To measure the performance of a trading strategy, annualized returns are often a common metric. However comparing two strategies based on annualised returns may not always be a logical way due to several reasons. Some strategies might be directional, some market neutral and some might be leveraged 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’. Sharpe Ratio takes care of risk assessment and the problem related to the comparison of strategies.

Sharpe ratio is a measure for calculating the risk-adjusted return. It is the ratio of the excess expected return of an investment (over risk-free rate) per unit of volatility or standard deviation.

**SHARPE RATIO = E(R****x**** – R****f****)** **/ StdDev(x)**

Where,

**x** is the investment

**R****x** is the average rate of return of x

**R****f** is the risk-free rate of return

**StdDev(x)** is the standard deviation of Rx

The assumption here is that the returns (Rx) are normally distributed and thus can be annualized. The risk-free rate of return (Rf) should be a suitable benchmark preferably as per the duration of the investment. In the case of dollar-neutral strategies/portfolios, the risk-free rate need not be deducted if it is self-financing. The Standard Deviation represents the volatility/risk. The accuracy of Sharpe Ratio depends on the statistical properties of the returns which can vary greatly from asset to asset and in portfolios and strategies over time.

Sharpe Ratio is used in many different contexts like performance measurement, risk management and to test market efficiency. For strategy performance measurement, as an industry standard, “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:

**SHARPE RATIO = ****√****N (E(R****x**** – R****f****)** **/ StdDev(x))**

**Trade Level Sharpe ratio for Intraday Strategies**

For an intraday trading strategy, instead of using the conventional sharpe calculation we can calculate the trade level sharpe to get a better view of the strategy’s performance. In this case, the risk-free rate can be considered to be 0 since we don’t roll over positions, there is no interest charge. Sharpe ratio can be calculated by following these simple steps:

Say the strategy does “N” number of trades in a day; calculate:

- The PnL for each trade (which is essentially what you make in excess of the brokerage you pay)
- The mean of PnLs for all trades
- The standard deviation for PnLs for all trades

Now as per sharpe ratio definition, the numerator, which is the mean of PnL gets multiplied by N when you consider all trades and the denominator which is the standard deviation gets multiplied by sqrt(N). So effectively, the sharpe ratio is sqrt(N) *(mean/std dev).

For high frequency strategies, a large number of small successful trades for specific amounts smoothen the PnL curve and the standard deviation approaches to zero which significantly spikes the sharpe, such that it might range in double digits.

**Practical range of Sharpe Ratio**

Any strategy with “annualized Sharpe ratio” less than 1 (after including execution costs) should be ignored. Most Quantitative hedge funds ignore strategies with annualized sharpe less than 2. For a retail algorithmic trader, an annualized sharpe greater than 2 is pretty good. For high frequency trading, as discussed, the sharpe ratio can go up in double digits as well, especially for opportunity driven but not highly scalable strategies.

**Limitations**

There are several limitations with the usage of Sharpe Ratio, due to certain assumptions and the way it has been defined. Some of the important limitations have been listed below:

- The calculation of Sharpe ratio pivots on the assumption that returns are normally distributed but in real market scenarios the distribution might suffer from kurtosis and fatter tails, which decreases the relevance of its use.
- Sharpe ratio cannot differentiate between intermittent and consecutive losses as the risk measure is independent of the order of various data points.
- Another notable drawback of sharpe is that it cannot distinguish between upside and downside, it focuses on Volatility but not it’s direction. Sharpe Ratio would penalize a system which exhibited sporadic sharp increases in equity, even if equity retracements were small.
- It is backward looking, accounts for historical returns and volatility. Decisions based on sharpe ratio assume future performance will be similar to the past.

Sharpe ratio suggests if a strategy is “statistically significant” or not, however, due to its limitations, sharpe alone cannot be used to judge a strategy’s performance. We have other performance evaluation ratios like “Sortino Ratio” (which only considers the downside deviation of volatility) and “Information Ratio” which can help in overcoming the drawbacks of Sharpe ratio.

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May 6, 2016

MilindHow is Sharpe ratio calculated in a leveraged trade?