In previous section, we have checked the explanatory ability of each selected model.
However, the estimation results perform well does not equivalently imply economically useful consequences. Thus, in this section, we follow Fleming et al. (2001) to evaluate the economic value of volatility timing by a dynamic asset allocation strategy. We use crude oil futures, USDX futures and three-month Treasury bill to construct a portfolio. First, the optimal portfolio weights of selected assets are constructed under the mean-variance framework.
Second, the quadratic utility function is employed to assess the performance of dynamic strategies based on different models and to quantify how personal opinion affects the performance. Finally, this framework establishes a concise approach to assess the significance and robustness of results.
4.1 Evaluation model
First we consider an investor who wants to minimize portfolio variance subject to achieving a particular expected return. Let rt be N 1 vector of returns on the risky assets, the investor solves the following optimization at each period t,
min vector of conditional expected returns and conditional covariance matrix of risk assets, respectively, rf is return on the riskless asset, p is the target conditional expected return of portfolio. The solution of the optimization problem is
which is the optimal weights on risky assets. The weight on the riskless asset is 1w 1t . In order to focus on the evaluation of volatility and dependence structures based on different models, we assume the conditional expected returns of selected risky assets at time t equal their unconditional means, i.e., t E r
t1 , and the optimal time-varying weights only rely on the one-step-ahead covariance matrix forecasts of selected risky assets,
( 1 )( 1 )
t Et rt rt
In order to measure the value of our models, we compare the performance of the dynamic strategies based on copula-based GARCH models to that of the static strategy based on sample covariance matrix. By the Taylor series, we can obtain the quadratic utility as a second-order approximation to the investor true utility function. Under this specification, the investor’s realized utility in period t 1 can be written as
relative risk aversion, the average realized utility can be used to estimate the expected utility generated by a given level of initial wealth W0, which is as follows
For the purposes of comparison between the static strategy and dynamic strategies based on selected models, we estimate the switching fees by equating the two average utility
On the other hand, transaction cost is important consideration for any dynamic strategy.
It is an important fact to impact our profitability of trading strategies. But making an accurate determination of the size of transaction costs is difficult because it involves many factors.
In the lack of reliable estimates of suitable transaction costs, we consider the break-even transaction cost, which is the maximum transaction cost. In comparing the dynamic strategy with the static strategy, an investor will prefer the dynamic strategy when the break-even transaction cost is high enough. Furthermore, the fact that the break-even transaction cost is much higher will make it easier to implement the dynamic strategy.
4.2 Out-of-sample evaluation result
In this section, we consider that a constant relative risk aversion investor can allocate wealth between the risk-free asset, crude oil futures and USDX futures based on different models. We involve rolling the five years sample data to compute the one-period-ahead forecasted in order to determine the series of optimal portfolio weights. The out-of-sample period for the dates covers five years ranging from January 2005 to December 2009. Then we measure the economic value of the short-term covariance forecasts between crude oil and exchange rate futures returns by a strategic asset allocation problem. We compare the out-of-sample performance of the dynamic strategies based on selected models with the static strategy based on a constant covariance matrix. In this part, our research focuses on the performance fees , which an investor is willing to pay for switching from the static strategy to the dynamic strategy. The fees display the economic value of each selected models relative to the static strategy with target return 5%, 10% and 15%. We present the fees with the
relative risk aversion level of 1 and 5.
Table 3 presents the out-of-sample performance fees and break-even transaction costs for the dynamic strategies based on selected models versus the static strategy under different target returns and risk aversion level with the minimum variance strategy. The most of dynamic strategy models have positive performance fee which demonstrate that the dynamic strategy is superior to the static strategy. For instance, when using the copula-based GARCH models, the investor is willing to pay form 50 to 407 annualized basis points (bps) for using that dynamic strategy instead of the static strategy. Next we compare the different dynamic models to verify their merits. We can find that GARCHGaussian is better than DCC everywhere. The discrepancy of the two models is produced by its residual distributions.
Because crude oil and exchange rate returns are not normality, the skewed-t distribution has better ability to describe the characterization and then leads to higher economic value.
Furthermore, comparing with three different marginal distributions, we find the GARCH model performs better than the others based on each copula function. This phenomenon is also concordant to the previous estimate result. We conclude the GARCH model is the best volatility model to explain the variation of crude oil and exchange rate. For example, using the copula-based GARCH dynamic strategy instead of the static strategy, the performance fee is between 40 and 104 basis points. Among them, GARCHFrank has excellent achievement.
In fact, Frank copula has better achievement on economic value among all selected copula functions no matter what marginal distributions. Finally, the survival Clayton has the poorest performance even worse than static strategy on some place.
The impact of transaction costs is an important consideration in constructing the profitability of trading strategies. Here we compute the break-even transaction costs tcbe as the minimum proportional cost. Because if the transaction costs are sufficiently high, the period-by-period changes in the dynamic weights of an optimal strategy will cause the
strategy too costly to implement relative to the static model. Comparing the dynamic strategy with the static strategy, an investor prefers the dynamic strategy when he pays a transaction costs lower than break-even transaction costs. The break-even transaction costs values are expressed in basis points per trade and are reported only when performance fee is positive.
Besides, we assume the transaction costs of crude oil and USDX futures are at the same level.
Under different relative risk aversion levels, the high level commonly accompanies a high break-even transaction costs. Results demonstrates that the tcbe value of GARCHFrank are positive and high; they tend to be higher almost 50 bps and can be as high as 59 bps. In a word, as the tcbe values are generally positive and reasonably high, we conclude that the performance fees we have reported is robust to reasonably high transaction costs for the dynamic strategy. After examining the forecast performance of all models by performance fee and break-even transaction cost, we can find that the GRACH marginal has excellent accomplishment in all respects. Among them, Frank copula is the most prominent.