Excess Volatility in Finance

Slide 1

Excess Volatility in Finance

$$ P_{t}=\sum_{s\ge1}\frac{\mathbb{E}_{t}(D_{t+s})}{R^{s}} $$ $$ P_{t}^{*}=\sum_{s\ge1}\frac{D_{t+s}}{R^{s}} $$
Slide 2
Graph comparing actual stock prices to a perfect foresight model.

Description of the Graph:

This is a time-series line graph comparing the actual stock market price with a theoretical "perfect foresight" price from 1982 to 2022. The x-axis represents Time. The y-axis is on a logarithmic scale, with values of 100, 300, 900, and 2700 marked.

There are two lines: a solid green line representing the actual price ($p_t$), and a dotted blue line representing the perfect foresight price ($p_t^*$), which is calculated based on knowing all future dividends. The blue line shows a smooth, steady upward trend. In contrast, the green line is highly volatile, fluctuating significantly around the smooth blue line. It shows major peaks, such as during the dot-com bubble around 2000, and sharp troughs, like during the 2008 financial crisis. The key takeaway is the "excess volatility" of the actual market price compared to the price justified by fundamental dividend information.

Slide 3-4

Excess Volatility in Finance

$$ P_{t}=\sum_{s\ge1}\frac{\mathbb{E}_{t}(D_{t+s})}{R^{s}} $$
Cochrane (2001) "...exuberance comes at the top of an unprecedented economic expansion, a time when the average investor is surely feeling less risk averse than ever..."
Slide 5

Excess Volatility in Finance

$$ P_{t}=\sum_{s\ge1}\frac{E_{t}^{\text{measured}}(D_{t+s})}{R^{s}} $$

Again, keep R constant. All the action is in the numerator.

Slide 6-7
Graph comparing actual, perfect foresight, and expectations-based stock prices.

Description of the Graph:

This graph is an extension of the one on slide 2, covering the same time period (1982-2022) and using the same axes. It includes the same green line (actual price, $p_t$) and dotted blue line (perfect foresight price, $p_t^*$).

A new solid red line has been added, labeled "expectations based index". This red line is constructed using measured analyst expectations of future earnings. The key observation is that this new red line tracks the volatile green line (actual price) much more closely than the smooth blue line does. It captures the major peaks and troughs, such as the run-up to 2000, the 2008 crash, and the subsequent recovery. This suggests that the measured expectations of analysts, rather than fundamentals alone, can account for a large portion of the observed stock market volatility.

Slide 8-9

Volatile Expectations Account for Shiller

Earnings Indexes $\Delta p$ $\Delta p^*$ $\Delta \tilde{p}$
Standard deviation 14.8% 0.7% 14.6%
95th Conf Interval 13.9%-15.9% 0.6%-0.7% 13.7%-15.6%
Slide 10

Predictable Returns and Market Efficiency

In efficient markets, expectations of cash flows are rational and average returns equal required returns

Joint hypothesis problem: without observing required returns, test of efficiency is also test of risk model

Our approach: keep required returns fixed, relax rationality of expectations

Slide 11

Predictable Returns and Market Efficiency

Strong growth $\rightarrow$ Excess optimism for LTG $\rightarrow$ price/dividend inflated $\rightarrow$ future disappointment $\rightarrow$ predictably low returns

Slide 12

Predictability of Aggregate Returns

$$ P_{t}=\mathbb{E}_{t}\left[\frac{D_{t+1}}{R}+\frac{D_{t+2}}{R^{2}}+\cdot\cdot\cdot\right] $$
Slide 13

LTG predicts future returns

$r_{t+1}$ $\sum_{j=1}^{3}\alpha^{j-1}r_{t+j}$ $\sum_{j=1}^{5}\alpha^{j-1}r_{t+j}$
Panel A: Returns and LTG
$LTG_t$ -0.2389b -0.4019a -0.4349a
(0.0928) (0.0944) (0.0831)
Observations 409 409 409
Adjusted $R^2$ 9% 24% 25%
Panel B: Returns and growth forecast for year 1
$\mathbb{E}_{t}^{O}[e_{t+1}-e_{t}]$ -0.0335 0.0467 0.1556a
(0.1027) (0.0716) (0.0587)
Observations 404 404 404
Adjusted $R^2$ 0% 0% 3%
Panel C: Returns and growth forecast for year 2
$\mathbb{E}_{t}^{O}[e_{t+2}-e_{t+1}]$ -0.0527 0.0408 0.2113
(0.0885) (0.1556) (0.1686)
Observations 404 404 404
Adjusted $R^2$ 0% 0% 6%
Slide 14

LTG, not proxies, predict returns

Might LTG spuriously capture variation in required returns?

