By performing this statistical and econometric analysis and comparing it with the long-term cyclical behavior, the following fundamental conclusions and results are obtained regarding the dynamics of the asset analyzed in this case, the COLCAP index.
When comparing different regression models, we observe the coefficients of determination (R² squared), which measure what percentage of the price variation is explained by time (start equation x):
Linear regression is poor; the long-term linear trend only captures the general direction of growth but completely ignores the market's upward and downward cycles.
Third-order polynomial regression (R² squared equals 89.96%) captures a basic curvature, significantly improving the explanation of the behavior.
Sixth-order polynomial regression (R² squared equals 94.69%) is the most accurate model. It explains almost 95% of the historical movements within the analyzed range. This demonstrates that the COLCAP's behavior is not linear, but rather responds to complex medium- and long-term cyclical oscillations.
The mathematical analysis of derivatives applied to the 6th-order polynomial allows us to "X-ray" the cycle as follows: the critical points (peaks and troughs). The model precisely defines the beginning and end of trends. The analyzed cycle lasts 1,042 days (approximately 4.1 years) from trough to trough (from Point B at x almost equal to 476 to Point D at x almost equal to 1518). Regarding momentum and trend changes (inflection points), the second derivatives (y'' = 0) reveal when the market begins to lose momentum before a visible price change occurs. Almost equal to 741.8, even though the price continues to rise, econometric analysis warns us that the upward momentum is waning, anticipating the major peak at Point C (x almost equal to 1029). By superimposing the mathematical curve (a 6th-degree polynomial) onto the actual COLCAP data series (the rapidly fluctuating, textured blue line):
Short-term noise smoothing and econometric analysis filter out the high-volatility daily fluctuations ("market noise") to reveal the clear structure of the underlying trend.
The danger of extrapolation (short vs. long term): high-degree polynomials (like the 6th-degree polynomial) are excellent for describing what has already happened (historical adjustment), but they tend to lose consistency at the extremes. At the end of the graph, it can be seen that the mathematical curve tends to rise almost vertically towards the end, while the real COLCAP index tends to sideways or correct.
The statistical analysis shows projections (e.g., a 360-day forecast of 2,154.68). However, the "Probabilities" section calculates that, under a normal distribution with historical mean (approximately 1,495) and standard deviation (approximately 350), reaching extremely high values (such as x = 2,292) has an associated mathematical probability (p = 98.86% cumulative), leaving a very low probability of exceeding historical extremes under normal conditions without a structural change.
The contrast demonstrates that the COLCAP market is highly cyclical. While a simple long-term linear trend would suggest that the market "always rises steadily," the sixth-order econometric analysis robustly demonstrates that the market operates under multi-year cycles of approximately four years. This allows investors to mathematically identify bullish exhaustion zones (turning points) and market bottoms (troughs) to optimize decision-making. The image presents a normal distribution of the COLCAP Index, a fundamental tool for understanding the behavior of this index and making informed investment decisions. A normal distribution, also known as a Gaussian bell curve, shows how the values of a random variable are distributed; in this case, the value of the COLCAP Index.
It is the arithmetic mean of all the values of the COLCAP Index. In the graph, the mean is represented by the dashed vertical line in the center, with a value of 1494.95.
The standard deviation (σ) is a measure that indicates how dispersed the COLCAP index values are around the mean. In the graph, the standard deviation is represented by the marks ±1σ, ±2σ, and ±3σ on the horizontal axis.
standard deviation of 1, or ±1σ, includes approximately 8.26% of all index values. In the graph, this ranges from 1144.60 to 1845.30.
A standard deviation of 2, or ±2σ, includes approximately 95.44% of all index values. In the chart, the range is approximately 1140 to 2000.
A standard deviation of 3, or ±3σ, encompasses approximately 99.73% of all index values. A larger standard deviation indicates greater volatility in the index, meaning that values can fluctuate more widely around the mean. A smaller standard deviation indicates lower volatility.
The normal curve allows us to calculate probabilities associated with different COLCAP index values. The area under the curve represents the total probability (which is 1, or 100%).
