Exploiting Volatility Timing �

EMILIO OSAMBELA y

May 28, 2006

Abstract

This paper proposes a stochastic realized volatility model (SRV-NIG) based on high frequency return data and studies the economicvalue of volatility timing, by analyzing how the dynamic portfoliochoice of investors in a multiperiod setting is a¤ected by volatility.The SRV-NIG model is consistent with volatility clustering, asymmet-ric and peaked distribution for returns, the mixture of distributionshypothesis, the time-varying risk premium hypothesis and the leveragee¤ect. I compute the hedging demands that arise in the time-varyinginvestment opportunity set, and measure the utility gains of exploitingvolatility timing both in a SRV-NIG and GARCH framework, relativeto strategies based on iid returns. Volatility-timing strategies basedon the proposed SRV-NIG model signi�cantly outperform strategiesbased on GARCH models, and both of these models signi�cantly out-perform strategies based on the iid returns model. The obtained resultshold out-of-sample, and are robust to short-selling and borrowing con-straints.

1 Introduction

This paper proposes a new framework for modeling stocks return and theirvolatility, based in high frequency data. The model allows for volatilityclustering, asymetric and peaked distribution for returns, the mixture ofdistributions hypothesis, the time-varying risk premium hypothesis and theleverage e¤ect. I measure the utility gains from dynamic portfolio rebal-ancing due to short time variation in volatility using the proposed model,and compare it with GARCH and iid models. The results show that theproposed model signi�cantly ourperforms both GARCH and iid models in aporto�io selection context.

�I thank my advisors Eric Jondeau and Michael Rockinger, and especially Tony Berradafor guidance and encouragement in the whole process of writing this paper. Helpfulcomments by Samuel Bieri, Tim Bollerslev and Olivier Scaillet are greatly aknowledged.

ySwiss Finance Institute and HEC Lausanne. Route de Chavannes 33, CH-1007 Lau-sanne - Switzerland. Email: emilio.osambela@s�-phd.ch

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Samuelson (1969) and Merton (1969, 1971, and 1973) have shown the-oretically that multi-period investors should alter their optimal portfolioholdings to take advantage of time-varying investment opportunities in adi¤erent way than single-period (myopic) investors. The way in which theseoptimal portfolio holdings should be altered depends crucially on prefer-ences, and in the way in which the investment opportunity set is character-ized. For general (non-log) utility functions the investor will be concernedabout hedging against shifts in the future investment opportunity set.

In the last decade, �nancial economists have explored empirically therelevance of these arguments. Most of the attention has been devoted to an-alyzing the implications of expected stock return predictability in portfolioallocation assuming heteroskedastic returns. This is surprising consideringthe documented important role of both risk and return in the portfolio de-cision making, and the strong empirical evidence supporting time-varyingand predictable volatility, especially in the context of high frequency stockreturn data. In fact, the empirical evidence in favor of time-varying volatil-ity is stronger and more conclusive than that of time-varying expected re-turn. This has been �rmly established both in the traditional generalizedautoregressive conditional heteroskedasticity (GARCH) framework, and ina recent new approach called realized variation, which exploits the informa-tion in high-frequency stocks return data.1 Thus, a pertinent question isto what extent risk averse investors could gain by properly modeling andtiming the volatility of stocks returns in a portfolio selection context. Inorder to answer this question, an investment opportunity set characterizedby volatility is required. In this paper I propose a stochastic variance modelbased on realized variation estimates from high frequency data labelled SRV-NIG model, and compare its economic performance with the GARCH andiid models.

I analyze the portfolio choice in discrete time for an investor with con-stant relative risk aversion utility over terminal wealth. The investor allo-cates wealth to a risky and a riskless asset. He calibrates the investmentopportunity set using intradaily data and rebalances his portfolio on a dailybasis, with investment horizons ranging from one day to one year. In this set-ting, I identify the out-of-sample proportional hedging demands and utilitygains associated with di¤erent volatility timing rules under realistic frictionssuch as short-selling and borrowing constraints.

Some distinctive features of the analysis in this paper should be noted.To begin, this is the �rst paper to analyze the economic value of volatilitytiming using realized volatility in a dynamic portfolio choice environment.In that sense, the precursor of this paper is the one by Fleming, Kirby and

1For detailed surveys in the GARCH framework see Bollerslev, Chou and Kroner(1992), Campbell, Lo and MacKinlay (1997) and Engle (2003). For a survey in therealized volatility framework see Andersen, Bollerslev, Diebold and Labys (2003).

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Ostdiek (2003), which addresses a similar issue considering a mean-varianceframework. In contrast, in this paper I allow the investor to rebalance hisportfolio as time varying investment opportunities change over time betweenthe trading period and the terminal date. Thus, we are concerned with �nd-ing not a single-period optimal portfolio, but the optimal dynamic portfoliostrategy or contingent plan that speci�es how to adjust the portfolio choicein response to the changing investment opportunity set.

Second, the SRV-NIG model proposed in this paper is new, althoughit is closely related to the ARCH-NIG model proposed by Bandor¤-Nielsen(1997) and later extended to allow for more �exible GARCH-type para-metrizations in Andersson (2001), Jensen and Lunde (2001) and Forsbergand Bollerslev (2002). The main di¤erences are that I allow for the impactof lagged realized variation (in addition to that of lagged squared shocksin returns) in the parametrization of the conditional variance of returns,and that I take realized variation as an observable (as opposed to latent)process. Further, the structure of the model is �exible enough to allow si-multaneously for volatility clustering, asymmetric and peaked distributionfor returns, the mixture of distributions hypothesis, the time-varying riskpremium hypothesis and the leverage e¤ect.

Third, in contrast to most previous literature assessing the economicvalue of return predictability (with the notable exception of Marquering andVerbeek (2005) and Han (2006)) the performance analysis is conducted out-of-sample. This issue is particularly important since in practice investors caninfer the parameters of any model from historical data only. Thus, althoughsome studies like Barberis (2000) have tried to bypass the out-of-sampleanalysis by introducing parameter uncertainty, even in that setting any in-sample performance analysis is irrelevant from the investor point of view.In particular, Neeley and Weller (2000) and Handa and Tiwari (2005) re-port that the VAR predictive model has very poor out-of-sample forecastingpower despite its signi�cant in-sample predictability.

The obtained results indicate that volatility-timing strategies based onthe proposed SRV-NIG model signi�cantly outperform strategies based onGARCH-type models and on iid return models. The median of the yearlycertainty equivalent gain of switching from the iid return model to the SRV-NIG model is 239 basis points, compared to only 173 in the TARCH case(TARCH is the speci�cation which gives the best �t for GARCH-type mod-els). In addition, the TARCHmodel generates positive horizon e¤ects, whichlast for �ve months. Accordingly, it generates positive intertemporal hedg-ing demands, which may account for proportions in the order of 32% of theout-of-sample optimal portfolio allocation. This is in contrast with the SRV-NIG model, whose associated intertemporal hedging demands are found tobe negligible. However, in both cases, the intertemporal hedging demandsrepresent a negligible contribution to the certainty equivalent gains.

The remainder of the paper is organized as follows. Section 2 contains a

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brief review of related literature. Section 3 introduces the SRV-NIG modeland describes the benchmark models of GARCH and iid returns. Section4 sets up the optimal portfolio choice problem of the multiperiod investor.The calibration follows in section 5. In section 6 the optimal portfolio choicesolution and its empirical implementation are analyzed. Finally, some con-cluding remarks are o¤ered in section 7.

2 Related Literature

This paper is related to di¤erent economic branches of the �nancial eco-nomics literature. First, it relates to a number of recent papers such asBandor¤-Nielsen (1997), Andersson (2001), Jensen and Lunde (2001) andForsberg and Bollerslev (2002) the goal of enriching the modeling of stocksreturn and their volatility in order to fully capture the negative skewness andexcess kurtosis of observed returns by assuming a normal inverse gaussian(NIG) conditional distribution for returns However, these models do notconsider realized volatility as a factor explaining the conditional varianceof returns, and do not provide an assessment of the economic value of theproposed models in a portfolio allocation context. Both of these features arecentral in this paper.

