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

David Feldman; Xin Xu

doi:10.1007/s10479-015-1972-8

Annals of Operations Research

March 2018, Volume 262, Issue 2, pp 493–518 | Cite as

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

Authors
Authors and affiliations

David FeldmanEmail author
Xin Xu

S.I.: Financial Economics

First Online: 02 September 2015

150 Downloads

Abstract

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.

Keywords

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

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.

JEL Classification

G17 G12 C58

This is a preview of subscription content, to check access.

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} $:

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

(25)

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

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

(26)

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

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

(27)

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,

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

(28)

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} $:⁴³

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

(29)

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 $:

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

(30)

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

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

(31)

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

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

(32)

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.⁴⁴ By the law of iterated expectations and after simplifying, the expected Hessian matrix is

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

(33)

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 $.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

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

(34)

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

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

(35)

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}} $.⁴⁶

References

Allen, L., Boudoukh, J., & Saunders, A. (2003). Understanding market, credit, and operational risk: The value at risk approach. Hoboken: Wiley.Google Scholar
Andersen, T. G., & Bollerslev, T. (1998). Answering the skeptics: Yes, standard volatility models do provide accurate forecasts. International Economic Review, 39, 885–905.CrossRefGoogle Scholar
Andersen, T. G., Bollerslev, T., Diebold, F. X., & Labys, P. (2003). Modeling and forecasting realized volatility. Econometrica, 71, 579–625.CrossRefGoogle Scholar
BCBS. (2011). Messages from the academic literature on risk measurement for the trading book. Working paper no. 19, Basel Committee on Banking Supervision.Google Scholar
Berndt, E. K., Hall, B. H., Hall, R. E., & Hausman, J. A. (1974). Estimation inference in nonlinear structural models. Annals of Economic and Social Measurement, 4, 653–665.Google Scholar
Bick, A. (1990). On viable diffusion price processes of the market portfolio. The Journal of Finance, 45, 673–689.CrossRefGoogle Scholar
Black, F. (1976). Studies in stock price volatility changes.’ In Proceedings of the 1976 business meeting of the business and economic statistics section (pp. 177–181). American Statistical Association.Google Scholar
Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. The Journal of Econometrics, 31, 307–327.CrossRefGoogle Scholar
Bollerslev, T., Chou, R. Y., & Kroner, K. F. (1992). ARCH modeling in finance: A review of the theory and empirical evidence. Journal of Econometrics, 52, 5–59.CrossRefGoogle Scholar
Bollerslev, T., Engle, R. F., & Wooldridge, J. M. (1988). A capital asset pricing model with time-varying covariances. The Journal of Political Economy, 96, 116–131.CrossRefGoogle Scholar
Boudoukh, J., Richardson, M., & Whitelaw, R. (1997). An investigation of a class of volatility forecasting models. Journal of Derivatives, 4, 63–71.CrossRefGoogle Scholar
Christoffersen, P. F., & Diebold, F. X. (2000). How relevant is volatility forecasting for financial risk management? The Review of Economics and Statistics, 82, 12–22.CrossRefGoogle Scholar
Christoffersen, P. F., Hahn, J., & Inoue, A. (2001). Testing and comparing value-at-risk measures. Journal of Empirical Finance, 8, 325–342.CrossRefGoogle Scholar
Clogg, C. C., Petkova, E., & Haritou, A. (1995). Statistical methods for comparing regression coefficients between models. American Journal of Sociology, 100, 1261–1293.CrossRefGoogle Scholar
Corradi, V. (2000). Reconsidering the continuous time limit of the GARCH(1,1) process. Journal of Econometrics, 96, 145–153.CrossRefGoogle Scholar
Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985a). An intertemporal general equilibrium model of asset prices. Econometrica, 51, 363–383.CrossRefGoogle Scholar
Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985b). A theory of the term structure of interest rates. Econometrica, 53, 385–407.CrossRefGoogle Scholar
Cox, J. C., & Ross, S. A. (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3, 145–166.CrossRefGoogle Scholar
Crouhy, M., Galai, D., & Mark, R. (2001). Risk management. New York: McGraw-Hill.Google Scholar
Diebold, F. X., & Mariano, R. S. (2002). Comparing predictive accuracy. Journal of Business and Economic Statistics, 20, 134–144.CrossRefGoogle Scholar
Dothan, M. U., & Feldman, D. (1986). Equilibrium interest rates and multiperiod bonds in a partially observable economy. The Journal of Finance, 41, 369–382.CrossRefGoogle Scholar
Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica, 50, 987–1007.CrossRefGoogle Scholar
