The Predictability and Volatility of Returns in the Presence of Rare Disasters

1. Introduction

I characterize the relationship between the variance premium at time $t$ and the excess returns over the next $h$ months in an economy with variable rare disasters as outlined in Gabaix (2011) using the parameter estimates given in Bollerslev, Tauchen and Zhou (2009). Hao Zhou has been very kind and posted his data¹ for me to use in this analysis.

First, in section 2, I relate the equity premium conditional on no disasters occurring to the value of a put option on the market given that a disaster does occur. Then, in section 3, I relate the variance premium–i.e., the difference between the Black-Scholes implied variance and the realized variance–to this equity premium over the next $h$ months. Finally, in section 4, I conclude by using the parameter estimates in Bollerslev et al. (2009) to compute the coefficient $\beta$ which links the equity premium to the variance premium predicted by Gabaix (2011).

2. The Put Option Premium

Intuitively, in an economy with rare disasters, the equity premium should be given by the sum of the premium due to normal times risk and the premium due to disaster risk:

(1) $\begin{align*} r_t - r_t^f &= \pi^{\mathtt{N}}_t + \pi^{\mathtt{D}}_t, \end{align*}$

where $\pi^{\mathtt{N}}_t$ is the premium due to Gaussian noise and $\pi^{\mathtt{D}}_t$ is the premium due to disasters. If we looked at an economy in which there was no Gaussian noise as in the main sections of Gabaix (2011), then the entire risk premium would be due to disaster risk. However, here I am going to study a world in which there exists normal times Gaussian noise. Specifically, consider an asset with price $P_t$ that evolves according to the rule,

(2) $\begin{align*} \frac{P_{t+1}}{P_t} &= e^\mu_t \times \begin{cases} e^{\sigma \cdot \varepsilon_{t+1} - \sigma^2/2} &\text{ w/o disaster } \\ F_{t+1} &\text{ else } \end{cases} \end{align*}$

where $\mu_t = g_D + \zeta(\hat{H}_t)$ as $P_t/D_t = a + b \cdot \hat{H}_t$ . The functional form of $\zeta$ is given in Gabaix (2011), but will not be important here.

Let $\hat{V}_t(K)$ be the value of a $1$ -period European put option on this asset with strike price $K$ ,

(3) $\begin{align*} \hat{V}_t(K) &= \mathbb{E}_t \left[ \frac{M_{t+1}}{M_t} \cdot \left( K - \frac{P_{t+1}}{P_t} \right)^+ \right] \end{align*}$

Proposition 3 in Gabaix (2011) tells us how to compute the value of this put option:

(4) $\begin{align*} \hat{V}_t(K) &= e^{- \delta + \mu_t} \cdot \left\{ \hat{V}^{\mathtt{N}}_t(K) + \hat{V}^{\mathtt{D}}_t(K) \right\} \\ \hat{V}^{\mathtt{N}}_t(K) &= (1 - p_t) \cdot V(K \cdot e^{- \mu_t}) \\ \hat{V}^{\mathtt{D}}_t(K) &= p_t \cdot \mathbb{E}^{\mathtt{D}}_t \left[ B_{t+1}^{-\gamma} \cdot \left( K \cdot e^{-\mu_t} - F_{t+1} \right)^+ \right] \end{align*}$

where $V(\cdot)$ is the Black-Scholes value of a put option with volatility $\sigma$ , initial price $1$ , maturity $1$ and interest rate $0$ . In the proposition below, I relate the equity premium due to disaster risk to the value of the disaster component of the put option:

Proposition (Disaster Risk Premium): For a put option with strike price $K = e^{\mu_t}$ , the disaster risk premium can be written as,

(5) $\begin{align*} \pi^{\mathtt{D}}_t &\approx \hat{V}^{\mathtt{D}}_t(e^{\mu_t}) \end{align*}$

This result reads that, given a sufficiently high strike price $K = e^{\mu_t}$ such that the fraction of dividend lost due to a disaster $F_{t+1}$ will always be less than the discounted strike price value of $1$ , the contribution of disaster risk to the put option value is equal to the disaster risk premium. More intuitively, the value of the disaster risk premium must equal the value of an asset that pays out $\$1$ in the event of a disaster; i.e., the value of $\hat{V}_t^{\mathtt{D}}(e^{\mu_t})$ .

