In our anxiety to weigh the possibility of high returns against the risk of losing money—or the “risk” of high variability—we investors have, over the years, come up with over a dozen measures of “risk-adjusted returns.” The most commonly used are the Sharpe ratio and Jensen’s alpha, but I have found a number of others equally or more useful, including the information ratio, median excess returns, and the omega ratio. It is this last that I’d like to turn my attention to today.
Omega was conceived in 2002 by Con Keating and William F. Shadwick; you can find one of their original papers http://www.all-in-or-out.com/An%20Introduction%20to%20Omega.pdf here. I’ll quote from it at length:
[quote] Omega . . . employs all the information contained within the return series. . . . It can be used to rank and evaluate portfolios unequivocally. All that is known about the risk and return of a portfolio is contained within this measure. . . . This function is, in a rigorous mathematical sense, equivalent to the returns distribution itself, rather than simply being an approximation to it. It therefore omits none of the information in the distribution and is as statistically significant as the returns series itself. As a result, Omega is ideally suited to the needs of financial performance measurement where what is of interest to the practitioner is the risk and reward characteristics of the return series. . . . In use, Omega will show markedly different rankings of funds, portfolios or assets from those derived using Sharpe ratios, Alphas or Value at Risk, precisely because of the additional information it employs.
Omega can be used to compare all sorts of returns, not just equity returns, but the returns have to have some variability. For a fund that simply returns a steady 0.1% per month, the omega function is useless.
There’s an omega ratio and an omega function. The ratio varies depending on your choice of a “hurdle rate,” or an expected rate of return. The omega function tracks the omega ratios for all possible hurdle rates.
The omega ratio is simple to calculate. You choose a hurdle rate of return. This could be zero, the risk-free rate, the median or mean return of a comparable benchmark, or simply a percentage return you’d really like to see. You then take the returns (monthly, weekly, or annual) of your portfolio, strategy, or fund and subtract that hurdle rate to get adjusted returns. Lastly, you sum up all the adjusted returns that are greater than zero, and divide by the absolute value of the sum of all the adjusted returns that are less than zero. (If the “hurdle rate” is zero, you get the gain-to-pain ratio, which was delineated by Jack Schwager in Appendix A of his 2012 book Hedge Fund Market Wizards.)
Through experimentation, I have found that setting the hurdle rate to the median return of whatever benchmark you’re using is a pretty good way to compare the future returns of a backtested strategy. I like this better than the simple gain-to-pain ratio. I also like to take the product of the ratio at three different hurdle rates: 0%, the median monthly return, and twice the median monthly return. A strategy whose omega ratio beats all others at all three of those points is a strategy that is highly likely to perform well going forward.
There is no one standard hurdle rate; the omega function is the continuous function of omega ratios as the hurdle rate increaces. Usually performance measures are associated with a single number, not a function or a series of numbers or a choice between various numbers depending on your expected rate of return. You can estimate the equation for the omega function, but an equation is necessarily more complicated than the single number that a Sharpe ratio or alpha consists of.
The omega function is basically a reconfigured histogram, but it’s far easier to read and process. You don’t have to worry about tails, skew, and kurtosis: all the information is right there in one simple curve. The Sharpe ratio and the information ratio are based on the assumption that returns are normally distributed; they can therefore fail when returns have asymmetric data. But omega takes all the moments of distribution into account.
There are many other things you can learn from the omega function. To illustrate, let’s compare the omega ratios of some mutual funds. I’ve chosen five at random here: XEXDX, Eaton Vance Tax-Advantage Bond; LAIEX, Lord Abbott International Opportunities Fund; HFQIX, Janus Henderson Global Equity Income Fund; HAIHX, Hartford Health Care HLS Fund; and MPGFX, Mairs and Power Growth Fund. Here is what their performance looked like over the last three years if you’d invested $100 in each. (I’m doing this for the simplicity of illustration alone; if I were actually thinking of investing in these funds I’d look at a lot more than three years of returns.)
[5 mutual fund returns]
From this chart, it looks like HAIHX performed the best and XEXDX the worst, but it also looks like HAIHX is the riskiest in terms of variability of returns.
Now let’s look at their omega function:
[omega function of 5 mutual funds]
As you can see, if the hurdle rate is 0%, the omega ratios (or gain-to-pain ratios) are 0.63, 1.70, 1.62, 2.01, and 2.22. This means, for instance, that the sum of the positive returns of HAIHX over the last three years is twice as large as the sum of the negative returns, while the sum of the negative returns of XEXDX is larger than the sum of the positive ones. As the hurdle rate increases, the omega ratios decrease, which is what you would intuitively expect. (Just a note: these are typical ratios for monthly returns; the ratios for yearly returns tend to be a lot higher and for weekly returns a lot lower.)
