# Risk, Return and Volatility Feedback: A Bayesian Nonparametric Analysis

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## Abstract

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## 1. Introduction

## 2. Return and Realized Variance Data

## 3. Risk Return and Volatility Feedback

## 4. Nonparametric Model of Market Excess Returns and Realized Variance

#### 4.1. Conditional Distribution of Returns Given Realized Variance

#### 4.2. Dirichlet Process Prior for the Infinite Number Of Unknowns

#### 4.3. Hierarchical Representation

#### 4.4. Posterior Simulation

- $\pi \left({\theta}_{j}\right|$$\mathit{r}$, $\mathit{RV}$$,S)\propto {g}_{0}\left({\theta}_{j}\right){\prod}_{\{t:{s}_{t}=j\}}f({r}_{t},\mathrm{log}\left(R{V}_{t}\right)|{\theta}_{j},{I}_{t-1})$, $j=1,\cdots ,K$.
- $\pi \left({v}_{j}\right|S)\propto \mathrm{Beta}\left({v}_{j}\right|{a}_{j},{b}_{j})$, $j=1,\cdots ,K$, with ${a}_{j}=1+{\sum}_{t=1}^{T}\mathbf{1}({s}_{t}=j),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{b}_{j}=\kappa +{\sum}_{t=1}^{T}\mathbf{1}({s}_{t}>j).$
- $\pi \left({u}_{t}\right|{\Omega}_{K},S)\propto \mathbf{1}(0<{u}_{t}<{\omega}_{{s}_{t}})$, $t=1,\cdots ,T$.
- Find the smallest K such that ${\sum}_{j=1}^{K}{\omega}_{j}>1-min\left\{U\right\}$.
- $P({s}_{t}=j|$$\mathit{r}$,
**RV**$,{\Theta}_{K},U,{\Omega}_{K})\propto {\sum}_{j=1}^{K}\mathbf{1}({u}_{t}<{\omega}_{j})f({r}_{t},\mathrm{log}\left(R{V}_{t}\right)|{\theta}_{{s}_{t}},{I}_{t-1})$.

## 5. Nonparametric Conditional Density Estimation

#### Nonparametric Conditional Mean Estimation

## 6. Empirical Findings

#### 6.1. Volatility Effect

#### 6.2. Risk and Return Trade-Off

#### 6.3. Conditional Quantiles and Contour Plots

#### 6.4. Parameter Estimates and Robustness

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1. | Ludvigson and Ng (2007) also utilize realized variance as a measure of conditional volatility. As we will show using realized variance provides additional flexibility in modeling the joint distribution and provides a better signal on volatility by using daily data to estimate monthly ex post variance. |

2. | A good summary of this research is found in Lettau and Ludvigson (2010). |

3. | |

4. | Harrison and Zhang (1999) also relaxes the normality assumption by applying Gallant and Tauchen (1989) semi- nonparametric estimator but only to the conditional distribution of excess returns. |

5. | For example, see Chib and Hamilton (2002); Burda et al. (2008); Conley et al. (2008); Delatola and Griffin (2013); Griffin and Steel (2004); and Chib and Greenberg (2010); Jensen and Maheu (2010, 2013, 2014) for recent applications of the DPM model. |

6. | The approximation is based on Campbell and Shiller (1988). Additional papers that build on this approach and find empirical support for volatility feedback include Turner et al. (1989); Kim et al. (2004); Kim et al. (2005); Bollerslev et al. (2006); and Calvet and Fisher (2007). |

7. | |

8. | A preliminary analysis showed the importance of a six-month component. |

9. | Several different functional forms for the conditional mean of ${r}_{t}$ given $\mathrm{log}\left(R{V}_{t}\right)$ result in similar findings and are discussed in Section 6.4. The current specification provides flexibility in modeling. |

10. | Alternative methods Escobar and West (1995) based on the hierarchical form of the model in Equation (14) are more difficult as our model and prior are non-conjugate. |

11. | Additional papers that also build on Muller et al. (1996) are Rodriguez et al. (2009); Shahbaba and Neal (2009); and Taddy and Kottas (2010). |

