Asset Allocation Problem

In this section, we consider the asset allocation problem for a risk-averse in- vestor. Following the stream of literature on volatility timing (e.g. Fleming, Kirby, and Ostdiek (2001) and Marquering and Verbeek (2004)), we assume that the investor has mean variance preferences, and rebalances the portfolio regularly according to the predicted conditional covariance matrix. One of the major problems of the mean variance based portfolio allocation is the difficulty to accurately predict the conditional mean, which can be very noisy when the number of assets is very large. Therefore, we follow Hautsch, Kyj, and Malec (2013) and consider the global minimum variance portfolio with the constraint that sum of weights equals to one.

Min

wt,t+1

w_t,t′ ₊₁Σˆt,t+1wt,t+1, s.t. wt,t′ +11= 1

where wt,t+1 is the vector of portfolio weights, 1 is the vector of ones, ˆΣt,t+1 is

the predicted conditional covariance matrix obtained as described in the previ- ous section.

The optimal weights can be solved for as

wt,t+1 = ˆ Σ−_t,t1₊₁1 1′_Σˆ−1 t,t+11 (3.19)

Since the optimal portfolio weight is only the function of the conditional covari- ance matrix, we avoid estimation error due to forecasting the conditional mean. We obtain the vector of optimal portfolio weights at each point in time. We also impose short selling constraint, hence weights are within the range from 0 to 1. Then we get the time series of daily ex post portfolio returns. To reflect

real world investment problems, we also consider weekly and monthly rebalanc- ing portfolios. We first obtain the daily portfolio weights, and then hold the

portfolio for one week and one month.3 _{The holding period portfolio return is}

r_t,tp _+h = w′

t,t+1rt,t+h, where rt,t+h is the vector of returns from t to t+h and

r_t,tp _+h is the ex post portfolio return for the holding periodh.

3.3.5

Performance Evaluations

Before we start to introduce the performance metrics, we first discuss our bench- mark strategies. In this paper, we consider two types of benchmark strategies, which directly relate to our research questions mentioned in the introduction part. The first benchmark strategy is a low frequency strategy. We employ the conventional GARCH (1,1) model of Bollerslev (1986). We use it to assess the potential benefit of high frequency data over low frequency data. The sec- ond benchmark strategy is a high frequency strategy. We employ the HAR-RV model by Corsi (2009) (RV only). We use it to assess the potential incremental benefit of decomposing realized volatility into components and using realized higher moments. For each of the low frequency and high frequency benchmark strategies, we consider two specifications, i.e. zero correlations and DCC corre- lations. To answer our main research questions, we fix the correlation structure for the benchmark strategy and the candidate strategy, for example we com- pare RV+zero with GARCH+zero or compare RV+DCC with GARCH+DCC to ensure fair comparisons and directly assess whether high frequency data con- tributes to portfolio performance. We follow the same procedures to assess the incremental improvement of realized volatility components and higher moments 3_{An alternative way is to conduct multi-horizon volatility forecasts and use the conditional}

covariance matrix for that horizon (e.g. daily, weekly, or monthly) to construct respective portfolio weights and ex post portfolio returns. We find the method we used is straightforward to implement and is also economic intuitive.

over total realized volatility. In the robustness checks part, we also compare high frequency strategies with alternative low frequency benchmarks.

We consider three performance metrics: Firstly, we use a conventional ex-post Sharp Ratio: SRp = r¯ p σp (3.20) where ¯rp _andσ

p are average portfolio return and portfolio volatility respectively

over the out-of-sample period.

Secondly, we use the turnover rate.

T O_tP = n X i wi,t+1−wi,t 1 +ri,t 1 +rp,t (3.21)

Liu (2009) and Hautsch, Kyj, and Malec (2013) also use turnover to assess portfolio performance. Before the rebalancing, the portfolio weights change to

wi,t₁₊1+_rrp,ti,t, therefore the absolute difference between it and the new weight can

measure the portfolio turnover caused by rebalancing.

Thirdly, to quantify the potential economic improvements relative to bench- mark strategies, we also use utility based criteria following Fleming, Kirby, and Ostdiek (2001) and Marquering and Verbeek (2004). Similar to Hautsch, Kyj, and Malec (2013), we assume that the investor has quadratic preferences. The

quadratic utility and the performance fees between two strategies are as follows, U(rp_t,t+h) = 1 +r_t,tp _+h− γ 2(1 +γ)(1 +r p t,t+h)2 (3.22) 1 T ₋h T_X−h t=0 [U(rp_{t,t+h −}∆)] = 1 T ₋h T−h X t=0 [U(rbm_t,t+h)] (3.23)

γ is the risk aversion parameter, and we consider different levels of 2, 7, and

10. ∆ refers to the performance fee investor willing to pay to switch from the benchmark strategy to the candidate strategy.