Now, the steeper an omega curve, the less “risky” it is, in the sense that it has fewer extreme gains and losses. This is because as you move the hurdle rate from one number to another, if the returns are mostly clustered around the median return, the drop-off is going to be pretty steep; if the returns are all over the place, moving the hurdle rate isn’t going to make such a big difference. So the mutual fund with the most extremes (the shallowest slope) is indeed HAIHX, and the one with the least is XEXDX. You can see this better if you look at the graph in logarithmic scale, where the curves become relatively straight lines.
[omega function of 5 mutual funds log scale]
There’s something else important on these graphs. Take a look at where each of the lines intersects y = 1 (on either graph). XEXDX does so at –0.42% (off the left side of the chart), LAIEX at 0.91%, HFQIX at 0.65%, HAIHX at 1.51%, and MPGFX at 1.15%. This is the average monthly return of each fund.
Now the omega function can be closely approximated by the exponential function y = b*emx, where b is the gain-to-pain ratio (the omega ratio at a hurdle rate of 0%) and m is the logarithmic slope of the curve (i.e. the linear slope if you use the logarithms of the omega ratios at each point). Here are the exponential functions approximated for the five mutual funds:
XEXDX: 0.63e–111.3x
LAIEX: 1.70e–56.5x
HFQIX: 1.62e–74.9x
HAIHX: 2.01e–48.0x
MPGFX: 2.22e–69.3x
This gives us additional information: if we want our returns to have low variability, we can choose not only on the basis of the omega ratio at our given hurdle rate(s), but also of the logarithmic slope, with more extreme numbers being better. Especially if you’re hedging or using margin, variability can be very important.
So I suggest taking the product of the omega ratio at 0%, the omega ratio at the median benchmark return, the omega ratio at twice the median benchmark return, and the logarithmic slope (which you can calculate quickly from just those three ratios without having to extend the line). I call this number the “ultimate omega.”
[omega scores 5 mutual funds]
You can see that the choice of mutual funds will be somewhat different if we take this approach. If we look at the omega ratio only at these three points, we’ll choose HAIHX; but if we also look at the logarithmic slope, using ultimate omega, MPGFX looks like a better bet.
If you’re using Excel, here’s how to calculate the ultimate omega given a sequence of returns, whether they be weekly, monthly, or yearly. Remember that this is a comparison tool, and that results will be very different if you’re using different time periods.
Let’s say your returns are in column A. In column B, subtract the median benchmark return from the column A returns. In column C, subtract the median benchmark return from the column B returns. Now in cell D1 type =SUMIF(A:A,">0")/-SUMIF(A:A,"<0"); in cell D2 type =SUMIF(B:B,">0")/-SUMIF(B:B,"<0"); and in cell D3 type =SUMIF(C:C,">0")/-SUMIF(C:C,"<0"). In cell E1, type 0; in cell E2, put in the median benchmark return; and in cell E3, put in twice the median benchmark return. Now in cell F1, type =PRODUCT(D1:D3,-SLOPE(LN(D:D),E:E)). That will be the ultimate omega.
I often perform what I call “correlation studies,” in which I backtest a number of different strategies using multiple in-sample and out-of-sample time periods, and then correlate the various performance measures during the in-sample periods with the actual returns of the out-of-sample periods. I recently performed such a study that I think may be the best one I’ve done. I took, at random, fifty different ranking systems from the over 7,000 public ranking systems that Portfolio123 [https://www.portfolio123.com/?sref=27297] users have created, and combined each with a different stock universe (whose rules screen out certain stocks, usually low-liquidity ones along with others based on various criteria). I then ran simulations of each one, holding the top fifty-ranked stocks, with a monthly rebalance and 0.25% slippage, from 1/1/1999 to today. I then compared the total returns of each strategy over six different “out-of-sample” five-year periods with the risk-adjusted returns of the same strategy over adjoining eight-year periods, as shown in the chart below.
[in sample out of sample]
I found that the risk-adjusted measures that correlated the most strongly with the out-of-sample total returns were, hands-down, the various omega ratios. I’ve put the results below, bolding the highest correlations for each period. (I used monthly returns, measured daily, for all the measures. “Omega 1” is the ratio after subtracting the median benchmark return, “omega 3” is the product of the first three ratios, “omega 1s” multiplies omega 1 by the logarithmic slope, and “omega 3s” is ultimate omega.)
[correlation study]
Based on these results, I think that in the future, “ultimate omega” will be the measure I will turn to most frequently when deciding upon the optimal investment strategy. After all, it contains more information about in-sample returns than any other performance measure, so no wonder that it correlates so well with out-of-sample returns.
