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A Supply and Demand Approach to Equity Pricing∗
Sebastien Betermier, Laurent E. Calvet, and Evan Jo
First version: July 4, 2019
This version: December 5, 2020
Abstract
We develop a tractable general equilibrium framework providing a direct mapping
between (i) the supply and demand for capital at the firm level and (ii) the
cross-section of stock returns. Investor behavioral tilts and hedging needs drive
capital supply, while firm profitability drives demand. Heterogeneity in supply
and demand factors determines the sign of the risk-return relation and generates
anomalies such as betting-against-beta, betting-against-correlation, size, value,
investment, and profitability. We estimate the supply and demand schedules of
over 4,000 U.S. firms and verify that the model accurately predicts the sign of
the risk-return relation conditional on characteristics.
Keywords: Asset pricing, anomalies, capital allocation, general equilibrium, factor-
based investing, production economy.
JEL Classification: C36, G11, G12.
∗
Betermier: Desautels Faculty of Management, McGill University, 1001 Sherbrooke St West, Montreal,
QC H3A 1G5, Canada; sebastien.betermier@mcgill.ca. Calvet: Department of Finance, EDHEC Business
School, 16 rue du Quatre-Septembre, 75002 Paris, France, and CEPR; laurent.calvet@edhec.edu. Jo: Desau-
tels Faculty of Management, McGill University, 1001 Sherbrooke St West, Montreal, QC H3A 1G5, Canada;
evan.zhou@mcgill.ca. We acknowledge helpful comments from Daniel Andrei, Ray Ball, Laurent Barras,
Jonathan Berk, Harjoat Bhamra, John Campbell, Yao Deng, Joost Driessen, Frank de Jong, Jean-Sebastien
Fontaine, Jens Kværner, Lars Loechster, Abraham Lioui, Francis Longstaff, Charles Kahn, Stefan Nagel,
Guillaume Rousselet, Enrique Schroth, Andreas Stathopoulos, Per Strömberg, Oren Sussman, Harald Uhlig,
Raman Uppal, Johan Walden, Mungo Wilson, Lu Zhang, and seminar participants at the Bank of Canada,
EDHEC Business School, McGill University, Oxford University, Tilburg University, Validus Risk Manage-
ment, the 2019 Annual Conference on Corporate Policies and Asset Prices at Cass University, and the
2020 European Finance Association Conference. This material is based upon work supported by the Social
Sciences and Humanities Research Council of Canada. We thank Calcul Québec (calculquebec.ca) and
Compute Canada (computecanada.ca) for access to the supercomputer Beluga.Understanding the relation between risk and return has proven to be a challenging task for
finance research. The standard Capital Asset Pricing Model (CAPM) predicts a linear and
positive relation between a stock’s average return and its exposure to market risk, yet the
empirical evidence reveals that this relation is non-linear and sometimes negative (Black
1972, Fama and French 1992, Hong and Sraer 2016). These anomalies are complemented by
the fact that hundreds of firm characteristics have been associated with persistent deviations
from the CAPM (Cochrane 2011, Harvey, Liu, and Zhu 2016).
While theoretical explanations of risk and return have long focused on building a stochas-
tic discount factor (SDF) from either investor or firm production decisions, a new strand of
the literature is shifting focus away from the SDF and deriving asset prices directly from
supply and demand effects (Ball, Sadka, and Tseng 2019, Garleanu, Pedersen, and Potesh-
man 2009, Greenwood, Hanson, Stein, and Sunderam 2020).1 In an influential study, Koijen
and Yogo (2019) exploit the growing availability of asset holdings data and develop a new
approach to estimate the price impact of large institutional investors on equity prices in a
setting where shares outstanding and firm characteristics are exogenous (see also Gabaix
and Koijen 2020a, 2020b, and Koijen, Richmond, and Yogo 2020).
The present paper builds on this emerging literature and develops a framework that
provides a direct mapping between (i) the supply and demand for capital at the firm level
and (ii) the cross-section of stock returns. We aim to answer several questions: Why is the
risk-return relation non-linear and sometimes negative? Why do CAPM anomalies come as
a pack? Is the cross-section of stock returns mainly driven by supply or demand factors?
To address these questions, we develop a tractable no-arbitrage general equilibrium model
in which investor behavioral tilts and hedging needs drive the supply of capital, while firm
profitability drives demand. We study how supply and demand factors jointly drive the
cross-section of firm sizes and risk premia and the SDF.
Our approach produces a fresh and useful perspective on equity pricing. We explain how
capital supply and demand factors determine whether the cross-sectional relation between
risk and return is positive or negative. We also uncover mechanisms through which het-
erogeneity in supply and demand forces jointly generates the betting-against-beta, betting-
against-correlation, size, book-to-market-ratio, investment, and profitability anomalies. Fur-
thermore, the model allows us to develop a novel estimation technique. We empirically
estimate the supply and demand schedules of over 4,000 U.S. firms by two- and three-stage
1
See also Cha and Lee 2001, Greenwood 2005, Hau, Massa, and Peress 2010, Petajisto 2009, Shleifer
1986, and Wurgler and Zhuravskaya 2002.
1least squares. Heterogeneity in firm demand schedules for financial capital is substantially
wider than heterogeneity in supply schedules, but both forms of heterogeneity help to explain
the cross-section of stock returns. Moreover, the model accurately predicts the sign of the
risk-return relation conditional on characteristics.
The detailed contributions of the paper are the following. We first develop a tractable
model of capital markets based on classic building blocks from the production- and investor-
based literatures. Each firm operates a decreasing-return-to-scale technology and produces
a risky cash flow at the terminal date. A firm has no initial resources and raises capital by
issuing stocks. Firms are heterogenous in profitability, so that each firm has its own demand
schedule for capital. The elasticity of the capital demand schedule is driven by the capital
elasticity of output.
The representative investor allocates her wealth to the firms and to a riskless asset in
zero net supply. Her capital allocation is driven both by the mean-variance properties of her
portfolio and by the hedging or behavioral properties of individual stocks. Relevant stock
properties may include exposures to the investor’s non-financial risk, subjective beliefs, or
investor preferences for environmentally or socially responsible companies. These motives
all cause the investor to tilt away from the mean-variance efficient portfolio and jointly
generate a deviation portfolio. Through her investment decisions, the representative investor
determines the supply of capital to each firm.2 A firm’s exposure to the deviation portfolio
is a factor driving the capital supply, whereas profitability is a demand factor.
