In the past decades several nonlinear time series models have been proposed and some of them have been used in practical applications. It is now fairly widely accepted stylized fact that financial time series, such as stock returns, exchange rates series and etc., exhibit strong signs of nonlinearity. The nonlinear phenomenon typically comes from the observation that many economic and financial time series are often characterized by regime specific behavior and asymmetric responses to shocks (see, Hansen (1996, 2000), Hsieh (1989, 1991, 1993), and Tsay (1989, 1998)). In addition, although the GARCH model can capture leptokurtosis and volatility clustering, structural breaks in the variance could lead to the spuriously high persistent in the GARCH model (see, Lamoureux and Lastrapes (1990) and Mikosch and Starica (2004)). Therefore, in the dissertation, we will use the threshold model to capture the nonlinear behavior in both returns and volatilities of financial assets.
In the threshold class of models, classical estimation of parameters is usually done by the maximum likelihood or the least squares methods with a joint grid search over the threshold values using information criteria such as AIC or BIC. However, Koop and Potter (2003) and So et al. (2005) showed that the inadequacy of these approaches is to fix the threshold parameters in advance before estimating the other parameters by least squares. Therefore, the uncertainty of the threshold parameters cannot be taken into account when implementing statistical inference for the other parameters. In addition, another problem is the large number of parameters in the threshold model must be estimated and the difficulty of estimation due to the positive definiteness restrictions of the covariance matrix. Thus, MLE will result in unstable estimates. Moreover, previous
literatures (such as Koop and Potter (1999)) noted that likelihood functions of many nonlinear models are non-smooth and multimodal. So, it may be inappropriate to chooses one point in the parameter space and to use the large sample property to infer.
Therefore, to moderate the above problems, we adopt a Bayesian approach to estimate the threshold parameters as well as the other parameters simultaneously. And Bayesian method, by using information from the entire parameter space, captures this finite sample uncertainty about the true parameter values. In addition to the above inference advantage, a Bayesian method incorporates the investor’s prior belief about the validity of the pricing model and managerial technique with the information in the data, and thus it will result in more appropriate conclusions (see, Pastor and Stambaugh (2000, 2002), and Jones and Shanken (2005)). The dissertation focuses on the application in several important issues in financial markets, including the mutual fund performance evaluation and the forecasting in conditional covariance matrix.
For the first issue in this dissertation, we propose three-regime Bayesian threshold models and use daily returns for each fund to examine the threshold effect and to test this effect resulting from the change of mutual fund managerial micro-forecasting ability and market exposures among different market conditions. Our results show that the effect of public information in the three-regime model is more significant than that in the two-regime model. The unconditional model will overestimate selectivity in the upside market, but underestimate selectivity in the downside market. It also underestimates fund market exposures whether in downside or upside markets. We also reveal clearly that much managerial timing ability decides by their skills to forecast the downside market. In addition, our results indicate that the conditional three-regime threshold model bring
most powerful detection for significant timing activity. Our empirical results also indicate that selectivity and timing are negatively associated and the effect is especially significant in the downside market. This is very important for investors to allocate their portfolio.
In the second issue, we present a robust threshold VAR (or VECM)-DCC-GARCH model and use the Metropolis-Hastings (MH) algorithm and the Gibbs sampling algorithm to estimate the parameters simultaneously. Our model extends existing approaches by admitting thresholds in conditional means, conditional volatilities and correlations of multivariate time series. Such an extension, allows us to account for rich asymmetric effects and dependencies of conditional means, volatilities and correlations, as they are often encountered in practical financial applications. In addition, we use the concept of Chen and So (2006) to define the threshold variables as the linear combination of endogenous variables. This setting can eliminate excessively subjective belief in threshold variable decision. Besides, the weight coefficient can serve as the proxy in deciding which market is the price leader and which market is the price follower.
Finally, threshold values in our model are not fixed ex ante, but they are estimated from the data, together with all other parameters in the model.
We investigate the empirical performance of our model in two data sets including daily S&P500 futures and spot prices, and S&P500 and Nasdaq100 spot prices. Our study attempts to use posterior odds ratio and Bayes factors as a formal tool for making comparison between competing models. We reduce our testing problem to a Bayesian model selection problem. We can then select the model with a higher posterior odds ratio. We also present the performance comparison results of the one-step-ahead forecast in the conditional covariance matrix. The forecast results are assessed by several
criteria which include the views of statistical loss and risk managers.