Also, this paper shares with a second line of research the goal of eval-uating the economic value of volatility-timing in the context of investmentdecisions (see West, Edison and Cho (1993), Fleming, Kirby and Ostdiek(2001, 2003), Marquering and Verbeek (2005) and Han (2006)). However,these studies focus on the mean-variance paradigm of Markowitz (1952).Although this approach captures both the diversi�cation aspect and thetrade-o¤ between expected returns and risk, it presents several limitations.First, the mean-variance problem is consistent with expected utility maxi-mization only for the particular case of quadratic utility function (which isnot monotonically increasing in wealth). Second, the mean-variance problemneglects any preferences towards higher-order return moments (like skewnessor kurtosis). Finally, the mean-variance problem is by construction a myopicsingle-period problem, while most real investment problems involve longerhorizons with intermediate portfolio rebalancing. This paper aims to ad-dress these issues by analyzing the economic bene�ts of volatility timing inthe dynamic portfolio choice of an investor, considering an intertemporalexpected utility maximization approach.

In addition, this paper relates to the research focused in evaluating theimplications of predictability of stocks returns in dynamic portfolio choice(see for example Brennan, Schwartz and Lagnado (1997), Brandt (1999),Barberis (2000) and Brandt, Goyal, Santa-Clara and Stroud (2005)). Whilethis literature restricts to the analysis of the predictability of stock expected

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returns assuming constant volatility, in this paper I analyze the implica-tions of predictability in the volatility stocks of returns in a dynamic assetallocation context.

Finally, this paper coincides with a few others in the literature on as-sessing the impact of volatility timing in a dynamic asset allocation setting.Larsson (2001) explores the horizon e¤ects on portfolio choice under timevarying and forecastable GARCH return volatility. Wang (2003) examinesthe e¤ect of a multivariate ARCH time-varying volatility in stocks returnand dividend yield. Gomes (2004) studies the dynamic portfolio alloca-tion problem assuming an AR(1)-GARCH(1,1) process. While these studiesconsider a deterministic volatility measure, characteristic in GARCH-typemodels, here I propose an investment opportunity set where volatility is sto-chastic, and therefore allows exploring for the joint dynamics of volatilityand returns, in line with the mixture of distribution hypothesis. Chackoand Viceira (2005) derive approximate analytical solutions for the dynamicoptimal consumption and portfolio choice in continuous time when stock re-turn volatility follows a mean-reverting square root process instantaneouslycorrelated with stock returns. In this paper I consider the realized volatilityapproach where intradaily returns are used to construct daily return volatil-ity estimates. The main argument of this approach from the point of view ofthe investor implementation is to use a higher frequency than that of trad-ing to construct volatility estimates. This would be impossible to do in thecontinuous time setting of Chacko and Viceira (2005), where by de�nitionthere cannot be a higher frequency than that of continuous trading.

3 Investment opportunity set

I assume that wealth consists of only two tradable assets: a riskless assetwith constant return and a risky asset with stochastic return. This sectioncharacterizes the distributional assumptions for the return on the risky asset.Subsection 3.1 sets a general framework for return and volatility modeling.Based on this framework, subsection 3.2 proposes a model for return andvolatility which is consistent with the empirical evidence and theoreticalarguments on stocks return and volatility. Subsection 3.3 brie�y discussestwo well known return models which will be used as benchmarks: GARCHand iid return models.

3.1 A Framework for Return and Volatility modeling

Denote by p (t) the log-price of a �nancial asset at time t. Assume the fol-lowing di¤usion process for p (t), expressed in stochastic di¤erential equationform

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dp (t) = � (t) dt+p�2 (t)dW (t) (1)

where �2 (t) is a càdlàg strictly positive stochastic variance process,W (t) is astandard Brownian motion stochastically independent of �2 (t), and � (t) is acontinuous and locally bounded drift process. In particular, consistent withthe time-varying risk premium hypothesis, take as drift � (t) = �+��2 (t) :2

Then the log-price process is de�ned by

dp (t) =��+ ��2 (t)

dt+

p�2 (t)dW (t) (2)

While theoretical pricing arguments are often most conveniently ex-pressed in terms of continuous-time models, any empirical investigation isbased on discretely sampled returns. Denote the corresponding discrete-timereturns de�ned by equation (2) as

rt =

tZt�1

dp (s) = p (t)� p (t� 1) ; t = 1; 2; ::: (3)

where I take as unit-time interval t a day (dt = 1). Then, regardless of theprocess for �2 (t), daily log-returns satisfy

rt j �2t � N��+ ��2t ; �

2t

�(4)

where

�2t =

tZt�1

�2 (s) ds = �2 (t)� �2 (t� 1) (5)

is the integrated variance, which can be interpreted as the ex-post returnvariance conditional on the sample path realization �2 (s) over the (daily)time interval ht� 1; t] : While in general the integrated variance is not di-rectly observable, the realized variation literature starting with Andersenand Bollerslev (1998), Andersen, Bollerslev, Diebold and Labys (2001) andBandor¤-Nielsen and Shepard (2002) proposes a non-parametric approachfor accurately measuring it using high-frequency data. Mainly, the ideaconsists in estimating the integrated variance by summing the squares ofintradaily returns sampled at very short intervals of length �.3 More specif-ically, assume that a day t can be divided into 1=� high-frequency equidis-tant periods. Then, daily realized variation is de�ned as

2A process is locally bounded if is is bounded at every instant t. Thus, if the stochasticvariance �2 (t) is well de�ned (does not go to in�nity) at every instant t, the process � (t)is locally bounded.

3As noted by Merton (1980), the integrated volatility of a Brownian motion can beapproximated to an arbitrary precision by using the intradaily sum of squared returns.

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RVt (�) =

1=��1Xj=0

r2t�j�;� (6)

By the theory of quadratic variation, RVt (�) provides a consistent es-timator of the daily integrated variance for the underlying process pt inequation (2). This implies that for � �! 0 we get

RVt (�)p�!

tZt�1

�2 (s) ds (7)

While the bene�ts of increasing the frequency 1=� is the gain in preci-sion, the cost of sampling at very high frequencies is that it may introducea bias in the variance estimate due to market microstructure e¤ects. Asa trade o¤ between both, Anderson, Bollerslev, Diebold and Labys (2001)propose to use �ve minutes intradaily frequency to compute daily realizedvolatility. When calibrating the model, I follow this advice.

An important advantage of the realized variation approach is that itprovides a model-free volatility measure, in the sense that using this methodwe obtain a time series of the daily ariance measure RVt whose dynamicsare not linked to any particular model. This is in contrast to GARCH typemodels, which may be seen as particular approximations to the expectationof the integrated variance, conditional on the information set available attime t� 1. Thus, if we denote by �t�1 the information set available at timet� 1, the GARCH variance may be expressed as

ht = Et�1

0@ tZt�1

�2 (s) ds j �t�1

1A (8)

As noted by Andersen, Bollerslev and Frederiksen (2006), in general thestochastic variance varies non trivially over the time-interval ht� 1; t], sothat returns standardized by the GARCH variance will result in a mixtureof distributions, with the mixture dictated by the integrated variance dis-tribution.

3.2 The SRV-NIG model

Based on the framework described in the previous subsection, I propose theSRV-NIG model as

Rt = �+ �RVt + "t, with "t =pRVtzt and zt � N (0; 1) (9)

where

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RVt j �t�1 � IG��t (�t�1; �) ;

q�2 � �2

�(10)

and

�t (�t�1; �) = �0 + �1�t�1 + �2RVt�1 + �3"2t�1 + �4If"t�1<0g (11)

In this setup, Rt is the daily log-return of the risky asset from day t� 1to t, while N and IG denote the normal and inverse gaussian distributions,respectively.4 By combining the distributional assumptions in (9) and (10),the conditional distribution for returns follows a normal inverse gaussian(NIG)

f (Rt j �t�1) =Zf (Rt j RVt) f (RVt j �t�1) dRVt = fNIG (Rt;�; �; �; �t (�t�1; �))

(12)where f denotes the density function. The NIG distribution was �rst usedfor modeling asset returns by Bandor¤-Nielsen (1997). It may be viewed as aparticular case of the generalized hyperbolic distribution of Bandor¤-Nielsen(1978). It is able to reproduce asymmetric distributions with possibly longtails in both directions, and it is analytically tractable.

