Separating Order from Disorder in a Stock Index Using Wavelet Neural Networks

In order to determine how much deterministic structure, if any, is present in the behavior of price yields for a stock index, the following practical investment problem is defined: Decide today whether to switch in or out of an S&P 500 index fund, where an initial $10,000 investment will be held for 13 weeks, after which the decision-making process is to be repeated. The alternative investment is a risk-free instrument yielding 5% compound annual rate of return. The performance of the buy-and-hold strategy is compared to the use of wavelet neural network (WNN) trading advisors. The decisions made by the WNNs are based on 13-week holding-period yield data available only up to the decision moment. It is shown that very simple WNNs are able to match or exceed the performance of buy-and-hold strategy, thus suggesting that a small but not inconsequential deterministic component may be separated from the sea of randomness in a stock index. I. IS THERE ORDER WITHIN DISORDER IN MARKET INDICES? The Efficient Market Hypothesis (EMH) (Casti 1990, Dobbins et al. 1994) states that all information available about a financial asset is already reflected in its price, and that therefore, no other information can be gathered that will lead to useful predictions. Claims of superior returns by some portfolio managers are conciliated in the EMH picture by the Capital Asset Pricing Model, stating that these managers are simply receiving, on average, a premium for accepting a higher level of risk. According to EMH, attempts to “time” the market are a waste of time. While empirical studies favor EMH to a large extent, there are clear “anomalies,” such as above-average performance of low P/E market segments, small size effect, January effect, monthly effect, etc., that cannot be explained by an increased level of risk. Suppose that EMH holds, say, to a 98% extent. Then, even the minute 2% difference would be enough to make a large monetary impact over the course of many years. Thus, the problem of separating order from disorder in financial time series is both very challenging (extremely difficult in light of EMH), and potentially very rewarding (economically and philosophically). In this paper, we address the issue by defining a practical investment problem: Decide today whether to switch in or out of a Standard & Poor 's 500 index fund, where an initial $10,000 investment will be held for 13 weeks before making the next decision. At the end of the 13-week holding period, the process is to be repeated. The alternative investment is risk-free and yields a generally available compound annual rate of return of ig = 5%. The performance of two strategies will be compared: (1) Buy-and-hold and (2) the use of wavelet neural network (WNN) trading advisors. With buy-and-hold (B&H), the initial investment is simply put in the fund index, and left untouched for the complete long-term holding period. Buy-and-hold is like a constantly bullish advisor. A reinterpretation of EMH is that it is impossible to go beyond the performance of B&H without incurring additional risk. This is like saying that the market is purely random and thus, our best predictor of performance in the least-squares sense is the mean value. A type of mean value is realized by B&H in the long term. With strategy (2), we train a different WNN every quarter with all data available up to the decision moment. The output of the network is a decision coded as 1 or 0, where 1 is a buy signal for S&P 500. The inputs to the network will be one or more historical 13-week holding-period yields. In this paper, the yield is a rate of return based on appreciation of index value only; it excludes any dividends or other payments that the fund may have distributed. The approach is illustrated in Fig. 1(a). The degree to which strategy (2) outperforms or underperforms strategy (1) will be taken, respectively, as evidence for or against the presence of orderly yield patterns in the stock index. (a) (b) Figure 1: (a) Trading approach for strategy (2). (b) Network inside the WNN Trading Advisor box. II. WAVELET NEURAL NETWORKS AS TRADING ADVISORS The holding-period yield (Dobbins et al. 1994) is the measure of rate of return defined as