Based on the estimation results, we find that the asymmetric dynamic structure is obvious in both the dynamic relationship between S&P500 futures and spot markets and between S&P500 and Nasdaq100 spot markets. We also detect that S&P500 futures market is the price leader between S&P500 futures and spot markets, and S&P500 spot market is the price leader between S&P500 and Nasdaq100 spot markets. Furthermore, based on several in-sample and out-of-sample performance measures in the conditional covariance matrix prediction, we find that the threshold model outperforms the linear model across most measure criteria.
Chapter 2.
Measuring Mutual Fund Asymmetric
Performance in Changing Market Conditions
1. INTRODUCTION
The scale of U.S. mutual funds, especially equity funds, has grown quite rapidly in the past several decades. The trend illustrates that more and more investors prefer investing their capital in mutual funds rather than directly investing in the equity market.
Due to the great number of funds in existence, it is significant to investigate whether managers of actively managed mutual funds rely on superior stock selection skills (micro-forecasting) and market timing capability (macro-forecasting) to outperform passive strategies. Furthermore, studies of relationships between mutual fund performances and characteristics are worthwhile for investors to reference when selecting funds. The investigation of mutual fund performance can serve as a guideline for fund investors and provide them detailed information and previous performance behavior about certain funds. In addition, the objective of identifying superior fund managers is also of interest to academia as a challenge to the efficient market hypothesis.
Therefore, over the past 30 years, numerous evaluation techniques have been proposed to examine the investment performance of mutual fund managers in the academic literature. For mutual funds’ managerial stock selection skills, starting with Jensen (1968), numerous studies employed regular proxies for the market portfolio such as Capital Asset Pricing Model (CAPM) benchmark to evaluate the performance of mutual funds. Later, Grinblatt and Titman (1994), Carhart (1997), and Wermers (2000) argued that the use of CAPM as a benchmark would result in inconsistent outcomes and were centered on examining managerial stock picking talents with several comparable passive benchmarks. They also showed that performance tests are entirely sensitive to the
chosen benchmark. In addition, Grinblatt and Titman (1989), Elton et al, (1993), Carhart (1997), and Wermers (2000) studied whether mutual fund managers possess superior talents for picking stocks with certain characteristics.
On the subject of managerial market timing ability, Treynor and Mazuy (1966), Merton and Henriksson (1981), and Ferson and Schadt (1996) used monthly fund returns to examine whether fund managers take advantage of superior information by increasing (decreasing) market exposure before the stock market turns bullish (bearish) and found that only a minority of fund managers have good market timing skills. Furthermore, Ferson and Schadt (1996) argued that traditional performance measures may be biased when fund managers use dynamic strategies resulting in time varying risk. Hence, Ferson and Schadt (1996) took public information into consideration and proposed conditional performance evaluation approach which is consistent with the semi-strong form of market efficiency hypothesis.
In addition, when the frequency of market timer (e.g. daily) is higher than that of measured fund returns (e.g. monthly), Goetzmann, Ingersoll, and Ivkovic (2000) found that widely used Henriksson-Merton (HM) parameteric test would result in weak and biased downward timing skill. Therefore, they adjusted HM type model to detect daily timing skill without requiring daily timer data. But simulations showed that the adjusted test was not as powerful as the classical HM test executed directly on daily timer returns.
Moreover, Busse (1999), and Bollen and Busse (2001) documented that daily data take account of more efficient estimates of time variation in systematic risk and are more powerful in testing timing ability than monthly data.
Although a variety of evaluation methods have been proposed and implemented to study stock selection and market timing performance to date, they have only examined managerial stock selection ability without distinguishing between good and bad market conditions. Moreover, they have merely investigated the difference in funds’ market
exposure between upside and downside markets, but have not examined the asymmetric risk adjustment between the transitions from neutral (i.e. the market tends to be neither downside nor upside) to upside markets and those from neutral to downside markets.
Additionally, in this essay, we take transaction costs into account when fund managers want to adjust their portfolios among different market conditions. Thus, managers may alter the overall risk composition of their portfolios in anticipation of excess market return being larger (smaller) than a certain positive (negative) level other than zero. We are also able to explore that managers’ overall timing ability mainly comes from downside or upside market timing ability.
