The Sortino ratio has emerged as a popular risk measure when evaluating investments. It is a modifcation of the Sharpe ratio, a workhorse indicator of mean/variance economics.
The Sharpe ratio is constructed like this:
where 
is the expected return, 
is a benchmark hurdle, and 
is the standard deviation of the returns. If you buy into a Gaussian mean/variance paradigm, then the Sharpe ratio tells you how many units of excess return you receive per unit of risk you take.
The Sortino ratio is constructed similarly:
Here, 
is the downside deviation, or the standard deviation of returns below the benchmark. Downside deviation is a specialized lower partial moment in which the reference point of the LPM is the average of the below-zero returns. An alternative is to use an LPM with a zero reference. The intuition of using this statistic is that people do not penalize investments for positive volatility (i.e. unpredictable but beneficial returns); they only care about negative volatility.
And here lies the rub: it's very easy to calculate a misleading Sortino ratio. The popular method - you'll see it floating around the web - is to take any positive (or above-benchmark) return, change it to a zero, and calculate a standard deviation as one normally would, across all returns.
To me, that's not right. You are artificially introducing a steady stream of zeros into your calculation, depressing the volatility calculation. A more proper way is to throw out any positive returns, and calculate the standard deviation of the negative returns (it should not be surprising that this method complies with the intuition for using the Sortino in the first place).
So the next time you're presented with a Sortino ratio, take care to understand whether it includes zeros or not - if it does, the denominator is necessarily biased toward zero, and the ratio is overstated.
{ 9 comments… read them below or add one }
I am pretty sure that I agree with you to some extent, but if you throw out all the upside returns you would be left with: (downside return – required rate of return)/std. dev. of downside returns. This seem like adding volatility because all returns are further skewed to the down side by deducting the required rate. However as long as you don’t start putting zeros in and use all returns net of the required rate, then you are following the intent of the formula.
Your numerator is still (expected return – required return), so it still includes the positive returns. Only the denominator is impacted by the change.
Another way to calculate a Sortino ratio is to use the lower partial second moment of your data with respect to zero, which is to say the square root of the average squared distance from zero (or from the required return). Downside deviation calculates the square root of the average squared distance from the average negative return, so the LPM would be a more conservative measure (and potentially more economically reasonable).
Under a measure like that, with a fixed reference, then subtracting a required rate would increase perceived volatility (though technically, in the denominator you wouldn’t subtract anything from your datapoints; rather you would add it to the required return). If the reference is the average of the negative points, as with the semi-deviation, then this will not impact volatility.
The key here is to note that this is a comparative metric; it doesn’t matter if you multiply the denominator by 100 so long as the other investments you are evaluating use a similar metric.
If you throw out positive returns then the ratio would hardly make sense. For example, portfolio A and portfolio B have same excess return, but portfolio A has 1 negative return of x% (downside deviation of x%, according to your method) , and portfolio B has 10 negative returns of x% (same downside deviation of x%, according to your method) . I think everyone would agree that B is riskier than A and should have lower Sortino ratio, so Sortino should be calculated with zeros for positive returns to avoid the above effect. However you are completely correct that Sortino ratio cannot be compared to Sharpe ratio, precisely for the reason you stated.
The Sortino ratio is not a measure of risk. It is a measure of risk-adjusted returns. I assume that in your example A and B are observed over the same period. We can therefore conclude that, because B has more negative outcomes but the same holding period return, it has larger positive outcomes. So yes, B has more negative returns in terms of absolute count, but compensates with much better positive returns when it has them.
In a Sortino framework, those positive returns make up for the greater count of negative returns and investors are indifferent between the two. Remember that the principal assumption of the Sortino ratio is that investors only care about downside volatility. They do not care about upside vol, and would welcome it to the extent it offsets losses.
Good point, J! And excellent blog, btw!
For my use, I think the zeros make sense for a ‘downside Sharpe’. If you simplify things about as much as you can, conisder a 20% annual return with a 40% annualized voaltility. For a risk-free rate 0f zero, the Sharpe is .5. Now for the same entity managed in a trading system that is in the market only half of the time, let us assume that the raturn is half, 10%. Let us assume the days in market are identical to those out of market, so the volatility of the in-market days is the same annualized 40%. Let us assume it is only 20% if zeroes are inserted into the volatility computation for the days out of market. The Sharpe computed without the zeros will be .25. Does it make sense that the wide-sense risk adjusted return is halved by staying out of the market half of the time? Out of market means zero risk for those days. The Sharpe computed with zeros in the SD is .5, the same as the computation for the always in the market instance. That makes sense to me.
I am using this Sortino in the same kind of wide-sense, in trading systems where the entities will not always be in play. For an integrated system, the deviations are zero on days out of market and on days when the entity is up. n is always the the total sample size, not the count of the in-market downside days. Admitted this understates the variance of downside in-market days. A Sortino-type ratio computed in this way will be about twice the Sharpe. Still, I think that is what I want for accounting the reduced risk from being out of market or from trading an entity with many more up days than down days. Any kind of SD is going to be a heavily smoothed estimate and in the real world without much validity in the tails of the density. In other words, it represents risk as only a very limited smoothed estimate. If I need a true picture of worst-case risk, which is what most people think of in terms of stops, pecent of equity at risk, etc.,, I am not going to use a stat that assumes a Gaussian density. Something with a historical maximum drawdown is probably the right risk-adjusted metric for a worst case scenario, and we know even that can fail to represent the true risk. If these risk-adjusted returns are used primarily for comparative purposes, and one wants to reward a trading system that keeps you out of the market during the periods of higher concentrations of downside movements, the Sortino with zeros is a lovely way to get a picture of the improvement, with the caveat that the true 95 or 99% short-term risk probability will never be reflected in any kind of Sharpe or Sorino metric.
To your first point, I agree that a Sharpe ratio should include zeros for days the strategy is uninvested; after all the Sharpe ratio uses the expected return over a period and the standard deviation of returns in that period – including zero returns.
But a Sortino ratio is defined not only over a period but also with respect solely to downside returns. Including positive returns or univested days (as zeros) distorts it by definition. That’s not to say there’s anything wrong with calculating what you describe as a “Sortino-type ratio”, but it is important to note that for a true Sortino ratio, the denominator sample size n is not the total number of trading days, but the number of down days.
You may find that the Sortino ratio does not capture the risk/reward profile you are aiming for, which is fine – in fact, it’s really great to be aware exactly of what these metrics do and do not convey. But a Sortino ratio is defined over the semi-deivation or second lower partial moment of the distribution, which does not include zeros.
I think the zeros should be accounted for when calculating the volatility. For example, if the negative returns are all -10%, then the downside deviation will be 0, which is not the true reflection that there are negative returns.
downside deviation won’t be zero. the author made a mistake. when computing the sortino, the reference point is not the average of the negative returns it’s either a risk free rate or a minimum acceptable return, same with the numerator. LOL
{ 1 trackback }