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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?

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The information ratio is the ratio of residual (active) return to residual (active) risk. This is usually expressed using annualized numbers for residual risk and residual return. Ex-post, you would take the mean return of the portfolio and subtract the mean return of the benchmark and divide it by the standard deviation of the (portfolio return - benchmark return). In the end, since the information ratio is a comparison tool, one needs to be consistent in the approach across all portfolios being evaluated.

Realized information ratios can and frequently are negative. As for their application to hedged portfolios (for this and any other metric) one must look to see if it assumptions underlying the information ratio hold for the portfolio. For example, for the residual (active) risk number to be a useful the benchmark has to be appropriate for the strategy. Is your benchmark similarily hedged? Also, more fundamentally, are the assumptions behind using standard deviation as a risk metric appropriate. Are the distribution of returns normally distributed? Most hedged portfolios are not. The design of such hedges are to mitigate risk over some range of outcomes and assumptions. As such the standard deviation is usually reduced at the expense of introducing fatter tails. At the extreme, say you hedged your portfolio to have a binary outcom. One where you have a larger probability of an expected return; and 1- that probability of a loss. Standard deviation will never produce either of these outcomes and is therefore a poor measure of risk for such a strategy. The use of such measures is often misapplied by investors when evaluating hedge fund strategies.

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  • $\begingroup$ Thanks for the reply - the benchmark is not similarly hedged, and the portfolio returns are not normally distributed. Nevertheless, I am being asked to report this metric so I want to make sure I fully understand your explanation. Are you saying I should take: (annualized portfolio returns - annualized index returns) / stdev of (portfolio returns - index returns) ? I don't have to annualize the denominator? $\endgroup$ – trock2000 Feb 23 at 22:46
  • $\begingroup$ @trock2000 Take the mean returns of the portfolios and benchmark and annualize them. Then take the strs deviation of the return differences from that same data set and annualize that as well. So in the denominator, I’d you used daily differences, multiply that std dev by the sq root of 262 to get an annualized volatility. $\endgroup$ – AlRacoon Feb 23 at 23:04
  • $\begingroup$ @trock2000 The annualization of st dev I gave assumes there were 262 trading days a year. Some assume 260 or 250. $\endgroup$ – AlRacoon Feb 23 at 23:15
  • $\begingroup$ @AIRacoon thanks for clarifying, I think I have it now: diff = dailyPortRets - dailyIndexRets trkError = np.std(diff)*np.sqrt(262) infRatio = (annualizedDailyPortfolioRets-annualizedDailyIndexRets) / trkError $\endgroup$ – trock2000 Feb 23 at 23:31

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