Presentation by Michael-Paul James on The Log-Linear Return Approximation, Bubbles, and Predictability by Tom Engsted, Thomas Q. Pedersen, and Carsten Tanggaard

1. The Log-Linear Return
Approximation, Bubbles, and
Predictability
Paper by Tom Engsted, Thomas Q. Pedersen, and Carsten Tanggaard
Presentation by Michael-Paul James
1

2. Table of contents
Introduction Stationarity
The Approximation Error under Stationarity
Story, Questions, Context, Issues,
Literature
Conclusion
Concluding Remarks
Predictability
Return Predictability under Bubbles,
Cochrane’s VAR Setup with a Bubble,
Return Predictability in the Simulated
Bubble Data, Share Repurchases
01 02
04 05
Explosive Bubble
The Approximation Error under an Explosive
Bubble, The Evans Bubble and the
Simulation Setup, Results from the
Simulation Study
03
2
Michael-Paul James

3. Introduction
01
story, questions, context, issues, literature
3
Michael-Paul James

4. Standards
4
Michael-Paul James
● Log-linear return approximation
○ Relates log stock returns linearly to log prices and log dividends
○ Commonly used in empirical research
■ Stock return predictability
■ Tests of present value models
■ Return variance decompositions
■ Discrete-time dynamic asset allocation
P: stock price D: dividend paid ρ: less than one

5. Goals
5
Michael-Paul James
● Investigate approximation error of log linear return approximation.
○ Both stationarity & explosiveness of log dividend price ratio
● Do rational bubbles explain stock return predictability based on δt
?
● Investigate finite sample properties of the log-linear approximation in
the presence of bubbles through simulations.
○ Expected returns are constant
○ Log dividends follow random walk with drift
○ Test to see if dividend price ratio predicts returns & dividend growth
■ They find that predictability only exists with bubbles.
● Investigate payout policy changes, substitute dividends w/ repurchases
○ δt
is highly persistent ○ Δdt
is unpredictable
○ No evidence of return predictability

6. Advantages
6
Michael-Paul James
● Time varying, stochastic returns can linearly relate prices & dividends,
allowing standard econometric techniques
● Taking expectations, solving for pt
, & imposing terminal conditions
leads to log-linear present value model (Dynamic Gordon Growth
Model)

7. Disadvantages
7
Michael-Paul James
● Log linear relation is an approximation from first order taylor expansion
of log gross stock returns around the dividend price ratio
● Dividend price ratio varies over time, thus the persistence and volatility
affect the approximation error
● Requires that δt
≡ dt
- pt
is stationary
○ Standard unit root tests stop rejecting the null hypothesis of
nonstationarity in empirical findings sing 1990s
○ Alternative models proposed when δt
is a random walk
○ Neither theory nor empirical evidence support such nonstationarity
in returns and dividend growth.
○ Unit root on δt
is only rationalized if Δdt
& pt
have unit roots

8. Bubbles
8
Michael-Paul James
● Bubbles lead to nonstationary dividend price ratio
● Explosive component in δt
● Explosive bubbles cannot be ruled out based on theory
○ Cochrane rules out bubbles based on a common sense argument
that P/E ratios will not go to 0 nor 1 million.

9. Goals
9
Michael-Paul James
● Investigate approximation error of log linear return approximation.
○ Both stationarity & explosiveness of log dividend price ratio
● Do rational bubbles explain stock return predictability based on δt
?
● Investigate finite sample properties of the log-linear approximation in
the presence of bubbles through simulations.
○ Expected returns are constant
○ Log dividends follow random walk with drift
○ Test to see if dividend price ratio predicts returns & dividend growth
■ They find that predictability only exists with bubbles.
● Investigate payout policy changes, substitute dividends w/ repurchases
○ δt
is highly persistent ○ Δdt
is unpredictable
○ No evidence of return predictability

