040 0.067
adj 20.000 0.166 0.215 -0.047 -0.006 0.022
61
Table 6: Testing the Future Cash Flow Channel for Leverage Volatility
This table reports the results of the Fama-Macbeth regression for testing the future cash ow channel for leverage volatility. Panels A and
B show the results of the AEM and HKM leverage volatility. The dependent variable CFO_sumis the operation cash ows summed over
∈ {1,2,3}years starting with quarter +1. The independent variable FCF is the free cash ow summed over ∈ {1,2,3}years starting with
quarter +1.
, and
, are the loadings on the AEM and HKM leverage volatility, respectively. We estimate Γfor both loadings. The
control variable, Accruals, is dened as GAAP earnings less cash from operations. Growth is the change in sales from quarter −4to . All
variables, except factor loadings, are winsorized at the 1% and 99% levels. The Newey-West -values with 12 lags are reported in parentheses.
All available samples are from 1977 to 2021.
Panel A: AEM
CFO_sum1 CFO_sum2 CFO_sum3 FCF_sum1 FCF_sum2 FCF_sum3
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
Γ−0.697 −0.27 −0.450 0.463 −0.799 0.169 −0.762 −0.68 −0.673 −0.39 −1.376 −1.32
(−1.434) (−0.650) (−0.462) (0.480) (−0.508) (0.108) (−2.257) (−2.182) (−0.925) (−0.557) (−0.589) (−0.659)
Growth 0.03 0.073 0.124 0.021 0.053 0.092
(3.975) (3.991) (3.912) (3.178) (3.151) (3.067)
Accurals −6.813 −13.078 −19.254 −4.772 −8.943 −12.903
(−7.099) (−6.424) (−6.154) (−6.014) (−5.360) (−5.198)
Constant 5.465 −0.942 10.613 −2.832 17.536 −3.282 2.661 −1.442 4.902 −3.773 8.842 −4.483
(12.921) (−1.681) (11.965) (−2.295) (12.742) (−1.620) (7.916) (−3.616) (7.206) (−4.221) (2.647) (−2.721)
Panel B: HKM
CFO_sum1 CFO_sum2 CFO_sum3 FCF_sum1 FCF_sum2 FCF_sum3
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
Γ−12.756 0.132 −81.926 −1.545 −98.439 −1.044 12.863 1.116 −24.833 0.544 −3.227 2.24
(−5.254) (0.380) (−5.357) (−1.725) (−5.236) (−0.780) (5.720) (3.324) (−5.144) (0.819) (−2.348) (1.681)
Growth 0.031 0.074 0.125 0.021 0.054 0.093
(3.974) (3.987) (3.905) (3.189) (3.156) (3.070)
Accurals −6.826 −13.11 −19.301 −4.789 −8.99 −12.974
(−7.003) (−6.347) (−6.087) (−5.942) (−5.322) (−5.164)
Constant −14.500 −1.177 −116.298 −4.874 −136.236 −5.808 21.828 −0.96 −34.595 −4.219 1.172 −4.372
(−3.869) (−1.966) (−4.907) (−3.264) (−4.680) (−2.573) (6.271) (−2.279) (−4.642) (−4.598) (0.622) (−2.648)
62
Table 7: Decomposition of Equity Term Structure: Leverage Volatility from Hidden
Markov Chain
This table reports the decomposition results for the equity term structure when leverage volatility follows a hidden
Markov model. The regressions in columns (1) and (2) are dened in Equation (16), and those in columns (3) and (4)
are dened in Equation (17), and the last two columns (5) and (6) are dened in Equation (18). We use the model-
implied equity yields provided by Giglio, Kelly, and Kozak (2024) to recover the returns on dividend futures and
oer more details in Appendix A. The results for the AEM measure are displayed in panel A, and those for HKM are
displayed in panel B. The independent variables include nancial leverage volatility measured by AEM, denoted as
, and measured by HKM, denoted as , as well as the relative tightness index, RT. The standard errors are
adjusted using the Newey-West correction with 12 lags and reported in parentheses. The sample covers 1973Q1 to
2020Q4 at a monthly frequency. The stars ∗,∗∗, and ∗∗∗denote 10%, 5%, and 1% levels of signicance, respectively.
Panel A: AEM
equity yield term premium change in yield
(1) (2) (3) (4) (5) (6)
−0.004 −0.002 0.423*** 0.424*** −0.020∗∗ −0.018∗∗
(0.005) (0.003) (0.126) (0.132) (0.009) (0.007)
RT −0.025∗∗∗ −0.011 −0.028∗∗
(0.007) (0.279) (0.014)
constant 0.008 0.009 −0.09 −0.089 0.003 0.004
(0.007) (0.006) (0.269) (0.272) (0.010) (0.009)
obs 563 563 551 551 551 551
20.005 0.221 0.031 0.031 0.05 0.152
Panel B: HKM
equity yield term premium change in yield
(1) (2) (3) (4) (5) (6)
−0.010∗-0.003 0.162 0.171 −0.019∗∗ −0.011∗∗
(0.005) (0.003) (0.140) (0.147) (0.008) (0.005)
RT −0.025∗∗∗ −0.031 −0.026∗
(0.006) (0.300) (0.014)
constant 0.008 0.009 −0.101 −0.10.004 0.004
(0.007) (0.006) (0.278) (0.279) (0.010) (0.010)
obs 563 563 551 551 551 551
20.035 0.222 0.005 0.005 0.047 0.127
63
Table 8: The Stochastic Volatility Model for Leverage Volatility
This table reports the estimates of the leverage volatility when we model it in a stochastic volatility model. The
results in Panels A and B, respectively, report the AEM and HKM leverage volatility. We estimate a stochastic
volatility model for both the rst and second moments of leverage through a Bayesian Markov Chain Monte Carlo
(MCMC) approach. We then report the mean, standard deviation, median, the 5th percentile, and the 95th percentile
for estimated parameters. The full sample period is from January 1970 to December 2023.
Panel A: AEM
Parameters mean sd median Q0.05 Q0.95
(Calibration) 0.002 0.000 0.002 0.002 0.002
0.052 0.029 0.047 0.012 0.107
0.651 0.182 0.685 0.291 0.884
0.293 0.086 0.280 0.171 0.453
−2.771 0.092 −2.773 −2.922 −2.602
Panel B: HKM
Parameters mean sd median Q0.05 Q0.95
(Calibration) -0.004 0.000 -0.004 -0.004 -0.004
0.078 0.039 0.072 0.024 0.152
0.446 0.152 0.450 0.192 0.693
0.364 0.072 0.356 0.258 0.484
−2.180 0.072 −2.178 −2.299 −2.067
64
Table 9: Fama-MacBeth Regressions: The Stochastic Volatility Model
This table shows the results of Fama-MacBeth regressions of monthly returns on lagged estimated risk loadings and
other control variables. The factor loadings on the rst and second moments of intermediary leverage come from
the 5-year quarterly rolling time series regression. We estimate a stochastic volatility model for both the rst and
second moments of leverage through a Bayesian Markov Chain Monte Carlo (MCMC) approach. The control variable
includes market equity, ME, Book-to-Market ratio, BM, 12-month momentum, MoM, 5-year idiosyncratic volatility,
IVOL, investment-to-asset, IA, and expected growth on IA, EG. Panels A and B respectively report the results of
AEM and HKM leverage. All coecients are in percent. The Newey-West t-statistics using 12 lags are reported in
parentheses. The period of the full sample is from January 1970 to December 2023.
ME BM MoM IVOL IA EG
Panel A: AEM
I 1.016
(1.145)
II −0.448
(−2.810)
III 0.278 −0.539
(0.201) (−3.142)
IV 0.445 −0.602 −7.086 23.124
(0.273) (−3.150) (−1.375) (3.137)
V−0.501 −0.656 17.330 42.389 0.291 15.512
(−0.233) (−2.488) (4.394) (7.292) (1.905) (5.625)
VI −0.407 −0.648 15.638 42.297 0.133 14.828 29.414 39.997
(−0.195) (−2.608) (4.212) (7.548) (0.914) (5.431) (4.791) (6.303)
Panel B: HKM
I 0.259
(0.515)
II −0.187
(−2.277)
III −1.353 −0.159
(−1.102) (−1.563)
IV 0.428 −0.140 −7.569 21.474
(0.343) (−1.447) (−1.482) (2.872)
V−2.415 −0.136 17.276 41.772 0.255 15.086
(−1.075) (−1.148) (4.402) (7.168) (1.642) (5.411)
VI −2.256 −0.047 15.549 41.635 0.101 14.546 29.778 40.177
(−0.962) (−0.358) (4.233) (7.499) (0.676) (5.307) (4.750) (6.066)
65
Table 10: Value-Weighted Portfolios and Risk-Adjusted Univariately Sorted on Lever-
age Volatility Risk: The Stochastic Volatility Model.
