III ENTROPY-BASED ASSET PRICING
III.4 Results and discussion
III.4.6 Short-term explanatory and predictive power
becomes visible, because we incorporate a valuable information to the estimation of expected risk premium. Based on the characteristics of risk-expected risk premium, we see a clear tendency to gain more return with a risky investment in upward trends and less return in downward trends. Compared to the full sample, the split of bullish and bearish periods result a good isolation of samples with different characteristics of asset pricing; therefore, we measure better explanatory powers in these samples. Based on that, we conclude that the identification of market regime has significant impact on asset pricing. However, we argue that the test results for the full sample give more relevant comparison opportunity, because a present moment the investor cannot decide whether there is an upward or a downward trend.
Table III.5 summarizes the explanatory power of the investigated risk measures for the various samples.
ˆ 1
P ,ˆP1, and ˆ 1
P show the results of the long-term analysis for the full period and during the upward and downward trends, respectively;ˆP i2 and ˆP o2 stand for the average accuracy measured for short term in- and out of sample, respectively; and R
ˆP i2
, and
ˆ 2
R P o
measure the relative standard deviation of the accuracy when applying the in-sample and out-of-sample test in short periods. In short term, entropy-based risk measures have significantly higher explanatory power (Shannon entropy: 0.1338; Rényi entropy: 0.1282) than standard deviation (0.0794) 20. We measure similar average explanatory power between CAPM beta (0.1331) and Shannon entropy. Standard deviation and entropy-based risk measures has statistically similar relative deviation (Standard deviation: 0.75, Shannon entropy: 0.69; Rényi entropy: 0.63), but significantly lower than CAPM beta (0.98). Detailed results for explanatory power in each 5-year period sample are summarized in Table III.6.
Table III.6. Explanatory power in short period samples
Train/Test Explanatory power
P2i H1 H2
1985 – 1989 0.032 0.094 0.051 0.037 1986 – 1990 0.017 0.036 0.026 0.036 1987 – 1991 0.041 0.048 0.060 0.079 1988 – 1992 0.055 0.050 0.067 0.068 1989 – 1993 0.035 0.042 0.089 0.072 1990 – 1994 0.096 0.071 0.234 0.201 1991 – 1995 0.160 0.136 0.281 0.219 1992 – 1996 0.163 0.178 0.244 0.205 1993 – 1997 0.075 0.249 0.154 0.136 1994 – 1998 0.070 0.301 0.158 0.126 1995 – 1999 0.166 0.512 0.283 0.273 1996 – 2000 0.086 0.282 0.180 0.208 1997 – 2001 0.021 0.153 0.072 0.097 1998 – 2002 0.002 0.025 0.015 0.024 1999 – 2003 0.061 0.075 0.081 0.098 2000 – 2004 0.019 0.001 0.015 0.014 2001 – 2005 0.151 0.056 0.175 0.184 2002 – 2006 0.179 0.089 0.223 0.231 Average 0.0794 0.1331 0.1337 0.1282
Rel. dev 0.75 0.98 0.69 0.63
20 The result of t-tests and F-tests in one-week-shifting samples are presented in Appendix A11 and Appendix A12.
Notes: This table summarizes the explanatory power of the investigated risk measures for expected risk premium in the first 5 years of 18 10-year periods (P2i) shifting by one year from period (1985-1994) to period (2002-2011).
We estimate and evaluate risk measures of 150 randomly selected securities from the S&P 500 index using standard deviation ( ), CAPM beta ( ), Shannon entropy (H1) and Rényi entropy (H2) risk estimation methods by daily risk premiums. Both types of entropy functions are calculated by histogram-based density function estimation, applying 175 bins for Shannon entropy and 50 bins for Rényi entropy.
In order to show the change of explanatory power over the time, we estimate and evaluate the explanatory power of investigated risk measures in “one-week-shifting” 10-year samples as well. In Figure III.7, a date point shows the end date of the 5-year training period (e.g. in point 1990, we see the explanatory power of risk measures evaluated in period (1985-1990)). Previously, we have shown that CAPM beta has good short-term explanatory power.
Based on Figure III.7, we think that one reason for that is CAPM beta shows outstanding explanatory power of the risk premiums before the dot-com collapse. We also see that Shannon- and Rényi entropy behaves quite similarly and the explanatory power of standard deviation is usually less or the same as the entropy. Overall, we find that all of the models are prominently better in the 90s than in the 2000s.
Figure III.7. Explanatory power of risk measures by week (5-year models)
Notes: This figure shows how the explanatory power of the investigated risk measures for expected risk premium changes over the time. We generate 887 10-year periods shifting by one week from period (1985-1994) to period (2002-2011). We estimate and evaluate risk measures of 150 randomly selected securities from the S&P 500 index using standard deviation (dark purple dashed curve) CAPM beta (light blue dashed curve) Shannon entropy (black curve) and Rényi entropy (grey curve) risk estimation methods by daily risk premiums on the first 5 years in each period. Both types of entropy functions are calculated by histogram-based density function estimation, applying 175 bins for Shannon entropy and 50 bins for Rényi entropy. Date shows the end of the training period, in the other words, the middle of each 10-year period.