Equation (9) is consistent with the mixture of distribution hypothesis(MDH). The MDH states that the distribution of discrete sampled returnsis normal conditional on some latent information arrival process. In general,the integrated variance process which serves as a mixture variable is assumedunobservable. However, as explained above, by using realized variation itis possible to obtain arbitrarily precise estimates of the integrated variance.Therefore, in the SRV-NIG model the realizations of the stochastic (real-ized) variance are considered observable, which is in line with the existingliterature on realized variation modeling.5 This is in contrast with most ofthe GARCH-NIG literature, which have assumed the variance of returns asa latent variable.6

In equation (10) I propose an IG distribution for the realized variationprocess. The IG distribution is de�ned over [0;1i and presents positiveskewness and positive excess kurtosis. The �rst parameter �t is proportionalto the mean and variance of the distribution, and negatively related with theskewness and kurtosis. The second parameter

p�2 � �2 is negatively related

4For the IG distribution to be well de�ned, it must hold that �t > 0 for all t and that0 <j � j< �: In order to ensure a positive and stationary process for �t, the followingconditions are imposed: �i > 0 for all i = f1; 2; 3; 4g and �1 + �2 + �3 < 1:

5See for example Corsi(2003), Andersen, Bollerslev and Diebold (2003) and Bollerslev,Kretschmer, Pigorsh and Tauchen (2005).

6To my knowledge, the only exception is the paper by Forsberg and Bollerslev (2002).

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to the �rst four moments of the distribution.7 Forsberg and Bollerslev (2002)have shown that the IG distribution provides an empirically good �t forrealized variation both unconditionally and conditionally.

Equation (11) is the proposed speci�cation for �t, which is proportionalto the conditional expectation of RVt. From equations (7) and (8), we knowthat as �! 0, the GARCH variance is the conditional expectation of RVt.Thus, a GARCH-type parametrization for �t seems reasonable. Accord-ingly, the �rst component is a constant related to the steady state value forthe parameter �: The second component is related to the the lagged con-ditional expectation of RVt. The next two components capture the docu-mented volatility clustering, using as proxies for lagged variance both laggedsquared shocks in returns (as is common practice in GARCH-type models)and lagged realized variation (which is a novel input in this setup). Whilethe GARCH parametrization assumes volatility as a latent process, in thesetup proposed here where volatility is considered as observable, it is in-tuitive to use past "observed" volatility as a component of the conditionalexpected volatility. Finally, the last component is related to the asymmetrice¤ect of news in volatility, corresponding to the leverage e¤ect.

The improved speci�cation for �t is one of the main contributions in thispaper, which extends the GARCH-NIG framework in empirically importantdirections. If �t is taken as a constant, we would get the model proposed byClark (1973).8 If we let � = � = �1 = �2 = �4 = 0, we get the ARCH-NIGmodel proposed by Bandor¤-Nielsen (1997). By letting � = � = �2 = �4 = 0we get the GARCH-NIG model proposed by Andersson (2001) and Forsbergand Bollerslev (2002). Finally, by considering �2 = �4 = 0 and imposingan APARCH structure for �t, we get the NIG-S&ARCH model proposed byJensen and Lunde (2001). As noted, the SRV-NIG model can be understoodas a generalization of all these models. Further, none of the previous modelhas being evaluated in a portfolio selection context. This will be done in thenext sections of the paper.

Equation (12) shows that the mixture of distributions in (6) and (7) leadsto the conditional NIG distribution of returns. The parameters (�; �; �; �t)are interpretable as follows: � and � are shape parameters which give steep-ness and asymmetry, respectively. Regarding � and �t, they are, respectively,a location and scale parameter.9 Accordingly, the normal distribution is aparticular case of the NIG distribution for � �!1, � = 0, and �t

� = �2 for

7The conditional mean, variance, skewness and kurtosis of realized volatility are, re-spectively, E = �tp

�2��2; V = �t

(�2��2)3=2; S = 3

p�t(�2��2)

1=4 and K = 3 + 15

�tp�2��2

:

8He assumes an iid distribution for volatility (which is inconsistent with volatilityclustering) but assumes a lognormal instead of an IG distribution.

9The conditional mean, variance, skewness and kurtosis of returns are, respec-tively, E = � + � �tp

�2��2; V = �2�t

(�2��2)3=2; S = 3�

�p�t(�2��2)

1=4 and K = 3 +

3h1 + 4

���

�2i 1

�tp�2��2

:

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all t. Similarly, the NIG distribution converges to the Cauchy distributionas � = 0, � = 0, � = 1; and �t = 0 for all t.

We should be careful in the interpretation of the parameter �. Fromequation (9) one would expect it to be positive consistent with the timevarying risk premium hypothesis. However, since it captures the skewnessof the conditional distribution of returns, it should be negative according tothe empirical evidence. Unfortunately, the proposed model does not allowto disentangle these e¤ects. Therefore the obtained parameter � will repre-sent the total e¤ect of time varying risk premium hypothesis and negativeskewness of returns.

It is important to notice that since �t de�nes the full distribution of bothreturns and realized variation, by testing volatility clustering and leveragee¤ects in this parameter, we are actually considering their global impact inthe distribution of both returns and realized variation.

From the investor perspective, the state variable observed at time tused for forecasting future stock return Rt+1 and realized variation RVt+1 isZSRVt = f�t+1g. Thus, after calibrating the parameters �SRV = f�; �; �; �0;�1; �2; �3; �4g, every period t he constructs the state variable according to

�t+1 = �0 + �1�t + �2RVt + �3(Rt � �� �RVt)2 + �4IfRt����RVt<0g (13)

Then he uses as predictive distributions for future stocks return andrealized variation

Rt+1 j �t+1 � NIG (�; �; �; �t+1) (14)

RVt+1 j �t+1 � IG��t+1;

q�2 � �2

�The predictive distribution for the future state variable is obtained by

leading one period equation (13), and taking Rt+1 and RVt+1 from (14).

3.3 Benchmark models

In this subsection I brie�y discuss two well known return models which willbe used as benchmarks: GARCH and iid return models.

3.3.1 GARCH models

The �rst autoregressive conditional heteroskedasticity (ARCH) model wasdeveloped by Engle (1982). Since then, several modi�cations have been pro-posed. In this section I present the GARCH (generalized ARCH), GARCH-

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M (GARCH in mean), the TARCH (threshold ARCH) model and the TARCH-M model, which is a mixture of the three.10 Consider the GARCH structure

Rt = �+ �ht + "t, with "t =phtzt and zt � N (0; 1) (15)

where

ht = �0 + �1ht�1 + �3"2t�1 + �4"

2t�1If"t�1<0g (16)

I label the general model presented as the TARCH-M model.11 If �4 = 0,this is the GARCH-M model, while in the case of � = 0 we get the TARCHmodel. Finally, the case where �4 = � = 0 corresponds to the GARCHmodel.12

In sharp contrast with the SRV-NIG model, GARCH-type models con-sider variance as an unobservable variable and therefore the model has tobe restricted to fully specify the conditional variance as a function of pastinnovations of returns and past variance itself. These restrictions requirethe GARCH variance measure to be deterministic by construction so thatthe only source of uncertainty comes from innovations to stock returns.

From the investor perspective, the state variable observed at time t usedfor forecasting future stock return Rt+1 is ZGARCHt = fht+1g. Thus, aftercalibrating the parameters �GARCH = f�; �; �0; �1; �3; �4g, every period t heconstructs the state variable according to

ht+1 = �0 + �1ht + �3(Rt � �� �ht)2 + �4IfRt����ht<0g (17)

Then he uses as predictive distribution for future stocks return

Rt+1 j ht+1 � N (�+ �ht+1; ht+1) (18)

The predictive distribution for the future state variable is obtained byleading one period equation (17), and taking Rt+1 from (18).

3.3.2 The iid return model

Finally we consider a model of identically and independently distributed(iid) returns

Rt = �+ "t, with "t � N�0; �2

�(19)

10The GARCH model was �rst proposed by Bollerslev (1986), the GARCH-in-mean wasdeveloped by Engle, Liliens and Robins (1987) and the TARCH was proposed by Glosten,Jaganathan, and Runkle (1993).11 In order to ensure a positive conditional variance and stationarity, the following con-

ditions are imposed: �i > 0 for all i = f1; 3; 4g and �1 + �3 < 1:12For simplicity I consider only one lag for past volatility as well as past squared returns.

