Return Predictability: Dividend Price Ratio versus Expected Returns

Return Predictability: Dividend Price Ratio versus Expected Returns Rambaccussing, Dooruj Department of Economics University of Exeter 08 May 2010 (Institute) 08 May 2010 1 / 17

Objective Perhaps one of the best predictor of realized returns is Expected Returns (Institute) 08 May 2010 2 / 17

Objective Perhaps one of the best predictor of realized returns is Expected Returns In the literature one of the best predictors of returns remains Price Dividend Ratios (Cohrane 2008) Campbell and Shiller (1988) posit a relationship between expected returns and expected dividend growth. If Dividend Growth is unpredictable, all variation in Price Dividend Ratio is caused by the returns and vice versa. (Institute) 08 May 2010 2 / 17

Definitions and Identity r t =ln( P t+1+d t+1 P t ) (1) (Institute) 08 May 2010 3 / 17

Definitions and Identity r t =ln( P t+1+d t+1 P t ) (1) pd t = P t D t (2) (Institute) 08 May 2010 3 / 17

Definitions and Identity r t =ln( P t+1+d t+1 P t ) (1) pd t = P t D t (2) d t+1 = ln( D t+1 D t ) (3) (Institute) 08 May 2010 3 / 17

Definitions and Identity r t =ln( P t+1+d t+1 P t ) (1) pd t = P t D t (2) d t+1 = ln( D t+1 D t ) (3) Campbell and Shiller Log-Linearized Identity (form 1,2 and 3): r t+1 = κ + ρpd t+1 + d t+1 pd t pd = E [log(pd t )], κ = log(1 + exp(pd)) ρpd andρ = pd t = κ 1 ρ + ρ pd + i=1 ρ i 1 ( d t+i r t+i ) exp(pd) 1 + exp(pd) (Institute) 08 May 2010 3 / 17

State Space Model State Equation µ t+1 δ 0 = δ 1 (µ t δ 0 ) + ε µ t+1 g t+1 γ 0 = γ 1 (g t γ 0 ) + ε g t+1 (Institute) 08 May 2010 4 / 17

State Space Model State Equation µ t+1 δ 0 = δ 1 (µ t δ 0 ) + ε µ t+1 Measurement Equation g t+1 γ 0 = γ 1 (g t γ 0 ) + ε g t+1 d t+1 = γ 0 + ĝ t + ε d t+1 A = pd t = A B µ t + Bĝ t κ 1 ρ + γ 0 δ 0 1 ρ, B 1 = 1 1 ρδ 1, B 2 = 1 1 ργ 1. (Institute) 08 May 2010 4 / 17

General Form of Model State Equation ĝ t+1 = γ 1 ĝ t + ε g t+1 (4) (Institute) 08 May 2010 5 / 17

General Form of Model State Equation ĝ t+1 = γ 1 ĝ t + ε g t+1 (4) Measurement Equation d t+1 = γ 0 + ĝ t + ε d t+1 (5) pd t+1 = (1 δ 1 )A B 2 (γ 1 δ 1 )ĝ t + δ 1 pd t B 1 ε µ t+1 + B 2ε g t+1 (6) (Institute) 08 May 2010 5 / 17

General Form of Model State Equation ĝ t+1 = γ 1 ĝ t + ε g t+1 (4) Measurement Equation d t+1 = γ 0 + ĝ t + ε d t+1 (5) pd t+1 = (1 δ 1 )A B 2 (γ 1 δ 1 )ĝ t + δ 1 pd t B 1 ε µ t+1 + B 2ε g t+1 (6) Θ = (γ 0, δ 0, γ 1, δ 1, σ g, σ µ, σ D, ρ g µ, ρ gd, ρ µd ) Θ = I 2 0x R 3 +x l 3 c x R 2 I 2 0 ( 1, 1) I 3 c [ 1, 1] (Institute) 08 May 2010 5 / 17

Matrix Structure X t = FX t 1 + Rε t Kalman Equations Y t = M 0 + M 1 Y t 1 + M 2 X t η t = Y t M 0 M 1 Y t 1 M 2 X t t 1 S t = M 2 P t t 1 M 2 K t = P t t 1 M 2S t 1 X t t = FX t 1 t 1 + K t η t P t t = (I K t M 2 )(FP t 1 t 1 F + RΣR ) (Institute) 08 May 2010 6 / 17