10. Stationarity
02
The Approximation Error under Stationarity
10
Michael-Paul James

11. Bubbles
11
Michael-Paul James
On period gross stock return:
Taking logs:
First order Taylor of f(1+eδt+1
):

12. Bubbles
12
Michael-Paul James
Approximation error:
Defining ρ and k:
Substitution:
Upper bound:

13. Bubbles
13
Michael-Paul James
Approximation log
dividend price ratio:
● Campbell and Shiller (pre-1990) find that the approximation error in:
○ log returns is on average < 10% of rt
○ Log dividend price ratios on average < 4% of δt
with SD < 10%
○ Correlation of log returns ~ 0.999
○ Correlation of dividend price ratios ~ 0.98
● After 1980 dividend price ratio fell, explosive stock prices by 1990

14. Explosive Bubble
03
The Approximation Error under an Explosive Bubble
The Evans Bubble and the Simulation Setup
Results from the Simulation Study
14
Michael-Paul James

15. Table
1:
Simulated
Distribution
of
the
Approximation
Error
15
Table 1 reports the mean, median, standard deviation, and correlation of exact and approximate log returns (rt
and r∗
t
) and exact and
approximate log dividend-price ratios (δt
and δ∗
t
), using the simulated data from the bubble model (10), (12), (13), and (14). Approximate log
returns are computed as r∗
t+1
= ρpt+1
+ (1 − ρ ) dt+1
−pt+k
, and approximate log dividend-price ratios are computed as δ∗
t
in equation (9); ρ is
calculated as ρ = (1 + exp(δ))−1, where δ is the average log dividend-price ratio in the particular simulation run; “Approx. Error” is obtained as
“Exact” minus “Approx”; “Percent Error, E1” gives the percentage average error, computed as “Approx. Error” divided by “Exact”; “Percent Error,
E2” gives the average percentage error, computed as the percentage error at each observation averaged over the T = 100 observations. The
numbers in the table are averages over 10,000 simulations.
TABLE 1: Simulated Distribution of the Approximation Error
(no-burst probability π = 0.85; bubble factor λ = 100; sample size T = 100)
Returns Size of Approximation Error
Approx. Percent Percent
Statistic Exact Approx. Error Error, E1 Error, E2
Panel A. Log Return
Mean 0.0262 0.025 0.0012 4.58% 7.21%
Median 0.0252 0.0239 0.0013 5.16% 4.06%
Std.dev. 0.2587 0.2586 0.0001 0.04% 0.07%
Corr(r,r∗) = 1.0000
Panel B. Log Dividend-Price
Mean –4.4810 –4.4306 –0.0504 1.12% 1.09%
Median –4.3925 –4.3458 –0.0467 1.06% 0.77%
Std.dev. 0.5342 0.5284 0.0058 1.09% 1.20%
Corr(δ,δ∗) = 0.9992
Accounts for 51.7%
of stock price

16. Figure
1:
Simulated
Periodically
Collapsing
Bubble
16
Figure 1 shows the fundamentals price (dashed line) and the bubble-inflated price (solid line) for one simulation.
One particular
simulation of
10,000

17. Figure
2:
Exact
and
Approximate
Log
Dividend-Price
Ratio
17
Figure 2 shows the exact (solid line) and the approximate (dashed line) log dividend-price ratio computed from the simulated periodically
collapsing bubble in Figure 1.
One particular
simulation of
10,000

18. Table
2:
Simulated
Distribution
of
the
Approximation
Error
18
Table 2 reports percentage approximation errors (E1 and E2) for the mean and standard deviation of log returns (“avg r” and “sd r”), and for the
mean and standard deviation of log dividend-price ratios (“avg δ” and “sd δ”). The table also reports the correlation between exact and
approximate values. Here, λB/P is the average size of the bubble relative to the total valuation of the stock. Otherwise, see the notes to Table 1.
TABLE 2: Approximation Error in the Simulated Data with Varying Bubble Size (λ)
(no-burst probability π = 0.85; sample size T = 100)
Bubble Factor (λ)
Statistic 0 1 50 100 150 200 250
Size of bubble: λB/P 0.00% 2.59% 38.2% 51.7% 59.7% 65.1% 69.0%
Panel A. Log Return
Corr (r, r∗) 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
E1 (avg r) 0.00% 0.25% 3.92% 4.58% 5.53% 6.02% 5.97%
E2 (avg r) 0.00% 0.61% 6.49% 7.21% 6.45% 7.75% 7.79%
E1 (sd r) 0.00% 0.00% 0.09% 0.04% 0.04% 0.03% 0.00%
E2 (sd r) 0.00% 0.02% 0.09% 0.07% 0.05% 0.04% 0.03%
Panel B. Log Dividend-Price
Corr(δ, δ∗) — 0.9998 0.9991 0.9992 0.9993 0.9993 0.9994
E1 (avg δ) 0.00% 0.11% 0.99% 1.12% 1.15% 1.13% 1.10%
E2 (avg δ) 0.00% 0.11% 0.93% 1.09% 1.11% 1.12% 1.11%
E1 (sd δ) 0.00% 0.78% 0.99% 1.09% 1.16% 1.17% 1.20%
E2 (sd δ) 0.00% 0.27% 1.10% 1.20% 1.24% 1.25% 1.25
Lets the bubble
factor λ vary

