# Sharpe Ratio

The Sharpe ratio is a way to determine how much return is achieved per each unit of risk. It is useful to, and can be computed by, all forms of capital market participants to evaluate their performance, from day traders to long-term buy-and-hold investors.

Naturally, when evaluating the performance of traders and investors it is not simply a matter of determining their overall return, but their return relative to their risk.

A 20% annual gain is very strong performance. However, if this comes from having an annualized volatility of 60% because of excess leveraging or trading very speculative instruments, this is actually relatively moderate performance when risk-adjusted. The Sharpe ratio would be considered 0.3. This is calculated as follows:

**Sharpe Ratio = (Return of Portfolio – Risk-Free Return) / Std Dev of Portfolio**

The risk-free rate of return is a user-based input. This is usually the equivalent of a safe risk-free bond. This might be the yield on a US Treasury, UK Gilt, German bund, or other safe instrument. Its duration is dependent on your time horizon.

For long-term investors or position traders who hold positions over long periods of time, they might choose a longer duration bond. Short-term investors or day traders who might hold positions only within the day might use a shorter bond duration, or virtually one equivalent to the countries overnight rate set by its central bank. This is usually approximated by a one-month or three-month government bond or by simply looking at the central bank’s overnight policy rate itself.

In the case of the 20% gain / 60% volatility portfolio mentioned above, if one uses a 10-year US Treasury (assume 3% yield), the Sharpe ratio would come to 0.283. If one uses a 3-month US Treasury (assume 2% yield), the Sharpe ratio would come to 0.300.

If the risk-free rate used is higher, this means the “excess return” is lower – namely a 17% return over the risk-free rate is not quite as good as an 18% excess return. Thus, a higher risk-free rate will result in a lower Sharpe ratio, holding all else equal.

## Ex-Ante vs. Ex-Post Sharpe Ratio

The Sharpe ratio can be considered either *ex-ante* (expected) or *ex-post* (backward-looking to evaluate past performance).

The ratio considered above is ex-post, in that its performance already occurred. The ex-ante Sharpe ratio takes into account expectations. Instead of portfolio returns and volatility, the calculation is instead the expected values of those things, denoted by “E” before the terms.

**Sharpe Ratio = E(Return of Portfolio – Risk-Free Return) / E(Std Dev of Portfolio)**

Therefore, if the S&P 500 is expected to generate 7% nominal annualized returns off 15% annualized volatility, with a risk-free rate of return of 3% (based on US Treasury yields far in the future), that produces a Sharpe ratio of 0.27.

Ex-post ratios can vary widely, especially among shorter timeframes. For example, the Sharpe ratio of the S&P 500 for 2017, due to higher returns at low volatility, was 4.78. For the year-to-date portion of 2018, it has been 0.23.

## Application in Finance

The Sharpe ratio are often used to determine the relative performance of portfolios, traders, and fund managers over time. The Sharpe ratios of individual asset classes are generally in the vicinity of 0.2 to 0.3 over the long-run.

A value between 0 and 1 signifies that the returns derived are better than the risk-free rate, but their excess risks exceed their excess returns. A value above 1 denotes that the returns are not only better than the risk-free rate, but excess returns are above their excess risks.

A negative Sharpe ratio means that the performance of a manager or portfolio is below the risk-free rate. For financial assets, negative Sharpe ratios won’t persist for indefinite periods of time. Capitalist economies would cease to function if this were true.

Negative Sharpe ratios can endure for long periods of time for specific asset classes, managers, or portfolios due to timing or the idiosyncratic risk associated with trading certain assets.

However, a negative Sharpe ratio is problematic to evaluate because a negative excess return with a large amount of volatility will actually make the Sharpe ratio *less *negative (because the denominator is larger), thus insinuating that its performance wasn’t as poor as expected. Likewise, a portfolio with a small negative excess return can be punished if the volatility associated with it is large, giving a smaller denominator and thus amplifying the negative value.

Therefore, negative Sharpe ratios can be exceedingly difficult to assess.

## Pros and Cons of the Sharpe Ratio

Like any statistical measure, it is only as good as its assumptions. In studies of financial risk assessment, it is often assumed that volatility is equal to risk or the best proxy of it. However, not all volatility is harmful and some is absolutely necessary to capture return.

Trading and investing is fundamentally about maximizing return per unit of risk. This is the central intention of the Sharpe ratio, but it does so in a simplistic manner.

Trading or investing strategies that properly balance risks or are able to accurately identify strong reward-to-risk opportunities will have high upside volatility. But considering that all volatility is penalized equally under the Sharpe ratio, the metric may not be the best for accurately identifying the risks associated with the portfolio.

Other risk-adjusted metrics, such as the Sortino ratio, may be a better fit for these types of portfolios and will generally be a more accurate reflection of their risk.

However, the Sharpe ratio is easily applicable and can be applied to any time series of returns without needing additional info regarding the sources of volatility or profitability.

Volatility of returns is also assumed to be normally distributed. Typically, financial variables tend to be more fat-tailed than those associated with the normal distribution and generally exhibit higher skewness and/or kurtosis.

And because the Sharpe ratio is typically used in the ex-post sense – to evaluate past performance – it can be flawed given past performance is not necessarily any prediction of what will occur in the future or over shorter time horizons.

Moreover, since the Sharpe ratio is not expressed in terms of a percentage or return, but rather as a simple number, its use is only valuable in comparison to other performances assessed through the Sharpe ratio.

As a rule of thumb, a Sharpe ratio above 0.5 is market-beating performance if achieved over the long run. A ratio of 1 is superb and difficult to achieve over long periods of time. A ratio of 0.2-0.3 is in line with the broader market. A negative Sharpe ratio, as aforementioned, is difficult to evaluate.