12. | This result makes use of expressing the numerator as $\int xp(x,y|\theta \left)p\right(\theta )d\theta dx=\int xp(x|y,\theta )p\left(y\right|\theta \left)p\right(\theta )d\theta dx=\int E[x|y,\theta ]p\left(y\right|\theta \left)p\right(\theta )d\theta $. |

13. | Note that the quantity $E\left[{r}_{t}\right|\mathrm{log}\left(R{V}_{t}\right),{I}_{t-1}]$ in (10) assumes parameters are known. In our case, they need to be estimated by the posterior density using the full sample of data r, $\mathit{R}\mathit{V}$. Therefore, our estimate implicitly conditions on the observed r and $\mathit{R}\mathit{V}$ in $E\left[r\right|\mathrm{log}\left(RV\right),{I}_{t-1}]$. |

14. | For convenience, our figures drop the conditioning set $\mathit{r},\mathit{R}\mathit{V}$. |

15. | In fact, averaging the curves from the nonparametric model would give something close to the parametric model in Figure 2. |

16. | From Table 1, average $\mathrm{log}\left(RV\right)$ is $-1.5602$ with a minimum of $-4.4595$ and maximum of $2.4245$. |

17. | $E\left[\mathrm{log}\left(RV\right)\right|{I}_{t-1},\mathit{r},\mathit{R}\mathit{V}]$ denotes the in-sample Bayesian estimate of the expectation of $\mathrm{log}\left(RV\right)$ given ${I}_{t-1}$. This conditions on regressors in the information set $t-1$ but uses the full posterior density based on $\mathit{r}$,$\mathit{RV}$ for the model parameters to integrate out parameter uncertainty. |

**Figure 2.**Expected excess return given log realized variance for the parametric model. This figure displays the expected excess return and 0.90 density intervals as a function of log realized variance for the parametric model.

**Figure 3.**Expected return given log realized variance for each of the information sets ${I}_{t-1}$, $t=2,\dots ,T$.

**Figure 4.**Expected excess return given log realized variance for the information set ${I}_{t-1}$ where volatility is low. This figure displays the expected excess return and 0.90 density intervals as a function of $\mathrm{log}\left(RV\right)$ conditional on the information set ${I}_{t-1}$, $t=1964:10$, which is a low volatility period. The expected log-realized volatility based on the model is blue, while the actual log-realized volatility for $t=1964:10$ is the black vertical line.

**Figure 5.**Expected excess return given log realized variance for the information set ${I}_{t-1}$ where volatility is near its average level. This figure displays the expected excess return and 0.90 density intervals as a function of $\mathrm{log}\left(RV\right)$ conditional on regressors in the information set from ${I}_{t-1}$, $t=1996:2$, which is an average volatility period. The expected log-realized volatility based on the model is blue while the actual log-realized volatility for $t=1996:2$ is the black vertical line.

**Figure 6.**Expected excess return given log realized variance for the information set ${I}_{t-1}$ where volatility is high. This figure displays the expected excess return and 0.90 density intervals as a function of $\mathrm{log}\left(RV\right)$ conditional on regressors in the information set from ${I}_{t-1}$, $t=2008:12$, which is a high volatility period. The expected log-realized volatility based on the model is blue while the actual log-realized volatility for $t=2008:12$ is the black vertical line.

**Figure 7.**Expected excess return given $\mathrm{log}\left(RV\right)$ for various periods. This figure displays the expected excess return as a function of $\mathrm{log}\left(RV\right)$ conditional on regressors ${I}_{t-1}$ taken from $t=1964:10$ “Low Log-RV”, $t=1996:2$, “Average Log-RV” and $t=2008:12$ “High Log-RV”.

**Figure 10.**Quantiles of excess returns given $\mathrm{log}\left(RV\right)$ for the parametric model. This figure displays the quantiles of the distribution of excess returns conditional on $\mathrm{log}\left(RV\right)$ for the parametric model. The green dotted line is the expected excess return given $\mathrm{log}\left(RV\right)$.

**Figure 11.**Quantiles of excess returns given $\mathrm{log}\left(RV\right)$ for low volatility. This figure displays the quantiles of the distribution of excess returns conditional on $\mathrm{log}\left(RV\right)$ for ${I}_{t-1}$, $t=1964:10$. The green dotted line is the expected excess return given $\mathrm{log}\left(RV\right)$.

