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Equilibrium-based volatility models of the market portfolio rate of return (peacock tails or stotting gazelles)


We introduce a theoretical and empirical method of studying equilibrium-consistent volatility models. We implement it with the market portfolio’s return, which is central to financial risk management. Within an equilibrium framework, we study two families of such models. One is deterministic volatility, represented by current popular models. The other is in the “constant elasticity of variance” family, in which we propose new models. Theoretically, we show that, together with constant expected returns, the latter family tends to have better ability to forecast. Empirically, our proposed models, while as easy to implement as the popular ones, outperform them in three out-of-sample forecast evaluations of different time periods, by standard predictability criteria. This is true particularly during high-volatility periods, whether the market rises or falls.

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  1. We define “volatility” as a rate of return’s instantaneous variance.

  2. See, for example, Zahavi and Zahavi (1997).

  3. Friedman (1953).

  4. Naturally, volatility models are also used to explain asset returns and growth. See, for example, Schorfheide et al. (2013).

  5. Following Bick (1990), He and Leland (1993), Xu (2013, Chapter 3) derived necessary and sufficient conditions for the market portfolio return process to be consistent with equilibrium under market clearing stochastic interest rates, supporting a larger family of equilibrium consistent specifications of mean/volatility structures.

  6. Interesting future research would be to identify implementations of our approach within the benchmark portfolio approach; see, for example, Heath and Platen (2002), Platen (2011), and Platen and Semmler (2012).

  7. Generalized Autoregressive Conditional Heteroskedasticity (Engle 1982; Bollerslev 1986). We use GARCH(1,1).

  8. We use the 1996 version (Morgan/Reuters 1996), which can be viewed as a special case of GARCH(1,1), not the 2006 version (Zumbach 2007).

  9. We thank Frederik Lundtofte for suggesting that we emphasize this point.

  10. The forecast evaluation methodology of volatility models is an active research area. In this paper we employ the two classic methodologies, and we might incorporate more recent ones in future studies; for example, Patton (2011).

  11. There are also other types of volatility models; for example, stochastic volatility models dated back at least to Hull and White (1987) and Heston (1993), which are beyond the scope of this paper.

  12. Xu (2013, Chapter 3) showed that “GARCH-in-Mean” model can indeed be viewed as an implementation of a simplified version of equilibrium conditions studied in this paper. However, it is unclear whether GARCH is the best method to model means. Hansen and Lunde (2005) demonstrated that “GARCH-in-Mean” did not appear to improve empirical performance of GARCH.

  13. Thus, \((\mu _t^E +\delta _t )\) is the instantaneous expected total rate of return of the market portfolio.

  14. In this case, in light of Eq. (2), \(r_M =\frac{\eta _t }{M(1-\eta )}\). Hence, r and \(\mu ^{E}\) can both be some logarithmic functions of \(M_t \).

  15. When interest rates are deterministic, as in these cases, the specifications also satisfy the conditions of He and Leland (1993).

  16. This stylized fact is that the volatility moves asymmetrically with the security value (or price), i.e., the volatility is higher when the value falls than when it rises. This phenomenon was explained by Black (1976) as the “leverage effect.” Nonetheless, to our best knowledge, it is unknown in the literature whether it can be observed when stochastic mean rates of return are modeled.

  17. We choose GARCH(1,1) over GARCH(p,q), p\(>\)1, q\(>\)1, for two reasons. First, in, both, academic literature and industry practice, GARCH(1,1) is, by far, the most popular specification in the GARCH family, possibly due to a simpler functional form and more robust forecasting performance. Second, while GARCH(p,q) might improve forecasting performance, we, focus here on identifying the best comparison of our new approach with current practice, rather than on achievieng the best forecasting. Moreover, our equilibrium framework is in continuous time, within which a time-lag of 1 is a natural choice. Unlike GARCH(1,1), the continuous-time limit of GARCH(p,q) is unclear. It might be possible to generalize our equilibrium framework in discrete time, which could be a topic for future studies.

  18. With appropriate initial conditions.

  19. The continuous time limit of the GARCH(1,1) process may not be unique. For example, the well-known result of Nelson (1990) showed that in the limit, it becomes a stochastic volatility process. We use Corradi’s (2000) result because the Euler discretization of Eq. (12) leads back to GARCH(1,1). For more detailed discussion, see Singleton (2006, Chapter 7, pp. 177–178).