Matrix Structure X t = FX t 1 + Rε t Kalman Equations Log Likelihood: Y t = M 0 + M 1 Y t 1 + M 2 X t η t = Y t M 0 M 1 Y t 1 M 2 X t t 1 S t = M 2 P t t 1 M 2 K t = P t t 1 M 2S t 1 X t t = FX t 1 t 1 + K t η t P t t = (I K t M 2 )(FP t 1 t 1 F + RΣR ) L = T t=1 log(det(s t )) T t=1 η t S t 1 η t (Institute) 08 May 2010 6 / 17

Estimation Results: The Estimate and S.E column reports the estimation results and the associated standard error from the net present value model given by equations 4,5 and 6.through optimization of the likelihood function using data between 1900 and 2008 on dividend growth (Institute) 08 May 2010 7 / 17 Optimization Results Parameter Coeffi cient Std error γ 0 0.0138 0.0108 δ 0 0.0524 0.016 γ 1 0.0717 0.1996 δ 1 0.9459 0.0401 σ g 0.0926 0.058 σ d 0.0139 0.0064 σ µ 0.0688 0.0772 ρ g µ 0.4802 0.4706 ρ µd -0.3815 1.3919 Table:

Predictive Regressions Univariate Regression r t = β 0 + β 1 µ t 1 + ε t (7) VAR r t = θ 0 + θ 1 PD t 1 + v t (8) Y t = [r t µ t 1 ] Y t = [r t PD t 1 ] Y t = C + p i=1 A i Y t i Measures of Predictive Accuracy - insample and out of sample. (Institute) 08 May 2010 8 / 17

Results of insample accuracy Periods β Std error t-ratio R-Squared 1 0.981 0.457 2.146 0.040 2 1.46 0.645 2.263 0.042 3 2.102 0.760 2.765 0.063 4 2.844 0.880 3.23 0.084 5 3.399 0.961 3.536 0.096 Periods β Std error t-ratio R-Squared 1-0.088 0.042-2.103 0.038 2-0.134 0.059-2.264 0.042 3-0.196 0.069-2.8 0.064 4-0.262 0.081-3.242 0.084 5-0.311 0.088-3.532 0.094 Table: Insample predictability with µ t 1. and pd t 1. (Institute) 08 May 2010 9 / 17

Out of Sample Forecast with lagged expected returns Horizon 1 Year 2 Year 3Year 4 Year 5 Year 1 0.0098 0.1791* 0.0472* 0.0012 0.00464 2 0.0222* 0.0657 0.0096 0.0027 0.03765 3 0.0006 0.0083 0.0007 0.0355* 0.09522* 4 0.00004** 0.0029 0.0037 0.0382 0.02861 5 0.0055 0.0052 0.0174 0.0038 0.07713 6 0.0010 0.00009** 0.0002** 0.0001** 0.00001** 7 0.0010 0.0071 0.0080 0.0043 0.00475 8 0.0010 0.02050 0.0221 0.0153 0.01522 Table: Out of Sample MSE. (µ t 1 )The table reflects the out of sample predictability over the period 2001-2008 for different horizon returns from 1 year to 5 years when returns are predicted by the filtered returns series.* represents the period where the mean squared error is highest ** represents the lowest mean squared error. (Institute) 08 May 2010 10 / 17

Out of sample forecast using price dividend ratio Horizon 1 Year 2 Year 3Year 4 Year 5 Year 1 0.05945 0.24030* 0.04674* 0.00144 0.00331 2 0.08510* 0.10404 0.00959 0.00221 0.03303 3 0.02627 0.02403 0.00071 0.03323 0.08657 4 0.01417 0.01265 0.00352 0.03557* 0.02383 5 0.00046** 0.00074** 0.01641 0.00521 0.08753* 6 0.00691 0.00405 0.00022** 0.00001** 0.00014** 7 0.00639 0.01874 0.00826 0.00518 0.00677 8 0.01998 0.03771 0.02250 0.01678 0.01858 Table: Out of Sample MSE. The table reflects the out of sample predictability over the period 2001-2008 when returns are predicted by pd t 1. (Institute) 08 May 2010 11 / 17