**Figure 12.**Quantiles of excess returns given $\mathrm{log}\left(RV\right)$ for average volatility. This figure displays the quantiles of the distribution of excess returns conditional on $\mathrm{log}\left(RV\right)$ for ${I}_{t-1}$, $t=1996:2$. The green dotted line is the expected excess return given $\mathrm{log}\left(RV\right)$.

**Figure 13.**Quantiles of excess returns given $\mathrm{log}\left(R{V}_{t}\right)$ for high volatility. This figure displays the quantiles of the distribution of excess returns conditional on $\mathrm{log}\left(RV\right)$ for ${I}_{t-1}$, $t=2008:12$. The green dotted line is the expected excess return given $\mathrm{log}\left(RV\right)$.

**Figure 14.**Predictive density for $r,\mathrm{log}\left(RV\right)$ for low volatility ${I}_{t-1}$, $t=1964:10$.

**Figure 15.**Predictive density for $r,\mathrm{log}\left(RV\right)$ for average volatility ${I}_{t-1}$, $t=1996:2$.

**Figure 16.**Predictive density for $r,\mathrm{log}\left(RV\right)$ for high volatility ${I}_{t-1}$, $t=2008:12$.

**Figure 17.**Posterior means of ${\alpha}_{0,{s}_{t}},{\alpha}_{1,{s}_{t}}$ and ${\eta}_{1,{s}_{t}}^{2}$.

**Figure 18.**Posterior means of ${\gamma}_{0,{s}_{t}},\cdots ,{\gamma}_{4,{s}_{t}}$ and ${\eta}_{2,{s}_{t}}^{2}$.

Mean | Variance | Skewness | Kurtosis | Min | Max | |
---|---|---|---|---|---|---|

${r}_{t}$ | 0.0514 | 0.3884 | −0.4047 | 10.0461 | −4.0710 | 4.1630 |

${r}_{t}^{2}$ | 0.3907 | 1.3474 | 9.7037 | 119.5948 | 0.0000 | 17.3300 |

$R{V}_{t}$ | 0.3790 | 0.5611 | 7.0305 | 69.4529 | 0.0116 | 11.3000 |

$\mathrm{log}\left(R{V}_{t}\right)$ | −1.5602 | 0.8846 | 0.8051 | 4.2910 | −4.4595 | 2.4245 |

$z={r}_{t}/\sqrt{R{V}_{t}}$ | 0.2296 | 1.0789 | 0.0030 | 2.6856 | −2.4080 | 2.8580 |

Mean | 0.95 Density Interval | |
---|---|---|

${\alpha}_{0}$ | 0.1922 | (0.1672, 0.2171) |

${\alpha}_{1}$ | −0.2801 | (−0.3895, −0.1748) |

${\eta}_{1}^{2}$ | 1.0177 | (0.9460, 1.0962) |

${\gamma}_{0}$ | −0.3319 | (−0.4151, −0.2470) |

${\gamma}_{1}$ | 0.3766 | (0.3179, 0.4329) |

${\gamma}_{2}$ | 0.4505 | (0.3817, 0.5180) |

${\gamma}_{3}$ | −0.1518 | (−0.1842, −0.1170) |

${\gamma}_{4}$ | 0.1258 | (0.0680, 0.1861) |

${\eta}_{2}^{2}$ | 0.3981 | (0.3702, 0.4278) |

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**MDPI and ACS Style**

Jensen, M.J.; Maheu, J.M.
Risk, Return and Volatility Feedback: A Bayesian Nonparametric Analysis. *J. Risk Financial Manag.* **2018**, *11*, 52.
https://doi.org/10.3390/jrfm11030052

**AMA Style**

Jensen MJ, Maheu JM.
Risk, Return and Volatility Feedback: A Bayesian Nonparametric Analysis. *Journal of Risk and Financial Management*. 2018; 11(3):52.
https://doi.org/10.3390/jrfm11030052

**Chicago/Turabian Style**

Jensen, Mark J., and John M. Maheu.
2018. "Risk, Return and Volatility Feedback: A Bayesian Nonparametric Analysis" *Journal of Risk and Financial Management* 11, no. 3: 52.
https://doi.org/10.3390/jrfm11030052