Figure 1: Average Conditional Empirical Pricing Kernels - Existing
Approaches
(a) Panel A
(b) Panel B
(c) Panel C
(d) Panel D
Notes: This figure shows the average of the conditional empirical
pricing kernels where the physical densities are estimated using the
historical data (Panel A), GARCH model (Panel B), stochastic volatility
model (Panel C), and stochastic volatility and jump model (Panel D).
This is a figure with four panels (A-D) showing line graphs of average conditional empirical pricing kernels. The x-axis in each panel shows monthly market return percentages ranging from approximately -10% to +10%. The y-axis shows pricing kernel values ranging from about 0 to 4.5. Panel A shows the kernel estimated using historical data, which exhibits a U-shaped pattern, decreasing until around 5% monthly return and then increasing. Panel B shows the kernel estimated using a GARCH model, with a similar U-shaped pattern. Panel C displays the kernel estimated with a stochastic volatility model, which decreases more consistently with a less pronounced increase in the right tail. Panel D shows the kernel estimated using a stochastic volatility and jump model, which is nonmonotonic but does not display the clear U-shaped pattern seen in Panels A and B.
Figure 2: Conditional Empirical Pricing Kernels - Existing Approaches
(a) Panel A: 09/2002
(b) Panel B: 09/2008
(c) Panel C: 09/2014
(d) Panel D: 03/2020
Notes: This figure shows the conditional empirical pricing kernels
estimated using the historical data, GARCH model, stochastic volatility
(SV) model, and stochastic volatility with jumps (SVJ) model on specific
months: 09/2002 (Panel A), 09/2008 (Panel B), 09/2014 (Panel C), and
03/2020 (Panel D).
This is a figure with four panels (A-D) showing line graphs comparing conditional empirical pricing kernels for specific months using different estimation approaches. The x-axis shows monthly market return percentages from approximately -10% to +10%. The y-axis shows pricing kernel values ranging from about 0 to 4.5. Each panel represents a different month: Panel A shows September 2002, Panel B shows September 2008 (during the financial crisis), Panel C shows September 2014, and Panel D shows March 2020 (during the COVID-19 pandemic). Each panel displays four different lines representing pricing kernels estimated using historical data, GARCH model, stochastic volatility (SV) model, and stochastic volatility with jumps (SVJ) model. The estimates vary substantially across methods within each time period, with particularly divergent patterns during the crisis periods in Panels B and D.
Figure 3: Unconditional Empirical Pricing Kernels
(a) Panel A: Linn et al., 2018
(b) Panel B: Unrestricted Polynomial EPK
(c) Panel C: Polynomial EPK with the P-Density Restriction and
Euler Equations
Notes: This figure shows the forward-looking unconditional empirical
pricing kernels (EPKs). Panel A follows the CDI method of , Panel B describes the
unrestricted polynomial pricing kernel estimation, and Panel C describes
the polynomial pricing kernel estimation considering the P-density
restriction and Euler equations. The left side of each panel presents
the empirical pricing kernel (solid blue line) with its point-wise
bootstrapped 90% confidence interval (dotted black lines). The right
side of each panel presents the histogram of the integral of the
physical densities over the support.
This is a figure with three panels (A-C), each containing two side-by-side graphs. The left graphs show line plots of unconditional empirical pricing kernels (EPKs) with the x-axis representing monthly market returns (-10% to +10%) and the y-axis showing pricing kernel values (0 to 2). The right graphs show histograms of the integral of physical densities over their support. Panel A shows results following the CDI method of Linn, Shive, and Shumway (2018). Panel B shows results from unrestricted polynomial EPK estimation. Panel C shows results from polynomial EPK with P-density restriction and Euler equations. All three left graphs show statistically downward-sloping pricing kernels with 90% confidence intervals (dotted black lines). The right histograms in Panels A and B show that many density integrals deviate from one, while Panel C shows all values clustered tightly around one.
Figure 4:
(a) Panel A: VIX
(b) Panel B: Sentiment
(c) Panel C: Term Spread
(d) Panel D: Industrial Production Growth
(e) Panel E: Credit Spread
(f) Panel F: Market Return
This is a figure with six panels (A-F), each containing two side-by-side graphs. Each panel represents a different conditioning variable: VIX (Panel A), sentiment (Panel B), term spread (Panel C), industrial production growth (Panel D), credit spread (Panel E), and market return (Panel F). The left graphs show average conditional pricing kernels (blue lines) with 90% confidence intervals (dotted black lines). The x-axis shows monthly market returns (-10% to +10%), and the y-axis shows pricing kernel values (0 to 2). The right graphs show the sensitivity of the pricing kernel to each conditioning variable, comparing low (5th percentile) versus high (95th percentile) values of the variable. All panels show more variation in the left tail (negative returns) than in the right tail. For example, in Panel A, when VIX is low, the pricing kernel is higher and steeper in the negative return region compared to when VIX is high, suggesting investors are more fearful of negative returns during periods of low volatility.