Proof (Equity Premium): From Proposition 1 in Gabaix (2011), we have that in a world with no Gaussian noise,

(6) $\begin{align*} r_t - r_t^f &= p_t \cdot \mathbb{E}^{\mathtt{D}}_t \left[ B_{t+1}^{-\gamma} \cdot \left( 1 - F_{t+1} \right) \right] \end{align*}$

If $K \cdot e^{\mu_t} \geq 1$ we can drop the $\max$ operator. Thus, for $K = e^{\mu_t}$ we have that $\hat{V}^{\mathtt{D}}_t(K)$ is equal to the disaster risk premium.

3. What’s Vol Got to Do With It?

Next, I want to use the result above to link the difference between in the implied variance given by the price of the $1$ period European put option $\hat{V}_t(e^{\mu_t})$ and the realized variance of the underlying asset to the excess rate of return on the market over the next $h$ months. I denote this variance premium as,

(7) $\begin{align*} \textit{vp}_t &= \tilde{\sigma}_t^2 - \sigma_t^2 \end{align*}$

I then want to be able to run the regression below where $r_{t \to (t+h)} - r^f_{t \to (t+h)}$ is the annualized excess return on the S&P 500 and $\eta_t(h)$ is an error term,

(8) $\begin{align*} r_{t \to (t+h)} - r^f_{t \to (t+h)} &= \alpha(h) + \beta(h) \cdot \textit{vp}_t + \eta_{t}(h) \end{align*}$

Proposition (Return Predictability): Let $\phi(\cdot)$ be the $\mathtt{pdf}$ of the standard Gaussian distribution:

(9) $\begin{align*} \phi(x) &= \frac{1}{\sqrt{2 \cdot \pi}} \cdot e^{-\frac{x^2}{2}} \end{align*}$

Then, given a strike price $K = e^{\mu_t}$ , we can relate the variance premium $\textit{vp}_t$ to the risk premium $r_{t \to (t+h)} - r^f_{t \to (t+h)}$ over the next $h$ months via the relationship below,

(10) $\begin{align*} r_{t \to (t+h)} - r_{t \to (t+h)}^f &\approx \alpha + \beta \cdot \textit{vp}_t \end{align*}$

with the coefficients,

(11) $\begin{align*} \alpha &= e^{\delta - \mu_t} \cdot \left\{ V(e^{\mu_t}) - e^{-\delta + \mu_t} \cdot (1 - p_t) \cdot V(1) \right\} + \pi^{\mathtt{N}}_t \\ \beta &\approx \frac{e^{\delta - \mu_t}}{2 \cdot \sigma} \cdot \phi \left( \frac{\delta + \frac{\sigma^2}{2}}{\sigma} \right) \end{align*}$

This result says that, conditional on the realized variance $\sigma_t^2$ , increasing an economy’s variance premium $\textit{vp}_t$ will increase the realized excess returns over the next $h$ months in a manner that is proportional to $e^{\delta - \mu_t} / (2 \cdot \sigma)$ scaled by the $z$ -statistic for $(\delta + \sigma^2/2)/\sigma$ . Note that the coefficient $\beta$ will be highly non-linear in the realized volatility $\sigma$ . On one hand, with regards to the $e^{\delta - \mu_t} / (2 \cdot \sigma)$ term, decreasing $\sigma$ will always increase the value of the equity premium. However, on the other hand, with regards to the Gaussian term, as $\sigma$ gets very small the input to the $\phi(\cdot)$ function gets large leading to a very unlikely realization.

Proof (Return Predictability): Suppose that we observe $\hat{V}(K)$ in the data. Then, we can approximate the implied volatility using the Black-Scholes $\nu$ (i.e., “vega”) via the relationship:

(12) $\begin{align*} \hat{V}(K) &= V(K) + \nu \cdot \left[ \ \tilde{\sigma} - \sigma \ \right] \\ \nu &= S \cdot \phi \left( \frac{\ln(S/K) + \delta + \frac{\sigma^2}{2}}{\sigma} \right) \end{align*}$

By definition, the starting price is $1$ . What’s more, since is near the money $e^{\mu_t} \approx 1$ , we have that $\ln(S/K) = 0$ .

