# Explanation of TLI Graphs

**Graph 1:**

Graph 1 is a dual axis line graph to show you the TLI versus the performance of the S&P 500 (shifted to the right by 12 trading days). The choice of two axes is to illustrate the relationship between the two variables with different magnitudes and scales of measurement, which will be explained below.

The y-axis on the left shows the TLI ratio of bullish to bearish insights: as previously stated, the critical thresholds are +/- 0.15. We can even observe moments where the TLI goes above these thresholds - signifying a more probable move in the S&P 500. The y-axis on the right shows the S&P 500’s change in value over 12 days expressed as a %. This explains why the SPX chart looks nothing like its price chart - the aim is to show the deviation in price over 12 days.

The difference in thickness between the TLI and SPX is because the TLI value is only updated once daily while the price of the SPX fluctuates throughout the trading day.

From the 1 January 2021 to the 7th June 2022, there is a forward correlation of 61% between the TLI and the S&P 500’s returns adjusted 12 days, meaning the two graphs have a 61% similarity to a 1:1 correspondence.

**Graph 2:**

Graph 2 is a scatter plot of the TLI value (ratio of bullish to bearish insights) against the projected SPX price range 12 days from today. The range is calculated by looking at historical data from 2021 and 2022.

The x-axis shows TLI values, and the y-axis shows projected SPX prices 12 days after. This is useful for option trading as it provides expected SPX levels given a TLI value.

We see strong positive correlation between the value of the TLI and SPX prices - as shown by the majority of the data points lie within the 80% confidence interval. Essentially, whenever the TLI had values greater than 0, the SPX rose in price while TLI values lesser than 0, were followed by a fall in price. Furthermore, we see nearly symmetrical distributions across the origin (magnitude in price movement increases proportionally as the TLI’s abs. value increases), showing the index works similarly in bullish and bearish positions.

The green bar at the top represents the top 10% of past SPX prices in relation to TLI values. and the red bar represents the lowest 10% of SPX prices. We see that all points past those lines were correctly forecasted by the TLI. Furthermore, we can see that the majority of incorrect TLI alerts were close to the x-axis - in other words, losses are minor when the TLI is wrong.

Why do the critical values of +/- 0.15 work? The y-axis can be interpreted as the change in price relative to 12 days prior, with the x-axis representing 0 (no change). All bullish TLI points below the X axis and bearish TLI points above the Y axis are instances where the market moved in a direction opposite to what the TLI stated. However, we easily notice that most ‘wrong’ cases occur around the y-axis, meaning the TLI’s value was close to 0. If we draw two vertical lines to remove the incorrect instances while preserving the most ‘correct’ points, we see that the +/- 0.15 lines are the most efficient: increasing the hit rate from 70% to 86% – while staying within thresholds 37% of the year.

What does hit rate mean? Similar to TOGGLE insights, the hit rate is a past looking measure - given the TLI was positive/negative, did the S&P 500 see the subsequent movement in price? It considers the SPX 12-day returns as a binary choice – positive (bullish TLI) and negative (bearish TLI). The high hit rate means that historically, the TLI has been outstandingly accurate in predicting market direction and thus has a low chance of predicting an outright wrong bearing (unless outliers present themselves).

**Graph 3:**

Granger Correlation investigates ‘Granger’s causality’ between variables in two time series by testing for *forecasting*: in other words, whether one time series is useful in forecasting another. It is a “bottom up” operation, meaning the data generating processes in the separate time series are assumed to be independent and are then tested to find a correlation.

In this context, Granger Correlation is applicable in our case because it tests whether the TLI *predicts* the SPX, and both time series are independent.

The null hypothesis refers to the belief that two variables have no statistical relationship and in this case - the TLI does not significantly predict the direction of the SPX 12 days later. The alternative hypothesis states the contrary - the TLI does significantly predict the direction of the SPX 12 days later.

To test these hypotheses, a p value must be generated (the probability of obtaining the observed result given the null hypothesis is true). The null hypothesis can only be rejected (and alternative hypothesis accepted) if the p value is less than the critical value (more statistically significant).

**Pearson’s r**, also known as the **correlation coefficient**, is a normalised measure of the covariance between two data sets. A positive value indicates a proportional relationship, while a negative value indicates an inversely proportional relationship. Since it is normalised, the value of pearson’s r ranges between positive and negative 1.

Looking back at chart 3, we can now define the y-axis as Pearson’s r and the x-axis as the days of lead against the SPX.

As an inversely proportional relationship between the TLI and SPX is logically inconsistent, we can assume that for any y values less than 0, there exists no correlation.

Graphically, the correlation (Pearson’s r) is highest at 12 days (the vertex of the orange parabola). As you can see, the gradient on either side of the vertex significantly drops for any horizon lesser or greater than a 12-day shift.

What does the black line represent? It represents the **autocorrelation** (the correlation of a time series to itself) of the SPX. In Graph 3, we see the autocorrelation of the SPX graphed from 12 to 24 days of lead. As stated previously, we can interpret negative r-values as showing no correlation, meaning the SPX price is not correlated to its future value 12 days later and onwards. The aim of the graph is to show that the TLI is not just capturing autocorrelation. In other words, the SPX’s autocorrelation is irrelevant except as a reference point to compare the TLI’s performance to.