We measure different results for predictive power 21. We find that predictive power of entropy-based risk measures (Shannon: 0.1015; Rényi: 0.0934) is significantly higher than CAPM beta (0.0645), furthermore, the Shannon entropy is also significantly better than standard deviation (0.0970). The relative standard deviation of predictive power of CAPM beta (1.02) is significantly higher, while the other risk measures has no statistical significance between their relative deviation (St. dev: 0.65, Shannon: 0.64; Rényi: 0.62). Although CAPM captures the systematic risk, entropy-based risk measures shows ~50% higher predictive power, which is a surprising result. Standard deviation-based risk measure has higher predictive power (0.0970) and lower relative standard deviation in 5-year samples (0.65) than CAPM beta.
Detailed results for predictive power in 10-year period samples are summarized in Table III.7.
Table III.7. Predictive power in short periods out of sample
Train Test Predictive power
P2i P2o H1 H2
1985 – 1989 1990 – 1994 0.073 0.028 0.130 0.100 1986 – 1990 1991 – 1995 0.170 0.041 0.193 0.181 1987 – 1991 1992 – 1996 0.215 0.059 0.226 0.175 1988 – 1992 1993 – 1997 0.098 0.079 0.146 0.132 1989 – 1993 1994 – 1998 0.079 0.165 0.135 0.116 1990 – 1994 1995 – 1999 0.100 0.239 0.166 0.149 1991 – 1995 1996 – 2000 0.090 0.141 0.091 0.090 1992 – 1996 1997 – 2001 0.113 0.147 0.117 0.118 1993 – 1997 1998 – 2002 0.142 0.047 0.127 0.108 1994 – 1998 1999 – 2003 0.247 0.027 0.175 0.198 1995 – 1999 2000 – 2004 0.036 0.068 0.003 0.005 1996 – 2000 2001 – 2005 0.080 0.000 0.036 0.030 1997 – 2001 2002 – 2006 0.103 0.004 0.061 0.045 1998 – 2002 2003 – 2007 0.078 0.032 0.064 0.058 1999 – 2003 2004 – 2008 0.015 0.032 0.019 0.021 2000 – 2004 2005 – 2009 0.047 0.012 0.050 0.051 2001 – 2005 2006 – 2010 0.022 0.019 0.032 0.040 2002 – 2006 2007 – 2011 0.040 0.023 0.054 0.065
Average 0.0970 0.0645 0.1014 0.0934
Relative deviation 0.65 1.02 0.64 0.62
Notes: This table summarizes the predictive power of the investigated risk measures for expected risk premium in the last 5 years of 18 10-year periods shifting by one year from period (1985-1994) to period (2002-2011). We estimate risk measures of 150 randomly selected securities from the S&P 500 index using standard deviation ( ), CAPM beta ( ), Shannon entropy (H1) and Rényi entropy (H2) risk estimation methods by daily risk premiums
21 The result of t-tests and F-tests in one-week-shifting samples are presented in Appendix A11 and Appendix A12.
in the first 5 years (P2i) and measure the predictive power on the next 5 years (P2o) by estimating the goodness of fit of linear regression (R2). Both types of entropy functions are calculated by histogram-based density function estimation, applying 175 bins for Shannon entropy and 50 bins for Rényi entropy.
To illustrate the change of predictive power by time, we estimate and evaluate it on
“one-week-shifting” 10-year samples and plot the results in Figure III.8. A date point in the figure shows the end date of the 5-year training period and thus the start date of the 5-year testing period (e.g. in point 1990, we see the risk estimated in period (1985-1990) and measured in period (1990-1995). Based on the figure, we see higher variance of the predictive power in time. The entropy based risk measures and standard deviation shows similar movement;
however, we have shown that the average predictive power of Shannon entropy is significantly better than standard deviation. We do not find any significant evidence that explains the phenomenon that standard deviation has higher predictive power than explanatory power;
however, standard deviation has high explanatory power before period of the dot-com collapse.
We show that the predictive power of the risk measures are better in the 90s than in the 2000s.
Figure III.8. Predictive power of risk measures by week (5-year models)
Notes: This figure shows how the predictive power of the investigated risk measures for expected risk premium changes over the time. We generate 887 10-year periods shifting by one week from period (1985-1994) to period (2002-2011). We estimate risk measures of 150 randomly selected securities from the S&P 500 index using standard deviation (dark purple dashed curve) CAPM beta (light blue dashed curve) Shannon entropy (black curve) and Rényi entropy (grey curve) risk estimation methods by daily risk premiums on the first 5 years and evaluate on the following 5 years in each period. Both types of entropy functions are calculated by histogram-based density function estimation, applying 175 bins for Shannon entropy and 50 bins for Rényi entropy. Date shows the end of the training period, in the other words, the middle of each 10-year period.
Based on these findings we can make the following statements. As for the standard metrics, while CAPM beta is more accurate on explaining returns for short term in sample, standard
deviation is better for predicting them. Although Shannon entropy has not significantly better explanatory power in short term than CAPM beta, the beta is significantly worse for predicting risk premiums than Shannon entropy; therefore, Shannon entropy-based risk measure offers the highest combined accuracy for both explaining and predicting returns. We find that standard deviation, Shannon- and Rényi entropy have statistically the same reliability (relative standard deviation or variance), CAPM beta has the highest relative standard deviation for both explaining and predicting risk premiums in short term.