  20. Note that in discretized versions, \(M_t \) becomes \(M_{t-1} \).

  21. Equation (13) might be used to identify the conditions under which the long-run mean of Model 1 exists; see “Appendix”. We also note that the volatility in all the models considered in this paper is conditionally deterministic. In other words, conditioning on the lagged information set, the (following) volatilty is known and the error terms are i.i.d. Therefore, the non-stationarity of the volatility process is of less concern when Maximum Likelihood method is used, as described in the next section.

  22. Note that we could have fine-tuned the parameter values of Model 2 because of its different functional form from RiskMetrics. Here we adopt the values from RiskMetrics for simplicity.

  23. Fornari and Mele (2006), for instance, used Nelson’s (1990) interpretation of GARCH(1,1) and, thus, required more sophisticated econometric methods such as Indirect Inference.

  24. However, we stress that our focus is on equilibrium-consistency of modeling practices rather than capturing “asymmetric volatility” (leverage effect) alone because the latter may have less significant implications. For example, it is still an open question whether leverage effect exists if stochastic conditional mean rates of return are modeled.

  25. The choice of N is arbitrary. We conjecture that it is less relevant to the model comparison here, because the models that we compare adopt the same number.

  26. In Model 1, we show \(\hat{{\sigma }}_{t+1}^2 \) instead of \(\hat{{S}}\) to draw an analog to the “Moving Average” model.

  27. “Forecast evaluation” is also frequently referred to in the literature as “back-testing.” We use this terminology following AB and Andersen et al. (2003).

  28. We give each model a sample of D daily (actual) returns immediately prior to each holdout observation (i.e., a rolling time window of size D). Each model then estimates the parameters using the D-trading day sample and produces a volatility forecast for the next day. We set \(D=2500\).

  29. The population \(b_0 \) and \(b_1 \) should be zero and one, respectively. However, “Errors-In-Variables” issues bias the regression coefficients estimators in MZ regressions, Eq. (24). Following AB, we use only the \(R^{2}\) criterion.

  30. The MSE of the out-of-sample forecasts (i.e., the forecasting errors), while commonly used in statistical predictive modeling, is not commonly used in the volatility literature.

  31. We thank Wayne Ferson for suggesting the motivation for using the MSE.

  32. We thank Jonathan Reeves for suggesting the DM tests.

  33. SPY has been considered the most liquid ETF and with relatively small tracking errors. We use the mid-point of bid-ask prices.

  34. We note that realized volatility measures using intraday data are more accurate but require high-frequency data, which is not always available. Thus, our empirical results for the second and third evaluation periods, which use intraday data, might be more reliable. Whenhigh-frequency data is not available, the standard practice is to use squared daily returns to measure realized volatility. We are unaware of material concerns in the literature regarding using this measure (see, for example, Boudoukh et al. 1997).

  35. Our result is consistent with that of AB in that \(R^{2}\)’s for the volatility models are higher when intraday returns are used for realized volatility. For our purpose, however, the absolute magnitude of the \(R^{2}\)’s are not of concern.

  36. We are not aware of any test in the volatility literature that can assess the statistical significance of the difference in \(R^{2}\)’s. One can interpret the reported differences in the \(R^{2}\)’s as the differences in economic significance of the volatility forecasts [see AB and Andersen et al. (2003)].

  37. They provide three pieces of evidence: (1) the t tests on \(b_0 \) and \(b_1 \)’s differences from 0 and 1, respectively, within the evaluation regressions give similar conclusions for our models and the corresponding reference models (although \(b_0 \) and \(b_1 \) may not accurately evaluate model bias for the reason discussed in Footnote 29); (2) the z tests on differences between \(b_0 \) and \(b_1 \) of our and corresponding models, assuming model independence (see Clogg et al. 1995), show no significant differences; and (3) the tests of Gujarati (2003, pp. 306–310) with heteroskedasticity robust standard errors show no significant differences between \(b_0 \) and \(b_1 \) of our model and corresponding models.

  38. The tables and graphs for other models are qualitatively similar.

  39. We define a day as “high-volatility” if, in holdout samples 2 and 3, the square root of an intraday realized volatility is \(\ge \)3 %, or, in holdout sample 1, the absolute value of a realized rate of return is \(\ge 5\,\% \). Results are robust if we slightly change the thresholds.