Although strictly speaking this is a GARCH(1,1), through the paper we refer to it simplyas GARCH.

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In this process returns are homoskedastic and there is no state variable whichcan be used for predicting future returns. Therefore, after calibrating theparameters �iid =

��; �2

, the investor uses as predictive distribution

Rt+1 � N��; �2

�(20)

which is unconditional, since this is a constant investment opportunity set.This case is taken as a benchmark, which will be compared with the time-varying investment opportunity sets presented before. Since this model canbe seen as a restricted version of the previous ones, we expect that theprevious models do at least as well as the iid model in forecasting returns.

4 Portfolio choice problem

This section analyzes the portfolio choice problem faced by the investor.Subsection 4.1 presents the general intertemporal expected utility maximiza-tion problem. Then, subsection 4.2 considers the speci�c case of constantrelative risk aversion (CRRA) preferences. Subsection 4.3 outlines the solu-tion method used to get the optimal portfolio shares in the risky asset foreach investment opportunity set. Subsection 4.4 describes the hedging de-mands computation. Finally, subsection 4.5 presents the methodology usedto evaluate the economic value of volatility timing.

4.1 Intertemporal Expected Utility Maximization

Consider the portfolio choice problem at time t of an investor who maximizesthe expected utility of wealth at some terminal date T , by trading in a riskyasset and a riskless asset at times t, t+1, ..., T � 1. The investor�s problemis

Vt (T � t;Wt; Zt) = maxf!sgT�1s=t

Et [u (WT )] (21)

subject to the budget constraint

Ws+1 =Ws [!s exp (Rs+1) + (1� !s) exp (Rf )] (22)

where f!sg is a sequence of portfolio weights on the risky asset chosen attime s, Rs+1 is the log return on the risky asset from time s to s + 1 andRf is the real risk-free log return on cash (assumed constant for simplicity).Short selling and borrowing are precluded so that !s � h0; 1i 8 s.13 In one13 I exclude the case where !s = 1 since for the CRRA preferences considered expected

utility is not bounded from bellow when agents are allowed to risk all their wealth. Iexclude the case where !s = 0 so that proportional hedging demands de�ned later (whichinclude !s in the denominator) are well de�ned.

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hand, this assumption imposes a realistic friction, since in general investorsconfront borrowing constraints and face legal constraints to short selling. Inthe other hand, for technical purposes it is intractable to simulate over alarge grid of possible portfolio weights (this will become more clear whenthe solving procedure is described in section 4.3).

The function u (�) measures the investor�s utility of terminal wealth WT ,and the subscript on the expectation denotes that it is conditional on theinformation set available at time t, that is, the state variable Zt.

In general, this is a multiperiod problem, so the solution involves thedynamic optimal rebalancing of the portfolio. At time t, the investor choosesthe portfolio weights !t conditional on having wealth Wt and observing thestate variable Zt. By doing this, the investor takes into account that at everyfuture time s, the portfolio weights will be optimally revised conditional onthe then available wealth Ws and the state variable Zs.

The value function Vt (T � t;Wt; Zt) denotes the investor�s expectationat time t of the utility of terminal wealthWT generated by current wealthWt

and the sequence of future optimal portfolio weights f!�sgT�1s=t , conditional

on the state variable Zt. It is a measure of the quality of the investor�sinvestment opportunities.

The multiperiod problem can be expressed as a single-period problemof next period�s value function Vt+1 (T � (t+ 1);Wt+1; Zt). Departing from(16) and using �rst the law of iterated expectations and then the de�nitionof the value function (as shown in the appendix) we get

Vt (T � t;Wt; Zt) = max!tEt [Vt+1 (T � (t+ 1);Wt+1; Zt)] (23)

This is the so-called Bellman equation, basic for any recursive solutionof the dynamic portfolio choice problem. The terminal condition of theBellman equation is given by

VT (0;WT ; ZT ) = u (WT ) 8 ZT (24)

The associated �rst order condition (FOC) is

0 = Et

�@Vt+1@Wt+1

[exp (Rt+1)� exp (Rf )]�

(25)

4.2 CRRA preferences

In this paper the investor�s preferences over terminal wealth are describedby a constant relative risk aversion (CRRA) utility function of the form

u (WT ) =W 1� T

1� (26)

13

The appendix shows that with these preferences the Bellman equationcan be simpli�ed to

Vt (T � t;Wt; Zt) = u (Wt)Qt (T � t; Zt) (27)

Thus, the current value function Vt (T � t;Wt; Zt) can be expressed asthe product of the utility of current wealth u (Wt) and a functionQt (T � t; Zt)of the horizon T � t and the state variable Zt. Further, by using this decom-position in the Bellman equation (23) current wealth Wt cancels out and weget a simpli�ed Bellman equation depending only in the horizon T � t andthe state variable Zt

Qt (T � t; Zt)=max!tEt

h[!t exp (Rt+1) + (1� !t) exp (Rf )]1� Qt+1 (T � (t+ 1); Zt)

i(28)

By comparing the terminal condition (24) with the simpli�cation in (27),the terminal condition of the simpli�ed Bellman equation is

QT (0; ZT ) = 1 8 ZT (29)

The associated FOC is given by

0 = Et�[!t exp (Rt+1) + (1� !t) exp (Rf )]� [exp (Rt+1)� exp (Rf )]Qt+1 (T � (t+ 1); Zt)

(30)

4.3 Solving procedure

Even with the simpli�cations achieved by considering CRRA preferences, theBellman equation can only be solved numerically, as has been usually thecase in previous dynamic portfolio choice literature. For example, Brennan,Schwartz and Lagnado (1997) solve numerically the PDE characterizing thesolution to the dynamic optimization problem. Campbell and Viceira (1999)and Viceira and Jurek (2006) log-linearize the FOCs and budget constraintsto obtain approximate closed-form solutions. Das and Sundaram (2000), Ko-gan and Uppal (2000) and Brandt, Goyal, Santa-Clara and Stroud (2005)perform di¤erent expansions of the value function to solve the problem an-alytically.

An alternative to these methods consists in discretizing the state spaceand then using backward induction. In this context, the evaluation of thevalue function can be done using quadrature integration as in Balduzzi andLynch (1999), non-parametric regressions as in Brandt (1999), simulationsas in Barberis (2000), or binomial discretizations as in Dammon, Spatt andZhang (2001).

In this paper, I adopt a simulation method based in Barberis (2000).The procedure used is the following:

14

1. Discretize the unconditional distribution of the state variable Zt intoJ equally spaced values, set to cover 100% of the in-sample realiza-tions of the state variable Zt. The discretized state variable is labeled�Zjt

�j=1;:::;J.

2. Starting from the last period T , use the terminal condition (28) to setthe initial value in the backward iteration QT (0; ZT ) = 1.

3. In a backward induction way (solving in the order s = T � 1; :::; t),for each

�Zjs�j=1;:::;Js=t;:::;T�1

draw a large samplehR(i)s+1; Z

(i)s+1

iIi=1

from the

predictive distributions p�Rs+1 j Zjs

�and p

�Zs+1 j Zjs

�.

4. Set Qs�T � s; Zjs

�equal to

max!s

1

I

IXi=1

h!s exp

�R(i)s+1

�+ (1� !s) exp (Rf )

i1� Qs+1

�T � (s+ 1) ; Z(i)s+1

�according to the simpli�ed Bellman equation (28). Since in general we

only knowQs+1 (T � (s+ 1) ; Zs+1) for Zs+1 = Zjs+1, Qs+1�T � (s+ 1) ; Z(i)s+1

�is approximated byQs+1

�T � (s+ 1) ; bZjs+1� where bZjs+1 = argmin

Zjs+1

n���Zjs+1 � Z(i)s+1���ois the closest element of the discretized space to Z(i)s+1.

5. The backward induction process in steps 3-4 give !�s�T � s; Zjs

�and

Qs

�T � s; Zjs

�for all j = 1; :::; J and s = t; :::; T � 1. Thus, by

keeping this procedure, eventually we reach Qt�T � t; Zjt

�and hence

the optimal portfolio choice !�t�T � t; Zjt

�.