We obtain simple and well-behaved schedules for each firm’s supply and demand of capital
under the following price and quantity metrics. Our price measure is a firm’s Sharpe ratio,
i.e. the stock return compensation per unit of volatility. The corresponding quantity metric
is risk capital, which we define as the dollar volatility of the firm’s market capitalization.
Under these definitions of price and quantity, the supply of risk capital to a firm is upward-
sloping while the demand schedule is downward-sloping. The allocation of capital, price of
risk, interest rate, and SDF are jointly determined in general equilibrium. The main insights
from the model can be summarized as follows.
First, the cross-sectional relation between the firms’ risk capital and price of risk depends
on whether supply factors or demand factors are the dominant source of heterogeneity. When
stocks have identical hedging or behavioral appeal, the supply schedule is the same for all
2
This classification may seem at odds with the view with the traditional view that investors are on the
demand side of the equity market. Both classifications are of course correct, but our terminology is preferable
for this paper given the focus on the allocation of capital to firms.
2firms. Equilibrium outcomes are all located on the common supply curve and cross-sectional
variation in profitability generates a positive cross-sectional relation between risk capital and
Sharpe ratio. By contrast, among firms with identical profitability and therefore identical
demand schedules, cross-sectional variation in supply factors generates a negative relation
between risk capital and Sharpe ratio.
Second, the cross-sectional price-quantity relation directly maps into the cross-sectional
relation between the stocks’ Sharpe ratio and market risk exposure. A stock’s correlation
with the market increases with its endogenous level of risk capital. In the special case
where the representative investor has no tilt from the mean-variance efficient portfolio, all
firms have the same capital supply schedule, and the cross-sectional relation between market
correlation and Sharpe ratio is linear and positive. However, when firms have heterogeneous
supply schedules, the relation between market correlation and Sharpe ratio is non-linear and
can take any sign. Consistent with our price and quantity findings, the risk-return relation
is positive if firms are primarily heterogeneous in profitability, and negative if heterogeneity
in exposure to the deviation portfolio dominates. The same logic applies to the relation
between market beta and expected return.
Third, the betting-against-beta (BAB) and betting-against-correlation (BAC) anomalies
documented in Frazzini and Pedersen (2014) and Asness et al. (2020) jointly arise in our
model. We show that low supply and low demand factors both lead to high CAPM alpha,
low market beta, and low market correlation. The main requirement to obtain these relations
is that the market has a positive exposure to the deviation portfolio. This condition holds,
for instance, if the market portfolio has positive exposure to aggregate non-financial income
shocks. Since a stock’s exposure to the deviation portfolio cannot be rewarded both directly
and via the market portfolio, the incremental reward for exposure to market risk is lower
than what the CAPM predicts. Hence any supply or demand factor that increases a firm’s
market correlation has a negative impact on alpha.
Fourth, anomalies based on firm characteristics such as size, investment, value, and
gross profitability (Fama and French 2015, Novy-Marx 2013) are also natural equilibrium
implications of supply and demand factors. Prior work based on an exogenous SDF (e.g.
Hou, Xue, and Zhang 2015) shows that these characteristics all relate to expected returns
through the firms’ first-order investment conditions. We also show that these relations have
implications for CAPM alpha in general equilibrium. Specifically, when the market has
positive exposure to the deviation portfolio, low supply and low demand factors both lead to
high alpha, low size, low investment, and low market-to-book ratio. In addition, low supply
3leads to high profitability conditional on a stock’s market-to-book ratio.
We next turn to the empirical implications of our model. We develop the structural
estimation of the supply and demand schedules of individual firms by instrumental variables.
Since the model imposes restrictions on the supply and demand slopes of each firm, the two-
and three-stage least squares estimators can be applied to a single cross-section or a panel
of individual stocks.
Our choice of instruments is grounded in economic theory. To uncover demand schedules,
we instrument supply variation by turnover, which Lo and Wang (2007) directly relate to
investor hedging demand in a general I-CAPM (Merton 1973). Turnover could also proxy for
investor sentiment (Baker and Stein 2004) and investor attention (Barber and Odean 2008,
Da, Engelberg, and Gao 2011). We empirically verify that stocks with high turnover tend
to have a high market correlation and a low Sharpe ratio, consistent with supply variation.
To uncover supply schedules, we instrument demand variation by the firms’ return on
equity, which Hou, Xue, and Zhang (2020) use as a measure of profitability. We verify
that the stocks of highly profitable firms generally have high market correlations and high
Sharpe ratios, consistent with demand variation. The selection of instruments is by no means
exhaustive, and future research could integrate additional instruments building on the work
of Ball, Sadka, and Tseng (2019), Koijen and Yogo (2019) and Gabaix and Koijen (2020a).
Our empirical methodology proceeds in several steps. We use time-series data to estimate
the Sharpe ratio, market correlation, and risk capital of 4,727 U.S. firms over five non-
overlapping five-year windows between 1994 and 2019. We then apply two- and three-stage
least squares to the panel of risk prices and quantities.
Supply slopes are positive and imply that the representative investor has a risk aversion
coefficient of about 1.2, which is consistent with standard utility theory. Demand slopes
are negative and imply mild but significant departures from constant returns to scale in
production. Even though we run the estimation at the stock level, regression coefficients
have reasonably high levels of significance. The fitted model is able to produce substantial
dispersion in equilibrium values of market correlation and Sharpe ratio. This dispersion
is dominated by cross-sectional variation in firm demand factors, which explains why the
relation between market correlation and Sharpe ratio is generally positive. Heterogeneity in
supply is nonetheless substantial and helps to explain key properties of the cross-section of
stock returns.
4The model accurately predicts the sign and magnitude of the risk-return relation across
sorted portfolios of stocks. We document a negative relation between a portfolio’s historical
Sharpe ratio and its market correlation when stocks are sorted by turnover, our instrument
for supply variation. Conversely, we measure a positive relation between risk and return
when stocks are sorted by profitability, our instrument for demand variation. Consistent
with model’s predictions, the empirical risk-return relation is therefore negative when supply
variation is the main source of heterogeneity across portfolios, and positive otherwise. We
verify that these results also hold when we use stock-level data to analyze the risk-return
relation within selected groups of stocks.
