Modern portfolio theory (MPT) proposes how rational investors use diversification to allocate their assets and optimize their portfolios. MPT models a portfolio as a weighted combination of assets so that the return of a portfolio can be expressed as a summation of the constituent asset’s returns. Additionally, the volatility of a portfolio (σ2portfolio) can be
shown as the function of the variance of each asset (σi2) and the correlation (ρij) of the component assets. One of the most influential concepts of MPT is Markowitz diversification.
Diversification in investment portfolio is a risk management technique that mixes a wide variety of assets within a portfolio. Because the fluctuations1 of a single asset have less impact on a diversified portfolio, diversification can eliminate the specific-risks or unsystematic-risks from any one investment portfolio. In order to reduce the specific-risks of a portfolio, one can invest multiple assets with varied risk levels, therefore, large losses in some assets are offset by others assets if their correlations are not equal to one (perfectly correlated). In other words, investors can reduce their exposure to individual asset risk by holding a diversified portfolio of assets. Although diversification minimizes the risk of a portfolio, it does not necessarily reduce the portfolio return. Consequently, well-diversified in assets is referred to as the free lunch in finance.
In Markowitz portfolio selection model, we can generalize the portfolio construction problem to the case of many risky assets and a risk-free asset. The first step is to determine the return-risk trade-off to the investor. There are summarized by the minimum-variance frontier. This frontier is a graph of the lowest possible portfolio variance that is attainable for a given portfolio expected return. Afterward we can easily find the weights of global
1 More volatile of fluctuations, more risk of assets. We use the standard deviation of the portfolio’s return to proxy the portfolio risk in this study.
minimum variance portfolio with the function of each asset’s variance (σi2) and the covariance (σij) between two constituent assets. The part of the frontier that lies above the global minimum-variance portfolio is called the efficient frontier of risky assets. The second part of the optimization includes the risk-free asset proxied by an investment in short-dated Government securities. The risk-free asset has zero variance in returns, and it is uncorrelated with any other asset. Accordingly, we search for the capital allocation line (CAL) with the highest reward-to-variability ratio (Sharpe ratio, S ), and the CAL must be p tangent to the efficient frontier. Sharpe ratio is a measure of the excess return (risk premium) per unit of risk. Therefore, the portfolio is the optimal risky portfolio with more than two risky assets and a risk-free asset. For the weights that result in risky portfolio with the highest Sharp ratio, the objective is to maximize the slope of the CAL for any possible portfolio.
A number of useful improvements have appeared since the moment of the classic theory creation. In static portfolio strategy, we can employ the unconditional expected returns (u ), variances (i σi), and correlations (ρij) for any target return (utarget) to acquire the optimal weights of risky and risk-free assets in our portfolio. The MPT uses the historical parameter “volatility” as a proxy for risk and assumes volatility never changes.
The optimal weights of the portfolio are not dynamic because we don’t take the time-varying character into account.
Recently, quantitative investment becomes popular in financial market. Investors and quantitative analysts begin using mathematical and statistical models to price stocks, bonds and derivatives. Dynamic investment strategies used in portfolio optimization would benefit investors because the financial markets are not entirely efficient and the phenomenon of volatility is changeable over time. In another word, dynamic investment strategies not only
reduce the risk but improve portfolio performance as well.
In the financial market, the financial economists found that the autocorrelation plays an important role in estimating volatility than in estimating return. Therefore, we can forecast the second moment such as volatility and correlation easily than the first moment (return).
Previous researchers had assumed constant volatility and used simple devices to approximate risk. Engle (1982) proposed the Autoregressive Conditional Heteroscedasticity (ARCH) model in which the variance at time t (σi t2, ) is modeled as a linear combination of the past q-period of squared errors (εt q2− ). Afterward, Bollerslev (1986) added lag lengths p
of variance (σi t p2,− ) to the model and advanced the GARCH (Generalized ARCH) model for measuring and forecasting financial market volatility. The GJR-GARCH model with asymmetry was introduced by Glosten, Jaganathan, and Runkle (1993) following the GARCH model. In the GJR-GARCH model, good news (εt q− > ) and bad news (0 εt q− < ) 0 have different effects on the conditional variance, the model suggests that bad news increases volatility more than good news in general.
The above-mentioned GARCH family models are based on the return data
(loge
(
ptclose ptclose−1)
). Recently, numerous studies have mentioned that the range data basedon the logarithmic difference of high and low prices in a fixed interval (loge
(
pthigh ptlow)
)make a superior estimation of volatility than the return data. Parkinson pioneered in estimating the variance of the rate of return (see Parkinson, 1980). The follow-up studies are Brandt and Jones (2006), Chou (2005, 2006) and Martens and Dijk (2007). Especially, Chou (2005) proposed a Conditional Autoregressive Range (CARR) model which provides sharper volatility estimates compared with the standard GARCH model.
In asset allocation and portfolio optimization, we not only take the conditional variance
of individual asset into account, but the conditional covariance and correlation as well.
Engle developed new econometric models of volatility that captured the tendency of more than two assets to move between high volatility and low volatility period. Engle (2002a) advanced the Dynamic Conditional Correlation (DCC) Model, which is derived from the GARCH family.
In the recent, researchers have noted that volatilities and correlations for financial markets rise more after negative returns shocks than after positive shocks. Namely, the asymmetric phenomena of volatility and correlation show that there are higher market volatility and correlation levels in market downswings than in market upswings. Cappiello, Engle, and Sheppard (2006) proposed the ADCC (Asymmetric DCC) model to capture the asymmetry in estimating dynamic correlations.
The existence of asymmetry has been widely studied and confirmed. It plays a vital role in risk management and asset allocation. From the viewpoints of investors, the main issue is whether asymmetric phenomenon can reduce the volatility, enhance risk-adjusted portfolio return and improve utility of investors.
In the mean-variance framework, investors acquire the different portfolio weights over time using varied models. Therefore, we can easily obtain the return and risk of the optimal portfolio. In order to measure the economic value of timing under uncertainty, we consider an investor with different risk-averse levels uses conditional volatility and correlation to allocate portfolio among cash, stock, and bond. Fleming, Kirby and Ostdiek (2001) extended West, Edison, and Cho’s (1993) utility criterion to measure the economic value of timing with different risk tolerance levels. This study shows that the CARR model may bring out a better performance to investor, that is to say, the investor might pay more annually to switch from the static strategy to the dynamic strategy.
The article is structured as follows. In Section Ⅱ, we introduce literature related to GARCH family and CARR with DCC and ADCC model. In addition, the literature resources related to the economic value of timing are also included in Section Ⅱ. Section
Ⅲ provides the methodology of asset allocation, the measurement of economic value over time, and the return-based (GARCH and GJR-GARCH) and range-based (CARR) models with DCC and ADCC. Section Ⅳ presents the data used in this paper, its summary statistics, and the details of the performance in the different strategies and risk aversion levels. Finally, Section Ⅴ is the conclusion to the paper.