The value of expected return persistence

Abstract

This work utilizes the fractional Black–Scholes model to estimate the option-implied Hurst exponents, interpreted as forward-looking expectations of return persistence. The focus of the paper is on how corresponding believes enter into factor based asset pricing models. Empirical analyses are carried out for the cross-section of S &P 500 stocks. We make the important observations that (i) stock returns show significant patterns of time-varying persistence and (ii) corresponding believes are reflected within option prices. Incorporating the Hurst exponents allows us to split up CAPM betas into pure market correlation risk (around 70–80%) and into excess persistence believes (about 20–30% of the risk loading). A direct comparison to standard CAPM shows that incorporating persistence believes significantly improves the predictability of future realized returns, and partially releases the beta anomaly. The effects become even stronger the greater the prediction horizon. Hence, the concept of fractal motions enables a deeper understanding of risk structures without the need of additional risk factors.

Appendix

1.1 A1. Expected return and horizon

See Figs. 8 and 9.

Fig. 8

Illustration of the expected rate of return $\langle \mu _i\rangle$ as a function of the time to maturity $\tau$. The setting is as follows: $\langle \mu _m\rangle =5\%$, $r_f=1\%$, $H_m=0.6$ and $\Delta H = \{-0.2, 0, 0.2\}$. Under classic Brownian Motion (i.e., $\Delta H=0$), $\langle \mu _i\rangle$ would be constant and thus independent of $\tau$. With the fractal generalization this independence is removed. As can be seen $\Delta H > 0$ coerces $\langle \mu _i\rangle$ to grow int the future horizon ahead. Differently, with $\Delta H <0$ the stock’s expected rate of return is believed to decline

1.2 A2. H-contribution

Analysis of individual stock’s expected market risk loading decomposed into classic BM correlation and into contribution from expected persistence (H-contribution):

Fig. 9

Histogram of single stock’s H contribution to predicted returns under fractal versus cBM CAPM, $\%H = \beta /\tilde{\beta } - 1$. Three conclusions are worth to mention. First, the contribution varies largely, hence expected persistence cannot be neglected. Second, contributions are mainly negative, which means that most stocks have H below the market portfolio’s H. Third, the negativity of $\%H$ means that cBM CAPM will thoroughly overestimate expected risk compared to fractal CAPM

1.3 A3. R/S analysis

The R/S analysis [cp. Hurst (1956)] is conducted as follows. Say X is the return series of $n_T$ observations for which one wants to compute the Hurst exponent. First, one hast to compute the sample mean $\hbox {m}(X)$ and the excess of mean series $\bar{X}$

$$\begin{aligned} \hbox {m}(X)=\tfrac{1}{n_T}\sum _{i=1}^{n_T}X_i \qquad \hbox {and}\qquad \bar{X}=X-\hbox {m}(X) \end{aligned}$$

(5.1)

by which the computation of the cumulative deviate series Z is straight forward,

$$\begin{aligned} Z_t = \sum _{i=1}^{t}\bar{X}_i \qquad t\in \{1,...,n_T\} \end{aligned}$$

(5.2)

The range R is then defined as the interval length of Z,

$$\begin{aligned} \hbox {R} = max(Z) - min(Z) \end{aligned}$$

(5.3)

and S is simply the sample standard deviation,

$$\begin{aligned} \hbox {S}=\sqrt{\frac{1}{n}\sum _{i=1}^{n}(X_i - \hbox {m}(X))^2} \end{aligned}$$

(5.4)

such that the R/S series is given by

$$\begin{aligned} (\hbox {R}/\hbox {S})_t=\frac{\hbox {R}_t}{\hbox {S}_t} \end{aligned}$$

(5.5)

The Hurst exponent is now defined by

$$\begin{aligned} \left\langle \frac{\hbox {R}}{\hbox {S}} \right\rangle = c\, n_T^H \end{aligned}$$

(5.6)

with c as a constant. Taking logs,

$$\begin{aligned} \log ((\hbox {R}/\hbox {S})_t) = \log (c)+ H \, \log (t) \end{aligned}$$

(5.7)

gives a linear equation that can be fitted to return data by OLS regression, where H corresponds to the regression’s coefficient.

1.4 A4. Expected volatility

As outlined before, option implied volatilities are per construction forward looking proxies of expected risk. Comparing physically realized returns with expected risk requires ${\mathbb {P}}$ implied volatilities—comparing physical returns with risk-neutral risk may delivers biased results. By Black–Scholes market model, volatility is deterministic such that Girsanov theorem applies and the market risk premium is incorporated via the drift term. However, by now the academic consensus agrees that volatility itself is stochastic [e.g., Heston (1993)]. If volatility is stochastic, then it is obvious that fluctuations in it is another source of risk, causing that investors require a premium for it. In the context of implied volatilities, this means that if one uses Black–Scholes formula to compute implied volatilities, the estimates will be derived under the ${\mathbb {Q}}$ measure. Those estimates are likely to be different from ${\mathbb {P}}$ expected volatilities as soon as volatility is stochastic. Just like in standard Black–Scholes there is a premium in the drift term, $\mu \ge r_f$, ($\mu$ is the ${\mathbb {P}}$ expected return, $r_f$ the ${\mathbb {Q}}$ expected one and $\tfrac{\mu - r_f}{\hat{\sigma }}$ is the risk premium), investors require a premium for volatility of volatility such that $\sigma _{{\mathbb {P}}}\le \sigma _{{\mathbb {Q}}}$. Therefore, $\sigma _{{\mathbb {Q}}}$ equals $\sigma _{{\mathbb {P}}}$ plus some variance risk premium [cp. Buss and Vilkov (2012)]. Empirically, this means that on average Black–Scholes implied volatilities slightly over-estimate future realized ones. Figure 10 displays the time-series of future realized volatility in comparison with Black–Scholes’s expected ${\mathbb {Q}}$ volatility