Furthermore, the majority of the earlier performance measure studies are based on least squares estimation or maximum likelihood estimation. However, a Bayesian approach of performance evaluation may be more adequate from an investor’s viewpoint.
It combines an investor’s prior belief about the accuracy of the pricing model and managerial skills with the information in the data and obtains posterior distribution of the model’s parameters. As a result, the conclusions based on the Bayesian method may be more informative for investors than those based on the traditional statistical approach.
To enable more powerful and efficient analyses of fund performance, we propose three-regime Bayesian threshold models and use daily returns for each fund to examine the threshold effect and to test this effect resulting from the change of mutual fund managerial micro-forecasting ability and market exposures among different market conditions. With regard to the chosen factor model, we select Carhart’s (1997) four-factor model, which contains market factor (MKT), size factor (SMB), book-to-market factor (HML), and the momentum factor (UMD).
Our results show that the effect of public information in the three-regime model is more significant than that in the two-regime model. The unconditional model will overestimate selectivity in the upside market, but underestimate selectivity in the
downside market. It also underestimates fund market exposures whether in downside or upside markets. We also reveal clearly that much managerial timing ability decides by their skills to forecast the downside market. In addition, our results indicate that the conditional three-regime threshold model bring most powerful detection for significant timing activity. Our empirical results also indicate that selectivity and timing are negatively associated and the effect is especially significant in the downside market. This is very important for investors to allocate their portfolio.
Subsequently, we adopt the annual estimation results based on the conditional three-regime threshold four-factor model and use a fixed effects panel data model to examine the relationships between investors’ behavior and past fund performances and various fund characteristics. We find strong evidence that funds with better past performances, except for downside market timing ability, will attract more investors and bring larger net cash flows. In addition, net cash flows are persistent and positively associated with fund sizes. We also find the inverse relationships between net cash flows and lagged turnover, load charges, and expenses.
We also use a fixed effects panel data model to explore the relationships between fund performances and characteristics. Although numerous prior studies have examined their relationships, they mostly focused on selectivity performance and found the following results. For the relationship between performance and fund size, Grinblatt and Titman (1994) and Carhart (1997) found that managerial stock selection performance is not apparently related to fund size, while, Chen et al. (2004) found that fund size corrodes mutual fund performance. Moreover, prior evidence on the relationship between turnover and performance is mixed. Elton et al. (1993) and Carhart (1997) found that funds trading more actively have worse stock selection talents than those that trade less frequently, while Grinblatt and Titman (1994) found a positive relation between turnover and net mutual fund returns.
Again, for the relationship between total load and stock picking skills, Elton et al.
(1993), Carhart (1997), and Dellva and Olson (1998) found that stock selection performance of the load fund is lower than that of the no-load fund. For the relation between expense and performance, Grinblatt and Titman (1994) and Carhart (1997) found that high expense funds have worse performances than low expense funds, while other extant literatures did not find any significant relations. In addition, Gruber (1996), Chevalier and Ellison (1997), Dellva and Olson (1998), and Sirri and Tufano (1998) found that funds with high cash positions and large net cash flows will produce superior overall stock selection performance.
Our empirical analyses using the fixed effects panel data model show the following findings. The selectivity and market timing performances are not long persistent. In addition, we find that fund investors have no good selection ability. Fund sizes will corrode selectivity performance in the upside market, while, they reduce fund performance loss in the downside market. Conversely, active managers will advance selectivity performance in the upside market, whereas increase the losses of abnormal returns in the downside market. Because managers of higher expense funds do not create more performances to recover their charged fees, investors prefer to select funds with lower expenses to maximize their net expense returns.
We also find that both managers with heavy trading as well as managers with low expenses have superior downside market timing ability, while they have worse upside market timing ability. In addition, net cash flows have negative impacts on timing the downside market but positive impacts on upside timing ability. This suggests that managers of large net cash flow funds might invest new cash in high beta stocks or call options to generate better timing ability in the upside market. Conversely, they might invest new cash in low beta stocks instead of put options and result in worse timing ability but better selectivity performances in the downside market. Finally, we also find
that funds with high expenses will have alert insights into the predictions of downside and upside markets.
The rest of the essay is organized as follows. Section 2 presents the theoretical methodology and the estimation procedure. Data and the performance estimation results for an individual fund and all funds are described in Section 3. Section 4 investigates the behavior of funds investors. Section 5 documents the relationships between fund performance measures and fund characteristics. Section 6 concludes the essay.