Figure 5: Bivariate Conditional Empirical Pricing Kernels
(a) Panel A: VIX, Sentiment
(b) Panel B: VIX, Term Spread
(c) Panel C: VIX, IP Growth
(d) Panel D: VIX, Credit Spread
(e) Panel E: VIX, Market Return
Notes: This figure shows the average forward-looking conditional
empirical pricing kernels with five sets of two conditioning variables:
the VIX and sentiment (Panel A), the VIX and term spread (Panel B), the
VIX and industrial production growth (Panel C), the VIX and credit
spread (Panel D), and the VIX and market return (Panel E). The dotted
black lines are the point-wise bootstrapped 90% confidence interval.
This is a figure with five panels (A-E) showing line graphs of average forward-looking conditional empirical pricing kernels using pairs of conditioning variables. The x-axis shows monthly market returns ranging from approximately -10% to +10%, and the y-axis shows pricing kernel values ranging from about 0 to 2. Panel A shows results using VIX and sentiment as conditioning variables. Panel B uses VIX and term spread. Panel C uses VIX and industrial production growth. Panel D uses VIX and credit spread. Panel E uses VIX and market return. Each panel includes dotted black lines representing 90% confidence intervals around the solid line of the average kernel. The shapes are similar across panels, especially in the negative return region, with some exhibiting U-shapes while others are monotonically decreasing within the -10% to +10% return range.
Figure 6: Characterizing Variation in the Bivariate Conditional Empirical
Pricing Kernel
(a) Panel A: Sensitivity to Each Conditioning
Variable
(b) Panel B: Time Series of the Conditional Pricing
Kernel
Notes: Panel A of this figure shows how the bivariate (the VIX and
the term spread) conditional empirical pricing kernel changes when the
VIX (left) or the term spread (right) varies from the 5th percentile to
the 95th percentile of its data while the other variable is fixed at its
median value. Panel B describes the time series of the conditional
pricing kernel for the January 1996 to December 2020 sample period.
This is a figure with two panels. Panel A contains two side-by-side graphs showing the sensitivity of the bivariate conditional empirical pricing kernel to each conditioning variable (VIX and term spread). The left graph shows how the pricing kernel changes with VIX (from 5th to 95th percentile) while holding term spread at its median value. The right graph shows how the pricing kernel changes with term spread while holding VIX constant. Both graphs show the x-axis as monthly market returns (-10% to +10%) and y-axis as pricing kernel values (0 to 2). Panel B shows a three-dimensional surface plot representing the time series of the conditional pricing kernel from January 1996 to December 2020. The x-axis represents time, the y-axis shows monthly market returns, and the z-axis (color and height) shows the pricing kernel value. The shape of the pricing kernel varies considerably over time, with particularly pronounced variation in the left tail (negative return region). During economic downturns like the 2008 financial crisis and 2020 COVID-19 crisis, the pricing kernel is notably lower in the negative return region.
Figure 7: Average Probability Distributions
Notes: This figure shows the average risk-neutral (solid blue line)
and physical probability densities. The dotted black line indicates the
physical density implied by the unconditional empirical pricing kernel,
and the dashed red line represents the one implied by the conditional
empirical pricing kernel with two conditioning variables, the VIX and
term spread.
This is a line graph showing three curves representing average probability distributions. The x-axis shows monthly market returns ranging from approximately -40% to +25%, and the y-axis shows probability density values. The solid blue line represents the average risk-neutral density, which has the highest peak and appears more negatively skewed. The dotted black line shows the physical density implied by the unconditional pricing kernel, and the dashed red line shows the physical density implied by the conditional pricing kernel with VIX and term spread as conditioning variables. Both physical densities have lower peaks and are less negatively skewed than the risk-neutral density, but all three distributions show negative skewness and fat tails (leptokurtosis).
Figure 8: Conditional Monthly Risk Premia
(a) Panel A: Equity Risk Premium
(b) Panel B: Variance Risk Premium
(c) Panel C: Skewness Risk Premium
(d) Panel D: Kurtosis Risk Premium
Notes: This figure describes the time series of the forward-looking
conditional monthly risk premia inferred from the unconditional (solid
blue line) and conditional (dashed red line) pricing kernel estimates.
