# portfolio information ratio calculation on daily returns including hedged strategy results interpretation

I am tasked with calculating the portfolio information ratio on ~15 years of daily portfolio returns and I am finding several approaches online which is quite confusing.

The first approach simply defines the tracking error (denominator) as the stdev of difference between the daily returns of the portfolio and the index and uses the mean of the difference as the numerator, but this yields extremely small results (ie 0.003)

diff = portfolio daily returns - benchmark daily returns
trkError = np.std(diff)
infRatio = np.mean(diff) / trkError


The second approach is the same as the above but annualizes both the mean difference and the stdev of the difference:

diff = portfolio daily returns - benchmark daily returns
retLngth = 252.0
anlDiff = ((np.mean(diff)+1)**retLngth)
trkError = np.std(diff)*np.sqrt(retLngth)
infRatio = anlDiff / trkError


The third approach defines the tracking error the same as the second approach and then simply subtracts the annualized portfolio return from the annualized index returns for the numerator:

diff = portfolio daily returns - benchmark daily returns
retLngth = 252.0
trkError = np.std(diff)*np.sqrt(retLngth)
infRatio = (annualized daily portfolio returns - annualized daily index returns) / trkError


The fourth approach I am seeing uses the same tracking error as examples 2 and 3 but uses the difference between the total ROI for the portfolio minus the index total ROI as the numerator, but this generates very large results (i.e. -3.07):

diff = portfolio daily returns - benchmark daily returns
retLngth = 252.0
trkError = np.std(diff)*np.sqrt(retLngth)
infRatio = (portfolio toal ROI-index total ROI) / trkError


I am also curious if information ratio is a fair metric for my portfolio approach, which employs a variable index hedge that on average has 60% of the total portfolio cash value shorted in the index. The correlation of these daily portfolio returns are typically around 0.4-0.6 vs the index depending on the market, and I am seeing very low and even negative numbers for my information ratio calculations. As expected, annualized returns are lower than the index for the hedged portfolio and this seems to be a central component of the information ratio calculation. I am curious if any hedged portfolios can have high information ratios considering by nature they have reduced systematic risk considerably? Is the information ratio metric only suitable for un-hedged portfolios?