## Getting Better at Sharpe Ratio

You might develop different strategies to balance various measures such as risk, volatility, expected returns etc. But how do you say one strategy is better than the other? The normal answer, its Overall Returns, is somewhat narrow in scope and does not help capture the big picture. Some strategies might be directional, some market neutral which makes annualized return alone a futile measure of performance measurement. Also, even if two strategies have comparable annual returns, the risk is still an important aspect that needs to be measured. A strategy with high annual returns is not necessarily very attractive if it has a high-risk component; we generally prefer better risk-adjusted returns over just ‘better returns’.

With this in mind, William Sharpe introduced a simple formula known as Sharpe Ratio to help compare different strategies and help us find the most feasible of them all. Let’s understand its mechanism in this article by covering the following topics in depth:

1. What is Sharpe Ratio?
2. Equations and Calculation of Sharpe Ratio
3. Interpretations
4. Drawbacks of Sharpe Ratio
5. Overcoming the Drawbacks

### Introduction

Sharpe ratio is a measure for calculating risk-adjusted return. It is the ratio of the excess expected return of investment (over risk-free rate) per unit of volatility or standard deviation. It indicates how well an equity investment performs in comparison to the rate of return on a risk-free investment, such as U.S. government treasury bonds or bills.

Most risk measures are best described graphically, a measure of return in the vertical axis and a measure of risk in the horizontal axis as shown below.

Ideally if investors are risk averse they should be looking for high return and low variability of return, in other words in the top left-hand quadrant of the graph. The Sharpe ratio simply measures the gradient of the line from the risk free rate (the natural starting point for any investor) to the combined return and risk of each strategy, the steeper the gradient, the higher the Sharpe ratio the better the combined performance of risk and return.

### Calculations

Mathematically, the Sharpe Ratio can be fundamentally expressed as:

The Sharpe ratio reveals the average investment return, minus the risk-free rate of return, divided by the standard deviation of returns for the investment. It’s all about maximizing returns and reducing volatility.

Sharpe Ratio can be used in many different contexts such as performance measurement, risk management and to test market efficiency. When it comes to strategy performance measurement, as an industry standard, the Sharpe ratio is usually quoted as “annualised Sharpe” which is calculated based on the trading period for which the returns are measured. If there are N trading periods in a year, the annualised Sharpe is calculated as:

Here trading period means the periods in your time-frame for which the markets remain open. Since crypto markets are open 24×7, N is taken the same as your time-frame and look-back period of your backtest. Mudrex does the calculations of Sharpe Ratio for you so you never have to worry about the equations of such performance metrics.

### Interpretations

Sharpe ratio indicates traders’ desire to earn returns which are higher than those provided by risk-free instruments like fixed deposits with guaranteed returns without any downside. It is based on standard deviation which in turn is a measure of total risk inherent in a trading strategy. It indicates the degree of returns generated by a strategy after taking into account all kinds of risks. It is the most useful ratio to determine the performance of a strategy.

The higher the Sharpe ratio of a strategy, the better is its risk-adjusted performance. However, if you obtain a negative Sharpe ratio, then it means that you would be better off investing in a risk-free asset than the one in which you are invested right now. Sharpe Ratio benchmarks as such that-

• Negative: Worse
• 2 to 3: Very Good
• Greater than 3: Excellent

It’s all about maximizing returns and reducing volatility. If an investment had an annual return of only 10% but had zero volatility, it would have an infinite (or undefined) Sharpe Ratio. Of course, it’s impossible to have zero volatility, even with a government bond (prices go up and down).  As volatility increases, the expected return has to go up significantly to compensate for that additional risk. Below is a summary of the exponential relationship between the volatility of returns and the Sharpe Ratio.

### Drawbacks of Sharpe Ratio

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.

1. Assumes Returns are Normally Distributed: By making use of standard deviation – a measure that assumes returns to be normally distributed – the Sharpe Ratio cannot adequately quantify behaviors observed in real world trading. In reality, rarely are trading strategy returns concentrated within fixed standard deviations around mean returns.
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.
1. Sensitive to Periodicity of Strategy Returns: Since the only component of the Sharpe Ratio‘s denominator is , 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 favorable Sharpe Ratio on one time frame, may possibly enforce a preference for that time frame and hence mislead the choice of your trading strategy due to time-frame bias.
1. Does not differentiate b/w Upside and Downside Volatility: Once again, owing to standard deviation 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 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 into consideration. It is a modification to the Sharpe Ratio, the approach penalizing “bad volatility” instead of all volatility.

Let’s investigate two very different instruments — Bitcoin and the S&P 500 index. The figure below tells the price stories for the two instruments between late 2015 and January 2020. One can see that (not surprisingly) Bitcoin has higher volatility than S&P 500, especially around 2018 — anyone holding Bitcoin at that time would need to have a mind of steel to withstand the roller coaster behaviours.

We are interested to see the Sharpe ratios for these two instruments. The resulting annualised Sharpe ratios from a simple Buy and Hold strategy are shown below:

If judging purely from the Sharpe ratio, Bitcoin is a better investment as it has a higher Sharpe ratio than the S&P 500. However, looking back at the price curves, Bitcoin went through a bubble bursting phase in 2018 — the Bitcoin price dropped 80% from its highest point! This is a dramatic drawdown from the original capital. All this was not reflected in the Sharpe ratio and this is where other metrics like Maximum Drawdown come into play which we’ll cover in the upcoming article.

As traders or investors we need to evaluate our trading strategy using a rigorous approach, at least not relying on a single performance metric alone. If one is not performing the due diligence or cross checking, the original trading capital may easily suffer. Furthermore, knowing the limitations of individual performance metrics and looking at the problem from different angles may sometimes save us from dangerous (aka capital-wiping) situations.

Stay tuned!