Introduction
Returns are at the heart of trading. In fact, you trade with the sole aim of ultimately generating returns. But, before you trade, you might like to analyse the potential returns of a trading strategy or an asset. You might even like to analyse the past performance of a fund. The way some funds quote their returns is not necessarily a transparent measure of their performance. In the below slides, we explain all you need to know to get started with understanding returns and the different ways by which they are measured.
A Brief Overview
- Returns are used to measure performance as well as to carry out compounding and discounting calculations.
- The most common types of returns are arithmetic, geometric and logarithmic. Logarithmic returns have appealing characteristics that make them more desirable.
- Arithmetic returns can be aggregated across assets but not time.
- Logarithmic returns can be aggregated across time but not assets.
What to Watch out for
- There is no one consensus on which method of calculating returns should be used.
- Some funds tend to use the mean arithmetic return to measure performance. We will show you why this is not a transparent measure of performance.
- Logarithmic returns are often used in the literature but capital returns are always calculated arithmetically.
- Mean arithmetic returns should not be calculated as an arithmetic average.
Lessons
In the following sections, we will be looking into asset returns in depth. The lessons on this page are just a taster of what is covered in the below course notes. As return methodologies are not implemented in the platform or spreadsheet yet, this tutorial will only cover theory. This is accomplished in five lessons whereby the theoretical foundation, as well as practical applications, of returns are explored.
I - Lesson 1: Definition
Returns form a central idea in trading, not least because you trade or invest with the intention of generating returns but also because they impact your decision about how much risk you are willing to assume. In fact, they are at the heart of every conscious or subconscious decision you make about trading or investment whether it is trading an individual asset, constructing a portfolio or investing in a fund or business. As this tutorial is focused on asset returns and capital/P&L generated by them, we will start by defining returns in this context with the aim of helping you understand why this concept so important. Slides 4-7 cover elementary definitions of return and how it can be used in trading and investment.
II - Lesson 2: Types of Returns
It is quite common to casually comment on the return of an asset or a trade deal. But, as we have seen above, there are several methods by which returns can be calculated. Lesson 1 has discussed briefly the most popular types of returns out there. This lesson takes the discussion one step further; it dives into the mathematical formulae behind these types of returns and starts the relevant analysis. Slides 8-10 present the formulae behind arithmetic, (mean) geometric and logarithmic returns for both long and short positions, followed by slides 11-12 that talk about the strengths and drawbacks of logarithmic returns. Whilst we will analyse arithmetic and geometric returns later, we have singled out logarithmic returns now due to their sheer importance.
II.1 – Arithmetic Returns: Formulae
This is perhaps the most ad-hoc definition of returns and the first one that springs to mind. Slide 8 displays the corresponding formula, how it should be aggregated across time and the (wrong) way of calculating the mean.
II.2 – Geometric Returns: Formulae
A more precise term to use here is geometric mean return. It is a sound replacement for the arithmetic mean return as we will see. Slide 9 presents the corresponding formula and a brief analysis.
II.3 – Logarithmic Returns: Formulae
This type of return has nice properties that make it a favourite in some circles. Slide 10 presents the corresponding formula, how it aggregates nicely through time and the subsequent mean.
II.4 – Logarithmic Returns: Strengths and Drawbacks
We have mentioned earlier that this type of return is a favourite so, before moving onto the next lesson, we will take some time to discuss its strengths and drawbacks. Amongst other things, logarithmic returns have nice statistical properties that make them easy to work with; they are time additive and facilitate continuous compounding. On the other hand, they are not transparent measures of terminal wealth or capital and can display downward bias. Slides 11 and 12 talk about the strengths and drawbacks of logarithmic returns in more details.
III – Lesson 3: Comparing Different Types of Returns
In this lesson, we will compare different types of returns. Slides 13-15 delve into the maths and theory behind it all, while slides 16-17 focus solely on the difference between arithmetic and logarithmic returns, giving an example in the form of a chart that illustrates how these two types of returns change as the underlying asset price changes. The next section in this lesson explains how returns can be aggregated across time. This is extensively covered in slides 18-23.
III.1 - Comparing Mean Returns
Slides 13-17 explain the relationship between all three types of returns supported by mathematical proof. They particularly place an emphasis on how logarithmic and arithmetic returns are linked mathematically; here, we also spend time analysing how each type of return fares vis-a-vis the other.
