Figure 1: The Equity Yield Slope and its Growth and Risk
Premium Components
(a) Slope: 10-1 year equity yield
(b) Slope: 10-1 year expected growth
(c) Slope: 10-1 year risk premium
We plot in Panel (a) the difference between the 10- and 1-year equity
yields for the market as well as the value and growth portfolios. We
plot the expected growth and risk premium components of the equity yield
slope in Panels (b) and (c), respectively. The sample period runs from
September 1974 to December 2019. Shaded areas are NBER
recessions.
This is a three-panel line chart showing quarterly data from 1974 to 2019, with NBER recessions shaded in gray. Panel (a) plots the 10-1 year equity yield slope (difference between 10-year and 1-year equity yields) for market, value, and growth portfolios, with the y-axis ranging from -0.3 to 0.4. The slopes fluctuate over time, typically positive but turning negative during recessions and certain non-recessionary periods (like 2019). Panel (b) shows the expected growth component of this slope, with values ranging from -0.5 to 0.5, exhibiting larger downward movements during recessions, particularly in 1990-1991 and 2007-2009 when it falls to approximately -0.3 to -0.4. Panel (c) displays the risk premium component with a narrower range from -0.1 to 0.1, showing less volatility than the expected growth component. During the dot-com bubble period (late 1990s), the risk premium slopes for value and growth moved in opposite directions. Note: Shaded areas indicate NBER recessions. Source: Giglio et al. (2024).
Figure 2: Realized Equity Term Premium in Data and
Models
(a) Data
(b) Models
Panels A shows the response of the realized market term premium (the
difference in return between the long- and short-term claim, \(r_{t}^{10}-r_{t}^{1}\)) to the long-run TFP
shock in the data. Panel B shows the impulse response in four models
that feature (a close equivalent of) the long-run growth shock: the
Bansal and Yaron, 2004 long-run
risk model, two reference asset pricing models for the equity term
structure: Lettau and Wachter, 2007 and Gormsen, 2021, and the Ai et al., 2018 production-based model. We
describe these models in Sections 4 and 5.
This is a two-panel line chart comparing responses of the realized market term premium to a long-run TFP shock over 10 years. Panel A shows the data response with the y-axis measuring percent from 0 to 6. The term premium starts near zero, increases sharply to approximately 3% within the first year, plateaus between years 1-3, then gradually decreases to about 1% by year 10. Panel B compares responses from four theoretical models: Bansal and Yaron (2004), Lettau and Wachter (2007), Gormsen (2021), and Ai et al. (2018). All models show similar patterns but with varying magnitudes. The Ai et al. model most closely matches the data with a peak of nearly 3%, while Bansal and Yaron (2004) and Gormsen (2021) show lower peaks of approximately 1.5%, and Lettau and Wachter (2007) peaks at about 2.2%. Source: Author calculations.
Figure 3: Impulse Responses to a News
Shock
(a) TFP
(b) Consumption
(c) Inflation
(d) Short rate
(e) Equity yield slope (10-1)
(f) 1- and 10-year equity yield
The TFP news shock is identified from a structural VAR as in Kurmann and Sims, 2020, with the
10-minus-1 year equity yield slope substituted in place of the term
spread. Solid black lines are the median estimates for the VAR estimated
with the 2020 vintage of adjusted TFP. The gray bands correspond to the
16–84 percentile bootstrapped confidence intervals. Panels (a) to (e)
report the impulse response function for the five variables included in
the VAR. Panel (f) shows the response of the 1 and 10 year yield
separately. The sample period runs from September 1974 to December
2019.
This is a six-panel line graph showing quarterly impulse responses to a TFP news shock over 10 years, with median estimates (solid black lines) and 16-84 percentile confidence intervals (gray bands). Panel (a) shows TFP gradually increasing to about 0.5% by year 5 then plateauing. Panel (b) shows consumption increasing more steadily to approximately 0.8-0.9% by year 5. Panel (c) shows inflation initially decreasing to about -0.5% before gradually returning toward zero. Panel (d) shows the short rate following a similar pattern, initially dropping to about -0.4%. Panel (e) displays the equity yield slope (10-1 year) increasing by approximately 1% within the first year before gradually declining to zero by year 3. Panel (f) compares the 1-year and 10-year equity yields separately, with the 1-year yield dropping more sharply (to about -1.2%) than the 10-year yield (approximately -0.4%). Note: The TFP news shock is identified from a structural VAR as in Kurmann and Sims (2020). Source: Author calculations.
Figure 4: Equity Yield Responses to a News Shock at the
1-Year Horizon.