The paper provides several take-aways for future research. A wide range of patterns
documented in the cross-section of equity returns have natural explanations once they are
recast as price-quantity relations. For instance, risk and return are negatively related in
general equilibrium and in the data when supply factors exhibit stronger heterogeneity than
demand factors. This result is noteworthy because the fact that some low-market beta stocks
generate higher average returns than high-market beta stocks has long been viewed as a core
violation of the fundamental trade-off between risk and return (Baker, Bradley, and Wurgler,
2011).
Classic predictors of CAPM deviations, such as firm size, investment, and market beta,
are natural implications of the heterogeneity of supply and demand. As a result, these
predictors therefore tend to come as a pack in general equilibrium. Multiple anomalies
should therefore be observed across asset classes, industries, and countries, consistent with
the empirical evidence (Asness, Moskowitz, and Pedersen 2013, Fama and French 2012).
Moreover, anomalies arise whether portfolio deviations from the mean-variance frontier have
a risk-based or behavioral origin. Our approach is therefore consistent with Kozak, Nagel,
and Santosh (2018), who demonstrate the observational equivalence of factor pricing and
behavioral asset pricing.
Heterogeneity in supply and demand schedules should also play a central role in the
debate on whether stock returns are mainly driven by firm-level characteristics or risk ex-
posures. We show that a firm’s profitability drives its demand for capital, exposure to
the aggregate deviation portfolio drives supply, and their interaction determines the stock’s
expected return. The debate about the respective roles of firm characteristics and risk ex-
posures can therefore be viewed as a debate about whether cross-sectional heterogeneity
primarily originates from demand or supply forces.
5Our paper complements the recent general equilibrium literature that generates the value
and size effects through a variety of risk channels (Garleanu, Kogan, and Panageas 2012,
Gomes, Kogan, and Zhang 2003, Kogan, Papanikolaou, and Stoffman 2018, Papanikolaou
2011, Parlour and Walden 2011). These models aim to pin down the underlying forms of risk
that generate the size and value factors and quantitatively match their risk premia. Since
we purposely remain agnostic on the drivers of aggregate deviations from mean-variance
efficiency, the mechanisms we uncover are likely to apply to a broad set of general equilibrium
economies.
Our framework closely relates to Koijen and Yogo (2019), who study the impact of port-
folio tilts on stock prices while taking the supply of stocks as fixed. We endogenize firm
production decisions and apply two- and three-stage least squares to extract information
from the joint variation in supply and demand instruments. The method can be combined
with the granular instrument methodology proposed by Koijen and Yogo (2019) and Gabaix
and Koijen (2020a). More broadly, the paper contributes to the growing empirical literature
documenting linkages between stock prices and the supply of financial capital from institu-
tions (Garleanu, Pedersen, and Poteshman 2009, Klinger and Sundaresan 2019, Koijen and
Yogo 2019) or households (Betermier, Calvet, and Sodini 2017, Betermier et al. 2012), and
to the literature documenting linkages between stock prices and firms’ demand for financial
capital (Liu, Whited, and Zhang 2009, Lowry 2003, Pastor and Veronesi 2005, 2009).
The structure of the paper is as follows. Section 1 sets up the economy. Section 2 derives
the supply and demand schedules for each firm and clears the markets. Section 3 studies
the equilibrium relation between firm sizes and risk prices and the relation between risk and
return. Section 4 develops the methodology for the estimation of supply and demand sched-
ules and presents empirical results. The Internet Appendix provides proofs and additional
empirical tests.
1. Definition of the Economy
1.1. Firms and Representative Investor
We consider an economy with two periods t ∈ {0, 1} and a unique good. All quantities at
dates 0 and 1 are expressed as units of this good. The production sector consists of N firms
indexed by n ∈ {1, · · · , N }. Firms have no initial resources at the beginning of period 0.
6A representative investor owns the N firms. She is endowed with exogenous wealth E0
at the beginning of period 0, which she uses to invest in the firms and in a riskless asset.
There is no consumption at date 0 and there are no trading constraints. The riskless asset
is in zero net supply and has net rate of return rf .
During period 0, each firm n raises financial capital Vn from the investor by issuing stocks.
It allocates In to uninstalled productive capital and pays out the economic profit Vn − In to
the investor. At the end of period 0, the total financial wealth of the investor is therefore:
N
X
W0 = E0 + (Vn − In ). (1.1)
n=1
In period 1, each firm n produces a random cash flow CFn , which the investor receives as
the firm’s owner. The representative investor consumes the aggregate cash flow, N
P
n=1 CFn .
The production process takes place as follows. Each firm n operates a specific decreasing
return to scale technology. An investment of In units of uninstalled capital at t = 0 results
in Kn (In ) units of installed capital in the same period, where Kn ( · ) is an increasing and
concave function of In that will be specified in Section 3. At date t = 1, the firm generates
the stochastic cash flow
CFn (Kn ) = (an + zn ) σ cf,n Kn . (1.2)
The random variable zn has zero mean and unit variance. We denote by z = (z1 , . . . , zN )0
the vector of cash flows shocks and by ρ = (ρi,j )1≤i,j≤N its correlation matrix. The variables
σ cf,n and an represent, respectively, the volatility and mean-to-volatility ratio of the firm’s
cash flow per unit of installed capital. The technology operated by each firm is exogenous
to the model.
1.2. Financial Markets
We assume that financial markets are competitive and that there exists a risk-adjusted
probability measure Q, which the investor uses to value financial assets. The risk-adjusted
measure will be endogenously determined in equilibrium. The market value of firm n’s equity
7at date t = 0 is
EQ (CFn ) an − λ n
Vn = = σ cf,n Kn , (1.3)
1 + rf 1 + rf
where λn = −EQ (zn ). By (1.2) and (1.3), the return on firm n’s equity is therefore
CFn 1 + rf
rn = −1= (an + zn ) − 1. (1.4)
Vn an − λn
We denote by r = (r1 , · · · , rn )0 the vector of returns, and by ω the N × 1 the vector of shares
of wealth invested in the firms by the investor. Her financial wealth at t = 1 is equal to
W1F = W0 [1 + rf + ω 0 (rr − rf 1 )], (1.5)
where 1 = (1, . . . , 1)0 .