2. UNCONDITIONAL AND CONDITIONAL BAYESIAN THRESHOLD MODEL
The model we proposed here is called the Bayesian threshold model, which allows the conditional variance to depend on its own previous realizations and to be drawn from different regimes. Since the purpose of this essay is to compare the stock selection and market timing ability among different market conditions, we assume the regression coefficients of other factor variables are identical in any market conditions. Therefore, the unconditional three-regime threshold model with conditional variances following GARCH(1,1) process based on Carhart’s (1997) four factor variables, is specified as follows:
whereRp,tis the excess return of the fund p at time t, RMKT,tis the market excess return at time t and is also an observed variable determining the switching points, and the three extra factors (RSMB t,,RHML t,,RUMD t, ) are size, book-to-market, momentum factors at time t.
The threshold parameters rp,1 and rp,2 satisfy −∞ <rp,1< <0 rp,2 < ∞ , and the distribution ofεp t, conditional on information up to t-1, denoted byℑp t−, 1, isN
(
0, hp t,)
.To allow heteroscedasticity in Rp,t we have a GARCH formulation in the conditional variance equation for ht . Standard restrictions on the GARCH parameters area( )pj,0 >0,a( )pj,1 ≥0,b( )p,1j ≥0, anda( )pj,1+b( )p,1j <1 for j =1,2,3.
The parameters
(
α( )p1 ,α( )p2 ,α( )p3)
are the abnormal returns for the fund p in the downside market, the neutral market, and the upside market, respectively. Moreover, the parameters(
( ) ( )2 ( )3,)
, 1
,MKT , pMKT , pMKT
p β β
β are the systematic risks for the fund p when the stock market conditions are downside, neutral, and upside, respectively. We motivate market timing from the fund managerial perspective, assuming that the manager attempts to time market exposure in the fund shareholder’s best interests. The successful market timer should increase the portfolio weight of highly-risky equities prior to the market upswings. Therefore, the fund investors will earn more return when the market return is larger than a certain positive level. Conversely, the successful market timer should decrease the portfolio weight of highly-risky equities prior to the market downturns.
Therefore, it will reduce the loss due to market factors when the market return is below a certain negative level. If a fund manager increases the market portfolio’s exposure prior to the market rise and decreases the market portfolio’s exposure prior to the market fall, thenβ( )p2,MKT −βp( )1,MKT,β( )p3,MKT −βp( )2,MKT, andβ( )p3,MKT −βp( )1,MKT will be significantly larger
than zero. In addition, we can view β( )p2,MKT −βp( )1,MKT andβp( )3,MKT −βp( )2,MKT as indices of managers’ timing ability to downside and upside markets, respectively. Furthermore, we can use αp to measure the fund managerial overall stock selection ability, where
( ) { , ,1} ( ) { ,1 , ,2} ( ) { , ,2}
is the weighted average of abnormal performance in the three different market conditions. If αp is significantly larger than zero, we say that the mutual fund manager, on average, has a superior selectivity performance.
Because there may be public information that is correlated with future market returns, managers who use just public information to time market should get no credit for superior ability. Therefore, in order to eliminate this naïve market timing ability and allow time-varying returns and risk, we postulate that beta is a linear function of a vector
−1
Zt of predetermined variables of information, as in Ferson and Schadt (1996).
Therefore, the conditional three-regime threshold model based on four factor variables can be expressed as follows:
( )
(
( ))
vector, Δpmeasures the response of the conditional beta to the information variables, and other notations are the same as equations (2.1) to (2.3). The criteria for measuring the fund’s performance are also identical to those described in the unconditional model.In the threshold class of models, classical estimation of parameters is usually done by the maximum likelihood or the least squares methods with a joint grid search over the
threshold valuesrp,1 and rp,2using information criteria such as AIC or BIC. (see, e.g., Tong (1990), Rabemananjara and Zakoian (1993), Li and Li (1996), Tsay (1998)) However, the inadequacy of these approaches is to fix the threshold parameters (rp,1,rp,2) in
threshold valuesrp,1 and rp,2using information criteria such as AIC or BIC. (see, e.g., Tong (1990), Rabemananjara and Zakoian (1993), Li and Li (1996), Tsay (1998)) However, the inadequacy of these approaches is to fix the threshold parameters (rp,1,rp,2) in