(a) Equity yield
(b) Expected growth
(c) Risk premium
The figure shows the response of equity yields, expected growth, and
risk premia for claims with maturities from 1 to 10 years to a TFP news
shock at the 1-year horizon. The responses of expected growth and risk
premia are derived from a VAR that includes both these variables. Each
dot represents the median estimate from the VAR and whiskers correspond
to the 16–84 percentile bootstrapped confidence intervals. The sample
period runs from September 1974 to December 2019.
This is a three-panel error bar chart showing responses to a TFP news shock at the 1-year horizon across asset maturities from 1 to 10 years. Panel (a) displays equity yield responses ranging from -2.5% to 0%, with yields for all maturities declining. The 1-year yield drops by approximately -1.4%, while the 10-year yield declines by about -0.4%, with response magnitude decreasing monotonically as maturity increases. Panel (b) shows expected growth responses ranging from 0 to 2%. The 1-year expected growth rises by approximately 1.2%, while the 10-year expected growth increases by about 0.3%. Panel (c) presents risk premium responses ranging from -0.4% to 0.4%, with values close to zero and confidence intervals consistently including zero. Source: Author calculations.
Figure 5: Impulse Responses of Expected Growth and Risk
Premium Components
(a) Expected growth slope (10-1)
(b) Expected growth, 1 and 10 year
(c) Risk premium slope (10-1)
(d) Risk premium, 1 and 10 year
The TFP news shock is identified as in Kurmann and Sims, 2020, with the 10-1 year maturity
difference in expected growth and risk premia included into the VAR (in
place of the yield spread). Solid black lines are the median estimates
and the gray bands correspond to the 16–84 percentile bootstrapped
confidence intervals. Panels (b) and (d) report separately the response
of the 1 and 10 year equity yield components. The sample period runs
from September 1974 to December 2019.
This is a four-panel line graph showing decomposed responses to a TFP news shock over 10 years. Panel (a) shows the expected growth slope (10-1 year) decreasing to approximately -1.4% within the first year before gradually reverting to zero by year 5. Panel (b) displays the 1-year and 10-year expected growth components separately, with the 1-year component increasing to about 1.2% before gradually declining, while the 10-year component increases by a smaller amount (approximately 0.4%). Panels (c) and (d) present the risk premium slope and its individual components, all showing minimal movement with confidence intervals that include zero. Source: Author calculations.
Figure 6: Responses to the Schorfheide et al., 2018 Long run
Shock
(a) Consumption
(b) Equity yield slope (10-1)
(c) Expected Growth slope (10-1)
(d) Risk premium slope (10-1)
(e) Equity yield slope (10-1): Growth
(f) Equity yield slope (10-1): Value
The long-run growth shock correspond to the innovation in the persistent
consumption component in the Schorfheide et al., 2018 model. The long-run
growth shock is then used in local projections to obtain the impulse
response functions. The local projection includes the constant, the
shock and the lags of the dependent variable (except for Panel (c) and
(d) where we use lags of the market slope, rather than its individual
components)). We use the same number of lags as in the VAR. Panel (a)
reports the impulse response function for consumption. Panel (b)-(d)
show the response for the market equity yield slope and its components.
Panel (e) and (f) report the impulse response function for the 10-1 year
equity yield slope for growth and value firms, respectively.
This is a six-panel line graph showing responses to a long-run consumption growth shock over 10 years. Panel (a) shows consumption steadily increasing to approximately 1.8% by year 10. Panel (b) displays the equity yield slope (10-1) increasing sharply to about 2-3% in the first year before gradually declining. Panel (c) presents the expected growth slope (10-1) showing a significant negative response of approximately -2% to -3% in the first year. Panel (d) shows the risk premium slope (10-1) with minimal response. Panel (e) displays the equity yield slope for growth firms, showing a stronger positive response of about 4-5% compared to the market. Panel (f) presents the equity yield slope for value firms, showing a more volatile response with confidence intervals including both positive and negative values. Source: Author calculations using the Schorfheide et al. (2018) model.
Figure 7: Responses of Components of Total Payout and
Investment.
(a) Cash payout
(b) Total payout
(c) Repurchases
(d) Investment
Panels (a) to (d) present impulse responses to the long-run TFP news
shock for cash payout, total payout, net repurchases, and investment –
all scaled by consumption. Investment is real chain-weighted gross
private domestic investment, whereas all other variables are taken from
Davydiuk et al., 2023. These
responses are derived from a five-variable VAR, where each variable is
substituted in place of consumption.