The risk and return characteristics of the stock follow from equations (1.2) to (1.4). The
stock’s risk premium µn − rf , and volatility σ n are given by
an σ cf,n Kn (1 + rf ) λn
µn − rf = − 1 − rf = , (1.6)
Vn an − λ n
σ cf,n Kn 1 + rf
σn = = . (1.7)
Vn an − λn
The Sharpe ratio is therefore:
µn − rf
λn = . (1.8)
σn
We stack expected returns into the vector µ = (µ1 , · · · , µN )0 and Sharpe ratios into λ =
(λ1 , · · · , λN )0 . We also denote by σ the diagonal matrix with elements σ 1 , ..., σ n , and by Σ
the variance-covariance matrix of returns. All these variables are endogenous in equilibrium.
The stock’s risk premium (1.6) and volatility (1.7) are both high when the firm has low
profitability and a high price of risk λn . The reason is that the value of equity, Vn , is then
low relative to the expected cash flow, which scales up the return distribution and therefore
produces a high risk premium and high volatility.
PN
The market portfolio is defined in the usual manner. Let Vm = n=1 Vn denote the
8aggregate value of the stock market. The market portfolio assigns the weight Vn /Vm to every
stock n. It earns the return rm = N
P
n=1 (Vn /Vm ) rn , which has mean µm and volatility σ m .
1.3. Equilibrium
We close the economy as follows.
Definition 1 (General Equilibrium). A general equilibrium consists of firms’ optimal in-
vestments {In }N
n=1 , the representative investor’s stock portfolio ω , a risk-adjusted probability
measure Q, Sharpe ratios {λn }N n=1 , and an interest rate rf such that:
(i) the market for each stock clears: W0 ω n = Vn for every n;
(ii) the market for the risk-free asset clears: 1 − 10 ω = 0.
In general equilibrium, the investor fully allocates the exogenous wealth to productive invest-
ment in the firms: E0 = N
P
n=1 In , consistent with the absence of consumption in period 0 and
the zero net supply of the risk-free asset. By (1.1), the aggregate value of the stock market
is equal to the investor’s wealth at the end of the initial period: Vm = W0 . Furthermore, the
stochastic discount factor
1 − λ0 ρ−1 z
SDF =
1 + rf
prices all financial assets, as we verify in the Appendix.3
In the next Section, we further specify the decision process of the investor and firms
and derive a closed-form characterization of Sharpe ratios and capital allocation in general
equilibrium.
2. General Equilibrium of Capital Markets
In this Section, we develop a specification of the economy that delivers a tractable character-
ization of equilibrium. Section 2.1 explains the units in which supply and demand schedules
are expressed. Section 2.2 specifies investor portfolio decisions, Section 2.3 specifies produc-
tion functions, and Section 2.4 presents the resulting general equilibrium.
3
The highest attainable Sharpe ratio is given by the Hansen-Jagannathan bound, SRmax =
σ (SDF )/EP (SDF ) = λ0 ρ−1 λ
P
λ.
92.1. Price and Quantity of Risk
We express supply and demand under the following price and quantity metrics, which are
closely tied to standard measures of risk and return. Our price measure is a firm’s Sharpe
ratio λn , i.e. the stock’s average compensation per unit of volatility in equation (1.8). The
corresponding quantity metric is risk capital, which we define as the dollar volatility of the
firm’s market capitalization:
Qn = σ n Vn = σ cf,n Kn = [V ar(CFn )]1/2 , (2.1)
where the last equalities follow from (1.7). The risk capital of the market portfolio is sim-
ilarly Qm = σ m Vm . These definitions are familiar in the risk management literature, which
increasingly measures risk in dollar amounts rather than in proportional changes (see Artzner
et. al. 1999).
The use of λn and Qn will help us map the firms’ capital supply and demand schedules
directly into the risk and return properties of stocks, as we now explain. By equations (1.6)
and (1.7), the endogenous price, λn , pins down the mean and volatility of the stock. The
endogenous risk capitals of all firms, Q1 , . . . , QN , determine the risk capital of the market
portfolio, Qm = σ m Vm = (Q Q0 ρ Q
Q)1/2 and the correlation between stock n and the market
portfolio:
PN P
i=1 ρn,i Qi Qn + i6=n ρn,i Qi
ρm,n = Corr(Rm , Rn ) = = , (2.2)
Qm Qm
where ρn,i is the exogenous correlation the cash flows of firms i and n. We refer the reader to
the Appendix for a derivation. A firm has high correlation with the market if it has large risk
capital Qn or if it correlates with other firms that take up the lion’s share of the economy’s
risk-bearing capacity.
2.2. Supply
The representative investor drives the supply of risk capital. We assume that her portfolio
of stocks is given by:
1
ω= ω max − ω δ , (2.3)
γ
10where γ quantifies the investor’s risk aversion and the vector
ω max = Σ−1 (µ
µ − rf 1 ) (2.4)
is a portfolio with maximum Sharpe ratio. The portfolio ω δ captures the investor’s tilt away
from ω max . This tilt can originate from a number of risk-based and behavioral channels that
we now examine.
Hedging Non Financial Wealth. Consider an investor exposed to background risk in the form
of an exogenous shock Rθ W0 to non-financial wealth at t = 1, where Rθ is an exogenous
random variable. Background risk may stem from human capital, private business holdings,
housing, inflation, or a combination thereof. The investor’s total wealth at t = 1 is the sum of
financial and nonfinancial wealth: W1 = W1F + Rθ W0 . We verify in the Appendix that (2.3)
holds exactly if the investor has rational expectations and quadratic utility: U = E(W1 ) −
γ V ar(W1 )/(2 W0 ), as in Merton (1987). The deviation portfolio ω δ = Σ−1 Cov(Rθ , r ) is the
portfolio with the highest correlation with Rθ .
In this example, investors use the stock market to hedge a fixed quantity of nonfinancial
risk, Rθ , per unit of initial wealth. As a result, the proportional tilt in risk capital allocated
to each stock for hedging purposes, qδ ≡ σ ω δ = ρ−1 Cov(Rθ , z ), is only determined by
parameters that are exogenous to the model.
ESG Preferences and Sentiment. Behavioral motives can also drive the deviation portfolio
ω δ . These motives may arise from preferences for certain types of firms, such as environ-
mentally or socially responsible companies. For instance, ω δ may be the ESG tilts in Pastor,
Stambaugh, and Taylor (2020).
Behavioral motives may also arise from subjective beliefs about expected stock returns.
Consider that the subjective assessment of the price of risk is λ + bsub , where bsub is a fixed
subjective bias. In the Appendix, we show that the investor with quadratic utility selects
ω δ = γ −1 σ −1 ρ−1 bsub as in the Black-Litterman (1992) model. The proportional tilt in risk
capital allocated to each stock, qδ ≡ σ ω δ , is once again exogenous to the model.