Vector Autoregression P =1 P =2 P =3 R t µ t 1 R t µ t 1 R t µ t 1 C 0.056 0.05959 0.0552 0.05994 0.04527 0.051 (0.021)* (0.002) (0)** (0.002) (0.018)* (0.036) R t 1 0.042 0.00027 0.07501 0.00401 0.075 0.003 (0.691) (0.977) (0.47) (0.432) (0.48) (0.442) µ t 1 0.594 0.92012-0.36116 0.942 0.513 1.079 (0.215) (0)** (0.706) (0)** (0.736) (0)** R t 2-0.23105-0.06505-0.226-0.0639 (0.026) (0)** (0.049) (0)** µ t 2 1.30222 0.03875 0.127-0.112 (0.203) (0.616) (0.952) (0.334) R t 3 0.109 0.0199 (0.488) (0.046) µ t 3 0.356 0.026 (0.786) (0.728) Adj R- Squared 0.0179 0.8398 0.0788 0.9355 0.0859 0.9503 Akaike 422.47 472.473 477.63 Schwartz 430.54 459.062 458.92 Table: VAR Results with µ t 1 variable as a predictor variable P refers to the number of lags in the VAR model. ** statistical significance at the 1 % level; * denotes significance at the 5 % level. The figures inside the brackets refer to the p-values (Institute) 08 May 2010 12 / 17

Forecasting VAR with PD Table: Results from VAR model with realized and expected dividend growth: Sample 1900-2008 P =1 P =2 P =3 R t PD t 1 R t PD t 1 R t PD t 1 C 0.0517 3.217 0.0497 3.23576 0.04848 3.239 (0.015)** (0)** (0)** (0)** (0.001)** (0)** R t 1 0.05274-0.276 0.0543-0.054 0.07822-0.0522 (0.617) (0.17) (0.609) (0.272) (0.464) (0.276) PD t 1 0.05966 0.551-0.0439 0.898-0.0785 1.1080 (0.039) (0.027) (0.331) (0) (0.649) (0) R t 2-0.219 0.7024-0.2337 0.70758 (0.034) (0) (0.036) (0) PD t 2 0.0418 0.0977-0.015-0.121 (0.105) (0) (0.928) (0.326) R t 3 0.1334-0.1499 (0.44) (0.25) PD t 3 0.00379 0.0008 (0.903) (0.975) Adj R-Squared 0.0299 0.4861 0.0764 0.9518 0.086 0.9546 Akaike 106.257 230.792 227.592 Schwartz 98.1831 217.381 208.882 Table: VAR Results with pd t 1 variable as a predictor variable P refers to the number of lags in the VAR model. ** statistical significance at the 1 % level; * denotes significance at the 5 % level. The figures inside the brackets refer to the p-values (Institute) 08 May 2010 13 / 17

Insample Forecast Accuracy Horizons and VAR order µ t 1 PD t 1 2 Years P = 1 0.220 0.234 P = 2 0.363 0.354 P = 3 0.420 0.443 3 Years P = 1 0.492 0.494 P = 2 0.601 0.599 P = 3 0.616 0.617 4 Year P = 1 0.551 0.548 P = 2 0.588 0.585 P = 3 0.602 0.606 5 Year P = 1 0.608 0.617 P =2 0.653 0.639 P = 3 0.653 0.651 Table: In Sample R- Squared. The left column illustrates the number of years of accumulated returns with the corresponding VAR order. The two other columns report the goodness of fit R-squared when µ t 1 and PD t 1 are used as predictors. (Institute) 08 May 2010 14 / 17