This is a four-panel line graph showing quarterly responses to a TFP news shock over 10 years. Panel (a) displays the cash payout response, which initially shows little change but gradually increases to approximately 0.05-0.1% after 3-4 years. Panel (b) shows the total payout response, which drops immediately by about -0.4% and further decreases to approximately -0.55% within the first year before gradually returning toward -0.1% by year 10. Panel (c) presents the repurchases response, dropping sharply to about -0.8% in the first year before slowly reverting to approximately -0.2% by year 10. Panel (d) shows investment increasing to approximately 0.2% within the first two years before gradually declining to about 0.1% by year 10. Note: All variables are scaled by consumption. Source: Davydiuk et al. (2023) for payout data; Author calculations.
Figure 8: Responses to a Long-Run Shock in Models versus
Data
(a) Equity yield, 1 year
(b) Equity yield, 10 year
(c) Expected growth, 1 year
(d) Expected growth, 10 year
(e) Risk premium, 1 year
(f) Risk Premium, 10 year
This figure shows the impulse responses (up to 10 years after the
long-run productivity shock) for model-implied equity yields as well as
their expected growth and risk premium components. To accommodate
comparison, we also report the estimated impulse responses in the data.
We focus on the 1- and 10-year maturity claim for the aggregate dividend
strip.
This is a six-panel line graph comparing model-implied responses to long-run productivity shocks with actual data responses over 10 years. Panels (a) and (b) compare equity yield responses for 1-year and 10-year maturities respectively. The data shows a 1-year yield decline of approximately -1.2% and a 10-year yield decline of about -0.4%, best matched by the Ai et al. (2018) and Gormsen (2021) models. Panels (c) and (d) compare expected growth responses, with data showing an increase of approximately 1.2% for the 1-year maturity, best matched by the Ai et al. model. Panels (e) and (f) compare risk premium responses, with data showing minimal response near zero, also best matched by the Ai et al. model while other models show larger negative responses. Source: Author calculations.
Figure 9: Responses of Cash and Total Payout in the
Model and Data.
(a) Cash payout
(b) Total payout
Panels (a) and (b) present impulse responses of cash and total payout to
the long-run shock in the data as well as in the modified Ai et al., 2018 model.
The responses in the data are derived from a five-variable VAR, where
cash and total payout replace consumption. Both series are sourced from
Davydiuk et al., 2023 and scaled
by consumption.
This is a two-panel line graph comparing responses to a long-run shock over 10 years. Panel (a) shows cash payout responses in both data and the Ai et al. (2018) model displaying a gradual increase to approximately 0.05-0.1% by year 3-4, with the model closely tracking the data pattern. Panel (b) presents total payout responses, with data showing an immediate negative response of about -0.4% that further decreases to approximately -0.5% before gradually reverting toward zero. The model shows a qualitatively similar pattern but with smaller magnitude, declining to only about -0.1% and showing a more persistent response. Note: Both series are scaled by consumption. Source: Davydiuk et al. (2023) for payout data; Author calculations.
Figure 10: Responses to a News Shock for Growth
Firms
(a) Equity yield slope (10-1)
(b) Equity yield, 1 and 10 year
(c) Expected growth slope (10-1)
(d) Expected growth, 1 and 10 year
(e) Risk premium slope (10-1)
(f) Risk Premium, 1 and 10 year
Focusing on the equity yield of growth firms, this
figure presents the impulse responses of the 10-1 year equity yield
slope (and its growth and risk premium components) to a TFP news shock.
Solid black lines are the median estimates and the gray bands correspond
to the 16–84 percentile bootstrapped confidence intervals. On the right,
we present the response of the 1- and 10-year claims separately. The
sample period runs from September 1974 to December
2019.
This is a six-panel line graph showing responses for growth firms to a TFP news shock over 10 years. Panel (a) displays the equity yield slope (10-1) increasing to approximately 2% in the first year before gradually declining to near zero by year 5. Panel (b) shows the 1-year yield decreasing substantially to about -2% and the 10-year yield decreasing by approximately -0.4%. Panel (c) presents the expected growth slope decreasing to approximately -2.5% in the first year before gradually reverting toward zero. Panel (d) shows the 1-year expected growth increasing to about 2% and the 10-year expected growth increasing by approximately 0.4%. Panels (e) and (f) display risk premium components showing minimal movement with confidence intervals including zero. Source: Author calculations.
Figure 11: Responses to a News Shock for Value
Firms
(a) Equity yield slope (10-1)
(b) Equity yield, 1 and 10 year
(c) Expected growth slope (10-1)
(d) Expected growth, 1 and 10 year
(e) Risk premium slope (10-1)
(f) Risk Premium, 1 and 10 year
This figure is identical to Figure 10 above, except that we now
present the impulse responses for value firms.
