Sharpe Ratio – A Reliable Measure of Performance?

8 March 2018
The Market Bull

This post describes some of the shortcomings of the Sharpe Ratio, that render it a non-optimal tool for measuring the performance of a trading strategy.

For your convenience, the remainder of this post is organized as follows:

  1. What is the Sharpe Ratio and what is it used for?
  2. Where and how does the Sharpe Ratio fall short?
  3. Alternatives for performance measurement.

1) What is the Sharpe Ratio?

Developed in 1966 by Nobel prize winner William Forsyth Sharpe, the Sharpe Ratio is a measure of the excess return of a portfolio or trading strategy relative to its underlying risk.

Originally termed the “reward-to-variability ratio”, it is commonly used as a risk/return measure in finance that describes how well asset returns compensate investors for risks undertaken.

Mathematically, the Sharpe Ratio can be expressed as:

\(Sharpe \space\ Ratio \space\ (p) = \frac{R_p \space\ – \space\ R_f}{\sigma_p}\)

\(where:\)

\( \space\ R_p \space\ = \space\ Expected \space\ excess \space\ return \space\ of \space\ investment \space\ or \space\ trading \space\ strategy  \)

\(\space\ R_f \space\ = \space\ Risk-free \space\ rate \space\ of \space\ return\)

\(\space\ \sigma_p \space\ = \space\ Standard \space\ deviation \space\ of \space\ excess \space\ returns\)

Note: For the remainder of this post, the risk-free rate (\(R_f\)) is assumed to be 0.00, i.e. 100% of the portfolio is assumed to be invested in risky assets.

While the Sharpe Ratio is certainly a decent measure to form initial impressions of risk/reward, the remainder of this post aims to demonstrate why – on its own – it is not adequate for evaluating a trading strategy’s performance, and hence unsuitable for guiding investment decisions.


2) Where does the Sharpe Ratio fall short?

The following is a comprehensive (but by no means exhaustive) list of problems with the Sharpe Ratio, in the context of evaluating trading strategy performance.

 

 2.1) Assumes Returns are Normally Distributed

Normal Distribution, Tail Risk, Fat Tails

By making use of standard deviation – a measure that assumes returns to be normally distributed – the Sharpe Ratio cannot adequately quantify behaviours observed in real world trading.

In reality, rarely are trading strategy returns concentrated within fixed standard deviations around mean returns.

In fact, most strategies will demonstrate non-zero skewness and kurtosis, whereas a normality assumption will hypothesize that skewness is zero and underestimate tail risk.

Therefore, if using the Sharpe Ratio, a trading strategy experiencing, e.g. a period of unexpected, very exceptional performance may lead to a significant deviation from normality, acutely biasing the performance evaluation and misleading investment decisions.

Over and above the assumption of normality, the Sharpe Ratio doesn’t give any further information about the strategy’s actual distribution of returns.


2.2) Sensitive to Periodicity of Strategy Returns

Since the only component of the Sharpe Ratio‘s denominator is \(\sigma_p\), varying the periodicity over the same time interval will affect the output Sharpe Ratio.

This is due to the fact that excess return over the same time interval will stay the same, but the standard deviation of returns over different periods (e.g. daily, weekly, monthly, etc) will be different.

In terms of performance evaluation, this is a risk since a more favourable Sharpe Ratio on one time frame, may possibly enforce a preference for that time frame and hence mislead an investment decision.


2.3) Insensitive to the “order” of excess returns.

Standard deviation as a measure, does not consider the order of data it is presented.

Two trading strategies with wildly different paths to the same excess return, could theoretically have the same standard deviation over the same time interval.

Using the Sharpe Ratio, it wouldn’t be possible to adequately evaluate which of the two is a more risk-efficient candidate for investment. Indeed, upon visual inspection of such strategies, the investor may have excluded one or the other regardless of its Sharpe Ratio.


2.4) Does not differentiate b/w Upside and Downside Volatility

Sharpe Ratio penalizes both upside and downside volatility

Once again, owing to standard deviation \(\sigma_p\) being the Sharpe Ratio’s denominator, large fluctuations in excess returns (even when they’re positive) can effectively lower the Sharpe Ratio.

This is due to the fact that both 1) an unusually large gain, and 2) a similar-sized drawdown, can potentially increase the value of the standard deviation more than that of the excess returns, over the same time interval, effectively penalizing both good and bad performers.

