(*) the Risk/Reward threshold applied by Laomedeia equals 0.5 (maximum). Unfavorable Risk/Reward translates into a ratio larger than 0.5

**References**

Applying Mathematics to Analyse Financial Markets - Part 1, Innovation

Applying Dynamic Chaos Theory to Analyse Financial Markets - Part 2, Good Investment Practice

**Chaos theory mathematics - A different perspective**

Consistently recurring characteristics are key ingredients of the mathematics behind dynamic chaos theory. The beauty of this action taking place simultaneously on multiple distinct levels is mind blowing. As a reader of research resulting from dynamic chaos theory mathematics, your mindset has to allow for an innovative concept regarding analyzing price action in financial markets. This concept is based on consistently recurring characteristics of complex polynomials, not on generally accepted economic or supply & demand criteria. (See Part 1, Innovation linked above)

Dynamic chaos theory mathematics represents a continuously evolving financial market system, which is generating time dependent (price) targets and time dependent price edges. A continuously evolving system cannot be back tested by definition, however, Laomedeia keeps track of an audit trail of all uniquely numbered price targets (see table 5).

**What does it mean to apply chaos theory mathematics on financial market data?**

When this method starts to work on multiple levels, it yields two time dependent price values for each distinct level. The time dependency of these price values is a direct consequence of the irregular shape of Julia-sets.

One of these price values represents the price-target for an underlying asset. This price-target equals the value an underlying asset will assume in future. The other price value is the price-edge and represents the lowest (or highest) value the underlying asset may assume without jeopardizing the price-target. In other words, the value of the underlying asset should stay within the boundaries of both time dependent price values generated by the method. This applies to all distinct levels.

There are two important characteristics of dynamic chaos theory mathematics:

- All calculations use time dependent price-target and time dependent price-edge information
- All research is relative to the actual price of an asset at the moment of publication

Although the information remains valid for medium and longer term, it is obvious that this research loses its relevance over time when observing shorter time windows.

**Outlook for S&P 500 index per January 31st, 2019**

All calculations are based on the actual value of the S&P 500 index (reflected in below tables). Due to time dependency of price targets as a result of a continuously evolving system, short term levels 8, 7 and 6 have most exposure to a change in value of the S&P 500 index over time.

Now, let's review the current status of the S&P 500 index and see if we can draw some conclusions from below information.

**Actual value and date of report**

**Dominant trend per distinct level**

Table 1 - Dominant trend per distinct level for S&P 500 index

Laomedeia keeps track of at least 8 distinct levels. Many readers ask to elaborate on the meaning of these levels, for which reason the following explanation and comparison may help understand.

Each distinct level reflects a set of consistently recurring characteristics as identified for the polynomials used in dynamic chaos theory mathematics. As soon as a consistently recurring characteristic pops up which does not belong to the same level, another level will be identified. This is analogous to zooming in and out and reflects the multi-dimensional character of financial markets.

The best comparison is a stack of non-aligned paper sheets where the edge of the sheet represents price target and price edge. Each distinct paper sheet shows the consistently recurring characteristics we are looking for. Asset prices usually move back and forth on a single sheet, but sometimes asset prices jump to the sheets above or sheets below.

From table 1 we can draw the following conclusions:

- Short term levels 8, 7 and 6 show a dominantly upward trend
- Medium term levels 6 and 5 show a dominantly upward trend which is conflicting with a dominantly downward trend in levels 4 and 3
- Long term levels 3, 2 and 1 show a dominantly downward trend
- The dominantly upward trend of medium term level 5 conflicts with a dominantly downward trend of medium term level 4
- In general, conflicting dominant trends between levels increases risk significantly

**Zooming in**

Let's now have a more detailed look at the levels where the action takes place. This relates to levels 6, 5, 4 and 3, as indicated in table 1 by means of the red rectangle.

**Price target and Price edge per level**

Table 2 reflects all relevant information when we zoom in on levels 6 to 3, including Laomedeia's unique identifier number, price target, price edge and price range per level. The price range is equal to the difference of price target and price edge. We will use this price range later in graph 1 to show the progress within the range relative to the actual value of the S&P 500 index.

But let's first focus on a few conclusions we can draw from table 2:

- Short term level 6 and medium term level 5 show a dominantly upward trend with a price target near 2,730
- Medium term level 4 shows a conflicting dominant downward trend with a price target near 2,085 which is far below the actual value of the S&P 500 index.
- Level 6 and level 5 show a price edge above the price targets of levels 4 and 3

In order to improve our understanding of risk and opportunities we will continue to peel this onion by sorting price targets (see table 3) and price edges (see table 4) relative to the actual value of the S&P 500 index.

Table 2 - Dominant trend, price target, price edge and range per distinct level for S&P 500 index

**Opportunities relative to price targets**

In case we sort price targets relative to the actual value of the S&P 500 index, it provides a better understanding of what each of the distinct levels are aiming for within their respective dominant trends (see table 3).