Throughout the paper, we take the tilt in risk capital per unit of initial wealth, qδ ≡ σ ω δ ,
as exogenous. We remain agnostic on its origins, so that the model can be used to study the
impact of any type of aggregate portfolio tilt. The assumption that the tilt in risk capital
does not vary with the price of risk is not an intrinsic limitation of our approach. If the
11deviation in risk capital is of the form qδ (λ)
(λ), a first-order Taylor expansion with respect to
λ allows us to obtain closed-form equilibrium conditions along the lines developed in this
Section, as we show in the Appendix. For expositional simplicity, we rule out this possibility
in the main text.
The return on the deviation portfolio, rδ = ω0δ rr, plays a central role in our analysis.
For every firm n, we consider the covariance between the stock’s normalized return and the
return of the deviation portfolio:
δ n = Cov(rδ , rn )/σ n . (2.5)
A high exposure to the deviation portfolio indicates that the stock strongly correlates with
stocks that are unappealing to the investor. We stack the covariances δ n into the column
vector δ = (δ 1 , ..., δ N )0 . By (2.5), this vector satisfies δ = ρ qδ . The covariances δ n are therefore
exogenous to the model.
The supply of risk capital follows directly from the optimal portfolio rule in (2.3). The
vector of supply curves is given by4
λs (Q) = γ δ + γW0−1ρ Q
Q. (2.6)
The investor requires a high Sharpe ratio to supply a high quantity of risk capital to firm
n because a high Qn entails high concentration of risk in the firm. She also requires a high
Sharpe ratio if the firm has high exposure to the deviation portfolio or high correlation with
other firms. Moreover, the investor tilt δ affects the functional form of the supply schedule
in the same way whether it has a risk-based or behavioral origin.
The supply equation (2.6) has immediate implications for the relation between risk and
return. A stock’s Sharpe ratio is the sum of a compensation for exposure to the deviation
portfolio and an incremental compensation for exposure to market risk:
λs,n = γ δ n + ρm,n (λm − γ δ m ), (2.7)
The term δ m = Cov(rδ , rm )/σ m = ( N
P
n=1 δ n σ n Vn )/(σ m Vm ) in the equation quantifies the
exposure of the market portfolio to the deviation portfolio.5 Once the stock’s reward for
deviation portfolio exposure is taken into account, the incremental reward for market ex-
4
We multiply (2.3) by W0 σ and use the properties that Σ = σ ρ σ , Q = W0 σ ω , and δ = ρ σ ω δ .
5
Equation (2.7) can be derived as follows. Since ρM,n is the nth component of the column vector ρ Q /Qm ,
12posure nets out the market portfolio’s compensation for deviation risk, as equation (2.7)
shows. In Section 3.3, we will show that these properties have important implications for
the cross-section for risk premia.
2.3. Demand
Firms drive the demand for risk capital. Installed capital at t = 0 is given by
η
η+1
η+1
Kn (In ) = In (2.8)
η
where η ∈ (0, +∞) is a fixed parameter. A high value of η implies near constant returns
to scale, whereas a low η implies fast decreasing returns to scale. We assume that η is
homogeneous across firms.6 The model converges to an endowment economy when η → 0.
The firm chooses the amount of investment that maximizes the economic profit Vn − In .
The optimal level of installed capital is
η
(an − λn ) σ cf,n
Kn = (2.9)
1 + rf
if λn ≤ acf,n and zero otherwise. The firm invests more if it is highly profitable and has a
low price of risk. The installed capital stock Kn is the book value of the firm.
By equation (2.9), the vector of demand schedules for risk capital is:
1/η 0
Q) = a1 − (1 + rf ) (Q1 /σ η+1 1/η
, . . . , aN − (1 + rf ) (QN /σ η+1
λd (Q cf,1 ) cf,N ) . (2.10)
More profitable firms have higher demand schedules.
2.4. Market Clearing
The exogenous parameters of the model consist of firm technologies and the investor’s initial
wealth E0 , risk aversion γ, and tilt vector qδ . All other variables are endogenous. For a
equations (2.2) and (2.6) imply that λS,n = γ δ n +γ ρm,n Qm / W0 . The aggregation of this relationship implies
λS,m = γ δ m + γ Qm / W0 ,
and equation (2.7) holds.
6
In the Appendix we extend the model to the case of firms with heterogeneous η.
13given interest rate rf , the firms’ risk capital and Sharpe ratio are determined by the market
clearing condition:
Q) = λd (Q
λs (Q Q). (2.11)
Since supply and demand schedules are available analytically, this condition is a closed-form
characterization of equilibrium and the linchpin of the theoretical analysis of the next section.
The equilibrium of the bond market pins down the interest rate rf .
Proposition 1 (Existence and Uniqueness). The economy with frictionless financial mar-
kets has a unique general equilibrium under the sufficient conditions stated in the Appendix.
In the rest of paper, we focus on a given equilibrium, so that rf is known.
We make several observations about equilibrium. Technological parameters, such as
profitability per unit of risk, an , and cash flow volatility, σ cf,n , drive the demand for capital
and will henceforth be called demand factors. By contrast, the comovement between the
normalized return and the deviation portfolio, δ n , drives the investor’s portfolio and will
henceforth be called a supply factor.
In the absence of background risk and behavioral tilts (δ n = 0 for all n), the equilibrium
reduces to
λn = ρm,n λm . (2.12)
The CAPM holds in this special setting, which can be immediately seen by multiplying (2.12)
by the stock’s volatility.
In the presence of background risk and behavioral tilts, the CAPM no longer holds.
By equation (2.7), the Sharpe ratio of a stock is the sum of a reward for market exposure
and a reward for exposure to the deviation portfolio, which is suggestive of a two-factor
equity pricing model. In the Appendix, we verify that the cross-section of equity returns
can indeed be represented by a two-factor model based on the market return, rm , and the
deviation portfolio return, rδ . We also show that the premium for market exposure satisfies
λm − γδ m = γ σ m . (2.13)
We now analyze the properties of this equilibrium.
143. Capital Allocation and Pricing
This section explains how supply and demand factors drive the equilibrium allocation of
capital, the cross-sectional relation between risk and return, and deviations from the CAPM.
