Multivariate risk models

As an outlook, in this subsection, we investigate the explanatory- and predictive power of multivariate risk models. For multivariate models, we use the extended methodology that has been introduced in Subsection III.3.8. First, we estimate and measure the accuracy of coefficients of Fama-French three-factor model, Carhart four-factor model and a two factor model that uses market returns and MOM factor (Beta-MOM). Second, we investigate whether the 3^rd and 4^th moments can be used to improve the accuracy of standard deviation and entropy-based risk measures. Third, we construct multi-factor models focusing on incorporating entropy as additional explanatory variable. Finally, we investigate how the explanatory power of multi-factor models change by diversification. Table III.9 summarizes the result of the analysis in three panels. Panel A collects CAPM and its extensions, Panel B shows moment-based risk models and Panel C summarizes additional custom multivariate models.

Table III.9. Comparison of accuracy of single- and multivariate risk models Panel A – CAPM and its extensions

Model # ˆ ₁

P ˆ 1

P ˆ 1

P ˆ_P2i ˆ_P2o Ri _Ro

CAPM beta 1 0.0617 0.3667 0.4369 0.1331 0.0645 0.98 1.02

Beta-MOM 2 0.3581 0.5115 0.4656 0.2604 0.1038 0.51 0.59

Fama-French 3 0.3023 0.5048 0.4463 0.2403 0.1526 0.46 0.42

Carhart 4 0.3801 0.5323 0.4705 0.3272 0.1689 0.30 0.37

Panel B – Moment-based risk models

Model # ˆ ₁

P ˆ_P1 ˆ_P1 ˆ_P2i ˆ_P2o Ri _Ro

St.deviation (Std) 1 0.0783 0.3390 0.3671 0.0794 0.0970 0.75 0.65 Variance (2^nd) 1 0.0764 0.3048 0.3305 0.0762 0.0927 0.72 0.68 Skewness (3^rd) 1 0.0079 0.0135 0.0004 0.0513 0.0249 0.91 1.14 Kurtosis (4^th) 1 0.0340 0.0805 0.0712 0.0401 0.0591 0.98 0.89

St. dev, 3^rd 2 0.1354 0.3533 0.3882 0.1110 0.1063 0.51 0.59

St. dev, 4^th 2 0.0786 0.3431 0.3768 0.1204 0.1084 0.53 0.59

St. dev, 3^rd, 4^th 3 0.1607 0.3543 0.3892 0.1499 0.1163 0.52 0.54 Shannon entropy (Sh) 1 0.1298 0.4345 0.3961 0.1338 0.1015 0.69 0.64 Rényi entropy (Ré) 1 0.1571 0.4236 0.3855 0.1282 0.0934 0.62 0.60 Shannon, 3^rd, 4^th 3 0.2016 0.4383 0.4153 0.1999 0.1288 0.49 0.52 Rényi, 3^rd, 4^th 3 0.2188 0.4244 0.4053 0.1865 0.1223 0.48 0.48

Panel C – Combined risk models

Model # ˆ ₁

P ˆ_P1 ˆ_P1 ˆ_P2i ˆ_P2o Ri _Ro

Fama-French (FF) 3 0.3023 0.5048 0.4463 0.2403 0.1526 0.46 0.42 Beta-MOM, Shannon 3 0.3987 0.5866 0.4843 0.3190 0.1635 0.37 0.41

FF, Shannon 4 0.3416 0.5599 0.4668 0.2549 0.1847 0.43 0.41

FF, Rényi 4 0.3600 0.5583 0.4652 0.2513 0.1765 0.43 0.41

Carhart (Ch) 4 0.3801 0.5323 0.4705 0.3272 0.1689 0.30 0.37

Carhart, Shannon 5 0.4133 0.5962 0.4848 0.3561 0.1978 0.28 0.36 Carhart, Rényi 5 0.4209 0.5861 0.4817 0.3510 0.1898 0.27 0.35 Ch, Sh, 3^rd, 4^th 7 0.4697 0.6333 0.4961 0.4052 0.2222 0.25 0.34 Ch, Ré, 3^rd, 4^th 7 0.4702 0.6242 0.4956 0.4012 0.2187 0.25 0.35 Notes: The table summarizes the explanatory power (in-sample R²) of various univariate and multivariate risk models in various samples (Panel A: CAPM and its extensions, Panel B: Moment-based models, Panel C:

Additional combined risk models). We estimate risk measures of 150 random securities using standard deviation, CAPM beta, Shannon- and Rényi entropy risk estimation methods for (1) long term, from 1985 to the end of 2011 (1985-2011); (2) long term on upward trends (bull market), (3) long term on downward trends (bear market), (4) 18 10-year periods shifting by one year from period (1985-1994) to period (2002-2011), split into two 5-5 year periods for each. The 2^nd column show the number of variables of the model. The 3^rd column shows the explanatory power of risk measures for long term. The 4^th and 5^th column are the explanatory power on upward and downward trends, respectively. The 6^th columnstands for the average explanatory power of risk measured in the first 5 years of 10-year shorter periods in sample. The 7^th stands for the average predictive power of risk measures (out-of-sample R²) calculated by estimating risk in the first 5 years and evaluating them on the other 5 years in each 10-year periods. The last two columns show the relative standard deviation of explanatory and predictive power of short-term samples.

Based on Panel A, we clearly see that the additional factors to CAPM significantly improve both explanatory- and predictive power; furthermore, we also find significant decrease in relative standard deviation in short-term samples. We find that the improvement of explanatory power in bear market is less significant. An interesting result is that the two-factor model (Beta and MOM) has better explanatory power than Fama-French three-factor model (Beta, MOM: 0.3581, 0.5115, 0.4656; Fama-French: 0.3023, 0.5048, 0.4463); however, the latter has significantly better predictive power (Beta, MOM: 0.1038; Fama-French: 0.1526).

This result allows us to conclude that MOM factor is better for explaining risk premium, SMB and HML factors are better for predicting that. Based on that, it is not surprising that Carhart four-factor model is the most accurate.

In Panel B, we collect the results of combination of moment-based risk measures.

Comparing standard deviation and variance (2^nd moment), we see that standard deviation has better linear explanatory- and predictive power than the latter one. The skewness (3^rd moment) and kurtosis (4^th moment) have significantly worse accuracy on its own than variance or standard deviation. We show that combining 3^rd and 4^th moment to standard deviation, Shannon- and Rényi entropy the accuracy increases significantly compared to the single risk

measures. This result allows us to conclude that higher moments has also impact in asset pricing. Although Rényi entropy may penalize higher moments based on the square on the probability density function, it still gains improvement in accuracy by combining higher moments. However, we measure higher gain of long-term explanatory power for Shannon entropy than for Rényi entropy by extending them with higher moments (Shannon: from 0.1289 to 0.2016, +56%; Rényi: from 0.1571 to 0.2188, +39%); therefore, we think that Rényi entropy captures more information about higher moments than Shannon entropy. One can see that the improvement is less significant if the market trend is identified.

Based on Panel C, we find that the three-variate combination of Beta-MOM and Shannon entropy outperforms the Fama-French model and almost the Carhart model in all samples. We find significant improvement in accuracy by extending Fama-French- and Carhart models by entropy-based risk measures. We measure the highest improve on predictive power and the lowest on bear market. We show that the relation between Shannon- and Rényi holds after combining them with the multi-factor model models, more specifically, while Rényi entropy has better long-term explanatory power, Shannon entropy is better in other samples.

Based on these results, entropy-based risk measures contain significant explanatory and predictive power allowing them to extend conventional asset pricing models. Using 3^rd and 4^th moments to extend these five-factor models, we construct seven-factor models that overperforms any other models. In Appendix A13, we summarize the empirical evaluation of several additional risk models.

We investigate how certain multivariate models behave if we increase the number of securities involved in portfolio. Figure III.9 measures the long-term explanatory power of coefficients of Fama-French three-factor model, Carhart four-factor model, their extensions with Shannon and Rényi entropy. We see that the explanatory power of Fama-French and Carhart model remains statistically the same, confirming that these models captures the systematic risk only. Comparing Fama-French and Carhart models with their extension by Shannon- and Rényi entropy, we find that performance gain reduces by the number of securities involved. This result allows us to conclude that idiosyncratic risk provides additional explanatory power for multi-factor models for less-diversified portfolios.

Figure III.9. Explanatory power of multivariate risk measures by diversification

Notes: This figure shows the explanatory power (R²) of the investigated risk measures for portfolios with various number of securities involved. We generate 10 million random equally weighted portfolios with various number of securities involved (at most 100,000 for each size) using daily risk premiums of 150 randomly selected securities. The risk of portfolios is estimated by Fama-French model (grey curve), Carhart model (black curve), Fama-French model with Shannon/Rényi entropy (solid/dashed purple curve), Carhart model with Shannon/Rényi entropy (solid/dashed light blue curve). 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.