Control for proxies of required returns in predictability regression

Dependent Variable: Five-year Return $\sum_{j=1}^{5}\alpha^{j-1}r_{t+j}$ $X_t = spc_t$ $X_t = cay_t$ $X_t = SVIX_t^2$
$LTG_t$ -0.4522a -0.5569a -0.3946a
(0.1033) (0.1179) (0.1016)
$X_t$ -0.1387 0.1894 $0.3852^b$
(0.1035) (0.1766) (0.1782)
Observations 409 137 193
Adjusted $R^2$ 27% 28% 47%
Slide 15-16

Key prediction

If LTG expectations over-react, then both forecast errors and returns are negatively predictable from:

Crucially, the predicted LTG forecast error should explain the contemporaneous return

boom and bust price pattern

Predictability of returns and forecast errors

Dependent Variable: $\Delta_{5}e_{t}/5-LTG_{t}$ $\sum_{j=1}^{5}\alpha^{j-1}r_{t+j}$ $\sum_{j=1}^{5}\alpha^{j-1}r_{t+j}$
$\Delta LTG_t$ $-0.8407^a$ $-0.6403^a$
(0.1533) (0.0764)
$LTG_{t-1}$ -0.2157 $-0.5252^a$
(0.1369) (0.0864)
$\Delta_5 e_{t+5}/5-LTG_t$ $0.8460^a$
(0.2501)
Observations 397 397 397
Adjusted $R^2$ 25% 31%
Montiel-Pflueger F-stat 10.9
Slide 17-18

Bottom line

Firm level evidence

Could the results be driven by outlier episodes? e.g. dot com bubble

Run firm level regressions

Slide 19

Overreaction at the firm level

$\Delta_{5}e_{i,t}/5-LTG_{i,t}$ $\sum_{j=1}^{5}\alpha^{j-1}r_{i,t+j}$ $\sum_{j=1}^{5}\alpha^{j-1}r_{i,t+j}$
$\Delta LTG_{i,t}$ $-0.3286^a$ $-0.1773^a$
(0.0248) (0.0409)
$LTG_{i,t-1}$ $-0.3626^a$ $-0.2163^a$
(0.0256) (0.0446)
$\Delta_5 e_{i,t}/5-LTG_{i,t}$ $0.5768^a$
(0.0919)
Observations 371,571 371,571 371,571
Adjusted $R^2$ 4% 1%
Kleibergen-Paap F-stat 101.8
Year FE / Firm FE Yes / Yes Yes / Yes Yes / Yes

Same results at the firm level (lower $R^2$ as expected)

Slide 20-21

Expectations and cross sectional returns

Returns reflect market risk (CAPM, Sharpe 1964), but significant predictable returns not tied to market exposure (Basu 1977, Banz 1981, Rosenberg et al 1985, Jegadeesh Titman 1993)

Our approach:

$$ r_{i,t+1} = (r_i - r) + \left(r + [g_{i,t+1} - \tilde{\mathbb{E}}_{t}^{e}(g_{i,t+1})] + \sum_{s\ge1}\alpha^{s}(\tilde{\mathbb{E}}_{t+1}^{e}-\tilde{\mathbb{E}}_{t}^{e})(g_{i,t+1+s})\right) $$

Where the first term is the risk premium relative to the market, and the second term is the Expectations based return, $EBR_{i,t+1}$.

Slide 22

EBRs explain the value premium

Graph comparing HML returns with HML EBRs.

Description of the Graph:

This is a time-series line graph comparing the HML (High Minus Low, or "value") factor return with its corresponding Expectations-Based Return (EBR). The x-axis shows the date, from 1980 to beyond 2020. The y-axis shows the return, ranging from -0.4 to 0.6.

There are two lines: a solid black line representing the actual HML return ("HML ret") and a solid red line representing the expectations-based return ("HML EBR"). The two lines track each other very closely over the entire 40-year period, showing similar peaks and troughs. This visual evidence strongly suggests that the systematic forecast errors and revisions captured by the EBR model can explain the well-known value premium anomaly.

Slide 23

Results extend to other standard factors

Four graphs showing EBRs explaining size, investment, profitability, and momentum factors.

Description of the Graphs:

This slide contains four separate time-series line graphs, each comparing an asset pricing factor return (in black) with its corresponding Expectations-Based Return (EBR, in red). The time period for all graphs is roughly 1980 to 2020.

