Estimating Risk-Return Relations with Price Targets Liuren Wu Baruch College March 29, 2016 Liuren Wu (Baruch) Equity risk premium March 29, 2916 1 / 13
Overview Asset pricing theories generate implications on the relation between risks and expected returns. Empirical tests often replace return expectation with ex post realization. Unfortunately, what you want is often quite different from what you get, resulting in disillusion (about the theory). One hopes that long-run averages of ex post realization become accurate enough estimates of long-run averages of expectation: Cross-sectional tests tend to perform better than time-series tests. Intertemporal tests often lead to puzzling or even negative findings: stocks, currencies, rates,... Common interpretations: Time-varying risk premium behaving weirdly... Reality: Short-term return realization tells very little about expectation. Bottom line: To test theoretical risk-return relations (especially intertemporally), one must obtain better, timely estimates on return expectations. Liuren Wu (Baruch) Equity risk premium March 29, 2916 2 / 13
Inferring expectations from market prices/surveys From market prices Yield to maturity for bonds, credit spreads (conditional on no default) Implied cost of equity from stock prices (with assumptions on cash flows/growth rates) From surveys/forecasts Expected currency returns from exchange rate forecasts Expected bond returns from interest rate forecasts Expected stock returns from analysts stock price targets From both forecasts and market prices Expected cost of debt from bond prices and default probability forecasts Implied cost of equity from stock prices and analysts forecasts on earnings/cashflows/growth rates. Common feature: All ex ante expectations instead of ex post realizations. You can nitpick on each one, but conclusions from any of these should be similar to each other, and could go a long way to resolve all those risk premium puzzles... Liuren Wu (Baruch) Equity risk premium March 29, 2916 3 / 13
Data and methodology Data: Sample period: 2003-2014 Universe: S&P 1500 (including 500 large, 400 medium, and 600 small), 90% of US market cap. Data source: OptionMetrics, Bloomberg, Capital IQ, French data library (on risk factors). Filtering: Data match, quarterly average dollar trading volume (> 100K), stock price level over the past year (> 5),... Construction: One-year expected stock return = log deviation between consensus price target and current stock price Ex post realization return Ex ante equity risk premium and ex post excess return are computed over libor rate Liuren Wu (Baruch) Equity risk premium March 29, 2916 4 / 13
Aggregate equity risk premium behavior A. Equal weighting B. Value weighting Expectation Realization Expectation Realization Horizon 12 12 3 1 12 12 3 1 Mean 0.103 0.112 0.124 0.123 0.101 0.084 0.088 0.090 Median 0.090 0.136 0.163 0.201 0.088 0.104 0.127 0.158 Std Dev 0.066 0.214 0.193 0.194 0.053 0.177 0.165 0.158 Skewness 1.466-0.405-0.797-0.966 1.425-1.014-1.090-1.228 Kurtosis 3.048 1.330 4.536 5.690 2.705 1.750 3.945 5.759 Minimum -0.007-0.528-0.480-0.364 0.014-0.488-0.427-0.310 Maximum 0.478 1.008 0.535 0.304 0.390 0.684 0.376 0.232 Correlation 0.233 0.075 0.053 0.254 0.055 0.050 Expectation largely matchs realization over the long run (12-year sample). Realization has 3-4 times larger standard deviation than expectation. Expectation is mostly positive, but realization can be hugely negative. Expectation predicts realization to some extent... Liuren Wu (Baruch) Equity risk premium March 29, 2916 5 / 13
Aggregate equity risk premium behavior Equal weighted equity risk premium, % 1.2 1 0.8 0.6 0.4 0.2 0 0.2 0.4 A. Equal weighting B. Value weighting Ex ante expectation Ex post realization Value weighted equity risk premium, % 0.8 0.6 0.4 0.2 0 0.2 0.4 Ex ante expectation Ex post realization 0.6 03 04 05 06 07 08 09 10 11 12 13 14 15 0.6 03 04 05 06 07 08 09 10 11 12 13 14 15 Many times, the surprises overwhelm the expectation in the realization... Tests with ex post returns can be more sample-period dependent... Liuren Wu (Baruch) Equity risk premium March 29, 2916 6 / 13
Estimating the intertemporal risk-return relation on the aggregate market using ex ante equity risk premium ERP t = a + bv t + e t Historical variance Option implied variance Horizon, Month 1 3 12 1 3 12 Slope 0.434 0.449 0.344 0.808 1.037 1.438 ( 6.66 ) ( 5.79 ) ( 2.26 ) ( 8.46 ) ( 8.46 ) ( 8.22 ) R 2 0.393 0.336 0.098 0.565 0.567 0.533 The slope is positive, and highly statistically significant. The R 2 is high. Using expected variance (from options) generate stronger results than using historical variance estimators. When using historical variance, using a shorter window generates more variation and stronger identification. Bottom line: Intertemporal risk-return relation is strong and positive as predicted by classic theory. Liuren Wu (Baruch) Equity risk premium March 29, 2916 7 / 13
