POLYNOMIAL MODELING OF CYCLES AND RISK-RETURN EFFICIENCY DIAGNOSIS

 

This study employs a multi-model approach, as it not only addresses the topic with simple linear regression but also applies polynomial models (of varying degrees), allowing it to capture trend changes, cycles, or nonlinear behaviors that a linear regression completely ignores.

The inclusion of the Sharpe Ratio and the Coefficient of Variation (CV) demonstrates that the study aims not only to predict value but also to understand the volatility and efficiency of the asset or variable under study.

By separating the sample into three different sizes—long-term with 1221 days of observations, medium-term with 730 days, and short-term with 180 days—the dynamics of each time frame can be observed.

In the long term, using the complete five-year historical analysis, a very weak least-squares trend line is observed (R² = 53.81%). However, the higher-degree polynomial model achieves an impressive R² of 95.09%. This indicates that CIBEST's historical behavior is strongly marked by complex cycles or oscillations.

The histogram and the Skewness (Asymmetry = 1.54) and Kurtosis (1.41) data show a clear positive skew. The data is concentrated at low values ​​(the peak is between 26,080 and 36,079) with a long tail to the right.

The Sharpe ratio of 2.20 is excellent, indicating a high return for each unit of risk assumed over this overall period.

In the medium term, using 730 days as the Intermediate Period, the linear trend becomes stronger (R² = 85.13%), and the polynomial rises to 97.13%. The series became much more predictable and uniform in this segment.

The Coefficient of Variation rises slightly (36.74%), and the Sharpe ratio falls to 1.45. This remains acceptable, but less efficient than in the overall picture.

In the short term (180 days, or recent period), the straight line fails spectacularly again in this shorter range (R² = 50.58%). Recent behavior is erratic or highly cyclical, requiring the polynomial (R² = 88.46%) for interpretation.

The change in behavior shows that the mean jumped to 74,824, a value drastically higher than the previous blocks (43,192 and 48,039). CIBEST has experienced massive growth recently, but with negative skewness (-0.15), suggesting that it is moving at the upper end of the distribution, with occasional sharp drops.

The exceptional long-term return of 193.94%, with a coefficient of variation (CV) of 35.05% and a standard deviation of 15,137.86, demonstrates a high level of efficiency in the Sharpe ratio of 2.20.

This period (representing the largest or historical sample) shows an extraordinary cumulative return of almost 194%. The most noteworthy aspect from a financial perspective is its Sharpe ratio of 2.20, which is considered excellent. This demonstrates that the asset generated a highly robust return for each unit of volatility assumed. Despite a considerable standard deviation, the consistency of the upward trend (captured in previous polynomial models) more than compensated for the risk.

Over the medium term (730 days), maturation and increased volatility are observed, with a return of 160.82%, a coefficient of variation (CV) of 36.74%, a standard deviation of 17,647.31, and a Sharpe ratio of 1.45. In this intermediate time frame, the return remains very high (160.82%). However, a structural change is observed: risk increased, and efficiency decreased. The standard deviation rises to 17,647.31, and the coefficient of variation increases slightly to 36.74%, meaning the asset became more volatile relative to its mean (48,039). As a direct consequence, the Sharpe ratio falls to 1.45. This remains an institutionally attractive performance (greater than 1.0), but it denotes a more turbulent market environment or one with more pronounced corrections.

This is still an institutionally attractive performance (greater than 1.0), but it indicates a more turbulent market environment or one with more pronounced corrections. Finally, in the short term (180 days) or recent phase of stabilization or commercial maturity, the return is 32.12%, with a coefficient of variation (CV) of 12.25% and a standard deviation of 9,166.91. The Sharpe ratio of 1.80 indicates good efficiency for CIBEST.

The shortest period (180 observations) shows a return of 32.12%. Although nominally lower compared to longer-term horizons, the underlying statistical behavior is extremely interesting. The coefficient of variation drops drastically to 12.25%, indicating that prices have compacted strongly around their new level The average price is high (74,824). Price dispersion decreased significantly.

With the reduction in volatility or standard deviation from 9,166.91, the Sharpe ratio rebounded to 1.80. This indicates that, in the recent short term, CIBEST is a much more predictable, safe, and efficient asset per unit of risk than in the medium term, moving within a plateau of high prices with controlled fluctuations.

The shift from the medium-term period to the short-term period reflects an asset that has moved from a phase of high volatility and aggressive growth to a stage of consolidation. The decrease in the coefficient of variation (CV) from 36.74% to 12.25% is a key technical indicator of stability in the recent period.

If historical consistency is assessed, the overall (long-term) scenario continues to show the best general efficiency metrics (Sharpe 2.20), suggesting that CIBEST has historically rewarded retention strategies or long-term analyses (Buy and Hold or structural investments), while intermediate periods experienced turbulence that temporarily affected the risk-premium relationship.

The graphs show polynomial equations of up to the 6th degree. Although mathematically an R² of 95% or 97% looks spectacular in the snapshot of the past, polynomials of such high degrees suffer from a serious prediction error at the extremes. If you use that 6th-degree equation to "forecast" tomorrow (data point 1222), the curve tends to spike unrealistically up or down.

The study is an excellent descriptive diagnosis of CIBEST's past performance. It clearly demonstrates that the variable is not linear and that it has recently gone through a period of strong short-term appreciation.