3.1. Cross-Sectional Relation Between Firm Sizes and Risk Prices
To gain insights into the cross-section of firm sizes and risk prices, we reduce dimensionality
by considering a symmetric economy in which firms have identical cash-flow correlations:
ρi,j = ρ̄ for every i 6= j. Firms can be heterogeneous along all other dimensions. In the
Appendix, we show that general equilibrium is available in closed form whether cash-flow
correlations are homogenous or heterogeneous, but we choose to focus on homogenous cor-
relations in this section for expositional simplicity.
Let Q̄ = N −1 N
P
n=1 Qn denote the average risk capital across firms. The supply schedule
of firm n ∈ {1, . . . , N } is:
λs,n = λ0 + γ δ n + ∆s Qn , (3.1)
where ∆s = (1 − ρ̄) γ/W0 and λ0 = γ ρ̄ N Q̄/W0 . This result follows directly from equa-
tion (2.6). The constant λ0 quantifies crowding out due to investment in other firms. It is
substantial if investor wealth is small compared to aggregate risk capital or if firm cash flows
are highly correlated to other firms.
The demand schedule of firm n is approximately equal to
λd,n = η −1 an − ∆d,n (an )Qn , (3.2)
where ∆d,n (an ) = (1 + rf )η /(η σ η+1 η−1
cf,n an ). We obtain this result by linearizing the demand
schedule (2.10) around the risk capital the firm would select under zero risk premia.7 We
henceforth assume that η > 1, which guarantees that a more profitable firm offers a higher
Sharpe ratio to the investor.
The intersection of the schedules (3.1) and (3.2) provides the firm’s outcome. Panel A
of Figure 1 illustrates the demand channel by comparing two firms that differ only in prof-
In the absence of risk premia (λd,n = 0), each firm demands the risk capital Q0n = (1 + rf )−η aηn σ η+1
7
cf,n .
We linearize the demand schedule around Q0n . The demand schedule (3.2) is exact if η = 1.
15itability. The more profitable firm demands more risk capital. Because the investor requires
compensation to supply the additional risk capital, the more profitable firm has a higher
Sharpe ratio than the less profitable firm in GE.
Panel B of Figure 1 illustrates the supply channel by comparing two firms that differ
only in their exposure to the deviation portfolio. The less exposed firm can raise capital
more cheaply than the other firm, which leads it to choose higher levels of investment and
production. Consequently, the firm with low exposure to the deviation portfolio has a higher
quantity of risk but a lower Sharpe ratio than the firm with high exposure.
The following proposition summarizes these results and provides equilibrium in closed
form.
Proposition 2 (Equilibrium Quantities and Prices). If firms have homogeneous cash-
flow correlations, the equilibrium risk capital and Sharpe ratio of firm n are approximately
given by
η −1 an − γ δ n − λ0
Qn (an , δ n ) ≡ , (3.3)
+ − ∆s + ∆d,n (an )
∆s η −1 an + ∆d,n (an ) γ δ n + ∆d,n (an )λ0
λn (an , δ n ) ≡ , (3.4)
+ + ∆s + ∆d,n (an )
where the + and − symbols respectively refer to increasing and decreasing relationships. Risk
capital increases with profitability and decreases with exposure to the deviation portfolio.
The Sharpe ratio increases in both profitability and exposure to the deviation portfolio.
The equilibrium cross-sectional relation between firm quantities and risk prices depends
on the heterogeneity of supply and demand factors. Among firms with identical deviation
portfolio exposures δ n and therefore identical supply curves, cross-sectional variation in de-
mand factors induces a positive cross-sectional relationship between quantities and prices
because firm outcomes are all located on the common supply curve. Similarly, among firms
with similar profitability and therefore identical demand schedules, cross-sectional variation
in supply factors induces a negative relationship between quantities and prices. Indeed,
among similarly profitable firms, large firms are the firms with a low price of risk. We
measure factor heterogeneity and the resulting price-quantity relationship in the empirical
analysis of Section 4.
163.2. Cross-Sectional Relation Between Risk and Return
We next turn to the equilibrium cross-relationship of risk and return. Under symmetric
correlation, the price-quantity results of Section 3.1 map directly into the cross-section of
stock returns. The reason is that a stock’s correlation to the market, ρm,n , is an affine
function of the firm’s risk capital:
ρm,n = [ρ̄ N Q̄ + (1 − ρ̄) Qn ]/Qm , (3.5)
as equation (2.2) implies. The equilibrium correlation between the stock and the market,
ρm,n (an , δ n ),
+ −
therefore increases with profitability and decreases with exposure to the deviation portfolio.
Since the affine function is the same for all firms, we can plot equilibrium in the market
correlation-Sharpe ratio plane.8 In Figure 2, we show that this relation is positive among
stocks that have different profitability but similar exposure to the deviation portfolio (Panel
A), because the cross-section produces equilibrium outcomes along the common supply curve.
However, it is negative among stocks that have similar profitability but different exposure to
the deviation portfolio (Panel B), because firm outcomes are now located along the common
demand curve. If firms differ in both profitability and exposure to the deviation portfolio
(Panels C and D), the sign of the relation between market correlation and Sharpe ratio
is driven by the dominant source of heterogeneity. The relation is positive if firms are
primarily heterogeneous in profitability (Panel C), and negative if heterogeneity in exposure
to the deviation portfolio dominates (Panel D). The exact slope of the cross-sectional relation
between market correlation and Sharpe ratio also depends on the correlation between supply
and demand factors. For instance in Panel C, the cross-sectional relation between risk and
return is relatively steep if firms with a high demand for capital face a low supply of capital
from investors, while firms with a low demand for capital face a high supply.
The empirical implication of our model is that a univariate regression of average perfor-
mance on market sensitivity can be downward sloping or reveal nonlinearities, consistent with
the empirical findings of Fama and French (1992), De Giorgi, Post, and Yalçın (2019), and
Hong and Sraer (2016). The negative relation between market correlation and Sharpe ratio
8
The supply schedule is given by (2.7) and the demand schedule is obtained by substituting out Qn
in (3.2).
17is noteworthy because, historically, the fact that some stocks with low market risk generate
higher returns than stocks with high market risk has been viewed as a core violation of the
trade-off between risk and return. In the Appendix, we show that, if production technologies
are linear (η = +∞) or fixed as in an endowment economy (η = 0), a downward-sloping
relation is possible only if it is hard-wired into the model. By contrast, a downward-sloping
relation arises naturally when firms exhibit decreasing returns to scale.
