# The Sharpe Ratio: Measuring Weight with a Ruler

Co-Author: Ahmed Bakhaty

In this post, we will explore one of the most widely (mis)used performance metrics in quantitative finance and why it fails to achieve what it was designed to do.

William Sharpe proposed the so-called “Sharpe Ratio” in 1966 as a measure of risk-adjusted return to assess the performance of an investment. Mathematically, it is defined as,

(MEAN RETURN – RISK FREE RATE) / RISK

In simple terms, it is the investment’s average return (less a risk-free rate) scaled down by the volatility. Will referred to it as the “reward-to-variability” ratio, since the numerator is the return (or reward) and the denominator is the variance (or variability).

The motivation for this metric comes from the randomness and unpredictability of the stock market. Suppose you have two investments: both have the same average return, but the second has much higher volatility (or variance). You are at a higher risk of losing more money with the latter strategy. So adjusting for risk seems like a more robust approach to evaluating investment strategies than looking at returns alone, hence, the Sharpe Ratio

Although the Sharpe Ratio sounds like a great idea in theory, closer examination reveals several limitations. In the past 50 years since the inception of the measure, our understanding of mathematics and statistics has drastically improved. This came with many criticisms of performance measures proposed during the old days of quantitative finance.

Let’s take a closer look at three main limitations of the Sharpe Ratio.

# Symmetry of Risk

The denominator of the Sharpe Ratio (the risk adjustment) is the variance of the returns. Variance is a statistical measure of the deviation from the mean. This deviation can be in either direction, so the “risk” is both downside and upside risk. This means you are penalized for winning!

To illustrate how this lack of distinction in the risk obfuscates the meaning of the ratio, consider the two hypothetical portfolio returns shown below. Portfolio A has only upside risk, while portfolio B has symmetric upside and downside risk. Portfolio A has a Sharpe Ratio of about 0.8 while portfolio B has a Sharpe Ratio of about 1.75, over twice that of portfolio A!

# Lack of Consideration for Varying Risk Preferences

The classical Markowitz risk-adjusted return is defined as

MEAN RETURN – (RISK FACTOR * RISK)

This is a pretty straightforward “scaling” of the average returns by the risk: the higher the risk, the lower the risk-adjusted return. The risk factor “tunes” how much the risk scales the risk: a more risk-averse investor will have a higher risk factor. If we take the mean return and risk to be equivalent to the numerator and denominator of the Sharpe Ratio, respectively, than a well-known theorem proves that the Markowitz risk-adjusted return is identical to the Sharpe Ratio, for one one particular risk factor. This suggests that the Sharpe Ratio is oblivious to investor risk preferences, and is an even less robust than the more antiquated Markowitz risk-adjusted return.

# Unrealistic Assumption of the Returns Distribution

William Sharpe (and other academics at the time) assumed, for convenience, that market returns are normally distributed or at least well-behaved, and turned to the associated statistics to perform analyses and design performance metrics. But empirical observations have shown that market returns are not so well-behaved. In fact, they are noisy, heavy-tailed, and skewed. Classical statistical measures such as mean and standard deviation fall apart under such distributions. That means the assumptions underlying antiquated measures, such as the Sharpe Ratio, do not hold in practice where it really matters. Consequently, such metrics are very misleading and they do not provide a fair and accurate judgment of the investment they are attempting to assess.

To see why this is an issue in practice, consider the simulated Sharpe ratios of two portfolios below. Portfolio A’s returns come from an ideal normal distribution, and portfolio B’s returns come from a more realistic fat-tailed distribution, both with the same theoretical Sharpe Ratio. Note how slowly portfolio B’s Sharpe Ratio converges to the true value. This makes the ratio impractical estimate precisely, especially if the true Sharpe ratio is changing and evolving with market conditions.

# Concluding Remarks

It is clear from the above discussions that the Sharpe Ratio is not quite as useful in practice as it may seem. With rudimentary knowledge, it is easy to draw premature conclusions by looking at the Sharpe Ratio, namely that a higher Sharpe Ratio is preferred. One can simply argue this is  by observing how passive strategies with low Sharpe Ratios, such as 0.5 for the S&P 500 index, have outperformed active hedge fund strategies that boast high Sharpe Ratios of 3 and above.

Recent and comprehensive readings have led to the conclusion that the Sharpe Ratio should not be considered for assessing portfolio performance or any financial data. It’s an archaic and unintelligent measure of risk-adjusted returns that was contrived in the 1960’s to justify a theory about an idealistic world where market returns are symmetric, normally-distributed, and purely random; a theory that is, at best, of little practical importance.  In fact, its own creator has conceded as early as the 1990’s.

We cannot overstate the dangers of considering the Sharpe Ratio in any analysis that is performed or implemented. We suggest instead considering measures, such as the Sortino or Omega ratio, that acknowledge that “ups” and “downs” are not the same.

Any paper offering a new ratio or an “improved” Sharpe Ratio briefly, but concisely, visits the drawbacks of the Sharpe Ratio. A few links from various corners of the web are provided as follows, the last of which is an example of a paper that offers a solution.

1. Barclays: “…the Sharpe ratio is always going to give the wrong answer on occasion for any given investor.”
2. Investopedia: “The Sharpe ratio has been around since 1966, but its life has not passed without controversy. Even its founder, William Sharpe, has admitted the ratio is not without its problems.”
3. Academia: “As the assumption of normality in return distributions is relaxed, classic Sharpe ratio and its descendants become questionable tools…”