This table reports the average returns and risk-adjusted of decile value-weighted portfolios based on . We
estimate the leverage volatility in a stochastic volatility model. The factor loading of Leverage volatility, , is from
the 5-year quarterly rolling time series regression. Factor models for calulating alphas include the CAPM, the Fama-
French three factor model (FF3), the Carhart four factor model (Carhart4), the Fama-French ve factor model (FF5),
the Fama-French ve factor model with a momentum factor (FF5+MoM), the -factor model (), and the augmented
-factor model(5). The last column shows the returns and risk-adjusted of a zero investment portfolio that is long
the "H" portfolio and short the "L" portfolio. The results in Panel A are using the leverage factor from AEM, and
those in Panel B are from HKM. All returns and alphas are in percent. The Newey-West t-statistics using 12 lags are
also reported in parentheses. The period of the full sample is from January 1970 to December 2023.
L 2 3 4 5 6 7 8 9 H H-L
Panel A: AEM
Average 2.581 1.827 1.603 1.604 1.443 1.454 1.493 1.462 1.671 2.103 −0.478
(8.118) (8.689) (7.985) (8.194) (7.891) (7.492) (8.325) (7.939) (8.275) (7.901) (−2.765)
CAPM alpha 1.668 1.102 0.937 0.937 0.817 0.771 0.814 0.774 0.940 1.202 −0.466
(8.035) (8.452) (7.815) (7.006) (8.630) (8.092) (9.230) (10.151) (8.266) (8.256) (−2.482)
FF3 alpha 1.760 1.034 0.872 0.864 0.767 0.724 0.788 0.755 0.937 1.254 −0.506
(8.535) (9.318) (8.876) (7.836) (9.224) (8.709) (9.837) (10.335) (8.897) (9.631) (−2.723)
Carhart4 alpha 1.775 1.145 0.914 0.849 0.796 0.762 0.828 0.762 0.895 1.266 −0.509
(9.294) (9.234) (9.552) (7.735) (9.187) (8.511) (9.201) (10.144) (8.804) (9.161) (−2.673)
FF5 alpha 2.096 1.042 0.816 0.812 0.679 0.702 0.707 0.664 0.863 1.357 −0.740
(9.310) (8.409) (8.064) (7.227) (8.823) (8.210) (9.888) (9.453) (8.332) (8.589) (−3.611)
FF5+MoM alpha 2.091 1.129 0.855 0.804 0.708 0.733 0.744 0.675 0.835 1.356 −0.735
(9.580) (9.098) (8.493) (7.205) (9.266) (7.956) (9.513) (9.323) (8.489) (8.451) (−3.476)
alpha 1.783 0.983 0.803 0.820 0.723 0.697 0.767 0.733 0.929 1.300 −0.483
(7.758) (8.371) (8.357) (7.573) (8.465) (8.010) (9.681) (10.311) (9.306) (8.975) (−2.369)
5alpha 2.161 1.204 0.871 0.797 0.726 0.708 0.682 0.608 0.704 1.256 −0.905
(8.850) (8.277) (8.562) (8.341) (7.607) (7.109) (9.180) (9.749) (7.893) (8.389) (−4.151)
Panel B: HKM
Average 2.499 1.912 1.745 1.497 1.476 1.440 1.458 1.571 1.663 2.100 −0.399
(8.140) (9.463) (8.850) (7.820) (8.210) (7.698) (7.231) (8.774) (7.989) (7.685) (−2.256)
CAPM alpha 1.541 1.155 1.029 0.815 0.817 0.792 0.788 0.919 0.952 1.230 −0.311
(8.042) (10.579) (8.128) (9.062) (9.399) (8.213) (8.461) (9.231) (7.935) (9.483) (−1.892)
FF3 alpha 1.672 1.178 1.032 0.785 0.795 0.748 0.737 0.875 0.895 1.262 −0.410
(8.472) (11.177) (7.956) (9.201) (9.531) (9.389) (8.773) (9.484) (8.469) (10.736) (−2.271)
Carhart4 alpha 1.719 1.176 1.013 0.795 0.847 0.770 0.787 0.845 0.883 1.319 −0.400
(9.502) (11.773) (9.009) (9.145) (8.513) (9.663) (9.015) (10.156) (8.080) (11.007) (−2.458)
FF5 alpha 2.053 1.253 1.057 0.763 0.720 0.700 0.628 0.771 0.809 1.338 −0.715
(9.975) (11.300) (7.603) (8.139) (8.203) (9.167) (8.215) (8.324) (9.159) (11.574) (−3.838)
FF5+MoM alpha 2.062 1.246 1.042 0.772 0.766 0.720 0.676 0.758 0.806 1.379 −0.683
(10.462) (11.766) (8.376) (8.150) (8.034) (9.430) (8.695) (8.784) (8.786) (11.736) (−3.959)
alpha 1.815 1.200 1.037 0.767 0.763 0.715 0.676 0.807 0.833 1.257 −0.557
(8.448) (10.997) (7.668) (8.490) (8.296) (8.897) (8.477) (8.869) (8.928) (10.428) (−3.048)
5alpha 2.011 1.237 0.981 0.748 0.770 0.666 0.615 0.707 0.771 1.520 −0.492
(9.500) (10.555) (8.889) (7.027) (6.919) (9.474) (7.276) (7.853) (9.262) (12.230) (−2.502)
66
Table 11: Decomposition of Equity Term Structure: The Stochastic Volatility Model
This table reports the decomposition results for the equity term structure when leverage volatility follows a stochastic
volatility model. The regressions in columns (1) and (2) are dened in Equation (16), and those in columns (3) and
(4) are dened in Equation (17), and the last two columns (5) and (6) are dened in Equation (18). We use the model-
implied equity yields provided by Giglio, Kelly, and Kozak (2024) to recover the returns on dividend futures and
oer more details in Appendix A. The results for the AEM measure are displayed in panel A, and those for HKM are
displayed in panel B. The independent variables include nancial leverage volatility measured by AEM, denoted as
, and measured by HKM, denoted as , as well as the relative tightness index, RT. The standard errors are
adjusted using the Newey-West correction with 12 lags and reported in parentheses. The sample covers 1973Q1 to
2020Q4 at a monthly frequency. The stars ∗,∗∗, and ∗∗∗denote 10%, 5%, and 1% levels of signicance, respectively.
Panel A: AEM
equity yield term premium change in yield
(1) (2) (3) (4) (5) (6)
−0.005 0.000 0.391** 0.402** −0.032∗∗ −0.028∗∗
(0.007) (0.006) (0.156) (0.177) (0.014) (0.012)
RT −0.026∗∗∗ −0.058 −0.024∗∗
(0.007) (0.294) (0.011)
constant 0.008 0.009 −0.104 −0.103 0.004 0.004
(0.007) (0.006) (0.274) (0.276) (0.009) (0.009)
obs 563 563 551 551 551 551
20.007 0.22 0.028 0.029 0.138 0.21
Panel B: HKM
equity yield term premium change in yield
(1) (2) (3) (4) (5) (6)
−0.019∗∗∗ −0.012∗0.23 0.256 −0.032∗∗ −0.024∗∗
(0.007) (0.006) (0.182) (0.179) (0.014) (0.010)
RT −0.021∗∗∗ −0.068 −0.021∗∗
(0.005) (0.294) (0.011)
constant 0.009 0.009 −0.105 −0.104 0.004 0.005
(0.006) (0.006) (0.277) (0.279) (0.009) (0.009)
obs 563 563 551 551 551 551
20.127 0.259 0.008 0.009 0.12 0.171
67
Table 12: The EGARCH Model for High-Frequency Leverage Volatility
This table reports the estimates of the EGARCH model for High-frequency Leverage Volatility. The leverage volatility
is from HKM and at a daily frequency. The results in Panel B use the robustness standard error. The full sample period
is from January 1st, 2000, to December 11th, 2018.
Panel A: Optimal estimation
Parameters estimation se -value -value
0.001 0.004 0.165 0.869
0.047 0.017 2.819 0.005
0.002 0.000 5.363 0.000
0.117 0.020 5.987 0.000
0.899 0.015 58.742 0.000
−0.077 0.016 −4.757 0.000
Panel B: Robustness standard error
Parameters estimation se -value -value
0.001 0.004 0.159 0.874
0.047 0.019 2.408 0.016
0.002 0.002 1.279 0.201
0.117 0.078 1.505 0.132
0.899 0.063 14.161 0.000
−0.077 0.057 −1.347 0.178
68
Table 13: Value-Weighted Portfolios and Risk-Adjusted Univariately Sorted on Lever-
age Volatility Risk: High-Frequency Data
This table reports aggregated monthly average returns and risk-adjusted of decile value-weighted portfolios based
on . The leverage volatility is estimated by an AR(1)-EGARCH(1,1) model using daily HKM leverage data. The
factor loading of Leverage volatility, , is from the 1-month daily rolling time series regression. The portfolio
returns are aggregated to a monthly frequency. Factor models for calulating alphas include the CAPM, the Fama-
French three factor model (FF3), the Carhart four factor model (Carhart4), the Fama-French ve factor model (FF5),
the Fama-French ve factor model with a momentum factor (FF5+MoM), the -factor model (), and the augmented
q-factor model (5). The last column shows the returns and risk-adjusted of a zero investment portfolio that is
long the "H" portfolio and short the "L" portfolio. All returns and alphas are in percent. The Newey-West t-statistics
using 12 lags are also reported in parentheses. The period of the full sample is from January 1st, 2000, to December
11th, 2018.
