Empirical Analysis

In order to deriving the optimal portfolio of two risky and risk-free assets, we employ the time-varying volatility models to estimate the conditional covariance and correlation. A static model does not specify the volatility over time, while a dynamic model does.

Dynamic models are typically represented with difference equations. In this paper, we use ordinary least square (OLS) model to stand for the static model, and the dynamic models are represented by GARCH-DCC, GJR-GARCH-DCC, CARR-DCC, GARCH-ADCC,

GJR-GARCH-ADCC, and CARR-ADCC. The performance of dynamic model in comparison with that of static model is the main purpose in our study.

< Table 2 is inserted about here >

In the Table 2, it is documented the empirical results of the estimation with the GARCH-DCC, CARR-DCC, GARCH-ADCC, and CARR-ADCC model over the sample period from 1990 to 2008. We divide the table into two parts corresponding to the two steps in the DCC and ADCC estimation. We use GARCH and CARR model in the first step so that we can obtain the parameters fitted for DCC and ADCC model. In the first stage of Table 1 (Panel A), we can utilize the GARCH and CARR model fitted by return and range data with individual assets for attaining standardized residuals. Then, these standardized residual series can be brought into the second stage for dynamic conditional correlation (DCC) and asymmetric dynamic conditional correlation (ADCC) estimation. Panel B of Table 2 shows the estimated parameters of DCC and ADCC under the quasi-maximum likelihood estimation (QMLE).

< Table 3 is inserted about here >

Table 3 reports the estimation results of GJR-GARCH-DCC and GJR-GARCH-ADCC model using the weekly data of S&P 500 futures and 10-year T-bond futures. The estimation of GJR-GARCH-DCC and GJR-GARCH-ADCC model is similar to GARCH-DCC and GARCH-ADCC. The difference between GJR-GARCH and GARCH model is the measure of variance equation. The GJR-GARCH model proposed by Glosten, Jagannathan and Runkle (1993) incorporates the asymmetric effect of good news and bad news in the GARCH process on duration.

< Figure 2 is inserted about here >

Figure 2 provides the dynamic volatility of the S&P 500 futures and the 10-year T-bond futures based on the GARCH, GJR-GARCH and CARR model. Panel A (GARCH fitted), Panel B (GJR-CARCH fitted), and Panel C (CARR fitted) show the volatility estimates for the S&P 500 futures are abnormally high (solid line) in several periods. The East Asian Financial Crisis was beginning in 1997 followed by Russian financial crisis in 1998. Although it initially happened in Asian, the impact of financial crisis also had put pressure on the S&P 500 futures market in the United States. Moreover, the Dot-Com Bubble Crisis (Internet Bubble Crisis) took place in 2000 which led to the collapse in the technology industry as well as the overall financial market. After the collapse of the Dot-Com Bubble, there are terrorist attacks in September 11, 2001. The attacks had a great impact on the economy of U.S. and financial markets, the major stock exchanges like New York Stock Exchange (NYSE), American Stock Exchange (AMEX), and NASDAQ did not open on September 11 and remained closed until September 17. Besides, the stock market downturn was the dramatic decline in stock prices during 2002. The downturn can be regarded as sharp correction in the stock price after a decade-long bull market. In the meantime, a wave of accounting scandals became known to the public in the U.S., including Enron, Arthur Andersen, and WorldCom. In the third quarter of 2007, the U.S. subprime mortgage financial crisis had a great amount of impact on financial market of U.S. as well as other countries. The influence of subprime crisis is still ongoing, it seems like investors in the stock market are unsure of where to go with the money.

< Figure 3 is inserted about here >

Figure 3 reports the correlation and covariance estimates between S&P 500 futures and 10-year T-bond futures for the return-based (GARCH) and range-based (CARR) DCC and

ADCC model as well as the DCC and ADCC with GJR-GARCH model.