Fig. 10

Time-series of Black–Scholes implied volatility with target maturity of one month vs. future realized volatility of same horizon. On average, risk-neutral implied volatilities ($\sigma _{\mathbb {Q}}$)—as investor expectations—slightly overestimate future realized ones ($\vec {\sigma }$) due to a variance risk premium $\lambda$: in mild times, risk averse investors pay the premium ($\vec {\sigma } < \sigma _{\mathbb {Q}}$); during crises times, realized volatilities overshoot implied ones ($\vec {\sigma } > \sigma _{\mathbb {Q}}$). Investors form believes about future volatility out of historical data and adjust them for the current market outlook. Thus, a slight lag pattern can be observed

Estimation of the expected variance risk premium $\langle \lambda _{v}\rangle$ enjoys its own discussion, for example Bollerslev et al. (2009) or Todorov (2010). Generally it is argued that $\langle \lambda _{v}\rangle$ is driven by the economic cycle and/or market phases. Therefore, we suggest it makes sense to model the variance risk premium relative to the level of $\sigma _{{\mathbb {Q}}}$. Hence, let $\vec {\sigma }(\tau )$ express the future realized volatility of horizon $\tau$, $\langle \lambda _{v}\rangle$ is incorporated by first computing

$$\begin{aligned} \lambda _{v,t}(\tau ) = \frac{\sigma _{{\mathbb {Q}},t}(\tau ) - \vec {\sigma }_{t}(\tau )}{\sigma _{{\mathbb {Q}},t}(\tau )} \qquad \hbox {and}\qquad \bar{\lambda }_v(\tau ) = \frac{1}{n_T} \sum _{t=1}^{n_T}\lambda _{v,t}(\tau )\qquad t\in \{1,...,n_T\} \end{aligned}$$

(5.8)

upon the S &P500 index and then transform ${\mathbb {Q}}$ to ${\mathbb {P}}$ measured expected volatilities by

$$\begin{aligned} \sigma _{{\mathbb {P}}, i, t}(\tau ) = \sigma _{{\mathbb {Q}}, i, t }(\tau ) (1-\bar{\lambda }_v(\tau )) \qquad \begin{array}{r} \forall t\in \{1,...,n_T\}\\ \forall i \in \{1,...,n\} \end{array} \end{aligned}$$

(5.9)

such that the same relative variance risk premium is applied among the cross-section to transform individual risk-neutral $\sigma _{{\mathbb {Q}}}$’s to physically expected $\sigma _{{\mathbb {P}}}$’s while keeping arbitrage-free principles. Estimated ${\lambda }_{v,t}(\tau )$’s are positive with mean values (i.e., $\bar{\lambda }_{v}(\tau )$) of 0.050, 0.075 and 0.100 at t-statistics of 1.39, 1.98, 2.63 ($\tau =30$, 90, and 180 days). For the rest of the analysis we use solely ${\mathbb {P}}$ expected volatilities (if not mentioned differently), hence we drop this subscript for simplification purposes.

To briefly motivate the use of implied volatilities instead of historically realized ones in the context of quantifying expectations, we draw a short comparison between those to future realized ones. The observed output is highlighted in Fig. 11 and Table 6. From the table one sees that for short time-horizons the prediction quality between historical and implied volatilities is almost identical; mean and standard deviations of residuals do not really differ but implied volatilities realize a slightly better goodness of fit (R$^2$’s of 0.60 vs. 0.54).⁸ Hence one could interpret this as option traders use historical volatilities to form their believes for future realizations. However, with an increasing horizon, we clearly observe that implied estimates outperform historical ones, which can be seen upon the R$^2$’s. The fact is while investors typically use historical volatilities to form believes for the future, they also adjust them for example to incorporate the current market situation or the mean reverting behavior of variance. This means that using longer time windows for historical volatilities causes predictions to react to slowly to current situations, hence especially in the aftermath of surprises (like 2009), historical volatilities are bad proxies for future realized risk. Therefore, it seems plausible that historical volatilities realize a very bad R$^2$ at the $\tau =180$ horizon, while implied volatilities—where investors adjust for economic phases and mean reversion—could partially keep its prediction quality. Thus, implied volatilities are not only theoretically, but also empirically better qualified candidates for expected risk.

Fig. 11

S &P500: Residuals of expected minus realized volatility as quantified by historical, ${\mathbb {P}}$ and ${\mathbb {Q}}$ implied volatilities. Both option implied estimates are on average better than the historical one in predicting future realized volatility. This is mainly attributable to the fact that historical volatilities per se are unable to adopt for current market phases: While investors indeed use historical volatilities to grasp an idea about future realizations, they correct them for current market outlooks (e.g. recovery) to adjust their estimates. This is why implied volatilities deliver on average better estimates for future realizations

Table 6 Volatility analysis of the S &P500: Residuals $\epsilon$ of future realized volatility predicted by expected volatilities (historical, ${\mathbb {Q}}$, and ${\mathbb {P}}$)