This is a six-panel line graph showing responses for value firms to a TFP news shock over 10 years. Panel (a) displays the equity yield slope (10-1) initially decreasing by about -0.8% before increasing to approximately 0.5% and then gradually declining. Panel (b) shows the 1-year yield initially increasing by about 0.8% before decreasing to approximately -0.6% and then gradually reverting toward zero, while the 10-year yield shows minimal movement. Panel (c) presents the expected growth slope initially showing a positive response of about 0.5% before turning negative to approximately -0.5%. Panel (d) shows the 1-year expected growth initially decreasing before gradually increasing toward zero, while the 10-year expected growth shows minimal movement. Panels (e) and (f) display risk premium components showing minimal responses with wide confidence intervals. Source: Author calculations.
Figure 12: Value and Growth Responses to a News Shock at
the 1-Year Horizon
(a) Equity yield: Growth
(b) Equity yield: Value
(c) Expected growth: Growth
(d) Expected growth: Value
(e) Risk premium: Growth
(f) Risk premium: Value
This figure shows the 1-year horizon
response of equity yields as well as its expected growth and risk
premium components for claims with maturities from 1 to 10 years to a
TFP news shock. The responses of expected growth and risk premia are
derived from a VAR that includes both of these variables. Each dot
represents the median estimate from the VAR and whiskers correspond to
the 16–84 percentile bootstrapped confidence intervals. The panels in
the left and right columns are for the growth and value portfolios,
respectively. The sample period runs from September 1974 to December
2019.
This is a six-panel error bar chart comparing responses at the 1-year horizon after a TFP news shock across maturities from 1 to 10 years. Panels (a) and (b) show equity yield responses for growth and value firms respectively, with growth firms exhibiting stronger negative responses (1-year yield approximately -3% versus -0.6% for value firms). Panels (c) and (d) present expected growth responses, with growth firms showing larger positive responses (1-year expected growth approximately 3% versus 0.5% for value firms). Panels (e) and (f) display risk premium responses, which are small for both portfolios but slightly larger for value firms at shorter maturities. Source: Author calculations.
Figure 13: Responses of the Value and Growth Portfolios
to a Long-Run Shock in the Ai et al., 2018 Model versus
Data
(a) Equity yield: Value
(b) Equity yield: Growth
(c) Expected growth: Value
(d) Expected growth: Growth
(e) Risk premium: Value
(f) Risk premium: Growth
This figure presents impulse responses for the value and
growth portfolios in the model and data. Following Ai et al., 2018, the cash
flows of value firms correspond to those from mature physical capital,
while the cash flows of growth firms correspond to those from intangible
capital or growth options, as defined in Equations (20) and
(21) in Section 5.
This is a six-panel line graph comparing model-implied responses with actual data responses over 10 years. Panels (a) and (b) compare equity yield responses for value and growth firms, with both model and data showing stronger negative responses for growth firms (approximately -2.3% in the model versus -2% in the data) than for value firms (about -0.8% in the model versus -0.6% in the data). Panels (c) and (d) compare expected growth responses, with both model and data showing stronger positive responses for growth firms (about 2.2% in the model versus 2% in the data) than for value firms (approximately 0.8% in the model versus 0.5% in the data). Panels (e) and (f) compare risk premium responses, with both model and data showing minimal responses close to zero for both firm types. Source: Author calculations; Ai et al. (2018) model.
Figure B.1: Impulse Responses to a News
Shock
(a) TFP
(b) TFP
(c) Equity yield slope (10-1)
(d) Equity yield slope (10-1)
(e) Equity Yield, 1 and 10 year
(f) Equity Yield, 1 and 10 years
We report impulse responses estimated from a VAR using the 2020 vintage
of adjusted TFP. Panels (a), (c), and (e) display responses of TFP, the
10-minus-1 year equity yield slope, and the 1- and 10-year equity yields
to a TFP news shock using the identification of Kurmann and Sims, 2020. Panels (b), (d), and (f) show
the corresponding responses using the identification of Barsky and Sims, 2011. The sample
period is September 1974 to December 2019.
This is a six-panel line chart comparing two identification methods for news shocks over 10 years, with percent on the y-axis. The left column shows responses using Kurmann and Sims (2020) identification, while the right column uses Barsky and Sims (2011). Panels (a) and (b) display TFP responses, both showing gradual increases to approximately 0.5% by year 5. Panels (c) and (d) show the equity yield slope (10-1 year), which increases to about 1% within the first year before gradually declining in both methods. Panels (e) and (f) compare the 1-year and 10-year equity yields separately, with the 1-year yield (solid line) dropping more sharply to approximately -1% compared to the 10-year yield (dashed line) at about -0.4%. Both identification methods produce similar patterns, indicating robustness of the results. Note: The sample period is September 1974 to December 2019. Source: Author calculations.