When the cumulative area is calculated—which is simply the area under the curve to the left of a given value—it represents the probability that the COLCAP index will be less than or equal to that value. The cumulative confidence area (p=0.9500), or shaded area, represents a 95% probability that the COLCAP index will fall within the range defined by the standard deviations. This is the range in which the index is expected to fluctuate under normal conditions.
The total cumulative area (p=0.9886), which includes both the blue and red areas, represents a 98.86% probability that the COLCAP index will be less than or equal to 2292.03. This means there is a very high probability that the index will remain below this value.
The right tail is the area under the curve to the right of a given value and represents the probability that the COLCAP index will be greater than that value.
The right tail (q=0.0114), shaded in red, represents a 1.14% probability that the COLCAP index will be greater than 2292.03. This probability is low, indicating that it is unlikely the index will exceed this value.
The confidence point is the value that defines a given level of confidence. In the chart, the 95% confidence limit is at 2073.03. It is the point beyond which values are considered unusual. The point of interest is a specific value that the investor wishes to analyze. In the chart, the point of interest is at 2292.03. The analysis shows that there is a very low probability (1.14%) that the index will exceed this value.
Now, the Z-score is a standardized measure that indicates how many standard deviations a given value is from the mean. It is calculated as Z = (X - μ) / σ.
This means that a Z-score of 1.65 for the confidence point and 2.28 for the point of interest indicates that these values are 1.65 and 2.28 standard deviations to the right of the mean, respectively. This confirms that these are unusual values.
The mean of 1494.95 and the range of plus or minus 1 sigma (1144.60 to 1845.30) provide a clear idea of the typical value of the COLCAP index and its normal volatility. Investors should expect the index to fluctuate within this range most of the time.
The 95% confidence limit at 2073.03 and the point of interest at 2292.03 are unusual values. Investors should be aware that the index is unlikely to exceed these values.
Analyzing the normal curve can help investors make informed decisions about when to buy or sell. For example, if the COLCAP index approaches the mean, investors might consider buying, as the index is at its typical value. If the index approaches the confidence limits, investors might consider selling, as the index is at an unusual value.
The normal distribution is a model; the normal curve is a simplification of reality. The behavior of the COLCAP index may not perfectly follow a normal distribution, especially during periods of high volatility or unforeseen events.
Technical and fundamental analysis are complementary; normal curve analysis is a form of technical analysis. It should be complemented by fundamental analysis, which focuses on the economic and financial factors that affect the COLCAP index.
Past performance is no guarantee of future performance; the historical data used to create the normal curve does not guarantee that the COLCAP index will behave the same way in the future.
ColCAP index normal curve analysis provides a valuable tool for investors to understand the index's behavior, identify unusual values, and make informed investment decisions. However, it is important to remember that this analysis is only one part of the investment process and should be complemented by other factors and consultation with professionals.
In the technical analysis, the trend appears extremely strong. The ADX(14) at 60.29 indicates that the current trend (which is bearish) has unusual and solid strength. Volatility is high. The ATR(14) at 7.70 suggests wide price ranges, requiring strict risk management.
The current price is trading below all the moving averages analyzed (from the short-term MA5 to the long-term MA200).
With the price below the MA200 (~2282 - 2302), the main long-term bias is entirely bearish. Any upward bounce will likely encounter resistance in this area.
Although the trend is strongly bearish, the fast oscillators show an important reading: oversold exhaustion.
The bearish continuation signals—the MACD (-2.48), the Ultimate Oscillator (39.17), and the ROC (-1.14)—confirm that the momentum remains with the sellers.
The oversold warning signals—the Stochastic Oscillator (0.447), the Williams %R (-93.57), and the CCI (-280.01)—are at extreme oversold levels.
The price has fallen so sharply and rapidly that a temporary technical bounce or a consolidation pause would not be unusual in the short term, before the decline resumes. Key price levels (pivot points for trading): Considering that the classic equilibrium price or pivot point (PP) is at 2270.66, these are the levels you should watch for the session. Support zones (floors to watch if the price continues to fall). Going against an ADX of 60 and a complete alignment of moving averages in a sell position is extremely risky (trading counter-trend bounces). In this scenario, conventional technical strategy suggests looking for sell (short) positions on failed tests of resistance zones (such as the Pivot Point at 2270.66 or R1), looking for a continuation of the downward trend towards support levels S1 and S2.
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