An important issue in the solving procedure is the accuracy of the simu-lation method used to obtain the optimal portfolios. In the method proposedin this section the accuracy is given by the size of the draw I and the numberof grid points J used to discretize the state variable. In an e¤ort to ensurea very high degree of accuracy, Barberis (2000) uses a very large numberof draws I = 1; 000; 000 from the corresponding distribution. However, inthe appendix he shows that using I = 10; 000 draws already provides a highdegree of accuracy. Regarding the number of grid points used to discretizethe state variable he sets J = 25. Based on these facts, in this paper Itake I = 100; 000 and J = 50. Thus, I double the number of grid pointsin the discretization of the corresponding state variables, while keeping thenumber of draws to a level where the degree of accuracy is high and thecomputation time is reasonable.

15

4.4 Hedging demands

The hedging demands represent the excess demand of stocks driven bythe desire to hedge against unexpected changes in the state variable and(through it) in future expected returns. In order to de�ne them, I �rstintroduce the notion of myopic optimal portfolio choice.

The optimal portfolio choice is said to by myopic whenever hedgingagainst shifts in the investment opportunity set is irrelevant.14 It is op-timal to invest myopically at least in the following cases. First, it is obviousthat when T � t = 1 hedging is irrelevant, since the multiperiod problemreduces trivially into a single-period one, where the investment opportunitydo not change. Second, if the investment opportunity set is constant (asin the iid return model) there would be no changes in the investment op-portunity set to hedge against. Third, even with a stochastic investmentopportunity set, if the state variable Zt is independent of returns, it couldnot be used to forecast the variation in the investment opportunity set, thushedging demand would not arise either. Finally, when the investor has alogarithmic utility (this corresponds to the limiting case when ! 1) theoptimal portfolio choice in the multiperiod setting converges to the myopicoptimal solution.15 To see this, note that (as shown in the appendix) in thiscase (27) turns into

Vt (T � t;Wt; Zt) = u (Wt)+ maxf!sgT�1s=t

Et

"T�1Xs=t

ln [!s exp (Rs+1) + (1� !s) exp (Rf )]#

(31)Since the sequence of optimal portfolio weights f!�sg

T�1s=t that maximize

the expectation of the sum is the same than those that maximize the sum ofthe expectations, the multiperiod setting is equivalent to solving a sequenceof single-period ones, thus we go back to the static case with no hedging.

The proportional hedging demand is the di¤erence between the dynamicoptimal portfolio choice in the multiperiod setting described in subsection3.1 and the myopic optimal portfolio choice, as a proportion of the former.I take as the myopic optimal portfolio choice the single-period case whereT � t = 1. Thus, the proportional hedging demand is given by

ht (T � t; Zt) =!�t (T � t; Zt)� !�T�1 (1; Zt)

!�t (T � t; Zt)(32)

14For the CRRA preferences assumed, the myopic optimal portfolio choice takes the

form 1

E[exp(Rt+1)jb�k;Zk;t]�exp(Rf )V [exp(Rt+1)jb�k;Zk;t] .

15Since for the log case ! 1, the myopic optimal portfolio choice takes the formE[exp(Rt+1)jb�k;Zk;t]�exp(Rf )

V [exp(Rt+1)jb�k;Zk;t] .

16

This measure captures the proportion of the portfolio choice which cor-responds to a hedging component. Since the optimal portfolio choice is byconstruction positive and lies between zero and one, the sign of this measureis associated to horizon e¤ects in the multiperiod portfolio choice problem.

4.5 Economic value of volatility timing

In order to understand the out-of-sample nature in the assessment of theeconomic value of volatility timing, this subsection starts by de�ning the es-timation sample and the evaluation sample. Departing from the full samples = 1; :::; T with length T , the estimation sample s = 1; :::; � with length� is �rst used by the investor in order to calibrate the parameters � whichcharacterize each investment opportunity set. Then, in each period of theevaluation sample s = � + 1; :::; T with length T � � the investor observesthe out-of-sample realizations of the state variable Zs, and optimally chooses!�s (T � s; Zs) which was solved based on the previously estimated parame-ters �, as described in subsection 4.3.16 Since the solution method used onlylet us observe !�s (T � s; Zs) for Zs = Z

js , we approximate !�s (T � s; Zs) by

!�s

�T � s; bZjs�.The most important question this paper aims to address is whether by

considering the SRV-NIG model, the corresponding dynamic optimal portfo-lio choice could generate any economic gains, with respect to the benchmarkmodels. To answer this question, we need to compute the out-of-sample util-ity gains associated with switching from the benchmark model of iid returnsto each corresponding time-varying investment opportunity set. In order todo this, we use the notion of certainty equivalent rate of return RCE . Thismeasure reports the risk-free rate of return that would make the investorgive up the selected portfolio choice !�s (T � s; Zs) and invest fully in theriskless asset. The appendix shows that in the context of a single-periodmodel (t = T � 1), RCET�1 can be computed as

RCET�1 =1

1� log�max!T�1

ET�1h([!T�1 exp (RT ) + (1� !T�1) exp (Rf )])1�

i�(33)

Since Samuelson (1969) has shown that in a constant investment op-portunity set of iid returns the optimal portfolio choice at any horizon isconsistent with that of a single-period investor, the RCEiid for the iid returnmodel is calculated as shown in (33).

16Clearly, it must be the case that 1 < � < T

17

For a multiperiod setting and a time-varying investment opportunityset as is the case for the GARCH and SRV-NIG models, we use the valuefunction instead of the utility function, and in this case the certainty equiva-lent rate of return will be time-varying and state-dependant. The appendixshows that in the multiperiod case RCEt (T � t; Zt) can be obtained as17

RCEt =1

1� log

8<: maxf!sgT�1s=t

Et

24 T�1Ys=t

[!s exp (Rs+1) + (1� !s) exp (Rf )]!1� 359=;(34)

Finally, the certainty equivalent gain from using the dynamic optimalportfolio choice corresponding to the time-varying investment opportunityset instead of the optimal rule implied by the constant investment opportu-nity set (iid return model) is de�ned as

(T � t; Zt) = RCEt �RCEiid (35)

This expression measures the out-of-sample incremental value of thetime-varying investment opportunity set optimal strategy over the constantinvestment opportunity set optimal strategy.

5 Calibration

This section presents the calibration of the parameters � for each investmentopportunity set described in section 3. Subsection 5.1 describes the data setused in the calibration of the parameters �. Subsection 5.2 presents thecalibrated parameters and discusses its implications.

5.1 Data description

I present the results for the US equity futures market (S&P 500 index). Thereturn and realized variation measurements are based on tick-by-tick trans-actions prices from the Chicago Mercantile Exchange (CME) augmentedwith overnight prices from the GLOBEX automated trade execution sys-tem, from January 1990 through December 2002. This same series has beenpreviously analyzed in Andersen, Bollerslev and Diebold (2005) and Ander-sen, Bollerslev, Diebold and Vega (2005).

After removing holidays and other inactive trading days, I get a totalof 3,213 observations. The daily return and realized variation are basedon linearly interpolated logarithmic �ve-minute returns, as in Müller et al.

17The appendix also shows that it can be expressed as a function of Qt�T � t; Zjt

�, so

that we can reconstruct the certainty equivalent directly from the algorithm described insection 3.3.

18

(1990) and Dacorogna et al. (1993). This results in a total of 1� = 97

�ve-minute intervals per day. Daily returns are computed as the sum oflogarithmic �ve-minute returns over the day, while daily realized variationis computed as the sum of squared logarithmic �ve-minute returns over theday.

In order to calibrate the SRV-NIG, GARCH and iid models, I take asthe estimation sample the period from January 1990 through December2001, with a total of 2967 observations. For evaluating the out-of-sampleeconomic performance of the models, I use the evaluation sample whichincludes the period from January 2002 through December 2002, with a totalof 246 observations.

