Sharpe Vs Sortino: Which Risk Adjusted Ratio Do We Prefer?

Ask any serious trader how she/he measures the effectiveness of any trading strategy, and you're bound to get the names of two popular ratios thrown your way: the Sharpe Ratio and the Sortino Ratio. While the Sharpe Ratio is undoubtedly the more popular among the two, the question still remains: which ratio actually does a better job at comparing risk-adjusted portfolio returns?

The truth is that both ratios do a good job at helping you assess the all-important question you must ask yourself when deciding upon any new investment portfolio/strategy:

Is the net return promised by this strategy worth the risk I am being asked to take on?

Before we get to our verdict, however, let's first provide some historical context into each ratio's development and distinguish how they differ from each other.

A Brief Chronology of Events

1952: Enter A.D. Roy and Harry Markowitz

32-year-old British economist A.D. Roy publishes "Safety First and the Holding of Assets" in Econometrica and introduces a risk management technique aimed at helping investors choose among different risk-adjusted portfolio returns by referencing a single "Safety-First" (SFRatio) variable.

The primary rationale behind Roy's introduction of the SFRatio was that investors often care more about whether or not a portfolio's return will fall below a minimum return (known as downside or tail risk) instead of the more "default" probability that a portfolio will exceed an expected return. Here's a reference to Roy's paper, where you can clearly see that he is working with a probability function related to maximizing a variable.

While this line of thinking may seem a bit more obvious to us in the year 2020, it's safe to say that most investors probably did not share this philosophy in 1952. The general line of thinking back then was that if a portfolio had a higher expected return, it was worth pursuing. The risk one was being asked to take on in order to produce the net return was not seamlessly factored into the decision making process, primarily because the tools required to make such a decision had not yet been developed.

In fact, this was a fairly novel tool for gauging investor expectations and Roy was clearly ahead of his time with his general line of thinking. It is therefore not at all surprising that despite the extreme similarities between the SFRatio and the Sharpe Ratio, the Sharpe Ratio would not be introduced for another 14 years, and didn't really enter mainstream quantitative finance until the 21th century, when the investing community had perhaps become a bit more warm to this new method of comparing portfolio returns.

The SFRatio formula is as follows:

SFRatio = (expected portfolio return - minimum threshold return) / standard deviation of excess portfolio returns

Let's demonstrate the genius of Roy's SFRatio with a simple example comparing two portfolios' expected returns and standard deviation of excess returns, assuming we have a minimum threshold requirement to generate at least a 3.75% return on our portfolio.

Portfolio A: Has an expected return of 25% and a standard deviation of excess returns of 27%

Portfolio B: Has an expected return of 14% and a standard deviation of excess returns of 20%

Therefore, after plugging in the numbers, we get

Portfolio A: SFRatio = (25% - 3.75%)/27% = 0.7870

Portfolio B: SFRatio = (14% - 3.75%)/20% = 0.5125

As we can clearly see, Portfolio A has the higher SFRatio, so Roy would have advocated Portfolio A. However, how does this ratio relate to the probability that the portfolio return will be less than our minimum threshold?

Well, the SFRatio assumes that the portfolio's returns are normally distributed, which is not exactly the most realistic scenario for many investment portfolios; however, it still provides a framework to work with. Under this assumption, the probability that the portfolio returns will be less than the minimum threshold is:

N (-SFRatio), where N is a Standard Normal Distribution. Therefore, in our case, we are looking at N (-.7870) and N(-0.5125). Using this handy Z-Score calculator, we get the following probabilities.

For Portfolio A, the probability is 21.56% that the portfolio return will not surpass the minimum threshold return, while for Portfolio B, it's 30.41%. Clearly, Portfolio A seems to be the better portfolio using Roy's method.

Interestingly, the well-known Nobel Prize winner Harry Markowitz (the father of the Modern Portfolio Theory) also independently published his famous dissertation Portfolio Selection in the year 1952, which resulted in him winning the Nobel Prize in Economics in 1990 (shared with William Sharpe the same year, featured next). Markowitz' paper is world renowned and argued that optimal portfolios could be constructed using expected returns and the variance of returns, which was a ground-breaking idea at the time and led to the "Efficient Frontier". However, Markowitz' paper focus was on the topic of diversification among securities and didn't specifically hone in on an easy to compare, all-encompassing ratio as Roy's SFRatio had.