Plugging in the expression for $\hat{V}(K)$ given in Proposition 3 of Gabaix (2011) and working backwards yields the desired result:

(13) $\begin{align*} \hat{V}(e^{\mu_t}) &= V(e^{\mu_t}) + \phi \left( \frac{\delta + \frac{\sigma^2}{2}}{\sigma} \right) \cdot \left[ \ \tilde{\sigma} - \sigma \ \right] \\ &= e^{- \delta + \mu_t} \cdot \left\{ \hat{V}^{\mathtt{N}}_t(e^{\mu_t}) + \pi_t^{\mathtt{D}} \right\} \\ &= e^{- \delta + \mu_t} \cdot \left\{ (1 - p) \cdot V(1) + \pi_t^{\mathtt{D}} \right\} \end{align*}$

I use the approximation:

(14) $\begin{align*} \frac{\tilde{\sigma}_t^2 - \sigma_t^2}{2 \cdot \sigma_t} &\approx \tilde{\sigma}_t - \sigma_t \end{align*}$

This proposition yields a corollary linking the variance premium to innovations in asset resilience $\hat{H}_t$ :

Corollary (Implied Volatility and Asset Resilience): Given that from Proposition 1 in Gabaix (2011), we know that,

(15) $\begin{align*} r_{t \to (t+1)} - r_{t \to (t+1)}^f &= \delta - H_t \end{align*}$

we know that the variance premium must be linearly related to innovations in asset resilience $\hat{H}_t$ with slope coefficient $\theta$ given below:

(16) $\begin{align*} \textit{vp}_t &= \zeta + \theta \cdot \hat{H}_t \\ \theta &= \frac{1}{\frac{e^{\delta - \mu_t}}{2 \cdot \sigma} \cdot \phi \left( \frac{\delta + \frac{\sigma^2}{2}}{\sigma} \right)} \end{align*}$

4. Matching Bollerslev et al. (2009)

In this section, I conclude by computing a model derived estimate of $\beta(1)$ using the parameter values given in Bollerslev et al. (2009) which estimates the regression given in Equation (8) above.

In their original regression results in Table 2, Bollerslev et al. (2009) use a somewhat unconventional choice of units. Namely, the excess returns are annualized while the variance premium is computed as a monthly estimate. To map the estimates from the original paper over to model implied values, I convert all of the data to annualized log values as outlined below,

(17) $\begin{align*} r_{t+1}^e &= r_{t+1}^{e,\mathtt{btz}} \times \frac{1}{10^2} \\ \textit{vp}_t &= \textit{vp}_t^{\mathtt{btz}} \times \frac{12}{10^4} \end{align*}$

After converting the variables to natural units, I find the summary statistics listed below where $\mathbb{S}$ is the standard deviation operator,

$\begin{align*} \begin{array}{l|c} & \text{Estimate} \\ \hline \hline \mathbb{E}[r^e_{t+1}] & 0.069 \\ \mathbb{S}[r^e_{t+1}] & 0.138 \\ \hline \mathbb{E}[\textit{vp}_t] & 0.022 \\ \mathbb{S}[\textit{vp}_t] & 0.0053 \end{array} \end{align*}$

The sample runs from Jan. 1990 to December 2007. These estimates read that the S&P 500 outperformed $3$ -month T-Bills by an average of $6.9\%$ a year. This gap had a volatility of $13.8\%$ a year. What’s more, the average distance between the VIX implied variance and the realized variance for the S&P 500 was $2.2\%^2$ .