5.2 Calibration results

Maximum likelihood estimates for the SRV-NIG and the benchmark modelsare presented in Table 1. I begin the discussion by �rst analyzing the SRV-NIG model. All the parameters estimates are signi�cant. The estimatedsteepness parameter � is low and equal to 1.4, indicating a leptokurtic con-ditional distribution for returns. The asymmetry parameter � is negativeand equal to -0.124. This implies that the e¤ect of negative skewness in thedistribution of returns is higher than the e¤ect of time varying risk premiumhypothesis, as argued in section 3. This result is consistent with the factthat in the GARCH speci�cation the time varying risk premium hypothesisis always rejected. The parameters related to the speci�cation of �t are atleast plausible, with a high impact of realized variation in �t, and givingsupport for the leverage e¤ect since �4 > 0:

The GARCH models show in general a very high and signi�cant persis-tence of conditional variance. The time varying risk premium hypothesis isalways rejected, while the leverage e¤ect is always accepted. Considering thelog-likelihood values as well as the AIC and BIC information criteria, theTARCH is the best model using the GARCH speci�cation. Using the samecriteria, the SRV-NIG model is found superior than the GARCH models,while the iid return model presents the worst performance.

Based on this results, the next section evaluates the economic value ofvolatility timing under the SRV-NIG and TARCH model, which providesthe best �t in the GARCH-type speci�cation.

6 Empirical analysis

This section analyzes the optimal portfolio choice of the investor for boththe SRV-NIG and TARCH models, as well as the results obtained fromimplementing them out-of-sample. Subsection 6.1 analyzes the obtained op-timal portfolio choice for the single-period investor. Subsection 6.2 does the

19

same for the multi-period investor. Finally, subsection 6.3 evaluates empiri-cally the out-of-sample portfolio choice, proportional hedging demands andcertainty equivalent gains for each investment opportunity set.

6.1 Single-period investor

Figure 1 presents the solution to the single-period problem as well as theassociated certainty equivalent gain for both the SRV-NIG and TARCHmodels. The blue, black and red lines denote risk aversion of two, �ve andten, respectively.

Consistent with Chacko and Viceira (2005) and Gomes (2004), volatil-ity timing has a signi�cant impact in the optimal portfolio allocation. Theresults are very similar for both models. At least four features can be in-ferred. First, the optimal allocation to stocks is decreasing in the statevariables capturing conditional expected volatility, which is consistent withrisk aversion of the investor. Second, the volatility timing in a single-periodcontext is a decreasing function of risk aversion. This is because as riskaversion increases, investors tend to invest less in the risky asset for all lev-els of the state variable. Third, the certainty equivalent gains of volatilitytiming are increasing in the state variables capturing conditional expectedvolatility. This is the case since as conditional expected volatility increases,the optimal portfolio rule allows the investor to reduce his investment in therisky asset, and therefore utility does not fall as much as it would if returnswere considered iid. Fourth, the certainty equivalent gain is decreasing inrisk aversion. This might seem counterintuitive at �rst sight, since we wouldexpect that more risk adverse investors bene�t more from a reduction in thevolatility of returns. However, the investors can only reduce volatility bydecreasing the investment in the risky asset, which also reduces the expectedreturn. The optimal trade-o¤ between both depends on risk aversion.

6.2 Multi-period investor

Consider now the analysis of the multiperiod investor. I consider horizonsranging from one day to one year, a risk aversion of 5 and daily rebalanc-ing. Figure 2 presents the optimal portfolio allocation for the multi-periodinvestor for both the SRV-NIG and TARCH models.

For the SRV-NIG model, we get that the optimal portfolio allocation isdecreasing in the state variable, which occurs for the same reasons as in thesingle-period context. In addition, the optimal portfolio allocation is �atthrough di¤erent investment horizons, implying a negligible role for hedgingdemands.

In the TARCH model we get as well an optimal portfolio allocation whichis decreasing in the state variable, which occurs for the same reasons as inthe single-period context. In contrast to the SRV-NIG model, we notice

20

that the optimal portfolio allocation exhibits positive horizon e¤ects whichlast for around 5 months in calendar time, giving a role for positive hedgingdemands in the sense of Merton (1973). The magnitude of these hedgingdemands will be quanti�ed in the out-of.-sample analysis performed in thenext section.

Figure 3 shows the certainty equivalent gains for the multiperiod investorfor both the SRV-NIG and the TARCH model. In both cases the certaintyequivalent is increasing in the state variables, which occurs for the samereasons as in the single-period context. This increasing e¤ect reduces forlonger horizons, converging in the long run to a certainty equivalent gainwhich shows little dependence on the state variables. Accordingly, we notethat when starting from a low level of the state variable, the certainty equiv-alent is increasing in the investment horizon, while when starting from a highlevel of the state variable, the certainty equivalent gain is decreasing in theinvestment horizon. This is consistent with the mean reverting pattern ofboth state variables and the de�nition of the certainty equivalent rate ofreturn in equation 34. The mean reverting pattern of both state variablesmakes the conditional expectation in (34) converge as the investment hori-zon increases. Since �t exhibits a lower persistence than ht, convergenceis reached �rst for the SRV-NIG model. Finally we note that the certaintyequivalent gain is higher for the TARCH model at shorter horizons (presum-ably, as a consequence of the hedging demands identi�ed in �gure 2) whilefor longer horizons the SRV-NIG converges to a higher level of certaintyequivalent gains.

6.3 Out-of-sample analysis

This subsection evaluates the out-of-sample multi-period optimal portfoliochoice, hedging demands and certainty equivalent gains for both the SRV-NIG and TARCH models.

In the SRV-NIG investment opportunity set, at each day t the investorobserves �t; Rt and RVt Based on this information, he constructs the statevariable �t+1 as indicated in (13). Then he chooses from the discretized space

the closest value to it b�jt+1. Finally, according to his investment horizon,he implements the solution !�t

�T � t;b�jt+1�. For the TARCH investment

opportunity set, at each day t the investor observes ht and Rt. Based onthis information, he constructs the state variable ht+1 as stated in (17)and then chooses from the discretized space the closest value to it bhjt+1.Finally, according to his investment horizon, he implements the solution

!�t

�T � t;bhjt+1� :By implementing this procedure every day through the investment hori-

zon, the corresponding out-of-sample optimal portfolio allocations as wellas the out-of-sample state variables are plotted in �gure 4 for both the

21

SRV-NIG and TARCH models. From the �gure it is clear that the out-of-sample optimal portfolio choice presents a negative correlation with thestate variable, which is more pronounced in the case of the SRV-NIG. Thisis consistent with the solution of the multiperiod optimal portfolio choice,shown in �gure 2. Table 2 presents descriptive statistics for the out-of-sample optimal portfolio choice for both the SRV-NIG and TARCH models.As noted from the table, the median of the SRV-NIG out-of-sample opti-mal portfolio choice was lower, but had a slightly higher standard deviationcompared to the out-of-sample optimal portfolio choice corresponding to theGARCH model. However, both series present in general a similar behavior,suggesting that the obtained results should be robust to transaction costs.

Figure 5 presents the out-of-sample proportional hedging demands, to-gether with the out-of-sample state variables for both the SRV-NIG andTARCH models. For the SRV-NIG model, the out-of-sample proportionalhedging demands are negligible. In contrast, the TARCH model presentssigni�cantly positive hedging demands, which may reach levels up to 32%of the optimal portfolio choice. This is consistent with the positive horizone¤ect present in the optimal portfolio choice for the TARCH model, shownin �gure 2. Table 2 presents descriptive statistics for the out-of-sample pro-portional hedging demand for both the SRV-NIG and TARCH models. Wenotice that the TARCH model is able to generate comparably much higherproportional hedging demand than the SRV-NIG model, with a median of15% for the TARCH model, compared to only 0% for the SRV-NIG model.The utility gains due to this hedging will be analyzed shortly.

Figure 6 presents the out-of-sample annualized certainty equivalent gainsof switching from an iid return model to the time-varying volatility models,together with the out-of-sample state variables for both the SRV-NIG andTARCH models. From the �gure it is clear that the SRV-NIG model signif-icantly outperforms the TARCH model at all the investment horizons con-sidered. Table 2 presents descriptive statistics for the out-of-sample annual-ized certainty equivalent gains for both the SRV-NIG and TARCH models.We note that the median of the certainty equivalent gain for the SRV-NIGmodel (2.39%) is much higher than that of GARCH models (1.73%). Thus,we conclude that the total out-of-sample economic gains of switching fromthe GARCH-type model giving the best �t to the proposed SRV-NIG modelare substantial. This results holds in a portfolio selection context performedout-of-sample and assuming borrowing and short-selling constraints.