  1. Size: Compares the size factor (SMB) return to its EBR. The two lines show significant co-movement.
  2. Investment: Compares the investment factor return to its EBR. Again, the red and black lines track each other closely.
  3. Profitability: Shows the profitability factor return and its EBR, demonstrating a similar strong correlation.
  4. Momentum: Compares the momentum factor return to its EBR. The two lines follow a very similar path.

The consistent finding across all four graphs is that the EBR model, based on analyst forecast errors and revisions, can closely replicate the returns of major cross-sectional asset pricing anomalies.

Slide 24-25

Why do EBRs explain the value premium?

Does the explanatory power of EBRs reflect market inefficiency?

Test 1. Predicted EBRs predict returns

$r_{HML,t+h}$ $h=1$ $h=12$
$\widehat{EBR}_{LMS,t,t+h}$ $1.3267^a$ $1.0300^a$
(0.4452) (0.3119)
$dp_{LMS,t}$ 0.0171 $0.1747^b$
(0.0126) (0.0853)
Constant -0.0039 $-0.0562^b$
(0.0040) (0.0267)
Obs 444 433
Adjusted R2 4% 16%
1st stage R2 14% 30%
Slide 26

Test 2. Characteristics predict EBRs

Regress firm level EBRs on characteristics

$EBR_{i,t,t+h}$ 1 month 1 year 5 years
$bm_{i,t}$$0.0056^a$$0.0573^a$$0.1434^a$
(0.0003)(0.0081)(0.0202)
$size_{i,t}$$0.0082^a$$-0.0717^a$$-0.3975^a$
(0.0002)(0.0055)(0.0306)
$Inv_{i,t}$$-0.0067^a$$-0.0705^a$$-0.0869^a$
(0.0004)(0.0056)(0.0107)
$op_{i,t}$$-0.0014^b$-0.0050-0.0098
(0.0006)(0.0110)(0.0313)
$r_{i,t-12-t-1}$$0.0176^a$$0.0806^a$$0.0305^b$
(0.0002)(0.0150)(0.0143)
Obs878,185775,234475,520
Adj R20%2%17%
Slide 27-28

Taking Stock

Economic and Financial Volatility

Slide 29-30

LTG and the Business Cycle

$$ y_{t+h}-y_{t+h-4}=\gamma_{0}+\gamma_{h}\Delta LTG_{t}+\gamma'X+u_{t+h} $$

For horizons $h=0,1,...10$ quarters

Slide 31

Local Projections For Other Business Cycle Indicators

Six graphs showing boom-bust cycles in economic indicators.

Description of the Graphs:

This slide presents six small line graphs, each showing the response of a different macroeconomic variable to a shock in Long-Term Growth (LTG) expectations. This is a local projection analysis. The x-axis for each graph is "t + h quarter projection", representing the number of quarters after the shock (from 0 to 10). The y-axis shows the percentage change or percentage point change in the variable.

The six variables are: investment-to-capital, GDP, consumption, employment, total wages, and 1-year inflation. All six graphs display a similar and distinct "boom-bust" pattern. Following a positive shock to LTG expectations at time 0, each variable rises for approximately 4 quarters (the boom), peaks, and then declines, falling below its initial level for the subsequent quarters (the bust). This consistent pattern across key economic indicators suggests that swings in long-term optimism and pessimism can generate business-cycle-like fluctuations in the real economy.

Slide 32-33

Takeaways

Credit Cycles

Graph of credit spreads from 1925 to 2015.

Description of the Graph:

This is a time-series line graph showing credit spreads from 1925 to 2015. The y-axis represents the spread in percentage points, from 0 to 8. The black line plots the credit spread over time. It is highly cyclical, with periods of low spreads followed by dramatic spikes. The light blue shaded vertical bars indicate recessions. The key observation is that credit spreads are typically low and stable before a recession and then spike to very high levels during the recessionary period. This is visible during the Great Depression (early 1930s), the early 1980s recession, the dot-com bust (early 2000s), and most prominently during the 2008 financial crisis.

Slide 34

Credit Cycles

Graph showing credit growth and issuer quality over time.

Description of the Graph:

This is a time-series line graph from 1962 to 2008 showing credit cycles. There are two y-axes. The left y-axis, for credit growth, ranges from -10% to 40%. The right y-axis, for issuer quality ($ISS^{EDF}$), ranges from -1.00 to 1.50.

