Positive Trade Ratio
It's ironic that Gardner says PTR is similar to the Sortino Ratio except that "there is no need for Downside Deviation"    and immediately afterwards, he calculates his own slightlymodified Downside Deviation (in the denominator).
The only difference between his denominator and Sortino's, is the fraction in front of the summation.
Gardner uses (1 / (#returns  1))
Sortino uses (1 / (#negativeReturns  1))
Whoop de do
The only difference between his denominator and Sortino's, is the fraction in front of the summation.
Gardner uses (1 / (#returns  1))
Sortino uses (1 / (#negativeReturns  1))
Whoop de do

 Roundtable Knight
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I may be wrong here but ...
Sortino uses a different numerator AND a different denominator. The denominator is different in two ways from Gardner's statistic:
Sortino numerator: cumulative average return (geometric return) less target return
Gardner Numerator: average (arithmetic) trade return.
Sortino denominator: root mean (i.e. 1/n) square deviation from the target of period returns that were below the target return. i.e. SQRT(SUM(Ri  T)^2/n) for all (Ri < T) where Ri is the ith period return, T is the Target, n is number of periods for which Ri < T.
Gardner Numerator: population estimate (i.e. 1 / (n+1)) of the deviation from the mean trade return of trades that were below the mean trade. i.e. SQRT((Ti  Tbar)^2 / (n+1)) for all (Ti < Tbar) where Ti is the ith trade, Tbar is the average trade and n is total number of trades.
Essentially, Gardner's Positive Trade Ratio uses the average trade as the target return  that's why he claims there's no need to select a minimum acceptable return (the target).
But, still, whoop de do!
Sortino uses a different numerator AND a different denominator. The denominator is different in two ways from Gardner's statistic:
Sortino numerator: cumulative average return (geometric return) less target return
Gardner Numerator: average (arithmetic) trade return.
Sortino denominator: root mean (i.e. 1/n) square deviation from the target of period returns that were below the target return. i.e. SQRT(SUM(Ri  T)^2/n) for all (Ri < T) where Ri is the ith period return, T is the Target, n is number of periods for which Ri < T.
Gardner Numerator: population estimate (i.e. 1 / (n+1)) of the deviation from the mean trade return of trades that were below the mean trade. i.e. SQRT((Ti  Tbar)^2 / (n+1)) for all (Ti < Tbar) where Ti is the ith trade, Tbar is the average trade and n is total number of trades.
Essentially, Gardner's Positive Trade Ratio uses the average trade as the target return  that's why he claims there's no need to select a minimum acceptable return (the target).
But, still, whoop de do!