Following Bali and Cakici (2004), this paper investigates the market factor (beta), firm size, BE/ME, and VaR to test whether the various company characteristics have significant explanatory power in relation to the average stock returns in Taiwan. The data set includes all stocks listed on the Taiwan stock exchange obtained from the Taiwan Economic Journal (TEJ) for the period from January 1990 through December 2004. This 15-year period is further divided by an estimation period and a test period. The estimation period extends from January 1990 through December 1994, while the test period extends from January 1995 to December 2004.
To be included in the sample, for a given month, a stock must satisfy certain criteria. First, its return over the previous 60 months must be available from the TEJ, and sufficient data must be available to calculate our variables, i.e. VaR and beta.
Second, data must be available from the TEJ to calculate the book-to-market ratio as of December of the previous year. Finally, we include only securities defined by the TEJ as ordinary common shares. This screening process yields averages of 133 stocks per month. Therefore the returns for companies that are calculated are listed throughout the whole period. Figure 1 characterizes the industry distributions and gives a brief overview of the overall sample.
[Insert Figure 1 here]
3.2 Variable Definitions
For each stock and for each month, the following control variables are calculated.
3.2.1 Systematic risk (Beta)
Beta measures a stock’s volatility, i.e. the degree to which its price fluctuates in relation to the overall market. Beta is a key component of the capital asset pricing model (CAPM) and it expresses the fundamental tradeoff between minimizing risk and maximizing return. Follow Fama and French (1992), this paper first sorts all the stocks by size (i.e. the market value of equity) to determine the stocks’ quintile breakpoints. The reason why portfolios are formed according to their size is based on the evidence of Chan and Chen (1998) and others, who found that size differences may be attributed to a wide range of average returns and βs. Next, based on the stocks’ quintile breakpoints, this paper subdivides each size quintile into five portfolios on the basis of pre-ranking betas for all the stocks. This paper estimates the pre-ranking betas for five years of monthly returns ending in December of year t-1.
After assigning each stock in the sample to one of five size quintiles and one of the pre-ranking beta quintiles, we then calculate the equally-weighted monthly returns of the resulting 25 portfolios2 for the next 12 months, from January of year t through December of year t. This procedure yields 108 post-ranking monthly returns for each of 25 portfolios from January 1996 to December 2004. Finally, this paper estimates the post-ranking betas by using a full sample of 108 post-ranking returns for each of the 25 portfolios, with the Taiwan stock value-weighted index serving as a proxy for the market. Following Allen and Cleary (1998), we estimate beta as the sum of the slopes in the regression of the returns on portfolios:
Ri,t =αi +β1,iRm,t +β2,iRm,t−1+εi,t, （1）
2 The choice of using portfolios instead of individual shares is dictated by the evidence of Griffin (2002) that this is the way the sampling error is reduced. Additionally, using portfolios also facilitates a comparison with past studies in the field, as the majority of these studies use portfolios instead of individual stocks. Further advantages of using portfolios instead of individual firms in the regressions include the following: (1) A pooled sample of individual firms used in CSR analysis allows us to eliminate the potential threat posed by temporal and firm-specific effects in terms of biasing the results. (2) There is significantly less computational effort in using portfolios instead of individual stocks in the regression analysis.
where Ri,t is the monthly return on stock i in period t,
α
i is the intercept term, is the monthly return on the TEJ value-weighted index in period t,t
Rm, β is the 1,i
synchronous covariance coefficient, β is the lagged one-period coefficient, and 2,i
t
ε is the residual series from the cross-sectional regression. i,
The beta estimate used to rank stocks (the pre-ranking β) is calculated as follows: lag term allows for a delay in the information process, which may be prevalent for small stocks or those that are infrequently traded (Scholes and Williams, 1977).
3.2.2 Size
Size refers to the value of a company, that is, the market value of its outstanding shares. This figure is found by the natural logarithm of the market value of equity (by taking the stock price and multiplying it by the total number of shares outstanding).
3.2.3 Book-to-Market Equity (BE/ME)
Book-to-market equity (BE/ME) is the natural logarithm of the ratio of the book value of equity plus deferred taxes over the market value of equity, which involves accounting- and market-based variables. The book value of equity, in turn, is the value of a company’s assets expressed on the balance sheet. This number is defined as the difference between the book value of assets and the book value of liabilities.
This paper uses a firm’s market equity at the end of December of the previous year to compute its BE/ME.
3.2.4 Value-at-Risk (VaR)
Value-at-Risk (VaR) has been widely promoted by the Bank for International Settlements (BIS) as well as central banks of all countries as a way of monitoring and managing market risk and as a basis for setting regulatory minimum capital standards.
The revised Basle Accord, implemented in January 1998, makes it mandatory for banks to use VaR as a basis for determining the amount of regulatory capital adequate for covering market risk. VaR measures the worst expected loss under normal market conditions over a specific time interval at a given confidence level. One of its definitions states: “VaR answers the question: How much can I lose with x%
probability over a pre-set horizon?” Another way of expressing this is to state that VaR is the lowest quantile of the potential losses that can occur within a given portfolio during a specified time period.
There are three major decision variables required to estimate VaR: the confidence level, a target horizon, and an estimation model. In this paper, we use three confidence levels (90%, 95%, and 99%) to check the robustness of VaR as an explanatory variable for expected stock returns. The time horizon is 1 month and the estimation model is based on the historical simulation method. 3 The estimation model is also based on the lower tail of the actual empirical distribution. We use 60 monthly returns to estimate the mean, and the cut-off return at the 90%, 95%, and 99% confidence levels from the empirical distribution. The 1%, 5%, and 10% VaRs are measured by the first-lowest, third-lowest, and sixth-lowest observations from the 60 monthly returns.
Once we have the VaR measures for each stock, we rank and place them into 5 quintile portfolios. Portfolio 1 has the lowest VaR and portfolio 5 has the highest VaR. The portfolio formation procedure is very similar to Fama and French (1992), except that they update their portfolios annually, whereas we update ours on a monthly basis. The estimation period for VaR starts in January 1990 and extends through December 1995 and the test period extends from January 1996 to December
3 We also use EWMA and the Monte Carlo method to check the performance of our model. The results are similar and are not reported here.
2004. For example, in January 1996 we estimate VaR for each stock based on the return history from January 1990 to December 1995 and rank all the stocks according to the estimated VaRs. Then five equally-weighted portfolios are formed based on the VaR rank. We then calculate the one-month-ahead portfolio returns in January 1996. For the next month, by rolling over one month ahead, we re-estimate VaR for each stock, rank them based on the updated VaR, and form new portfolios. This procedure is repeated until December 2004 when we have no more data left.
Therefore, we have 108 time series for the 5 equally-weighted portfolios based on their VaRs. In Figure 2-4, we graph the relationship between the 1%, 5% and 10%
VaR levels and the average returns for the quintile portfolios. It is clear that the portfolios for the higher VaR tend to produce rates of return that are greater than the returns from the portfolios of lower VaR companies.