Sortino Ratio
It’s been a while, but we’ll finally finish up the portfolio performance metrics that I intended to introduce during the whole portfolio performance series that I’ve been writing on and off over the last few weeks. If you remember, the last post was on the Sharpe Ratio. The Sharpe ratio is a slightly more exacting portfolio risk calculation in comparison to the beta and alpha combination, but, for all intents and purposes, delivers similar information in that it attempts to quantify how much a portfolio strategy made in excess of the volatility (risk) it assumes.
The Sortino Ratio is an adjustment on the Sharpe Ratio in that it only penalizes downside volatility. This is done by creating a value known as downside deviation which is based on some minimum acceptable return (MAR) which is a rate of return that an investor can set. This could be 0%, if you want to judge your portfolio on how well it operates with respect to never losing money. Or, .43% which is the approximate monthly rate of return you would receive on a 5% annual risk free asset (unrealistic given current interest rates).
The equation for the Sortino Ratio is actually quite easy:
Sortino Ratio = (Compound Period Return – MAR)/Downside Deviation
Compound monthly return is calculated by taking the total return over a given number of periods then calculating the rate of return that would have to be compounded in each period to yield such return. An equation might look someting like this:
Compound Period Return = (1+Total Return)^(1/N) – 1
N is the number of periods. Total Return is simply the percent return over a number of periods. For example, 20% return over one year would yield a compound monthly return of 1.2^(1/12)-1 = 1.53%.
If you’re really lazy, you can usually just average the period returns for a relatively decent approximation provided that your portfolio returns are not overly volatile.
Downside Deviation is the most difficult number to calculate. Here are the steps:
- Subtract MAR from each period’s return.
- If negative, record the value. If positive, set value to 0.
- Square all the period returns and sum them.
- Divide by the number of periods.
- Take the square root of your result.
There you have it, all the tools to calculate Sortino Ratio. Give it a try. I’ll upload an excel file in a final post which will do calculations of the metrics I mentioned in this post series so don’t worry if my instructions are hard to follow.
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Comments
The zeros are part of the calculation. The sum of squares includes the 0s and you will still divide by total number of periods.
With respect to Alexander’s question of Nov. 9th, I, too, was interested in your answer, but from a different perspective. More to the point is that there may be periods (depending on the look-back period) when downside deviation itself is zero, which causes the denominator of the formula to be zero, which then of course causes the whole ratio to resolve to an incalculable number, which is infinity. I’ve never seen an adequate explanation or work-around to this problem, except a forced modification of the formula to accommodate these exceptions. I was wondering what your view was on this.
I haven’t actually seen a situation where Sortino ratio is modified to account for a 0 downside deviaion so if you have an example you’d like to link. I’d be curious to see it.
In my mind, a downside deviation of 0 ought to correspond to a Sortino ratio of infinity. After all, the Sortino ratio is like alpha in that it tells you how you performed relative to some measure of risk. In this case, it assumes that the only kind of risk is risk to the downside as most of us would be okay with volatility that gives upside performance. In that sense, the Sortino ratio tells us that minimizing downside risk is every bit as important as return.
Dan, you are correct in that a downside deviation of 0 would correspond to a Sortino Ratio of infinity. Conceptually, this is fine, however, when computing this mathematically, using computer software, this results in an error (#DIV/0! in Excel).
I, personally, have created adaptive-formula workarounds in Excel that vary the look-back period to the nearest historical point in time where the calculation of downside deviation yields a positive number. However, I haven’t yet formed an opinion as to whether I trust this or not because, in many respects, it’s like comparing apples to oranges. This is obvious because normally for historical-based data comparisons to be valid, certain things, such as look-back periods, need to be held to a constant. Normally, this might not be a problem unless you’re dealing with relatively short look-back periods that may actually have a 0 downside deviation.
Another “forced” modification of a formula is frequently used when calculating RSI, where there, too, a 0 in the denominator (representing an average down period of 0) would “force” the formula to resolve to a value of 100. So, conceptually, there are ways of dealing with this. I’m just sure the result is valid (except for things like RSI calcs, which by definition should result in “100″ when the average down period is 0).
I read your article from 2007 (very clear the way you noted the difference between the ratios) and have been attempting to implement either the Sortino or Sharpe ratio on Excel but have not been able to do it. Do you have and Excel or Lotus version which you could point me to?
I would immensely appreciate your help.
Thanks and Sincerely.
JaederPP
I actually did a follow up post on this and put it here: http://thecuriousinvestor.com/2007/11/13/portfolio-performance-excel-file/. Hope it helps!
Reading your formulas, I noticed you are not substracting the mean of the negative returns, to get a proper standard deviation of them when exposing the detailed Downside Deviation formulas. The mean is clearly not going to be zero on negative-only returns. Is that intentional and the real definition of the Sortino ratio or an oversight?
I believe that downside deviation is calculated using MAR as the “mean” in the formula. To argue semantically, it is not so much an exact “standard deviation of negative returns” as it is the “deviation of returns lower than target.” Let me know if this makes sense or if you think otherwise.
I think you are right, one should substract the MAR, not the mean (which is obviously negative in this case). I wasn’t sure before, just wanted to know if there was a clearcut why. I am pretty sure now, given the formulas I’ve seen using continuous data and not discrete, which point to substracting MAR.
Thanks anyway.
I am calculating the Sortino ratio (a yearly one) based on monthly returns and they are negative when returns are negative – is this possible?
Dan, very helpful description of Sortino. What is your view of taking monthly or even daily figures. For example we have analyzed 170,000 investment strategies daily for 23 years and selected 11 for our public library. As of end July these are up an average of 46% and I can process Sortino daily (if we can agree on the formula). Would it be more accurate? What do professionals typically do?
I believe it is ‘n’ total periods. Just using the periods when return is less than MAR would skew results in an unpredictable manner.
I haven’t tried to simulate figures on a monthly versus daily basis. I believe that you should use a time period long enough to determine a reasonable MAR. My thought is that few people operate strategies which provide such a consistent daily performance as to warrant analysis on this specific a level. Monthly is how I analyze my portfolio performance. Then again, maybe a day trader would disagree. Though, a day trader is less interested in risk/return as opposed to just the pursuit of return.
Please tell me that as sortino ratio is a Risk Quant for measuring Fund’s downward volatility, my question is that for what kind of funds’ category is this can i use, i.e: equity, Asset Allocation, Income or money market or either for all funds
WHen you calculate the volatility for Sortino, do your periods of zero’s become part of the calculation. Or are they excluded completely ?