L23456789HH-L
Average 2.275 1.496 1.395 1.116 1.260 1.293 1.131 1.114 1.074 1.590 −0.685
(5.320) (4.958) (4.846) (4.060) (4.798) (4.721) (4.357) (4.667) (3.631) (4.006) (−2.354)
CAPM alpha 1.790 1.094 1.026 0.772 0.937 0.966 0.800 0.774 0.681 1.096 −0.694
(4.434) (4.363) (4.752) (4.611) (4.411) (4.745) (4.529) (5.285) (4.399) (4.934) (−2.328)
FF3 alpha 1.725 1.022 0.998 0.747 0.887 0.920 0.791 0.739 0.677 1.097 −0.628
(4.673) (4.515) (5.085) (5.617) (5.391) (5.553) (5.102) (5.901) (4.449) (4.960) (−2.230)
Carhart4 alpha 1.863 1.066 1.013 0.761 0.906 0.930 0.802 0.742 0.670 1.163 −0.700
(5.410) (4.903) (5.316) (5.597) (5.158) (5.634) (5.081) (5.848) (4.440) (5.512) (−2.561)
FF5 alpha 1.883 1.012 0.955 0.660 0.850 0.823 0.696 0.654 0.581 1.196 −0.687
(5.149) (5.009) (5.204) (5.342) (5.349) (6.166) (4.784) (5.246) (4.215) (5.201) (−2.522)
FF5+MoM alpha 1.942 1.033 0.963 0.669 0.860 0.830 0.703 0.658 0.581 1.223 −0.719
(5.917) (5.427) (5.473) (5.625) (5.359) (6.437) (4.952) (5.420) (4.252) (5.678) (−2.740)
alpha 1.690 0.996 0.995 0.744 0.886 0.926 0.807 0.726 0.660 1.120 −0.570
(4.715) (4.652) (4.921) (5.151) (4.916) (5.009) (5.004) (5.454) (4.498) (4.831) (−2.154)
5alpha 1.995 0.982 0.979 0.661 0.825 0.796 0.624 0.679 0.586 1.324 −0.671
(5.710) (6.174) (5.593) (6.013) (4.977) (5.773) (4.522) (5.112) (4.017) (5.634) (−2.408)
69
Table 14: Decomposition of Equity Term Structure: High-Frequency Data
This table reports the decomposition results for the equity term structure based on daily HKM leverage volatility
risk. The regressions in columns (1) and (2) are dened in Equation (16), and those in columns (3) and (4) are dened
in Equation (17), and the last two columns (5) and (6) are dened in Equation (18). We use the model-implied equity
yields provided by Giglio, Kelly, and Kozak (2024) to recover the returns on dividend futures and oer more details
in Appendix A. , represents the economic tracking portfolio for the daily leverage volatility, in which we
aggregate it at a monthly frequency. RT refers to the relative tightness index. The standard errors are adjusted
using the Newey-West correction with 12 lags and reported in parentheses. The sample covers January 1st, 2000, to
December 11th, 2018, at a daily frequency, which we aggregate to monthly data eventually. The stars ∗,∗∗, and ∗∗∗
denote 10%, 5%, and 1% levels of signicance, respectively.
equity yield term premium change in yield
(1) (2) (3) (4) (5) (6)
, 0.628 −0.245 −29.755 −32.279 0.803 −1.872∗
(0.492) (0.622) (23.918) (29.453) (0.772) (1.089)
RT −0.025 −0.072 −0.076∗∗
(0.020) (0.419) (0.032)
constant 0.050*** 0.003 −1.916 −2.05 0.054* −0.088
(0.016) (0.035) (1.286) (1.635) (0.031) (0.063)
obs 228 228 228 228 228 228
20.047 0.16 0.067 0.068 0.02 0.285
70
Table 15: The Markov Model for Intermediary Capital Ratio Volatility
This table reports the estimates of the Markov model for the volatility of the intermediary equity capital ratio.
The results in Panel A use the intermediary equity capital ratio from AEM, and those in Panel B use HKM. The
standard errors are also reported in parentheses. The full sample period is from January 1970 to December 2023. The
parameters in mean and volatility are standardized by multiplying by 10.
Panel A: AEM
Parameters ×10
0.083 −0.137 0.972 0.435
0.079 0.086 0.057 0.038
Transition Probabilies
0.512 0.482 0.438 0.590
0.398 0.370 0.132 0.114
Panel B: HKM
Parameters ×10
0.446 −0.799 1.476 0.528
0.075 0.149 0.107 0.042
Transition Probabilies
0.671 0.293 0.455 0.552
0.071 0.123 0.139 0.131
71
Table 16: Fama-MacBeth Regressions: Intermediary Equity Capital Ratio
This table shows the results of Fama-MacBeth regressions of monthly returns on lagged estimated risk loadings and
other control variables. The factor loadings on the intermediary equity capital ratio and its volatility factors come
from the 5-year quarterly rolling time series regression. We model the intermediary equity capital ratio volatility
in a hidden Markov model. The control variable includes market equity, ME, Book-to-Market ratio, BM, 12-month
momentum, MoM, 5-year idiosyncratic volatility, IVOL, investment-to-asset, IA, and expected growth on IA, EG.
Panel A uses the leverage factor from AEM, while Panel B from HKM. All coecients are in percent. The Newey-
West t-statistics using 12 lags are reported in parentheses. The period of the full sample is from January 1970 to
December 2023.
ME BM MoM IVOL IA EG
Panel A: AEM
I−0.303
(−0.335)
II −0.301
(−2.752)
III 0.080 −0.307
(0.059) (−2.173)
IV 0.143 −0.336 −6.777 23.622
(0.093) (−2.150) (−1.311) (3.192)
V 1.493 −0.398 17.442 42.385 0.312 15.530
(0.756) (−2.047) (4.442) (7.221) (2.096) (5.549)
VI 1.247 -0.367 15.821 42.281 0.142 14.869 29.280 39.792
(0.670) (−1.943) (4.252) (7.474) (1.013) (5.352) (4.658) (6.157)
Panel B: HKM
I 0.145
(0.297)
II −0.037
(−1.977)
III 1.254 −0.044
(1.276) (−2.178)
IV 0.367 −0.053 −7.51 21.699
(0.374) (−2.217) (−1.476) (2.902)
V 1.934 −0.046 17.55 42.591 0.263 15.198
(1.261) (−1.705) (4.447) (7.217) (1.700) (5.425)
VI 1.903 −0.041 15.835 42.328 0.105 14.601 29.754 39.979
(1.223) (−1.574) (4.280) (7.511) (0.707) (5.303) (4.694) (6.021)
72
Table 17: Value-Weighted Portfolios and Risk-Adjusted Univariately Sorted on Inter-
mediary Equity Capital Ratio Volatility Risk.
This table reports the average returns and risk-adjusted of decile value-weighted portfolios based on . We
estimate the volatility of the intermediary equity capital ratio in a hidden Markov model. The factor loading of the
intermediary equity capital ratio volatility, , is from the 5-year quarterly rolling time series regression. Factor
models for calulating alphas include the CAPM, the Fama-French three factor model (FF3), the Carhart four factor
model (Carhart4), the Fama-French ve factor model (FF5), the Fama-French ve factor model with a momentum
factor (FF5+MoM), the -factor model (), and the augmented -factor model(5). The last column shows the returns
and alphas of a zero investment portfolio that is long the "H" portfolio and short the "L" portfolio. The results in Panel
A are using the volatility of the intermediary equity capital ratio from AEM, and those in Panel B are from HKM. All
returns and alphas are in percent. The Newey-West t-statistics using 12 lags are also reported in parentheses. The
period of the full sample is from January 1970 to December 2023.
