Introduction

How to describe the return and risk is central in speculation activity.

Some definitions

Our examination of the strategy primarily focuses on

1. the daily equity series of the strategy . The length of the series is .
2. trade-by-trade records . The length of the series is . can be much smaller than , or much larger than .

Two derivatives from :

1. return rate series . It's the daily return rate recorded every day (the value on today's equity curve divided by yesterday's value minus one). The length of the series is .
2. high water mark . The high watermark level is the percentage deviation from the previous high. It is 0 on days when a new high is reached, and it's less than 0 on days with drawdowns. Generally, this indicator informs us about the extent of the account's drawdown on any given day. The length of the series is .

From these series we have:

1. Compound Annual Growth Rate (CAGR). Set

   as the final compound return of the strategy. where is the last value in the equity series and is the first value in the equity series. Then

2. Max drawdown (MaxDD), Max underwater days (MaxUD), and underwater ratio (DPER). Since we know that if the equity creates a new all time high, then , otherwise . Let ​ represent the indices of these occurrences when , where . We use a two-dimension to describe a particular drawdown : (, ), is the strength of the drawdown, which can be represented as

   The max drawdown (MaxDD) is defined as
   The Max underwater days (MaxUD) is defined as
   The underwater ratio (DPER) is defined as

3. Sharpe Ratio (SR) and Sortino Ratio (SoR). SR is defined as

   This is saying the Sharpe ratio is the ratio of the first moment (mean) to the second moment (standard deviation) of the discrete sequence of return rates. Since the first moment represents the average return rate and the second moment represents volatility, the Sharpe ratio can be seen as a measure of how much return we are getting for a unit of risk (volatility). SoR is defined as
   Both the Sharpe ratio and the Sortino Ratio help assess the risk-adjusted return of an investment or strategy and provide a way to compare different strategies based on their risk and return profiles.

4. MAR Ratio, Sterling Ratio, and Calmar Ratio.

   Sterling ratio and modified sterling ratio are defined as, respectively
   And the Calmar ratio uses a slightly modified Sterling ratio – average annual rate of return for the last 36 months divided by the maximum drawdown for the last 36 months – and calculates it on a monthly basis, instead of the Sterling ratio's yearly basis. All three ratios and related modifications are all use average return and the drawdowns to provide a glance on how much we can gain per unit drawdown. The difference between these ratios with the sharpe ratio is that, instead of using the deviation as the risk, these ratios are using drawdowns as a profile of risks in speculation activity.

5. Win rate and odds. These parameters can defined in equity series or trade series.

6. Correlation with the SPY, QQQ, IWM, and DIA.

Some remarks on Sharpe ratio

We usually use annualized SR to compare different strategies. However, our strategies typically do not hold positions on an annual basis, so sometimes we need to make adjustments to the Sharpe ratio. One approach is to upgrade the Sharpe ratio calculated from the discrete sequence of daily returns to an annualized Sharpe ratio, and then compare it with the values mentioned above. Generally speaking,

corresponds to the Sharpe ratio of longer periods or sparse data; is the Sharpe ratio calculated from the daily return discrete series. T represents the number of small periods contained in a larger period. For upgrading the daily Sharpe ratio to an annualized one, T = 252.

However, this approach has some limitations. For instance, Lo discusses in his 2002 article that if the daily return series is not independently and identically distributed (iid), using the above formula to upgrade to annualized Sharpe ratio may lead to biases. According to his reports (specific references in Lo 2002, Pav 2021, Riondato 2018), this issue is addressed.

where represents the autocorrelation coefficient of the daily return series with a lag of k. In other words, it quantifies the correlation between r[:-k] and r[k:] of the return series.

Reference numbers

The annualized Sharpe ratio is generally categorized as (see here):

Below 1: Considered poor.

Between 1 and 2: Considered good.

Between 2 and 3: Considered very good.

Above 3: Considered excellent.

Another approach is to map the above values to the requirements for the daily Sharpe ratio. Therefore, for daily Sharpe ratio:

Below 0.063: Considered poor.

Between 0.063 - 0.126: Considered good.

Between 0.126 - 0.189: Considered very good.

Above 0.189: Considered excellent.

This method avoids some issues that may arise when mapping our calculated values to annual terms.

Some remarks on MAR

The MAR is repeatedly mentioned in "Diary of a Professional Commodity Trader," specifically on page 10 in Table 1.1 and on page 30. The approximate range of values for this ratio is 2:1 to 1:1. The author's own strategies fall in the range of 3:1 to 2:1, as mentioned on page 30.

Another instance where a similar indicator is discussed is in ValeryN's post on page 41 (https://tenyears.us/content/valeryn-elitetrader-post/). In his post, he mentions: "Individual strategies, typically, don't offer ARR/MaxDD better than 1:1 or 2:1. And even if they do 2:1 I would still expect them to have 1:1 live."

A rough estimate suggests that the modified Sterling ratio for top-tier commodity firms in "Diary of a Professional Commodity Trader" is likely to be in the range of 1 to 0.5.