2016issue C088-12
Ranking systems with a geometric reward-to-risk average
A single arithmetic average is a poor stand-in for mixed reward-to-risk scores. A geometric mean, with optional metric weights, yields one composite used to rank complete trading procedures and to state how far their profiles sit apart.
- A single arithmetic average of mixed reward-to-risk scores is a poor stand-in for a system's overall reward-to-risk profile.
- Averaging unlike units, or changing a unit's scale, can reverse which candidate looks better under an arithmetic mean, and the same disagreement remains after conversion to unit-free ratios.
- The geometric mean of positive reward-to-risk scores is a coherent composite that lies between the lowest and highest component scores and stays consistent under inversion.
- Applying metric weights to the same scores can reorder which system ranks first, so a coherent composite is computed before a candidate is chosen.
System evaluation as a ranking problem
Tradeoff analysis compares complete trading procedures by folding several reward-to-risk scores into one ranking instead of treating each score as a separate contest. System ranking is that ordering: complete procedures are placed by a single composite reward-to-risk profile rather than by one isolated statistic.
A risk-reward ratio is a positive score that compares a system's return or gain measure with a loss, drawdown, or risk measure. Several such scores can describe the same procedure, so they have to be combined before one ranking can be stated.
Why an arithmetic mean is a poor stand-in
A single arithmetic average of mixed reward-to-risk scores is a poor stand-in for a system's overall reward-to-risk profile. The arithmetic mean is the sum of values divided by their count. It is appropriate when quantities share a scale and equal absolute changes have the same meaning across those quantities.
Averaging unlike units, or changing a unit's scale, can reverse which candidate looks better under an arithmetic mean. That mean is suited to same-scale absolute quantities, not inverted ratios.
The inversion paradox for ratios
Converting values to unit-free ratios still leaves an inversion paradox: the arithmetic mean of A-versus-B ratios need not agree with the arithmetic mean of the inverted B-versus-A ratios.
Reward-to-risk scores live in the positive numbers, cannot be zero or negative, and inversion clusters large values into the interval between zero and one.
The geometric mean as a coherent average
A coherent average for such ratios is obtained by taking logarithms, averaging in that space, and converting back with the exponential, which is the geometric mean. That construction is the coherent average for positive ratios that must stay consistent under inversion.
The geometric mean of several positive reward-to-risk scores is the corresponding root of their product and is undefined if any score is zero or negative. The composite is a hybrid number lying between the lowest and highest component scores. It is used to rank systems and to state how far apart their reward-to-risk profiles are.
Weights can reorder the field
In an unweighted five-system illustration with four reward-to-risk scores each, system X ranked first at 3.97 and system Z last at 1.87.
A weighted geometric mean gives each reward-to-risk metric a chosen importance weight before averaging. Applying metric weights of 3, 2, 1, and 2 to the same scores reordered the field so system V led at 3.71 and system Z remained last at 1.74.
System optimization, in this workflow, means choosing among candidate systems only after a coherent composite reward-to-risk score has been computed, including optional metric weights.