Should (Co)jump variation be included in asset allocation?

ABSTRACT

This study explores whether to introduce jump variation into asset allocation. Using the high-frequency data of constituents of the Dow Jones Industrial(DJI) 30 index, we construct global minimum variance portfolios based on two different covariation estimators: pre-averaged and pre-averaged truncated Hayashi-Yoshida estimators (PAYHE and PATHYE, respectively). By comparing the performance of two different portfolios consider jump or not, we find that eliminating jump variation yields a significantly higher Sharpe ratio, and results in a lower turnover and superior positions compared to incorporating the jump variation. Thesve improvements are also reflected in economic gains, and the huge economic gain mainly comes from the reduction in turnover.

KEYWORDS:

• Jump variation
• global minimum variance portfolio
• covariation matrix
• turnover

JEL CLASSIFICATION:

• C22
• C51
• C53

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data that support the findings of this study are available from TAQ Database. Restrictions apply to the availability of these data, which were used under license for this study. Data are available from https://wrds-www.wharton.upenn.edu/ with the permission of WHARTON RESEARCH DATA SERVICES.

A.2 Composite Maximum Likelihood

Following the approach proposed by Pakel et al. (2020), let $V_t$ denote the $N\times N$ conditional covariance matrix for the $N\times 1$ vector of returns $r_t$. The standard quasi-likelihood obtained under the auxiliary assumption of conditional normality then takes the form:

$logL(\theta ;r)=\sum _{i=1}^{T}l(\theta ;r_t),$

(A.1) $l(\theta ;r_t)=-\frac{1}{2}log|V_t|-\frac{1}{2}{r}_{t{ }^{\mathrm{\prime }}}{V}_t^{-1}{r}_t.$(A.1)

If $N$ is of large dimensions, this can be difficult and time-consuming to implement. The composite likelihood approach sidesteps these problems by instead approximating the likelihood with a number of lower dimensional marginal densities, so that the dimension of the problem is reduced from N to 2. We rely on contiguous pairs, $Y_{1t}=(r_{1,t},r_{2,t})$, $Y_{2t}=(r_{2,t},r_{3,t})$, $\dots$, $Y_{d-1,t}=(r_{d-1,t},r_{d,t})$, resulting in the composite likelihood,

(A.2) $CL(\theta )=\frac{1}{T(N-1)}\sum _{j=1}^{N-1}\sum _{t=1}^T{l}_{jt}(\theta ;Y_{jt}).$(A.2)

Maximizing equation (A.2) yields a consistent and asymptotically normal estimate for $\theta$.

A.3 Assessment Statistics

• Average return and accumulated return: average return is the out-of-sample average portfolio return, and it is calculated as: ${\mu }_p=\frac{1}{T-t_0}{\sum }_{t=t_0+1}^T{r}_{pt}=\frac{1}{T-t_0}{\sum }_{t=t_0+1}^T{\hat{w}}_t^{\prime }{r}_t$. Accumulated return is the accumulated portfolio return over the out-of-sample period, and it is calculated as: ${\prod }_{t=t_0+1}^T(1+r_{pt})$,

• Portfolio standard deviation: this is the portfolio return standard deviation over the out-of-sample period, and it is calculated as: ${\sigma }_p=\sqrt{\frac{1}{T-t_0}{\sum }_{t=t_0+1}^T(r_{pt}-\frac{1}{T-t_0}{\sum }_{t=t_0+1}^T{r}_{pt})}$.

• Sharpe ratio: $SR=\frac{{\mu }_p}{{\sigma }_p}$. The larger the Sharpe ratio, the better is the portfolio, as it delivers higher ratios of return over risk.

• Average turnover: the turnover measures the average change in the portfolio weights, following the literature (see, e.g. Han (2006), Liu (2009), Hautsch, Kyj, and Malec (2015)), the total portfolio turnover from day t to day t+1 is measured by:

(A.3) $T{O}_t=\sum _{i=1}^N|w_{i,t+1}-w_{i,t}\frac{1+r_{i,t}}{1+{w{ }^{\mathrm{\prime }}}_t{r}_t}|.$(A.3)

the average turnover is then calculated as: $AT=\frac{1}{T-t_0}{\sum }_{t=t_0+1}^{T}T{O}_t$. Also, we assume that the investor faces a fixed transaction costs $c$ proportional to the turnover rates in portfolios, the portfolio excess return net of transaction cost at time t is given as: $r_{pt}=w{{ }^{\mathrm{\prime }}}_t{r}_t-cT{O}_t$.

• Portfolio concentration: $PC=\frac{1}{T-t_0}{\sum }_{t=t_0+1}^T({\sum }_{i=1}^N{w}_{i,t}^2{)}^{1/2}$. It quantifies the concentration of resulting portfolio weights.

• Proportion of leverage: $PL=\frac{1}{T-t_0}{\sum }_{t=t_0+1}^T{\sum }_{i=1}^{n}I({\hat{w}}_{i,t}<0)$. The proporation of leverage is the fraction of negative weights computed from all assets and all time periods.

• Economic value: To evaluate the economic significance of the different covariance matrix estimation, we consider the utility-based framework of Fleming, Kirby, and Ostdiek (2001). The investor’s utility at each point in time is given as:

(A.4) $U(r_{pt})=(1+r_{pt})-\frac{\gamma }{2(1+\gamma )}(1+r_{pt}{)}^2.$(A.4)

where $\gamma$ is the investor’s risk aversion coefficient. The larger the value of $\gamma$, the more risk averse is the investor.

The economic value if the value of $\mathrm{\Delta }$ such that, for different portfolios $p_1$ and $p_2$, we have:

(A.5) $\sum _{t=t_0+1}^{T}U(r_{p_{1}t},\gamma )=\sum _{t=t_0+1}^{T}U(r_{p_{2}t}-\mathrm{\Delta },\gamma ).$(A.5)

It represents the maximum return the investor would be willing to sacrifice each period to get the performance gains associated with switching to the second portfolio. In our analysis, $p_1$ will always be the covariance matrix without considering the jump effect. We report the value of ${\mathrm{\Delta }}_{\gamma }$ as a basis point fee.

Notes

¹ The reason of the choice of HAR-DRD and HEAVY models are as follows: The HAR-DRD model directly forecasts the daily covariances. Compared to other models which directly forecasting daily covariances, like HAR, the HAR-DRD model estimates the variances and correlation separately, and greatly reduces the computational burden. Also, the HAR-DRD can ensure the positive definiteness of forecasted covariance matrix by applying a simple constraint on estimated parameters. Unlike HAR-DRD, the HEAVY model is designed to characterize the daily return distribution, and it has been proved to be more robust to certain types of structural breaks and be able to adjust rapidly to changes in the level of volatility.

Additional information

Funding

We acknowledge financial support from the National Natural Science Foundation of China through grant number 72073096;National Natural Science Foundation of China [72073096];