  40. We also performed another robustness check by using averages of squared daily returns in the following 5 days as a measure of realized volatility, following Boudoukh et al. (1997). The results (not shown here) are similar.

  41. The tables and graphs for other models [see Xu (2013, Appendix 4B)] are qualitatively similar.

  42. For example, we could test the “10-(trading) days-ahead” volatility forecasts (see, for example, Christoffersen and Diebold 2000).

  43. In this study, we consider only the first-order relationships, i.e., \((\mu _{t+1} ,\sigma _{t+1}^2 )\) depend only on \((\mu _t ,\sigma _t^2 )\).

  44. We implement the BHHH algorithm by supplying an “Outer Product of Gradient” estimator of the Hessian matrix to a built-in optimization procedure in MATLAB.

  45. In the implementation of GARCH(1,1), we set the initial value as \(\sigma _1^2 =\frac{\alpha _0 }{1-\alpha _1 -\alpha _2 }\), which is standard practice.

  46. We only need a rough approximation here because the initial value is not essential for estimation or forecasting due to the large sample size D.


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Correspondence to David Feldman.

Additional information

This paper was drafted at the University of New South Wales, Australia, as part of the preparation for Xin Xu’s Ph.D. thesis, Xu (2013). The views expressed in this paper are those of the author and do not necessarily reflect those of the employer. We thank Tony Berrada, Avi Bick, Wayne Ferson, Giampaolo Gabbi, Dan Galai, Hayette Gatfaoui, Ufuk Gucbilmez, Robert Kohn, Delphine Lautier, Jaehoon Lee, Frederik Lundtofte, Valentyn Panchenko, Linda Pesante, Robert Faff, Haim Reisman, Jonathan Reeves, Konark Saxena, Tzu Tai, Jing Xu, and Jin Zhang for helpful discussions, and we thank seminar participants at UNSW School of Banking and Finance, University of Zurich, Australasian Finance and Banking Conference, French Finance Association Conference, International Risk Management Conference, Pacific Basin Finance Economics Accounting and Management Conference, the Bicentenary Conference of the Italian Academy of Management (AIDEA), North American Winter Meetings of the Econometric Society joint with the American Economic Association, International Finance and Banking Society Conference, Portuguese Finance Network International Conference, The Econometric Society Australasian Meeting, Auckland Finance Meeting, the International Finance Conference, and the European Financial Management Association Meeting. We especially thank two anonymous referees for comments and suggestions that improved this article.

Appendix: MLE for Model 1 and Model 3

Appendix: MLE for Model 1 and Model 3

For Model 3, we obtain MLEs for \((\mu ,S)\) as follows. We first write down the process for the rate of return, \(y_{t+1} \):

$$\begin{aligned} y_{t+1}= & {} \mu +\sigma _{t+1} \varepsilon _{t+1} ;\quad \varepsilon _{t+1} \sim N(0,1);\quad E(\varepsilon _i \varepsilon _j )=0,\quad \forall \, i\ne j \nonumber \\ \sigma _{t+1}^2= & {} \frac{S}{M_t }. \end{aligned}$$

Ignoring the constant terms, the log-likelihood functions for the parameters given one observation and for the parameters given the N-day sample, respectively, are

$$\begin{aligned} l^{s}= & {} -\frac{1}{2}\log \sigma _s^2 -\frac{(y_s -\mu )^{2}}{2\sigma _s^2 } \nonumber \\= & {} \frac{1}{2}\log M_{s-1} -\frac{1}{2}\log S-\frac{(y_s -\mu )^{2}M_{s-1} }{2S}, \nonumber \\ L= & {} -\frac{N}{2}\log S-\frac{\sum \limits _{s=1}^N [ (y_s -\mu )^{2}M_{s-1} ]}{2S}. \end{aligned}$$

From the first-order condition of maximizing L, we have

$$\begin{aligned} \hat{{\mu }}=\frac{\sum \limits _{s=t-N+1}^t {y_s M_{s-1} } }{\sum \limits _{s=t-N+1}^t {M_{s-1} } },\quad \quad \hat{{S}}=\frac{\sum \limits _{s=t-N+1}^t {(y_s -\hat{{\mu }})^{2}M_{s-1} } }{N}, \end{aligned}$$

and, consequently, obtain Model 3, described in Eq. (19). We could have fine-tuned N (for instance, using cross-validation) to produce better forecasts, but in this study we set \(N=10\) for simplicity. We note that Model 3 can be thought of as a “weighted moving average” model, weighted by the past prices, and thus is very similar to the MLE of its reference model, the “moving average” model (i.e., constant \(\sigma ^2\) and \(\mu \)).