III.2 – Aggregating Across Time
In this section, we attempt, as comprehensively as possible, to explain how returns are aggregated across time. In fact, this is where we elaborate on our earlier statement that arithmetic returns do not aggregate across time. An example is given to illustrate this point, unconventionally followed by the supportive theoretical background. You may like to study the material in slides 18-23 if you would like to find out more about this aspect.
IV - Lesson 4: Example
This lesson illustrates the difference, as well as the relationship, between logarithmic and arithmetic returns through an example. Slides 24-30 will take you through a tabulated example and explain every element in the corresponding cells, which makes it easy for you to replicate it. Finally, a detailed analysis of the example is drawn to conclude this lesson.
V - Lesson 5: Return on Terminal Wealth or Capital
We conclude this tutorial by discussing terminal wealth returns, as well as profit and loss (P&L) profile, whilst shedding some light on how these quantities relate to logarithmic and arithmetic returns. Slides 31-52 take you through this journey and provide you with exhaustive examples that we hope will clarify any confusion you may have around the subject.
V.1 – Definitions: Capital and P&L
Slide 31 draws a succinct discussion on the basic definitions of P&L and capital by way of introduction into the lesson and to ensure that the fundamentals are clear, which is why this is section is brief.
V.2 - Long vs Short Positions
Anyone who has attempted to calculate the arithmetic return for a short position understands how tricky it can be, not least because there is no general consensus, nor a wealth of material, on that. In slide 32, we explain and analyse available alternatives that would hopefully help you in deciding on an approach. We move on to discuss the subsequent implications for capital in slides 33 and 34.
V.3 – Examples: Step-by-Step Calculation
In this part of the tutorial, we provide an extensive example that explains the above concepts. Slides 35-40 describe in numbers how to calculate returns, including those on initial capital, for long and short positions or a combination of the two with particular attention being paid to the short position.
V.4 – Examples: Total Returns for the Period – Long Position
We expand the example in this case to incorporate a trading strategy return over a period of time with the aim of illustrating how various types of returns compare to those on capital. Slides 41-43 take on the first example corresponding to a long position through a tabulated display followed by a comprehensive analysis.
V.5 – Examples: Total Returns for the Period – Short Position
We continue further with tabulated examples in this section too. Slides 44-48 illustrate an example for a short position with an extensive analysis particularly focused on how returns are linked.
V.6 – Examples: Total Returns for the Period – Long + Short Strategy
We finally conclude the lesson (and tutorial) with one last tabulated example on calculating the returns for a long + short strategy and similarly draw an in-depth analysis of this. Slides 49-52 take you through this final example in details which will come in handy when you work through the strategy backtesting tutorial.
Theory Notes
In this section, you may explore the below slides to study lessons 1 to 5 material in more depth. This part covers the theory behind, and practical use of, return calculations. Just click on the below slides to enlarge and you can scroll freely through the lessons. We strongly suggest that you divide your eLearning into sections, moving from one concept to the next only after you have grasped the first one. (Authored by Dr Jolnar Assi)
Conclusion
In this tutorial, we have discussed three types of calculating returns: logarithmic, geometric and arithmetic. We have explained the theory and mathematical formulae behind each methodology. We have also drawn a comparison between all three types of returns, with particular emphasis on logarithmic vs arithmetic returns, giving extensive examples where possible. We have finally compared these returns to those on initial capital with step-by-step illustrations, drawing conclusions that will hopefully assist you in measuring returns, particularly when optimising a strategy or evaluating an investment.
FAQs
Have a question on returns? Check out answers to some popular questions below. Alternatively, write to us!
Do I have to be an expert trader before tackling this tutorial?
Whilst being adept at trading is not a requirement before you make your way through the tutorial, possessing some familiarity with the financial markets does not harm. However, this does not mean that you cannot go through the tutorial and write to us with any questions that you may come up with along the way!
I have been through this tutorial but it is still unclear to me how I can use returns. Can you please explain?
Sure! The obvious application is to use returns to measure the performance of your trading strategy or individual assets in your portfolio. You may also like to compare different assets or different investments, in which case returns come in as one handy tool. Alternatively, you can use this knowledge to understand return numbers quoted by funds, which will hopefully help you in making informed investment decisions. Last but not least, returns can help you in defining your risk/return profile.
Before You Go...
Beginning Trader
I am all confused; I am only just getting started. Please take me back to the basics! I would like to start from the very beginning.
Expert Trader
I have been trading for a while and I would like further suggestions on topics that are similarly covered in depth on this website.
Stock Selection
I like the idea of using returns to decide on investments. I would like to find out more though how I can proceed with selecting stocks.