19. TABLE
3:
Approximation
Error
in
the
Simulated
Data
with
Varying
Bubble
Size
(λ)
19
Table 3 reports percentage approximation errors (E1 and E2) for the mean and standard deviation of log returns (“avg r” and “sd r”), and for the
mean and standard deviation of log dividend-price ratios (“avg δ” and “sd δ”). The table also reports the correlation between exact and
approximate values. Here, λB/P is the average size of the bubble relative to the total valuation of the stock. Otherwise, see the notes to Table 1.
TABLE 3: Approximation Error in the Simulated Data with Varying Bubble Size (λ)
(no-burst probability π = 0.85; sample size T = 1,000)
Bubble Factor (λ)
Statistic 0 1 50 100 150 200 250
Size of bubble: λB/P 0.00% 0.70% 9.62% 13.5% 16.0% 17.9% 19.4%
Panel A. Log Return
Corr (r, r∗) 1.0000 1.0000 0.9998 0.9997 0.9997 0.9996 0.9996
E1 (avg r) 0.00% 0.25% 2.67% 4.16% 5.68% 6.96% 7.96%
E2 (avg r) 0.00% 0.14% 2.99% 4.92% 6.45% 7.73% 8.84%
E1 (sd r) 0.00% 0.00% 0.18% 0.22% 0.26% 0.20% 0.19%
E2 (sd r) 0.00% 0.02% 0.17% 0.20% 0.21% 0.22% 0.22%
Panel B. Log Dividend-Price
Corr(δ, δ∗) — 0.9981 0.9890 0.9873 0.9864 0.9858 0.9854
E1 (avg δ) 0.00% 0.05% 1.13% 1.90% 2.53% 3.08% 3.57%
E2 (avg δ) 0.00% 0.05% 1.08% 1.79% 2.38% 2.89% 3.34%
E1 (sd δ) 0.00% 5.30% 10.2% 11.3% 11.7% 12.00% 12.00%
E2 (sd δ) 0.00% 1.59% 9.17% 11.1% 12.0% 12.50% 12.80%
Sample size
T = 1,000

20. TABLE
4:
Approximation
Error
in
the
Simulated
Data
with
Varying
No-Burst
Probability
(π)
20
Table 4 reports percentage approximation errors (E1 and E2) for the mean and standard deviation of log returns (“avg r” and “sd r”), and for the
mean and standard deviation of log dividend-price ratios (“avg δ” and “sd δ”). The table also reports the correlation between exact and
approximate values. Here, λB/P is the average size of the bubble relative to the total valuation of the stock. Otherwise, see the notes to Table 1.
TABLE 4: Approximation Error in the Simulated Data with Varying No-Burst Probability (π)
(bubble factor λ = 100; sample size T = 100)
No-Burst Probability (π)
Statistic 0.65 0.75 0.85 0.95 0.99
Size of bubble: λB/P 48.2% 49.3% 51.7% 60.0% 72.3%
Panel A. Log Return
Corr (r, r∗) 1.0000 1.0000 1.0000 0.9999 0.9997
E1 (avg r) 3.49% 3.86% 4.58% 5.80% 4.26%
E2 (avg r) 4.59% 5.54% 7.21% 8.74% 6.15%
E1 (sd r) 0.12% 0.12% 0.04% 0.04% 1.41%
E2 (sd r) 0.11% 0.10% 0.07% 0.06% 0.15%
Panel B. Log Dividend-Price
Corr(δ, δ∗) 0.9994 0.9993 0.9992 0.9991 0.9995
E1 (avg δ) 0.70% 0.84% 1.12% 1.55% 1.21%
E2 (avg δ) 0.68% 0.82% 1.09% 1.49% 1.22%
E1 (sd δ) 1.02% 1.07% 1.09% 0.51% 1.22%
E2 (sd δ) 1.06% 1.13% 1.20% 0.93% 0.72%
Burst probability

