# Performance Metrics — Distributions and the Sharpe Ratio (Part 4 of 4)

**Welcome**

Hello and welcome to our thirteenth article. This issue discusses the importance of Distributions and the Sharpe Ratio. All articles are saved at our Medium page and on the IQT website, which has a clickable version of the map of all articles: https://theiqt.com/blog.

These articles are based on my experience from consulting and product development at The IQT. Do let us know if there are topics you would like us to cover, questions you would like to resolve, or if there are insights you would like to share from your own experience. Hit that reply button!

**Last Time**

In Article 12 — “ Performance Metrics — Data Quality (Part 3 of 4)”, we explored key metrics used to define and assess a strategy’s data quality with reference to backtesting reports in MT4.

**Overview**

In the previous articles on performance metrics, we discussed measures which are available within an MT4 backtesting report. This article differs in that it discusses measures to do with the probability distribution of returns, which are not provided by MT4, but which a trader should calculate for themselves.

1) The distribution of a strategy’s returns shows the frequency with which each return value occurs. The most important measures beyond a histogram are the mean, the variance, the standard deviation, and skewness.

2) The Sharpe Ratio (SR) is a measure of a strategy’s excess return per unit of risk. Risk is measured by the standard deviation of returns. A higher SR is better.

3) Alternative measures exist based on highlighting a strategy’s downside risk, such as the Calmar Ratio, which incorporates the Maximum Drawdown Duration instead of the standard deviation.

**Main Points**

**1) Distributions**

The distribution of a strategy’s returns shows the frequency with which each return value occurs. It provides an idea of the typical range of outcomes for returns. There are three especially important measures:

**1) Mean:** this is the arithmetic average of returns, i.e. the sum of returns divided by the number of returns. It shows the centre of the distribution. It is commonly termed as the expected value or the “average”, but there are other types of average, such as the mode (most common value) and the median (middle sorted value). Formally,

where *µ* denotes the mean, *R_i* is the *i*th return and *N* is the number of returns.

**2) Variance:** this is (proportional to) the mean of the squared differences between returns and the mean return. It shows how spread out or dispersed the distribution is. Formally,

where *σ^*2 denotes the (sample) variance.

**Standard Deviation** is the square root of variance. It is often called volatility in financial markets.

**3) Skewness:** this is (proportional to) the mean of the cubed differences between returns and the mean return, divided by the cube of the standard deviation. It shows how asymmetric a distribution is. Formally,

where *G* denotes the (sample) skewness.

Wikipedia have a helpful diagram of skewness types at: https://en.wikipedia.org/wiki/Skewness. We reproduce it here:

a. A skewness value of 0 (middle section of the diagram) indicates a perfectly symmetrical distribution, such as the Normal distribution.

b. A skewness value greater than 0 (left section of the diagram) indicates a distribution which seems to lean to the left, with a tail extending to the right. Strategies with positive skewness typically have many low values of returns (possibly even negative) which are bounded (e.g. due to capped stoploss rules), but a long range of high-valued returns.

c. A skewness value less than 0 (right section of the diagram) indicates a distribution which seems to lean to the right, with a tail extending to the left. Strategies with negative skewness typically have many high values of returns which are bounded (e.g. due to capped takeprofit rules), but a long range of low-valued (possibly negative) returns.

**2) The Sharpe Ratio**

The Sharpe Ratio (SR) is commonly used to compare strategies. It is calculated as the difference between the mean return and a benchmark rate of return, all divided by the standard deviation of returns. A higher SR indicates higher excess return per unit of risk.

The benchmark rate of return should correspond to the market index of the strategy’s asset class. E.g. a trading strategy which concentrates on large stocks in the US should use the S&P 500 to calculate the benchmark rate of return. Using the treasury bill rate as the benchmark instead would artificially inflate the SR.

To ensure comparability among strategies with different trading frequencies and time lengths, the annualised SR is calculated. Essentially, the SR over a particular interval e.g. a month, needs to be multiplied by the square root of the number of times the interval divides into a year, e.g. √12.

Note that this assumes that the returns are independent and identically distributed (IID). As Professor Andrew Lo has shown (https://alo.mit.edu/wp-content/uploads/2017/06/The-Statistics-of-Sharpe-Ratios.pdf), annualising is more complex if returns are non-IID, e.g. if they are serially correlated.

Assuming that (annualised) SRs are calculated correctly, Dr Ernest Chan, a famous quantitative trader, states the following rules of thumb on SR values in his book “Quantitative Trading”:

1: returns are positive overall but the strategy is “not suitable as a stand-alone strategy”. An implication is that the strategy may still be combined with other low SR strategies in a portfolio, then the standard deviation will fall as a consequence of diversification, boosting the (portfolio) SR.

>2: returns are positive almost every month

>3: returns are positive almost every day

In this context, HFT can achieve very large SRs e.g. I knew a HFT trader claiming an SR of 30, implying positive returns at a short intra-day time interval e.g. one minute. However, it is not clear how sustainable HFT edges are; the same trader subsequently changed profession.

**3) Alternative Measures based on Downside Risk**

The SR compares strategies based on the standard deviation, a measure of risk which places equal importance on upside and downside risk. However, traders appreciate upside risk and dislike downside risk. This has given rise to various measures which focus on downside risk instead.

The main one we will mention is the Calmar Ratio. This is the Compound Annual Growth Rate (CAGR) / Maximum Drawdown Duration. This has a similar interpretation to the SR, but now accounts for drawdown during the strategy. A strategy with a higher SR may not be as valuable to a trader as a strategy with a higher Calmar Ratio if they dislike long drawdown periods.

Other measures focusing on downside risk are Semivariance, Semideviation, Value at Risk, Expected Shortfall, the Sterling Ratio, Sortino Ratio, Upside Potential Ratio, Risk Return Ratio, and Burke Ratio. We may review these in a further article.

**Further Reading**

This article marks the end of The IQT’s regular, foundational articles. In the next article, we will review the articles to date, and explain how future articles will fit in.

Thanks and happy trading!