In Panel A, it appears that the dynamic conditional correlations become negative (ρ_12,t < ) at the end of 1997 no matter what the dynamic model we apply. In Panel B, the 0 time-varying correlations characteristic of GARCH-ADCC and CARR-ADCC are similar to the above-mentioned case in Panel A. Correlation is expressed by numbers ranging from -1 to +1. To eliminate diversifiable portfolio risk completely, we needs an intra-portfolio correlation approaches perfect inverse correlation (ρ_12,t = − ). Therefore, diversification 1 minimizes the risk of our portfolio well because S&P 500 futures prices have very low dynamic correlations with 10-year T-bond futures prices after the end of 1997, but it does not necessarily lower our expected return of portfolio.

Here we assume that investors use short-horizon mean-variance strategies to create portfolios from stock market (S&P 500 futures), bond market (10-year T-bond futures) and risk-free asset (T-bill rates). We construct the static portfolio (built by ordinary least square, OLS) using the unconditional mean (μ_i), variance (σ_i²) and covariance (σ_ij). Under the minimum variance framework, the weights of the portfolio are computed by the given target return, expected return, and conditional covariance matrices estimated by the GARCH-DCC and GARCH-ADCC, the GJR-GARCH-DCC and GJR-GARCH-ADCC, and the CARR-DCC and CARR-ADCC. Consequently, we can compare the economic value of the volatility models on 12 different target annualized return (5%~16%, 1% in an interval).

< Table 4 is inserted about here >

Table 4 reports how the economic values vary with the different target returns and the different constant relative risk aversions (CRRA). Panel A shows the annualized expected returns (μ), volatilities (σ) and Sharpe ratios (S_p) of the portfolios estimated from the

OLS, GARCH-DCC, and CARR-DCC model. For a quick look, the annualized Sharpe ratio (reward-to-variability ratio) calculated from the CARR-DCC (0.640) and GARCH-DCC (0.588) are higher than the OLS model (0.498). The advantages of using dynamic conditional correlation model to construct the portfolio are their better performance and smaller risk. Panel B shows the average annualized performance fees (△r) among the three models that a risk-averse investor would be willing to pay to switch from the static to the dynamic forecasting models. The values of CRRA are set to 1, 5, and 10, respectively. Roughly speaking, the switching fees raise consistently with higher target returns and higher constant relative risk aversions. Besides, Panel B also reports the performance fees if an investor change from the GARCH-DCC to the CARR-DCC model.

Positive values for all cases show that CARR-DCC model dominates the GARCH-DCC model in forecasting conditional variances and correlations.

< Table 5 is inserted about here >

< Table 6 is inserted about here >

Table 5 gives a representation of the portfolio performance and switching fees that an investor would be willing to pay to switch from the symmetric GARCH-DCC and CARR-DCC to the asymmetric GARCH-DCC and CARR-DCC forecasting model. The results of GJR-GARCH-DCC and GJR-GARCH-ADCC model are displayed in Table 6.

< Table 7 is inserted about here >

Table 7 shows the incremental values of time-varying volatility among the OLS, GJR-GARCH-DCC, GJR-GARCH-ADCC, GARCH-DCC, CARR-DCC, GARCH-ADCC, and CARR-ADCC model using 5%, 10% and 16% target return respectively. In this paper,

we propose the asymmetric effect on the DCC model for the better performance. Panel A shows the volatility value with 5% target return. In the Panel A, the CARR-ADCC model has no superior as a dynamic forecasting model though there is no significant difference between CARR-DCC and CARR-ADCC model. Panel B and Panel C with the target return of 10% and 16% respectively show the opposite results of incremental value of volatility timing on model selection. In the Panel B as well as Panel C, the CARR-fitted models are better than the GARCH-fitted models, and the CARR-DCC is superior to the CARR-ADCC model.

<Figure 4 is inserted about here>

Figure 4 plots the weights of minimum volatility portfolio derived from static and dynamic models while setting the target return equal to 10%. The charts from Panel A to Panel F show the dynamic portfolio weights that minimize conditional volatility. In addition, Panel G has the constant portfolio weights for cash (-0.854), stock (1.360), and bond (0.494).