21. TABLE
5:
Approximation
Error
in
the
Simulated
Data
with
Varying
No-Burst
Probability
(π)
21
Table 5 reports percentage approximation errors (E1 and E2) for the mean and standard deviation of log returns (“avg r” and “sd r”), and for the
mean and standard deviation of log dividend-price ratios (“avg δ” and “sd δ”). The table also reports the correlation between exact and
approximate values. Here, λB/P is the average size of the bubble relative to the total valuation of the stock. Otherwise, see the notes to Table 1.
TABLE 5: Approximation Error in the Simulated Data with Varying No-Burst Probability (π)
(bubble factor λ = 100; sample size T = 1,000)
No-Burst Probability (π)
Statistic 0.65 0.75 0.85 0.95 0.99
Size of bubble: λB/P 11.8% 12.3% 13.5% 18.8% 41.5%
Panel A. Log Return
Corr (r, r∗) 0.9998 0.9998 0.9997 0.9994 0.9996
E1 (avg r) 3.01% 3.30% 4.16% 9.51% 29.6%
E2 (avg r) 3.37% 3.82% 4.92% 10.8% 32.1%
E1 (sd r) 0.23% 0.23% 0.22% 0.14% 0.05%
E2 (sd r) 0.21% 0.21% 0.20% 0.14% 0.03%
Panel B. Log Dividend-Price
Corr(δ, δ∗) 0.9924 0.9908 0.9873 0.9759 0.9529
E1 (avg δ) 1.22% 1.41% 1.90% 5.35% 37.1%
E2 (avg δ) 1.15% 1.33% 1.79% 4.98% 32.4%
E1 (sd δ) 10.4% 10.6% 11.3% 13.6% 9.50%
E2 (sd δ) 10.1% 10.4% 11.1% 13.5% 11.6%
Burst probability
Large sample size

22. Predictability
04
Return Predictability under Bubbles
Cochrane’s VAR Setup with a Bubble
Return Predictability in the Simulated Bubble Data
Share Repurchases
22
Michael-Paul James

23. TABLE
6:
Predictability
Regressions
on
the
Simulated
Bubble
Data
(sample
size
T
=
100)
23
Table 6 reports estimates of br , bd, and φ (and associated standard errors, σ) in the system (15)–(17), using the simulated data from bubble
model (10), (12), (13), and (14). The numbers are averages of regressions over 10,000 simulated series with T = 100 observations in each. “Implied”
denotes the calculated coefficient based on the other two coefficients and identity (18), using ρ = (1+ exp (δ ))−1. The values of ρ in Panels A, B, C,
and D are 0.9888, 0.9935, 0.9872, and 0.9954, respectively; λ is the bubble multiplication factor; 1 − π is the probability that the bubble will burst
every period; λB/P is the average size of the bubble relative to the total valuation of the stock.
TABLE 6: Predictability Regressions on the Simulated Bubble Data (sample size T = 100)
Coefficients
Variable b*,φ* σ* Implied b*,φ* σ* Implied
Panel A. λ = 100, Panel B. λ = 250,
π = 0.85, λB/P = 52% π = 0.85, λB/P = 69%
r 0.117 0.065 0.118 0.113 0.057 0.113
Δd –0.026 0.039 –0.027 –0.019 0.026 –0.020
δ 0.866 0.062 0.867 0.873 0.060 0.874
Panel C. λ = 100, Panel D. λ = 100,
π = 0.65, λB/P = 48% π = 0.99, λB/P = 72%
r 0.167 0.090 0.169 0.023 0.041 0.022
Δd –0.037 0.045 –0.038 –0.014 0.027 –0.013
δ 0.805 0.087 0.806 0.968 0.041 0.967
Panel A similar to
Cochrane