This issue was addressed to an extent with the development of the Sortino Ratio, which takes only downside deviation (let’s call it \(\sigma_{downside}\)) into consideration. It is a modification to the Sharpe Ratio, the approach penalizing “bad volatility” instead of all volatility.

However, many of the problems with the Sharpe Ratio as discussed in this post also apply to similar variations.


2.5) What about periods of inactivity?

By design (excess return over standard deviation), a strategy’s Sharpe Ratio will decline over periods of trading inactivity despite no positions (i.e. zero return) being taken during that time.

To make this concrete,

Let’s assume a strategy so far has 12 months of non-zero returns:

1.5%, 2.0%, -1.0%, 1.2%, 3.5%, -2.5%, 0.2%, 4.1%, 1.5%, 2.0%, -1.0%, 1.2%

 

Therefore, assuming \(R_f\) = 0.00,

Expected excess return \(R_p\) = 0.01058333

Standard Deviation \(\sigma_p\) = 0.01889905

Sharpe Ratio = \(\frac{0.01058333 – 0.00}{0.01889905}\) = 0.5599928

Let’s now assume that the strategy does not trade for the next month.

The new returns series is now:

1.5%, 2.0%, -1.0%, 1.2%, 3.5%, -2.5%, 0.2%, 4.1%, 1.5%, 2.0%, -1.0%, 1.2%, 0.00%

 

Once again, assuming \(R_f\) = 0.00,

Revised expected excess return \(R_p\) = 0.009769231

Revised Standard Deviation \(\sigma_p\) = 0.018331

Revised Sharpe Ratio = \(\frac{0.009769231 – 0.00}{0.018331}\) = 0.5329349

 

What we’ve just observed is the Sharpe Ratio penalizing trading inactivity, the Sharpe Ratio declining by 4.83% without the strategy taking any trading decisions over the last month.

This tendency therefore renders it non-optimal as a performance measure.


2.6) Ignores Serial Correlation

Serial Correlation

Serial correlation describes the relationship between individual returns over successive intervals of time.

When present, it can smooth strategy returns, consequently biasing standard deviation calculations of the time interval under study.

In such situations, there is a risk of the Sharpe Ratio being considerably OVERSTATED in the case of positive serial correlation, and UNDERSTATED in the case of negative serial correlation.


2.7) Standard Deviation \(\sigma_p\) does not equal Risk

This is perhaps best explained using the following scenario.

Let’s suppose a strategy:

  1. Trades very small movements,
  2. Over small periods of time,
  3. During low volatility hours,
  4. ..but with HUGE leverage.

If the market barely moves, the standard deviation \(\sigma_p\) will be minimal.

But the strategy will still have traded with very high leverage and hence very high risk.


2.8) It is only applicable to strategies with low VaR (value-at-risk)

The Sharpe Ratio can really only be applied to trading strategies operating with low to mid-sized VaR, e.g. lower than 20% to 30% at maximum.

Let’s suppose a trader:

  1. Has an amazing Win Ratio of 100:1
  2. But risks all his capital on every single trade.

Given these dynamics, it is inevitable that the trader will lose all his capital at some point.

The Sharpe Ratio up until this strategy’s inevitable demise would have been quite spectacular 🙂

More alarmingly though, it could have mislead investors using it as a performance measure, to make ill-fated investment decisions.


However, if this same trader risked only 10% of his capital per trade (i.e. lower strategy VaR), he would likely make consistently spectacular returns without self-combusting.

At Darwinex, we measure Return/Risk (since inception) for DARWINSa metric similar to the Sharpe Ratio – because a DARWIN’s risk is capped at 10% VaR, for which such a ratio is practical and adequately applicable.

In summary,

Because the Sharpe Ratio depends on the underlying strategy’s risk, on its own it cannot adequately judge a strategy’s performance.


3) Alternatives for Performance Measurement

If the objective is to measure trading strategy performance, it makes more sense to estimate:

  • A trading strategy’s performance compared to random strategies trading with the same risk profile.
  • How much risk a strategy can handle compared to random strategies trading the same asset profile.

In the Darwinex Analytical Toolkit, the Performance (Pf) investment attribute measures the former, while Return/Risk (since inception) measures the latter.

Darwinex Analytical Investment Toolkit

For more details on the same, please visit the following links:

1) Performance (Pf)Click Here.

2) Return/Risk (since inception)Click Here.


Conclusion

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