In addition to the analysis performed, it would be interesting to quantifyhow much of the economic gains are driven by the intertemporal hedgingdemands found in �gure 5, specially for the case of the TARCH model. Inorder to do this, �gure 7 plots the certainty equivalent gains associated onlyto the intertemporal hedging demands. These are computed as the di¤er-ence between the total certainty equivalent gains (those plotted in �gure 6)and the certainty equivalent gains that would hold if the investor behaved

22

myopically.18 In the case of the SRV-NIG model, these gains are negligi-ble as the hedging demands it produces. For the TARCH models we �ndhigher economic gains driven by the intertemporal hedging demands, butstill very small, in the order of 0.025%. Thus we conclude that although theTARCH model is able to generate signi�cant and positive hedging demands,the economic gains associated with these hedging demands are negligible.The descriptive statistics presented in table 2 con�rm these �ndings.

Finally, �gure 13 plots the out-of-sample wealth process for the SRV-NIG, TARCH and iid models, normalizing the initial value of wealth toone. The three models generate a very similar pattern for the out-of-samplewealth process. However the wealth in the terminal period (which is theobject of interest for the investor) is the highest for the SRV-NIG model(1.0129), followed by the TARCH model (1.0126) and the lowest level cor-responds to the iid model (1.0120). This is consistent with the results forcertainty equivalent gains analysis, and further con�rms that the proposedSRV-NIG model is superior to the GARCH and iid models in a portfolioselection context.

7 Conclusions

This paper proposes a new stochastic volatility (SRV-NIG) model based onhigh frequency data, provides a framework for dynamic asset allocation un-der volatility timing and conducts an out-of-sample analysis of the potentialeconomic gains it could generate.

The certainty equivalent gains associated to volatility timing using theproposed SRV-NIG model instead of the iid return model are higher thanthose reported by GARCH models for all the investment horizons consid-ered. Thus, we conclude that the SRV-NIG model outperforms both theGARCH and the iid return model. Although the TARCH models seem tobe better suited for capturing intertemporal hedging demands, the e¤ect ofthese hedging demands in the utility of investors is found to be negligible.

I have tried to use the simplest possible structure to illustrate these�ndings. The framework considered allows for several extensions. One pos-sibility is to include additional frictions in the portfolio choice problem, suchas transaction costs or parameter and model uncertainty. Another potentialavenue is to extend the analysis to multivariate risky assets and analyzethe impact of volatility-timing including conditional covariances in optimalportfolio choice. Finally, it would be interesting to analyze how allowing fora jump component of volatility would a¤ect the results.

18As in section 4.4, I consider as the myopic optimal portfolio choice the single-periodcase when T � t = 1:

23

Appendix A: Intertemporal expected utility maximization ex-pressions

In order to get equation (18) we depart from (16). In the second linewe apply the law of iterated expectations, while in the third we just use thede�nition of the value function.

Vt (T � t;Wt; Zt)= maxf!sgT�1s=t

Et [u (WT )]

= max!tEt

"max

f!sgT�1s=t+1

Et+1 [u (WT )]

#= max

!tEt [Vt+1 (T � (t+ 1);Wt+1; Zt+1)]

The simpli�cation in (22) is derived here. Essentially, we plug in theutility function, and use the law of iterated expectations and the budgetconstraint. Thus, we get that the value function Vt (T � t;Wt; Zt) is givenby

Vt (T � t;Wt; Zt) = max!tEt

"max

f!sgT�1s=t+1

Et+1

"W 1� T

1�

##

= max!tEt

26664 maxf!sgT�1s=t+1

Et+1

26664�Wt

T�1Qs=t

[!s exp (Rs+1) + (1� !s) exp (Rf )]�1�

1�

3777537775

=W 1� t

1� maxf!sgT�1s=t

Et

24 T�1Ys=t

[!s exp (Rs+1) + (1� !s) exp (Rf )]!1� 35

= u (Wt)Qt (T � t; Zt)

For getting equation (26), we just follow the same steps as before, butusing the log utility function, and using the basic log properties.

24

Appendix B: Certainty equivalent expressions

For the single-period investor (t = T � 1), denote by eU the utility func-tion when there is no risky asset available. Then, the RCE de�nition implies

eUT�1 �1;WT�1; RCET�1

�= max!T�1

ET�1 [UT (0;WT�1)]

By using the CRRA utility function, we get

�exp

�RCET�1

�WT�1

�1� 1� = max

!T�1ET�1

"([!T�1 exp (RT ) + (1� !T�1) exp (Rf )]WT�1)

1�

1�

#

exp�RCET�1

�=

�max!T�1

ET�1h([!T�1 exp (RT ) + (1� !T�1) exp (Rf )])1�

i� 1(1� )

RCET�1 =1

1� log�max!T�1

ET�1h([!T�1 exp (RT ) + (1� !T�1) exp (Rf )])1�

i�In the case of the multiperiod investor, denote by eV the value function

when there is no risky asset available. Then, the RCEt de�nition implies

eVt �T � t;Wt; Zt; RCEt

�= Vt (T � t;Wt; Zt)

By using the decomposition in (22), and the CRRA utility function, weget

�exp

�RCEt

�Wt

�1� 1� = u (Wt)Qt (T � t; Zt)

exp�RCEt

�= [Qt (T � t; Zt)]

1(1� )

RCEt =1

1� log [Qt (T � t; Zt)] (36)

From the de�nition of Qt (T � t; Zt) in the previous appendix, we canalso express it as

RCEt =1

1� log

8<: maxf!sgT�1s=t

Et

24 T�1Ys=t

[!s exp (Rs+1) + (1� !s) exp (Rf )]!1� 359=;

From this last expression, it becomes clear that equation (28) is just aparticular case where t = T � 1.

25

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29

Table 1: Parameter estimates of the SRV-NIG and benchmarkmodels

This table reports the estimates for the parameters of the SRV-NIG, GARCHand iid models presented in section 3. The maximum likelihood estimates andstandard errors (in parenthesis) are reported. log-L refers to the value of the log-likelihood, while AIC and BIC refer to the standard models of information criteria.The sample used was 02/01/1992 - 31/12/2001.

iid GARCH­M GARCH TARCH­M TARCH SRV­NIG1.4002

(0.0178)0.0402 0.0172 ­0.0929

(0.0322) (0.0294) (0.0186)0.0291 0.019 0.044 0.0148 0.0252 0.127(0.017) (0.0247) (0.0149) (0.023) (0.0149) (0.0123)0.9611

(0.0249)0.0076 0.0074 0.0131 0.0126 0.2593

(0.0014) (0.0014) (0.0018) (0.0016) (0.0118)0.9268 0.9281 0.9196 0.921 0.4525

(0.0059) (0.0058) (0.0066) (0.0062) (0.0023)0.4555(0.011)

0.0676 0.0664 0.0134 0.0129 0.002(0.0052) (0.0051) (0.0067) (0.0066) (0.0004)

0.1048 0.1055 0.072(0.0086) (0.0086) (0.0159)

log­L ­4151.09 ­3837.17 ­3838.17 ­3803.34 ­3803.53 ­2238.33AIC 8306.17 7684.33 7684.34 7618.67 7617.06 4492.65BIC 8318.16 7714.31 7708.33 7654.64 7647.04 4540.61

W

J

K

a2

_0

_1

_2

_3

_4

30

Table 2: Descriptive statistics of the out-of-sample results forthe TARCH and SRV-NIG models

This table reports the descriptive statistics for the out-of-sample optimal port-folio choice, proportional hedging demand, total certainty equivalent gain, certaintyequivalent gain associated with hedging demand, and wealth process for the SRV-NIG and TARCH models. The wealth process is constructed assuming an initiallevel of wealth equal to one.