Estimating the intertemporal risk-return relation on the aggregate market using ex post realized excess return for comparison ER t+1 = a + bv t + e t+1 Historical variance Option implied variance Horizon, Month 1 3 12 1 3 12 Ex post one-year excess return Slope 0.421 0.661 1.226 0.861 1.224 2.026 ( 1.02 ) ( 1.43 ) ( 1.86 ) ( 1.34 ) ( 1.48 ) ( 1.74 ) R 2 0.037 0.071 0.118 0.062 0.076 0.101 Ex post one-month excess return Slope -0.064-0.062 0.096-0.021-0.008 0.039 ( -1.43 ) ( -1.17 ) ( 1.21 ) ( -0.29 ) ( -0.08 ) ( 0.28 ) R 2 0.012 0.009 0.010 0.001 0.000 0.001 Slope is positive (but insignificant) when using longer-term averages, but become negative when using short-term averages. R 2 is close to zero, by nature. Bottom line: The negative finding does not reject the theory, but rejects the methodology. Liuren Wu (Baruch) Equity risk premium March 29, 2916 8 / 13
Estimating the cross-sectional risk-return relation ERP i,t = a + γσ it + e t, ERP i,t = a + γβ it + e t Regressor σ im,t β i,t β i,t σ im,t β i,t β i,t σ im,t β i,t β i,t Window 1 3 12 Panel A: Ex ante expected equity risk premium Slope 1.263 0.029 0.038 2.270 0.047 0.057 3.762 0.074 0.086 Average t ( 4.88 ) ( 4.88 ) ( 6.05 ) ( 6.49 ) ( 6.49 ) ( 7.57 ) ( 8.87 ) ( 8.87 ) ( 9.91 ) NW t ( 4.94 ) ( 4.57 ) ( 5.47 ) ( 5.66 ) ( 5.44 ) ( 6.28 ) ( 6.61 ) ( 6.98 ) ( 7.86 ) R 2 0.034 0.034 0.042 0.047 0.047 0.055 0.067 0.067 0.076 Slope is positive and highly statistically significant Relative risk aversion: 1.26-3.76. Average risk premium 3.8-8.6% Bottom line: Classic asset pricing theory generates the right implications on both intertemporal and cross-sectional relations. Liuren Wu (Baruch) Equity risk premium March 29, 2916 9 / 13
Estimating the cross-sectional risk-return relation with ex post excess returns: ER i,t+1 = a + γσ it + e t+1, ER i,t+1 = a + γβ it + e t+1 Regressor σ im,t β i,t β i,t σ im,t β i,t β i,t σ im,t β i,t Window 1 3 12 Panel B: Ex post one-year excess return Slope -0.329 0.016 0.025-0.548 0.018 0.028-0.346 0.030 Average t ( -0.41 ) ( -0.41 ) ( -0.09 ) ( -0.60 ) ( -0.60 ) ( -0.23 ) ( -0.42 ) ( -0.42 ) ( -0 NW t ( -0.73 ) ( 0.85 ) ( 1.12 ) ( -0.67 ) ( 0.67 ) ( 0.92 ) ( -0.39 ) ( 0.80 ) ( 0 R 2 0.011 0.011 0.013 0.017 0.017 0.020 0.018 0.018 Panel D: Ex post one-month excess return Slope -0.885 0.016 0.022-1.864 0.016 0.022-1.462 0.016 Average t ( -0.13 ) ( -0.13 ) ( -0.04 ) ( -0.18 ) ( -0.18 ) ( -0.08 ) ( -0.32 ) ( -0.32 ) ( -0 NW t ( -1.09 ) ( 0.72 ) ( 0.80 ) ( -1.33 ) ( 0.46 ) ( 0.54 ) ( -0.86 ) ( 0.31 ) ( 0 R 2 0.018 0.018 0.022 0.027 0.027 0.031 0.031 0.031 Negative/insignificant Attribute the success to the test, attribute the failure to the theory. Liuren Wu (Baruch) Equity risk premium March 29, 2916 10 / 13
Different beta estimators Now that we know the relation is strong using equity risk premium, we can explore further the role of different beta estimators Regressor HB HSB HTB OIB ISB ITB Slope 0.074 0.086 0.044 0.118 0.135 0.074 Average t ( 8.87 ) ( 9.91 ) ( 11.18 ) ( 11.88 ) ( 13.06 ) ( 14.97 ) NW t ( 6.98 ) ( 7.86 ) ( 8.21 ) ( 11.47 ) ( 12.56 ) ( 12.00 ) Average R 2 0.067 0.076 0.091 0.103 0.116 0.140 Implied variance can be a better measure of risk than historical variance. Historical correlation estimates are noisy,... or all risks are priced SB correlation averaged within sector TB total beta Liuren Wu (Baruch) Equity risk premium March 29, 2916 11 / 13
Risk premium on other risk factors Constant Market Size BM Momentum R 2 Panel A. Fama-French Three-Factor model Slope 0.031 0.064 0.034-0.008 0.097 Average t ( 2.55 ) ( 5.93 ) ( 6.53 ) ( -1.65 ) NW t ( 3.28 ) ( 5.40 ) ( 6.30 ) ( -2.35 ) ( 0.086 ) Panel A. Fama-French Three-Factor + Momemtum Slope 0.034 0.062 0.034-0.010 0.019 0.107 Average t ( 2.82 ) ( 5.45 ) ( 6.36 ) ( -1.98 ) ( 3.01 ) NW t ( 3.86 ) ( 5.20 ) ( 6.56 ) ( -2.79 ) ( 3.07 ) ( 0.087 ) Market beta is strongly significant and generates the largest premium ( 6.2-6.4%). Size is also highly significantly and leads to about 3.4% premium. Book-to-market effect is negative/small. Momentum is significant, with 1.9% risk premium contribution. Bottom line: These other risk factors may be useful, but they do not replace the role of market beta. Liuren Wu (Baruch) Equity risk premium March 29, 2916 12 / 13
Concluding remarks Ex ante expectation and ex post realization can be quite different. Using ex post realization to test theories on ex ante expectation can lead to weak/negative/puzzling results. One can learn market expectation from market prices and/or surveys/forecasts A lot can be done here, across different asset classes, to address all types of risk premium puzzles identified from ex post return realizations. Somewhat more indirectly, one can also infer expectations from actions: Attention: google search Positions: 13F filings Liuren Wu (Baruch) Equity risk premium March 29, 2916 13 / 13