In our anxiety to weigh the possibility of high returns against the risk of losing money—or the “risk” of high variability—we investors have, over the years, come up with over a dozen measures of “risk-adjusted returns.” The most commonly used are the Sharpe ratio and Jensen’s alpha, but I have found a number of others equally or more useful, including the information ratio, median excess returns, and the omega ratio. It is this last that I’d like to turn my attention to today.
Omega was conceived in 2002 by Con Keating and William F. Shadwick; you can find one of their original papers http://www.all-in-or-out.com/An%20Introduction%20to%20Omega.pdf here. I’ll quote from it at length:
[quote] Omega . . . employs all the information contained within the return series. . . . It can be used to rank and evaluate portfolios unequivocally. All that is known about the risk and return of a portfolio is contained within this measure. . . . This function is, in a rigorous mathematical sense, equivalent to the returns distribution itself, rather than simply being an approximation to it. It therefore omits none of the information in the distribution and is as statistically significant as the returns series itself. As a result, Omega is ideally suited to the needs of financial performance measurement where what is of interest to the practitioner is the risk and reward characteristics of the return series. . . . In use, Omega will show markedly different rankings of funds, portfolios or assets from those derived using Sharpe ratios, Alphas or Value at Risk, precisely because of the additional information it employs.
Omega can be used to compare all sorts of returns, not just equity returns, but the returns have to have some variability. For a fund that simply returns a steady 0.1% per month, the omega function is useless.
There’s an omega ratio and an omega function. The ratio varies depending on your choice of a “hurdle rate,” or an expected rate of return. The omega function tracks the omega ratios for all possible hurdle rates.
The omega ratio is simple to calculate. You choose a hurdle rate of return. This could be zero, the risk-free rate, the median or mean return of a comparable benchmark, or simply a percentage return you’d really like to see. You then take the returns (monthly, weekly, or annual) of your portfolio, strategy, or fund and subtract that hurdle rate to get adjusted returns. Lastly, you sum up all the adjusted returns that are greater than zero, and divide by the absolute value of the sum of all the adjusted returns that are less than zero. (If the “hurdle rate” is zero, you get the gain-to-pain ratio, which was delineated by Jack Schwager in Appendix A of his 2012 book Hedge Fund Market Wizards.)
Through experimentation, I have found that setting the hurdle rate to the median return of whatever benchmark you’re using is a pretty good way to compare the future returns of a backtested strategy. I like this better than the simple gain-to-pain ratio. I also like to take the product of the ratio at three different hurdle rates: 0%, the median monthly return, and twice the median monthly return. A strategy whose omega ratio beats all others at all three of those points is a strategy that is highly likely to perform well going forward.
There is no one standard hurdle rate; the omega function is the continuous function of omega ratios as the hurdle rate increaces. Usually performance measures are associated with a single number, not a function or a series of numbers or a choice between various numbers depending on your expected rate of return. You can estimate the equation for the omega function, but an equation is necessarily more complicated than the single number that a Sharpe ratio or alpha consists of.
The omega function is basically a reconfigured histogram, but it’s far easier to read and process. You don’t have to worry about tails, skew, and kurtosis: all the information is right there in one simple curve. The Sharpe ratio and the information ratio are based on the assumption that returns are normally distributed; they can therefore fail when returns have asymmetric data. But omega takes all the moments of distribution into account.
There are many other things you can learn from the omega function. To illustrate, let’s compare the omega ratios of some mutual funds. I’ve chosen five at random here: XEXDX, Eaton Vance Tax-Advantage Bond; LAIEX, Lord Abbott International Opportunities Fund; HFQIX, Janus Henderson Global Equity Income Fund; HAIHX, Hartford Health Care HLS Fund; and MPGFX, Mairs and Power Growth Fund. Here is what their performance looked like over the last three years if you’d invested $100 in each. (I’m doing this for the simplicity of illustration alone; if I were actually thinking of investing in these funds I’d look at a lot more than three years of returns.)
[5 mutual fund returns]
From this chart, it looks like HAIHX performed the best and XEXDX the worst, but it also looks like HAIHX is the riskiest in terms of variability of returns.
Now let’s look at their omega function:
[omega function of 5 mutual funds]
As you can see, if the hurdle rate is 0%, the omega ratios (or gain-to-pain ratios) are 0.63, 1.70, 1.62, 2.01, and 2.22. This means, for instance, that the sum of the positive returns of HAIHX over the last three years is twice as large as the sum of the negative returns, while the sum of the negative returns of XEXDX is larger than the sum of the positive ones. As the hurdle rate increases, the omega ratios decrease, which is what you would intuitively expect. (Just a note: these are typical ratios for monthly returns; the ratios for yearly returns tend to be a lot higher and for weekly returns a lot lower.)