Figure C.1: Responses to a News Shock using Option-Implied
Yields and in Subsamples
(a) Equity yield slope (10-1)
(b) Equity yield, 1 year
(c) Equity yield, 10 year
(d) Equity yield slope (10-1)
(e) Equity yield, 1 year
(f) Equity yield, 10 year
(g) Equity yield slope (10-1)
(h) Equity yield, 1 year
(i) Equity yield, 10 year
The TFP news shock is identified as in Kurmann and Sims, 2020, with the 10-1 year equity yield
slope substituted into the VAR in place of the term spread. The news
shock is then used in local projections to obtain the impulse response
functions. The local projection includes the same number of lags and
same control variables as in the VAR. Panels (a) to (f) use the
model-implied equity yields from Giglio et al., 2024 for both the 1 and 10 year
maturity. The bottom row of panels replace the 1-year yield with the
option-implied short-term yield from Cassella et al., 2023.
This is a nine-panel line chart showing responses to news shocks over 10 quarters, with percent on the y-axis. The top row displays responses using model-implied equity yields from the full sample. The middle row shows responses using the same yields but restricted to a recent sample. The bottom row uses option-implied short-term yields from the recent sample. Each row displays three panels: the equity yield slope (10-1), the 1-year yield, and the 10-year yield. All panels show qualitatively similar patterns, with the equity yield slope rising to 1-2% and the 1-year yield declining more sharply than the 10-year yield. The responses are stronger in the recent sample rows, with the 1-year yield dropping to approximately -2.5% compared to -1.5% in the full sample. The option-implied yields in the bottom row show similar patterns to the model-implied yields in the middle row. Note: The TFP news shock is identified using local projections with the same control variables as in the VAR. Source: Giglio et al. (2024); Cassella et al. (2023); Author calculations.
Figure D.1: Responses to a Long-Run Shock in Belo et al., 2015
We report median impulse responses up to 10 years after a long-run
productivity shock in the model of Belo et al., 2015. The left panel
reports the responses of the log-value of debt (\(\log(B_t)\)) and the leverage ratio (\(l_t\)), while the right panel shows the
responses of the log-value of dividends (\(d_t\)) and EBIT (\(y_t\)).
This is a two-panel line chart showing median impulse responses to a long-run productivity shock over 10 years in the Belo et al. (2015) model. The left panel displays the log-value of debt (solid line) and leverage ratio (dashed line), with the leverage ratio initially dropping by approximately -1.5 percentage points before gradually reverting toward zero. The log-value of debt initially decreases slightly before increasing steadily to about 0.5%. The right panel shows the log-value of dividends (solid line) and EBIT (dashed line), with dividends initially jumping by approximately 0.4% before gradually increasing to about 0.6%, while EBIT rises more gradually to about 0.5% by year 10. Source: Author calculations based on Belo et al. (2015) model.
Figure D.2: Responses to a Long-Run Shock in Models versus
Data
(a) Equity yield, 1 year
(b) Equity yield, 10 year
(c) Expected growth, 1 year
(d) Expected growth, 10 year
(e) Risk premium,1 year
(f) Risk premium, 10 year
This figure shows the impulse response function (up to 10 years after
the long-run productivity shock) of the model-implied equity yields,
expected growth rates and risk premia for the aggregate dividend strip
with a 10-year (red solid lines) and a 1-year maturity (red dashed
lines) from Belo et al., 2015. For an easier
comparison, we also report the estimated impulse responses from the VAR
model, represented by black solid lines for the 10-year maturity and
black dashed lines for the 1-year maturity.
This is a six-panel line chart comparing responses to a long-run productivity shock over 10 years in the Belo et al. (2015) model versus VAR data. The panels show equity yields (top row), expected growth (middle row), and risk premium (bottom row) for 1-year claims (left column) and 10-year claims (right column). The model responses (red lines) diverge significantly from the data responses (black lines). For equity yields, the model shows increases of approximately 0.3-0.5% while the data shows decreases of -0.5% to -1.2%. Expected growth in the model initially rises before quickly turning negative, while data shows persistent positive responses of about 0.4-1.2%. Risk premia in the model increase by about 0.5-0.7%, while data shows minimal responses close to zero. Source: Author calculations based on Belo et al. (2015) model and VAR data.
Figure E.1: Heterogeneity of Equity Yields Over the Firm’s
Lifecycle
The figure shows the firm-specific component, \(\frac{1}{n} \log\left( \frac{s_{t}^i}{s_{t,n}^i}
\right)\), that drives the heterogeneity in equity yields of
individual firms in Lettau and Wachter, 2007. We show four firms in
different stages of their lifecycle. Their lifecycle length is 50 years.
