Omega Ratio Formula
The Omega ratio is the ratio of probability-weighted gains above a threshold to probability-weighted losses below it. Unlike Sharpe or Sortino, it uses the entire return distribution rather than just the first two moments, so it captures skew and fat tails automatically.
Formula
Copy the exact expression or work through it step by step below.
Omega(T) = ( integral from T to +inf of (1 - F(r)) dr ) / ( integral from -inf to T of F(r) dr )
Discrete: Omega(T) = ( sum of max(0, R_i - T) ) / ( sum of max(0, T - R_i) ) Variables
T
Threshold return
The dividing line between a gain and a loss for the investor, often zero or the risk-free rate. Returns above T feed the numerator; returns below feed the denominator. As T rises, Omega falls.
F(r)
Cumulative distribution function
The CDF of returns. The numerator integrates the upper tail (1 - F) above the threshold; the denominator integrates the lower tail (F) below it. This is the continuous form of the gain-to-loss accounting.
R_i
Period return
An individual observed return in the discrete (sample) version of the formula.
max(0, R_i - T)
Excess gain
The amount by which a return clears the threshold, floored at zero. Summed across all periods this estimates the area of expected gains above T.
Step By Step
- 1
Choose the threshold T that separates acceptable from unacceptable returns.
Set T = 0 so any positive return is a gain and any negative return is a loss.
- 2
For each return, compute the excess above the threshold, flooring negatives at zero, and sum these to get total weighted gains.
Returns +5%, +2%, -3%, +1%, -4% against T = 0 give gains 0.05, 0.02, 0, 0.01, 0; sum = 0.08.
- 3
For each return, compute the shortfall below the threshold, flooring negatives at zero, and sum these to get total weighted losses.
The same returns give losses 0, 0, 0.03, 0, 0.04; sum = 0.07.
- 4
Divide total weighted gains by total weighted losses.
0.08 / 0.07 = 1.143.
Worked Example
Discrete Omega over five monthly returns, threshold T = 0
Monthly returns
+5%, +2%, -3%, +1%, -4%
Threshold T
0%
Gains above T: max(0, 0.05) + max(0, 0.02) + max(0, -0.03) + max(0, 0.01) + max(0, -0.04) = 0.05 + 0.02 + 0 + 0.01 + 0 = 0.08. Losses below T: max(0, -0.05)... evaluated as max(0, T - R): 0 + 0 + 0.03 + 0 + 0.04 = 0.07. Omega = 0.08 / 0.07 = 1.143.
Omega of about 1.14 at a zero threshold, meaning the probability-weighted gains exceed the losses by roughly 14%. An Omega above 1.0 indicates the distribution favors the investor at that threshold; at exactly T equal to the mean return, Omega equals 1.0 by construction.
Common Variations
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Risk-Adjusted Returns Calculator
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Sharpe vs Sortino Calculator
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Sources & References
- A Universal Performance Measure — Con Keating and William F. Shadwick, Journal of Performance Measurement (2002)
- An Introduction to Omega — Con Keating and William F. Shadwick, The Finance Development Centre (2002)
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