Out of sample forecast ability model :mu(t-1) Period 1 period Return 2 period Return 3 period Return 4 period Return 5 period Return Year P = 1 P = 2 P = 3 P = 1 P = 2 P = 3 P = 1 P = 2 P = 3 P = 1 P = 2 P = 3 P = 1 P = 2 P = 3 1 0.0396 0.0233 0.0210 0.1231 0.0587 0.0555 0.0093 0.0522 0.0500 0.0031** 0.0016 0.0051 0.0021** 0.0005** 0.0024 2 0.0851 0.0718 0.0618 0.0030 0.0162 0.0368 0.0246 0.0096 0.0094 0.0118 0.0100 0.0048 0.0142 0.0473 0.0584 3 0.0211 0.0328 0.0482 0.0630 0.0612 0.0637 0.0726 0.0837 0.0406 0.1139 0.1309 0.1258 0.0339 0.1041 0.1172 4 0.0011 0.0007** 0.0071** 0.0026 0.0005** 0.0075 0.0000** 0.0029 0.0009** 0.0085 0.0057** 0.0020 0.4810 0.2757 0.2598 5 0.0008** 0.0067 0.0156 0.0009 0.0664 0.0511 0.0009 0.0235 0.0051 0.3164 0.1974 0.1703 0.0637 0.0068 0.0077** 6 0.0034 0.0104 0.0181 0.0002** 0.0101 0.0013** 0.2478 0.1517 0.2221 0.0288 0.0024 0.0000** 0.0460 0.0156 0.0272 7 0.0015 0.0000 0.0002 0.3435 0.2139 0.2481 0.0678 0.0005** 0.0030 0.0687 0.0578 0.0246 0.0913 0.0348 0.0321 8 0.2975 0.2314 0.2128 0.1232 0.0193 0.0107 0.0427 0.0242 0.0233 0.0411 0.0299 0.0267 0.0540 0.0123 0.0103 Mean Squared Error for out of sample forecasts. The predictor variable is µ t 1. * illustrates the recursive mean squared error which is largest. ** illustrates the mean squared error which is lowest. P relates to the order of the VAR model. (Institute) 08 May 2010 15 / 17

Out of Sample Pd(t-1) Period 1 period Return 2 period Return 3 period Return 4 period Return 5 period Return Year P = 1 P = 2 P = 3 P = 1 P = 2 P = 3 P = 1 P = 2 P = 3 P = 1 P = 2 P = 3 P = 1 P = 2 P = 3 1 0.0264 0.0195 0.0189 0.0833 0.0506 0.0505 0.0120 0.0521 0.0456 0.0010** 0.0034 0.0060 0.0050** 0.0033 0.0041** 2 0.0629 0.0693 0.0583 0.0154 0.0136 0.0406 0.0297 0.0039 0.0087 0.0063 0.0121 0.0044 0.0635 0.0495 0.0631 3 0.0342 0.0283 0.0479 0.1016 0.0389 0.0464 0.0811 0.0442 0.0505 0.0953 0.1264 0.1359 0.1003 0.0792 0.1198 4 0.0045 0.0009 0.0051 0.0001** 0.0019 0.0012 0.0003** 0.0021 0.0012** 0.0142 0.0084 0.0003 0.3276 0.3513 0.2604 5 0.0000 0.0023 0.0109 0.0055 0.0272 0.0198 0.0004 0.0010** 0.0224 0.3386 0.2252 0.1489 0.0296 0.0022** 0.0049 6 0.0073 0.0108 0.0113 0.0038 0.0081** 0.0002** 0.2375 0.1778 0.1436 0.0357 0.0020** 0.0000** 0.0161 0.0168 0.0281 7 0.0001** 0.0000** 0.0001** 0.2897 0.2499 0.2709 0.0625 0.0057 0.0025 0.0784 0.0616 0.0185 0.0475 0.0526 0.0359 8 0.2711 0.2428 0.2236 0.0955 0.0326 0.0168 0.0391 0.0456 0.0112 0.0477 0.0328 0.0233 0.0249 0.0247 0.0105 Mean Squared Error for out of sample forecasts.in the case where the predictor is pd t 1.* illustrates the recursive mean squared error which is largest. ** illustrates the mean squared error which is lowest. P relates to the order of the VAR model. (Institute) 08 May 2010 16 / 17

Conclusion Evidence of Long Run Predictability for both Expected Returns and the Price Dividend Ratio (Institute) 08 May 2010 17 / 17

Conclusion Evidence of Long Run Predictability for both Expected Returns and the Price Dividend Ratio Price Dividend Ratio is marginally a better Predictor than Expected returns. Both Series are persistent. There may be a small amount of information present in Dividend Growth that makes the Price Dividend Ratio a marginally better predictor than expected returns. (Institute) 08 May 2010 17 / 17