Stage 1 refers to a newly-born firm (in Year 0). A Stage 2 firm is a
firm in the middle of its dividend share growth period. Stage 3 refers
to a firm which has reached the highest level of dividend share and is
expected to shrink in future. Lastly, a Stage 4 firm is a firm in the
middle of its shrinking period.
This is a line chart showing how equity yields vary across a firm's 50-year lifecycle in the Lettau and Wachter (2007) model. The x-axis displays years from 0 to 50, and the y-axis shows the firm-specific component that drives yield heterogeneity, ranging from approximately -25% to 15%. Four firm stages are plotted: Stage 1 (newly-born firm at Year 0), Stage 2 (firm in middle of dividend share growth at approximately Year 8), Stage 3 (firm at highest dividend share at Year 18), and Stage 4 (firm in middle of shrinking period at Year 37). The chart shows that firms early in their lifecycle (Stages 1-2) have yields 15-25% lower than market yields due to high expected dividend share growth, while mature firms (Stages 3-4) have yields 5-10% higher than market yields due to decreasing future dividend shares. Source: Author calculations based on Lettau and Wachter (2007) model.
Figure F.1: Responses to a News Shock for Alternative
Market Term Premia.
(a) Market return minus 1-year claim: rt − rt1
(b) Market return: rt
This figure is analogous to Figure 2,
but now we present responses in the data for an alternative definition
of the realized term premium based on the total market return, \(r_t-r_t^{1}\), as well as for the market
return in isolation, \(r_t\).
This is a two-panel line chart showing responses to a TFP news shock over 10 years, with percent on the y-axis ranging from -1 to 6. Panel (a) displays the market return minus 1-year claim (rt - r¹t), which increases from 0% to approximately 3-4% within the first year before gradually declining to about 2% by year 10. Panel (b) shows the market return (rt) in isolation, which follows a similar pattern but with slightly higher values, reaching about 4-5% at its peak before declining to approximately 2-3% by year 10. Both panels include confidence intervals (gray bands) that widen as the horizon increases. Source: Author calculations.
Figure F.2: Impulse Responses of Consumption and Dividends
to a News Shock
(a) Consumption (Benchmark case)
(b) Dividends (VAR excluding C)
(c) Consumption (VAR with C and D)
(d) Dividends (VAR with C and D)
The news shock is identified as in Kurmann and Sims, 2020. The VAR features TFP,
Consumption (C) and / or Dividends (D), Inflation, Short Rate and Equity
Yields. Solid black lines are the median estimates for the VAR estimated
with the 2020 vintage of adjusted TFP. The gray bands correspond to the
16–84 bootstrapped confidence intervals. Sample is 1974/09–2019/12. We
see that a TFP news shock impacts dividends more than consumption at its
peak (Panel (b) versus (a)). The response of dividends is poorly
estimated when both consumption and dividends are included, however
(Panel (d) versus (b)), potentially due to collinearity issues.
This is a four-panel line chart showing responses to a TFP news shock over 10 years, with percent on the y-axis. Panel (a) shows consumption in the benchmark case, gradually increasing to approximately 0.8-0.9% by year 5. Panel (b) displays dividends from a VAR excluding consumption, showing a stronger response that increases to about 1.5-1.8% by years 3-4. Panel (c) shows consumption from a VAR including both consumption and dividends, with a pattern similar to panel (a). Panel (d) presents dividends from the same VAR, showing a more volatile and less precisely estimated response with wide confidence bands that include both positive and negative values, likely due to collinearity issues when both consumption and dividends are included. Note: The VAR features TFP, Consumption (C) and/or Dividends (D), Inflation, Short Rate and Equity Yields. Source: Author calculations.
Figure F.3: Forecast Error Variance Decomposition for a
News Shock
(a) Equity yield slope (10-1)
(b) Equity yield
(c) Expected growth slope (10-1)
(d) Expected Growth
(e) Risk premium slope (10-1)
(f) Risk premium
The TFP news shock is identified as in Kurmann and Sims, 2020, with the 10-1 year equity yield
slope (or its growth and risk premium components) substituted into the
VAR in place of the term spread. Solid black lines are the median
estimates for the VAR estimated with the 2020 vintage of adjusted TFP.
The gray bands correspond to the 16–84 percentile bootstrapped
confidence intervals. The left panels present the FEVD for the slope for
horizons up to 10 years. The right panels present the FEVD at the \(h=40\) quarter horizon from a VAR that
includes as last variable the equity yield for maturities from 1 to 10
years. The sample period runs from September 1974 to December
2019.