Now, the steeper an omega curve, the less “risky” it is, in the sense that it has fewer extreme gains and losses. This is because as you move the hurdle rate from one number to another, if the returns are mostly clustered around the median return, the drop-off is going to be pretty steep; if the returns are all over the place, moving the hurdle rate isn’t going to make such a big difference. So the mutual fund with the most extremes (the shallowest slope) is indeed HAIHX, and the one with the least is XEXDX. You can see this better if you look at the graph in logarithmic scale, where the curves become relatively straight lines.
[omega function of 5 mutual funds log scale]
There’s something else important on these graphs. Take a look at where each of the lines intersects y = 1 (on either graph). XEXDX does so at –0.42% (off the left side of the chart), LAIEX at 0.91%, HFQIX at 0.65%, HAIHX at 1.51%, and MPGFX at 1.15%. This is the average monthly return of each fund.
Now the omega function can be closely approximated by the exponential function y = b*emx, where b is the gain-to-pain ratio (the omega ratio at a hurdle rate of 0%) and m is the logarithmic slope of the curve (i.e. the linear slope if you use the logarithms of the omega ratios at each point). Here are the exponential functions approximated for the five mutual funds:
XEXDX: 0.63e–111.3x
LAIEX: 1.70e–56.5x
HFQIX: 1.62e–74.9x
HAIHX: 2.01e–48.0x
MPGFX: 2.22e–69.3x
This gives us additional information: if we want our returns to have low variability, we can choose not only on the basis of the omega ratio at our given hurdle rate(s), but also of the logarithmic slope, with more extreme numbers being better. Especially if you’re hedging or using margin, variability can be very important.
So I suggest taking the product of the omega ratio at 0%, the omega ratio at the median benchmark return, the omega ratio at twice the median benchmark return, and the logarithmic slope (which you can calculate quickly from just those three ratios without having to extend the line). I call this number the “ultimate omega.”
[omega scores 5 mutual funds]
You can see that the choice of mutual funds will be somewhat different if we take this approach. If we look at the omega ratio only at these three points, we’ll choose HAIHX; but if we also look at the logarithmic slope, using ultimate omega, MPGFX looks like a better bet.
If you’re using Excel, here’s how to calculate the ultimate omega given a sequence of returns, whether they be weekly, monthly, or yearly. Remember that this is a comparison tool, and that results will be very different if you’re using different time periods.
Let’s say your returns are in column A. In column B, subtract the median benchmark return from the column A returns. In column C, subtract the median benchmark return from the column B returns. Now in cell D1 type =SUMIF(A:A,">0")/-SUMIF(A:A,"<0"); in cell D2 type =SUMIF(B:B,">0")/-SUMIF(B:B,"<0"); and in cell D3 type =SUMIF(C:C,">0")/-SUMIF(C:C,"<0"). In cell E1, type 0; in cell E2, put in the median benchmark return; and in cell E3, put in twice the median benchmark return. Now in cell F1, type =PRODUCT(D1:D3,-SLOPE(LN(D:D),E:E)). That will be the ultimate omega.
I often perform what I call “correlation studies,” in which I backtest a number of different strategies using multiple in-sample and out-of-sample time periods, and then correlate the various performance measures during the in-sample periods with the actual returns of the out-of-sample periods. I recently performed such a study that I think may be the best one I’ve done. I took, at random, fifty different ranking systems from the over 7,000 public ranking systems that Portfolio123 [https://www.portfolio123.com/?sref=27297] users have created, and combined each with a different stock universe (whose rules screen out certain stocks, usually low-liquidity ones along with others based on various criteria). I then ran simulations of each one, holding the top fifty-ranked stocks, with a monthly rebalance and 0.25% slippage, from 1/1/1999 to today. I then compared the total returns of each strategy over six different “out-of-sample” five-year periods with the risk-adjusted returns of the same strategy over adjoining eight-year periods, as shown in the chart below.
[in sample out of sample]
I found that the risk-adjusted measures that correlated the most strongly with the out-of-sample total returns were, hands-down, the various omega ratios. I’ve put the results below, bolding the highest correlations for each period. (I used monthly returns, measured daily, for all the measures. “Omega 1” is the ratio after subtracting the median benchmark return, “omega 3” is the product of the first three ratios, “omega 1s” multiplies omega 1 by the logarithmic slope, and “omega 3s” is ultimate omega.)
[correlation study]
Based on these results, I think that in the future, “ultimate omega” will be the measure I will turn to most frequently when deciding upon the optimal investment strategy. After all, it contains more information about in-sample returns than any other performance measure, so no wonder that it correlates so well with out-of-sample returns.
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