Panels A through D present the equity, variance, skewness, and kurtosis
risk premia, respectively. The sample period is from January 1996 to
December 2020.
This is a figure with four panels (A-D) showing time series line graphs of forward-looking conditional monthly risk premia from January 1996 to December 2020. Each panel compares estimates based on unconditional pricing kernel (solid blue line) and conditional pricing kernel (dashed red line). Panel A shows the equity risk premium. The two estimates differ substantially during economic downturns, with the unconditional estimate showing larger spikes up to nearly 2% higher during the 2008 financial crisis and 2020 COVID-19 crisis. Panel B shows the variance risk premium, with values ranging from about -60%² to 0%². Both estimates track each other closely through the sample period. Panel C shows the skewness risk premium ranging from approximately 0 to 1.5, and Panel D shows the kurtosis risk premium ranging from about -25 to 1. In both Panels C and D, the unconditional and conditional estimates show similar patterns throughout the period.
Figure 9: The Sources of the Conditional Equity Risk Premium
(a) Panel A: Full Sample Period
(b) Panel B: Low VIX Period
(c) Panel C: High VIX Period
Notes: This figure shows the sources of the conditional equity risk
premium during the full sample period (Panel A), low-VIX period (Panel
B), and high-VIX period (Panel C). The low-VIX (high-VIX) period
consists of the subsample where the VIX is lower (higher) than or equal
to the 10th (90th) percentile of the VIX data. In each panel, the
average contributions to the ERP based on the unconditional (solid blue
line) and conditional (dashed red line) pricing kernel estimates are
plotted.
This is a figure with three panels (A-C) showing line graphs of the cumulative contribution of different monthly return states to the conditional equity risk premium. The x-axis shows monthly market returns ranging from -80% to +30%, and the y-axis shows the percentage contribution to the equity risk premium (0% to 120%). Panel A shows results for the full sample period, comparing contributions based on unconditional pricing kernel (solid blue line) and conditional pricing kernel (dashed red line). The conditional kernel shows slightly higher contributions from negative returns. Panel B shows results during low-VIX periods (when VIX is at or below its 10th percentile), with smaller differences between the two estimates. Panel C shows results during high-VIX periods (when VIX is at or above its 90th percentile), with dramatic differences between estimates. Using the conditional pricing kernel, 99% of the contribution comes from returns below -20%, while using the unconditional kernel, a 99% contribution is achieved around +15% returns.
Figure 10: The Sources of the Conditional Equity Risk Premium: Time
Variation
(a) Panel A: 10% Contribution
(b) Panel B: 50% Contribution
(c) Panel C: 99% Contribution
Notes: This figure describes the time series of the sources of the
conditional equity risk premium based on the unconditional (solid blue
line) and conditional (dashed red line) pricing kernel estimates. Panels
A through C present the return levels for the 10%, 50%, and 99%
contributions to the ERP, respectively. The sample period is from
January 1996 to December 2020.
This is a figure with three panels (A-C) showing time series line graphs from 1996 to 2020. Each panel compares return levels that achieve specific percentage contributions to the equity risk premium based on unconditional (solid blue line) and conditional (dashed red line) pricing kernel estimates. Panel A shows the return level for 10% contribution, with both approaches showing similar patterns except during crises like 2008 and 2020. Panel B shows the return level for 50% contribution, with moderate differences between the approaches. Panel C shows the return level for 99% contribution, with substantial differences between the approaches, especially during crisis periods. During the 2008 financial crisis, the conditional pricing kernel shows that the 99% contribution comes from approximately -30% returns, while the unconditional approach shows it coming from +20% returns. Similar large divergences appear during other crises (1998 LTCM collapse, 2002 market crash, 2012 European debt crisis, and 2020 COVID-19 crisis).
Figure A1: Validating the Risk-Neutral Density Estimates
Notes: This figure shows the histogram of the integrated values of
the conditional risk-neutral density estimates over the support, where
the integrated value is denoted by CDF(\(\infty\)) on the \(x\)-axis. The number of samples in the
histogram is 300, covering the monthly data from January 1996 to
December 2020.
This is a histogram showing the distribution of integrated values of conditional risk-neutral density estimates over their support (denoted as CDF(∞) on the x-axis). The x-axis ranges from approximately 0.999 to 10001. The distribution is tightly centered around 1.0, with most values falling between 0.9998 and 1.0001. This indicates that the risk-neutral density estimates are well-calibrated, as a properly specified probability density should integrate to exactly 1.0 over its entire support.