We further investigate the statistical significance of our performance fee mea- sure. Motivated by Engle and Colacito (2006) and Bandi, Russell, and Zhu (2008), we use the Diebold and Mariano (1995) test for it. We view perfor- mance fee as the loss differential of two alternative but nested forecasts. We compute daily spot realized utility for main and benchmark strategies and then compute daily spot performance fees. We then project the time series of perfor- mance fees on a vector of ones using Newey-West standard errors; the resulting t-statistic allows us to make inference about the statistical significance of per- formance fees with the null hypothesis of a zero performance fees

3.4

Empirical Findings

3.4.1

Realized Volatility Forecasting

In this section, we discuss in and out-of-sample volatility forecasting results. Ta- ble 3.2 documents the summary statistics of different realized measures for the cross-section of 30 assets from 2001 to 2009. We only report mean and standard deviations for each realized measure to save space. A few observations should be

noted. Firstly, SPY has a much lower mean and standard deviation compared to individual stocks, which can be explained by the diversification feature of the index, as the individual stocks contain idiosyncratic components. Secondly, although there is no clear cut pattern whether upside volatilities or downside volatilities are larger; it seems that the downside ones are less volatile across time for almost all assets. Moreover, jump variation is much smaller than the bipower variation, confirming jumps are rare. Finally, we find positive skewness and high excess kurtosis for almost all assets. The non-normality in the data suggests that realized measures beyond variances may play an important role in our following empirical analysis.

Table 3.3 reports the one day ahead in-sample volatility forecasting results us- ing models with different realized measures. Due to the relative large number of assets, we focus exemplary on the index ETF SPY. Main findings can be summarized as follows. Firstly, the RV only model already fits data well with

adjusted R2 _{of 57.73%. While both the weekly and monthly lagged RV co-}

efficients are statistically significant, we show the daily lagged RV coefficient is insignificant. Secondly, when we decompose RV into upside and downside components, we observe a clear improvement in model fit, namely the adjusted

R2 _{increases to 65.15%. Although the coefficient of the daily upside volatil-}

ity component remains insignificant, the coefficient of daily downside volatility component becomes highly significant, suggesting that the downside part is more important. We also show that weekly and monthly upside and downside components are insignificant, implying recent downside information is more im- portant for forecasting short horizon volatility. Similarly, when we decompose RV into jump and diffusive components, we observe an increase in the adjusted

R2 _{to 60.58%, however the magnitude of improvement is smaller compared to}

that for upside and downside decomposition. We also show that the coefficient for the daily diffusive component is significant while the one for the daily jump component is insignificant, suggesting the diffusion component is more impor- tant. Different from results in upside and downside volatilities, we also suggest that weekly and monthly volatility components are also important. Our findings regarding volatility forecasting using different volatility components are gener- ally consistent with previous studies (Patton and Sheppard 2013, Andersen, Bollerslev, and Diebold 2007). We show that separating total realized volatil- ity into different components improves volatility forecasting performance, and hence may also potentially improve volatility timing based portfolio allocation strategies. Thirdly, we assess the usefulness of realized higher moments as ad- ditional volatility predictors. We show that realized skewness is negative and significant at daily level while realized kurtosis is negative and significant at

daily and weekly levels. However, improvements in adjusted R2 _{are less than}

1%, indicating they may possess only limited incremental information for future volatility prediction.

Since we are interested in portfolio allocation with a relative large number of assets, we then report the volatility forecasting performance for the cross-

section of 30 assets. Table 3.4 reports in-sample adjusted R2_{s for the 30 assets.}

Our findings are generally consistent with the results for SPY discussed above.

Initially, the RV only model generates adjusted R2_{s of over 50% on average.}

Secondly, both the upside and downside components model and the jump and

diffusive components model yield higher adjusted R2_{s in general, however the}

improvements are generally small compared to results for SPY. The model using

upside and downside components can still generate the highest improvements

in adjusted R2_{s for both the median and the mean value. Thirdly, the models}

using realized higher moments can still outperform the RV only model, however the improvements are small.

We then report the out-of-sample volatility forecasting results, which will then be used to construct our out-of-sample portfolio allocation analysis. We use the first 1000 days as in-sample period to estimate the models, and then use the rest of the observations to conduct out-of-sample forecasts. Table 3.5 reports out-of- sample mean squared errors (MSEs) for the set of 30 assets. We find that MSEs for SPY are generally smaller than for individual stocks. Models using upside and downside components, jump and diffusive components, and skewness can generate smaller MSEs than the model using RV only. For individual stocks, both mean and median values of MSEs suggest that the model using upside and downside components can have lower MSEs than the model using RV only. However, for models using jump and diffusive components and higher moments, there are more cross sectional variations, and results are more mixed. The differ- ent empirical results for index and individual stocks may again be explained by higher microstructure noise and higher idiosyncratic risks for individual stocks compared to the more liquidly traded index ETF SPY. To summarize, both in-sample and out-of-sample results suggest that by using high frequency data, one can measure and forecast volatility precisely, and using realized volatility components and higher moments can further improve forecasting performance.