# Back Testing of TRE

We back-tested portfolios constructed using Tsallis Relative Entropy (TRE) and Beta vs. two benchmarks: SPY for the SP500 index and ONEQ for the Nasdaq index, for periods of 1 year and 6 years. To build up a sufficient number of samples to calculate statistics, we computed overlapping periods each shifted by one month, e.g., January 2010 to January 2016, then February 2010 to February 2016, etc., up to January 2014 to January 2020 for 6 year periods. Similarly for 1 year periods. This results in 49 samples for 6 year periods and 109 samples for 1 year periods.

For each sample, we compute total return over 6 year and 1 year periods (not annual return). From that, we find the median return of all samples, the overall minimum and maximum return of all samples, percentage of samples with negative return, and percentage of samples that performed worse than the benchmark.

Portfolios were constructed of 16 stocks centered on a Beta or TRE risk value. Each portfolio was rebalanced every 12 months, i.e., a new set of 16 stocks was determined and the portfolio value at that time was evenly distributed among the 16 stocks. The center risk value and the average plus/minus risk range is given in the tables.

Table 1: Total Return, 6 Year Periods, 49 Samples, 16 Stocks in Portfolio

## Table 2:Total Return, 1 Year Periods, 109 Samples, 16 Stocks in Portfolio

The above portfolios constructed with both TRE and Beta are high risk, by design, as we picked high values of TRE and Beta when constructing the portfolios. As such, their volatilities are greater than the benchmark volatilities, i.e., we expect to see better returns than the benchmarks when the market is up, but we may also see worse returns when the market is down. Even so, by holding the portfolios for several years, and reselecting and rebalancing the stocks in the portfolios every 12 months, we get much better total return from these high risk portfolios than we do from the benchmark ETFs.

As expected, the TRE portfolios perform better than the Beta portfolios.

The portfolios over 6 year periods almost always beat their benchmarks: 100% of the time for TRE vs. SPY and 98% of the time for TRE vs. ONEQ. Even over 1 year periods, TRE will be better than SPY and ONEQ two-thirds of the time.