This is a six-panel chart showing forecast error variance decomposition for a TFP news shock. The left panels (a, c, e) display the fraction of variance explained for the 10-1 year slope of equity yield, expected growth, and risk premium over horizons up to 10 years, with values ranging from 0.05 to 0.5. The news shock explains approximately 18-20% of variation in the equity yield slope and expected growth slope, and about 5-10% for risk premium slope. The right panels (b, d, f) show variance decomposition at the 40-quarter horizon across maturities from 1 to 10 years. Panel (b) shows the shock explains 25-40% of equity yield variance, with higher contributions for longer maturities. Panel (d) indicates the shock explains about 25% of expected growth variance across maturities. Panel (f) shows lower contributions (5-10%) to risk premium variance. Note: The TFP news shock is identified as in Kurmann and Sims (2020). Source: Author calculations.
Figure F.4: On-impact Responses to a News
Shock
(a) Equity Yield
(b) Expected Growth
(c) Risk premium
The figure shows the instantaneous response of equity yields, expected
growth, and risk premia for maturities from 1 to 10 years to a news
shock. The responses of expected growth and risk premia are derived from
a VAR that includes both these variables. Each dot represents the median
estimate from the VAR and whiskers correspond to the 16–84 percentile
bootstrapped confidence intervals. The sample period runs from September
1974 to December 2019.
This is a three-panel error bar chart showing instantaneous responses to a TFP news shock across maturities from 1 to 10 years. Panel (a) displays equity yield responses ranging from -2.5% to 0%, with yields for all maturities declining and shorter maturities showing stronger responses (approximately -1.8% for 1-year maturity versus -0.5% for 10-year maturity). Panel (b) shows expected growth responses ranging from 0 to 2.5%, with shorter maturities displaying stronger positive responses (approximately 1.8% for 1-year maturity versus 0.4% for 10-year maturity). Panel (c) presents risk premium responses ranging from -0.8% to 0.8%, with values close to zero across all maturities and confidence intervals consistently including zero. Note: The sample period runs from September 1974 to December 2019. Source: Author calculations.
Figure F.5: Impulse Responses to a News Shock: Slope of
Forward Equity Yields:
(a) TFP
(b) Consumption
(c) Inflation
(d) Short rate
(e) Forward equity yield slope (10-1)
This figure is analogous to Figure 3 of the paper, but now we use the 10-1
year forward equity yield slope (i.e., the equity yield adjusted for
nominally risk-free government bond rates).
This is a five-panel line chart showing responses to a TFP news shock over 10 years, with percent on the y-axis. Panel (a) shows TFP gradually increasing to about 0.5% by year 5. Panel (b) displays consumption steadily rising to approximately 0.8-0.9%. Panel (c) shows inflation initially decreasing to about -0.5% before gradually returning toward zero. Panel (d) presents the short rate following a similar pattern to inflation, initially dropping to approximately -0.4%. Panel (e) displays the forward equity yield slope (10-1 year, adjusted for nominally risk-free government bond rates), which increases to about 1.2-1.4% in the first year before gradually declining toward zero over 2-3 years. All panels include confidence intervals (gray bands). Source: Author calculations.
Figure F.6: On-impact Responses of Forward Equity Yields
to a News Shock This figure is analogous to Figure F.4 but uses forward equity yields.
This is an error bar chart showing instantaneous responses of forward equity yields (equity yields adjusted for nominally risk-free government bond rates) to a TFP news shock across maturities from 1 to 10 years. The y-axis shows percent, ranging from -2.5% to 0%. Forward equity yields for all maturities show negative responses, with stronger negative responses for shorter maturities. The 1-year forward yield decreases by approximately -2.2%, while the 10-year forward yield decreases by about -0.4%. The magnitude of responses decreases monotonically as maturity increases, and confidence intervals (indicated by whiskers) are narrower for shorter maturities than for longer maturities. Source: Author calculations.
Figure F.7: Forecast Error Variance Decomposition for the
Contemporaneous TFP Shock
(a) Equity yield slope (10-1)
(b) Equity yield
(c) Expected growth
(d) Risk premium
The shock is identified through a Choleski decomposition with TFP
ordered first in the VAR. Solid black lines are the median estimates for
the VAR estimated with the 2020 vintage of adjusted TFP. The gray bands
correspond to the 16–84 bootstrapped confidence intervals. Panels
(b)-(d) present the forecast error variance decomposition at horizon
\(h=40\) quarters.