1966: Enter William Sharpe

Also aged 32 at the time of his paper's publication, William Sharpe (pictured below) introduced a "reward-to-variability", or R/V Ratio, aimed at simplifying the decision making process for comparing mutual fund portfolio returns. This ratio, of course, would eventually come to be known as the Sharpe Ratio, named after its infamous founder.

Interestingly, Sharpe's primary contribution was not the Sharpe Ratio, but the much more famous Capital Assets Pricing Model (CAPM), which most of us have been taught at some point in time in school, thus illustrating the profoundness of the model. Sharpe was one of the originators of the CAPM Model.

Sharpe introduced his work on CAPM while pursuing a PhD at UCLA, and perhaps not surprisingly, his direct advisor at UCLA for his dissertation was none other than Markowitz!

The original Sharpe Ratio formula is as follows:

Sharpe Ratio = (expected portfolio return - risk-free rate of return) / standard deviation of excess portfolio returns

That tiny difference - substituting a minimum threshold return with a risk-free rate of return - in actuality, made all the difference in the world. With Sharpe's ratio, you could now compare apples to apples across asset classes in a more meaningful manner since the risk-free rate should be a commonly agreed upon variable by all market participants; on the other hand, the SFRatio proposed by Roy was dependent on each investor's unique minimum threshold requirement.

This original version of the Sharpe Ratio is what most investors, traders, and probably even fund management companies refer to today due to the extreme ease of use associated with it; however, Sharpe himself acknowledged in 1994 in this paper that from a practicality perspective, a more realistic approach would be to compare a fund's performance to a benchmark based on a similar asset class/profile. Hence, he introduced the Ex Ante and Ex Post versions of the Sharpe, thereby throwing out the universality component behind the original Sharpe.

The Ex Ante (ante means "before" Latin, so we are forecasting into the future) Sharpe Ratio is looking to approximate the future Sharpe for a portfolio by comparing a fund's expected return to an expected benchmark return for a similar asset class/profile, and then dividing that numerator by the standard deviation of differential returns. Namely,

Ex-Ante Sharpe Ratio = (expected portfolio return - expected benchmark return)/standard deviation of differential returns

The Ex Post Sharpe Ratio is what we are probably more concerned with, namely due to routine activity of back-testing, which is the activity of gauging past performance and deducing into the future by seamlessly comparing risk-adjusted ratios in a meaningful manner. The formula is

Ex-Post Sharpe Ratio = (historical average portfolio return - historical average benchmark return)/(standard deviation of differential returns)

So really, all we've done is:

• Replace the risk-free rate with a benchmark return (say I'm trading US equities, I could use the S&P500 as the benchmark)
• Instead of computing the standard deviation of excess returns from a risk-free rate, we are taking the standard deviation of returns from a benchmark rate

What goes missing from Sharpe's Ratios, which did exist in a preliminary manner in Roy's SFRatio, is a "Safety-First" principle - meaning, where downside risk is fundamentally viewed and addressed differently than upside risk. Sharpe's didn't necessarily differentiate between the two sources of risk. Ultimately, Sharpe's original ratio prevailed because it allowed for an apples-to-apples comparison between portfolios across asset classes seamlessly (and even after his revisions 28 years later, his original is still the most referenced version today). It wouldn't be until another 14 years before a ratio was introduced which blended the best of both worlds' intentions.

1980: Enter Frank Sortino

Suppose we have a trading strategy generating the following returns.

Now, suppose we were to have a blockbuster 10th period and generate a 30% return during the period. Surely, our Sharpe must go up, right?

Wrong! The Sharpe actually decreases since the extreme increase in variability (and hence the standard deviation, making up the denominator in the Sharpe formula) offsets the increase in expected return (numerator).

