invest(igations)

A Systematic Evaluation of Ten Stocks

I use an automated evaluation system to choose what stocks to buy and sell. It has worked well for me, producing returns of 45% in 2016, 58% in 2017, and 28% so far this year (annualized, that’s 56%). My own system is designed for microcap investing, but I have produced a variation of the system for general use, the Stock Evaluator, which you can read about here and here and subscribe to here.

In this article I want to give ten practical examples of how the system works, using ten stocks, five of which I own and five of which I’d advise you to sell quickly if you own them. The five stocks I own, which the Stock Evaluator ranks 99 or 100, are:

• Audiocodes (AUDC), an Israeli tech company that specializes in VOIP (note: I began writing this article before AUDC’s fabulous earnings announcement on 7/24, and everything in this article reflects its previous numbers);
• Finjan Holdings (FNJN), which owns a lot of cybersecurity patents;
• LifeVantage (LFVN), which develops and markets “neutraceutical” dietary supplements and skin-care products;
• Points International (PCOM), a Canadian company that manages loyalty programs; and
• PFSWeb (PFSW), an IT consulting firm.

The five stocks whose price I think will tank, which the Stock Evaluator ranks 0, are:

• Aurora Cannabis (ACBFF), a Canadian producer and distributor of medical marijuana;
• Akoustis Technologies (AKTS), which manufactures radio frequency filters for mobile phones;
• Carvana (CVNA), a used-car online marketplace;
• GDS Holdings (GDS), a Chinese data-center manufacturer; and
• Vivint Solar (VSLR), a solar energy provider.

Let me say at the outset that I don’t really care about the quality of these companies’ products. I personally think LifeVantage’s products are garbage, but the stock has doubled in price since I bought it and I think it’s going to go up some more; and I’m a strong supporter of medical marijuana and solar energy, but I think Aurora Cannabis and Vivint Solar are overpriced companies with bad prognoses. Other authors have written about these companies in depth, and look at many more aspects of their businesses than I do. I’m only crunching numbers here. I don’t go into as much depth into these companies’ merits and faults as other Seeking Alpha contributors normally do. But my practice of trusting in numbers gleaned from earnings reports and other sources, and made available to me through Portfolio123, has served me well.

For the sake of simplicity, I’m going to break down the main factors I use into four categories: growth, quality, sentiment/technical, and value. Let’s tackle growth first, since it comes first alphabetically. Growth factors look at the change in a company’s earnings and sales over time.

In the first column is each stock’s estimated EPS growth, which I calculate by taking the current quarter’s estimated EPS, subtracting the reported EPS for the same quarter last year, and dividing by the absolute value of the latter. There’s a pretty big divide between the good stocks and the bad stocks here, with the exceptions of GDS, which is showing healthy growth, and LFVN, for which no EPS estimate is available. In the second column is actual income growth, which I calculate by taking the most recent quarter’s net income, subtracting that of the same quarter last year, and dividing by the absolute value of the latter (I adjust the income figures by subtracting a fraction of the “special items” reported). Once again, you can see a major difference between the bad stocks and the good ones. In the third column is a figure representing change in operating margin. This is the median difference between each of the last twelve quarters and the same quarter the previous year. The best option would be a number in the range of 0% to 3%. As you can see, ACBFF, AKTS, VSLR, and FNJN give me alarmingly high and low numbers here. In three of these companies, income growth is outpacing sales growth to an alarming degree, and in the case of AKTS, income is falling far faster than sales. In the fourth column is sales acceleration. This is a rather complicated factor. You first measure the sales growth by comparing this quarter to the same quarter last year. Then you measure it by comparing the last twelve months to the twelve months before that. Lastly, you take the difference between those two numbers and divide it by the second. All the good stocks have accelerating sales, while three of the bad stocks have decelerating sales. In the last column is the rank, by sector, from 0 to 100, of the sales growth over the last three years. What I’m looking for here is a number close to 70. Sales growth that is extremely high—like that of CVNA, GDS, VSLR, and FNJN—is likely going to result in some disappointment down the line, and poor sales growth like that of ACBFF, AKTS, AUDC, and LFVN is not very promising. The best stocks in this column are PFSW and PCOM.

Now let’s turn to quality. These are factors that have nothing to do with a stock’s price, trading volume, or change in earnings or sales.