24. 24
TABLE
7:
Predictability
Regressions
on
the
Simulated
Bubble
Data
(sample
size
T
=
1,000)
Table 7 reports estimates of br, bd, and φ (and associated standard errors, σ) in the system (15)–(17), using the simulated data from bubble model
(10), (12), (13), and (14). The numbers are averages of regressions over 10,000 simulated series with T = 1,000 observations in each. “Implied”
denotes the calculated coefficient based on the other two coefficients and identity (18), using ρ = (1 + exp (δ))−1. The values of ρ in Panels A, B, C,
and D are 0.9781, 0.9811, 0.9771, and 0.9960, respectively; λ is the bubble multiplication factor; 1 − π is the probability that the bubble will burst
every period; λB/P is the average size of the bubble relative to the total valuation of the stock.
TABLE 7: Predictability Regressions on the Simulated Bubble Data (sample size T = 1,000)
Coefficients
Variable b*,φ* σ* Implied b*,φ* σ* Implied
Panel A. λ = 100, Panel B. λ = 250,
π = 0.85, λB/P = 13% π = 0.85, λB/P = 19%
r 0.059 0.021 0.067 0.048 0.014 0.055
Δd –0.009 0.017 –0.017 –0.005 0.010 –0.012
δ 0.945 0.017 0.953 0.958 0.014 0.965
Panel C. λ = 100, Panel D. λ = 100,
π = 0.65, λB/P = 12% π = 0.99, λB/P = 86%
r 0.077 0.034 0.083 0.017 0.008 0.015
Δd –0.012 0.020 –0.018 –0.0015 0.004 –0.0002
δ 0.926 0.032 0.932 0.987 0.006 0.986

25. 25
TABLE
8:
Predictability
Regressions
on
the
Simulated
Data
with
Repurchases
(sample
size
T
=
100)
Table 8 reports estimates of br , bd, and φ (and associated standard errors, σ) in the system (15)–(17), using the simulated data from the model in
Section IV.C with T = 100 observations. Here, θ < 1 is the scaling factor that is multiplied onto dividends from time t = 76 and onwards. The
numbers are averages of regressions over 10,000 simulated series. “Implied” denotes the calculated coefficient based on the other two
coefficients and identity (18), using ρ = (1 + exp(δ ))−1. The values of ρ in Panels A, B, and C are 0.9770, 0.9782, and 0.9797, respectively.
TABLE 8: Predictability Regressions on the Simulated Data with Repurchases (sample size T = 100)
Coefficients
Variable b*,φ* σ* Implied
Panel A. θ = 0.5
r 0.020 0.042 0.023
Δd –0.032 0.066 –0.035
δ 0.967 0.045 0.970
Panel B. θ = 0.4
r 0.018 0.032 0.021
Δd –0.025 0.049 –0.028
δ 0.975 0.033 0.978
Panel C. θ = 0.3
r 0.016 0.024 0.021
Δd –0.020 0.036 –0.025
δ 0.979 0.023 0.984
A periodically collapsing explosive bubble, which looks stationary in finite
samples, may generate return predictability when expected returns are constant

26. Conclusion
05
Concluding Remarks
26
Michael-Paul James

27. Remarks
27
Michael-Paul James
● Found upper bound of the mean approximation error, given
stationarity of the log dividend price ratio which equals the
undcontiona mean of δt
● Bubbles unless very large do not induce large approximation errors in
the log linear relation
● Using the simulated bubble data from a constant expected returns
model, log returns, rt+1
, appear significantly predictable from δt
, and that
δt
appears stationary
● Approximation error in Campbell-Shiller log linear approximation is
negligible

28. Take Away
28
Michael-Paul James
● Campbell-Shiller (1988a) approximation appears
○ Highly accurate and robust
○ Even when the log dividend-price ratio is highly volatile and
contains nonstationary components
○ Periodically collapsing rational bubbles may generate return
predictability even when expected returns are constant.

29. You are Amazing
Ask me all the questions you desire. I will do my best to answer honestly
and strive to grasp your intent and creativity.
29
Michael-Paul James