Port. choice Hedging CE gain (%) Hedge CE gain (%) Wealth Mean 0.0728 ­0.0033 2.3800 ­0.0002 1.0864 Median 0.0728 0.0000 2.3921 ­0.0001 0.9963 Maximum 0.1231 0.0690 2.5840 0.0082 3.7424 Minimum 0.0293 ­0.0690 0.2993 ­0.0244 0.9042 Std. Dev. 0.0192 0.0172 0.1682 0.0022 0.3027 Skewness ­0.2207 ­2.3211 ­9.0877 ­5.9721 5.9756 Kurtosis 2.2585 12.0371 103.1895 67.2966 48.5872

Port. choice Hedging CE gain (%) Hedge CE gain (%) Wealth Mean 0.0897 0.1517 1.7166 ­0.0010 1.1189 Median 0.0838 0.1563 1.7261 ­0.0043 0.9939 Maximum 0.1328 0.3203 2.0927 0.0515 4.7055 Minimum 0.0739 0.0000 0.3982 ­0.0162 0.8823 Std. Dev. 0.0144 0.0772 0.1678 0.0108 0.4183 Skewness 1.5878 0.3482 ­2.7871 1.7082 5.5622 Kurtosis 4.6451 2.5424 21.5875 6.2165 41.2044

SRV­NIG model

TARCH model

31

Figure 1: Optimal portfolio choice and certainty equivalentgains for the single-period investor

This �gure shows the optimal portfolio choice and certainty equivalent gains forthe single-period investor as a function of the state variable. In the left hand-side,the SRV-NIG model is considered, while in the right hand-side the TARCH modelis analyzed.

2 4 6 8 10 12 140

0.05

0.1

0.15

0.2

0.25

Single­period optimal portfolio for the SRV­NIG model

State (delta)

Sha

re in

vest

ed in

 the 

risky

 ass

et

gama=2gama=5gama=10

2 4 6 8 10 12 140

2

4

6

8

10Single­period CE gain for the SRV­NIG model

State (delta)

Cer

tain

ty e

quiv

alen

t gai

n

gama=2gama=5gama=10

2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

Single­period optimal portfolio for the TARCH model

State (TARCH variance)

Sha

re in

vest

ed in

 the 

risky

 ass

et

gama=2gama=5gama=10

2 4 6 8 100

2

4

6

8

10Single­period CE gain for the TARCH model

State (TARCH variance)

Cer

tain

ty e

quiv

alen

t gai

n

gama=2gama=5gama=10

32

Figure 2: Optimal portfolio choice for the multi-period investor

This �gure shows the optimal portfolio choice for the multiperiod investor as afunction of the state variable and the investment horizon. In the left hand side theSRV-NIG model is considered, while in the right hand side the TARCH model isanalyzed.

0100

200

05

1015

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Remaining investment horizon (T­t)

Multi­period optimal portfolio choice for the SRV­NIG model

State (delta)

Sha

re in

vest

ed in

 the 

risky

 ass

et

0100

200

05

10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Remaining investment horizon (T­t)

Multi­period optimal portfolio choice for the TARCH model

State (TARCH variance)

Sha

re in

vest

ed in

 the 

risky

 ass

et

33

Figure 3: Certainty equivalent gains for the multi-period in-vestor

This �gure shows the proportional hedging demand for the multiperiod investoras a function of the state variable and the investment horizon. In the left hand sidethe SRV-NIG model is considered while in the right hand side the TARCH modelis analyzed.

0 50 100 150 2000

10

­0.5

0

0.5

1

1.5

2

2.5

3

3.5

Remaining investment horizon (T­t)

Multi­period CE gains for the SRV­NIG model

State (delta)

Cer

tain

ty e

quiv

alen

t gai

n

0 50 100 150 20005

10

­0.5

0

0.5

1

1.5

2

2.5

3

3.5

Remaining investment horizon (T­t)

Multi­period CE gains for the TARCH modelState (TARCH variance)

Cer

tain

ty e

quiv

alen

t gai

n

34

Figure 4: Out-of-sample optimal portfolio choice for the multi-period investor

This �gure shows the out-of-sample optimal portfolio choice of the multiperiodinvestor, as well as the corresponding state variable. In the left hand side theSRV-NIG model is considered, while in the right hand side the TARCH model isanalyzed.

50 100 150 2000

0.02

0.04

0.06

0.08

0.1

0.12

0.14Out­of­sample SRV­NIG optimal portfolio choice

Remaining investment horizon (T­t)

Sha

re in

vest

ed in

 the 

risky

 ass

et

50 100 150 2000

0.02

0.04

0.06

0.08

0.1

0.12

0.14Out­of­sample TARCH optimal portfolio choice

Remaining investment horizon (T­t)

Sha

re in

vest

ed in

 the 

risky

 ass

et

50 100 150 2000

2

4

6

8

10

12

14

16

18Out­of­sample state (delta)

Remaining investment horizon (T­t)

delta

50 100 150 2000

1

2

3

4

5

6

7

8Out­of­sample state (TARCH variance)

Remaining investment horizon (T­t)

TAR

CH

 var

ianc

e

35

Figure 5: Out-of-sample proportional hedging demand for themulti-period investor

This �gure shows the out-of-sample proportional hedging demand of the mul-tiperiod investor, as well as the corresponding state variable. In the left hand sidethe SRV-NIG model is considered, while in the right hand side the TARCH modelis analyzed.

50 100 150 200­0.4

­0.3

­0.2

­0.1

0

0.1

0.2

0.3

0.4Out­of­sample SRV­NIG proportional hedging demand

Remaining investment horizon (T­t)

Pro

porti

onal

 hed

ging

 dem

and

50 100 150 200­0.4

­0.3

­0.2

­0.1

0

0.1

0.2

0.3

0.4Out­of­sample TARCH proportional hedging demand

Remaining investment horizon (T­t)

Pro

porti

onal

 hed

ging

 dem

and

50 100 150 2000

2

4

6

8

10

12

14

16

18Out­of­sample state (delta)

Remaining investment horizon (T­t)

delta

50 100 150 2000

1

2

3

4

5

6

7

8Out­of­sample state (TARCH variance)

Remaining investment horizon (T­t)

TAR

CH

 var

ianc

e

36

Figure 6: Out-of-sample total certainty equivalent gains for themulti-period investor

This �gure shows the out-of-sample total certainty equivalent gains of the mul-tiperiod investor, as well as the corresponding state variable. In the left hand sidethe SRV-NIG model is considered, while in the right hand side the TARCH modelis analyzed.

50 100 150 2000

0.5

1

1.5

2

2.5

3Out­of­sample SRV­NIG CE gain

Remaining investment horizon (T­t)

CE

 gai

n

50 100 150 2000

0.5

1

1.5

2

2.5

3Out­of­sample TARCH CE gain

Remaining investment horizon (T­t)

CE

 gai

n

50 100 150 2000

2

4

6

8

10

12

14

16

18Out­of­sample state (delta)

Remaining investment horizon (T­t)

delta

50 100 150 2000

1

2

3

4

5

6

7

8Out­of­sample state (TARCH variance)

Remaining investment horizon (T­t)

TAR

CH

 var

ianc

e

37

Figure 7: Out-of-sample certainty equivalent gains from hedg-ing for the multi-period investor

This �gure shows the out-of-sample certainty equivalent gains from hedgingfor the multiperiod investor, as well as the corresponding state variable. In theleft hand side the SRV-NIG model is considered, while in the right hand side theTARCH model is analyzed. The out-of-sample certainty equivalent gains fromhedging are computed as the di¤erence between the certainty equivalent gains forthe multiperiod investor and the certainty equivalent gains for the myopic (single-period) investor.

50 100 150 200­0.05

0

0.05Out­of­sample SRV­NIG CE gain from hedging

Remaining investment horizon (T­t)

CE

 gai

n fro

m h

edgi

ng

50 100 150 200­0.05

0

0.05Out­of­sample TARCH CE gain from hedging

Remaining investment horizon (T­t)

CE

 gai

n fro

m h

edgi

ng

50 100 150 2000

2

4

6

8

10

12

14

16

18Out­of­sample state (delta)

Remaining investment horizon (T­t)

delta

50 100 150 2000

1

2

3

4

5

6

7

8Out­of­sample state (TARCH variance)

Remaining investment horizon (T­t)

TAR

CH

 var

ianc

e

38

Figure 8: Out-of-sample wealth process for the multi-periodinvestor

This �gure shows the out-of-sample wealth process of the multiperiod investorfor the SRV-NIG, GARCH and iid models, when the initial value of wealth isnormalized to one.

50 100 150 2000.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5Out­of­sample wealth process

Investment period

Wea

lth

SRV­NIGGARCHiid

39