So far we have, for each return r between the Minimum and Maximum returns (L and U): We interpret this as follows: Suppose that f(x) and F(x) are the probability density and cumulative probability for some set of returns. Pick some threshold return r between the Minimum and Maximum returns, L and U. We consider the numerator in the above expression for Omega: 1 - F(r) is the probability that a randomly selected return is greater than r. (r,U) is the average of returns which are greater than r. (r,U) - r is the how much this average exceeds r. The numerator is then (the probability that a return is greater than r) * (average excess of returns ) and is a measure of Gains with respect to the selected return r. Now, the denominator: F(r) is the probability that a randomly selected return is less than r. (L,r) is the average of returns which are less than r. r - (L,r) is how much r exceeds this average. The denominator is then : (the probability that a return is less than r) * (average deficit of returns) and is a measure of Loss with respect to the selected return r. Omega then is a measure of Gains to Losses for the stock in question. >A stock .. or an entire portfolio of stocks? Either. Note that we look at returns above some threshold return r and see if we're in that neighbourhood.   ... as measured by 1 - F(r) Then we look at returns below r and see if we're in that neighbourhood   ... as measured F(r) Since each of these neighbourhood has an associated average return ... >Yeah, okay. Are you finished? Hardly. Let's look carefully at the average of the "excess" and "deficit" returns, compared to some risk-free rate rf: "Excess Returns" = (rf,U) - rf     ... which we associate with a Reward   "Deficit Returns" = rf - (L,rf)     ... which we associate with a Risk   This gives a neat Risk/Reward Ratio, namely: Ω = Risk/Reward = ( rf - (L,rf) ) / ((rf,U) - rf ) >Huh? Ω is omega ... in Greek. >Greek? Why am I not surprised ... but isn't the Sharpe Ratio a Risk / Reward Ratio? No, it's a Reward / Risk ratio. The Sharpe Ratio is: Sharpe Ratio = ( - rf) / V where is the average of all returns (from L to U) and V is the Volatility (or Standard Deviation) of all returns and getting an average return greater than the Risk-free Rate (that's - rf) is your Reward and (following a popular, tho' silly interpretation), V is your "Risk". >So, which is best? Define "best". >Which do you like best? Personally, I like Ω ... but what do I know. For the Sharpe Ratio I guess one would look at historical returns and extract just two numbers, and V ... ignoring the distribution of returns. On the other hand, to evaluate Ω you need to investigate the entire distribution of returns. >But the proof ... Is in the pudding? Exactly ... so we'll investigate this stuff. Let's look at the weekly returns for GE stock, over a moving 10-year period ... and a Risk-free Rate of 4%. >4% per week? No, we'll use Risk-free = (1.04)1/52 - 1 or about 0.076% per week. Anyway, a comparison of the Sharpe Ratio and Omega is here >What about Ω ? That's here: Note that Ω is a kind of "Risk / Reward" ratio whereas Omega is a ratio which reflects "Gains / Losses". Their relationship is : Ω = (1/F(r) - 1) / Omega   where "r" is the threshold return (which may be taken as a Risk-free Rate). >So, which is best? Define "best"   >Okay, which would have given you the best portfolio performance over, say 1950 to 2000? You mean which stock or which allocation or which ...? >Which allocation of assets, choosing from ... uh ...? Okay, here's what we'll do: We choose from Large Cap Growth (LG) and Small Cap Growth (SG) and Small Cap Value (SV) and, say T-bills. We look at the annual returns of these four equity classes from 1940 to 1960 to see how they performed. We pick a particular allocation, with annual rebalancing (say 40% LG + 20% SG + 30%SV + 10% T-bills). We determine the annualized return of such a portfolio (using the data from this time period). We repeat steps 1 and 2 to see which allocation gives the largest Sharpe Ratio and Omega and smallest Ω. We compare the annualized returns for the three "best" portfolios (largest Sharpe, largest Omega and smallest Ω) >The "best" is 100% SV. Am I right? Yes. Very clever. Congratulations. Here's the result for 1940 - 1960: "Best" LG SG LV T-bills Return Sharpe 35% 0% 65% 0% 17.0% Omega 40% 0% 60% 0% 16.7% Ω 0% 35% 65% 0% 17.6% For this time period, Small Cap Value had an annualized return of 19.5% and the S&P500 was 13.4% >I still like100% SV! Pay attention! Having decided upon the "best" allocations, we consider the next 20 years, using those "best" allocations. Here's what we get : 1960-2000 LG SG LV T-bills Return Sharpe using 35% 0% 65% 0% 14.6% Omega using 40% 0% 60% 0% 14.4% Ω using 0% 35% 65% 0% 13.8% For this time period, Small Cap Value had an annualized return of 16% and the S&P500 was 11.6% >Which would YOU use ... to predict future allocations? Me? What do I know ... however, I'd probably choose this   >Very funny. So where's the spreadsheet? Well, it looks like this with an explanation which looks like this. To download the Excel spreadsheet, RIGHT-click here and Save Target. >And you guarantee the accuracy? No. >Okay, but does it make sense to calculate Omega for a single stock? I have no idea, but there's a spreadsheet that'll do that (I think). It looks like this and can be downloaded by RIGHT-clicking here ... then Save Target. Note: Other references to Omega can be found here.