We obtain the standard error of the estimators, from their asymptotic covariance matrix, \((N\hat{{\mathcal {I}}})^{-1}\), where \(\hat{{\mathcal {I}}}\) is the estimator of the information matrix. We obtain \(\hat{{\mathcal {I}}}\) from the sample version of the expected Hessian matrix,

$$\begin{aligned} \hat{{\mathcal {I}}}\mathop {=}\limits ^{a}\left[ {{\begin{array}{ll} {\frac{\sum \limits _{s=t-N+1}^t {M_{s-1} } }{N\hat{{S}}}}&{} 0 \\ 0&{} {\frac{1}{2\hat{{S}}^{2}}} \\ \end{array} }} \right] , \end{aligned}$$

where \(\mathop {=}\limits ^{a}\) denotes asymptotic equivalence. We note the similarity of the above asymptotic covariance matrix, Eq. (28), to that of the constant volatility model. From the standard error of \(\hat{{S}}\), we can find the standard error of \(\hat{{\sigma }}_{{t+1}^2}^\mathrm{MLE}\), which is \(\left( {\sqrt{\frac{2}{N}}} \right) \hat{{\sigma }}_{{t+1}^2}^\mathrm{MLE}\).

For Model 1, we basically follow the estimation procedures developed in Engle (1982) and Bollerslev (1986). In a general form, denoting \(\Theta \) as the unknown parameters to be estimated, we have the process for the rate of return, \(y_{t+1} \):Footnote 43

$$\begin{aligned}&y_{t+1} =\mu _{t+1} +\sigma _{t+1} \varepsilon _{t+1} ;\quad \varepsilon _{t+1} \sim N(0,1);\quad E(\varepsilon _i \varepsilon _j )=0,\quad \forall \, i\ne j, \nonumber \\&\mu _{t+1} =m(\Theta ,\mu _t ,\sigma _t^2 ), \nonumber \\&\sigma _{t+1}^2 =s(\Theta ,\mu _t ,\sigma _t^2 ), \end{aligned}$$

where \(m(\cdot )\) and \(s(\cdot )\) are differentiable functions. Using Eq. (29), we can take the first derivative of \(\mu _{t+1} \) and \(\sigma _{t+1}^2 \) w.r.t. \(\Theta \):

$$\begin{aligned} \frac{\partial \mu _{t+1} }{\partial \Theta }= & {} \frac{\partial m}{\partial \Theta }+\frac{\partial m}{\partial \mu _t }\frac{\partial \mu _t }{\partial \Theta }+\frac{\partial m}{\partial \sigma _t^2 }\frac{\partial \sigma _t^2 }{\partial \Theta }, \nonumber \\ \frac{\partial \sigma _{t+1}^2 }{\partial \Theta }= & {} \frac{\partial s}{\partial \Theta }+\frac{\partial s}{\partial \mu _t }\frac{\partial \mu _t }{\partial \Theta }+\frac{\partial s}{\partial \sigma _t^2 }\frac{\partial \sigma _t^2 }{\partial \Theta }. \end{aligned}$$

Ignoring the constant terms, the conditional log-likelihood functions for \(\Theta \) given one observation and for \(\Theta \) given the whole sample, respectively, are

$$\begin{aligned} l^{t}=-\frac{1}{2}[\log \sigma _t^2 +(y_t -\mu _t )^{2}(\sigma _t^2 )^{-1}],\quad \quad \quad L=\sum _{t=1}^D {l^{t}} , \end{aligned}$$

where D is the sample size. Taking the partial derivative of \(l^{t}\) w.r.t. \(\Theta \) gives the score (vector) for one observation:

$$\begin{aligned} \frac{\partial l^{t}}{\partial \Theta }=-\frac{1}{2}\left( \frac{\partial \sigma _t^2 }{\partial \Theta }\right) \left[ (\sigma _t^2 )^{-1}-(\sigma _t^2 )^{-2}(y_t -\mu _t )^{2}\right] +\left( \frac{\partial \mu _t }{\partial \Theta }\right) \left( \sigma _t^2 )^{-1}(y_t -\mu _t \right) . \end{aligned}$$

Equations (30) and (32) together define an iterative procedure to find the score (vector) for the whole sample.