Table 3 - Price target sorted per level (from low to high) relative to actual value of S&P 500 index

We can now draw the following conclusions:

- Levels 6 and 5 show a short/medium term recovery aiming for a price target near 2,730
- Medium term we may expect significantly lower price levels with price targets below 2,100
- Only after reviewing Risk/Reward we may draw any additional conclusions

In case we sort price-edge per level relative to the actual value of the S&P 500 index, we may be able to better understand where to expect support or resistance (see table 4).

Table 4 - Price edge sorted per level (from low to high) relative to actual value of S&P 500 index

When reviewing the levels sorted for price edge, relative to the actual value we can draw the following conclusions:

- Levels 4 and 3 show a price-edge slightly above the price target for levels 6 and 5, so we may expect to find resistance between 2,738 and 2,742
- Levels 5 and 6 show a price-edge below the actual price but well above the price targets of levels 4 and 3, so we may expect support near 2,565
- Only after reviewing Risk/Reward we may draw any additional conclusions

**Opportunities relative to progress within a price range**

Reviewing the progress of a dominant trend within the price range of a level helps to better understand potential opportunities. The price range has been defined as the difference between price target and price edge. For this purpose we use table 1 which reflects the progress within a price range per level. Progress within a price range is inversely related to opportunities for that level.

From below graph we observe the following:

- Level 6 and 5 show a dominantly upward trend which arrived just above 70% of their price ranges, in other words there is sufficient remaining upward potential available.
- All remaining levels show a dominantly downward trend and all of them show significant remaining potential in their price ranges (keep in mind this relates to downward price evolution).
- Only after reviewing Risk/Reward we may draw any additional conclusions

Graph 1 - Progress within a price range per distinct level (yellow color represents a dominantly upward trend of a distinct level; pink color represents a dominantly downward trend of a distinct level)

Remaining potential within a level, either with a dominantly upward trend or a dominantly downward trend does not mean much without reviewing potential gain and potential loss (see graph 2), which ultimately will bring us to the Risk/Reward per distinct level (see graph 3).

**Opportunities relative to potential gain and potential loss**

Potential gain is the difference between price-target and actual value. Potential loss reflects the difference between actual value and price-edge. Potential gain per level and potential loss per level are visualized in below graph and should be interpreted relative to the dominant trend for each level:

- For dominantly upward trends as in levels 6 and 5, we observe limited potential gain, while potential loss seems larger. This should be interpreted as a risk regarding a potential price decline of approximately 4% compared to an opportunity for a potential price increase of 2%.
- For dominantly downward trending levels 4, 3, 2 and 1, we observe a potential gain of over 20% compared to a potential loss of less than 4%. This should be interpreted as an opportunity related to a potential price decrease of 20% from current levels and a risk of a potential price increase of 4%.
- Level 1 shows that Laomedeia still expects further trouble ahead, after all level 1 represents a dominantly downward trend

Graph 2 - Potential gain versus potential loss per distinct level (green color represents the potential gain for a dominantly upward trend in a distinct level; blue color represents potential gain for a dominantly downward trend)

**Risk and Reward per distinct level**

Graph 3 represents Risk versus Reward per level. Please, carefully keep in mind the dominant trend for each level. Laomedeia applies a Risk/Reward threshold equal to 0.5 (maximum). All conclusions with respect to joining the market or exit the market are based on this threshold.

- We have learned that only short term level 6 and medium term level 5 show a dominantly upward trend, but from graph 3 we conclude an unfavorable Risk/Reward ratio.
- Medium and long term levels 4, 3, 2 and 1 reflect dominantly downward trends showing favorable Risk/Reward ratios.

Graph 3 - Risk/Reward ratio per distinct level (Laomedeia applies a max threshold of 0.5)

**Overall conclusions:**

- We observe an unfavorable Risk/Reward ratio for levels 6 and 5, both representing dominantly upward trends for short and medium term. Progress within their respective price ranges allows for further upward potential, however, it is not worth the risk as the potential gain is significantly less compared to the potential loss.
- We also observe medium and long term dominantly downward trends with price targets far below actual value, which puts short/medium term levels 6 and 5 in the position of a continued recovery within a longer term dominantly downward trend.
- We currently expect resistance between 2,738 and 2,742, which is just above the current price targets for levels 6 and 5
- We currently expect support near 2,565 within the longer term dominantly downward trend.

**Audit trail**

Financial markets apparently represent a continuously evolving system which generates time dependent price targets and time dependent price edges by means of dynamic chaos theory mathematics. Laomedeia keeps track of all time dependent price targets by means of a unique identifier number per price target per distinct level (see table 2).

Readers may have noticed different identifier numbers in table 2, compared to the identifier numbers in our initial publication of January 3rd, 2019 (See Mathematics and Markets - S&P 500 Index Outlook Per January 3rd, 2019).

The evolution of price targets per level including the date upon which these price targets have been hit is represented in below table. Status per January 31st, 2019, for your information.

Table 5 - Audit trail of time dependent price targets per distinct level

Thank you for reading and your feedback.

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.