Asset Pricing II: Second (Individual) Assignment Predictability
Dr. Jiri Knesl
Due: Friday, March 13 2020 by 12 noon via online submission system. Weighting: 20% of Asset Pricing overall grade.
Please enclose your computer code with your solutions.
Data: All data for this assignment is contained in the spreadsheet “Assignment2Data.xlsx” which is available on the course website.
Spreadsheet Q1 contains the data relevant for this part of the assignment. The dataset includes annual total returns (R) on the NYSE/Nasdaq/AMEX value-weighted index of all listed shares; growth in aggregate dividends on these shares (DG), aggregate dividend yield on this portfolio (DP), and the Payout Ratio (DE) all adjusted for changes in the CPI, starting in 1927 and ending in 2019.
Conduct your analysis using variables adjusted for changes in the CPI. Assignment
1. Report in a table the mean and standard deviation of total returns and of net dividend growth for the periods 1927-2019 and the more recent period 1960-2019, together with the correlations of returns and dividend growth over these periods.
[15 points]
2. Does the dividend yield predict equity index annual total returns in the periods 1927- 2019 and the more recent period 1960-2019? Answer by reporting coefficients, t-statistics and adjusted-R2s for these two periods for the regression
Rt+1 = a+b(Dt/Pt)+εt+1. (1)
[15 points]
3. Does the dividend yield predict annual real dividend growth in the periods 1927-2019 and the more recent period 1960-2019? Answer by reporting coefficients, t-statistics and adjusted-R2s for these two periods for the regression
(Dt+1 −Dt)/Dt = a+b(Dt/Pt)+εt+1. (2)
[15 points]
4. Does anything else in your data predict dividend growth in these periods?
[15 points]
1
5. Campbell and Shiller (1988) propose an approximation for log total returns around the mean of the log dividend yield:
Pt + 1 + D t + 1
rt+1 =ln P ≈k−(pt −dt)+∆dt+1+ρ(pt+1−dt+1) (3)
t
and they also show that this approximation is very accurate for observed values of volatility of the dividend yield. Here lower case letters denote logs and ∆dt+1 is log gross dividend growth ln(Dt+1/Dt). ρ = (1+eE[d−p])−1 and in annual data is about 0.95.
Rearranging this gives
pt −dt ≈ k+∆dt+1 −rt+1 +ρ(pt+1 −dt+1). (4) Then Cochrane (2005) shows
Var[pt −dt]≈Cov[pt −dt,∆dt+1]−Cov[pt −dt,rt+1]+Cov[pt −dt,ρ(pt+1−dt+1)] and dividing by the left-hand side gives
1 ≈ Cov[pt −dt,∆dt+1]−Cov[pt −dt,rt+1]+Cov[pt −dt,ρ(pt+1−dt+1)] (5) Var[pt −dt] Var[pt −dt] Var[pt −dt]
≈ βpd,∆d − βpd,r + ρβpd,pd+1
where the quantities βpd, j are the OLS coefficients of a forecasting regression of j (∆dt+1,
rt+1, (pt+1 −dt+1)) on pt −dt.
Since equation (3) comes from an (approximate) accounting identity, Cochrane argues that the correct null hypothesis is not that pt − dt does not predict returns, but that if it does not predict returns it therefore predicts a combination of dividend growth and itself, discounted at the constant rate ρ. I want you to test the hypothesis of return predictibility against the correct null using a restricted VAR.
Estimate the following system of seemingly unrelated regressions
Pt
rt+1 = βpd,rlog D +ε1,t+1 (6)
t
Pt
∆dt+1 = βpd,∆dlog D +ε2,t+1 (7)
Pt+1 D
where
and Σ is the covariance matrix of residuals ε.
t
Pt
log
=
βpd,pd+1 log D t
+ ε3,t+1 (8)
t+1
εt+1 ∼N(0,Σ)
2
Estimate the coefficients βpd,∆d and βpd,pd+1 equation by equation, and then construct the residuals for the first equation by forcing βNull = −(1 − βpd,∆d − ρβpd,pd ). You can esti-
pd,r +1
mate the intercept under this correct null by calculating the mean of rt+1 −βNull(pt −dt).)
pd,r
Report your results in a table. In 1-2 lines (more will be neither read nor marked), explain
why your answer matters.
[20 points]
6. Now let Xt = rt ∆dt pt − dt ′, and estimate the following VAR: Xt+1 = a3×1+ΓXt+ut+1
ut+1 ∼ N(0,Ω)
Use your estimate of the transition matrix Γ to carry out a variance decomposition of stock market returns. Report the share of stock market return variance between 1927 and 2019 due to news about future cash-flows, news about future returns and covariance between the two. What is the correlation between cash-flow news and discount-rate news?
[20 points]
[BONUS QUESTION]
Predictability of announcement and non-announcement days’ returns.
Spreadsheet BonusQuestion contains the data relevant for this part of the assignment. The data for this question are quarterly log returns on the NYSE/Nasdaq/AMEX value-weighted index of all listed shares, quarterly cumulative returns accrued on announcement and non- announcement days for that index, aggregate quarterly dividend yield on this portfolio and an average number of announcement and non-announcement days in each quarter.
AN
Let Rd and Rd′ stand for the daily gross return on a scheduled announcement day d and non-
announcement day d′, respectively. I define the cumulative return accrued on scheduled announcement days and non-announcement days in quarter t + 1 as a sum of the relevant daily log returns:
AANN rt+1 = ∑ logRd rt+1 = ∑ logRd′.
d∈t+1 d′∈t+1
Thus, the total return in quarter t + 1 can be expressed as r = rA
returns provided in the spreadsheet.
+ rN t+1
. These are the
3
t+1 t+1
Using the sample period between 01/01/1960 – 31/12/2016 estimate the following relation- ships:
rt+1 =α+βlog(Pt )+εt+1 (9) Dt
rA =αA+βAlog(Pt)+εA (10) t+1 Dt t+1
rN =αN+βNlog(Pt)+εN . (11) t+1 Dt t+1
For each of the models report in a table the values of the estimated coefficients, their t- statistics, model’s adjusted R2, and the average of the dependent variable. Finally, compute the average daily return for each of the three specifications. Comment on your results.
[10 points]
References
• Campbell, John Y., and Robert J. Shiller. “The dividend-price ratio and expectations of future dividends and discount factors.” Review of financial studies 1, no. 3 (1988): 195-228.
• Cochrane, John H. “The dog that did not bark: A defense of return predictability.” Review of Financial Studies 21, no. 4 (2008): 1533-1575.
4