Engle, R. F. (2002). New frontiers for ARCH models. Journal of Applied Econometrics, 17, 425–446.CrossRefGoogle Scholar
Feldman, D. (1989). The term structure of interest rates in a partially observable economy. The Journal of Finance, 44, 789–812.CrossRefGoogle Scholar
Feldman, D. (2007). Incomplete information equilibria separation theorems and other myths. Annals of Operations Research, Special Issue on Financial Modeling, 151, 119–149.CrossRefGoogle Scholar
Fornari, F., & Mele, A. (2006). Approximating volatility diffusions with CEV-ARCH models. Journal of Economic Dynamics and Control, 30, 931–966.CrossRefGoogle Scholar
Friedman, M. (1953). Essays in positive economics. Chicago: The University of Chicago Press.Google Scholar
Gujarati, D. N. (2003). Basic econometrics (4th ed.). New York: McGraw-Hill.Google Scholar
Hansen, P. R., & Lunde, A. (2005). A forecast comparison of volatility models: Does anything beat a GARCH(1, 1)? Journal of Applied Econometrics, 20(7), 873–889.CrossRefGoogle Scholar
Heath, D. P., & Platen, E. (2002). Pricing and hedging of index derivatives under an alternative asset price model with endogenous stochastic volatility. In J. Yong (Ed.), Recent developments in mathematical finance (pp. 117–126). Singapore: World Scientific.Google Scholar
He, H., & Leland, H. (1993). On equilibrium asset price processes. The Review of Financial Studies, 6, 593–617.CrossRefGoogle Scholar
Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies, 6, 327–343.CrossRefGoogle Scholar
Hull, J., & White, A. (1987). The pricing of options on assets with stochastic volatility. Journal of Finance, 3, 281–300.CrossRefGoogle Scholar
Jorion, P. (2007). Value at risk: The new benchmark for managing financial risk (3rd ed.). New York: McGraw-Hill.Google Scholar
Liptser, R. S., & Shiryaev, A. N. (2001). Statistics of random processes, I. General theory (2nd ed.). Berlin: Springer.CrossRefGoogle Scholar
Litterman, R., & Winkelmann, K. (1998). Estimating covariance matrices, risk management series. New York: Goldman, Sachs & Co.Google Scholar
Lucas, R. E. (1978). Asset prices in an exchange economy. Econometrica, 46, 1429–1445.CrossRefGoogle Scholar
Merton, R. (1971). Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3, 373–413.CrossRefGoogle Scholar
Merton, R. (1973). An intertemporal capital asset pricing model. Econometrica, 41, 867–887.CrossRefGoogle Scholar
Mincer, J., & Zarnowitz, V. (1969). The evaluation of economic forecasts. In J. Mincer (Ed.), Economic forecasts and expectations. Cambridge, MA: National Bureau of Economic Research.Google Scholar
Morgan/Reuters, J. P. (1996). RiskMetrics $^{TM}$–Technical document (4th ed.). New York: Morgan Guaranty Trust Company and Reuters Ltd.Google Scholar
Nelson, D. B. (1990). ARCH models as diffusion approximation. Journal of Econometrics, 45, 7–38.CrossRefGoogle Scholar
Park, B.-J. (2014). Time-varying, heterogeneous risk aversion and dynamics of asset prices among boundedly rational agents. Journal of Banking and Finance, 43, 150–159.CrossRefGoogle Scholar
Patton, A. J. (2011). Volatility forecast comparison using imperfect volatility proxies. The Journal of Econometrics, 160, 246–256.CrossRefGoogle Scholar
Phelan, M. J. (1995). Probability and statistics applied to the practice of financial risk management: The case of JP Morgan’s riskmetrics $^{TM}$. Working paper, Wharton School, the University of Pennsylvania.Google Scholar
Platen, E., & Semmler, W. (2012). A dynamic portfolio approach to asset markets and monetary policy. Stochastic processes, finance and control: A festschrift in honor of Robert J. Elliott, pp. 347–373, Advances in statistics, probability and actuarial sciences, Vol. 1. Singapore: World Scientific.Google Scholar
Platen, E. (2011). A benchmark approach to investing and pricing. In L. MacLean, E. Thorp, & W. T. Ziemba (Eds.), The Kelly capital growth investment criterion. World Scientific handbook in financial economics series (Vol. 3, pp. 409–426). Singapore: World Scientific.CrossRefGoogle Scholar
Schorfheide, F., Song, D., & Yaron, A. (2013). Identifying long-run risks: A Bayesian mixed-frequency approach. Working paper, University of Pennsylvania.Google Scholar
Singleton, K. J. (2006). Empirical dynamic asset pricing: Model specification and econometric assessment. Princeton: Princeton University Press.Google Scholar
Wiener, Z. (1999). Introduction to VaR, risk management and regulation in banking. New York: Kluwer Academic.Google Scholar
Xu, X. (2013). Modeling risk of the multi-period market portfolio: An equilibrium-based approach. PhD thesis, University of New South Wales, Australia. The link for the thesis is: http://primoa.library.unsw.edu.au/primo_library/libweb/tiles/lrs/unsworks/datastream.jsp?pid=UNSWorks_ALMA11687.
Zahavi, A., & Zahavi, A. (1997). The handicap principle: A missing piece of Darwin’s puzzle. New York: Oxford University Press.Google Scholar
Zieling, D., Mahayni, A., & Balder, S. (2014). Performance evaluation of optimized portfolio insurance strategies. Journal of Banking and Finance, 43, 212–225.CrossRefGoogle Scholar
Zumbach, G. (2007). The RiskMetrics 2006 methodology. Technical report, RiskMetrics Group.Google Scholar

Copyright information

Authors and Affiliations

David Feldman
- 1
Email authorView author's OrcID profile
Xin Xu
- 2

1.Banking and Finance, Australian School of BusinessThe University of New South Wales (UNSW)UNSW SydneyAustralia
2.Fair Isaac (FICO)SydneyAustralia

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

Abstract

Keywords

JEL Classification

References

Copyright information

Authors and Affiliations

Personalised recommendations

Cite article