L 2 3 4 5 6 7 8 9 H H-L
Panel A: AEM
Average 2.434 1.673 1.654 1.510 1.613 1.546 1.412 1.558 1.790 2.061 −0.374
(8.171) (8.259) (8.894) (8.345) (8.322) (7.685) (7.815) (8.163) (8.580) (7.414) (−1.981)
CAPM alpha 1.539 0.981 1.026 0.868 0.961 0.863 0.725 0.851 1.025 1.158 −0.381
(8.786) (7.643) (8.227) (8.922) (7.676) (8.608) (9.234) (7.706) (10.265) (7.576) (−2.008)
FF3 alpha 1.631 0.941 0.970 0.816 0.919 0.823 0.686 0.826 1.028 1.191 −0.439
(9.451) (8.173) (9.007) (9.594) (8.631) (9.475) (8.929) (8.235) (10.421) (9.002) (−2.342)
Carhart4 alpha 1.654 0.978 0.970 0.833 0.923 0.866 0.731 0.841 1.001 1.234 −0.420
(9.921) (8.633) (9.100) (9.285) (9.205) (9.129) (8.915) (8.281) (10.128) (8.489) (−2.146)
FF5 alpha 1.945 0.931 0.920 0.747 0.870 0.810 0.605 0.769 1.035 1.233 −0.712
(10.276) (7.585) (9.136) (9.695) (8.133) (10.070) (8.248) (8.638) (9.206) (8.072) (−3.273)
FF5+MoM alpha 1.945 0.962 0.926 0.764 0.878 0.846 0.647 0.784 1.012 1.262 −0.684
(10.337) (7.854) (8.932) (9.558) (8.520) (9.467) (9.031) (8.702) (9.200) (7.827) (−3.060)
alpha 1.674 0.899 0.913 0.788 0.879 0.802 0.648 0.815 1.041 1.207 −0.467
(8.637) (7.315) (8.641) (8.853) (8.349) (9.147) (8.622) (8.208) (10.336) (8.415) (−2.321)
5alpha 2.079 1.007 0.954 0.794 0.803 0.772 0.580 0.725 0.866 1.206 −0.873
(10.020) (8.804) (8.934) (9.175) (8.168) (9.509) (7.542) (9.351) (8.376) (8.378) (−4.167)
Panel B:HKM
Average 2.442 1.965 1.636 1.638 1.411 1.45 1.424 1.605 1.724 2.161 −0.281
(8.205) (9.626) (8.493) (8.392) (7.925) (8.147) (7.392) (7.951) (8.653) (7.888) (−1.771)
CAPM alpha 1.505 1.188 0.937 0.956 0.747 0.819 0.772 0.918 1.011 1.264 −0.24
(7.587) (10.176) (9.145) (9.885) (8.106) (7.448) (7.248) (9.393) (8.792) (10.357) (−1.656)
FF3 alpha 1.609 1.206 0.925 0.946 0.725 0.768 0.727 0.895 0.947 1.32 −0.289
(8.577) (10.553) (9.096) (9.483) (8.142) (8.077) (8.167) (8.835) (9.540) (11.773) (−1.931)
Carhart4 alpha 1.632 1.233 0.949 0.955 0.706 0.799 0.757 0.857 0.963 1.355 −0.277
(9.524) (10.539) (9.203) (8.926) (8.231) (7.883) (8.388) (9.413) (8.904) (11.644) (−1.999)
FF5 alpha 1.844 1.287 0.94 0.942 0.724 0.663 0.667 0.81 0.823 1.412 −0.431
(9.556) (10.225) (8.480) (8.278) (7.082) (7.858) (8.873) (7.871) (9.740) (13.023) (−2.662)
FF5+MoM alpha 1.847 1.3 0.96 0.952 0.708 0.695 0.694 0.791 0.844 1.435 −0.412
(10.143) (10.397) (8.476) (7.822) (7.443) (7.866) (8.946) (8.395) (9.049) (12.652) (−2.709)
alpha 1.675 1.246 0.908 0.918 0.717 0.722 0.692 0.839 0.879 1.329 −0.345
(8.160) (10.219) (8.334) (8.567) (7.543) (7.456) (7.664) (8.774) (10.208) (11.376) (−2.244)
5alpha 1.768 1.216 0.952 0.948 0.727 0.639 0.647 0.796 0.81 1.578 −0.19
(8.875) (9.065) (9.038) (7.087) (7.826) (6.690) (8.648) (9.424) (8.251) (12.188) (−1.038)
73
Table 18: Decomposition of Equity Term Structure for Intermediary Equity Capital Ratio
Volatility
This table reports the decomposition results of the equity term structure for the volatility of the intermediary equity
capital ratio. The regressions in columns (1) and (2) are dened in Equation (16), and those in columns (3) and (4) are
dened in Equation (17), and the last two columns (5) and (6) are dened in Equation (18). We use the model-implied
equity yields provided by Giglio, Kelly, and Kozak (2024) to recover the returns on dividend futures and oer more
details in Appendix A. The results for the AEM measure are displayed in panel A, and those for HKM are displayed
in panel B. The independent variables include the volatility of the AEM intermediary equity capital ratio, denoted as
,, and that measured by HKM, denoted as ,, as well as the relative tightness index, RT. The standard
errors are adjusted using the Newey-West correction with 12 lags and reported in parentheses. The sample covers
1973Q1 to 2020Q4 at a monthly frequency. The stars ∗,∗∗, and ∗∗∗denote 10%, 5%, and 1% levels of signicance,
respectively.
Panel A: AEM
equity yield term premium change in yield
(1) (2) (3) (4) (5) (6)
, −0.004 −0.002 0.430*** 0.431*** −0.019∗∗ −0.017∗∗
(0.005) (0.004) (0.125) (0.132) (0.009) (0.007)
RT −0.025∗∗∗ −0.017 −0.028∗∗
(0.007) (0.281) (0.014)
constant 0.008 0.009 −0.089 −0.089 0.003 0.004
(0.007) (0.006) (0.269) (0.272) (0.010) (0.009)
obs 563 563 551 551 551 551
20.006 0.221 0.032 0.032 0.048 0.148
Panel B: HKM
equity yield term premium change in yield
(1) (2) (3) (4) (5) (6)
, −0.009∗−0.003 0.23 0.237 −0.016∗∗ −0.010∗∗
(0.005) (0.003) (0.155) (0.160) (0.007) (0.004)
RT −0.025∗∗∗ −0.033 −0.027∗
(0.006) (0.292) (0.014)
constant 0.008 0.009 −0.095 −0.094 0.003 0.004
(0.007) (0.006) (0.276) (0.278) (0.010) (0.010)
obs 563 563 551 551 551 551
20.025 0.222 0.009 0.01 0.033 0.124
74
Table 19: Calibrated Parameters
This table displays the parameters for the benchmark calibration, mainly relying on Li and Xu (2024). The parameters
of the nancial leverage volatility shock are from Table 1for both the AEM and HKM measures.
Preference Parameters
Discount Factor 0.9985
Risk Aversion 10
Intertemporal Elasticity of Substitution 1.5
Consumption Dynamics
Mean of Consumption Growth 0.5
Volatility of Consumption Growth 1.25
Dividend Dynamics
Mean of dividend growth 0.35
Dividend loading on consumption growth shocks 3.5
Dividend loading on dividend specic shocks 4
Financial Intermediary Parameters
Asset Divertible Fraction 0.4
Fraction of Wealth Transferred to New Intermediaries 0.01
Exit Probability of Intermediaries 0.08
Parameter of Intermediaries Loan 0.14
of Government Bond 0.1
Financial Leverage Volatility
High Volatility [0.00987, 0.01476]
Low Volatility [0.00447, 0.00528]
Transition Probabilities (high regime to high regime) [0.434, 0.455]
Transition Probabilities (low regime to low regime) [0.595, 0.552]
75
Table 20: Aggregate Moments
This table presents simulated aggregate moments from the benchmark calibrated intermediary asset pricing model
and an augmented model of predictable growth. The data for aggregate quantities are real values, sampled at an
annual frequency, and cover the sample period from 1930 to 2020, whenever the data are available. Equity yields are
from Giglio, Kelly, and Kozak (2024), covering the period from 1974Q3 to 2020Q3. Bond premium data cover the sam-
ple period of 2004–2020. The columns Benchmark and Predictable report simulated moments based on a very long
simulation (10000 years of articial data) from our proposed benchmark and augmented models. For aggregate quan-
tities, we investigate average consumption growth (Δ), volatility of consumption growth (Δ), average dividend
growth (Δ), volatility of dividend growth (Δ), correlation of consumption and dividend (Δ,Δ), interme-
diary capital ratio (/), book-to-market ratio of nancial intermediary (1/), relative volatility of log(+1
)
(Δlog())
(Δlog()) , and sensitivity of 1-year dividend growth
1,+1 on standardized . For asset prices, we examine equity
premiums (−), volatility of equity return (), average log price-dividend ratio E(p-d), volatility of log price-
dividend ratio (−), mean of equity yield slope (
5−
1), volatility of equity yield slope (
5−
1), and 2-year
bond premium.