We next examine how supply and demand factors drive stock expected returns and market
betas in general equilibrium. The analysis builds on the identities:
µn = rf + λn (an , δ n ) σ n (an , δ n ), (3.6)
+ + − +
β n = ρm,n (an , δ n ) σ n (an , δ n ) / σ m . (3.7)
+ − − +
The stock volatility, σ n (an , δ n ), decreases with profitability and increases with exposure to the
deviation portfolio, as equations (1.7) and (3.4) imply. The volatility channel can therefore
offset the impact of supply and demand factors on market correlation and Sharpe ratio. In
the Appendix, we verify that, under mildly decreasing returns to scale, the volatility channel
is weak. Hence a stock’s exposure to the deviation portfolio reduces its market beta, while
profitability increases both the market beta and expected return. In particular, the impact of
profitability on expected return is consistent with the evidence in Haugen and Baker (1996)
and Novy-Marx (2013). To sum up, supply and demand factors have similar effects on beta
and expected return as they do on market correlation and Sharpe ratio. The theoretical
relations are cleaner when we scale returns by volatility, which provides guidance for the
next sections.
3.3. Drivers of Alpha
The stock’s alpha, αn = µn − rf − β n (µm − rf ), measures the average return earned on the
stock in excess of the compensation for market risk exposure. By (2.7), the alpha-to-volatility
ratio, or normalized alpha, satisfies
αn
= γ δ n − δ m ρm,n (an , δ n ) . (3.8)
σn + −
Normalized alpha is driven by both supply and demand factors in general equilibrium. It
has two components: (i) the stock’s compensation for exposure to the deviation portfolio,
18γ δ n , and (ii) the compensation impact of the stock’s exposure to the market, −γ δ m ρm,n .
The investor is rewarded for holding stocks with high exposure to the deviation portfolio
because these stocks are unappealing (for hedging or sentiment reasons) and therefore trade
at a discount compared to the CAPM. Market correlation also drives normalized alpha
because the incremental reward for exposure to market risk in (2.7) differs from the market
Sharpe ratio implied by the CAPM. For instance, if the market has positive exposure to
the deviation portfolio (δ m > 0), the incremental reward for market exposure is lower than
the market’s Sharpe ratio, consistent with the fact that a stock cannot be compensated for
deviation portfolio exposure both via γ δ n and via ρm,n . Therefore, a high market correlation
leads to a relatively small Sharpe ratio and a negative alpha.
The supply factor δ n drives alpha positively both directly and indirectly. As equa-
tion (3.8) shows, a high exposure to the deviation portfolio has (i) a direct positive effect
on the stock’s alpha and (ii) an indirect and equally positive effect through the stock’s low
market correlation. In comparison, the demand factor an contributes to alpha only through
the correlation effect. A stock with high profitability has high market correlation and, con-
sequently, negative alpha. We now show that supply and demand factors produce a wide
range of deviations from the CAPM captured by firm characteristics.
Betting Against Beta and Betting Against Correlation. Portfolios holding low-
market sensitivity stocks and shorting high-market sensitivity stocks tend to generate positive
alphas, whether market sensitivity is measured by beta or correlation (Asness et al. 2020,
Black 1993, Black, Jensen, and Scholes 1972, Hong and Sraer 2016). These observations
prompted the development of betting-against-beta (BAB) and betting-against-correlation
(BAC) strategies (Asness, Moskowitz, and Pedersen 2013, Frazzini and Pedersen 2014).
BAB and BAC arise naturally in our frictionless representative-agent economy.9 Assume
for instance that the market has positive exposure to the deviation portfolio (δ m > 0).
As equation (3.8) shows, a strategy that goes long low-correlation stocks and shorts high-
correlation stocks generates positive alpha. Furthermore, this effect should appear more
clearly if the long and short ends of the portfolio have the same average deviation portfolio
exposure δ n . This condition is likely to hold if the portfolio is constructed from a set of stocks
9
Previous explanations of BAB and BAC have relied on the presence of market frictions and investor
heterogeneity. For example, Black (1993) and Frazzini and Pedersen (2014) generate BAB via a leverage
constraint, while Hong and Sraer (2016) consider both heterogeneous beliefs and short-sales constraints. In
our model, BAC and BAB naturally arise in GE when the investor’s portfolio deviates from the mean-variance
efficient portfolio.
19with similar hedging or sentiment characteristics, such as stocks in the same industry. These
implications of the model are consistent with the existence of anomalies across and within
industries (Asness, Frazzini, and Pedersen 2014, Banko, Conover, and Jensen 2006).
Size, Value, and Investment. We classify a stock with a high market-to-book ratio,
Vn /Kn , as a growth stock, and a stock with a low market-to-book ratio as a value stock.
By equations (1.3) and (2.9), the market-to-book ratio encodes information about the firm’s
opportunities:
Vn (an − λn ) σ cf,n 1
= = Knη . (3.9)
Kn 1 + rf
A growth stock is highly profitable and has a low price of risk. Equation (3.9) implies that
Vn = (Vn /Kn )η+1 . Hence large firms tend to be growth firms. This result is consistent with
Fama and French (1992) and Loughran (1997), who document empirically that stocks with
high market capitalizations have on average high market-to-book ratios.10
Stocks with high capitalization, high market-to-book ratio, and high investment persis-
tently generate negative alpha (see Fama and French 1993 and Cooper, Gulen, and Schill
2008, among many others). These patterns can arise in our model from both supply and
demand channels. Firms with strong profitability and low exposure to the deviation portfolio
tend to be large growth firms with high investment. By equation (3.8), these stocks produce
negative alpha when the market has positive exposure to the deviation portfolio.
Moreover, because stocks that are larger, invest more, and have high market-to-book ratio
tend to have higher market correlation and higher beta, the model predicts that the BAC
and BAB strategies should be negatively related to size, value, and investment anomalies.
This prediction is supported by the data (De Giorgi, Post, and Yalçın 2019, Frazzini and
Pedersen 2014, Liu 2018).
Profitability Anomaly. Novy-Marx (2013) documents a gross profitability premium un-
explained by the CAPM that co-exists with the value premium (see also Fama and French
2015, and Hou, Xue, and Zhang 2015). Among stocks with the same market-to-book ratio,
those with high gross profitability empirically generate positive alphas, while stocks with low
profitability generate negative alphas.
10
Jurek and Viceira (2011) also find that growth stocks have accounted for 70% to 85% of the U.S.
aggregate stock market since 1927.
