## Abstract for "The Risk-Return Tradeoff and Leverage Effect in a Stochastic Volatility-in-Mean Model" by Bent Jesper Christensen

Asset pricing theory predicts a positive risk-return tradeoff. Empirically, conditional first and second moments of asset returns are time-varying, and this must be accounted for when testing the risk-return relation. Empirical research also documents a strong leverage effect that potentially makes it more difficult to identify the risk-return relation. Thus, according to Fisher Black's explanation of the leverage effect, a price drop increases the debt-equity ratio and hence expected risk. The increase in risk would in turn increase expected returns in case of a positive risk-return relation. Depending on whether the empirical researcher associates the increase in risk with the initial price drop (negative return) or manages to link it to the higher subsequent (expected) returns, the apparent empirical risk-return relation may be of either sign. This highlights the identification issue, and may suggest why the empirical literature tends to show mixed results on the significance and sign of the risk-return relation. Evidently, further analysis should be carried out in a model that in addition to the risk-return relation explicitly accommodates a separate leverage effect. In the present paper, we use the Barndorff-Nielsen and Shephard (JRSS-B, 2001) (henceforth BNS) model since this includes a leverage effect, along with a risk-return relation and time-varying conditional moments, and is consistent with a number of additional stylized facts that characterize asset returns. These include volatility clustering, semi-heavy-tailed non-normal return distributions with skewness and excess kurtosis, and aggregational Gaussianity, i.e., the normal approximation improves as the observation frequency is reduced. Furthermore, the model allows explicit calculation of joint conditional and unconditional moments, thus facilitating explicit estimating equations, closed form parameter estimators, highly tractable option pricing formulas, etc.. In spite of this, the model has rarely been applied, and it has never been used in a systematic study of the risk premium and leverage effect jointly. The model is of the continuous-time stochastic volatility-in-mean variety, and we present explicit closed-form estimators and implement the relevant hypothesis tests in an application to the Standard and Poor's 500 index.Recent literature documents strong persistence, to the point of possibly long memory or fractional integration, in volatility. Christensen and Nielsen (Review of Economics and Statistics, 2007) use a model with long memory in volatility and find a strong leverage effect in weekly data, using either realized volatility or option implied volatility, whereas the risk premium is positive and borderline significant in some of the specifications. Since long memory in volatility would spill over into long memory in returns through a risk-return relation of the Merton type, and since long memory in returns is not empirically warranted, the authors modify the risk-return relation to depend on changes in volatility rather than volatility levels, following Ang, Hodrick, Xing and Zhang (Jrnl of Finance, 2006). The BNS model on the other hand is able to capture strong serial dependence in volatility without being of the long memory type, and so in this paper we use the BNS model with the unmodified Merton risk-return relation.

The volatility feedback effect is that an increase in volatility should increase the discount rate in case of a positive risk-return relation, and thus in turn reduce the stock price, again leading to a negative association between risk and return in the short run. This suggests that empirical results on the risk-return relation may depend on the observation frequency, whereas the leverage effect may depend less on this. Thus, we implement our empirical procedures using both daily, weekly, and monthly data.

The BNS model is a continuous-time model in which the return equation is a stochastic volatility-in-mean specification that also accommodates a leverage effect, whereas the volatility equation is a mean-reverting (or Ornstein-Uhlenbeck) specification with unconditional mean or target for mean reversion of the instantaneous variance. The BNS specification is of a non-Gaussian OU-process, i.e., the shock is not a standard Wiener process, but another time-homogeneous Levy process, i.e., a subordinator, or a process with independent and stationary increments. A Wiener process for the shock in the volatility equations would imply negative instantaneous variances with positive probability, which is of course not warranted. By the subordinator specification, the volatility shock has positive jumps, and although the drift of volatility can be negative, it becomes positive when the instantaneous variance sufficiently small, and the volatility never becomes negative.

By running the subordinator according to time index λt instead of just t (e.g., large λ means running the process faster), the OU structure implies that the unconditional or invariant distribution of volatility is independent of λ, and only depends on the choice of subordinator process. In fact, it is convenient to identify the volatility subordinator by the invariant distribution of σ_{t}. For example, this could be the gamma or the inverse Gaussian distribution. If ρ=0, the inverse Gaussian distribution for σ_{t}² implies a normal-inverse Gaussian (NIG) distribution for returns, which has proved empirically relevant in some cases. For other parameter values and subordinators, more general return distributions are obtained, all consistent with volatility clustering, non-normal returns, and leverage (if ρ≠0).

We test the following hypothese: (i) Is the slope (or relative risk aversion) positive and significant? We establish below that this is indeed a test on the conditional risk-return relation. (ii) Is the correlation (leverage effect) negative and significant? (iii) We also consider an additional overidentifying test motivated by Merton's equilibrium risk-return relation, essentially the zero intercept restriction on this.

From the Introduction, the risk-return relation in asset pricing theory actually does not relate expected return to integrated variance, but to the conditional variance of return. We thus need the relation between the parameters from the risk-return relation and the parameters of the BNS model. To this end, we write down the exact discrete time version of the BNS model and in this calculate the conditional mean and variance of the discrete time returns. We then seek conditions under which the two satisfy the risk-return relation.

The exact discrete time model implies that the conditional mean return is an affine function of the conditional mean of integrated volatility. Furthermore, the conditional variance of return derivations uses that integrated variance is driven by error terms that may also be related to in the returm equation, so the two terms inside the variance operators are uncorrelated. The first variance is a constant, not depending on state variables. The second is computed by conditioning, as the conditional mean and the conditional variance of return are affine in the conditional mean of integrated volatility.

These findings and further developments lead to the following theorem. The detailed proof is in the appendix.

Theorem 1m: The relation between the parameters from the risk-return relation and the parameters of the continuous-time stochastic volatility-in-mean model is given explicitly. Consequently, in the continuous-time stochastic volatility-in-mean model, the equilibrium risk-return relation holds up to a constant intercept that may be calculated in terms of model parameters. Merton's overidentifying zero condition on the intercept may thus be tested as a cross-restriction on the model parameters. The proportionality parameter γ in the equilibrium risk-return relation actually coincides with the slope parameter from the drift specification of the continuous-time model. Thus, the search for sign and significance of the risk-return relation γ in asset pricing becomes a test on the γ^{*} parameter in the BNS model. Finally, the test for the leverage effect is on the parameter ρ, and the presence of this parameter facilitates an interpretation of any finding of a risk-return relation as being free of contamination by leverage.