Table 3. Predicting power in short periods out of sample.
| P2i | P2o | 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 | 1990–1994 | 7.3% | 2.8% | 13.0% | 10.0% | 35.2 | 80.4 | 15.4 | 52.0 | ***/*** | ***/*** |
| 1986–1990 | 1991–1995 | 17.0% | 4.1% | 19.3% | 18.1% | 12.8 | 101.4 | 5.6 | 85.8 | ***/*** | ***/*** |
| 1987–1991 | 1992–1996 | 21.5% | 5.9% | 22.6% | 17.5% | 7.1 | 99.8 | −23.0 | 71.8 | ***/*** | /*** |
| 1988–1992 | 1993–1997 | 9.8% | 7.9% | 14.6% | 13.2% | 34.9 | 43.9 | 27.8 | 38.1 | ***/*** | ***/*** |
| 1989–1993 | 1994–1998 | 7.9% | 16.5% | 13.5% | 11.6% | 65.8 | −21.3 | 45.5 | −35.8 | ***/ | ***/ |
| 1990–1994 | 1995–1999 | 10.0% | 23.9% | 16.6% | 14.9% | 63.2 | −57.1 | 48.0 | −70.1 | ***/ | ***/ |
| 1991–1995 | 1996–2000 | 9.0% | 14.1% | 9.1% | 9.0% | 0.1 | −44.0 | −0.7 | −45.0 | / | / |
| 1992–1996 | 1997–2001 | 11.3% | 14.7% | 11.7% | 11.8% | 1.7 | −20.8 | 2.0 | −21.0 | */ | **/ |
| 1993–1997 | 1998–2002 | 14.2% | 4.7% | 12.7% | 10.8% | −11.0 | 66.8 | −26.3 | 54.4 | /*** | /*** |
| 1994–1998 | 1999–2003 | 24.7% | 2.7% | 17.5% | 19.8% | −54.6 | 154.7 | −36.3 | 173.0 | /*** | /*** |
| 1995–1999 | 2000–2004 | 3.6% | 6.8% | 0.3% | 0.5% | −59.0 | −90.3 | −55.0 | −87.1 | / | / |
| 1996–2000 | 2001–2005 | 8.0 | 0.0% | 3.6% | 3.0% | −47.1 | 67.9 | −54.5 | 62.7 | /*** | /*** |
| 1997–2001 | 2002–2006 | 10.3% | 0.4% | 6.1% | 4.5% | −38.9 | 91.9 | −56.1 | 78.8 | /*** | /*** |
| 1998–2002 | 2003–2007 | 7.8% | 3.2% | 6.4% | 5.8% | −13.4 | 40.8 | −19.7 | 35.0 | /*** | /*** |
| 1999–2003 | 2004–2008 | 1.5% | 3.2% | 1.9% | 2.1% | 5.5 | −18.5 | 7.7 | −16.3 | ***/ | ***/ |
| 2000–2004 | 2005–2009 | 4.7% | 1.2% | 5.0% | 5.1% | 4.8 | 62.9 | 5.9 | 63.1 | ***/*** | ***/*** |
| 2001–2005 | 2006–2010 | 2.2% | 1.9% | 3.2% | 4.0% | 17.8 | 20.9 | 28.9 | 31.5 | ***/*** | ***/*** |
| 2002–2006 | 2007–2011 | 4.0% | 2.3% | 5.4% | 6.5% | 21.5 | 50.1 | 35.2 | 62.2 | ***/*** | ***/*** |
| Average | 9.70% | 6.45% | 10.14% | 9.34% | |||||||
| Relative deviation | 0.65 | 1.02 | 0.64 | 0.62 | |||||||
Note: This table summarizes the predicting 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&P500 index using standard deviation (σ), CAPM beta (β), Shannon entropy (H1) and Rényi entropy (H2) risk estimation methods by daily risk premiums in the first 5 years (P2i) and measure the predicting power on the next 5 years (P2o) by estimating the goodness of fit of linear regression (R 2). 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.