Data Benchmark Predictable
Panel A: Aggregate Quantities
(Δ)1.90 1.98 1.99
(Δ)2.25 2.51 2.48
(Δ)1.39 1.29 1.37
(Δ)12.54 13.34 11.21
(Δ,Δ)0.47 0.66 0.78
(/)0.21 0.24 0.30
(1/)0.65 0.57 0.63
(Δlog())
(Δlog()) 1.67 1.88 2.31
−1.15 —−1.34
Panel B: Asset Prices
(−)6.09 6.28 6.98
()19.12 16.28 18.45
E(p-d) 3.41 2.75 2.65
(−)0.47 0.09 0.13
(
5−
1)0.37 1.18 2.27
(
5−
1)6.69 1.79 3.93
(
+1)0.60 0.60 2.07
76
Table 21: Conditional Moments
This table presents simulated conditional moments across dierent phases of the business cycle with both the bench-
mark model and a model of predictable growth. For aggregate quantities, we investigate average consumption growth
(Δ)and relative volatility of log(+1
),/(Δ())
(Δ()) . The growth rate of our relative tightness index determines
the growth rate of the marginal value of intermediary net worth. For asset prices, we intend to match average ex-
cess return (−)and integrated volatility, (), which is calculated as the sum of squared, demeaned
quarterly returns. The sample period covers 1970 to 2020. We also emphasize 1-year equity yield (
1)and the
slope of equity yield, (
5−
1), where their data are from Giglio, Kelly, and Kozak (2024) and cover the period from
1974Q3 to 2020Q3. The columns labeled Benchmark and Predictable show corresponding moments derived from
a long-term simulation (10,000 years of articial data) based on the benchmark model and a model of predictable
growth. Both the models and the empirical data categorize periods when the marginal value of net worth (relative
tightness index) is in the top 15% of its distribution as recessions. This 15% recession frequency aligns with the 16%
NBER-dened recession frequency from 1970 to 2020. Panel B decomposes the equity yield slope into the discount
rate component and dividend growth component (i.e., expected dividend growth slope (
5−
1)) during periods
of economic expansion and recession in a model of predictable growth.
Panel A: Conditional Moments
Expansion Recession
Data Benchmark Predictable Data Benchmark Predictable
(Δ)2.48 2.93 2.95 −1.14 −3.36 −3.48
/(Δ())
(Δ()) 0.79 0.66 0.47 1.67 1.90 2.27
(−)9.28 11.33 13.28 −12.99 −22.32 −28.76
()15.57 17.19 19.25 18.34 17.86 23.26
(
1) −3.22 2.56 1.40 10.89 8.62 14.27
(
5−
1)1.59 1.79 3.54 −5.74 −2.33 −4.91
Panel B: Discount Rate Slope vs. Dividend Growth Slope: A Model of Predictable Growth
Expansion Recession Dierence
(
5−
1)3.54 −4.91 8.44
(5−1)3.11 −4.07 7.18
(
5−
1) −0.42 0.84 −1.26
77
Table 22: Value-Weighted Portfolios Univariately Sorted on Leverage Volatility Risk: Ob-
served Data vs. Simulated Data.
This table reports the average returns of decile value-weighted portfolios based on for both the observed and
articial data. We estimate the leverage volatility in a hidden Markov model (HMM). The empirical factor loading of
Leverage volatility is from the 5-year quarterly rolling time series regression. The period of the full sample is from
January 1970 to December 2023. For the articial data, we then simulate a panel of 1000 economies and 1000 years.
We solve for the equilibrium leverage ratio for each economy and compute their average. We nd the volatility of
the average equilibrium leverage ratio as simulated leverage volatility and compute its changes, where we obtain the
factor loading for its changes. The last column shows the returns of a zero investment portfolio that is long the "H"
portfolio and short the "L" portfolio. The results in Panels A and B are using the mimicking leverage volatility factor
from AEM and HKM, and those in Panel C are from simulated data. All returns are in percent. The Newey-West
t-statistics using 12 lags are also reported in parentheses.
L23456789HH-L
Panel A: AEM
Average 2.499 1.716 1.606 1.517 1.661 1.477 1.467 1.54 1.757 2.075 −0.423
(8.292) (8.019) (8.675) (8.489) (8.574) (7.634) (8.268) (7.979) (8.350) (7.691) (−2.198)
Panel B: HKM
Average 2.493 1.898 1.749 1.635 1.441 1.418 1.341 1.576 1.656 2.135 −0.358
(8.304) (8.792) (9.284) (7.715) (8.251) (7.577) (6.554) (8.442) (7.996) (7.298) (−1.787)
Panel C: Simulated
Average 1.893 1.929 1.975 2.015 1.979 1.976 1.969 1.895 1.817 1.616 −0.277
(100.1) (108.3) (104.2) (103.3) (102.3) (101.9) (99.29) (104.9) (105.3) (98.71) (−11.37)
78
Table 23: Decomposition of Equity Term Structure: Simulated Data
This table reports the decomposition results for the equity term structure for articial leverage volatility data follow-
ing a hidden Markov chain model. We simulate a panel of 1000 economies and 1000 years, in which we utilize the
rst 51 years in our tests. The regressions in columns (1) and (2) are dened in Equation (16), and those in columns
(3) and (4) are dened in Equation (17), and the last two columns (5) and (6) are dened in Equation (18). We use the
model-implied equity yields provided by Giglio, Kelly, and Kozak (2024) to recover the returns on dividend futures
and oer more details in Appendix A. The independent variables include nancial leverage and its volatility. The
standard errors are adjusted using the Newey-West correction with 12 lags and reported in parentheses. The stars ∗,
∗∗, and ∗∗∗denote 10%, 5%, and 1% levels of signicance, respectively.
Simulated Data and Equity Term Structure
equity yield term premium change in yield
(1) (2) (3) (4) (5) (6)
−0.019∗∗∗ −0.001 ∗∗∗ 0.011*** 0.001*** −0.049∗∗∗ −0.004∗∗
(0.0001) (0.0001) (0.0001) (0.0001) (0.014) (0.0003)
−0.012∗∗∗ 0.007*** −0.032∗∗
(0.00005) (0.00002) (0.0001)
constant 0.023*** 0.066*** 0.009*** −0.015∗∗∗ 0.037*** 0.148***
(0.0001) (0.006) (0.00004) (0.0001) (0.0002) (0.0004)
obs 204, 000 204, 000 204, 000 204, 000 204, 000 204, 000
20.130 0.359 0.148 0.393 0.134 0.369
79
Internet Appendix for "Leverage Volatility Risk"
A Recovering Returns on Dividend Futures
In this section, we illustrate how to recover the one-period holding log returns on dividend futures
alone based on equity yields and ex-dividend returns on the Standard Poor 500 index (SPX). We
also show that the recovered returns have the same cyclical pattern as the returns that are directly
constructed by dividend futures (Li and Xu,2024;Bansal, Miller, Song, and Yaron,2021).
A.1 Methodology
Constructing Returns via Dividend Strips. First, we recover the one-period holding returns
on dividend strips. As stated in Giglio, Kelly, and Kozak (2024), the dividend yield is dened as:
≡ln(1+
)(52)
which yields ln(
)= ln(exp(
)−1).(53)
We can rewrite the equity yield, ,, as:
, ≡1
ln(
,) = 1
ln(
,)
=1
[ln(
)−ln(,
)]
We use the identity (53) and obtain
, =exp{ln(exp(
)−1)−,}(54)
80
Accordingly, the price of dividend strips is a function of dividend yields, stock prices, and equity
yields. For convenience, we denote , as exp{ln(exp(
)−1)−,}. Recall that the denition of
a one-period holding return on dividend strips is ln(+1,−1
, ). Using equation (54), we immediately
have, +1, ≡ln(+1,−1
, ) =ln(+1
)+ln(+1,−1)−ln(,).(55)
where ln(+1,−1
, )is the log change in stock prices and can be substituted by the log gross return
on the stock without dividends.
Constructing Returns via Dividend Futures. We then derive the one-period holding re-
turns on dividend futures from returns on dividend strips. Note that, under no arbitrage, the
return on dividend strips can be decomposed into the return on its associated dividend future,
denoted as
+1,, and the return on the nominal bond with the same maturity, denoted as
+1,,
that is
+1, ≡ln(+1,−1
, ) =ln( +1,−1 exp{,}
, exp{(−1)+1,−1})
=ln(+1,−1
, )+ln(+1,−1
, ) =
+1, +
+1, (56)
where , denotes the zero-coupon bond price and can be expressed by , =1
exp{,}.We obtain
the return on dividend futures from Equation (56)
+1, =+1, −
+1,.(57)
A.2 Testing Returns on Dividend Futures
In this section, we show that the recovered returns on dividend futures, which are based only
on estimated equity yields, have the same cyclical pattern as that found in Li and Xu (2024). For
this reason, we can use the articial data directly to test the eect of leverage volatility on equity
81
term structure.
Data. Giglio, Kelly, and Kozak (2024) provide a full statistical method to obtain the equity
yields on SPX without dividend strips. They use a rich ane model that captures the dynamics
of equity returns, dividends, and prices to estimate the equity yields on portfolios and the market
index. Although the model-implied equity yields do not involve any information on dividend
futures, Giglio, Kelly, and Kozak (2024) shows that their model-generated equity yields closely
match the equity yields formed on the traded dividend futures.
