## The Range Ratio

The Range Ratio metric provides signals that help identify turning points in stock prices, particularly recovery from price declines. The metric is based on predicted high/low price data and can be shown to conformally map the movement of closing prices after adjustments to the RR and the price series.

The Range Ratio (RR) is the ratio of the predicted k-day price range, divided by the predicted k-trading low price. Thus, when k = 5, the Range Ratio is

where is the predicted high price for the 5-trading day period beginning on trading day t, and is the predicted low price for the same 5-trading day period.

Figure 1 charts a RR based on predicted five-trading day high and low prices for the S&P 500 over 8/29/2018 to 9/28/2020. The top panel shows the raw data for both time series, and the lower panel displays adjusted values. Raw and adjusted daily values of the RR metric are highlighted in red. The blue lines chart daily closing prices of the S&P 500 with the lower panel showing adjusted prices, after subtracting 30-trading day moving averages of previous closing prices. The right vertical axis in the upper and lower panel is for the S&P 500 Index, and the left vertical axis in both panels provides the metric for the RR.

The prominent feature of both charts is the correction February 2020, strongly indicated by the RR indicator. Overall, the conformation of the adjusted RR with adjusted closing prices is notable.

Figure 2 shows more than two decades of price data – 2000 to September 2020. The “spikes” in the RR metric reliably indicate major local minima of the daily S&P 500 closing price series. The February 2020 correction previously noted and other local minima are closely correlated with spikes in the RR. The major price collapse in 2008 October 2008 is linked with the largest RR spike in the chart. While this massive spike precedes the date of the actual minimum price reversal , which occurred March 3, 2009, the RR metric sustains values higher than many other periods in the historic record through to that point until prices start to recover and move up.

These charts are persuasive that the RR signals turning points in the daily closing price series when the RR value rapidly increases to higher values or “spikes.”

Can the RR indicator also signal local maxima in the closing price series, providing guidance for trades?

The answer is a conditional “yes.” Thus, it is possible to identify a single set of maximum and minimum thresholds for the adjusted RR which apply to the whole, more than two decade period shown in Figure 2, and which could support profitable trades – if, in fact, it were institutionally possible to trade the S&P 500.

Figure 3 gives a sense of what this partitioning of the trading calendar looks like. Light blue shading indicates trading days for which the value of the adjusted RR indicates a price increase. Periods of price decrease are indicated by white bands.

Figure 4 shows the hypothetical cumulative gain from applying a trading strategy based on the Range Ratio, modified by adjustments. From the end of 2009 to September 2020, this gain is on the order of twice the cumulative gain from a buy and hold approach.

The trading strategy is straight-forward. The RR is shifted down and flipped in sign, as in Figure 1, and, then, a criterion series is generated by calculating a transformation of this adjusted RR on a rolling basis. Long trades are initiated when this criterion series turns positive and are exited the day the criterion series turns negative. Short trades occur in the other trading days in the calendar.

Trading profits based on this modified RR typically avoid drawdowns at major price corrections and dips. This is intrinsic to a technical stock price indicator which functions best in identifying price declines and the point at which they are likely to recover.

### Discussion

The Range Ratio metric presented here is based on predictions of high and low prices for various periods. A specialized finance and econometrics literature provides computational approaches to producing such predictions. Without going too far afield, it is relevant to note that predictions of high and low prices over various periods, like a day, several days, or a minute are consistent with icons of financial theory like “rational expectations” and “efficient markets.” In fact, a random walk, considered over specific time periods, produces maximum and minimum values, since otherwise the random walk would be a flat line.

Another facet of the Range Ratio is its connection to volatility. Volatility is a latent characteristic of a price series, reflected in its variability over time. There is an extensive technical literature on utilizing the price range as an alternate metric of volatility, rather than the conventional standard deviation of daily returns.

Finally, a caveat to the preceding application of the RR is that one needs to know the proper maximum and minimum thresholds for the adjusted RR in order to produce the results shown in Figure 4. A rolling, adaptive algorithm for identifying appropriate thresholds is the only out-of-sample method for identifying the appropriate thresholds.