20In our model, the profitability anomaly naturally results from the supply channel. As-
sume for simplicity that we double sort firms by the market-to-book ratio (3.9) and cash-flow
volatility, σ cf,n . In a given bucket, the more profitable firms have high exposure to the de-
viation portfolio while the less profitable firms bear low exposure to the deviation portfolio,
1
which follows directly from the property that Vn /Kn = [Qn (an , δ n )/σ cf,n ] η is constant. Sort-
ing firms by profitability is equivalent to sorting them by their exposure to the deviation
portfolio.11 Consequently, stocks with high profitability in the bucket have high exposure to
the deviation portfolio and generate positive alpha, consistent with the data.12
Overall, the model captures qualitatively a wide set of stylized facts about capital al-
location and the cross-section of equity returns. In general equilibrium, the cross-sectional
relation between risk and return is positive if demand factors are more dispersed than sup-
ply factors, but is negative otherwise. The model closely links firm-level supply and demand
factors to the endogenous characteristics and risk-return profile of stocks. When the market
portfolio and the deviation portfolio co-move positively, a stock with high deviation portfolio
exposure and low profitability has a high alpha, low market correlation, low market beta, low
investment, low market-to-book ratio, and low size. Furthermore, within a group of stocks
with similar market-to-book ratio, the most profitable stocks have a high alpha. In the next
section, we estimate the model specification and verify that its quantitative implications are
supported by the data.
4. Empirical Results
This section investigates the empirical implications of our model. In Section 4.1, we verify
that the cross-sectional relation between market values and book values is consistent with the
specification of firm technologies. Section 4.2 develops the system of simultaneous equations
defining the firm-level supply and demand for capital. Section 4.3 defines instruments for
estimating this system. Section 4.4 applies this method to a large sample of U.S. publicly
traded firms. Section 4.5 plots empirical estimates of supply and demand schedules and
discusses which factors mainly drive the cross-sectional relation between risk and return.
In Section 4.6, we verify that our model accurately predicts the sign and magnitude of the
11
Hou, Xue, and Zhang (2015), Kogan and Papanikolaou (2013), and Zhang (2017) obtain a similar result.
Controlling for the firm’s level of investment, firms with high profitability have higher expected returns
because they also have a higher discount rate.
12
The market correlation (3.5) is constant in the bucket, so that alpha is only driven by deviation portfolio
exposure.
21risk-return relationship in selected groups of stocks.
4.1. Data and Distribution of Book and Market Values
The empirical analysis is based on U.S. stocks from the merged CRSP and Compustat
database over the 1984 to 2019 period. We apply to firms the following set of filters. As in
Fama and French (1992, 1993), we eliminate financial and utility stocks (SIC codes 4900-4949
and 6000-6999). As in Hou, Xue, and Zhang (2015), we calculate book equity as the sum of
shareholders’ equity, deferred taxes, and investment tax credit, minus preferred stock. We
exclude stocks with either negative book equity, less than a year of quarterly earnings, or
less than a year of daily returns (252 days). We obtain an unbalanced panel containing 4,727
stocks over the period. Data on daily market factors and risk-free rates are retrieved from
Ken French’s data library.
We calculate a stock’s Sharpe ratio, market correlation, and risk capital in annual units
from daily excess returns. A stock’s risk capital, Qn , is measured by the annualized volatility
of its dollar daily excess return.13 Details of variable construction are provided in the
Appendix.
Table 1 presents summary statistics of the sample. Over the 1994 to 2014 period, the
average firm has a market value of $4.2 billion, a book value of $1.4 billion, a risk capital
of $1.2 billion, a market correlation of 30%, and an annualized Sharpe ratio of 0.37. By
comparison, the market portfolio has a Sharpe ratio of 0.51 during the period.
The specification of firms’ technologies and demand for capital defined in Section 2.3
implies that the market value and book value of a firm satisfy the relationship:
Vn = Kn1+1/η , (4.1)
as per equation (3.9). In the Appendix, we develop a dynamic model of production, in which
the firm solves a multi-period problem. We verify that the relation (4.1) still holds every
period, so that it can be tested empirically in a single or a pooled cross-section.
In Table 2, we regress the log market value on the log book value of firms in our sample.
Consistent with the functional form (4.1), we set the intercept equal to 0. We run the
13
We measure the dollar excess return as the product of the stock’s market capitalization (lagged by one
day) and daily excess returns. Similarly, the risk capital of the market portfolio Qm is calculated from the
daily dollar excess returns of the value weighted market portfolio.
22regressions over the following sample periods: (1) end of 1994, (2) end of 1999, (3) end of
2004, (4) end of 2009, (5) end of 2014, and (6) all five years combined. The R2 coefficient is
close to 100% in all regressions. The slope coefficient, which estimates 1+1/η, is above unity
with very high levels of significance (p value less than 0.001) in all regressions, which confirms
that firms exhibit decreasing returns to scale. The slope coefficient is stable between 3.5%
and 5%, which corresponds to a coefficient η between 20 and 33. Thus, firms exhibit mild
but significant departure from constant returns to scale.
Figure 3 illustrates the densities of the firms’ log market values and log book values over
the full sample. The densities are estimated non-parametrically via an Epanechnikov kernel
with optimal bandwidth.14 We also report the density of the market value predicted from the
observed book value, V̂n = Kn1+η . Since 1 + 1/η is greater than unity, the distribution of log
market values is shifted to the right. Perhaps more strikingly, the predicted market value,
V̂n , and the observed market value, Kn , have remarkably similar densities. In particular, the
heterogeneity of the coefficient η across firms, which we consider in a version of the model
discussed in the Appendix, does not seem required to explain the cross-section of book values
and market values.
The results of Table 2 and Figure 3 confirm the validity of the technologies and demand
for capital defined in Section 2.3. Firms exhibit mildly decreasing returns to scale and our
production setup accurately predicts the relation between market value and book value.
4.2. Econometric Specification of Supply and Demand Schedules
We now consider the joint implications of supply and demand in the general version of the
model developed in Section 2. Firms can be heterogeneous along all dimensions, including
the pairwise correlations of cash flows.
The empirical analysis is based on the following econometric specification of supply and
14
Furthermore, since the Epanechnikov kernel K(u) = 3/4(1 − u2 )1{|u|≤1} has a finite support, the tail
behavior of the density estimate is not predetermined by the shape of the kernel in the tails but is instead
strongly driven by the data.
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