To recover the returns on dividend futures, we also need yields on the nominal bonds and
the ex-dividend returns on the SPX. The nominal bond yields come from Gürkaynak, Sack, and
Wright (2007) and are available on the website of the Federal Reserve Board. The ex-dividend
returns on SPX come from CRSP. After merging all data sets, we obtain a sample at a monthly
frequency from 1973Q1 to 2020Q4.
Regression Models. Li and Xu (2024) decompose equity yield through Equation (14) and
ascertain the determinants of an equity yield slope. They nd that the eect of expected changes
in yields plays a larger role than equity term premiums. To test the reliability of the model-implied
returns, we follow Li and Xu (2024) and run three regressions:
, −
,1
slope
=1
0+1+1,(58)
1
=1(
+1, −
+1,)
equity term premium
=2
0+2+2,(59)
1
=1(−1)(
+1,−1 −
,−1)
changes in yields
=3
0+3+3.(60)
where is the regressor used to test the cyclical patterns of equity term structure. We follow Li
82
and Xu (2024) and use the RT index and the dividend-price (henceforth dp) ratio as the regressor.
There are two main empirical ndings from Li and Xu (2024) on equity yields: (i) the term of
expected changes in yields is procyclical, but the equity term premium is countercyclical, and (ii)
the expected changes in yields dominate the equity yield slope. These two statements imply that
the coecients of the regressor in the regressions (58) to (60) should satisfy
1≈3,2∗3<0.(61)
Note that both the RT index and the dp ratio are countercyclical, yielding
2>0,3<0.(62)
Thus, testing the reliability of the articial data is equivalent to testing the relations among co-
ecients described in Equations (61) and (62). Moreover, the RT index, which is dened as a
measure for intermediary constraints, is considered to be better than the dp ratio in describing
the equity yield slope. We then expect the RT index to capture more information on changes in
yields and to be inuenced less by equity term premiums (compared to the dp ratio).
Results. We report estimates for regressions (58), (59), and (60) in Table A1. As shown in
Table A1, the patterns of the coecients are perfectly consistent with our expected outcomes
described in (61) and (62). Specically, in the three regressions for the dp ratio and RT index, the
eect on the equity term premium, captured by 2, is always positive. The eect on expected
changes in yields, captured by 3, is always negative. Clearly, this pattern is consistent with the
rst statement that the two eects move in the opposite direction. Further, the standard errors of
2are too great for both regressors. As a result, the associated Newey-West t-values are less than
1 (and are not statistically signicant), implying that the eect of the equity term premium is not
dominant. In addition, for both regressors, the coecient 3is (at least) statistically signicant
at the 10% level and is close to the value of 1. Accordingly, the expected changes in yields
83
dominate the equity yield slope.
The result also echoes the statement in Li and Xu (2024) that the RT index is a better measure
than the dp ratio. To see this, the 2of the RT index is close to 0 (with 2= 0.0001)33 and
this implies the RT index is inuenced less by equity term premiums, resulting in little dierence
between the values of 2and 1. In sum, all the model-implied patterns match the estimation
results using the true traded dividend futures, showing that our data construction is reliable.
B Additional Tests
B.1 Quantile Regression
Our theoretical model suggests that output growth and dividend growth depend on the same
state variable. Once the leverage volatility, mainly the HKM proxy, aects dividend growth via
the cash ow channel, it should aect the output growth as well. To test this conjecture, we
follow Adrian, Boyarchenko, and Giannone (2019) and use quantile regressions to capture the
association between conditional leverage volatility and output growth. We denote +to be the
quarterly average output growth rate between and +, and our vector of regressors includes
leverage volatility and an intercept. Our quantile regression yields a regression slope that
minimizes the quantile-weighted absolute value of errors:
=argmin
∈
−
+1 1(+≥)|+−|+(1−)1(+≥)|+−|(63)
where 1(.)is the indicator function, yielding the predicted value,
+|(|), as:
+|(|)=
(64)
33And the F-test also fails in this regression.
84
where
+|(|)is a consistent linear estimator proved by Koenker and Bassett (1978).
Figure B1 and Table B1 present the implications of innovations to leverage volatility on eco-
nomic growth via quantile regressions. We nd no associations between the AEM measure and
output growth. In contrast, leverage volatility formed on the HKM measure moves inversely to
output growth with an estimated coecient of −0.84. More specically, we nd that this associ-
ation mainly exists across the left tail of output growth, including the 1 percent, 2 percent, 3 per-
cent, and 25 percent on the left side. We nd similar patterns of nancial volatility documented in
Adrian, Boyarchenko, and Giannone (2019) using the HKM leverage volatility. Although there is
no association between the AEM measure and output growth, the AEM leverage volatility moves
positively to the NBER recession index and is countercyclical.
Our ndings align with the asset pricing mechanisms we inspected before. The leverage
volatility based on the AEM proxy aects asset prices mainly in the discount rate channel but
contributes less to the cash ow channel. Instead, the leverage volatility estimated from the HKM
proxy determines asset prices via the cash ow channel but less in the discount rate channel.
C Convenience Yields in the Intermediary Model
Another important impact related to intermediary asset pricing models is the convenience yield,
which is dened as the dierence in yields between a non-collateralized risk-free asset (hereafter
referred to as a safe asset) and a money-like risk-free asset. The convenience yield measures the
liquidity and collateral value of the risk-free assets (Li and Xu,2024;van Binsbergen, Diamond,
and Grotteria,2022) and is, of course, considered to be aected by the intermediary constraints.
In this section, we investigate whether shocks to conditional leverage volatility can aect
convenience yields, both empirically and quantitatively. Specically, we focus on the eects of
leverage volatility on the term structure of convenience yields and future bond returns. Our
results show that only leverage volatility has a weak impact on convenience yield, and only the
85
AEM measure shows the predictability of bond returns.
C.1 Empirical Results
We show the empirical results rst.
Convenience Yields. A convenience yield is dened as the dierence between the return
on safe assets and the return on money-like assets. We use the yield on government bonds to
approximate the return on money-like assets, and the return on safe assets needs to be calculated
using the option strategy provided by van Binsbergen, Diamond, and Grotteria (2022). Let
,
denote the government bond yield with maturity , and
, denote the yield on a safe asset with
the same maturity. 34 The maturity-convenience yield, denoted as
,, is given by:
, ≡
, −
, .(65)
Li and Xu (2024) also apply the same decomposition method of equity yield to a convenience
yield to analyze its cyclical patterns. We follow their method to investigate the eect of leverage
volatility on the convenience yield slope. As in Li and Xu (2024), we mainly focus on the slope
that is constructed by the dierence between the 1-year maturity convenience yield and the half-
year maturity convenience yield. Using proposition 2 in Li and Xu (2024), this convenience yield
slope can be expressed as
12, −
6,1=1
2(𝔼[
12,+6]−𝔼[
6,+6])
term premium of convenience yield
+1
2(𝔼[
6,+6]−
6,)
expected change in convenience yields
(66)
where
, denotes the one-period-holding log return on the convenience yield and can be calcu-
lated using the dierence between the one-period-holding log return on the safe assets and that
34The return on a safe asset can be regarded as the interest rate without collateral value, and van Binsbergen,
Diamond, and Grotteria (2022) and Li and Xu (2024) also call it the box rate.
86
on the government bond, that is
, =
, −
, .(67)
Data. The monthly yields on safe assets (or box rates) and convenience yields are available
on the personal website of William Diamond. 35 Li and Xu (2024) also provide this data among
their online materials. The period of the data is from 2004Q1 to 2020Q3.
Regression Models. We run the following three regressions via a similar approach used in
testing the eect of leverage volatility on equity yields, which is given by:
12, −
6,1
convenience yield slope
=1
0+1+1,(68)
1
2(𝔼[
12,+6]−𝔼[
6,+6])
term premium of convenience yield
=2
0+2+2,(69)
1
2(𝔼[
6,+6]−
6,)
expected change in convenience yields
=3
0+3+3.(70)
where is the chosen regressor. To investigate the impact of leverage volatility on the con-
venience yield slope, we choose two types of leverage volatility, and , as the single
regressor for the regressions (68) to (70) in turn. By considering the channel that leverage volatil-
ity may aect the convenience yield slope through leverage constraints, we re-run these models
but add a control variable, the RT index. To compare the performance of the nancial leverage
volatility to other factors used in Li and Xu (2024), we also choose the dp ratio and RT index as
the regressor in turn and re-run these regressions.
In addition, we check whether the volatility of nancial leverage can predict future bond
returns. This idea comes from the empirical ndings in Li and Xu (2024) and van Binsbergen,
Diamond, and Grotteria (2022), in that (i) the convenience yields can be used to predict future
35More details can be found at https://williamdiamond.weebly.com/papers.html
87
bond returns, and (ii) the risk source of convenience yields comes from leverage constraints and
can also be used to predict bond returns. We conrm that nancial leverage volatility can be used
to predict bond returns as long as it is correlated to leverage constraints. To check this channel,
we follow Li and Xu (2024) and run the regression below
+12,2=+∗++12.(71)
where
+12,2denotes the 12-month ahead excess return on 2-year government bonds, and
denotes the regressor. For the same reason that we mentioned before, we choose two types of
nancial leverage volatility as the benchmark and use the 6-month convenience yield and RT
index as the control variables to re-run the models. We also provide comparable results using the
6-month convenience yield and RT index as the single regressor.