Using these newly converted variables, I estimate a $\beta(1)$ of $3.205$ with a $t$ -stat of $1.781$ . Below I report the remainder of the regression results where the values in square brackets represent standard errors,

*** QuickLaTeX cannot compile formula:
\begin{align*}
\begin{array}{l|ccccc}
& h=1 & h=3 & h=6 & h=9 & h=12
\\
\hline
\hline
\alpha & -0.0024 & -0.023 & 0.0097 & 0.033 & 0.042
\\
\text{s.e.} & <sup class='footnote'><a href='#fn-1150-2' id='fnref-1150-2' onclick='return fdfootnote_show(1150)'>2</a></sup> & <sup class='footnote'><a href='#fn-1150-3' id='fnref-1150-3' onclick='return fdfootnote_show(1150)'>3</a></sup> & <sup class='footnote'><a href='#fn-1150-4' id='fnref-1150-4' onclick='return fdfootnote_show(1150)'>4</a></sup> & <sup class='footnote'><a href='#fn-1150-5' id='fnref-1150-5' onclick='return fdfootnote_show(1150)'>5</a></sup> & <sup class='footnote'><a href='#fn-1150-6' id='fnref-1150-6' onclick='return fdfootnote_show(1150)'>6</a></sup>
\\
\text{s.e.}_{H1992} & <sup class='footnote'><a href='#fn-1150-7' id='fnref-1150-7' onclick='return fdfootnote_show(1150)'>7</a></sup> & <sup class='footnote'><a href='#fn-1150-8' id='fnref-1150-8' onclick='return fdfootnote_show(1150)'>8</a></sup> & <sup class='footnote'><a href='#fn-1150-9' id='fnref-1150-9' onclick='return fdfootnote_show(1150)'>9</a></sup> & <sup class='footnote'><a href='#fn-1150-10' id='fnref-1150-10' onclick='return fdfootnote_show(1150)'>10</a></sup> & <sup class='footnote'><a href='#fn-1150-11' id='fnref-1150-11' onclick='return fdfootnote_show(1150)'>11</a></sup>
\\
\hline
\beta & 3.21 & 4.00 & 2.54 & 1.53 & 1.17
\\
\text{s.e.} & <sup class='footnote'><a href='#fn-1150-12' id='fnref-1150-12' onclick='return fdfootnote_show(1150)'>12</a></sup> & <sup class='footnote'><a href='#fn-1150-13' id='fnref-1150-13' onclick='return fdfootnote_show(1150)'>13</a></sup> & <sup class='footnote'><a href='#fn-1150-14' id='fnref-1150-14' onclick='return fdfootnote_show(1150)'>14</a></sup> & <sup class='footnote'><a href='#fn-1150-15' id='fnref-1150-15' onclick='return fdfootnote_show(1150)'>15</a></sup> & <sup class='footnote'><a href='#fn-1150-16' id='fnref-1150-16' onclick='return fdfootnote_show(1150)'>16</a></sup>
\\
\text{s.e.}_{H1992} & <sup class='footnote'><a href='#fn-1150-17' id='fnref-1150-17' onclick='return fdfootnote_show(1150)'>17</a></sup> & <sup class='footnote'><a href='#fn-1150-18' id='fnref-1150-18' onclick='return fdfootnote_show(1150)'>18</a></sup> & <sup class='footnote'><a href='#fn-1150-19' id='fnref-1150-19' onclick='return fdfootnote_show(1150)'>19</a></sup> & <sup class='footnote'><a href='#fn-1150-20' id='fnref-1150-20' onclick='return fdfootnote_show(1150)'>20</a></sup> & <sup class='footnote'><a href='#fn-1150-21' id='fnref-1150-21' onclick='return fdfootnote_show(1150)'>21</a></sup>
\\
\hline
\textit{Adj. } R^2 & 0.010 & 0.070 & 0.057 & 0.027 & 0.018
\end{array}
\end{align*}

*** Error message:
You can't use `macro parameter character #' in math mode.
leading text: \end{align*}

Alternatively, using the data Drechler and Yaron (2011), I find the coefficient estimates below:

*** QuickLaTeX cannot compile formula:
\begin{align*}
\begin{array}{l|ccccc}
& h=1 & h=3 & h=6 & h=9 & h=12
\\
\hline
\hline
\alpha & -0.017 & -0.030 & 0.012 & 0.039 & 0.042
\\
\text{s.e.} & <sup class='footnote'><a href='#fn-1150-22' id='fnref-1150-22' onclick='return fdfootnote_show(1150)'>22</a></sup> & <sup class='footnote'><a href='#fn-1150-23' id='fnref-1150-23' onclick='return fdfootnote_show(1150)'>23</a></sup> & <sup class='footnote'><a href='#fn-1150-24' id='fnref-1150-24' onclick='return fdfootnote_show(1150)'>24</a></sup> & <sup class='footnote'><a href='#fn-1150-4' id='fnref-1150-4' onclick='return fdfootnote_show(1150)'>4</a></sup> & <sup class='footnote'><a href='#fn-1150-26' id='fnref-1150-26' onclick='return fdfootnote_show(1150)'>26</a></sup>
\\
\text{s.e.}_{H1992} & <sup class='footnote'><a href='#fn-1150-27' id='fnref-1150-27' onclick='return fdfootnote_show(1150)'>27</a></sup> & <sup class='footnote'><a href='#fn-1150-28' id='fnref-1150-28' onclick='return fdfootnote_show(1150)'>28</a></sup> & <sup class='footnote'><a href='#fn-1150-29' id='fnref-1150-29' onclick='return fdfootnote_show(1150)'>29</a></sup> & <sup class='footnote'><a href='#fn-1150-10' id='fnref-1150-10' onclick='return fdfootnote_show(1150)'>10</a></sup> & <sup class='footnote'><a href='#fn-1150-7' id='fnref-1150-7' onclick='return fdfootnote_show(1150)'>7</a></sup>
\\
\beta & 6.30 & 7.18 & 4.09 & 2.17 & 1.31
\\
\text{s.e.} & <sup class='footnote'><a href='#fn-1150-32' id='fnref-1150-32' onclick='return fdfootnote_show(1150)'>32</a></sup> & <sup class='footnote'><a href='#fn-1150-33' id='fnref-1150-33' onclick='return fdfootnote_show(1150)'>33</a></sup> & <sup class='footnote'><a href='#fn-1150-34' id='fnref-1150-34' onclick='return fdfootnote_show(1150)'>34</a></sup> & <sup class='footnote'><a href='#fn-1150-35' id='fnref-1150-35' onclick='return fdfootnote_show(1150)'>35</a></sup> & <sup class='footnote'><a href='#fn-1150-36' id='fnref-1150-36' onclick='return fdfootnote_show(1150)'>36</a></sup>
\\
\text{s.e.}_{H1992} & <sup class='footnote'><a href='#fn-1150-37' id='fnref-1150-37' onclick='return fdfootnote_show(1150)'>37</a></sup> & <sup class='footnote'><a href='#fn-1150-38' id='fnref-1150-38' onclick='return fdfootnote_show(1150)'>38</a></sup> & <sup class='footnote'><a href='#fn-1150-39' id='fnref-1150-39' onclick='return fdfootnote_show(1150)'>39</a></sup> & <sup class='footnote'><a href='#fn-1150-40' id='fnref-1150-40' onclick='return fdfootnote_show(1150)'>40</a></sup> & <sup class='footnote'><a href='#fn-1150-41' id='fnref-1150-41' onclick='return fdfootnote_show(1150)'>41</a></sup>
\\
\hline
\textit{Adj. } R^2 & 0.0098 & 0.054 & 0.035 & 0.011 & 0.018
\end{array}
\end{align*}

*** Error message:
You can't use `macro parameter character #' in math mode.
leading text: \end{align*}

I want to compute the $\beta(1)$ estimates implied by the variable rare disasters model given in Gabaix (2011) which requires that I have estimates for $\sigma$ , $\delta$ and $\mu_t$ . I take the annualized values of $\delta = 0.05$ and $\mu_t = 0.025$ from Gabaix (2011). I take the estimate of the annualized realized variance from Bollerslev et al. (2009) of $\sigma_t^2 = 0.133$ . Using these values, I find that,

(18) $\begin{align*} \beta &= \frac{e^{\delta - \mu_t}}{2 \cdot \sigma} \cdot \phi \left( \frac{\delta + \frac{\sigma^2}{2}}{\sigma} \right) \\ &= \frac{e^{0.05 - 0.025}}{2 \cdot 0.133} \cdot \phi \left( \frac{0.05 + \frac{0.133^2}{2}}{0.133} \right) \\ &= 1.394 \end{align*}$

This estimate implies that if the variance premium rises by $1\%^2$ per year, then the equity premium will rise by $1.394\%$ per year.

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