This is a four-panel chart showing forecast error variance decomposition for a contemporaneous TFP shock. Panel (a) displays the fraction of variance explained for the 10-1 year equity yield slope across quarters 5-40, showing values between 0.05-0.3. Panels (b), (c), and (d) show the fraction of variance explained at the 40-quarter horizon across maturities 1-10 years for equity yield, expected growth, and risk premium respectively. Panel (b) indicates the contemporaneous shock explains 5-10% of equity yield variance for most maturities. Panel (c) shows similar contributions of 5-10% for expected growth variance. Panel (d) shows slightly lower contributions of approximately 5% for risk premium variance. The variance contributions are substantially lower than those observed for the news shock in Figure F.3. Note: The shock is identified through a Cholesky decomposition with TFP ordered first in the VAR. Source: Author calculations.
Figure F.8: Impulse Responses to a Contemporaneous TFP
Shock
(a) TFP
(b) Consumption
(c) Inflation
(d) Short rate
(e) Equity yield slope (10-1)
(f) Equity yield, 1 and 10 year
The shock is identified through a Choleski decomposition with TFP
ordered first in the VAR. Solid black lines are the median estimates for
the VAR estimated with the 2020 vintage of adjusted TFP. The gray bands
correspond to the 16–84 percentile bootstrapped confidence intervals.
Panels (a) to (e) report the impulse response function for the five
variables included in the VAR. Panel (f) shows the response of the 1 and
10 year yield separately. The sample period runs from September 1974 to
December 2019.
This is a six-panel line chart showing responses to a contemporaneous TFP shock over 10 years, with percent on the y-axis. Panel (a) shows TFP immediately jumping to 0.4-0.5% and gradually increasing to about 0.8% by year 10. Panel (b) displays consumption steadily rising to approximately 0.8-1.0%. Panel (c) shows inflation fluctuating between -0.1% and 0.1% with confidence intervals including zero. Panel (d) presents the short rate initially decreasing to about -0.2% before gradually reverting toward zero. Panel (e) displays the equity yield slope (10-1), which increases to about 0.8-1.0% within the first year before slowly declining. Panel (f) shows the 1-year yield (solid line) decreasing more sharply (to about -1.2%) than the 10-year yield (dashed line, approximately -0.4%). Note: The shock is identified through a Cholesky decomposition with TFP ordered first. Source: Author calculations.
Figure F.9: Realized Value-Minus-Growth Term
Premium
(a) (rV, t10 − rV, t1) − (rG, t10 − rG, t1)
We present the response of the realized value-minus-growth term premium
observed in the data (solid black lines representing the median
estimates from the VAR model) as well as from the Ai et al., 2018 model (line with
triangles).
This is a line chart showing the response of the realized value-minus-growth term premium to a TFP news shock over 10 years, with percent on the y-axis ranging from -4 to 3. The chart compares VAR data estimates (solid black line with gray confidence bands) with the Ai et al. (2018) model prediction (line with triangles). Both the data and model show small, mostly negative responses ranging from approximately -0.7% to -1.0% in the first few years, gradually moving toward zero by year 10. The data estimates include wide confidence intervals that encompass zero throughout the entire period, suggesting the response is not statistically significant. This response is substantially smaller in magnitude than the realized market term premium response shown in Figure 2. Source: Author calculations; Ai et al. (2018) model.
Figure F.10: Responses of the Value and Growth Portfolios
to a Long-Run Shock in the Bansal and Yaron (2004) Model versus
Data
(a) Equity yield: Value
(b) Equity yields: Growth
(c) Expected growth: Value
(d) Expected growth: Growth
(e) Risk premium: Value
(f) Risk premium: Growth
This figure is similar to Figure 8, but
presents results for the value and growth portfolios that we define
using the definition of cross-sectional payouts in Breugem et al., 2024 and applied to
the model of Bansal and Yaron (2004, see Section 6.2.b for more detail).
This is a six-panel line chart comparing model-implied responses with actual data responses over 10 years. Panels (a) and (b) compare equity yield responses for value and growth firms, with the model (red lines) showing stronger negative responses for value firms (approximately -0.8% versus -0.4% for growth firms). This pattern contradicts the data (black lines), which shows stronger negative responses for growth firms. Panels (c) and (d) compare expected growth responses, with the model showing stronger positive responses for value firms, again contradicting the data pattern. Panels (e) and (f) compare risk premium responses, with the model showing negative responses for both firm types while the data shows minimal responses close to zero. Overall, the model predictions directly contradict the empirical patterns observed in the value versus growth responses to long-run shocks. Note: Cross-sectional payouts are defined following Breugem et al. (2024) applied to the Bansal and Yaron (2004) model. Source: Author calculations; Bansal and Yaron (2004) model.