Naturally, you could empirically look at the set of data points and make a case that you're just not all that comfortable with that 30% return because you realize such a strategy could be accompanied with extreme downside risk as well. That's your call to make - but what if you are OK with it, and you're just interested in limiting potential extreme downside risk, but with the help of a single reference ratio that can easily compare portfolio returns apples to apples?

And this is again where Markovitz enters the picture - in 1959, 7 years after he and Roy independently introduced their respective pieces of work to the world, he made a mention that "semi-variance is the more plausible measure of risk" than his mean-variance theory, meaning that just addressing the variance on the downside may be more meaningful than considering the full variance. However, in his groundbreaking Modern Portfolio Analysis work, he solely referenced full variance, probably because it was not computationally practical to use semi-variance at the time.

Dr. Frank Sortino, however, was going to change that when he emerged two decades later (pictured below).

A finance professor at San Francisco State University, Sortino saw the potential to improve upon the Sharpe, and leveraging the fact that computing resources were now freely available to sort through large sets of data and return results using just downside variance, the Sortino Ratio was first published in 1980. The Sortino differs from the Sharpe as it only penalizes deviations which fall under our target rate of return.

The Sortino Ratio formula is:

Sortino Ratio= (expected portfolio return - target rate of return) / standard deviation of downside portfolio returns from the average deviation

The Sortino is often calculated incorrectly, so we're going to first go through an entire example in order to clearly demonstrate how to calculate the ratio. Let's use the same example as before and show the Sortino and Sharpe Ratios before and after the 10th period is introduced.

Before introduction

After introduction (the formula bar is included for reference to the Sortino Formula)

Not only does the Sortino not go down with the introduction, but it goes up! Why is that? Because in the denominator, the Sortino is looking at the standard deviation of returns from the target return only penalizing downside returns; with the introduction of the 10th element, our average goes down (average of column E; we need to include the 0% deviations in the average).

And so, right off the bat, we can easily see why many do consider the Sortino an improvement over the Sharpe.

What's RAIN's verdict?

Sharpe Vs Sortino: Which do we prefer?

The Sharpe has its advantages. Namely - the universality of the original version allows for easy comparisons for investment strategies across asset classes. The Sharpe does penalize strategies which don't prioritize consistency of returns. It also only rewards strategies which beat a risk-free rate of return.

But, it has its disadvantages as well.

Namely, in a practical setting, fund managers, traders, and investors would not consider a risk-free rate of return as the target rate of return; in actuality, the target rate of return would be a similar product offering in the same asset class. Secondly, the Sharpe can penalize strategies which long-tail distributions of rewards; for example, a trend following strategy may end up holding on big winners and generating large one-off winning trades; why should those returns be penalized by increasing the standard deviation of (excess) returns?

Therefore, RAIN prefers the Sortino. That doesn't mean the Sharpe doesn't have its uses. Ideally, if you're seriously looking to make a decision with no regrets, it's important to work with as much as data as possible. We recommend that you use both the Sharpe and Sortino in some shape or form as a part of the decision making process to start a new strategy.

The bottom line

You can't go terribly wrong with either the Sortino or the Sharpe. They both do a good job in putting you in a place where you can compare investing and trading strategies within an asset class, and if you're using the original Sharpe, across asset classes. Neither does a perfect job at assessing the quality of a strategy which does not a follow a Standard Normal distribution of returns. However, with fund management, a big determining factor towards success is the ability to crystallize an all-important truth when it comes to the general nature of probabilities and statistics:

It's not about being right, or being wrong, right now. It's about being more right, than wrong, most of the time.

51% over 49%. Totality of wins slightly outpacing the totality of losses. Squeezing out an edge by having the probabilities slightly favor you. What it all amounts to is this: if you're working with ratios such as the Sharpe and Sortino and you are accurately measuring your data and conducting your analysis, you are already on the path to success and profitability.

After all, success with quantitative finance is all about putting in the work to put yourself in a position to succeed.

Statistics takes care of the rest.

The author is a Co-Founder at?RAIN, a technology company obsessed with increasing investor participation rates at the retail level.