There are three lines: two solid lines tracking "Credit Growth" (from two different sources, FOF and Compustat) and one dashed gray line tracking issuer quality. The credit growth lines are cyclical, showing periods of expansion and contraction. The issuer quality line generally moves in the opposite direction of credit growth: when credit growth is high, it often means lending standards are looser, so the quality of new issuers is lower. This is particularly evident in the run-up to crises.

Slide 35-37

Link between credit expansion and economic crises?

Minsky's financial instability hypothesis

Diagnostic expectations generate such booms and busts

Slide 38

Credit expansion and low bond returns

Graph comparing issuer quality with future high-yield bond returns.

Description of the Graph:

This is a time-series line graph from 1962 to 2008. The left y-axis shows "Issuer Quality $ISS^{EDF}$" and the right y-axis shows "2-year Excess HY Returns (%)", which is inverted (higher values are at the bottom).

The solid blue line represents issuer quality. The dashed red line represents the subsequent two-year excess return on high-yield bonds. There is a striking inverse relationship. Periods of high issuer quality (meaning a higher share of risky firms are issuing debt, indicating optimism) are systematically followed by periods of very poor bond returns. Conversely, periods of low issuer quality (when only the safest firms can issue debt) are followed by high returns. This suggests that credit booms fueled by optimism lead to predictable losses for investors.

Slide 39

Credit expansion and weakening economic activity

Scatter plot of GDP growth vs. past credit-market sentiment.

Description of the Graph:

This is a scatter plot examining the relationship between credit market sentiment and future economic growth. The x-axis represents "Credit-market sentiment at t-2" (two years prior). The y-axis represents "Growth in real GDP per capita at t".

Each black dot is an observation. A solid red regression line is drawn through the data points, showing a clear downward slope. This indicates a negative correlation: periods of high credit-market sentiment (optimism) are followed two years later by periods of lower real GDP growth. This supports the idea that credit booms, driven by optimism, lead to subsequent economic downturns.

Slide 40-41

Why do credit booms create risk for the economy?

Real Credit Cycles

Slide 42

Firm level boom-bust cycles

Scatter plot and regression table showing predictable forecast errors.

Description of the Graph and Table:

The slide contains a scatter plot and a regression table that demonstrate predictable forecast errors at the firm level.

Scatter Plot (top left): The x-axis is "Current Investment, % from Mean" and the y-axis is "Future Forecast Error, % from Mean". The red dots represent data points, and a solid blue line shows the line of best fit. The line slopes steeply downwards, indicating a strong negative relationship. Firms that are currently investing heavily (a sign of optimism) tend to have large, negative forecast errors in the future, meaning their initial optimism was systematically wrong.

Regression Table (bottom right): This table quantifies the relationship. The dependent variable is the future forecast error. Column (2) shows that "Investment" has a large, statistically significant negative coefficient (-0.457). This confirms the visual from the scatter plot: higher investment predicts more negative future forecast errors.

Slide 43

Credit expansion and financial crises

Graph showing actual vs. fundamental credit spreads around a crisis.

Description of the Graph:

This line graph shows the behavior of credit spreads in the period surrounding a financial crisis. The x-axis is "time", where 0 marks the beginning of the crisis. The period shown is from 5 years before to 5 years after the crisis.

There are two lines: a blue line for the actual "Spread" and a red line for the "Fundamental Spread" (the spread justified by economic fundamentals). Before the crisis (from t=-5 to t=-1), the actual spread is consistently below the fundamental spread, indicating excessive optimism and underpricing of risk. At the onset of the crisis (t=0), the actual spread explodes upwards, dramatically overshooting the fundamental spread. It then gradually declines in the following years. This pattern suggests that crises are preceded by a period of risk underpricing, which then corrects violently.

Slide 44

Credit expansion and financial crises

Graph showing the probability of a crisis increases after a credit boom.

Description of the Graph:

This graph shows the estimated probability of a financial crisis occurring over the next five years. The x-axis is "Year" from 1 to 5. The y-axis is "P(Crisis)", the probability of a crisis, from 0 to 0.7.

There are two colored bands with lines running through them, representing predictions under different conditions. The upper, yellow band shows a much higher probability of a crisis than the lower, green band. Both bands slope upwards, indicating that the probability of a crisis increases in the years following certain initial conditions. The context implies that the yellow band represents the high probability of a crisis following a credit boom (high credit growth and low spreads), while the green band represents the lower probability in normal times. The graph demonstrates that credit booms are strong predictors of future financial crises.

Slide 45

Conclusion