Results. We report the decomposition results of convenience yield slopes in Table C1. First,
the results in panel C conrm the ndings from Li and Xu (2024) and van Binsbergen, Diamond,
and Grotteria (2022), in that (i) the convenience yield slope is close to being cyclical, and (ii)
this is mainly due to the high expected change in convenience yields and high term premiums of
convenience yields. Second, the results in panels A and B show that the eect of leverage volatility
on the convenience yield slope has the same pattern as the RT index and the dp ratio but is much
weaker. The leverage volatility has a positive eect on the term premium of convenience yield
and a negative eect on the change in yields. However, these eects are too weak, so most of
their estimates are insignicant from zero. 36 The low expected change in convenience yields
and low equity term premiums also lead to a zero-constant slope. Lastly, unlike the ndings
from the equity yield slope, the eects of leverage volatility on the two parts of the convenience
yield slope become zero when controlling for the RT index. This nding implies that nancial
leverage volatility can only weakly and indirectly aect the term premium and expected changes
36The performance of leverage volatility based on the AEM leverage index is better than that based on HKM,
according to the data results. This results may be because the AEM leverage contains more information on leverage
constraints than the HKM and can also be inuenced more by the RT index.
88
in convenience yields through changes in leverage constraints.
The results of the eects of leverage volatility on bond returns are shown in Table C2. First,
the results in panel C successfully replicate the ndings in Li and Xu (2024), which conrms
their statement that the equity term structure and convenience yield have a common risk source
related to leverage constraints. Second, only the leverage volatility based on the AEM measure
shows signicant predictability in relation to bond returns. This is consistent with the nding
in Table C1 that only the AEM measure has a signicantly positive eect on the term premium.
37 Finally, we nd that the coecient of the AEM measure becomes half its previous value and
is insignicant from zero, which implies that some predictability of bond returns for the AEM
measure is due to the changes in leverage constraints. Note that the above results also imply that
the AEM measure contains more information on leverage constraints than the HKM measure.
C.2 Model-Implied Convenience Yield
The previous section introduced the denition and empirical ndings regarding convenience
yields. In this section, we present the model-implied patterns regarding convenience yields.
As discussed in Li and Xu (2024), returns on assets dier according to their collateralizabil-
ity. Consequently, we extend the model by incorporating two additional assets: interbank loans
and government bonds. Furthermore, we assume that government bonds face a higher leverage
constraint than interbank loans. Accordingly, the banks participation equation is modied as
follows: ≥
+
+
(72)
in which
and
are the fraction of loans among nancial institutions and government bonds
held by nancial intermediaries, respectively, and
and
follow shocks to conditional nancial
leverage volatility. Li and Xu (2024) assume that the net supplies of these two assets are zero, so
37Note that the construction of the term premium of convenience yields is related to bond returns
89
the additional two assets cannot aect the original equilibrium. The risk-free rate is ,, the
government bond rate is ,, and the interbank rate , is a proxy to denote returns from loans
among nancial institutions, which are given by
, =1
(+1)
, =, +
[+1{+(1−)++1}]
, =, +
[+1{+(1−)++1}]
(73)
The convenience yield in the model can be dened as the dierence between the non-collateralizable
and collateralizable assets. In our model, we introduce the convenience yield as the dierence
between the interbank rate and the government bond rate,
−
.
C.2.1 Quantitative Results for Convenience Yields: Moments
In this section, we illustrate the conditional and unconditional moments in our model related
to the convenience yield. Table C3 shows the aggregate moment of the baseline model. For
the convenience yield, this baseline model reasonably replicates the mean of convenience yields
(0.14% vs. 0.35% in the data) and estimates the volatility of convenience yields precisely (0.18).
In contrast to the benchmark model, our augmented model replicates a lower mean of con-
venience yields (0.10 percent) and a higher volatility for convenience yields (0.20 percent).
Table C4 reports the model-implied moments related to the convenience yields conditional
on business cycle uctuations. In a recession, our model replicates the convenience yields closely
to the data (0.48 vs. 0.58 in the data) and is not far from the data observed in the expansion (0.07
vs. 0.31).
In Table C4, the convenience yields are estimated to be lower both in expansion and recession
than in the benchmark model.
90
C.2.2 Quantitative Results for Convenience Yields: Decision Rules
Figure C1 displays the decision rules for leverage volatility on convenience yields, in which we
include the convenience yields, conditional slopes of convenience yields, expected change in
convenience yields, term premiums of convenience yields, and government bond excess returns.
Since states high persistence for leverage volatility, the model is now in recession. Panel A in
Figure C1 shows that the convenience yield increases when the nancial leverage stays in a high
volatility regime. Panel B depicts a slight increase in both the conditional slope of convenience
yields and the expected change in convenience yields under a high-leverage volatility regime. In
panel C, the excess return on government bonds decreases with a high-leverage volatility regime.
In panel D, the term premium of convenience yields is slightly procyclical. This is in line with
our empirical part that posits that leverage volatility can only weakly aect the term premium
and expected changes in convenience yields.
91
0.0 0.2 0.4 0.6 0.8 1.0
−20 −10 0 10 20
Constant
0.0 0.2 0.4 0.6 0.8 1.0
−10 −5 0 5 10 15
Leverage volatility based on AEM
(a)
0.0 0.2 0.4 0.6 0.8 1.0
−10 −5 0 5 10 15 20
Constant
0.0 0.2 0.4 0.6 0.8 1.0
−10 −5 0
Leverage volatility based on HKM
(b)
Figure B1: Estimates Under Dierent Quantiles
This gure shows the estimated constants and coecients of leverage volatility using the quan-
tile regression
=argmin
∈∑−
+1 1(+≥)|+−|+(1−)1(+≥)|+−|and its
predicted value
+|(|)=
is a consistent linear estimator from quantiles from 1 to 98.
Panels (a) and (b) show the estimates based on AEM and HKM measures, respectively. The hor-
izontal axis represents each quantile from 1 to 98, and the vertical axis denotes the estimated
slopes. For each quantile regression, the black points represent the point estimates, and the gray
areas represent the associated condence sets. The red line represents the coecient of the Or-
dinary Least Squares (OLS) estimate, and the red dashed line represents the OLS estimates con-
dence sets.
92
0.0 0.2 0.4 0.6 0.8 1.0
0.00244
0.00246
0.00248
0.00250
0.00252
0.00254
Convenience Yield
Convenience Yield
0.0 0.2 0.4 0.6 0.8 1.0
0.00152
0.00151
0.00150
0.00149
0.00148
0.00147
0.00146
Term Structure of Convenience Yield: Decomposition
Conditional slope of convenience yield
Expected change in convenience yield
0.0 0.2 0.4 0.6 0.8 1.0
0.03275
0.03280
0.03285
0.03290
0.03295
0.03300
0.03305
0.03310
0.03315 Government Bond Excess Return
2-year Gov Bond Excess Return
0.0 0.2 0.4 0.6 0.8 1.0
3.695
3.690
3.685
3.680
3.675
3.670 1e 5 Term premium of convenience yield
Term premium of convenience yield
Figure C1: Convenience Yields in Intermediary Model
Figure C1 displays the convenience yields in the intermediary model as functions of : transition probabilities
(from high regime to high regime) under benchmark calibration in Table 19. Panel A in Figure C1 displays the
convenience yield, panel B displays the 2-year government bond excess returns, and panels C and D show the
decomposition of the term structure slope of convenience yields.
93
Table A1: Cyclical Patterns of Equity Term Structure
This table reports the estimation results of the decomposition of equity term structure when using model-implied
data. The regressions in columns (1) and (2) are dened in Equation (58), and those in columns (3) and (4) are dened
in Equation (59), and those in the last two columns (5) and (6) are dened in Equation (60). We use the model-implied
equity yields provided by Giglio, Kelly, and Kozak (2024) to recover the returns on dividend futures, and we provide
more details in Appendix A. We use the dp ratio and RT index as the regressor. The standard errors are adjusted
by the Newey-West correction with 12 lags and reported in parentheses. The sample covers 1973Q1 to 2020Q4 at a
monthly frequency. The stars ∗,∗∗, and ∗∗∗denote 10%, 5%, and 1% levels of signicance, respectively.