In the first three columns are various accrual measures. I want these to be negative, with the lower the better, and indeed all the good stocks have negative accruals across the board. The first two columns measure the difference between operating cash flow and net income, divided by total assets; the third measures the difference between the last twelve months’ net operating assets and that of the previous twelve months, divided by average total assets. CVNA is too young a company for me to have much data, but its one-year accruals are terrible. The difference between the balance-sheet accruals for the bad and good stocks is particularly notable. Ideally, the change in the cash conversion cycle between one year and another shouldn’t be more than three months, and you can see that VSLR, AUDC, PCOM, and PFSW are within that range. The rest of the stocks are presenting somewhat worrisome numbers here. Free cash ROA is the most recent quarter’s free cash flow (net cash flow from operations plus net cash flow from investments) divided by total assets. The bad stocks all have negative numbers, the good stocks all positive. The reason for all the N/As in the gross margin column is that all these bad stocks are less than five years old. The number here is the company’s gross margin minus the industry average—a great measure of a company’s “moat.” I’m looking for positive numbers here, so the figures for PCOM and PFSW are less than ideal. Net operating assets to total assets is my number-one quality ratio (I’ve written about it here). I want the number here to be low—definitely under 0.5. Indeed it is for all the good stocks, and it’s too high for all the bad ones. Lastly, I’d rather invest in a company that’s reducing its share count after splits than one that’s increasing it. This measure is the current share count divided by the share count a year ago, adjusted for splits. The amount by which ACBFF, CVNA, and GDS have increased their share count is troubling.

Now we turn to technical and sentiment measures.

I would rather invest in a stock in an industry with some momentum, so the numbers in the first column for CVNA, GDS, FNJN, PCOM, and PFSW are encouraging, while those for ACBFF and VSLR are dismal. Because most of these are small and microcap stocks, price volatility, as measured by the standard deviation of the daily returns over the last three months, is pretty high. AUDC, though, is a nice exception. I wrote about share turnover here, and once again AUDC is posting some nice low numbers, while CVNA is damn scary. And the short interest in all of the bad stocks is way too high for my liking—though CVNA, with short interest at 55% of its shares outstanding, could be ripe for a short squeeze were its other numbers not so dismal.

Now we turn to value.

EVA is economic value added, a measure devised by Joel Stern in 1983 for maximizing value at firms for which he consulted. Basically, EVA is the company’s net operating profit after taxes minus the product of the weighted average cost of capital and the economic capital employed. I like to measure the EVA of companies I invest in and divide that by their market cap. Most of these companies have negative EVAs, but FNJN’s EVA is outstanding compared to the company’s price. Forward earnings yield and forward sales yield are basically the current year’s earnings and sales estimates divided by the market cap. You can see the numbers are far better for the good stocks than the bad stocks on the whole, with the exception of CVNA’s forward sales yield. The price to sales ratio (trailing twelve months) for the bad stocks is astonishingly high, with the exception of CVNA, while for the good stocks it’s less than 2. Gross profit to enterprise value is one of my mainstays, and you can see it’s far better for the good stocks than the bad. AUDC’s ratio of R&D to market cap is quite strong, while that of all the “bad stocks” except AKTS is pretty terrible. Price to tangible book value is a somewhat problematic ratio, but in the extremes it’s a good measure of overpriced and underpriced companies. It’s essential to measure this against other companies in the industry, and by this measure FNJN is particularly strong and AKTS and CVNA particularly weak. Unlevered free cash flow to enterprise value is the foundation of discounted cash flow analysis, and you can see quickly that all the bad firms fall down on this measure, while AUDC, FNJN, and PCOM are posting some very healthy ratios. I also look at dividend yield and payout ratios, but none of these ten stocks pays dividends (though AUDC announced one after this article was mostly written).

By weighting these various factors, along with others, and comparing the values to those of other companies (sometimes in the same sector or industry), I come up with a final score from 0 to 100. As you can see, even the best companies can post some bad numbers on some factors, and even the worst can post some good ones. What’s important is the aggregate score; the diversity of factors ensures that a diversity of companies will score well.

I’ve written before that buying a stock is like buying a used car (though perhaps not from Carvana), in that the most important thing to do is to look under the hood. That’s what the evaluation process can be compared with. Evaluating stocks using proven measures like the ones I’ve highlighted in this article should be an essential tool for all stock market traders and investors. These are, of course, not the only measures one should use, and savvy accountants and analysts may use entirely different ones. But no matter which measures you use, I think it wouldn’t be too difficult to come to an agreement that five of the stocks I’m highlighting in this article are a lot better buys than the other five.

CAGR since 1/1/16: 52%.

My top ten holdings right now: LFVN, FNJN, IRMD, PCOM, PERI, MGIC, NTWK, ZYXI, TRIB, MSON.

Ultimate Omega: The Best Risk-Adjusted Performance Measure?

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 here. I’ll quote from it at length:

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 eight years if you’d invested $100 in each.

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:

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.

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*e^mx, 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.”

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 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.

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.)

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.

CAGR since 1/1/2016: 53%

Top ten holdings right now: FNJN, LFVN, PCOM, PERI, NTWK, TRIB, ZYXI, MCHX, MNDO, GSB.

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*e^mx, 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*e^mx, 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.