We use a numerical procedure to maximize the log-likelihood function L in Eq. (31). As in Engle (1982) and Bollerslev (1986), we apply the Beetrndt, Hall, Hall and Hausman (1974, BHHH) algorithm to find the MLEs.Footnote 44 By the law of iterated expectations and after simplifying, the expected Hessian matrix is

$$\begin{aligned} \hbox {E}H=-D\hbox {E}\left[ \frac{(\sigma _t^2 )^{-2}}{2}\left( \frac{\partial \sigma _t^2 }{\partial \Theta }\right) \left( \frac{\partial \sigma _t^2 }{\partial \Theta }\right) '({\partial \sigma _t^2})^{-1}\left( \frac{\partial \mu _t }{\partial \Theta }\right) \left( \frac{\partial \mu _t }{\partial \Theta }\right) '\right] \end{aligned}$$

where “\({\prime }\)” is the “transpose” operator. The sample version of (33) is \(\hat{{H}}=-\sum \nolimits _{t=1}^{D}\left[ \frac{(\sigma _t^2 )^{-2}}{2}\left( \frac{\partial \sigma _t^2 }{\partial \Theta }\right) \left( \frac{\partial \sigma _t^2 }{\partial \Theta }\right) '({\partial \sigma _t^2})^{-1}\left( \frac{\partial \mu _t }{\partial \Theta }\right) \left( \frac{\partial \mu _t }{\partial \Theta }\right) '\right] \). In the implementation, we use the robust asymptotic covariance matrix (i.e., the “sandwich” form). Denoting \(\hat{{V}}=\sum \nolimits _{t=1}^{D}\left( \frac{\partial l^{t}}{\partial \Theta })(\frac{\partial l^{t}}{\partial \Theta }\right) '\) as the “Outer Product of Gradient” estimator of the Hessian matrix, the robust asymptotic covariance matrix is \(\hat{{H}}^{-1}\hat{{V}}\hat{{H}}^{-1}\).

We set the initial value, \(\sigma _1^2 \), to be the long-run mean of \(\sigma _t^2 \), \(\hbox {E}\sigma _t^2 \).Footnote 45 By the assumptions of i.i.d. \(\varepsilon _t ,\;\;\forall t\) and continuous compounding, we take the unconditional expectation on both sides of Eq. (13) and obtain

$$\begin{aligned} \hbox {E}\sigma _t^2 =\frac{\alpha _0 }{M_0 }\exp \left[ (-t+1)\mu +\frac{1}{2}\sum _{s=1}^{t-1} \hbox {E} \sigma _s^2\right] +(\alpha _1 +\alpha _2 +\mu ^{2}-\mu )\hbox {E}\sigma _{t-1}^2 +\hbox {E}(\sigma _{t-1}^2 )^{2}. \end{aligned}$$

Further assuming that \(M_0 \) is finite, that the unconditional variance of \(\sigma _t^2 \) exists, and that \(\hbox {E}\sigma _t^2 <2\mu \), asymptotically (i.e., \(t\rightarrow \infty )\) we have

$$\begin{aligned} \hbox {E}\sigma _t^2 \mathop {=}\limits ^{a} \;\frac{\hbox {E}[(\sigma _t^2 )^{2}]}{1+\mu -\mu ^{2}-\alpha _1 -\alpha _2 }. \end{aligned}$$

In the implementation, we use Eq. (35) as \(\sigma _1^2 \), where \(\hbox {E}[(\sigma _t^2 )^{2}]\) is approximated by the forth moment of the sample, \(\frac{1}{D}\sum \nolimits _{t=1}^D {\left( y_t -\frac{1}{D}\sum \nolimits _{s=1}^D {y_s } \right) ^{4}} \).Footnote 46

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Feldman, D., Xu, X. Equilibrium-based volatility models of the market portfolio rate of return (peacock tails or stotting gazelles). Ann Oper Res 262, 493–518 (2018).

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  • Market risk
  • Volatility model
  • Systematic risk
  • Market portfolio
  • Predictive power
  • Equilibrium
  • RiskMetrics
  • Piecewise constant volatility
  • Constant elasticity of variance

JEL Classification

  • G17
  • G12
  • C58