Table 2. Explanatory power in short period samples.
| P2i | σ | β | H 1 | H 2 | T-test H 1,σ | T-test H 1,β | T-test H 2,σ | T-test H 2,β | sig H 1 (σ/β) | sig H 2 (σ/β) |
| 1985–1989 | 3.2% | 9.4% | 5.1% | 3.7% | 19.3 | −32.8 | 6.7 | −46.0 | ***/ | ***/ |
| 1986–1990 | 1.7% | 3.6% | 2.6% | 3.6% | 11.8 | −12.8 | 22.7 | 0.4 | ***/ | ***/ |
| 1987–1991 | 4.1% | 4.8% | 6.0% | 7.9% | 13.9 | 7.6 | 27.8 | 21.8 | ***/*** | ***/*** |
| 1988–1992 | 5.5% | 5.0% | 6.7% | 6.8% | 7.7 | 8.5 | 8.1 | 8.8 | ***/*** | ***/*** |
| 1989–1993 | 3.5% | 4.2% | 8.9% | 7.2% | 40.6 | 32.9 | 28.9 | 21.4 | ***/*** | ***/*** |
| 1990–1994 | 9.6% | 7.1% | 23.4% | 20.1% | 66.1 | 75.8 | 48.2 | 57.7 | ***/*** | ***/*** |
| 1991–1995 | 16.0% | 13.6% | 28.1% | 21.9% | 67.7 | 56.3 | 32.9 | 30.8 | ***/*** | ***/*** |
| 1992–1996 | 16.3% | 17.8% | 24.4% | 20.5% | 41.8 | 29.1 | 24.4 | 12.4 | ***/*** | ***/*** |
| 1993–1997 | 7.5% | 24.9% | 15.4% | 13.6% | 38.1 | −40.0 | 32.7 | −55.2 | ***/ | ***/ |
| 1994–1998 | 7.0% | 30.1% | 15.8% | 12.6% | 86.1 | −101.1 | 55.6 | −124.3 | ***/ | ***/ |
| 1995–1999 | 16.6% | 51.2% | 28.3% | 27.3% | 88.3 | −166.4 | 77.8 | −167.8 | ***/ | ***/ |
| 1996–2000 | 8.6% | 28.2% | 18.0% | 20.8% | 63.7 | −67.7 | 85.0 | −49.7 | ***/ | ***/ |
| 1997–2001 | 2.1% | 15.3% | 7.2% | 9.7% | 39.3 | −60.0 | 58.2 | −41.7 | ***/ | ***/ |
| 1998–2002 | 0.2% | 2.5% | 1.5% | 2.4% | 28.2 | −16.0 | 41.8 | −0.3 | ***/ | ***/ |
| 1999–2003 | 6.1% | 7.5% | 8.1% | 9.8% | 16.9 | 6.2 | 29.2 | 18.8 | ***/*** | ***/*** |
| 2000–2004 | 1.9% | 0.1% | 1.5% | 1.4% | −8.0 | 35.5 | −9.2 | 34.8 | /*** | /*** |
| 2001–2005 | 15.1% | 5.6% | 17.5% | 18.4% | 16.0 | 93.5 | 22.4 | 102.2 | ***/*** | ***/*** |
| 2002–2006 | 17.9% | 8.9% | 22.3% | 23.1% | 24.6 | 89.6 | 30.2 | 98.8 | ***/*** | ***/*** |
| Average | 7.94% | 13.31% | 13.37% | 12.82% | ||||||
| Rel. dev | 0.75 | 0.98 | 0.69 | 0.63 |
Note: This table summarizes the explanatory power of the different 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&P500 index using standard deviation (σ), CAPM beta (β), 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, with 175 bins for Shannon entropy and 50 bins for Rényi entropy. We apply t-statistics by bootstrapping method to measure whether differences in R 2s are significant. We use *s to designate that the entropy based risk measure is significantly higher than the standard deviation and CAPM beta;
***, ** and * stands for 1%, 5% and 10% significance level respectively.