Equity Yield Slope Term Premium Changes in Equity Yield
(1) (2) (3) (4) (5) (6)
dp ratio −0.018*** 0.384 −0.026**
(0.007) (0.413) (0.010)
RT −0.026** 0.017 −0.029*
(0.007) (0.291) (0.015)
Constant −0.102*** 0.009 2.278 −0.096 −0.155** 0.004
(0.039) (0.006) (2.476) (0.280) (0.061) (0.010)
Obs. 563 563 551 551 551 551
20.031 0.22 0.008 0.0001 0.026 0.111
94
Table B1: Quantile Regressions
This table shows univariate quantile regressions of current real GDP growth on current leverage volatility based on
AEM and HKM. The real GDP growth data comes from the Federal Reserve Economic Data (FRED). For each panel,
model (1) shows the OLS result, and models (2) to (10) show the quantile regression results with conditional 1, 2, 3,
4, 5, 25, 50, 75, and 95 percent quantiles, respectively. The standard errors are reported in parentheses. The sample
covers 1970Q1 to 2023Q4 at the quarterly frequency.
Panel A: AEM
OLS quantile regression
0.01 0.02 0.03 0.04 0.05 0.25 0.5 0.75 0.95
(1) (2) (3) (4) (5) (6) (2) (3) (4) (5)
−0.005 1.188 0.594 −0.163 −0.274 −0.622 −0.297 0−0.209 −0.308
(0.30) (4.63) (2.46) (2.39) (0.59) (0.55) (0.39) (0.23) (0.22) (1.52)
Constant 2.845*** −7.808 −6.185 −4.42 −3.738∗∗∗ −2.787∗∗ 1.380*** 3.000*** 4.363*** 7.856***
(0.30) (10.62) (5.65) (5.53) (1.35) (1.27) (0.30) (0.20) (0.30) (1.00)
Observations 216 216 216 216 216 216 216 216 216 216
20
Panel B: HKM
OLS quantile regression
0.01 0.02 0.03 0.04 0.05 0.25 0.5 0.75 0.95
(1) (2) (3) (4) (5) (6) (2) (3) (4) (5)
−0.841∗∗∗ −5.964∗∗∗ −6.058∗∗∗ −6.352∗−5.009 −1.94 −0.941∗∗ −0.454 0.053 −0.326
(0.30) (0.34) (1.07) (3.65) (3.21) (3.10) (0.45) (0.28) (0.43) (0.41)
Constant 2.845*** −8.229∗∗∗ −7.915∗∗∗ −6.934∗∗∗ −5.519∗∗∗ −2.933∗1.404*** 2.905*** 4.426*** 7.988***
(0.30) (1.09) (1.05) (2.11) (1.80) (1.74) (0.29) (0.19) (0.29) (0.79)
Observations 216 216 216 216 216 216 216 216 216 216
20.036
95
Table C1: Decomposition of Convenience Yield Slopes
This table reports the estimation results of the decomposition of the term structure of convenience yield. The regres-
sions in columns (1) and (2) are dened in (68), and those in columns (3) and (4) are dened in (69), and those in the
last two columns (5) and (6) are dened in (70). The regressors in these models are the nancial leverage volatility
based on the AEM and HKM measurements, the dividend-price (dp) ratio, and the RT index. The convenience yield
data comes from the personal website of William Diamond and is also available in the online materials of Li and Xu
(2024). The standard errors are adjusted by the Newey-West correction with 12 lags and reported in parentheses.
The sample covers 2004Q1 to 2020Q3 at a monthly frequency. The stars ∗,∗∗, and ∗∗∗denote 10%, 5%, and 1% levels
of signicance, respectively.
Panel A: AEM
Convenience Yield Slope Term Premium of Convenience Yield Changes in Convenience Yields
(1) (2) (3) (4) (5) (6)
0.05 0.065 0.191* 0.075 −0.141 −0.003
(0.084) (0.107) (0.116) (0.107) (0.146) (0.104)
RT −0.034 0.265 −0.315**
(0.099) (0.176) (0.139)
Constant 0.128 0.132 0.149 0.129 −0.034 −0.01
(0.096) (0.098) (0.148) (0.140) (0.154) (0.124)
Obs. 199 199 193 193 193 193
20.003 0.005 0.023 0.097 0.013 0.124
Panel B: HKM
Convenience Yield Slope Term Premium of Convenience Yield Changes in Convenience Yields
(1) (2) (3) (4) (5) (6)
−0.025 −0.02 0.053 −0.101 −0.11 0.049
(0.074) (0.075) (0.105) (0.158) (0.083) (0.142)
RT −0.01 0.325 −0.336*
(0.089) (0.209) (0.174)
Constant 0.112 0.112 0.098 0.098 −0.005 −0.005
(0.091) (0.091) (0.139) (0.131) (0.120) (0.108)
Obs. 199 199 193 193 193 193
20.001 0.002 0.003 0.102 0.013 0.126
Panel C: dp ratio and RT
Convenience Yield Slope Term Premium of Convenience Yield Changes in Convenience Yields
(1) (2) (3) (4) (5) (6)
dp ratio 0.031 0.473 −0.437
(0.145) (0.374) (0.269)
RT −0.019 0.283* −0.316**
(0.085) (0.168) (0.132)
Constant 0.314 0.114 3.122 0.108 −2.789 −0.009
(0.919) (0.091) (2.449) (0.133) (1.774) (0.109)
Obs. 199 199 193 193 193 193
20.0002 0.001 0.026 0.094 0.023 0.124
96
Table C2: The Predictability of Bond Returns
This table reports the prediction results for future bond returns. The regression model is shown in Equation (71).
The response is the 12-month ahead excess return on 2-year government bonds. The regressors in these models are
the nancial leverage volatility based on the AEM and HKM measurements, the dividend-price (dp) ratio, the RT
index, and the 6-month convenience yield (cy). The bond yield data comes from the personal website of William
Diamond and is also available in the online materials of Li and Xu (2024). The standard errors are adjusted by the
Newey-West correction with 12 lags and reported in parentheses. The sample covers 2004Q1 to 2020Q3 at a monthly
frequency. The stars ∗,∗∗, and ∗∗∗denote 10%, 5%, and 1% levels of signicance, respectively.
Panel A: AEM Panel B: HKM Panel C: RT, cy and Equity Slope
(1) (2) (3) (4) (5) (6) (8) (9) (10)
0.004** 0.002 0.003**
(0.001) (0.002) (0.001)
0.001 −0.001 0.001
(0.001) (0.001) (0.001)
RT 0.003* 0.004** 0.004**
(0.002) (0.002) (0.002)
cy_6m 2.442*** 2.541*** 2.587***
(0.866) (0.820) (0.822)
Equity Slope −0.045*
(0.026)
Constant 0.008*** 0.008*** −0.0004 0.007** 0.007*** −0.002 0.007*** −0.002 0.007***
(0.003) (0.003) (0.004) (0.003) (0.003) (0.004) (0.003) (0.004) (0.003)
Obs. 199 199 199 199 199 199 199 199 199
20.047 0.118 0.183 0.013 0.105 0.162 0.104 0.155 0.071
97
Table C3: Aggregate Moments (Including Bonds)
This table supplements simulated moments for convenience yields in the benchmark calibration. The observed mean
and volatility of convenience yields, (
12)and (
12), cover the sample period of 2004–2020. Other variables are
the same as those in Table 20.
Data Benchmark Predictable
Panel A: Aggregate Quantities
(Δ)1.90 1.98 1.99
(Δ)2.25 2.51 2.48
(Δ)1.39 1.29 1.37
(Δ)12.54 13.34 11.21
(Δ,Δ)0.47 0.66 0.78
(/)0.21 0.24 0.30
(1/)0.65 0.57 0.63
(Δlog())
(Δlog()) 1.67 1.88 2.31
−1.15 —−1.34
Panel B: Asset Prices
(−)6.09 6.28 6.98
()19.12 16.28 18.45
E(p-d) 3.41 2.75 2.65
(−)0.47 0.09 0.13
(
5−
1)0.37 1.18 2.27
(
5−
1)6.69 1.79 3.93
(
12)0.35 0.14 0.10
(
12)0.18 0.18 0.20
(
+1)0.60 0.60 2.07
98
Table C4: Conditional Moments (Including Bonds)
This table supplements simulated moments for convenience yields across dierent phases of the business cycle with
benchmark calibration. The observed convenience yields, (
12), cover the sample period of 2004–2020. Other
variables are the same as those in Table 21.
Panel A: Conditional Moments
Expansion Recession
Data Benchmark Predictable Data Benchmark Predictable
(Δ)2.48 2.93 2.95 −1.14 −3.36 −3.48
/(Δ())
(Δ()) 0.79 0.66 0.47 1.67 1.90 2.27
(−)9.28 11.33 13.28 −12.99 −22.32 −28.76
()15.57 17.19 19.25 18.34 17.86 23.26
(
1) −3.22 2.56 1.40 10.89 8.62 14.27
(
5−
1)1.59 1.79 3.54 −5.74 −2.33 −4.91
(
12)0.31 0.07 0.04 0.58 0.48 0.42
Panel B: Discount Rate Slope vs. Dividend Growth Slope: A Model of Predictable Growth
Expansion Recession Dierence
(
5−
1)3.54 −4.91 8.44
(5−1)3.11 −4.07 7.18
(
5−
1) −0.42 0.84 −1.26
99
